Magnetic Relaxation of Lanthanide-Based Molecular Magnets

Chapter 1

Magnetic Relaxation of Lanthanide-Based Molecular Magnets
*, Ana Arauzo†,‡, Javier Luzo´n‡,§, Juan Bartolome
‡,¶ Elena Bartolome ‡,¶,1
and Fernando Bartolome *

Escola Universitària Salesiana de Sarrià (EUSS), Barcelona, Spain Servicio de Medidas Fı´sicas, Universidad de Zaragoza, Zaragoza, Spain ‡ Instituto de Ciencia de Materiales de Arago´n, Universidad de Zaragoza—CSIC, Zaragoza, Spain § Centro Universitario de la Defensa, Academia General Militar, Zaragoza, Spain ¶ Departamento de Fı´sica de la Materia Condensada, Universidad de Zaragoza, Zaragoza, Spain 1 Corresponding author: e-mail: [email protected] †

Chapter Outline 1 Introduction 2 Electronic Structure, Magnetism, and Relaxation of Lanthanide Ions 2.1 Electronic Structure of the Lanthanides 2.2 Mechanisms of Magnetic Relaxation of Molecules 2.3 Magnetic Interactions 2.4 Beyond the Anisotropy Barrier: Energy Spectrum and Relaxation Times 2.5 Summary 3 Computational Tools and Theoretical Methods 3.1 Introduction 3.2 Electrostatic Models 3.3 Quantum Chemistry Methods 3.4 CASSCF–CASPT2/RASSI-SO Method

2

6 6 12 24

31 33 35 35 35 41 45

4 Experimental Methods for Investigation of Relaxation Mechanisms in Lanthanide Molecular Magnets 4.1 Introduction 4.2 DC Magnetization and AC Susceptibility Experiments 4.3 Heat Capacity Measurements 4.4 Spectroscopic Methods 4.5 Conclusions 5 Lanthanide-Based SIMs 5.1 Introduction 5.2 Influence of the Lanthanide Ion 5.3 Crystal Field Environment 5.4 Conclusions and Outlook 6 Dinuclear Lanthanide-Based SMMs 6.1 Homo Dinuclear [Ln2] SMMs

Handbook of Magnetic Materials, Vol. 26. https://doi.org/10.1016/bs.hmm.2017.09.002 © 2017 Elsevier B.V. All rights reserved.

60 60 61 76 84 101 101 101 103 117 137 139 139

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6.2 Heterodinuclear SMMs 6.3 Conclusions 7 Polynuclear Lanthanide-Based SMMs 7.1 Introduction 7.2 Homonuclear Ln Clusters (Lnx, x > 2): SMTs and SMMs 7.3 Heteronuclear Ln–3d Clusters 7.4 Conclusions 8 1D, 2D, and 3D Extended Systems 8.1 1D Molecular Magnets

163 167 168 168

168 182 194 196 197

8.2 Higher Dimensionality 8.3 Conclusions 9 Molecular Magnets on Surfaces 9.1 Selected Molecular Systems 9.2 Experimental Techniques 9.3 Surfaces and Substrates 9.4 Devices 9.5 Conclusions and Outlook 10 Conclusions and Perspectives Acknowledgments References

215 226 226 232 233 235 240 245 246 251 251

1 INTRODUCTION Upon the advent of nanomagnetism, the quest for finding small magnetic objects that could fulfill high-density information storage or acting as quantum qubits in quantum computing, soon evidenced that reducing the size of nanoparticles by physical procedures, the “top-down” strategy, led to inhomogeneous particle size, an unwanted trade. The alternative “bottom-up” strategy offered by Molecular Chemistry results very appealing since molecules of equal size can be synthesized in a reproducible manner, and they self-organize naturally. As a consequence, the basic and applied research on molecular magnetic materials has received a very large impulse in the last years because of the possibility of finding new quantum phenomena in tailored molecules that may be optimized for high-density information storage, quantum computing, or spintronics. In fact, the possibility of reading and writing on single atoms has been proven recently on Ho atoms supported on MgO (Natterer et al., 2016). The magnetic relaxation of a collection of magnetic atoms embedded in a solid consists in the time evolution of this spin system toward thermal equilibrium, after being perturbed by an external field. In the research field of molecular magnetic materials for the aforementioned technological applications, the slowness of the magnetic relaxation is one of the most relevant figures of merit since it underlines some required specific physical properties as magnetic bistability, large quantum coherence time, or compatibility with other systems, like molecular superconducting hybrid circuits. Therefore, new complexes with original characteristics that allow to investigate the mechanisms governing the magnetic relaxation are at the core of the research field. The main process for the magnetic relaxation of the spin system is through the energy exchange with the solid lattice vibrations, the phonon bath. In this spin–lattice relaxation process, the electronic quantum transitions may emit or absorb one or more phonons with the lattice, or conversely, the atomic fluctuations of the crystal or ligand field acting on the atomic orbital states induce

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electronic transitions that, in turn, modify the magnetic quantum state of the atom so that the system tends to thermal equilibrium (Abragam and Bleaney, 1970). Spin–lattice relaxation involving one phonon, direct process, or two phonons, Raman and Orbach processes (Orbach, 1961a) explained most experimental results till the discovery of quantum tunneling in magnetism. Indeed, the discovery of magnetic bistability in Mn12-acetate below the so-called blocking temperature TB (Sessoli et al., 1993) set the start of a new paradigm in magnetic dynamics, since other types of relaxation processes were active in this paradigmatic compound. They were related to quantum tunneling of the magnetization (QTM), either by pure tunneling through the hindering barrier to spin reversal via the ground doublet, which does not need to exchange energy with the lattice (Barbara et al., 1995), or by thermally activated resonant tunneling of the magnetization (TAQTM) (Friedman et al., 1996; Herna´ndez et al., 1996; Thomas et al., 1996), which is spin phonon assisted. These findings were followed by intensive work with the goal of increasing the hindering barrier energy, and finding the interactions that facilitate or forbid tunneling (Bartolom
e et al., 2014b; Ferna´ndez et al., 1998; Luis et al., 1997). However, few improvements were found beyond the properties of Mn12-acetate as a high spin (S ¼ 10) combined with a high-energy barrier (U ¼ 61 K). Hence an exciting race started with the objective of obtaining new bistable magnetic molecules below the blocking temperature. Such a molecule should have a magnetic anisotropy energy barrier separating the two states with opposite magnetic moments at the ground state and depict a nonzero coercivity that would maintain the information recorded in one of the states for as long as possible, allowing to use it as a molecular information bit. The term Single-Molecule Magnet (SMM) has been adopted to describe the magnetic bistability of polynuclear magnetic clusters upon application of an external magnetic field; i.e., below TB the cluster shows slow relaxation, leading to an open-loop magnetic hysteresis curve in the absence of short- or long-range intercluster interaction (Christou et al., 2000; Sessoli et al., 1993). Till 2003, most of the work done on this subject was related to transition metal clusters. However, the discovery of slow magnetic relaxation in terbium double-decker phthalocyanine [TbPc2] (Ishikawa et al., 2003b) triggered the investigation of bistability in monoatomic complexes, which were to be denominated single-ion magnets thereafter (SIMs). As a consequence of this discovery the interest on lanthanide-based compounds grew exponentially since some of them have a large spin and orbital moment and high anisotropy, as requested for SIMs and SMMs. Actually, before 2003, there were already early studies of the spin–lattice magnetic relaxation of lanthanide ions diluted in diamagnetic salts done by means of electron paramagnetic resonance (EPR) methods, for example, on La2Mg3(NO3)1224H2O doped with Ce(III) and Nd(III) (Brya and Wagner, 1966; Ruby et al., 1962), with other lanthanides and for rare earth-doped ethylsulfates (Larson and Jeffries, 1966).

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At present, lanthanide compounds play a major role in the quest to find optimum bistable magnetic compounds. Lanthanide-based SIMs have the advantage of being simpler than SMMs, they present high versatility for the modification of the spin Hamiltonian via the rational design of the local coordination, and they can be made more robust against decoherence. Moreover, an additional degree of freedom plays a distinguished role in the magnetic relaxation processes, namely the lanthanide nuclear angular momentum. A new domain on the magnetic dynamics studies opened, where tunneling takes place via coupled states of the electronic and nuclear angular momenta (Ishikawa et al., 2005b). This new approach to tunneling in lanthanide ions owes much to pioneering work on the relaxation processes found in rare earths highly diluted in a fluoride single crystal. For example, for LiY0.998Ho0.002F4 a stair-shaped hysteresis loop hinted to tunneling occurring at avoided level crossings of the energy spectrum of a single Ho(III) ion in the presence of hyperfine interactions (Giraud et al., 2001). In addition, quantum effects in SIMs are much more pronounced than those in polynuclear SMMs; hence these systems are ideal to study quantum phenomena such as tunneling, relaxation, and coherence (Bartolom
e et al., 2014b). In addition to SIMs and SMMs, lanthanide molecular systems can be also employed for the synthesis and design of single-chain magnets (SCMs). These are ferro-, ferri-, or antiferromagnetic chains in which the slow magnetic relaxation is caused by magnetic domain wall motion. The flipping of spins at the domain walls generates an imbalance of magnetic moments that can be related to the finite chain coherence length. The activation process of the chain relaxation is related to the intrachain spin–spin interaction, and to single-ion anisotropy. The expanding interest in lanthanide-containing molecular magnetism has caused an explosion of chemical, physical, and theoretical works addressing the various research aspects which are open at present. There have been a number of review papers dealing with lanthanide containing complexes that discuss partial aspects of magnetic dynamics in these compounds. The chemical strategies to synthesize SMMs based on lanthanide ions with high nuclearity clusters, giving rise to large moments in the ground state were reviewed in Sessoli and Powell (2009), including the effect of weak intracluster interactions and geometrical factors affecting the relaxation processes. Another chemical review paper specialized on Dy-containing complexes deals with the formation of homonuclear Dy SMMs with different ligands and how the different atomic geometrical arrays affect the slow magnetic relaxation phenomena by modification of the hindering barrier (Zhang et al., 2013). The synthesis, characterization, and applications of lanthanide SIM and SMM complexes are reviewed from the chemical, physical, theoretical, and calculational point of views (Layfield and Murugesu, 2015; Tang and Zhang, 2015). The combination of ab initio calculations, advanced angledependent magnetometry, and optical spectroscopies has allowed to interpret

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magnetic dynamics in lanthanide-containing molecules where the Ln ion sits in a low-symmetry site (Luzon and Sessoli, 2012). Moreover, specific aspects of heterometallic rare earth-transition metal clusters have been described in Gao (2015). A comprehensive review on lanthanide SIMs as well as homo- and heteronuclear SMMs is contained in Rosado Piquer and San˜udo (2015), including theoretical models for the understanding of the magnetic dynamics of these systems, as well as some applications. The roles of magnetic anisotropy and exchange interaction in the magnetization dynamics in lanthanide-containing complexes have been discussed with a view to possible application as qubits (Gatteschi et al., 2016). A perspective paper dealing on the concept of scalable architecture for quantum computation, as qubits or quantum gates, built with molecular nanomagnets, including lanthanide-based complexes indicates possible arrow lines in the future development of these systems (Jenkins et al., 2016). Though to date there are some lanthanide-based examples of SCM relaxational behavior in lanthanide-containing complexes, reviews cover just partial aspects of this interesting class of systems (Bogani et al., 2008; Li et al., 2010; Wang et al., 2009). A recent review by Sieklucka and Pinkowicz (2017) encompasses from SIMs and SMMs to qubits and some interesting materials such as molecular magnetic sponges, surveys optical, luminescent, and conduction properties, including also two dedicated sections to theory. In this work, we have collected the most recent highlights (2014–2017) in the field of magnetic relaxation processes present in lanthanide-containing molecular magnets, in an order of increasing complexity, with introductory sections that address theoretical and experimental tools to understand the content of the chapter consistently. The emphasis of this review is to focus on the understanding of the magnetic relaxation processes of lanthanide-based compounds, mainly observable at low and very low temperatures. In this respect, the relatively few works performed below 1 K have been reported in some detail. Besides, the review delves on the consideration of the dependence on temperature of the magnetic dimensionality of the compounds, a fact that is seldom taken into account. Today an interplay of experimental and theoretical calculations is the current state of the art to interpret the magnetic static and dynamics properties of the molecular magnets. The theoretical and computational methods, their advantages, and limits of application have also been discussed in detail. The content of the chapter is as follows: Section 2 collects the theoretical basis and available physical models in a self-contained document that enable the reader to understand the descriptions contained in the review without recourse to other works. Section 3 describes in detail the theoretical and computational tools most frequently used in this research field, with a particular emphasis on relativistic ab initio calculations, with their contributions and limits. In Section 4 the experimental methods applied in the determination of the thermal, optical, and magnetic properties are described with an emphasis toward the magnetic relaxation studies. Section 5 is devoted to the all-important SIM compounds, citing the most informative examples together with the most

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recent ones. In Sections 6 and 7, the most recent homo- and heteronuclear dimeric and polynuclear compounds, respectively, showing SMM behavior have been reviewed, including those compounds linked by a magnetic radical. Section 8 extends the review to compounds with a higher magnetic dimension in the interactions, i.e., chains, with an emphasis in SCMs, planar, and threedimensional compounds, many of them metal organic framework (MOF) systems. Such a review would not be complete today if the relaxation studies are not extended to molecules deposited on a substrate. The most advanced applications are deemed to consist in single or finite number of molecules supported on a metallic, semiconductor, or insulator supporting media, with a special trend toward carbon-based substrates. Section 10 describes our conclusions on the state of the art in magnetic relaxation studies in lanthanide-based molecular magnetism and future perspectives. Concluding, the increase of hindering barrier height to magnetic moment reversal by several orders of magnitude, the increase of blocking temperature to tens of Kelvin, and the long coherence time of milliseconds already achieved in one decade of intensive research is just an overture to the great opus that lays ahead, the achievement of molecular devices, including classical or quantum memories and computers.

2 ELECTRONIC STRUCTURE, MAGNETISM, AND RELAXATION OF LANTHANIDE IONS Magnetism of lanthanide-containing materials is diverse, interesting, and useful. This is so thanks to the interplay of several ingredients stemming from the origin of the magnetic moment in lanthanide ions: the incomplete 4f shell. From lanthanum (Z ¼ 57, n ¼ 0) to lutetium (Z ¼ 71, n ¼ 14) the electronic configuration of the atomic ground state of the 15 lanthanide elements is [Xe]4fn15d16s2 for La, Ce, Lu, and Gd, and [Xe]4fn6s2 for the rest. When forming part of molecules and compounds, lanthanide atoms usually loose the outer 5d16s2 electrons, exhibiting a valence Ln3+. The 4f shell occupies an inner space within the atom: it is localized and well screened by outer atomic levels 5s and 5p, which are full along the series. These two factors give to the lanthanide magnetic moments a very localized and isolated character, modulating the intensity of interactions with neighboring atoms.

2.1 Electronic Structure of the Lanthanides The Hamiltonian operator acting on the n electrons bound to a nucleus in a lanthanide ion can be written as the sum of different terms, each of them to be treated as a perturbation of the previous, more intense ones:  X n  n n X 1 2 Z e2 e2 X + + xðri Þ li si + HE + HB + HN , (1) pi  H¼ ri r 2m i
Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

Electronic configuration

Coulomb repulsion

Spin-orbit coupling

Crystal electric field

Magnetic interactions

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7

Hyperfine interactions

4f n−15d1

104 K

4f n

2S+1

103 K

102 K

L

0.1 – 1 K

2S+1

LJ

⏐ψ 〉

1 – 10 K ⏐± ψ 〉

⏐± ψ 〉⏐I 〉

FIG. 1 Typical energetic structure of a lanthanide ion evidencing the effect of progressively weaker perturbations present in the Hamiltonian of Eq. (1). The magnetic term is estimated assuming a field of the order of 1 T.

where the first term includes the kinetic term and the Coulomb interaction between electrons (located at ri) and the atomic nucleus; the second is the Coulomb repulsion between electrons, responsible of electronic correlations; and the third is the spin–orbit interaction. HE, HB, and HN include electric, magnetic, and hyperfine interactions. Fig. 1 shows schematically the electronic structure of a lanthanide ion after the consecutive application of all these terms.

2.1.1 Single Ion: Configuration, Levels, and Terms The Hamiltonian, including Coulomb repulsion, is separated into a central 0 part, H0, and a noncentral perturbation. The single-particle eigenfunctions will be hydrogen like. Introducing the spin, the states are described by the well-known quantum numbers jnlmlmsi . Electronic correlations are included P within the Russell–Saunders scheme in lanthanides. Total orbital L ¼ ni¼1 li Pn and total spin S ¼ i¼1 si moments are good quantum numbers. First and second Hund’s rules describe the 2S+1L energy ordering of the levels, with splittings of the order of 104 K. Spin–orbit interaction (HLS ¼ lLS) splits the 2S+1 L levels into 2S+1LJ terms with total angular momentum: J ¼ L + S. The l parameter is the Russell–Saunders spin–orbit coupling. The energy of the 2S+1LJ term is given by ELSJ ¼ (l/2)[J(J + 1)  L(L + 1)  S(S + 1)] and the ground-state term is determined by the third Hund’s rule (J ¼ jL  Sj for n < 7, and J ¼ L + S for n > 7).

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The interaction with the electric field created by neighboring atoms is much less intense than the spin–orbit coupling (typical CF splitting DECF/ kB  102 K, compared to spin–orbit, typically DELS/kB  103 K). Experimental values are tabulated in Dieke (1968). Except for Eu3+ and Sm3+ ions, DELS/kB is notably higher than room temperature, so the magnetism of lanthanides is determined by the ground 2S+1LJ term. The excited terms are, however, relevant in determining the optical emission and absorption energies, which are needed for a proper determination of the crystal field Hamiltonian, HCF.

2.1.2 Magnetism of the Isolated Ion The magnetic moment associated to the total angular momentum J is: 3 SðS + 1Þ  LðL + 1Þ : mJ ¼ mB gJ J ¼ mB ðL + gS SÞ, with gJ ¼ + 2 2J ðJ + 1Þ

(2)

A magnetic field along a given axis z, H ¼ Hz^z, interacts with the magnetic moment mJ removing the (2J + 1) degeneracy, by a Zeeman energy: EZ ¼ hJ mJ jHZ jJ mJ i ¼ hJ mJ jmJ Hz jJ mJ i ¼ mB gJ Hz mJ :

(3)

The magnetization of n noninteracting mBgJJ magnetic moments per unit volume as a function of field is given by the Brillouin function BJ(x) of a dimensionless variable x ¼ mBgJJH/kBT (the Zeeman to thermal energy ratio):      2J + 1 2J + 1 1 1 coth x  coth x , (4) M ¼ Ms BJ ðxÞ ¼ Ms 2J 2J 2J 2J where Ms ¼ nmBgJJ is the saturation magnetization of the ensemble. The Curie law for the temperature dependence of the magnetic susceptibility (w ¼ dM/dH) is easily derived: w¼

C Ng2J J ðJ + 1Þm2B : ¼ 3kB T T

(5)

The statistical mechanics of the paramagnet, a textbook system very properly describing some lanthanide magnetism, are easy to calculate and very well known (Blundell, 2001). Indeed, the “Curie constant” and the product wT give the “effective” moment of an ion with the ground-state term pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S+1 LJ: meff ¼ gJ mB J ðJ + 1Þ. This “isolated ion” behavior is only verified if the term 2S+1LJ is fully degenerated. This is the case only in Gd3+, whose L ¼ 0 ground state is not acted by crystalline electric fields. For lanthanide ions with L 6¼ 0, the assumption is no longer valid, even at room temperature. The contribution to the magnetic moment and susceptibility of excited terms 2S+1LJ0 are only relevant in some particular cases, when the splitting between terms is small. In Eu3+, with a diamagnetic ground state (L ¼ S ¼ 3, J ¼ L  S ¼ 0) the participation of excited terms 7FJ with J  1 is the only source of magnetism. This contribution was determined by Van Vleck (1932). It is

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also relevant to describe the room temperature effective moment in many Sm3+ compounds, given the small energy gap between terms, DJJ . The susceptibility, including the Van Vleck contribution, takes the form: 0

J¼L X+ S

ð2J + 1Þ wðJ Þ exp ½lJ ðJ + 1Þ=2kB T 

J¼LS wðT Þ ¼ J¼L + S X

,

(6)

ð2J + 1Þ exp ½lJ ðJ + 1Þ=2kB T 

J¼LS

where w(J) is the susceptibility as given in Eq. (5) once corrected by the Van Vleck temperature-independent contribution arising from the coupling between ground and excited states, of energy E(J) ¼ lJ(J + 1)/2, through the Zeeman perturbation: wð J Þ ¼

NgJ m2B J ðJ + 1Þ 2N ðgJ  1ÞðgJ  2Þm2B + : 3l 3kB T

(7)

Van Vleck terms are a rather small correction in compounds of Ln3+ other than Eu3+ or Sm3+.

2.1.3 Crystal Field Interaction When an atom or an ion is embedded in a molecular or crystalline environment, it is acted by the crystal electric field (CF) created by its environment, which has a symmetry lower than spherical. Depending on the intensity of the CF interaction with respect to the electronic correlation and spin–orbit coupling, there are three “schemes” of applying perturbation theory: strong field, applicable to 4d and 5d ions with a very unscreened unfilled electronic shell; intermediate field, for those cases where the effect of the environment is comparable to the electronic repulsion, applicable to 3d ions; and weak field, applicable to the lanthanides, because due to electronic screening and localization of the 4f shell, the environment can be treated as a perturbation of the electronic correlation and spin–orbit coupling. The original CF theory (Bethe, 1929; Hutchings, 1964) considered the potential at the ith electron position ri created by j neighbor ions, considering those as Zj point charges at fixed lattice positions Rj: V ðr Þ ¼

X e Zj  :   j ri  Rj

(8)

The factor j ri  Rj j1 can be developed as a series of Legendre polynomials, and therefore in terms of spherical harmonics, Yk,q(y, ’) (Arfken et al., 2012). The use of Yk,q(y, ’) has two important advantages. First, the symmetry of the system reduces the number of terms included in the sum. Second, the matrix elements of the Hamiltonian are easily calculated, as the

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jLSJMJi states are linear combinations spherical harmonics. The potential can be written as rffiffiffiffiffiffiffiffiffiffiffiffi ∞ X X 4p k Yk, q ðy’Þ, VCF ðr Þ ¼ Ak, q r (9) 2k + 1 k¼0 q¼k…k where, once the reference system is fixed, the parameters Ak,q are fully determined by the positions of the ligands and its charges. Albeit the success of the original theory, it had some quantitative limitations, which were overcome by ligand field theory, its successful evolution. The electric potential is assumed to be created by a charge density originated at the ligands r(r), which admits a multipolar expansion: charges, dipoles, poles, etc. associated to the ions or atoms forming the hosting matrix. The ligand field theory only fixes the symmetry of the problem, letting the potential to be adjustable, through the parameters Bk,q ¼ Ak,qrk, which are to be fit with available experimental data. VCF(r) verifies the Poisson equation, it is continuous and real. Moreover, VCF(r) has to be invariant under the symmetry operations of the point group of the ion in the solid. Within a given 4fn configuration, parity is a good quantum number, and the parity of Yk,q is (1)k, so only terms with even k are relevant. For 4f electrons only k ¼ 2, 4, and 6 are needed. The k ¼ 0 term is spherically symmetric, and it is normally included in the central Hamiltonian 0 H0 (the so-called nephelauxetic effect) (Tchougr
eeff and Dronskowski, 2009). There have been several parametrizations of the CF potential. Stevens’ “equivalent operator method” (Stevens, 1952) and Wybourne’s formalism are especially important. Stevens formulated a very useful “equivalent operator method” (Stevens, 1952). If the physics can be restricted to the ground 2S+1LJ term, the Hamiltonian can be written as polynomials of total angular momentum operators, J2, Jz, and J, which are tabulated. The Hamiltonian is written as HSt CF ¼

X k¼2, 4, 6

rk

k X

 q ^ , Aqk r k O k

(10)

q¼k

^ q are the Stevens equivalent operawhere Aqk is an adjustable parameter, and O k tors (Abragam and Bleaney, 1970). This method eliminates the use of singleelectron eigenfunctions by the use of angular momentum operators, which act direct and simply on the angular part of the wave function of the coupled system (Elliott and Stevens, 1953a,b; Judd, 1955; Stevens, 1952). The method is useful for easily obtaining estimates of the energy splittings and the crystalfield parameters from experimental measurements on isolated multiplets (from experiments on magnetism, EPR, and inelastic neutron scattering (INS) transitions), and it is widely used. However, the Stevens formalism does not allow to calculate optical transitions involving different initial |LSJmJi and final |L0 S0 J 0 m0J i terms. Optical

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spectroscopy is the only way to obtain energy splittings in an extended energy range, allowing parameter determination. In low-symmetry situations, the number of adjustable parameters may be rather high, and the input from optical spectroscopy is necessary. To include this input in the physical description of lanthanide-containing molecules, the full Hamiltonian has to be used, instead of the Stevens’ equivalent one. Wybourne’s formalism includes calculating the matrix elements of the ligand field Hamiltonian: HCF ¼

n X 6 X k X 4pð1Þq i¼1 k¼2 q¼k

2k + 1

Bk,q ðri Þ Yk, q ðy, ’Þ:

(11)

The radial common factor on the CF matrix elements hR4f j Bk,q(ri)j R4fi  Bqk is left as adjustable parameters, giving up to its calculation. The Hamiltonian is therefore effective: Heff CL ¼

6 X k X k¼2 q¼k

Bqk Cek,q , where Cek,q ¼

n X 4pð1Þq i¼1

2k + 1

Yk, q ðy, ’Þ,

(12)

whose matrix elements are D E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 0 0 n 0L S J M ln τLSJMj Heff ¼dSS0 ð2l + 1Þ ð2J + 1Þð2J 0 + 1Þ ð1Þl + L + S + J + J M , CF j l τ    

 J k J0 l k l X S L J n k q l τLSjj U^k jj ln τ0 L0 S0 , ð1Þ Bk M q M0 0 0 0 k, q k J 0 L0 (13) where use is made of the well-known 3j symbols by application of the Wigner–Eckart theorem. τ and τ0 are the seniority numbers, included by Racah to differentiate coincident triads (L, S, J) from different Hund couplings which may coincide in a matrix element. The tensor operators Cek, q ^ k, transform as the qth component of the Racah unitary tensors of range k, U whose reduced matrix elements were tabulated by Nielson and Koster (1963). The matrix elements given in Eq. (13) are easily computed, as a function of the parameters Bqk , as the rest of the factors are well known (Edmonds, 1957), or tabulated (Nielson and Koster, 1963).

2.1.4 The Role of Symmetry The CF Hamiltonian must be invariant under any symmetry operations of the point group of the lanthanide site. This imposes limitations on which are the nonzero Bqk (Abragam and Bleaney, 1970). In molecular magnets, as in oxides, the point symmetry of the lanthanide site can be rather low, and the number of parameters to be adjusted, quite high. It is customary to obtain a reasonably good “first guess,” which can be taken with reasonable guarantee not too

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far away of the best solution in the parametric space. Otherwise, fitting experimental data with confidence when the number of parameters is as high as 9, 14, or even 25 is hopeless. In general, the theoretical guidance has been found necessary to ensure a correct parametrization.

2.1.5 The Kramers Theorem Time reversal symmetry leads to an important result: in a system consisting of an odd number of fermions whose Hamiltonian is invariant under time reversal, the energy levels are always degenerated (Sakurai, 1964): in a lanthanide with an odd number of electrons acted only by electric fields (which are invariant under time reversal), each energy level must be at least twofold degenerate. This is called Kramers degeneracy. The two states in a Kramers doublet are time reversed of each other (see Section 3). A perturbation breaking the time-reversal invariance of the Hamiltonian lifts Kramers degeneracy, notably external magnetic fields. The lowtemperature magnetism of Kramers ions is strongly influenced by this degeneracy: in general, the CF splits the 2S+1LJ term in (2J + 1)/2 doublets (except for Gd, whose eight states are nearly degenerated). At sufficiently low temperature, the occupation probability of the excited doublets is so low that the system can be described just by considering the ground doublet, i.e., as an effective spin S* ¼ 1/2. This simplifies the interpretation of experiments. Kramers degeneracy plays an important role on QTM, one of the magnetic relaxation channels for molecular magnets. 2.1.6 The Griffith Theorem An interesting result by Griffith (1963) concerns non-Kramers “accidental” doublets, such that no other independent state has the same energy as the doublet. In that case, the states of the doublet define a unique axis, which is the principal one of the g-tensor, and if denoted as the z-axis, then g?  gx ¼ gy ¼ 0. Recent examples of non-Kramers doublets S* ¼ 1/2 where Griffith theorem applies are the acetylacetonate derivatives Ln(acac)3(H2O)2, with Ln ¼ Ho and Tb (Vieru et al., 2016b), the encapsulated single-ion magnet HoSc2N@C80 (Dreiser et al., 2014b), or the “butterfly” heteronuclear Fe3Ln cluster molecules [Fe3Ln(m3-O)2 (CCl3COO)8(H2O)(THF)3], with Ln ¼ Tb and Ho (Badı´a-Romano et al., 2015). On the other hand, several attempts have been performed to formalize symmetry-adapted effective S* ¼ 1 formalisms for non-Kramers ions, which behave properly under different symmetries (Mueller, 1968).

2.2 Mechanisms of Magnetic Relaxation of Molecules The applications of magnetic molecules do depend not only on their equilibrium properties but also on how fast (or preferably slow) the system recovers

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equilibrium once some perturbation has been removed. This way back to equilibrium can be described by a simple differential equation: dMðtÞ ðMðtÞ  Mð0ÞÞ ¼ , dt τ

(14)

where τ is a characteristic time constant, called the relaxation time, which governs the changes in magnetization. Experimentally, the relaxation time for a given molecule as a function of temperature and magnetic field will be determined, as well as the physical process responsible for the magnetic relaxation. The goal would be to design molecular systems with very long living metastable magnetic states, to allow quantum coherence to take place between such states, or to maintain written data unaltered, as long as desired. In this section the different mechanisms for magnetic relaxation are presented. The subject has been studied thoroughly for decades. Classical references are Abragam and Bleaney (1970), Kronig (1939), Orbach (1961a,b), Stevens (1967), and Van Vleck (1941a,b) for relaxation via the thermal phonon bath, as well as Garanin (1991), Gatteschi et al. (2006), Luis et al. (1998), and Prokof’ev and Stamp (2000) for quantum tunneling-assisted relaxation, among many others. In general, the lanthanide ion relaxes from an excited state j 2i to a ground state j1i, being the difference in energy between the two levels D ¼ E2  E1. The time derivatives of the population numbers of each level are n_ 1 ¼ n_ 2 ¼ o12 n1 + o21 n2 , where o12 and o21 are the rates at which the ion undergoes transitions from j 1i to j 2i and vice versa. The equilibrium is reached when n_ 1 ¼ n_ 2 ¼ 0. Defining the equilibrium populations as N1 and N2 (thus n1 + n2 ¼ N1 + N2), we have 0 ¼ o12N1  o21N2 with N2 ¼ N1 exp(D/kBT), being T the temperature of the thermal reservoir for the phonon system. After some elemental algebra, the following differential equation is obtained: d ð n1  n2 Þ ¼ ðo21 + o12 Þ½ðN1  N2 Þ  ðn1  n2 Þ, dt

(15)

whose solution is

ðn1  n2 Þ ¼ ðN1  N2 Þ + ðn1  n2 Þ0  ðN1  N2 Þ0 exp ðt=τÞ,

(16)

with τ the relaxation time for this simple model: τ ¼ ðo21 + o12 Þ1 :

(17)

This is an important and general result: To predict the spin–lattice relaxation times, one has to calculate the probabilities of transition between the initial and final states of the whole system, oJ F , including the magnetic and phonon subsystems.

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2.2.1 The Spin-Phonon Interaction Once the magnetic molecule has been perturbed, the return to equilibrium takes place either by energy exchange with the lattice or by quantum tunneling. Letting aside for the moment this latter possibility, the energy exchange with the lattice requires a mechanism of spin–phonon interaction. The spin system is affected by oscillatory electromagnetic fields with precise frequencies, created by the charges and dipoles of its surroundings, oscillating by lattice vibrations (phonons) with wave vector k ¼ o/n (the velocity n may be longitudinal or transverse). The coupling between the lattice phonons and the spin system has been treated (Orbach, 1961a) by expanding the CF potential as a Taylor series in powers of the atomic displacements, dRi of the ith neighbor of the lanthanide ion from its equilibrium position. The displacements dRi cause local strains Ei, which generate vibrations of the lattice. The dynamic CF potential is therefore: V ¼ V0 + EV1 + E2 V2 + ⋯ XX ∂ q 1 X X ∂2 Vk ∂Ri + Vkq ∂Ri ∂Rj + ⋯, ¼ V0 + ∂R ∂R ∂R 2 i i j i k, q ij k, q

(18)

where the unperturbed (static) CF potential, V0, is as given in Eq. (9), and V1, V2 are the first and second derivatives of the CF potential with respect to the displacements from the equilibrium positions Ri, Rj of the i, jth neighbor atoms. The strain can be expressed in terms of the phonons, i.e., lattice vibration quanta. These are quantified by phonon creation and annihilation operators, a+ and a: rffiffiffiffiffiffiffiffiffiffi X  ℏ  (19) ak + ak+ eikRi , dRi  Ei ¼ 2Mo k where M is the mass of the crystal, o is the frequency of the phonon, and a+k (ak) is the operator that creates (annihilates) a phonon of wave vector k. The potential described in Eq. (18) has two resonant terms, from V1 and V2, which correspond to the emission (or absorption) of one or two phonons, respectively. The one- and two-phonon terms are usually relevant in the interpretation of experimental results. The subsequent terms are usually neglected, although they have been studied with some detail (Orbach, 1961a). With this approach, several processes are relevant to the interpretation of experiments in magnetic molecules, which are schematized in Fig. 2.

2.2.2 Direct Process The magnetic system jumps from the initial j 2i to the final statej 1i with the emission (or absorption) of one single phonon, whose energy ℏo coincides with the energy difference between the two magnetic states, ℏo12 ¼ D12 ¼ jE2  E1j.

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⎮u〉 Phonon cutoff ⎮3〉 hw 2→3 ⎮2〉

hw = D2→1

⎮2〉

hw s

hw i

⎮1〉 Direct process

⎮2〉

hw 3→1

⎮1〉 Raman process

⎮1〉 Orbach process

FIG. 2 Level scheme for spin–lattice relaxation theory. (a) Direct process, (b) Orbach process, and (c) Raman process.

If longitudinal and transverse modes are not distinguished, and we integrate along frequencies, denoting the matrix elements of jh1 j V1 j 2ij2  jV1 j2, we get (Abragam and Bleaney, 1970):    1 3 ℏok 2 3 : (20) V o tanh τ¼ j j 1 k 2kB T 2pℏrn5 The value of j V1 j2 in Eq. (20) is different if the lanthanide is a Kramers or a non-Kramers ion. In the case of a non-Kramers ion, j 1i and j 2i are orbitally different, and j V1 j2 are temperature and field independent. The only effect of a field on τ will be through Zeeman splitting, D21 ∝ H, and therefore, the frequency of the emitted phonon must adapt: ℏok ∝ H in the direct process. This implies τ ∝ H3 tanh(mH/2kBT). In the high-temperature regime kBT ≫ ℏo, tanh(mH/2kBT)  (mH/2kBT), so τ  H2T1. In a Kramers ion, h+i j V1 j ii ¼ 0, i ¼ 1, 2. We have to consider explicitly a magnetic field to lift the Kramers degeneracy, allowing second-order perturbation including excited states. If we denote by D12 the CF splitting between j1i and j2i, the ground-state wave functions are

 h2jJ j1i 0 j2i + ⋯ : j1i ¼ j1i + gJ mB H D12 The wave functions are linear on H, and the relaxation time becomes: " #1   3ð2mB gJ Þ2 jH h2jJ j1ij2 ℏok 2 3 : (21) V o tanh τ¼ j j 1 k 2pℏrn5 2kB T D212

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FIG. 3 Plot of the function x3/(ex  1).

If we take again ℏok ¼ mH, in the high-temperature limit, we obtain τ ∝ H4T1. To evaluate the efficacy of the direct process, the phonon density with frequency within the range o and o + do as a function of temperature has to be evaluated: rðoÞdo ¼

3ℏo3 do : 2 3 2p n exp ðℏo=kB T Þ  1

(22)

Fig. 3 shows the function x3/(ex  1). The arrow o12 in the figure points to the number of phonons available for a transition j2i Ð j 1i with energy ratio x ¼ 0.2 (a realistic value above T  20 K). The phonon density of states at low x is very small, strongly reducing the effectivity of the direct process at high temperature.

2.2.3 Raman Process The magnetic system may undergo a transition from the initial to the final magnetic state by the nonresonant simultaneous absorption of an incident phonon of energy ℏoi and the emission of a scattered one with energy ℏos, provided that D12 ¼ E2  E1 ¼ ℏ(os  oi). This is usually regarded as the ion reaching a virtual, nonstationary intermediate state at higher in energy than the phonon cutoff, the energy corresponding to the highest phonon frequency allowed in the crystal. The importance of the Raman process is evident by inspection of Fig. 3. The arrows labeled oi and os point to a couple of phonons allowing a twophonon Raman process, with the restriction j os  oi j ¼ o12. Evidently, the Raman process “scans” the whole spectrum, and the number of phonons potentially involved in a j 2i Ð j 1i transition through 2-phonon Raman channel is much higher than those involved in a direct process. Above

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T0  20–30 K, the Raman relaxation channel is usually dominant with respect to the direct one. A calculation procedure similar to the one developed for the direct process, to second order, allows to obtain a general expression for τ for the Raman process (Abragam and Bleaney, 1970; Shrivastava, 1972, 1983). The probability of Raman scattering allowing a transition from the initial state j J i ¼ j 2ij n1, …(n + 1)l, …nm…noci to j F i ¼ j1ij n1, …nl…(n + 1)m…noci includes every possible virtual, intermediate state j ci with energy D above the continuum of allowed phonon frequencies, as indicated in Fig. 2. If j 1i and j2i are not Kramers doublets, the relevant probabilities are:  2 2p  X hF jEl V1 jcihcjEm V1 jJ i PJ !F ¼ 2  (23)  dðℏðol  om Þ + E1  E2 Þ, ℏ c, l, m ℏol  D + E1  E2  This allows to obtain τ by calculating the squared strains E2l E2m, with ol and om scanning the whole continuum from 0 to the phonon frequency cutoff, oc. The Raman relaxation time for non-Kramers ions (Abragam and Bleaney, 1970) results:      9  6! 1 D2 ℏ 7 : (24) τ¼ kB T V14 4ℏ3 r2 n10 In Kramers ions, the terms in the sum of Eq. (23) appear in pairs, with the same absolute value but opposite sign, canceling the whole sum except for a subtle difference in the denominator, which results in a residue of order j 2ℏo V21/D2 j2  o2. The Raman relaxation time for Kramers ions results:   1  4   9! D ℏ 9 : (25) τ ¼ 3 2 10 p rn kB T V14 A rough calculation of the order of magnitude of the Raman relaxation times gives estimates of τ  106T7 for non-Kramers ions and τ  105T9 for Kramers ions, so this process would be able to predominate at temperatures above T  20 – 30 K.

2.2.4 Orbach Process The relaxation from the initial to the final magnetic state may involve a twostep process, through a real intermediate state j 3i which can be reached by the absorption of one phonon, with the adequate frequency energy o23, and the subsequent emission of another phonon with frequency o31 from the intermediate level to the final state. The energy conservation ensures D21 ¼ jE2  E1 j ¼ ℏ(o23  o31). The number of phonons available for these two processes may be much larger than those available for the direct process, thus dominating the relaxation. The relevant probabilities are formally the same as those considered in

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Handbook of Magnetic Materials

the Raman case in Eq. (23), but involving only the intermediate j3i state. Moreover, it is not possible anymore to neglect ℏol with respect to D in the denominator. Orbach shows that the relaxation time in this case is (Orbach, 1961a) " #  1 3 f exp ðD=kB T Þ  1g jh2jV1 j3ij2 + jh3jV1 j1ij2 , (26) τ¼ D3 2pℏ4 rn5 jh2jV1 j3ih3jV1 j1ij2 This relaxation time varies very strongly with temperature: in the intermediate temperature regime, D ≫ kBT ≫ ℏol, τ varies as D3 exp(D/kBT). Typical values give τ  104D3 exp(D/kBT). In many cases, the presence of an adequate state j3i leads Orbach processes to dominate the magnetic relaxation in lanthanide molecular compounds at high temperature.

2.2.5 The Bottleneck Effect Spin–lattice relaxation rates have been calculated for an excited spin relaxing to a phonon system which remains unaltered by the interaction. This approach breaks down if the phonons of the required frequency are not able to transfer the energy to the bath fast enough. This induces a warming up of the lattice (at least within a frequency band around the spin excitation energy), which approaches the temperature of the spin system. This effect is known as “phonon bottleneck” (PB) (Van Vleck, 1941a,b). The emitted phonons are resonant along the spin system, and prone to be reabsorbed, giving rise to a coherent emission followed by absorption process (resonant phonon trapping, RPT) before the lattice-only mean free path for phonons is attained. RPT induces PB and strongly enhances the observed relaxation time. RPT is effective in large, defect-free crystals. Although there have been many attempts to clarify the microscopic origin of PB (Garanin, 2007, 2008; Giordmaine and Nash, 1965; Huber, 1965; Pines-Rojansky and Fain, 1990; Standley and Vaughan, 1969; Stoneham, 1965), a full explanation of the observed complex behavior is still lacking: PB encompasses micro- and macroscopic effects of both intrinsic and extrinsic origins, such as the phonon density of states, the establishment of RPT, the size of the crystals, and the details of the experimental setup and measuring conditions. In order to get some quantitative results, we shall look first at the problem with a thermodynamical approach, as described by Stoneham (1965). We regard the experimental system as composed of three subsystems: the spins, the band of phonons relevant to spin–lattice relaxation, and the bath. The bath is usually treated as a subsystem of infinite heat capacity, which remains at a constant temperature TB. The lanthanide ion ground state is j1i, and it has a spin-excited state at energy d, j 2i. To study PB in Orbach processes, we will also consider a higher energy state j 3i with energy D. Until now, it has been assumed that the phonons were always in equilibrium with the bath, but now we release this constraint. As schematized in Fig. 4 the

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FIG. 4 Block scheme of the three subsystems in which the magnetic measurement setup is divided in the simplest treatment of phonon bottleneck.

spins and the lattice are characterized by their own internal energies and temperatures, UL, TL, and US, TS, defining finite specific heats cL and cS, respectively. Both TL and TS are in principle different from the bath temperature, TB. 2.2.5.1 PB in High-Temperature Regime We first will assume D ≫ kBT ≫ d. The observed relaxation time is defined by the linearized differential equation, T_ S ¼ τ1 obs ðTS  TB Þ, which depends on the rate at which the spins transfer its energy to the phonon system, and on the rate at which the lattice transfers energy to the bath. Those rates do define the spin– lattice relaxation equation, T_ S ¼ τ1 SL ðTS  TL Þ, and the lattice–bath relaxation ð T  T Þ. equation T_ L ¼ τ1 L B LB The temperatures, internal energies, and specific heats in Fig. 4 allow to establish the following differential equations: U_ S ¼ cS T_ S ¼ ðcS =τSL ÞðTL  TS Þ, U_ L ¼ cL T_ L ¼ ðcS =τSL ÞðTS  TL Þ + ðcL =τLB ÞðTB  TL Þ:

(27)

Typically, U_ L is smaller (by a factor of 105) than all the other terms in Eq. (27). If U_ L is neglected, the observed relaxation time is: τobs ¼ τSL + τLB ðcS =cL Þ,

(28)

which coincides with the unbottlenecked spin–lattice relaxation time τSL enhanced by the lattice–bath relaxation time τLB times the ratio between the spin and the lattice specific heats. The specific heat of a two-level spin system in the range kBT ≫ d is cS ’ NkB(d/2kBT)2, with N being the density of spins. The lattice specific heat is proportional to the width of the resonant phonon band (G) and to the density of resonant phonons r(d). Therefore, in the direct process, the observed relaxation time is (Stoneham, 1965):   N d 2 A D  + 2: (29) τobs ¼ τSL + τLB rðdÞG 2kB T T T

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Handbook of Magnetic Materials

In the Raman process the whole phonon spectrum participates in the energy transfer, so the whole lattice specific heat is involved, and the effects of the bottleneck may be expected to be much smaller at the range kBT ≫ d. In the Orbach process the state j 3i enters into play. The lattice specific heat is:     d 2 D where now TL exp c0L ¼ rðDÞ G0 kB kB TL kB TB ðd=2kB Þ

N ðDÞ + N ðD + dÞ : N ðDÞ  N ðD + dÞ

(30)

With this lattice specific heat, the observed relaxation has an exponential temperature dependence when bottlenecked in the high-temperature regime:   N D , (31) exp τobs ¼ τSL + τLB 2rðdÞG0 kB TB which is essentially the same exponential dependence found in the unbottlenecked case. 2.2.5.2 PB in Low-Temperature Regime: RPT In the low-temperature regime, kBT ≫ d does not hold anymore. When analyzing the RPT as the cause for PB at low temperature, Huber (1965) enumerates three conditions: (1) the transfer of energy via spin–spin interactions should be negligible, (2) the temperature of the crystal must be well below d/kB, so thermal phonons may be ignored, and (3) the wavelength of the trapped phonons should still be larger than the interatomic distances, kresr12 ≪ 1. In general, lanthanide-containing molecular systems do fulfill these conditions in the very low-temperature regime, due to the weakness of the magnetic interactions. If the phonon wavelength exceeds the interatomic distance, the phonon may repeatedly and coherently be absorbed and emitted, establishing RPT. Giordmaine and Nash (1965) studied the diffusion of “imprisoned phonons.” The analysis includes both the spin resonance linewidth and the phonon lineshape. τobs strongly depends on the lineshape: for a rectangular spectrum they obtain τobs ∝ L2T4 B , being L the dimension of the crystal. For 1/2 1 a Gaussian line, a τobs ∝ LT2 B dependence is found but τ obs ∝ L TB for Lorb entzian resonances. In general, a low-temperature relaxation time τobs ∝ T B with 1 < b 4 can be attributed to PB through RPT. The effect of the crystal size has been directly studied recently in nonlanthanide-containing molecules by Tesi et al. (2016), measuring the relaxation time as a function of the crystallite dimension of three series of vanadyl b-diketonate complexes. Tesi and coworkers find that τobs enhanced

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

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at low temperatures by up to two orders of magnitude in a sample formed by millimeter-sized crystals with respect to what is found in intermediate grinded samples and finely crushed microcrystalline powders. A poor thermal contact between the sample, the sample holder, and the bath (thermal reservoir of the experimental setup), by means of thermal contact grease (Schenker et al., 2005) or helium gas pressure, has been shown to induce PB, as will be discussed in Section 4.

2.2.6 Quantum Tunneling of the Magnetization Up to now, phonon-assisted magnetic relaxation has been discussed. Alternatively, relaxation is attainable by QTM. The classical expression for the energy of a magnetic moment with uniaxial !!

anisotropy is transformed from E ¼ Ucl cos 2 y m H (where Ucl is the classical energy barrier) into the quantum Hamiltonian H ¼  DS2z  gmBSzHz. The classical potential barrier to magnetic moment reversal is a double well (symmetric for H ¼ 0) as a function of y. It is schematically depicted in the right side of Fig. 5. In the quantum regime, the discrete energies of the j S,  mi…jS, +mi eigenstates replace the continuous double well. The eigenstates are stationary, and therefore tunneling between those eigenstates is impossible, even when Hz ¼ 0 and the j S,  mi are degenerated. To open a door for quantum tunneling, the Hamiltonian has to include terms which do not commute with Sz, such as nonuniaxial anisotropy, or the action of a transverse field. In the simplest case, S ¼ 1/2, the Hamiltonian includes a bias Hz and a transverse Hx field, H ¼  DS2z  gmB(HzSz + HxSx). Forgetting the constant term depending on D, the Hamiltonian has two eigenvalues, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ ðgmB =2Þ Hz2 + Hx2 . The eigenstates are never degenerate (except in the trivial case Hx ¼ Hz ¼ 0). Fig. 5 (left and center) shows the eigenvalues and eigenstates of the Hamiltonian. The energy difference E+  E has a minimum DT ¼ gmBHx, the “tunneling splitting.” The eigenstates j ’i are coherent superpositions of j+i and ji. When Hz 0 the eigenstates j ’i are the symmetric and the antisymmetric combination of the “localized” ji states, periodically oscillating with frequency o ¼ DT/ℏ, a process denominated coherent tunneling. As soon as the energies j gmBHz j and j gmBHx j are notably different, the eigenstates approach the localized “spin-up” and “spin-down” ones: coherent tunneling is only possible in a rather narrow energy window for which the Zeeman energy due to the bias field is smaller than the tunnel splitting. An important corollary is that quantum tunneling is forbidden in a pure Kramers doublet under zero field, as the eigenstates are stationary and degenerated. A weak transverse field is able to open the door to QTM, including effective hyperfine or dipolar fields. The so-called internal bias fields play a central role for QTM in Kramers systems.

FIG. 5 Left: Eigenvalues E of a two-level system as described in the text. The corresponding eigenstates, shown at the center of the figure, are coherent combinations of the “up” and “down” states. Right: Scheme showing the three paths for relaxation described in the text.

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The Landau–Zener (LZ) model (Zener, 1932) gives the tunneling probability PLZ for a two-level model such as the scheme in the left panel of Fig. 5. The system is initially prepared in one of the two states (“j+i” in Fig. 5, with probability P0LZ ¼ 1) as the field goes from negative to positive at a finite rate of change with time, n. Landau and Zener found that there is a finite probability PLZ of tunneling from j+i to ji (and of course 1  PLZ of staying j+i), with:   p D2T for S ¼ 1=2, (32) PLZ ¼ 1  exp  2ℏgmB n and a similar expression for a general S or J. In such a case, DT can be calculated perturbatively by the so-called chain rule (Garanin, 1991) for high spin numbers. PLZ is exponentially depending on the inverse of the field sweep rate: a slow sweeping rate ensures the tunneling, while a fast one is able to “freeze” the original state through the HZ 0 passage, in good agreement with hysteresis experiments. PLZ depends also on the tunneling splitting D2T. Finally, PLZ is independent of T, and so will be the associated relaxation time. p D2

DT is usually very small, and the LZ probability is PLZ 2ℏgmT n. B In a general case with S > 1/2 the applied magnetic field moves the levels staircase up and down depending on the relative orientation of the moment and the field. In order to relax, a spin j m ¼  Si under a field +Hz may follow three different paths: 1. Thermal relaxation “over the barrier” is schematized in Fig. 5 (1) by green arrows, involving successive phonon absorption and emission events, as previously analyzed in this section. 2. Pure quantum tunneling (blue arrow (2) in Fig. 5) is possible as discussed previously within a window of energy for which the bias Zeeman energy is smaller than the tunnel splitting created by a transverse field or anisotropy. The ground state jm ¼  Si reverses by resonant tunneling when it coincides with one of the other states j S  ni (with n S) under the effect of a field Hz that shifts the two staircases in opposite directions. Such a pure quantum tunneling process is possible when the external plus the internal (dipolar plus hyperfine) bias fields tune two states within the energy window determined by the tunneling splitting, DT. The maximum tunneling rate (thus minimum relaxation time τT) takes place when the energies satisfy the energy window condition  DT/2 xdip DT/2 (dashed horizontal lines in Fig. 5). Pure QTM connects states at both sides of the well only through high-order perturbation, and consequently, DT is very small. The distribution of dipolar bias energies P(xdip) with xdip ¼ gzmBHdip,z is much larger than DT in real crystals, and only a fraction of the spins will be able to tunnel. In the LZ model, the tunneling relaxation time is (Prokof’ev and Stamp, 1998; Wernsdorfer et al., 1999) τT ℏ/D2TP(xdip). At Hz 6¼ 0, spins are driven out of this condition, so only those spins for which xdip and the hyperfine energy xhyp ¼ gNmNHhyp,z

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field “compensate” xZeeman have a total energy tuned within the tunneling window. The relaxation time becomes very sensitive to the strength of the local bias field. As the magnetic field is swept, different levels at both sides of the well are tuned within the tunneling window at field values for which relaxation is accelerated, and magnetization abruptly changes. Between two consecutive “matching” field values, the magnetization has “plateaus” whose slope depends on the field sweeping velocity, giving rise to the characteristic magnetization staircase hysteresis loops of molecular magnets in the quantum regime (Thomas et al., 1996; Urdampilleta et al., 2013), as shown in Fig. 6, where the hysteresis of a single crystal of [(Tb0.02Y0.98)Pc2]TBA+ (with TBA+ ¼ tetrabutylammonium ion) is shown (Ishikawa, 2010). Internal bias fields are relevant in Kramers ions to destroy the time reversal symmetry, allowing QTM to take place between otherwise QT-unconnected states. In the vicinity of the matching field, the general expression for the tunneling probability, proposed in the nineties (Villain et al., 1997), allows to express the quantum tunneling relaxation time as τQTM ¼ (1 + B2H2)/B1, where all the unknowns are grouped into fit parameters (Zadrozny et al., 2013). As the parameter B1 includes D2T, the QTM is extremely sensitive to detuning by magnetic field: QTM easily quenches, enabling the slowing down of the relaxation. 3. Thermally activated QTM (TAQTM; orange and blue arrows defining path (3) in Fig. 5) is a combination of thermal and quantum relaxation: the size of DT is strongly dependent on the order in the perturbation theory necessary to connect the states contributing to j ’i. In general, Snx,y are combinations of Sn+,, connecting j S, mzi with j S, mz  ni. The lower the values of mz and mz  n, the lower the order of perturbation needed to connect those states, and the larger DT, facilitating QTM. The relaxation time of this will be mostly imposed by the thermal branches of the path, as QTM is much faster than phonon-assisted relaxation TAQTM, but notably shorter than “over-thebarrier” relaxation.

2.3 Magnetic Interactions The magnetism of lanthanide-containing molecules, due to the localized character of the 4f electrons, is in first approximation a magnetism of isolated molecules. However, interactions within and between molecules are crucial to understand the low-temperature magnetism and magnetic relaxation of molecular compounds. The magnetic term of the Hamiltonian, HB, includes several contributions which are relevant to describe the slow relaxation of lanthanide-containing molecules: the Zeeman term, which describes the interaction with an applied magnetic field, the interactions with other electronic magnetic moments through exchange and dipolar interactions, and finally, the hyperfine term,

A

C

1 4

0.5

7

9 8

6

3

0

0.280 T/s 0.140 T/s 0.070 T/s 0.035 T/s 0.017 T/s 0.008 T/s 0.004 T/s 0.002 T/s 0.001 T/s

2 3 4 5

−0.5

6 7 8 9

−1 −1.2 B

−0.8

−0.4

0 m 0 H (T)

0.4

0.8

E/kB (K)

7

6

−644.0

9

⏐+6〉⏐−3/2〉

−644.2 −644.4

⏐−6〉⏐−3/2〉

−644.6

⏐−6〉⏐−1/2〉

−644.8

⏐−6〉⏐+3/2〉

⏐−6〉⏐+1/2〉

1

0.5

5

0

4 3 2

M/Ms

1

0

−0.5

−0.5 −1 −0.06

⏐+6〉⏐+1/2〉 ⏐+6〉⏐−1/2〉

D

8

⏐+6〉⏐+3/2〉

−643.8

1.2

1

0.5 M/Ms

−643.6

1

5

1

M/Ms

2

−1 −0.04

−0.02

0 0.02 m 0 H (T)

0.04

0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

m0 H (T)

FIG. 6 (A) Hysteresis loops at 0.04 K for a single crystal of [(Tb0.02Y0.98)Pc2]TBA+ measured at several field scan rates. (B) Enlarged view of the low-field zone of the hysteresis loops in (A). (C) Zeeman diagrams calculated including hyperfine and nuclear quadrupole interactions. (D) Hysteresis loop measured at T ¼ 0.04 K measured at 1 mT/s, evidencing the plateaus and the abrupt jumps at matching fields. Reproduced with permission from Ishikawa, N., 2010. Phthalocyanine-based magnets. In: Functional Phthalocyanine Molecular Materials, Jiang J. (Ed.), Structure and Bonding, vol. 135, 2010, Springer: Berlin, Heidelberg, 211–228. https://doi.org/10.1007/978-3-642-04752-7_7. Copyright © 2010, Springer-Verlag Berlin Heidelberg.

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describing the interaction of the electronic moment with the nuclear magnetic moment.

2.3.1 Zeeman Interaction The interaction of a magnetic field B with a magnetic moment mJ is given by the well-known Zeeman Hamiltonian HZ ¼  mJH ¼  mBgJJH. This HamilP tonian acts on CF states j ai ¼ MJaMJ j JmJi, and the details depend on the specific linear combination of base states. If the direction of the application of the field defines the axis of quantization, z, we can write X  2    X  2 am  mJ JZ mJ0 ¼ mB gJ Bz am  mJ (33) EZ ¼ hajHZ jai ¼ mB gJ Bz J J mJ , mJ 0 mJ The gJ is characteristic of the whole 2S+1LJ term, but CF splitting is much larger than thermal energy, even at room temperature. If only a limited number of CF levels are thermally available, it is useful to build an effective spin Hamiltonian within the available subspace (Abragam and Bleaney, 1970; Chibotaru, 2013). The archetypical example of this low-temperature projection is the Kramers doublet. At T below a certain threshold, the lowest doublets are the only populated states. In those cases, the system is treated as an effective spin S* ¼ 1/2, with an effective g-tensor which depends on the particular CF combination of the ground doublet ji. The anisotropic effective g-values are gk ¼ 2gJh+j Jz j+i and g? ¼ gJh+j J+ji can be notably different from gJ. Within the doublet, the energies are Ek ¼  mBgk Hz/2 and E? ¼  mBg? H?/2.

2.3.2 Exchange Interactions Exchange is a subtle combination of electrostatic interactions and the Pauli exclusion principle. The total wave function for electrons must be antisymmetric. To be so, the spatial and the spin parts must by symmetric and antisymmetric with respect to the exchange of two electrons. The energy difference between the symmetric and antisymmetric spin states is the exchange energy. The evaluation of the exchange in a molecule would involve complex integrals of overlapping electron wave functions. In practice, all those are summarized into empirical exchange constants. For two spins the exchange energy would be given by Hex ¼  2JijSi  Sj. The sign of Jij determines if the coupling is ferromagnetic (Jij > 0, FM) or antiferromagnetic (Jij < 0, AF). Exchange is originated in the spatial overlapping between electron wave functions. Therefore, it is a very short-range interaction, usually taken only up to nearest neighbors (n.n.). The spatial dimensionality of the spin ranges from 1 to 3. The terminology for the different models is

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

Ising for one  dimensional spin : Hex ¼ 2

X

1

27

Jij Siz  Sjz ,

i>j

 X  Jij Six  Sjx + Siy  Sjy XY for a two  dimensional spin space : Hex ¼ 2 i>j

and Heisenberg for a three  dimensional one : Hex ¼ 2

X

Jij Si  Sj ,

i>j

where the sums run to n.n. unless stated otherwise, and i > j counts every pair just once. The Heisenberg Hamiltonian can be simplified by assuming the Jij equal P between the different neighbors, such that Hex ¼ 2 i>j J Si  Sj . The exchange constants may be also allowed to be anisotropic, i.e., different for each direction:  X Jx Six  Sjx + Jy Siy  Sjy + Jz Siz  Sjz : Hex ¼ 2 (34) i>j

Exchange takes place between electrons at different levels in a material. Exchange between different shells in an atom, usually referred to as intraatomic exchange, is very important in lanthanides. The 4f shell is so localized that the exchange path always involves intraatomic exchange, usually 4f–5d: exchange in lanthanides is always indirect. Moreover, in molecules, both intra- and intermolecular interactions are indirect exchange interactions mediated by orbitals of nonmagnetic neighboring atoms (superexchange). In molecular clusters with more than one magnetic atom per molecule, it is central to understand if the molecule behaves as a coupled, single magnetic moment, denoted SMM. The paradigmatic SMM has been the Mn12Ac cluster (Caneschi et al., 1991; Lis, 1980; Sessoli et al., 1993), which consists of eight Mn ions in the 3 + oxidation state (S ¼ 2) and other four in the 4 + state (S ¼ 3/2). Strong ferro- and antiferromagnetic exchange interactions between similar and different spins, respectively, stabilize a total ground state spin S ¼ 10. In contrast, some clusters may have a more complex behavior. For example, in [Fe3LnO2] “butterfly” clusters the Ln magnetic moment is not so strongly coupled to the unity formed by the three iron S ¼ 5/2 (Badı´aRomano et al., 2015) and an applied magnetic field is capable of changing the relative orientation of the moments. The extent and intensity of intramolecular exchange interactions are, therefore, central to the phenomenon of magnetic relaxation. Intermolecular exchange is very weak in lanthanide-containing molecules. It only would be relevant at very low temperatures, but dipolar interactions use to be dominant. At any rate, the dimensionality of the lattice is a relevant

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parameter: while in three-dimensional lattices long-range magnetic order may be established within Heisenberg, XY, and Ising systems, in two-dimensional ones only the Heisenberg exchange is able to induce it (XY systems may in contrast undergo a topological Kosterlitz–Thouless phase transition in 2D). No long-range magnetic order is possible in purely one-dimensional systems, although the crossover from a one- to three-dimensional behavior is not unusual due to weak interchain interactions (exchange or dipolar) at sufficiently low temperature (de Jongh and Miedema, 1974).

2.3.3 Dipolar Interactions Unlike exchange, dipolar interactions are purely magnetic in nature. These are weak interactions, in general, whose effects are only evident at very low temperatures (usually below T ¼ 1 K). However, in lanthanide-containing molecules, we are dealing with very localized and well screened 4f electrons, and therefore quite weak exchange interactions, too. The Hamiltonian for the dipolar interaction in a solid is: 1 0   mi rij mj rij m0 X @mi mj A, (35) 3 Hd ¼ 4p rij3 rij5 i, j i6¼j

where r12 ¼ r1  r2 is the vector joining both magnetic moments, with i and j running, in principle, over every magnetic moment in the crystal. The dipole–dipole Hamiltonian depends on the relative orientation of the moments, and it is a long-range interaction, scaling with the inverse of the cube of the distance. Depending on the details of the structure, the convergence of the sum may not be easily attained. In contrast to the situation in the vast majority of inorganic crystalline magnets, dipolar interactions cannot be safely neglected in molecular magnets, in particular in lanthanide-containing molecules. First, due to their nontrivial spatial dependence, dipolar interactions are able to inject nonzero off-diagonal terms in the spin Hamiltonian, whose presence allows anticrossing of the levels, as a function of applied field, opening a path for QTM in some cases. Second, dipolar interactions may lead to the settlement of longrange order in molecular solids at sufficiently low temperature.

2.3.4 Hyperfine Interactions The atomic nucleus often has a magnetic moment, resulting from the intrinsic angular momentum of its components coupled to a nonzero spin. Each isotope has a nuclear spin quantum number, I in units of ℏ, which may be 0, 1/2, 1, 3/2, and so on. The nuclear magnetic moment is mn ¼ gnmNI, with mN ¼ eℏ/2mp the nuclear magneton, and gn the nuclear gyromagnetic ratio, which reflects the structure of the isotope, and whose typical values are of the order of 1. Nuclear

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moments are much smaller than electronic ones, as the proton to electron mass ratio is 1836. The coupling scheme in which the electronic moment J and the nuclear moment I couple into a resultant total moment F is not applicable, as the splittings due to CF and Zeeman electronic interactions are much larger than the hyperfine interaction. The precise mechanism of the hyperfine interaction is complex, having three main contributions. First, contact interaction is due to the field created by the nuclear moment acting on the electrons at the nucleus position. Only s orbitals have a nonzero probability to be found at the nucleus, so p, d, and f electrons do not feel contact hyperfine interaction. Second, dipolar interaction between the nuclear magnetic moment and the electronic spin summed over all electrons. This interaction only vanishes for s electrons, due to its spherical symmetry. And third, the dipolar interaction between the nuclear moment and the electronic orbital moment, which is in general nonzero in lanthanides. The interaction can be quantified as (Abragam and Bleaney, 1970; Blundell, 2001).   m 8p L S 3rðS  rÞ , (36) Hhyp ¼ 0 2gn mB mN I 5 SdðrÞ + 3  3 + 4p r r r r5 but it is usually expressed using an effective Hamiltonian approach: Hhyp ¼ Adip I  J:

(37)

Indeed, if a field Bz is applied in the quantization direction, we can consider Hhyp ¼ AdipIzJz. Otherwise, I  J ¼ IzJz + (I+ J + I J+)/2. Defining an effective hyperfine field HH, the total Zeeman  energy on the electrons can P 2 be summarized as EZ ¼ mB gJ mJ jcmJ j mJ ðHz + HH Þ, where the hyperfine and the applied field can be taken together into account, acting on the electronic ground state. The effective field HH is proportional to the third component of the nuclear moment Iz, so the j ’eij ’Ii state splits in (2I + 1) levels. The nuclear electric quadrupole interaction adds a term to the energy: EZqn ¼ Pquad(I2z + I(I + 1)/3), and as a result, the (2I + 1) levels in which the j ’eij ’Ii ground state splits are not equally spaced anymore as HH is not simply proportional to Iz. The importance of nuclear interaction in molecular magnetism is exemplified in the role that hyperfine interaction has in quantum tunneling relaxation. An archetypical example is the very much studied TbPc2 family of compounds (Ishikawa, 2010; Ishikawa et al., 2005b).

2.3.5 Single-Chain Magnets A particularly interesting system which may have slow relaxation through the interplay between local anisotropy and exchange interaction is the Ising chain with exchange interactions to first neighbors. This system does not magnetically order at finite temperature. However, an interacting chain

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may have magnetic relaxation processes which are not unique due to the anisotropy of individual spins, but to correlations among them. These slowly relaxing one-dimensional systems are coined “SCMs.” Relaxation in a SCM is due to the presence of domain walls separating finite-length ordered sections. A specially interesting case is the Ising anisotropic chain, which is defined by H1D ¼ 2J

∞ X

Si Si + 1 + D

∞

∞ X ∞

S2 i  2JS2

∞ X

si si + 1 + DS2

∞

∞ X ∞

s2i ,

(38)

where J is the exchange constant between adjacent spins, the Ising variables are simply si ¼  1, and the anisotropy constant is D < 0 if the system has an easy axes. A scheme of the main ingredients of the 1D chain is shown in Fig. 7: The distance between adjacent spins is a; the distance between two domain walls is twice the correlation length, 2x; and L is the average distance between defects in the chain. The simplest description of the SCM dynamics is based on stochastic spin flips: the spin values are functions of time, with a transition probability Wi(si) which depends on the neighboring spins and the applied field. The only requirement to the Wi(si) is the fulfillment of the detailed balance relation, which ensures equilibrium after infinite time. This lets some freedom to the detailed form of Wi(si), being Glauber’s the simplest choice (Glauber, 1963):  1  g 1  si ðsi1 + si + 1 Þ , (39) Wi ðsi Þ ¼ 2τT 2 where τT is the characteristic time of spin flipping in the absence of interaction at a given temperature and g ¼ tanh(4JS2/kBT). τT is expected to follow an activated law τT(T) ¼ τ0 exp(DA/kBT) with an energy gap, DA, that, in the Ising limit (for narrow domain walls), is equal to DS2. If we write a master equation with Wi(si), a dynamic differential equation is obtained: τ0

d hsi i + hsi ið1  gÞ ¼ 0, dt

(40)

with h i indicating average on the chain. The low-temperature equilibrium state consists of domains of size 2x separated by narrow domain walls, well

FIG. 7 Scheme of the magnetically coupled Ising chain.

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isolated from each other, following a Boltzmann statistics. Their number decreases (and therefore x increases) exponentially as exp( Dx/kBT), where Dx ¼ 4JS2 is the creation energy of a domain wall. The density of defects defines two different regimes, depending on the effective size of the chain, L: (i) The infinite chain regime, when L > 2x: The intrinsic dynamics of the Ising model apply, and the relaxation time is given by τ(q0) ¼ (τ0/2) exp[(2Dx + DA)/kBT], with q0 ¼ 0 in a ferromagnetic chain, and in a two-sublattice antiferromagnetic chain. (ii) The finite chain limit, when 2x > L: The finite chain regime has been discussed by Luscombe et al. (1996). The energy required to create a domain wall next to a defect is divided by two, and in the finite chain regime, this “new” Dx/2 activation energy dominates the dynamics. Therefore, the relaxation time when 2x ≫ L tends to: τ¼

τ0 L exp ½ðDx + DA Þ=kB T , 2a

(41)

and therefore, at the temperature at which L 2x, τ experiences a crossover. Experimentally, the slope of a ln(τ) vs 1/T (quite abruptly) changes, allowing to deduce the parameters of the system from Glauber’s dynamics. There are two very different realizations of the model: the ferro- or ferrimagnetic chain, and the purely AF chain. The ferrimagnetic case is similar to the ferromagnetic one, because the spin unit of the Glauber model can be identified with a unit formed by different AF-coupled moments. Indeed, the first examples of Glauber dynamics in the literature are of this kind (Caneschi et al., 2002; Coulon et al., 2004). The purely AF chain is somehow different. The infinite AF chain has no net magnetization except at the domain walls, but the presence of defects in the chain may create finite segments with a nonzero net magnetic moment, depending on the odd or even number of spins. The spins reverse direction stochastically, giving rise to SCM behavior in the finite length chain limit (Bartolom
e et al., 2016).

2.4 Beyond the Anisotropy Barrier: Energy Spectrum and Relaxation Times Recent works (see Pedersen et al., 2015 and references therein) show a profound disagreement between spectroscopic levels measured in lanthanide materials and the anisotropy energy barrier Ueff obtained from the fit of τ to an Arrhenius law τ ¼ τ0eUeff /kBT. Typically, Ueff has been thought to be originated in the gap D associated to an Orbach process with relaxation time τ ∝ eD/kBT with D given by the first CF level measured spectroscopically.

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For example, Er(trensal) shows a mismatch between Dspec/kB ¼ 76  2 K, obtained from luminescence spectroscopy (Flanagan et al., 2001) or INS (Pedersen et al., 2014) and Dmag/kB ¼ 30  2 K (Lucaccini et al., 2014; Pedersen et al., 2014) from the temperature dependence of magnetic relaxation times. In the Yb(trensal) derivative, the disagreement is enhanced: from measurements of near-infrared spectroscopy, Pedersen and coworkers determined the value for the first excited state values of the order of Dspec/ kB ¼ 668 K, while the fit of the relaxation times assuming an Orbach deexcitation process is DOrbach/kB ¼ 55 K, one order of magnitude lower. The left panel of Fig. 8 shows the time relaxation data of Yb(trensal), obtained from AC magnetic susceptibility with a generalized Debye model, as usual (see Section 4). The red line is the Arrhenius law corresponding to the Dspec/kB ¼ 668 K barrier. The blue line is a simulation of the expected behavior for the smallest energy difference in their spectroscopic data, Dspec/kB ¼ 245 K, (if incorrectly assigned as first excited doublet). Both are in evident disagreement with data. The green solid line is the fit to an Arrhenius of the high-temperature data, corresponding to DOrbach/kB ¼ 55 K.

FIG. 8 Left: Arrhenius plot (reciprocal temperature axis) showing the temperature dependence of the relaxation time at HDC ¼ 2000 Oe for Yb(trensal) together with the calculated slopes for Orbach processes as described in the text. Right: Double-logarithmic plot of the temperature dependence of the relaxation time at HDC ¼ 2000 Oe, and the fits as described in the text. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. Adapted with permission from Pedersen, K.S., Dreiser, J., Weihe, H., Sibille, R., Johannesen, H. V., Sørensen, M.A., Nielsen, B.E., Sigrist, M., Mutka, H., Rols, S., Bendix, J., Piligkos, S., 2015. Design of singlemolecule magnets: insufficiency of the anisotropy barrier as the sole criterion. Inorg. Chem. 54, 7600–7606. https://doi.org/10.1021/acs.inorgchem.5b01209. Copyright 2015 American Chemical Society.

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From this comparison, it is clear that the interpretation of relaxation data as a simple Orbach process through an excited state at energy D may not be correct, in general. It has been proposed theoretically and experimentally that first- and second-order Raman processes may be significant. Pedersen and coworkers propose that direct and Raman processes should be taken into account. The right panel of Fig. 8 shows fits to a more general expression, which here we adapt to explicitly include bottleneck effects: τ ¼ AH 2 T 1 +

  1 + B2 H 2 + CT n + CT m + τ0 exp Ueff =kB T , B1

(42)

where the different terms correspond to the direct process, quantum tunneling, Raman (n  4), bottlenecked (2 m 4), and Orbach relaxation mechanisms. The right panel of Fig. 8 shows clearly the power dependence τ ∝ T7 corresponding to a Raman mechanism at high temperature and a bottlenecked direct process (τ ∝ T2) at low temperatures in measurements on single crystal. PB is prevented by reducing the size of the crystals, in a powdered sample where the transition from Raman to direct process (τ ∝ T1) fits the data. The details of the phonon spectrum may also be relevant to solve, partially, the discrepancies between the effective energy barrier and the spectroscopic energies, due to the presence of phonon dissipation and anharmonic phonons. Anharmonicity has been invoked by Lunghi et al. (2017) to play an important role in mediating the energy exchange between spins and the bath. Under-barrier relaxation would be allowed by spin interacting with lowenergy phonon modes, whose finite lifetime and corresponding nonzero energy width would open a channel for nonresonant direct and Orbach relaxation process to take place.

2.5

Summary

The magnetism of lanthanide-containing molecules is an interesting subject because it opens new ways on molecular spintronics, including candidates to the development of nanosized memory units, quantum devices, etc. In order to advance in the development of new devices, the dynamics of the magnetization have to be slow enough to allow any operations needed for the device to work to happen prior to the demagnetization of the system. Slow relaxation is therefore crucial. The identification of the mechanisms of magnetic relaxation is a fundamental step for the advancement in the molecular design with the required characteristics. Along this section, the dependence of the relaxation time on temperature T, magnetic field H, and crystal size L for different relaxation mechanisms, processes, and ions has been reviewed, and the main results are reunited in Table 1. We have tried to present a revision of the fundamental concepts that will be used in the rest of the chapter.

TABLE 1 Temperature Dependence of the Magnetic Relaxation Time for Different Physical Mechanisms in Different Regimes τ

Relaxation Mechanism Direct

Ion

Kramers 4

T

1

Intrinsic

H

PB—high T

T 2

PB—low T

lorentz:

Raman Non-K H

2

T

1

Kramers T

9

T  n, 2 n 4

pffiffiffi 1 LT , gauss: LT 2, rect.: L2T 4

QTM Non-K T

7

Orbach 



D kB T   D D3 exp kB T

D3 exp

Kramers

Non-Kramers

Forbidden

Temperature independent

1 + B2 H 2 B1

PB, phonon bottleneck. High T and low T refer to the regimes in which kBT ≫ d is a valid approximation or not. d and D are the energy gaps between j1i and j2i, and j3i states, used as in the text. The labels “lorentz,” “gauss,” and “rect.” refer to lineshapes of the phonon resonances, respectively.

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COMPUTATIONAL TOOLS AND THEORETICAL METHODS Introduction

Knowing the electronic structure of lanthanide ions in SMMs is indispensable in order to rationalize their magnetic behavior and to shed light on the factors than can enhance their energy barriers for the magnetic relaxation. From an experimental point of view, this can be done by using Hamiltonian models to fit experimental data (SQUID magnetometry, spectroscopy methods, INS, etc.). These Hamiltonian models must include a CF term, which is usually formulated using either the Wybourne (1965) or the Stevens (1952) notations, both explained in Section 2. However, a lanthanide ion in a low-symmetry surrounding, as is generally the case in molecular systems, will require a large number of CF parameters, e.g., 27 in a C1 symmetry using the Stevens notation. In such case, even a good quality set of experimental data cannot uniquely determine either the CF or the electronic structure, complicating the analysis of the magnetic properties. Although a reduction in the number of CF parameters can be done by approximating the real symmetry to an ideal one, such a simplification can produce significant effects on the electronic structure of the lanthanide ions. Because of that, theoretical studies are essential in order to analyze, fit, and understand experimental data. An additional benefit of theoretical models is the possibility to perform computational experiments, searching for structure–electronic correlations and determining which factors can influence the magnetic behavior of the compounds, paving the way for development of improved materials. In the last years, there has been a vast effort in the development and application of theoretical approaches in this research field, ranging from simple electrostatic models to high level and computationally expensive quantum chemistry methods. In this section, we will review the most relevant ones, either for their already contribution to a better understanding of the lanthanide-based SMMs or for their potential to improve such understanding in the future.

3.2

Electrostatic Models

The 4f valence electrons of trivalent lanthanide ions are located in the inner area of the atom, screened by the 4d and 5s outer electrons; therefore, they are well isolated and not involved in chemical bonding. This is the justification of the electrostatic models, which consider that the electrostatic interaction between the 4f electrons and the charge distributed along the surrounding ligands is the main source of the crystal field of the lanthanide ion. Here, we are going to focus on two of these electrostatic models, both of them based on the point charge electrostatic model (PCEM) (Hutchings, 1964), in which the crystal field term is generated by point charges located at the atomic positions of the surrounding ligand molecules. In the first one

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(Chilton et al., 2013), the 4f-multielectronic wave function is replaced by its corresponding electron density distribution in order to evaluate the effect of the crystal field, resulting in a classical model which, despite its simplicity, provides an intuitive picture of the magnetic anisotropy of the lanthanide ions. The second one (Baldovı´ et al., 2012) introduces effective parameters in the PCEM in order to take into account covalent effects and displacements of the electron density from the center of the atomic positions of the ligand atoms.

3.2.1 “Electron Density” Electrostatic Model Rinehart and Long (2011) provided a design criterion for maximizing the uniaxial magnetic anisotropy of lanthanide ions in molecular systems by taking into consideration the aspherical electron density distribution of the mJ states of the 2S+1LJ ground multiplet of the lanthanide ions (Sievers, 1982). A desirable condition in order to get a uniaxial magnetic anisotropy is that the ground doublet state should be mainly composed by the states with the largest mJ value (Ising states). Therefore, since the different mJ states have different aspherical electron density distributions (see Fig. 9), the CF must be designed, and even chosen in order to stabilize the electron density of the Ising states, these states becoming the ground doublet. So, the previous design criterion proposes that lanthanide ions in which the electron density of the Ising states has an oblate shape (Ce3+, Pr3+, Nd3+, Tb3+, Dy3+, Ho3+)

Ce3+ 4f 1

Pr3+ 4f 2

Nd3+ 4f 3

Pm3+ 4f 4

Sm3+ 4f 5

Eu3+ 4f 6

Gd3+ 4f 7

Tb3+ 4f 8

Dy3+ 4f 9

Ho3+ 4f 10

Er3+ 4f 11

Tm3+ 4f 12

Yb3+ 4f 13

Lu3+ 4f 14

⏐± 12 〉

⏐± 32 〉

⏐± 52 〉

⏐± 72 〉

⏐± 92 〉

⏐±11 2 〉

⏐±13 2 〉

15

⏐± 2 〉

FIG. 9 First two rows: Angular dependence of the 4f charge-density of the Ising states (mJ ¼  J) for all the lanthanide ions. Last row: Angular dependence of the 4f charge density of the mJ states from the ground J-multiplet for the Dy3+ ion, going from a prolate shape (mJ ¼  1/2) to an oblate one (mJ ¼  15/2). Reprinted with permission from Jiang, S.-D., Qin, S.-X., 2015. Prediction of the quantized axis of rare-earth ions: the electrostatic model with displaced point charges. Inorg. Chem. Front. 2, 613–619. https://doi.org/10.1039/C5QI00052A. Published by The Royal Society of Chemistry.

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should have an axial ligand charge distribution, whereas lanthanide ions in which the electron density of the Ising states has a prolate shape (Pm3+, Sm3+, Er3+, Tm3+, Yb3+) should have a planar ligand charge distribution. Based on this criterion, Chilton has proposed a simple electrostatic model (Chilton et al., 2013) predicting the orientation of the uniaxial magnetic axis of dysprosium ions by searching which orientation of the aspherical f-electron density with respect to the CF potential minimizes the electrostatic energy. This approach considers the ground doublet state of the dysprosium ions as a pure mJ ¼  15/2 one, which produces an f-electron density r(a,b) 15/2(y, ’), where a and b are polar angles for the quantization axis and y and ’ polar angles expressing the angular dependence of the aspherical f-electron density. For trivalent lanthanide ions, the aspherical electron density of pure mJ states can be expressed as a lineal combination of three spherical harmonics, Y2,0(y, ’), Y4,0(y, ’), and Y6,0(y, ’) (Sievers, 1982). As for the construction of the CF potential, this approach uses a very simple point charge model in which fractional charges are located in the positions of some ligand atoms around the lanthanide ion. These charges are obtained from a valence bond model in which the charge on a ligand atom is calculated as a weighted sum of its atomic charge on all the resonating Lewis structures of the ligand, resulting in a distribution of fractional charges on a few atoms, while all the other atoms of the ligand remain neutral. Then, following the Stevens notation, the CF potential of the lanthanide ion can be written as a summation of spherical harmonics up to rank 6 (Hutchings, 1964): VCF ðy, ’Þ ¼

k X X X Yk, m ðyn , ’ Þ 4pð1Þm  k n r Yk,m ðy, ’Þ qn , 2k + 1 Rkn + 1 n k¼2, 4, 6 m¼k

(43)

where (Rn, yn, ’n) are the spherical coordinates of the nth ligand atoms with a qn charge and hrki is the radial average of the lanthanide ions, which is tabulated in the literature (Freeman and Watson, 1962). The previous electron density and CF potential are combined in order to compute an electrostatic energy integral for generating an electrostatic potential map around the lanthanide ion: Z p Z 2p ða, bÞ VCF ðy, ’Þr15=2 ðy, ’Þsin ðyÞdyd’: (44) E15=2 ða, bÞ ¼ y¼0

’¼0

The electron density will be oriented as indicated by the minimum of this electrostatic potential map, allowing to straightforwardly determine the orientation of the uniaxial anisotropy axis. Application of this simple approach to several dysprosium compounds has given surprising qualitative agreement with computationally expensive quantum chemistry calculations for the orientation of the uniaxial magnetic anisotropy (Chilton et al., 2013), confirming that purely electrostatic effects

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are one of the main sources of the magnetic anisotropy on lanthanide ions and validating the Rinehart and Long’s proposal for the rational search of lanthanide-based SMMs (Rinehart and Long, 2011). This approach is not intended to provide a full description of the electronic structure of the lanthanide ions. Despite of that, this method can be complementary to computational expensive quantum chemistry calculations in order to rationalize in a simple way the magnetic anisotropy of lanthanide ions, and in particular the orientation of the uniaxial anisotropy axis, giving an intuitive picture of the role played by each ligand molecule surrounding the lanthanide ion. In this respect, it is interesting to remark that this model does not depend on the fitting of experimental data, since it does not have any adjustable parameter. A drawback of this approach is that it neglects any covalent effect on the ligand field and related to that, it does not take into consideration neutral ligands, which due to either charge polarization or covalency can influence the CF (Cucinotta et al., 2012). A recent proposal (Jiang and Qin, 2015) to include covalent effects inside this model is based on the simple-overlap model (Malta, 1982), introducing a displacement of the negative charges on the ligand atoms which are coordinated to the lanthanide ions. By assuming that the charges are in the middle of the coordination bond, the charge displacement is done along the ligand s or p orbitals involved in the bonding and with a value equal to the corresponding average atomic orbital radius, which can still be determined and tabulated without the need of any fitting parameter. A graphical representation of the charge displacement is shown in Fig. 10. This model has been tested in several lanthanide ions vs ab initio calculations or experimental data, showing a net improvement in the determination of the uniaxial magnetic axis with respect to the model without the charge displacements (Jiang and Qin, 2015; Meng et al., 2016b). A

B raver O raver

Rn

Ln3+

Ln3+

Rn

FIG. 10 The two types of charge displacement in s- and p-coordination bonds. The displacement in the s-bond is along the ligand–metal vector (A), and the shift in the p-bond is along the normal of the aromatic plane (B). Reprinted with permission from Jiang, S.-D., Qin, S.-X., 2015. Prediction of the quantized axis of rare-earth ions: the electrostatic model with displaced point charges. Inorg. Chem. Front. 2, 613–619. https://doi.org/10.1039/C5QI00052A. Published by The Royal Society of Chemistry.

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Finally, it is important to remark that this model supposes that the ground state is an Ising one, a pure mJ ¼  J state, and therefore is not applicable to the frequent case in which the ground state is a mixture of mJ states.

3.2.2 The Radial Effective Charge and the Lone-Pair Effective Charge Models These two models (Baldovı´ et al., 2012), implemented in the SIMPRE package (Baldovı´ et al., 2013a), introduce effective charges and displacements as adjustable parameters in order to take into account covalent effects. As the previous Chilton’s model, they consider the CF as generated by a set of point charges (Eq. 43). However, there are several relevant differences with Chilton’s. First, in these approaches the CF is applied on the electronic wave functions and not on the electron density of the Ising states. Second, now, there are free parameters in order to fit experimental data. These two differences, although slightly reducing the simplicity and the intuitive description of the magnetic anisotropy, allow to obtain the mJ composition of the electronic states and their corresponding energy levels, both required for a proper analysis of the magnetic behavior. In the radial effective charge (REC) approximation each ligating atom around the lanthanide ion is modeled by an effective point charge (Zeff) located at a distance Dr from the center of the ligating atom along the lanthanide–ligand axis. These two effective parameters have a clear chemical foundation, accounting for covalent effects in the ligand charge value and its position. This approximation is well adapted for describing spherical ligating atoms with bonding lone pairs pointing toward the lanthanide ion. For the frequent case in organic ligands in which the ligating atom has a sp2 hybridation, resulting in a charged nonbonding lone pair not pointing toward the lanthanide ion, a modification of the REC has been proposed, the lone-pair effective charge (LPEC) model. In this model, in addition to the effective charge (Zeff) and the radial displacement (Dr), another charge displacement (Dh) is introduced in order to reflect the effective position of the charge of the nonbonding lone pair. This displacement is in the plane defined by the ligating atom and its two covalently bonded atoms and along the bisection line of the angle defined by them (Fig. 11).

Dh

N

Dr

CI

Ln3+

Dr

Ln3+

FIG. 11 Graphical representation for the effective distances in the REC (left) and LPEC (right) approaches (Gatteschi et al., 2016).

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A clear example in which the LPEC model improves the results from the REC ones is the double-decker [TbPc2] complex (Pc ¼ phthalocyaninato) (Baldovı´ et al., 2012). Experimental data for this compound indicate a highspin doublet ground state. However, the REC model predicts a diamagnetic single ground state. Using LPCE for modeling the nonbonding lone pairs of nitrogen atoms of the phthalocyaninato ligand results in a high-spin doublet ground spin, in agreement with experiments. Here, it is also interesting to point out a computational experiment using quantum chemistry methods on the [Dy(DOTA)] compound (Cucinotta et al., 2012), which showed a dependence of the nature of the doublet ground state with the orientation of a ligand water molecule in an apical position. A REC model could not account for such effect. However, the orientation of the charged nonbonding lone pair of the water molecule could be the source of that surprising computational result. These REC and LPEC models have been applied to several families of compounds (Baldovı´ et al., 2012, 2016a; Lim et al., 2016; Qian et al., 2015), usually with homoleptic ligands, providing in most cases a good agreement with experimental data. Although these approaches require the fitting of free parameters, they present a clear advantage with respect to a direct fit of the CF parameters, since the free parameters have a clear chemical significance, allowing to choose the set of fitting parameters which are chemically meaningful. In addition, for the REC model, it has been proposed that the effective charge and the radical displacement can be related to chemical factors, what could also reduce the number of fitting parameters (Baldovı´ et al., 2015). One of the actual interests of the REC/LPEC approach is building a reusable library of ligands with effective REC/LPEC parameters (Baldovı´ et al., 2014). The existence of such library would reduce the number of free parameters for compounds in which one or several ligands are already included in the library. More important, the library would favor the rational design of lanthanide SMMs by screening at a very low computational cost potential combination of ligand molecules and lanthanide ions with improved SMM behavior. The construction of a library would require a good ratio between experimental data and free REC/LPEC parameters. Thus, it implies to get a highquality set of experimental data (magnetic data, spectroscopy data, INS, etc.). Moreover, it would be desirable the study of homoleptic compounds in order to reduce the number of effective parameters and studies along series of compounds in which the same ligand molecules are coordinated to different lanthanide ions. Thereby, adding new ligands to the library with a trustable quality can be a challenging task. In addition, it must be remarked that the use of a library of REC/LPEC effective parameters makes several assumptions: first, the additivity of the effects of different ligands on the CF; second, the transferability of the REC/LPEC from one compound to another compound with different lanthanide ions or ligand combination; and third, and

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the most important, the validity of the REC/LPEC models for an accurate reproduction of the CF including any covalent effect. For the last point, the validity of the REC/LPEC models, an important drawback is that they only consider the first coordination sphere, although charges beyond the first coordination sphere can produce relevant effects on the CF of the lanthanide ion (Cucinotta et al., 2012; Pedersen et al., 2014; see Section 5 for details).

3.3

Quantum Chemistry Methods

A correct quantum chemistry treatment of lanthanide ions and their compounds is a considerable challenge due to three main factors: the high degeneracy of states arising from the partially filled 4f shell, the large electron correlation, and, finally, the existence of significant relativistic effects. Moreover, in the theoretical study of lanthanide SMMs, the treatment of the relativistic effects is crucial, since one of them, the spin–orbit coupling, is the main source of their magnetic anisotropy. In the next subsections, we will expose the basis of the relativistic quantum chemistry methods employed for the study of molecular systems, focusing in the treatment of relativity (Section 3.3.1) and the electron correlation (Section 3.3.2). A more detailed and deeper description of relativistic quantum chemistry methods can be found in Reiher and Wolf (2009) and references inside.

3.3.1 Relativity in Quantum Chemistry The formal way to introduce relativity in quantum chemistry is starting from the one-electron Dirac equation (Dirac, 1930). The solution of the Dirac equation is a four-component wave function: 0 1 c +"   B c +# C c+ C B c¼ (45) @ c" A, c c# where c+, called large component, is a two-component wave function associated to one electron including its spin coordinates, up (") or down (#). Similarly, c, called small component, is a two-component spinor associated to one-electron antiparticle, positron. Solutions of the Dirac equation for positive energies usually have a large c+ component and a small, but not zero, c component, i.e., solutions for positive energies are not pure electronic wave functions, but they also have a small positronic component. Going from the Dirac equation for one particle to a many-body quantum chemistry equation is not straightforward. One of the main difficulties is that the classical Coulomb potential is not well suited for describing the

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electron–electron interaction in a relativistic framework since it considers instant interactions, what is not allowed in Relativity. In the Dirac–Coulomb– Breit Hamiltonian (Bethe and Salpeter, 1957), this difficulty is solved by a relativistic correction to order 1/c2 of the classical Coulomb interaction, derived by the perturbation theory within the framework of quantum electrodynamics. Such relativistic correction is already adequate for treating almost all quantum chemistry systems. However, the Dirac–Coulomb–Breit Hamiltonian is computationally too expensive for the study of molecular systems, with 4N-component spinors for a system with N electrons. Fortunately, nowadays, there are several theoretical approaches to decouple the large (c+) and the small (c) components for the positive energy solutions in order to obtain an effective Hamiltonian acting only on the electrons, without any positronic contribution. The Hamiltonian of the Dirac equation, HDirac, can be expressed as a 4  4 matrix acting on the four-component wave function. A procedure to decouple the large and small components is to find a unitary transformation, which transforms the Dirac Hamiltonian in a block diagonal Hamiltonian:   h+ 0 , (46) UH Dirac U { ¼ Hblockdiag , H blockdiag ¼ 0 h thereby decoupling the large component and the small components: h + c + ¼ Ec + ; h c ¼ Ec :

(47)

In most approaches the unitary transformation is not done in a unique step, but by n successive transformations (U ¼ U1U2 ⋯ Un), each one reducing the nondiagonal blocks up to order n in a given parameter. In the Douglas– Kroll–Hess (DKH) protocol (Douglas and Kroll, 1974; Hess, 1986), the unitary transformation is expanded in powers of the external potential acting on the electron (V), where the order of the expansion defines the order of the DKH Hamiltonian: H DKHn ¼ U1 ⋯Un H Dirac Un{ ⋯U1{ :

(48)

Although only the infinite order produces an exact decoupling of the electronic and positronic states, the second-order DKH Hamiltonian (DKH2) already correctly describes all the nonnegligible relativistic effects for the chemical/physical properties of molecular systems. Therefore, the secondorder DKH Hamiltonian is the one usually implemented and employed in quantum chemistry packages. The DKH and all the other decoupling protocols generate an electronic Hamiltonian which recovers the nonrelativistic time-independent Schr€odinger equation plus additional terms due to relativistic effects, which can be classified into two types, spin free and spin dependent. The spin-free terms, also called scalar relativistic effects, do not alter the nonrelativistic spatial and spin

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symmetries and they can be easily implemented in nonrelativistic quantum chemistry methods. On the other hand, the treatment of spin-dependent terms, also called spin–orbit coupling effects, is more challenging since they couple the degrees of freedom associated with spin and space, making impossible to treat separately spatial and spin symmetries. As a result, the simultaneous treatment of both spin–orbit coupling and electron correlation is very complex, and at present, it is not affordable for molecular systems. An alternative approach, with less complexity and lower computational cost, is the so-called two-step approach in which accounting for electron correlation, scalar relativistic effects, and spin–orbit coupling effects is done in a two-step procedure. In the first step, spin-free quantum chemistry computation is performed including both electron correlation and scalar relativistic effects. In the second step, the spin–orbit coupling effects are included by computing the spin–orbit coupling terms in the basis of the spin-free wave functions obtained in the first step.

3.3.2 Nonrelativistic Quantum Chemistry and Electron Correlation In addition to the decrease in complexity and the computational cost, the twostep approach allows to take advantage of the highly developed nonrelativistic quantum chemistry methods for treating the electronic correlation. In the next paragraphs, we will introduce some basic concepts about nonrelativistic quantum chemistry methods and what exactly is meant by electron correlation. Most nonrelativistic quantum chemistry methods are based on the Born– Oppenheimer approximation, which assumes that the nuclear and electronic motions can be separated. Then, a multielectronic Hamiltonian, describing the possible states of N-electrons in the field created by M nuclei, can be written: ! N M N X 1 2 X Zk 1X 1 ri  , (49) + H¼ r 2 2 r i¼1 k¼1 ik i6¼j¼1 ij associated to the time-independent Schr€ odinger equation: Hcl ¼ El cl :

(50)

The simplest way to solve this equation is the well-known Hartree–Fock approximation. This approximation assumes that the electron positions are not correlated, and the multielectronic wave function (cl) can be expressed as a determinant, the so-called Slater determinant, of N one-electron wave functions (fi):    f 1 ðr1 Þ ⋯ f N ðrN Þ    1 (51) cl ðr1 ⋯rn Þ ¼ pffiffiffiffiffi  ⋮ ⋱ ⋮ : N!  f ðr Þ ⋯ f ðr Þ  N N 1

N

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Handbook of Magnetic Materials

The one-electronic wave functions (molecular orbitals) are usually expressed as a linear combination of a set of atom-centered functions (atomic orbitals), {wk}, which are known as basis set: X f i ðri Þ ¼ cik wk ðri Þ: (52) k

There are two main approximations in the Hartree–Fock approach: first, to assume that the multielectronic wave function is well represented by only one Slater determinant, implying that the electronic state is well described by a unique electronic configuration, i.e., how electrons are arranged in the different molecular orbitals; second, to assume that the electron moves in a timeaveraged potential created by the rest of the electrons, instead of suffering an instantaneous electron–electron interaction. Due to these two approximations, the energy obtained from a Hartree–Fock calculation is higher than the actual energy of the system. The difference between these two energies, in the so-called Hartree–Fock limit using an infinite basis set, is called energy correlation: Ecorr ¼ E  Elimit HF :

(53)

Considering the two main approximations of the Hartree–Fock method, the electron correlation can be divided into two types: static and dynamic. The static correlation appears when the electronic state does not correspond to only one electronic configuration but several nearly degenerated ones. The dynamic correlation is due to the instantaneous nature of the electron– electron Coulomb interaction. Therefore, for a given instant, the positions of the electrons must be correlated, the probability in the position of one electron being affected by the positions of all the other electrons. The most straightforward way to treat electron correlation is the configuration interaction (CI) approach in which the multielectronic wave function is expressed as a lineal combination of Slater determinants. Each Slater determinant represents an electronic configuration and it is constructed by how the electrons are distributed in a given set of monoelectronic molecular orbitals. In a full configuration interaction method (full CI), all the possible electronic configurations obtained by distributing the electrons of the quantum system on the different molecular orbitals are considered. Since the molecular orbitals are expressed as a lineal combination of atomic orbitals (the basis set), the number of generated molecular orbitals is the same as the number of atomic orbitals used as the basis set. Using an enough large basis set, the full CI method would properly describe the multielectronic wave function, correctly accounting for both static and dynamic correlations. However, such an approach is too computationally expensive and not applicable for real molecular systems. Therefore, it is a practical requisite to truncate the full CI expansion by considering only a number of Slater determinants. Choosing the right Slater

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determinants, corresponding to the most relevant electronic configurations, would allow to correctly treat the static electronic correlation. Then, there are two different modes to include the dynamic electron correlation. In the variational mode, additional Slater determinants are added to the lineal expansion providing more flexibility to the multielectronic wave function. In the perturbative mode, the contribution of other electronic configurations in both the energy and the wave function is accounted using a perturbative approach. The main difference among the different quantum chemistry methods is how to choose the contributing Slater determinants for the lineal expansion of the multielectronic wave functions and how the dynamic correlation is included.

3.4

CASSCF–CASPT2/RASSI-SO Method

Up to the date, almost all the published quantum chemistry studies on lanthanide-based SMMs are performed using the two-step approach called CASSCF–CASPT2/RASSI-SO (Roos and Malmqvist, 2004) as implemented in the MOLCAS quantum chemistry package (Aquilante et al., 2016; Karlstr€ om et al., 2003). Therefore, here, we are going to describe this method and some details of its implementation in the MOLCAS software. In this method, the spin-free step consists on a complete active space self-consistent field (CASSCF) calculation (Roos et al., 1980) to treat the static electronic correlation followed by the introduction of the dynamic electron correlation through the evaluation of the single and double excitation contributions in a second-order perturbative approach (CASPT2) (Andersson et al., 1990, 1992). The idea of the CASSCF method is to reduce the number of electronic configurations, and their corresponding Slater determinants, with respect to a full CI by splitting up the molecular orbitals in three groups, called orbital spaces: inactive orbitals will be always doubly occupied; virtual orbitals will be always unoccupied; and finally, active orbitals can be occupied by 0, 1, or 2 electrons. These electrons, distributed among the active orbitals, are called active electrons. The different electronic configurations are obtained by arranging the active electrons in the molecular orbitals which belong to the active space, i.e., the active orbitals. From a chemical point of view, inactive and active orbitals are usually related, respectively, with core and valence orbitals. Due to their localized nature close to the nucleus, 4f orbitals are not involved in covalent bonding. Because of that, the active space in the CASSCF calculations is usually limited to only the 4f orbitals, with no need to include molecular orbitals from the ligand molecules. Since the CASSCF/CASPT2 states and energies are used to compute a spin–orbit coupling Hamiltonian term, a large number of CASSCF states must be computed in order to include all the spin-free states in the spin–orbit coupling Hamiltonian matrix which can have a relevant contribution on the lowest-energy spin–orbit states. Because of that, individual CASSCF calculations for each spin-free state are not affordable. Instead, state-averaged

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CASSCF calculations are performed: several states are computed at once by optimizing the same monoelectronic wave functions for all the states instead of using individually optimized monoelectronic wave functions for each state. Scalar relativistic terms are included in the first step of the computation. An important effect of these terms is that they affect significantly the radial shape of the electronic orbitals, producing a contraction of the s and p orbitals and an expansion of the d and f orbitals. In most of the available basis sets for ab initio calculations, their atomic orbitals have been optimized using a nonrelativistic electronic Hamiltonian. Hence, they are not well adapted for reproducing the changes in the orbital shape due to the scalar relativistic effects. Therefore, a suitable basis set for relativistic ab initio calculations should have been optimized with a Hamiltonian including these terms. This is the case of the ANO-RCC basis set (Roos et al., 2004, 2005) as implemented in MOLCAS. This basis set has been generated using the scalar relativistic terms of the DKH2 Hamiltonian and, therefore, it is suitable for relativistic calculations with that Hamiltonian. The methodology here exposed has some size limitations: first, the computation time increases with the size of the quantum cluster. Therefore, a not too large quantum cluster must be extracted from the crystal structure but without compromising the accuracy of the computation. In order to do that, sometimes it is useful to replace some molecular fragments which are not close to the lanthanide ion with chemically sounded and simpler molecular ones. Second, because of the nowadays limit in the size of the active space, only one lanthanide ion can be treated in the computation. Therefore, in polynuclear systems only the 4f orbitals of one of the lanthanide ions will be included in the active space. Therefore, other lanthanide ions are usually replaced by some appropriate ion with no unpaired electrons. For instance, for late trivalent lanthanide ions, the Y3+ has a similar ionic radius, a close electronic structure (except for the 4f ions), and a close chemistry.

3.4.1 Computational Data Analysis Relativistic quantum chemistry calculations provide both the energy-level structure and the composition of the multielectronic states. These computational data can be used in order to model and analyze experimental results for a better understanding of the static and dynamical magnetic behavior of the compounds. In this respect, Ungur and Chibotaru have implemented a module in the MOLCAS software package, Single ANISO (Aquilante et al., 2016), which allows to compute a large number of physical magnitudes, such as magnetic susceptibility, magnetization, CF parameters, gyromagnetic factors, or transition magnetic moments. The next subsections will be focused on some physical magnitudes and information that can be extracted from ab initio calculations which have proven to be very helpful for the comprehension of the magnetism of the

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lanthanide molecular systems, in particular their slow magnetic relaxation behavior. Actually, although these magnitudes are almost always obtained in the literature from ab initio calculations, they could also be get from any other method which provide a right description of the energy levels and the multielectronic wave functions for the lanthanide ions. 3.4.1.1

g-Tensor

A well-established requirement for observing slow relaxation in a lanthanidebased SMM is the presence of a doublet ground state with a large magnetic anisotropy. A way to quantify the magnetic anisotropy is through the concept of the g-tensor. Moreover, g-tensor is valuable for not only quantifying the magnetic anisotropy but also the quantum tunneling rate between the two states of the doublet. In the next paragraphs, we are going to explain the relationship among the g-tensor, the magnetic anisotropy, the quantum tunneling, the energy barrier, and the composition of the multielectronic states. The Zeeman Hamiltonian describing the effect of a magnetic field can be projected on a doublet state by using an effective spin Hamiltonian: !

!

HZE ¼ mB B  ge  S 0 ,

(54) !

where ge is a real symmetric 3  3 matrix, known as g-tensor, and S 0 is an 0 effective doublet spin operator (S ¼ 1/2). In the case of the ground doublet state of a lanthanide ion, this g-tensor can be computed from a relativistic quantum chemistry calculation. The g-tensor is obtained by imposing that the effective spin Hamiltonian must reproduce the energy splitting of the two states of the doublet given by the Zeeman Hamiltonian for any external magnetic field (Bolvin, 2006). The practical procedure to obtain the g-tensor matrix results in its diagonalization, which provides the main magnetic axes (x, y, z). In these axes, the effective Zeeman Hamiltonian can be written as X X mB gi S0i Bi ¼ mi Bi (55) HZE ¼ i¼x, y, z i¼x, y, z where gi are the mean values of the g-tensor and mi ¼ mBgiSi0 are the magnetic moment operators along the main magnetic axes. Appling a magnetic field along one of the main magnetic axes (x, y, z) will break the spin degeneration resulting in two nondegenerated states with magnetic moments along that axis equal to hmi i ¼  12 mB gi , where i ¼ x, y, z. A doublet state with a large magnetic anisotropy corresponds to a large magnetic moment along one of the main magnetic axes (taken as z by convention and named as easy anisotropy axis) much larger than along the two other main axes (hmzi ≫ hmxi, hmyi), which clearly implies that the g mean value along the z-axis must be much larger than along the two other axes (gz ≫ gx, gy). Therefore, as stated earlier, the theoretical determination of the g-tensor and its

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mean values helps to establish the strength of the magnetic anisotropy of the ground doublet state. In the previous discussion, we have considered a Kramers doublet. NonKramers ions with a quasi-doublet ground state fulfill the Griffith theorem, and gx ¼ gy ¼ 0. In this case, the smallness of gx and gy is not an indication of the strength of the magnetic anisotropy of the doublet. However, the computation of the g-tensor and its diagonalization is valuable since it provides the orientation of the easy anisotropy axis and the value of gz. A requirement in order to observe slow magnetic relaxation in a molecular system is the quenching of QTM between the two states of the ground doublet state. The QTM rate is proportional to the square of the tunneling gap (D2T) between the two states (Gatteschi et al., 2006). For non-Kramers systems the CF already produces a splitting between the two states of the pseudodoublet, DT ¼ DCF, splitting that can be obtained from an ab initio calculation. As for Kramers systems, the isolated electronic systems must be degenerated and, therefore, the tunneling should be quenched. However, external stimuli (magnetic field, hyperfine interaction, dipolar interaction, super-exchange magnetic interaction) can break up the degeneracy and allow the quantum tunneling. Supposing that the external stimuli can be treated as an effective exter!

nal field (B eff ), the tunneling gap splitting can be written as 1=2 1  : DT ¼ mB g2x B2eff , x + g2y B2eff , y 2

(56)

Therefore, for Kramers ions, small values of gx and gy favor both the uniaxial magnetic anisotropy and the quenching of the QTM between the ground doublet states, being an essential requirement for observing a slow magnetic relaxation behavior on them. At this point, it is interesting to analyze the relationship between the smallness of the gx and gy values and the composition of the ground doublet in order to make clear its interest for increasing the magnetic anisotropy and quenching the QTM. By supposing that there is not mixing among different J-multiplets, the two states of the ground doublet can be written in the function of j J, mJi states of the ground J-multiplet, taking the ab initio easy magnetic anisotropy axis as the quantization axis. Now, let us consider a Kramers doublet, the two degenerated states will be a lineal combination of j J, mJi states: X X (57) cmJ jJ, mJ i;j1i ¼ Yj1 +i ¼ ð1ÞJ + mJ c∗mJ jJ,  mJ i, j 1 +i ¼ mJ

mJ

where Y is the time-inversion operator. Among the infinite lineal combinations of two degenerated eigenstates, j 1 +i and j 1i are chosen in order to maximize the modulus of the magnetic moment along the easy anisotropy axis. This corresponds to j 1+i mainly composed by positive mJ states and,

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similarly, j1i mainly composed by negative mJ states, resulting in both states mainly located in opposite sides of the energy barrier. In the Ising limit, these two states will be j 1 +i ¼ j J, Ji and j1 i ¼ j J,  Ji, with gz ¼ 2JgJ and gx, gy ¼ 0. In this ideal case, applying a magnetic field perpendicular to the z-axis will not break up the degeneration and there will not be QTM. However, in the case of nonzero transversal g-factors (gx, gy 6¼ 0), applying a magnetic field perpendicular to the z-axis will break the degeneration and produce an energy splitting which will enhance the QTM. In this case h1 j Jx j 1+i 6¼ 0 and h1j Jy j 1 +i 6¼ 0, and therefore, the transversal magnetic field will produce an admixing of the j 1i and j 1 +i states. The larger the splitting, the higher the admixing of the j 1 i and j 1+i, and the higher the delocalization of the nondegenerated states in both sides of the anisotropy energy barrier, enhancing the tunneling between both states due to spin–spin interactions. To sum up, for Kramers ions large magnetic anisotropy requires gz ≫ gx, gy what, in the ideal case, corresponds to the ground-doublet states being mainly j 1+i ¼ jJ, Ji and j1i ¼ j J,  Ji. On the other hand, quenching the QTM requires gx, gy 0, which corresponds to the ground-doublet states being pure mJ states, or, at least, states well localized in one side of the energy anisotropy barrier in order to avoid the admixing of the states by the Jx and Jy operators. Although the previous discussion has been done considering the doublet ground state, the same procedure can be applied to excited Kramer doublets for obtaining their g-tensors, their mean g-factors, and their easy anisotropy axes (Ungur and Chibotaru, 2011). Moreover, the degree of noncollinearity between the easy magnetic axes of the ground doublet state and an excited doublet state has been proposed as an indicator of the probability of transition from one side to the other of the energy anisotropy barrier through an Orbach process involving that excited doublet state (Ungur and Chibotaru, 2011). This is because the larger the noncollinearity of the easy anisotropy axes, the larger the admixing of mJ states in both sides of the energy anisotropy barrier for the excited doublet states and the larger the transition probability from one side to the other of the energy barrier due to spin–phonon interactions. 3.4.1.2 Transition Magnetic Moments In contrast with 3d-cluster SMMs, in most lanthanide-based SMMs the thermally assisted magnetic relaxation is mainly through TAQTM. However, a potential quenching of the TAQTM through the first excited doublet state should produce an increase of the energy barrier for the magnetic relaxation. Related to that, Chibotaru has recently proposed a computational procedure to qualitatively determine the state pathway for the magnetic relaxation between the two ground states of lanthanide-based SMMs (Ungur and Chibotaru, 2011). Such procedure, which starts from ab initio electronic wave functions

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and uses the concept of transition magnetic moments, will be described in the next paragraphs. In addition to the quantum tunneling (QTM and TAQTM), the main processes that are involved in the magnetic relaxation are the direct, the Raman, and the Orbach ones, which have been discussed in detail in Section 2. These three processes are phonon assisted since there is an energy interchange between the electronic system and the lattice by the creation, absorption, or scattering of one or two phonons. The theoretical analysis of these processes by treating altogether the lattice and the electronic system presents an unaffordable complexity. Fortunately, as presented in Section 2, the total system can be treated as the electronic systems in the presence of a phonon bath, as described in Eq. (18). This is a good approximation because the lattice dynamics are much faster than the magnetic ones. This additional term to the Hamiltonian, usually called spin–phonon coupling, takes into account the effect of the lattice vibrations (phonons) on the electronic system, inducing transitions between different multielectronic states. The dynamical CF is expanded in a power series of the atomic displacements or strains (e): dynamical ¼ V0 + eV1 + e2 V2 + ⋯, HCF

(58)

where V0 is the static CF, and V1 and V2 are the first and second derivatives of the CF with respect to the strain, respectively. The expressions for the transition probabilities and/or the relaxation times between two electronic states for the direct, Raman, and Orbach processes are given in Section 2, where the power expansion has been considered only up to first order in the strain. Although the expressions are complex, it can be observed that there is a direct dependence between the transition probability between two states, ci and cj, and their corresponding matrix elements hci j V1 j cji. The proposal of Chibotaru supposes that an estimation of the hci j V1 j cji matrix elements can be done by computing the transition magnetic moments hci j ma j cji with ma ¼ mBgJJa and a ¼ x, y, z, which connect mJ states with DmJ ¼ 0,  1. Moreover, for Kramers doublets, it can easily be deduced (see the previous subsection) that the transition magnetic moments will also be proportional to the tunneling rate between the two degenerated states inside a doublet. Therefore, Chibotaru proposes  to use the   average sum of the transition magnetic moments 13 jhmx ij +  my  + jhmz ij between two states as an indicator of the transition probability between them. Consequently, the calculation of these transition magnetic moments allows to determine the state pathway followed by the magnetic relaxation between the two states of the ground doublet: the shortest pathway of states connected by nonnegligible transition magnetic moments (see an example in Fig. 12). The application of this procedure to a hypothetical DyO+ molecule (Ungur and Chibotaru, 2011) predicted an energy barrier for the magnetic relaxation

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1

DyI (1) 1400

2000 1800

1200 1600 1400 2.9 800

−1

4.5 × 10

3− 600 2.3

−1

4.2

0 ×1

0

1.8

3+

−2

−3

1 5×

3.3

1000

10

800

2.9 × 10 0

200

× 2.1

−3

2−

400

1200

4+

4−

2.5

×

2+

400

9.

200

1− −6

600

−4

10

1.5 × 10−4 −10 −8

Energy / K

Energy / cm−1

1000

−4

−2 0 4 2 Magnetic moment

6

8

10

1+ 0 / mB

FIG. 12 Magnetization blocking barrier of (NNTBS)DyI(THF)2, [NNTBS ¼ fc(NHSitBuMe2)2, fc ¼ 1,10 -ferrocenediyl] (Harriman et al., 2017a). The thick black lines represent the Kramers doublets as a function of their magnetic moment along the easy anisotropy axis. Arrows depict the most probable path for magnetic relaxation (red), QTM (blue), and Orbach relaxation (green). The numbers at each arrow stand for the mean absolute value of the corresponding matrix element of transition magnetic moment. Reprinted with permission from Harriman, K.L.M., Brosmer, J.L., Ungur, L., Diaconescu, P.L., Murugesu, M., 2017a. Pursuit of record breaking energy barriers: a study of magnetic axiality in diamide ligated Dy III single-molecule magnets. J. Am. Chem. Soc. 139, 1420–1423. https://doi.org/10.1021/jacs.6b12374. Copyright 2017 American Chemical Society.

higher than 3000 K, indicating that large uniaxial CFs can produce not only large CF splittings but also the quenching of the TAQTM through the lowest doublet states. Indeed, this method has been applied to several compounds with a very strong axial CF in order to rationalize their high experimental energy barriers (Blagg et al., 2013; Chilton et al., 2015; Ding et al., 2016a; Gregson et al., 2016; Gupta et al., 2016a; Harriman et al., 2017a; Liu et al., 2016d; Pugh et al., 2016). In these compounds the calculation of the transition magnetic moments has helped to rationalize the values of the experimental energy barriers pointing out that magnetic relaxation is not through the usual quantum tunneling of the first doublet states but through the quantum tunneling of higher excited states. In the last years, this procedure has being extensively used in the ab initio analysis of lanthanide SMM magnetic relaxation. Because of that, it is important to expose several observations about its applicability.

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First, this procedure approximates the spin–phonon coupling Hamiltonian, or equivalently the dynamical CF, to the average of the transition magnetic moments. A similar approximation has already proven to be useful in the study of the magnetic relaxation in 3d transition metal clusters (Luis et al., 1998). However, in most transition metal clusters the static spin Hamiltonian is mainly of second order in the spin operators ( DS2z ); therefore, it is expected that the first derivative of the spin Hamiltonian with respect to lineal strains could be approximated to a lineal combination of SxSz and SySz operators and its permutations (Gatteschi et al., 2006), connecting states with DmJ ¼  1. The situation for lanthanide ions is much more complex, and V1 should fulfill the same conditions as the static CF. In particular, it can be expressed as a lineal combination of Steven operators Okq with k ¼ 0, 2, 4, 6, connecting mJ states with DmJ ¼ 0,  1,  2,  3,  4,  5,  6 depending on which Steven operators are the most relevant ones in V1. Despite this discouraging complexity of the dynamic CF, its approximation to the average of the transition magnetic moments could be well sounded for strong axial CFs. In this case, by supposing a similar relative weight of the Stevens operators for the static and dynamic CFs, the most relevant nondiagonal (q 6¼ 0) Stevens operators of the dynamic CF could be the O21, connecting mJ states with DmJ ¼  1, as they usually are one of the most relevant for the static CF (Chen et al., 2016b; Liu et al., 2016d). Second, the hci j V1 j cji matrix elements are the transition probabilities between the two states but proportional to them. As can be seen in Section 2, these transition probabilities depend on other factors as the energy difference between the states or the population of the phonons involved in the transition. Indeed, the expression for the transition probability also depends on the type of the relaxation process; therefore, transition magnetic moments cannot provide a measurement of the relative probability among different types of processes. Moreover, in the Raman process the relaxation is through a virtual electronic state; therefore the matrix coupling between the initial and the final states should be more complex than a term proportional to the transition moments, which could fail in predicting nonnegligible Raman transition probabilities. Although the previous observations must be taken into account when using the transition magnetic moments, they can still be very helpful for the analysis of the relaxation pathway in lanthanide SMMs for the following reasons: First, while the transition magnetic moments cannot provide the relative probability of transition among different processes, they can provide the effectiveness of the quantum tunneling in the different doublets, allowing to determine the lowest energy doublet in which the quantum tunneling will be significant. Second, the spin–phonon coupling Hamiltonian can only be approximated as the average of transition magnetic moments for lanthanide ions in which its nondiagonal dynamic CF should be mainly composed by the O21 Stevens operators. Fortunately, strong axial CFs, which are likely to fulfill this

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requirement, are one of the most suitable environments for lanthanide SMMs with high-energy barriers. 3.4.1.3 Electron and Magnetization Spin Densities Recently, Autschbach (Aquilante et al., 2016; Autschbach, 2016) has implemented a formalism in the MOLCAS software package to compute the electron, the spin, and the angular momentum densities of the ab initio multielectronic wave functions once the spin–orbit coupling has been accounted for. For a given spin–orbit multielectronic wave function, c, the electron and the spin magnetization densities can be defined as Z rðr Þ ¼ N c{ cdτ, Z (59) ma ðr Þ ¼ 2 c{ Sa cdτ, where the integration is over all but one spatial coordinates and over all the spin degrees of freedom. In addition, N is the number of electrons and a ¼ x, y, z. The previous spin density can be expressed in a matrix notation in the same basis set of atomic orbitals in which the wave function is written. Diagonalizing this matrix will result in a set of orthonormal molecular orbitals {’p}, called natural orbitals (NO), with eigenvalues np corresponding to the occupation of each NO: X X

2 rðrÞ ¼ np ’p ðrÞ ; N ¼ np : (60) p

p

Equivalently, the spin magnetization density along a ¼ x, y, z can also be expressed as a function of orthonormal orbitals {’ap(r)}, now called natural spin orbitals (NSOs), with eigenvalues nap, corresponding to the spin population of each NSO: i2  X X h nap ’ap ðrÞ ; 2 Sa ¼ nap : (61) m a ðrÞ ¼ p

p

In both cases, due to the multiconfigurational nature of the multielectronic wave function and to the spin–orbit coupling the np and nap populations may be noninteger numbers, with np ranging from 0 to 2 (for a doubly occupied NO) and nap from 1 to 1. Although up to date this formalism has only been applied to a few actinide and lanthanide compounds (Chow et al., 2015; Gendron et al., 2014, 2015; Martı´n-Ramos et al., 2016), we consider that it can be very useful in order to get a visualization of the electron and the spin density. Let us remind that after the work of Rinehart and Long (2011), now it is clear that one of the main factors controlling the magnetic anisotropy and the energy anisotropy barrier is the aspherical electron distribution of the 4f electrons. With this

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respect, although the NO and the NSOs and their populations are a simplification of the very complex multiconfigurational orbital structure of the lanthanide ions, they can provide a helpful and intuitive representation of the correlation between the ligand environment and its corresponding CF with the shape of the electronic cloud of the 4f electrons. An example of the visualization of the NSOs for a lanthanide molecular system is shown in Fig. 13.

FIG. 13 NSOs for the [Ga4Dy2(shi3)4(Hshi2)2(H2shi)2(C5H5N)4(CH3OH)x(H2O)x] xC5H5NxCH3OHxH2O (where H3shi ¼ salicylhydroxamic acid) determined along the direction corresponding to the orientation of the magnetization axis. The corresponding spin occupation for each NSO is indicated below each plot. The SI conversion factor is 1 cm1 ¼ 1.986  1023 J and 1 K ¼ 1.381  1023 J for energy. Reprinted from Chow, C.Y., Bolvin, H., Campbell, V.E., Guillot, R., Kampf, J.F., Wernsdorfer, W., Gendron, F.V., Autschbach, J., Pecoraro, V.L., Mallah, T., 2015. Assessing the exchange coupling in binuclear lanthanide(III) complexes and the slow relaxation of the magnetization in the antiferromagnetically coupled Dy2 derivative. Chem. Sci. 6, 4148–4159, https://doi.org/10.1039/C5SC01029B. Published by The Royal Society of Chemistry under a Creative Commons Attribute license.

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3.4.1.4 Magnetic Interactions Although magnetic interactions involving lanthanide ions in molecular systems are usually very weak, there is a high interest in understanding and controlling them since they can be very valuable in two aspects. First, magnetic interactions can produce an energy gap between the two ground states located in both sides of the energy barrier, moving them out the quantum tunneling resonance, and therefore quenching the QTM. Second, strong enough magnetic interactions could promote collaborative effects for the magnetic relaxation. In such case, the energy barrier could be increased since it would not be associated to the flipping of the magnetic moment of one isolated ion but to the flipping of two or more of them. As already stated in Section 2, the two main types of magnetic interactions in lanthanide molecular systems are the dipolar and the exchange ones. The dipolar interaction, see Eq. (35), can be straightforwardly computed once the magnetic moments of the different states have been obtained from ab initio calculations. Moreover, for highly uniaxial anisotropic states, the dipolar interaction can be cast on an Ising Hamiltonian and its corresponding magnetic coupling constant, Jdip, can be easily calculated. As for the exchange interaction, it can be decomposed into two main components, kinetic and potential, both of them related to bielectronic integrals including the localized molecular orbitals of the two interacting electrons (Anderson, 1959; Kahn and Briat, 1976). In the case of lanthanide ions, the existence of several 4f unpaired electrons, the multiconfigurational nature of the electronic states, and the strong spin–orbit coupling makes very complex the theoretical and computational treatment of the exchange interaction. An especial case is the Gd3+ ion, since it has a well-isolated, orbitally nondegenerated 8S7/2 ground state, with null spin–orbit coupling and without a multiconfigurational nature, making the analysis of its magnetic interactions not as complex as for other trivalent lanthanide ions. Indeed, although Gd3+ ion is not suitable for synthesizing SMMs, the studies of the gadolinium magnetic interactions have allowed to shed light on the mechanisms of the exchange interaction of lanthanide ions. Here, it is worth to mention seminal computational analysis of the magnetic interaction in Gd–Cu dimers (Cirera and Ruiz, 2008; Cremades et al., 2012; Paulovicˇ et al., 2004; Rajaraman et al., 2009), which have evidenced several relevant facts for the exchange interaction that can be extrapolated to other lanthanide ions. First, the exchange interaction directly through the 4f lanthanide orbitals is very weak due to their localized nature, resulting in a negligible overlap with the magnetic molecular orbitals of the other magnetic centers. Second, spin polarization from the 4f orbitals toward empty shells, in particular to the 5d one, can enable an exchange interaction through the orbitals of the empty shell, which are more expanded and more involved in covalent bonding than the 4f orbitals, favoring the magnetic interaction. Finally, magnetic interaction can also

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TC

)C

II u(

C

5X

Gd(III) 5d

Gd(III) 4f delocalization

2

Cu(II) 3dx2–y2

JF

X JA F

5 X JF

Gd(III) 4f FIG. 14 A schematic representation of the different mechanisms for the magnetic coupling in Gd(III)–Cu(II) pairs (Rajaraman et al., 2009). Reprinted with permission from Rajaraman, G., Totti, F., Bencini, A., Caneschi, A., Sessoli, R., Gatteschi, D., 2009. Density functional studies on the exchange interaction of a dinuclear Gd(III)-Cu(II) complex: method assessment, magnetic coupling mechanism and magneto-structural correlations. Dalton Trans. 3153–3161, https://doi. org/10.1039/b817540c. Published by The Royal Society of Chemistry.

be the result of a metal–metal charge transfer to the empty 5d lanthanide orbitals. For instance, in the Cu–Gd dimers, the Cu2+ 3d electron can jump to the 5d shell only if its spin is parallel to the spin of the lanthanide ion. Therefore, the excited charge transfer state will stabilize the ferromagnetic Cu–Ln state against the antiferromagnetic one. A graphical representation of the different processes for the exchange interactions in a Cu–Gd dimer is shown in Fig. 14. Gd3+ ions are also used for obtaining an ab initio estimation of the magnetic interactions of other lanthanide ions by replacing these ions by the Gd3+ ones (Langley et al., 2013). This is usually done using the density functional theory (DFT), an ab initio approach which is not so computationally expensive as the CASSCF–CASPT2/RASSI-SO quantum chemistry approach. DFT is well suited for studying the pure-spin ground state of Gd3+ ions, but not for studying the magnetic interactions of other lanthanide ions due to the multiconfigurational nature of their electronic states and, even more important, their strong spin–orbit coupling effects. The estimation of the magnetic interaction of an anisotropic lanthanide ion by replacing it by the isotropic Gd3+ ion before performing the ab initio calculation supposes that the obtained magnetic coupling constant can be simply rescaled to reflect the exchange Hamiltonian with the spin of the original lanthanide ion. Therefore, this approach neglects any dependency of the magnetic interaction with the actual electron distribution of the lanthanide ions, what is a very drastic assumption. The already mentioned complexity of the magnetic exchange interactions in lanthanide molecular systems is the reason that, with the exception of

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gadolinium molecular systems and up to our knowledge, the CASSCF– CASPT2/RASSI-SO quantum chemistry approach has only been employed for computing 2p–4f magnetic interactions (Ortu et al., 2017). In the case of 3d–4f, 4d–4f, or 4f–4f magnetic interactions, the size of the required active space and the number of their corresponding active electrons make unaffordable such computations. Still, ab initio calculations can be employed in order to model the magnetic interaction of the lanthanide ions. That is done using a magnetic interaction Hamiltonian applied to the wave function states of the interacting ions, which have been obtained by fragment ab initio calculations, one for each interacting ion. Then, the magnetic coupling constants are used as free parameters in order to fit experimental data, usually magnetic susceptibility and magnetization. Even in this case, a formal treatment of the magnetic interaction is a hard task, since the magnetic interaction Hamiltonian for orbitally degenerated states would be a very complex expression (Iwahara and Chibotaru, 2015; Levy, 1964). An usual simplification, known as the Lines model (Lines, 1971), is to consider an isotropic exchange Heisenberg Hamiltonian with a unique magnetic coupling constant, J, for each pair of interacting magnetic centers. Although this approximation is only strictly exact when the two interacting magnetic centers can be represented by either isotropic or Ising spins, it has proven to be valuable for modeling the magnetic interaction of lanthanidebased SMMs (Chibotaru et al., 2008; Ungur et al., 2009). The reason is that due to the usual weakness of the exchange interaction, much lower than the CF splitting in lanthanide SMMs, the main contribution of the magnetic interaction to the magnetic signal is due to the ground doublet state. This ground state is highly uniaxial, close to an Ising state, one of the two limit cases in which the Lines model is exact. For strong magnetic interactions comparable to the CF splitting, the Lines approach would not be appropriated and a much more complex expression for the magnetic interaction should be employed (Vieru et al., 2016a).

3.4.2 Validity of the CASSCF–CASPT2/RASSI-SO Method The final part of this section is devoted to a discussion on the validity of the CASSCF/RASSI-SO method and its accuracy when compared with experimental data. One of the usual confrontations of the CASSCF/RASSI-SO method with experimental results is the orientation of the easy magnetic anisotropy axis of the ground doublet state, experimentally obtained by angle-resolved magnetometry (Bernot et al., 2009a). In most cases (Bernot et al., 2009a,b; Boulon et al., 2013; Cucinotta et al., 2012; Da Cunha et al., 2013; Jung et al., 2014b,c; Meng et al., 2016b; Perfetti et al., 2014), comparison between the experimental and the computational easy anisotropy axes of lanthanide

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SMMs has resulted in a qualitative agreement, with angle differences below 10—15°, indicating that the CASSCF/RASSI-SO method usually provides a good description of the ground doublet state. Moreover, such angle differences can be attributed not only to the accuracy of the theoretical computation but also to the precision of the experimental technique. However, there have been some interesting cases in which experiment and computation have not matched. For instance, an experimental/computational study (Boulon et al., 2013) along the series of compounds Na[LnDOTA(H2O)]4H2O, where DOTA is the anion of 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid, showed qualitative agreement among all the derivatives in which the experimental easy anisotropy axis was determined, except for the erbium one. The paper indicated that the erbium derivative was the only one showing an important contribution of the electron dynamical correlation in the CASPT2 step. Therefore, failing in accounting all the dynamical electron correlation due to its complexity can be the reason of the disagreement between experiment and ab initio in the orientation of the easy anisotropy axis for this compound. In addition to the correct orientation of the easy magnetic anisotropy axis, another fact that supports that CASSCF/RASSI-SO produces a good description of the ground state is the smallness of the ab initio gx and gy components of the g-factor, which usually is in good agreement with the existence or not of QTM for the ground doublet state (Aravena and Ruiz, 2013). In an Orbach process through an excited doublet state, the energy barrier for the magnetic relaxation corresponds, in principle, to the energy gap between that excited state and the ground one. Therefore, in the literature it is customary to compare the experimentally determined energy barrier for the magnetic relaxation with the energy-level structure from ab initio computations. Although in some cases there is a good agreement between them, in most cases the ab initio energy gap is larger than the experimental energy barrier (Aravena and Ruiz, 2013). However, this cannot be simply attributed to a tendency of the ab initio method to the overestimation of the energy-level splitting but also to a not so direct correspondence between the experimental energy barrier and the energy of an excited state. First, recently a theoretical and computational work has evidenced that anharmonic phonons can play a relevant role in the Orbach process for the magnetic relaxation of SMMs, resulting in an experimental energy barrier lower than the energy of the excited state involved in the Orbach process (Lunghi et al., 2017). Second, in addition to the Orbach one, there can be other processes taking part in the magnetic relaxation. In such case, the energy barrier obtained by fitting the magnetic relaxation time with a simple Orbach process would be wrong. Actually, although the apparent overestimation of the energy barrier by the CASSCF/RASSI-SO method has usually been attributed to the existence of QTM, recently it has been shown that the Raman process can also be effective in the magnetic relaxation and should be considered in the fitting of the experimental magnetic relaxation times (Pedersen et al., 2015).

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In addition to the determination of the easy anisotropy axis and to the estimation of the energy barrier, the CASSCF/RASSI-SO method is also used for modeling DC magnetic measurements. The ab initio method usually reproduces qualitatively these experimental data; however, sometimes a scale factor must be introduced to the ab initio magnetic simulation (Antal et al., 2016; Marx et al., 2014), scale factor without a clear chemical justification. Finally, the best benchmark for the CASSCF/RASSI-SO method is the energy-level structure obtained from spectroscopic methods. Although the ab initio method usually produces an energy-level scheme coarsely similar to the experimental one, in particular for the lowest energy levels (Bi et al., 2016; Chen et al., 2017; Cucinotta et al., 2012; Pointillart et al., 2015c), there are also cases with important deviations from the experimental data. In particular, there are several studies along a series in which there is a qualitative agreement between the experimental and the ab initio energy-level structure for most of the lanthanide ions in the series, except for the Er3+ one (Baldovı´ et al., 2016b; Giansiracusa et al., 2016; Vonci et al., 2016, 2017). In some publications, an arbitrary scale factor has been introduced in order to recover the qualitative agreement between experiment and computation (Long et al., 2011; Marx et al., 2014), which is usually justified by the limited size of the basis set, by the use of a not accurate nuclear structure in the computation or by an unaccounted dynamical correlation. Despite the nonnegligible deviations between experiment and calculations in some lanthanide-based SMMs, the CASSCF/RASSI-SO method is, in our opinion, the best computational tool at present time for modeling, analyzing, and predicting the static and dynamic magnetic behavior of lanthanide molecular systems. Moreover, some of the previous deviations could be corrected by improved CASSCF–CASPT2/RASSI-SO computations. Some of the improvement that could provide more accurate ab initio results are:  Accurate calculations of a large number of spin-free states to be used as basis set for the diagonalization of the spin-orbit operator.  Inclusion of dynamical electronic correlation by performing the CASPT2 correction on the spin-free states and by using a large active space.  Large basis set for providing more flexibility to the molecular orbitals: This large basis set should not be only for the lanthanide ion but also for the ligand atoms in its first shells in order to avoid problems with an unbalanced basis set.  The use of a precise nuclear structure, in particular with accurate positions for the neighbor hydrogen atoms, which have shown a large influence on the electronic structure of the lanthanide ion (Cucinotta et al., 2012; Jung et al., 2014a). Unfortunately, all these improvements, in particular the three first ones, cannot be performed at the same time with the present computational power and some compromise must be found. For instance, in most computations the CASPT2

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correction is not applied since the huge computational cost of its application on a large number of spin-free states from different multiplets. The same is for the size of the active space, usually considering only the 4f shell. However, it has been recently shown for an Er-trensal complex (H3trensal ¼ 2,20 ,200 -tris(salicylideneimido)trimethylamine) (Ungur and Chibotaru, 2016a) that both applying the CASPT2 correction and using a larger active improved the agreement with the spectroscopic and magnetic experimental data. Future increase in the computational power or future improvement on the efficiency of the computational method would allow the application of all the improvement at the same time what should enhance the accuracy of the CASSCF–CASPT2/RASSI-SO method.

4 EXPERIMENTAL METHODS FOR INVESTIGATION OF RELAXATION MECHANISMS IN LANTHANIDE MOLECULAR MAGNETS 4.1 Introduction The physical description of the SMM properties needs from different experimental techniques to determine the most important parameters into play: magnetic exchange interactions, magnetic anisotropy, and ligand field energy splittings. In this section, we give an overview of the most relevant techniques used in the study of slow relaxation in molecular compounds, focusing on the experimental aspects which are less common and have been proven in the recent years to be of special relevance. Particular attention is given to lowtemperature techniques (T < 1 K), where many interesting physical phenomena related to magnetic relaxation are encountered. Experimental techniques where the characteristics of a SMM are evidenced are magnetization and AC susceptibility measurements, through experiments as a function of temperature, magnetic field, and frequency. Complementary, static heat capacity measurements may provide valuable information about electronic levels and energy of phonons. Global thermomagnetic characterization serves as well to determine the magnetic interactions and the physical model of the SMMs. Spectroscopic methods, on its turn, have become very important in the last few years to determine experimentally the electronic structure of the ground multiplet. This renewed interest in the CF splitting of f-elements, which was a matter of study in the 1960s and later on, is caused by the need to corroborate theoretical studies and to explain observed relaxation processes. Magnetization obtained as a function of temperature and magnetic field is the most used macroscopic measurement, normally on polycrystalline samples, where an averaged value is obtained. Measurements on single crystals are less common, due to the difficulties in synthesis of sufficiently big samples and additional experimental complexity.

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DC Magnetization and AC Susceptibility Experiments

The commercialization of SQUID (Superconducting Quantum Interference Device) magnetometers of high sensitivity by the company Quantum Design (QD) has triggered an exponential increase of studies of magnetic properties of different kinds of materials. The extended use of the model MPMS® magnetometer (MPMS®) in research laboratories worldwide gives a trademark signature for measurements and comparison of results in the field of magnetic materials (MPMS®, n.d.). A SQUID sensor (Fagaly, 2006) is the most sensitive device to measure magnetic flux, which in combination with a superconducting detection coil allows measuring magnetic moments down to 1012 Am2. These instruments can measure magnetization and AC magnetic susceptibility as a function of magnetic field frequency in a wide frequency window, from 0.01 to 1400 Hz. The use of the SQUID sensor offers the great advantage of measuring with a frequency-independent sensitivity, enabling very low frequency measurements. Magnetic properties are commonly obtained in between 300 and 1.8 K. Measurements are mostly done on polycrystalline samples, where random orientation of crystals is considered. For SQUID measurements, a volume of 60 mL is enough to obtain a good signal-to-noise ratio even at low magnetic fields and high temperatures. Obtained data can be complemented with measurements at higher frequencies and/or lower temperatures using other scientific instruments. Occasionally, AC susceptibility measurements are carried out in the ACMS (AC Measurement System) option of conventional PPMS® (PPMS®, n.d.), with a 10 Hz to 10 kHz frequency range.

4.2.1 DC Magnetization When a constant magnetic field is applied, the sample is magnetized and the equilibrium value of the magnetic moment of the sample is measured. By normalizing to the sample volume or to the number atoms, the static magnetization of the sample, M, is obtained in macroscopic (A/m) or atomic units (mB/f.u.). In DC measurements, it is considered that the sample is in thermodynamic equilibrium after the application of the magnetic field. Experimental time, τexp, is larger than sample relaxation time, τ, τexp ≫ τ. Magnetic field is settled and the resulting steady-state magnetization is measured. Experimental measuring time in DC measurements depends on the instrument. Usually it varies in between 1 and 100 s. In the case the sample response is measured as the field is swept, out-of-equilibrium magnetization can be measured when the magnetic field rate is comparable to the sample magnetization relaxation time, τexp  τ (see below). In the case of lanthanide (Ln) molecular magnets, DC magnetization measurement allows to obtain the paramagnetic properties of the magnetic ions at high temperature, and the transition to a blocked magnetic state at low

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temperatures, where a magnetized state of pure molecular origin is retained for very long times. The manifestation of SMM behavior at temperatures below a blocking temperature, TB, is the appearance of an open hysteresis loop in the magnetic field dependence of the magnetization, M(H). The temperature dependence of the magnetization is a direct measurement of the DC magnetic susceptibility, defined as wDC ¼ M =H :

(62)

The high-temperature limit of the product wDCT tends to the Curie constant of the paramagnetic behavior of the complex under study. Magnetization as a function of temperature decreases on cooling following the depopulation of the ligand field split levels. In some lanthanides, this variation can be fitted within the zero-field splitting model as a function of Stevens operators (see Section 2 for further details). This approach, in the case of powder samples, leads to overparametrized functions which yield partial or inaccurate information of the electronic levels. The use of ab initio calculations instead allows the simulation of the predicted curve, which can be compared to experiment, validating the theoretical model (see Fig. 15).

FIG. 15 The DC susceptibility temperature product (wDCT) of a powdered sample of {Dy2Ba (a-fur)8} and theoretical curve calculated using the ab initio calculated eigenfunctions and eigenvalues. The wDCT product per mol tends to C ¼ 2gJ2J (J + 1)/8 ¼ 28.58 mB2 above 100 K. This yields an effective paramagnetic moment of meff ¼ 10.69 mB, very close to the free ion limit. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility. Reprinted with permission from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https://doi.org/10.1039/ C4DT00538D. Published by The Royal Society of Chemistry.

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In Ln-coordinated compounds, magnetic interaction between SMMs is weak, but it is clearly manifested at low temperature, where ferromagnetic and antiferromagnetic interactions between individuals are seen at different levels: dimerization, 1D chains, and short-range and 3D long-range magnetic order. Magnetic interaction contribution to wDC(T) has to be taken into consideration. Investigation on diluted samples may discard the magnetic interaction factor in the study of relaxation processes in SMM. As T is lowered, the static magnetization ideally exhibits the effects of thermal and magnetic field hysteresis characteristics of molecular magnets. In a zero field-cooled (ZFC) and field-cooled (FC) experiment, magnetization (or magnetic susceptibility) presents a divergence in both curves below the blocking temperature. The ZFC curve shows a maximum at a temperature below TB (see, e.g., Fig. 16A). The definition of the blocking temperature is not clearly stated in DC measurements, as the reported values depend on the experimental conditions. Below TB, butterfly hysteresis loops are obtained, with small or reduced remanent magnetization caused by the often fast QTM of lanthanide SMMs at zero field. The hysteresis loop shape is dependent on the temperature and magnetic field sweep rate. The variation with magnetic field ramping speed depends on the magnetization relaxation time. In this sense, TB is defined as the temperature below which an open hysteresis loop is measured at the zero field. As this condition depends on the field-ramping speed, no consensus is found in the literature so far. Hysteresis loops are reported for field sweeping rates ranging from 2 to 22 mT/s when measured in conventional instruments (SQUID magnetometers and PPMS magnetometers). Larger field rates can be achieved in last generation SQUID magnetometers (up to 70 mT/s in MPMS® 3) and other instruments resulting in higher blocking temperatures and coercive fields. As an example, in Fig. 16 the remarkable properties of a Dy(III) SIM reported by Liu et al. (2016d) are shown. The observed divergence in the ZFC and FC curves occurs at 9.5 K, while open hysteresis loops are obtained up to TB ¼ 14 K. This apparent discrepancy is explained by the difference of the scan speed of the hysteresis loops (0.02 T/s) and the relaxation times. As already stated, magnetic characterization is usually done in polycrystalline or powered samples. To prevent orientation of the grains with magnetic field, crystals must be soaked or embedded in a magnetically inert liquid or grease to fix the random grain orientation at low temperatures. This orientation at high fields is otherwise nonnegligible and uncontrolled in high anisotropic Ln3+ ions. See, for instance, the effect in M(H) in Fig. 17 (Bartolom
e et al., 2014a). Examples of substances commonly used in the literature are eicosanne, N-apiezon, vaseline, Daphne Oil, hexane, etc. Investigation of magnetic properties in polycrystalline samples is sometimes not sufficient to provide a complete picture of the magnetic anisotropy of Ln-based SMMs. These studies can be complemented with magnetic

2.25

B

2 HDC = 2 kOe

0.04

0.8

0.02

0.6

0.00

0.4

−0.02

0.2

−0.04 −1000

14 K 13 K 12 K 11 K 10 K 9K 8K 7K 6K 5K 4K 3K 2K

14 K

FC

13 K 12 K

1.5

M / Ms

cm /cm3/mol

1.75

1.0

M / Ms

A

ZFC 1.25

0.0

−500

0 H / Oe

500

1000

−0.2 −0.4

1

−0.6 0.75

−0.8

0.5 0

2

4

6

8

10 12 T/K

14

16

18

20

−1.0 −30 −25 −20 −15 −10 −5 0 5 H / kOe

10

15

20

25

30

FIG. 16 Magnetic characterization of a powder sample of Dy(bbpen)Br SIM. (A) Magnetic susceptibility vs temperature during FC (blue) and ZFC (red) measurements. (B) Variable field magnetization data in the 2–14 K. Data taken at the sweep rate of 0.02 T/s (Liu et al., 2016d). The SI conversion factor is 1 cm3/mol ¼ 4p  106 m3/mol for molar susceptibility and 1 Oe ¼ 103/4p A/m for magnetic field strength. Reprinted with permission from Liu, J., Chen, Y.C., Liu, J.L., Vieru, V., Ungur, L., Jia, J.H., Chibotaru, L.F., Lan, Y., Wernsdorfer, W., Gao, S., Chen, X.M., Tong M.L., 2016d. A stable pentagonal bipyramidal Dy(III) single-ion magnet with a record magnetization reversal barrier over 1000 K. J. Am. Chem. Soc. 138, 5441–5450, https://doi.org/10.1021/jacs.6b02638. Copyright 2016 American Chemical Society.

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FIG. 17 DC magnetization as a function of the applied field, M(H), measured at 1.8 K for complex {Dy(a-fur)3}n, with oil (red circles) and without oil (blue diamonds). Without oil grains are oriented by the magnetic field. The in-oil data are compared to the predicted curve using ab initio calculations for a random distribution of orientations (dashed green line). The SI conversion factor e, E., is 1 Oe ¼ 103/4p A/m for magnetic field strength. Reprinted with permission from Bartolom
Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https://doi.org/10.1039/C4DT00538D. Published by The Royal Society of Chemistry.

measurements on oriented single crystals, where a direct observation of the magnetic anisotropy is performed. Angle-resolved magnetometry is therefore the recommended technique to determine unequivocally the easy axis of magnetization (EAM). In this technique a small single oriented crystal is placed in a rotating sample holder device and magnetization as a function of the relative orientation of the main axes of the crystal and the magnetic field is measured. These measurements can be performed using the automated horizontal rotator option of the MPMS SQUID magnetometer. Angle-resolved magnetometry has been extensively used by Sessoli and coworkers (Gatteschi et al., 2016). However, this technique presents some limitations to the study of SMMs. Individual magnetic centers may present noncollinear anisotropy tensors and the symmetry of the ion is usually lower than the symmetry of the crystal. The same authors present torque magnetometry as an alternative technique to the study of magnetic anisotropy in molecular magnets. Torque magnetometry is based on the basic idea that a magnetic moment m, under a uniform magnetic field H, generates a torque τ, given by τ ¼ m0m  H. In cantilever torque magnetometry (CMT), sample is mounted in a millimeter-scale cantilever and the deflection of the cantilever is measured, which gives a direct measure of the exerted torque. Some nice examples of the potentiality of this

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technique, particularly in the case of lanthanide ions, can be found in the recent review of Sessoli and coworkers (Gatteschi et al., 2016).

4.2.2 Frequency Domain Measurements: Magnetic Relaxation Besides the static susceptibility introduced earlier, it becomes of fundamental interest in the study of SMMs the measurement of the dynamic of differential susceptibility, dM/dH, by applying an oscillating magnetic field, HAC ¼ h0cos (ot), and measuring the sample response as a function of time. The system of magnetic molecules may or may not be capable of following the changes of this external magnetic field. Molecule magnetic moments are excited to different energy levels and relax to the ground energy state via different relaxation processes. When relaxation time is very fast as compared to the frequency of the driving field, magnetic moment of the magnetic ions follows field oscillation. Conversely, when spins relax slowly, the induced magnetic moment may lag behind the drive field. In the AC susceptibility technique, a small oscillating magnetic field is applied to the sample and the instrument measures the susceptibility in-phase or real component, w0 and the out of phase, or imaginary component, w00 : w0 ¼ w  cos ’, w00 ¼ w  sin ’,

(63)

where ’ is the phase shift. The variation of ’ with the temperature and frequency allows the study of relaxation processes within the experimental frequency window, which is normally in between 0.1 Hz and 10 kHz. AC measurements with SQUID magnetometers are able to extend the measurement window in the low-frequency range down to 0.01 Hz, with some limitations regarding sensitivity. Although SQUID sensor sensitivity is frequency independent, low-frequency measurements imply very long measurements, and additional very low frequency noises like electronic drifts are coupled to the measurement. Magnetic susceptibility in the limit of o ! 0 is the isothermal susceptibility w0, which corresponds to the equilibrium susceptibility or wDC. In the limit of high frequency, o ! ∞, when spins are not responding to magnetic field, the adiabatic susceptibility, denoted by w∞, is found. In the latter limit, the spin system is isolated from the thermal reservoir. When SMM relaxation proceeds through a single path, with just one characteristic time, τ, corresponding to a Debye relaxation process, the complex magnetic susceptibility as a function of frequency is given by the expression (Casimir and du Pr
e, 1938): w∗ ðoÞ ¼ w0 + iw00 ¼ w∞ +

ð w 0  w∞ Þ , 1 + ioτ

(64)

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

w00 ¼

ðw0  w∞ Þoτ ðw  w Þ ; w0 ¼ w∞ + 0 2 ∞2 : 1 + o2 τ 2 1+o τ

1

67

(65)

The compliance of the Kramers–Kronig theorem allows to estimate the relaxation time as: τ¼

w00 : ðw0  w∞ Þo

(66)

In many cases, a distribution of relaxation times is found, and the complex magnetic susceptibility should be written as (Cole and Cole, 1941): w∗ ðoÞ ¼ w0 + iw00 ¼ w∞ +

ð w0  w ∞ Þ

1 + ðioτÞð1aÞ

,

(67)

where a denotes the width of the distribution, ranging from a ¼ 0 for a single relaxation time to a ¼ 1, where the number of relaxation times tends to infinite. When the frequency is varied so that the relation oτ ¼ 1 is accomplished, a maximum in w00 (o) and an inflection in w0 (o) curve is observed. In this way, the magnetic relaxation time, τ, can be determined from the position of the maximum of the w00 (o) curve, according to τ ¼ 1/o. The experimental frequency window, f ¼ o/2p ¼ 0.1 Hz to 10 kHz, allows obtaining SMM relaxation times, in the range of 1.6  105 s to 1.6 s. The presence of two different relaxation processes is often encountered; in that case, the complex AC susceptibility data can be fit to the sum of two modified Debye functions: w∗ ðoÞ ¼ w∞1 +

ðw01  w∞1 Þ

ð1a1 Þ

1 + ðioτ1 Þ

+ w∞2 +

ðw02  w∞2 Þ

1 + ðioτ2 Þð1a2 Þ

:

(68)

Sometimes, though the AC susceptibility is clearly frequency dependent, the w00 peaks fall out of the experimental measurement range, preventing the determination of τ through the condition τo ¼ 1. In these cases, it is still possible to estimate the relaxation time by data processing. One possibility is to double fit the experimental w0 and w00 curves to a (generalized) Debye model. Another procedure can be applied when relaxation time temperature dependence follows an Arrhenius law: τ ¼ τ0  exp ðEa =kB T Þ:

(69) 00

0

The activation energy, Ea, can be estimated from w /w vs 1/T plots with a semilogarithmic scale (Bartolom
e et al., 2009):  00  w (70) log 0 ¼ log ðoτ0 Þ + Ea =a kB T: w At low frequencies where the condition oτ ≪ 1 is met, temperature dependence of the relaxation rate can be estimated from Eq. (66), considering w∞ as

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the measured susceptibility at the maximum frequency, w0 (omax) and restricting the calculation to o ≪ omax: τ ðT Þ

w00 ðT Þ ðw0 ðT Þ  w0 ðo

max , T ÞÞo

:

(71)

On the other hand, low frequencies establish a limit to measure long relaxation times, which can be determined by DC relaxation magnetization measurements. After the removal of a high saturating magnetic field, remnant magnetization decays exponentially with time. These kinds of magnetic relaxation measurements extend the temperature dependence of relaxation times to temperatures below the blocking temperature (Chen et al., 2016b). Even though it becomes possible to experimentally observe relaxation at high frequencies, and being able to determine very low relaxation rates, the question that arises is which processes may be labeled as “slow relaxation.” Obtaining short relaxation times is very useful to model the relaxation process by expanding the temperature and/or magnetic field dependence range. This helps in the understanding of the physical processes which are contributing to the spin system to relax to equilibrium. There is not a figure of merit or threshold of what we can consider “slow” or “fast.” In the SMM literature, the search for magnetic relaxation has been the leit motif of many works and the claim of “slow relaxation” has become the requisite to be part of the research field. Keeping this in mind, we might try to set some limits for the definition of “slow relaxation” in the field of molecular magnets, but this is a delicate point. Taking into account an experimental criteria, the most usual frequency measurement window (see Fig. 18), we could propose that a magnetic system shows slow relaxation, in a given temperature and magnetic field range, when its relaxation time is longer than 0.1 ms. In the experimental characterization of a SMM, it is desirable to obtain the temperature and magnetic field dependence of the different relaxation processes taking place. In this respect, the measurement of the AC susceptibility as a function of temperature at different frequencies at zero DC magnetic field is the first experiment to be performed where thermally activated processes are clearly manifested with a maximum of the w00 (T) which shifts to lower temperatures as the frequency decreases. The determination of the relaxation time is given by the frequency dependence of AC susceptibility (see Eqs. 64 and 65); therefore, once the temperature range of interest is determined, a measurement of the susceptibility as a function of frequency for different temperatures is recommended. Moreover, in the representation of susceptibility as a function of frequency, relaxation processes with none or little dependence with temperature become discernible (see Fig. 19). If no relaxation is observed at zero field, most likely due to quantum tunneling of the magnetization, the application of an external magnetic field may allow the suppression of the QTM and the observation of slow relaxation.

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69

FIG. 18 Frequency and temperature experimental window for the determination of magnetic relaxation times. Stripped pattern shows the conventional instrumental frequency range for AC susceptibility measurements. The lowest temperature achieved by SQUID magnetometers, 1.8 K is marked in blue as the limit for very low T experiments. Two hypothetical sets of relaxation times following an Arrhenius law T dependence are depicted; stars: τ0 ¼ 1.0  109 s, Ea/kB ¼ 100 K; triangles: τ0 ¼ 1.0  107 s, Ea/kB ¼ 20 K. The red line sets the yardstick 100 s relaxation time for blocked molecules.

In those cases where field-induced magnetic relaxation is detected, the maximum of the w00 (o) at a given T is measured as a function of DC magnetic field. The “optimum field” is determined either by the highest maximum intensity or by the longest relaxation time. If more than one process is revealed, the optimal field may be chosen as the one where different processes are visible. Characterization of relaxation processes as a function of temperature at one or several DC fields is commonly performed in the SMM literature. The blocking temperature, TB, is usually benchmarked as the temperature at which the relaxation time becomes 100 s (Gatteschi et al., 2006). This would correspond to the maximum of w00 (T) measured at 0.0016 Hz, below the low-frequency limit of experimental window (see Fig. 18). Therefore, TB is usually estimated by extrapolating the temperature dependence of the relaxation time, τ(T). The temperature and field dependence of the relaxation time τ(H,T) are used to interpret the type of relaxation mechanisms through which the system relaxes, according to Table 1 given in Section 2. A typical example of what can be extracted from AC susceptibility measurements is shown in Fig. 20, for Nd(III) in cyanoacetate polymers {[Nd2(CNCH2COO)6(H2O)4]2H2O}n (Arauzo et al., 2014). Measurement of AC susceptibility as a function of temperature, frequency, and external magnetic field allows the determination of

FIG. 19 Observation of different relaxation processes in {Dy2Ba(a-fur)8} polymer at H ¼ 2 kOe (Bartolom
e et al., 2014a). (A) w0 and w00 temperature dependence at different frequencies shows an Orbach process. (B) The frequency dependence at different temperatures shows additionally the presence of a T-independent very slow process. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility and 1 Oe ¼ 103/4p A/m for magnetic field strength. Reprinted with permission from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https://doi.org/10.1039/ C4DT00538D. Published by The Royal Society of Chemistry.

FIG. 20 AC susceptibility data for complex {[Nd2(CNCH2COO)6 (H2O)4]2H2O}n. Left: Imaginary component of the AC susceptibility as a function of frequency, w00 (f ); (A): w00 (T) at a constant field of H ¼ 1.5 kOe at different temperatures; (B) at a constant temperature of T ¼ 2.0 K, at different applied fields. Right: Relaxation time of relaxation processes found for the Nd complex. Top: Temperature dependence at H ¼ 1.5 kOe, τf(1/T) and τs(1/T). The high T region is fitted to an Arrhenius law with activation energy Ueff/kB ¼ 26.6 K. Bottom: Field dependence of the relaxation time for the two processes at T ¼ 2.0 K. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility and 1 Oe ¼ 103/4p A/m for magnetic field strength. Adapted with permission from Arauzo, A., Lazarescu, A., Shova, S., Bartolom
e, E., Cases, R., Luzo´n, J., Bartolom
e, J., Turta C., 2014. Structural and magnetic properties of some lanthanide (Ln ¼ Eu(III), Gd(III) and Nd(III)) cyanoacetate polymers: field-induced slow magnetic relaxation in the Gd and Nd substitutions. Dalton Trans. 43, 12342–12356, https://doi.org/10.1039/c4dt01104j. Published by The Royal Society of Chemistry.

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the relevant relaxation processes in this system: two field-induced relaxation processes are clearly identified, a faster one (τf ¼ 105–102 s) and a slower one (τs ¼ 0.1–3 s), assigned to an Orbach process and a direct relaxation process, respectively. Another important aspects to take into account in magnetic relaxation measurements are experiment thermal contact conditions. Magnetization relaxation of a spin system is accomplished by the coupling of the spin system to the lattice. Energy absorbed by lattice modes, on its turn, has to be released to the thermal bath. As a consequence, the environment plays an important role in low-frequency relaxation experiments. PB effects due to poor thermal contact between the sample and the heat bath have been often observed, mostly in direct processes (Arauzo et al., 2014; Bartolom
e et al., 2016). In most AC susceptibility measurements, the thermal contact with the sample is provided by helium residual gas in the sample chamber. In typical experiments in SQUID magnetometers and PPMS systems, the sample is immersed in a sample chamber with a low pressure of helium, which is of less than 0.1 Torr at 2 K. By modifying the pressure of the helium gas atmosphere, variations in the relaxation process of the spin system can be observed. Phonon-bottleneck effects in spin–lattice relaxation and the influence of experiment thermal contact conditions have been extensively studied in the past (Gerritsma et al., 1978; Flokstra, 1974). See Section 2 for further details. In Bartolom
e et al. (2016), the field-induced slow relaxation observed at 2–3 K and attributed to a direct process is highly affected by the pressure of the He residual gas. Relaxation time is increased by two orders of magnitude when the pressure of the exchange gas is increased (Fig. 21).

4.2.3 Sub-Kelvin Temperature Measurements At present, understanding of the physics underlying relaxation processes has motivated the research of SMMs at very low temperatures, where a plethora of interesting quantum effects, low energy interactions, and new physical phenomena appear. Extending the measuring temperatures to the mK range allows extending the experimental T window more than two orders of magnitude (see Fig. 18). Most important is the appearance of dominant quantum tunneling processes at sub-Kelvin temperatures and the possibility to visualize the transition of thermally activated relaxation to quantum tunneling. At very low temperatures, magnetic interactions within magnetic clusters or within individual moieties play an important role in magnetic behavior. The exchange and dipolar interaction energy, usually low in lanthanides, becomes comparable to the thermal energy and the transition from slow magnetic relaxation behavior to magnetic ordering can be studied in detail (see Section 8). Magnetic measurements at temperatures in the sub-Kelvin range are normally restricted to 3He–4He dilution refrigerator (DR) systems. Custom-made

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FIG. 21 {Tb(a-fur)3}n polymer. w00 (f ) at low T ¼ 2–3 K, H ¼ 3 kOe measured using the SQUID susceptometer under different pressure conditions: sample chamber purged and vented. Effect on the effective relaxation time which is increased two orders of magnitude by increasing the thermal contact through the residual gas. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility and 1 Oe ¼ 103/4p A/m for magnetic field strength. Adapted with permission from Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050, https://doi.org/10.1039/C6TC00919K. Published by The Royal Society of Chemistry.

susceptometers and other dedicated instruments are built to be placed in a DR with the particular constraints of the very low temperature experiments with respect to volume and thermal load. Experiments are not fully automated and relative data are obtained. Absolute values are obtained after scaling the relative data with absolute values measured by SQUID in a common range of temperatures. For DC magnetization measurements at very low temperatures, magnetometers based on microSQUID sensors or micro-Hall-effect devices are normally used. Micro-Hall effect devices are made of semiconductor heterostructures, like GaAs/GaAsAl, which is a realization of a 2D electron gas and shows an enhanced Hall effect at very low T. These devices present an active area in the range 10 10 to 100  100 mm2 for measuring single crystals of 10–100 mm size with the highest sensitivity. Hall sensors are available in the market and can be implemented in a DR or in the helium-3 option of the Quantum Design PPMS commercial instrument extending magnetization measurements down to 0.35 K (Magnetometry by means of Hall micro-probes in the Quantum Design PPMS, 2008). As an example, Fig. 22 shows hysteresis loops, measured between T ¼ 0.4 and 2 K, of a strong easy-axis anisotropy Zn–Dy–Zn complex (Oyarzabal

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1.0

M / Msaf

0.5

0.0

−0.5 0.4 K - 18 mT/s 0.4 K - 6 mT/s −1.0 −1.0

−0.5

0.0 B/T

0.5

1.0

FIG. 22 Hysteresis loops for Zn–Dy–Zn complex, [ZnCl(m-L)Dy(m-L)ClZn][ZnCl3(CH3OH)] 3CH3OH, at T ¼ 0.4 K for different magnetic field sweep rates. Measurements performed with a home-made high-sensitivity micro-Hall-effect magnetometer seated in the helium-3 insert of a Quantum Design PPMS instrument. Reprinted from Oyarzabal, I., Ruiz, J., Seco, J.M., Evangelisti, M., Camo´n, A., Ruiz, E., Aravena, D., Colacio E., 2014. Rational electrostatic design of easy-axis magnetic anisotropy in a ZnII-DyIII-ZnII single-molecule magnet with a high energy barrier. Chem. A Eur. J. 20, 14262–14269, https://doi.org/10.1002/chem.201403670 with permission from John Wiley & Sons, Inc.

et al., 2014). This compound exhibits the typical butterfly-shaped hysteresis loops of a lanthanide SMM. MicroSQUID or nanoSQUID sensors are designed to achieve a good coupling to submicron-sized samples by placing the sample directly on the sensor (Wernsdorfer, 2009). These sensors can be fabricated by electron beam lithography and are commercially available. The use of these kinds of sensors allows the observation of single crystal effects, as for instance, the weak intermolecular interactions measured in the pentagonal bipyramidal Dy SIM with the record magnetization reversal barrier (Ueff/kB ¼ 1025 K). Hysteresis loops of single crystals for different orientations of the magnetic field evidence the presence of AF interactions between Dy(III) ions (Liu et al., 2016d). For AC susceptibility measurements at very low T, the use of ad hocdesigned susceptometers and microSQUID-based susceptometers is reported in the literature. AC susceptibility measurements in the range of 15 mK to 3.0 K using a homemade microSQUID (Drung et al., 2014) seated in a DR have been crucial in the observation of SCM slow relaxation occurring in {Tb(a-fur)3}n polymer (see Fig. 23). SCM slow relaxation in AF chains is enabled by the presence of 2%–4% defects, which break the AF chains into segments (Bartolom
e et al., 2016).

FIG. 23 Left: w0 (T) of {Tb(a-fur)3}n polymer at 0.3 Hz down to 15 mK. Blue line: Prediction for an Ising S* ¼ 1/2 chain with intrachain coupling J*/kB ¼ 0.135 K; red line: susceptibility driven by a concentration of 4% defects in the chain. Right: w0 (T) and w00 (T) measured down to 15 mK at zero bias field for different frequencies showing the SCM slow relaxation. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility. Adapted with permission from Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050, https://doi.org/10.1039/ C6TC00919K. Published by The Royal Society of Chemistry.

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The same authors used a home-built susceptometer (90–13,330 Hz) to observe in the {Dy2Ba(a-fur)8}n furoate-based polymer how very fast relaxation due to QTM slows down when approaching the transition to a 3D longrange AF order at TN ¼ 0.25 K (see Fig. 24). This critical slowing down of magnetic relaxation time is due to the competition of short-range correlations with quantum tunneling fluctuations (Bartolom
e et al., 2014a). In summary, without pretending in this section to give an overview of all the research done in low-temperature studies of Ln SMMs, it can be asserted that the study of magnetic properties at very low temperatures enables the discovery of new interesting physics. Access to mK range is crucial to understand the different parameters influencing magnetic relaxation in SMMs. The appearance of new relaxation processes, the effects of magnetic interaction and the competition with magnetic relaxation, the study of molecular magnets in its blocked state, the observation of quantum effects, and the study of single molecular crystals are some of the examples which point out the potentiality of low-temperature physics experiments.

4.3 Heat Capacity Measurements The equilibrium heat capacity (HC) of a compound is defined as the amount of heat absorbed by this compound when its temperature is changed. Usually, in the case of solids, the heat capacity at constant pressure is measured, which scaled to the mass of the sample results in the specific heat at constant pressure given by:     ∂Q dS ¼T , (72) CP ¼ ∂T P dT P where dS is the associated change of entropy. Although the measurement for a polycrystalline sample is a rather soft curve and this is not a high-resolution technique, the analysis of heat capacity is of very valuable help in a multitechnique approach for the detailed understanding of the relaxation processes in SMMs. In addition, information of magnetic interactions and magnetic transitions can also be extracted from the very low T heat (Arauzo et al., 2014; Bartolom
e et al., 2013, 2016; Evangelisti et al., 1999). Heat capacity is not an extensively used technique in the field of SMMs mainly due to a limited accessibility to low-temperature heat capacity measurements and the difficulty of the interpretation of the results.

4.3.1 Heat Capacity Contributions Heat capacity is related to the internal energy of a system and contains information about the lattice (phonons) and the electrons. The different contributions to

FIG. 24 {Dy2Ba(a-fur)8}n compound. Left: w0 (T) and w00 (T) measured down to 85 mK at zero bias field for different frequencies in a DR. Magnetic relaxation is observed when approaching the magnetic order transition at TN ¼ 0.25 K. Right: Obtained relaxation time as a function of T showing critical slowing downs of the QTM when approaching TN (Bartolom
e et al., 2014a). The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility and 1 Oe ¼ 103/4p A/m for magnetic field strength. Adapted with permission from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https:// doi.org/10.1039/C4DT00538D. Published by The Royal Society of Chemistry.

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the heat capacity are lattice contribution, electronic contribution, hyperfine contribution, and magnetic contribution: CP ¼ Clatt + Cm + Ce + Chf ¼ AT 3 + Cm + gT +

a : T2

(73)

The lattice specific heat, Clatt, can be approximated by the Debye model, which treats the solid as an isotropic continuum medium. This approximation is questionable in the case of molecular solids, where most of the times a T n with n < 3 is observed. Nevertheless, the Debye model allows the determination of the Debye frequency, oD, which represents the maximum acoustic phonon frequency of the crystal. The so-called Debye characteristic temperature TD ¼ ħoD/kB is obtained from the lattice contribution to the heat capacity at low temperatures:  3 Clatt 12 4 T ¼ p n for T≪ TD , NA kB 5 TD

(74)

where n is the number of atoms per formula unit. In the Debye model, the density of phonon states depends quadratically on the phonon frequency up to the Debye frequency. The Debye frequency sets the maximum energy threshold for acoustic phonons which participate in spin relaxation processes. The low-temperature region T < 4 K is of special interest in magnetic systems because the lattice contribution is very small and allows the study of magnetic, electronic, and nuclear degrees of freedom. In many cases, the magnetic contribution to the specific heat, and the associated magnetic entropy, is related to the energy levels of the magnetic ions. CF splitting of the ground state level is manifested in the so-called Schottky anomaly. Most of the lanthanides can be modeled at T < 4 K by a two-level energy system, which can be split by the CF in the case of non-Kramers ions. This splitting may be obtained and modeled as due to CF and/or magnetic interactions by means of heat capacity measurements. The formula for the Shottky contribution for a two-level system is of the form: CP =R ¼

ðD=kB T Þ2 ðx0 =x1 Þ exp ðD=kB T Þ ½1 + ðx0 =x1 Þ exp ðD=kB T Þ2

,

(75)

where D is the energy difference in between level 0 and level 1, and x0 and x1 are the degeneracies of the two levels, respectively (see, e.g., Fig. 25). The SI value of the gas constant is R ’ 8.3145 J/K/mol. More generally, the depopulation of CF split levels at low temperature is manifested in the heat capacity contribution. The specific heat of a compound can be obtained from the system partition function and contains the contribution of all the energy levels of the system.

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79

PcDyPcTbPc* Schottky PcYPcYPc*

10 Specific heat C / R

1

1

0.1

0.01 1

10

Temperature T / K FIG. 25 Specific heat measured on 0.5 mg of microcrystals of [Dy,Tb]Pc complex (black open circle) and nonmagnetic [Y,Y]Pc (blue open squares). The red line is the calculated Schottky anomaly contribution for an effective energy gap of 3.8 K. Reprinted from Lan, Y., Klyatskaya, S., Ruben, M., Fuhr, O., Wernsdorfer, W., Candini, A., Corradini, V., Lodi Rizzini, A., del Pennino, U., Troiani, F., Joly, L., Klar, D., Wende, H., Affronte, M., 2015. Magnetic interplay between two different lanthanides in a tris-phthalocyaninato complex: a viable synthetic route and detailed investigation in the bulk and on the surface, J. Mater. Chem. C 3, 9794–9801, https://doi.org/10.1039/C5TC02011E. Published by The Royal Society of Chemistry. The SI value of the gas constant is R ’ 8.3145 J/K/mol.

"

# Z 00 ∗Z  Z 0 2 dZ CP =R ¼ b ; Z 0 ¼ ; b ¼ 1=kB T, Z2 db 2

(76)

where the partition function is the sum of the contribution of all energy levels with energy Ei and degeneracy xi: Z¼

n X

xi  exp ðEi bÞ:

(77)

i¼1

In a magnetic molecule, the interaction of magnetic moments and the interaction with an external field modify the energy levels of the system and therefore the contribution to the specific heat.

4.3.2 HC Measurement Methods There exist different methods to measure the heat capacity of a compound, adiabatic and semiadabatic. In the last years, a semiadiabatic method based on thermal relaxation (Bachmann et al., 1972) has become the most used technique, based on the availability of a fully automated commercial equipment (HC option of PPMS system from Quantum Design) and the advantage of using small samples (Lashley et al., 2003). In these measurements the thermal relaxation of the sample after the application of a heat pulse is fitted to an

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C(H)–C(0) (J/K)

1.5 × 10−8 1 × 10−8 4.75 K

5 × 10−9

4.1 K

0

3.6 K −5 × 10−9 −8

−1 × 10

−1

T=2.7 K −0.5

0 H (T)

−0.5

1

FIG. 26 Heat capacity of a 20 mg Mn12O12-acetate monocrystal as a function of an applied magnetic field along the easy axis. Reprinted figure with permission from Fominaya, F., Villain, J., Gandit, P., 1997. Heat capacity anomalies induced by magnetization quantum tunneling in a Mn12O12-acetate single crystal, Phys. Rev. Lett. 79, 1126, https://doi.org/10.1103/PhysRevLett. 79.1126. Copyright 1997 by the American Physical Society.

exponential law with a time constant proportional to the heat capacity of the sample. The use of nonadiabatic or dynamic techniques presents the additional advantage of measuring time-dependent phenomena related to magnetic relaxation. In Fominaya’s work (Fominaya et al., 1997), an HC AC-steady state method (Sullivan and Seidel, 1968) was used to measure tunneling processes in Mn12O12–acetate. A modulated power at a pulsation o was applied to the sample as a function of magnetic field in different temperature regimes. Quantum tunneling has been proven to produce observed anomalies in the heat capacity as a function of magnetic field (Ferna´ndez et al., 1998; Fominaya et al., 1997). Peaks in the HC are observed at the magnetic field values corresponding to the crossing of CF levels (Fig. 26). Time-dependent heat capacity of Mn12 clusters has been further analyzed by Sales et al. using a relaxation method (Sales et al., 1999). HC peaks at resonant fields were detected depending on the experimental time window and temperature. Very similar results were more recently obtained for Fe8 SMM (Gaudin et al., 2002). However, calorimetric studies of SMMs are scarce in the literature (Sorai et al., 2006) and no example is found for time-dependent heat capacity of Ln-based SMMs.

4.3.3 Examples of HC Studies on Ln-Based SMMs In this Section, we would like to highlight the interest and usefulness of the low-temperature heat capacity measurements, by illustrating the resolution of problems with the help of this technique.

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10 H

Cm /R

1

0.1

0.01

1E-3

1

10 T (K)

FIG. 27 {[Gd2(CNCH2COO)6(H2O)4]2H2O}n. Magnetic HC, Cm(T, H), and fit to the predicted HC for a randomly oriented system of Gd(III) ions with parameters D/kB ¼ 0.11  K, E ¼ 0, and g ¼ 2 (Arauzo et al., 2014). The SI value of the gas constant is R ’ 8.3145 J/K/mol. Reproduced from Arauzo, A., Lazarescu, A., Shova, S., Bartolom
e, E., Cases, R., Luzo´n, J., Bartolom
e, J., Turta C., 2014. Structural and magnetic properties of some lanthanide (Ln ¼ Eu(III), Gd(III) and Nd(III)) cyanoacetate polymers: field-induced slow magnetic relaxation in the Gd and Nd substitutions. Dalton Trans. 43, 12342–12356, https://doi.org/10.1039/c4dt01104j with permission from The Royal Society of Chemistry.

4.3.3.1 Determination of CF Splitting When the CF splitting is of the order of a few K, heat capacity may be very effective in determining the energy levels. For instance, in the case of Gd(III) in lanthanide(III) cyanoacetate complex, {[Gd2(CNCH2COO)6(H2O)4]2H2O}n, where the CF splitting is of very low energy, Cp allowed the determination of the axial component of the ZFS (Arauzo et al., 2014). Fig. 27 shows the measured HC at different fields, together with the simulation taking into account the ZFS and the Zeeman terms. In the same work, the heat capacity measurement at H ¼ 0 for the Nd compound, {[Nd2(CNCH2COO)6(H2O)4]2H2O}n, was modeled considering a magnetic interaction between Nd ions, allowing the determination of an average value for the anisotropic exchange interaction j Jz j/kB ¼ 0.15 K. 4.3.3.2 Observation of Magnetic Ordering HC allows detecting the (possible) establishment of long-range ordering, as spin relaxation critically slows down by decreasing the temperature. Fig. 28 shows the signature in the HC of the 3D AF long-range order transition at Neel’s temperature of TN ¼ 0.668 K measured in complex {Dy(a-fur)3}n, enabled by the AF dipolar interchain coupling between ferromagnetic polymeric chains.

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10 0.8

In 2

S/R

0.6

1

0.4

TN = 0.668 K

0.2

CmLT

C/R

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T (K)

CmHT

CLT

0.1 TN = 0.668 K 0.01 1

10 T (K)

FIG. 28 Heat capacity vs temperature for complex, {Dy(a-fur)}n, and fits of the lattice contribution, CLT/R ¼ 2.5  106 T3, and the magnetic contribution above TN: CmHT/R ¼ 0.1566 T2, and below TN: CmLT/R ¼ 3.9228 T3 (T < TN). Inset: Entropy vs temperature obtained from CP(T) data. The SI value of the gas constant is R ’ 8.3145 J/K/mol. Reproduced from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Luis, F., Turta, C., 2013. {Dy(a-fur)3}n: from double relaxation single-ion magnet behavior to 3D ordering. Dalton Trans. 42, 10153–10171, https://doi.org/10.1039/c3dt51080h with permission from The Royal Society of Chemistry.

4 Tb2Cu3

Cm /R

Ho2Cu3

2

0 0

1

2

3 T (K)

4

5

6

FIG. 29 Magnetic molar specific heat of Tb2Cu3 (Tc ¼ 0.81 K) and Ho2Cu3 molecular magnets. Reprinted from Evangelisti, M., Bartolom
e, J., Mettes, F., Jongh, L.J.D., Kahn, M.L., Mathonie, C., Kahn O., 2001. Specific heat of spin ladder lanthanide and transition-metal-based molecular magnets, Polyhedron 20, 1447–1450, https://doi.org/10.1016/S0277-5387(01)00634-9. Copyright 2001, with permission from Elsevier.

4.3.3.3 Determination of Magnetic Interactions Many examples have shown the utility of combining HC and magnetic studies to unveil magnetic interactions of molecular compounds at low temperatures (Das et al., 2015a; Evangelisti et al., 2001, 2004; Prins et al., 2007). Fig. 29

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illustrates the case of Tb2[Cu(opba)]3S compound, which presents a typical l-type anomaly indicating a phase transition to a long-range ordered state at Tc ¼ 0.81 K. 4.3.3.4

Hyperfine Interaction

Cm /R

Most lanthanide ions possess isotopes of different natural abundance with nuclear magnetic moment (Bruker, 2011). Hyperfine interactions or interactions between nuclear spin and electron spin become relevant at low temperatures and they produce the splitting of the ground state (see Fig. 30). The determination of the hyperfine field is of special relevance in the modeling of the magnetic states which participate in the magnetic relaxation and in the relaxation processes themselves. Hyperfine coupling is partially responsible of the effective QTM in Ln ions. From previous examples, we may assert that very low temperature specific heat has a very relevant position among the techniques informing on the magnetic behavior on lanthanide complexes. In some cases, it has opened new avenues on certain phenomena. We illustrate this statement with the example of [Gd(hfac)3NITEt] (Bartolom
e et al., 1996). This compound can be described as a quasi-1-dimensional (1D) molecular magnet with alternating

0.1

0 Oe 1 kOe 2 kOe 3 kOe 4 kOe 6 kOe fit 1 T (K)

FIG. 30 Magnetic contribution to the heat capacity at different fields in {Tb(a-fur)3}n complex. The upturn observed at low temperatures is ascribed to the presence of hyperfine contributions. The fit of the 6 kOe data gives a dipolar hyperfine constant of A/kB ¼ 25(2) mK. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. The SI value of the gas constant is R ’ 8.3145 J/K/mol. Reproduced from Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050, https://doi.org/10.1039/C6TC00919K with permission from The Royal Society of Chemistry.

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spins and competing n.n. and next nearest neighbors (n.n.n.) exchange interactions. The spins are perpendicular to the chain axis, and rotated by an angle y with respect to the adjacent spin, giving rise to two different chiralities. Early adiabatic heat capacity measurements (Bartolom
e et al., 1996), confirmed later with pulsed heat capacity measurements (Cinti et al., 2008), detected two anomalies at TN ¼ 1.88 K and T0 ¼ 2.19 K. Curiously, the transition at TN could be associated to long-range ordering to an helimagnetic structure along the chain; however, no feature accompanied the anomaly at T0 in magnetization or m-SR measurements. It was assigned to the transition from a paramagnetic state at T > T0 to a chiral spin-liquid phase, where the spin rotation angles are disordered along the chain. These quasi-1D chains, [Ln(hfac)3 NITR], with the use of different lanthanides and radicals R, served as a playground to observe a large variety of magnetic phenomena, ranging from 3D magnetic ordering to 1D slow dynamics (SCM behavior).

4.4 Spectroscopic Methods In the field of molecular magnetism the study of the lanthanide electronic structure has become an active field. The determination of energy levels by spectroscopic methods helps in the assignment of the relaxation processes taking part in the SMMs. Specially in the case of Orbach mechanism, where the energy barrier for the reversal of the magnetization is compared to the energy of the CF split levels. The direct techniques to determine energy levels are high-resolution optical absorption and luminescence spectroscopies, far infrared spectroscopy (FIR), X-ray magnetic circular dichroism (XMCD) spectroscopy, EPR spectroscopy, M€ ossbauer spectroscopy (MS), and inelastic neutron scattering (INS) (Table 2).

4.4.1 Optical Absorption and Emission Spectroscopy Absorption and luminescence spectroscopies in lanthanides address the optical transitions between the ground multiplet and the excited levels. Absorption spectroscopy determines the spectrum of excitation energies of the complex under study. It allows identifying the most adapted irradiation for excitation of the emitting levels of a lanthanide in a compound. Absorption from the ligands and the f–f transitions of the lanthanide ion are normally observed (H€ ufner, 1978). Luminescence spectroscopy, which is usually of high efficiency in most lanthanides, provides multiline emission spectra, which allows the determination of the energy levels of the ground multiplet. Provided that the ligand field splitting is very sensitive to the coordination environment, luminescence studies may give information about the presence of different sites for the same ion, and the effect of a subtle change of the ion coordination sphere (Blackburn et al., 2016). As an example, the luminescence spectra

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TABLE 2 Spectroscopic Techniques Used in the Study of Ln Electronic Structure

Technique

Physical Process

Radiation Energy (cm21)

M€ ossbauer

Nuclear transitions

108–1010

Ground state

Inner electrons ionization

10 –10

Ground state

Luminescence

Valence electrons transitions

10 –10

Ground and first excited multiplet

FIR

Electronic transitions and molecular vibrations

10–400

Ground multiplet

EPR

Electronic Zeeman effect

0.3–15

Low-lying levels ground multiplet

INS

Atomic and molecule vibrational, magnetic, and lattice excitations

40–800

Ground multiplet

XMCD

6 4

8 6

Ln Energy Levels

Electromagnetic radiation energy involved and lanthanide energy levels which can be determined with each technique. The SI conversion factor is 1 cm1 ¼ 1.986  1023 J for energy.

of Eu3+ ion in the study of cyanoacetate complexes of the formula {[Ln2(CNCH2COO)6(H2O)4]2H2O}n, where Ln ¼ Eu, Gd, Nd has clearly shown the presence of two distinct sites for the Eu(III) ions (Arauzo et al., 2014). The presence of two 5D0 ! 7F0 bands and the neatly resolved spectra of the 5D0 ! 7F1 into two triplets are only compatible with the presence of two nonequivalent crystallographic sites for Eu in the lattice (see Fig. 31). Lanthanide(III) luminescence bands originate from the very sharp electronic transitions within their valence 4f orbitals, which range in between the visible and near infrared spectral regions, with different wavelengths unzli and Piguet, 2005). See Fig. 32 where the main depending on the Ln3+ (B€ luminescent levels of different lanthanides are shown, and Table 3, where the corresponding energies of the transitions are summarized. The direct excitation of these transitions is in some cases inefficient, since most of these f–f transitions are forbidden due to electric dipole selection rules and they present low molar absorption coefficients. In applications where Ln luminescence properties are used, this problem is overcome by the use of a highly absorption chromophore ligand which serves as antenna transferring very efficiently the absorbed energy to the Ln(III)-emitting level. Additionally, to obtain high quantum yields, the ligands must protect the Ln3+ ion from nonradiative deactivation pathways due to the presence of OH, NH, and CH vibrations. In this context, some studies have been devoted to the search

FIG. 31 Luminescence spectra at RT and 77 K of the Eu complex in cyanoacetate polymers {[Ln2(CNCH2COO)6(H2O)4]2H2O}n. The presence of two bands in the 5D0 ! 7F0 emission spectra (A), and two triplets in the 5D0 ! 7F1 spectra at LNT (B), indicates the presence of two Eu sites. Reprinted with permission from Arauzo, A., Lazarescu, A., Shova, S., Bartolom
e, E., Cases, R., Luzo´n, J., Bartolom
e, J., Turta C., 2014. Structural and magnetic properties of some lanthanide (Ln ¼ Eu(III), Gd(III) and Nd(III)) cyanoacetate polymers: field-induced slow magnetic relaxation in the Gd and Nd substitutions. Dalton Trans. 43, 12342–12356, https://doi.org/10.1039/c4dt01104j. Published by The Royal Society of Chemistry.

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Gd

40

Tb

Dy

Ho

Er

Tm

Yb 3

35 Pr

Nd

Sm

Eu 6

30

30

P7/2

E / 103 cm−1

1

25

15

5

P0 4

G5/2

1

D2

G4

3

5

5

D

20

G4

F9/2

D4

4

S2

S3/2

15

4

F3/2

F4

25

1

4

5

1 0

D2

D3

2 3

1

10

40

P0

35

20

87

1

2

5

F5/2

10 5

4

I13/2

0

0 3

H4

4

I9/2

6

H5/2

7

F0.1

8

S7/2

7

F6

6

H15/2

5

I8

4

I15/2

3

H6

2

F7/2

FIG. 32 Partial energy diagrams for the lanthanide aquo ions. The main luminescent levels are drawn in red, while the fundamental level is indicated in blue. The SI conversion factor is 1 cm1 € ¼ 1.986  1023 J for energy. Reproduced from Bunzli, J.-C.G., Piguet, C., 2005. Taking advantage of luminescent lanthanide ions. Chem. Soc. Rev. 34, 1048, https://doi.org/10.1039/ b406082m with permission from The Royal Society of Chemistry.

of multifunctional materials, with exceptional optical properties and SMM behavior (Coutinho et al., 2015; Goswami et al., 2016). In the study of the energy levels of Ln(III) as SMM, it is not very common to have in conjunction coordination ligands which fulfill the symmetry requirements to optimize SMM properties, and which serve as efficient sensitizing moiety for the Ln3+ ions. As a consequence, high-intensity emission spectra are not always obtained for every Ln SMM. Luminescence studies of SMM are reported for many different lanthanide ions, but mostly are focused on Dy, Tb, and Yb ions. Spectra can be normally obtained at room temperature, but in order to observe the splitting of the different transitions, high-resolution cryogenic temperatures spectra are required. The spectra resolution and the number of energy levels and relative splitting are a key factor to obtain the complete set of energy levels. In some cases of high complexity, only a rational comparison with obtained ab initio calculations can be done (Bartolom
e et al., 2016; see Fig. 33). When luminescence spectra show enough resolution, the spectra can be fitted using Gaussian functions for the different transition components and a complete energy diagram of the Stark sublevels can be derived (see Fig. 34). This analysis has been done for Dy ion, where a minimum of eight transitions are expected from the basic level of the 4F9/2 excited multiplet to the 6H15/2 ground multiplet (Al Hareri et al., 2016; Mamontova et al., 2016;

TABLE 3 Main Luminescent Transitions of Trivalent Lanthanide Aquo Ions Ln

Excited Statea

Pr

1

G4

1

D2

3

P0

τ Rad/msb

End Statec

n.a.

3

n.a.

3

n.a.

3

HJ FJ HJ

Lumin. Typed

l/nme

Emission Color

4–6

P

1300

NIR

2–4

P

890, 1060

NIR

4–6

F

525–680

Orange

Nd

4

0.42

4

9/2–15/2

F

1060

N1R

Sm

4

6.26

6

5/2–15/2

P

590

Orange

Eu

5

9.67

7

0–6

P

620

Red

Gd

6

10.9

8

P

312

UV

Tb

5

9.02

7

6–0

P

550

Green

Dy

4

1.85

6

15/2–5/2

P

570

Yellow-orange

Ho

5

n.a.

5

8–4

F

970, 1450

N1R

5

0.37

5

8–4

F

540

Green

4

0.66

4

15/2–9/2

F

4

1530

NIR

980

NIR

F3/2 G5/2 D0 P7/2 D4 F9/2 F5 S2

Er

S3/2

IJ HJ FJ S7/2 FJ HJ IJ IJ IJ

n.a.

4

Tm

1

n.a.

3

Yb

2

f

2

I13/2

a

G4 F5/2

1.2

I15/2 HJ F7/2

F 6–4

P F

Most luminescent excited states. Radiative lifetime of the excited state for aquo ions. Radiative lifetimes vary substantially from one compound to another. Range of J-values indicated on the right-hand side. d F: fluorescence; P: phosphorescence. e Approximate wavelength of most intense emission lines (or emission range). f For [Yb(dtpa)]2. Data from B€ unzli, J.-C.G., Piguet, C., 2005. Taking advantage of luminescent lanthanide ions. Chem. Soc. Rev. 34, 1048, https://doi.org/10.1039/b406082m and references therein. b c

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l (nm) 496 0.07

5

D4

488

492 7

484

480

F6

Emission (a.u.)

0.06 Tb(A) Tb(B)

0.05 0.04 0.03 0.02 0.01 0.00

20,200

20,400

20,600

20,800

−1

E (cm ) FIG. 33 The emission spectrum of {Tb(a-fur)3}n at 77 K compared to the electronic energylevel scheme predicted by ab initio calculations for the two types of Tb sites (A and B) in the complex. The SI conversion factor is 1 cm1 ¼ 1.986  1023 J for energy. Reproduced from Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050, https:// doi.org/10.1039/C6TC00919K with permission from The Royal Society of Chemistry.

Ren et al., 2015). In the case of Yb3+, a more simple spectrum is expected, as we are dealing with the transitions from the 2F5/2 to the 2F7/2 (Yi et al., 2014), even though there can also be emissions from the second Stark component of the excited state and vibronic contributions (Pedersen et al., 2015). For Yb3+, luminescence spectra at 77 K have enough resolution to distinguish the main four transitions, and a rationalization of magnetic data and/or ab initio calculations can be performed ( Jung et al., 2014b; Pointillart et al., 2015b,c; Ruiz et al., 2014). For Tb(III) the emission spectrum corresponds to the transition from the 5D4 to the 7F6 ground multiplet, conformed of 13 lines. Low T luminescence experiments are recommended to obtain a well-resolved multiline spectrum. Luminescence spectroscopy under strong magnetic fields presents a Zeeman effect, and it can be used as a complementary method to study the lowest magnetic sublevels. For instance, Bi et al. (2016) have recorded the luminescence spectra of a [CsDyL4] compound under a strong pulsed magnetic field of 36 T observing a shift of the emission peaks. Analysis of the spectra using a reduced Zeeman model provided an estimation of the average value of the gyromagnetic factor (Fig. 35). Luminescence studies are not only useful to support and improve ab initio calculations, but they can also be used as a direct determination of the energy

A

B

D

9

ΔE 21,000

20,450

4

5

8

10

11

7 3

6 8 9

1

Energy (cm−1)

Normalized intensity

2

20,650

20,850

Energy / cm−1

4

F9/2

500 400 300

100

20,910 20,945 20,980 21,015 21,050 C Energy / cm−1 0.1 0.0 −0.1 20,650 20,850 21,050 21,050 20,450

21,017 20,982

20,900

200

10 11

11 7 5 10 9 8 6 4 3 2 1

0

432 382 301 238

6

H15/2

130 61 0 ΔE

Independent variable

FIG. 34 [Dy(NO3)3(H2O)4]2H2O complex. (A, B) Detail of the 4F9/2 ! 6H15/2 transition at 14 K and excited at 385 nm. Multi-Gaussian functions envelope fit (circles) and the components arising from the (orange shadow) first and (purple shadow) second 4F9/2 Stark sublevels to the 6H15/2 multiplet; (C) fit regular residual plot; and (D) schematic diagram of the radiative transitions between the Stark sublevels of the 4F9/2 and 6H15/2 multiplets of the Dy(III) ion. Reprinted from Mamontova, E., Long, J., Ferreira, R., Botas, A., Luneau, D., Guari, Y., Carlos, L., Larionova, J., 2016. Magneto-luminescence correlation in the textbook dysprosium(III) nitrate single-ion magnet, Magnetochemistry 2, 41, https://doi.org/10.3390/magnetochemistry2040041, published in Magnetochemistry under the Creative Commons Attribution License. The SI conversion factor is 1 cm1 ¼ 1.986  1023 J for energy.

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A 0T 1T 5T 10 T 15 T 20 T 25 T 30 T 36 T

Intensity (a.u.)

15.0k

10.0k

5.0k

Peak 2

Peak 1

0.0 500

495

490

485

480

475

470

465

l (nm)

B

4

F9/2 Bext

6

y

H15/2

1,±

y

0,±

y

600

1,± y 0,±

y y

B

0,−

E (cm-1)

A

Ediff (cm-1)

120 110

{DyCs} {DyCs}sub

500

1,−

400 300 200 100

100

0 90 0

5

10

15

20

25

30

35

40

Mag

Lumi

Ab initio

Bext (T)

FIG. 35 Left: (A) Luminescence spectrum at 5 K under a pulsed magnetic field up to 36 T for the compound [CsDyL4]: [CsDy(8-mCND)4(CH3OH)(Me2CO)]2[Me2CO]2. Black arrows point out the trend in peak shift. (B) Zeeman splitting scheme. Right: Energies for the magnetic low-lying states of the ground term obtained by AC susceptibility, luminescence spectra, and ab initio calculations. Data for {DyCs} compound and the sublimed sample, {DyCs}sub. The SI conversion factor is 1 cm1 ¼ 1.986  1023 J for energy. Reprinted from Bi, Y., Chen, C., Zhao, Y.-F., Zhang, Y.-Q., Jiang, S.-D., Wang, B.-W., Han, J.-B., Sun, J.-L., Bian, Z.-Q., Wang, Z.-M., Gao, S., 2016. Thermostability and photoluminiscence of Dy(III) single-molecule magnets under a magnetic field. Chem. Sci. 7, 5020–5031. Published by The Royal Society of Chemistry under the Creative Commons Attribution License.

levels, to assign or discard a thermally activated relaxation process to an Orbach type. In the work of Pedersen et al. (2015), they conclude that the observed relaxation for Yb(trensal) presents an energy barrier of 54 K which is not compatible with the measured energy separation between the ground and first excited Kramers doublet, 663 K. Conversely, Gregson and coworkers are able to assign the relaxation process of Dy(III) in a bis(methanediide) complex with strong axial CF to an Orbach process with energy barriers of 721 and 813 K (Gregson et al., 2016). These energies, which are among the largest reported for a Dy SIM, correspond to the fourth and fifth KD in agreement with ab initio calculations and luminescence measurements.

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When the complexity of the spectrum hampers the determination of all the energy levels, only the high-energy region of the spectrum is analyzed in order to estimate the energy difference between the ground level and the first Stark component. This energy difference is then compared with the effective energy barrier obtained in the magnetic relaxation analysis (Gavey et al., 2015; Long et al., 2015).

4.4.2 FIR Spectroscopy The splitting of the ground multiplet is of the order of a few hundredths of cm1, and it is directly observable by far infrared (IR) spectroscopy (10–400 cm1) (Bloor and Copland, 1972). The main difficulty in IR spectroscopy is to distinguish electronic transitions from phononic and vibronic ones. The identification of the electronic transitions is done in many cases by the observation of a Zeeman shift when applying a magnetic field (Haas et al., 2014; Marx et al., 2014; Moreno Pineda et al., 2014). The CF splitting in the archetypal LnPc2 double deckers, (NBu4)+[LnPc2] 2dmf (Ln ¼ Dy, Ho, Er; dmf ¼ N,N-dimethylformamide), was investigated by FIR (Marx et al., 2014; see Fig. 36). In the case of Ho derivative, the observation of three independent peaks enabled the determination of CF-splitting parameters and comparison to CASSCF calculations. In ab initio determined energy levels, a semiempirical scaling factor of 0.68 was used to account for different effects, including the possible deviations of low-temperature structure from the high-temperature one used in the calculations. The agreement between experimental spectra and calculated by CASSCF-based methods was very good. However, assuming a j6i ground doublet instead of a j5i leads to equally good fits. Therefore, the use of FIR spectroscopy alone does not allow determining unequivocally the eigenstates. 4.4.3 Electron Paramagnetic Resonance EPR technique has been extensively used to study energy-level splitting of paramagnetic systems in the presence of a static magnetic field. It measures the transitions between electronic levels by the absorption of radiation in the microwave region. Main observed transitions are between electronic levels which fulfill the selection rules Dms ¼  1, DmI ¼ 0 (where ms and mI are the electronic and nuclear spin quantum numbers, respectively). CF splittings that can be accessed by EPR techniques are very small. For most lanthanide ions, only the ground state or low-lying excited states can be studied with this technique. Conventional EPR uses microwave energies in the 9 GHz (X-band) and 35 GHz (Q-band) range, which may address energy differences of crystal splitting of the order of 0.4 K (0.3 cm1) and 1.7 K (1.2 cm1), respectively. Additionally, non-Kramers ions are considered as EPR silent, because of the complete removal of the electronic degeneracy by the ligand field with energies out of the scope of conventional

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FIG. 36 Experimental FIR spectra recorded on a pressed powder sample of (NBu4)+[HoPc2]2dmf at 10 K and different magnetic fields. Simulations based on CF parameters (Marx et al., 2014): (A) for a j5i ground doublet; (B) for a j6i ground doublet; (C) derived from CASSCF calculations, after rescaling of the energies of the states by 1.31; (D) reported by Baldovı´ et al. (2012); (E) reported by Ishikawa et al. (2003b). The SI conversion factor is 1 cm1 ¼ € M., Ungur, L., 1.986  1023 J for energy. Reprinted by permission from Marx, R., Moro, F., Dorfel, Waters, M., Jiang, S.D., Orlita, M., Taylor, J., Frey, W., Chibotaru, L.F., van Slageren J., 2014, Spectroscopic determination of crystal field splittings in lanthanide double deckers. Chem. Sci. 5, 3287, https://doi.org/10.1039/c4sc00751d. Published by The Royal Society of Chemistry.

EPR frequencies or forbidden transitions, as this is the case of Tb(III) or Ho(III). High-field high-frequency EPR (HF-EPR) has been more recently developed to expand the study of EPR spectroscopy to higher energies. It uses superconducting magnets up to 25 T and frequencies in the range of 100–500 GHz (3.3–17 cm1). Multifrequency experiments, at different temperatures, provide a more complete characterization of systems. HF-EPR has been shown to be a unique experimental tool to determine CF parameters in large spin clusters, allowing the determination of anisotropy parameters up to second order in powder samples and to fourth order in single

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crystals. Indeed the determination of fourth-order spin operators in Mn12Ac was used to explain the mechanism of quantum tunneling in this system (Barra et al., 1997). In these 3d clusters, all transitions in the ground multiplet are accessible with HF-EPR. This is however not the case for anisotropic Ln SIMs with splittings in the range of 102 cm1. In general, within the field of Ln SMM, EPR technique has been used to complement other spectroscopic techniques with the main objective of defining the ground state model (Lucaccini et al., 2014, 2017). Measurements in single crystals as a function of orientation may allow the determination of the principal values of the g-tensor, hyperfine interactions, and ligand field parameters (Pedersen et al., 2015). The searched ideal conditions for a SMM, namely a strong axial anisotropy with a high mJ ground state, results in a ground state doublet with negligible EPR transition probability. The observation of an EPR signal is an indication of off-diagonal terms which are mixing the mJ levels. In a recent work of Amjad et al. (2016), a model of anisotropy of different Ln ions in LnZn heterodinuclear complexes could be obtained by means of EPR spectroscopy. EPR of Kramers ions was interpreted in terms of an effective S ¼ 1/2 spin ground state with an anisotropy which does not follow expectations according to a simple electrostatic model. The lack of EPR spectra for the Dy derivative is in agreement with the high-energy barrier slow relaxation observed. HF-EPR is often used in the study of non-Kramers ions, as is the case of an Ho(III) SMM encapsulated in a POM cage, where a mJ ¼  4 doublet ground state is observed (Ghosh et al., 2012). Also very interesting is the dedicated study of magnetic interactions in a Dy2 complex by multifrequency EPR, by Moreno Pineda et al. (2014). Magnetic interaction is quantified and they propose the design criteria for the relative orientation of individual spins, which have to be parallel to avoid quenching of the SMM behavior. Rechkemmer et al. (2015) have recently used EPR and HF-EPR to determine with precision the ground-state KD of an Er SIM. They conclude that EPR is crucial for the unambiguous determination of CF parameters, underlining the importance of spectroscopic studies for the research of SMMs (see Fig. 37). All in all, EPR spectroscopy of Ln SMMs can provide valuable information about the anisotropy and composition of the ground state, magnetic interactions, and ligand field parameters. It helps in the interpretation of magnetization dynamics and relaxation mechanisms and supports theoretical calculations. In addition to the classical continuous wave EPR spectroscopy, the use of the pulsed-EPR or EPR in the time domain has emerged as a very powerful technique to study spin dynamics in molecular nanomagnets. The measurement of the phase memory time, or coherence time, of the entangled ground spin states of a molecule gives the substantial information for its applicability as qubit. Most importantly, the application of microwave frequency pulses can be used to control quantum operations in quantum gates, which are the basic elements of a quantum computer.

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FIG. 37 Er(III) SIM: {C(NH2)3}5[Er(CO3)4]11 H2O. Experimental (blue) and simulated (red) X-band EPR (left) and high-frequency EPR (right). Reprinted with permission from € Rechkemmer, Y., Fischer, J.E., Marx, R., Dorfel, M., Neugebauer, P., Horvath, S., Gysler, M., Brock-Nannestad, T., Frey, W., Reid, M.F., van Slageren J., 2015. Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet. J. Am. Chem. Soc. 137, 13114–13120, https://doi.org/10.1021/jacs.5b08344. Copyright 2015 American Chemical Society.

€ 4.4.4 Mossbauer Spectroscopy MS technique is based on the M€ ossbauer effect, in which the nuclear decay of a radioactive nucleus emits a gamma ray which carries information about the nucleus state and, indirectly, about the interaction between the nucleus and the electrons. The most suitable nucleus and most used is 57Fe; therefore, M€ ossbauer studies are restricted to SMM-containing iron (Carretta and Lascialfari, 2007). The iron nuclei are local probes at the molecule to study its magnetic properties. Spin fluctuations influence MS spectrum through the hyperfine interaction. The spectra consist of doublets for paramagnetic species and sextets for blocked spins (within the MS frequency window). In SMMs using 3d/4f combinations of highly anisotropic lanthanide and transition metals, the often fast quantum tunneling of the magnetization can be suppressed by the presence of 3d metals, favoring SMM behavior. In the case where the transition metal is Fe, MS spectroscopy allows to determine at a microscopic level the dynamical state of the Fe spin by the spectrum broadening, the number of different Fe sites by the number and relative weight of the spectral components, and their valence state by the isomer shift. For magnetic characterization, the hyperfine field Bhf acting on the Fe nucleus is the most relevant parameter. The hyperfine field has three components, the contact, the self-orbital, and the dipolar term, the latter responsible for the influence of neighboring Ln ions. In an applied field B, the effective field acting on the nucleus is the vector sum: Beff ¼ Bhf + B:

(78)

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Handbook of Magnetic Materials

Thus, when an external field is applied, the character of the Fe–Fe interaction can be assessed, and indirectly the local anisotropy at the Fe nucleus originated by the dipolar field of the Ln ions. One of the advantages of MS spectroscopy is that it provides a 107–108 s time window and can give detailed microscopic information of the Fe spin and the Ln spin dynamics on a much faster timescale than conventional AC susceptibility. Specifically, the time window for 57Fe MS spectroscopy is given by the Larmor frequency of the iron nuclear magnetic moment around the magnetic hyperfine field. It varies between 8.4  108 s and 1.7  108 s for a hyperfine field between 10 and 50 T, respectively. An example of the usefulness of MS spectroscopy is the extended study of “butterfly” {Fe3LnO2} SMMs (Bartolom
e et al., 2009). At B ¼ 0, spin blocking at T ¼ 3 K was found in the Tb and Dy compounds within the MS observation window, while the Gd showed just paramagnetic behavior. Although the sample is powdered, the evolution of Beff with the applied field B proved the antiferromagnetic character of the Fe–Fe interactions and showed the polarization of the Fe moments in the field direction and the anisotropy of the Dy (parallel to the cluster main axis) or Tb (more isotropic) ions (Mereacre, 2012). In another example, Abbas et al. (2013) used MS spectroscopy in Ln3Fe7 clusters (Ln ¼ Gd(III), Tb(III), Dy(III), and Er(III)) to evidence the influence of the lanthanide ions on the Fe7 subcluster. High anisotropic Tb, Dy, and Er ions induced a slowing of the spin relaxation of all iron sites. Additionally, the application of a magnetic field produced changes in the internal magnetic field values consistent with the effect of spin polarization along the field direction, indicating AF coupling in the Fe7 subcluster (see Fig. 38). The role of 57Fe MS in understanding lanthanide anisotropy in Fe–Dycontaining molecular clusters has been recently demonstrated by Peng et al. (2016). In this work, MS spectroscopy suggests that the fast relaxation observed at the zero field is not only caused by QTM, and other relaxation processes are taking place.

4.4.5 X-Ray Magnetic Circular Dichroism XMCD uses polarized synchrotron radiation to measure the different X-ray absorption spectra between right and left circularly polarized light in the presence of a magnetic field, IXMCD(Eph) ¼ I+(Eph)  I(Eph), where Eph is the X-ray photon energy. This difference gives information on the spin and angular momentum of magnetic absorbing atoms. XMCD is an element-selective technique, and by employing of the magneto-optical sum rules derived by Thole et al. (1992) and Carra et al. (1993), it allows the determination of the spin and orbital moments. These sum rules work well for the heavier 3d transition metal elements and in the case of M4,5 edges (3d3/2,5/2 ! 4f5/2,7/2 transitions) of lanthanide elements, but fail to analyze the L2,3 edges

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FIG. 38 M€ ossbauer spectra of Ln3Fe7 clusters obtained at 3 K, at zero field (left) and 5 or 6 T magnetic field (right). Color lines are the assignments to the different Fe sites. Reprinted with permission from Abbas, G., Lan, Y., Mereacre, V.M., Buth, G., Sougrati, M.T., Grandjean, F., € spectroLong, G.J., Anson, C.E., Powell, A.K., 2013. Synthesis, magnetism, and 57Fe Mossbauer scopic study of a family of [Ln3Fe7] coordination clusters (Ln ¼ Gd, Tb, and Er). Inorg. Chem. 52, 11767–11777, https://doi.org/10.1021/ic401011d. Copyright 2013 American Chemical Society.

(2p1/2,3/2 ! 5d3/2,5/2 transitions) of lanthanide elements with 4f moments, where the considered approximations are no longer valid (Dartyge et al., 1998). XMCD element selectivity in combination with macroscopic magnetic measurements provides valuable information of intramolecular interactions and individual contributions in SMM clusters. In a recent work of Badı´a-Romano et al. (2015) for butterfly Fe3LnO2 molecules with Tb(III) and Ho(III) ions, intracluster exchange interactions could be determined by XMCD. The combination of macroscopic magnetometry with XMCD microscopic magnetometry enabled to evaluate the effect of the Ln ion and the Fe3 subcluster contributions separately and to evaluate the effect of Ln anisotropy on the magnetic interactions within the molecule (see Fig. 39). XMCD has been recently applied to the study of endohedral fullerenes. DySc2N@C80 was studied by both XMCD and SQUID magnetometry for

12 Quasi-symmetry plane

Z

10

M{Fe3TbO2} complex

8 X O(1)

O(2) Fe(3)

Fe(1)

Y

M(μB /fu)

Tb

Fe(2)

6

MTb

4

MFe3

Tb

2 Fe(1)

{Fe3TbO2}

J

Fe(3)

J Fe(2) J′

0 T = 2.7 K

−2 0

2

4

6

8

10

12

14

16

18

Magnetic field (T) FIG. 39 Left: Structure of the {Fe3TbO2} “butterfly” complex, [Fe3Ln(m3-O)2(CCl3COO)8(H2O)(THF)3]. Arrows indicate the magnetic moment direction for different ions in the cluster. Inset: Exchange interaction model. Right: Macroscopic M(H) curve for the complex (black line) and microscopic magnetization curves obtained from XMCD at the L2 edge for Tb (blue triangles). Magnetization of the Fe3 cluster is obtained by subtraction. Theoretical predictions are depicted as blue lines. Reprinted figures with permission from Badı´a-Romano, L., Rubı´n, J., Bartolom
e, F., Bartolom
e, J., Luzo´n, J., Prodius, D., Turta, C., Mereacre, V., Wilhelm, F., Rogalev A., 2015. Intracluster interactions in butterfly {Fe3LnO2} molecules with the non-Kramers ions Tb(III) and Ho(III), Phys. Rev. B 92, 64411, https://doi.org/10.1103/PhysRevB.92.064411. Copyright 2015 by the American Physical Society.

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99

comparison. Open hysteresis loops below 6 K were observed in the two cases. However, differences in the M(H) cycles were noticed, which were attributed to the different field scan rates, 1.3 mT/s for SQUID and 17 mT/s for XMCD, and by a slightly higher sample temperature during XMCD experiments (Westerstr€ om et al., 2012; see Fig. 40A). The same authors (Dreiser et al., 2014b) have shown recently that the shrinking of the hysteresis loop observed by XMCD cannot be explained by a homogeneous temperature rise due to X-ray absorption (see Fig. 40B). XMCD on the endohedral SMM DySc2N@C80 shows that the exposure of the molecules to X-rays resonant with the Dy M5 edge accelerates the relaxation of magnetization more than off-resonant X-rays. It is therefore concluded that high-energy photon irradiation leads to an increased magnetization relaxation rate. It is possible to detect SMM hysteresis with XMCD, but at least in some cases, and X-ray-induced demagnetization which depends on the radiation dose has been observed. The element selectivity and ultrahigh sensitivity of XMCD spectroscopy make it relevant for the study of SIMs and SMMs on surfaces (Dreiser, 2015; see Section 9). The main limitation of this powerful technique, apart from the need to access to a Synchrotron radiation facility, is the potential radiation damage of the sample. The latter factor has to be accurately controlled during the experiment and may preclude the study of certain kinds of molecules by XMCD.

4.4.6 Inelastic Neutron Scattering Since the seminal work of Cacciuffo (Caciuffo et al., 1998), INS has become a powerful tool for the study of the low-lying levels of transition metal SMMs (Amoretti et al., 2008; Baker and Mutka, 2012). This is because neutrons, due to their S ¼ 1/2 spin, can be magnetically scattered, inducing spin transition between different states of the sample (DS ¼ 0, 1, DmS ¼  1). The intensity of the scattered neutrons is measured as a function of the momentum and the energy exchanged with the sample, resulting in a direct probe of the relevant energy levels. Moreover, the scattered intensity profile can be fitted using spin Hamiltonian models, also providing information about the nature of the states involved in the spin transitions. As for lanthanide SMMs, INS has only been employed in a few published works due to two main reasons. First, the large neutron absorption cross section of Dy in its natural abundance makes INS experiments on Dy compounds very problematic (while Dy is the most predominant ion in lanthanide SMMs). Second, the kinematic range of most INS instruments prevents the scan of large CF energy splittings. In addition, there are other difficulties inherent to the INS technique as the large incoherent background produced by the hydrogen atoms, which usually requires deuterated samples for molecular systems, or the low scattering intensities, requiring a large sample mass.

0.6

B E1

T = 2 K, m0 H = 6 T XMCD

Dy M5

0.4 E2

I+ I−

0.4 0.2

XMCD (a.u.)

0.0 0.4

I+ − I−

1.0 × 10−7

SQUID

Dy M4 XMCD asymmetry

I (a.u.)

2.0 1.8 1.6 1.4 1.2 1.0 0.8

0.2

0.5

0.0

0.0

−0.2

−0.5

−0.4

−1.0

Magnetic moment (Am2)

I tot (a.u.)

A

0.2 0.0 1270 1280 1290 1300 1310 1320 1330 1340 Photon energy (eV)

−5

−4

−3

−2

−1

0

1

2

3

4

5

m 0 H (T)

FIG. 40 DySc2N@C80. (A) Sum of X-ray absorption spectra of both X-ray helicities recorded at the Dy M4,5 edge at 6 T (top); polarization-dependent spectra (middle); XMCD spectrum, I+–I (bottom). (B) Hysteresis loops recorded by XMCD (17 mT/s) and SQUID (1.3 mT/s) at 2 K. Adapted with permission from Wes€ R., Dreiser, J., Piamonteze, C., Muntwiler, M., Weyeneth, S., Brune, H., Rusponi, S., Nolting, F., Popov, A., Yang, S., Dunsch, L., Greber T., 2012. An terstrom, endohedral single-molecule magnet with long relaxation times: DySc2N@C80. J. Am. Chem. Soc. 134, 9840–9843, https://doi.org/10.1021/ja301044p. Copyright 2013 American Chemical Society.

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Despite its scarce use, INS on lanthanide compounds has proven to be very valuable in two aspects: first, for the analysis of the magnetic interaction of the lanthanide ions with other magnetic units as radicals (Baker et al., 2015), transition metals (Kettles et al., 2014; Kofu et al., 2013), or lanthanide ions (Prsˇa et al., 2016); second, as a direct probe of the lowest CF energy levels of the lanthanide ion (Giansiracusa et al., 2016; Marx et al., 2014; Pedersen et al., 2014; Vonci et al., 2016, 2017). Indeed, although INS experiments cannot be used to get the full energy-level splitting of the lanthanide ground multiplet, its precise direct determination of the lowest energy levels, in combination with other experimental and computational methods, can be extremely useful for the determination of the CF of the lanthanide ion.

4.5

Conclusions

Experimental investigation of lanthanide-based SMMs and a comprehensive analysis of the magnetic relaxation mechanisms need from a multitechnique approach. Different aspects are addressed and complemented by different techniques with the goal to model the observed behavior at the microscopic level and rationalize the design of SMMs with improved properties. Spectroscopic methods are revealed as very powerful tools in the determination of energy levels participating in the magnetic relaxation processes. Additionally, they are crucial in validating theoretical models and ab initio calculations. Very low temperature measurements in the sub-Kelvin regime are regarded as the way for understanding the physics of the ground state in SMMs. The appearance of fascinating underlying physical phenomena, like quantum effects, competing magnetic interactions of different kind, vortex-like ground states, etc., endorses the prominent role of low temperatures studies. Access to mK range is crucial to understand the different parameters influencing magnetic relaxation in SMMs.

5 5.1

LANTHANIDE-BASED SIMs Introduction

The discovery by Ishikawa and coworkers in 2003 of magnetic hysteresis in double-decker [LnPc2] molecules (Ishikawa et al., 2003b) ignited a huge interest in the dynamics of the magnetization of lanthanide-based mononuclear complexes, also referred to as SIMs. The origin of the peculiar magnetic behavior of lanthanide ions lies in their strong magnetic anisotropy, which stems from the combined action of the spin–orbit coupling and the CF induced by the donor atoms of the ligands, and by the large total angular momentum J (for the second half of the 4f series). When the two states with the largest projection of J are the

102 Handbook of Magnetic Materials

ground state, this results in an anisotropy energy barrier for the reversal of the magnetization. In the absence of other efficient, quantum relaxation paths, it is then possible to observe slow relaxation of the magnetization. Lanthanide-based mononuclear complexes are specially suited for studying slow relaxation and quantum phenomena because they present high versatility for the modification of the spin Hamiltonian via the rational design of the local coordination, and quantum effects are much more pronounced than in polynuclear SMMs. Moreover, Ln-mononuclear moieties offer promising possibilities for the storage and processing of information at the quantum level. Generally speaking, these molecular entities may be used as spintronic devices in two different ways, requiring different conditions: (i) Classical memory SIMs: The observation of magnetic hysteresis opens the possibility of using SIMs as classical memories, encoding the information in the form of Boolean 0 and 1 bits. With this application in mind, main efforts are currently directed toward enhancing the height of the energy barrier for magnetization reversal between the two bistable spin states (Ueff). This requires a high MJ ground state to create an energy barrier, leading to slow spin relaxation and low mixing of the wave functions to minimize the fast spin relaxation through QTM. (ii) Quantum computing qubits: Quantum computing refers to the explicit use of quantum mechanical phenomena (quantum superposition and entanglement) for information technology. The basic element of a quantum computer is the qubit, which can be in j 0i, j 1i, or any linear combination of these two, j ’i ¼ a j0i + b j 1i. The material realization of a qubit is any quantum twolevel system (Aromı´ et al., 2015). Two-level mononuclear Ln complexes may be used as qubits (Ding et al., 2016b), provided the CF can be tailored so as to control the mixing composition of the wave functions, achieve sufficient isolation of the two-level subset from the rest of the spectrum, and minimize decoherence, i.e., loss of quantum coherence due to uncontrolled entanglement with its environment. In principle, the favorable conditions for the design of Ln-complex qubits are (Clemente-Juan et al., 2015): (i) the use of non-Kramers ions (Tb or Ho) with large tunneling splitting, (ii) a coordination geometry and symmetry allowing extradiagonal CF terms, and (iii) to avoid decoherence due to interaction with other nuclei, all atoms in the first coordination sphere should be oxygen and the sample should be deuterated. Single-lanthanide complexes that have been considered for qubits are polyoxometalates (POMs) (Baldovı´ et al., 2013a,b; Martı´nez-P
erez et al., 2012a), the famous TbPc2 (Thiele et al., 2014), and Yb[trensal] (Pedersen et al., 2016). Recent advances in this field have been recently reviewed (Aromı´ et al., 2015; Ding et al., 2016b; Sessoli, 2015). The practical implementation of Ln-based SIMs as high-density information storage memory units relies heavily on increasing both the energy barrier to spin reversal (Ueff) and the magnetic blocking temperature (TB). Research

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 103

in the past years has been directed toward maximizing these two parameters through careful choice of the magnetic ions and tuning of the ligand field environments, in order to increase single-ion anisotropy while preventing QTM, which is seen as a nuisance. These works have shown that the key factors influencing the anisotropy are the type of ion and the coordination environment, determined by the symmetry, the number, and the type of coordinating atoms, and the ligands in the first and even the second coordination sphere (Lee and Ogawa, 2017; Meng et al., 2016a). Studying the magnetic anisotropy of a lanthanide under a given ligand environment remains challenging, due to the difficulty of synthetically controlling all the above factors, some of which are in addition interconnected. In this section, we summarize the most relevant, recent works that pushed our understanding on the role of the different parameters on the slow relaxation phenomena of SIMs, with perspectives of their application as memory units (see Table 4).

5.2

Influence of the Lanthanide Ion

The vast majority of SIMs are based on trivalent lanthanides Ln(III) with 4fn orbitals. They can be generally categorized as Kramers (Ce(III), Nd(III), Sm(III), Gd(III), Dy(III), Er(III), Yb(III), Lu(III)) or non-Kramers (Pr(III), Pm(III), Eu(III), Tb(III), Ho(III), Tm(III)), depending on the half-integer or integer nature of J, respectively.

5.2.1 Electron Density Distribution (Prolate vs Oblate) The free-ion electron density distribution of Ln(III) ions is strongly angular dependent and has a preferred orientation under the electrostatic potential generated by the ligand donor atoms. Therefore, it has been argued that charge distribution of the ligand field can provide an efficient way to control the magnetic anisotropy. In 2011, Long and coworkers proposed a simple model according to which Ln(III) ions having an oblate (equatorially extended) 4f charge density would have larger anisotropy in an axial environment, whereas inversely for prolate (axialy extended) an equatorially coordinate geometry is preferred (Rinehart and Long, 2011; see Section 3 and Fig. 9 for details). This general principle has been verified, e.g., for the oblate ions of the archetypal family of Pc complexes [Bu4N][LnPc2], Ln ¼ Dy, Tb, and also for complexes like Er[N(SiMe3)2]3 based on prolate Er ion in an equatorial field (Zhang et al., 2014b; Fig. 41A). In the later work, two classes of complexes were compared, Ln[N(SiMe3)2]3 with Ln ¼ Dy, Er, in which the lanthanide ion was equatorially coordinated in triangle geometry; while for the Er compound the ground state was the highest, MJ ¼  15/2 and the order of the rest of mJ states was from highest to lowest, for Dy complex the ground state was MJ ¼  1/2 and the order of energy levels was just the opposite, confirming Long’s prediction.

TABLE 4 Relaxation Characteristics of Glossed SIM Complexes Complex

#

Er[N(Si Me3)2]3

1b

Ln Er

Symmetry C3

Ueff/kB (K) 122

τ 0 (s)

τ QTM (s)

H (Oe)

References

9.33  10

9

C

n

0

2.05  10

5

0

Zhang et al. (2014b)

Dy(NHPhiPr2)3(THF)2

2a

Dy

TBP

34

[Ln(tta)3(L)]C6H14

Dy

Dy

 D4d

—a

Yb

Yb

Ln[N(SiMe3)2]3ClLi(THF)3

1-Er

Er

Pseudo-C3

63.3

8.88  108

0

Ln[N(SiMe3)2]3

2-Er

Er

C3

70.5

4.48  108

0

[(C6Me6)Ln(AlCl4)3]

Dy Tb

Dy Tb

Quasi-D5h

101

5.1  1010

Jung et al. (2014b)

0.28  103

0

Liu et al. (2014c)

2000

Lucaccini et al. (2017)

1000

Le Roy et al. (2015) Wada et al. (2017)

No SIM [Ln(trenovan)]

Ce, Nd, Dy, Er, Gd, Yb

Trigonal C3

a Not Orbach

 C8

21

5.5  106

[Li(DME)3][NdIII (COT00 )2]

3

Nd

[Ln(NO3)3(18-crown-6)]

1

Ce

30.3

2.2  1011

0.108

5

1000

[Ln(NO3)3(1,10diaza-18-crown-6)]

3

Nd

30.9

2.2  109

4.1

5

1000

4

Ce

45

2.6  1010

0.52

5

1000

10.7

9

1000

6 

[L2Nd(H2O)5] [I]3 L2(H2O)

1

Nd Nd

73 Pseudo-D5h

1.4  10

8

Zhang et al. (2016c)

16.08

2.64  10

4

0

24.69

5.03  106

0

Gupta et al. (2016b)

[(Tp)Tm(COT)]

1

Tm

Pseudo-C3v

111

4.79  107 2.42  10

[(Tp*)Tm(COT)]

2

[Ln(Murex)3]11H2O

Yb

Yb

C4v

Yb(trensal)

1

Yb

Trigonal

[Cp*Yb(DAD) (THF)]C7H8

1

Yb

20.1

[LnL3]CH3OH

1

Yb

[LnL2(tmh)(CH3OH)] nH2OmCH3OH

1b

[LnL2(tta)(CH3OH)] CH3OH [Ln(OETAP)2]

15.6

2000

Meng et al. (2016b)

2000

Yi et al. (2014)

6

2.73  106 0.15

6.2

2000

Pedersen et al. (2015)

1.74  106

0.00208

9

1500

Trifonov et al. (2015)

11.8

4.6  106

82.1

3.6

1000

Yb

29.7

3.5  107

0.2

6.5

1000

Jim
enez et al. (2016)

1c

Yb

30.3

2.0  107

0.3

6.5

1000

Tb

Tb

383

1.3  108

0

6

0

D4d

1.4  10

Dy

Dy

D4d

32

[Pc2Tb][TBA]+

1

Tb

D4d

840

0

[Pc2Tb][TBA]+  143[TBA]Br

3

Tb

D4d

922

0

[Ln{(Mo5O13) (OMe)4NNC6H4-pNO2}2]3

DyMo10

Dy

D4d

38.5

6.6  108

23.3

6

[LnIII(COT)2]

YbMo10

Yb

D4d

Er

D4d

—

Er

Er

D8d

286

Dy

D8h

Francesca Branzoli et al. (2009a)

1000

Baldovı´ et al. (2016a)

3.7  109

0

5

0

Ungur et al. (2014a)

6.7  10

a

ErMo10

Dy

Gim
enezAgullo´ et al. (2014)

11

2.2  10

Continued

TABLE 4 Relaxation Characteristics of Glossed SIM Complexes—cont’d Complex [L2Dy(H2O)5] [I]3L2(H2O)

#

Ln

1

Dy

10

Dy:Y

Symmetry D5h

D5h

Ueff/kB (K)

τ 0 (s)

τ QTM (s)

H (Oe)

References

5.63  10

12

0

705.3

5.83  10

12

2000

Gupta et al. (2016a)

735.4

1.56  1012

651

C

n

0

12

0 0

[Dy(Cy3PO)2(H2O)5]Cl3 (Cy3PO)H2OEtOH

1

Dy

D5h

472

8.7  10

[Dy(Cy3PO)2(H2O)5]Br3 2(Cy3PO)2H2O2EtOH

2

Dy

D5h

543

2.0  1011

[Dy(bbpen)X] X ¼ Cl

1

Dy

D5h

708

9.46  10

11

0

12

0

X ¼ Br

2

Dy

D5h

1025

4.21  10

[Zn2(L1)2DyCl3]2H2O

1

Dy

C2

430

7.4  1011

1

[Zn2(L )2Dy(MeOH)Br3]

2

Dy

C2

233

2.5  10

0

7

0

[Zn2(L )2Dy(H2O)Br2] [ZnBr4]0.5

3

Dy

C2

121

8.5  10

[Zn2(L2)2DyCl3]2H2O

4

Dy

C2

398

3.5  105

[Dy(BIPMTMS)2] [K(18C6)(THF)2]

2Dy

Dy

721

1.1  106

1

[Cp*2Ln(BPh4)]

813 1

0

2’

Tb Dy

C2

344 498

0

8

5.65  10

Chen et al. (2016b)

Liu et al. (2016d)

Sun et al. (2016b)

0

13

3.07  1011

8

9

6

3.55  10

0

Gregson et al. (2016) Demir et al. (2014)

1.0  10

10

0.0091

0.006

4.1

0

1.0  10

9

1.2

0.0032

3.42

0

DyNCN

Pure

Dy

C2v

335.2

6  1010

0

Guo et al. (2014)

[Ln(hfac)3(NITIn)]

4

Tb

Distorted SAP

1.4

2.4  107

3000

Li et al. (2017a)

r ¼ Indazole (s ¼ 1/2)

5

Dy

0.4

9.9  106

0

5

Tb

D4dTb

a

—

0

Li et al. (2016b)

[DyLz2(o-vanilin)2] Xsolvent

1

Dy

Distorted D4d

221

0

Wu et al. (2016)

X ¼ Br

2

Dy

615

0

[Tb(Phtfac)3(MeTrzNIT)]2 C7H143H2O

.

r ¼ MeTrzNIT (s ¼ 1/2)

X ¼ NO3



2

0

Dy:Y

696

X ¼ CF3SO3

3

Dy

120

5.75  1011

Dy(TFI)3(bpy)

2

Dy

48.8

3.99  106 6

D2d

0 0 Zhu et al. (2014a)

[Dy(TFI)3(Phen)] 0.02ChCl3

3

Dy

D4d

57.9

3.10  10

Dy(EIFD)3(H2O)CH2Cl2 Dy(EIFD)3(DMF)CH2Cl2 Dy(EIFD)3(DMSO)

1

Dy

DistortedC3v-capped

56.7

3.7  107

0

Dy(EIFD)3(TPPO)

2

Dy

Octah.

28.7

2.5  107

2000

6

2000

3

Dy

27.8

4

Dy

No SIM

1.0  10

Dong et al. (2015)

Continued

TABLE 4 Relaxation Characteristics of Glossed SIM Complexes—cont’d Complex

#

Ln

Symmetry

Ueff/kB (K)

τ 0 (s)

H (Oe)

References

[Dy(H4daps) (H2O)3(NO3)] (NO3)2(H2O)

1

Dy

Muffin

32.7

1.82  106

τ QTM (s)

2000

Mondal et al. (2015)

[Dy(H3daps) (H2O)3(NO3)] (NO3)2(MeOH)

2

Dy

Muffin

23.8

9.14  105

2000

Dy(EIFD)3-(bpy)CH3CN

1

Dy

D4d

28.8

1.1  105

0

6

0

Dy(EIFD)3(phen)CH2Cl2

2

Dy

D4d

41.8

7.5  10

Dy(EIFD)3(dpq)CH2Cl2

3

Dy

D4d

41.7

4.7  106

32.4

5

Dy(EIFD)3(dppz)2H2O

4

D4d

1.1  10

C

n

Dong et al. (2016)

450 1500

Dy [Dy(LOMe)2(H2O)2](PF6) Dy(LOMe)2(NO3)

1

No SIM

2

73.6

2.92  10

10

3.44  10

5

0.04547

5

0

Dy(III) N,N-bis(R) methylene-1, 8-diamino-3, 6-dioxaoctane R ¼ imine-2-yl R ¼ amine-2-yl

Lim et al. (2016) Campbell et al. (2014)

1 2

Dy Dy

DB-SAP DB-SAP

50 34

6.8  107

1000

6

2.51  10

1000

Er(trensal) X¼Y¼H

1

Er

X ¼ CH3

2

Er

Y¼I

3

Er

1.93

0.173

9

1.93

0.173

9

Pedersen et al. (2014)

[Hex4N][Dy(DBM)4]

1, pure

Dy

1,dil Dy:Y

27

1.3  107

0

56.6

6.6  1010

300

68.1

3.4  1011

1500

63.8 (LT) 79.8 (HT) [Dy(OH)8]3+(benzo15C5)3

4

Dy

SAP

37.4 (A)

[LnH4LRRRRRR/ SSSSSS(SCN)2] (SCN)2xCH3OHyH2O

R-Dy1

Dy

DCT

34.5

[Dy(PLN)2(HPLN)Cl (EtOH)]

1

Dy

Monocapped trigonal

43.8

1.6  10

10

1.5  10

9

Sun et al. (2014a)

0

800

Al Hareri et al. (2016)

1.1  106

200

Lin et al. (2015b)

3.3  106

0

Lan et al. (2016)

# Complex number as in original reference. Relaxation time according to Orbach τ1 ¼ τ0 1 e Ueff =kB T , or Raman τ1 ¼ τQTM 1 + CTn processes. DB-SAP, distorted bicapped square antiprism; DCT, distorted capped trigonal; TBP, trigonal bipyramide. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. a Slow relaxation not reported.

A

1000 800 E (cm−1)

N Er Er N Si C

B Th Exp

±15/2 ±13/2

C3v symmetry

±11/2

DyIII

600 ±9/2

±1/2 ±3/2 ±5/2

±7/2

±7/2 ±9/2

400 200 0

±5/2 ±3/2 ±1/2

56cm−1

Dy (1a)

82cm−1

YbIII Th

Exp

±11/2 ±13/2 ±15/2

Er (1b)

FIG. 41 (A Compounds Ln[N(SiMe3)2]3. Left: Molecular structure of the Ln ¼ Er compound (1a). Center: Energy levels for Ln ¼ Dy (1a) and Er (1b) compounds (Zhang et al., 2014b). (B) Experimental (dark green) and theoretical (orange) ground-state anisotropy axis for [Ln(tta)3(L)]C6H14, Ln ¼ Dy (left) and Yb (right) (Jung et al., 2014b). Adapted with permission from Zhang, P., Zhang, L., Wang, C., Xue, S., Lin, S.Y., Tang, J., 2014b. Equatorially coordinated lanthanide single ion magnets. J. Am. Chem. Soc. 136, 4484–4487, https://doi.org/10.1021/ja500793x. Copyright 2014 American Chemical Society. Adapted with permission from Jung, J., da Cunha, T.T., Le Guennic, B., Pointillart, F., Pereira, C.L.M., Luzon, J., Golhen, S., Cador, O., Maury, O., Ouahab, L., 2014b. Magnetic studies of redox-active tetrathiafulvalene-based complexes: dysprosium vs. ytterbium analogues. Eur. J. Inorg. Chem. 2014, 3888–3894, https://doi.org/10. 1002/ejic.201400121. Copyright 2014 John Wiley and Sons.

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Several experimental works have shown the different relaxational behavior of isostructural complexes depending on the oblate/prolate character of the used lanthanide. For example, Jung et al. (2014b) compared the magnetic properties of two isostructural complexes, [Ln(tta)3(L)]C6H14, with the lanthanide ion in the distorted-square-antiprism coordination, for Ln ¼ Dy3+ (oblate) and Yb3+ (prolate). While the Dy3+ compound behaved as SIM, no slow relaxation was measured for Yb3+. Angular-resolved magnetometry on single crystals enabled the experimental determination of the anisotropy axes and comparison with ab initio calculations. It was concluded that for Dy complex the easy axis lies along the most negatively charged direction, while for Yb it lies along the less negatively charge direction, almost perpendicular to that found for Dy, in agreement with the oblate and prolate charge distributions of the Dy, Yb ions, respectively (Fig. 41B). Besides, ab initio calculations showed that ground state for Dy was close to gz ¼ 20 (Ising system MJ ¼ 15/2), while for Yb the principal gz ¼ 5.60 differed significantly from the gz ¼ 8 expected for an Ising MJ ¼ 7/2 ground state, possibly explaining the absence of SIM behavior. Nevertheless, subsequent works have evidenced that the “oblate” or “prolate” nature of the ions and the apparent shape of the ligands is not enough to explain differences in the relaxational behavior of SIMs, so subtle differences in the coordination environment have to be considered (Brown et al., 2015; Lucaccini et al., 2014; Zhang et al., 2016c). A recent example was put forward by Zhang et al. (2016c), who compared the magnetic relaxation, ab initio, and electrostatic potential maps of four synthesized complexes, Ln[N(SiMe3)2]3ClLi(THF)3 (Ln ¼ 1-Dy, 1-Er), Ln[N(SiMe3)2]3 (Ln ¼ 2-Dy, 2-Er), and one model complex 1-Dy0 formed by removing the ClLi(THF)3 moiety in 1-Dy complex. In agreement with Long’s predictions, in the predominantly equatorial ligand field, 2-Er showed SIM behavior, while 2-Dy did not. Small changes in the equatorially coordinating ligand field upon the coordination of Cl-ion led to a reduced Ueff of 1-Er (63.3 K) with respect to 2-Er (70.5 K). Surprisingly, in 1-Dy the calculated magnetic easy axis was located in the equatorial plane, away from the pseudo C3 axis of the molecule (Fig. 42, left). Comparison of the electrostatic potential maps for complexes 1-Dy, 1-Dy0 , and 2-Dy (Fig. 42, right) allowed concluding that differences in the magnetic behavior were to be attributed to structural modifications in the equatorial plane leading to critical deviations from the C3 symmetry, rather than to the presence of an axial ClLi(THF)3 moiety. In addition, Gendron et al. (2015) have recently discussed the limitations to a density-only-based rationalization of the magnetic properties of f-element complexes. The authors emphasized that the 4f electron density does not directly contain information about the magnetic moment (only indirectly via the underlying wave function). An analysis of the total magnetization and of its orbital and spin contributions extracted from ab initio calculations was performed on the Ln(COT) 2 series (Ln ¼ Ce, Pr, Nd, Pm,

FIG. 42 Left: Orientation of the calculated gz direction associated with the ground-state KD for complexes Ln[N(SiMe3)2]3ClLi(THF)3 (Ln ¼ 1-Dy, 1-Er), Ln[N (SiMe3)2]3 (Ln ¼ 2-Dy, 2-Er). Right: Electrostatic potential maps for complex 1-Dy, model 1-Dy0 , and 2-Dy, in the equatorial ligand field plane perpendicular to the pseudo-C3 axis. The maps represent the individual contributions of the atomic charges (A), dipole moments (B), and quadrupole (C) moments. The black axes stand for the almost symmetry axes found in the different electrostatic potential maps. Adapted with permission from Zhang, P., Jung, J., Zhang, L., Tang, J., Le Guennic B., 2016c. Elucidating the magnetic anisotropy and relaxation dynamics of low-coordinate lanthanide compounds. Inorg. Chem. 55, 1905–1911, https:// doi.org/10.1021/acs.inorgchem.5b02792. Copyright 2016 American Chemical Society.

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Sm, Gd). The analysis of the “shape” of the 4f electronic density and of the spin magnetizations showed, in agreement with previous findings, that the spin magnetization reflects the electron density in the first half of the series, whereas it approximately represents the unpaired electrons in the second half of the series. However, the spin density may not be the main contributor to the total magnetic moment, given that the orbital contribution in lanthanides may be significant. Therefore, the authors suggest the analysis of magnetizations obtained from ab initio calculations to guide the research in the search for adequate SIM environments.

5.2.2 Kramers vs Non-Kramers Ions In the absence of further interactions, QTM through the ground state is formally forbidden for Kramers ions, due to time-reversal symmetry. For this reason, Kramers ions would be a priori a better choice for SIM-memory devices than non-Kramers ions. In the practice, however, the presence of dipolar and hyperfine interactions can give rise to transversal fields which break the degeneracy of the Kramers doublets (KDs), enabling the occurrence of fast relaxation. Suppression of QTM can be achieved by diluting the paramagnetic complexes within an isostructural diamagnetic matrix, which reduces dipolar interactions, and/or by applying a static magnetic field. Complexes exhibiting slow relaxation of the magnetization only under the application of an external magnetic field are commonly referred to as “field-induced SIMs.” For a similar geometry and use of “oblate” ions, the different Kramers or non-Kramers nature of the ion can yield to different relaxational behavior. There are numerous examples in which isostructural complexes of Dy exhibit SIM behavior, while the Tb counterpart does not. For example, Liu et al. (2014c) compared two half-sandwich complexes [(C6Me6)Ln(AlCl4)3], Ln ¼ Dy, Tb, where the Ln ion is coordinated with a p-bonded arene and presents a quasi-D5h symmetry. The Dy complex deploys SIM behavior (Ueff/ kB ¼ 101 K) and a hysteresis loop up to 3 K; according to ab initio calculations, the GS is highly anisotropic (gz ¼ 19.6985) and the EAM lies coincident with the Dy-HMB direction. In contrast, no SIM behavior is observed in the Tb congener, in which the calculated gz ¼ 16.45 is far from the Ising limit and the EAM significantly deviates from the Ln-HMB direction (37.5°). The most prolific family among Ln SIMs is based on Kramers Dy(III) ion because the large J ¼ 15/2 and KD ground state favor a large magnetic anisotropy. Go´mez-Coca et al. (2015) have recently investigated, using ab initio calculations, the necessary conditions for the presence of zero-field and field-induced SIM behavior in mononuclear Dy complexes. It was concluded that the oblate/prolate shape of the electron density is crucial for anisotropy. The presence of a spin-free first excited state with a similar shape of the electron density as in the ground state leads to a spin–orbit ground state with high

114 Handbook of Magnetic Materials

anisotropy if higher excited states make a low contribution (large second excitation energy). Thus, Dy complexes with a large figure of merit DE2  DE1/DE1 usually show zero-field SIM behavior, while those with small DE1 and also relatively small DE2 are usually field-induced SIMs. The spatial distribution of the electron density is also influenced by the electrostatic potential caused by the ligands to reduce the electrostatic repulsion. Thus, the metal cloud of Dy tries to be accommodated in the low electrostatic regions, avoiding the charged ligands with the shortest Ln–ligand distances. A screening of Dy complexes in 41 different coordination modes allowed concluding that the highest spin relaxation barriers and the highest axiality are found in the lowest coordination number systems. Ab initio calculations offer nowadays an attractive approach to compare the anisotropy properties of complexes with different types of lanthanides in identical coordination environments. E.g., to shed light on the influence of the type of ion on the activation energy, Gupta and Rajaraman (2014) performed a series of ab initio calculations on model complex [LnIII(NO3)6]3, where Ln(III) was 12-coordinated in a slightly distorted octahedron coordination environment, for different Kramers (Dy, Ce, Nd, and Sm) and non-Kramers (Tb, Pr) lanthanides. For Pr complex, the energy spectrum of the ground multiplet consisted of three low-lying single states and two pseudo-doublets, precluding SIM behavior. For all studied complexes (except for Pr) a significant Ueff was computed, in decreasing order Ce > Sm > Dy > Nd > Tb. However, the nature of anisotropy computed suggests that substantial tunneling between the ground state KDs may occur in the Ce, Nd, and Sm complexes. Remarkably, Ce(III) ion possesses oblate electron density and according to Long’s model would be expected to yield only small Ueff values; however, calculations predict the largest Ueff of the whole series. Very recently, Lucaccini et al. (2017) have reported the magnetic properties of the complete Ln(trenovan) series, related to the extensively studied Ln(trensal) family (Pedersen et al., 2015). Remarkably, it is demonstrated that the magnetic relaxation in these complexes is not related to the magnetic anisotropy of the complexes, since both compounds with easy axis and easy plane anisotropy (as obtained from EPR) show a nonzero out-of-phase susceptibility, but it is rather determined by the number of unpaired electrons: only compounds containing Kramers ions (Ce, Nd, Gd, Dy, Er, and Yb) exhibited field-induced relaxation, not attributable to an Orbach process but rather to Raman and direct processes (Fig. 43).

5.2.3 SIMs Based on Atypical Lanthanide Ions The scope of this review is not to provide a chemical survey of all reported SIMs, since excellent reviews are already available (Edelmann, 2016; Feltham and Brooker, 2014). We highlight here only a few recent, eyecatching works dedicated to SIM complexes based on atypical lanthanides.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

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FIG. 43 (A) Structure of the Ln(trenovan) complex. (B) Temperature dependence of the magnetic relaxation time for the samples of the Ln(trenovan) family. Relaxation is not attributable to an Orbach process. Adapted with permission from Lucaccini, E., Baldovı´, J.J., Chelazzi, L., Barra, A.L., Grepioni, F., Costes, J.P., Sorace L., 2017. Electronic structure and magnetic anisotropy in lanthanoid single-ion magnets with C3 symmetry: the Ln(trenovan) series. Inorg. Chem. 56, 4728–4738, https://doi.org/10.1021/acs.inorgchem.7b00413. Copyright 2017 American Chemical Society.

5.2.3.1

Light Ln(III) Ions

The number of SIMs based on light lanthanides is relatively scarce. To the best of our knowledge, slow relaxation in mononuclear complexes based on Pr(III), Pm(III), Sm(III), or Eu(III) has never been reported. For the lightest lanthanide, Ce(III), field-induced SIM behavior has been recently demonstrated in two different crown-ether based complexes, [Ln(NO3)3(18crown-6)] and [Ln(NO3)3(1,10-diaza-18-crown-6)] by Wada et al. (2017), and in Ce(trenovan) (Lucaccini et al., 2017). On the other hand, field-induced slow relaxation of the magnetization has been reported only in a few Nd(III) SIMs (Le Roy et al., 2015; Lucaccini et al., 2017; Rinehart and Long, 2012; Wada et al., 2017), SMMs (Arauzo et al., 2014), and 1D systems ( Jassal et al., 2015), with the largest activation energy reported for mononuclear Na9[Nd(W5O18)232H2O, with Ueff/kB ¼ 73.95 K (at 1000 Oe). Remarkably, unprecedented zero-field SIM behavior has been recently reported by Gupta et al. (2016b) in [L2Nd(H2O)5][I]3L2(H2O) (Ueff/kB ¼ 24.7 K), which was achieved thanks to the pseudo-D5h symmetry of the complex leading to the stabilization of the mJ ¼ j9/2i state. 5.2.3.2 Gd(III) Ions Gd represents a special lanthanide, due to its quenched angular magnetic moment (L ¼ 0). The electronic wave function possesses spherical symmetry and the magnetic anisotropy vanishes in first order. However, CF effects can produce very small orbital distortions leading to a weak, but measurable, anisotropy that strongly depends on the local coordination, and therefore some mononuclear Gd complexes, like GdW10, GdW30 POMs, have been studied as model crystal-field probes (Martı´nez-P
erez et al., 2012a). However, since relaxation becomes soon dominated by QTM, Gd mononuclear complexes

116 Handbook of Magnetic Materials

are rather being investigated as qubits (Aromı´ et al., 2015; Martı´nez-P
erez et al., 2012a) or as promising refrigerants to attain ultra-low temperatures based on the magnetocaloric effect (MCE) (Luis and Evangelisti, 2015; Martı´nez-P
erez et al., 2012b). 5.2.3.3 Heavy Ln(III) Ions Examples of non-Kramers SIM complexes other than Tb are extremely rare. In principle, the prolate shape of MJ ¼  6 enables Tm(III) to be a candidate for designing SIMs, though not many examples have been reported, the main difficulty being its non-Kramers nature and the difficulty in maintaining high uniaxial symmetry to reduce the mixing of MJ states. Recently, two halfsandwich complexes based on Tm showing field-induced SIM behavior, [(Tp)Tm(COT)] 1 and [(Tp*)Tm(COT)] 2, and even zero-field SIM behavior in diluted congeners, have been reported (Meng et al., 2016b). Ab initio calculations evidenced the importance of high symmetry for obtaining Tm-based SIMs: indeed, 1 possessed a high uniaxial C3v axis and the ground state was 0.96 j6i + 0.03 j3i, while the distortion of 2 from the C3v symmetry induced more nonaxial CF parameters and further MJ mixing of the GS (0.92 j6i + 0.04 j4i + 0.02 j2i), thus resulting in a higher energy barrier for 1 (Ueff/kB ¼ 111 K) than for 2 (Ueff/kB ¼ 46 K). A small number of Yb(III)-based complexes behaving as SIMs have been reported in the past few years (Boulon et al., 2013; Feltham et al., 2011; Li et al., 2015a; Lin et al., 2012; Yi et al., 2014). It has been often observed that the temperature dependence of the relaxation time presents two regimes with different slopes (Liu et al., 2012; Pedersen et al., 2015; Zeng et al., 2016). Many initial works simply reported an “effective” barrier resulting from the fit of the data at larger temperatures to an Arrhenius law. However, recent studies have evidenced that in some of those complexes, the Arrhenius energy was dramatically smaller than the energy gap between the ground and first states calculated by ab initio (Liu et al., 2012), or directly measured from luminescence/absorption spectra (Pedersen et al., 2015), therefore excluding the presence of an Orbach process. In Yb(III) complexes recently reported like Yb(trensal) (Pedersen et al., 2015), Yb(trenovan) (Lucaccini et al., 2017), [Cp*Yb(DAD)(THF)]C7H8 (Trifonov et al., 2015), “half-sandwich” YbIII-metallacrowns complexes (Liu et al., 2014c), or YbIII b-diketonates (Jim
enez et al., 2016), the τ1 (T) dependence is better described by the sum of two power laws, corresponding to an admixture of Raman and direct processes (e.g., Fig. 43B). 5.2.3.4

Divalent Ln(II) Complexes

The recent discovery of the divalent state in the entire lanthanide series in compounds of the type [K(2.2.2-cryptand)][Cp30 Ln] (Fieser et al., 2015; Macdonald et al., 2013) has opened up the possibility of considering new

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

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magnetic centers for the design of molecular magnets with unprecedented large magnetic moments. In support of recent spectroscopic and computational results, Meihaus et al. (2015) presented DC magnetic susceptibility data suggesting a 4fn configuration for the series SmII, EuII, and TmII, while a coupled 4fn5d1 configuration was more likely for YII, LaII, GdII, TbII, DyII, HoII, ErII, and for CeII, PrII, NdII, the electronic structure could not be deciphered on the basis of DC susceptibility alone. Remarkably, DyII and HoII exhibited the highest room T magnetic moments ever reported for a single metal ion (meff ¼ 11.35 mB, 11.41 mB, respectively). Although no slow relaxation has been observed for any of these complexes, the extremely large moments accessible hold promise in the pursuit of SIMs with enhanced properties.

5.3

Crystal Field Environment

Although the unpaired 4f electrons of the lanthanide ions are well shielded by the outer 5d and 6s electrons, it has been shown that both the coordination geometry and the strength of the ligand field have a great impact on the performance of Ueff and TB of SIMs. Given that the CF has a crucial role in the relaxation and tunneling properties of the mononuclear complexes, considerable effort has been done in the past years toward determining the CF parameters, following ab initio or phenomenological models (see Section 3 for details). In particular, ab initio calculations (Chibotaru, 2015) combined with the methodology proposed by Ungur et al. (2013) have been extensively useful to rationalize the relaxation performance of Ln complexes in terms of the properties of the blocking barrier structure (Andreiadis et al., 2012; Antal et al., 2016).

5.3.1 Symmetry A highly axial molecular symmetry is expected to suppress the transversal components of the magnetic anisotropy which facilitate QTM within magnetic doublets. Experimental and theoretical studies have corroborated that in axial symmetry environments, such as D∞h, S8, D4d, and D5h, D3h, D2d, D5h, C5, and C∞, mononuclear lanthanide compounds often show high Ueff. At the same time, gradual deviations from ideal symmetry result in the diminishing of axiality, which induces increased QTM (Feltham et al., 2011). A high degree of axiality of the ligand field can be achieved in several ways: one is to build only a very short, strong chemical bond with the metal ion dominating over all other chemical bonds of the metal site (Blagg et al., 2013; Ungur and Chibotaru, 2011). Another chemical route for obtaining an axial ligand field is to employ highly symmetrical ligands with no bonding atoms on the symmetry axis (Ishikawa et al., 2003b). High local symmetry is not easy to achieve, due to the intrinsic high coordination modes of the lanthanide ions; however, some successful architectures have been demonstrated. We highlight here several symmetries of complexes where high symmetry plays a crucial role.

118 Handbook of Magnetic Materials

5.3.1.1 Square-Antiprismatic D4d Symmetry Complexes 5.3.1.1.1 Sandwich-Type LnPc2 Complexes The well known molecule TbPc2, reported as the first SIM (Ishikawa et al., 2003b) is based on this architecture. Ever since LnP2 systems, particularly with TbIII, have been extensively studied (Layfield and Murugesu, 2015), and one of them possesses one of the largest energy barriers reported to date (Ueff/kB ¼ 922 K) (Branzoli et al., 2009a). The success of this geometry relies on the rigid phthalocyanine ligands with four strong N donors that help fixing the central lanthanide ions within ideal D4d symmetry. Recently, a new family of [Ln(OETAP)2] double-decker complexes with D4d symmetry has been reported by Gim
enez-Agullo´ et al. (2014), with SIM behavior exhibiting high Ueff/kB ¼ 383 K, 32 K, and high blocking temperatures over 50 K and 10 K for the Tb(III), Dy(III) complexes, respectively. In contrast with their Pc counterparts, these complexes are highly soluble in nonpolar organic solvents and are sublimable under relatively mild conditions, opening perspectives for on-substrate deposition. 5.3.1.1.2 b-Diketonate DyIII Derivatives Different groups have reported b-diketonate-based square antiprismatic (SAP) mononuclear SIMs with local symmetry D4d. Deviations from the local symmetry induced by the capping ligands with different conjugation degrees have great influence on the anisotropy barriers (Bi et al., 2011; Jiang et al., 2010; Tong et al., 2015; Zhang et al., 2016d). 5.3.1.1.3 POM-Based Complexes Mononuclear Ln complexes encapsulated by POMs, like LnW10 or LnW30, have produced key examples of spin qubits for quantum computing (Cardona-Serra et al., 2012; Shiddiq et al., 2016). Recently, two new families of polyoxomolybdate complexes (Fig. 44) have been reported: [Ln(b-Mo8O26)2]5 (LnMo16), soluble in organic solvents, and the functionalized [Ln{Mo5O13(OMe)4NNC6H4-p-NO2}2]3 (LnMo10), where the Ln ¼ Dy, Yb, and Er complexes exhibit field-induced relaxation (Baldovı´ et al., 2016a). Although polyoxomolybdates display poorer behavior as SIMs than analogous polyoxotungstates, the solubility and functionalization properties would facilitate their grafting onto surfaces and the incorporation of a second property coming from the organic ligand. 5.3.1.2 Complexes Using Cyclic Ligands With High Rotational Symmetry 5.3.1.2.1 C8-Symmetric Cyclooctatetraene (COT) Ligands Ungur et al. (2014a) synthesized two complexes [Er(COT)2] and [Dy(COT)2] using highly symmetric dianionic COT2 rings as ligands. The Er complex was close to a D8d symmetry, while the Dy compound was more eclipsed, close to D8h. Ab initio calculations were performed to calculate the CF split energy levels and rationalize the relaxation behavior in terms of the constructed magnetization blocking barriers (Fig. 45).

FIG. 44 Structure of POM-based (A) LnMo16 (projection of the coordination sphere showing the SAP coordination site) and (B) LnMo10. Adapted from Baldovı´, J.J., Duan, Y., Bustos, C., Cardona-Serra, S., Gouzerh, P., Villanneau, R., Gontard, G., Clemente-Juan, J.M., Gaita-Arin˜o, A., Gim
enez-Saiz, C., Proust, A., Coronado, E., 2016a. Single ion magnets based on lanthanoid polyoxomolybdate complexes, Dalton Trans. 45, 16653–16660, https://doi.org/10.1039/ C6DT02258H with permission. Published by The Royal Society of Chemistry.

FIG. 45 Molecular structure of [LnIII(COT)2], Ln ¼ Er (left), Dy (right). Dashed lines show the calculated orientation of the main magnetic axis on Ln ions in the GS (1) and first excited (2) KDs, and ab initio calculated magnetization blocking barriers. The thick black lines represent the Kramers doublets as a function of their magnetic moment along the axis connecting the centers of COT rings. The green dashed lines correspond to diagonal quantum tunneling of magnetization (QTM); the blue dashed lines represent Orbach relaxation processes. The numbers at each arrow stand for the mean absolute value of the corresponding matrix element of transition magnetic moment (ImxI + ImyI + ImzI)/3. The path shown by the red arrows represents the most probable path for magnetic relaxation. Reprinted from Ungur, L., Leroy, J.J., Korobkov, I., Murugesu, M., Chibotaru, L.F., 2014a. Fine-tuning the local symmetry to attain record blocking temperature and magnetic remanence in a single-ion magnet, Angew. Chem. Int. Ed. 53, 4413–4417, https://doi.org/10.1002/anie.201310451 with permission. Copyright 2014 John Wiley & Sons, Inc. The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J.

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[Er(COT)2] exhibited SIM behavior with high Ueff/kB ¼ 286 K, and a hysteresis loop with remarkably large coercive field, 7000 Oe at 1.8 K (previously, large coercive fields had been only observed in polynuclear metal complexes). The reason for this remarkable SIM performance was associated to the almost perfectly rotational electronic density from the ring felt by the Er ion. The ab initio-calculated CF axial parameters B02 < 0, B04, B06 were the largest, while nonaxial parameters were one or more orders of magnitude smaller. This resulted in a strong magnetic anisotropy of the GS (gx,y ¼ 3.5  106, gz ¼ 17.96), the wave function being almost of 15/2 type, and of the first excited KD (gx,y ¼ 5.4  104, gz ¼ 15.53), with almost collinear gz axes (y < 1°). The negligible value of the transversal magnetic moment in the GS (106) explained the absence of QTM at low temperatures; the transversal magnetic moment was also small for the first KD state (104), and therefore, TAQTM via this KD was expected to be negligible. In addition, owing to the collinearity of the GS and first excited KD the off-diagonal matrix element of the transversal magnetic components of opposite magnetization was also small (105), and the corresponding Orbach process was suppressed. Therefore, magnetic relaxation in the Er compound could only occur via the second excited state, explaining the high observed Ueff. In contrast, a much smaller Ueff/kB ¼ 11 K with no obvious opening of the hysteresis at 1.8 K was observed for the Dy compound. The ultimate reason was found to be the opposed sign of the B02, B04 axial parameters, producing a wave function of 9/2 type with larger mixing with other J projections. As a result, the g-tensor for the GS (gx,y ¼ 1.6  101, gz ¼ 12.64) and first KD (gx,y ¼ 5.8  102, gz ¼ 13.84) had larger transverse contributions, and the angle between the corresponding gz axes was larger, y < 21°, therefore enabling tunneling via both the GS and the first excited state. This example evidences how slight deviations from symmetry can have a drastic effect in the relaxational processes. 5.3.1.3

Complexes With Pentagonal–Bipyramidal D5h Symmetry

Remarkable achievements have been recently obtained for complexes with pseudo-D5h symmetry. Gupta et al. (2016a) reported the complex [L2Dy(H2O)5][I]3L2(H2O) with a large anisotropy barrier Ueff/kB ¼ 735.4 K and a magnetization blocking up to 12 K (30 K) with a large coercivity 0.9 T (1.5 K) at a field sweep rate of 0.0018 T/s (0.02 T/s). Ab initio calculations established that the high activation energy owes to relaxation of the magnetization via the second excited KD, owing to the pseudo-high-order symmetry which quenches QTM (Fig. 46). Indeed, the computed g-tensor of the GS KD was purely Ising in nature (gxx, gyy ¼ 0.4  104, gzz ¼ 19.86), with the gzz axis lying nearly along the pseudo-C5 axis, and the GS state was found to be a pure mJ ¼ j15/2i; the practical absence of transverse anisotropy at the GS quenched QTM. Besides, the first excited KD was found

FIG. 46 (A) Molecular structure of complex [L2Dy(H2O)5][I]3L2(H2O) and calculated gzz orientation of the GS KD. (B) Possible relaxation paths. (C) Fielddependent magnetization data collected at a sweep rate of 0.0018 T/s. Adapted with permission from Gupta, S.K., Rajeshkumar, T., Rajaraman, G., Murugavel, R., 2016a. An air-stable Dy(III) single-ion magnet with high anisotropy barrier and blocking temperature. Chem. Sci. 7, 5181–5191, https://doi.org/10.1039/ C6SC00279J. Published by The Royal Society of Chemistry. The SI conversion factor for energy is 1 K ¼ 1.381  1023 J.

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to lie 461.6 K above the ground state and this state was a pure mJ ¼ j13/2i with a negligible transverse anisotropy (gzz ¼ 17.08, gxx, gyy ¼ 0.02), therefore quenching TAQTM via this excited level also. The gzz axis of the first KD was practically coincident with that of the GS, suggesting that relaxation does not likely occur via this first excited state. In contrast the second KD was found to lie at 688.3 K above the GS and here the g-tensor had a significant transverse component (gzz ¼ 16.53, gxx ¼ 0.58, gyy ¼ 3.13) and this state was estimated to be an admixture of mJ ¼ j1/2i and mJ ¼ j5/2i. The gzz axis was found to be tilted by 94° with respect to the GS KD orientation. This significant deviation in the angle and the observance of a transverse component at this level suggested that the relaxation likely happened via this second excited state. Very recently, Tong and coworkers have demonstrated breakthrough activation energies in Dy complexes with D5h geometry: [Dy(Cy3PO)2(H2O)5]Cl3(Cy3PO) H2OEtOH (Ueff/kB ¼ 472 K), [Dy-(Cy3PO)2(H2O)5]Br32(Cy3PO)2H2O2EtOH (Ueff/kB ¼ 543 K) (Chen et al., 2016b), and specially, [Dy(bbpen)X] (Liu et al., 2016d) with X ¼ Cl (Ueff/kB ¼ 708K), Br (Ueff/kB ¼ 1025 K) (see Fig. 47). The later compound holds at present the record activation energy among all Ln SIMs

FIG. 47 (A) Molecular structure of [Dy(bbpen)Br] complex; (B) ab initio computed magnetization blocking barrier scheme. Bottom: Magnetization vs magnetic field hysteresis loops for pure (C) and diluted (D) complexes at T ¼ 0.03 K. Adapted with permission from Liu, J., Chen, Y.C., Liu, J.L., Vieru, V., Ungur, L., Jia, J.H., Chibotaru, L.F., Lan, Y., Wernsdorfer, W., Gao, S., Chen, X.M., Tong, M.L., 2016d. A stable pentagonal bipyramidal Dy(III) single-ion magnet with a record magnetization reversal barrier over 1000 K. J. Am. Chem. Soc. 138, 5441–5450, https:// doi.org/10.1021/jacs.6b02638. Copyright 2016 American Chemical Society. The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J.

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and presents pronounced hysteresis loops up to 14 K. Ab initio calculations were performed to understand the larger Ueff in the Br complex compared to the Cl one. It was found that the CF parameter B(2,0) was larger in Br than in Cl, explaining the stronger CF splitting in this compound. At the same time, the nonaxial parameters, notably the B(2,q) parameters were smaller in Br complex, resulting in more axial KDs. These slight differences in the CF parameters were caused by subtle structural distances between the two compounds; namely the Dy–O distance was slightly shorter and the O–Dy–O angle was closer to 180° in the Br; furthermore, Cl is closer to Dy than Br, therefore making a greater contribution to the transverse CF parameters B(k,q). Analysis of the magnetization blocking barriers calculated by ab initio revealed that the extremely high Ueff observed for the Br complex could be explained by slow relaxation occurring via the third KD.

5.3.2 Low Symmetry, Highly Axial CF High local symmetry may be hard to achieve due to the intrinsic high coordination numbers and variable coordination modes of the lanthanide atoms. A strategy to obtain large anisotropy in low-symmetry systems has been to produce complexes with an axial high charge distribution along the magnetic axis. Sun et al. (2016b) used this approach to synthesize four Zn–Dy–Zn SIMs, [Zn2(L1)2DyCl3]2H2O 1, [Zn2(L1)2Dy(MeOH)Br3]3H2O 2, [Zn2(L1)2Dy (H2O)Br2][ZnBr4]0.5 3, and [Zn2(L2)2DyCl3]2H2O 4 (Fig. 48), all displaying slow magnetic relaxation with relatively high barrier (Ueff/kB ¼ 430 K, 233 K, 121 K, 398 K for 1–4, respectively) and hysteresis temperature (8 K, 6 K, 4 K, 8 K for 1–4), despite having low local symmetry of the Dy(III) ion (only a C2 axis). The remarkable SIM performance was achieved through the combination of a magnetic easy axis oriented nearly parallel to the high negative charge distribution along the Zn–Dy–Zn direction, and five oxygen atoms with a lower electron density constituting a hard plane with a pseudo C5 axis surrounding the Dy(III) center. Deviations of the five coordination atoms from their leastsquare planes were recognized as the key factor introducing transverse anisotropy enabling QTM: therefore complexes 2, 3, displaying relatively more apparent deviation from the hard plane than 1, 4, showed a smaller Ueff. A strong and purely axial ligand potential, stabilizing the maximal angular momentum projections of oblate lanthanides, can be achieved by a linear arrangement of negatively charged donor atoms. This idea has been demonstrated, e.g., by Gregson et al. (2016) in complex [Dy(BIPMTMS)2][K (18C6)(THF)2] (Fig. 48B): AC magnetic measurements in H ¼ 0 show slow magnetic relaxation via two relaxation processes; in both of them, Orbach relaxation dominates at high T, but at lower temperatures a second-order Raman process dominates. The complex exhibits two thermally activated energy barriers of 721 K and 813 K, the record Ueff value for monometallic Dy(III) complexes. Ab initio calculations indicate the lowest three KDs of the

FIG. 48 (A) Core structure for Zn–Dy–Zn complexes: ([Zn2(L1)2DyCl3]2H2O 1, [Zn2(L1)2Dy(MeOH)Br3]3H2O 2, [Zn2(L1)2Dy(H2O)Br2][ZnBr4]0.5 3, and [Zn2(L2)2DyCl3]2H2O 4). The pseudo-pentagonal hard plane is highlighted (Sun et al., 2016b); (B) [Dy(BIPMTMS)2] [K(18C6)(THF)2] complex, and natural logarithm of the relaxation times for the two barriers observed in the complex as a function of the reciprocal temperature (Gregson et al., 2016). Adapted with permission from Sun, W.-B., Yan, P.-F., Jiang, S.-D., Wang, B.-W., Zhang, Y.-Q., Li, H.-F., Chen, P., Wang, Z.-M., Gao, S., 2016b. High symmetry or low symmetry, that is the question—high performance Dy(III) single-ion magnets by electrostatic potential design. Chem. Sci. 7, 684–691, https://doi.org/10.1039/ C5SC02986D; Gregson, M., Chilton, N.F., Ariciu, A.-M., Tuna, F., Crowe, I.F., Lewis, W., Blake, A.J., Collison, D., McInnes, E.J.L., Winpenny, R.E.P., Liddle, S.T., 2016. A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour, Chem. Sci. 7, 155–165, https://doi.org/10.1039/C5SC03111G. Both published by The Royal Society of Chemistry.

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ground 6H15/2 multiplet are essentially pure, well-isolated j15/2i, j13/2i, and j11/2i quantized along the C ¼ Dy ¼ C axis. Thermal relaxation occurs via the fourth and fifth doublets, the highest experimentally observed so far. SIM behavior has been in fact observed in very low-symmetry complexes, provided the coordination environment provides a sufficiently axial CF. For example, Demir et al. (2014) reported low-symmetry [Cp*2Ln(BPh4)] complexes, with energy barriers among the highest reported for Tb and Dy (Ueff/kB ¼ 344 K and 498 K for Tb and Dy, respectively), in which the axis of preferred alignment extends between the most separated carbon atoms of the two Cp* rings (Fig. 49A), while the weak ligand field imposed by the BPh 4 anion leads to a favorable weak transverse anisotropy. Alternatively, a pincer low-symmetry environment can be used to reach high Ueff barriers by forcing relaxation to occur through high-energy states. This idea has been exploited by Guo et al. (2014) in a DyNCN complex, with the Dy–C defining only a C2v low-symmetry axis (Fig. 49B); ab initio calculations showed that due to the D2v symmetry the GS and first excited KD are strongly axial and possess anisotropy axes almost parallel to the symmetry axis, whereas the second excited state (at 377 K) is also very axial (15/2), but its magnetic axis is perpendicular to the complex symmetry axis of the complex, passing through the two chlorine atoms. The observed SIM barrier Ueff/kB ¼ 336.7 K is well above the first excited KD, implying that relaxation occurs via the second excited state.

5.3.3 Lanthanide–Radical SIMs The use of organic radicals with a nonzero spin, acting on the lanthanide’s ligand field or causing magnetic interaction, is also being considered as a strategy for obtaining new SIMs. Previously demonstrated examples of mononuclear lanthanide–radical [Ln–r] SIMs include the phthalocyanine and nitroxide radical complexes, Gd benzosemiquinonate and Yb radical metallocenes surveyed in Demir et al. (2015). Recently, some additional examples have been reported, like complex [Ln(hfac)3(NITIn)], with S ¼ 1/2 r ¼ indazole radical (Li et al., 2017a), and [Tb(Phtfac)3(MeTrzNIT)]2C7H143H2O with the S ¼ 1/2 r ¼ MeTrzNIT radical (Li et al., 2016b), exhibiting very low activation energies, below 2 K. 5.3.4 Coordination Number The coordination number (C.N.) around the lanthanide ion alters significantly the SIM behavior. Indeed, an increase in the C.N. produces concomitant changes in the electrostatic potential of the ligands and the coordination symmetry, which in turn influence the anisotropy, as discussed earlier. Systematic ab initio calculations in combination with Ungur’s methodology (Ungur et al., 2013) have been recently used to analyze the effect, on model complexes of different Ln(III) ions, of gradually increasing the C.N. on the magnetic anisotropy and energy barrier Ueff.

FIG. 49 (A) Orientation of the main anisotropy axis in [Cp*2Ln(BPh4)]. Green: Ln; purple: B (Demir et al., 2014). (B) Left: Anisotropy axes in lowest KDs; the second excited state (KD3) lies perpendicular to the C2v symmetry axes. (B) Right: Ab initio calculated magnetization blocking scheme (Guo et al., 2014). The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J. Adapted with permission from Demir, S., Zadrozny, J.M., Long, J.R., 2014. Large spin-relaxation barriers for the low-symmetry organolanthanide complexes [Cp*2Ln(BPh4)] (Cp* ¼ pentamethylcyclopentadienyl; Ln ¼ Tb, Dy). Chem. A Eur. J. 20, 9524–9529, https://doi.org/10.1002/chem.201403751. Copyright 2014 John Wiley & Sons, Inc.; and from Guo, Y.-N., Ungur, L., Granroth, G.E., Powell, A.K., Wu, C., Nagler, S.E., Tang, J., Chibotaru, L.F., Cui, D., 2014. An NCN-pincer ligand dysprosium single-ion magnet showing magnetic relaxation via the second excited state. Sci. Rep. 4, 5471, https://doi.org/10.1038/srep05471. Published by Springer Nature under the Creative Commons Attribution License.

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For example, Gupta and Rajaraman (2014) calculated by ab initio the Ueff, direction, and anisotropy of a series of monometallic Ln(III) (Ln ¼ Dy, Tb, Ce, Nd, Pr, Sm) complexes surrounded by nitrate ligands, where the C.N. was gradually increased from 2 to 12. In each case the energy barrier, g-tensor, and angle between the direction of the excited state gzz and the ground state were calculated. Large values indicate that relaxation predominantly happens via that particular KD, while smaller values reveal that the anisotropy is collinear and thus relaxation of magnetization likely proceeds via higher excited states, leading to larger Ueff values. For example, results of the study for Dy(III) are shown in Fig. 50. Barring a few exceptions, an inverse correlation between the C.N. and Ueff and gz values was demonstrated. Therefore, results suggested that synthetic efforts should be targeted toward achieving low-coordinate complexes toward larger Ueff SIMs. Singh et al. (2014a) performed similar ab initio studies on [Er(OH)n] models based on Er(III) prolate ion. By varying the C.N. from 1 to 12, they showed that the presence of ligand interaction on the equatorial plane and high symmetry are the two favorable conditions to obtain large Ueff values. The absence of axial ligands was much more crucial than symmetry conditions. Among all the tested models, the three-coordinated D3h model and the four-coordinated D4h model were found to possess the largest barrier height, followed by a moderate Ueff for models C4v (C.N.5), D5h (C.N.7), and D4d (C.N.8), and lowest Ueff (in decreasing order) observed for models D3h (C.N.11), D4d (C.N.10), D3h (C.N.9), D3h (C.N.5), Oh (C.N.6), Oh (C.N.12), D∞ h (C.N.2), and C∞ v (C.N.1).

5.3.5 Coordination Environment Not only the geometry but also the coordination environment, i.e., (i) the type of coordinating atoms, (ii) the identity and nature of the ligands, and (iii) cis–trans isomerism can influence the relaxation behavior. For example, Wu et al. (2016) reported a series of mononuclear Dy(III) complexes, [DyLz2(o-vanilin)2]Xsolvent, with X ¼ Br (1), NO3 (2), CF3SO3 (3), with Dy in a distorted D4h symmetry. In 1 and 3 the two Lz ligands were in cis position to each other, while in 2 the ligands were in trans position. In the trans configuration, contrary to the cis configuration, the p-stacking interactions were nonoperative so that the energy splitting was much larger in 2 than in 1, 3, and magnetic relaxation through a larger, third excited state was enabled, yielding a high anisotropy barrier of Ueff/kB ¼ 615 K and opening of the hysteresis loop up to 7 K (Fig. 51). 5.3.5.1

Second Coordination Environment

Several works have revealed the extreme sensitivity of the magnetic relaxation to subtle changes in the coordination environment involving minor distortions even beyond the first coordination sphere.

FIG. 50 Left: Model structures where C.N. varied from 2 to 11 around Dy(III) ion, along with the computed direction for GS and first excited KD states. Table: calculated energy spectrum, g-tensors, and angles of principal anisotropy axes of the ground KDs of low valent Dy(III) model complexes with respect to their respective excited state KDs. Dependence of the effective energy barrier and ground-state anisotropy direction along the Z-axis with the C.N. Reprinted with permission from Gupta, T., Rajaraman G., 2014, How strongly are the magnetic anisotropy and coordination numbers correlated in lanthanide based molecular magnets. J. Chem. Sci. 126, 1569–1579, https://doi.org/10.1007/s12039-014-0691-z. Springer. Copyright © 2014, Indian Academy of Sciences.

FIG. 51 (A) Schematic drawing of absolute configurations with top views (right) for the cis and trans configurations of the [DyLz2(o-vanilin)2]+ units; (B) the magnetization blocking barriers and relaxation pathways with highest probability in complex [DyLz2(o-vanilin)2]X solvent, with X ¼ CF3SO3 (3). Adapted from Wu, J., Jung, J., Zhang, P., Zhang, H., Tang, J., Le Guennic, B., 2016. cis-trans isomerism modulates the magnetic relaxation of dysprosium single-molecule magnets. Chem. Sci. 7, 3632–3639, https://doi.org/10.1039/c5sc04510j. Published by The Royal Society of Chemistry. The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J.

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5.3.5.1.1 Auxiliary Ligands For a constant number and type of coordinating atoms, the introduction of auxiliary ligands significantly changes the relaxation characteristics. For example, Zhu et al. (2014a) reported two b-diketone-based Dy complexes, in which the Dy(III) ion was eightcoordinated by six O atoms from three TFI ligands and two O/N atoms from different auxiliary ligands: the use of (bpy) led to complex Dy(TFI)3(bpy) (2), where Dy ion is in a distorted dodecahedral geometry, while auxiliary ligand (phen) led to the formation of [Dy(TFI)3(Phen)]0.02ChCl3 (3) with distorted square antiprismatic geometry. A slower relaxation was observed in the (phen) compared to the (bpy) complex, which was attributed to a combination of the large aromatic SAP CF of phen around Dy enabling the 11/2 and 13/2 sublevels in the GS, and the increased symmetry (D4d screw axis instead of D2d axis) in the phen complex (Fig. 52). The auxiliary ligands can introduce more subtle changes in the symmetry, also leading to changes in the relaxation. As an example, Dong et al. (2015) presented a series of Dy-based b-diketone complexes with atypical sevencoordination: Dy(EIFD)3(H2O)CH2Cl2 1, Dy(EIFD)3(DMF)CH2Cl2 2, Dy(EIFD)3(DMSO) 3, Dy(EIFD)3(TPPO) 4. In all cases the central Dy(III) was coordinated to six oxygen atoms, three from EIFD ligands and one from H2O, DMF, DMSO, and TPPO molecules for 1–4, respectively. As a result of the progressive distortion of the C3v-capped coordination octahedron introduced by the auxiliary ligands, SIM behavior under H ¼ 0 was observed for 1, while complexes 2–3 exhibited slow relaxation only under H 6¼ 0, and 4 did not show w00 frequency dependence at all (Fig. 53A). Mondal et al. (2015) showed that a slight difference of local symmetry simply due to deprotonation of a ligand, and the change from monodentate to bidentate binding mode of one of the peripheral nitrate anions lead to a higher activation energy in deprotonated complex [Dy(H3daps)(H2O)3(NO3)] (NO3)2(MeOH) 2, compared to [Dy(H4daps)(H2O)3(NO3)](NO3)2(H2O) 1 (Fig. 53B), with Ueff/kB ¼ 32.7 K and 23.8 K (at 2000 Oe), respectively. Nevertheless, distortion from symmetry by the auxiliary ligands is not solely influencing the relaxation. This was shown by Dong et al. (2016) who studied b-diketone complexes (related to the above mentioned ones) with Dy in SAP geometry, eight-coordinated by three EIFD ligands and two nitrogen atoms from different auxiliary ligands: Dy(EIFD)3-(bpy) CH3CN 1, Dy(EIFD)3(phen)CH2Cl2 2, Dy(EIFD)3(dpq)CH2Cl2 3, and Dy(EIFD)3(dppz)2H2O 4. On the basis of Shape calculations the symmetry sequence of these complexes from high to low was 3 (0.666) > 1 (0.721) > 2 (0.878) > 4 (0.962), the values in parentheses indicating the distortion from the ideal D4d symmetry; in contrast, the trend of anisotropy barriers was 2 (Ueff/kB ¼ 41.8 K @ 0 Oe) > 1 (Ueff/kB ¼ 28.8 K) > 3 (Ueff/kB ¼ 41.7 @ 450 Oe) > 4 (Ueff/kB ¼ 32.4 K @ 1500 Oe). This example shows that the strength of the ligand field may dominate over the coordinating symmetry in determining the slow relaxation properties.

FIG. 52 Molecular structures, local coordination geometries, and temperature dependence of the relaxation time for complexes Dy(TFI)3(bpy) and [Dy(TFI)3 (Phen)]0.02ChCl3. Adapted with permission from Zhu, J., Wang, C., Luan, F., Liu, T., Yan, P., Li, G., 2014a. Local coordination geometry perturbed b-diketone dysprosium single-ion magnets. Inorg. Chem. 53, 8895–8901, https://doi.org/10.1021/ic500501r. Copyright 2014 American Chemical Society.

A O6

O7

O4

O14

O5

Dy1 O1

O12

O2 O6 O4

O6

O4

O2

O3

O1

O2 O3

O5 O1

O7

O5

O4 O7 Dy1 O3

O2

O13 O11

O9

O10 O8

O3

O13 O12

O6

O5

O7 Dy1

Dy1

O7

O1

O5

O2 O3

O5 O1

O4

O7

Dy1 O4

Dy1

O14

O1

O8

O11

O6 O9

O3

O6

O10

O2

B O3

O11

O3 N1 N2

O1

H3

N4

O1 H5

N3 C17

Dy1

N5 O6

O7 O2

O11 N1

C10

N2 N3 C17

Dy1 O7

N4

H4 N5

O6

C10

O2

FIG. 53 (A) Crystal structures and local coordination geometry for complexes Dy(EIFD)3(H2O)CH2Cl2 1, Dy(EIFD)3(DMF)CH2Cl2 2, Dy(EIFD)3(DMSO) 3, Dy(EIFD)3(TPPO) 4 (Dong et al., 2015). (B) Crystal structures of [Dy(H4daps)(H2O)3(NO3)](NO3)2(H2O) 1 and [Dy(H3daps)(H2O)3(NO3)](NO3)2(MeOH) 2 (Mondal et al., 2015). Reproduced from Mondal, A.K., Goswami, S., Konar, S., 2015. Influence of the coordination environment on slow magnetic relaxation and photoluminescence behavior in two mononuclear dysprosium(III) based single molecule magnets. Dalton Trans. 44, 5086–5094, https://doi.org/10.1039/ c4dt03620d. Published by The Royal Society of Chemistry. Reproduced with permission from Dong, Y., Yan, P., Zou, X., Liu, T., Li, G., 2015. Exploiting single-molecule magnets of b-diketone dysprosium complexes with C3v symmetry: suppression of quantum tunneling of magnetization. J. Mater. Chem. 3, 4407–4415. Published by The Royal Society of Chemistry.

134 Handbook of Magnetic Materials

Moreover, the introduction of a charged ligand over one of the positions of the coordination sphere, maintaining a similar coordination environment, can dramatically change the slow relaxation. Lim et al. (2016) prepared two similar square-antiprismatic complexes, [Dy(LOMe)2(H2O)2](PF6) 1 and Dy(LOMe)2(NO3) 2, including either two neutral water molecules (1) or an anionic nitrate ligand (2) and demonstrated that slow relaxation switched from off in 1 to on in 2 by the introduction of a charged ligand stabilizing the easy axis of magnetization along the nitrate direction, as calculated by the electrostatic program MAGELLAN (Fig. 54A). 5.3.5.1.2 Second Coordination Sphere Even slight differences in the second coordination sphere seem to produce significant changes in the slow relaxation. For example, Campbell et al. (2014) compared two Dy(III) with a minor change in the second coordination sphere: Dy(III) N,N-bis(R)methylene-1,8diamino-3,6-dioxaoctane (R ¼ imine-2-yl 1, amine-2-yl 2) (Fig. 54B). The stronger ligand field due to this slight modification provoked a larger energy barrier for field-induced slow relaxation for the imine complex 1 (Ueff/kB ¼ 50 K) than for the amine complex 2 (Ueff/kB ¼ 34 K). Another example was provided by Pedersen et al. (2014), who reported the ligand-field splittings and magnetic properties of three Er(III) SIMs differing only in the second, peripheral ligand sphere: Er(trensal), 1: X ¼ Y ¼ H; 2: X ¼ CH3, Y ¼ I; 3: X ¼ Cl, Y ¼ H (Fig. 54C). CF parameters were deduced by combining ab initio with INS measurements. The introduction of substituent groups far from the first coordination sphere led to drastic modifications of the low-lying energy spectra. A manifestation of the sensitivity of the CF to peripheral changes is the observation of multiple relaxation paths associated to different ligand conformations around a single Ln-site. For example, Sun et al. (2014a) reported the complex [Hex4N][Dy(DBM)4] in which the existence of benzene disorder within the conjugated system of the b-diketone ligand produced two different conformations, resulting in two distinct thermal relaxation pathways with Ueff/kB ¼ 68.1 K and Ueff/kB ¼ 88.0 K (at 1500 Oe) (Fig. 55). Ab initio calculations confirmed the different energy gap between the ground and first states (E1a ¼ 80 K, E1b ¼ 102 K) of the two disordered structures.

5.3.6 Relaxation Through High States In the majority of Ln SIMs spin–lattice relaxation occurs via the first excited  mJ state. At present, one of the current strategies toward achieving long relaxation times in a wide temperature range is searching for SIMs relaxing through higher states. Examples of SIMs relaxing through second (Guo et al., 2014; Gupta et al., 2016a), third (Al Hareri et al., 2016; Liu et al., 2016d), fourth (Singh et al., 2014b), and even fifth (Gregson et al., 2016; Wu et al., 2016) states have been demonstrated. The existence of such high

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1 135

FIG. 54 Influence of second coordination sphere: (A) SIM relaxation switched from off (1) to on (2) by the introduction of a charged ligand ([Dy(LOMe)2(H2O)2](PF6) 1 and Dy(LOMe)2(NO3) 2) (Lim et al., 2016). (B) Schematic representation of complexes Dy(III) N,N-bis(R)methylene-1,8-diamino-3,6-dioxaoctane (R ¼ imine-2-yl 1, amine-2-yl 2) (Campbell et al., 2014). (C) Pictorial representation of Er(trensal) complexes, 1: X ¼ Y ¼ H; 2: X ¼ CH3, Y ¼ I; 3: X ¼ Cl, Y ¼ H. Adapted with permission from Campbell, V.E., Bolvin, H., Rivie`re, E., Guillot, R., Wernsdorfer, W., Mallah, T., 2014. Structural and electronic dependence of the single-molecule-magnet behavior of dysprosium(III) complexes. Inorg. Chem. 53, 2598–2605, https://doi.org/10.1021/ic402950j. Copyright 2014 American Chemical Society. Adapted with permission from Pedersen, K.S., Ungur, L., Sigrist, M., Sundt, A., Schau-Magnussen, M., Vieru, V., Mutka, H., Rols, S., Weihe, H., Waldmann, O., Chibotaru, L.F., Bendix, J., Dreiser, J., 2014. Modifying the properties of 4f single-ion magnets by peripheral ligand functionalization. Chem. Sci. 5, 1650–1660, https://doi.org/10.1039/c3sc53044b. Published by The Royal Society of Chemistry. Adapted with permission from Lim, K.S., Baldovı´, J.J., Lee, W.R., Song, J.H., Yoon, S.W., Suh, B.J., Coronado, E., Gaita-Arin˜o, A., Hong, C.S., 2016. Switching of slow magnetic relaxation dynamics in mononuclear dysprosium(III) compounds with charge density. Inorg. Chem. 55, 5398–5404, https://doi.org/10.1021/acs.inorgchem.6b00410. Copyright 2016 American Chemical Society.

relaxation paths requires the presence of intermediate states for which QTM is blocked, which is often related to symmetry attained in low-coordinated molecules, as shown in some of the examples discussed earlier. Ab initio simulations are demonstrating to be a powerful tool to rationalize the existence of relaxation pathways in correlation with the structure. For example, Singh et al. (2014a) have studied ab initio the differences on the

FIG. 55 Center: Two disorder structures for complex [Hex4N][Dy(DBM)4] and ab initio calculated EAM directions. Left: T dependence of AC susceptibility for diluted complex under H ¼ 0. Right: Arrhenius plots. Adapted with permission from Sun, W.-B., Yan, B., Zhang, Y.-Q., Wang, B.-W., Wang, Z.-M., Jia, J.-H., Gao, S., 2014a. The slow magnetic relaxation regulated by ligand conformation of a lanthanide single-ion magnet [Hex4N][Dy(DBM)4]. Inorg. Chem. Front. 1, 503, https://doi.org/10.1039/C4QI00057A. Published by The Royal Society of Chemistry. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

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relaxation in four different three-coordinate complexes (Zhang et al., 2014b): [LnIII(N(SiMe3)2)3], where Ln ¼ Er(1a), Dy(1b), with C3v symmetry and Ln located slightly above the trigonal plane of the donor atoms; and [LnIII(NHPhiPr2)3(THF)2] (where Ln ¼ Er(2a), Dy(2b)) possessing trigonal bipyramidal geometry with two tetrahydrofuran ligands coordinated in axial positions (Fig. 56). For 1a (Er in C3v), the g-tensor is purely axial and stabilizes mJ ¼  15/2 as the GS; due to the presence of symmetry, major relaxation proceeds via the fourth KD, although partial TAQTM is operative through all the four excited states, and Orbach path is also (weakly) operational. In contrast, for 1b (Dy in C3v) the g-tensor has high transversal components, and mJ ¼  1/2 is stabilized, followed by other higher mJ excited states. Consequently, QTM is the major relaxation pathway, and SMM is wiped out. For 2a (Er in trigonal bipiramidal geometry), the GS g-tensor shows axiality but lacks purely Ising character; the first KD possesses high transversal anisotropy and lies at 109 K from GS, and the corresponding axis is tilted at 55.39°. (Field-induced) SIM occurs through the first excited state via Orbach/TAQTM processes due to noncollinear magnetic moments. Significant QTM is predicted, which explains this being a field-induced SMM. Finally, for 2b (Dy in trigonal bipiramidal geometry), the GS g-tensor presents a higher degree of axiality. The first excited state is at 286.3 K, with the axis tilted 18.67° with respect to GS KD axis. Slow relaxation proceeds also through first excited level via Orbach and TAQTM, while QTM is weak. Overall the results show that the equatorial ligand field favors Er(III) ions, while axial ligand field favors Dy(III), and slow relaxation via the fourth excited state is achieved for the Er(III) complex in equatorially field (1a).

5.4

Conclusions and Outlook

Mononuclear lanthanide complexes constitute simple model systems allowing to understand the parameters influencing spin relaxation. Research in the past years has evidenced that even small deviations in the symmetry around the ion, or ligand changes even beyond the first coordination sphere can drastically influence slow relaxation, making the pursuit for better performing SIMs a challenging task. Nevertheless, thanks to progress in the understanding of the effects of CF and its chemistry control, SIMs with impressively high activation energies 1000 K have been achieved. While ever-increasing values of Ueff have been reached in the past years, rising the blocking temperature remains elusive, and more investigation is needed in this line. In the search for new multifunctional molecular materials, a field of growing interest is the investigation of SIMs with other coexisting physical properties, such as ferroelectricity, chirality (Lin et al., 2015b), or luminescence (Coutinho et al., 2015; Goswami et al., 2016; Zhao et al., 2015a). In this section, we have summarized works focused on the correlation between anisotropy and SIM

400

1a

1b

400

Energy (cm−1)

Energy (cm−1)

500

300 200

300 200 100

100 0

0 −3

−2

−1

500

3.5

Energy (cm−1)

2

−4

3

900

−3

−2

−1

0 1 M(mB)

2

3

4

800

0. 25

1200

C)

1

0.13

Energy (cm−1)

2a

0 M(mB)

3.5

600

2b

600 400 200

300 0

0 −4

−3

−2

−1

0 M(mB)

1

2

3

4

−4

−2

0 M(mB)

2

4

FIG. 56 Comparison of complexes [LnIII(N(SiMe3)2)3], Ln ¼ Er(1a), Dy(1b), and [LnIII(NHPhiPr2)3(THF)2], Ln ¼ Er(2a), Dy(2b): Structure, ab initio computed orientation of the principal magnetization axis of the ground-state KDs, and computed magnetization blocking barrier. The thick black line indicates the KDs as a function of magnetic moment. The green lines show the possible pathway of the Orbach process. The blue lines show the most probable relaxation pathways for magnetization reversal. The dotted red lines represent the presence of QTM/TAQTM between the connecting pairs. The numbers provided on each arrow are the mean absolute values for the corresponding matrix elements of the transition magnetic moment). Adapted with permission from Singh, S.K., Gupta, T., Rajaraman, G., 2014a. Magnetic anisotropy and mechanism of magnetic relaxation in Er(III) single-ion magnets. Inorg. Chem. 53, 10835–10845, https://doi. org/10.1021/ic500772f. Published by The Royal Society of Chemistry. The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J.

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behavior in isolated mononuclear complexes. In crystals, however, interactions between ions of exchange or dipolar origin, which become increasingly important at low temperatures, may start competing with SIM relaxation and even lead to 3D ordering. Works discussing the effect of these interactions will be reviewed in Sections 6–8. An indispensable step toward the implementation of practical SIM memories is developing methods to graft and address molecules deposited into surfaces. Considerable effort is being done at present to enlarge the family of sublimable SIMs (Bi et al., 2016; Gao et al., 2016a; Gim
enez-Agullo´ et al., 2014; Lan et al., 2016) and understand how the original magnetic properties of unabsorbed molecules are changed when deposited onto different substrates. This will be discussed in detail in Section 9.

6

DINUCLEAR LANTHANIDE-BASED SMMs

Dinuclear clusters are the simplest entities allowing to study how both singleion anisotropy and magnetic interactions affect slow magnetic relaxation, a necessary step toward the understanding of the behavior of more complex, multilanthanide clusters. In this section, selected recent works on dinuclear SMMs, both 4f–4f and 4f–3d, are summarized.

6.1

Homo Dinuclear [Ln2] SMMs

Dinuclear complexes formed by two equal lanthanides [Ln2] represent ideal models to put to the test new bridging moieties, leading potentially to higher Ueff and TB performances, and investigate the influence of the nature and intensity of 4f–4f interactions on the Ln2 relaxation. Works in this line have been therefore guided by these two leit motifs.

6.1.1 Dinuclear 4f Clusters With New Ln–Ln Bridges [Ln2] clusters with a broad variety of bridging motifs, yielding to a range of effective energy barriers, have been reported (Habib and Murugesu, 2013). Recently, Han et al. (2016) have reviewed dinuclear SMMs up to January 2016, classified by the bridge type, including monoatomic hydrogen, carbon, nitrogen, oxygen, chloride and sulfur bridges, and polyatomic bridges like m1,3-carboxilate, pyrazine, and radical bridges. We highlight here some notable and more recent examples. Whenever a new Ln2 family is reported, the Gd2 complex is often analyzed first as a way to determine the exchange interaction without the influence to the first order of spin–orbit coupling; however, we omit those complexes from our survey, since they are not slow-relaxing. 6.1.1.1

Oxygen-Based Bridges (Table 5)

Complexes with O-based bridges have been extensively utilized in SMM chemistry because these ligand systems can be easily designed and

TABLE 5 Glossed Homodinuclear [Ln2] SMMs Using Oxygen Bridges Complex

C (s21/Kn)

#

J/kB (K)

Ueff/kB (K)

τ 0 (s)

1

FM

150 (FR)

2.3  108

198 (SR)

9

n

H (kOe)

References

0

Guo et al. (2011)

0

Yi et al. (2012)

Oxygen-based bridges [Dy2(ovph)2Cl2(MeOH)3].MeCN

[Dy(hfac)3(PyNO)]2

AF

7.3  10

166.9

3

98

2.3  108

0

Holynska et al. (2015)

[Dy2(hfac)4L2]

4

50.33

1.05  108

0

Shen et al. (2015)

[Dy2(dbm)4(OQ)2(CH3OH)2]

3

109.5

4.23  109

3

Shen et al. (2016)

[Ln2(NO3)2(saph)2(DMF)4]

5

25.03 (1)

3.0  106

0

Anastasiadis et al. (2015)

0

Lin et al. (2015c)

[Dy2(naphthsaoH)2(acac)4 H(OH)]0.85CH3CN1.58H2O

FM

4.4  107 23.31 (2) 2  107

[Dy2(HL)2(PhCOO)2(DMF)2]4DMF

2

FM

61

[Dy2(HL)2-(CH3COO)2(DMF)2]

1

FM

—a

0

[Ln2(H3L)2(PhCOO)6]

2

—a

0.3

3

a

1.3

[Dy2(H4L0)2(PhCOO)4]2CH3CN

, two processes

Lin et al. (2016)

Dy2(Hhms)2(NO3)4]MeCN [Dy2(Hhms)2(H2O)3] (NO3)2MeCN(H2O)2 [Ln2(ovgrd)2(acac)2(H2O)2]2EtOH

72.1

1.15  108

4, Er

21

5, Yb

1

1.2

Li et al. (2017b)

3.7

2.75  106

1.2

22

2.13  106

1.8

Jiang et al. (2016)

9.5

1.08  106

1.5

Gorczynski et al. (2016)

4, Dy

67

4.8  106

0

5, Ho

—a

Zhang et al. (2016b)

No SMM

0

Moreno Pineda et al. (2014)

—a

3.

Baniodeh et al. (2017)

3

[Er2L3](NO)3 [Ln2L2(NO3)2(C2H5OH)2]0.5py

12.8

AF

1.67

1.66

[hqH2][Dy2(hq)4(NO3)3]MeOH

3

[Dy2(H2tea)2(O2CPh)4]2H2O

1

[Dy2(valdien)2(NO3)2]

4

3.777

76

6.04  106

0

Long et al. (2011)

[Dy2(Lx)2(L’)2(CH3OH)2].yG]

1a

0.065

69

1.37  106

0

51

6

0

Mukherjee et al. (2016)

2a 3a (Dy(acac)2(CH3OH)2(m-HMq)2 [Dy(DBM)2]2(m-HMq)2(n-C6H14)

1 2

FM

0.237 0.070 0.324 1.51

1 75.6 18.6

4.54  10 1  10

4

0

2.1  10

8

0.76

3.4

2.

2.9  10

6

47.3

1.4

1.9

6

2.6

2.1

1.9

53.2

1.7

1.5

[Dy(hmac)2]2(m-HMq)2

3

4.89

25.8

6.1  10

[Dy(hfac)3]2(m-HMq)2

4

0.7

26.9

4.7  107

Zhang et al. (2016e)

Continued

TABLE 5 Glossed Homodinuclear [Ln2] SMMs Using Oxygen Bridges—cont’d Complex [Dy2(NO3)4 (sacbH)2(H2O)2(MeCN)2] [Ga4Ln2(shi3)4(Hshi2)2 (H2shi)2(C5H5N)4(CH3OH)x (H2O)x]xC5H5NxCH3OHxH2O

# 1

J/kB (K) 0.002

Ueff/kB (K) 109.3 128.2

3, Dy2 6, YDy

26 (HT) 18 (LT) 31

C (s21/Kn)

τ 0 (s)

H (kOe)

References

1.4  10

7

n

0

1.1  10

7

1

Mazarakioti et al. (2017)

6.8  10

6

0

3.6  10

6

0

7.6  10

6

Chow et al. (2015)

0

# Complex number as in the original reference. The intramolecular coupling constant is given in 2JS1S2 formalism. Relaxation time according to τ ¼ τ0 1 e Ueff =kB T + CTn, where the first and second terms correspond to Orbach and Raman relaxation mechanisms, respectively. LT/HT, low-/high-temperature processes. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. a Slow relaxation but no Ueff reported. 1

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 143

provide direct superexchange pathways. Numerous examples of Ln2 with monoatomic O-bridges exist, with record activation energies approaching 200 K, achieved for [Dy2(ovph)2Cl2(MeOH)3]MeCN, where the FM-coupled Dy ions are bridged by the phenoxido groups of two ovph2 ligands (Ueff/kB ¼ 150 K, 198 K for the two observable processes), and [Dy (hfac)3(PyNO)]2 with Ueff/kB ¼ 166.9 K and an opening of the hysteresis loop at 1.4 K (Yi et al., 2012). In more recent years, Holynska et al. (2015) have reported the [Dy2(naphthsaoH)2(acac)4H(OH)]0.85CH3CN1.58H2O complex based on oxime ligands, showing SMM behavior with a significant Ueff/kB ¼ 100 K. Also, Shen et al. (2015) presented complex [Dy2(hfac)4L2] using two bidentate hfac and two m-phenol-bridging ligands, showing slow magnetic relaxation with Ueff/kB ¼ 50.33 K, in zero applied field. The same authors presented in a different work (Shen et al., 2016) complex [Ln2(dbm)4(OQ)2(CH3OH)2], where each Dy ion is eight-coordinated with two bidentate dbm and two m-phenol-bridging OQ ligands and one methanol molecule, showing field-induced relaxation of Ueff/ kB ¼ 109 K (at 3000Oe). Schiff ligands are being employed for the synthesis of numerous dinuclear complexes. For example, Anastasiadis et al. (2015) have reported [Dy2(NO3)2(saph)2(DMF)4] based on the use of a tridentate Schiff base, in which the Dy ions are doubly bridged by the deprotonated iminophenolato oxygen atoms of two nearly planar Z1:Z1:Z2:msaph2 ligands. A probable FM exchange interaction was claimed, and AC susceptibility measurements revealed two relaxation processes (Ueff/kB ¼ 25.03 K, 23.31 K), assigned to SMM via excited states. Lin et al. (2015c) have presented two novel centrosymmetrical dinuclear Dy(III) compounds, [Dy2(HL)2-(CH3COO)2(DMF)2] (1) and [Dy2(HL)2(PhCOO)2(DMF)2]4DMF (2) with differing carboxylate groups, which introduce different dynamic behavior (Fig. 57B). SMM relaxation is observed for complex 2 (Ueff/kB ¼ 61 K), thanks to the m2-chelating benzoate ligands and stronger FM interaction between the Dy2 atoms, while no relaxation is evident in complex 1 with acetate-bridging groups with a syn–syn Z1:Z1:m2 mode. Two different symmetrical Schiff-base supported complexes were recently presented by Lin et al. (2016): in [Dy2(H3L)2(PhCOO)6] (2) the ninecoordinated Dy(III) ions are bridged by two syn–syn Z1:Z1-m2-benzoate groups, while in [Dy2(H4L0)2(PhCOO)4]2CH3CN (3) the eight-coordinated Dy ions are bridged by two alkoxido oxygen atoms from additional arms of H5L0 . Field-induced behavior was reported for both compounds, although the Ueff was not given. Under the optimal field, two relaxational processes were observed. Li et al. (2017b) have recently shown that fine-tuning of the ligand field through changes to the backbone of the Schiff ligand and the m-O bridges

O-bridges A N1

N4 O13

O3

O6

O2

Dy1A

O14 O8

O8

O1 O1A

O9 Dy1

O2A

O3A

O7

O1

O7

O5

O4

N1

O9

O2

O10 O3

Dy2

O5

O2

N1

Dy1

O7

Dy1A Dy1 O2A

O8 O5 O3

O9

O3

O1A

O11

N1A

N1

3

1

4

B

N4 Dy1A O1

O3

N3

N6 Dy1

O3A

O1A

O1B

Dy1 O2 O2B N1B

Dy1B

N4A

1

2

FIG. 57 Dinuclear O-bridged complexes. (A) [Dy2(Hhms)2(NO3)4].MeCN (1), [Dy2(Hhms)2(H2O)3](NO3)2MeCN(H2O)2 (3), and [Dy2(Hhmc)2(NO3)4] THFMeCN (4) reported by Li et al. (2017b). (B) [Dy2(HL)2-(CH3COO)2(DMF)2] (1), [Dy2(HL)2(PhCOO)2(DMF)2]4DMF (2) from Lin et al. (2015c). Adapted with permission from Li, M., Wu, H., Zhang, S., Sun, L., Ke, H., Wei, Q., Xie, G., Chen, S., Gao, S., 2017b. Fine-tuning ligand fields with Schiff-base ligands in Dy2 compounds. Eur. J. Inorg. Chem. 2017, 811–819, https://doi.org/10.1002/ejic.201601188. Copyright 2017 John Wiley and Sons. Adapted with permission from Lin, S.-Y., Wu, J., Wang, C., Zhao, L., Tang, J., 2015c. Modulating relaxation dynamics of Dy2 compounds through carboxylate coordination modes. Eur. J. Inorg. Chem. 2015, 5488–5494, https://doi.org/10.1002/ejic.201500956. Copyright 2015 John Wiley and Sons.

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provides an elegant way of studying the influence of ligand-field effects on the SMM behavior of Dy2 systems. Comparing the behavior of complex [Dy2(Hhms)2(NO3)4]MeCN (1) and [Dy2(Hhms)2(H2O)3](NO3)2MeCN (H2O)2 (3), they established that an additional bridge in 3 compared with in 1 breaks the hula-hoop-like coordination geometry and induces smaller Dy–O–Dy angles, resulting in SMM behavior for 1 and absence of it in 3. Moreover, maintaining the coordination environment while changing the extent of conjugation in the backbone of the ligand gives rise to different dynamic properties, as shown by the absence of SMM behavior in the previously reported complex [Dy2(Hhmc)2(NO3)4]THFMeCN (4) (Kuo et al., 2015) in contrast to 1 (Fig. 57A). The vast majority of dinuclear SMMs are based on Dy(III) ion (e.g., a summary of m2-bridged Dy2 complexes can be found in J. Zhang et al. (2016b). However, a few examples including other lanthanides have been recently reported. Jiang et al. (2016) have presented the centrosymmetric complexes [Ln2(ovgrd)2(acac)2(H2O)2].2EtOH, Ln ¼ Dy(3), Er(4), Yb(5) based on the new Schiff base ligand (H2ovgrd), where the Ln ions are bridged by two oxygens from alkoxide groups. Differently to what is commonly observed, in this family of compounds the Dy2 complex does not present SMM behavior, while the Er2 and Yb2 counterparts do show field-induced slow relaxation. Another curious example is the erbium triple-stranded helicate [Er2L3] (NO)3 with the {Er2O3}3+ phenoxo core presented by Gorczynski et al. (2016). The Er2 complex presents field-induced SMM behavior (with small Ueff/kB ¼ 9.5 K at 1500 Oe), while the Dy2 counterpart does not. Helical species are interesting since they may display inherent chirality; the unit cell of this compound was found to include 12 independent helical species equally distributed between D and L isomers. On the other hand, J. Zhang et al. (2016b) have recently demonstrated the first Ho2 compound exhibiting slow magnetic relaxation (though no peaks could be found to determine the Ueff, even after application of DC field). The Dy2 analogue presented SMM behavior with a relatively high Ueff among the m2-O-bridged Dy2 complexes. 6.1.1.2 Carboxylic Bridges (Table 6) Wang et al. (2016c) have recently reported the first example of a carboxylatebridged dinuclear Dy2 system. Interestingly, the authors presented three complexes ([Dy2(m2-anthc)4(anthc)2(L)2], where the first coordination sphere of the Dy(III) ions was fixed, while terminal neutral ligands L were sequentially modified (L ¼ 2,20 -bipyridyl (1), 1,10-phenanthroline (2), and 4,7dimethyl-1,10-phenanthroline (3)), see Fig. 58, left). The increasing size of the neutral ligands from 1 to 3 led to larger charge distribution along the magnetic axis and lower charge distribution in the equatorial (hard) plane formed

TABLE 6 Glossed Homodinuclear [Ln2] SMMs Using Carboxylate, Nitrogen, and C-Aromatic Bridges Complex

#

J (K)

Ueff/kB (K)

τ 0 (s)

C (s21/Kn)

1

6.56

51.2

3.2  108

3.2  108

9

9

n

H (kOe)

References

0

Wang et al. (2016c)

Carboxylate-based bridges [Dy2(m2-anthc)4 (anthc)2(L)2]

2 3

49.4 5.64

4.6  10

8

4.6  10

0

8

31.6

3.4  10

3.4  10

120 (LT)

5.6  109

0.066

5.1

0

6.4  1010

0.018

4.1

0

11

0.035

4.6

0

0

5.52 Nitrogen-based bridges [(DBM)6Dy2BPYM] 2CHCl3

1

AF

[(DBM)6Dy2BPYM]. MeCN

2

AF

266 (HT)

2.1  10

[(15C5)4PcTb (15C5)4PcTb(Pc)]

1

FM

229.9 K

7.1  109

0

[(15C5)4PcTb(15C5)4Pc] Y(Pc)

2

6.7  107

0

[(15C5)4Pc]Y[(15C5)4Pc] Tb(Pc)

3

2.7  107

0

129.8 K

Sun et al. (2016a)

Holmberg et al. (2016b)

168.1 K

C-aromatic bridges [Er2(COT00 )3]

1

332.4

Le Roy et al. (2014)

[(NNTBS) Dy]2(m-biphenyl)[K (solvent)]2

Dy2— biph

1.02

34.5

[KEr2(C7H7)(N(SiMe3)2)4]

3, Er

1.72

58

2.9  108

0.8

Harriman et al. (2017b)

1

FM

13.2

2.3  108

0

Poneti et al. (2007)

FM

326.6

8.2  109

0

Rinehart et al. (2011)

56.1

5.08  105

Guo and Layfield (2017)

Huang et al. (2015a)

Radical bridges [Dy(hfac)3(NIT4Py)]2 [K(18-C-6)] {[(Me3Si)2(THF) Ln]2(m-Z2:Z2-N2)} [{(Z5-Cp*)2Ln}2(m-ind)]n

1Dy 2Dy

50.4

3Dy

No SMM

8

1.6  10

1.72  103

5.2

0

3

8.2

0

2.27  10

# Complex number as in the original reference. The intramolecular coupling constant is given in 2JS1S2 formalism. Relaxation time according to τ1 ¼ τ0 1 e Ueff =kB T + CTn, where the first and second terms correspond to Orbach and Raman relaxation mechanisms, respectively. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength

148 Handbook of Magnetic Materials

FIG. 58 Left: Carboxylate-bridged complexes: [Dy2(m2-anthc)4(anthc)2(L)2], L ¼ 2,20 -bipyridyl (1), 1,10-phenanthroline (2), 4,7-dimethyl-1,10-phenanthroline (3) from Y. L. Wang et al. (2016c). Right: N-bridged complex [(DBM)6Dy2B-PYM]2CHCl3 (1) from (Sun et al., 2016a). Reprinted with permission from Wang, Y.L., Han, C.-B., Zhang, Y., Liu, Q.-Y., Liu, C.-M., Yin, S.-G., 2016c. Fine-tuning ligand to modulate the magnetic anisotropy in a carboxylate-bridged Dy2 single-molecule magnet system. Inorg. Chem. 55, 5578–5584. Copyright 2016 American Chemical Society. Reprinted from Sun, W.-B., Yan, B., Jia, L.-H., Wang, B.-W., Yang, Q., Cheng, X., Li, H.-F., Chen, P., Wang, Z.-M., Gao, S., 2016a. Dinuclear dysprosium SMMs bridged by a neutral bipyrimidine ligand: two crystal systems that depend on different lattice solvents lead to a distinct slow relaxation behavior. Dalton Trans. 45, 8790–8794, https://doi.org/10.1039/ c6dt01082b. Published by The Royal Society of Chemistry.

by five coplanar coordination atoms around each Dy(III), leading to increasing anisotropy energy barriers Ueff(1) > Ueff(2) > Ueff(3). 6.1.1.3

Nitrogen-Based Bridges (Table 6)

Layfield and coworkers presented pioneering studies on different N-bridged complexes, which helped rationalizing why some dinuclear species show SMM behavior, while others do not (Layfield et al., 2010). New types of N-bridges have been recently investigated. For example, Sun et al. (2016a) have reported Dy2 complexes bridged by neutral bipyrimidine (BPYM) ligands, [(DBM)6Dy2B-PYM]2CHCl3 (1) (Fig. 58, right) and [(DBM)6Dy2BPYM]MeCN (2). BPYM possesses a rigid coplanar and delocalized aromatic conjugated electronic structure. By performing experiments on diluted samples, it was shown that relaxation in these systems stems from relaxation of the individual Dy ions. The single and double relaxation processes observed in complexes 1 and 2, respectively, were assigned to one and two types of Dy(III) environments in the two types of dimers. Hence in these systems the weak magnetic coupling was not a dominant factor for relaxation. This work is in contrast with the previously reported complex [(Cp*2Ln)2(m-bpym.)]+ (Demir et al., 2012) in which the Ln2 ions were bridged by the radical BPYM. anion, and in which the AF coupling constant favored the Dy2 system to display a SMM energy barrier of Ueff/kB ¼ 126.3 K and magnetic hysteresis up to 6.5 K (at sweep rate of 0.002 T/s).

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 149

6.1.1.4 C-aromatic Bridges (Table 6) The use of arene bridges proved useful in the past to build dinuclear species like [Ln2COT00 3], Ln ¼ Dy, Er (Le Roy et al., 2014) or [(NNTBS) Dy]2(m-biphenyl)[K(solvent)]2 (Huang et al., 2015a), where the aromatic ligand played a major role in providing a superexchange pathway for interaction. Harriman et al. (2017b) have recently reported new, rare dinuclear complexes containing the m7-C7R7 bridge: ([KLn2(C7H7)(N(SiMe3)2)4] (Ln ¼ Dy (2), Er(3)) and [K(THF)2Er2(C7H7)(N(SiMe3)2)4]) 4 (Fig. 59). Significant exchange constants of Jex/kB ¼ + 1.99 K, +2.587 K, and +4.53 K overwhelming dipolar contributions (Jdip/kB ¼  0.867 K, 0.865 K, and 0.683 K) were found by ab initio for compounds 2–4. Ab initio calculations also revealed drastic changes in the orientations of the EAM between the Ln ions in the Er2 and Dy2 complexes. While for the Dy(III) sites the EAM was oriented in the plane of the N–Dy–N atoms, almost parallel to the N–N direction, in the Er compounds 3 and 4 the EAM was oriented almost perpendicular to the N–Er–N planes. Two different field-induced relaxation processes were observed, which were ascribed to single-ion behavior, given the noncentrosymmetric nature of the molecules. 6.1.1.5 Radical Bridges The synthesis of magnetic clusters involving organic radicals that contribute with a nonzero spin to the cluster moment has been effective in obtaining new SMMs, since in some cases they enhance magnetic interaction or influence the ligand field acting on the Ln ion. Radical-bridged dinuclear systems Ring C-bridge A

B Si1 Si1

Si2

N1

Si2

Er1

N1

Si5 Si5

Dy1

Si3 Er2 N3

N2

Si6

Si3 N3

Dy2

N2

Si6

Si4 N4 Si7

Si4

N4

K1

Si7

K1

Si8 Si8

Dy2 (2)

Er2 (3)

FIG. 59 C-ring-bridged complexes ([KLn2(C7H7)(N(SiMe3)2)4]: (A) Ln ¼ Dy(2), (B) Er(3). Reprinted from Harriman, K.L.M., Le Roy, J.J., Ungur, L., Holmberg, I.K., Murugesu, M., 2017b. Cycloheptatrienyl trianion: an elusive bridge in the search of exchange coupled dinuclear organolanthanide single-molecule magnets. Chem. Sci. 8, 231–240, https://doi.org/10.1039/ c6sc01224h. Published by The Royal Society of Chemistry.

150 Handbook of Magnetic Materials

hold great promise for achieving strong interaction between the metal centers, due to the diffuse spin orbitals present in radical ligands that can penetrate the core electron density of the lanthanide ions. The first radical-bridged dinuclear system was based upon the pyridine functionalized nitronyl nitroxide ligand (NIT-4Py), the complex [Dy(hfac)3(NIT-4Py)]2, exhibiting an activation energy of Ueff/kB ¼ 13.24 K (Poneti et al., 2007). Demir et al. (2015) presented a survey of radical ligand-containing SMMs including complexes bridged by nitronyl nitroxide, tetrathiafulvalene, thiazyl, 2,2-bipyrimididine, tetra-2-pyridinylpyrazine, and N3 2 radicals. Particularly important within the last family is the N3 2 -bridged Tb2 dinuclear complex [K(18-crown-6)] {[(Me3Si)2(THF)Ln]2(m-Z2:Z2-N2)}, which holds at present the highest barrier for a multinuclear SMM (Ueff/kB ¼ 326.6 K), and a striking hysteresis loop widely opens up to 14 K, which represents one of the highest magnetic blocking temperature for any molecular species (Rinehart et al., 2011). In Table 7 the most recent radical-bridged Ln-based dinuclear SMMs appearing in the literature are summarized. From inspection, one observes that only for the Ln ¼ Tb and Dy substitutions slow relaxation has been detected. A Dy–r–Dy cluster with the r ¼ MeTrzNIT radical connecting two Dy(hfac)3 units has been recently presented (Li et al., 2016b). Examples of biradical-bridged clusters (Ln–2r–Ln) have also been reported. For example, in the U-shaped [Dy2(hfac)6(IPhIN)(H2O)2] cluster the presence of the biradical bis(imino nitroxide, IPhIN), with S ¼ 1/2 for each r, is peculiar since one needs to take account of the interaction between the radicals within the biradical Jrr/kB ¼ 4.7 K to explain the static magnetic properties; a relatively large Ueff/kB ¼ 27 K is reported for this cluster (Reis et al., 2016). The active modulation of the magnetic relaxation may be achieved, for example, by using a photoresponsive coupler between the Ln ions in the dimer. In [Ln(tta)3(4-D1pyO)]2 (Fig. 60A), Ln ¼ Tb and Dy the slow relaxation characteristics are modified drastically after irradiating (Murashima et al., 2016). Indeed, when the 4-D1py is photolyzed, carbene groups are formed within the triplet state that interact ferromagnetically with the Ln ions through the pyridine-N-oxide, albeit weakly. For the Ln ¼ Tb substitution irradiation gives the cluster the SMM character (with two relaxing species), while for the Ln ¼ Dy case the strong SMM behavior, with a very high Ueff in the nonirradiated case, decreases in Ueff by a factor of 3 when irradiated (see Fig. 60B). It is puzzling that such a strong effect is produced by the generation of carbene, since the magnetic interactions between radical and Ln are very weak. Another alternative is that the ligand field is modified by irradiation and, consequently, the Dy single-ion anisotropy is strongly reduced. Guo and Layfield (2017) have recently analyzed the magnetostructural properties of indigo-bridged Ln2 complexes, [{(Z5-Cp*)2Ln}2(m-ind)]n, where the indigo ligand can be accessed in three different oxidation states, n ¼ 0, 1, 2 (complexes 1Dy, 2Dy, 3Dy, Fig. 61). Complexes 1Dy and 2Dy exhibited SMM behavior, with Ueff (1) > Ueff (2), while 3Dy did not. The later was

TABLE 7 Glossed Homodinuclear [Ln2] SMMs Using Radical Bridges Cluster

Complex

Radical (r)

#

Ueff/ kB (K)

τ 0 (s)

H (kOe)

References

Dy–2R– Dy

[Dy(hfac)3(NITpPy)]2

NITpPy

1

13.2

2.3  108

0

Poneti et al. (2007)

Dy–R– Dy

[K(18-C-6)]{[(Me3Si)2(THF) Ln]2(m-Z2:Z2-N2)}

N3 2

1

326.6

8.2  109

0

Rinehart et al. (2011)

Dy–R– Dy

[Dy2(hfac)6(MeTrzNIT) (H2O)2].1/2CH2Cl2

MeTrzNIT

3(Dy)

6.0

4  106

0

Li et al. (2016b)

Dy–R– R–Dy

[Dy2(hfac)6(IPhIN)(H2O)2]

Bis(imino nitroxide)

2(Dy)

27

1.4  108

2

Reis et al. (2016)

2d(Tb)

No SMM

—

—

Murashima et al. (2016)

Irradiated

31, 9.1

5.9  108, 6.5  106

0

3d(Dy)

102

3.6  107

0

Irradiated

39

1.5  108

0

(s ¼ 1/2)

(s ¼ 1/2)

(s ¼ 1/2) Tb-2RTb

Dy–2R– Dy

Dy–R– Dy

[Tb(tta)3(4-D1pyO)]2

4-D1pyO (s ¼ 1)

[Dy(tta)3(4-D1pyO)]2

5

n

[{(Z -Cp*)2Dy}2(m-ind)]

[ind]

3

1Dy(Dy) 2Dy(Dy)

56.1 50.4

5

5.08  10

8

1.6  10

0 0

Guo and Layfield (2017)

3Dy(Dy) The intramolecular coupling constant is given in 2JS1S2 formalism. Relaxation time according to τ1 ¼ τ0 1 e Ueff =kB T . The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength

152 Handbook of Magnetic Materials

FIG. 60 (A) Molecular structure of radical-bridged [Dy(tta)3(4-D1pyO)]2. C, gray; N, blue; O, red; S, yellow, Dy, pink. (B) Relaxation time temperature dependence before irradiation (red), after irradiation (blue). Adapted with permission from Murashima, K., Karasawa, S., Yoza, K., Inagaki, Y., Koga, N., 2016. 3- and 4-(a-diazobenzyl)pyridine-N-oxides as photoresponsive magnetic couplers for 2p–4f heterospin systems: formation of carbene–TbIII and carbene–DyIII single-molecule magnets. Dalton Trans. 45, 7067–7077, https://doi.org/10.1039/c6dt00420b. Published by The Royal Society of Chemistry.

ascribed to the large formal charge on the ligand (4) in complex 3Dy, giving rise to strong electrostatic interactions between the indigo ligand and the Dy3+ centers in the hard plane. The authors concluded that direct coupled radical ligands in SMMs do not necessarily result in high magnetic blocking temperatures and open hysteresis, and that the hard/soft nature of the donor atoms and their formal charge should be accounted for.

6.1.2 Influence of the Intradimer Coupling on Magnetic Relaxation Magnetic relaxation of Ln2 complexes is both mediated by the single-ion anisotropy of the individual Ln ions and Ln–Ln interactions. However, the effect of such interactions is intriguing, as they have shown to hinder or enhance magnetic relaxation in different examples, and therefore, understanding their influence is crucial. A Design of Experiments (DOE) approach would be desirable in order to determine how the ion anisotropy, the sign (FM/AF), nature (dipolar vs exchange), and strength of interactions affect relaxation, but unfortunately, such systematic synthetic approach is at present out of reach. Nevertheless, some of the cases have been treated (see, e.g., Table 8), evidencing a rich diversity of phenomenology. It is worth mentioning that ab initio calculations are gaining importance in the interpretation of relaxation processes in dimers, as they allow determining both the single-ion properties (SINGLE-ANISO), and the dimer energy states and the sign and intensity of both dipolar and exchange interactions under the Lines model (POLY-ANISO) (Harriman et al., 2017b; Jiang et al., 2016; Mukherjee et al., 2016; Zhang et al., 2016b; see Section 3 for details). On the other hand, EPR is also proving to be an excellent technique to determine

FIG. 61 Radical-bridged complexes [{(Z5-Cp*)2Ln}2(m-ind)]n, with n ¼ 0, 1, 2 (complexes 1Dy, 2Dy, 3Dy). Adapted with permission from Guo, F.-S., Layfield, R.A., 2017. Strong direct exchange coupling and single-molecule magnetism in indigo-bridged lanthanide dimers. Chem. Commun. 53, 3130–3133, https://doi.org/10.1039/c7cc01046j. Published by The Royal Society of Chemistry.

TABLE 8 Design of Experiments (DOE)-Like Matrix, Summarizing the Results of Some Recent Works Studying the Effect of the Single-Ion Anisotropy (Quantified Here by the GS Single-Ion gz Value), the Nature (FM: +/AF: 2) and the Strength (From High ++++ to Low +) of Dipolar, Exchange, and Total Intradimer Interactions on the Slow Relaxation Energy Barrier Complex

#

gz

Jdip/kB (K)

[Dy2(Lx)2(L )2(CH3OH)2]yG, x¼2

2a

19.683

FM

++

0.14

FM

++

0.09

FM

++++

[Dy(hfac)3]2(m-HMq)2

4

14.22

FM

++

0.05

FM

++

0.67

FM

(Dy(acac)2(CH3OH)2(m-Mq)2

1

19.577

FM

++

0.86

AF



0.54

[Dy2(Lx)2(L0 )2(CH3OH)2]yG, x¼1

1a

19.655

FM

++

0.13

AF



0.06

[Dy2(Lx)2(L0 )2(CH3OH)2]yG, x¼3

3a

19.629

FM

++

0.14

AF



[Dy2(m2-anthc)4(anthc)2(L)2]

1

19.654

FM

++

2.00

AF

2

19.629

FM

++

2.31

3

19.672

FM

++

[Dy(DBM)2]2(m-HMq)2(n-C6H14)

2

19.339

AF

[Dy2(NO3)4 (sacbH)2(H2O)2(MeCN)2]

1

19.79

AF

[KEr2(C7H7)(N(SiMe3)2)4]

3

17.763 (Er1)

0

Jex/kB (K)

Jtot/kB (K)

Ueff/kB (K)

References

0.24

51

1

++++

0.72

26.9

2

FM

+

0.32

75.6

2

FM

+

0.06

69

1

0.07

FM

+

0.07

1

1



0.36

FM

+

1.64

51.2

3

AF



0.90

FM

+

1.41

49.4

3

1.92

AF



0.54

FM

+

1.38

31.6

3



1.87

FM

+

0.36

AF



1.51

18.6

2



0.0085

FM

+

0.0065

AF



0.002

109.3

4

AF



0.86

FM

++

2.59

FM

+

1.72

58

5

AF



1.73

AF



3.14

AF



4.87

25.8

2

0

L ¼ 2,2 -bipyridyl [Dy2(m2-anthc)4(anthc)2(L)2] L ¼ 1,10-phenanthroline [Dy2(m2-anthc)4(anthc)2(L)2] L ¼ 4,7-dimethyl-1,10phenanthroline

17.978 (Er2) [Dy(hmac)2]2(m-HMq)2

3

19.581

Coupling constants are expressed in 2JS1S2 effective spin Hamiltonian. References: 1, Mukherjee et al. (2016); 2, Zhang et al. (2016e); 3, Wang et al. (2016c); 4, Mazarakioti et al. (2017); 5, Harriman et al. (2017b).

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 155

experimentally the {g} and anisotropic {J}-tensors, calculate the dimer’s low energy eigenstates, and therefore rationalize the occurrence or not of SMM behavior in {Ln2} systems (Baniodeh et al., 2017; Moreno Pineda et al., 2014). Strong magnetic exchange coupling provided by radical-bridging ligands has led to remarkable magnetic hysteresis at T ¼ 14 K (Rinehart et al., 2011); this result has motivated the research of radical-bridged Ln2 complexes with yet stronger exchange pathways, as commented in the previous section. Absence of collinearity between the Ln–Ln axes hinders the observation of SMM behavior. Indeed, Moreno Pineda et al. (2014) have demonstrated in the asymmetric complex [hqH2][Dy2(hq)4(NO3)3]MeOH (Fig. 62A) that the noncollinearity of the principal axis of the Dy(III) ions, tilted by 44°, and the associated anisotropic ferromagnetic exchange interaction leads to quenching of the SMM behavior. The EPR-obtained lowest energy states consist of a ferromagnetic GS doublet and two singlets (Fig. 62B); at zerofield fast relaxation between these doublet states and the excited states is very efficient; as field is applied, the component of the doublet that is deestabilized begins to mix with the first singlet excited state, so further efficient relaxation pathways become available, and SMM is not observed even under applied field. It is remarkable though that certain dimers with relatively low small uniaxial anisotropy can still support field-induced SMM behavior. E.g., Baniodeh et al. (2017) have recently shown that complex [Dy2(H2tea)2(O2CPh)4] 2H2O, in which Ln–Ln dipolar (exchange) coupling constants are Jdip x / dip ex dip kB ¼  1.132 K (Jex x /kB ¼ 0.891 K), Jy /kB ¼ 0.891 K (Jy /kB ¼  0.004 K), Jz / ex kB ¼ 0.361 K (Jz /kB ¼  0.888 K), exhibits SMM (under 3000 Oe), despite the large deviation of local anisotropy from uniaxiality of the Dy(III) ions (gx ¼ 11, gy ¼ 8.2, gz ¼ 1). In other compounds, weak exchange Ln–Ln interactions have shown to shift the zero-field quantum tunneling step to a finite field (the so-called exchange biasing), with different effects on the M(H) cycles measured at very low T depending on the sign of interaction. For example, Guo et al. (2011) reported for complex [Dy2ovph2Cl2(MeOH)3]MeCN a shift in the magnetization loops measured at very low T ¼ 0.04 K, attributed to the exchange bias field arising from weak ferromagnetic interactions, mainly of dipolar origin (Jdip/kB ¼ 3.85 K, Jex/kB ¼ 0.374 K), originating from the nearly parallel alignment of the EAM and the line connecting the Dy ions (Fig. 63A). On the other hand, Long et al. (2011) reported for complex [Dy2(valdien)2(NO3)2] “S-shaped” hysteresis loops with large steps at H ¼  3 T, which were associated to the spin-flip of the AF-coupled Dy spins (Jdip/kB ¼  3.626 K, Jex/ kB ¼  0.151 K) (Fig. 63B). By performing experiments on {DyY} diluted compounds with 50%, 10%, and 5% Dy, Habib et al. (2011) further showed that the increase in the percentage of Dy2 served to eliminate the QTM at zero field, shifting it to a higher bias field (Fig. 63C).

FIG. 62 (A) Crystal structure and (B) schematic of magnetic model for EPR simulation for complex [hqH2][Dy2(hq)4(NO3)3].MeOH; (C) lowest energy states calculated from EPR as a function of magnetic field coincident with the principal axis of Dy(1); wave functions c1 ¼ j""i, c2 ¼ j##i, c3 ¼ 1/√2(j"#i + j#"i), c4 ¼ 1/√2(j#"i + j"#i); arrows correspond to EPR resonances observed for this orientation (Moreno Pineda et al., 2014). The SI conversion factor for energy is 1 cm1 ¼ 1.986  1023 J. Reprinted by permission from Macmillan Publishers Ltd: Nature Communications (Moreno Pineda, E., Chilton, N.F., Marx, R., Dorfel, € M., Sells, D.O., Neugebauer, P., Jiang, S., Collison, D., van Slageren, J., McInnes, E.J.L., Winpenny, R.E.P., 2014. Direct measurement of dysprosium(III)dysprosium(III) interactions in a single-molecule magnet. Nat. Commun. 5, 5243, https://doi.org/10.1038/ncomms6243), copyright 2014.

A

B

C

Y1

Y1a Y

1

1 0.07 T/s 5%

0.04 K 0.5 K 1K 2K 3K 4K 5K

1

0.04 K

0.5

0 0.280 T/s 0.140T/s 0.070 T/s 0.035 T/s 0.017 T/s 0.008 T/s 0.004 T/s 0.002 T/s 0.001 T/s

−0.5 Hbias −0.4

−0.2

0 m 0 H (T)

0.2

0.4

0.04 K 0.5 K 1K 2K 3K 4K

−0.5 −1

−0.6 −0.4 −0.2

0 0.2 m 0 H (T)

0.4

0.04 K 0.5 K 1K 2K 3K 4K 5K

−1

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 m 0 H (T)

1

0.14 T/s

0.5

100 %

−0.5 0.6

0

50 %

0

−1

0.04 K 0.5 K 1K 2K 3K 4K 5K

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 m 0 H (T)

Dy1a

10 %

−0.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 m 0 H (T)

Dy1

0.07 T/s

1 0.07 T/s 0.5

0

M/Ms

M/Ms

M/Ms

0.5

−1

−1

0.14 T/s

M/Ms

0

−0.5 1

0.5

M/Ms

M/Ms

0.5

Dy

0

−0.5 −1

0.04 K 0.5 K 1K 2K 3K 4K

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 m 0 H (T)

FIG. 63 Structure and ab initio calculated EAM, and hysteresis cycles measured at T ¼ 0.04 K for (A) complex [Dy2ovph2Cl2(MeOH)3]MeCN (Guo et al., 2011); (B) complex [Dy2(valdien)2(NO3)2] (Long et al., 2011); and (C) the same valdien pure complex (100%) and diluted {DyY} complexes with 50%, 10%, 5% Dy (Habib et al., 2011). Adapted with permission from Guo, Y.N., Xu, G.F., Wernsdorfer, W., Ungur, L., Guo, Y., Tang, J., Zhang, H.J., Chibotaru, L.F., Powell, A.K., 2011. Strong axiality and Ising exchange interaction suppress zero-field tunneling of magnetization of an asymmetric Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 11948–11951, https://doi.org/10.1021/ja205035g; Long, J., Habib, F., Lin, P.H., Korobkov, I., Enright, G., Ungur, L., Wernsdorfer, W., Chibotaru, L.F., Murugesu, M., 2011. Single-molecule magnet behavior for an antiferromagnetically superexchange-coupled dinuclear dysprosium(III) complex. J. Am. Chem. Soc. 133, 5319–5328, https://doi.org/10.1021/ja109706y; and Habib, F., Lin, P.-H., Long, J., Korobkov, I., Wernsdorfer, W., Murugesu, M., 2011. The use of magnetic dilution to elucidate the slow magnetic relaxation effects of a Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 8830, https://doi.org/10. 1021/ja2017009. Copyright 2011 American Chemical Society.

158 Handbook of Magnetic Materials

Small structural differences fine-tuned by the ligand architecture may have significant influence in the dimeric coupling. For example, Mukherjee et al. (2016) showed by studying three m2-oxo-bridged Dy2 complexes, [Dy2(Lx)2(L0 )2(CH3OH)2]yG, x ¼ 1(1a), 2(2a), 3(3a), Fig. 64A, that here subtle changes in the linker functionality influence the relaxational properties. Ab initio calculations together with DFT were employed to estimate the dipolar and exchange contributions to the total dimer coupling and construct dinuclear relaxation blockade diagrams rationalizing the different slow relaxation observed for complexes 1a–3a. Interestingly, the total and dipolar coupling constant was FM for the three complexes, while the exchange contribution was AF for 1a, 3a but FM for 2a. Moreover, a larger tunneling splitting associated to small structural changes was calculated for complex 3a, yielding a smaller energy barrier compared to complexes 1a, 2a (see Table 8, orange shaded rows). On the other hand Zhang et al. (2016e) have discussed, based on a series of HMq-bridged Dy2 complexes, (Dy(acac)2(CH3OH)2(m-HMq)2 1, [Dy(DBM)2]2(m-HMq)2(n-C6H14) 2, [Dy(hmac)2]2(m-HMq)2 3, and [Dy (hfac)3]2(m-HMq)2 4 (Fig. 64B), how changes in the periphery of the b-diketonate ligands coordinating the two Dy(III) ions affect the SMM behavior. Distinct anisotropy axis of the individual Dy(III) ions and dimer couplings was derived for the different substituent on the b-diketone terminal, leading to significant different relaxational behaviors (see Table 8, gray shaded rows). AF dimer interactions led to relatively low activation energies for 2 and 3, while contrarily complex 1, with FM interaction, exhibited the highest energy barrier (Ueff/kB ¼ 76 K). Moreover, the anisotropy of individual ions also proved important, since for 4 only a moderate energy barrier (26.9 K) was observed despite the FM dimer interaction, due to the small anisotropy of the individual ions. Very recently, Mazarakioti et al. (2017) have shown the establishment of a remarkably high-energy barrier of Ueff/kB ¼ 109 K and large magnetic hysteresis up to 5 K (0.14 T/s) for the oxygen-bridged dinuclear complex [Dy2(NO3)4(sacbH)2(H2O)2(MeCN)2], enabled by the high anisotropy of the nine-coordinated metal ions, each residing in a rare spherical tricapped trigonal coordination geometry, and very small AF intradimer coupling (Jtot ¼  0.002 K), the dipolar AF interaction (Jdip/kB ¼  0.017 K) opposing the FM exchange one (Jex/kB ¼ +0.0065 K) (see Table 8, blue shaded rows). Depending on the relative strength of the intradimer interaction compared to the single-ion anisotropy, the observed slow relaxation may arise from SMM behavior of the Ln2 as an entity or SIM relaxation of the individual ions. Several works have reported the presence of two relaxation processes in Ln2 complexes (e.g., Anastasiadis et al., 2015; Chow et al., 2015; Harriman et al., 2017b; Holynska et al., 2015; Lin et al., 2016; Shen et al., 2016; Sun et al., 2016a). In noncentrosymmetric molecules with two kinds of coordination environments around the Ln(III) ions, the observation of

FIG. 64 (A) Structures of complexes [Dy2(Lx)2(L0 )2(CH3OH)2]yG, x ¼ 1(1a), 2(2a), 3(3a) (Mukherjee et al., 2016). (B) Structures and ab initio calculated EAM of (Dy(acac)2(CH3OH)2(m-HMq)2 1, [Dy(DBM)2]2(m-HMq)2(n-C6H14) 2, [Dy(hmac)2]2(m-HMq)2 3, and [Dy(hfac)3]2(m-HMq)2 4 (Zhang et al., 2016e). Panel (A) adapted with permission from Mukherjee, S., Lu, J., Velmurugan, G., Singh, S., Rajaraman, G., Tang, J., Ghosh, S.K., 2016. Influence of tuned linker functionality on modulation of magnetic properties and relaxation dynamics in a family of six isotypic Ln2 (Ln ¼ Dy and Gd) complexes. Inorg. Chem. 55, 11283–11298, https://doi.org/10.1021/acs.inorgchem.6b01863. Copyright 2016 American Chemical Society. Panel (B) adapted with permission from Zhang, W.-Y., Tian, Y.-M., Li, H.-F., Chen, P., Sun, W.-B., Zhang, Y.-Q., Yan, P.-F., 2016e. A series of dinuclear Dy(III) complexes bridged by 2-methyl-8-hydroxylquinoline: replacement on the periphery coordinated b-diketonate terminal leads to different single-molecule magnetic properties. Dalton Trans. 45, 3863–3873, https://doi.org/10.1039/C5DT04449A. Published by The Royal Society of Chemistry.

160 Handbook of Magnetic Materials

two relaxation processes is often attributed to relaxation of the individual ions (Harriman et al., 2017b; Sun et al., 2016a), while in centrosymmetric complexes it is assigned to relaxation through excited levels (Fig. 62B) (Anastasiadis et al., 2015). Studies on diluted {LnY} samples have proved useful to discern SMM from SIM relaxation of the uncoupled ions (Chow et al., 2015). For example, in the BPYM-bridged Dy2 complexes reported by Sun et al. (2016a), experiments on dilute samples demonstrated unambiguously that relaxation in the complexes stemmed from relaxation of the individual Dy ions. The single and double relaxation processes observed in complexes 1 and 2, respectively, were assigned to one and two types of Dy(III) environments in the two different types of dimers (Fig. 65A). Doping studies on triple-decker phthalocyanine complexes including two dissimilar Ln sites have played an important role in understanding the magnetic properties of dinuclear 4f complexes. Ishikawa and coworkers performed pioneering studies on homodinuclear PcLnPcLnPc* {LnLn}, Fig. 66A, and diluted {LnY}, {YLn} complexes of different lanthanide ions, which allowed rationalizing that, owing to the dipolar nature of the interaction between the metal centers (directly resulting from the elongation of the closed surfaces representing the magnetic susceptibility tensors either in the z or in the xy direction), the magnetic interaction is ferromagnetic for the (DyIII, TbIII, HoIII) dinuclear complexes, AF for the {Er2} complex, and negligible small for {Yb2} (Ishikawa et al., 2002b, 2003a, 2005a). Tb2Pc3 compounds are especially attractive owing to their tendency toward high energetic barriers, and potential customization for surface adhesion. Recently, Holmberg et al. (2016b) have reported the properties of a new Tb2Pc3 homodinuclear compound, [(15C5)4PcTb(15C5)4PcTb (Pc)] (1), and two novel TbIII/YIII complexes: [(15C5)4PcTb(15C5)4Pc]Y(Pc) (2), and [(15C5)4Pc]Y[(15C5)4Pc]Tb(Pc) (3). In the Tb2 complex, thanks to the nonnegligible dipolar ferromagnetic interaction between the lanthanide ions, zero-field SMM behavior with a high Ueff/kB ¼ 229.9 K (1) is observed, while the TbIII/YIII complexes exhibit field-induced SIM behavior, with larger activation energy observed for (3) Ueff/kB ¼ 169.1 K with TbIII in the heteroleptically substituted environment than for complex (2) Ueff/kB ¼ 129.8 K with Tb in homoleptic position (Fig. 63B). Another way of studying the increasing influence of interactions is by decreasing sufficiently the temperature. Indeed, when intradimer interactions become relevant, a crossover from SIM relaxation of the Ln(III) ions to SMM of the Ln2 may occur. An example of this is observed in the previously presented asymmetric complex [Dy2ovph2Cl2(MeOH)3]MeCN (Guo et al., 2011). At high temperatures, double relaxation occurs due to the individual relaxation of the two inequivalent ions (Fig. 67), whereas at low temperatures, blockage of magnetization and SMM behavior take place, enabled by the strong Ising exchange interaction between the Ln–Ln ions.

B 1.0

1.0

(1) c ″ / cm3/mol

0.6 0.4 0.2

(2)

0.8 0.6 0.4 0.2

0.0

0.0 5

10

15

20

25

30

5

10

T/K

15

20

−8

−8

−11

35

For 1, RLT at zero field For 1, RLT at 1500 Oe field For 1, RHT at zero field For 1, RHT at 1500 Oe field For 2, At zero field For 2, At 1500 Oe field Orbach and Raman simulation

0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 1/T (K−1)

380 356 174 85 41.7 20.4 10.0

(5)

1.0

0.5

0.0 2

4

6

0.1

0.2

8 T/K

10

12

0.3

0.4

0.5

14

-4

(Diluted) −7

In(t)

In(t)

(Undiluted)

−10

30

1.5

T/K

−7

−9

25

1488 1218 1008 830 679 561 462

2.0

-5

For 1a, RLT at zero field

−9

For 1a, RLT at 1500 Oe field For 1a, RHT at zero field

In(1/2pu)

c ″ / cm3/mol

0.8

100 Hz 178 Hz 316 Hz 421 Hz 562 Hz 749 Hz 1000 Hz 1334 Hz 1780 Hz 2731 Hz 3160 Hz 4216 Hz 5620 Hz 7499 Hz 10,000 Hz

c ″m / emu/mol

A

-6 -7

−10

For 1a, RHT at 1500 Oe field

-8

−11

For 2a, At zero field For 2a, At 1500 Oe field Orbach and Raman simulation

-9

0.04

0.06

0.08

0.10

1/T (K−1)

0.12

0.14

T -1 / K-1

FIG. 65 Ln2 with two slow relaxation processes. Top: Temperature Dependence of the out-of-phase susceptibility signals. Bottom: Plots of ln(t) vs T1. (A) Relaxation arising from single ions (for complexes [(DBM)6Dy2B-PYM]2CHCl3 (1) and [(DBM)6Dy2BPYM]MeCN (2), undiluted and diluted), from Sun et al. (2016a); (B) assigned to relaxation through higher states, complex [Dy2(NO3)2(saph)2(DMF)4] (5) by Anastasiadis et al. (2015). The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. Adapted with permission from Sun, W.-B., Yan, B., Jia, L.-H., Wang, B.-W., Yang, Q., Cheng, X., Li, H.-F., Chen, P., Wang, Z.-M., Gao, S., 2016a. Dinuclear dysprosium SMMs bridged by a neutral bipyrimidine ligand: two crystal systems that depend on different lattice solvents lead to a distinct slow relaxation behavior. Dalton Trans. 45, 8790–8794, https://doi.org/10.1039/c6dt01082b; Anastasiadis, N.C., Kalofolias, D.A., Philippidis, A., Tzani, S., Raptopoulou, C.P., Psycharis, V., Milios, C.J., Escuer, A., Perlepes, S.P., 2015. A family of dinuclear lanthanide(III) complexes from the use of a tridentate Schiff base. Dalton Trans. 44, 10200–10209, https://doi.org/10.1039/c5dt01218j. Both published by The Royal Society of Chemistry.

FIG. 66 (A) PcLnPcLnPc* from Ishikawa et al. (2002a); (B) structure of [(15C5)4PcTb(15C5)4PcTb(Pc)] (1), and diluted TbIII/YIII complexes [(15C5)4PcTb (15C5)4Pc]Y(Pc) (2) and [(15C5)4Pc]Y[(15C5)4Pc]Tb(Pc) (3); ln(τ) vs T for 1, 2, 3 (Holmberg et al., 2016b). Panel (A) reprinted with permission from Ishikawa, N., Iino, T., Kaizu, Y., 2002a. Interaction between f-electronic systems in dinuclear lanthanide complexes with phthalocyanines. J. Am. Chem. Soc. 124, 11440–11447, https://doi.org/10.1021/ja027119n. Copyright 2002 American Chemical Society. Panel (B) reprinted with permission from Holmberg, R.J., Polovkova, M.A., Martynov, A.G., Gorbunova, Y.G., Murugesu, M., 2016b. Impact of the coordination environment on the magnetic properties of single-molecule magnets based on homo- and hetero-dinuclear terbium(III) heteroleptic tris(crownphthalocyaninate). Dalton Trans. 45, 9320–9327, https://doi.org/10.1039/ c6dt00777e. Published by The Royal Society of Chemistry.

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FIG. 67 Relaxation of complex [Dy2ovph2Cl2(MeOH)3]MeCN (inset): (A) T-dependence of the out-of-phase susceptibility showing two (SIM) relaxation processes; (B) ln(τ) vs 1/T showing the two SIM processes at high temperature, crossing over to SMM behavior at lower T. Adapted with permission from Guo, Y.N., Xu, G.F., Wernsdorfer, W., Ungur, L., Guo, Y., Tang, J., Zhang, H.J., Chibotaru, L.F., Powell, A.K., 2011. Strong axiality and Ising exchange interaction suppress zerofield tunneling of magnetization of an asymmetric Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 11948–11951, https://doi.org/10.1021/ja205035g. Copyright 2011 American Chemical Society.

Another very interesting example has been recently shown by Chow et al. (2015), who investigated a family of Ln2Ga4 complexes (Tb 2, Dy 3, Er 4, and the diluted species YIII0.9DyIII0.1 6), where the Ln ions were bridged by oxygen atoms forming the diamond-like Ln2O2 core (Fig. 68). For the Tb and Dy complexes, AF coupling due to exchange interactions was shown, while no evidence of AF coupling was observed for the Er complex. Analysis of the NSOs, which are related to the magnitude of the AF exchange, showed that the coupling was about 10 times larger for the Dy2 than for the Er2 complex. Remarkably, micro-SQUID measurements below T ¼ 2 K revealed that despite the presence of AF coupling, the Dy2 complex exhibited an opening of the magnetic hysteresis cycle, which was explained by the presence of a residual amount of molecules in the FM state. AC susceptibility studies of Dy2 revealed the presence of two relaxation process, a high-T process with Ueff/ kB ¼ 26 K and a lower-T one with Ueff/kB ¼ 18 K at zero field, assigned, respectively, to the dimer FM excited state and to the uncoupled ions. The single-ion nature of the high-T process was confirmed by the study of the diluted mononuclear complex.

6.2

Heterodinuclear SMMs

6.2.1 [Ln–Ln0 ] Complexes The synthesis of heterodinuclear lanthanide SMMs with perspectives of application as memories is very restricted. Namely, we find the triple-decker

164 Handbook of Magnetic Materials

FIG. 68 Relaxation of Ln2Ga4 complex (Chow et al., 2015). (A) Structure of the complex. (B) Field-dependent energy diagram showing the different relaxation processes. (C) Normalized magnetization cycle at 0.03, 0.5, and 1 K for DC field sweep rate of 0.035 T/s; (D) ln(τ) vs 1/T showing two relaxation processes (HT: SIM of uncoupled ions; LT: SMM). Adapted from Chow, C.Y., Bolvin, H., Campbell, V.E., Guillot, R., Kampf, J.F., Wernsdorfer, W., Gendron, F.V., Autschbach, J., Pecoraro, V.L., Mallah, T., 2015. Assessing the exchange coupling in binuclear lanthanide(III) complexes and the slow relaxation of the magnetization in the antiferromagnetically coupled Dy2 derivative. Chem. Sci. 6, 4148–4159, https://doi.org/10.1039/C5SC01029B. Published by The Royal Society of Chemistry.

phthalocyanine compounds discussed earlier (Holmberg et al., 2016b), exhibiting very high energy barriers, and the polyoxometallates TBA8H4 [{(Lnm2-OH)2Ln0 }(g-SiW10O36)2], where the [Dy(m2-OH)2Ln]4+ dimeric groups are sandwiched by two [g-SiW10O36]8 and Ln0 ¼ Lu, Yb, Dy, and Lu. These complexes have increasing anisotropy barriers in the trend of decreasing atomic radius of the Ln0 adjoint to the Dy atom, an effect essentially due to the modification of the single-ion Dy anisotropy axis by the n.n. ligand (Sato et al., 2013). On the other hand, asymmetric [Ln–Ln0 ] are being investigated as promising candidates for the realization of quantum gates. Indeed, the realization of universal quantum gates is a necessary step for quantum computing. It has been shown that any algorithm in quantum computing can be realized by combing single qubits with CNOT (two qubits) or SWAP quantum gates,

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 165

which necessitate two coupled qubits. CNOT quantum gates based on lanthanides can be realized by dinuclear [Ln–Ln0 ] complexes, with the two metal centers magnetically different from each other and only weakly coupled (Aguilà et al., 2014; Luis et al., 2011). More recently, the [LnLn0 (HL)2(H2L)(NO3)(py)(H2O)] complexes, allowing the synthesis of both homo- and heterodimers, have been presented (Repoll
es, 2016). Two different Ln ions may enter into the molecule, since each site has a different coordination and size. The ErLa, DyLa substitutions show two types of magnetic relaxation at temperatures below 1 K: a temperature dependent (Arrhenius law) with activation energy compatible with hyperfine interaction broadening of the ground state, and an independent component due to QTM. In the Er2 and Dy2 substitution the intradimer exchange interaction contributes to the broadening. The CeEr dimer is deemed to perform as a quantum gate since all stable Ce isotopes have nonmagnetic nuclei, while only 22.9% of the stable isotopes of Er carry a nuclear spin, and consequently, the decoherence caused by hyperfine interactions is reduced. Actually, the relaxation time in this compound has not been reported so far; however, time-resolved EPR experiments showed that it exhibits coherent spin dynamics, with the important figure of merit of a decoherence time T2 of about 400 ns (Aguilà et al., 2013, 2014).

6.2.2 Hetero [Ln–3d] Complexes The heteronuclear dimeric complexes reported recently include a trivalent lanthanide and a divalent transition metal atom M (Table 9). In all cases the coupling between the two atoms is via an O atom belonging to the bridging ligand. The interaction is of ferromagnetic character, except for the [Ni(L)Ln(NO3)3(H2O)] case (Wen et al., 2015), where the Ni(II) is diamagnetic and consequently does not couple with the Ln magnetic moment. Unluckily, there is no quantitative determination of the Ln–M exchange (or dipolar) interaction, except for [Dy{M(mH2L)}piv2(OH2)]ClO4, M ¼ Ni and Co (Bender et al., 2015). In these compounds the magnetization and susceptibility measurements were simulated with ab initio methods, allowing to interpret the results (Fig. 69A). In particular, the slow magnetic relaxation of Orbach characteristics present in the Dy–Ni dimer is found to be correlated with the strong uniaxial anisotropy axis generated by the coincidence in anisotropy axis of the ground and first excited electronic states. In contrast, in the CoDy case the QTM seems to dominate the relaxation process, since the Co anisotropy axis is perpendicular to the Dy one, and transverse anisotropy allows a more effective tunneling, which is in detriment of the slow magnetic relaxation process. We mention here the heteronuclear dimeric complex [DyFe(CN)6 (hep)2(H2O)4], where the transition metal is in a low-spin trivalent state (S ¼ 1/2), the intradimer interaction is antiferromagnetic and detrimental to

TABLE 9 Glossed Heterodinuclear [Ln-M] SMMs Compound [Dy{Ni(mH2L)}piv2(OH2)]ClO4

[Ln-M] DyNi

J/kB (K) 0.67

Ueff/kB (K) 19.86

τ 0 (s) 7

7.81  10

7

[Ni(N3)(H2O)(valpn)Dy (hfac)2(H2O)]H2O

DyNi

FM

16.9

1.02  10

[Ni(L)Ln(NO3)3(H2O)]

TbNi

Ni dia

29.12

3.21  109

DyNi

H (kOe)

References

1.8

Bender et al. (2015)

1

Lu et al. (2015) Wen et al. (2015)

6

18.40

7.39  10

[CuL(MeOH)Tb(NO3)3]

TbCu

FM

24.6

2.1  108

Chiboub Fellah et al. (2016)

[CoDy (L)(DBM)3]

DyCo

FM

—a

—a

Xie et al. (2014)

[DyFe(CN)6(hep)2(H2O)4]

DyFe

AF

23

8

9.7  10

The intramolecular coupling constant is given in 2JS1S2 formalism. The relaxation time is given by the formula τ 1 Oe ¼ 103/4p A/m for magnetic field strength. H applied field in kOe. a Slow relaxation but no Ueff nor τ0 reported.

2 1

¼ τ0

Zhang et al. (2015c)

1 Ueff

e

=kB T . The SI conversion factor is

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 167

FIG. 69 (A) [Dy{Ni(mH2L)}piv2(OH2)]ClO4: [NiDy] orientation of the main magnetic axes of Ni(II) and Dy(III) centers with less than 5° deviation (thick green and blue lines), and [CoDy] magnetic axes of Co(II) and Dy(III) centers with a deviation of 92° (thick violet and blue lines). Reprinted with permission from Bender, M., Comba, P., Demeshko, S., Großhauser, M., Muller, € D., Wadepohl, H., 2015. Theoretically predicted and experimentally observed relaxation pathways of two heterodinuclear 3d-4f complexes. Zeitsch. Anorg. Allg. Chem. 641, 2291–2299, https://doi.org/10.1002/zaac.201500595. Copyright 2015 John Wiley & Sons, Inc. (B) Groundstate KD orientation for both Dy(III) ions of Dy2@665-(C79N). Reprinted from Singh, M.K., Yadav, N., Rajaraman, G., 2015. Record high magnetic exchange and magnetization blockade in Ln2@C79N (Ln ¼ Gd(III) and Dy(III)) molecules: a theoretical perspective. Chem. Commun. 51, 17732–17735, https://doi.org/10.1039/C5CC06642E. Published by The Royal Society of Chemistry.

the cluster anisotropy with respect to the isomorphous clusters, where the transition metal is diamagnetic (Zhang et al., 2015c).

6.3

Conclusions

Though slow relaxation in lanthanide dimeric compounds is in principle regulated by both the single-ion anisotropy and the interion coupling, the weakness of Ln–Ln interactions makes them almost irrelevant, within the SQUID range of temperatures. As is almost systematic along lanthanide molecule magnets, it is the Dy substitutions that are relevant in the Ln–Ln (4f–4f ) and Ln–M (4f–3d) dimers as SMMs. The Ln–M(II) interaction, with M ¼ Ni, Co, is more intense and couples the Ln and M moments ferromagnetically, but again, it is the single-ion anisotropy that enhances the dimer’s barrier to moment reversal and renders the compound interesting as SMM. A recent theoretical work on the Dy2@C79N fullerene may open a different perspective on this question (Singh et al., 2015). The fullerene bucky ball contains two Dy atoms, each of which is strongly coupled to the C79N radical (Fig. 69B). The magnetic Dy-radical interaction JDyr/kB ¼ 205.7 K as compared to the Dy–Dy interaction JDyDy/kB ¼  0.22 K is dominant. The calculated barrier height of the Dy2@C79N would amount to Ueff/ kB ¼ 1023 K thanks to quenched QTM by the intramolecular exchange interaction. This compound, if could be synthesized, would settle a record of effective magnetic exchange and magnetization blockade, as desirable for a SMM.

168 Handbook of Magnetic Materials

7 POLYNUCLEAR LANTHANIDE-BASED SMMs 7.1 Introduction The constant search for SMMs with high effective energy barriers and high blocking temperatures in Ln-based SMMs has been successful in many mononuclear and dinuclear complexes. However, a systematic investigation of polynuclear SMMs is more difficult to achieve, as the structure and magnetic interactions of many Ln centers result in a complex scenario. Theoretically, the more spin centers in one metal cluster, the larger spin ground state when ferromagnetic interactions between the centers are present. This condition is however difficult to fulfill in f-elements, because of the shielding of the valence electrons and the commonly weak exchange interactions with other metal ions. In most of the Ln polynuclear SMMs, relaxation processes are of single-ion origin, and magnetic interactions play a secondary role. Magnetic coupling may influence the relaxation of magnetization, either by promoting or by quenching the quantum tunneling of the magnetization. However, many outstanding achievements have been reported in SMMs as compared to SIM complexes, in particular, the anisotropic barrier records of Ueff/kB ¼ 528 K for a squared-based Dy5 pyramid, or the Ueff/kB ¼ 692 K and TB ¼ 5 K of a Dy4K2 octahedron (Blagg et al., 2013). Generally, high-energy barrier and blocking temperatures for polynuclear clusters have been obtained for different Lnx core topologies (Tang and Zhang, 2015). Therefore, the study of structural details of highly performant SMMs is an active field of research in molecular magnetism. Ligand design is a crucial element in modulating the nuclearity and structural topology of Ln clusters. Magnetic properties of different synthetized complexes are of special interest not only for their potential applications but also from the chemical and physical point of view. In this section, we present the most recent relevant examples of cluster SMMs, including those of homo- and heteronuclear type, attending to their design novelty, SMM characteristics achievement, and remarkable physical properties.

7.2 Homonuclear Ln Clusters (Lnx, x > 2): SMTs and SMMs Latest contributions of homopolynuclear lanthanide complexes where magnetic relaxation has been observed and characterized are given in Table 10. The main parameters determining the performance of a molecule as SMM are the magnitude of the energy barrier for the reversal of the magnetization by thermal activation, Ueff, and the blocking temperature TB. Fast quantum tunneling of the magnetization, QTM, is the main inhibition mechanism for polynuclear SMMs to exhibit high TB values. At this point, it is worth to distinguish complexes exhibiting a toroidal magnetic moment in the ground state, originated by the noncollinear arrangement

TABLE 10 Ln-Based SMMs With Nuclearities Larger Than 2 τ 0 (s)

Complex

Dy3

[Dy3L12(H2O)9](Cl)56H2O

Almost linear butterfly

11.2

3.1  10

Dy3

[(Pc)2Dy3(L)(OAc)(OCH3)2]

Sandwich

60

Dy3

[{(thd)3Dy}3HAN]

Triangular

Dy3

(Dy3[BPh4])

Topology

Ueff/kB (K)

Core

(Dy3[Dy(NO3)6])

Dy4

[Dy4(L)4(m2-Z1Z1Piv)4]6CH3OH4H2O

[Dy4(m4-OH)(HL)(H2L)3(H2O)4] Cl2(CH3OH)4(H2O)8

References Adhikary et al. (2014)

1.5  106

FM

Gao et al. (2016b)

60 (1 kOe)

3.8  106

1.9 to 2.0

74 (1 kOe)

5.6  10

6

Grindell et al. (2016)

49.7 (1 kOe)

7.5  107

0.11a

H€ anninen et al. (2014)

27.9 (1 kOe)

3.7  107

27.7 (1 kOe)

4.7  106

Distorted cubane

73 (1 kOe)

4.4  108

47 (1 kOe)

5.0  10

7

Square

100

1.2  108

Triangular

Triangular

[Dy3(m3-OCH3)2(m-HL)3(NO3)3][Dy (NO3)6]0.33CH3OH6H2O Dy4

6

Interactions J/ kB (K) 0.02

[Dy3(m3-OCH3)2(m-HL)3(NO3)3] BPh44CH3OHH2O

Dy3

TB

29

a

 0.15

0.12a  0.16

1.3  10

7

0.09a

Das et al. (2014a)

0.06a

Das et al. (2015a)

J1 ¼ + 0.014 J2 ¼  0.014

Dy4

Dy4

[Dy4(acac)4L6(m3-OH)2]

[Dy4(dbm)4L6(m3-OH)2]

Butterfly

Butterfly

48

2.2  107 8

121

2.8  10

56

2.6  107

Gao et al. (2016c) Gao et al. (2016d)

Continued

TABLE 10 Ln-Based SMMs With Nuclearities Larger Than 2—cont’d Core

Complex

Topology

Ueff/kB (K)

τ 0 (s)

Dy4

[Dy4(OH)2(bpt)4(NO3)4(OAc)2]

Square grid

133

4.1  109

204

2.8  109

61.3

6.9  106

Dy4

[Dy4(bzhdep-2H)4(H2O)4(NO3)4] 6CH3OH6H2O

Square grid

Dy4

[Dy4(DFMP)2(H2L)2(HL)2] 6CH3OH2H2O

Dy4

TB

6

Interactions J/ kB (K)

References Guo et al. (2015)

16 K (500 Oe/s)

Huang et al. (2016)

96.7 (1 kOe)

2.0  10

Parallelogram

50.2

3.6  106

Weak AF

Lin et al. (2015a)

[Dy4Cl2(m3-OH)2(m-OH)2(2, 2-bpt)4(H2O)4]Cl22H2O4EtOH

Defect dicubane

190

2.2  108

Weak AF

Liu et al. (2015e)

Dy4

[Dy4Br2(m3-OH)2(m-OH)2(2, 2-bpt)4(H2O)4]Br22H2O4EtOH

Defect dicubane

197

1.4  108

Weak AF

Dy4

[Dy4(L)2(HL)2(NO3)2(OH)2](NO3)24H2O

Coplanar rhombic

48

6.5  106

FM

Luan et al. (2015b)

Dy4

[Dy4(L)2(HL)2Cl2(m3OH)2]2Cl2(OH)23CH3CH2OHH2O

Defect dicubane

55.7

7.5  106

FM

Luan et al. (2015a)

Dy4

[Dy4(m4-O)(OMe) (HOMe)2(CH3COO)3(L3)2]2H2O

Tetrahedral

3.8

6.4  106

0.07a

Sheikh et al. (2014)

Dy4

{[Dy4(m4-O)(HL1)4(H2O)4]2(NO3)3(OH)} 2H2O2CH3OH

Square

6.5

2.7  105

AF

Wu et al. (2017)

Dy4

[Dy4(m4-O)(HL2)4(SCN)2]2H2O4CH3OH

Square

57.6

1.4  107

AF

Dy4

[Dy4(m3-OH)(L2 )4(H2O)6](ClO4)36H2O

Y-shaped

>1.8 K

4

5.7

1.2  10

83.7

9.5  108

7.6

5.2  103

<1.9 K

AF

Xue et al. (2014a)

Dy4

{[Dy4(m4-OH)(L)2(acac)4(MeOH)2(EtOH) (H2O)](NO3)2(MeOH)3(EtOH)}

Square grid

51.5

3.0  108

Dy4

[Dy4L2(m3-OH)2(m4-NO3)(NO3)4(OCH3) (H2O)]xMeCNyMeOH

Edge-sharing bitriangular

28

1.7  107

Dy5

Dy5(H2O)(OH)4(NO3)3(BZA)4L

Goblet

4.1 (6 kOe)

3.45  105

Dy6

[Dy6L22(HCO2)4(m3-OH)4(DMF)6(H2O)2] (Cl)24H2O

Four-distorted hemicubanes

9.7

6.4  106

Adhikary et al. (2014)

Dy6

[{Dy6(L)2(LH)2}(m3-OH)4] [MeOH]2[H2O]6[Cl]48H2O4CH3OH

Triangular trimeric motifs

46.2

2.85  107

Das et al. (2014c)

Dy6

[Dy6(L1)6(L0 )6(OCH3)6(2CH3OH)]

Wheels

12.2

5.0  106

11.5

5.1  10

6

Weak AF

9

Weak AF

Dy6

0

[Dy6(L2)6(L )6(OCH3)6(2CH3OH)]

Wheels

<2 K

AF

Yadav et al. (2015)

AF

Zou et al. (2014)

FM

Wang et al. (2016a)

Weak FM

Dy6

[Dy6L2(m3-OH)4(m2-OH)2(SCN)8(H2O)4] 6CH3CN2CH3OHH2O

Linked triangles

181

8.1  10

Dy6

[Dy6L2(m3-OH)4(m2-OH)2(NO3)6(H2O)6] 2NO310H2O

Linked triangles

116

1.1  107

Weak AF

Dy6

[Dy6(m4-O)2(HCOO)2L4(HL0 )2(dmf )2]

Edge-sharing tetrahedrons

85

3.9  107

0.06a

Dy8

([DyIII 8 (m3-OH)4(L1)4(DEA)4Cl4])

Butterfly shaped

49.3

9.0  1010

Tb8

([TbIII 8 (m3-OH)4(L1)4(DEA)4Cl4])

Butterfly shaped

33.9

7.9  108

Dy8

([DyIII 8 (m3-OH)4(L2)6(DMF)4(H2O)8])

Staircase arrangement

36.5

8.5  108

Joarder et al. (2014) Li et al. (2015b)

Zhang et al. (2017) Bala et al. (2015)

Continued

TABLE 10 Ln-Based SMMs With Nuclearities Larger Than 2—cont’d Core

Complex

Topology

Ueff/kB (K)

τ 0 (s)

Dy9

[Dy9L4(m4-OH)2(m3OH)8(NO3)8(CH3OH)2(H2O)2](NO3) 4CH3OH9H2O

Sandglass

15.3

3.2  10

Dy10

[Dy10(LH)10(k2-Piv)10]9 CHCl34CH3CNPH2OQMeOH

Macrocycle

16 (4 kOe)

Dy12

[DyIII12Na3(m3OH)2(hmmp)6(piv)12(CO3)6(MeOH)6] OH5MeOH

Triangular shelf-shape

Dy15

[{Dy15(OH)20(PepCO2)10(DBM)10Cl}Cl4]

Dy17

TB 8

Interactions J/ kB (K)

References

AF

Zou et al. (2015)

3.3  105

0.11a

Das et al. (2015c)

3.5

1.2  107

Weak AF

Li et al. (2016c)

Pentagonal cyclic

9.2 (2 kOe)

3.5  106

FM

Thielemann et al. (2014)

[Dy(m3-OH)8][Dy16(m4-O)(m4-OH)(m3OH)8(H2O)8(m4-dcd)8][(m3-dcd)8]22H2O

Cage

7.6

1.8  105

AF

Zhou et al. (2016b)

Dy27

[ClO4@Dy27(m3OH)32(CO3)8(CH3CH2COO)20(H2O)40] (ClO4)12(H2O)50

Cage

3.8

3.55  107

AF

Zheng et al. (2016)

Dy72

Dy72(mda)24(mdaH)8(OH)120(O)8(NO3)16] (NO3)816CH3OH168H2O

Dy24 wheel units in a saddle-like tubular cluster

6.5

9.5  107

<2 K

Qin et al. (2016)

References for 2014–2017 where relaxation processes are characterized. The intramolecular coupling constant is given in 2JSiSj formalism. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. a Maximum magnetic interaction constants obtained from the modeling of the Gd(III) analogue.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 173

of the individual magnetic moments of the Ln(III) centers. The first and archetypal example of a nonmagnetic ground state due to the toroidal arrangement of anisotropic axes was reported for a triangular Dy3 (Luzon et al., 2008). Such a vortex P configuration of spins is at the origin of the toroidal-like moment, T ¼ iri  Si, with two states corresponding to the clockwise and anticlockwise arrangement of spins, Si. Interestingly, these systems normally present SMM behavior, which is associated with a thermally excited spin state of the molecule. This subset of SMMs has been denominated single-molecule toroics (SMTs) and they have been the subject of numerous experimental and theoretical studies, in view of the intriguing physics and potential applications (Ungur et al., 2014b). Magnetic measurements indicating a nonmagnetic ground state are not a direct proof of the presence of a toroidal moment. Ab initio calculations are required to prove the toroidal arrangement of magnetic moments. Gysler et al. (2016) have recently revisited the Dy3 archetypal triangular SMT, providing more experimental evidences and ab initio calculations of the electronic structure of this singular complex (see Fig. 70). Exchange interaction in this system was calculated to be between J/kB ¼  5 K and 6 K (H ¼  2JSiSj formulation). They have demonstrated that HF-EPR and in-plane torque measurements can be explained by means of a very simple model, considering Ising-type coupling between three S ¼ 1/2 pseudospins.

FIG. 70 (A) Molecular structure of archetypal Dy3 triangle ([Dy3(m3-OH)2L3Cl(H2O)5]Cl3). (B) Scheme of local anisotropy axes according to ab initio calculations of the ground-state KD of each Dy(III) ion. Adapted from Gysler, M., El Hallak, F., Ungur, L., Marx, R., Hakl, M., Neugebauer, P., Rechkemmer, Y., Lan, Y., Sheikin, I., Orlita, M., Anson, C.E., Powell, A.K., Sessoli, R., Chibotaru, L.F., van Slageren, J., 2016. Multitechnique investigation of Dy3— implications for coupled lanthanide clusters. Chem. Sci. 7, 4347–4354, https://doi.org/10.1039/ C6SC00318D. Published by The Royal Society of Chemistry under the Creative Commons Attribution License.

174 Handbook of Magnetic Materials

However, far-infrared (FIR) and CTM are very sensitive to the CF splitting of individual ions and to the details of the magnetic coupling. Therefore, a comprehensive study of polynuclear lanthanide-based SMMs needs from very specific spectroscopic and magnetometric studies. A large variety of Ln clusters, with many different structure topologies, exhibiting slow relaxation are found in the literature. Magnetic interactions are usually weak and AF in nature. Therefore, in most of the cases, SMM behavior is attributed to single-ion processes within the cluster. The presence of different types of Ln(III) local coordination geometry gives as a result the observation of several relaxation processes and a distribution of relaxation times. Verification that the observed processes are attributed to single-ion behavior can be carried out by studying doped compounds with nonmagnetic ion, as Y(III) (Guo et al., 2015; Huang et al., 2016). To date, the highest effective energy barrier reported in a polynuclear Ln SMM has been the case of the Dy4K2 compound, [Dy4K2O(OtBu)12], studied by Winpenny and coworkers (Blagg et al., 2013). A strongly axial CF is the dominant factor which blocks the relaxation via the first excited state, presenting a preferential relaxation via the second excited Kramers doublet instead. The consequence is an increased effective energy of more than 800 K in diluted compounds and a blocking temperature of TB ¼ 5 K (0.14 T/s). Magnetic properties of polynuclear clusters of different lanthanides are reported in the literature, but the majority of the studies are centered in Dy(III) derivatives, where SMM behavior is most likely encountered. Tb(III) analogues are often studied, but very few exhibit SMM behavior (Bala et al., 2015). The study of Gd(III) in different complexes is intended, apart from the magnetocaloric properties, as a magnetic probe of magnetic interactions within the cluster which result to be AF and of very low energy (less than 0.3 K). Depending on the core geometry and nuclearity, magnetic exchange models of different complexity are used. Generally, a Heisenberg Hamiltonian with two exchange constants, J1, for n.n. and, J2, for n.n.n. is considered: X X Si Sj  2J2 Si Sj : (79) H ¼ 2J1 nn

nnn

Exchange constants are determined by fitting experimental magnetization as a function of temperature and field. Ab initio calculations are performed to study magnetic interactions between highly anisotropic Ln ions, like Dy(III). Interactions are modeled within an Ising exchange Hamiltonian, taking into consideration intracluster dipolar and exchange interactions. This Hamiltonian has the general form:  X  dip Jij + Jijex Si, zi Sj, zj : (80) H ¼ 2 i, j i
Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 175

Calculated main anisotropy axes for different Ln(III) ions in the cluster are considered in the simulation of the exchange spectrum. Comparison between simulated curves and experimental magnetic data allows the determination of the exchange parameters.

7.2.1 Dy3 Triangles Among all the Ln clusters, Dy3 triangles have raised special attention because of the observation of a nonmagnetic ground state. SMTs have been identified in several dysprosium compounds with different nuclearities, many of them triangular Dy3 complexes. Nevertheless, this triangular topology does not guarantee a vortex-like arrangement of spins. A necessary condition is that the individual anisotropy axes are coplanar. Many new complexes containing Dy3 triangles have been synthetized and studied to shed light into the structural features influencing the magnetic relaxation in these systems and the magnetic properties of the ground state. H€anninen et al. (2014) have presented two new examples of Dy3 triangles, Dy3[BPh4] and Dy3[Dy(NO3)6], which behave as SMM with a magnetic ground state. The direction of the shortest Dy–O bond controls the magnetic moment orientation which lies perpendicular to triangle plane, discarding toroidal alignment. These complexes are compared with other triangular Dy3 complexes where magnetic or nonmagnetic ground state is obtained (see table 2 in H€anninen et al., 2014). Another interesting example is presented by Layfield and coworkers, where magnetic frustration is revealed by ab initio calculations in a triangular Dy3 complex, [{(thd)3Dy}3HAN] (Grindell et al., 2016). Two field-induced magnetic relaxation processes are characterized, with effective energies of 60 K and 74 K. The main magnetic axes are oriented approximately perpendicular to the triangle plane, which precludes SMT properties. Total magnetic interactions are calculated, considering Ising exchange and dipolar interactions. The overall interactions between pairs of Dy ions in the triangular core are AF (Jex/kB ¼  3.3 K to 3.6 K; Jdip/kB ¼  0.4 K). Furthermore, the resulting energy exchange states are quasi-degenerated, denoting frustration of the magnetic ground state. An illustrative case of triangular arrangement where anisotropy axes do not form a toroidal moment, despite axis coplanarity, is presented by Giansiracusa et al. (2016). These authors have studied a different Ln(III) triangular arrangement bridged by a carbonate ligand and sandwiched between two trilacunar Keggin POM ligands. Dy and Er analogues reveal the onset of field-induced magnetic relaxation at the lowest temperatures. Ab initio calculations indicate noncollinear magnetic axes in 1-Er42H2O (Na11[{Er(OH2)}3CO3(PW9O34)2] 42H2O), that are coplanar with the erbium triangle, and radially arranged with respect to the triangle’s centroid (see Fig. 71). The absence of magnetic coupling, as observed in magnetic and INS data, is argued by mutual cancellation

176 Handbook of Magnetic Materials

FIG. 71 Left: Representation of [{Er(OH2)}3CO3(PW9O34)2] complex. Right: Schematic representation of the direction of the three principal axes for Er1 (red arrow), Er2 (blue arrow), and Er3 (Green arrow) in compound 1-Er. Magnetic axes are nearly coplanar with the triangle and present almost radial orientation. Adapted with permission from Giansiracusa, M.J., Vonci, M., Van Den Heuvel, W., Gable, R.W., Moubaraki, B., Murray, K.S., Yu, D., Mole, R.A., Soncini, A., Boskovic, C., 2016. Carbonate-bridged lanthanoid triangles: single-molecule magnet behavior, inelastic neutron scattering, and ab initio studies. Inorg. Chem. 55, 5201–5214, https:// doi.org/10.1021/acs.inorgchem.6b00108. Copyright 2016 American Chemical Society.

of dipolar and AF superexchange interactions. The estimation of the exchange interaction, needed to cancel dipolar interactions in the triangular Er(III) ring, is of Jex/kB ¼  0.67 K. Magnetic measurements below 2 K would be necessary to determine magnetic interactions more accurately.

7.2.2 Tetranuclear Dy SMMs and SMTs A recent review of tetranuclear dysprosium SMMs with diverse core topologies has been presented by Lin and Tang (2014). From the survey, it has been concluded that the axiality of Ln ions is a key feature enhancing SMM properties. Strong axial ligand field can lead to thermal relaxation via higher excited states. Reinforcing magnetic coupling is the other factor that may lead to higher blocking temperatures, as demonstrated in radical-bridged lanthanide complexes. Among the different geometries, there exist numerous examples of Dy4 SMMs with a [2  2] metallogrid structure. Remarkably, in a recent work of Huang et al. (2016) the case of a Dy4 metallogrid, [Dy4(bzhdep2H)4(H2O)4(NO3)4]6CH3OH6H2O, showing a hysteresis loop up to TB ¼ 16 K (at 0.5T/s) is reported (see Fig. 72). In table 2 of that work, previous examples of similar compounds showing magnetic relaxation are presented. It is worth noting that the effective energy, Ueff/kB ¼ 61.3 K at H ¼ 0 Oe, is not among the highest in Dy4 SMMs. Similar hysteresis loops are obtained in a Dy4@Y4 diluted sample, indicating single-ion behavior. Magnetic exchange interaction is weak or negligible within the metallogrid. In this compound, QTM has been

FIG. 72 Left: Molecular structure of metallogrid Dy4 compound, [Dy4(bzhdep-2H)4(H2O)4(NO3)4]6CH3OH6H2O. Right: (A) Sweep rate dependence of hysteresis loops of reduced magnetization vs m0H at 2  0.002 K. (B) Hysteresis loops at different temperatures collected at a sweep rate of 0.05 T/s. Adapted with permission from Huang, W., Shen, F.X., Wu, S.Q., Liu, L., Wu, D., Zheng, Z., Xu, J., Zhang, M., Huang, X.C., Jiang, J., Pan, F., Li, Y., Zhu, K., Sato, O., 2016. Metallogrid single-molecule magnet: solvent-induced nuclearity transformation and magnetic hysteresis at 16 K. Inorg. Chem. 55, 5476–5484, https:// doi.org/10.1021/acs.inorgchem.6b00500. Copyright 2016 American Chemical Society.

178 Handbook of Magnetic Materials

successfully suppressed by employing organic-bridging ligands rather than radicals. The record in effective energy barrier for tetranuclear complexes has been reported by Guo et al. (2015). Square grid Dy4 cluster based on a polypyridyl triazolate ligand (Hbpt), [Dy4(OH)2(bpt)4(NO3)4(OAc)2], shows a slow relaxation with 206 K energy barrier. No information is given about the magnetic hysteresis performance of this compound. Remarkably high-energy barriers of 190 K (1) and 197 K (2) have been achieved in defect-dicubane Dy4 clusters, [Dy4X2(m3-OH)2(m-OH)2(2,2bpt)4(H2O)4]X22H2O4EtOH (X ¼ Cl (1) and Br (2)), by perturbing the ligand field with alkaline metals (Liu et al., 2015e). The reduced interaction between the ligands and f-electron charge clouds in the transversal plane enhances the axiality of the CF of individual Dy(III) in these compounds. Modest energy barriers are found for Ln4 salen-type SMMs. Luan et al. (2015a) have reported the highest effective energy barrier among the so far reported salen-type tetranuclear dysprosium SMMs, with Ueff/kB ¼ 55.7 K. Molecular Dy4 squares are interesting by their potential to exhibit a toroidal magnetic moment. Rajamaran and coworkers have presented a comprehensive analysis of magnetic relaxation processes and toroidal ground state in the Dy4 square complex [Dy4(m4-OH)(HL)(H2L)3(H2O)4]Cl2 (CH3OH)4(H2O)8 (Das et al., 2015a). Detailed ab initio calculations reveal the existence of a toroidal magnetic moment in the ground state which is confirmed by the S-shaped magnetization curve at 2.0 K. Observed magnetic relaxation processes are rationalized by means of analysis of singleion relaxation mechanisms in view of ab initio computational results. The high-temperature process with 100 K energy barrier is assigned to Dy4 site, and the observed low-temperature process with 29 K energy barrier is due to the Dy1, Dy2, and Dy3 sites. Magnetization reversal takes place through the first excited state in both cases. Magnetic interaction between Dy centers has been determined to be of J1/kB ¼ 0.014 K for n.n. and J2/kB ¼  0.014 K for n.n.n., in agreement with values obtained from low-temperature heat capacity experiments, JSE/kB ¼ 0.013 K. The calculated exchanged-coupled states present a small magnetic moment matrix element for the GS exchange doublet and the low-lying states, which results in the suppression of QTM and TAQTM. As a consequence the relaxation of the Dy4 cluster occurs via spin– phonon mechanism through the excited states, at energies at approximately 197 K above the GS. As the exchange interactions are extremely small, at low temperatures, individual Dy sites are expected to have independent orientation. More recently, Tang and coworkers have reported SMM and SMT properties of three complexes presenting a Dy4 square core (Wu et al., 2017). The use of the Schiff base ligand H3L1, bridging the four Dy ions in an antiparallel fashion, results in a toroidal arrangement of spins for complex 1 (see Fig. 73, left (A)). Modifications of the Schiff-base ligand make the Dy ions bridge in a

1 179

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

A

24 Complex 1 Complex 2 Complex 3

B

16

8

H/T −3

−2

−1

0

1

2

3

−8

C

−16 −24

M / mB

FIG. 73 Left: Ground-state magnetic anisotropy axes for Dy ions of complexes [Dy4(m4-O) (HL1)4(H2O)4]2(NO3)3(OH)2H2O2CH3OH (1) (A), [Dy4(m4-O)(HL2)4(SCN)2]2H2O4CH3OH (2) (B), and [Dy4(m4-O), (H2L3)2(SCN)2]6H2O (3) (C). Right: S-shaped hysteresis loop of complex 1 (red line), confirming toroidal magnetic moment at low temperature. Adapted with permission from Wu, J., Lin, S.-Y., Shen, S., Li, X.-L., Zhao, L., Zhang, L., Tang, J., 2017. Probing the magnetic relaxation and magnetic moment arrangement in a series of Dy4 squares. Dalton Trans. 46, 1577–1584, https://doi.org/10.1039/C6DT04456E. Published by the Royal Society of Chemistry.

parallel configuration resulting in a parallel alignment of the anisotropy axes. It is shown that small changes in the coordination environment induce drastic changes in the SMM and SMT properties. SMT behavior is experimentally evidenced by a clear step in M(H) curve at 1.9 K. Slow magnetic relaxation is observed in complex 1, with a low effective energy barrier of 6.5 K and in complex 2, where two relaxation processes can be distinguished, with activation energies of 5.7 K and 57.6 K, respectively.

7.2.3 Hexanuclear Dy SMMs A family of hexanuclear complexes has been analyzed by Das et al. (2014c), where only the Dy6 analogue, [{Dy6(L)2(LH)2}(m3-OH)4][MeOH]2 [H2O]6[Cl]48H2O4CH3OH, presents SMM behavior. A moderate energy barrier or 46.2 K is found for this complex. The authors include an interesting comparison to other Ln6 complexes previously reported in literature (table 3 in Das et al., 2014c), where most of the cores are formed by O-capped triangular trimeric motifs connected to each other, either at the vertices or at the edges. The assembling of triangular Dy3 structures seems to be a successful approach to form the Dy6 complexes reported by Li et al. (2015b) (see Fig. 74). They use two discrete Dy3 triangles aggregated in a planar Dy3 + Dy3

8 7 1.9 K

c′¢ / cm3/mol

6

N6 N4 N7 O8

O3

O5

Dy3

Dy2 O9

3

0

Dy1 N8

N9

4

1

N10

O6

5

2

O4

O7

25 K

0.1

1

2

O1

O2

10 n / Hz

100

1000

0

N3

-2 In(t / S)

N1

-4

Direct process

-6

Raman process Obarch process

-8

Fit

-10 -12 0.0

0.1

0.2

0.3

0.4

0.5

0.6

T -1 / K-1

FIG. 74 Left: Molecular structure of Dy6-SCN compound. [Dy6L2(m3-OH)4(m2-OH)2(SCN)8(H2O)4]6CH3CN2CH3OHH2O. Right: Frequency dependence of the out-of-phase AC susceptibility for Dy6-SCN. Fitting of the Arrhenius plot. Adapted with permission from Li, X.-L., Li, H., Chen, D.-M., Wang, C., Wu, J., Tang, J., Shi, W., Cheng, P., 2015b. Planar Dy3 + Dy3 clusters: design, structure and axial ligand perturbed magnetic dynamics. Dalton Trans. 44, 20316–20320, https://doi.org/10.1039/c5dt03931b. Published by The Royal Society of Chemistry. The SI conversion factor is 1 cm3/mol ¼ 4p  106 m3/mol for molar susceptibility.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 181

fashion encapsulated by a multidentate Schiff-base ligand. A relatively highenergy barrier is obtained for Dy6-SCN reaching 181 K, whereas Dy6-NO3 complex presents a lower energy barrier, 116 K. In both complexes, Dy3 cores are encapsulated inside the coordination pockets of two ligands, but with different axial coordination anions, which is affecting the coordination geometries of individual Dy ions, influencing the dynamics of magnetic relaxation. These results point to a promising synthetic strategy of new clusters with modulated SMM properties by incorporating triangular Dy3 units which introduce different auxiliary ligands.

7.2.4 High Nuclearity Ln SMMs For high nuclearity Ln SMMs, there are scarce reported results with remarkable performance. It becomes difficult to control the anisotropy of individual centers and the presence of many different sites in the cluster may result in a too wide distribution of relaxation rates which broadens magnetic susceptibility curves and prevents the experimental observation of slow magnetic relaxation. Within this review, we highlight the results obtained for Dy8 and Tb8 cage complexes reported by Bala et al. (2015). QTM is suppressed by applying a static field of 5 kOe, and slow magnetic relaxation for two different Dy8 complexes and a Tb8 complex is clearly observed (see Table 10). This is one of the few examples where a multinuclear Tb(III) complex exhibits SMM behavior. 7.2.5 New Synthetic Strategies The main synthetic strategy to obtain polynuclear complexes has been to use multidentate ligands, with hard N and O donors which present strong affinity for Ln ions. More recently, the synthesis of organometallic Ln SMMs has been proved to achieve stronger CF environments and larger magnetic interactions resulting in SMMs with enhanced properties. In this case, the coordination of soft donors provides a favorable electronic configuration for the Ln center. However, large organometallic clusters are still very rare within the SMM field and compounds are normally sensitive to oxygen and moisture (Layfield, 2014). Another synthetic strategy which is giving interesting results is the building block approach, where high SMM performant units are assembled to build larger polynuclear complexes. This strategy has been successfully employed, e.g., by Le Roy et al. (2014) in Er–cyclooctatetraenyl complexes, where Er dimers display enhanced SMM properties over the mononuclear analogues. A similar building approach for designing multinuclear SMMs has been followed by Pointillart et al. (2015a). Within the frame of organizing SMMs into extended molecular networks, they have assembled different SMM unit

182 Handbook of Magnetic Materials

FIG. 75 Crystal structure of [Dy4(tta)12(L)2] (Dy–Dy2–Dy) and open magnetic hysteresis loop at T ¼ 0.5 K. Reprinted with permission from Pointillart, F., Guizouarn, T., Lefeuvre, B., Golhen, S., Cador, O., Ouahab, L., 2015a. Rational design of a lanthanide-based complex featuring different single-molecule magnets. Chem. A Eur. J. 21, 16929–16934, https://doi.org/10.1002/chem. 201502416. Copyright 2015 John Wiley & Sons, Inc. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

blocks with the objective of combining their different properties. They have self-assembled two known mononuclear SMMs bridged by a known dinuclear SMM into a tetranuclear Dy–Dy2–Dy SMM. In the original dinuclear Dy(III)based SMM, QTM is suppressed by the dimer AF interaction and a double butterfly-shaped hysteresis loop is observed. Besides, in the Dy-based SIM, the hysteresis loop opens by the application of an external field. In the assembled Dy4 complex, the individual properties of the monomer and the dimer are maintained (see Fig. 75). Another building block proposal is that of Harriman and Murugesu (2016). The authors have used the sandwich complexes of Dy(III) and Er(III) with COT metallocenes, which present outstanding SMM performance, to generate new dinuclear SMMs with remarkable properties. The record-breaking TB ¼ 8 K of the Er double-decker is astonishingly increased to TB ¼ 12 K in the Er triple-decker compound. Ideally, stacking monomers in a linear chain-like array should yield a material with axial orientation of anisotropy, giving rise to increased Ueff values and potentially larger TB values (see Fig. 76).

7.3 Heteronuclear Ln–3d Clusters The quest for lanthanide complexes with high cluster anisotropy and reduced QTM has been explored by the inclusion of transition metal atoms in the cluster core, in the hope that the 3d metals provide magnetic moment to the cluster, and the rare earths a large anisotropy. The organic ligands employed play a small role since the Ln–Ln interaction through the ligand path is very small, although not negligible. It seems that their role in achieving SMM behavior is to quench QTM by increasing the intramolecular exchange field and breaking down the fast tunneling relaxation existent in the monomeric Ln moiety. As in

FIG. 76 Left: Building block approach to create double-, triple-, and quadruple-decker complexes. Right: Hypothetical chain-like arrangement of Ln(COT)2 monomers. Reprinted with permission from Harriman, K.L.M., Murugesu, M., 2016. An organolanthanide building block approach to single-molecule magnets. Acc. Chem. Res. 49, 1158–1167, https://doi.org/10.1021/acs.accounts.6b00100. Copyright 2016 John Wiley & Sons, Inc.

184 Handbook of Magnetic Materials

the previous section, the revision is centered on the compounds that show SMM almost exclusively, with increasing complexity in number of core atoms.

7.3.1 Heteronuclear Trimers There are tens of compounds with 3d–4f–3d cluster core configuration. In many cases the 3d atom is nonmagnetic, like Zn (Costes et al., 2015; Das et al., 2015b; Liu et al., 2016c; Shan et al., 2015; Takehara et al., 2015; Wen et al., 2017) or Mg (Das et al., 2015b), where the SMM behavior that may appear is exclusively due to SIM behavior of the Ln ion. In fact, those complexes are overviewed in this review. The complexes where both the 3d metal and the Ln atom are magnetic have been listed in Table 11, including their relaxation characteristics. At any rate, only those compounds that have outstanding properties are glossed later. The 3d–4f–3d {Co(II)–Dy(III)–Co(II)} trimer with the largest energy barrier so far reported belongs to the series [Co2Dy(LBr)2(H2O)]NO33H2O (1.3H2O), [Co2Dy(LBr)2(H2O)]NO3H2O (1H2O) and [Co2Dy(LBr)2(H2O)] NO3 (1), namely, the latter compound, whose barrier rises up to Ueff/kB ¼ 600 K (Liu et al., 2015d). On one hand, a large single crystal anisotropy at the Dy site is favored by the quasi D5h, compressed pentagonal bipiramidal coordination. Two asymmetric six-coordinate Co(II) are linked at the two sides of the Dy atom (see Fig. 77, left). Upon solvation from the 1.3H2O compound toward 1, the five equatorial atoms become more planar located, and one of the axial Dy–O distances shorter. As a consequence, the anisotropy axis is in the axial direction, and the CF effect is larger in compound 1, since the covalent (shorter bonds) and electrostatic (larger negative charge) effects cooperate in this case. As a result, the high-temperature Orbach-type relaxation is largest for 1, and as shown in Fig. 77, right, the trend to QTM at lower temperatures indicates a tunneling frequency two orders of magnitude slower in 1 with respect to the 1.3H2O compound. Thus, apparently, the presence (or absence) of the three water molecules enhances (or reduces) the vibrational coupling between the {Co(II)–Dy(III)–Co(II)} trimers in the lattice, aiding (or quenching) the magnetic relaxation via tunneling. This compound holds the record barrier in a 3d–4f system up to date. 7.3.2 Heteronuclear Tetrameric Clusters Quite a number of tetrameric 3d–4f have been synthesized; however, those with a nonmagnetic 3d metal have been overviewed since their relaxation can be directly related to the SIM relaxation of the Ln atoms in the cluster, thus not contributing to the understanding of the 3d–4f magnetism, for example, with the diamagnetic M ¼ Co(III) (Langley et al., 2012, 2014b) or Cu(I) (Han et al., 2015a). A 5d–4f complex, with M ¼ W, has been reported to behave as SIM for the Dy substitution (Dickie and Nippe, 2016). In this

TABLE 11 Heterometallic Polynuclear Compounds With Magnetic Relaxation Processes Compound

Topology

Ueff/ kB (K)

τ 0 (s)

Co–Dy– Co

[Co2Ln(LH3)4]3NO3

Linear

—a

—a

Co–Dy– Co

[Co2Dy(LBr)2(H2O)]NO33H2O (1.3H2O)

Linear

422

2.4  1011

[Co2Dy(LBr)2(H2O)]NO3H2O (1H2O)

(FR) 462

3.5  1010

[Co2Dy(LBr)2(H2O)]NO3 (1)

(SL) 522

1.8  1010

600

1.4  1011

Core

H (KOe)

TB (K) (T/s)

J3d4f, J3d3d (K)

References

Trimers

Co–Gd– Co

[LCoGdCoL]NO3

Co–Tb– Co

[LCoTbCoL]NO3

Co–Dy– Co

[LCoDyCoL]NO3CH3OH

Ni–Dy– Ni

[Ni2Dy(LH3)4]3NO3

Linear

21.3

0

FM

Chandrasekhar et al. (2014)

0.62, 0.40

Liu et al. (2015d)

0.45, 0.27

0

1

0.65, 0.36

14.5

1.1

0.8, 0.25

14

1.1

1.1, 0.18

Ungur et al. (2013)

Linear

—a

—a

AF

Das et al. (2014b)

FM

Upadhyay et al. (2016)

3MeOHH2OCH3CN

Ni–Tb– Ni

[Ni2Tb(L)6](NO3)0.5(Cl)0.5

Linear

—a

—a

CrIII–Dy– CrIII

[CrDy2(OCH3)4(dpm)5(CH3OH)] CH3OH

Nonlinear

19.7

8.4  108

0

FeII–Dy– FeII

[Fe2Dy(L)2(H2O)]ClO42H2O

Nonlinear

459

1.1  1010

0

Car et al. (2015) 0.27, 0.05

Liu et al. (2014a)

Continued

TABLE 11 Heterometallic Polynuclear Compounds With Magnetic Relaxation Processes—cont’d Core

Compound

Topology

Ueff/ kB (K)

τ 0 (s)

H (KOe)

8.8

2  107

0

7.8

3.9  107

1

47.9

2.75  107 4.23  107

TB (K) (T/s)

J3d4f, J3d3d (K)

References

Tetranuclear clusters Co2Dy2

[Co2(L)2(PhCOO)2Ln2(hfac)4] BTP

Zn2Dy2

[Ni2Dy2]3

Coplanar rhombic

AF

Abtab et al. (2014)

0

FM

Ahmed et al. (2014)

0

FM

Upadhyay et al. (2016)

[Ni2Ln2(CH3CO2)3(HL)4(H2O)2] (NO3)3DBTP

Bracket shape

19

[Ni2Dy2(L )2(o-vanillin)2(CO3)2 (NO3)2(MeOH)2]

Tilted parallelogram

—a

Fe3-Dy

[Fe3(m3-O)(inicH)6(H2O)3][Dy (NO3)5(H2O)](NO3)5n(H2O)

Isolated Fe3 and Dy

23

3  107

1

No 3d–4f interaction

Nayak et al. (2014)

Fe3Gd

[Fe3Ln(m3-O)2(CCl3COO)8(H2O) (THF)3]

Butterfly

—

—

—

0.25, 50

Bartolom
e et al. (2009)

Fe3Dy

(z ¼ 8)

12.34

3.9  107

5.5

0.4, 50

Badı´aRomano et al. (2013)

Fe3Tb

2.8

6  108

0

0.13, 50

Badı´aRomano et al. (2015)

Fe3Ho

8.1

1.1  109

0

0.18, 50

2.2  10

7

0

25.6, 0

16.2

1.8  10

7

1.5

77

5.1  108

+

[Ni2Dy2]2 +

Cu3Tb Cu3Dy Cr2Dy2

[LnCu3(H2edte)3(NO3)] [NO3]20.5MeOH

Star

[Cr2Dy2(OMe)2(O2CPh)4(mdea)2 (NO3)2]

Coplanar rhombic

17.3

3.5 (0.003)

23.6, 0 3.5 (0.003)

0.75, 10.2

Kettles et al. (2014) Langley et al. (2013)

Cr2Ln2

Cr2Ln2(OMe)2(O2CPh)4(mdea) 2(NO3)2], Ln ¼ Tb

Coplanar rhombic

Ln ¼ Ho Cr2Dy2

[Cr2Dy2(OMe)2(RN{(CH2) 2OH}2)2(acac)4(NO3)2]

Coplanar rhombic

(R ¼ Me, Et, nBu)

Cr2Ln2

[Cr2Ln2(OMe)2x(OH).

64

17  109

2.5

52

1.1  109

1.8

34.6

1.2  107

41.6 37.5

Weak FM

Langley et al. (2015)

1.8

1.32, 5.62

Langley et al. (2014c)

9.2  108

2.2

1.38, 5.91

3.1  107

2.2

0.89, 5.24

Coplanar rhombic

Weak FM

Langley et al. (2016)

(2-Cl-4,5-F-benz)4(mdea)2 (NO3)2]MeOH Ln ¼ Tb Dy Ho

63.36

7.7  109

3.5

87.84

2.1  10

7

4.7

6.8  10

9

2.6

7.7  10

8

3.1

51.84 Coplanar rhombic

64.8

Star

—a

2 Dy

[Mn3Dy(L2)2(HL2)2(naph)(NCS) (H2O)(MeOH)1.8](0.5NO3) (0.5ClO4) 1.8MeOH0.6H2O (naph: naphthaldehyde)

Co2Dy2

Dy2Co2L10(bipy)2

Linear tetramer

118

1.8  1011

105

11

Cr2Dy2

[Cr2Dy2(OMe) (OH)(4tBubenz)4(tBudea)2(NO3)2] MeOH2Et2O

MnIVMnIII III

Ni2Dy2

Dy2Ni2L10(bipy)2

Langley et al. (2016)

Tziotzi et al. (2016)

1.8  10

0

Zhao et al. (2014)

Continued

TABLE 11 Heterometallic Polynuclear Compounds With Magnetic Relaxation Processes—cont’d Topology

Ueff/ kB (K)

Core

Compound

Fe2Tb2

[FeIII 2 Tb2(H2L)4(Z2-NO3)2]

Zigzag tetramer

—

Fe2Dy2

[DyFeIII 2 Dy(m3-OH)2(pmide)2 (p-Me-PhCO2)6]

Coplanar rhombic

16.2

Irregular Ln cubane with two linked Co

—a

Windmil

—a

τ 0 (s)

H (KOe)

TB (K) (T/s)

J3d4f, J3d3d (K)

a

References Bag et al. (2014)

2ClO42CH3OH2H2O

2.6  106

1

AF, 5

Peng et al. (2016)

1

AF

Goura et al. (2015)

AF

Biswas et al. (2016a)

Hexanuclear clusters Co2Dy4

[Co2Ln4(m3-OH)4(L)2(piv)8(mOH2)] wCH3CxCH2Cl2yCH3OHzH2O, (DTD, DMCSA)

Other polynuclear clusters Ni4Ln4

[Ln4Ni4(H3L)4(m3-OH)4(m2-OH)4] 4ClxH2OyCHCl3 Dy3+, x ¼ 30.6, y ¼ 2 (1); Tb3+, x ¼ 28, y ¼ 0 (2)

Cr (III)4Dy4

[Cr4Dy4(m-F4)(m3-OMe)1.25 (m3-H)2.75(O2CPh)8(mdea)4]

Square grid

55

0

Cr (III)4Dy4

[Cr4Dy4(m3-OH)4(m-N3)4 (mdea)4(O2CC(CH3)3)4]

Square grid

10.4

0

Ni6Dy3

[Ni6Dy3(OH)6(HL)6(NO3)3] 2MeCN2.7Et2O2.4H2O

Trigonal prismatic

23.84

Mn6Dy6

III [MnIII 6 Ln6 (OH)7(H2O)3(O2CPh)11 (L)3(HL)4(NO3)]6.5MeCNH2O

Cubane star

—a

Cu6Dy12

[Cu6Dy12(OH)20(N3)6 (NO3)8(dapdo)6(H2O)18](OH)2

Windmill

17

3.63  108

3.5

Vignesh et al. (2017) Rinck et al. (2010)

0

18.62, 1.15

Canaj et al. (2015) Tziotzi et al. (2015)

3  1011

0

1.1

Alexandropoulos et al. (2017)

The cluster topology is given, as well as the Orbach-type relaxation Ueff and τ0, at the field applied in the measurement. The blocking temperature is given when available. The intramolecular coupling constants are given in 2JS1S2 formalism. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. a Reported slow relaxation below 2 K.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 189

−2 1.3H2O

In (t / s)

−4

1 Fit for 1.3H2O

−6

Fit for 1

−8

Orbach process for 1.3H2O Raman process for 1.3H2O QTM process for 1.3H2O

−10 −12 0.0

Orbach process for 1 Raman process for 1 QTM process for 1

0.1

0.2

0.3

0.4

0.5

T −1 / K−1

FIG. 77 Left: Molecular structure of [Co2Dy(LBr)2(H2O)]NO3 (1). Dy, gold; Co, violet; Br, brown, O, red; N, blue; silver. Green vectors are the calculated magnetic moments at the ground state. Right: Magnetic relaxation dynamics for the 1.3 H2O and 1 compounds. Adapted with permission from Liu, J.-L., Wu, J.-Y., Huang, G.-Z., Chen, Y.-C., Jia, J.-H., Ungur, L., Chibotaru, L.F., Chen, X.-M., Tong, M.-L., 2015d. Desolvation-driven 100-fold slow-down of tunneling relaxation rate in Co(II)-Dy(III) single-molecule magnets through a single-crystal-to-single-crystal process. Sci. Rep. 5, 16621, https://doi.org/10.1038/srep16621 published by Springer Nature under the Creative Commons Attribution License.

section, just those tetrameric clusters with magnetic 3d metal atoms in the core are considered in Table 11, and those with remarkable contributions are glossed later. Two Ga–Dy compounds have also been recently reported to show SMM behavior (Chen et al., 2016a). A clarifying view on the magnetic moment reversal process involved in the slow relaxation of the magnetization in [LnCu3(H2edte)3(NO3)] [NO3]20.5MeOH, Ln ¼ Tb and Dy, has been gained with an INS study (Kettles et al., 2014). The molecule has a three-pronged star shape, with the Ln ion at the center surrounded by the three Cu(II) atoms (see Fig. 78). The Ln coordination is a distorted trigonal dodecahedron or biaugmented trigonal prism, while each of the Cu ions is in a distorted five-coordination, sharing two oxygen atoms with the Ln polyhedron. Both the Tb and Dy ions lend to the cluster a high uniaxial anisotropy, and the compounds show an Orbach type of relaxation with similar Ueff/kB 17 K at high temperature (of a few K), the former at H ¼ 0, while the latter needs the application of an external field (H ¼ 1.5 kOe) for its observation. However, the single crystal magnetization vs field hysteresis loops are quite different (see Fig. 78 for constant temperature M(H) at different sweep rates). The {Cu3Tb} hysteresis loop is characteristic of a SMM, albeit undergoing QTM relaxation at T ¼ 0.03 K, as indicated by the fall of magnetization at H ¼ 0. In contrast, for {Cu3Dy} there is a lack of coercivity, and fast relaxation at H ¼ 0, as corresponds to a strong QTM relaxation. Since the total moment of the cluster is a half integer in {Cu3Tb}, thus has a Kramers doublet as ground state, QTM is less efficient than in {Cu3Dy}, which with an integer total moment has a non-Kramers

190 Handbook of Magnetic Materials

FIG. 78 Upper: {Cu3Ln} core of the in [LnCu3(H2edte)3(NO3)][NO3]20.5MeOH. Ln, cyan; Cu, orange; N, blue; O, red. Lower: Single crystal vs field hysteresis loops for (left) {Cu3Tb} and (right) {Cu3Dy}, at a constant temperature of 0.03 K with different sweep rates between 0.04 and 0.280 T/s. Adapted with permission from Kettles, F.J., Milway, V.A., Tuna, F., Valiente, R., Thomas, L.H., Wernsdorfer, W., Ochsenbein, S.T., Murrie, M., 2014. Exchange interactions at the origin of slow relaxation of the magnetization in {TbCu3} and {DyCu3} single-molecule magnets. Inorg. Chem. 53, 8970–8978, https://doi.org/10.1021/ic500885r published by The American Chemical Society under the Creative Commons Attribution License.

ground state. The INS measurements show excitations corresponding to the energy-level scheme consistent with the Ueff determined from AC susceptibility measurements. They could be assigned to spin flips of the Cu magnetic moments hindered by the magnetic coupling to the central Ln ion, which is rather strong and ferromagnetic (average JCuLn/kB 25 K). Therefore, the magnetic relaxation found in these compounds is governed by the Cu subsystem, modulated by the quantum mechanical characteristic with respect to time reversal. The “butterfly” molecules [Fe3Ln(m3-O)2(CCl3COO)8(H2O)(THF)3] have a tetrameric magnetic core, consisting of a Fe(1)–Fe(2)–Fe(3) nonlinear triad, and a Ln atom linked to the Fe atoms via m-oxo bridges (see Fig. 79). Within the Fe3 subcluster the Fe(2)–Fe(1,3) interaction is antiferromagnetic (JFeFe/ kB ¼  50 K), constituting a robust S ¼ 5/2 unit (Bartolom
e et al., 2009). This unit is antiferromagnetically coupled with the Ln ion, as could be proven by means of XMCD spectroscopy performed at the L2,3 edges of the Ln atom, that could measure the Ln subcluster magnetization, combined with a

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 191

A {Fe3LnO2} Dy(1)

10 O(1) Fe(1)

H = 0 Oe

O(2) Fe(3)

C/R

Fe(2)

Gd Tb

B

Ho

Dy J¢¢

J¢¢

Dy

Fe(3)

J

CLT α T 3

0.1

J¢¢¢ Fe(1)

Y

1

J

Fe(2)

0.1 J¢

1

10 T (K)

FIG. 79 Left: (A) Molecular structure of [Fe3Dy(m3-O)2(CCl3COO)8(H2O)(THF)3]. (B) The {Fe3Dy} core of the compound and the defined interatomic exchange parameters. Right: Lowtemperature heat of the {Fe3Ln} series. Dashed line, lattice contributions. Note the nonequilibrium behavior of the {Fe3Dy} sample (see text) (Badı´a-Romano, 2014; Badı´a-Romano et al., 2013, 2015). The SI value of the gas constant is R ’ 8.3145 J K–1 mol–1. Adapted with permission from Badı´a-Romano, L., Bartolom
e, F., Bartolom
e, J., Luzo´n, J., Prodius, D., Turta, C., Mereacre, V., Wilhelm, F., Rogalev, A., 2013. Field-induced internal Fe and Ln spin reorientation in butterfly {Fe3LnO2} (Ln ¼ Dy and Gd) single-molecule magnets, Phys. Rev. B 87, 184403, https://doi.org/10.1103/PhysRevB.87.184403; Badı´a-Romano, L., Rubı´n, J., Bartolom
e, F., Bartolom
e, J., Luzo´n, J., Prodius, D., Turta, C., Mereacre, V., Wilhelm, F., Rogalev, A., 2015. Intracluster interactions in butterfly {Fe3LnO2} molecules with the non-Kramers ions Tb(III) and Ho(III). Phys. Rev. B 92, 64411, https://doi.org/10.1103/PhysRevB.92.064411. Copyright 2013 and 2015 respectively by the American Physical Society.

conventional SQUID magnetization study for the total magnetization (Badı´aRomano et al., 2013, 2015). The intracluster interaction was found to be very small for the studied Ln substitutions, Gd as isotropic and Dy anisotropic Kramers ions, Tb and Ho as anisotropic non-Kramers ions (see Table 11). In a subsequent study of their relaxation properties below 1 K, it was found that the Fe3 subcluster and the {Fe3Gd} clusters flip at a fast rate compatible with QTM, while the anisotropic clusters undergo an Orbach type of relaxation at relatively high temperature and tend toward QTM at lower temperatures, with activation energies compatible with thermally activated quantum tunneling via the excited electronic states of the whole cluster. The time reversal characteristics of the whole cluster is determined by the coupling of the electronic and nuclear wave functions, where the electronic part has an integer spin for the {Fe3Dy} case and a half-integer for the {Fe3Tb} and {Fe3Ho} cases, since 50% of the Dy nuclei and 100% of the Tb and Ho nuclei have a half-integer nuclear spin, in the Ln natural abundance isotopes. Therefore, 50% of the {Fe3Dy} has a total Kramers ground state, hindering QTM, while for {Fe3Tb} and {Fe3Ho} the total wave function is non-Kramers and has a

192 Handbook of Magnetic Materials

faster QTM. In fact, it becomes so slow that the {Fe3Dy} heat capacity does not reach equilibrium at an experimental characteristic time of 2 s, as shown in Fig. 79 (Badı´a-Romano, 2014). The quest for lanthanide complexes with high cluster anisotropy is correlated with the presence of open magnetic hysteresis curves at as high a temperature as possible. The inclusion of transition metal atoms in the cluster core has been successfully employed to this effect. The compound [Cr2Dy2(OMe)2(O2CPh)4(mdea)2(NO3)2], where two Cr(III) atoms with S ¼ 3/2 are coupled with two Dy(III) atoms in a coplanar rhombohedral cluster (see Fig. 80, left), behaves as a SMM with a barrier as high as Ueff/kB ¼ 77 K with a relatively high TB ¼ 3.5 K (Langley et al., 2013). Below that temperature the hysteresis curve, measured with a field average sweep rate of 0.003 T/s, is very broad, with a coercive field of 2.8 T, and with just a small decay near H ¼ 0. This feature indicates that the QTM relaxation mode has been quite effectively quenched, and the cluster is a good SMM. A careful analysis of its magnetic properties, aided by ab initio calculations, has led to conclude that the Cr–Dy interaction is antiferromagnetic and dominant with respect to the Dy–Dy or Cr–Cr (see Table 11). The cluster energy levels have been calculated, and by means of the magnetization matrix method, it was argued that the tunneling gaps of the cluster at the ground state and at the excited level are too small to allow for QTM. The relaxation of magnetization is expected to occur through a spin–phonon mechanism through the excited states, giving rise to a calculated effective energy barrier of 72 K, in good agreement with the experimental 77 K. The intracluster magnetic interaction has played the essential role of avoiding QMT. The substitution of Dy by other Ln atoms in the same complex led to two rare examples of SMM with Tb and Ho (Langley et al., 2015). However, the large coercivity found in the Dy substitution is strongly reduced and a large loss of magnetization at H ¼ 0 indicates a much more effective QTM in these two compounds at the expense of a lower Ueff. The variation of the amine based polyalcohol modulates the Cr–Dy exchange interaction and as a consequence the exchange-dependent effective barrier, albeit to lower values than in the [Cr2Dy2(OMe)2(O2CPh)4(mdea)2(NO3)2] (Langley et al., 2014c). The replacement of the bridging benzoate ligand by 2-chloro-4,5-fluorobenzoate introduces halogen atoms in the bridge which are electron withdrawing. As an effect, the Cr–Dy interaction increases, and the single-ion anisotropy also increases, giving as a result the highest Ueff and TB and coercivity in these series for [Cr2Dy2(OMe)2x(OH)x(2-Cl-4,5-F-benz)4(mdea)2(NO3)2]xMeOH (see Fig. 80, left). A remarkable work using MS has been published on the nonlinear Fe(III)– Ln–Ln–Fe(III) tetramer [Fe2Ln2(H2L)4(Z2-NO3)2]2ClO42CH3OH2H2O, where the different relaxation rates for the different Ln substitutions were probed against the natural Larmor frequency window of the MS technique (Bag et al., 2014). Along the Ln series a transition in relaxation rates was

FIG. 80 Left: Molecular structure of [Cr2Dy2(OMe)2(O2CPh)4(mdea)2(NO3)2]. The core cluster is in coplanar rhombic topology. The calculated Ln and Cr(III) magnetic moment vectors are depicted. Reprinted with permission from Langley, S.K., Wielechowski, D.P., Vieru, V., Chilton, N.F., Moubaraki, B., Abrahams, B.F., Chibotaru, L.F., Murray, K.S., 2013. A {CrIII2DyIII2} single-molecule magnet: enhancing the blocking temperature through 3d magnetic exchange. Angew. Chem. Int. Ed. 52, 12014–12019, https://doi.org/10.1002/anie.201306329. Copyright 2013 John Wiley and Sons Right: Comparison of the M(H) hysteresis curves with a magnetic field sweep of 0.003 T/s. Adapted with permission from Langley, S.K., Wielechowski, D.P., Moubaraki, B., Murray, K.S., 2016. Enhancing the magnetic blocking temperature and magnetic coercivity of {CrIII2LnIII2} single-molecule magnets via bridging ligand modification. Chem. Commun. 52, 10976–10979, https://doi.org/10.1039/C6CC06152D. Published by The Royal Society of Chemistry.

194 Handbook of Magnetic Materials

found from fast for Gd (>109 s1) to intermediate for Dy (between 108 and 109 s1) and very slow for Tb (≪108 s1). In the latter case, AC susceptibility showed hints of low barrier SMM below 1.8 K. MS has been also useful in detecting three relaxational regimes appearing as a function of applied field in [DyFeIII2Dy(m3-OH)2(pmide)2(p-Me-PhCO2)6], which are ascribed to the effect of dipolar fields originating from the magnetic substates of the Dy ions (Peng et al., 2016).

7.3.3 Clusters with Higher Nuclearity As in the previous sections, the references included in Table 11 report on SMM processes with 3d nonmagnetic metals, like Co(III) (Tian et al., 2014; Xue et al., 2014b), which will not be reviewed. However, there is an interesting compound with the exotic nonmagnetic 5d atom, Os, in the pentanuclear (Bu4N)5[Ln{Os(NO)(m-ox)-Cl3}4(H2O)n] [Ln ¼ Y and Dy when n ¼ 0; Ln ¼ Dy, Tb, and Gd when n ¼ 1] with SIM properties. Below some interesting cases are discussed. The octonuclear Cr(III)4Dy(III)4 clusters provide a family of compounds where the Cr atoms sit at the corners of a square and the Dy ones between them. The four Dy atoms are bridged by four m atoms and four m3 groups, and each m3 group to a single Cr ion. In [Cr4Dy4(m-F4)(m3-OMe)1.25(m3OH)2.75(O2CPh)8(mdea)4] the m atom is a fluoride F. The presence of this F has a bearing on the Cr–Dy interaction enhancing the antiferromagnetic component (JCrDy/kB ¼  0.9 K), while the Dy–Dy interaction becomes ferromagnetic and dominant (JDyDy/kB ¼ 1.4 K). The barrier height is also increased by the F atoms (Vignesh et al., 2017). The QTM is quenched at the first and second excited states, leading to the relaxation via the third excited state. Thus the presence of the Cr(III) and F contributes additively to the quenching of the QTM and an associated increased barrier height and relaxation times with respect to the case where F is substituted by OH. The enneanuclear [Ni6Dy3(OH)6(HL)6(NO3)3]2MeCN2.7Et2O2.4H2O complex has a {Ni6Dy3} core formed by a trigonal prism of Ni(II) atoms, with three Ln ions near the prism faces (see Fig. 81, left; Canaj et al., 2015). The dominant interaction is the ferromagnetic Ni–Ni one, followed by the ferromagnetic Ni–Gd, at least in that case. The Dy-substituted cluster has a distinct SMM behavior with a relatively high Ueff/kB ¼ 23.84 K (see Fig. 81, right).

7.4 Conclusions Homonuclear lanthanide SMMs with more than two metal centers present many interesting properties to be highlighted. On the one hand, exchange interactions are weak and mostly AF and magnetic relaxation properties are related to single-ion anisotropy. The successful achievement of outstanding SMM properties is related to the enhancement of the axial symmetry and the reduction of QTM. In this sense, the ligand design

9

Ni2

Ni1

In(1/t)

8

Er2

Er1

7 Ni6

Ni3 Ni4 Ni5

Er3

6 0.34

0.35

0.36

0.37

0.38

0.39

0.40

0.41

0.42

1/Tmax (K−1) FIG. 81 Left: [Ni6Dy3(OH)6(HL)6(NO3)3]2MeCN2.7Et2O2.4H2O molecule, showing the Ni6Dy3 cage. Ni, green; Dy, purple; O, red; N, blue. Right: Inverse relaxation time with an Orbach type process with Ueff/kB ¼ 23.84 K. Adapted with permission from Canaj, A.B., Tzimopoulos, D.I., Siczek, M., Lis, T., Inglis, R., Milios, C.J., 2015. Enneanuclear [Ni6Ln3] cages: [LnIII3] triangles capping [NiII6] trigonal prisms including a [Ni6Dy3] single-molecule magnet. Inorg. Chem. 54, 7089–7095, https://doi.org/10.1021/acs.inorgchem.5b01149. Copyright 2015 American Chemical Society.

196 Handbook of Magnetic Materials

strategy in different core topologies plays the role of modulating symmetry of individual ions. Particular structure topologies and ligands give rise to comparable SMM behaviors. Additionally, the presence of different orientations of the anisotropy axis in the different centers of the cluster weakens intercluster dipolar interaction, reducing QTM. On the other hand, magnetic interactions in Ln polynuclear clusters, albeit weak, are of relevance in the low-temperature magnetic behavior and very interesting underlying physical phenomena appear. Ab initio calculations together with a multitechnique approach, including low-temperature measurements, are needed to understand the electronic structure of the different ions involved and its relation to the static and dynamic properties. There are still open questions regarding the effect of magnetic interaction within the cluster in the relaxation processes and the mechanisms involved in the relaxation. Although some interesting compounds of 3d–4f heteronuclear compounds have been synthesized showing SMM behavior, this strategy has not been too successful. It was deemed that combining 3d metals, to contribute with magnetic cluster moment and by including a highly anisotropic Ln to aid its anisotropy, such as Dy or Tb, would result in high moment, high anisotropy clusters. Unluckily, the weakness of the 3d–Ln interaction does not allow this expected synergy to work positively. At any rate, it can be concluded from this section that the research of polynuclear lanthanide SMM remains an active research field, where different design strategies together with a comprehensive theoretical and experimental analysis may contribute to the development of new complexes with extraordinary properties.

8 1D, 2D, AND 3D EXTENDED SYSTEMS Magnetic interaction between lanthanide ions has two components, shortrange spin–spin exchange interaction and long-range dipolar interaction. Both interactions depend on the atom–atom distances in a magnetic compound; thus the exchange interaction extends to n.n., while the dipolar one extends to several coordination spheres. The dimensionality of a magnetic system describes the topology of the dominant interaction in a sample, in a definite temperature domain. In one-dimensional (1D) magnetic systems, such as chains, zigzag chains, double chains, etc., the interacting moments extend along a topological line, not necessarily straight. Likewise, in a 2D and a 3D magnetic system the interacting moments remain in a topological plane or tridimensional network, respectively. Magnetic linear chains have been thoroughly studied for decades, since they provide excellent models to test theoretical predictions for the phenomena associated to collective spin excitations such as spin waves in ideal topological 1D systems. Since the magnetic chains are supported by the 3D crystal structure, the intrachain magnetic coupling must be dominant with respect to

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 197

interchain coupling to maintain the 1D characteristics. However, when the temperature is decreased so that the kinetic energy kBT is of the order of the interchain coupling, a crossover to higher dimensionality, 2D or 3D depending on the dimensionality of the interchain coupling may take place. In many cases, this crossover gives rise to a long-range ordering transition, Curie temperature TC for ferro- or ferrimagnetic ordering, or Neel temperature TN for AF ordering. The associated critical behavior when approaching TC (TN) from high or low temperatures is governed by the topological dependent (1D, 2D, 3D) universality rules that have been developed for the equilibrium state of collective excitations in macroscopic systems (de Jongh and Miedema, 1974). At low temperatures, the relaxation time toward equilibrium, τ, due to thermal or magnetic perturbations is of the order or higher than the characteristic experimental time τex, and as a consequence it is measurable. Besides the magnetic fluctuations caused by intrachain and interchain interactions, and in competition with them, are those provoked by single-ion phenomena such as magnetic and electronic excitations (and desexcitations) to real or virtual electronic states, magnetic quantum tunneling, and thermally activated magnetic quantum tunneling, that render the low temperature range spin dynamics as a fascinating field of research. Moreover, the presence of finite size magnetic domains in the chains separated by mobile walls has provided a new paradigmatic behavior in the so-called SCMs. As a consequence of all these interactions, time-dependent relaxation phenomena are observed at low temperature.

8.1

1D Molecular Magnets

The object of this section is to review recent developments in the field of relaxation processes in 1D molecular magnetic systems. The 1D magnetic molecular systems are formed by bridging ligands that favor linking of the molecules that contain the Ln ion. The magnetic coupling between these ions in the case of lanthanides is either exchange coupling between n.n., albeit weak because of the shielded character of the 4f electron orbitals, and longrange dipolar interaction, which may be relevant at very low temperatures because of the large orbital moment in many of the lanthanide substitutions. 1D lanthanide-based complexes represent an ideal playground to study how slow dynamics and tunneling depend on the type of ion and its coordination sphere, the relative importance of the ion’s anisotropy vs exchange, and the nature, sign, and intensity of intrachain/interchain competing interactions.

8.1.1 Polymers With SIM Behavior The interest on the lanthanide containing chains in the last few years has been almost restricted to polymers. These systems self-organize the molecules which contain the Ln single ions, or homo- and heteroatomic clusters along preferential directions. Some of these magnetic polymers show magnetic relaxation at low temperature. Below, the recently reported compounds

198 Handbook of Magnetic Materials

showing this characteristic magnetic behavior are reviewed in an order of increasing complexity. Many of the homometallic polymers have very weak intermolecular magnetic interaction. Even in cluster type of molecules, their Ln–Ln intramolecular interactions may be very weak. In spite of their 1D well-formed structure, in the temperature region above 2 K, where most of the measurements have been done, their magnetic properties correspond to their single-ion characteristics. In some Ln substitutions, slow magnetic relaxation has been detected, at zero magnetic applied field in some cases, and at H 6¼ 0 in the majority of the compounds. Magnetic dilution, i.e., substitution of magnetic ions by nonmagnetic ones is a standard procedure, by weakening any possible interion interaction, to check whether the relaxation is SIM. In Table 12 the most recent reports belonging to this class are collected, and in this section just a few outstanding papers from the point of view of magnetic relaxation are discussed. For each complex, only those Ln substitutions which undergo slow relaxation are included in the table. To describe the SIM behavior, we may use the relaxation time-dependent expression:   m 1 n τ1 ¼ τ1 QMT + AT + τ 0 exp Ueff =kB T + CT ,

(81)

where the first, second, third, and fourth terms correspond to quantum magnetic tunneling, direct, Orbach, and optical acoustic Raman processes, respectively. It is quite remarkable that in most complexes it is only the Dy substitution that gives rise to SIM behavior. In all cases an Arrhenius temperature dependence τ1 ¼ τ1 0 exp( Ueff/kBT) has been found, which in general has been discussed in terms of the presence of a relaxation time ascribed to an Orbach process (see Table 12). Besides the Ln ¼ Dy cases, Orbach-type processes could be detected just for the Ln substitutions with Tb, Er (Girginova et al., 2014), and Yb (Castro et al., 2016; Li et al., 2015d; Yatoo et al., 2016). As the temperature is lowered, the general trend of the relaxation time is to curve down to a lower slope: see Fig. 82B, for the case of {Dy(a-fur)3}n, where two such processes are present due to the presence of two different Dy sites in the structure (Bartolom
e et al., 2013). This trend has been interpreted in terms of the Ln single ions undergoing a crossover to dominating QTM at low temperature, that, indeed, is quite probable in lanthanides as Dy. However, more recently, it has been realized that a direct process within the ground doublet, split by external fields, may also give rise to this plateaulike behavior. In Fig. 82B, a slow relaxation process is also depicted, which is associated to a direct process affected by spin–phonon bottleneck effect. Now we focus on the direct process, whose relaxation time is characterized by parameters A and m. In a helix chain compound [Dy(HNA) (NA)2(NO3)]n (Liu et al., 2015f), for Dy it has been found A ¼ 208.29 s1/K,

TABLE 12 Recently Reported Lanthanide Coordination Polymers Formula of the Polymer {[Dy(phen)(NDA)1.5]0.5H2NDA}n

Ln Dy

Coordination DBTP

Dy [DyNa(valdien)((PhO)2PO2)Cl]n

Dy

Dy

PB

DTD

Yb [Dy(HNA)(NA)2(NO3)]n (helix) {[Ln(DMTDC)1.5(H2O)2]0.5DMF0.5H2O}n [Ln(3,5-DNBz)3(H2O)2](H2O)

Dy Dy Dy

29.0 55.5(2)

[Dy(valdien)((PhO)2PO2)]n [Ln(m3-OH)(na)(pyzc)]n

Ueff/kB (K)

DTD DBTB SAP-BTP

71.9

{[Dy(L)3(H2O)]5H2O}n

Dy

Dy

DBTP

DTD

{[Dy0.5Y0.5(L)3(H2O)]5H2O}n [Ln(bptc)(phen)(H2O)]n [Ln(L)(MeO)(MeOH)0.5]nMeOH [Ln(L)(N3)0.75(MeO)0.25(MeOH)]n 0

[Dy2(L )3(MeOH)]2MeOH

Dy

DSAP

Dy

Z ¼ 7–8

7

4.47  10

10

4.0(0)  10

8

1.50  10

1.03  10

39.6

1.52  108 2.13  10

37

5.27  108

77.8

(A) 29

7

4.38  10

Fang et al. (2015)

1.2 1

Huang et al. (2015b)

3

Li et al. (2015d)

0.9

Liu et al. (2015f)

2

Wang et al. (2016b)

0.3

Zhu et al. (2014c)

11

5.5  10

2

8

0

3.4  10

5  10

11

6  10

(A) 38

2  108

Bartolom
e et al. (2013)

4

11

(B) 79

7  10

48.2

1.1  109

1

Han et al. (2015b)

1

Zhao et al. (2015a)

10

46.1

9.4  10

49.3

1.5  106

—

2

10

(B) 80

a

References

7

14.1

48.29

H (kOe)

8

76.68

90.9 {[Ln(a-C4H3OCOO)(m-(a-C4H3OCOO))2 (H2O)3]}n

τ 0 (s)

Canaj et al. (2016)

DTD Z ¼ 7–8 Continued

TABLE 12 Recently Reported Lanthanide Coordination Polymers—cont’d Formula of the Polymer

Ln

Coordination

Ueff/kB (K)

[Ln(CymCOO)(acac)2(H2O)]n

Dy, Er

PB

—

Ln(glu)(pic)(H2O)2

Gd

DL

11.43

Na[Ln(EDTA)(H2O)3](H2O)3]n

[K{Er(hfac)4}]

H (kOe)

7.17  106 4.70  10

Er

14.7

4.30  106

55.5

4  1010

DMSAP

Dy

5.7 DSAP

Er

23.21 20.21

{[Yb(L)(H2O)3(DMF)](HL)(H2O)}n

Yb

DSAP

28.1

{[Ln2(DTE)(HDTE)(MeOH)2] 2H2O}n

Yb

CTP

85

2

3.3  10

2 1

1.02  10

7

3.3(9)  10

7

21.7

8.1  10

{[Dy2Ba(a-C4H3OCOO)8(H2O)4]2H2O}n

Dy

DSAP

68

1.0  109

[Ln3L4(phen)4(H2O)4](ClO4)2H2O

Dy

DSAP


—

Holmberg et al. (2016a)

8

DSAP

BTP

1.2 3

3.91  10

Dy

Dy

Girginova et al. (2014)

9

[Ln2(DTE)3(bipyridine)2(H2O)2]n

[Ln4L4(OH)2](OAc)2xH2O [Dy4L4(NO3)] (NO3)22CH3OHH2O

1

8

66.9

Dy

References Gavrikov et al. (2015)

Dy

Tb [K{Dy(hfac)4}]

τ 0 (s)

a

Zeng et al. (2014)

1

Castro et al. (2016)

2

Yatoo et al. (2016)

1

Cosquer et al. (2014) Bartolom
e et al. (2014a) Hu et al. (2015b)

6

1.9  10

1

Mondal et al. (2016)

The coordination type, Arrhenius-type activation energy Ueff, prefactor τ0, and field applied in the AC susceptibility measurement H are included. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. DBTP, distorted bicapped trigonal prism; DSAP, distorted square antiprism; BTP, biaugmented trigonal prism; CTP, capped trigonal prism; DL, dimeric ladder; DMSA, distorted monocapped square antiprism; DTD, distorted triangular dodecahedron; PB, pentagonal bipiramidal; SAP-BTP intermediate, square antiprism, and bicapped trigonal prism. a Reported slow relaxation below 2 K.

FIG. 82 (A) Center: Distorted bicapped trigonal O8 coordination prism of a Dy ion in {Dy(a-fur)3}n. Crystal structure along with atom-labeling schemes (green: Dy, red: O, gray: C, blue: H); the different position of one of the furoate ligands results in two coordination environment types, Dy(A) and Dy(B); (B) relaxation time as a function of the inverse temperature, τ(1/T), for the Dy(A) (red) and Dy(B) (blue) types, in an applied field m0H ¼ 4 kOe (bold symbols). Adapted with permission from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Luis, F., Turta, C., 2013. {Dy(a-fur)3}n: from double relaxation single-ion magnet behavior to 3D ordering, Dalton Trans. 42, 10153–10171, https://doi.org/10.1039/c3dt51080h. Published by The Royal Society of Chemistry. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

202 Handbook of Magnetic Materials

m ¼ 1. Besides, in the quasi-D5h Dy coordination symmetry compound [DyNa (valdien)((PhO)2PO2)Cl]n (Huang et al., 2015b), the Raman constant C ¼ 100.9 s1/K3.5, n ¼ 3.5, and 505.0 s1/K2, n ¼ 2, have been proposed. Since the direct and Raman processes have the same temperature dependence, it is questionable whether they can be resolved with just temperature-varying experiments. The dependence of τ with an applied field should be measured to discern which process is dominant. However, it is interesting to note the increasing concern on processes, beyond the Orbach one, to explain the observed relaxation times in high anisotropic lanthanides such as Dy. In Table 12, we have included the parameters Ueff and τ0. In the Ln ¼ Dy, it ranges from Ueff/kB ¼ 21 K for a distorted square antiprism coordination (Cosquer et al., 2014), to a maximum of Ueff/kB ¼ 90 K in an intermediate square antiprism and bicapped trigonal prism coordination (Zhu et al., 2014c). It is essentially dependent not only on the symmetry but on the charges that surround the ion. Among many 1D Ln-based polymers, the tetranuclear quadrupole-stranded helicate compound [Dy4L4(NO3)](NO3)22CH3OHH2O has caught our attention because of its interesting and beautiful crystallographic structure (Mondal et al., 2016). In the helical structure the central molecular entity ([Ln4L4(OH)2]+2 ) contains four deprotonated ligands (L), two hydroxy (OH) bridges, each connected to adjacent Ln ions, and other four ligand backbones connecting four lanthanide centers that create the tetranuclear quadruple-stranded helicates (Fig. 83). Out of the tetranuclear molecular 1D systems described in that work, only the two compounds with Ln ¼ Dy substitution present evidence for slow relaxation, albeit in an applied field of 1 kOe.

FIG. 83 (A) The tetranuclear quadrupole stranded helicate [Dy4L4(NO3)](NO3)22CH3OHH2O; (B, C) different modes of ligand strand. Adapted with permission from Mondal, A.K., Jena, H.S., Malviya, A., Konar, S., 2016. Lanthanide-directed fabrication of four tetranuclear quadruple stranded helicates showing magnetic refrigeration and slow magnetic relaxation. Inorg. Chem. 55, 5237–5244, https://doi.org/10.1021/acs.inorgchem.6b00177. Copyright 2016 American Chemical Society.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 203

Thus, it corresponds to SIM behavior similar to all other samples mentioned in this section.

8.1.2 Crossover From SIM to Collective Fluctuations Access to very low temperatures has enabled to observe the crossover from SIM behavior to spin collective processes associated to spin wave dynamics as temperature is decreased and approaches the long-range antiferromagnetic transition temperature TN. In the low-dimensional magnetic polymer {[Dy(a-C4H3OCOO) (m-(a-CH4H3OCOO))2(H2O)3]}n, abbreviated {Dy(a-fur)3}n, based on the a-furoate ligands, two of the furoate groups link the Dy ion to the adjacent one consolidating the chains along the c-axis, while H-bonds link them in the c-axis forming 2D supramolecular layers. The layers are further linked one to each other along the a-axis by H-bonds. In {Dy(a-fur)3}, each Dy is in a distorted bicapped trigonal prism. Disorder in the carboxylate ligand gives rise to two different types of Dy coordination, A and B, which has a bearing on the dynamic magnetic behavior (see Fig. 82). The ligand field splitting of the Dy electronic states yields to a large gap between the ground state Kramers doublet and the next excited states for the two types of Dy sites. It was found that above 2 K the dominant process is of Orbach type at H ¼ 0, unlike most other examples. The activation energy Ueff at the two different sites differs strongly, an effect associated to differences in the second sphere of coordination, as was proven by comparison to ab initio calculations of the electronic energy-level scheme. Magnetic dilution was employed to verify that the SIM behavior above T ¼ 2 K is practically independent of the intrachain interactions. As the temperature decreases, one observes a reduction of the ln τ(1/T) slope, which can be rationalized in terms of a crossover to QTM, or dominance of a direct process (see Fig. 82B). Magnetic AC susceptibility on loose powder allowed to establish that the intrachain Dy–Dy interaction is ferromagnetic. Moreover, heat capacity below 1 K determined a crossover to 3D long-range ordering at a critical temperature TN ¼ 0.66 K (Bartolom
e et al., 2013). Within an effective spin S* ¼ 1/2 Ising model the intrachain interaction was found Jc/kB ¼ 0.755 K, while the ratio between interchain and intrachain interaction was found to be J∗ab/J∗c ¼  0.135. As the temperature approaches TN in the paramagnetic phase, short-range ordering within the chain establishes correlations within the chain that give rise to a slowing down of the QTM relaxation process (see Fig. 84B). A different type of polymeric magnetic chain is provided by the furoate {[Dy2Ba(a-C4H3OCOO)8(H2O)4]2H2O}n, brief {Dy2Ba(a-fur)8}. In this compound, two Dy ions are coupled via two furoate ligands forming a dimer, while the Ba atom is linked to two different dimers along the c-axis, consolidating a zigzag chain (Fig. 84A). In this case the Dy is in a distorted square antiprismatic coordination, and from ab initio calculations, it is predicted that

J′′′

A

B

34° J′

101

Bold symbols: H = 0 Open symbols: H ≠ 0

100

SQUID

10−1

t (s)

J″ 10−2

2B 2(A - B)

10−3 tT (2) 10−4

1

2A c

tT (1) b

a

10−5

c ¢/c ≤ window 10−6 0.0

0.2

0.4

0.6

0.8

1.0 1 / T (K−1)

2

3

1/TN(2) 0

4

5

1/TN(1) 00

FIG. 84 (A) Schematics showing the assembly of dimer zigzag chains in {Dy2Ba(a-fur)8}n, and the definition of the intradimer (J ), interdimer (J ), and interchain (J000 ) coupling constants. (B) Comparison between the relaxation data, τ(1/T), determined for complex {Dy2Ba(a-fur)8}n (Bartolom
e et al., 2014a) (green), and the previously studied complex {Dy(a-fur)3}n (Bartolom
e et al., 2013) containing two different Dy sites (A: red) and (B: blue). Black symbols correspond to an average relaxation time for sites A and B. Data were collected at H ¼ 0 (bold symbols) and H 6¼ 0 (2–4 kOe) (open symbols). The grated yellow and plain yellow areas correspond to the frequency windows for SQUID, PPMS, and very low temperature AC susceptometers, respectively. Reprinted with permission from Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Luis, F., Turta, C., 2013. {Dy(a-fur)3}n: from double relaxation singleion magnet behavior to 3D ordering. Dalton Trans. 42, 10153–10171, https://doi.org/10.1039/c3dt51080h; (Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba (a-fur)8}n furoate polymers, Dalton Trans. 43, 10999–11013, https://doi.org/10.1039/C4DT00538D. Published by The Royal Society of Chemistry.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 205

the first excited electronic levels lay about 73.4 K from the ground state Kramers doublet. A field-induced Orbach process with Ueff/kB ¼ 68.3 K was found above 2 K, in good agreement with ab initio calculations. Besides, the AC magnetic susceptibility measured down to 85 mK confirmed that the intrachain interaction is ferromagnetic, while the interchain interaction is AF. An average intrachain interaction value J0 /kB ¼ 0.528 K was proposed, while the interchain interaction was far weaker, J000 /kB ¼  0.02 K (see Fig. 84A; Bartolom
e et al., 2014a). These weak AF interchain interactions enabled the establishment of a 3D long-range ordering at TN ¼ 0.25 K, which was detected by heat capacity and AC susceptibility measurements. In these two last examples the temperature-dependent studies were pushed down to temperatures below 1 K. Since the long-range temperature is lower for {Dy2Ba(a-fur)8}n than for {Dy(a-fur)3}n, the temperature region where one observes competition between the short-range ordering above TN and the QTM fluctuations is more extensive, so the slowing down of QTM caused by the increasing intensity of the intrachain mean field is more clearly distinguished (see Fig. 84B; Bartolom
e et al., 2014a). The cause is the competition between high-frequency QTM fluctuations, and the increasing internal mean field as short-range ordering sets on, that tends to polarize the moments and damp their fluctuations. For most complexes these very low temperature studies are not existent, so there may be an unexplored rich realm of peculiar relaxation behaviors.

8.1.3 Single-Chain Magnets In a SCM slow magnetic relaxation processes are caused by the presence of domain walls separating finite chain stretches of length (L). The first reported case was on a Co nitronil nitroxide chain (Caneschi et al., 2001), which proposed this chain to be a good representative for the Glauber model of finite chain magnetic dynamics (Coulon et al., 2004). Recent reviews on the theoretical models explaining SCM relaxations, including the Glauber model and recent developments for non-Ising and weak ferromagnetic systems can be found in Coulon et al. (2015) and Gatteschi and Vindigni (2014). Moreover, a revision of recently found SCMs is contained in Sessoli and Bernot (2015). Below, our review deals on recently published work on SCMs. It is collected in Table 13, and those works with significant new information are discussed. In SCMs with dominating ferromagnetic interactions the correlated spins in a segment change their magnetic orientation with time due to external thermal or magnetic field perturbations that give rise to domain-wall motion. The corresponding relaxation time dependence with chain size, infinite and finite sizes, and interactions is given in Section 2.3.5. In the case of AF interactions, thus with no net magnetization, the unavoidable presence of defects in the chain creates finite segments with a nonzero magnetic moment that may

TABLE 13 Linear Chains Formed by: (a) NITR 5 2-(R)-4,4,5,5-Tetramethyl-4,5-Dihydro-1H-Imidazolyl-1-Oxy 3-Oxide, (b) Hnic 5 Nicotinic Acid, and (c) a-Furoic Acid r

Ln

Ordering Temperature (K)

JLnr/kB (K)

Jrr/kB (K)

JLnLn/ kB (K)

Isopropyl

Gd

Not observed

0.97

2.7

0.17

Champion et al. (2003)

6.8

1.6

0.8

Bartolom
e et al. (1996)

Dj (K)

Dion (K)

τ 0 (s)

References

Ln-2p 1D compounds Gd(hfac)3NITR

Methoxyphenyl

Benelli et al. (1995) Ethyl

Gd

1.88(2) 2.19(2)

2.52

[Ln (hfac)3(NITR)]n

3BrPhOMe

Tb

19.9

AF

[Dy (hfac)3(NITR)]n

Ph(OMe)2

Dy

Not observed

[Ln (hfac)3(NITR)]

C6H4OPh

Dy

Not reported

3.83

0.49

Cinti et al. (2008) 58.75

54.71

2.25  107

Hu et al. (2013)

70.31

66.93

8.68  1015

Wang et al. (2015b)

Dx0 ¼ 42 (1)

5.6  1010

Bogani et al. (2005)

Dx00 ¼ 69 (1)

1.9  1012

Haas et al. (2014)

Tb

Dx00 ¼ 69

9.6  109

Bernot et al. (2006)

Ho

Dx0 ¼ 18 (2)

4.4  108

Bernot et al. (2009a,b)

Dx00 ¼ 34 (2)

2.6  1011

9.5

3.7

2

[Ln3Cu(hfac)11 (NITR)4]

PhOAll

Cu, Tb

—a

Not reported

PhOAll

4.02  10

21.4

9

Zhu et al. (2014b)

Tb

Not reported

2.58

48.90

9.69  1010

Li et al. (2016a)

[Ln(Hnic) (nic)2(NO3)]n (helix)

Dy

Not reported

0.46

DΑ ¼ 47.2

3  109

Mihalcea et al. (2016)

{Tb (a-fur)3(H2O)3}n

Tb

[Ln (hfac)3(NITT)]n

Ph2OEt

Other SCMs

YDy

3d–4f–4(5)d— SCM [{Cu2(valpn)2Ln} {M(CN)8}] nH2OmCH3CN

0.135

Not reported

9

DΒ ¼ 46.2

2.7  10

DΑ ¼ 0.87

7.6  109

DΒ ¼ 1.89

2.6  109

Bartolom
e et al. (2016)

JCuMo/ kB (K) Cu2Tb– Mo

Not reported

11.08

Visinescu et al. (2015)

M ¼ Mo

20.55

5.5  107

M¼W

15.1

1.5.5  107

The intramolecular coupling constant is given in 2J0 S1S2 formalism.

208 Handbook of Magnetic Materials

Jrr r

r

JrLn Ln

Ln

Ln

JLnLn FIG. 85 Magnetic interaction paths in Ln(hfac)3NITR. Based on Champion, G., Lalioti, N., Tangoulis, V., Arrio, M.A., Sainctavit, P., Villain, F., Caneschi, A., Gatteschi, D., Giorgetti, C., Baudelet, F., Verdaguer, M., Dit Moulin, C.C., 2003. XMCD for monitoring exchange interactions. The role of the Gd 4f and 5d orbitals in metal-nitronyl nitroxide magnetic chains. J. Am. Chem. Soc. 125, 8371–8376, https://doi.org/10.1021/ja034608u.

reverse direction with time, giving rise to SCM behavior in the finite length chain limit. A special class of magnetic linear chains is provided by the family of compounds with general formula Ln(hfac)3NITr, where hfac ¼ hexafluoroacetylacetonate, NITr ¼ 2-(r)-4,4,5,5,tetramethyl-4,5-dihydro-1Himidazolyl-1-oxy3-oxyde, with different radical r substitutions (see Table 13). These complexes form lanthanide–radical spin magnetic chains, with the NITR radical contributing s ¼ 1/2 and the Ln with S, alternating along the chain (see Fig. 85). The interaction intensity between radical–radical Jrr, radical–Ln JrLn, and Ln–Ln JLnLn depends strongly on the substituting radical. The radical provides an exchange pathway that enhances the average interaction within the chain giving rise even to long-range ordering at a nonzero temperature, as in Ln(hfac)3NITEt (Benelli et al., 1995). The transition metal-substituted NITR compounds have also provided paragon examples of SCMs (Coulon et al., 2014); therefore an exploration of Ln substitutions seemed pertinent. The compound Dy(hfac)3NIT(C6H4OPh) has received much attention since it provides an alternating chain with high Ln cation anisotropy. In this chain the single-ion strong anisotropy of the Dy ion leads to canting of the Dy spin moments with respect to the direction of the radical spins, yielding to a net magnetization along the chain. A careful magnetic anisotropy study on a single crystal, ab initio calculations on the Dy electronic-level scheme, and a classical chain model for the thermodynamic properties of the sample allowed to conclude that antiferromagnetic Dy–Dy n.n.n. interactions played a predominant role in explaining the so-called anisotropy inversion phenomenon, in which the easy and hard axes of the magnetization change as a function of increasing temperature (Bernot et al., 2009b). Far-infrared optical transition experiments in magnetic field allowed to determine experimentally the ligand field split low-lying electronic levels that did compare satisfactorily with ab initio calculated ones. Besides, weaker transitions caused by coupling to neighboring spins, and spin flip processes could be observed. The SCM-type activation energy expressed in terms of the Glauber model (see Section 2) fell below the excitation energy identified for this process in the

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 209

spectroscopic measurements, a fact attributable to shortcomings in the theoretical model (Haas et al., 2014). It is noteworthy to mention the interesting work on XMCD at the M4,5 and L2,3-edges on the Gd(hfac)3NITEt compound (Champion et al., 2003). This technique allowed to inspect the magnetism of the Gd atom independently of the radical. It could be concluded that the electronic density of Gd depends on its surrounding neighbors; however, the Gd orbitals are not involved in the exchange interaction between radicals. Molecular compounds with chiral optical properties have a long tradition in applied chemistry. More recently the magnetochiral effect (MChE) has attracted attention so as to design possible multifunctional materials. The observation of MChE is possible only when the parity reversal and timereversal symmetry are simultaneously broken. It is necessary to have high mixing between orbitals of different parities, allowed in noncentrosymmetric structures, and a strong magnetic anisotropy, which may be provided by Ln substitution. In particular the chiral [Ln(Hnic)(nic)2(NO3)]n polymer, that crystallizes in a helix structure, has been reported in detail (Mihalcea et al., 2016). Despite the absence of chiral ligands, spontaneous resolution of the two helicities occurs through crystallization in the enantiomeric P61 and P65 space groups. The Ln ¼ Tb-, Er-, and Dy-substituted compounds are isotypical compounds. In particular, the Ln ¼ Dy substitution is of interest in this context since only for this ion slow relaxation behavior was observed. The local anisotropy of the Ln sites was resolved by means of CTM aided by calculations with an electrostatic model. Curiously, the single-ion anisotropy for the Dy ion is planar, in contrast to most other cases (see Fig. 86A). It was interpreted in terms of the SCM model with weak ferromagnetic intrachain

FIG. 86 (A) Schematic representation of the spin helicity of the [Dy(Hnic)(nic)2(NO3)]n polymer; (B) temperature dependence of the wT product for the pure and diluted compounds. (Inset) Relaxation times for both compounds. Adapted with permission from Mihalcea, I., Perfetti, M., Pineider, F., Tesi, L., Mereacre, V., Wilhelm, F., Rogalev, A., Anson, C.E., Powell, A.K., Sessoli, R., 2016. Spin helicity in chiral lanthanide chains. Inorg. Chem. 55, 10068–10074, https://doi.org/10.1021/acs.inorgchem.6b01010. Copyright 2016 American Chemical Society. The SI conversion factor is 1 emu/mol ¼ 4p  106 m3/mol for molar susceptibility.

210 Handbook of Magnetic Materials

interaction and strong local anisotropy components in the activation energy. The diluted YDy substitution, where the intrachain interaction is weakened, has similar activation energy but an order of magnitude shorter, probably due to an increase in the number of available phonons which favor the relaxation process. A different class of SCM consists of systems where intrachain antiferromagnetic coupling is dominant. Such is the case for {Tb(a-fur)3}n polymer, which is isostructural to the Dy substitution previously discussed (see Section 8.1.2). The ligand field interaction on Tb with the distorted bicapped trigonal prism coordination also yields a ground state quasi-doublet and to a large gap to the first excited levels. However, Tb is a non-Kramers doublet and there is a zero-field splitting energy between the two states of the quasidoublet. No long-range order feature could be detected either by heat capacity or by AC magnetic susceptibility measurements (down to 15 mK). The intrachain interaction is Jc/kB ¼  0.135 K, and interchain interaction is negligibly small. The presence of defects within the chains breaks them into finite sections, inhibiting further the establishment of long-range order (Bartolom
e et al., 2016). Indeed, below 0.1 K, the static and AC susceptibility reflect the presence of defects in the chains. The slow relaxation of magnetization observed in temperature-dependent measurements could be assigned to two different SCM behavior in two different types of AF chains (A and B), characterized by different surrounding ligands in the second sphere of coordination. The slow relaxation is triggered by the existence of defects breaking the chains into finite length chains which, if containing an odd number of spins, may bear an uncompensated moment (see in Fig. 87A). As a consequence, there is a nonzero parallel susceptibility contribution proportional to the concentration of defects (2–4 at.%), which induces finite size effects on the static susceptibility. The ln(wT) vs T1 plot shows a nonlinear dependence with T1, as expected in an antiferromagnetic SCM. The relaxation time temperature dependence could be explained in terms of an Arrhenius relation caused by SCM relaxation, with an activation energy consisting of an intrachain interaction energy term (wall energy) Dx and the single-ion anisotropy term DA,B, where A and B correspond to the two types of chains in this compound (see Table 13). In this case, zero-field splitting due to single-ion anisotropy and SCM excitations has precluded the onset of 3D long-range ordering, in contrast to the Dy substitution. At temperatures below 0.1 K the SCM relaxation crosses over to individual relaxation of the Dy moments through the direct process (Fig. 87B).

8.1.4 1D Cluster Compounds This class of 1D molecular systems consists of homo- or heteronuclear clusters that are linked along a 1D topological dimension. Among the first members of ferromagnetic molecular materials discovered were some spin-ladder

FIG. 87 (A) Schematics of the two types of chains in {Tb(a-fur)3}n polymer, broken into segments with even and odd number of spins. (B) Relaxation times of the two processes determined at H ¼ 0. Reprinted with permission from Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta, C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050, https://doi.org/10.1039/C6TC00919K. Published by The Royal Society of Chemistry.

212 Handbook of Magnetic Materials

FIG. 88 Ladder-like structure of L2[M(opba)]3xDMSOyH2O: (A) projection of the ladders in the ac plane; (B) projection in the bc plane. Reprinted figures with permission from Evangelisti, M., Kahn, M.L., Bartolom
e, J., de Jongh, L.J., Meyers, C., Leandri, J., Leroyer, Y., Mathoniere, C., 2003. Magnetic and thermal properties of 4f-3d ladder-type molecular compounds. Phys. Rev. B 68, 184405, https://doi.org/10.1103/PhysRevB.68.184405. Copyright 2003 by the American Physical Society.

types of compounds with shorthand notation Ln2Cu3, Ln ¼ Gd, Tb, Dy, and Ho. The crystal structure consists of discrete, infinite ladder-like motifs, with Ln occupying the nodes and Cu located between the L atoms (see Fig. 88). These 1D ladders give rise to short-range magnetic ordering at high temperature, with ferromagnetic Ln–Cu interaction. However, depending on the interladder magnetic interaction, the low-temperature behavior may be of different characters. In the case of Gd2(ox)[Cu(pba)]3[Cu(H2O)]20H2O long-range order sets on at Tc ¼ 1.05 K, because of the interladder coupling via the oxalate bridges that associate adjacent ladder into double honeycomb layers. As a consequence, above Tc the short-range order is 2D (Evangelisti et al., 1999). In the similar ladder complexes Ln2[M(opba)]3xDMSOyH2O, where M ¼ Cu and Ln ¼ Gd, Tb, and Dy, the interladder interactions are much weaker, and therefore their magnetic behavior as 1D chains is maintained till interladder interaction causes long-range ordering at Tc ¼ 1.78 K (Bartolom
e et al., 1995), 0.81 K and 0.75 K (Evangelisti et al., 2001), respectively. Combining very low temperature magnetic susceptibility and heat capacity measurements, it could be proven that the Ln–Cu intraladder interaction is ferromagnetic in the three substitutions; however, the long-range ordering is most probably of antiferromagnetic character for Tb and Dy, while it is quite probably ferromagnetic for Ln ¼ Gd (Evangelisti et al., 2003). In all cases the Cu atoms intercalated in the sides and connecting the L atoms in the rungs act as effective exchange pathways, since when substituted by Zn the Ln–Ln interaction is inhibited. These compounds merit being mentioned since they provide the ladder-like ferromagnetic molecular magnets with the highest ordering temperature (Table 14).

TABLE 14 1D Chains Formed by Polynuclear Units, Activation Energy Ueff, Prefactor τ 0, and long-range ordering critical temperature Tc Formula

Cluster

Interaction

Ln2(ox)[Cu(pba)]3[Cu(H20)5]20H2O

Gd–Cu

Ln2[Mopba]3xDMSOyH2O

[Dy4(m3-OH)2(L)10(bipy)2(H2O)2]n

Ueff/ kB (K)

Tc (K)

References

FM

1.05

Evangelisti et al. (1999)

Gd–Cu

FM

1.78

Tb–Cu

FM

0.81

Bartolom
e et al. (1995)

Dy–Cu

AF

Dy4

FM

0.75 23.6

{[Dy10(m3-OH)8(L)22(bipy)2(H2O)2]5H2O}n

Dy10

Unknown

[Gd(NO3)3(H2O)4][Cr4Gd(hdpta)2(OH)4(H2O)5] (NO3)32H2O

Gd–Cr4

FM

[Mn2Ln2 (OH)(OMe) (hmp)4(NO3)4(O3SC6H4CH3)2]nMeCNMeOHH2O

Dy2Mn2

Unknown

—a

[Ln2Cu6(ipO)6(H2O)12]

Dy2-Cu6

FM

63.68

[(Hphen)2.5[Ln0.5(phen)(H2O)][MoIV(CN)8] 1.5CH3CN

DyMo2

Unknown

—a

AF

14

[{(NiL)2Ln(NO3)3}pyz]n1.5nH2O a

Reported slow relaxation below 2 K.

τ 0 (s)

3.2

6

2.57  10

6

1.32  10

Liu et al. (2016e) Langley et al. (2014a) 3.77  106

Wang et al. (2015a) Zhou et al. (2016a)

ErMo2 Ni–Dy– Ni

Wu et al. (2014)

9.9  109

Ghosh et al. (2016)

214 Handbook of Magnetic Materials

The trinuclear [Ni(II)L]2Dy cores linked by pyrazine ligands form a zigzag chain in the compounds [{(NiL)2Ln(NO3)3}pyz]n1.5nH2O, where Ln ¼ Gd, Tb, and Dy. While the cores are chemically unstable, the pyrazine stabilizes the chain compounds. The Dy compound shows SMM behavior at an applied external field, apparently dominated by the Dy single-ion anisotropy (Ghosh et al., 2016). In Wu et al. (2014) two series of linear chains formed by polynuclear Dy4 and Dy10 clusters are described, where the clusters are linked by two bridging carboxylate groups in the Z1:Z1:Z2 mode. In the Dy4 compound ferromagnetic coupling within the chain is present, while it is too weak in the Dy10 to be detected. The relaxation time in both compounds obeys an Arrhenius law, probably dominated by the SMM behavior of the clusters. Two more 1D cluster compounds, the {Mn2Dy2} clusters linked by two sulfonate bridges, and the {LnMo2} clusters that originate from helical chains by prolonging the diffusion reaction time (Langley et al., 2014a), have been recently reported, showing relaxation behavior in an applied field at low temperatures. However, they do not show the telltale maximum in the temperature-dependent AC susceptibility that would allow deriving an activation energy from the experiment (Fig. 89B). Nevertheless, they are interesting from the structural point of view. The rare heterotrimetallic compounds [{Cu2(valpn)2Ln}{M(CN)8}] nH2OmCH3CN, M ¼ Mo, W, resulting from the association of the trinuclear {CoII2LnIII} moiety and the {Mo/WV(CN8)} anion (M with S ¼ 1/2 spin), have been reported to behave as a 3d–4f–4(5)d heterotrimetallic molecule linked as a chain and presenting SCM behavior for the Ln ¼ Tb substitution (Visinescu et al., 2015).

FIG. 89 (A) (Hphen)2.5[Ln0.5(phen)(H2O)][MoIV(CN)8]1.5CH3CN; (B) in-phase and out-ofphase components of the AC susceptibility at m0H ¼ 600 Oe. Adapted with permission from Zhou, H., Chen, Q., Zhou, H., Yang, X., Song, Y., Yuan, A., 2016a. Structural conversion and magnetic studies of low-dimensional LnIII/MoV/IV(CN)8 (Ln ¼ Gd-Lu) systems: from helical chain to trinuclear cluster. Cryst. Growth Des. 16, 1708–1716, https://doi.org/10.1021/acs.cgd.5b01782. Copyright 2016 American Chemical Society. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

8.2

1 215

Higher Dimensionality

This section reviews on 2D and 3D magnetic systems exhibiting relaxation processes that have been reported in the last years. Most of these systems are MOF compounds, formed by metal ions or clusters coordinated to organic ligands, that form 2D or 3D structures. They may consist of interpenetrating lattices, with porous channels or voids in some cases, or coordination polymer (CD) chains coupled in 2D or 3D networks. Lanthanide–organic frameworks are very interesting for gas storage applications and as magnetoluminiscent materials (Roy et al., 2014). From the magnetic relaxation point of view, they belong mostly to the SIM class of relaxation behavior. For this reason, Table 15 for 2D- and Table 16 for 3D-coordinated polymers and MOFS have been generated, including atomic coordination, activation energies, and relaxation times, when reported. In most cases, only the Dy substitution, with its large single-ion anisotropy, gives rise to slow relaxation; however, some peculiar cases with Er (Zeng et al., 2016), Ho (Pinkowicz et al., 2014), Tb (Zhu et al., 2015), or Yb (Zeng et al., 2016) have been studied. In all cases the SIM behavior dominates apparently the magnetic dynamic response.

8.2.1 2D Compounds The Dy-layered hydroxide compound Dy8(OH)20Cl46H2O (or LDyH) (Monteiro et al., 2013), its diluted isomorph LYH:0.04Dy, and the interstitial LDyH-2.6-NDC (Monteiro et al., 2015) conform a very interesting set of compounds since each one belongs to a different dimensionality in magnetic behavior. Thus, LDyH is formed by 3D coupled layers, in the interstitial LDyH-2.6-NDC the layers are further separated by the deprotonated 2,6naphthalene carboxylate groups and the 2D is approached (see Fig. 90, upper panel), while the dilution with nonmagnetic Y renders the compound as 0D. In the 3D and 2D cases, the magnetic relaxation is Arrhenius like, with a single, average relaxation time at H ¼ 0 (see Table 15). It is argued that ferromagnetic coupling within the layers contributes to the magnetic anisotropy barrier of the Dy ion, being much lower in the LDyH-2.6-NDC compound since the interlayer distance is much larger. The diluted case is different: for H ¼ 0 just one relaxation process is observed (see Table 15), while for H ¼ 1 kOe, two relaxation times could be resolved, one caused by the Dy1 ion, with z ¼ 7, dodecahedral coordination (Ueff/kB ¼ 56 K, τ0 ¼ 2.79  1010 s), and the other by the Dy2 and Dy3 ions with z ¼ 9 monocapped square antiprism coordination (Ueff/kB ¼ 105 K, τ0 ¼ 4.6  1010 s) (Fig. 90, lower panel). These relaxations have been explained as due to single-ion Orbach processes that were alternatively interpreted in terms of a Raman process (see Fig. 82), with a temperature dependence given by Eq. (81). The polymorphic layered lanthanide phosphonates Ln(2-qpH)(SO4)(H2O)2 (Zeng et al., 2015, 2016) have a layered structure constituted by {Dy2O2}

TABLE 15 Recently Reported Lanthanide 2D MOFS Ln

Coordination

MOF grid

Ueff/ kB (K)

τ 0 (s)

H (kOe)

{[Ln(NNO)(glu)]0.25H2O}n

Dy

DSAP

2D-CP

44.2

2.3  108

2

[Ln(Hdppd)(H2O)3H2O]n

Dy

DBTP

2D-CP

—a

Ln(2-qpH)(SO4)(H2O)2

aDy

DD

2D-dimeric

32

Formula of the Polymer

References

Ln 2D MOFS

You et al. (2014) 7.5  107

2

8

bDy

83.0

1.3  10

bYDy

89.5

2.910-9

bEr

17.8

1.9  106

bYb

Liu et al. (2015b)

Zeng et al. (2016)

6

11.2

8.6  10

Zeng et al. (2015)

{([Ln2L3(H2O)2]n2nCH3OH) 2nH2O}

Dy

DTTP

2D–CP

19.2

2.5  106

0

Biswas et al. (2015)

[Dy2L2(H2O)5]n

Dy

DSAP

2D–2D entangled

12.5

6  107

2

Liu et al. (2014b)

{[Dy2(HCAM)3(H2O)4] 2H2O}n

Dy

DTTP

(4,4) ferro

63.5

5  109

2

Liu et al. (2015a)

DSAP

57.1

10

2.1  10

[{Ln2(dae)3(DMSO)3(MeOH)} 10MeOH]n

Dy8(OH)20Cl46H2O (or LDyH)

Dy

14.1

1.8  108

1.5

Pinkowicz et al. (2014)

9

DyUV

14.2

7.25  10

Dyvis

14.1

7.25  109

Ho

14.4

7.49  108

DMSA

36.1

1.21  1010

0

Monteiro et al. (2013)

DD

31

6.84  109

0

11.63

3.86  106

0

Monteiro et al. (2015)

22.63

9.35  108

3

11.89

1.14  107

Dy

LYH:0.04Dy LDyH-2.6-NDC Ln–2p–3d 2D compound {Tb(hfac)3[Cu (hfac)2]3(NITPhPyrim)2} 2CH2Cl2

Tb

DD

Tb–Cu– rad.

Zhu et al. (2015)

The coordination type, Arrhenius-type activation energy Ueff, prefactor τ0, and field applied in the AC susceptibility measurement H are included. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

TABLE 16 Recently Reported Lanthanide 3D MOFs Ln

Coordination

MOF Grid or Cage

Ueff/ kB (K)

τ 0 (s)

H (kOe)

References

[Ln(C4O4)(C2O2)0.5(H2O)2]n

Dy

DSAP

Cuboctahedral

10.8

2.0  106

1.8

Biswas et al. (2014a)

[Ln(phen)-(L)]n

Dy

DBP

(3,6)-rutile

131

1.6  106

0

Chen et al. (2014)

[Ln(vspc)(Hvspc)(H2O)]n

Dy

DD

(3,6)

1.82

2.11  106

0

Hou et al. (2014)

{[Dy(C2O4)1.5phen]0.5H2O}n

Dy

DSAP

3-connected (72.9)

35.5

3.1  1010

1.2

32.6

1.0  10

Liu et al. (2016b)

1.4  107

1

Liu et al. (2016a)

Formula of the Polymer Ln homometallic MOFs

[Ln(BTB)H2O]n

Dy

ETD

Cross-folding

17.8

{Ln(BCPBA)(H2O)}n

Dy

Z¼7

(41061583)

—a

9

Wei et al. (2014)

(4 6 ) 5

2

[Ln2(pam)3(DMF)2(H2O)2]nnDMF

Dy

DSAP

Dumbell

17.72

[Ln2(PIP)3(H2O)4] 2DMF3H2O

Dy

Z¼9

(412.63) pcu

—a

{[Dy2(1H-5-Cl-6-Opy-3-CO2)2 (OX)2(HO)]2H2O}n

Dy

DMSA

(32.52)

37.6

PB

5

(3.5 )

9.75  107

3.5

Biswas et al. (2014b) Li and Du (2015)

5.5  106

2

Liu et al. (2015c)

{[Dy2(PA)3(H2O)(DMF)] (DMF)2(H2O)}n

Dy

Dy2(INO)4(NO3)22 solvent (Dy2-solvent)

Dy

DTD

(412.63)

2.16

8.8  106

0

Zhang et al. (2015a)

0

Zhang et al. (2015b)

0

Biswas et al. (2016b)

DBTP MSAP

Rectangle

(Dy2-DMF)

—a

(Dy2-CH3CN)

110

3.4  101

(Dy2-A) no solvent

97

1.81  1011

Rectangle

—a

Z¼8

HTA

—a

Dy

DSAP-

Lantern

13.6

8.82  107

0

Ren et al. (2014)

Dy

DSAP

Dy2-Mn CP (42.82)

63.3

2.89  109

0

Sun et al. (2014b)

Dy

DTTP

Ln5-K

5.69

6.53  106

2

[Ln5(m3-OH)5(m3-O) (CO3)2(HCO2)2-(C4O4)(H2O)2]

Dy

[Ln7(DPA)5(NA)3(m3OH)8(H2O)3]2.5H2O

Dy

[Na4Ln12(stp)8(OH)16(H2O)12] 10H2O

DTP& BTP

Hu et al. (2015a)

Ln–3d heterometallic MOFs [Ln2Mn(Hbidc)2(SO4)28H2O]n

[K5Ln5(pztc)5(H2O)19]7H2O

Yb

[LnCo1.5(L)2(H2O)5]n

Dy

[Ln2Cu3(2,3-pydc)6(H2O)10] 8H2O

Dy

(4 6 8) (456)2

5.78

DTTP

[DyO2Co]–Co CP

—a

DSAP

(482)2

83.7

15

12

(4 8 10) 2

11

6

1.71  10

Zhang et al. (2014a) Zhao et al. (2015b)

4  109

2

Zhang et al. (2016a)

Continued

TABLE 16 Recently Reported Lanthanide 3D MOFs—cont’d Formula of the Polymer

Ln

Coordination

MOF Grid or Cage

Ueff/ kB (K)

τ 0 (s)

H (kOe)

References

Dy

DBTP

Ln–rad.

15

1.25  106

3

Li et al. (2015c)

Dy2 motif

—a

Ln–radical 3D compound [Ln(hfac)3(NITFumbis)]2

DTD Double-chain MOF [Ln(TZI)(H2O)4] 3H2O

Dy

Z¼8

Zhang et al. (2016f )

The coordination type, MOF description with Schl€afli symbol (when available), activation energy Ueff, prefactor τ0, and field applied in the AC susceptibility measurement, H, are included. 2D-CP, 2D coordination polymers; 2D-Dim, 2D dimeric; BTP, biaugmented trigonal prism; CTP, capped trigonal prism; DBTP, distorted bicapped trigonal prism; DD, Z ¼ 8 distorted dodecahedral; DL, dimeric ladder; DMSA, distorted monocapped square antiprism; DPB, distorted pentagonal–bipyramidal; DSAP, distorted square antiprism; DTD, distorted triangular dodecahedron; DTTP, distorted tricapped trigonalprismatic; ETB, elongated triangular bipyramid; HTA, heptanuclear trigonal antiprismatic; MSAP, Z ¼ 9 monocapped square antiprism; PB, pentagonal bipiramidal; SAP-BTP, intermediate square antiprism, and bicapped trigonal prism. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength. a Reported slow relaxation below 2 K.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 221

Ln3 Ln2 Ln1

c

b

a

a

c a

A

B

LDyH

−4

4.5

−6

4.0

−12 −14

(−1/9)

−10

3.5

t

−8 Ln(t)

LDyH-2,6-NDC

3.0

Ln(t) HDC = 0 G

2.5

Ln(t1) HDC = 1000 G

2.0

t (−1/9) HDC = 0 G t1(−1/9) HDC = 1000 G t2(−1/9) HDC = 1000 G

Ln(t2) HDC = 1000 G

−16 0.0

0.1

0.2

0.3

0.4

1/T (K−1)

0.5

0.6

2

4

6

8

10

T (K)

FIG. 90 Top: Ln8(OH)20Cl4nH2O crystal structure viewed along the c-axis (left) and b-axis (center) and the intercalated compound LDyH-2,6-NDC (right) (Monteiro et al., 2015). Bottom: (A) Temperature-dependent relaxation time interpreted in terms of Orbach process. (B) The same data interpreted in terms of a Raman process. Reprinted with permission from Monteiro, B., Coutinho, J.T., Pereira, C.C.L., Pereira, L.C.J., Marc¸alo, J., Almeida, M., Baldovı´, J.J., Coronado, E., Gaita-Arin˜o, A., 2015. Magnetic properties of the layered lanthanide hydroxide series Yx Dy8-x(OH)20Cl46H2O: from single ion magnets to 2D and 3D interaction effects. Inorg. Chem. 54, 1949–1957, https://doi.org/10.1021/ic502839c. Copyright 2015 American Chemical Society.

dimers and phosphonate ligands, while quinoline groups fill the interlayer spaces. Depending on the arrangement of the dimers within the layers two structures exist: in the a-Ln phase the dimers are arranged in a zigzag form along the c-axis, while in the b-Ln the dimers are arranged in parallel within the layer. The magnetic relaxation of the Ln ion depends on the actual structure. For the Ln ¼ Dy substitution both structures show slow magnetic relaxation of the Orbach type; however, Ueff in the b-Dy phase is more than double that of the a-Dy one (Zeng et al., 2015), most likely due to different single-ion anisotropy produced by the slightly different coordination and relative arrangement of the molecules within the layers. This statement is confirmed by the diluted b-YDy, which has practically the same activation energy, albeit with a prefactor τ0 one order of magnitude shorter. Curiously, none of the Ln ¼ Gd, Tb, Ho, Er substitutions in the a-Ln type have observable slow

222 Handbook of Magnetic Materials

relaxation. In the b-Ln, Ln ¼ Gd, Tb, Ho, Er, and Yb, only the Er and Yb substitutions show slow relaxation in an applied field of 2 kOe. The absence of slow relaxation in the non-Kramers Ln ¼ Tb and Ho is probably related to the more efficient tunneling process in these two substitutions, in comparison to the Kramers cases Dy, Er, and Yb, where tunneling is theoretically forbidden or, at least, in real systems, is less efficient than for the non-Kramers cases. Besides, the presence of slow relaxation in b-Er and b-Yb is justified by the parallel arrangement of the easy anisotropy axes in those two compounds that favor ferromagnetic coupling and a consequential increase in the Ln anisotropy barrier to magnetic moment reversal. The [{Ln2(dae)3(DMSO)3(MeOH)}10MeOH]n compound crystal structure consists of 2D layers of Ln2 dimeric units connected by the dae2 single or double bridges (see Fig. 91A; Pinkowicz et al., 2014). The magnetic relaxation of the Dy substitution obeys a high-temperature Arrhenius behavior that tends to a smaller slope at lower temperatures, as expected in a SIMdominated behavior (see Fig. 91B). The irradiation with light provokes photochromic response that affects (slightly) the dynamic response. Indeed, with UV or visible irradiation the activation energy remains constant, but the prefactor decreases strongly. It appeared that the SIM behavior is affected by the photoisomerization of the ligand; thus small changes in the coordination sphere of the Dy ion may modify the crystal anisotropy or modify the weak magnetic interactions within the dimer. Finally, for completeness sake, we mention the only 2D MOF, including a NITr (s ¼ 1/2) radical, reported. Namely, {Tb(hfac)3[Cu(hfac)2]3 (NITPhPyrim)2}2CH2Cl2 provides an interesting 2D compound, with A

B 1a

−8

dae

double dae

Dy2

In t

sin

gle

1a-vis −9

1a-UV

−10 0.3

0.4

0.5

T −1 / K−1

FIG. 91 (A) 2D structural diagram showing a dae2-bridged coordination layer of [{Ln2(dae)3(DMSO)3(MeOH)}10MeOH]n. The Dy2 nodes are represented by circles. (B) Dependence of the relaxation time as a function of irradiation with UV or visible light. Reprinted with permission from Pinkowicz, D., Ren, M., Zheng, L.M., Sato, S., Hasegawa, M., Morimoto, M., Irie, M., Breedlove, B.K., Cosquer, G., Katoh, K., Yamashita, M., 2014. Control of the single-molecule magnet behavior of lanthanide-diarylethene photochromic assemblies by irradiation with light. Chem. A Eur. J. 20, 12502–12513, https://doi.org/10.1002/chem. 201402647. Copyright 2014 John Wiley & Sons, Inc.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 223

ferromagnetic interaction between the radical and the metal atoms (Zhu et al., 2015). In spite of this interaction, the SIM behavior dominates in this case.

8.2.2 3D Compounds It is interesting to note that among the 3D MOF compounds, there is no compound where a magnetic long-range order transition has been reported above 1.8 K. The obvious reason is that the connecting ligands, H-bonds or ions creating the network, couple weakly the lanthanide ions. The main magnetic coupling mechanism in these families is dipolar interaction which may manifest below 1 K, while most studies are limited in temperature. A singular case is the 3D network formed by dodecanuclear Dy clusters [Na4Ln12(stp)8(OH)16(H2O)12]10H2O (Ren et al., 2014). Its structure is formed by Dy12 clusters, which can be described as four vertex connected Dy cubane-like units, forming a “lantern”-shaped cluster. The Dy12 clusters are connected along the c-axis by two m2-oxigen atoms forming chains. The stp3 ligand connects the chains into a 3D network (see Fig. 92). Although one could expect a total super-spin cluster behavior, in fact the magnetic relaxation is actually dominated instead by a single-ion anisotropy-driven Orbach process. The weak magnetic interactions within the cluster actually only seem to break down the conditions for magnetic quantum tunneling, thus favoring the SIM behavior. A rich variety of different relaxation phenomena are provided by the 3D network of linked Dy2 dimers, Dy2(INO)4(NO3)22 solvent, where the different solvents encaged in the MOF voids (see Fig. 93) give rise to modified relaxation processes (Zhang et al., 2015b). So, in the (Dy2-DMF) magnetic A

B

Dy4C Dy4B O16B

O16C Dy3C

Dy3B

O15C O15B

O18B

O18C Dy1 O17

Dy2

Na2E

O19 Na2D

O17A

O19A

Dy2A

Dy1A O18A

O18 Dy3

O16

O15

O15A

Dy4

Dy4A

Dy3A O16A

FIG. 92 (A) “Lantern” cage formed by two Na ions and four cubane units linked, sharing vertex; (B) 3D network viewed along the c-axis. The Dy12 cluster is depicted as the blue ball, and the stp3 ligands as pink balls. Reprinted with permission from Ren, Y.X., Zheng, X.J., Li, L.C., Yuan, D.Q., An, M., Jin, L.P., 2014. Three-dimensional frameworks based on dodecanuclear Dy-hydroxo wheel cluster with slow relaxation of magnetization. Inorg. Chem. 53, 12234–12236, https://doi.org/10.1021/ic502042h. Copyright 2014 American Chemical Society.

224 Handbook of Magnetic Materials

FIG. 93 Crystal structures of the 3D MOFS (Dy2-DMF) and (Dy2-CH3CN). The nodes are constituted by the Dy2 dimers. Reprinted with permission from Zhang, X., Vieru, V., Feng, X., Liu, J.L., Zhang, Z., Na, B., Shi, W., Wang, B.W., Powell, A.K., Chibotaru, L.F., Gao, S., Cheng, P., Long, J.R., 2015b. Influence of guest exchange on the magnetization dynamics of dilanthanide single-moleculemagnet nodes within a metal-organic framework. Angew. Chem. Int. Ed. 54, 9861–9865, https://doi. org/10.1002/anie.201503636. Copyright 2015 John Wiley & Sons, Inc.

relaxation is described in terms of a low-temperature direct process, while in the (Dy2-CH3CN) an Arrhenius law, ascribed to Orbach process via excited electronic levels of the ligand field split level scheme, is clearly dominant at higher temperature. The Dy ions are ferromagnetically coupled within the dimer, while no hint of dimer–dimer interaction was observed. The barrier is quite high (Ueff/kB 109 K), since there seems to be an effective ferromagnetic coupling within the dimer. The case (Dy2-A) in which no solvent was included has the same behavior as the latter case, albeit with a lower barrier. Therefore, the main differences causing the modification in magnetic dynamics can be associated to the guest ligand (or absence of guest ligand) in the MOF that affects the dipole interaction which modifies the Dy tunneling gap; indeed, it has been calculated that the tunneling energy gap is DT 1.4  102 K and 1.4  103 K for the (Dy2-DMF), and (Dy2-CH3CN) or (Dy2-A), respectively. Since the incoherent tunneling relaxation time τQTM is proportional to D2 T (Prokof’ev and Stamp, 1998), it will be two orders of magnitude larger for (Dy2-CH3CN) or (Dy2-A) than for (Dy2-DMF). Therefore, magnetic quantum tunneling pathway is less effective in (Dy2-CH3CN) or (Dy2-A) and the slow relaxation characterized by the Orbach process is observed neatly in those two cases. A different class of 3D networks with lanthanide ions includes a 3d transition metal compound in the molecule. In particular the [Dy2Mn (Hbidc)2(SO4)28H2O]n structure is constructed by Dy2 dimers coupled by coordination polyhedral of Mn(II) (see Fig. 94; Sun et al., 2014b). The two Dy ions in the dimer are magnetically coupled, and as in the previously described compound, it shows slow relaxation with a high activation energy that is related to the barrier to dimer total spin reversal. However, the Mn ions do not seem to play any role in the relaxation process.

FIG. 94 (A) 3D network of [Dy2Mn(Hbidc)2(SO4)28H2O]n, where the Mn(II) coordination octahedral linkers are clearly shown; (B) w00 (T) showing its frequency dependence. Reprinted with permission from Sun, Y.G., Zong, W.H., Xiong, G., Guo, M.Y., Ding, F., Wang, S.J., You, L.X., Ren, B.Y., Xu, Z.H., Gao, E.J., 2014b. Synthesis, structure, photoluminescence and magnetism of 3d-4f heterometallic coordination polymers bearing benzimidazole-5,6-dicarboxylate, Polyhedron 83, 68–76, https://doi.org/10.1016/j.poly.2014.04.029. Copyright 2014 Elsevier Ltd.

226 Handbook of Magnetic Materials

As mentioned earlier in this section for the 1D and 2D systems, magnetic radicals with s ¼ 1/2 may also be involved in the construction of 3D networks. In the molecule [Ln(hfac)3(NITFumbis)]2 the Ln(III) ion and the directly bonding bis-nitronyl nitroxide radical based on furan ring (NIT-Fumbis) interact antiferromagnetically (Li et al., 2015c). The radical–radical interaction is also antiferromagnetic (Jrr/kB ¼  36.8 K). Only the Dy substitution shows field-induced slow relaxation of the Orbach type. Thus, magnetic interactions in this case promote slow relaxation, as we have found all along this section.

8.3 Conclusions As conclusion, in 1D, 2D, and 3D network systems above 2 K the magnetic relaxation is essentially dominated by the single-ion anisotropy of the lanthanide ion. Very frequently, the Dy compounds show that the slow relaxation time at high temperature is described by an Arrhenius law that has been interpreted to as an Orbach process, or as a power-law assigned to a Raman process. At very low temperature direct processes enter into play. To observe this slow relaxation by means of AC susceptibility experiments, one needs an internal or external field that breaks down the magnetic quantum tunneling pathway. Indeed, in its absence the QTM process has a relaxation time too small to fall within the SQUID or PPMS frequency window, and as a consequence no relaxation is observed down to 1.2 K. This simple description is incomplete when temperature is decreased below that temperature in dilution refrigerators. Then the kinetic energy becomes so small that weak interactions are competitive and a crossover to different processes, like SCM or critical slowing down near the long-range ordering temperature, that become observable, thus opening a wider panorama to relaxation processes in molecular magnets.

9 MOLECULAR MAGNETS ON SURFACES Many proposals of applications of molecular magnets imply miniaturization and, in most cases, preparation of nano- or microstructures onto a surface. The deposition of molecules on a surface, with appropriate structuring, should allow to act on the molecule, somehow writing and reading its state, etc. The nature of the different applications implies very different approaches to the material’s science problem of deposition and structuring. Some applications require a thick film, as in magnetic cooling, while others, such as spin valves, molecular transistors, or qubits, do involve a single layer or even a single molecule per device. Although molecular magnetism has been maturing for more than two decades, the inclusion of lanthanide ions in mono- and polynuclear molecules and, even more, the development of surface-deposited materials are clearly in the beginning of a rapidly growing branch, as evidenced in this work and previous revisions of the subject. In Table 17, we have reunited part

TABLE 17 Recently Ln-Based SMMs Sublimated on Surfaces and Nanoparticles Complex

Surface

Measurement

TB (K) and Ueff/kB

H (kOe)

τ (s)

References

LnPc2—double deckers TbPc2

Cu(100)

Single molecules—1 monolayer

STM, dI/dV, XMCD

Stepanow et al. (2010)

TbPc2

Cu(111)

Single molecules—1 monolayer

STM, dI/dV

Vitali et al. (2008)

TbPc2, TbNPcPc, YPc2, Y2Pc3

Au(111)

Single molecules —1 monolayer

STM, dI/dV

Komeda et al. (2013)

Submonolayer

XAS, XMCD

2K

0–50

—

Malavolti et al. (2013b)

Submonolayer

XAS, XMCD

8K

0–15

—

Klar et al. (2013)

Submonolayer and monolayer

STM, XNLD, XMCD

Up to 100 K

 103 s

Lodi Rizzini et al. (2014)

Submonolayer

XAS, XMCD

5K

TbPc2

Co/Cu(100)

Komeda (2014)

LSMO/STO TbPc2

Co/Cu(100) Ni/Cu(100)

MnPc, TbPc2, Tb2Pc3

Co/Cu(100) CoO/Ag(100)

TbPc2, DyPc2, ErPc2

Ni(111)

8T

Candini et al. (2016)

Continued

TABLE 17 Recently Ln-Based SMMs Sublimated on Surfaces and Nanoparticles—cont’d Complex

Surface

TbPc2

Fe/Cu(100)

Measurement

TB (K) and Ueff/kB

H (kOe)

Submonolayer

STM, XMCD

5–100 K

5T

τ (s)

References Nistor et al. (2015)

FeMn/Cu (100) Li + Fe/Cu (100) Hexyl-6-pyrene-substitutedTbPc2

CNT

Single molecules by drop casting

Single-electron tunneling

20 mK

B jj 14 kOe B? 5 kOe

250 mT/s

Ganzhorn et al. (2013)

Hexyl-6-pyrene-substitutedTbPc2

CNT

Single molecules by drop casting

Magnetoconductance

150 mK

B jj 14 kOe B?5 kOe

 100 mT/s

Urdampilleta et al. (2013)

Hexil-6-pyrene-substitutedTbPc2

CNT

Single molecules by drop casting

Magnetoconductance

40 mK

12 kOe transverse

20 mT/s

Urdampilleta et al. (2015)

Single TbPc2 molecule

Au electromigrated terminals

Supported molecule

Transport measurements

150 mK

B jj up to 600 Oe

13 s

Thiele et al. (2013)

Spin transistor

25 s 17 s

Transport measurements Spin transistor

34 s

Thiele et al. (2014)

TbPc2

TbPc2

SiO2/Si(111)

AFM, MOKE, NEXAFS, XMCD

8K

Up to 70 kOe

Open hyst. on PTCDA and SiO

Robaschik et al. (2015)

PTCDA/SiO2/ Si(111)

Multilayer

Co/SiO2/Si(111)

8–87 nm

Cu(111), Fe/Cu (111)

Monolayer

STM, XMCD

Moreno Pineda et al. (2016)

Lumetti et al. (2016)

Mn/Ag(100) CNT Graphene HOPG MgO TbPc2

Notched electroburnt Gr

Single molecules by drop casting

I/V curves

NdPc2

Au(111), Cu(111), Fe/Ir (110)

Submonolayer

STM, SP-STM

5K

Fahrendorf et al. (2013, 2014)

DyPc2

Au(111)

Sub- to 1 monolayer

STM dI/dV

4.2 K

Zhang et al. (2015d)

TbPc2(OC11H21)8

Chemically grafted Si

1 Monolayer

XNLD, XMCD

Open hysteresis T < 10 K

TbPc2

SiO2, PTCDA

20, 50 nm

MFM, XMCD

Enhanced Opening 10

500 Oe/s

40

Mannini et al. (2014) Serri et al. (2017)

Continued

TABLE 17 Recently Ln-Based SMMs Sublimated on Surfaces and Nanoparticles—cont’d Complex

Surface

Measurement

TbPc2

MgO/Ag(100) Ag(100)

1 Monolayer, multilayer

XMCD

TbPc2, LuPc2

Au NPs

Decoration

AC susceptibility

TbPc2

HOPG

Submonolayer

XMCD

τ (s)

References

“Giant” hysteresis

5

840

W€ackerlin et al. (2016)

Open hysteresis

5

10 Oe/s

Noda et al. (2014)

OH @ 2 K

HOPG

Monolayer (from drop casting)

STM

Poly(methyl methacrylate) (PMMA)

Thick layer

MCD, SQUID

1.8 K

Au, HOPG

Submonolayer

XAS, XMCD

1.8 K

Gr @ Ir(111)

1–10 monolayers

STM, XAS, XMCD

Monolayer

STM, XAS, XMCD

Tb(III)-bisdodecylporphyrin TbPc2, YPc2

H (kOe)

OH @ 0.5 K

DyPc2 Tb(III)-Octaethylporphyrin

TB (K) and Ueff/kB

2880 s

Klar et al. (2014) Inose et al. (2014) Malavolti et al. (2013a)

LnLn0 Pc3—triple deckers Octahexyl-substituted Pc-TbDyPc3

—

Open hysteresis on HOPG

Lan et al. (2015)

Ln(trensal) Er(trensal)

Gr @ Ru(0001)

Dreiser et al. (2016)

Ru(0001) Er(trensal)

Ni/Cu(111) Au(111)

No hysteresis Weak-AF Ni–Er

0–60

—

Dreiser et al. (2014a)

Endohedral cages Dy2ScN@C80

Rh(111)

Multilayer and submonolayer

τ  30s

XAS and XMCD Dy M4,5

4 K: hysteresis

Westerstr€ om et al. (2015)

AFM + STM

No hysteresis

0–60

Corradini et al. (2014)

No hysteresis

0–90

Lorusso et al. (2013)

No hysteresis

0–50

Roubeau et al. (2017)

1.5, Ueff ¼ 167

0

Magnetic cooling molecules [Gd4Ni8(OH)8(hmp)8(O2CEt)8

Au(111), HOPG

(MeOH)6][ClO4]4

XAS, XMCD, XPS

[Gd4Zn8 (OH)8(hmp)8 (O2CiPr)8][ClO4]4 [Gd2(CH3COO)6(H2O)4]4H2O

OHfunctionalized Si(110)

[Gd2(CH3COO)6(H2O)4]

MW-CNTs

AFM–MFM

4H2O Other molecules 8.5  106

[Dy(hfac)3(PyNO)]2

Poly-Au, Teflon

200 nm

mSR

Dy(tta)3(H2O)2

Au(111)

sML

STM, NEXAFS, XMCD

Stoll et al. (2016) Zhang (2016)

Gd(tta)3(H2O)2 Eu(COT)2

Cu(111)

Theoretical

[Dy2(Hhmb)3(NCS)3].2MeOHPy

Au NPs

SEM, TEM, HRTEM

Ueff/kB ¼ 2.4

Dy(III)

SiO2 NPs

XMCD

Open hysteresis

1

Kiefl et al. (2016)

0.16

Holmberg et al. (2013)

16 Oe/s

Allouche et al. (2017)

HOPG, highly oriented pyrolithic graphite; L–B, Langmuir–Blodgett films; NPs, nanoparticles; Py, pyridine; PyNO, pyridine -N-oxide; hfac, hexafluoroacetylacetonate; hmp ¼, 2-(hydroximethyl)pyridine. The SI conversion factor is 1 Oe ¼ 103/4p A/m for magnetic field strength.

232 Handbook of Magnetic Materials

of the most recent examples of evaporated lanthanide-containing magnetic molecules on surfaces. Three main areas are being foreseen for molecular magnets on surfaces: spintronics, quantum computing and information, and on-chip cooling.

9.1 Selected Molecular Systems Only a limited number of lanthanide-containing molecules have been evaporated on surfaces. This requires structural robustness upon evaporation, and low interaction, or hybridization with the substrate for the original magnetic properties to survive. The Ln-“double-decker” family, bis(Phthalocyaninato)-lanthanide(III) (denoted LnPc2), is a remarkable structure consisting of a Ln3+ ion that is sandwiched between two organic phthalocyanine (Pc) ligands (see Fig. 95, left). LnPc2 is the most studied family of SMMs deposited on surfaces. TbPc2, in particular, has been the subject of numerous experiments: it has been shown to offer a happy conjunction of stability upon evaporation, and a very interesting magnetic behavior. The LnPc2 molecule may involve the Ln(III) electronic magnetic moment, the nuclear magnetic moment of the Ln isotopes (in some lanthanides), and, depending on the chemical environment, a S ¼ 1/2 caused by an electron delocalized over the p-orbitals of the two Pc units on the radical. The ground term of Tb3+ has L ¼ S ¼ 3; J ¼ 6. The antiprismatic environment around the Tb creates a CF with symmetry D4d. CF parameters for

1−

[TbPc2]− FIG. 95 Left: TbPc2 molecule (anion; gray: C; blue: N; red: Tb). Center: Er(trensal) (gray: C; blue: N; red: O; green: Er). Right: Chemical structure of DySc2N@C80 (gray frame: C; green, N; purple, Sc; orange, Dy). Left: Adapted with permission from Meihaus, K.R., Fieser, M.E., Corbey, J.F., Evans, W.J., Long, R., 2015. Record high single-ion magnetic moments through 4f electron configurations in the divalent lanthanide complexes [(C5H4SiMe3)3Ln]. J. Am. Chem. Soc. 137, 9855–9860, https://doi.org/10.1021/jacs.5b03710. Published by The Royal Society of Chemistry. Center: Adapted with permission from Holmberg, R.J., Murugesu, M., 2015. Adhering magnetic molecules to surfaces. J. Mater. Chem. C 3, 11986–11998. https://doi.org/10.1039/ C5TC03225C. Published by The Royal Society of Chemistry. Right: Adapted with permission from € R., Dreiser, J., Piamonteze, C., Muntwiler, M., Weyeneth, S., Brune, H., Rusponi, S., Westerstrom, Nolting, F., Popov, A., Yang, S., Dunsch, L., Greber, T., 2012. An endohedral single-molecule magnet with long relaxation times: DySc2N@C80. J. Am. Chem. Soc. 134, 9840–9843, https:// doi.org/10.1021/ja301044p. Copyright 2012 American Chemical Society.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 233

TbPc2 have been calculated (Ishikawa, 2010) to fit NMR spectra and magnetic susceptibility measurements. The diagonalization of the CF reveals a degenerate quasi-doublet j J, MJi ¼ j 6,  6i as the electronic ground state, separated from the first excited doublet by approx. D ¼ 600 K. Tb3+ is an effective S* ¼ 1/2 j6i, below room temperature, and because it is nonKramers, the Griffith theorem grants that it behaves as an Ising system. In addition, bare [TbPc2] is a radical, with an unpaired electron delocalized over the two phthalocyanine ligands, which carries a radical spin S ¼ 1/2. In many experiments, the radical form is not the molecule in use, but neutral forms which can be more efficiently grafted to surfaces, nanoparticles, or NTCs, such as hexa-hexyl-pyrene-substituted TbPc2 group. Therefore, this extra radical spin is not always present in TbPc2 specimens. Moreover, Tb carries a nuclear spin I ¼ 3/2, with natural abundance of 100%, and a nuclear moment gn ImN 2mN. The Ln(trensal) series, where H3trensal ¼ 2,20 ,200 -tris(salicylideneimino)trimethylamine, is a recently discovered family of SMMs with a threefold symmetry around the central lanthanide (see Fig. 95, center). It is a robust trigonal pyramidal structure with several adsorption conformations. Both Ln ¼ Er3+ and Dy3+ show slow relaxation of the magnetization, and the Er derivative has strong uniaxial anisotropy (Lucaccini et al., 2014). Another group of SMMs with promising perspectives to be evaporated on surfaces are the cluster fullerenes, potentially interesting in spintronics, quantum computing, and high-density storage devices (Treier et al., 2009; Westerstr€ om et al., 2012, 2015). The magnetic properties are determined by the inner cluster, while the Cn cage protects the magnetic center both structurally and electronically, reducing the scattering and hybridization with the surface. Other innovative examples of magnetic molecules that have been deposited on surfaces: double-decker molecules Ln(C8H8)2 (Zhang, 2016), Ln-dimers [Dy(hfac)3(PyNO)]2 with hfac ¼ hexafluoroacetylacetonate and PyNO ¼ pyridine-N-oxide (Kiefl et al., 2016), Ln-tris(1,1,1-trifluoro-4-(2thienyl)-2,4-butanedionate) (Dy(tta)3 and Gd(tta)3) complexes (Stoll et al., 2016), among others. These novel systems are listed in Table 17.

9.2

Experimental Techniques

The path from the bulk to the surface implies a change in the experimental techniques available to study magnetic relaxation. In the bulk, relaxation studies are dominated by AC susceptibility, as evidenced in the previous sections. Usually, the signal from the sample is larger than that from the vial or the sample holder in use. In contrast, when characterizing a very thin-film sample (from the submonolayer to about 100 nm), the signal from the substrate (diamagnetism and spurious paramagnetism) is orders of magnitude larger and fully hinders that from the sample. The magnetic flux changes induced by

234 Handbook of Magnetic Materials

the film cannot be detected in commercial SQUID susceptometers; thus, the AC magnetic susceptibility has to be abandoned. Magnetic relaxation time cannot be directly measured for samples smaller than the thick film (well above 10 nm). Luckily, XMCD (see Section 4) has become the magnetometry of choice in surface-deposited molecular magnetism. The only remaining witness of slow magnetic relaxation is bistability, giving rise to the opening of the hysteresis loop. The information of the process giving rise to SMM behavior is, mostly, lost. In those lucky cases where magnetoresistance is present, transport properties offer another way to indirectly detect magnetic hysteresis. In general, magnetic characterization of films of molecular materials onto surfaces has been made possible by the development of three kinds of experiments: X-ray absorption techniques, including XMCD, scanning probe techniques, such as scanning tunneling microscopy (STM) and spectroscopy (STS), including spin-dependent STM, and other resonance techniques, among them muon spin resonance (mSR) for which thicker films are needed. Scanning probe techniques, mainly STM and STS, are able to perform almost atom-resolved studies of the molecules deposited on a surface. These are unique tools to gain information on the molecular structure at surfaces but also to measure electronic structure of the molecular orbitals thanks to tunnel spectroscopy. A particularly interesting feature is the Kondo resonance, observed, for example, on an asymmetric member of the LnPc2 family: TbNPcPc (where NPc stands for naphthalocyaninato), caused by unpaired p electron upon adsorption on the Au(111) surface. Kondo resonance, which originates in interactions between conduction electrons and a localized spin, is strongly dependent on the supramolecular organization (Komeda, 2014; Komeda et al., 2014; Moreno Pineda et al., 2016): the Kondo temperatures observed on the NPc-up and the Pc-up are different, but other arrangements show no Kondo peak, such as the 1D chain composed of only NPc-up molecules, or the alternate 2D pattern with Pc-up molecules surrounded by NPc-up ones. A general result from the contributions by Komeda and coworkers, and others, is that the Tb 4f states in TbPc2 cannot be reached by STS, due to its localization and the obstruction placed by the ligand molecular orbitals. Interestingly, the Nd4f states can be reached by STS, as shown by Fahrendorf et al. (2013), which would open a very important research line to study how the deposition in different surfaces affects the 4f electrons. Unfortunately, NdPc2 is quite fragile upon deposition, showing very high decomposition probability, regardless of the surface (from near 50% for Fe/W(110) up to 100% in Au(111) (Fahrendorf et al., 2014). mSR is also able to study the magnetism of films of molecular materials deposited on surfaces (Hofmann et al., 2012; Kiefl et al., 2016). As XAS and XMCD, mSR can be performed on films and bulk samples (Branzoli et al., 2009a,b), allowing a direct comparison between bulk, thick, and thin films. Even more, mSR allows to obtain the relaxation time (one or several)

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of films of molecular magnets, within a high-frequency window, which in bulk is somehow complementary to AC susceptibility. However, the thickness of the films studied by mSR is limited to a minimum of at least 10–20 nm, far from the monolayer level. mSR requires a knowledge of the structure of the studied material, and therefore, a proper interpretation relies on a detailed sample characterization.

9.3

Surfaces and Substrates

Not surprisingly, the structure and the magnetic properties of deposited SMMs are dependent on the structure, electronic, and magnetic nature of the surface on which those are deposited, because the interaction between molecule and surface is able to change both the atomic and the electronic structures. The most important interactions governing the behavior of molecules on surfaces are adsorption, whose typical stabilization energies are of the order of 0.5–5 eV, strongly depending on substrate reactivity and the electronic exchange between the molecule and the surface. The adsorption may involve a chemical reaction between the surface and the molecule (chemisorption, quite strong) or not (physisorption, much weaker). The energetic landscape for surface migration and rotations of the molecules once those are adsorbed determine the formation of 2D structures. Even in physisorbed molecules, charge transfer between the molecule and the surface can be expected up to some point, and the magnetic properties of the molecule (magnetic moment, anisotropy) may change abruptly. The only lanthanide-containing family of magnetic molecules which has been deposited on a wide variety of surfaces is LnPc2, in particular TbPc2 and its derivatives (see Table 17).

9.3.1 Nonmagnetic Metals Neutral [TbPc2]0 has been evaporated onto Cu(100) (Stepanow et al., 2010) and Au(111) (Komeda, 2014; Komeda et al., 2013), and also deposited on Cu(111) by dry imprinting (Vitali et al., 2008), using a soft applicator formed by a fiber-glass bundle coated with a fine-grained powder of the SMM crystals. Simultaneously, Stepanow et al. have reported the magnetic properties of isolated TbPc2 molecules supported on a Cu(100) surface deposited by thermal evaporation (Stepanow et al., 2010). The molecular structure is preserved in all those cases, as it is the magnetic moment of the Tb ion and its very strong Ising anisotropy. Isolated molecules and monolayers tend to orient themselves with the Pc planes parallel to the metal surface. STM studies of TbPc2 on Cu(111) clearly reveal the expected number of lobes from the double-decker structure in the lying configuration, in agreement with DFT simulations (see, e.g., Fig. 96B). The anisotropy of the molecule, with the magnetic moment along its easy axis, which lies perpendicular to the Pc planes, is preserved as shown in

236 Handbook of Magnetic Materials

D L

B

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0° 45° 70°

XMCD (a.u.)

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A

−4

−2

0 B (T)

2

4

FIG. 96 (A) Topographic image at constant current of two isolated TbPc2 molecules on Cu(111) surface deposited by dry imprinting. (B) STM simulation image of an isolated TbPc2 molecule. (C) Schematic representation of TbPc2 on Cu(111) with the spin (S) and orbital angular momentum (L). (D) Magnetization curves obtained from the Tb M5 XMCD intensity for TbPc2 on Cu(100) at T ¼ 8 K (bottom). Adapted with permission from Stepanow, S., Honolka, J., Gambardella, P., Vitali, L., Abdurakhmanova, N., Tseng, T.C., Rauschenbach, S., Tait, S.L., Sessi, V., Klyatskaya, S., Ruben, M., Kern, K., 2010. Spin and orbital magnetic moment anisotropies of monodispersed bis(phthalocyaninato)terbium on a copper surface. J. Am. Chem. Soc. 132, 11900–11901, https://doi.org/10.1021/ja105124r. Copyright 2010 American Chemical Society. Adapted with permission from Vitali, L., Fabris, S., Conte, A.M., Brink, S., Ruben, M., Baroni, S., Kern, K., 2008, Electronic structure of surface-supported bis(phthalocyaninato) terbium (III) single molecular magnets. Nano Lett. 8, 3364–3368, https://doi.org/10.1021/ nl801869b. Copyright 2008 American Chemical Society.

Fig. 96D by the angular dependence of the XMCD measured on Cu(100), which shows how the magnetization cancels along the plane of the surface. The S ¼ 1/2 on the radical is quenched in Cu(111) and Cu(100), while it is preserved in the p orbitals of the Pc ligands on Au(111), showing that the TbPc2 with the surface in this case is not as strong as with Cu. The open magnetization loop present in bulk TbPc2 is completely closed in TbPc2 on Cu(100) at T ¼ 8 K, as shown in Stepanow et al. (2010); and only a very small butterfly hysteresis is found in Au(111) at 2 K (Margheriti et al., 2010). It is quite interesting that a clear open loop is found at temperatures as high as 15 K in a 200-nm thick film of TbPc2 on Al foil, with the easy axes lying parallel to the substrate. This is a consequence of the thickness of the layer: the molecules evolve from lying to standing configuration above a certain threshold which is around tens of monolayers, showing longer relaxation times in the standing configuration (Hofmann et al., 2012), consistent with the XMCD results. Hofmann and coworkers attribute the different magnetization behavior to the different relaxation times (measured by mSR) originated by the packing of the molecules in both configurations. The quenching of the magnetic moments from p orbitals by interaction with Cu(111) surface is a quite general fact, although in some cases it enhances the magnetic moment of the whole molecule: The double-decker 4f7 molecule Eu2+(C8H8)2 originally has an antiferromagnetic arrangement between the Eu2+ and the radical moments, and therefore, the quenching of the radical spin is positive to the magnetism of the whole molecule (Zhang, 2016).

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The so-called endohedral molecules are protected of the electronic scattering and the hybridization with the surface by the fullerene-like cage surrounding the lanthanide-containing cluster. Moreover, the local magnetic easy axis for the encapsulated Ln ions in nonspherical cages with the preferential adsorption geometry of the endohedral cluster induces surface alignment of the 4f moments and a macroscopic anisotropy. This is observed by Westerstr€ om et al. (2015) on a sub-ML of Dy2ScN@C80 on Rh(111), revealing a one-to-one correspondence between structural and magnetic ordering. The XMCD hysteresis in the sub-ML regime at T ¼ 4 K shows that “anisotropic structural protection” strategies for the 4f magnetic moments offer a good path toward useful systems.

9.3.2 Magnetic Substrates Monolayers of TbPc2 have been grown on different ferromagnetic metallic substrates. When evaporated on metallic magnets with easy axes perpendicular to the plane, the coupling between the Tb moment of TbPc2 is antiferromagnetically coupled with Co (Lodi Rizzini et al., 2014; Malavolti et al., 2013a), Ni (Klar et al., 2013), and Fe (Nistor et al., 2015). The AF coupling is also found between the Tb moment and the Fe-uncompensated moments of the antiferromagnetic system FeMn in TbPc2/FeMn/Cu(100). In contrast, no evident magnetic interaction was found in a previous work (Malavolti et al., 2013b) between a submonolayer of TbPc2 and ferromagnetic substrates on TbPc2/La0.3Sr0.7MnO3/SrTiO3 and TbPc2/Co/Cu(100), with the molecules on standing and lying configuration, respectively. Moreover, while no hysteresis loop has been observed in TbPc2/Co systems, a quite clear open loop was observed in both Tb and Ni in TbPc2/Ni/ Cu(100) for fields below H ¼ 0.1 T. This system maintains a Tb open loop at temperatures up to 100 K, opening the possibility to maintain Tb magnetization at interesting temperatures through surface engineering. In contrast, TbPc2/Ni/Ag(100), where Ni presents in-plane anisotropy, shows no Tb hysteresis even at the lowest temperatures. The exchange coupling between the Tb moment and that of the magnetic substrates is mediated by the radical S ¼ 1/2 due to the delocalized electron on the Pc ligand. It is interesting that doping the Fe/Cu(100) system with Li induces the Tb coercivity to change sign in TbPc2/Li + Fe/Cu(100), evidencing a ferromagnetic coupling. The change in charge transfer between the surface and the molecule induced by Li doping changes the exchange coupling from AFM to FM. The hysteresis in TbPc2/Fe/Cu(100) and TbPc2/Li + Fe/Cu(100) shows an open loop on Tb at T ¼ 8 K, as shown in Fig. 97, from Nistor et al. (2015), with considerable remanence and coercivity (of the order of hundreds of mT). TbPc2 films on antiferromagnetic surfaces may have the advantage of a reduced magnetization from the surface, which could be advantageous when

238 Handbook of Magnetic Materials

FIG. 97 (A, B) Fe and Tb magnetization curves measured on TbPc2/Fe/Cu(100). (C, D) Fe and Tb magnetization curves measured on TbPc2/Li + Fe/Cu(100), at oblique incidence after field cooling to T ¼ 8 K in the zero field. The insets show details of the low-field region Adapted with permission from Nistor, C., Krull, C., Mugarza, A., Stepanow, S., Stamm, C., Soares, M., Klyatskaya, S., Ruben, M., Gambardella, P., 2015. Exchange bias of TbPc2 molecular magnets on antiferromagnetic FeMn and ferromagnetic Fe films, Phys. Rev. B 92, 184402, https://doi. org/10.1103/PhysRevB.92.184402. Copyright 2015 American Physical Society.

detecting (very small) TbPc2 magnetization. Studies of LnPc2 films and CoO and FeMn show that Tb moment couples antiferromagnetically to the uncompensated moments in the surface (Lodi Rizzini et al., 2014; Nistor et al., 2015). The results show a very small Tb coercivity. Strong interactions between the substrate and the molecules destroy the butterfly hysteresis of the TbPc2.

9.3.3 Nonmetallic Substrates Both magnetic and nonmagnetic metal substrates lead to a general diminution of the hysteresis of TbPc2, except in those cases where Tb is coupled to the substrate magnetization. To minimize the interaction and the charge transfer with the substrate, there have been efforts recently published by several groups to grow LnPc2 films on nonmetallic substrates, such as SiO2 (Robaschik et al., 2015; Serri et al., 2017), MgO, hBN (W€ackerlin et al., 2016), H-terminated Si (Mannini et al., 2014), and perylene tetracarboxylic dianhydride (PTCDA) (Robaschik et al., 2015; Serri et al., 2017), kapton, and quartz (Malavolti et al., 2013a), as shown in Table 17. In general, TbPc2 shows much larger and more robust hysteresis when deposited on insulators than in metals. Clearly, an insulating substrate acts

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FIG. 98 (A) Schematics of a TbPc2 molecule on an MgO film on Ag(100); (B) STM image revealing self-assembled arrays of TbPc2 on MgO. The image sizes and parameters (bias V ¼ 2 V, I ¼ 20 pA); (C, D) XMCD hysteresis loops at 3 K for 0.3 ML TbPc2 adsorbed directly on Ag(100); (C) compared with 0.6 ML TbPc2 on 5 ML MgO; (D) field sweep rate 2 T/min, normal incidence. Reprinted with permission from W€ ackerlin, C., Donati, F., Singha, A., Baltic, R., Rusponi, S., Diller, K., Patthey, F., Pivetta, M., Lan, Y., Klyatskaya, S., Ruben, M., Brune, H., Dreiser, J., 2016. Single-molecule magnets: giant hysteresis of single-molecule magnets adsorbed on a nonmagnetic insulator. Adv. Mater. 28, 5142, https://doi.org/10.1002/adma.201670180. Copyright 2016 John Wiley & Sons, Inc.

as a barrier for quantum tunneling and this results in widely more open loops in TbPc2 films over insulators than those obtained on metals. It is particularly interesting the giant hysteresis of monolayers of TbPc2 on MgO, which only closes at H ¼ 3 T at T ¼ 3 K. A quite striking comparison of the TbPc2 magnetization between the molecule on Ag(100) and on MgO is shown in Fig. 98 after W€ackerlin et al. (2016). The remanence and the hysteresis opening of TbPc2 on MgO are favored by the suppression of scattering of conduction electrons from the metal at the molecule and, second, the much smaller molecule-surface hybridization. As the electron tunneling rate depends exponentially on the barrier thickness, narrower hysteresis loops are observed by W€ackerlin et al. for thinner MgO interlayers, opening a path toward SMM-based tunnel devices.

9.3.4 Carbon Substrates The quest to develop spintronic devices makes desirable to combine the intrinsic quantum nature and the magnetic and transport properties of SMMs with the unique electronic and mechanical properties of sp2-carbon substrates,

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including single-walled carbon nanotubes (CNTs), graphene, and highly oriented pyrolytic graphite (HOPG). In order to ameliorate the grafting of SMMs on carbon surfaces, and in particular CNTs, without destroying the CNT transport properties, the molecules have to be engineered. STM studies have shown that alkyl chains have strong affinity for carbon surfaces (Inose et al., 2014). Some TbPc2 derivatives have been synthesized to ameliorate its anchoring to carbon surfaces (Gonidec et al., 2011; Ivanov et al., 2006; Kyatskaya et al., 2009). For example, hexa-hexyl-4-(tetrapyren-1-ylbutoxy)-TbPc2 (called TbPc2*), shown in Fig. 99, left, was used to improve the anchoring of the TbPc2 unit to CNTs by Kyatskaya et al. (2009). The magnetization of the molecules linked to CNTs at 40 mK with varying sweeping rates allowed to find an improved opening of the magnetization in the TbPc2* with respect to the original molecule. The most interesting results with TbPc2 systems on carbon substrates are two prototypes developed by Wernsdorfer, Ruben, and their groups: a spin transistor, linking TbPc2 on graphene to two Au nanocontacts (Thiele et al., 2013, 2014; Vincent et al., 2012) and a supramolecular spin valve with TbPc2* on CNT (Urdampilleta et al., 2011, 2013). Also very promising is the perpendicular magnetic anisotropy of the Er(trensal) system when deposited on Ir(111), Ru(0001), and graphene grown on those surfaces (Er(trensal)@Gr/Ir(111) and Er(trensal)@Gr/Rh(0001)) (Dreiser et al., 2016). When deposited on the bare surfaces, the threefold geometry of Er(trensal) has not a preferred axes; thus the resulting system has not global magnetic anisotropy. In contrast, when deposited on graphene, the molecules are fully oriented with easy axes normal to the substrate surface, forming self-assembled islands. A large contrast between normal and grazing magnetization is observed in XMCD, in agreement with the out-ofplane alignment of the C3 axes of the molecule. This is a very promising result with a very robust SMM family.

9.4 Devices After about 25 years of continuous success in materials science basic research, molecular magnetism is mature enough, and maybe needed as a field of research, to produce a device or at least a prototype of a disruptive advance in which magnetic molecules are a critical ingredient (Cl
erac and Winpenny, 2017). Magnetic spintronics and magnetic cooling are two lines of research where lanthanide-containing molecules are close to reach the prototype stage.

9.4.1 TbPc2 Spintronics Both the spin transistor and the mentioned spin valve based on TbPc2 share the subtle relationship between the nuclear and the electronic magnetic moments of Tb in TbPc2 coupled by hyperfine interactions. These two systems open a new way to molecular spintronics (Candini et al., 2011;

FIG. 99 Left: Zeeman diagrams of the ground electronic doublet calculated with hyperfine and nuclear quadrupole interaction terms. Right: Hysteresis loop measured on a single crystal of [(Pc)2Tb0.02Y0.98]TBA+ at T ¼ 40 mK with a sweeping rate of 1 mT/s. The SI conversion factor is 1 K ¼ 1.381  1023 J for energy. Adapted with permission from Ganzhorn, and Wernsdorfer, Molecular quantum spintronics using single-molecule magnets, In: Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https://doi.org/10.1039/C4DT00538D. Copyright © Springer-Verlag Berlin Heidelberg 2014.

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Ganzhorn et al., 2013, 2016; Thiele et al., 2013, 2014; Urdampilleta et al., 2011, 2013, 2015; Vincent et al., 2012). The electronic ground Griffith quasi-doublet is split by Zeeman, hyperfine, and nuclear quadrupole interactions into two unequally spaced quartets, as shown in Fig. 99 (center), obtained by numerical diagonalization of the full Hamiltonian. The hyperfine Adip/kB ¼ 24.5 mK and the nuclear quadrupolar parameters Pquad/kB ¼ 14.4 mK are determined by fitting the steps in low-field magnetization, as corresponds to quantum tunneling taking place at field values driving to (anti)crossing of levels in a single crystal of [(Pc)2Tb0.02Y0.98]TBA+, as shown in Fig. 99 (center and right) (Ishikawa et al., 2005b). The measurements were performed at T ¼ 40 mK and a field sweeping rate of 1 mT/s. In an isolated single molecule, only the four low-field avoided level crossing corresponding to j DmJ j ¼ 12; DIz ¼ 0 have an associated nonzero tunnel splitting (Thiele et al., 2014). Contrary to the isolated molecule, in an ensemble such as a crystal the magnetization hysteresis shows jumps at field values which evidence multispin tunneling taking place at crossings where the third component of the nuclear moment is not conserved. This is evidenced in Fig. 99 (right) and also in Ishikawa (2010) (see Fig. 6), but the mechanism responsible for this effect remains to be fully clarified. The isolated molecule tunneling channels are evidenced, for example, when a single TbPc2 is placed in a molecular transistor configuration (Vincent et al., 2012), directly connected to source and drain gold electrodes obtained by electromigration, and a back-gate underneath. The conductance between the electrodes allows to measure the nuclear spin, as the bias field at which QTM takes place depends on the particular nuclear state of Tb. Enough statistics, obtained at bath temperature T ¼ 40 mK, allowed to obtain nuclear spin lifetimes of tens of seconds, long enough to perform coherent manipulations (Rabi oscillations) of the Tb nuclear spin with an RF antenna mounted in close proximity to the device (Thiele et al., 2014). It is also very promising the CNT spin valve created by grafting a couple of TbPc2* on a CNT, as shown in Fig. 9 (left) by Wernsdorfer and Ruben groups. When two molecules are grafted to one CNT, their spin may be parallel or antiparallel. The weak exchange interaction between the Tb electronic spin and the S* ¼ 1/2 radical spin, delocalized on the p orbitals of the Pc moieties, slightly changes the conductivity between the two electrodes through the decorated CNT. Fig. 100 (center) evidences “butterfly” hysteresis loops on the field dependence of the conductance of the device. When the spin state in molecule A is reversed with respect to that of molecule B, as shown in Fig. 100 (right, up), the energy mismatch between levels with identical spin results in a current blockade. In contrast, in the parallel spin configuration of both molecules A and B, the alignment of levels with the same spin allow electron transport through the CNT, evidencing the electronic spin state of the molecules.

FIG. 100 Left: Artistic view of carbon nanotube-based supramolecular spin valve. Center: Spin valve behavior of a supramolecular spintronic device based on a carbon nanotube quantum dot with two TbPc2 SMMs. Butterfly hysteresis loop at T ¼ 40 mK. Right: Scheme of the antiparallel (up) and parallel (down) spin configurations. Left: Courtesy of M. Urdampilleta. Center and Right: Adapted with permission from Ganzhorn and Wernsdorfer, Molecular quantum spintronics using single-molecule magnets, In: Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´a-Romano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013, https://doi.org/ 10.1039/C4DT00538D. Copyright © Springer-Verlag Berlin Heidelberg 2014.

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Moreover, the (quantum) mechanical vibrations of a suspended nanotube have been also evidenced, measured, and controlled at ultralow temperatures in a series of experiments (Ganzhorn et al., 2013, 2016), where strong spin– phonon coupling between a single TbPc2* and a CNT is used to fabricate a CNT-SMM-based nanoelectromechanical system (NEMS). The bias magnetic field is used to avoid the QTM relaxation, only allowing direct relaxation processes for the spin to relax. Direct processes, with one-phonon exchange between the spin and the vibrational state of the CNT, are detected using the hyperfine-based splitting of the Tb quasi-doublet, opening the way to ultrasensitive mass sensing or magnetic torque nanodetectors.

9.4.2 Cooling Devices Magnetic cooling is an emerging technology with applications in high and low temperature. Lanthanide-containing molecules (mostly Gd) have a realistic niche in 3He-free sub-Kelvin refrigeration, as they combine high magnetic moment and paramagnetism down to extremely low temperatures. Magnetic cooling using Gd molecular materials is a competitive candidate for sub-Kelvin refrigeration. In particular, on-chip cooling is an inexpensive alternative for cryogenic solutions, such as sensors, detectors, and low-temperature sample holders, which would benefit of a built-in refrigerator. On-chip cooling with magnetic molecules is challenging mainly because the amount of mass which can be deposited is very small, and therefore, the entropy content per mole liberated at the temperature of interest must be maximized. Thermal binding of the cooling film to the chip is another difficulty. To implement molecular nanomagnet refrigerators on Si-based thermal sensors, Lorusso and coworkers (2013) deposited [Gd2(CH3COO)6(H2O)4]4H2O molecules (called Gd2-ac) on polished Si(100) (from Si wafers) with boron doping (type p) by dip-pen nanolithography (for a review on the technique, see Domingo et al., 2012). A thin layer of native oxide enables the adsorption of molecules through hydrogen bonding with hydroxyl groups naturally present at the surface of oxides. Drops of a 5 mg/mL solution of Gd2-ac generate 10 nm thick, 1.5 mm ovals which once dry were studied by magnetic force microscopy. The estimated maximum magnetization is 2  108 mB at saturation. The as-deposited Gd2-ac molecules hold intact their magnetic properties, paving the way toward the realization of molecule-based microrefrigerating devices. Another approach to maintain single-molecule properties (in particular MCE) is to use more complex cage-like molecules where the Gd ions are protected by the external molecular structure when deposited on HOPG or even Au(111), as proven by Corradini and coworkers (2014) by dispersing [Gd4Ni8(OH)8(hmp)8(O2CEt)8(MeOH)6][ClO4]4 (“Gd4Ni8”), using a liquid phase method to deposit it on substrates, with good results. One of the challenges of the idea is to optimize the thermal contact between the deposited mass of magnetic molecules and the chip carrying the

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device. To do so, Roubeau and coworkers (2017) have grafted Gd2-ac to a much better thermal conductor, such as multiwalled carbon nanotubes (MWCNT). Functionalization of MWCNT surface with molecular coolers is done by covalent or strong noncovalent interactions. Molecular coolers are ideally suited to form this composite, due to the flexibility of their binding to surfaces. Moreover, the reduction in crystalline order disfavors magnetic order, which lowers the base temperature attainable. A composite formed by large pieces of the grafted MWCNT-[Gd2ac] in a parallel disposition and embedded in a resin for mechanical support notably improves the thermal conductivity of the material by a factor of 2, validating this composite strategy.

9.5

Conclusions and Outlook

Surface implementation is probably the newest branch of the relatively young (but very fast growing) area of lanthanide-containing SMMs. A very fast pace of developments in chemistry, physics, and surface science allows to prepare, characterize, and even control systems which were not possible not so long ago. Evidence has been established that bulk SMMs retain its properties when brought into contact with adequate substrates. A variety of surfaces (metallic, magnetic and nonmagnetic, graphitic, or insulating) and other supports such as CNTs and nanoparticles are used to grow films, from the mono- or submonolayer level to thin and thick films. A variety of deposition techniques, from in vacuum sublimation to dry and wet imprinting, are established nowadays, each with its own range of applications. It is quite clear that hybridization and electron scattering resulting from the direct contact of the molecules with noble metal surfaces are, in general, detrimental to the magnetic properties of molecules, even for the well-screened 4f electrons of a lanthanide. New deposition techniques and new substrates (semiconducting, insulators, carbon-based) as well as the use of naturally protected molecules (cages, fullerenes, double-deckers, etc.) have shown that implementation of SMMs can be optimized for the envisaged applications, although a lot of work remains to be done. Functionalization of the surfaces and the SMMs are two alternative and complementary paths which widely open the range of possibilities. Moreover, the use of SMM-decorated CNTs is extremely promising in several areas, from nanomechanics to spintronics and magnetic cooling. More examples of robust, sublimable molecules with stable magnetic moments and long relaxation times are desirable to extend a range of possibilities which today we only reach to envisage. Indeed, this is already taking place, although the hegemony of TbPc2 in number, originality, and interest of the results is evident in the literature, the lessons learnt in TbPc2 will be for sure translated to other families. The advances made by Ruben and Wernsdorfer groups on creating, building, and controlling TbPc2-based molecular spintronics are really exciting,

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actually showing that new paths of science and applications are open in the area. This set of works should act as a boost in the field. High-temperature applications are of course desirable, but if disruptive applications are found, the inherent difficulty involved in cooling to the mK regime may be a price which technology would be happy to pay, at least as a first step. To be really useful, information has to be written and read, ideally from a single SMM on a surface onto which molecules could be organized, or from an array of CNTs with anchored molecules. The necessary weak interaction links between 4f magnetic moments which would allow, for example, the already designed qubits and quantum gates to interact, may open the way to scalable architectures in quantum information. The involved interactions may be magnetic, but also optical, EM, or even mechanic. Molecular coolers, a prerequisite for on-chip magnetic refrigeration, are also an area with fast development. Although the useful molecules appear to be already optimized, there are several materials science issues to improve (thermal conductivity, cooling power, among others) prior to reach the stage of device production.

10 CONCLUSIONS AND PERSPECTIVES Molecular magnetism has been the subject of intense investigation in the last 20 years, which has led to the discovery of new astonishing physics at the edge between the classical and the quantum world, and has opened the possibility of storing or processing information at the single-molecule level. To fulfill the technical requirements for information storage or quantum computing, the dynamics of the molecular magnet has to allow the designed operation: bistability, as well as long coherence and long relaxation times are mandatory, relative to the characteristic times of the phenomenon of interest. Therefore, understanding and tailoring magnetic relaxation mechanisms in the molecules through adequate chemical design are at the core of this multidisciplinary research field. In view of using molecular magnets as memory units, we have assisted in the last decade to a race toward achieving compounds with higher energy barriers to spin reversal (Ueff) and magnetic blocking temperatures (TB). Fig. 101 summarizes a collection of the record Ueff and TB values for several families of molecules reported until today (May 2017). The use of lanthanide SIMs was boosted in 2003 by the impressive properties of TbPc2 (Ishikawa et al., 2003b), a double-decker compound exhibiting an effective barrier Ueff one order of magnitude larger than in Mn12, probably the most studied SMM up to that moment, and the playground where QTM was evidenced beyond any doubt (Gatteschi et al., 2006). The single-ion anisotropy, governed by CF and characteristic of lanthanides, was translated to the molecule. At that point, a race started to increase Ueff using molecular design to optimize anisotropy. The absolute record until 2013 was held by

2020

2020 DyOSc@C82

2000 1000

[Dy(bbpen)Br]

[Tb((O–(C6H4)-p-tBu)8Pc)(Pc′)]

[Ln]N SMMs [Ln] SIMs EMF-SMMs

[Dy(bbpen)Br]

2000 1000

[Tb((O–(C6H4)-p-tBu)8Pc)(Pc′)]

800

[Dy4K2O]

[Dy(bbpen)CI]] [DyL2(H2O)5]

[TbPc2]°

600 400

[Dy5O(OiPr)13]

[Dy4K2O]

[Dy(bbpen)CI]] [DyL2(H2O)5]

600 [Dy(Cy3PO)2]

[Dy5O(OiPr)13]

[Tb2K(18-crown-6)]

[Dy4(μ3-OH)2] [Dy2(hmi)2]

2004

800

400

[TbPc2]−

200

[Dy(Cy3PO)2]

Ueff /kB (K)

Ueff /kB (K)

[TbPc(OEt)2]°

2007

[Tb2K(18-crown-6)] [Dy2(ovph)2]

[Dy2(ovph)2] [(Cp*2Dy)2] [Dy (bzhdep-2H) ] 4 4 DySc2N@C80

2010 2013 Years

2016

2019

200 DySc22N@C80

0

5

[Dy4(μ3-OH)2] [(Cp*2Dy)2] [Dy4(bzhdep-2H)4]

10

15 20 TB (K)

25

30

35

FIG. 101 Remarkable energy activation energies (Ueff) and blocking temperatures (TB) reached in the past years. Abbreviations and references: Mononuclear Ln SIMs (black symbols): [TbPc2] ¼ [TbPc2]TBA+ (Ishikawa et al., 2003b); [TbPc2]0 (Ishikawa et al., 2004); [TbPc(OEt)8]+ ¼ [TbPc(OEt)8]+(SbCl6) (Takamatsu et al., 2007); [Tb((O-(C6H4)-p-tBu)8Pc)(Pc’)] (Ganivet et al., 2013); [Dy(Cy3PO)2] ¼ [Dy(Cy3PO)2(H2O)5]Br32(Cy3PO)2H2O2EtOH, TB@200 Oe/s (Chen et al., 2016b); [Dy(bbpen)X], X ¼ Cl, Br, TB@200 Oe/s (Liu et al., 2016d); [Dy(H2O)5L2] ¼ [L2Dy(H2O)5][I]3L2(H2O), TB ¼ 12 K (18 Oe/s), TB ¼ 30 K (200 Oe/s) (Gupta et al., 2016a). Polynuclear Ln SMMs (red symbols): [Dy2(hmi)2] ¼ [Dy2(hmi)2(NO3)2(MeOH)2]∞MeCN (Lin et al., 2008); [Dy4(m3-OH)2] ¼ [Dy4(m3OH)2(bmh)2(msh)4Cl2], TB@1400 Oe/s (Lin et al., 2009); [Dy5O(OiPr)13] (Blagg et al., 2011); [Dy2(ovph)2] ¼ [Dy2(ovph)2Cl2(MeOH)3]MeCN (Guo et al., 2011); [Tb2K(18-crown-6)] ¼ [K(18-crown-6)(THF)2][([Me3Si)2N]2(THF)Tb)2(m-Z2:Z2-N2)], TB@9 Oe/s (Rinehart et al., 2011); [(Cp*2Dy)2] ¼ [(Cp*2Dy)2](mbpym)](BPh4), TB@20 Oe/s (Demir et al., 2012); [Dy4K2O] ¼ [Dy4K2O(OtBu)12]C6H14 ¼ [Dy4K2O], TB@1400 Oe/s (Blagg et al., 2013); [Dy4(bzhdep-2H)4] ¼ [Dy4(bzhdep-2H)4(H2O)4(NO3)4]6CH3OH6H2O, TB@500 Oe/s (Huang et al., 2016). EMFs (blue symbols), bold: measured DySc2N@C80 (Westerstr€om et al., 2012); open symbols: ab initio-predicted DyOSc@C82, TB@2 T/min (Singh and Rajaraman, 2016). The SI conversion factor is 1 K ¼ 1.381  1023 J for energy.

248 Handbook of Magnetic Materials

Ln(III) SIMs derived of the TbPc2 family (Fig. 101, left, black symbols), improving in a decade the Ueff by a factor of 3, from about 300 K to nearly 900 K in the asymmetric double-decker [Tb((O-(C6H4)-p-tBu)8Pc)(Pc0 )] (Ganivet et al., 2013). In 2016, Liu and coworkers synthesized two Dy pentagonal bipiramid compounds, [Dy(bbpen)X] with X ¼ Cl and Br. The latter shows an impressive Ueff over 1000 K (Liu et al., 2016d) and holds at present the experimental Ueff record for any SMM. As shown along the chapter, promising routes toward achieving SIMs with yet higher anisotropy barriers include: (i) designing a ligand field with the desirable charge distribution to enhance the axial anisotropy of the lanthanide center, with the help of ab initio and electrostatic model predicting tools; (ii) elaborating a strongly uniaxial ligand field to force the magnetization reversal via higher excited energy levels. The synthesis of polynuclear lanthanide SMMs is a more recent achievement, so the first complexes with remarkable Ueff/kB > 100 K date only from 2008 (Fig. 101, left, red symbols). The rate of improvement is however very high, reaching a barrier near 800 K for [Dy4K2O]. In this complex, both TAQTM and Orbach relaxation are blocked through the first excited doublet thanks to the smallness of the transverse g values, and the parallel alignment of the easy axes for the ground and first excited doublets (Ungur and Chibotaru, 2011). This is a nice example of a “program” of optimization of the physical properties in Materials Science through successive rounds involving chemical design, a complete physical characterization, and the understanding of the physical mechanisms governing relaxation, to inject new ideas as input into the next round. The size of Ueff is directly related to the relaxation times of processes obeying an Arrhenius law, such as Orbach relaxation. However, Orbach processes are not the only possible relaxation paths, and therefore, the maximization of the anisotropy barrier is insufficient if taken as the only parameter to be tuned, as it has been pointed out recently by Pedersen et al. (2015). Therefore, consideration of additional design criteria that address the presence of alternative relaxation processes beyond the traditional double-well picture is required. In the right panel of Fig. 101, we plot the Ueff as a function of the blocking temperature, TB, of the same “Ueff record molecules.” It has to be clarified that TB has been taken as the highest at which open hysteresis is claimed to be observed, but that’s a rather ill-defined parameter, due to the lack of consensus in the field rate at which it should be reported, e.g., for [Dy(H2O)5L2] the blocking temperature spans from TB ¼ 12 K (measured with a sweeping rate of 18 Oe/s) to TB ¼ 30 K (at 200 Oe/s) (Gupta et al., 2016a). In any case, the figure is striking; the scattering of data is difficult to reconcile with the existence of an increasing function linking Ueff and TB (in fact, there seems to be a very rough, inverse correlation). More efforts should be devoted to clarify the correlation between the two magnitudes in view of optimizing both simultaneously.

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Theoretical understanding of the relevant processes relies, at present, on computational ab initio estimations of the electronic structure, anisotropy, and magnetic interactions of molecules. DFT approaches are not so computationally expensive as more refined routes, such as the CASSCF–CASPT2/ RASSI-SO quantum chemistry approach, which is, in our opinion, the best tool for modeling and predicting the static and dynamic magnetic behavior of lanthanide-containing molecules. The improvement of several aspects of the calculation will be very important in the near future: a large number of spin-free states; to implement dynamical electronic correlations; the use of a large basis set for both the lanthanide and the ligands; and an increased structural precision, notably of hydrogen atoms. A relevant enhancement of the computational power or of the computational efficiency of the method would allow and enhancement of the accuracy of the CASSCF–CASPT2/RASSI-SO method, and the improvement of the whole chemical design program. Preventing relaxation through QTM and TAQTM seems to be the key issue, and several chemical design strategies have been proposed to do so: using Kramers ions, while minimizing dipolar and nuclear interactions (which are two interactions difficult to control), but mostly Raman relaxation modes, which are probably more relevant than suspected until very recently, should be taken into account in the chemical design. A “disconnection” of the magnetic ions from the electronic and phonon modes of the solid would be desirable. Sublimation on surfaces may be helpful in some cases, and that would be at the origin of the giant coercivity (3 T at 3 K) recently observed in a monolayer of TbPc2 on top of an insulating MgO layer (W€ackerlin et al., 2016). A new strategy along similar lines has been proposed involving endohedral metallofullerens containing different types of Ln–M clusters, like DySc2N@C80 with Ueff/kB ¼ 24 K (at 0.3 T) (Westerstr€om et al., 2012) or TbNC with Ueff/kB ¼ 12 K (at 0.12 T) (Liu et al., 2017). These materials have recently emerged as an interesting new family of SMMs. The carbon cage provides robustness, prevents dipolar interaction, and allows sublimation onto surfaces, as remarkably shown for Dy2Sc2N@C80 on Rh(111) (Westerstr€om et al., 2015), exhibiting a hysteresis cycle up to 4 K (2 T/min). Although the demonstrated activation energies to date are still modest, recently reported ab initio simulations have predicted that record values around Ueff/kB > 2000 K may be attained in cages containing linear arrangements of magnetic units, like {DyOSc}@C82 (Singh and Rajaraman, 2016) or {DyOSc}@C80 (Ungur and Chibotaru, 2016b), opening up new avenues in the search for new generation SMMs. Another optimization strategy considers the idea of including 3d and 4f ions in dimers and polynuclear compounds. This idea was borrowed from the hard-magnet design (Herbst, 1991), and the design path was started by the late O. Kahn as early as the 1990s decade (Bartolom
e et al., 1995; Kahn, 1993). It was expected that the more intense strength of the Ln–M(II) interaction (with M ¼ Ni, Co), coupling the Ln and M moments ferromagnetically and

250 Handbook of Magnetic Materials

the Ln single-ion anisotropy, enhancing the energy barrier to moment reversal, would render the system interesting as SMM. This is sometimes the case (mostly with Ln ¼ Dy). Unluckily, the weakness of the 3d–Ln interaction does not allow this expected synergy to work very positively. Recently, a clever use of symmetry and topology at the design of the molecule in polynuclear clusters has led to improved SMM properties. First, the enhancement of the axial symmetry has been related to strong enhancements of the Ueff and the reduction of QTM. Second, the ligand design strategy in different core topologies plays the role of modulating the symmetry of individual ions. It has been understood that particular structure topologies and ligands give rise to SMM behavior. In addition, the presence of different orientations of the anisotropy axis in the different centers of the clusters weakens intercluster dipolar interaction, also reducing QTM. The exploration of magnetic coupling effects within and between toroidal molecules is of critical importance for the fundamental understanding of ferrotoroidicity, and further coupling those SMT molecules into 2D/3D ordering materials seems to be a more challenging task. Investigating the role of Ln–Ln interactions and their interplay with the single-ion anisotropy is a crucial aspect toward a full understanding of the relaxation phenomena, and optimization of the SMM properties. Model dimeric Ln–Ln systems and extended lanthanide-based 1D, 2D, and 3D systems are ideal playgrounds for this purpose. Up to now, the vast majority of SMMs have been only characterized within the SQUID’s range of temperatures (>1.6 K), at which the magnetic behavior is essentially dominated by the single-ion anisotropy, given the weakness of 4f–4f interactions. However, at very low temperatures weak intra- and intermolecular interactions become competitive with the kinetic energy, allowing to observe very interesting physics, like the crossover between different spin relaxation regimes, SIM to SCM crossover, critical slowing-down of quantum tunneling, or the establishment of long-range ordering (Bartolom
e et al., 2013, 2016). Thus, we deem that multitechnique characterization of molecular magnets, including magnetometry down to sub-Kelvin temperatures, will open a wider panorama to relaxation processes. We believe indeed that “There is plenty of room at the bottom” (after Feynman), but also that it’s gonna be “Hot in the Fridge.” Surface implementation, toward materials optimization and device design, is another area where much progress is foreseen. Evidence has been established that bulk SMMs may retain or even ameliorate its properties when brought in contact with adequate substrates. A variety of surfaces (including CNTs and nanoparticles) and deposition techniques have been tried in a quite limited number of lanthanide-containing systems, yet. The area is still in its beginnings, as only TbPc2 has been treated systematically. Hybridization and electron scattering are detrimental to the magnetic properties of molecules, even for well-screened 4f electrons. More examples of robust, sublimable molecules with stable magnetic moments and long

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relaxation times will come; in particular naturally protected molecules (cages, fullerenes, double-deckers, etc.) are of high interest. The use of SMMdecorated CNTs is very promising, including disruptive research areas such as nanomechanics, spintronics, and magnetic cooling. We agree with Cl
erac and Winpenny (2017) that molecular magnetism as a research area needs some form of device to continue to thrive. But the advances in some areas, the most notable being magnetic cooling, probably depict this goal as not so far away. Moreover, amazing experiments, such as those performed by the groups of Wernsdorfer and Ruben on TbPc2, may open new lines in Spintronics, Nanomechanics, and Quantum Information. It is indeed inspiring that TbPc2, the first molecule for which a really systematic, global study has been performed as deposited on surfaces, shows such a wonderful range of possibilities and results, from giant hysteresis to quantum devices. Undoubtedly, many molecules are still to appear and new strategies, both chemical and physical, have to be followed. Research paths appear to be fully open for next future.

ACKNOWLEDGMENTS The financial support of the Spanish MINEICO, through projects MAT2014-53921-R and MAT2015-68200-C2-2-P, the Aragonese DGA, through IMANA E34 project (co-funded by the European Social Fund), and the European Union FEDER funds is acknowledged. The authors have profited of fruitful discussions with Dr. Prodius, Melnic, Shova, and late Prof. Turta, from the Institute of Chemistry of the Moldavian Academy of Science at Chisenau, Moldova.

REFERENCES Abbas, G., Lan, Y., Mereacre, V.M., Buth, G., Sougrati, M.T., Grandjean, F., Long, G.J., Anson, C.E., Powell, A.K., 2013. Synthesis, magnetism, and 57Fe M€ossbauer spectroscopic study of a family of [Ln3Fe7] coordination clusters (Ln ¼ Gd, Tb, and Er). Inorg. Chem. 52, 11767–11777. https://doi.org/10.1021/ic401011d. Abragam, A., Bleaney, B., 1970. Electron Paramagnetic Resonance of Transition Ions. Oxford University press, Oxford. Abtab, S.M.T., Majee, M.C., Maity, M., Titisˇ, J., Bocˇa, R., Chaudhury, M., 2014. Tetranuclear hetero-metal [Co(II)2Ln(III)2] (Ln ¼ Gd, Tb, Dy, Ho, La) complexes involving carboxylato bridges in a rare M4-M2:M2 mode: synthesis, crystal structures, and magnetic properties. Inorg. Chem. 53, 1295–1306. https://doi.org/10.1021/ic401484d. Adhikary, A., Sheikh, J.A., Biswas, S., Konar, S., 2014. Synthesis, crystal structure and study of magnetocaloric effect and single molecular magnetic behaviour in discrete lanthanide complexes. Dalton Trans. 43, 9334–9343. https://doi.org/10.1039/c4dt00540f. Aguilà, D., Barrios, L.A., Velasco, V., Arnedo, L., Aliaga-Alcalde, N., Menelaou, M., Teat, S.J., Roubeau, O., Luis, F., Aromı´, G., 2013. Lanthanide contraction within a series of asymmetric dinuclear [Ln2] complexes. Chem. A Eur. J. 19, 5881–5891. https://doi.org/10.1002/ chem.201204451.

252 Handbook of Magnetic Materials Aguilà, D., Barrios, L.A., Velasco, V., Roubeau, O., Repoll
es, A., Alonso, P.J., Ses
e, J., Teat, S.J., Luis, F., Aromi, G., Pablo, J., 2014. Heterodimetallic [LnLn’] lanthanide complexes: towards a chemical design of 2-qubit molecular spin quantum gates. J. Am. Chem. Soc. 136, 14215–14222. https://doi.org/10.1021/ja507809w. Ahmed, N., Das, C., Vaidya, S., Langley, S.K., Murray, K.S., Shanmugam, M., 2014. Nickel(II)lanthanide(III) magnetic exchange coupling influencing single-molecule magnetic features in {Ni2Ln2} complexes. Chem. A Eur. J. 20, 14235–14239. https://doi.org/10.1002/ chem.201404393. Al Hareri, M., Gavey, E.L., Regier, J., Ras Ali, Z., Carlos, L.D., Ferreira, R.A.S., Pilkington, M., 2016. Encapsulation of a [Dy(OH2)8]3+ cation: magneto-optical and theoretical studies of a caged, emissive SMM. Chem. Commun. 52, 11335–11338. https://doi.org/10.1039/C6CC03578G. Alexandropoulos, D.I., Poole, K.M., Cunha-Silva, L., Ahmad Sheikh, J., Wernsdorfer, W., Christou, G., Stamatatos, T.C., 2017. A family of “windmill”-like {Cu6Ln12} complexes exhibiting single-molecule magnetism behavior and large magnetic entropy changes. Chem. Commun. 53, 4266–4269. https://doi.org/10.1039/C7CC01382E. Allouche, F., Lapadula, G., Siddiqi, G., Lukens, W.W., Maury, O., Le Guennic, B., Pointillart, F., Dreiser, J., Mougel, V., Cador, O., Cop
eret, C., 2017. Magnetic memory from site isolated Dy(III) on silica materials. ACS Cent. Sci. 3, 244–249. https://doi.org/10.1021/ acscentsci.7b00035. Amjad, A., Madalan, A.M., Andruh, M., Caneschi, A., Sorace, L., 2016. Slow relaxation of magnetization in an isostructural series of zinc–lanthanide complexes: an integrated EPR and AC susceptibility study. Chem. A Eur. J. 22, 12849–12858. https://doi.org/10.1002/chem.201601996. Amoretti, G., Caciuffo, R., Carretta, S., Guidi, T., Magnani, N., Santini, P., 2008. Inelastic neutron scattering investigations of molecular nanomagnets. Inorg. Chim. Acta 361, 3771–3776. https://doi.org/10.1016/j.ica.2008.03.047. Anastasiadis, N.C., Kalofolias, D.A., Philippidis, A., Tzani, S., Raptopoulou, C.P., Psycharis, V., Milios, C.J., Escuer, A., Perlepes, S.P., 2015. A family of dinuclear lanthanide(III) complexes from the use of a tridentate Schiff base. Dalton Trans. 44, 10200–10209. https://doi.org/ 10.1039/c5dt01218j. Anderson, P.W., 1959. New approach to the theory of superexchange interactions. Phys. Rev. 115, 2–13. https://doi.org/10.1103/PhysRev.115.2. Andersson, K., Malmqvist, P., Roos, B.O., Sadlej, A.J., Wolinski, K., 1990. Second-order perturbation theory with a CASSCF reference function. J. Phys. Chem. 94, 5483–5488. https://doi. org/10.1021/j100377a012. Andersson, K., Malmqvist, P.-A., Roos, B.O., 1992. Second-order perturbation theory with a complete active space self-consistent field reference function. J. Chem. Phys. 96, 1218. https:// doi.org/10.1063/1.462209. Andreiadis, E.S., Imbert, D., P
ecaut, J., Demadrille, R., Mazzanti, M., 2012. Self-assembly of highly luminescent lanthanide complexes promoted by pyridine-tetrazolate ligands. Dalton Trans. 41, 1268. https://doi.org/10.1039/c1dt11627d. Antal, P., Drahosˇ, B., Herchel, R., Tra´vnı´cˇek, Z., 2016. Muffin-like lanthanide complexes with an N5O2-donor macrocyclic ligand showing field-induced single-molecule magnet behavior. Dalton Trans. 45, 15114–15121. https://doi.org/10.1039/C6DT02537D. Aquilante, F., Autschbach, J., Carlson, R.K., Chibotaru, L.F., Delcey, M.G., De Vico, L., Fdez. Galva´n, I., Ferr
e, N., Frutos, L.M., Gagliardi, L., Garavelli, M., Giussani, A., Hoyer, C.E., ˚ ., M€uller, T., Nenov, A., Olivucci, M., Li Manni, G., Lischka, H., Ma, D., Malmqvist, P.A Pedersen, T.B., Peng, D., Plasser, F., Pritchard, B., Reiher, M., Rivalta, I., Schapiro, I., Segarra-Martı´, J., Stenrup, M., Truhlar, D.G., Ungur, L., Valentini, A., Vancoillie, S.,

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 253

Veryazov, V., Vysotskiy, V.P., Weingart, O., Zapata, F., Lindh, R., 2016. Molcas 8: new capabilities for multiconfigurational quantum chemical calculations across the periodic table. J. Comput. Chem. 37, 506–541. https://doi.org/10.1002/jcc.24221. Arauzo, A., Lazarescu, A., Shova, S., Bartolom
e, E., Cases, R., Luzo´n, J., Bartolom
e, J., Turta, C., 2014. Structural and magnetic properties of some lanthanide (Ln ¼ Eu(III), Gd(III) and Nd(III)) cyanoacetate polymers: field-induced slow magnetic relaxation in the Gd and Nd substitutions. Dalton Trans. 43, 12342–12356. https://doi.org/10.1039/c4dt01104j. Aravena, D., Ruiz, E., 2013. Shedding light on the single-molecule magnet behavior of mononuclear Dy(III) complexes. Inorg. Chem. 52, 13770–13778. https://doi.org/10.1021/ic402367c. Arfken, G.B., Weber, H.J., Harris, F.E., 2012. Mathematical Methods for Physicists: A Comprehensive Guide, seventh ed. Elsevier, Amsterdam. Aromı´, G., Luı´s, F., Roubeau, O., 2015. Lanthanide complexes as realization of qubits and quates for quantum computing. In: Layfield, R.A., Murugesu, M. (Eds.), Lanthanides and Actinides in Molecular Magnetism. Wiley-VCH, Weinheim, Germany, pp. 185–221. Autschbach, J., 2016. Orbitals for analyzing bonding and magnetism of heavy-metal complexes. Comments Inorg. Chem. 36, 215–244. https://doi.org/10.1080/02603594.2015.1121874. Bachmann, R., DiSalvo, F.J., Geballe, T.H., Greene, R.L., Howard, R.E., King, C.N., Kirsch, H.C., Lee, K.N., Schwall, R.E., Thomas, H.-U., Zubeck, R.B., 1972. Heat capacity measurements on small samples at low temperatures. Rev. Sci. Instrum. 43, 205–214. https://doi.org/10.1063/1.1685596. Badı´a-Romano, L., 2016. Nanomagnetism of thin films and molecules: a spectroscopic study, 2014. Estudios de Fı´sica121Prensas de la Universidad de Zaragoza (Ph.D. dissertation of the University of Zaragoza). Badı´a-Romano, L., Bartolom
e, F., Bartolom
e, J., Luzo´n, J., Prodius, D., Turta, C., Mereacre, V., Wilhelm, F., Rogalev, A., 2013. Field-induced internal Fe and Ln spin reorientation in butterfly {Fe3LnO2} (Ln ¼ Dy and Gd) single-molecule magnets. Phys. Rev. B. 87, 184403. https:// doi.org/10.1103/PhysRevB.87.184403. Badı´a-Romano, L., Rubı´n, J., Bartolom
e, F., Bartolom
e, J., Luzo´n, J., Prodius, D., Turta, C., Mereacre, V., Wilhelm, F., Rogalev, A., 2015. Intracluster interactions in butterfly {Fe3LnO2} molecules with the non-Kramers ions Tb(III) and Ho(III). Phys. Rev. B. 92, 64411. https://doi.org/10.1103/PhysRevB.92.064411. Bag, P., Goura, J., Mereacre, V., Novitchi, G., Powell, A.K., Chandrasekhar, V., 2014. Synthesis, magnetism and M€ossbauer studies of tetranuclear heterometallic {Fe(III)2Ln2}(Ln ¼ Gd, Dy, Tb) complexes: evidence of slow relaxation of magnetization in the terbium analogue. Dalton Trans. 43, 16366–16376. https://doi.org/10.1039/c4dt01451k. Baker, M.L., Mutka, H., 2012. Neutron spectroscopy of molecular nanomagnets. Eur. Phys. J. Spec. Top. 213, 53–68. https://doi.org/10.1140/epjst/e2012-01663-6. Baker, M.L., Tanaka, T., Murakami, R., Ohira-Kawamura, S., Nakajima, K., Ishida, T., Nojiri, H., 2015. Relationship between torsion and anisotropic exchange coupling in a Tb(III)radical-based single-molecule magnet. Inorg. Chem. 54, 5732–5738. https://doi.org/10.1021/ acs.inorgchem.5b00300. Bala, S., Bishwas, M.S., Pramanik, B., Khanra, S., Fromm, K.M., Poddar, P., Mondal, R., 2015. Construction of polynuclear lanthanide (Ln ¼ DyIII, TbIII, and NdIII) cage complexes using pyridine-pyrazole-based ligands: versatile molecular topologies and SMM behavior. Inorg. Chem. 54, 8197–8206. https://doi.org/10.1021/acs.inorgchem.5b00334. Baldovı´, J.J., Borra´s-Almenar, J.J., Clemente-Juan, J.M., Coronado, E., Gaita-Arin˜o, A., 2012. Modeling the properties of lanthanoid single-ion magnets using an effective point-charge approach. Dalton Trans. 41, 13705–13710. https://doi.org/10.1039/c2dt31411h.

254 Handbook of Magnetic Materials Baldovı´, J.J., Cardona-Serra, S., Clemente-Juan, J.M., Coronado, E., Gaita-Arin˜o, A., Palii, A., 2013a. SIMPRE: a software package to calculate crystal field parameters, energy levels, and magnetic properties on mononuclear lanthanoid complexes based on charge distributions. J. Comput. Chem. 34, 1961–1967. https://doi.org/10.1002/jcc.23341. Baldovı´, J.J., Cardona-Serra, S., Clemente-Juan, J.M., Coronado, E., Gaita-Arin˜o, A., Prima-Garc´ıa, H., 2013b. Coherent manipulation of spin qubits based on polyoxometalates: the case of the single ion magnet [GdW30P5O110]14–. Chem. Commun. 49, 8922–8924. https://doi.org/ 10.1039/c3cc44838j. Baldovı´, J.J., Clemente-Juan, J.M., Coronado, E., Duan, Y., Gaita-Arin˜o, A., Gim
enez-Saiz, C., 2014. Construction of a general library for the rational design of nanomagnets and spin qubits based on mononuclear f-block complexes. The polyoxometalate case. Inorg. Chem. 53, 9976–9980. https://doi.org/10.1021/ic501867d. Baldovı´, J.J., Gaita-Arin˜o, A., Coronado, E., 2015. Modeling the magnetic properties of lanthanide complexes: relationship of the REC parameters with Pauling electronegativity and coordination number. Dalton Trans. 44, 12535–12538. https://doi.org/10.1039/ c5dt01839k. Baldovı´, J.J., Duan, Y., Bustos, C., Cardona-Serra, S., Gouzerh, P., Villanneau, R., Gontard, G., Clemente-Juan, J.M., Gaita-Arin˜o, A., Gim
enez-Saiz, C., Proust, A., Coronado, E., 2016a. Single ion magnets based on lanthanoid polyoxomolybdate complexes. Dalton Trans. 45, 16653–16660. https://doi.org/10.1039/C6DT02258H. Baldovı´, J.J., Duan, Y., Morales, R., Gaita-Arin˜o, A., Ruiz, E., Coronado, E., 2016b. Rational design of lanthanoid single-ion magnets: predictive power of the theoretical models. Chem. A Eur. J. 22, 13532–13539. https://doi.org/10.1002/chem.201601741. Baniodeh, A., Mondal, A., Galeev, R., Sukhanov, A., Eremina, R., Voronkova, V., Anson, C.E., Powell, A.K., 2017. How far can the anisotropy deviate from uniaxiality in a Dy-based single-molecule magnet? Dinuclear Dy(III) complex study. Appl. Magn. Reson. 48, 101–113. https://doi.org/10.1007/s00723-016-0852-y. Barbara, B., Wernsdorfer, W., Sampaio, L.C., Park, J.G., Paulsen, C., Novak, M.A., Ferr
e, R., Mailly, D., Sessoli, R., Caneschi, A., Hasselbach, K., Benoit, A., Thomas, L., 1995. Mesoscopic quantum tunneling of the magnetization. J. Magn. Magn. Mater. 140–144, 1825–1828. https://doi.org/10.1016/0304-8853(94)00585-0. Barra, A.L., Gatteschi, D., Sessoli, R., 1997. High frequency EPR spectra of a molecular nanomagnet provide a key to understand quantum tunneling of the magnetization. Phys. Rev. B 56, 8192–8198. https://doi.org/10.1103/PhysRevB.56.8192. Bartolom
e, F., Bartolome, J., Oushoorn, R.L., Guillou, O., Kahn, O., 1995. 3D ordering of the rare earth-molecular based magnets RE2(Ox)[Cu(Pba)]3[Cu(H2O)5]20H2O (Re ¼ Gd, La). J. Magn. Magn. Mater. 140, 1711–1712. https://doi.org/10.1016/0304-8853(94)01180-X. Bartolom
e, F., Bartolom
e, J., Benelli, C., Caneschi, A., Gatteschi, D., Paulsen, C., Pini, M.G., Rettori, A., Sessoli, R., Volokitin, Y., 1996. Effect of chiral domain walls on the specific heat of Gd(hfac)3NITR (R equals ethyl, isopropyl, methyl, phenyl) molecular magnetic chains. Phys. Rev. Lett. 77, 382–385. https://doi.org/10.1103/PhysRevLett.77.382. Bartolom
e, J., Filoti, G., Kuncser, V., Schinteie, G., Mereacre, V., Anson, C.E., Powell, A.K., Prodius, D., Turta, C., 2009. Magnetostructural correlations in the tetranuclear series of {Fe3LnO2} butterfly core clusters: magnetic and M€ossbauer spectroscopic study. Phys. Rev. B 80, 14430. https://doi.org/10.1103/PhysRevB.80.014430. Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Luis, F., Turta, C., 2013. {Dy(a-fur)3}n: from double relaxation single-ion magnet behavior to 3D ordering. Dalton Trans. 42, 10153–10171. https://doi.org/10.1039/c3dt51080h.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 255

Bartolom
e, E., Bartolom
e, J., Melnic, S., Prodius, D., Shova, S., Arauzo, A., Luzo´n, J., Badı´aRomano, L., Luis, F., Turta, C., 2014a. Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(a-fur)8}n furoate polymers. Dalton Trans. 43, 10999–11013. https://doi.org/ 10.1039/C4DT00538D. Molecular Magnets: Physics and Applications. Bartolom
e, J., Luis, F., Ferna´ndez, J.F. (Eds.), 2014b. Springer-Verlag, Berlin, Heidelberg, Germany. https://doi.org/10.1007/978-3-64240609-6. Bartolom
e, E., Bartolom
e, J., Arauzo, A., Luzo´n, J., Badı´a, L., Cases, R., Luis, F., Melnic, S., Prodius, D., Shova, S., Turta, C., 2016. Antiferromagnetic single-chain magnet slow relaxation in the {Tb(a-fur)3}n polymer with non-Kramers ions. J. Mater. Chem. C 4, 5038–5050. https://doi.org/10.1039/C6TC00919K. Bender, M., Comba, P., Demeshko, S., Großhauser, M., M€uller, D., Wadepohl, H., 2015. Theoretically predicted and experimentally observed relaxation pathways of two heterodinuclear 3d-4f complexes. Zeitsch. Anorg. Allg. Chem. 641, 2291–2299. https://doi.org/10.1002/ zaac.201500595. Benelli, C., Gatteschi, D., Sessoli, R., Rettori, A., Pini, M.G., Bartolom
e, F., Bartolom
e, J., 1995. Gd(hfac)3NITEt molecular magnetic chain. J. Magn. Magn. Mater. 140–144, 1649–1650. https://doi.org/10.1016/0304-8853(94)01100-1. Bernot, K., Bogani, L., Caneschi, A., Gatteschi, D., Sessoli, R., 2006. A family of rare-earth-based single chain magnets: playing with anisotropy. J. Am. Chem. Soc. 128, 7947–7956. https:// doi.org/10.1021/ja061101l. Bernot, K., Luzon, J., Bogani, L., Etienne, M., Sangregorio, C., Shanmugam, M., Caneschi, A., Sessoli, R., Gatteschi, D., 2009a. Magnetic anisotropy of dysprosium(III) in a low-symmetry environment: a theoretical and experimental investigation. J. Am. Chem. Soc. 131, 5573–5579. https://doi.org/10.1021/ja8100038. Bernot, K., Luzon, J., Caneschi, A., Gatteschi, D., Sessoli, R., Bogani, L., Vindigni, A., Rettori, A., Pini, M.G., 2009b. Spin canting in a Dy-based single-chain magnet with dominant next-nearest-neighbor antiferromagnetic interactions. Phys. Rev. B 79, 134419. https://doi. org/10.1103/PhysRevB.79.134419. Bethe, H.A., 1929. Splitting of terms in crystals. Ann. Phys. 3, 133–206. Bethe, H.A., Salpeter, E.E., 1957. Quantum mechanics of one- and two-electron systems. In: Atoms I/Atome I. Encyclopedia of Physics/Handbuch der Physik, vol. 7/35. Springer, Berlin, Heidelberg, pp. 88–436. https://doi.org/10.1007/978-3-642-45869-9_2. Bi, Y., Guo, Y.N., Zhao, L., Guo, Y., Lin, S.Y., Jiang, S.D., Tang, J., Wang, B.W., Gao, S., 2011. Capping ligand perturbed slow magnetic relaxation in dysprosium single-ion magnets. Chem. A Eur. J. 17, 12476–12481. https://doi.org/10.1002/chem.201101838. Bi, Y., Chen, C., Zhao, Y.-F., Zhang, Y.-Q., Jiang, S.-D., Wang, B.-W., Han, J.-B., Sun, J.-L., Bian, Z.-Q., Wang, Z.-M., Gao, S., 2016. Thermostability and photoluminiscence of Dy(III) single-molecule magnets under a magnetic field. Chem. Sci. 7, 5020–5031. Biswas, S., Jena, H.S., Adhikary, A., Konar, S., 2014a. Two isostructural 3D lanthanide coordination networks (Ln ¼ Gd3+, Dy3+) with squashed cuboid-type nanoscopic cages showing significant cryogenic magnetic refrigeration and slow magnetic relaxation. Inorg. Chem. 53, 3926–3928. https://doi.org/10.1021/ic4030316. Biswas, S., Jena, H.S., Goswami, S., Sanda, S., Konar, S., 2014b. Synthesis and characterization of two lanthanide (Gd3+ and Dy3+)-based three-dimensional metal organic frameworks with squashed metallomacrocycle type building blocks and their magnetic, sorption, and fluorescence properties study. Cryst. Growth Des. 14, 1287–1295. https://doi.org/10.1021/ cg401804e.

256 Handbook of Magnetic Materials Biswas, S., Jena, H.S., Sanda, S., Konar, S., 2015. Proton-conducting magnetic coordination polymers. Chem. A Eur. J. 21, 13793–13801. https://doi.org/10.1002/chem.201501526. Biswas, S., Goura, J., Das, S., Topping, C.V., Brambleby, J., Goddard, P.A., Chandrasekhar, V., 2016a. Octanuclear heterobimetallic {Ni4Ln4} assemblies possessing Ln4 square grid [2  2] motifs: synthesis, structure, and magnetism. Inorg. Chem. 55, 8422–8436. https:// doi.org/10.1021/acs.inorgchem.6b01019. Biswas, S., Mondal, A.K., Konar, S., 2016b. Densely packed lanthanide cubane based 3D metalorganic frameworks for efficient magnetic refrigeration and slow magnetic relaxation. Inorg. Chem. 55, 2085–2090. https://doi.org/10.1021/acs.inorgchem.5b02486. Blackburn, O.A., Kenwright, A.M., Jupp, A.R., Goicoechea, J.M., Beer, P.D., Faulkner, S., 2016. Fluoride binding and crystal-field analysis of lanthanide complexes of tetrapicolyl-appended cyclen. Chem. A Eur. J. 22, 8929–8936. https://doi.org/10.1002/chem.201601170. Blagg, R.J., Muryn, C.A., McInnes, E.J.L., Tuna, F., Winpenny, R.E.P., 2011. Single pyramid magnets: Dy5 pyramids with slow magnetic relaxation to 40K. Angew. Chem. Int. Ed. 50, 6530–6533. https://doi.org/10.1002/anie.201101932. Blagg, R.J., Ungur, L., Tuna, F., Speak, J., Comar, P., Collison, D., Wernsdorfer, W., McInnes, E.J.L., Chibotaru, L.F., Winpenny, R.E.P., 2013. Magnetic relaxation pathways in lanthanide single-molecule magnets. Nat. Chem. 5, 673–678. https://doi.org/10.1038/ nchem.1707. Bloor, D., Copland, G.M., 1972. Far infrared spectra of magnetic ions in crystals. Rep. Prog. Phys. 1173, 1173–1264. https://doi.org/10.1088/0034-4885/35/3/304. Blundell, S., 2001. Magnetism in Condensed Matter, Oxford Master Series in Condensed Matter Physics. OUP, Oxford. Bogani, L., Sangregorio, C., Sessoli, R., Gatteschi, D., 2005. Molecular engineering for singlechain-magnet behavior in a one-dimensional dysprosium-nitronyl nitroxide compound. Angew. Chem. Int. Ed. 44, 5817–5821. https://doi.org/10.1002/anie.200500464. Bogani, L., Vindigni, A., Sessoli, R., Gatteschi, D., 2008. Single chain magnets: where to from here? J. Mater. Chem. 18, 4750. https://doi.org/10.1039/b807824f. Bolvin, H., 2006. An alternative approach to the g-matrix: theory and applications. Chemphyschem 7, 1575–1589. https://doi.org/10.1002/cphc.200600051. Boulon, M.E., Cucinotta, G., Luzon, J., Degl’Innocenti, C., Perfetti, M., Bernot, K., Calvez, G., Caneschi, A., Sessoli, R., 2013. Magnetic anisotropy and spin-parity effect along the series of lanthanide complexes with DOTA. Angew. Chem. Int. Ed. 52, 350–354. https://doi.org/ 10.1002/anie.201205938. Branzoli, F., Carretta, P., Filibian, M., Zoppellaro, G., Graf, M.J., Galan-Mascaros, J.R., Fuhr, O., Brink, S., Ruben, M., 2009a. Spin dynamics in the negatively charged terbium (III) bisphthalocyaninato complex. J. Am. Chem. Soc. 131, 4387–4396. https://doi.org/10.1021/ ja808649g. Branzoli, F., Filibian, M., Carretta, P., Klyatskaya, S., Ruben, M., 2009b. Spin dynamics in the neutral rare-earth single-molecule magnets [TbPc2]0 and [DyPc2]0 from mSR and NMR spectroscopies. Phys. Rev. B 79, 220404R. https://doi.org/10.1103/PhysRevB.79.220404. Brown, A.J., Pinkowicz, D., Saber, M.R., Dunbar, K.R., 2015. A trigonal-pyramidal Erbium(III) single-molecule magnet. Angew. Chem. Int. Ed. 54, 5864–5868. https://doi.org/10.1002/ anie.201411190. Bruker, 2011. Bruker Almanac 2011: Analytical Tables and Product Overview. Bruker, Karlsruhe, Germany. Brya, W.J., Wagner, P.E., 1966. Direct, Orbach and Raman relaxation in dilute cerous magnesium nitrate. Phys. Rev. 147, 239. https://doi.org/10.1103/PhysRev.147.239.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 257

B€ unzli, J.-C.G., Piguet, C., 2005. Taking advantage of luminescent lanthanide ions. Chem. Soc. Rev. 34, 1048. https://doi.org/10.1039/b406082m. Caciuffo, R., Amoretti, G., Murani, A., Sessoli, R., Caneschi, A., Gatteschi, D., 1998. Neutron spectroscopy for the magnetic anisotropy of molecular clusters. Phys. Rev. Lett. 81, 4744–4747. https://doi.org/10.1103/PhysRevLett.81.4744. Campbell, V.E., Bolvin, H., Rivie`re, E., Guillot, R., Wernsdorfer, W., Mallah, T., 2014. Structural and electronic dependence of the single-molecule-magnet behavior of dysprosium(III) complexes. Inorg. Chem. 53, 2598–2605. https://doi.org/10.1021/ic402950j. Canaj, A.B., Tzimopoulos, D.I., Siczek, M., Lis, T., Inglis, R., Milios, C.J., 2015. Enneanuclear [Ni6Ln3] cages: [LnIII3] triangles capping [NiII6] trigonal prisms including a [Ni6Dy3] single-molecule magnet. Inorg. Chem. 54, 7089–7095. https://doi.org/10.1021/acs. inorgchem.5b01149. Canaj, A.B., Siczek, M., Otręba, M., Lis, T., Lorusso, G., Evangelisti, M., Milios, C.J., 2016. Building 1D lanthanide chains and non-symmetrical [Ln2 ] “triple-decker” clusters using salen-type ligands: magnetic cooling and relaxation phenomena. Dalton Trans. 45, 18591–18602. https://doi.org/10.1039/C6DT02932A. Candini, A., Klyatskaya, S., Ruben, M., Wernsdorfer, W., Affronte, M., 2011. Graphene spintronic devices with molecular nanomagnets. Nano Lett. 11, 2634–2639. https://doi.org/10.1021/ nl2006142. Candini, A., Klar, D., Marocchi, S., Corradini, V., Biagi, R., De Renzi, V., del Pennino, U., Troiani, F., Bellini, V., Klyatskaya, S., Ruben, M., Kummer, K., Brookes, N.B., Huang, H., Soncini, A., Wende, H., Affronte, M., 2016. Spin-communication channels between Ln(III) bis-phthalocyanines molecular nanomagnets and a magnetic substrate. Sci. Rep. 6, 21740. https://doi.org/10.1038/srep21740. Caneschi, A., Gatteschi, D., Lalioti, N., Sangregorio, C., Sessoli, R., Venturi, G., Vindigni, A., Rettori, A., Pini, M.G., Novak, M.A., 2001. Cobalt(II)-nitronyl nitroxide chains as molecular magnetic nanowires. Angew. Chem. Int. Ed. Engl. 40, 1760–1761. Caneschi, A., Gatteschi, D., Sessoli, R., Barra, A.-L., Brunel, L.-C.C., Guillot, M., 1991. Alternating current susceptibility, high field magnetization, and millimeter band EPR evidence for a ground S ¼ 10 state in [Mn12O12(Ch3COO)16(H2O)4].2CH3COOH.4H2O. J. Am. Chem. Soc. 113, 5873–5874. https://doi.org/10.1021/ja00015a057. Caneschi, A., Gatteschi, D., Lalioti, N., Sangregorio, C., Sessoli, R., Venturi, G., Vindigni, A., Rettori, A., Pini, M.G., Novak, M.A., 2002. Glauber slow dynamics of the magnetization in a molecular Ising chain. Europhys. Lett. 58, 771–777. https://doi.org/10.1209/epl/i200200416-x. Car, P.-E., Favre, A., Caneschi, A., Sessoli, R., 2015. Single molecule magnet behaviour in a rare trinuclear {Cr III DyIII2} methoxo-bridged complex. Dalton Trans. 44, 15769–15773. https:// doi.org/10.1039/C5DT02459E. Cardona-Serra, S., Clemente-Juan, J.M., Coronado, E., Gaita-Arin˜o, A., Camo´n, A., Evangelisti, M., Luis, F., Martı´nez-P
erez, M.J., Ses
e, J., 2012. Lanthanoid single-ion magnets based on polyoxometalates with a 5-fold symmetry: the series [LnP5W30O110]12– (Ln3+ ¼ Tb, Dy, Ho, Er, Tm, and Yb). J. Am. Chem. Soc. 134, 14982–14990. https://doi.org/10.1021/ ja305163t. Carra, P., Thole, B.T., Altarelli, M., Wang, X., 1993. X-ray circular dichroism and local magnetic fields. Phys. Rev. Lett. 70, 694–697. https://doi.org/10.1103/PhysRevLett.70.694. Carretta, P., Lascialfari, A., 2007. NMR-MRI, mSR and M€ossbauer Spectroscopies in Molecular Magnets, NMR-MRI, mSR and M€ossbauer Spectroscopies in Molecular Magnets. Springer Milan, Milano. https://doi.org/10.1007/978-88-470-0532-7.

258 Handbook of Magnetic Materials Casimir, H.B.G., du Pr
e, F.K., 1938. Note on the thermodynamic interpretation of paramagnetic relaxation phenomena. Phys. Ther. 5, 507–511. https://doi.org/10.1016/S0031-8914(38) 80164-6. Castro, A.B., Jung, J., Golhen, S., Guennic, B.L., Ouahab, L., Cador, O., Pointillart, F., 2016. Slow magnetic relaxation in unprecedented mono-dimensional coordination polymer of ytterbium involving tetrathiafulvalene-dicarboxylate linker. Magnetochemistry 3, 26. https://doi. org/10.3390/magnetochemistry2020026. Champion, G., Lalioti, N., Tangoulis, V., Arrio, M.A., Sainctavit, P., Villain, F., Caneschi, A., Gatteschi, D., Giorgetti, C., Baudelet, F., Verdaguer, M., Dit Moulin, C.C., 2003. XMCD for monitoring exchange interactions. The role of the Gd 4f and 5d orbitals in metal-nitronyl nitroxide magnetic chains. J. Am. Chem. Soc. 125, 8371–8376. https://doi.org/10.1021/ ja034608u. Chandrasekhar, V., Das, S., Dey, A., Hossain, S., Kundu, S., Colacio, E., 2014. Linear, edgesharing heterometallic trinuclear [CoII-LnIII-CoII] (LnIII ¼ GdIII, DyIII, TbIII, and HoIII) complexes: slow relaxation of magnetization in the DyIII derivative. Eur. J. Inorg. Chem. 397–406. https://doi.org/10.1002/ejic.201301171. Chen, M., San˜udo, E.C., Jim
enez, E., Fang, S.M., Liu, C.S., Du, M., 2014. Lanthanide-organic coordination frameworks showing new 5-connected network topology and 3D ordered array of single-molecular magnet behavior in the Dy case. Inorg. Chem. 53, 6708–6714. https:// doi.org/10.1021/ic500490x. Chen, S., Mereacre, V., Anson, C.E., Powell, A.K., 2016a. First heterometallic GaIII–DyIII singlemolecule magnets: implication of GaIII in extracting Fe–Dy interaction. Dalton Trans. 45, 9336–9344. https://doi.org/10.1039/C6DT01364C. Chen, Y.C., Liu, J.L., Ungur, L., Liu, J., Li, Q.W., Wang, L.F., Ni, Z.P., Chibotaru, L.F., Chen, X.M., Tong, M.L., 2016b. Symmetry-supported magnetic blocking at 20 K in pentagonal bipyramidal Dy(III) single-ion magnets. J. Am. Chem. Soc. 138, 2829–2837. https://doi. org/10.1021/jacs.5b13584. Chen, Y.-C., Liu, J.-L., Lan, Y., Zhong, Z.-Q., Mansikkam€aki, A., Ungur, L., Li, Q.-W., Jia, J.-H., Chibotaru, L.F., Han, J.-B., Wernsdorfer, W., Chen, X.-M., Tong, M.-L., 2017. Dynamic magnetic and optical insight into a high performance pentagonal bipyramidal Dy(III) singleion magnet. Chem. A Eur. J. 23, 5708–5715. https://doi.org/10.1002/chem.201606029. Chibotaru, L.F., 2013. Ab initio methodology for pseudospin Hamiltonian of anisotropic magnetic complexes. Adv. Chem. Phys. 153, 397. https://doi.org/10.1002/9781118571767.ch6. Chibotaru, L.F., 2015. Theoretical understanding of anisotropy in molecular nanomagnets. Struct. Bond. 164, 185–230. https://doi.org/10.1007/430. Chibotaru, L.F., Ungur, L., Soncini, A., 2008. The origin of nonmagnetic Kramers doublets in the ground state of dysprosium triangles: evidence for a toroidal magnetic moment. Angew. Chem. Int. Ed. 47, 4126–4129. https://doi.org/10.1002/anie.200800283. Chiboub Fellah, F.Z., Boulefred, S., Chiboub Fellah, A., El Rez, B., Duhayon, C., Sutter, J.P., 2016. Binuclear CuLn complexes (LnIII ¼ Gd, Tb, Dy) of alcohol-functionalized bicompartmental Schiff-base ligand. Hydrogen bonding and magnetic behaviors. Inorg. Chim. Acta 439, 24–29. https://doi.org/10.1016/j.ica.2015.09.032. Chilton, N.F., Collison, D., McInnes, E.J.L., Winpenny, R.E.P., Soncini, A., 2013. An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes. Nat. Commun. 4, 2551. https://doi.org/10.1038/ncomms3551. Chilton, N.F., Goodwin, C.A.P., Mills, D.P., Winpenny, R.E.P., 2015. The first near-linear bis(amide) f-block complex: a blueprint for a high temperature single molecule magnet. Chem. Commun. 51, 101–103. https://doi.org/10.1039/C4CC08312A.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 259

Chow, C.Y., Bolvin, H., Campbell, V.E., Guillot, R., Kampf, J.F., Wernsdorfer, W., Gendron, F.V., Autschbach, J., Pecoraro, V.L., Mallah, T., 2015. Assessing the exchange coupling in binuclear lanthanide(III) complexes and the slow relaxation of the magnetization in the antiferromagnetically coupled Dy2 derivative. Chem. Sci. 6, 4148–4159. https://doi.org/ 10.1039/C5SC01029B. Christou, G., Gatteschi, D., Hendrickson, D.N., Sessoli, R., 2000. Single-molecule magnets. MRS Bull. 25, 66–71. https://doi.org/10.1557/mrs2000.226. Cinti, F., Rettori, A., Pini, M.G., Mariani, M., Micotti, E., Lascialfari, A., Papinutto, N., Amato, A., Caneschi, A., Gatteschi, D., Affronte, M., 2008. Two-step magnetic ordering in quasi-one-dimensional helimagnets: possible experimental validation of Villain’s conjecture about a chiral spin liquid phase. Phys. Rev. Lett. 100, 057203. https://doi.org/10.1103/ PhysRevLett.100.057203. Cirera, J., Ruiz, E., 2008. Exchange coupling in CuIIGdIII dinuclear complexes: a theoretical perspective. C. R. Chim. 11, 1227–1234. https://doi.org/10.1016/j.crci.2008.04.012. Clemente-Juan, J.M., Coronado, E., Gaita-Arin˜o, A., 2015. Mononuclear lanthanide complexes: use of the crystal field theory to design single-ion magnets and spin qubits. In: Layfield, R.A., Murugesu, M. (Eds.), Lanthanides and Actinides in Molecular Magnetism. Wiley-VCH, Weinheim, pp. 27–59. Cl
erac, R., Winpenny, R.E.P., 2017. Single-molecule magnets and related phenomena. Struct. Bond. 172, 35–48. https://doi.org/10.1007/430_2015_198. Cole, K.S., Cole, R.H., 1941. Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341. https://doi.org/10.1063/1.1750906. Corradini, V., Ghirri, A., Candini, A., Biagi, R., Del Pennino, U., De Renzi, V., Dotti, G., Otero, E., Hooper, T.N., Inglis, R., Brechin, E.K., Affronte, M., 2014. Surface investigation on Gd4M8 (M ¼ Zn, Ni) single molecule coolers. Adv. Funct. Mater. 24, 4782–4788. https://doi.org/10.1002/adfm.201400460. Cosquer, G., Pointillart, F., Jung, J., Le Guennic, B., Golhen, S., Cador, O., Guyot, Y., Brenier, A., Maury, O., Ouahab, L., 2014. Alkylation effects in lanthanide complexes involving tetrathiafulvalene chromophores: experimental and theoretical correlation between magnetism and near-infrared emission. Eur. J. Inorg. Chem. 69–82. https://doi.org/10.1002/ ejic.201301358. Costes, J.P., Titos-Padilla, S., Oyarzabal, I., Gupta, T., Duhayon, C., Rajaraman, G., Colacio, E., 2015. Analysis of the role of peripheral ligands coordinated to ZnII in enhancing the energy barrier in luminescent linear trinuclear Zn-Dy-Zn single-molecule magnets. Chem. A Eur. J. 21, 15785–15796. https://doi.org/10.1002/chem.201501500. Coulon, C., Cl
erac, R., Lecren, L., Wernsdorfer, W., Miyasaka, H., 2004. Glauber dynamics in a single-chain magnet: from theory to real systems. Phys. Rev. B 69, 132408. https://doi.org/ 10.1103/PhysRevB.69.132408. Coulon, C., Pianet, V., Urdampilleta, M., Cl
erac, R., 2014. Single-chain magnets and related systems. In: Gao, S. (Ed.), Molecular Nanomagnets and Related Phenomena. Struct. Bond. 164, 143–184. https://doi.org/10.1007/430_2014_154. Coutinho, J.T., Pereira, L.C.J., Martı´n-Ramos, P., Ramos Silva, M., Zheng, Y.X., Liang, X., Ye, H.Q., Peng, Y., Baker, P.J., Wyatt, P.B., Gillin, W.P., 2015. Field-induced single-ion magnetic behaviour in a highly luminescent Er3+ complex. Mater. Chem. Phys. 160, 429–434. https://doi.org/10.1016/j.matchemphys.2015.04.059. Cremades, E., Go´mez-Coca, S., Aravena, D., Alvarez, S., Ruiz, E., 2012. Theoretical study of exchange coupling in 3d-Gd complexes: large magnetocaloric effect systems. J. Am. Chem. Soc. 134, 10532–10542. https://doi.org/10.1021/ja302851n.

260 Handbook of Magnetic Materials Cucinotta, G., Perfetti, M., Luzon, J., Etienne, M., Car, P.-E.E., Caneschi, A., Calvez, G., Bernot, K., Sessoli, R., 2012. Magnetic anisotropy in a dysprosium/DOTA single-molecule magnet: beyond simple magneto-structural correlations. Angew. Chem. Int. Ed. 51, 1606–1610. https://doi.org/10.1002/anie.201107453. Da Cunha, T.T., Jung, J., Boulon, M.E., Campo, G., Pointillart, F., Pereira, C.L.M., Le Guennic, B., Cador, O., Bernot, K., Pineider, F., Golhen, S., Ouahab, L., 2013. Magnetic poles determinations and robustness of memory effect upon solubilization in a DyIII-based single ion magnet. J. Am. Chem. Soc. 135, 16332–16335. https://doi.org/10.1021/ja4089956. Dartyge, E., Baudelet, F., Giorgetti, C., Odin, S., 1998. X-ray magnetic circular dichroism as a tool for magnetic studies of 4f compounds. J. Alloys Compd. 275–277, 526–532. https:// doi.org/10.1016/S0925-8388(98)00384-3. Das, S., Dey, A., Biswas, S., Colacio, E., Chandrasekhar, V., 2014a. Hydroxide-free cubaneshaped tetranuclear [Ln4] complexes. Inorg. Chem. 53, 3417–3426. https://doi.org/10.1021/ ic402827b. Das, S., Dey, A., Kundu, S., Biswas, S., Mota, A.J., Colacio, E., Chandrasekhar, V., 2014b. Linear {NiII-LnIII-NiII} complexes containing twisted planar Ni(m-phenolate)2Ln fragments: synthesis, structure, and magnetothermal properties. Chem. Asian J. 9, 1876–1887. https://doi. org/10.1002/asia.201402062. Das, S., Hossain, S., Dey, A., Biswas, S., Sutter, J.P., Chandrasekhar, V., 2014c. Molecular magnets based on homometallic hexanuclear lanthanide(III) complexes. Inorg. Chem. 53, 5020–5028. https://doi.org/10.1021/ic500052v. Das, C., Vaidya, S., Gupta, T., Frost, J.M., Righi, M., Brechin, E.K., Affronte, M., Rajaraman, G., Shanmugam, M., 2015a. Single-molecule magnetism, enhanced magnetocaloric effect, and toroidal magnetic moments in a family of Ln4 squares. Chem. A Eur. J. 21, 15639–15650. https://doi.org/10.1002/chem.201502720. Das, S., Bejoymohandas, K.S., Dey, A., Biswas, S., Reddy, M.L.P., Morales, R., Ruiz, E., TitosPadilla, S., Colacio, E., Chandrasekhar, V., 2015b. Amending the anisotropy barrier and luminescence behavior of heterometallic trinuclear linear [MII-LnIII-MII] (LnIII ¼ Gd, Tb, Dy; MII ¼ Mg/Zn) complexes by change from divalent paramagnetic to diamagnetic metal ions. Chem. A Eur. J. 21, 6449–6464. https://doi.org/10.1002/chem.201406666. Das, S., Dey, A., Kundu, S., Biswas, S., Narayanan, R.S., Titos-Padilla, S., Lorusso, G., Evangelisti, M., Colacio, E., Chandrasekhar, V., 2015c. Decanuclear Ln10 wheels and vertex-shared spirocyclic Ln5 cores: synthesis, structure, SMM behavior, and MCE properties. Chem. A Eur. J. 21, 16955–16967. https://doi.org/10.1002/chem.201501992. de Jongh, L.J., Miedema, A.R., 1974. Experiments on simple magnetic model systems. Adv. Phys. 23, 1–260. https://doi.org/10.1080/00018739700101558. Demir, S., Zadrozny, J.M., Nippe, M., Long, J.R., 2012. Exchange coupling and magnetic blocking in bipyrimidyl radical-bridged dilanthanide complexes. J. Am. Chem. Soc. 134, 18546–18549. https://doi.org/10.1021/ja308945d. Demir, S., Zadrozny, J.M., Long, J.R., 2014. Large spin-relaxation barriers for the low-symmetry organolanthanide complexes [Cp*2Ln(BPh4)] (Cp* ¼ pentamethylcyclopentadienyl; Ln ¼ Tb, Dy). Chem. A Eur. J. 20, 9524–9529. https://doi.org/10.1002/chem.201403751. Demir, S., Jeon, I.R., Long, J.R., Harris, T.D., 2015. Radical ligand-containing single-molecule magnets. Coord. Chem. Rev. 289–290, 149–176. https://doi.org/10.1016/.ccr.2014.10.012. Dickie, C.M., Nippe, M., 2016. Magnetization dynamics of a heterometallic Dy-isocarbonyl complex. Inorg. Chem. Front. 3, 97–103. https://doi.org/10.1039/C5QI00224A. Dieke, G.H., 1968. Spectra and Energy Levels of Rare Earth Ions in Crystals. Interscience Publishers, New York.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 261

Ding, Y.-S., Chilton, N.F., Winpenny, R.E.P., Zheng, Y.-Z., 2016a. On approaching the limit of molecular magnetic anisotropy: a near-perfect pentagonal bipyramidal dysprosium(III) single-molecule magnet. Angew. Chem. Int. Ed. 55, 16071–16074. https://doi.org/10.1002/ anie.201609685. Ding, Y.-S., Deng, Y.-F., Zheng, Y.-Z., 2016b. The rise of single-ion magnets as spin qubits. Magnetochemistry 2, 40. https://doi.org/10.3390/magnetochemistry2040040. Dirac, P.A.M., 1930. Principles of Quantum Mechanics. Oxford University Press, Oxford, UK. Domingo, N., Bellido, E., Ruiz-Molina, D., 2012. Advances on structuring, integration and magnetic characterization of molecular nanomagnets on surfaces and devices. Chem. Soc. Rev. 41, 258. https://doi.org/10.1039/c1cs15096k. Dong, Y., Yan, P., Zou, X., Liu, T., Li, G., 2015. Exploiting single-molecule magnets of b-diketone dysprosium complexes with C3v symmetry: suppression of quantum tunneling of magnetization. J. Mater. Chem. 3, 4407–4415. Dong, Y., Yan, P., Zou, X., Yao, X., Hou, G., Li, G., 2016. Auxiliary ligand field dominated single-molecule magnets of a series of indole-derivative b-diketone mononuclear Dy(III) complexes. Dalton Trans. 45, 9148–9157. Douglas, M., Kroll, N.M., 1974. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. (NY). 82, 89–155. https://doi.org/10.1016/0003-4916(74)90333-9. Dreiser, J., 2015. Molecular lanthanide single-ion magnets: from bulk to submonolayers. J. Phys. Condens. Matter 27, 183203. https://doi.org/10.1088/0953-8984/27/18/183203. Dreiser, J., Westerstr€om, R., Piamonteze, C., Nolting, F., Rusponi, S., Brune, H., Yang, S., Popov, A., Dunsch, L., Greber, T., 2014a. X-ray induced demagnetization of single-molecule magnets. Appl. Phys. Lett. 105, 18–21. https://doi.org/10.1063/1.4891485. Dreiser, J., Westerstr€om, R., Zhang, Y., Popov, A.A., Dunsch, L., Kr€amer, K., Liu, S.-X., Decurtins, S., Greber, T., 2014b. The metallofullerene field-induced single-ion magnet HoSc2N@C80. Chem. A Eur. J. 20, 13536–13540. https://doi.org/10.1002/chem. 201403042. Dreiser, J., Pacchioni, G.E., Donati, F., Gragnaniello, L., Cavallin, A., Pedersen, K.S., Bendix, J., Delley, B., Pivetta, M., Rusponi, S., Brune, H., 2016. Out-of-plane alignment of Er(trensal) easy magnetization axes using graphene. ACS Nano 10, 2887–2892. https://doi.org/ 10.1021/acsnano.5b08178. Drung, D., Storm, J.-H., Ruede, F., Kirste, A., Regin, M., Schurig, T., Repolles, A.M., Sese, J., Luis, F., 2014. Thin-film microsusceptometer with integrated nanoloop. IEEE Trans. Appl. Supercond. 24, 160026. https://doi.org/10.1109/TASC.2014.2318322. Edelmann, F.T., 2016. Lanthanides and actinides: annual survey of their organometallic chemistry covering the year 2015. Coord. Chem. Rev. 318, 29–130. https://doi.org/10.1016/j. ccr.2016.04.001. Edmonds, A.R., 1957. Angular Momentum in Quantum Mechanics. Princeton University Press, New Jersey. Elliott, R.J., Stevens, K.W.H., 1953a. The theory of magnetic resonance experiments on salts of the rare earths. Proc. R. Soc. A Math. Phys. Eng. Sci. 218, 553–566. https://doi.org/ 10.1098/rspa.1953.0124. Elliott, R.J., Stevens, K.W.H., 1953b. The magnetic properties of certain rare-earth ethyl sulphates. Proc. R. Soc. Lond. A Math. Phys. Sci. 219, 387–404. https://doi.org/10.1098/ rspa.1953.0155. Evangelisti, M., Bartolom
e, F., Bartolom
e, J., Kahn, M.L., Kahn, O., 1999. Specific heat and magnetic interactions in spin ladder gadolinium and copper-based molecular ferromagnets. J. Magn. Magn. Mater. 196, 584–585. https://doi.org/10.1016/S0304-8853(98)00846-4.

262 Handbook of Magnetic Materials Evangelisti, M., Bartolom
e, J., Mettes, F., Jongh, L.J.D., Kahn, M.L., Mathonie, C., Kahn, O., 2001. Specific heat of spin ladder lanthanide and transition-metal-based molecular magnets. Polyhedron 20, 1447–1450. https://doi.org/10.1016/S0277-5387(01)00634-9. Evangelisti, M., Kahn, M.L., Bartolom
e, J., de Jongh, L.J., Meyers, C., Leandri, J., Leroyer, Y., Mathoniere, C., 2003. Magnetic and thermal properties of 4f-3d ladder-type molecular compounds. Phys. Rev. B 68, 184405. https://doi.org/10.1103/PhysRevB.68.184405. Evangelisti, M., Luis, F., Mettes, F.L., Aliaga, N., Aromı´, G., Alonso, J.J., Christou, G., De Jongh, L.J., 2004. Magnetic long-range order induced by quantum relaxation in single-molecule magnets. Phys. Rev. Lett. 93, 117202. https://doi.org/10.1103/PhysRevLett.93.117202. Fagaly, R.L., 2006. Superconducting quantum interference device instruments and applications. Rev. Sci. Instrum. 77, 101101. https://doi.org/10.1063/1.2354545. Fahrendorf, S., Atodiresei, N., Besson, C., Caciuc, V., Matthes, F., Bl€ugel, S., K€ogerler, P., B€ urgler, D.E., Schneider, C.M., 2013. Accessing 4f-states in single-molecule spintronics. Nat. Commun. 4, 2425. https://doi.org/10.1038/ncomms3425. Fahrendorf, S., Matthes, F., B€urgler, D.E., Schneider, C.M., Atodiresei, N., Caciuc, V., Bl€ugel, S., Besson, C., K€ ogerler, P., 2014. Structural integrity of single bis(phthalocyaninato)-neodymium(III) molecules on metal surfaces with different reactivity. Spine 4, 1440007. https:// doi.org/10.1142/S2010324714400074. Fang, M., Li, X., Cui, P., Zhao, B., 2015. Ferromagnetic interactions and slow magnetic relaxation behaviors of two lanthanide coordination polymers bridged by 2,6-naphthalenedicarboxylate ligand. J. Solid State Chem. 223, 138–143. https://doi.org/10.1016/j.jssc.2014.07.020. Feltham, H.L.C., Brooker, S., 2014. Review of purely 4f and mixed-metal nd-4f single-molecule magnets containing only one lanthanide ion. Coord. Chem. Rev. 276, 1–33. https://doi.org/ 10.1016/j.ccr.2014.05.011. Feltham, H.L.C., Lan, Y., Kl€ower, F., Ungur, L., Chibotaru, L.F., Powell, A.K., Brooker, S., 2011. A non-sandwiched macrocyclic monolanthanide single-molecule magnet: the key role of axiality. Chem. Eur. J. 17, 4362–4365. Ferna´ndez, J.F., Luis, F., Bartolom
e, J., 1998. Time dependent specific heat of a magnetic quantum tunneling system. Phys. Rev. Lett. 80, 5659–5662. https://doi.org/10.1103/ PhysRevLett.80.5659. Fieser, M.E., MacDonald, M.R., Krull, B.T., Bates, J.E., Ziller, J.W., Furche, F., Evans, W.J., 2015. Structural, spectroscopic, and theoretical comparison of traditional vs recently discovered Ln2+ ions in the [K(2.2.2-cryptand)][(C5H4SiMe3)3Ln] complexes: the variable nature of Dy2+ and Nd2 +. J. Am. Chem. Soc. 137, 369–382. https://doi.org/10.1021/ja510831n. Flanagan, B.M., Bernhardt, P.V., Krausz, E.R., L€uthi, S.R., Riley, M.J., 2001. Ligand-field analysis of an Er(III) complex with a heptadentate tripodal N4O3 ligand. Inorg. Chem. 40, 5401–5407. https://doi.org/10.1021/ic0103244. Flokstra, J., 1974. PhD Thesis. Technical University of Twente. Fominaya, F., Villain, J., Gandit, P., 1997. Heat capacity anomalies induced by magnetization quantum tunneling in a Mn12O12-acetate single crystal. Phys. Rev. Lett. 79, 1126. https:// doi.org/10.1103/PhysRevLett.79.1126. Freeman, A.J., Watson, R.E., 1962. Theoretical investigation of some magnetic and spectroscopic properties of rare-earth ions. Phys. Rev. 127, 2058–2075. https://doi.org/10.1103/ PhysRev.127.2058. Friedman, J.R., Sarachik, M.P., Ziolo, R., Tejada, J., 1996. Macroscopic measurement of resonant magnetization tunneling in high-spin molecules. Phys. Rev. Lett. 76, 3830–3833. https://doi. org/10.1103/PhysRevLett.76.3830.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 263

Ganivet, C.R., Ballesteros, B., De La Torre, G., Clemente-Juan, J.M., Coronado, E., Torres, T., 2013. Influence of peripheral substitution on the magnetic behavior of single-ion magnets based on homo- and heteroleptic TbIII bis(phthalocyaninate). Chem. A Eur. J. 19, 1457–1465. https://doi.org/10.1002/chem.201202600. Ganzhorn, M., Klyatskaya, S., Ruben, M., Wernsdorfer, W., 2013. Strong spin–phonon coupling between a single-molecule magnet and a carbon nanotube nanoelectromechanical system. Nat. Nanotechnol. 8, 165–169. https://doi.org/10.1038/nnano.2012.258. Ganzhorn, M., Klyatskaya, S., Ruben, M., Wernsdorfer, W., 2016. Quantum Einstein-de Haas effect. Nat. Commun. 711443https://doi.org/10.1038/ncomms11443. Gao, S. (Ed.), 2015. Molecular Nanomagnets and Related Phenomena. In: Structure and Bonding. Springer-Verlag, Berlin, Heidelberghttps://doi.org/10.1007/430_2014_150. Gao, C., Yang, Q., Wang, B.-W., Wang, Z.-M., Gao, S., 2016a. Evaporable lanthanide single-ion magnet. Cryst. Eng. Comm. 18, 4165–4171. Gao, F., Feng, X., Yang, L., Chen, X., 2016b. New sandwich-type lanthanide complexes based on closed-macrocyclic Schiff base and phthalocyanine molecules. Dalton Trans. 45, 7476–7482. https://doi.org/10.1039/C6DT00683C. Gao, H.-L., Jiang, L., Liu, S., Shen, H.-Y., Wang, W.-M., Cui, J.-Z., 2016c. Multiple magnetic relaxation processes, magnetocaloric effect and fluorescence properties of rhombus-shaped tetranuclear rare earth complexes. Dalton Trans. 45, 253–264. https://doi.org/10.1039/C5DT03790E. Gao, H.L., Jiang, L., Wang, W.M., Wang, S.Y., Zhang, H.X., Cui, J.Z., 2016d. Single-moleculemagnet behavior and fluorescence properties of 8-hydroxyquinolinate derivative-based rare-earth complexes. Inorg. Chem. 55, 8898–8904. https://doi.org/10.1021/acs. inorgchem.6b01420. Garanin, D.A., 1991. Spin tunnelling: a perturbative approach. J. Phys. A. Math. Gen. 24, L61–L62. https://doi.org/10.1088/0305-4470/24/2/002. Garanin, D.A., 2007. Towards a microscopic understanding of the phonon bottleneck. Phys. Rev. B 75, 094409. https://doi.org/10.1103/PhysRevB.75.094409. Garanin, D.A., 2008. Phonon bottleneck in the low-excitation limit. Phys. Rev. B 77, 024429. https://doi.org/10.1103/PhysRevB.77.024429. Gatteschi, D., Vindigni, A., 2014. Single-chain magnets. In: Bartolom
e, J., Luis, F., Ferna´ndez, J.F. (Eds.), Molecular Magnets: Physics and Applications. Springer-Verlag, Berlin, Heidelberg, Germany, pp. 191–220. Gatteschi, D., Sessoli, R., Villain, J., 2006. Molecular Nanomagnets. Oxford University Press, Oxford. Gatteschi, D., Sessoli, R., Sorace, L., 2016. Magnetic bistability in lanthanide-based molecular systems: the role of anisotropy and exchange interactions. In: Bunzli, J. C.-G., Pecharski, V.K. (Eds.), Handbook on the Physics and Chemistry of Rare Earths, ch. 286, vol. 50. Elsevier, Amsterdam, pp. 91–139. (Chapter 286). https://doi.org/10.1016/bs.hpcre.2016.09.003. Gaudin, G., Gandit, P., Chaussy, J., Sessoli, R., 2002. Magnetic field dependent thermodynamics of Fe8 single crystal in the thermally activated regime. J. Magn. Magn. Mater. 242–245, 915–920. https://doi.org/10.1016/S0304-8853(01)01323-3. Gavey, E.L., Al Hareri, M., Regier, J., Carlos, L.D., Ferreira, R.A.S., Razavi, F.S., Rawson, J.M., Pilkington, M., 2015. Placing a crown on Dy III—a dual property Ln III crown ether complex displaying optical properties and SMM behaviour. J. Mater. Chem. C 3, 7738–7747. https:// doi.org/10.1039/C5TC01264C. Gavrikov, A.V., Koroteev, P.S., Dobrokhotova, V.Z., Ilyukhin, A.B., Efimova, N.N., Kirdyankin, D.I., Bykovb, M.A., Ryumina, M.A., Novotortsev, V.M., 2015. Novel

264 Handbook of Magnetic Materials heterometallic polymeric lanthanide acetylacetonates with bridging cymantrenecarboxylate groups—synthesis, magnetism and thermolysis. Polyhedron 102, 48–59. https://doi.org/ 10.1016/j.poly.2014.09.040. Gendron, F., Pa´ez-Herna´ndez, D., Notter, F.P., Pritchard, B., Bolvin, H., Autschbach, J., 2014. Magnetic properties and electronic structure of neptunyl(VI) complexes: wavefunctions, orbitals, and crystal-field models. Chem. A Eur. J. 20, 7994–8011. https://doi.org/10.1002/ chem.201305039. Gendron, F., Pritchard, B., Bolvin, H., Autschbach, J., 2015. Single-ion 4f element magnetism: an ab-initio look at Ln(COT)2. Dalton Trans. 44, 19886–19900. https://doi.org/10.1039/ c5dt02858b. Gerritsma, G.J., Flokstra, J., Hartemink, G.A., Scholten, J.J.M., Vermeulen, A.J.W.A., van der Marel, L.C., 1978. The influence of thermal conduction in spin-lattice relaxation experiments. Physica B + C 95, 173–182. https://doi.org/10.1016/0378-4363(78)90089-X. Ghosh, S., Datta, S., Friend, L., Cardona-Serra, S., Gaita-Arino, A., Coronado, E., Hill, S., 2012. Multi-frequency EPR studies of a mononuclear holmium single-molecule magnet based on the polyoxometalate [HoIII(W5O18)2]9. Dalton Trans. 41, 13697–13704. https://doi.org/ 10.1039/C2DT31674A. Ghosh, S., Mahapatra, P., Kanetomo, T., Drew, M.G.B., Ishida, T., Ghosh, A., 2016. Syntheses, crystal structure and magnetic properties of an unprecedented one-dimensional coordination polymer derived from an {(NiL)2Ln} node and a pyrazine spacer (H2L ¼ N,N0 -bis(salicylidene)-1,3-propanediamine, Ln ¼Gd, Tb and Dy). ChemistrySelect 1, 2722–2729. https:// doi.org/10.1002/slct.201600637. Giansiracusa, M.J., Vonci, M., Van Den Heuvel, W., Gable, R.W., Moubaraki, B., Murray, K.S., Yu, D., Mole, R.A., Soncini, A., Boskovic, C., 2016. Carbonate-bridged lanthanoid triangles: single-molecule magnet behavior, inelastic neutron scattering, and ab initio studies. Inorg. Chem. 55, 5201–5214. https://doi.org/10.1021/acs.inorgchem.6b00108. Gim
enez-Agullo´, N., Sa´enz de Pipao´n, C., Adriaenssens, L., Filibian, M., Martı´nez-Belmonte, M., Escudero-Ada´n, E.C., Carretta, P., Ballester, P., Gala´n-Mascaro´s, J.R., 2014. Single-moleculemagnet behavior in the family of [Ln(OETAP)2] double-Decker complexes (Ln ¼ Lanthanide, OETAP ¼ Octa(ethyl)tetraazaporphyrin). Chem. A Eur. J. 20, 12817–12825. Giordmaine, J.A., Nash, F.R., 1965. Phonon bottleneck and frequency distribution in paramagnetic relaxation at low temperatures. Phys. Rev. 138, A1510–A1523. https://doi.org/ 10.1103/PhysRev.138.A1510. Giraud, R., Wernsdorfer, W., Tkachuk, A.M., Mailly, D., Barbara, B., 2001. Nuclear spin driven quantum relaxation in LiY0.998Ho0.002F4. Phys. Rev. Lett. 87, 57203. https://doi.org/ 10.1103/PhysRevLett.87.057203. Girginova, P.I., Pereira, L.C.J., Coutinho, J.T., Santos, I.C., Almeida, M., 2014. Slow magnetic relaxation in lanthanide ladder type coordination polymers. Dalton Trans. 43, 1897–1905. https://doi.org/10.1039/c3dt52748d. Glauber, R.J., 1963. Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307. https://doi.org/10.1063/1.1703954. Go´mez-Coca, S., Aravena, D., Morales, R., Ruiz, E., 2015. Large magnetic anisotropy in mononuclear metal complexes. Coord. Chem. Rev. 289–290, 379–392. https://doi.org/10.1016/j. ccr.2015.01.021. Gonidec, M., Biagi, R., Corradini, V., Moro, F., De Renzi, V., Del Pennino, U., Summa, D., Muccioli, L., Zannoni, C., Amabilino, D.B., Veciana, J., 2011. Surface supramolecular organization of a terbium(III) double-decker complex on graphite and its single molecule magnet behavior. J. Am. Chem. Soc. 133, 6603–6612. https://doi.org/10.1021/ja109296c.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 265

Gorczynski, A., Kubicki, M., Pinkowicz, D., Petka, R., Patroniak, V., Podgajny, R., 2016. The first example of erbium triple-stranded helicates displaying SMM behaviour. Dalton Trans. 44, 16833–16839. https://doi.org/10.1039/c5dt02554k. Goswami, S., Biswas, S., Tomar, K., Konar, S., 2016. Tuning the Magnetoluminescence behavior of lanthanide complexes having Sphenocorona and cubic coordination geometries. Eur. J. Inorg. Chem. 2016, 2774–2782. https://doi.org/10.1002/ejic.201600152. Goura, J., Chakraborty, A., Walsh, J.P.S., Tuna, F., Chandrasekhar, V., 2015. Hexanuclear 3d–4f neutral CoII2 LnIII4 clusters: synthesis, structure, and magnetism. Cryst. Growth Des. 15, 3157–3165. https://doi.org/10.1021/acs.cgd.5b00588. Gregson, M., Chilton, N.F., Ariciu, A.-M., Tuna, F., Crowe, I.F., Lewis, W., Blake, A.J., Collison, D., McInnes, E.J.L., Winpenny, R.E.P., Liddle, S.T., 2016. A monometallic lanthanide bis(methanediide) single molecule magnet with a large energy barrier and complex spin relaxation behaviour. Chem. Sci. 7, 155–165. https://doi.org/10.1039/C5SC03111G. Griffith, J.S., 1963. Spin Hamiltonian for even-electron systems having even multiplicity. Phys. Rev. 132, 316. https://doi.org/10.1103/PhysRev.132.316. Grindell, R., Vieru, V., Pugh, T., Chibotaru, L.F., Layfield, R.A., 2016. Magnetic frustration in a hexaazatrinaphthylene-bridged trimetallic dysprosium single-molecule magnet. Dalton Trans. 45, 16556–16560. https://doi.org/10.1039/C6DT01763K. Guo, F.-S., Layfield, R.A., 2017. Strong direct exchange coupling and single-molecule magnetism in indigo-bridged lanthanide dimers. Chem. Commun. 53, 3130–3133. https://doi.org/ 10.1039/c7cc01046j. Guo, Y.N., Xu, G.F., Wernsdorfer, W., Ungur, L., Guo, Y., Tang, J., Zhang, H.J., Chibotaru, L.F., Powell, A.K., 2011. Strong axiality and Ising exchange interaction suppress zero-field tunneling of magnetization of an asymmetric Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 11948–11951. https://doi.org/10.1021/ja205035g. Guo, Y.-N., Ungur, L., Granroth, G.E., Powell, A.K., Wu, C., Nagler, S.E., Tang, J., Chibotaru, L.F., Cui, D., 2014. An NCN-pincer ligand dysprosium single-ion magnet showing magnetic relaxation via the second excited state. Sci. Rep. 4, 5471. https://doi.org/10.1038/ srep05471. Guo, P.-H., Liu, J., Wu, Z.-H., Yan, H., Chen, Y.-C., Jia, J.-H., Tong, M.-L., 2015. Singlemolecule-magnet behavior in a [2  2] grid DyIII4 cluster and a dysprosium-doped YIII4 cluster. Inorg. Chem. 54, 8087–8092. https://doi.org/10.1021/acs.inorgchem.5b01322. Gupta T., Rajaraman, G., 2014. How strongly are the magnetic anisotropy and coordination numbers correlated in lanthanide based molecular magnets. J. Chem. Sci. 126, 1569–1579. https:// doi.org/10.1007/s12039-014-0691-z. Gupta, S.K., Rajeshkumar, T., Rajaraman, G., Murugavel, R., 2016a. An air-stable Dy(III) singleion magnet with high anisotropy barrier and blocking temperature. Chem. Sci. 7, 5181–5191. https://doi.org/10.1039/C6SC00279J. Gupta, S.K., Rajeshkumar, T., Rajaraman, G., Murugavel, R., 2016b. An unprecedented zero field neodymium(III) single-ion magnet based on a phosphonic diamide. Chem. Commun. 52, 7168–7171. https://doi.org/10.1039/C6CC03066A. Gysler, M., El Hallak, F., Ungur, L., Marx, R., Hakl, M., Neugebauer, P., Rechkemmer, Y., Lan, Y., Sheikin, I., Orlita, M., Anson, C.E., Powell, A.K., Sessoli, R., Chibotaru, L.F., van Slageren, J., 2016. Multitechnique investigation of Dy3—implications for coupled lanthanide clusters. Chem. Sci. 7, 4347–4354. https://doi.org/10.1039/C6SC00318D. Haas, S., Heintze, E., Zapf, S., Gorshunov, B., Dressel, M., Bogani, L., 2014. Direct observation of the discrete energy spectrum of two lanthanide-based single-chain magnets by far-infrared spectroscopy. Phys. Rev. B 89, 174409. https://doi.org/10.1103/PhysRevB.89.174409.

266 Handbook of Magnetic Materials Habib, F., Murugesu, M., 2013. Lessons learned from dinuclear lanthanide nano-magnets. Chem. Soc. Rev. 42, 3278–3288. https://doi.org/10.1039/c2cs35361j. Habib, F., Lin, P.-H., Long, J., Korobkov, I., Wernsdorfer, W., Murugesu, M., 2011. The use of magnetic dilution to elucidate the slow magnetic relaxation effects of a Dy2 single-molecule magnet. J. Am. Chem. Soc. 133, 8830. https://doi.org/10.1021/ja2017009. Han, S.-D., Wang, Q.-L., Xu, J., Bu, X.-H., 2015a. Anion-triggered modulation of structure and magnetic properties of copper(I)-dysprosium(III) complexes derived from 1-hydroxybenzotriazolate. Eur. J. Inorg. Chem. 2015, 5379–5386. https://doi.org/10.1002/ ejic.201500799. Han, T., Leng, J.-D., Ding, Y.-S., Wang, Y., Zheng, Z., Zheng, Y.-Z., 2015b. Field and dilution effects on the magnetic relaxation behaviours of a 1D dysprosium(III)-carboxylate chain built from chiral ligands. Dalton Trans. 44, 13480–13484. https://doi.org/10.1039/C4DT03980G. Han, T., Ding, Y.Z., Zheng, Y.-Z., 2016. Lanthanide Clusters Toward Single-Molecule Magnets. Springer International Publishing, Switzerland. https://doi.org/10.1007/430_2016_9. H€anninen, M.M., Mota, A.J., Aravena, D., Ruiz, E., Sillanp€a€a, R., Camo´n, A., Evangelisti, M., Colacio, E., 2014. Two C3-symmetric DyIII3 complexes with triple di-m-methoxo-m-phenoxo bridges, magnetic ground state, and single-molecule magnetic behavior. Chem. A Eur. J. 20, 8410–8420. https://doi.org/10.1002/chem.201402392. Harriman, K.L.M., Murugesu, M., 2016. An organolanthanide building block approach to single-molecule magnets. Acc. Chem. Res. 49, 1158–1167. https://doi.org/10.1021/acs. accounts.6b00100. Harriman, K.L.M., Brosmer, J.L., Ungur, L., Diaconescu, P.L., Murugesu, M., 2017a. Pursuit of record breaking energy barriers: a study of magnetic axiality in diamide ligated Dy III single-molecule magnets. J. Am. Chem. Soc. 139, 1420–1423. https://doi.org/10.1021/ jacs.6b12374. Harriman, K.L.M., Le Roy, J.J., Ungur, L., Holmberg, I.K., Murugesu, M., 2017b. Cycloheptatrienyl trianion: an elusive bridge in the search of exchange coupled dinuclear organolanthanide single-molecule magnets. Chem. Sci. 8, 231–240. https://doi.org/10.1039/c6sc01224h. Herbst, J.F., 1991. R2Fe14B materials: intrinsic properties and technological aspects. Rev. Mod. Phys. 63, 819–898. https://doi.org/10.1103/RevModPhys.63.819. Herna´ndez, J.M., Zhang, X.X., Luis, F., Bartolom
e, J., Tejada, J., Ziolo, R., 1996. Field tuning of thermally activated magnetic quantum tunnelling in Mn12—Ac molecules. Europhys. Lett. 35, 301–306. https://doi.org/10.1209/epl/i1996-00570-7. Hess, B.A., 1986. Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Phys. Rev. A 33, 3742–3748. https://doi.org/10.1103/PhysRevA.33.3742. Hofmann, A., Salman, Z., Mannini, M., Amato, A., Malavolti, L., Morenzoni, E., Prokscha, T., Sessoli, R., Suter, A., 2012. Depth-dependent spin dynamics in thin films of TbPc2 nanomagnets explored by low-energy implanted muons. ACS Nano 6, 8390–8396. https://doi.org/ 10.1021/nn3031673. Holmberg, R.J., Hutchings, A.-J., Habib, F., Korobkov, I., Scaiano, J.C., Murugesu, M., 2013. Hybrid nanomaterials: anchoring magnetic molecules on naked gold nanocrystals. Inorg. Chem. 52, 14411–14418. https://doi.org/10.1021/ic4026975. Holmberg, R.J., Korobkov, I., Murugesu, M., 2016a. Enchaining EDTA-chelated lanthanide molecular magnets into ordered 1D networks. RSC Adv. 6, 72510–72518. https://doi.org/ 10.1039/C6RA09831B. Holmberg, R.J., Polovkova, M.A., Martynov, A.G., Gorbunova, Y.G., Murugesu, M., 2016b. Impact of the coordination environment on the magnetic properties of single-molecule magnets based

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 267

on homo- and hetero-dinuclear terbium(III) heteroleptic tris(crownphthalocyaninate). Dalton Trans. 45, 9320–9327. https://doi.org/10.1039/c6dt00777e. Holynska, M., Clerac, R., Rouzieres, M., 2015. Lanthanide complexes with multidentate oxime ligands as single-molecule magnets and atmospheric carbon dioxide fixation systems. Chem. A Eur. J. 21, 13321–13329. https://doi.org/10.1002/chem.201501824. Hou, Y.-L., Cheng, R.-R., Xiong, G., Cui, J.-Z., Zhao, B., 2014. Structures, luminescent and magnetic properties of a series of (3,6)-connected lanthanide-organic frameworks. Dalton Trans. 43, 1814–1820. https://doi.org/10.1039/c3dt52305e. Hu, P., Wang, X., Ma, Y., Wang, Q., Li, L., Liao, D., 2013. A new family of Ln-radical chains (Ln ¼ Nd, Sm, Gd, Tb and Dy): synthesis, structure, and magnetic properties. Dalton Trans. 43, 2234–2243. https://doi.org/10.1039/c3dt52959b. Hu, F.L., Jiang, F.L., Zheng, J., Wu, M.Y., Pang, J.D., Hong, M.C., 2015a. Magnetic properties of 3D heptanuclear lanthanide frameworks supported by mixed ligands. Inorg. Chem. 54, 6081–6083. https://doi.org/10.1021/acs.inorgchem.5b00917. Hu, M., Zhao, H., San˜udo, E.C., Chen, M., 2015b. Four lanthanide-carboxylate coordination polymers with mixed 2,3-naphthalenedicarboxylate and phen ligands: syntheses, structures, luminescent and magnetic properties. Polyhedron 101, 270–275. https://doi.org/10.1016/j. poly.2015.08.030. Huang, W., Le Roy, J.J., Khan, S.I., Ungur, L., Murugesu, M., Diaconescu, P.L., 2015a. Tetraaionic biphenyl lanthanide complexes as single-molecule magnets. Inorg. Chem. 54, 2374–2382. Huang, X.-C., Zhang, M., Wu, D., Shao, D., Zhao, X.-H., Huang, W., Wang, X.-Y., 2015b. Single molecule magnet behavior observed in a 1-D dysprosium chain with quasi-D 5h symmetry. Dalton Trans. 44, 20834–20838. https://doi.org/10.1039/C5DT04147C. Huang, W., Shen, F.X., Wu, S.Q., Liu, L., Wu, D., Zheng, Z., Xu, J., Zhang, M., Huang, X.C., Jiang, J., Pan, F., Li, Y., Zhu, K., Sato, O., 2016. Metallogrid single-molecule magnet: solvent-induced nuclearity transformation and magnetic hysteresis at 16 K. Inorg. Chem. 55, 5476–5484. https://doi.org/10.1021/acs.inorgchem.6b00500. Huber, D.L., 1965. Trapping of a resonant phonon by a pair of paramagnetic ions. Phys. Rev. 139, A1684–A1686. https://doi.org/PhysRev.139.A1684. H€ ufner, S., 1978. Optical Spectra of Transparent Rare Earth Compounds. Academic Press, New York, USA. Hutchings, M.T., 1964. Point-charge calculations of energy levels of magnetic ions in crystalline electric fields. Solid State Phys. Adv. Res. Appl. 16, 227–273. https://doi.org/10.1016/S00811947(08)60517-2. Inose, T., Tanaka, D., Tanaka, H., Ivasenko, O., Nagata, T., Ohta, Y., De Feyter, S., Ishikawa, N., Ogawa, T., 2014. Switching of single-molecule magnetic properties of TbIII-porphyrin double-decker complexes and observation of their supramolecular structures on a carbon surface. Chem. A Eur. J. 20, 11362–11369. https://doi.org/10.1002/chem.201402669. Ishikawa, N., 2010. Phthalocyanine-based magnets. In: Jiang, J. (Ed.), Functional Phthalocyanine Molecular Materials. Structure and Bonding, vol. 135. Springer, Berlin, Heidelberg, pp. 211–228. https://doi.org/10.1007/978-3-642-04752-7_7. Ishikawa, N., Iino, T., Kaizu, Y., 2002a. Interaction between f-electronic systems in dinuclear lanthanide complexes with phthalocyanines. J. Am. Chem. Soc. 124, 11440–11447. https://doi. org/10.1021/ja027119n. Ishikawa, N., Iino, T., Kaizu, Y., 2002b. Determination of ligand-field parameters and f-electronic structures of hetero-dinuclear phthalocyanine complexes with a diamagnetic yttrium(III) and a paramagnetic trivalent lanthanide ion. J. Phys. Chem. A 106, 9543–9550. https://doi.org/ 10.1021/jp0209244.

268 Handbook of Magnetic Materials Ishikawa, N., Iino, T., Kaizu, Y., 2003a. Study of 1H NMR spectra of dinuclear complexes of heavy lanthanides with phthalocyanines based on separation of the effects of two paramagnetic centers. J. Phys. Chem. A 107, 7879–7884. https://doi.org/10.1021/jp034971n. Ishikawa, N., Sugita, M., Ishikawa, T., Koshihara, S.Y., Kaizu, Y., 2003b. Lanthanide doubledecker complexes functioning as magnets at the single-molecular level. J. Am. Chem. Soc. 125, 8694–8695. https://doi.org/10.1021/ja029629n. Ishikawa, N., Sugita, M., Tanaka, N., Ishikawa, T., Koshihara, S.Y., Kaizu, Y., 2004. Upward temperature shift of the intrinsic phase lag of the magnetization of bis(phthalocyaninato)terbium by ligand oxidation creating an S ¼ 1/2 spin. Inorg. Chem. 43, 5498–5500. https://doi. org/10.1021/ic049348b. Ishikawa, N., Otsuka, S., Kaizu, Y., 2005a. The effect of the f-f interaction on the dynamic magnetism of a coupled 4f8 system in a dinuclear terbium complex with phthalocyanines. Angew. Chem. Int. Ed. 44, 731–733. https://doi.org/10.1002/ange.200461546. Ishikawa, N., Sugita, M., Wernsdorfer, W., 2005b. Quantum tunneling of magnetization in lanthanide single-molecule magnets: bis(phthalocyaninato)terbium and bis(phthalocyaninato)dysprosium anions. Angew. Chem. Int. Ed. 44, 2931–2935. https://doi.org/10.1002/ anie.200462638. Ivanov, A.V., Svinareva, P.A., Zhukov, I.V., Tomilova, L.G., Zefirov, N.S., 2006. New diphthalocyanine complexes of rare earth metals based on 4,5 isopropylidenedioxyphthalonitrile. Russ. Chem. Bull. Int. Ed. 55, 281–286. https://doi.org/10.1007/s11172-006-0249-4. Iwahara, N., Chibotaru, L.F., 2015. Exchange interaction between J multiplets. Phys. Rev. B 91, 174438. https://doi.org/10.1103/PhysRevB.91.174438. Jassal, A.K., Aliaga-Alcalde, N., Corbella, M., Aravena, D., Ruiz, E., Hundal, G., 2015. Neodymium 1D systems: targeting new sources for field-induced slow magnetization relaxation. Dalton Trans. 44, 15774–15778. https://doi.org/10.1039/C5DT02533H. Jenkins, M.D., Zueco, D., Roubeau, O., Aromi, G., Majer, J., Luis, F., 2016. A scalable architecture for quantum computation with molecular nanomagnets. Dalton Trans. 45, 16682–16693. https://doi.org/10.1039/c6dt02664h. Jiang, S.-D., Qin, S.-X., 2015. Prediction of the quantized axis of rare-earth ions: the electrostatic model with displaced point charges. Inorg. Chem. Front. 2, 613–619. https://doi.org/10.1039/ C5QI00052A. Jiang, S.D., Wang, B.W., Su, G., Wang, Z.M., Gao, S., 2010. A mononuclear dysprosium complex featuring single-molecule-magnet behavior. Angew. Chem. Int. Ed. 49, 7448–7451. https://doi.org/10.1002/anie.201004027. Jiang, Y., Holmberg, R.J., Habib, F., Ungur, L., Korobkov, I., Chibotaru, L.F., Murugesu, M., 2016. Probing the structural and magnetic properties of a new family of centrosymmetric dinuclear lanthanide complexes. RSC Adv. 6, 56668–56673. Jim
enez, J.R., Dı´az-Ortega, I.F., Ruiz, E., Aravena, D., Pope, S.J.A., Colacio, E., Herrera, J.M., 2016. Lanthanide tetrazolate complexes combining single-molecule magnet and luminescence properties: the effect of the replacement of tetrazolate N3 by b-diketonate ligands on the anisotropy energy barrier. Chem. A Eur. J. 22, 14548–14559. https://doi.org/10.1002/ chem.201601457. Joarder, B., Mukherjee, S., Xue, S., Tang, J., Ghosh, S.K., 2014. Structures and magnetic properties of two analogous Dy6 wheels with electron-donation and -withdrawal effects. Inorg. Chem. 53, 7554–7560. https://doi.org/10.1021/ic500875m. Judd, B.R., 1955. The magnetic and spectroscopic properties of certain rare-earth double nitrates. Proc. R. Soc. Lond. A. Math. Phys. Sci. 232, 458–474. https://doi.org/10.1098/ rspa.1955.0231.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 269

Jung, J., Cador, O., Bernot, K., Pointillart, F., Luzon, J., Guennic, B.L., 2014a. Influence of the supramolecular architecture on the magnetic properties of a DyIII single-molecule magnet: an ab initio investigation. Beilstein J. Nanotechnol. 5, 2267–2274. https://doi.org/10.3762/ bjnano.5.236. Jung, J., da Cunha, T.T., Le Guennic, B., Pointillart, F., Pereira, C.L.M., Luzon, J., Golhen, S., Cador, O., Maury, O., Ouahab, L., 2014b. Magnetic studies of redox-active tetrathiafulvalenebased complexes: dysprosium vs. ytterbium analogues. Eur. J. Inorg. Chem. 2014, 3888–3894. https://doi.org/10.1002/ejic.201400121. Jung, J., Le Natur, F., Cador, O., Pointillart, F., Calvez, G., Daiguebonne, C., Guillou, O., Guizouarn, T., Le Guennic, B., Bernot, K., 2014c. Experimental and theoretical evidence that electrostatics governs easy-axis orientation in Dy(III)-based molecular chains. Chem. Commun. (Camb.) 50, 13346–13348. https://doi.org/10.1039/c4cc05062b. Kahn, O., 1993. Molecular Magnetism. VCH Publishers, New York. Kahn, O., Briat, B., 1976. Exchange interaction in polynuclear complexes. Part 1. Principles, model and application to the binuclear complexes of chromium(III). J. Chem. Soc. Faraday Trans. 72, 268. https://doi.org/10.1039/f29767200268. ˚ ., Roos, B.O., Ryde, U., Veryazov, V., Widmark, P.-O., Karlstr€ om, G., Lindh, R., Malmqvist, P.-A Cossi, M., Schimmelpfennig, B., Neogrady, P., Seijo, L., 2003. MOLCAS: a program package for computational chemistry. Comput. Mater. Sci. 28, 222–239. https://doi.org/10.1016/ S0927-0256(03)00109-5. Kettles, F.J., Milway, V.A., Tuna, F., Valiente, R., Thomas, L.H., Wernsdorfer, W., Ochsenbein, S.T., Murrie, M., 2014. Exchange interactions at the origin of slow relaxation of the magnetization in {TbCu3} and {DyCu3} single-molecule magnets. Inorg. Chem. 53, 8970–8978. https://doi.org/10.1021/ic500885r. Kiefl, E., Mannini, M., Bernot, K., Yi, X., Amato, A., Leviant, T., Magnani, A., Prokscha, T., Suter, A., Sessoli, R., Salman, Z., 2016. Robust magnetic properties of a sublimable singlemolecule magnet. ACS Nano 10, 5663–5669. https://doi.org/10.1021/acsnano.6b01817. Klar, D., Klyatskaya, S., Candini, A., Krumme, B., Kummer, K., Ohresser, P., Corradini, V., de Renzi, V., Biagi, R., Joly, L., Kappler, J.P., Pennino, U.d., Affronte, M., Wende, H., Ruben, M., 2013. Antiferromagnetic coupling of TbPc2 molecules to ultrathin Ni and Co films. Beilstein J. Nanotechnol. 4, 320–324. https://doi.org/10.3762/bjnano.4.36. Klar, D., Candini, A., Joly, L., Klyatskaya, S., Krumme, B., Ohresser, P., Kappler, J.-P., Ruben, M., Wende, H., 2014. Hysteretic behaviour in a vacuum deposited submonolayer of single ion magnets. Dalton Trans. 43, 10686–10689. https://doi.org/10.1039/ C4DT01005A. Kofu, M., Yamamuro, O., Kajiwara, T., Yoshimura, Y., Nakano, M., Nakajima, K., OhiraKawamura, S., Kikuchi, T., Inamura, Y., 2013. Hyperfine structure of magnetic excitations in a Tb-based single-molecule magnet studied by high-resolution neutron spectroscopy. Phys. Rev. B 88, 64405. https://doi.org/10.1103/PhysRevB.88.064405. Komeda, T., 2014. Spins of adsorbed molecules investigated by the detection of Kondo resonance. Surf. Sci. 630, 343–355. https://doi.org/10.1016/j.susc.2014.07.012. Komeda, T., Isshiki, H., Liu, J., Katoh, K., Shirakata, M., Breedlove, B.K., Yamashita, M., 2013. Variation of kondo peak observed in the assembly of heteroleptic 2,3-naphthalocyaninato phthalocyaninato Tb(III) double-decker complex on Au(111). ACS Nano 7, 1092–1099. https://doi.org/10.1021/nn304035h. Komeda, T., Katoh, K., Yamashita, M., 2014. Double-decker phthalocyanine complex: scanning tunneling microscopy study of film formation and spin properties. Prog. Surf. Sci. 89, 127–160. https://doi.org/10.1016/j.progsurf.2014.03.001.

270 Handbook of Magnetic Materials Kronig, R.d.L., 1939. On the mechanism of paramagnetic relaxation. Physica 6, 33. https://doi. org/10.1016/S0031-8914(39)80015-5. Kuo, C.-J., Holmberg, R.J., Lin, P.-H., 2015. Slight synthetic changes eliciting different topologies: synthesis, structure and magnetic properties of novel dinuclear and nonanuclear dysprosium complexes. Dalton Trans. 44, 19758–19762. https://doi.org/10.1039/C5DT02899J. Kyatskaya, S., Gala´n Mascaro´s, J.R., Bogani, L., Hennrich, F., Kappes, M., Wernsdorfer, W., Ruben, M., 2009. Anchoring of rare-earth-based single-molecule magnets on single-walled carbon nanotubes. J. Am. Chem. Soc. 131, 15143–15151. https://doi.org/10.1021/ ja906165e. Lan, Y., Klyatskaya, S., Ruben, M., Fuhr, O., Wernsdorfer, W., Candini, A., Corradini, V., Lodi Rizzini, A., del Pennino, U., Troiani, F., Joly, L., Klar, D., Wende, H., Affronte, M., 2015. Magnetic interplay between two different lanthanides in a tris-phthalocyaninato complex: a viable synthetic route and detailed investigation in the bulk and on the surface. J. Mater. Chem. C 3, 9794–9801. https://doi.org/10.1039/C5TC02011E. Lan, Y., Magri, A., Fuhr, O., Ruben, M., 2016. Phenalenyl-based mononuclear dysprosium complexes. Beilstein J. Nanotechnol. 7, 995–1009. https://doi.org/10.3762/bjnano.7.92. Langley, S.K., Chilton, N.F., Ungur, L., Moubaraki, B., Chibotaru, L.F., Murray, K.S., 2012. Heterometallic Tetranuclear [LnIII2CoIII2] complexes including suppression of quantum tunneling of magnetization in the [DyIII2CoIII2] single molecule magnet. Inorg. Chem. 5111873. https:// doi.org/10.1021/ic301784m. Langley, S.K., Wielechowski, D.P., Vieru, V., Chilton, N.F., Moubaraki, B., Abrahams, B.F., Chibotaru, L.F., Murray, K.S., 2013. A {CrIII2DyIII2} single-molecule magnet: enhancing the blocking temperature through 3d magnetic exchange. Angew. Chem. Int. Ed. 52, 12014–12019. https://doi.org/10.1002/anie.201306329. Langley, S.K., Moubaraki, B., Murray, K.S., 2014a. Heterometallic tetranuclear {MnIII2LnIII2}n 1D coordination polymers: employing sulfonate ligands as connecting groups. Aust. J. Chem. 67, 1601–1606. https://doi.org/10.1071/CH14385. Langley, S.K., Ungur, L., Chilton, N.F., Moubaraki, B., Chibotaru, L.F., Murray, K.S., 2014b. Single-molecule magnetism in a family of {CoIII2DyIII2} butterfly complexes: effects of ligand replacement on the dynamics of magnetic relaxation. Inorg. Chem. 53, 4303–4315. https://doi.org/10.1021/ic4029645. Langley, S.K., Wielechowski, D.P., Vieru, V., Chilton, N.F., Moubaraki, B., Chibotaru, L.F., Murray, K.S., 2014c. Modulation of slow magnetic relaxation by tuning magnetic exchange in {Cr2Dy2} single molecule magnets. Chem. Sci. 5, 3246–3256. https://doi.org/10.1039/ c4sc01239a. Langley, S.K., Wielechowski, D.P., Chilton, N.F., Moubaraki, B., Murray, K.S., 2015. A family of {CrIII2LnIII2} butterfly complexes: effect of the lanthanide ion on the single-molecule magnet properties. Inorg. Chem. 54, 10497–10503. https://doi.org/10.1021/acs. inorgchem.5b01999. Langley, S.K., Wielechowski, D.P., Moubaraki, B., Murray, K.S., 2016. Enhancing the magnetic blocking temperature and magnetic coercivity of {CrIII2LnIII2} single-molecule magnets via bridging ligand modification. Chem. Commun. 52, 10976–10979. https://doi.org/10.1039/ C6CC06152D. Larson, G.H., Jeffries, C.D., 1966. Spin-lattice relaxation in some rare earth salts. I. Temperature dependence. Phys. Rev. 141, 461–478. https://doi.org/PhysRev.141.461. Lashley, J.C., Hundley, M.F., Migliori, A., Sarrao, J.L., Pagliuso, P.G., Darling, T.W., Jaime, M., Cooley, J.C., Hults, W.L., Morales, L., Thoma, D.J., Smith, J.L., Boerio-Goates, J., Woodfield, B.F., Stewart, G.R., Fisher, R.A., Phillips, N.E., 2003. Critical examination of

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 271

heat capacity measurements made on a quantum design physical property measurement system. Cryogenics (Guildf ) 43, 369–378. https://doi.org/10.1016/S0011-2275(03)00092-4. Layfield, R.A., 2014. Organometallic single-molecule magnets. Organometallics 33, 1084–1099. https://doi.org/10.1021/om401107f. Layfield, R.A., Murugesu, M., 2015. Lanthanides and Actinides in Molecular Magnetism. WileyVCH, Weinheim, Germany. Layfield, A., McDouall, J.W., Sulway, S.A., Tuna, F., Collison, D., Winpenny, R.E.P., 2010. Influence of the N-bridging ligand on magnetic relaxation in an organometallic dysprosium single-molecule magnet. Chem. Eur. J. 16, 4442–4446. Lee, S., Ogawa, T., 2017. Molecular deisgn for single-molecule magnetism of lanthanide complexes. Chem. Lett. 46, 10–18. https://doi.org/10.1246/cl.160800. Le Roy, J.J., Ungur, L., Korobkov, I., Chibotaru, L.F., Murugesu, M., 2014. Coupling strategies to enhance single-molecule magnet properties of erbium-cyclooctatetraenyl complexes. J. Am. Chem. Soc. 136, 8003–8010. https://doi.org/10.1021/ja5022552. Le Roy, J.J., Gorelsky, S.I., Korobkov, I., Murugesu, M., 2015. Slow magnetic relaxation in uranium(III) and neodymium(III) cyclooctatetraenyl complexes. Organometallics 34, 1415–1418. https://doi.org/10.1021/om501214c. Levy, P.M., 1964. Rare-earth-iron exchange interaction in the garnets. I. Hamiltonian for anisotropic exchange interaction. Phys. Rev. 135, A155. https://doi.org/10.1103/PhysRev.135.A155. Li, Q., Du, S., 2015. A family of 3D lanthanide organic frameworks with tunable luminescence and slow magnetic relaxation. RSC Adv. 5, 9898–9903. https://doi.org/10.1039/c4ra14939d. Li, M., Lan, Y., Ako, A.M., Wernsdorfer, W., Anson, C.E., Buth, G., Powell, A.K., Wang, Z., Gao, S., 2010. A family of 3d-4f octa-nuclear [MnIII4LnIII4] wheels (Ln ¼ Sm, Gd, Tb, Dy, Ho, Er, and Y): synthesis, structure, and magnetism. Inorg. Chem. 49, 11587–11594. https://doi.org/10.1021/ic101754g. Li, Q.W., Liu, J.L., Jia, J.H., Chen, Y.-C., Liu, J., Wang, L., Tong, M.-L., 2015a. “Half-sandwich” YbIII single-ion magnets with metallacrowns. Chem. Commun. 51, 10291–10294. https://doi. org/10.1039/C5CC03389F. Li, X.-L., Li, H., Chen, D.-M., Wang, C., Wu, J., Tang, J., Shi, W., Cheng, P., 2015b. Planar Dy3 + Dy3 clusters: design, structure and axial ligand perturbed magnetic dynamics. Dalton Trans. 44, 20316–20320. https://doi.org/10.1039/c5dt03931b. Li, X., Li, T., Tian, L., Liu, Z.Y., Wang, X.G., 2015c. A family of rare earth complexes with chelating furan biradicals: syntheses, structures and magnetic properties. RSC Adv. 5, 74864–74873. https://doi.org/10.1039/C5RA13386F. Li, Y., Yu, J.W., Liu, Z.Y., Yang, E.C., Zhao, X.J., 2015d. Three isostructural one-dimensional LnIII chains with distorted cubane motifs showing dual fluorescence and slow magnetic relaxation/magnetocaloric effect. Inorg. Chem. 54, 153–160. https://doi.org/10.1021/ic501952b. Li, C., Sun, J., Yang, M., Sun, G., Guo, J., Ma, Y., Li, L., 2016a. From monomeric species to onedimensional chain: enhancing slow magnetic relaxation through coupling mononuclear fragments in ln-rad system. Cryst. Growth Des. 16, 7155–7162. https://doi.org/10.1021/acs. cgd.6b01369. Li, T., Zhou, S.Y., Li, X., Tian, L., Liao, D.Z., Liu, Z.Y., Guo, J.H., 2016b. Modulating spin dynamics of LnIII-radical complexes by using different coligands. RSC Adv. 6, 3058–3067. https://doi.org/10.1039/c5ra21991d. Li, Y.-M., Kuang, W.-W., Zhu, L.-L., Xu, Y., Yang, P.-P., 2016c. Two discrete Ln12 shelf-shaped clusters: magnetic studies reveal a significant cryogenic magnetocaloric effect and slow magnetic relaxation. Eur. J. Inorg. Chem. 2016c, 4996–5003. https://doi.org/10.1002/ ejic.201600556.

272 Handbook of Magnetic Materials Li, X., Li, T., Shi, X.J., Tian, L., 2017a. A family of 2p-4f complexes based on indazole radical: syntheses, structures and magnetic properties. Inorgan. Chim. Acta 456, 216–223. https://doi. org/10.1016/j.ica.2016.11.002. Li, M., Wu, H., Zhang, S., Sun, L., Ke, H., Wei, Q., Xie, G., Chen, S., Gao, S., 2017b. Fine-tuning ligand fields with Schiff-base ligands in Dy2 compounds. Eur. J. Inorg. Chem reference. 2017, 811–819. https://doi.org/10.1002/ejic.201601188. Lim, K.S., Baldovı´, J.J., Lee, W.R., Song, J.H., Yoon, S.W., Suh, B.J., Coronado, E., Gaita-Arin˜o, A., Hong, C.S., 2016. Switching of slow magnetic relaxation dynamics in mononuclear dysprosium(III) compounds with charge density. Inorg. Chem. 55, 5398–5404. https://doi. org/10.1021/acs.inorgchem.6b00410. Lin, S.Y., Tang, J., 2014. Versatile tetranuclear dysprosium single-molecule magnets. Polyhedron 83, 185–196. https://doi.org/10.1016/j.poly.2014.06.013. Lin, P.H., Burchell, T.J., Cl
erac, R., Murugesu, M., 2008. Dinuclear dysprosium(III) singlemolecule magnets with a large anisotropic barrier. Angew. Chem. Int. Ed. 47, 8848–8851. https://doi.org/10.1002/anie.200802966. Lin, P.H., Burchell, T.J., Ungur, L., Chibotaru, L.F., Wernsdorfer, W., Murugesu, M., 2009. A polynuclear lanthanide single-molecule magnet with a record anisotropic barrier. Angew. Chem. Int. Ed. 48, 9489–9492. https://doi.org/10.1002/anie.200903199. Lin, P.H., Sun, W.B., Tian, Y.M., Yan, P.F., Ungur, L., Chibotaru, L.F., Murugesu, M., 2012. Ytterbium can relax slowly too: a field-induced Yb2 single-molecule magnet. Dalton Trans. 41, 12349–12352. https://doi.org/10.1039/C2DT31677C. Lin, S.-Y., Li, X.-L., Ke, H., Xu, Z., 2015a. A series of tetranuclear lanthanide compounds constructed by in situ polydentate ligands: synthesis, structure, and SMM behaviour of the Dy 4 compound. CrystEngComm 17, 9167–9174. https://doi.org/10.1039/C5CE01818H. Lin, S.-Y., Wang, C., Zhao, L., Wu, J., Tang, J., 2015b. Chiral mononuclear lanthanide complexes and the field-induced single-ion magnet behaviour of a Dy analogue. Dalton Trans. 44, 223–229. https://doi.org/10.1039/c4dt02889a. Lin, S.-Y., Wu, J., Wang, C., Zhao, L., Tang, J., 2015c. Modulating relaxation dynamics of Dy2 compounds through carboxylate coordination modes. Eur. J. Inorg. Chem. 2015, 5488–5494. https://doi.org/10.1002/ejic.201500956. Lin, S.-Y., Wang, C., Xu, Z., 2016. Syntheses, structures and magnetic properties of symmetrical Schiff-base supported dinuclear Ln(III) compounds. RSC Adv. 6, 103035. https://doi.org/ 10.1039/C6RA16669E. Lines, M.E., 1971. Orbital angular momentum in the theory of paramagnetic clusters. J. Chem. Phys. 55, 2977–2984. https://doi.org/10.1063/1.1676524. Lis, T., 1980. Preparation, structure, and magnetic properties of a dodecanuclear mixed-valence manganese carboxylate. Acta Crystallogr. B36, 2042–2046. https://doi.org/10.1107/ S0567740880007893. Liu, J., Yuan, K., Leng, J., Ungur, L., Wernsdorfer, W., Guo, F., Chibotaru, F.L., Tong, M., 2012. A six-coordinate ytterbium complex exhibiting easy-plane anisotropy and field-induced single-ion magnet behavior. Inorg. Chem. 51, 8538–8544. https://doi.org/10.1021/ic301115b. Liu, J.L., Wu, J.Y., Chen, Y.C., Mereacre, V., Powell, A.K., Ungur, L., Chibotaru, L.F., Chen, X.M., Tong, M.L., 2014a. A heterometallic FeII-DyIII single-molecule magnet with a record anisotropy barrier. Angew. Chem. Int. Ed. 53, 12966–12970. https://doi.org/ 10.1002/anie.201407799. Liu, C., Zhang, D., Zhu, D., 2014b. A 2D ! 2D polyrotaxane lanthanide–organic framework showing field-induced single-molecule magnet behaviour. RSC Adv. 4, 36053. https://doi. org/10.1039/C4RA06802E.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 273

Liu, S.-S., Ziller, J.W., Zhang, Y.-Q., Wang, B.-W., Evans, W.J., Gao, S., 2014c. A half-sandwich organometallic single-ion magnet with hexamethylbenzene coordinated to the Dy(III) ion. Chem. Commun. (Camb.) 50, 11418–11420. https://doi.org/10.1039/c4cc04262j. Liu, C.-M., Xiong, J., Zhang, D.-Q., Wang, B.-W., Zhu, D.-B., 2015a. Multiple thermal magnetic relaxation in a two-dimensional ferromagnetic dysprosium(III) metal-organic framework. RSC Adv. 5, 104854–104861. https://doi.org/10.1039/c5ra23638j. Liu, C.-M., Zhang, D.-Q., Hao, X., Zhu, D.-B., 2015b. Luminescence and slow magnetic relaxation of isostructural 2D lanthanide metal-organic frameworks derived from both nicotinate N-oxide and glutarate. RSC Adv. 5, 92980–92987. https://doi.org/10.1039/c5ra19682e. Liu, C.-M., Zhang, D.-Q., Zhu, D.-B., 2015c. Slow magnetic relaxation of a three-dimensional metal-organic framework featuring a unique dysprosium(III) oxalate layer. RSC Adv. 5, 63186–63192. https://doi.org/10.1039/c5ra11621j. Liu, J.-L., Wu, J.-Y., Huang, G.-Z., Chen, Y.-C., Jia, J.-H., Ungur, L., Chibotaru, L.F., Chen, X.-M., Tong, M.-L., 2015d. Desolvation-driven 100-fold slow-down of tunneling relaxation rate in Co(II)-Dy(III) single-molecule magnets through a single-crystal-to-single-crystal process. Sci. Rep. 5, 16621. https://doi.org/10.1038/srep16621. Liu, J., Chen, Y.-C., Jiang, Z.-X., Liu, J.-L., Jia, J.-H., Wang, L.-F., Li, Q.-W., Tong, M.-L., 2015e. Efficient enhancement of magnetic anisotropy by optimizing the ligand-field in a typically tetranuclear dysprosium cluster. Dalton Trans. 44, 8150–8155. https://doi.org/10.1039/ c5dt00880h. Liu, S., Li, L.-L., Li, H., Gao, H.-L., Cui, J.-Z., Cheng, P., 2015f. Slow magnetic relaxation in a lanthanide helix chain compound [Dy(HNA)(NA)2(NO3)]n (HNA ¼ nicotinic acid). Dalton Trans. 44, 6169–6174. https://doi.org/10.1039/C5DT00205B. Liu, C.-M., Zhang, D.-Q., Zhao, Y.-S., Hao, X., Zhu, D.-B., 2016a. Two-step warming solvothermal syntheses, luminescence and slow magnetic relaxation of isostructural dense LnMOFs based on nanoscale 3-connected linkers. Inorg. Chem. Front. 3, 1076–1081. https://doi.org/ 10.1039/C6QI00137H. Liu, C.-M., Zhang, D.-Q., Zhu, D.-B., 2016b. A 3D MOF constructed from dysprosium( III ) oxalate and capping ligands: ferromagnetic coupling and field-induced two-step magnetic relaxation. Chem. Commun. 52, 4804–4807. https://doi.org/10.1039/C6CC00498A. Liu, F., Gao, C.-L., Deng, Q., Zhu, X., Kostanyan, A., Westerstr€om, R., Wang, S., Tan, Y.-Z., Tao, J., Xie, S.-Y., Popov, A.A., Greber, T., Yang, S., 2016c. Triangular monometallic cyanide cluster entrapped in carbon cage with geometry-dependent molecular magnetism. J. Am. Chem. Soc. 138, 14764–14771. https://doi.org/10.1021/jacs.6b09329. Liu, J., Chen, Y.C., Liu, J.L., Vieru, V., Ungur, L., Jia, J.H., Chibotaru, L.F., Lan, Y., Wernsdorfer, W., Gao, S., Chen, X.M., Tong, M.L., 2016d. A stable pentagonal bipyramidal Dy(III) single-ion magnet with a record magnetization reversal barrier over 1000 K. J. Am. Chem. Soc. 138, 5441–5450. https://doi.org/10.1021/jacs.6b02638. Liu, S.-Q., Dong, C.-L., Zhao, H., Zhang, H., Ni, J., Liu, X.-Y., Zhang, J.-J., 2016e. A one dimensional 3d–4f heterometallic chain based on Gd3+ nodes and tetranuclear {Cr4(hdpta)2} complex ligands: synthesis, structure and magnetic properties. J. Clust. Sci. 27, 883–894. https://doi.org/10.1007/s10876-015-0958-7. Liu, F., Wang, S., Gao, C.L., Deng, Q., Zhu, X., Kostanyan, A., Westerstrom, R., Jin, F., Xie, S.Y., Popov, A.A., Greber, T., Yang, S., 2017. Mononuclear clusterfullerene singlemolecule magnet containing strained fused-pentagone stabilized by a nearly linear metal cyanine cluster. Angew. Chem. Int. Ed. 56, 1830–1834. https://doi.org/10.1002/anie.201611345. Lodi Rizzini, A., Krull, C., Mugarza, A., Balashov, T., Nistor, C., Piquerel, R., Klyatskaya, S., Ruben, M., Sheverdyaeva, P.M., Moras, P., Carbone, C., Stamm, C., Miedema, P.S.,

274 Handbook of Magnetic Materials Thakur, P.K., Sessi, V., Soares, M., Yakhou-Harris, F., Cezar, J.C., Stepanow, S., Gambardella, P., 2014. Coupling of single, double, and triple-decker metal-phthalocyanine complexes to ferromagnetic and antiferromagnetic substrates. Surf. Sci. 630, 361–374. https://doi.org/10.1016/j.susc.2014.07.008. Long, J., Habib, F., Lin, P.H., Korobkov, I., Enright, G., Ungur, L., Wernsdorfer, W., Chibotaru, L.F., Murugesu, M., 2011. Single-molecule magnet behavior for an antiferromagnetically superexchange-coupled dinuclear dysprosium(III) complex. J. Am. Chem. Soc. 133, 5319–5328. https://doi.org/10.1021/ja109706y. Long, J., Rouquette, J., Thibaud, J.M., Ferreira, R.A.S., Carlos, L.D., Donnadieu, B., Vieru, V., Chibotaru, L.F., Konczewicz, L., Haines, J., Guari, Y., Larionova, J., 2015. A hightemperature molecular ferroelectric Zn/Dy complex exhibiting single-ion-magnet behavior and lanthanide luminescence. Angew. Chem. Int. Ed. 54, 2236–2240. https://doi.org/ 10.1002/anie.201410523. Lorusso, G., Jenkins, M., Gonza´lez-Monje, P., Arauzo, A., Ses
e, J., Ruiz-Molina, D., Roubeau, O., Evangelisti, M., 2013. Surface-confined molecular coolers for cryogenics. Adv. Mater. 25, 2984–2988. https://doi.org/10.1002/adma.201204863. Lu, X.-Y., Liu, Y.-Q., Deng, X.-W., Zhu, Z.-X., Yao, M.-X., Jing, S., 2015. Synthesis, structures and magnetism of heterodinuclear Ni–Ln complexes: field-induced single-molecule magnet behavior in the dysprosium analogue. New J. Chem. 39, 3467–3473. https://doi.org/ 10.1039/C4NJ02162B. Luan, F., Liu, T., Yan, P., Zou, X., Li, Y., Li, G., 2015a. Single-molecule magnet of a tetranuclear dysprosium complex disturbed by a salen-type ligand and chloride counterions. Inorg. Chem. 54, 3485–3490. https://doi.org/10.1021/acs.inorgchem.5b00061. Luan, F., Yan, P., Zhu, J., Liu, T., Zou, X., Li, G., Feng, S., Chibotaru, L.F., Powell, A.K., Hendrickson, D.N., Jones, R.A., Cheng, P., 2015b. A salen-type Dy4 single-molecule magnet with an enhanced energy barrier and its analogues. Dalton Trans. 44, 4046–4053. https://doi. org/10.1039/C4DT02607A. Lucaccini, E., Sorace, L., Perfetti, M., Costes, J.-P., Sessoli, R., 2014. Beyond the anisotropy barrier: slow relaxation of the magnetization in both easy-axis and easy-plane Ln(trensal) complexes. Chem. Commun. (Camb.) 50, 1648–1651. https://doi.org/10.1039/ c3cc48866g. Lucaccini, E., Baldovı´, J.J., Chelazzi, L., Barra, A.L., Grepioni, F., Costes, J.P., Sorace, L., 2017. Electronic structure and magnetic anisotropy in lanthanoid single-ion magnets with C3 symmetry: the Ln(trenovan) series. Inorg. Chem. 56, 4728–4738. https://doi.org/10.1021/acs. inorgchem.7b00413. Luis, F., Evangelisti, M., 2015. Magnetic refrigeration and spin-lattice relaxation in gadolinium-based molecular nanomagnets. Struct. Bond. 164, 431–460. https://doi.org/ 10.1007/430_2014_152. Luis, F., Bartolom
e, J., Ferna´ndez, J.F., Tejada, J., Herna´ndez, J.M., Zhang, X.X., Ziolo, R., 1997. Thermally activated and field-tuned tunneling in Mn12Ac studied by ac magnetic susceptibility. Phys. Rev. B 55, 11448–11456. https://doi.org/10.1103/PhysRevB.55.11448. Luis, F., Bartolom
e, J., Ferna´ndez, J.F., 1998. Resonant magnetic quantum tunneling through thermally activated states. Phys. Rev. B 57, 505–513. https://doi.org/10.1103/ PhysRevB.57.505. Luis, F., Repoll
es, A., Martı´nez-P
erez, M.J., Aguilà, D., Roubeau, O., Zueco, D., Alonso, P.J., Evangelisti, M., Camo´n, A., Ses
e, J., Barrios, L.A., Aromı´, G., 2011. Molecular prototypes for spin-based CNOT and SWAP quantum gates. Phys. Rev. Lett. 107, 117203. https://doi. org/10.1103/PhysRevLett.107.117203.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 275

Lumetti, S., Candini, A., Godfrin, C., Balestro, F., Wernsdorfer, W., Klyatskaya, S., Ruben, M., Affronte, M., 2016. Single-molecule devices with graphene electrodes. Dalton Trans. 45, 16570–16574. https://doi.org/10.1039/C6DT02445A. Lunghi, A., Totti, F., Sessoli, R., Sanvito, S., 2017. The role of anharmonic phonons in underbarrier spin relaxation of single molecule magnet. Nat. Commun. 8, 14620. https://doi.org/ 10.1038/ncomms14620. Luscombe, J., Luban, M., Reynolds, J., 1996. Finite-size scaling of the Glauber model of critical dynamics. Phys. Rev. E 53, 5852–5860. https://doi.org/PhysRevE.53.5852. Luzon, J., Sessoli, R., 2012. Lanthanides in molecular magnetism: so fascinating, so challenging. Dalton Trans. 41, 13556. https://doi.org/10.1039/C2DT31388J. Luzon, J., Bernot, K., Hewitt, I.J., Anson, C.E., Powell, A.K., Sessoli, R., 2008. Spin chirality in a molecular dysprosium triangle: the archetype of the noncollinear ising model. Phys. Rev. Lett. 100, 247205. https://doi.org/10.1103/PhysRevLett.100.247205. Macdonald, M.R., Bates, J.E., Ziller, J.W., Furche, F., Evans, W.J., 2013. Completing the series of +2 ions for the lanthanide elements: synthesis of molecular complexes of Pr2 +,Gd2 +,Tb2 +, and Lu2 +. J. Am. Chem. Soc. 135, 9857–9868. https://doi.org/10.1021/ja510831n. Magnetometry by means of Hall micro-probes in the Quantum Design PPMS [WWW Document], 2008. Quantum Des. Appl. Note. URL https://www.qdusa.com/sitedocs/appNotes/ppms/ 1084-701.pdf. Malavolti, L., Mannini, M., Car, P.-E., Campo, G., Pineider, F., Sessoli, R., 2013a. Erratic magnetic hysteresis of TbPc2 molecular nanomagnets. J. Mater. Chem. C 1, 2935. https://doi.org/ 10.1039/c3tc00925d. Malavolti, L., Poggini, L., Margheriti, L., Chiappe, D., Graziosi, P., Cortigiani, B., Lanzilotto, V., de Mongeot, F.B., Ohresser, P., Otero, E., Choueikani, F., Sainctavit, P., Bergenti, I., Dediu, V.a., Mannini, M., Sessoli, R., 2013b. Magnetism of TbPc2 SMMs on ferromagnetic electrodes used in organic spintronics. Chem. Commun. 49, 11506. https://doi.org/10.1039/ c3cc46868b. Malta, O.L., 1982. A simple overlap model in lanthanide crystal-field theory. Chem. Phys. Lett. 87, 27–29. https://doi.org/10.1016/0009-2614(82)83546-X. Mamontova, E., Long, J., Ferreira, R., Botas, A., Luneau, D., Guari, Y., Carlos, L., Larionova, J., 2016. Magneto-luminescence correlation in the textbook dysprosium(III) nitrate single-ion magnet. Magnetochemistry 2, 41. https://doi.org/10.3390/magnetochemistry2040041. Mannini, M., Bertani, F., Tudisco, C., Malavolti, L., Poggini, L., Misztal, K., Menozzi, D., Motta, A., Otero, E., Ohresser, P., Sainctavit, P., Condorelli, G.G., Dalcanale, E., Sessoli, R., 2014. Magnetic behaviour of TbPc2 single-molecule magnets chemically grafted on silicon surface. Nat. Commun. 5, 4582. https://doi.org/10.1038/ncomms5582. Margheriti, L., Chiappe, D., Mannini, M., Car, P.E., Sainctavit, P., Arrio, M.A., De Mongeot, F.B., Cezar, J.C., Piras, F.M., Magnani, A., Otero, E., Caneschi, A., Sessoli, R., 2010. Xray detected magnetic hysteresis of thermally evaporated terbium double-Decker oriented films. Adv. Mater. 22, 5488–5493. https://doi.org/10.1002/adma.201003275. Martı´n-Ramos, P., Pereira, L.C.J., Coutinho, J.T., Koprowiak, F., Bolvin, H., Lavı´n, V., Martı´n, I.R., Martı´n-Gil, J., Silva, M.R., 2016. Structure, luminescence and magnetic properties of an erbium(III) b-diketonate homodinuclear complex. New J. Chem. 40, 8251–8261. https://doi.org/10.1039/C6NJ01598K. Martı´nez-P
erez, M.J., Cardona-Serra, S., Schlegel, C., Moro, F., Alonso, P.J., Prima-Garcı´a, H., Clemente-Juan, J.M., Evangelisti, M., Gaita-Arin˜o, A., Ses
e, J., van Slageren, J., Coronado, E., Luis, F., 2012a. Gd-based single-ion magnets with tunable magnetic

276 Handbook of Magnetic Materials anisotropy: molecular design of spin qubits. Phys. Rev. Lett. 108, 247213. https://doi.org/ 10.1103/PhysRevLett.108.247213. Martı´nez-P
erez, M.J., Montero, O., Evangelisti, M., Luis, F., Ses
e, J., Cardona-Serra, S., Coronado, E., 2012b. Fragmenting gadolinium: mononuclear polyoxometalate-based magnetic coolers for ultra-low temperatures. Adv. Mater. 24, 4301–4305. https://doi.org/ 10.1002/adma.201200750. Marx, R., Moro, F., D€orfel, M., Ungur, L., Waters, M., Jiang, S.D., Orlita, M., Taylor, J., Frey, W., Chibotaru, L.F., van Slageren, J., 2014. Spectroscopic determination of crystal field splittings in lanthanide double deckers. Chem. Sci. 5, 3287. https://doi.org/10.1039/ c4sc00751d. Mazarakioti, E.C., Regier, J., Cunha-Silva, L., Wernsdorfer, W., Pilkington, M., Tang, J., Stamatatos, T.C., 2017. Large energy barrier and magnetization hysteresis at 5K for a symmetric {Dy2} complex with spherical tricapped trigonal prismatic DyIII ions. Inorg. Chem. 56, 3568–3578. https://doi.org/10.1021/acs.inorgchem.7b00026. Meihaus, K.R., Fieser, M.E., Corbey, J.F., Evans, W.J., Long, R., 2015. Record high single-ion magnetic moments through 4f electron configurations in the divalent lanthanide complexes [(C5H4SiMe3)3Ln]. J. Am. Chem. Soc. 137, 9855–9860. https://doi.org/10.1021/ jacs.5b03710. Meng, Y.-S., Jiang, S.–.D., Wang, B.W., Gao, S., 2016a. Understanding the magnetic anisotropy toward single-ion magnets. Acc. Chem. Res. 49, 2381–2389. https://doi.org/10.1021/acs. accounts.6b00222. Meng, Y.S., Qiao, Y.S., Zhang, Y.Q., Jiang, S.-D., Meng, Z.S., Wang, B.W., Wang, Z.M., Gao, S., 2016b. Can non-Kramers TmIII mononuclear molecules be single-molecule magnets (SMMs)? Chem. A Eur. J. 22, 4704–4708. https://doi.org/10.1002/chem.201600023. Mereacre, V., 2012. Differentiation of highly anisotropic Tb III and DyIII with 57Fe M€ossbauer spectroscopy. Angew. Chem. Int. Ed. 51, 9922–9925. https://doi.org/10.1002/ anie.201204838. Mihalcea, I., Perfetti, M., Pineider, F., Tesi, L., Mereacre, V., Wilhelm, F., Rogalev, A., Anson, C.E., Powell, A.K., Sessoli, R., 2016. Spin helicity in chiral lanthanide chains. Inorg. Chem. 55, 10068–10074. https://doi.org/10.1021/acs.inorgchem.6b01010. Mondal, A.K., Goswami, S., Konar, S., 2015. Influence of the coordination environment on slow magnetic relaxation and photoluminescence behavior in two mononuclear dysprosium(III) based single molecule magnets. Dalton Trans. 44, 5086–5094. https://doi.org/10.1039/ c4dt03620d. Mondal, A.K., Jena, H.S., Malviya, A., Konar, S., 2016. Lanthanide-directed fabrication of four tetranuclear quadruple stranded helicates showing magnetic refrigeration and slow magnetic relaxation. Inorg. Chem. 55, 5237–5244. https://doi.org/10.1021/acs. inorgchem.6b00177. Monteiro, B., Pereira, C.C.L., Coutinho, J.T., Pereira, L.C.J., Marc¸alo, J., Almeida, M., 2013. A 2D layered lanthanide hydroxide showing slow relaxation of magnetization—Dy8(OH) 20Cl46H2O. Eur. J. Inorg. Chem. 2013, 5059–5063. https://doi.org/10.1002/ejic.201300793. Monteiro, B., Coutinho, J.T., Pereira, C.C.L., Pereira, L.C.J., Marc¸alo, J., Almeida, M., Baldovı´, J.J., Coronado, E., Gaita-Arin˜o, A., 2015. Magnetic properties of the layered lanthanide hydroxide series Yx Dy8-x(OH)20Cl46H2O: from single ion magnets to 2D and 3D interaction effects. Inorg. Chem. 54, 1949–1957. https://doi.org/10.1021/ic502839c. Moreno Pineda, E., Chilton, N.F., Marx, R., D€orfel, M., Sells, D.O., Neugebauer, P., Jiang, S., Collison, D., van Slageren, J., McInnes, E.J.L., Winpenny, R.E.P., 2014. Direct measurement

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 277

of dysprosium(III)dysprosium(III) interactions in a single-molecule magnet. Nat. Commun. 5, 5243. https://doi.org/10.1038/ncomms6243. Moreno Pineda, E., Komeda, T., Katoh, K., Yamashita, M., Ruben, M., 2016. Surface confinement of TbPc2-SMMs: structural, electronic and magnetic properties. Dalton Trans. 45, 18417–18433. https://doi.org/10.1039/C6DT03298B. MPMS®, n.d. MPMS (Magnetic Properties Measurement System) Model. Quantum Des. Inc. California, USA. Mueller, K.A., 1968. Effective-spin Hamiltonian for “non-kramers” doublets. Phys. Rev. 171, 350, https://doi.org/PhysRev171.350. Mukherjee, S., Lu, J., Velmurugan, G., Singh, S., Rajaraman, G., Tang, J., Ghosh, S.K., 2016. Influence of tuned linker functionality on modulation of magnetic properties and relaxation dynamics in a family of six isotypic Ln2 (Ln¼Dy and Gd) complexes. Inorg. Chem. 55, 11283–11298. https://doi.org/10.1021/acs.inorgchem.6b01863. Murashima, K., Karasawa, S., Yoza, K., Inagaki, Y., Koga, N., 2016. 3- and 4-(a-diazobenzyl) pyridine-N-oxides as photoresponsive magnetic couplers for 2p–4f heterospin systems: formation of carbene–TbIII and carbene–DyIII single-molecule magnets. Dalton Trans. 45, 7067–7077. https://doi.org/10.1039/c6dt00420b. Natterer, F.D., Yang, K., Paul, W., Willke, P., Choi, T., Greber, T., Heinrich, A.J., Lutz, C.P., 2016. Reading and writing single-atom magnets. Nature 543, 226–228. https://doi.org/ 10.1038/nature21371. Nayak, S., Novitchi, G., Hoły
nska, M., Dehnen, S., 2014. Two heterometallic ionic compounds with isolated [3d] and [4f] complex units: field-induced single-ion magnet (SIM) behavior observed from a mononuclear dysprosium(III) complex. Eur. J. Inorg. Chem. 2014, 3065–3071. https://doi.org/10.1002/ejic.201402114. Nielson, C.W., Koster, G.F., 1963. Spectroscopic Coefficients for the Pn, Dn, and Fn Configurations. The MIT Press, Cambridge, Massachussetts. Nistor, C., Krull, C., Mugarza, A., Stepanow, S., Stamm, C., Soares, M., Klyatskaya, S., Ruben, M., Gambardella, P., 2015. Exchange bias of TbPc2 molecular magnets on antiferromagnetic FeMn and ferromagnetic Fe films. Phys. Rev. B 92, 184402. https://doi.org/ 10.1103/PhysRevB.92.184402. Noda, Y., Noro, S., Akutagawa, T., Nakamura, T., 2014. Gold nanoparticle assemblies stabilized by bis(phthalocyaninato)lanthanide(III) complexes through van der Waals interactions. Sci. Rep. 4, 3758. https://doi.org/10.1038/srep03758. Orbach, R., 1961a. Spin-lattice relaxation in rare-earth salts. Proc. R. Soc. A Math. Phys. Eng. Sci. 264, 458–484. https://doi.org/10.1098/rspa.1961.0211. Orbach, R., 1961b. Spin-lattice relaxation in rare earth salts: field dependence of the two-phonon process. Proc. R. Soc. Lond. A 264, 485–495. https://doi.org/10.1098/rspa.1961.0212. Ortu, F., Liu, J., Burton, M., Fowler, J.M., Formanuik, A., Boulon, M.-E., Chilton, N.F., Mills, D.P., 2017. Analysis of lanthanide-radical magnetic interactions in Ce(III) 2,20 bipyridyl complexes. Inorg. Chem. 56, 2496–2505. https://doi.org/10.1021/acs. inorgchem.6b02683. Oyarzabal, I., Ruiz, J., Seco, J.M., Evangelisti, M., Camo´n, A., Ruiz, E., Aravena, D., Colacio, E., 2014. Rational electrostatic design of easy-axis magnetic anisotropy in a ZnII-DyIII-ZnII single-molecule magnet with a high energy barrier. Chem. A Eur. J. 20, 14262–14269. https://doi.org/10.1002/chem.201403670. Paulovicˇ, J., Cimpoesu, F., Ferbinteanu, M., Hirao, K., 2004. Mechanism of ferromagnetic coupling in copper(II)-gadolinium(III) complexes. J. Am. Chem. Soc. 126, 3321–3331. https:// doi.org/10.1021/ja030628k.

278 Handbook of Magnetic Materials Pedersen, K.S., Ungur, L., Sigrist, M., Sundt, A., Schau-Magnussen, M., Vieru, V., Mutka, H., Rols, S., Weihe, H., Waldmann, O., Chibotaru, L.F., Bendix, J., Dreiser, J., 2014. Modifying the properties of 4f single-ion magnets by peripheral ligand functionalisation. Chem. Sci. 5, 1650–1660. https://doi.org/10.1039/c3sc53044b. Pedersen, K.S., Dreiser, J., Weihe, H., Sibille, R., Johannesen, H.V., Sørensen, M.A., Nielsen, B.E., Sigrist, M., Mutka, H., Rols, S., Bendix, J., Piligkos, S., 2015. Design of single-molecule magnets: insufficiency of the anisotropy barrier as the sole criterion. Inorg. Chem. 54, 7600–7606. https://doi.org/10.1021/acs.inorgchem.5b01209. Pedersen, K.S., Ariciu, A.M., McAdams, S., Weihe, H., Bendix, J., Tuna, F., Piligkos, S., 2016. Toward molecular 4f single-ion magnet qubits. J. Am. Chem. Soc. 138, 5801–5804. https:// doi.org/10.1021/jacs.6b02702. Peng, Y., Mereacre, V., Anson, C.E., Powell, A.K., 2016. Multiple superhyperfine fields in a {DyFe2Dy} coordination cluster revealed using bulk susceptibility and 57Fe M€ossbauer studies. Phys. Chem. Chem. Phys. 18, 21469–21480. https://doi.org/10.1039/C6CP02942F. Perfetti, M., Cucinotta, G., Boulon, M.-E., El Hallak, F., Gao, S., Sessoli, R., 2014. Angularresolved magnetometry beyond triclinic crystals. Part II: torque magnetometry of Cp*ErCOT single-molecule magnets. Chem. A Eur. J. 20, 14051–14056. https://doi.org/10.1002/ chem.201404218. Pines-Rojansky, D., Fain, B., 1990. Phonon-assisted energy-transfer process in a bottlenecked lattice. Phys. Rev. B 41, 2704–2711. https://doi.org/PhysRevB.41.2704. Pinkowicz, D., Ren, M., Zheng, L.M., Sato, S., Hasegawa, M., Morimoto, M., Irie, M., Breedlove, B.K., Cosquer, G., Katoh, K., Yamashita, M., 2014. Control of the singlemolecule magnet behavior of lanthanide-diarylethene photochromic assemblies by irradiation with light. Chem. A Eur. J. 20, 12502–12513. https://doi.org/10.1002/chem.201402647. Pointillart, F., Guizouarn, T., Lefeuvre, B., Golhen, S., Cador, O., Ouahab, L., 2015a. Rational design of a lanthanide-based complex featuring different single-molecule magnets. Chem. A Eur. J. 21, 16929–16934. https://doi.org/10.1002/chem.201502416. Pointillart, F., Jung, J., Berraud-Pache, R., Le Guennic, B., Dorcet, V., Golhen, S., Cador, O., Maury, O., Guyot, Y., Decurtins, S., Liu, S.X., Ouahab, L., 2015b. Luminescence and single-molecule magnet behavior in lanthanide complexes involving a tetrathiafulvalenefused dipyridophenazine ligand. Inorg. Chem. 54, 5384–5397. https://doi.org/10.1021/acs. inorgchem.5b00441. Pointillart, F., Le Guennic, B., Cador, O., Maury, O., Ouahab, L., 2015c. Lanthanide ion and tetrathiafulvalene-based ligand as a “magic” couple toward luminescence, single molecule magnets, and magnetostructural correlations. Acc. Chem. Res. 48, 2834–2842. https://doi. org/10.1021/acs.accounts.5b00296. Poneti, G., Bernot, K., Bogani, L., Caneschi, A., Sessoli, R., Wernsdorfer, W., Gatteschi, D., 2007. A rational approach to the modulation of the dynamics of the magnetisation in a dysprosium–nitronyl-nitroxide radical complex. Chem. Commun. 1807–1809. https://doi.org/ 10.1039/B617898G. PPMS®, n.d. PPMS (Physical Properties Measurement System). Quantum Des. Inc. California, USA. Prins, F., Pasca, E., De Jongh, L.J., Kooijman, H., Spek, A.L., Tanase, S., 2007. Long-range magnetic ordering in a TbIII-MoV cyanido-bridged quasi-one-dimensional complex. Angew. Chem. Int. Ed. 46, 6081–6084. https://doi.org/10.1002/anie.200701847. Prokof’ev, N.V., Stamp, P.C.E., 1998. Low-temperature quantum relaxation in a system of magnetic nanomolecules. Phys. Rev. Lett 80, 5794–5797. https://doi.org/10.1103/ PhysRevLett.80.5794.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 279

Prokof’ev, N.V., Stamp, P.C.E., 2000. Theory of the spin bath. Rep. Prog. Phys. 63, 669–726. https://doi.org/10.1088/0034-4885/63/4/204. Prsˇa, K., Nehrkorn, J., Corbey, J., Evans, W., Demir, S., Long, J., Guidi, T., Waldmann, O., 2016. Perspectives on neutron scattering in lanthanide-based single-molecule magnets and a case study of the Tb2(m-N2) system. Magnetochemistry 2, 45. https://doi.org/10.3390/ magnetochemistry2040045. Pugh, T., Chilton, N.F., Layfield, R.A., 2016. A low-symmetry dysprosium metallocene singlemolecule magnet with a high anisotropy barrier. Angew. Chemi. Int. Ed. 55, 11082–11085. https://doi.org/10.1002/anie.201604346. Qian, K., Baldovı´, J.J., Jiang, S.-D., Gaita-Arin˜o, A., Zhang, Y.-Q., Overgaard, J., Wang, B.-W., Coronado, E., Gao, S., 2015. Does the thermal evolution of molecular structures critically affect the magnetic anisotropy? Chem. Sci. 6, 4587–4593. https://doi.org/10.1039/C5SC01245G. Qin, L., Yu, Y.Z., Liao, P.Q., Xue, W., Zheng, Z., Chen, X.M., Zheng, Y.Z., 2016. A “molecular water pipe”: a Giant tubular cluster (Dy72) exhibits fast proton transport and slow magnetic relaxation. Adv. Mater. 10772–10779. https://doi.org/10.1002/adma.201603381. Rajaraman, G., Totti, F., Bencini, A., Caneschi, A., Sessoli, R., Gatteschi, D., 2009. Density functional studies on the exchange interaction of a dinuclear Gd(III)-Cu(II) complex: method assessment, magnetic coupling mechanism and magneto-structural correlations. Dalton Trans. 3153–3161. https://doi.org/10.1039/b817540c. Rechkemmer, Y., Fischer, J.E., Marx, R., D€orfel, M., Neugebauer, P., Horvath, S., Gysler, M., Brock-Nannestad, T., Frey, W., Reid, M.F., van Slageren, J., 2015. Comprehensive spectroscopic determination of the crystal field splitting in an erbium single-ion magnet. J. Am. Chem. Soc. 137, 13114–13120. https://doi.org/10.1021/jacs.5b08344. Reiher, M., Wolf, A., 2009. Relativistic Quantum Chemistry. Advances in Quantum Chemistry. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. https://doi.org/ 10.1002/9783527627486. Reis, S.G., Briganti, M., Martins, D.O.T.A., Akpinar, H., Calancea, S., Guedes, G.P., Soriano, S., Andruh, M., Cassaro, R.A.A., Lahti, P.M., Totti, F., Vaz, M.G.F., 2016. First coordination compounds based on a bis(imino nitroxide) biradical and 4f metal ions: synthesis, crystal structures and magnetic properties. Dalton Trans. 45, 2936–2944. https://doi.org/10.1039/ C5DT04469C. Ren, Y.X., Zheng, X.J., Li, L.C., Yuan, D.Q., An, M., Jin, L.P., 2014. Three-dimensional frameworks based on dodecanuclear Dy-hydroxo wheel cluster with slow relaxation of magnetization. Inorg. Chem. 53, 12234–12236. https://doi.org/10.1021/ic502042h. Ren, M., Bao, S.-S., Wang, B.-W., Ferreira, R.a.S., Zheng, L.-M., Carlos, L.D., 2015. Lanthanide phosphonates with pseudo-D5h local symmetry exhibiting magnetic and luminescence bifunctional properties. Inorg. Chem. Front. 2, 558–566. https://doi.org/10.1039/C4QI00242C. Repoll
es, A.M., 2016. Quantum Computing with Molecular Magnets, Estudios de Fı´sica, 131, Prensas de la Universidad de Zaragoza, Zaragoza (Ph.D. dissertation of the University of Zaragoza). Rinck, J., Novitchi, G., Van den Heuvel, W., Ungur, L., Lan, Y., Wernsdorfer, W., Anson, C.E., Chibotaru, L.F., Powell, A.K., 2010. An octanuclear [CrIII4DyIII4] 3d-4f single-molecule magnet. Angew. Chem. Int. Ed. 49, 7583–7587. https://doi.org/10.1002/anie.201002690. Rinehart, J.D., Long, J.R., 2011. Exploiting single-ion anisotropy in the design of f-element single-molecule magnets. Chem. Sci. 2, 2078. https://doi.org/10.1039/c1sc00513h. Rinehart, J.D., Long, J.R., 2012. Slow magnetic relaxation in homoleptic trispyrazolylborate complexes of neodymium(III) and uranium(III). Dalton Trans. 41, 13572. https://doi.org/10.1039/ c2dt31352a.

280 Handbook of Magnetic Materials Rinehart, J.D., Fang, M., Evans, W.J., Long, J.R., 2011. Strong exchange and magnetic blocking in N₂3-radical-bridged lanthanide complexes. Nat. Chem. 3, 538–542. https://doi.org/ 10.1038/nchem.1063. Robaschik, P., Fronk, M., Toader, M., Klyatskaya, S., Ganss, F., Siles, P.F., Schmidt, O.G., Albrecht, M., Hietschold, M., Ruben, M., Zahn, D.R.T., Salvan, G., 2015. Tuning the magneto-optical response of TbPc2 single molecule magnets by the choice of the substrate. J. Mater. Chem. C 3, 8039–8049. https://doi.org/10.1039/C5TC01520K. Roos, B.O., Malmqvist, P.A., 2004. Relativistic quantum chemistry: the multiconfigurational approach. Phys. Chem. Chem. Phys. 6, 2919. https://doi.org/10.1039/b401472n. Roos, B.O., Taylor, P.R., Siegbahn, P.E.M., 1980. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 48, 157–173. https://doi.org/10.1016/0301-0104(80)80045-0. ˚ ., Veryazov, V., Widmark, P.O., 2004. Main group atoms Roos, B.O., Lindh, R., Malmqvist, P.A and dimers studied with a new relativistic ANO basis set. J. Phys. Chem. A 108, 2851–2858. https://doi.org/10.1021/jp031064+. ˚ ., Veryazov, V., Widmark, P.O., 2005. New relativistic Roos, B.O., Lindh, R., Malmqvist, P.A ANO basis sets for transition metal atoms. J. Phys. Chem. A 109, 6575–6579. https://doi. org/10.1021/jp0581126. Rosado Piquer, L., San˜udo, E.C., 2015. Heterometallic 3d–4f single-molecule magnets. Dalton Trans. 44, 8771–8780. https://doi.org/10.1039/C5DT00549C. Roubeau, O., Natividad, E., Evangelisti, M., Lorusso, G., Palacios, E., 2017. A magnetocaloric composite based on molecular coolers and carbon nanotubes with enhanced thermal conductivity. Mater. Horiz. 4, 464–476. https://doi.org/10.1039/C6MH00533K. Roy, S., Chakraborty, A., Maji, T.K., 2014. Lanthanide-organic frameworks for gas storage and as magneto-luminescent materials. Coord. Chem. Rev. 273–274, 139–164. https://doi.org/ 10.1016/j.ccr.2014.03.035. Ruby, R.H., Benoit, H., Jeffries, C.D., 1962. Paramagnetic resonance below 1°K: spin-lattice relaxation of Ce3+ and Nd3+ in lanthanum magnesium nitrate. Phys. Rev. 127, 51–56. https://doi.org/10.1103/PhysRev.127.51. Ruiz, J., Lorusso, G., Evangelisti, M., Brechin, E.K., Pope, S.J.A., Colacio, E., 2014. Closelyrelated ZnII2LnIII2 complexes (lnIII ¼ Gd, Yb) with either magnetic refrigerant or luminescent single-molecule magnet properties. Inorg. Chem. 53, 3586–3594. https://doi.org/ 10.1021/ic403097s. Sakurai, J.J., 1964. Modern Quantum Mechanics. Addison-Weasley, Reading, UK. Sales, M., Hernandez, J.M., Tejada, J., Martinez, J.L., 1999. Time-dependent heat capacity of Mn12 clusters. Phys. Rev. B 60, 14557–14560. https://doi.org/10.1103/PhysRevB.60.14557. Sato, R., Suzuki, K., Sugawa, M., Mizuno, N., 2013. Heterodinuclear lanthanoid-containing polyoxometalates: stepwise synthesis and single-molecule magnet behavior. Chem. A Eur. J. 19, 12982–12990. https://doi.org/10.1002/chem.201302596. Schenker, R., Leuenberger, M.N., Chaboussant, G., Loss, D., G€udel, H.U., 2005. Phonon bottleneck effect leads to observation of quantum tunneling of the magnetization and butterfly hysteresis loops in (Et4N)3Fe2F9. Phys. Rev. B 72, 184403. https://doi.org/10.1103/ PhysRevB.72.184403. Serri, M., Mannini, M., Poggini, L., Velez-Fort, E., Cortigiani, B., Rovai, D., Caneschi, A., Sessoli, R., 2017. Low temperature magnetic force microscopy on single molecule magnetbased microarrays. Nano Lett. 17, 1899–1905. https://doi.org/10.1021/acs.nanolett.6b05208. Sessoli, R., 2015. Toward the quantum computer: magnetic molecules back in the race. ACS Cent. Sci. 1, 473–474.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 281

Sessoli, R., Bernot, K., 2015. Lanthanides in extended molecular networks. In: Layfield, R., Murugesu, M. (Eds.), Lanthanides and Actinides in Molecular Magnetism. Wiley-VCH, Weinheim, Germany, pp. 89–123. Sessoli, R., Powell, A.K., 2009. Strategies towards single molecule magnets based on lanthanide ions. Coord. Chem. Rev. 253, 2328–2341. https://doi.org/10.1016/j.ccr.2008.12.014. Sessoli, R., Gatteschi, D., Caneschi, A., Novak, M.A., 1993. Magnetic bistability in a metal-ion cluster. Nature 365, 141–143. https://doi.org/10.1038/365141a0. Shan, P.Y., Li, H.F., Chen, P., Tian, Y.M., Sun, W.B., Yan, P.F., 2015. Synthesis, crystal structure, and single-molecule magnetic properties of a salen-type Zn-Dy-Zn complex. Zeitsch. Anorg. Allg. Chem. 641, 1119–1124. https://doi.org/10.1002/zaac.201400610. Sheikh, J.A., Adhikary, A., Konar, S., 2014. Magnetic refrigeration and slow magnetic relaxation in tetranuclear lanthanide cages (Ln ¼ Gd, Dy) with in situ ligand transformation. New J. Chem. 38, 3006–3014. https://doi.org/10.1039/c4nj00349g. Shen, H.-Y., Wang, W.-M., Bi, Y.-X., Gao, H.-L., Liu, S., Cui, J.-Z., 2015. Luminescence, magnetocaloric effect and single-molecule magnet behavior in lanthanide complexes based on a tridentate ligand derived from 8-hydroxyquinoline. Dalton Trans. 44, 18893–18901. https:// doi.org/10.1039/C5DT02894A. Shen, H.-Y., Wang, W.-M., Gao, H.-L., Cui, J.-Z., 2016. Near-infrared luminescence and SMM behaviors of a family of dinuclear lanthanide 8-quinolinolate complexes. RSC Adv. 6, 34165–34174. https://doi.org/10.1039/C6RA02656G. Shiddiq, M., Komijani, D., Duan, Y., Gaita-Arin˜o, A., Coronado, E., Hill, S., 2016. Enhancing coherence in molecular spin qubits via atomic clock transitions. Nature 531, 348–351. https://doi.org/10.1038/nature16984. Shrivastava, K.N., 1972. Raman relaxation times in electronic systems. Phys. Status Solidi (b) 51, 377–387. https://doi.org/10.1002/pssb.2220510138. Shrivastava, K.N., 1983. Theory of spin-lattice relaxation. Phys. Status Solidi (b) 117, 437–458. https://doi.org/10.1002/pssb.2221170202. Sieklucka, B., Pinkowicz, D. (Eds.), 2017. Molecular Magnetic Materials. Wiley-VCH Verlag GmbH, Weinheim, Germany. Sievers, J., 1982. Asphericity of 4f-shells in their Hund’s rule ground states. Zeitsch. Phys. B Condens. Matter 45, 289–296. https://doi.org/10.1007/BF01321865. Singh, M.K., Rajaraman, G., 2016. Acquiring a record barrier heigh for magnetization reversal in lanthanide encapsulated fullerene molecules using DFT and ab initio calculations. Chem. Commun. 52, 14047–14050. https://doi.org/10.1039/C6CC08232G. Singh, S.K., Gupta, T., Rajaraman, G., 2014a. Magnetic anisotropy and mechanism of magnetic relaxation in Er(III) single-ion magnets. Inorg. Chem. 53, 10835–10845. https://doi.org/ 10.1021/ic500772f. Singh, S.K., Gupta, T., Shanmugam, M., Rajaraman, G., 2014b. Unprecedent magnetic relaxation via the fourth excited state in low-coordinate lanthanide single-ion magnets: a theoretical approach. Chem. Commun. 5015513https://doi.org/10.1039/c4cc05522e. Singh, M.K., Yadav, N., Rajaraman, G., 2015. Record high magnetic exchange and magnetization blockade in Ln2@C79N (Ln ¼ Gd(III) and Dy(III)) molecules: a theoretical perspective. Chem. Commun. 51, 17732–17735. https://doi.org/10.1039/C5CC06642E. Sorai, M., Nakano, M., Miyazaki, Y., 2006. Calorimetric investigation of phase transitions occurring in molecule-based magnets. Chem. Rev. 106, 976–1031. https://doi.org/10.1021/cr960049g. Standley, J., Vaughan, R.A., 1969. The phonon bottleneck. In: Electron Spin Relaxation Phenomena in Solids. Monographs on Electron Spin Resonance. Springer, Boston, MA. https://doi. org/10.1007/978-1-4899-6539-4_4.

282 Handbook of Magnetic Materials Stepanow, S., Honolka, J., Gambardella, P., Vitali, L., Abdurakhmanova, N., Tseng, T.C., Rauschenbach, S., Tait, S.L., Sessi, V., Klyatskaya, S., Ruben, M., Kern, K., 2010. Spin and orbital magnetic moment anisotropies of monodispersed bis(phthalocyaninato)terbium on a copper surface. J. Am. Chem. Soc. 132, 11900–11901. https://doi.org/10.1021/ ja105124r. Stevens, K.W.H., 1952. Matrix elements and operator equivalents connected with magnetic properties of rare earth ions. Proc. Phys. Soc. A 65, 209. https://doi.org/10.1088/0370-1298/65/ 3/308. Stevens, K.W.H., 1967. The theory of paramagnetic relaxation. Rep. Prog. Phys. 30, 189. https:// doi.org/10.1088/0034-4885/30/1/305. Stoll, P., Bernien, M., Rolf, D., Nickel, F., Xu, Q., Hartmann, C., Umbach, T.R., Kopprasch, J., Ladenthin, J.N., Schierle, E., Weschke, E., Czekelius, C., Kuch, W., Franke, K.J., 2016. Magnetic anisotropy in surface-supported single-ion lanthanide complexes. Phys. Rev. B 94, 224426. https://doi.org/10.1103/PhysRevB.94.224426. Stoneham, A.M., 1965. The phonon bottleneck in paramagnetic crystals. Proc. Phys. Soc. 86, 1163. https://doi.org/10.1088/0370-1328/86/6/302. Sullivan, P.F., Seidel, G., 1968. Steady-state, Ac-temperature calorimetry. Phys. Rev. 173, 679–685. https://doi.org/10.1103/PhysRev.173.679. Sun, W.-B., Yan, B., Zhang, Y.-Q., Wang, B.-W., Wang, Z.-M., Jia, J.-H., Gao, S., 2014a. The slow magnetic relaxation regulated by ligand conformation of a lanthanide single-ion magnet [Hex4N][Dy(DBM)4]. Inorg. Chem. Front. 1, 503. https://doi.org/10.1039/ C4QI00057A. Sun, Y.G., Zong, W.H., Xiong, G., Guo, M.Y., Ding, F., Wang, S.J., You, L.X., Ren, B.Y., Xu, Z.H., Gao, E.J., 2014b. Synthesis, structure, photoluminescence and magnetism of 3d-4f heterometallic coordination polymers bearing benzimidazole-5,6-dicarboxylate. Polyhedron 83, 68–76. https://doi.org/10.1016/j.poly.2014.04.029. Sun, W.-B., Yan, B., Jia, L.-H., Wang, B.-W., Yang, Q., Cheng, X., Li, H.-F., Chen, P., Wang, Z.M., Gao, S., 2016a. Dinuclear dysprosium SMMs bridged by a neutral bipyrimidine ligand: two crystal systems that depend on different lattice solvents lead to a distinct slow relaxation behaviour. Dalton Trans. 45, 8790–8794. https://doi.org/10.1039/c6dt01082b. Sun, W.-B., Yan, P.-F., Jiang, S.-D., Wang, B.-W., Zhang, Y.-Q., Li, H.-F., Chen, P., Wang, Z.M., Gao, S., 2016b. High symmetry or low symmetry, that is the question—high performance Dy(III) single-ion magnets by electrostatic potential design. Chem. Sci. 7, 684–691. https:// doi.org/10.1039/C5SC02986D. Takamatsu, S., Ishikawa, T., Koshihara, S.Y., Ishikawa, N., 2007. Significant increase of the barrier energy for magnetization reversal of a single-4f-ionic single-molecule magnet by a longitudinal contraction of the coordination space. Inorg. Chem. 46, 7250–7252. https://doi.org/ 10.1021/ic700954t. Takehara, C., Then, P.L., Kataoka, Y., Nakano, M., Yamamura, T., Kajiwara, T., 2015. Slow magnetic relaxation of light lanthanide-based linear LnZn2 trinuclear complexes. Dalton Trans. 44, 18276–18283. https://doi.org/10.1039/C5DT03148F. Tang, J., Zhang, P., 2015. Lanthanide Single Molecule Magnets. Springer Berlin Heidelberg, Berlin, Heidelberghttps://doi.org/10.1007/978-3-662-46999-6. Tchougr
eeff, A.L., Dronskowski, R., 2009. Nephelauxetic effect revisited. Int. J. Quantum Chem. 109, 2606–2621. https://doi.org/10.1002/qua.21989. Tesi, L., Lunghi, A., Atzori, M., Lucaccini, E., Sorace, L., Totti, F., Sessoli, R., 2016. Giant spinphonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence. Dalton Trans. 45, 16635–16645. https://doi.org/10.1039/C6DT02559E.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 283

Thiele, S., Vincent, R., Holzmann, M., Klyatskaya, S., Ruben, M., Balestro, F., Wernsdorfer, W., 2013. Electrical readout of individual nuclear spin trajectories in a single-molecule magnet spin transistor. Phys. Rev. Lett. 111, 037203. https://doi.org/10.1103/ PhysRevLett.111.037203. Thiele, S., Balestro, F., Ballou, R., Klyatskaya, S., Ruben, M., Wernsdorfer, W., 2014. Electrically driven nuclear spin resonance in single-molecule magnets. Science 344 (80), 1135–1138. https://doi.org/10.1126/science.1249802. Thielemann, D.T., Wagner, A.T., Lan, Y., On˜a-Burgos, P., Ferna´ndez, I., R€osch, E.S., K€ olmel, D.K., Powell, A.K., Br€ase, S., Roesky, P.W., 2014. Peptoid-ligated pentadecanuclear yttrium and dysprosium hydroxy clusters. Chem. A Eur. J. 21, 2813–2820. https://doi.org/ 10.1002/chem.201405569. Thole, B.T., Carra, P., Sette, F., Van Der Laan, G., 1992. X-ray circular dichroism as a probe of orbital magnetization. Phys. Rev. Lett. 68, 1943–1946. https://doi.org/10.1103/ PhysRevLett.68.1943. Thomas, L., Lionti, F., Ballou, R., Gatteschi, D., Sessoli, R., Barbara, B., 1996. Macroscopic quantum tunnelling of magnetization in a single crystal of nanomagnets. Nature 383, 145–147. https://doi.org/10.1038/383145a0. Tian, C.-B., Yuan, D.-Q., Han, Y.-H., Li, Z.-H., Lin, P., Du, S.-W., 2014. Synthesis, structures, and magnetic properties of a series of new heterometallic hexanuclear Co2Ln4 (Ln ¼ Eu, Gd, Tb and Dy) clusters. Inorg. Chem. Front. 1, 695–704. https://doi.org/10.1039/ C4QI00116H. Tong, Y.Z., Gao, C., Wang, Q.L., Wang, B.W., Gao, S., Cheng, P., Liao, D.Z., 2015. Two mononuclear single molecule magnets derived from dysprosium(III) and tmphen (tmphen ¼ 3,4,7,8tetramethyl-1,10-phenanthroline). Dalton Trans. 44, 9020–9026. https://doi.org/10.1039/ c5dt00869g. Treier, M., Ruffieux, P., Fasel, R., Nolting, F., Yang, S., Dunsch, L., Greber, T., 2009. Looking inside an endohedral fullerene: inter- and intramolecular ordering of Dy3N @ C80 (Ih) on Cu(111). Phys. Rev. B 80, 81403. https://doi.org/10.1103/PhysRevB.80.081403. Trifonov, A.A., Shestakov, B., Long, J., Lyssenko, K., Guari, Y., Larionova, J., 2015. An organoytterbium(III) complex exhibiting field-induced single-ion magnet behavior. Inorg. Chem. 54, 7667–7669. https://doi.org/10.1021/acs.inorgchem.5b01318. Tziotzi, T.G., Tzimopoulos, D.I., Lis, T., Inglis, R., Milios, C.J., 2015. Dodecanuclear [MnLn] species: synthesis, structures and characterization of magnetic relaxation phenomena. Dalton Trans. 44, 11696–11699. https://doi.org/10.1039/c5dt01695a. Tziotzi, T.G., Siczek, M., Lis, T., Inglis, R., Milios, C.J., 2016. Two unique star-like [MnIVMnIII2LnIII] clusters: magnetic relaxation phenomena. RSC Adv. 6, 45326–45329. https://doi.org/10.1039/C6RA09066D. Ungur, L., Chibotaru, L.F., 2011. Magnetic anisotropy in the excited states of low symmetry lanthanide complexes. Phys. Chem. Chem. Phys. 13, 20086–20090. https://doi.org/10.1039/ c1cp22689d. Ungur, L., Chibotaru, L.F., 2016a. Ab initio crystal field for lanthanides. Chem. A Eur. J. 3708–3718. https://doi.org/10.1002/CHEM.201605102. Ungur, L., Chibotaru, L.F., 2016b. Strategies toward high-temperature lanthanide-based singlemolecule magnets. Inorg. Chem. 55, 10043–10056. https://doi.org/10.1021/acs. inorgchem.6b01353. Ungur, L., Van den Heuvel, W., Chibotaru, L.F., 2009. Ab initio investigation of the non-collinear magnetic structure and the lowest magnetic excitations in dysprosium triangles. New J. Chem. 33, 1224. https://doi.org/10.1039/b903126j.

284 Handbook of Magnetic Materials Ungur, L., Thewissen, M., Costes, J.P., Wernsdorfer, W., Chibotaru, L.F., 2013. Interplay of strongly anisotropic metal ions in magnetic blocking of complexes. Inorg. Chem. 52, 6328–6337. https://doi.org/10.1021/ic302568x. Ungur, L., Leroy, J.J., Korobkov, I., Murugesu, M., Chibotaru, L.F., 2014a. Fine-tuning the local symmetry to attain record blocking temperature and magnetic remanence in a single-ion magnet. Angew. Chem. Int. Ed. 53, 4413–4417. https://doi.org/10.1002/anie.201310451. Ungur, L., Lin, S.-Y., Tang, J., Chibotaru, L.F., 2014b. Single-molecule toroics in Ising-type lanthanide molecular clusters. Chem. Soc. Rev. 43, 6894–6905. https://doi.org/10.1039/c4cs00095a. Upadhyay, A., Das, C., Langley, S.K., Murray, K.S., Srivastava, A.K., Shanmugam, M., 2016. Heteronuclear Ni(II)-Ln(III) (Ln ¼ La, Pr, Tb, Dy) complexes: synthesis and single-molecule magnet behaviour. Dalton Trans. 45, 3616–3626. https://doi.org/10.1039/C5DT04102C. Urdampilleta, M., Klyatskaya, S., Cleuziou, J.-P., Ruben, M., Wernsdorfer, W., 2011. Supramolecular spin valves. Nat. Mater. 10, 502–506. https://doi.org/10.1038/nmat3050. Urdampilleta, M., Klyatskaya, S., Ruben, M., Wernsdorfer, W., 2013. Landau-Zener tunneling of a single Tb3+ magnetic moment allowing the electronic read-out of a nuclear spin. Phys. Rev. B 87, 195412. https://doi.org/10.1103/PhysRevB.87.195412. Urdampilleta, M., Klayatskaya, S., Ruben, M., Wernsdorfer, W., 2015. Magnetic interaction between a radical spin and a single-molecule magnet in a molecular spin-valve. ACS Nano 9, 4458–4464. https://doi.org/10.1021/acsnano.5b01056. Van Vleck, J.H., 1932. The Theory of Electric and Magnetic Susceptibilities. Oxford University Press, Oxford, UK, 226. Van Vleck, J.H., 1941a. Calculation of energy exchange between lattice oscillators. Phys. Rev. 59, 730–736. https://doi.org/10.1103/PhysRev.59.730. Van Vleck, J.H., 1941b. Paramagnetic relaxation and the equilibrium of lattice oscillators. Phys. Rev. 59, 724–729. https://doi.org/10.1103/PhysRev.59.724. Vieru, V., Iwahara, N., Ungur, L., Chibotaru, L.F., 2016a. Giant exchange interaction in mixed lanthanides. Sci. Rep. 6, 24046. https://doi.org/10.1038/srep24046. Vieru, V., Pasatoiu, T.D., Ungur, L., Suturina, E., Madalan, A.M., Duhayon, C., Sutter, J.-P., Andruh, M., Chibotaru, L.F., 2016b. Synthesis, crystal structures, magnetic properties, and theoretical investigation of a new series of NiII–LnIII–WV heterotrimetallics: understanding the SMM behavior of mixed polynuclear complexes. Inorg. Chem. 55, 12158–12171. https:// doi.org/10.1021/acs.inorgchem.6b01669. Vignesh, K.R., Langley, S.K., Murray, K.S., Rajaraman, G., 2017. Quenching the quantum tunneling of magnetization in heterometallic octanuclear {TMIII4DyIII4} (TM ¼ Co and Cr) single-molecule magnets by modification of the bridging ligands and enhancing the magnetic exchange coupling. Chem. A Eur. J. 1654–1666. https://doi.org/10.1002/chem.201604835. urger, A., Fort, A., Rettori, A., 1997. Effet tunnel dans les syste`mes magn
etiques: de Villain, J., W€ la description microscopique et d
eterministe à l’
equation maıˆtresse. J. Phys. I Fr. 7, 1583–1594. https://doi.org/10.1051/jp1:1997156. Vincent, R., Klyatskaya, S., Ruben, M., Wernsdorfer, W., Balestro, F., 2012. Electronic read-out of a single nuclear spin using a molecular spin transistor. Nature 488, 357–360. https://doi. org/10.1038/nature11341. Visinescu, D., Alexandru, M.-G., Madalan, A.M., Pichon, C., Duhayon, C., Sutter, J.-P., Andruh, M., 2015. Magneto-structural variety of new 3d–4f–4(5)d heterotrimetallic complexes. Dalton Trans. 44, 16713–16727. https://doi.org/10.1039/C5DT01738F. Vitali, L., Fabris, S., Conte, A.M., Brink, S., Ruben, M., Baroni, S., Kern, K., 2008. Electronic structure of surface-supported bis(phthalocyaninato) terbium (III) single molecular magnets. Nano Lett. 8, 3364–3368. https://doi.org/10.1021/nl801869b.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 285

Vonci, M., Giansiracusa, M.J., Gable, R.W., Van den Heuvel, W., Latham, K., Moubaraki, B., Murray, K.S., Yu, D., Mole, R.A., Soncini, A., Boskovic, C., 2016. Ab initio calculations as a quantitative tool in the inelastic neutron scattering study of a single-molecule magnet analogue. Chem. Commun. 52, 2091–2094. https://doi.org/10.1039/C5CC07541F. Vonci, M., Giansiracusa, M.J., Van den Heuvel, W., Gable, R.W., Moubaraki, B., Murray, K.S., Yu, D., Mole, R.A., Soncini, A., Boskovic, C., 2017. Magnetic excitations in polyoxotungstate-supported lanthanoid single-molecule magnets: an inelastic neutron scattering and ab initio study. Inorg. Chem. 56, 378–394. https://doi.org/10.1021/acs. inorgchem.6b02312. W€ackerlin, C., Donati, F., Singha, A., Baltic, R., Rusponi, S., Diller, K., Patthey, F., Pivetta, M., Lan, Y., Klyatskaya, S., Ruben, M., Brune, H., Dreiser, J., 2016. Single-molecule magnets: giant hysteresis of single-molecule magnets adsorbed on a nonmagnetic insulator. Adv. Mater. 28, 5142. https://doi.org/10.1002/adma.201670180. Wada, H., Ooka, S., Yamamura, T., Kajiwara, T., 2017. Light lanthanide complexes with crown ether and its aza derivative which show slow magnetic relaxation behaviors. Inorg. Chem. 56, 147–155. https://doi.org/10.1021/acs.inorgchem.6b01764. Wang, B., Jiang, S., Wang, X., Gao, S., 2009. Magnetic molecular materials with paramagnetic lanthanide ions. Sci. China Ser. B-Chem. 52, 1739–1758. https://doi.org/10.1007/s11426009-0275-9. Wang, K., Chen, Z.L., Zou, H.H., Zhang, Z., Sun, W.Y., Liang, F.P., 2015a. Two types of Cu-Ln heterometallic coordination polymers with 2-hydroxyisophthalate: syntheses, structures, and magnetic properties. Cryst. Growth Des. 15, 2883–2890. https://doi.org/10.1021/acs. cgd.5b00327. Wang, X., Li, Y., Hu, P., Wang, J., Li, L., 2015b. Slow magnetic relaxation and field-induced metamagnetism in nitronyl nitroxide-Dy(III) magnetic chains. Dalton Trans. 44, 4560–4567. https://doi.org/10.1039/c4dt01878h. Wang, G., Wei, Y., Wu, K., 2016a. Goblet-shaped pentanuclear lanthanide clusters assembled with a cyclen derivative ligand exhibiting slow magnetic relaxation. Dalton Trans. 45, 12734–12738. https://doi.org/10.1039/C6DT02062C. Wang, S., Cao, T., Yan, H., Li, Y., Lu, J., Ma, R., Li, D., Dou, J., Bai, J., 2016b. Functionalization of microporous lanthanide-based metal-organic frameworks by dicarboxylate ligands with methyl-substituted thieno[2,3-b]thiophene groups: sensing activities and magnetic properties. Inorg. Chem. 55, 5139–5151. https://doi.org/10.1021/acs.inorgchem.5b02801. Wang, Y.L., Han, C.-B., Zhang, Y., Liu, Q.-Y., Liu, C.-M., Yin, S.-G., 2016c. Fine-tuning ligand to modulate the magnetic anisotropy in a carboxylate-bridged Dy2 single-molecule magnet system. Inorg. Chem. 55, 5578–5584. Wei, X.-H., Yang, L.-Y., Liao, S.-Y., Zhang, M., Tian, J.-L., Du, P.-Y., Gu, W., Liu, X., 2014. A series of rare earth complexes with novel non-interpenetrating 3D networks: synthesis, structures, magnetic and optical properties. Dalton Trans. 43, 5793. https://doi.org/10.1039/ c3dt53112k. Wen, H.R., Liu, S.J., Xie, X.R., Bao, J., Liu, C.M., Chen, J.L., 2015. A family of nickellanthanide heterometallic dinuclear complexes derived from a chiral Schiff-base ligand exhibiting single-molecule magnet behaviors. Inorg. Chim. Acta 435, 274–282. https://doi.org/ 10.1016/j.ica.2015.07.009. Wen, H.-R., Dong, P.-P., Liu, S.-J., Liao, J.-S., Liang, F.-Y., Liu, C.-M., 2017. 3D–4F heterometallic trinuclear complexes derived from amine-phenol tripodal ligands exhibiting magnetic and luminescent properties. Dalton Trans. 46, 1153–1162. https://doi.org/10.1039/ C6DT04027F.

286 Handbook of Magnetic Materials Wernsdorfer, W., 2009. From micro- to nano-SQUIDs: applications to nanomagnetism. Supercond. Sci. Technol. 22, 64013. https://doi.org/10.1088/0953-2048/22/6/064013. Wernsdorfer, W., Ohm, T., Sangregorio, C., Sessoli, R., Mailly, D., Paulsen, C., 1999. Observation of the distribution of molecular spin states by resonant quantum tunneling of the magnetization. Phys. Rev. Lett. 82, 3903. https://doi.org/10.1103/ PhysRevLett.82.3903. Westerstr€ om, R., Dreiser, J., Piamonteze, C., Muntwiler, M., Weyeneth, S., Brune, H., Rusponi, S., Nolting, F., Popov, A., Yang, S., Dunsch, L., Greber, T., 2012. An endohedral single-molecule magnet with long relaxation times: DySc2N@C80. J. Am. Chem. Soc. 134, 9840–9843. https://doi.org/10.1021/ja301044p. Westerstr€ om, R., Uldry, A.C., Stania, R., Dreiser, J., Piamonteze, C., Muntwiler, M., Matsui, F., Rusponi, S., Brune, H., Yang, S., Popov, A., B€uchner, B., Delley, B., Greber, T., 2015. Surface aligned magnetic moments and hysteresis of an endohedral single-molecule magnet on a metal. Phys. Rev. Lett. 114, 87201. https://doi.org/10.1103/PhysRevLett.114.087201. Wu, Z.-L., Dong, J., Ni, W.-Y., Zhang, B.-W., Cui, J.-Z., Zhao, B., 2014. pH-induced Dy4 and Dy10 cluster-based 1D chains with different magnetic relaxation features. Dalton Trans. 43, 16838–16845. https://doi.org/10.1039/C4DT02427C. Wu, J., Jung, J., Zhang, P., Zhang, H., Tang, J., Le Guennic, B., 2016. cis-trans isomerism modulates the magnetic relaxation of dysprosium single-molecule magnets. Chem. Sci. 7, 3632–3639. https://doi.org/10.1039/c5sc04510j. Wu, J., Lin, S.-Y., Shen, S., Li, X.-L., Zhao, L., Zhang, L., Tang, J., 2017. Probing the magnetic relaxation and magnetic moment arrangement in a series of Dy4 squares. Dalton Trans. 46, 1577–1584. https://doi.org/10.1039/C6DT04456E. Wybourne, B.G., 1965. Spectroscopic Properties of Rare Earths. John Wiley & Sons, Inc., New York Xie, Q.-W., Wu, S.-Q., Shi, W.-B., Liu, C.-M., Cui, A.-L., Kou, H.-Z., 2014. Heterodinuclear M (II)-Ln(III) single molecule magnets constructed from exchange-coupled single ion magnets. Dalton Trans. 43, 11309–11316. https://doi.org/10.1039/c4dt00740a. Xue, S., Guo, Y.-N., Zhao, L., Zhang, P., Tang, J., 2014a. Unique Y-shaped lanthanide aggregates and single-molecule magnet behaviour for the Dy4 analogue. Dalton Trans. 43, 1564–1570. https://doi.org/10.1039/C3DT52444B. Xue, S., Ungur, L., Guo, Y., Tang, J., Chibotaru, L.F., 2014b. Field-induced multiple relaxation mechanism of CoII2DyIII compound with the Dy ion in a low symmetrical environment. Inorg. Chem. 53, 12658. https://doi.org/10.1021/ic502443g. Yadav, M., Mondal, A., Mereacre, V., Jana, S.K., Powell, A.K., Roesky, P.W., 2015. Tetranuclear and pentanuclear compounds of the rare-earth metals: synthesis and magnetism. Inorg. Chem. 54, 7846–7856. https://doi.org/10.1021/acs.inorgchem.5b00899. Yatoo, M.A., Cosquer, G., Morimoto, M., Irie, M., Breedlove, B.K., Yamashita, M., 2016. 1D chains of lanthanoid ions and a dithienylethene ligand showing slow relaxation of the magnetization. Magnetochemistry 2, 21. https://doi.org/10.3390/magnetochemistry2020021. Yi, X., Bernot, K., Pointillart, F., Poneti, G., Calvez, G., Daiguebonne, C., Guillou, O., Sessoli, R., 2012. A luminescent and sublimable DyIII-based single-molecule magnet. Chem. A Eur. J. 18, 11379–11387. https://doi.org/10.1002/chem.201201167. Yi, X., Bernot, K., Le Corre, V., Calvez, G., Pointillart, F., Cador, O., Le Guennic, B., Jung, J., Maury, O., Placide, V., Guyot, Y., Roisnel, T., Daiguebonne, C., Guillou, O., 2014. Unraveling the crystal structure of lanthanide-murexide complexes: use of an ancient complexometry indicator as a near-infrared-emitting single-ion magnet. Chem. A Eur. J. 20, 1569–1576. https://doi.org/10.1002/chem.201303833.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 287

You, L.X., Wang, S.J., Xiong, G., Ding, F., Meert, K.W., Poelman, D., Smet, P.F., Ren, B.Y., Tian, Y.W., Sun, Y.G., 2014. Synthesis, structure and properties of 2D lanthanide coordination polymers based on N-heterocyclic arylpolycarboxylate ligands. Dalton Trans. 43, 17385–17394. https://doi.org/10.1039/c4dt02517b. Zadrozny, J.M., Atanasov, M., Bryan, A.M., Lin, C.-Y., Rekken, B.D., Power, P.P., Neese, F., Long, J.R., 2013. Slow magnetization dynamics in a series of two-coordinate iron(II) complexes. Chem. Sci. 4, 125–138. https://doi.org/10.1039/c2sc20801f. Zener, C., 1932. Non-adiabatic crossing of energy levels. Proc. R. Soc. A Math. Phys. Eng. Sci. 137, 696–702. https://doi.org/10.1098/rspa.1932.0165. Zeng, D., Ren, M., Bao, S.S., Zheng, L.M., 2014. Tuning the coordination geometries and magnetic dynamics of [Ln(hfac)4]- through alkali metal counterions. Inorg. Chem. 53, 795–801. https://doi.org/10.1021/ic402111v. Zeng, D., Ren, M., Bao, S.-S., Feng, J.-S., Li, L., Zheng, L.-M., 2015. pH-controlled polymorphism in a layered dysprosium phosphonate and its impact on the magnetization relaxation. Chem. Commun. 51, 2649–2652. https://doi.org/10.1039/C4CC09341K. Zeng, D., Ren, M., Bao, S.S., Cai, Z.S., Xu, C., Zheng, L.M., 2016. Polymorphic lanthanide phosphonates showing distinct magnetic behavior. Inorg. Chem. 55, 5297–5304. https://doi.org/ 10.1021/acs.inorgchem.6b00280. Zhang, Y., 2016. Surface enhanced single-molecule magnetism involving 4f spin. Appl. Phys. Lett. 108, 132407. https://doi.org/10.1063/1.4945088. Zhang, P., Guo, Y.N., Tang, J., 2013. Recent advances in dysprosium-based single molecule magnets: structural overview and synthetic strategies. Coord. Chem. Rev. 257, 1728–1763. https:// doi.org/10.1016/j.ccr.2013.01.012. Zhang, F., Yan, P., Zou, X., Zhang, J., Hou, G., Li, G., 2014a. Novel 3D alkali–lanthanide heterometal–organic frameworks with pyrazine-2,3,5,6-tetracarboxylic acid: synthesis, structure, and magnetism. Cryst. Growth Des. 14 (4), 2014–2021. https://doi.org/10.1021/ cg5001254. Zhang, P., Zhang, L., Wang, C., Xue, S., Lin, S.Y., Tang, J., 2014b. Equatorially coordinated lanthanide single ion magnets. J. Am. Chem. Soc. 136, 4484–4487. https://doi.org/10.1021/ ja500793x. Zhang, L., Lu, S., Zhang, C., Du, C., Hou, H., 2015a. Highly pH-dependent synthesis of two novel three-dimensional dysprosium complexes with interesting magnetic and luminescence properties. Cryst. Eng. Comm. 17, 846–855. https://doi.org/10.1039/C4CE02023E. Zhang, X., Vieru, V., Feng, X., Liu, J.L., Zhang, Z., Na, B., Shi, W., Wang, B.W., Powell, A.K., Chibotaru, L.F., Gao, S., Cheng, P., Long, J.R., 2015b. Influence of guest exchange on the magnetization dynamics of dilanthanide single-molecule-magnet nodes within a metalorganic framework. Angew. Chem. Int. Ed. 54, 9861–9865. https://doi.org/10.1002/ anie.201503636. Zhang, Y., Guo, Z., Xie, S., Li, H.-L., Zhu, W.-H., Liu, L., Dong, X.-Q., He, W.-X., Ren, J.-C., Liu, L.-Z., Powell, A.K., 2015c. Tuning the origin of magnetic relaxation by substituting the 3d or rare-earth ions into three isostructural cyano-bridged 3d–4f heterodinuclear compounds. Inorg. Chem. 54, 10316–10322. https://doi.org/10.1021/acs.inorgchem.5b01763. Zhang, Y., Liao, P., Kan, J., Yin, C., Li, N., Liu, J., Chen, Q., Wang, Y., Chen, W., Xu, G.Q., Jiang, J., Berndt, R., Wu, K., 2015d. Low-temperature scanning tunneling microscopy study on the electronic properties of a double-decker DyPc2 molecule at the surface. Phys. Chem. Chem. Phys. 17, 27019–27026. https://doi.org/10.1039/C5CP03925H. Zhang, J.-W., Kan, X.-M., Li, X.-L., Liu, Y., Liu, B.-Q., 2016a. A series of copper-lanthanide heterometallic coordination polymers derived from pyridine-2,3-dicarboxylic acid and in situ

288 Handbook of Magnetic Materials generated succinic acid. Eur. J. Inorg. Chem. 2016a, 1060–1067. https://doi.org/10.1002/ ejic.201501328. Zhang, J., Zhang, H., Chen, Y., Zhang, X., Li, Y., Liu, W., Dong, Y., 2016b. A series of dinuclear lanthanide complexes with slow magnetic relaxation for Dy2 and Ho2. Dalton Trans. 45, 16463–16470. https://doi.org/10.1039/C6DT02962K. Zhang, P., Jung, J., Zhang, L., Tang, J., Le Guennic, B., 2016c. Elucidating the magnetic anisotropy and relaxation dynamics of low-coordinate lanthanide compounds. Inorg. Chem. 55, 1905–1911. https://doi.org/10.1021/acs.inorgchem.5b02792. Zhang, S., Ke, H., Shi, Q., Zhang, J., Yang, Q., Wei, Q., Xie, G., Wang, W., Yang, D., Chen, S., 2016d. Dysprosium(III) complexes with a square-antiprism configuration featuring mononuclear single-molecule magnetic behaviors based on different b-diketonate ligands and auxiliary ligands. Dalton Trans. 45, 5310–5320. https://doi.org/10.1039/C6DT00219F. Zhang, W.-Y., Tian, Y.-M., Li, H.-F., Chen, P., Sun, W.-B., Zhang, Y.-Q., Yan, P.-F., 2016e. A series of dinuclear Dy(III) complexes bridged by 2-methyl-8-hydroxylquinoline: replacement on the periphery coordinated b-diketonate terminal leads to different singlemolecule magnetic properties. Dalton Trans. 45, 3863–3873. https://doi.org/10.1039/ C5DT04449A. Zhang, X.-M., Li, P., Gao, W., Liu, F., Liu, J.-P., 2016f. Construction of three lanthanide metalorganic frameworks: synthesis, structure, magnetic properties and highly selective sensing of metal ions. J. Solid State Chem. 244, 6–11. https://doi.org/10.1016/j.jssc.2016.09.009. Zhang, H., Liu, R., Zhang, J., Li, Y., Liu, W., Dong, Y., 2017. Five rare m 4-O-centered Ln6 clusters with single-molecule magnet behavior for Dy6. Chem. Asian J. 12, 507–514. https://doi. org/10.1002/asia.201601412. Zhao, F.H., Li, H., Che, Y.X., Zheng, J.M., Vieru, V., Chibotaru, L.F., Grandjean, F., Long, G.J., 2014. Synthesis, structure and magnetic properties of Dy2Co2L10(bipy)2 and Ln2Ni2L10(bipy)2, Ln ¼ La, Gad, Tb, Dy and Ho: slow magnetic relaxation in Dy2Co2L10(bipy)2 and Dy2Ni2L10(bipy)2. Inorg. Chem. 10, 9785–9799. https://doi.org/10.1021/ic501374q. Zhao, J., Zhu, G.-H., Xie, L.-Q., Wu, Y.-S., Wu, H.-L., Zhou, A.-J., Wu, Z.-Y., Wang, J., Chen, Y.-C., Tong, M.-L., 2015a. Magnetic and luminescent properties of lanthanide coordination polymers with asymmetric biphenyl-3,20 ,50 -tricarboxylate. Dalton Trans. 44, 14424–14435. https://doi.org/10.1039/c5dt01894c. Zhao, X.-Q., Liu, X.-H., Kang, X.-M., Zhao, B., 2015b. Self-assembly of heterometallic Ln III– Co II coordination polymers: syntheses, structures, and magnetic studies. Dalton Trans. 44, 18856–18863. https://doi.org/10.1039/C5DT02280K. Zheng, X.-Y., Peng, J.-B., Kong, X.-J., Long, L.-S., Zheng, L.-S., 2016. Mixed-anion templated cage-like lanthanide clusters: Gd27 and Dy27. Inorg. Chem. Front. 3, 320–325. https://doi. org/10.1039/C5QI00249D. Zhou, H., Chen, Q., Zhou, H., Yang, X., Song, Y., Yuan, A., 2016a. Structural conversion and magnetic studies of low-dimensional LnIII/MoV/IV(CN)8 (Ln ¼ Gd-Lu) systems: from helical chain to trinuclear cluster. Cryst. Growth Des. 16, 1708–1716. https://doi.org/10.1021/acs. cgd.5b01782. Zhou, Y.Y., Geng, B., Zhang, Z.W., Guan, Q., Lu, J.L., Bo, Q.B., 2016b. New family of octagonal-prismatic lanthanide coordination cages assembled from unique Ln17 clusters and simple cliplike dicarboxylate ligands. Inorg. Chem. 55, 2037–2047. https://doi.org/10.1021/ acs.inorgchem.5b02367. Zhu, J., Wang, C., Luan, F., Liu, T., Yan, P., Li, G., 2014a. Local coordination geometry perturbed b-diketone dysprosium single-ion magnets. Inorg. Chem. 53, 8895–8901. https://doi. org/10.1021/ic500501r.

Magnetic Relaxation of Lanthanide-Based Molecular Magnets Chapter

1 289

Zhu, M., Hu, P., Li, Y., Wang, X., Li, L., Liao, D., Durga Prasad Goli, V.M.L., Ramasesha, S., Sutter, J.P., 2014b. Hetero-tri-spin [2p-3d-4f] chain compounds based on nitronyl nitroxide lanthanide metallo-ligands: synthesis, structure, and magnetic properties. Chem. A Eur. J. 20, 13356–13365. https://doi.org/10.1002/chem.201403581. Zhu, W.-H., Zhang, Y., Guo, Z., Wang, S., Wang, J., Huang, Y.-L., Liu, L., Fan, Y.-Q., Cao, F., Xiang, S.-W., 2014c. A simple route to a 1D ferromagnetic Dy-containing compound showing magnetic relaxation behaviour. RSC Adv. 4, 49934–49941. https://doi.org/10.1039/ C4RA08729A. Zhu, M., Wang, J., Yang, M., Ma, Y., Li, L., 2015. Slow magnetic relaxation in two-dimensional 3d-4f complexes based on phenyl pyrimidyl substituted nitronyl nitroxide radicals. Dalton Trans. 44, 9815–9822. https://doi.org/10.1039/c4dt03330b. Zou, H.-H., Wang, R., Chen, Z.-L., Liu, D.-C., Liang, F.-P., 2014. Series of edge-sharing bi-triangle Ln4 clusters with a m4-NO3- bridge: syntheses, structures, luminescence, and the SMM behavior of the Dy4 analogue. Dalton Trans. 43, 2581–2587. https://doi.org/10.1039/ c3dt52316k. Zou, H.-H., Sheng, L.-B., Chen, Z.-L., Liang, F.-P., 2015. Lanthanide nonanuclear clusters with sandglass-like topology and the SMM behavior of dysprosium analogue. Polyhedron 88, 110–115. https://doi.org/10.1016/j.poly.2014.12.024.