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Quenching points of dimeric single-molecule magnets: Exchange interaction effects
Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
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Quenching points of dimeric single-molecule magnets: Exchange interaction effects ˜ ez b,,1, P. Vargas a,,1 J.M. Florez a,,1, A´lvaro S. Nu´n a b
Departamento de Fı´sica, Universidad Te´cnica Federico Santa Marı´a, P.O. Box 110-V, Valparaı´so, Chile ´ticas, Universidad de Chile, Casilla 487-3, Santiago, Chile Departamento de Fı´sica, Facultad de Ciencias Fı´sicas y Matema
a r t i c l e in fo
abstract
Article history: Received 8 June 2010 Available online 14 July 2010
We study the quenched energy-splitting ðDE Þ of a single-molecule magnet (SMM) conformed by two exchange coupled giant-spins. An assessment of two nontrivial characteristics of this quenching is presented: (i) The quenching-points of a strongly exchange-coupled dimer differ from the ones of their respective giant-spin modeled SMM and such a difference can be well described by using the Solari– Kochetov extra phase; (ii) the dependence on the exchange coupling of the magnetic field values at the quenching-points when DE passes from monomeric to dimeric behavior. The physics behind these exchange-modified points, their relation with the DE oscillations experimentally obtained by the Landau–Zener method and with the diabolical-plane of a SMM, is discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Single-molecule magnets Spin tunneling Spintronic
1. Introduction Single-molecule magnets (SMMs) attract great interest from the scientific community due to both their fundamental significance in the context of quantum dynamics and their potential applications in the context of information processing and storage [1–7]. Recent experimental developments [2–5] provide a great step forward in the continuing struggle for the use and manipulation of molecular quantum-properties. Among the basic phenomenology associated with SMM-dynamics there is a variety of phenomena that can be accounted for in terms of a simple qualitative picture provided by the giant-spin model. Within this model the dynamics of the system is regarded as confined within the sub-space generated by projecting out states with overall spin different from the ground state spin. The quantum tunneling of the magnetization (QTM) as a description of the reversion process in SMM is a major success in the class of phenomenology that can be described rather accurately from the viewpoint of the giant-spin model [6,7]. The transition from QTM in magnetic nanoparticles to SMMs was promoted when it was known that the tunneling process in magnetic nanoparticles, which can be approximately described with a magnetic viscosity in an effective Arrhenious-like rate transition [6], presents a plateau with a width comparable to the characteristic magnetization-steps of a SMM at low temperatures [8,9]. Since SMMs are mostly molecules containing transition–
Principal corresponding author. Tel.: + 56 32 87435885; fax: +56 32 2797656. Corresponding authors.
E-mail addresses: juanmanuel.fl[email protected], jmfl[email protected] ˜ ez), [email protected] (J.M. Florez), alnunez@dfi.uchile.cl (A´.S. Nu´n (P. Vargas). 1 http://www.magnetismo.cl 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.07.002
metal ions bridged by organic ligands and with spins strongly exchange coupled, at low temperature each molecule behaves effectively like a single-domain particle with fixed total spin [6,7]. As a consequence, several studies of the magnetization tunneling in SMMs have been based on giant-spin Hamiltonians that account for different magnetic interactions [6,7,10–12]. Nevertheless, recent reports [13–18] have shown that the influence of giant-spin fluctuations besides the assorted non-reduced behavior given by dimer-conformed SMMs, i.e., molecules based on two effective-spins interacting which behave separately like a SMM, should be taken into account in order to explain further the features so far briefly considered, e.g., the effects of S-mixing within and between different multiplets. In the QTM problem of SMM the cornerstone describing the quantum reversion process is the energy-splitting DE . This DE is the energy-difference between two low-lying energy levels, which can involve excited states depending upon the external longitudinal magnetic field, and its presence leads us to the finite ‘‘direction’’-probability on both sides of the double-spin-well conformed with an anisotropic barrier; this probability determines the tunneling probability of the magnetization. The tunneling probability is nontrivial related to DE and it has been the target of QTM formulations based on semiclassical methods [6], where the macroscopic features of spin tunneling are evidenced. Most first theoretical predictions on QTM in nanoparticles and posteriorly in SMM were developed using spin path integrals (PI) [6], while the advantages of Landau–Zener (L–Z) theory in SMM were principally appreciated when implementing in the experimental method necessary to measure DE [7,19,20]. There are also strategies to calculate features of giant-spins in the tunneling regime [21–23], which in principle can include a variety of effects like the dipole–dipole interaction [24,25]. However, the
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oscillations observed in DE as a function of the external transverse field, which are related with the topological features of spin tunneling [19,20,26–33], have been investigated so far by using single giant-spin considerations. Therefore, several questions remain about the general behavior of a dimer-conformed SMM when the exchange interaction (J) and the transversal Zeeman field are switched on giving rise to S-mixing into selected S-multiplets and other fundamental characteristics. It was predicted that DE as a function of a transversal magnetic field would present oscillations due to the interference of quantum paths, described onto the coherent-state sphere by spin tunneling, which are symmetric with respect to the hard axes [27]. This interference process was introduced, in the context of magnetic tunneling, when defining the half-spin rules for coherent quantum tunneling in magnetic nanoparticles [6,26]. Investigations of the coherence in QTM processes have included studies of nuclear-spins effects [28] and dissipation due to environmental-coupling [29,30]. QTM phenomena in antiferromagnetic particles and SMMs [31,32] as well as generalizations of the L–Z theory accounting for thermal relaxation [33], have also been explored. In principle, it is possible to have a hardware based on an array of SMMs and where DE plays an important role in controlling the energy-difference and times decoherence between the ðj0S,j1SÞmolecular states [1]. In this way, the features of DE that might be somehow tunable by experiments, are interesting to manipulate the molecular states useful to SMMs-based electronics. This was the case of the paradox about the quenching-points of DE in Fe8 molecule. The so-called quenching refers to the vanishing of DE when the transversal spinoperators, modulated by Zeeman fields or anisotropic terms, are modified. Experimental measurements of Fe8 [19,20] showed just half of the quenching-points predicted by instantons-based PI method [27]. This problem was understood when according to inclusion of weak fourth-order anisotropy in the simulation of inelastic neutron scattering [20], a new instantons technique was developed [34,35] to demonstrate that after some transversal field these new instantons do not allow anymore quenching [36]. Additional considerations for DE , e.g., the inclusion of inhomogeneities in the magnetic field [37] and of small quantum fluctuations of the giant-spin instantons [38], were also proposed. These quenching-points belong to a subspace of the so-called diabolical-plane of a SMM, i.e., to the collection of points determined by the couple of values of two Hamiltonian parameters (Zeeman fields, anisotropies) at the crossings lying between different energylevels [39–41]. The Hamiltonian equation (1) represents the usual giant-spin model of a SMM. The quenching-points for this model can be calculated using the instantons-based PI [27,42]. Such a method has been explored for different anisotropy orders and symmetries of a giant-spin Hamiltonian [11,12,34]. Nevertheless, the instantons method has not been used to evaluate the Berry phase of coupled spin-paths. The difficulty of using instantons for a dimeric problem lies in the dynamical nature of the Euclidean action, which leads us to dynamical calculations of the multiple instantons for each monomer and to nontrivial quantum fluctuations around saddle points of the energy density, and in many-spins problem these calculations are still an open question. Known approximated strategies for this problem reduce the Hilbert Space (HS) through construction of effective instantons that represent the collective dynamic of the system, and this is not adequate to generally describe S-mixing [15,41]. As far as the authors know, there are no theoretical descriptions of the quenching-points dependence on the exchange interaction of a dimer-conformed SMM. In order to do that, different strategies can be used. Studies of the degeneracy of lying energy spin-levels [10], the arguments of Bruno based on the general diabolical points [41] as well as the calculations of the diabolical-plane of a quartic anisotropy model [11], were developed using simple methods and mixed strategies. In this paper we are
going to use different scopes of calculation to do an assessment of nontrivial properties of such quenching. This paper is distributed as follows: in Section 2 we highlight the quantum model for the dimerconformed SMM. In Section 3 we show how the quenching-points of a strongly exchange coupled dimer differs from the ones of the respective giant-spin modeled SMM and such difference can be approximately calculated using the Solari–Kochetov (SK) extra phase [43,44]. In Section 4 we simulate the quenching-points of a dimeric SMM and calculate an expression for the quenching-points’ dependence on the exchange parameter J when the molecule moves from monomeric to dimeric behavior. In Section 5 we gave a discussion about the results and ideas of the previous sections.
2. Spin Hamiltonians In this section we introduce the model and highlight the main features it conveys. Let us consider the Hamiltonian: ^ local ¼ K1 S^ 2 þK2 S^ 2 g m HS^ z , H B z y
ð1Þ
where K1 bK2 are anisotropies and H stands for the Zeeman field. For H ¼ 0, this Hamiltonian has two degenerated classical minima, ^ As H grows, the locations of the minima lies in namely ~ S ¼ 7 x. the x2z plane. To each of the classical ground states we can ^ associate a spin coherent state j 7 xS. The quantum fluctuations induce these ground states to be hybridized into two states of ^ ^ Their respective energies are opposite parity jc 7 S ¼ jxS7 jxS. modified, and the degeneracy is lifted by an energy splitting DE . The main qualitative feature that it manifests is the oscillations of its magnitude as H is modified, as we mentioned above. Now, we consider two local moments, each one with spin S and obeying the same dynamics, dictated by the Hamiltonian in Eq. (1). To this Hamiltonian we add a coupling term of the form ^ ¼H ^ local ðS^ 1 Þ þ H ^ local ðS^ 2 ÞJ S^ 1 S^ 2 H
ð2Þ
Our goal is to understand the behavior of the DE oscillations as a function of H and J, namely, to find the quenching-points when the external magnetic field varies and also their dependence on the exchange coupling. J ranges from zero (two independent monomers of spin value S) to a big enough value at which a single giant-spin behavior with spin Sð2Þ ¼ 2S, is reached.
3. Path integral approach: strong coupling Let us analyze what happens if a giant-spin modeled dimericSMM is ‘‘split up’’ into its spin-monomers such that Eq. (2) (with strong coupling J) represents the SMM instead of Eq. (1). Our basic approach starts from the SU(2) PI representation of the partition function [6]. The coherent-state representation of the Euclidean action can be obtained by two different ways, namely, using polar or stereographic coordinates. In the case of explicit time depending calculation, divergences related to instantons dynamic are avoided using stereographic coordinates as well as the polar ones in a rotated frame. In this section we use stereographic coordinates. The partition function reads as Z K¼ D2 z1 D2 z2 expfSðz1 ,z 1 ; z2 ,z 2 Þg ð3Þ The full action in the last equation is given by: S ¼ S 1 ðz1 ,z 1 Þ þ S 1 ðz2 ,z 2 Þ þ S 2 ðz1 ,z 1 ; z2 ,z 2 Þ, where S 1 that represents the single spin action and S 2 ðz1 ,z 1 ; z2 ,z 2 Þ are given by ) Z b( _ z zz_ z Hðz,zÞ dt S S 1 ðz,zÞ ¼ 1þ zz 0
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
S 2 ðz1;2 ,z 1;2 Þ ¼
ð1z 1 z1 Þð1z 2 z2 Þ þ 2ðz1 z 2 þ z 1 z2 Þ ðJS2 Þð1Þ ð1þ z 1 z1 Þð1 þz 2 z2 Þ
ð4Þ
In Eq. (5), z is a complex number associated with a trajectory in the unit sphere according to the standard stereographic representation (we have chosen the pole of the representation at the ^ north pole of the sphere), Hðz,zÞ stands for /zjHjzS=/zjzS, and jzS is a coherent spin state along the direction defined by z. A careful evaluation of the matrix element of Eq. (1) gives us: Hðz,zÞ ¼ k~ 1 S2
2 2 ~ ð1zzÞ2 lðzzÞ2 2hð1z z Þ
ð1 þ zzÞ2
ð5Þ
where we have defined k~ 1 ¼ ðS12ÞK1 =S, h~ ¼ g mB H=2K1 ðS12Þ, l ¼ K2 =K1 , by following Ref. [42]. To separate the physics of the giant-spin model from the S-mixing contributions we introduce the change of variables z1 ¼ z þ dz=2, z2 ¼ zdz=2 and their respective conjugates. In the zeroth order term the only effect is an overall factor of 2 multiplying the single spin action. If we absorb the factor in the S value of the first term of Eq. (5), which is the Berry-phase, we obtain the effective action of a spin with double size, described by a Hamiltonian like the one in Eq. (1), but with effective parameters: Kið2Þ ¼ Ki =2 and Sð2Þ ¼ 2S. In the classical limit, ðS b1Þ we see how, as expected, the spin Hamiltonian goes into one of double spin magnitude with half anisotropic constants. In the large-J limit we can regard the amplitude of the direction deviations associated with dz to be small; in terms of stereographic variables it is convenient to use dz ¼ ð1 þ zzÞZ, with JZJ 51. We then, represent the partition function in terms of such variables, and restrict the contributions up to second order in Z. We obtain Z Z S dt Cy F ðz,zÞC K D2 z D2 Zexp S 0 ðz,zÞ ð6Þ 2 In the last expression we have used the spinor notation Cy ¼ ðZ , ZÞ, C ¼ ðZ, Z Þt and S 0 represents the effective giant-spin action. F stands for a rank two spinor of the form ! Bðz,zÞ @t þJSð2Þ þ Aðz,zÞ F ðz,zÞ ¼ ð7Þ @t þ JSð2Þ þ Aðz,zÞ Bðz,zÞ (2)
In this last matrix S is the effective giant-spin and the contributions denoted by A, B and B, need to be calculated starting from the specific action in Eq. (5). In this way, they contain the contributions from the Berry-phase term and from the Hamiltonian. We remark that, besides the explicit term next to the derivative in Eq. (7), there are no further contributions from the exchange term into F . This fact is essential in the manipulations associated with the J-1 limit. In general we have Aðz,zÞ ¼
z_ zz_ z 1 ð1 þ zzÞ2 @2zz Hðz,zÞ þ 1 þzz 2S
Bðz,zÞ ¼ 2
z_ z 1 ð1 þ zzÞ2 @2zz Hðz,zÞ þ 1 þ zz 2S
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The action in Eq. (10) contains, formally, the physics associated with the interplay of the giant-spin sub-space and the S-mixing fluctuations. For a strongly exchange coupled dimer the non-local contribution in Eq. (10) will be dominated by the physics described with the partial kernel F 0 ¼ s3 @t þ JSð2Þ (s3 the Pauli matrix). We can evaluate that contribution in the required limit and this has the final form Z 1 dt dtu trðGðt, tu ÞdF ðtu , tÞÞ dS 1 ¼ ð11Þ 2 u where Gðt, tu Þ ¼ F 1 0 ðt, t Þ. Since the contribution dF is local in time, and after evaluating the propagator G, we obtain the simple result: dS 1 ¼ S SK . This last contribution does not have any dependence on J. In this way even at zeroth order in 1/J we would have had a renormalization of the bare giant-spin action associated with S-mixing. However, this contribution is exactly canceled out by the SK extra phase. The consequences of the vanishing of this extra phase on the dimeric quenching-points can be analyzed after calculating the phase of the DE oscillations for each case, i.e., the phase with (PhaseðDE Þ þ SK ) and without the extra phase contribution (PhaseðDE ÞSK ). Remember that the quenching-points are determined by the values of some parameters of the Hamiltonian at the vanishing points of DE . In this case such parameters are the magnetic fields, namely, the longitudinal magnetic field, which is fixed at the zeroth resonance Eq. (1) ðHx ¼ 0Þ, and the transversal field, which is a modifiable parameter. The quenching-points are then given by the transversal field values at the vanishing of PhaseðDE Þ 7 SK . We find such quenching, respectively: pffiffiffiffiffiffiffiffiffiffi 1l ð2Þ 1 mu hmu ¼ S þ þ SK 2 Sð2Þ
hmu SK ¼
pffiffiffiffiffiffiffiffiffiffi 1l Sð2Þ
2
2
Sð2Þ
1 1 mu Sð2Þ 4 2
with mu ¼ 1,2,3, . . . ,Sð2Þ
ð12Þ
and where h ¼ g mB H=2K1 S . In Fig. 1 we have plotted the quenching-points equation (12). In order to check our approach, we have included in Fig. 1 the quenching-points obtained by an exact simulation of DE . Two set of points are simulated, namely, ð2Þ
hmSK
0.1
ð8Þ 0.2
ð9Þ
Evaluating the Gaussian integral associated with the fluctuation-Z functions we obtain an effective action for the giant-spin degrees of freedom. Formally, we have Z 1 1 dt AðzðtÞ,zðtÞÞ ð10Þ S eff ¼ S 0 TrðlogF Þ þ 2 2 The third term in the action is known as the Solari–Kochetov S SK extra phase [42–45]. We need to remark that in general this contribution is not a phase. It bears its name from its behavior in some simple cases (whenever the Hamiltonian is a combination of the SU(2)-algebra generators). Its mathematical origin is clear, arising as the correction of an anomaly created by the ambiguity of the continuous representation of the SU(2)-coherent states path integral. However its physical interpretation is more involved.
0.2
0.4
hmSK
0.1 2
0.2
9
0.3
0.4 Fig. 1. Polar plot of quenching-points equation (12) (white-small-points). hmu SK and hmu þ SK are joined by dashed and solid guiding-lines, respectively. The exact quenching-points of a dimer strongly coupled by exchange interaction and of a giant-spin modeled dimeric SMM are also plotted (red-big-points). The para(2) meters used here are J=K1 ¼ 78 ¼10, correspond10, K1 ¼ 0.321, K2 ¼ 0.229, and S ingly. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
ΔE (K)
1000
10
0.1
0.001 0.0
0.1
0.2
0.3
0.4
0.5
h
0.4
Log (ΔE(K)
0.2
–5 –10 –15 –20 –25
0.0
0.0 0.2 h
0.4
Fig. 2. Upper panel: energy-splitting of a giant-spin modeled dimeric SMM equation (1), dimer strongly exchange-coupled, and weakly exchange-coupled equation (2). Phase ðDE Þ 7 SK for S(2) ¼10, and phase ðDE Þ þ SK with S(2) ¼ 5 are also plotted. Lower panel: energy-splitting of a dimeric SMM equation (2) as a function of d ¼ J=K1 . In the upper panel the exchange values are shown in the picture and for both plots K1 ¼ 0.321, K2 ¼0.229 ðl ¼ K2 =K1 ¼ 0:7Þ.
the ones of a strongly exchange coupled dimer and the ones of a giant-spin modeled dimeric SMM (that we develop in next section). Moreover, in Fig. 2 (upper panel) we show DE for these two mentioned systems as well as PhaseðDE Þ 7 SK . Such comparisons are performed for molecular parameters similar to the corresponding to Fe8 molecule [42]. As Figs. 1 and 2 show, differences between the quenching behavior of this two systems are actually well described, for a reasonable strong value of J, by the approach presented here. Differences between these quenching-points increase with increasing of h and decrease with higher values of l. Also, moderate spin values (Sð2Þ 10) as reported in the recent experimental studies [16–18,46,47] would allow to have, in principle, appreciable differences.
4. Diabolical quenching: up to moderate coupling In this section our target is to develop a strategy that allows us to study the exact mechanism involved in the generation and the behavior of such quenching-points when the exchange coupling is modified. A many-body instantons-based PI method constitutes a complex tool which still conveys many questions because we have to control several contributing-instantons with adequate intervals of integration [11] and their possible jumps at the end points [11,34]. These features are strongly affected by interactions that produce S-mixing. In the case of overlapped paths connecting different sets of coherent-states the problem cannot be handled by the usual instantons-based PI [7,41]. A general possible method based on PI is then out of the scope of this work.
Nevertheless, the quantum mechanics nature of typical SMMs represented by Eqs. (1) and (2) allow us to explore other scopes. Therefore, in order to understand what happens with the influence of the exchange interaction on the quenched energysplitting of a dimeric SMM, let us just switch on the exchange J and sweep it up to a strong enough value where the giant-spin behavior is reached, and in this process calculate the energysplitting as well as its quenching-points (QP from now on). Starting with the dimer already discussed in the previous section, we perform exact simulations of DE which we picture in Fig. 2 (Lower panel) where the parameters employed in generation of Fig. 1 have been used as well as the reduced exchange d ¼ J=K1 . Simulations in Fig. 2 show clearly how switching on J generates a splitting of the initial five QP, which correspond to the monomers behavior, into 10 QP. These 10 QP are shifted as a function of J until some finals magnetic field values that are actually the ones studied in Section 3, and which differ from the corresponding to the giant-spin model, approximately, by the values shown in Fig. 1. In the case of d 0 the oscillations of a dimeric SMM are determined by the ones of an isolated monomer, as expected from Eqs. (2) and (5), and in Fig. 2 (Upper panel) we extract such splitting from the full-simulation in there. To visualize better how the exchange coupling modifies the position of the QP, we have extracted from Fig. 2 each magnetic field value at the quenching as a function of d, and the results are shown in Fig. 3. In this last simulation two things are quite interesting: Magnetic field values at the QP are quickly affected for small exchange coupling and moderated values of d allow us to reach a giant-spin-like behavior; the last quenching, with respect to increasing h value, does not change when the reduced exchange d is modified. These two features would be important as long as the physics they convey does not depend upon the l value. Additional simulations for different l values are developed in Fig. 4 and they corroborate these two mentioned features. In order to understand the behavior of the diabolical subspace determined by the QP studied here (DQP), which are given by the points at the axe of the zeroth resonant field, in the diabolicalplane, with couples fHx ¼ 0,Hz-QP g [11,39–41], let us to perform a theoretical assessment of these QP by using and simple strategy that can be extended to another problems with similar characteristics and larger HS. Such a strategy that we use is based on a Van Vleck transformation (VT) [48–50]. Although the Van Vleck method has been used in several problems, it is seldom described in the literature and, surprisingly, its advantages compared with another methods and which are based on their power to handle
Quenching at h for ΔE ()
3626
0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 3. Diabolical quenching-points as a function of reduced coupling d as extracted from Fig. 2. Red-dashed lines show the quenching-points hmu þ SK with S(2) ¼ 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
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in e from the matrix elements of Hu connecting any particular unperturbed energy level with the other distinct unperturbed n levels, we choose Qmn ¼ i/mjHujnS=ðEm 0 E0 Þ and Qmmu ¼ Qnnu ¼ 0. These matrix elements we just took allow two things: V does not affect the m or n blocks and they make Gumn ¼ 0. The remaining coupling terms in the considered block are of second order or higher, and since they are off-diagonal they cannot contribute to the energies until the fourth order. If the approach is taken to second or third order, this decoupling of the blocks in the perturbation allows to separate the treatment of each block until these orders, which is usually adequate in practice. The previous procedure leads us the formulas for the mmu elements of each block (Appendix A), which conform to a problem of lower dimension. In the case of one dimensional blocks VT is also applicable (A.3) as well as fourth order transformations. Recently, dissipative dynamics of a qubit coupled to a nonlinear environment was studied, and advantages of VT were used [50]. Now, the Hamiltonian equation (2) can be disassembled as follows: H ¼ H0 þHe þ Ha
ð13Þ
where He corresponds to the xy-component of the exchange interaction and we have separated the anisotropy such that: 0 1 2 X z2 z lX j j A ^ ^ @ dSz1 Sz2 SS H0 ¼ S i 2ShS i þ 4j¼ þ i i i¼1 He ¼
d 2
Ha ¼
Fig. 4. DE of a dimeric SMM equation (2) as a function of d. Upper panel: l ¼ 0:12. Lower panel: l ¼ 0:0.
þ ðS1þ S 2 þ S1 S2 Þ
2; l X
4 i;j ¼ 1; þ
Sji
2
ð14Þ
ð15Þ
where we normalize H by K1 and j ranges within ( + , ). The eigenstates of Sz1,2 of the two monomers diagonalize H0 and the P P eigenvalues are E0 ðm1 ,m2 Þ ¼ 2i ¼ 1 ðm2i 2Shmi þðl=4Þ j¼ þ j j Cmi Cmi þ ðjÞ1 Þdm1 m2 , with mi ¼ 1,2 the corresponding quantumffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j number along z-direction and where 2Cm ¼ 34mi ðmi þðjÞ1Þ. i On the other hand, the trickier physics behind the QP behavior illustrated in Figs. 2–4 does not strictly depend upon the l value. Moreover, in Hamiltonian equation (1) the DQP reflect ^ local jmð2Þ 7 1S ¼ 0, where jmð2Þ S are the selection rule /mð2Þ jH ð2Þ ^ the eigenstates of S z , and these crossings of energy-levels are not avoided modifying the l value, as Fig. 4 confirms. Therefore, we develop our approach by applying the VT to Eq. (14), for which the matrix representation of the perturbation, in the base jmS ¼ jm1 ,m2 S, is found to be 2/mujHe jmS ¼ dðy ðmÞdmu,m þ
highly degenerated block systems, have been untapped in manyspins-conformed SMM, which naturally represent just these kinds of problems. We give a short description of the matrix transformation that we use. Let us consider the Hermitian matrix Q which defines the unitary transformation VðeÞ such that VðeÞ ¼ eieQ ¼ 1 þieQ e2 Q 2 =2 þ , where e stands for a parameter that expands the perturbative series of H ¼ H0 þHu, H0 being an unperturbed Hamiltonian whose solutions are known and Hu represents the perturbation matrix as usual. We want to construct VðeÞ such that the first order term in the Hamiltonian has vanishing matrix elements between the blocks of interest and the other ones, i.e, Gumn ¼ 0, where G is defined by the unitary transformation: GðeÞ ¼ Vy HV ¼ G0 þ eGu þ e2 G00 þ , and where the fm . . . mug and fn . . . nug elements label the states within the Hu blocks of interest and the outside blocks, correspondingly. Expanding both the sides of GðeÞ and equating the coefficients of like powers of e, we get the expressions for each one of the contributions G0 ,Gu,G00 , . . ., and to eliminate the terms of first order
þ y þ ðmÞdmu,m Þ, with the coefficients given by 4y 8 ðmÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 93ðW 8 ðmÞ þX 7 ðmÞÞ þW 8 ðmÞX 7 ðmÞ. The respective functions in the last equation and in the elements /mujHe jmS are 8
given by dmu,m ¼ dmu1 m1 8 1 dmu2 m2 7 1 , W 8 ðmÞ ¼ 4m1 ðm1 8 1Þ and X 7 ðmÞ ¼ 4m2 ðm2 7 1Þ. In order to do the calculations we need to discriminate the energies involved on the DQP. Energies E0(m), with m ¼ ðm1 ,m2 Þ, which have ðmi ,mi 81Þ as well as the energies with (mi,mi), generate such DQP. Moreover, in the first case the energies are degenerated due to the invariance ðmi ,mi 8 1Þ2ðmi 8 1,mi Þ which leads us to two of these levels for each initial monomeric DQP. This last energy-characterization can be visualized in Fig. 6, where we plot E0(m) and the quantum numbers of the lowest lying energy levels conforming DE when h is modified. These numbers increase when h does and the five DQP that characterize the monomer behavior, before including the xy-exchange terms, are actually the crossings between four levels as Fig. 6 shows. These points become doubly degenerated when all the exchange interactions are considered and the interceptions of the levels
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shifted by the xy-exchange terms with the m ¼ ðmi ,mi Þ ðmi ¼ 1,2 ¼ 0, 7 ð1,2,3,4,5ÞÞ levels are then the ddepending DQP. We now perform the VT. By using the equation for nondegenerated blocks (A.3) and solving the degenerated-subspaces with (A.1), the magnetic field values determining the DQP are found to be ð2n1Þ l 2 1 5ðld2Þ 2 l ð29nðn1ÞÞ 1 2 þ sd 5ðld2Þ
hns ðl, dÞ ¼
sd2 5ðld2Þ
M3=2s,n
ð16Þ
where for the sake of simplicity we have extended the VT up to second order; s ¼ 7 1=2 and the matrix M ¼ ff330,315,259,174, 81g,f301,234,144,55,0gg, with n ¼ 1, . . . ,5, correspondingly. The approach given by Eq. (16) has been compared with the exact values obtained in our simulations and these comparisons are presented in Fig. 5. As the results show, Eq. (16) is in good agreement with the exact values and although l is considered small in this approach, this is often the case in SMM realizations [7,46]. Extending this approach to larger d values is not difficult (see Appendix A) and we discuss this in the final section. The
Quenching at h for ΔE ()
0.8
0.6
0.4
0.2
0.00
0.02
0.04
0.06
0.08
0.10
0.06
0.08
0.10
inclusion of the remaining anisotropic terms in Eq. (15) is also possible by extending the VT but as we already demonstrated, the main physics behind the DQP of a dimeric SMM is explicated by our approach. Finally, let us to explain with simple arguments the origin of the constant QP (CQP) in all the previous calculations. In the CQP, the levels involved are labeled by the maximal (m1,m2) values, namely, ((S 1,S 1),(S,S),(S 1,S),(S,S 1)) as Fig. 6 displays. The z-exchange term dm1 m2 in H0 shifts all these levels but the ((S,S 1),(S 1,S)) ones remain degenerated, as initially, while the (S,S) and (S 1,S-1) levels are shifted unequally, the modified (S,S) being the lowest one. The key from now on is to understand that a CQP just can be possible if all nonlinear energy corrections on d, which arise from the xy-exchange term, are in fact zero for this CQP and that the linear ones summarized to the described above have to match exactly to cancel out each other the final linear term corresponding to the (S,S) energy level and the one of the lowest level in the ((S,S 1),(S 1,S)) couple, when it is lifted by the xy-coupling. Surprisingly, the above description turns out to be true. The facts that makes it so are the following: in the nonlinear expansions of (A.1)–(A.3) the levels that contribute to the corrections are related to the initial ones by ðmi ,mi 7 1Þ-ðmi 72,ðmi 7 1Þ 8 2Þ for (A.1) in the degenerated two rank sub-spaces and ðmi ,mi Þ-ðmi 81,mi 7 1Þ for (A.3) in the nondegenerated levels. These rules are the reflect of the symmetry in the initial Hamiltonian equation (1) which is invariant under a rotation in p degrees with respect to the z-axis and that symmetry separates its HS into two disjointed sub-spaces labeled by the numbers fS,S2, . . .g and fS1,S3, . . .g [30]. The exchange interaction mixes the sub-spaces corresponding to the two spin-monomers and the numbers in the above rules appear in a simple way. Starting from the sub-spaces that contain (mi,mi) (A.3) we take the levels whose z-numbers belong to the local opposed sub-spaces, correspondingly, and which are neighboring to the initial ones and with the same total z-projection. In the case of ðmi ,mi 7 1Þ (A.1) levels, we start from the sub-spaces that contain the initial numbers, then we take the levels with numbers within the same sub-spaces and which are neighbors to the initial ones with the same total z-projection. The aforementioned selection rules evidence the fact that in the Hamiltonian equation (1) the levels that generate the initial monomeric QP have neighboring numbers corresponding to opposed sub-spaces. The levels crossing that belong to the same sub-spaces are avoided by the inclusion of transversal anisotropy.
Quenching at h for ΔE ()
0.8
0.6
0.4
0.2
0.00
0.02
0.04
Fig. 5. Upper panel: DQP of Eq. (14) for l ¼ 0:12. Hard points (with dashed guiding lines) determine the exact simulation and solid-lines represent hns ðl, dÞ in Eq. (16). (2) Horizontal lines are given for hmu ¼ 5. Lower panel: same calculation as þ SK with S upper panel but for l ¼ 0:0. These DQP correspond to the simulations in Fig. 4.
Fig. 6. Lowest levels of E0(m1, m2). Solid-lines show levels with m1 ¼ m2 ; dashedlines show levels with m1 ¼ m2 7 1. Quantum numbers of the lowest lying levels as h increases are also shown. Multiplicity is determined by the degenerated subspaces at the five monomeric initial QP, which are presented by QP symbols. Small arrows indicate to which line the couple (m1,m2) belong. The exchange coupling used in this picture is d ¼ 0:06.
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
From previous discussion it is straightforward to deduce that for the CQP there are no levels that meet the requirements and for this reason the nonlinear expansions in (A.1)–(A.3) do not contribute. Now, the difference between the z-exchange terms (H0 ) of the (S,S) and (S,S 1) levels is dS. On the other hand, the linear terms in (A.1) and (A.3) are found to be ðmi þSÞ ðmi ðS þ 1ÞÞd=2 and 0, respectively; at the CQP the first previous term has mi ¼ S such that it is finally equal to dS, which exactly cancels out the z-exchange difference between the levels involved in the CQP as we anticipated.
5. Discussion In this paper we have assessed one of the actual problems in the area of molecular magnets, namely, we have calculated the quenching points of the energy-splitting of a dimeric SMM as a function of the exchange interaction and implemented simple techniques that can be used to extract such a quenching with good approximation and for large molecular Hilbert spaces. The importance of the quenching points studied here lies in the possible realization of a molecular-based hardware that implements the advantages of such quantum devices in information processing, besides, fundamental reasons as trying to discover new phenomena that this nano-systems can manifest. In this way, understanding the role of quantum selection rules in molecular magnets behavior [51] is an important step that has to be further studied, and the discussions about DQP that we developed here go along this direction. Differences between QP of a strongly exchange coupled dimer and the ones of a giant-spin modeled dimeric SMM are more evident for moderated spin values and high magnetic fields. Such characteristics are actually more relevant when we take into account that the magnitude of the energy-splitting is often larger in higher magnetic fields making that field region more propitious for reliable detection of the energy-splitting. The aforementioned difference between the two sets of quenching-points in Eq. (12) is in agreement with the fact that SK-phase influence should be similar to the difference between the principal and the Weyl symbols of a quantum spin Hamiltonian [52]. The constant value of the highest QP for all values of anisotropy and exchange coupling is explained in terms of the symmetry of the monomeric sub-spaces and specific values of m1 and m2 involved in the calculation. Such a CQP can represent a useful tool in future manipulations of the energy-splitting as computational mechanics due to its behavior against magnetic field and exchange interaction; this means we can modulate the position of the DQP in order to tailor computational processes, and at the same time keep the information (maybe partially) carried up to the initial time when (J) was modified. CQP can be used somehow as a setup-point for the energy-splitting of a J-modifiable quantum device SMM-based. The quenching values in Eq. (16) describe satisfactorily the behavior of our simulations. Due to the generality of their behavior with respect to l as well as the importance of small d values to have a rich physics in a dimeric energy-splitting, these DQP would apply widely in the context of dimeric SMM-based molecular electronic. In case of strong coupled dimers, the giantspin-like behavior can be reached at moderate couplings and in that effective case Eq. (16) seems to be a good approach for low magnetic field values. Besides, the expansion in d can be improved just by using the terms all in (A.1)–(A.3) or by extending the transformation up to fourth order. We believe that this work can be widely used in studying new single-molecular magnets that are conformed by effective spin-monomers and in which an isotropic exchange coupling
3629
dominates the spin–spin interaction. Moreover, recent studies have proposed the modification of quantum properties in a molecular system through experimental procedures which seem to be feasible from different points of view, namely, chemical substitution in an effective three-spin molecule [5], magnetic dilution of a molecular crystal [46] and application of uniaxial stress along molecular axes [11] (and references therein). The study that we present here can be very useful for future molecular realization where the exchange parameter d or anisotropies l would be easily modifiable in order to modulate the energysplitting behavior and therefore the quantum reversal process of the molecular spins. In the case of SMMs the reversion is dominated by quantum tunneling processes at low temperature, and from L–Z theory it is easy to see that the quenching points of the energy-splitting are the points of zero reversal probability through magnetic tunneling. Usually, in SMMs the relaxation times are in years [7] which means that operational quantum processes would last an attainable time for experimental developments [1–5], opening the door to the study presented here.
Acknowledgments We acknowledge the financial support from the Conicyt scholarship to graduate students and from PIIC 2009 of the DGIP Universidad Te´cnica Federico Santa Marı´a, and the funds from Nu´cleo Cientı´fico Milenio Magnetismo Ba´sico y Aplicado P06022F. A.S.N. work was partially funded by Proyecto de Iniciacio´n en investigacio´n Fondecyt 11070008 and Proyecto Bicentenario de Ciencia y Tecnologı´a ACT027, P.V. acknowledges Fondecyt Grant 1100508.
Appendix A. Van Vleck transformation In this appendix we give the formulas obtained after the procedure described in Section 4. The expressions are presented up to third order which is enough for the calculation in Section 4. In the case of a degenerated block and if the expansion of the Hamiltonian is represented as H ¼ H0 þ eHu þ e2 H00 þ , the components of this degenerated block can be calculated as follows: Gmmu ðeÞ ¼ E0 ðmÞdmmu þ eHummu þ
l e3 X
2
l e2 X
2
ð1Þ ½U ð1Þ ðm,mu,nÞ þPU ðm,mu,nÞ
n
V ð1Þ ðn,m,muÞ
ðA:1Þ
n
with l X ð3Þ ð2Þ ð2Þ ½U ð3Þ ðm,mu,n,m00 Þ þ PU ðm,mu,n,m00 Þ ½U ðm,mu,nÞ þPU ðm,mu,nÞ
V ð1Þ ðn,m,muÞ ¼
m00
2
l X
U ð4Þ ðm,mu,n,nuÞ
nu
where the matrix elements involved in the summations above are given by U ð1Þ ðm,mu,nÞ ¼
Humn Hunmu E0 ðmÞE0 ðnÞ
U ð2Þ ðm,mu,nÞ ¼
Humn H00 nmu E0 ðmÞE0 ðnÞ
U ð3Þ ðm,mu,n,m00 Þ
Humm00 Hum00 n Hunmu ðE0 ðm00 ÞE0 ðnÞÞðE0 ðmuÞE0 ðnÞÞ
3630
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
U ð4Þ ðm,mu,n,nuÞ ¼
Humn Hunnu Hunumu ðE0 ðmÞE0 ðnÞÞðE0 ðmuÞE0 ðnuÞÞ
ðA:2Þ
and where P is an operator which does the following changes in the matrix elements: 8 First m2mu > > > < Second Himn 2Hinm PU ¼ > whereHi ¼ Hu,H00 > > : Third the matrix element is evaluated The notation presented here is general and in our case studied in Section 4 we have used m ¼ ðm1 ,m2 Þ, as the quantum numbers corresponding to the eigenstates jmS ¼ jm1 ,m2 S that diagonalize H0 with energies E0 ðmÞ ¼ E0 ðm1 ,m2 Þ. Now, in the case of nodegenerated block we give the result by considering H ¼ H0 þ eHu þ , for the sake of simplicity: Gmm ðeÞ ¼ E0 ðmÞ þ eHumm þ e2
l X
3 U ð1Þ ðm,m,nÞ e
n
l X
V ð2Þ ðn,mÞ ,
n
with V ð2Þ ðn,mÞ ¼
l X m00
dmm00 U ð3Þ ðm,m,n,m00 Þ
l X
U ð4Þ ðm,m,n,nuÞ
ðA:3Þ
nu
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Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Quenching points of dimeric single-molecule magnets: Exchange interaction effects ˜ ez b,,1, P. Vargas a,,1 J.M. Florez a,,1, A´lvaro S. Nu´n a b
Departamento de Fı´sica, Universidad Te´cnica Federico Santa Marı´a, P.O. Box 110-V, Valparaı´so, Chile ´ticas, Universidad de Chile, Casilla 487-3, Santiago, Chile Departamento de Fı´sica, Facultad de Ciencias Fı´sicas y Matema
a r t i c l e in fo
abstract
Article history: Received 8 June 2010 Available online 14 July 2010
We study the quenched energy-splitting ðDE Þ of a single-molecule magnet (SMM) conformed by two exchange coupled giant-spins. An assessment of two nontrivial characteristics of this quenching is presented: (i) The quenching-points of a strongly exchange-coupled dimer differ from the ones of their respective giant-spin modeled SMM and such a difference can be well described by using the Solari– Kochetov extra phase; (ii) the dependence on the exchange coupling of the magnetic field values at the quenching-points when DE passes from monomeric to dimeric behavior. The physics behind these exchange-modified points, their relation with the DE oscillations experimentally obtained by the Landau–Zener method and with the diabolical-plane of a SMM, is discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Single-molecule magnets Spin tunneling Spintronic
1. Introduction Single-molecule magnets (SMMs) attract great interest from the scientific community due to both their fundamental significance in the context of quantum dynamics and their potential applications in the context of information processing and storage [1–7]. Recent experimental developments [2–5] provide a great step forward in the continuing struggle for the use and manipulation of molecular quantum-properties. Among the basic phenomenology associated with SMM-dynamics there is a variety of phenomena that can be accounted for in terms of a simple qualitative picture provided by the giant-spin model. Within this model the dynamics of the system is regarded as confined within the sub-space generated by projecting out states with overall spin different from the ground state spin. The quantum tunneling of the magnetization (QTM) as a description of the reversion process in SMM is a major success in the class of phenomenology that can be described rather accurately from the viewpoint of the giant-spin model [6,7]. The transition from QTM in magnetic nanoparticles to SMMs was promoted when it was known that the tunneling process in magnetic nanoparticles, which can be approximately described with a magnetic viscosity in an effective Arrhenious-like rate transition [6], presents a plateau with a width comparable to the characteristic magnetization-steps of a SMM at low temperatures [8,9]. Since SMMs are mostly molecules containing transition–
Principal corresponding author. Tel.: + 56 32 87435885; fax: +56 32 2797656. Corresponding authors.
E-mail addresses: juanmanuel.fl[email protected], jmfl[email protected] ˜ ez), [email protected] (J.M. Florez), alnunez@dfi.uchile.cl (A´.S. Nu´n (P. Vargas). 1 http://www.magnetismo.cl 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.07.002
metal ions bridged by organic ligands and with spins strongly exchange coupled, at low temperature each molecule behaves effectively like a single-domain particle with fixed total spin [6,7]. As a consequence, several studies of the magnetization tunneling in SMMs have been based on giant-spin Hamiltonians that account for different magnetic interactions [6,7,10–12]. Nevertheless, recent reports [13–18] have shown that the influence of giant-spin fluctuations besides the assorted non-reduced behavior given by dimer-conformed SMMs, i.e., molecules based on two effective-spins interacting which behave separately like a SMM, should be taken into account in order to explain further the features so far briefly considered, e.g., the effects of S-mixing within and between different multiplets. In the QTM problem of SMM the cornerstone describing the quantum reversion process is the energy-splitting DE . This DE is the energy-difference between two low-lying energy levels, which can involve excited states depending upon the external longitudinal magnetic field, and its presence leads us to the finite ‘‘direction’’-probability on both sides of the double-spin-well conformed with an anisotropic barrier; this probability determines the tunneling probability of the magnetization. The tunneling probability is nontrivial related to DE and it has been the target of QTM formulations based on semiclassical methods [6], where the macroscopic features of spin tunneling are evidenced. Most first theoretical predictions on QTM in nanoparticles and posteriorly in SMM were developed using spin path integrals (PI) [6], while the advantages of Landau–Zener (L–Z) theory in SMM were principally appreciated when implementing in the experimental method necessary to measure DE [7,19,20]. There are also strategies to calculate features of giant-spins in the tunneling regime [21–23], which in principle can include a variety of effects like the dipole–dipole interaction [24,25]. However, the
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J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
oscillations observed in DE as a function of the external transverse field, which are related with the topological features of spin tunneling [19,20,26–33], have been investigated so far by using single giant-spin considerations. Therefore, several questions remain about the general behavior of a dimer-conformed SMM when the exchange interaction (J) and the transversal Zeeman field are switched on giving rise to S-mixing into selected S-multiplets and other fundamental characteristics. It was predicted that DE as a function of a transversal magnetic field would present oscillations due to the interference of quantum paths, described onto the coherent-state sphere by spin tunneling, which are symmetric with respect to the hard axes [27]. This interference process was introduced, in the context of magnetic tunneling, when defining the half-spin rules for coherent quantum tunneling in magnetic nanoparticles [6,26]. Investigations of the coherence in QTM processes have included studies of nuclear-spins effects [28] and dissipation due to environmental-coupling [29,30]. QTM phenomena in antiferromagnetic particles and SMMs [31,32] as well as generalizations of the L–Z theory accounting for thermal relaxation [33], have also been explored. In principle, it is possible to have a hardware based on an array of SMMs and where DE plays an important role in controlling the energy-difference and times decoherence between the ðj0S,j1SÞmolecular states [1]. In this way, the features of DE that might be somehow tunable by experiments, are interesting to manipulate the molecular states useful to SMMs-based electronics. This was the case of the paradox about the quenching-points of DE in Fe8 molecule. The so-called quenching refers to the vanishing of DE when the transversal spinoperators, modulated by Zeeman fields or anisotropic terms, are modified. Experimental measurements of Fe8 [19,20] showed just half of the quenching-points predicted by instantons-based PI method [27]. This problem was understood when according to inclusion of weak fourth-order anisotropy in the simulation of inelastic neutron scattering [20], a new instantons technique was developed [34,35] to demonstrate that after some transversal field these new instantons do not allow anymore quenching [36]. Additional considerations for DE , e.g., the inclusion of inhomogeneities in the magnetic field [37] and of small quantum fluctuations of the giant-spin instantons [38], were also proposed. These quenching-points belong to a subspace of the so-called diabolical-plane of a SMM, i.e., to the collection of points determined by the couple of values of two Hamiltonian parameters (Zeeman fields, anisotropies) at the crossings lying between different energylevels [39–41]. The Hamiltonian equation (1) represents the usual giant-spin model of a SMM. The quenching-points for this model can be calculated using the instantons-based PI [27,42]. Such a method has been explored for different anisotropy orders and symmetries of a giant-spin Hamiltonian [11,12,34]. Nevertheless, the instantons method has not been used to evaluate the Berry phase of coupled spin-paths. The difficulty of using instantons for a dimeric problem lies in the dynamical nature of the Euclidean action, which leads us to dynamical calculations of the multiple instantons for each monomer and to nontrivial quantum fluctuations around saddle points of the energy density, and in many-spins problem these calculations are still an open question. Known approximated strategies for this problem reduce the Hilbert Space (HS) through construction of effective instantons that represent the collective dynamic of the system, and this is not adequate to generally describe S-mixing [15,41]. As far as the authors know, there are no theoretical descriptions of the quenching-points dependence on the exchange interaction of a dimer-conformed SMM. In order to do that, different strategies can be used. Studies of the degeneracy of lying energy spin-levels [10], the arguments of Bruno based on the general diabolical points [41] as well as the calculations of the diabolical-plane of a quartic anisotropy model [11], were developed using simple methods and mixed strategies. In this paper we are
going to use different scopes of calculation to do an assessment of nontrivial properties of such quenching. This paper is distributed as follows: in Section 2 we highlight the quantum model for the dimerconformed SMM. In Section 3 we show how the quenching-points of a strongly exchange coupled dimer differs from the ones of the respective giant-spin modeled SMM and such difference can be approximately calculated using the Solari–Kochetov (SK) extra phase [43,44]. In Section 4 we simulate the quenching-points of a dimeric SMM and calculate an expression for the quenching-points’ dependence on the exchange parameter J when the molecule moves from monomeric to dimeric behavior. In Section 5 we gave a discussion about the results and ideas of the previous sections.
2. Spin Hamiltonians In this section we introduce the model and highlight the main features it conveys. Let us consider the Hamiltonian: ^ local ¼ K1 S^ 2 þK2 S^ 2 g m HS^ z , H B z y
ð1Þ
where K1 bK2 are anisotropies and H stands for the Zeeman field. For H ¼ 0, this Hamiltonian has two degenerated classical minima, ^ As H grows, the locations of the minima lies in namely ~ S ¼ 7 x. the x2z plane. To each of the classical ground states we can ^ associate a spin coherent state j 7 xS. The quantum fluctuations induce these ground states to be hybridized into two states of ^ ^ Their respective energies are opposite parity jc 7 S ¼ jxS7 jxS. modified, and the degeneracy is lifted by an energy splitting DE . The main qualitative feature that it manifests is the oscillations of its magnitude as H is modified, as we mentioned above. Now, we consider two local moments, each one with spin S and obeying the same dynamics, dictated by the Hamiltonian in Eq. (1). To this Hamiltonian we add a coupling term of the form ^ ¼H ^ local ðS^ 1 Þ þ H ^ local ðS^ 2 ÞJ S^ 1 S^ 2 H
ð2Þ
Our goal is to understand the behavior of the DE oscillations as a function of H and J, namely, to find the quenching-points when the external magnetic field varies and also their dependence on the exchange coupling. J ranges from zero (two independent monomers of spin value S) to a big enough value at which a single giant-spin behavior with spin Sð2Þ ¼ 2S, is reached.
3. Path integral approach: strong coupling Let us analyze what happens if a giant-spin modeled dimericSMM is ‘‘split up’’ into its spin-monomers such that Eq. (2) (with strong coupling J) represents the SMM instead of Eq. (1). Our basic approach starts from the SU(2) PI representation of the partition function [6]. The coherent-state representation of the Euclidean action can be obtained by two different ways, namely, using polar or stereographic coordinates. In the case of explicit time depending calculation, divergences related to instantons dynamic are avoided using stereographic coordinates as well as the polar ones in a rotated frame. In this section we use stereographic coordinates. The partition function reads as Z K¼ D2 z1 D2 z2 expfSðz1 ,z 1 ; z2 ,z 2 Þg ð3Þ The full action in the last equation is given by: S ¼ S 1 ðz1 ,z 1 Þ þ S 1 ðz2 ,z 2 Þ þ S 2 ðz1 ,z 1 ; z2 ,z 2 Þ, where S 1 that represents the single spin action and S 2 ðz1 ,z 1 ; z2 ,z 2 Þ are given by ) Z b( _ z zz_ z Hðz,zÞ dt S S 1 ðz,zÞ ¼ 1þ zz 0
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
S 2 ðz1;2 ,z 1;2 Þ ¼
ð1z 1 z1 Þð1z 2 z2 Þ þ 2ðz1 z 2 þ z 1 z2 Þ ðJS2 Þð1Þ ð1þ z 1 z1 Þð1 þz 2 z2 Þ
ð4Þ
In Eq. (5), z is a complex number associated with a trajectory in the unit sphere according to the standard stereographic representation (we have chosen the pole of the representation at the ^ north pole of the sphere), Hðz,zÞ stands for /zjHjzS=/zjzS, and jzS is a coherent spin state along the direction defined by z. A careful evaluation of the matrix element of Eq. (1) gives us: Hðz,zÞ ¼ k~ 1 S2
2 2 ~ ð1zzÞ2 lðzzÞ2 2hð1z z Þ
ð1 þ zzÞ2
ð5Þ
where we have defined k~ 1 ¼ ðS12ÞK1 =S, h~ ¼ g mB H=2K1 ðS12Þ, l ¼ K2 =K1 , by following Ref. [42]. To separate the physics of the giant-spin model from the S-mixing contributions we introduce the change of variables z1 ¼ z þ dz=2, z2 ¼ zdz=2 and their respective conjugates. In the zeroth order term the only effect is an overall factor of 2 multiplying the single spin action. If we absorb the factor in the S value of the first term of Eq. (5), which is the Berry-phase, we obtain the effective action of a spin with double size, described by a Hamiltonian like the one in Eq. (1), but with effective parameters: Kið2Þ ¼ Ki =2 and Sð2Þ ¼ 2S. In the classical limit, ðS b1Þ we see how, as expected, the spin Hamiltonian goes into one of double spin magnitude with half anisotropic constants. In the large-J limit we can regard the amplitude of the direction deviations associated with dz to be small; in terms of stereographic variables it is convenient to use dz ¼ ð1 þ zzÞZ, with JZJ 51. We then, represent the partition function in terms of such variables, and restrict the contributions up to second order in Z. We obtain Z Z S dt Cy F ðz,zÞC K D2 z D2 Zexp S 0 ðz,zÞ ð6Þ 2 In the last expression we have used the spinor notation Cy ¼ ðZ , ZÞ, C ¼ ðZ, Z Þt and S 0 represents the effective giant-spin action. F stands for a rank two spinor of the form ! Bðz,zÞ @t þJSð2Þ þ Aðz,zÞ F ðz,zÞ ¼ ð7Þ @t þ JSð2Þ þ Aðz,zÞ Bðz,zÞ (2)
In this last matrix S is the effective giant-spin and the contributions denoted by A, B and B, need to be calculated starting from the specific action in Eq. (5). In this way, they contain the contributions from the Berry-phase term and from the Hamiltonian. We remark that, besides the explicit term next to the derivative in Eq. (7), there are no further contributions from the exchange term into F . This fact is essential in the manipulations associated with the J-1 limit. In general we have Aðz,zÞ ¼
z_ zz_ z 1 ð1 þ zzÞ2 @2zz Hðz,zÞ þ 1 þzz 2S
Bðz,zÞ ¼ 2
z_ z 1 ð1 þ zzÞ2 @2zz Hðz,zÞ þ 1 þ zz 2S
3625
The action in Eq. (10) contains, formally, the physics associated with the interplay of the giant-spin sub-space and the S-mixing fluctuations. For a strongly exchange coupled dimer the non-local contribution in Eq. (10) will be dominated by the physics described with the partial kernel F 0 ¼ s3 @t þ JSð2Þ (s3 the Pauli matrix). We can evaluate that contribution in the required limit and this has the final form Z 1 dt dtu trðGðt, tu ÞdF ðtu , tÞÞ dS 1 ¼ ð11Þ 2 u where Gðt, tu Þ ¼ F 1 0 ðt, t Þ. Since the contribution dF is local in time, and after evaluating the propagator G, we obtain the simple result: dS 1 ¼ S SK . This last contribution does not have any dependence on J. In this way even at zeroth order in 1/J we would have had a renormalization of the bare giant-spin action associated with S-mixing. However, this contribution is exactly canceled out by the SK extra phase. The consequences of the vanishing of this extra phase on the dimeric quenching-points can be analyzed after calculating the phase of the DE oscillations for each case, i.e., the phase with (PhaseðDE Þ þ SK ) and without the extra phase contribution (PhaseðDE ÞSK ). Remember that the quenching-points are determined by the values of some parameters of the Hamiltonian at the vanishing points of DE . In this case such parameters are the magnetic fields, namely, the longitudinal magnetic field, which is fixed at the zeroth resonance Eq. (1) ðHx ¼ 0Þ, and the transversal field, which is a modifiable parameter. The quenching-points are then given by the transversal field values at the vanishing of PhaseðDE Þ 7 SK . We find such quenching, respectively: pffiffiffiffiffiffiffiffiffiffi 1l ð2Þ 1 mu hmu ¼ S þ þ SK 2 Sð2Þ
hmu SK ¼
pffiffiffiffiffiffiffiffiffiffi 1l Sð2Þ
2
2
Sð2Þ
1 1 mu Sð2Þ 4 2
with mu ¼ 1,2,3, . . . ,Sð2Þ
ð12Þ
and where h ¼ g mB H=2K1 S . In Fig. 1 we have plotted the quenching-points equation (12). In order to check our approach, we have included in Fig. 1 the quenching-points obtained by an exact simulation of DE . Two set of points are simulated, namely, ð2Þ
hmSK
0.1
ð8Þ 0.2
ð9Þ
Evaluating the Gaussian integral associated with the fluctuation-Z functions we obtain an effective action for the giant-spin degrees of freedom. Formally, we have Z 1 1 dt AðzðtÞ,zðtÞÞ ð10Þ S eff ¼ S 0 TrðlogF Þ þ 2 2 The third term in the action is known as the Solari–Kochetov S SK extra phase [42–45]. We need to remark that in general this contribution is not a phase. It bears its name from its behavior in some simple cases (whenever the Hamiltonian is a combination of the SU(2)-algebra generators). Its mathematical origin is clear, arising as the correction of an anomaly created by the ambiguity of the continuous representation of the SU(2)-coherent states path integral. However its physical interpretation is more involved.
0.2
0.4
hmSK
0.1 2
0.2
9
0.3
0.4 Fig. 1. Polar plot of quenching-points equation (12) (white-small-points). hmu SK and hmu þ SK are joined by dashed and solid guiding-lines, respectively. The exact quenching-points of a dimer strongly coupled by exchange interaction and of a giant-spin modeled dimeric SMM are also plotted (red-big-points). The para(2) meters used here are J=K1 ¼ 78 ¼10, correspond10, K1 ¼ 0.321, K2 ¼ 0.229, and S ingly. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
ΔE (K)
1000
10
0.1
0.001 0.0
0.1
0.2
0.3
0.4
0.5
h
0.4
Log (ΔE(K)
0.2
–5 –10 –15 –20 –25
0.0
0.0 0.2 h
0.4
Fig. 2. Upper panel: energy-splitting of a giant-spin modeled dimeric SMM equation (1), dimer strongly exchange-coupled, and weakly exchange-coupled equation (2). Phase ðDE Þ 7 SK for S(2) ¼10, and phase ðDE Þ þ SK with S(2) ¼ 5 are also plotted. Lower panel: energy-splitting of a dimeric SMM equation (2) as a function of d ¼ J=K1 . In the upper panel the exchange values are shown in the picture and for both plots K1 ¼ 0.321, K2 ¼0.229 ðl ¼ K2 =K1 ¼ 0:7Þ.
the ones of a strongly exchange coupled dimer and the ones of a giant-spin modeled dimeric SMM (that we develop in next section). Moreover, in Fig. 2 (upper panel) we show DE for these two mentioned systems as well as PhaseðDE Þ 7 SK . Such comparisons are performed for molecular parameters similar to the corresponding to Fe8 molecule [42]. As Figs. 1 and 2 show, differences between the quenching behavior of this two systems are actually well described, for a reasonable strong value of J, by the approach presented here. Differences between these quenching-points increase with increasing of h and decrease with higher values of l. Also, moderate spin values (Sð2Þ 10) as reported in the recent experimental studies [16–18,46,47] would allow to have, in principle, appreciable differences.
4. Diabolical quenching: up to moderate coupling In this section our target is to develop a strategy that allows us to study the exact mechanism involved in the generation and the behavior of such quenching-points when the exchange coupling is modified. A many-body instantons-based PI method constitutes a complex tool which still conveys many questions because we have to control several contributing-instantons with adequate intervals of integration [11] and their possible jumps at the end points [11,34]. These features are strongly affected by interactions that produce S-mixing. In the case of overlapped paths connecting different sets of coherent-states the problem cannot be handled by the usual instantons-based PI [7,41]. A general possible method based on PI is then out of the scope of this work.
Nevertheless, the quantum mechanics nature of typical SMMs represented by Eqs. (1) and (2) allow us to explore other scopes. Therefore, in order to understand what happens with the influence of the exchange interaction on the quenched energysplitting of a dimeric SMM, let us just switch on the exchange J and sweep it up to a strong enough value where the giant-spin behavior is reached, and in this process calculate the energysplitting as well as its quenching-points (QP from now on). Starting with the dimer already discussed in the previous section, we perform exact simulations of DE which we picture in Fig. 2 (Lower panel) where the parameters employed in generation of Fig. 1 have been used as well as the reduced exchange d ¼ J=K1 . Simulations in Fig. 2 show clearly how switching on J generates a splitting of the initial five QP, which correspond to the monomers behavior, into 10 QP. These 10 QP are shifted as a function of J until some finals magnetic field values that are actually the ones studied in Section 3, and which differ from the corresponding to the giant-spin model, approximately, by the values shown in Fig. 1. In the case of d 0 the oscillations of a dimeric SMM are determined by the ones of an isolated monomer, as expected from Eqs. (2) and (5), and in Fig. 2 (Upper panel) we extract such splitting from the full-simulation in there. To visualize better how the exchange coupling modifies the position of the QP, we have extracted from Fig. 2 each magnetic field value at the quenching as a function of d, and the results are shown in Fig. 3. In this last simulation two things are quite interesting: Magnetic field values at the QP are quickly affected for small exchange coupling and moderated values of d allow us to reach a giant-spin-like behavior; the last quenching, with respect to increasing h value, does not change when the reduced exchange d is modified. These two features would be important as long as the physics they convey does not depend upon the l value. Additional simulations for different l values are developed in Fig. 4 and they corroborate these two mentioned features. In order to understand the behavior of the diabolical subspace determined by the QP studied here (DQP), which are given by the points at the axe of the zeroth resonant field, in the diabolicalplane, with couples fHx ¼ 0,Hz-QP g [11,39–41], let us to perform a theoretical assessment of these QP by using and simple strategy that can be extended to another problems with similar characteristics and larger HS. Such a strategy that we use is based on a Van Vleck transformation (VT) [48–50]. Although the Van Vleck method has been used in several problems, it is seldom described in the literature and, surprisingly, its advantages compared with another methods and which are based on their power to handle
Quenching at h for ΔE ()
3626
0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 3. Diabolical quenching-points as a function of reduced coupling d as extracted from Fig. 2. Red-dashed lines show the quenching-points hmu þ SK with S(2) ¼ 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
3627
in e from the matrix elements of Hu connecting any particular unperturbed energy level with the other distinct unperturbed n levels, we choose Qmn ¼ i/mjHujnS=ðEm 0 E0 Þ and Qmmu ¼ Qnnu ¼ 0. These matrix elements we just took allow two things: V does not affect the m or n blocks and they make Gumn ¼ 0. The remaining coupling terms in the considered block are of second order or higher, and since they are off-diagonal they cannot contribute to the energies until the fourth order. If the approach is taken to second or third order, this decoupling of the blocks in the perturbation allows to separate the treatment of each block until these orders, which is usually adequate in practice. The previous procedure leads us the formulas for the mmu elements of each block (Appendix A), which conform to a problem of lower dimension. In the case of one dimensional blocks VT is also applicable (A.3) as well as fourth order transformations. Recently, dissipative dynamics of a qubit coupled to a nonlinear environment was studied, and advantages of VT were used [50]. Now, the Hamiltonian equation (2) can be disassembled as follows: H ¼ H0 þHe þ Ha
ð13Þ
where He corresponds to the xy-component of the exchange interaction and we have separated the anisotropy such that: 0 1 2 X z2 z lX j j A ^ ^ @ dSz1 Sz2 SS H0 ¼ S i 2ShS i þ 4j¼ þ i i i¼1 He ¼
d 2
Ha ¼
Fig. 4. DE of a dimeric SMM equation (2) as a function of d. Upper panel: l ¼ 0:12. Lower panel: l ¼ 0:0.
þ ðS1þ S 2 þ S1 S2 Þ
2; l X
4 i;j ¼ 1; þ
Sji
2
ð14Þ
ð15Þ
where we normalize H by K1 and j ranges within ( + , ). The eigenstates of Sz1,2 of the two monomers diagonalize H0 and the P P eigenvalues are E0 ðm1 ,m2 Þ ¼ 2i ¼ 1 ðm2i 2Shmi þðl=4Þ j¼ þ j j Cmi Cmi þ ðjÞ1 Þdm1 m2 , with mi ¼ 1,2 the corresponding quantumffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j number along z-direction and where 2Cm ¼ 34mi ðmi þðjÞ1Þ. i On the other hand, the trickier physics behind the QP behavior illustrated in Figs. 2–4 does not strictly depend upon the l value. Moreover, in Hamiltonian equation (1) the DQP reflect ^ local jmð2Þ 7 1S ¼ 0, where jmð2Þ S are the selection rule /mð2Þ jH ð2Þ ^ the eigenstates of S z , and these crossings of energy-levels are not avoided modifying the l value, as Fig. 4 confirms. Therefore, we develop our approach by applying the VT to Eq. (14), for which the matrix representation of the perturbation, in the base jmS ¼ jm1 ,m2 S, is found to be 2/mujHe jmS ¼ dðy ðmÞdmu,m þ
highly degenerated block systems, have been untapped in manyspins-conformed SMM, which naturally represent just these kinds of problems. We give a short description of the matrix transformation that we use. Let us consider the Hermitian matrix Q which defines the unitary transformation VðeÞ such that VðeÞ ¼ eieQ ¼ 1 þieQ e2 Q 2 =2 þ , where e stands for a parameter that expands the perturbative series of H ¼ H0 þHu, H0 being an unperturbed Hamiltonian whose solutions are known and Hu represents the perturbation matrix as usual. We want to construct VðeÞ such that the first order term in the Hamiltonian has vanishing matrix elements between the blocks of interest and the other ones, i.e, Gumn ¼ 0, where G is defined by the unitary transformation: GðeÞ ¼ Vy HV ¼ G0 þ eGu þ e2 G00 þ , and where the fm . . . mug and fn . . . nug elements label the states within the Hu blocks of interest and the outside blocks, correspondingly. Expanding both the sides of GðeÞ and equating the coefficients of like powers of e, we get the expressions for each one of the contributions G0 ,Gu,G00 , . . ., and to eliminate the terms of first order
þ y þ ðmÞdmu,m Þ, with the coefficients given by 4y 8 ðmÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 93ðW 8 ðmÞ þX 7 ðmÞÞ þW 8 ðmÞX 7 ðmÞ. The respective functions in the last equation and in the elements /mujHe jmS are 8
given by dmu,m ¼ dmu1 m1 8 1 dmu2 m2 7 1 , W 8 ðmÞ ¼ 4m1 ðm1 8 1Þ and X 7 ðmÞ ¼ 4m2 ðm2 7 1Þ. In order to do the calculations we need to discriminate the energies involved on the DQP. Energies E0(m), with m ¼ ðm1 ,m2 Þ, which have ðmi ,mi 81Þ as well as the energies with (mi,mi), generate such DQP. Moreover, in the first case the energies are degenerated due to the invariance ðmi ,mi 8 1Þ2ðmi 8 1,mi Þ which leads us to two of these levels for each initial monomeric DQP. This last energy-characterization can be visualized in Fig. 6, where we plot E0(m) and the quantum numbers of the lowest lying energy levels conforming DE when h is modified. These numbers increase when h does and the five DQP that characterize the monomer behavior, before including the xy-exchange terms, are actually the crossings between four levels as Fig. 6 shows. These points become doubly degenerated when all the exchange interactions are considered and the interceptions of the levels
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J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
shifted by the xy-exchange terms with the m ¼ ðmi ,mi Þ ðmi ¼ 1,2 ¼ 0, 7 ð1,2,3,4,5ÞÞ levels are then the ddepending DQP. We now perform the VT. By using the equation for nondegenerated blocks (A.3) and solving the degenerated-subspaces with (A.1), the magnetic field values determining the DQP are found to be ð2n1Þ l 2 1 5ðld2Þ 2 l ð29nðn1ÞÞ 1 2 þ sd 5ðld2Þ
hns ðl, dÞ ¼
sd2 5ðld2Þ
M3=2s,n
ð16Þ
where for the sake of simplicity we have extended the VT up to second order; s ¼ 7 1=2 and the matrix M ¼ ff330,315,259,174, 81g,f301,234,144,55,0gg, with n ¼ 1, . . . ,5, correspondingly. The approach given by Eq. (16) has been compared with the exact values obtained in our simulations and these comparisons are presented in Fig. 5. As the results show, Eq. (16) is in good agreement with the exact values and although l is considered small in this approach, this is often the case in SMM realizations [7,46]. Extending this approach to larger d values is not difficult (see Appendix A) and we discuss this in the final section. The
Quenching at h for ΔE ()
0.8
0.6
0.4
0.2
0.00
0.02
0.04
0.06
0.08
0.10
0.06
0.08
0.10
inclusion of the remaining anisotropic terms in Eq. (15) is also possible by extending the VT but as we already demonstrated, the main physics behind the DQP of a dimeric SMM is explicated by our approach. Finally, let us to explain with simple arguments the origin of the constant QP (CQP) in all the previous calculations. In the CQP, the levels involved are labeled by the maximal (m1,m2) values, namely, ((S 1,S 1),(S,S),(S 1,S),(S,S 1)) as Fig. 6 displays. The z-exchange term dm1 m2 in H0 shifts all these levels but the ((S,S 1),(S 1,S)) ones remain degenerated, as initially, while the (S,S) and (S 1,S-1) levels are shifted unequally, the modified (S,S) being the lowest one. The key from now on is to understand that a CQP just can be possible if all nonlinear energy corrections on d, which arise from the xy-exchange term, are in fact zero for this CQP and that the linear ones summarized to the described above have to match exactly to cancel out each other the final linear term corresponding to the (S,S) energy level and the one of the lowest level in the ((S,S 1),(S 1,S)) couple, when it is lifted by the xy-coupling. Surprisingly, the above description turns out to be true. The facts that makes it so are the following: in the nonlinear expansions of (A.1)–(A.3) the levels that contribute to the corrections are related to the initial ones by ðmi ,mi 7 1Þ-ðmi 72,ðmi 7 1Þ 8 2Þ for (A.1) in the degenerated two rank sub-spaces and ðmi ,mi Þ-ðmi 81,mi 7 1Þ for (A.3) in the nondegenerated levels. These rules are the reflect of the symmetry in the initial Hamiltonian equation (1) which is invariant under a rotation in p degrees with respect to the z-axis and that symmetry separates its HS into two disjointed sub-spaces labeled by the numbers fS,S2, . . .g and fS1,S3, . . .g [30]. The exchange interaction mixes the sub-spaces corresponding to the two spin-monomers and the numbers in the above rules appear in a simple way. Starting from the sub-spaces that contain (mi,mi) (A.3) we take the levels whose z-numbers belong to the local opposed sub-spaces, correspondingly, and which are neighboring to the initial ones and with the same total z-projection. In the case of ðmi ,mi 7 1Þ (A.1) levels, we start from the sub-spaces that contain the initial numbers, then we take the levels with numbers within the same sub-spaces and which are neighbors to the initial ones with the same total z-projection. The aforementioned selection rules evidence the fact that in the Hamiltonian equation (1) the levels that generate the initial monomeric QP have neighboring numbers corresponding to opposed sub-spaces. The levels crossing that belong to the same sub-spaces are avoided by the inclusion of transversal anisotropy.
Quenching at h for ΔE ()
0.8
0.6
0.4
0.2
0.00
0.02
0.04
Fig. 5. Upper panel: DQP of Eq. (14) for l ¼ 0:12. Hard points (with dashed guiding lines) determine the exact simulation and solid-lines represent hns ðl, dÞ in Eq. (16). (2) Horizontal lines are given for hmu ¼ 5. Lower panel: same calculation as þ SK with S upper panel but for l ¼ 0:0. These DQP correspond to the simulations in Fig. 4.
Fig. 6. Lowest levels of E0(m1, m2). Solid-lines show levels with m1 ¼ m2 ; dashedlines show levels with m1 ¼ m2 7 1. Quantum numbers of the lowest lying levels as h increases are also shown. Multiplicity is determined by the degenerated subspaces at the five monomeric initial QP, which are presented by QP symbols. Small arrows indicate to which line the couple (m1,m2) belong. The exchange coupling used in this picture is d ¼ 0:06.
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
From previous discussion it is straightforward to deduce that for the CQP there are no levels that meet the requirements and for this reason the nonlinear expansions in (A.1)–(A.3) do not contribute. Now, the difference between the z-exchange terms (H0 ) of the (S,S) and (S,S 1) levels is dS. On the other hand, the linear terms in (A.1) and (A.3) are found to be ðmi þSÞ ðmi ðS þ 1ÞÞd=2 and 0, respectively; at the CQP the first previous term has mi ¼ S such that it is finally equal to dS, which exactly cancels out the z-exchange difference between the levels involved in the CQP as we anticipated.
5. Discussion In this paper we have assessed one of the actual problems in the area of molecular magnets, namely, we have calculated the quenching points of the energy-splitting of a dimeric SMM as a function of the exchange interaction and implemented simple techniques that can be used to extract such a quenching with good approximation and for large molecular Hilbert spaces. The importance of the quenching points studied here lies in the possible realization of a molecular-based hardware that implements the advantages of such quantum devices in information processing, besides, fundamental reasons as trying to discover new phenomena that this nano-systems can manifest. In this way, understanding the role of quantum selection rules in molecular magnets behavior [51] is an important step that has to be further studied, and the discussions about DQP that we developed here go along this direction. Differences between QP of a strongly exchange coupled dimer and the ones of a giant-spin modeled dimeric SMM are more evident for moderated spin values and high magnetic fields. Such characteristics are actually more relevant when we take into account that the magnitude of the energy-splitting is often larger in higher magnetic fields making that field region more propitious for reliable detection of the energy-splitting. The aforementioned difference between the two sets of quenching-points in Eq. (12) is in agreement with the fact that SK-phase influence should be similar to the difference between the principal and the Weyl symbols of a quantum spin Hamiltonian [52]. The constant value of the highest QP for all values of anisotropy and exchange coupling is explained in terms of the symmetry of the monomeric sub-spaces and specific values of m1 and m2 involved in the calculation. Such a CQP can represent a useful tool in future manipulations of the energy-splitting as computational mechanics due to its behavior against magnetic field and exchange interaction; this means we can modulate the position of the DQP in order to tailor computational processes, and at the same time keep the information (maybe partially) carried up to the initial time when (J) was modified. CQP can be used somehow as a setup-point for the energy-splitting of a J-modifiable quantum device SMM-based. The quenching values in Eq. (16) describe satisfactorily the behavior of our simulations. Due to the generality of their behavior with respect to l as well as the importance of small d values to have a rich physics in a dimeric energy-splitting, these DQP would apply widely in the context of dimeric SMM-based molecular electronic. In case of strong coupled dimers, the giantspin-like behavior can be reached at moderate couplings and in that effective case Eq. (16) seems to be a good approach for low magnetic field values. Besides, the expansion in d can be improved just by using the terms all in (A.1)–(A.3) or by extending the transformation up to fourth order. We believe that this work can be widely used in studying new single-molecular magnets that are conformed by effective spin-monomers and in which an isotropic exchange coupling
3629
dominates the spin–spin interaction. Moreover, recent studies have proposed the modification of quantum properties in a molecular system through experimental procedures which seem to be feasible from different points of view, namely, chemical substitution in an effective three-spin molecule [5], magnetic dilution of a molecular crystal [46] and application of uniaxial stress along molecular axes [11] (and references therein). The study that we present here can be very useful for future molecular realization where the exchange parameter d or anisotropies l would be easily modifiable in order to modulate the energysplitting behavior and therefore the quantum reversal process of the molecular spins. In the case of SMMs the reversion is dominated by quantum tunneling processes at low temperature, and from L–Z theory it is easy to see that the quenching points of the energy-splitting are the points of zero reversal probability through magnetic tunneling. Usually, in SMMs the relaxation times are in years [7] which means that operational quantum processes would last an attainable time for experimental developments [1–5], opening the door to the study presented here.
Acknowledgments We acknowledge the financial support from the Conicyt scholarship to graduate students and from PIIC 2009 of the DGIP Universidad Te´cnica Federico Santa Marı´a, and the funds from Nu´cleo Cientı´fico Milenio Magnetismo Ba´sico y Aplicado P06022F. A.S.N. work was partially funded by Proyecto de Iniciacio´n en investigacio´n Fondecyt 11070008 and Proyecto Bicentenario de Ciencia y Tecnologı´a ACT027, P.V. acknowledges Fondecyt Grant 1100508.
Appendix A. Van Vleck transformation In this appendix we give the formulas obtained after the procedure described in Section 4. The expressions are presented up to third order which is enough for the calculation in Section 4. In the case of a degenerated block and if the expansion of the Hamiltonian is represented as H ¼ H0 þ eHu þ e2 H00 þ , the components of this degenerated block can be calculated as follows: Gmmu ðeÞ ¼ E0 ðmÞdmmu þ eHummu þ
l e3 X
2
l e2 X
2
ð1Þ ½U ð1Þ ðm,mu,nÞ þPU ðm,mu,nÞ
n
V ð1Þ ðn,m,muÞ
ðA:1Þ
n
with l X ð3Þ ð2Þ ð2Þ ½U ð3Þ ðm,mu,n,m00 Þ þ PU ðm,mu,n,m00 Þ ½U ðm,mu,nÞ þPU ðm,mu,nÞ
V ð1Þ ðn,m,muÞ ¼
m00
2
l X
U ð4Þ ðm,mu,n,nuÞ
nu
where the matrix elements involved in the summations above are given by U ð1Þ ðm,mu,nÞ ¼
Humn Hunmu E0 ðmÞE0 ðnÞ
U ð2Þ ðm,mu,nÞ ¼
Humn H00 nmu E0 ðmÞE0 ðnÞ
U ð3Þ ðm,mu,n,m00 Þ
Humm00 Hum00 n Hunmu ðE0 ðm00 ÞE0 ðnÞÞðE0 ðmuÞE0 ðnÞÞ
3630
J.M. Florez et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 3623–3630
U ð4Þ ðm,mu,n,nuÞ ¼
Humn Hunnu Hunumu ðE0 ðmÞE0 ðnÞÞðE0 ðmuÞE0 ðnuÞÞ
ðA:2Þ
and where P is an operator which does the following changes in the matrix elements: 8 First m2mu > > > < Second Himn 2Hinm PU ¼ > whereHi ¼ Hu,H00 > > : Third the matrix element is evaluated The notation presented here is general and in our case studied in Section 4 we have used m ¼ ðm1 ,m2 Þ, as the quantum numbers corresponding to the eigenstates jmS ¼ jm1 ,m2 S that diagonalize H0 with energies E0 ðmÞ ¼ E0 ðm1 ,m2 Þ. Now, in the case of nodegenerated block we give the result by considering H ¼ H0 þ eHu þ , for the sake of simplicity: Gmm ðeÞ ¼ E0 ðmÞ þ eHumm þ e2
l X
3 U ð1Þ ðm,m,nÞ e
n
l X
V ð2Þ ðn,mÞ ,
n
with V ð2Þ ðn,mÞ ¼
l X m00
dmm00 U ð3Þ ðm,m,n,m00 Þ
l X
U ð4Þ ðm,m,n,nuÞ
ðA:3Þ
nu
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