Theory of photoinduced phase transitions in itinerant electron systems

Physics Reports 465 (2008) 1–60

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Theory of photoinduced phase transitions in itinerant electron systems Kenji Yonemitsu a,∗ , Keiichiro Nasu b a

Institute for Molecular Science, Graduate University for Advanced Studies, Okazaki, Aichi 444-8585, Japan

b

Solid State Theory Division, Institute of Materials Structure Science, KEK, Graduate University for Advanced Studies, CREST JST, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

article

info

Article history: Accepted 28 April 2008 Available online 13 May 2008 editor: D.K. Campbell PACS: 78.20.Bh 71.10.Fd 71.10.Hf 71.30.+h 71.38.-k 71.45.-d 63.20.Kr 78.47.+p Keywords: Photoinduced phase transition

a b s t r a c t Theoretical progress in the research of photoinduced phase transitions is reviewed with closely related experiments. After a brief introduction of stochastic evolution in statistical systems and domino effects in localized electron systems, we treat photoinduced dynamics in itinerant-electron systems. Relevant interactions are required in the models to describe the fast and ultrafast charge-lattice-coupled dynamics after photoexcitations. First, we discuss neutral-ionic transitions in the mixed-stack charge-transfer complex, TTF-CA. When induced by intrachain charge-transfer photoexcitations, the dynamics of the ionicto-neutral transition are characterized by a threshold behavior, while those of the neutralto-ionic transition by an almost linear behavior. The difference originates from the different electron correlations in the neutral and ionic phases. Second, we deal with halogenbridged metal complexes, which show metal, Mott insulator, charge-density-wave, and charge–polarization phases. The latter two phases have different broken symmetries. The charge-density-wave to charge–polarization transition is much more easily achieved than the reverse transition. This is clarified by considering microscopic charge-transfer processes. The transition from the charge-density-wave to Mott insulator phases and that from the Mott insulator to metal phases proceed much faster than those between the lowsymmetry phases. Next, we discuss ultrafast, inverse spin-Peierls transitions in an organic radical crystal and alkali-TCNQ from the viewpoint of intradimer and interdimer chargetransfer excitations. Then, we study photogenerated electrons in the quantum paraelectric perovskite, SrTiO3 , which are assumed to couple differently with soft-anharmonic phonons and breathing-type high-energy phonons. The different electron–phonon couplings result in two types of polarons, a ‘‘super-paraelectric large polaron’’ with a quasi-global parity violation, and an ‘‘off-center-type self-trapped polaron’’ with only a local parity violation. The former is equivalent to a charged and conductive ferroelectric domain, which greatly enhances both the quasi-static electric susceptibility and the electric conductivity. Finally, we outline the development of time-resolved X-ray diffraction experiments, which directly accesses the dynamics of electronic, atomic and molecular motions in photoexcited materials. They are extremely useful when a three-dimensional structural long-range order is established and changes the symmetry. © 2008 Elsevier B.V. All rights reserved.

Contents 1. 2.

∗

Introduction............................................................................................................................................................................................. Stochastic theories and localized electron systems.............................................................................................................................. 2.1. Polydiacetylenes .........................................................................................................................................................................

Corresponding author. E-mail addresses: [email protected] (K. Yonemitsu), [email protected] (K. Nasu).

0370-1573/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2008.04.008

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K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

2.2. 2.3.

3.

4.

5.

6.

Linear chain systems .................................................................................................................................................................. Extension to general dimensions............................................................................................................................................... 2.3.1. Relation to the Ising model ......................................................................................................................................... 2.3.2. Stability in the mean-field approximation ................................................................................................................ 2.3.3. Self-consistency condition and transition probabilities ........................................................................................... 2.3.4. Rate equation and stochastic dynamics ..................................................................................................................... 2.3.5. Related approaches...................................................................................................................................................... 2.4. Domino effect.............................................................................................................................................................................. Photoinduced phase transitions in quasi-one-dimensional itinerant-electron systems ................................................................... 3.1. Mixed-stack charge-transfer complex, TTF-CA......................................................................................................................... 3.1.1. Initial-condition sensitivity......................................................................................................................................... 3.1.2. Adiabatic relaxation path ............................................................................................................................................ 3.1.3. Coherent motion of a macroscopic domain ............................................................................................................... 3.1.4. Transition dynamics due to electron–lattice interactions ........................................................................................ 3.1.5. Photoinduced phase transition dynamics in TTF-CA................................................................................................. 3.1.6. Difference between I-to-N and N-to-I transitions ..................................................................................................... 3.1.7. Electronic-state-dependent transition dynamics...................................................................................................... 3.1.8. Interchain coupling effects on the phase diagram .................................................................................................... 3.1.9. Interchain coupling effects on transition dynamics .................................................................................................. 3.2. Halogen-bridged metal-chain compounds ............................................................................................................................... 3.2.1. Insulator-to-metal phase transition in one-band models......................................................................................... 3.2.2. Insulator-to-metal phase transition in MX chains .................................................................................................... 3.2.3. CDW-to-Mott phase transition ................................................................................................................................... 3.3. Halogen-bridged binuclear metal complexes ........................................................................................................................... 3.3.1. CDW-CP phase transition ............................................................................................................................................ 3.4. Spin chains coupled with phonons............................................................................................................................................ 3.4.1. Organic radical crystal, TTTA....................................................................................................................................... 3.4.2. Organic charge-transfer compound, alkali-TCNQ ..................................................................................................... 3.4.3. Ultrafast photoinduced inverse spin-Peierls transition ............................................................................................ 3.4.4. Adiabatic picture for melting of molecular dimerization ......................................................................................... Photoinduced macroscopic parity violation and ferroelectricity ........................................................................................................ 4.1. Large polaron, self-trapped polaron, linear and quadratic couplings ..................................................................................... 4.2. Photoinduced phenomena in SrTiO3 ......................................................................................................................................... 4.2.1. Quantum dielectric, soft-anharmonic T1u mode and quadratic coupling ................................................................ 4.2.2. Possible scenario.......................................................................................................................................................... 4.2.3. Model Hamiltonian...................................................................................................................................................... 4.2.4. Variational method for polaron .................................................................................................................................. 4.2.5. Continuum approximation and super-paraelectric large polaron ........................................................................... 4.2.6. Numerical results......................................................................................................................................................... 4.2.7. Bipolaron ...................................................................................................................................................................... 4.2.8. Phonon softening and dielectric enhancement ......................................................................................................... 4.2.9. Polaron transport ......................................................................................................................................................... 4.2.10. Concluding remarks..................................................................................................................................................... Probing by time-resolved X-ray diffraction .......................................................................................................................................... 5.1. Structural changes seen by X-ray scattering............................................................................................................................. 5.2. Fast time-resolved X-ray diffraction for molecular materials ................................................................................................. 5.2.1. Neutral-ionic transition in TTF-CA ............................................................................................................................. 5.3. Ultrafast time-resolved X-ray diffraction for hard materials................................................................................................... 5.3.1. Nonthermal surface melting in semiconductors ....................................................................................................... 5.3.2. Insulator-to-metal transition in vanadium oxide...................................................................................................... 5.3.3. Coherent optical phonons in bismuth ........................................................................................................................ Summary ................................................................................................................................................................................................. 6.1. Differences between one- and two-dimensional systems ....................................................................................................... 6.2. Melting of charge order in quasi-two-dimensional organic conductors ................................................................................ 6.3. Insulator-to-metal transitions in manganese oxides ............................................................................................................... Acknowledgments .................................................................................................................................................................................. References................................................................................................................................................................................................

5 7 7 7 8 9 9 9 11 11 12 12 13 15 16 16 17 21 23 25 25 28 31 32 34 36 36 38 39 39 42 42 43 43 44 44 45 46 47 48 49 50 50 51 52 53 53 54 54 55 55 55 57 57 57 58 58

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1. Introduction A variety of electronic phases exist in condensed matters. Changing temperature, pressure, and other parameters in thermal equilibrium cause transitions between different phases. To experimentally realize and/or theoretically understand such phases, we need to consider appropriate interactions and dimensionality. For example, many condensed molecular materials are characterized by their low-dimensional electric conductivity originating from the anisotropy in molecular orbitals and molecular arrangements. Electronic low-dimensionality brings about various instabilities toward spontaneously broken symmetries and allows the appearance of many different phases. Small overlaps between neighboring molecular orbitals lead to the severe competition between the kinetic energy favoring the itinerant character and the interaction energy often favoring the localization of electrons. Now many organic conductors are regarded as strongly correlated electron systems [1–3]. In addition, relatively easy modulation of molecular arrangements by changing the environment implies strong electron–phonon interactions as well. Now we want to modify their electronic phases, not by changing a parameter characterizing thermal equilibrium, but dynamically or even transiently by an external field through nonequilibrium processes. Among them, photoirradiation is a very hopeful method. When excited by a photon, an electron in an insulating crystal generally induces a local lattice distortion around itself. The optical excitation changes charge distribution so that the lattice system reaches a new stable position in the excited state, i.e. lattice relaxation. In most of the cases, only a few molecules and electrons are involved. Then, this phenomenon is microscopic. In some cases, the excited state moves a large number of molecules and electrons collectively. It results in a macroscopic domain with new structural and electronic states quite different from the original states. This phenomenon is called a photoinduced (structural) phase transition and has attracted much attention [4]. Initially local structural deformations trigger a macroscopic change in transport, optical, dielectric, and/or magnetic properties. The transition dynamics are recently studied in different materials, revealing the origin of cooperativity responsible for the proliferation of photoinduced electronic states [5–7]. Photoinduced phase transitions are achieved on different energy/time scales in different materials. Here we focus on the cases where electronic properties such as electric conductivity, dielectric permittivity, and/or magnetic susceptibility are largely altered. In other words, we treat those phenomena which are described by itinerant-electron models. The excitation density of absorbed photons is generally much lower than the density of molecules. The photons can move a much larger number of electrons in a collective manner. The laws of thermodynamics are, of course, not violated. The energy provided by photoexcitation is roughly given by the magnitude of the optical gap, which is much larger than the energy scale of thermal fluctuations. Because of the high energies accessible by photoexcitations, the induced phase can be different from any phase realized in thermal equilibrium by changing temperature, pressure, or elements constituting the material. Even though its life time is finite, it is often much longer than the time scale of measurements. The observed photoinduced dynamics are very different from simple relaxation processes in most of the cases. Photoirradiation of insulators and semiconductors generally produce pairs of electrons and holes accompanied with structural deformations. The deformations can be small and wide forming large polarons, or large and local forming small polarons, which are often self-trapped. Electrostatic attraction between an electron and a hole can bind them to form a neutral exciton. Excitons can be formed by sharing the deformation by an electron and a hole. Such oppositely charged particles can be unbound by absorbing extra energy from the pump light. Thus, a variety of initial states are possible after the pump light is turned off. Near a thermal phase transition, the initially local structural deformations can grow into macroscopic ones, changing the electronic properties drastically. The growth process may be regarded as a domino effect. Note, however, that not all photoinduced phase transitions are accompanied by structural deformations. To develop the initially local structural deformation spatially, cooperativity is crucial. The cooperativity can be brought about by different types of interactions. At an early stage of theoretical studies, in order to clarify how local structural distortions lead to global ones, a theory based on stochastic processes was proposed. Employed models are statistical like Ising models, which qualitatively describe thermal phase transitions. Transition probabilities are so (but not uniquely) determined as to satisfy detailed balance. This guarantees that a thermally equilibrium phase is finally reached. However, short-time dynamics including charge-transfer processes are not described by such stochastic processes. Such processes are justified when the states of interest represented by Ising variables are weakly coupled with environments consisting of many degrees of freedom. Typical examples are spin-crossover complexes where spin states are coupled with lattice displacements including ligands, so that effective interactions among spin variables are weak and their transitions are slow. Transitions are largely influenced by thermal fluctuations of surroundings. The corresponding time evolutions are not really dynamics, but rather kinematics, so we do not deal with them in detail. In short, statistical models are useful to describe coarse-grained and thus long-time evolutions in which exchanges of energies are regarded as stochastic processes. Even if the above condition is satisfied, very short-time evolutions would not be described stochastically. Thanks to the recent development of time-resolved spectroscopy, time evolutions of different quantities are being directly observed. Typical time resolutions of pump-probe spectroscopy have changed rather rapidly from nanoseconds to picoseconds and now to about 100 femtoseconds, which are short enough to analyze photoinduced oscillations due to phonons. A single material can show different dynamics depending on the energy and strength of pump light, i.e. the initial state. The diversity of materials that show photoinduced phase transitions has dramatically increased. Many types

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of transitions are realized among various electronic phases. Now, photoinduced dynamics may be classified according to the nonlinearity, directional property, relaxation time or stability of the induced phase, etc. Some photoreflectance data are almost linear, and others show a threshold behavior, as a function of the absorbed photon density. Some transitions are photoinduced in one direction, and others in both directions. In the latter case, the efficiency and photoinduced dynamics may strongly depend on the direction. Some photoinduced phases quickly decay with a very short relaxation time, and others are long-lived. In order to understand these dynamics, relevant models are required to describe the electron or electron–phonon systems properly. In this context, itinerant-electron models are often used to approach the evolution of physical properties in a direct manner. Then, it becomes possible to discuss how the interactions and dimensionality in each material are responsible for the characteristics of its photoinduced dynamics. From the viewpoint of the variety in photoinduced dynamics, we somehow need to get down into specifics. There is, however, some systematics concerning the relations among such characteristics as nonlinearity, directional property, and stability of photoinduced phases. In this sense, some universality may exist, although the relations among them are not so strict. In any case, we need to accumulate both experimental and theoretical data because the phenomena are essentially nonlinear and the systems are strongly correlated in terms of electrons and phonons. Furthermore, some specific example may be developed from the viewpoint of potential application to devices or potential control of desired properties. Note that, compared with the variety in thermal phase transitions, that in photoinduced phase transitions is still much less. In this article, we review the progress in photoinduced phase transitions mainly from the theoretical aspect, focusing on itinerant electron systems. It would be useful in future in designing functional materials for optical switching or optical control of electronic properties in nonequilibrium environments. 2. Stochastic theories and localized electron systems In most of the cases, electrons/holes or excitons introduced by photoirradiation are accompanied by local structural deformation. To develop it spatially, some cooperativity is crucial. One needs to know first what kind of interaction possesses such cooperativity. In order to clarify how the local structural distortions lead to global ones, a theory based on stochastic processes was proposed at first. Here we briefly review researches at an early stage. 2.1. Polydiacetylenes Historically, a first theory for photoinduced structural transformations is applied to polydiacetylenes. There are different kinds of polydiacetylene crystals according to their side chains. Generally, they are classified into A and B forms, which correspond to the low-temperature and high-temperature phases, respectively. The transition between them is of firstorder and accompanied with hysteresis [8]. From the absorption energy in the A form, that in the B form, and the Stokes shift in the emission of light in the B form, the parameters EFC , S, and K in the model introduced later are evaluated [9]. The coupling K turns out to be strong, which is consistent with the strong nonlinearity observed in experiments. Reversible changes between the A and B phases are shown to be induced by photoexcitation in single crystals of polydiacetylenes with alkyl-urethanes, (CH2 )4 OCONH(CH2 )n−1 CH3 , as the side-group (abbreviated as poly-4Un) [10]. As is common in many other polydiacetylenes, two spectroscopically distinct A and B phases exist in the poly-4Un series. At various steps of a thermal cycle shown in Fig. 1, the reflectance and Raman spectra are measured. The free energy for this crystal is represented by a curve with double local minima corresponding to the A and B phases, which are separated by an energy barrier. At point 2 in the warming cycle, the sample was irradiated with a single shot of 2.81 eV from a pulsed dye laser with 20-ns width. The optical absorption and Raman spectra after the photoexcitation clearly show almost full conversion of the A phase into the B phase. At point 4 in the cooling cycle, the sample was excited by a single pulse of 3.18 eV. The spectra after the excitation show that about 50% of the B phase is converted into the A phase at the excitation photon density of 7 × 1018 cm−3 , which corresponds to the absorption of one photon in a fraction of the polymer crystal composed of about 140 repeat units. The converted fractions are estimated from the photoinduced changes in the integrated intensity of Raman peaks due to the C=C stretching mode. The photoinduced changes are nonlinear in the excitation intensity, as shown in Fig. 2, with a threshold at about (2–3) × 1018 cm−3 . It suggests that some cooperative interaction is necessary for local photoexcited species to evolve into the macroscopic phase conversion, as discussed theoretically [11,9]. The photon energy of the exciting laser pulse needs to exceed a certain value: about 2.4 eV for the A-to-B and about 2.7 eV for the B-to-A transitions. These values are located 0.3–0.5 eV higher than the exciton absorption peak for both the A-to-B and B-to-A phase changes. Therefore, the conversion efficiencies are very low near the exciton absorption peak where the excitation light is strongly absorbed. The exciting photon energy dependence of the converted fraction is similar to the photocurrent action spectra. This similarity indicates that the photogenerated charge carriers, not excitons, are necessary for the photoinduced phase transitions to take place. It is speculated that photogenerated polaronic species play an important role in the transient formation of domain walls separating the A and B phase regions in the polymer backbones. Dynamical processes of the photoinduced phase transition in polydiacetylene crystals are investigated by time-resolved spectroscopy [12]. It is found that the primary process of the transition is mostly completed within 50 ns.

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Fig. 1. Temperature dependence of reflectivity in a poly-4U3 crystal at 1.95 eV, with schematic diagrams of free energy with minima corresponding to A and B phases. At points 2 and 4, the crystal surface was irradiated with a single shot of a pulsed laser as denoted by dashed lines, and the photoinduced effect were measured. Reprinted with permission from [10]. Copyright APS.

Fig. 2. Absorbed photon density dependence of converted fraction for A-to-B transition induced by 2.81-eV light at 390 K and for B-to-A transition induced by 3.18-eV light at 370 K. Reprinted with permission from [10]. Copyright APS.

2.2. Linear chain systems The relaxation of displacements to thermal equilibrium ceases on a time scale of the inverse Debye frequency. This relaxation process was ignored as a first step, so that the displacements are assumed to be distributed always around the bottom of a diabatic potential. Hanamura and Nagaosa introduced a simple model [11]: H =

X

√ |el i(EFC −

SQl )hel | +

l

1X 2

l

Ql2 −

1X 2

Kll0 Ql Ql0 ,

(1)

ll0

and

|gl ihgl | + |el ihel | = 1,

(2)

where EFC is the Franck–Condon excitation energy, Ql the relevant displacement that dominantly interacts with an electron √ in the photoexcited |el i state at the lth molecule and is treated as a classical variable, and SQl the Stokes shift in this state. The coupling strength between the displacements of the lth and l0 th molecules is denoted by Kll0 with its diagonal element Kll set at zero.

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K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

Fig. 3. Energy diagrams of 0th molecule, which is closest to cluster of m excited molecules, in ground and excited states [11,9].

They have investigated linear chain systems. Suppose that electrons on m consecutive molecules (l ∈ E with E = {1, 2, . . . , m}) are in the excited state, and others are in the ground state. In the case of nearest-neighbor couplings, Kl,l0 = K (δl+1,l0 + δl,l0 +1 ), the energy of the local minimum can be written as mEFC − SEst (m, {Kl,l0 }). The positive quantity Est (m, {Kl,l0 }) is shown to be proportional to m for small K , and to m2 for K close to 1/4. Therefore, the cluster of m excited molecules is stable for large S and K close to 1/4 satisfying the inequality, mEFC − SEst (m, {Kl,l0 }) < 0. (m)

When each molecule is located on the respective equilibrium position, the state is described by the Hamiltonian Hg for a cluster of m neighboring excited molecules (l = 1, 2, . . . , m). Suppose that an electron on one of the neighboring molecules (l = 0) is photoexcited. The corresponding molecule is now away from the equilibrium position. This state is governed by (m+1) the Hamiltonian He defined as He(m+1) = Hg(m) + EFC −

√

SQ0 .

(3)

Introducing an interaction mode q0 , which is a linear combination of fluctuating components of Ql around the equilibrium (m) (m) positions, we decompose the Hamiltonian Hg into the contribution from the interaction mode q0 and the rest Eg as Hg(m) =

1 2

q20 + Eg(m) .

(4)

The Hamiltonian for the system with a photoexcited molecule can then be rewritten as

√ (m+1)

He

=

1 2

q20

(m)

+ Eg

S

+ EFC −

ω¯

√ q0 −

(m)

S ∆0 ,

where ω ¯ is an average of the renormalized frequency ωk , with ωk2 ≡ 1 − 2 (m)

(m+1)

(5)

P

(m)

j6=0

Kj0 eikj , and ∆0

is the stationary component

of Q0 . The diabatic potentials for Hg and He are displaced parabolas, as shown in Fig. 3, where the energy and q0 axes are shifted for later purposes. In general, for displaced parabolas, their relative stability is determined by the minima of the two parabolas Eg ≡ (m)

(m+1)

(q0 = qe ), where qg and qe give the respective minimum points, and by EgV ≡ Hg(m) (q0 = qe ) (m+1) and EeV ≡ He (q0 = qg ). If the inequality Ee > EgV (Eg > EeV ) is satisfied, the excited (ground) state is unstable and would Hg (q0 = qg ) and Ee ≡ He

decay into the ground (‘‘excited’’) state. Otherwise, the smaller one of Eg and Ee corresponds to the stable state, and the larger one to the metastable state. √ In the present model, the minimum points are qg = 0 and qe = S /ω ¯ . The vertical energy differences are EeV − Eg = EFC −

√

(m)

S ∆0 ,

(6)

and EgV − Ee = S /ω ¯2 +

√

(m)

S ∆0

− EFC ,

(7)

while the equilibrium energy difference is Ee − Eg = EFC − S /(2ω ¯ 2) −

√

(m)

S ∆0 .

(8)

These equations are on which√we can discuss how a√ cluster of photoexcited molecules is stabilized. The critical √ the(mbases ) values of EFC are S ∆0 , S /(2ω ¯ 2 ) + S ∆(0m) , and S /ω¯ 2 + S ∆(0m) . The key parameters here are the electron–lattice interaction, whose dimensionless strength is the lattice relaxation energy S divided by the Franck–Condon excitation energy EFC , and by the coupling K among the displacements. The model is used to describe mechanisms of photopolymerization of diacetylene and diolefin crystals, photoisomerization of polydiacetylene crystals, photochromism and photo-chemical hole burning [11]. A similar analysis also explains how a molecular excitation far from the cluster is attracted to the cluster.

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2.3. Extension to general dimensions 2.3.1. Relation to the Ising model Photoinduced structural deformations are not one-dimensional in real materials. Even if electronic conduction occurs in some restricted dimensions, structural distortions take place in three dimensions. Then, Nagaosa and Ogawa have extended the above theory to general dimensions and considered various transition processes [9]. When the relaxation of the displacement Ql is rapid, the system is almost always in equilibrium with a given electronic state. These displacements can be integrated out to reduce the original model (1) to the Ising model, where the spin-up (σl = 1/2) and spin-down (σl = −1/2) states correspond to the excited and ground states: H = −h

X

1X

σl −

2

l

Jll0 σl σl0 ,

(9)

l6=l0

where the field h and the exchange Jll0 are given by h = −EFC +

S 2(1 − K )

,

(10)

with the coupling strength now defined by K = S X cos[k · (Rl − Rl0 )]

Jll0 =

N

ωk2

k

P

Kll0 , and

l0

,

(11)

with N being the total number of sites, and the phonon dispersion defined by ωk2 = 1 − l0 Kll0 exp[ik · (Rl − Rl0 )]. Here Rl denotes the lattice vector of the lth molecule, and ωk2=0 = 1 − K . P At the mean-field level, which is equivalent to the replacement of Jll0 by the average (1/N ) l0 6=l Jll0 = J /N with J ≡ SK /(1 − K ), the transition temperature Tc in the absence of the field h is given by kB Tc = J /4. At zero temperature, the total energy has double minima for |h| < J /2, where the uniform state is stable against the single spin flip. The inequality |h| < J /2 is rewritten in terms of the original model parameters as

P

0<

1 − K EFC − S /2 K

S

< 1.

(12)

2.3.2. Stability in the mean-field approximation Returning to the original model (1), we will discuss the diabatic potentials again. Instead of introducing an interaction mode, we apply the mean-field approximation [9]. The displacements surrounding Ql are replaced by a mean field,

∆l =

X

Kll0 Ql0 .

(13)

l0 6=l

Then, the diabatic potential for Ql in the |gl i state is given by HgMF =

1 2

1

(Ql − ∆l )2 − ∆2l ,

(14)

2

and that in the |el i state by HeMF =

1 2

(Ql − ∆l −

√

S )2 + EFC −

1 2

(∆l +

√

S )2 .

(15)

They are displaced parabolas, as shown in Fig. 3 with q0 replaced by Ql . The minimum points are Qg = ∆l and Qe = ∆l + The vertical energy differences are EeV − Eg = EFC −

√ S.

√

S ∆l ,

(16)

and EgV − Ee = S +

√

S ∆l − EFC ,

(17)

while the equilibrium energy difference is Ee − Eg = EFC − S /2 −

√

S ∆l .

(18)

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K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

Fig. 4. Four types of relative locations of two parabolas. Arrows in (a) and (d) indicate spontaneous emissions. (b) and (c) are optically stable for T  S. Reprinted with permission from [9]. Copyright APS.

√

√

For the √ purpose of stability analysis, these equations are sufficient. The critical values of EFC are S ∆, S /2 + S ∆, and S + S ∆, for uniform ∆l = ∆ (Fig. 4). √ In order to discuss thermal transition probabilities, we need to know the crossing point of the two parabolas, Qc = EFC / S, and the activation energies,

√ 2



Ec − Eg = ∆l − EFC / S

/2,

(19)

 √ √ 2 Ec − Ee = ∆l + S − EFC / S /2.

(20)

2.3.3. Self-consistency condition and transition probabilities We consider the self-consistency condition for the displacement. Replacing the influence Pfrom the surrounding sites by 0 the average, ∆l takes a uniform value ∆ = K hQ i with the coupling strength defined by K = l0 Kll and the average value of Ql , hQ i. For the total number of |el i sites, Ne , among N sites, the density of excited sites is given by ne = Ne /N, determining the average value as

hQ i = ne Qe + (1 − ne )Qg √ = ne (∆ + S ) + (1 − ne )∆ √ √ = ∆ + Sne = K hQ i + Sne .

(21)

Solving Eq. (21) for Ql , we obtain

√ ∆ = K hQ i =

S

K

1−K

ne ,

(22)

which shows that the state is specified only by ne in the mean-field approximation. With the help of the above equation, the activation energies are rewritten as Ec − Eg =

Ec − Ee =

S



2

2

1−K

2 S

K



K

2

1−K

(ne − nU )2 ,

(23)

(ne − nL )2 ,

(24)

where the critical concentration nU and nL are defined by the relations Ec − Eg = 0 and Ec − Ee = 0, respectively, (i.e., by

√

√

the relations EFC = S ∆ and EFC = S + S ∆, respectively), as nU = (1 − K )/K · EFC /S, and nL = (1 − K )/K · (EFC − S )/S. The thermal transition probabilities are assumed to be the product of the attempt frequency and the activation factor,

i h α = τ −1 exp − (ne − nL )2 ,

(25)

  h α i Ec − Eg exp − = τ −1 exp − (ne − nU )2 ,

(26)



Ec − Ee Peth→g = τ −1 exp − kB T Pgth→e

=τ

−1

kB T



2

2

with the coefficient α defined as

α=

S kB T



K 1−K

2

.

(27)

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

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Fig. 5. Critical duration time tcr as a function of I0 . Reprinted with permission from [9]. Copyright APS.

2.3.4. Rate equation and stochastic dynamics The time evolution is described by a rate equation for the density of the excited sites ne (t ), which is a function of time t. Although more general cases √ are originally discussed√ [9], we will treat the case where stable and metastable states are present, nL < ne (t ) < nU (i.e., S ∆(t ) < EFC < S + S ∆(t )), at low temperatures to ignore the optical transitions for simplicity. The rate equation is written as d

ne (t ) = I (t ) + Pgth→e [1 − ne (t )] − Peth→g ne (t ), (28) dt where I (t ) is the optical pumping term, being positive to create an |ei site or negative to create a |g i state. The thermal transition probabilities are already given by Eqs. (25) and (26), whose exponents are quadratic functions of ne (t ) in the mean-field approximation. Let us rewrite the right hand side of Eq. (28) as I (t ) + u(ne ) = I (t ) − dU (ne )/dne . The function U (ne ) is regarded as a potential because the system relaxes to a minimum of U (ne ) without the optical pumping. Generally, U (ne ) has two minima separated by a barrier. The two minima are at ne = 0 and ne = 1 at low temperatures. The location of the maximum is denoted by ncr , which is the solution to u(ne ) = 0 lying between nL and nU . It is straightforward to derive 1 − K EFC − S /2 (nL + nU ) = , (29) 2 K S at low temperatures. The condition for the total energy having double minima (12) is then equivalent to the inequality 0 < ncr < 1. Now, we consider the stochastic dynamics under and after the optical pumping I (t ) = I0 θ (t0 − t ) with I0 > 0 from the initial value ni = 0 at t = 0. The rate equation (28) can be integrated to show the following [9]. When I0 is less than the threshold value Ith = nL /τ , ne (t ) does not exceed a certain value however long the optical pumping is applied. After the pumping is switched off at t = t0 , ne (t ) relaxes to the initial value ni = 0. When I0 is greater than Ith , on the other hand, ne (t ) monotonically increases without saturation for 0 < t < t0 . After the pumping is switched off, ne (t ) relaxes to the final value nf = 0 for ne (t0 ) < ncr and to nf = 1 for ne (t0 ) > ncr . The critical value of the duration time tcr is given by ne (t0 = tcr ) = ncr . Thus, for t0 > tcr , the system is switched from ni = 0 to nf = 1 (Fig. 5). As a function of I0 , tcr is inversely proportional to I0 for large I0 . For I0 just above the threshold Ith , tcr is proportional to (I0 − Ith )−1/2 . ncr =

1

2.3.5. Related approaches Such statistic theories based on diabatic potentials as described above are generally applied to systems with weakly interacting components to which thermal fluctuations are so dominant that transition probabilities are crucial. Spincrossover complexes are also such systems that spins indirectly interact with each other through couplings with lattice displacements. An extension to systems with internal degrees of freedom, e.g., spin-crossover complexes that are composed of dimers and show two-step transitions [13–17], can be made in a straightforward manner [18]. More generally, stochastic dynamics can be treated after mapping into classical models by solving their master equations analytically [19–24] or numerically through Monte Carlo simulations [19,22,25,26,23,28,27,24], as applied to spin-crossover complexes. They are composed of localized electrons coupled with lattice displacements. 2.4. Domino effect Photoinduced dynamics are sometimes regarded as domino effects. This description gives a hint for the mechanisms of photoinduced phase transitions. Koshino and Ogawa theoretically show a domino-like structural transformation in a onedimensional model composed of localized electrons and classical lattice displacements [29,30]: H =

" X  − 2γ uj + u2 j j

t0



t0 2uj +

u2j

 +

∂ uj ∂t

2 # +

X (i,j)

kij (ui − uj )2 ,

(30)

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Fig. 6. Adiabatic potentials ± (uj ). The upward (downward) arrow represents the electronic transition accompanied with absorption (emission) of a photon. From [29]. Reprinted with permission from JPS.

where uj and ∂ uj /∂ t denote the jth displacement and velocity, respectively, kij the elastic constants,  the energy difference between the two electronic states at uj = 0, t0 the overlap integral between them, and γ the electron–lattice coupling constant. The adiabatic approximation is valid if the overlap integral t0 is much larger than S −1/2 with the so-called Huang-Rhys factor S, which is given by the ratio of the (single-site) stabilization energy to the (single-site) phonon energy. Consider the bistable case, where the structures A and B are metastable and stable, respectively ( < 0). In the absence of intersite interactions kij , the adiabatic potentials ± (uj ) are shown in Fig. 6, for the ground state |−(uj )ij and the excited state |+(uj )ij . The equation of motion for the jth displacement is given by

∂ 2 uj ∂ ∂ uj =− ± (uj ) − Γ , 2 ∂t ∂ uj ∂t

(31)

where Γ is the dimensionless friction constant, and the lattice temperature is assumed to be absolute zero. The electronic transition between the ground and excited states is accompanied with absorption or emission of a photon, represented by the vertical arrows in Fig. 6. We ignore thermal or quantum processes and treat optical processes only. The intersite interaction is assumed to be parameterized by the strength k and the force range µ as kij =

k 2



1 − e−1/µ exp −




 |i − j| − 1 . µ

(32)

It approaches kij = (k/2)δ|i−j|,1 in the µ → 0 limit, while kij = k/N in the µ → ∞ limit with N being the total number of sites. After one-site excitation at the 0th site, the system first relaxes in the excited-state adiabatic potential, (1)

E+ ({uj }) =

X

− (uj ) + + (u0 ) +

X

kij (ui − uj )2 ,

(33)

(i,j)

j6=0

(1)

(1)

to the minimum of E+ ({¯uj }), according to Eq. (31) with ± (uj ) replaced by E+ ({uj }). After the spontaneous emission of a photon, the system relaxes from {¯uj } in the ground-state adiabatic potential, E− ({uj }) =

X j

− (uj ) +

X

kij (ui − uj )2 ,

(34)

(i,j)

according to Eq. (31) with ± (uj ) replaced by E− ({uj }). When the system starts from the A structure, three qualitatively different evolution patterns are shown to exist. (a) When the coupling strength k is too small, the local structural distortion remains locally. (b) Only when k is intermediate and the force range µ is short, the initial local distortion triggers a global structural transformation. (c) When k is too large, the initial local distortion is pulled back by surrounding lattice displacements to the position before the absorption. When the system starts from the B structure, every site always goes back to the B structure. In the case (b), the B structure is extended step by step through the domino effect. This is more effectively induced in the case that the local distortion around the excited site exerts strong influence on the neighboring sites (for small µ) than in the case that it exerts weak influence on many remote sites (for large µ). Once the domino dynamics proceeds, the boundaries between the A and B structures move at a constant velocity, which is estimated in the small-Γ and large-Γ cases [29,30]. These domino processes in the adiabatic regime are deterministic. The diabatic approximation is valid if the overlap integral t0 is much smaller than S −1/2 with the Huang-Rhys factor S. In this case also (we consider the bistable case again), three qualitatively different evolution patterns are shown to exist [31]. (a) When the coupling strength k is small, every site is in the lower electronic state. It is stable against the radiative decay, so that the local distortion does neither grow nor disappear. (b) When k is intermediate and the force range µ is short, the

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11

Fig. 7. Schematic electronic and lattice structures of TTF-CA.

excited site’s neighbors are in the higher electronic state. It is unstable and radiatively decays emitting a photon. As the spontaneous emission is a stochastic process, we do not know which site emits a photon earlier. After the emission and relaxation, another site becomes unstable against a radiative decay in the newly relaxed configuration. After all, the new structure is extended through such sequence of radiative decays. Therefore, the domino processes in the diabatic regime are stochastic and take a much longer time than the nonradiative structural deformation. (c) When k is large, the excited site is in the higher electronic state. It is unstable and radiatively decays emitting a photon. After the spontaneous emission, the system relaxes to the initial structure. It is noted that the crossover regime between the adiabatic and diabatic limits, where the overlap integral t0 is comparable to S −1/2 , is also discussed [32]. Quite recently, a two-dimensional model consisting of quantum phonons coupled with localized electrons is numerically studied [33]. The photoinduced nucleation process is triggered only when a certain amount of excitation energy is supplied in a narrow part of the system. This means the existence of a smallest cluster of excited molecules that realizes the nucleation and subsequent growth of a domain. As a consequence, the fraction of cooperatively converted molecules nonlinearly depends on the photoexcitation strength. 3. Photoinduced phase transitions in quasi-one-dimensional itinerant-electron systems Photoinduced phase transitions are often related to multistability, where different electronic phases are stable and metastable. Such stability may be manifested by the presence of a first-order phase transition induced by changing temperature or pressure. The photoinduced phase transition dynamics depend on the electronic state and especially on how the initial state is prepared. For this purpose, relevant itinerant-electron models are necessary. They have finite transfer integrals between neighboring molecular orbitals and are essential in theoretical studies on the physical properties of molecular materials [1]. They are off-diagonal elements of the model Hamiltonian, giving transition amplitudes. In contrast, the stochastic dynamics within classical statistical models are governed by transition probabilities, which are often given by Boltzmann factors satisfying detailed balance. 3.1. Mixed-stack charge-transfer complex, TTF-CA Organic molecular tetrathiafulvalene-p-chloranil (TTF-CA) crystals are the most intensively studied. The TTF donor and CA acceptor molecules are alternately stacked along the most conducting axis, as shown in Fig. 7 in which the highest occupied molecular orbital (HOMO) of the donor and the lowest unoccupied molecular orbital (LUMO) of the acceptor are taken into account [34]. At room temperature, TTF-CA is in the neutral state. By lowering temperature, it undergoes a neutral-to-ionic phase transition at TNI = 81 K [35]. The effective ionization energy of a donor–acceptor pair is equal to ID − EA , where ID is the ionization potential of the donor molecule and EA the electron affinity of the acceptor molecule. The transition is induced also by applying pressure because it is caused by the energy gain due to the long-range Coulomb attractive interaction overcoming the effective ionization energy of donor–acceptor pairs. The degree of charge transfer between the donor and acceptor molecules ρ , often called ionicity, changes from 0.3 to 0.7 at TNI . In the ionic phase at ambient pressure, each molecule has spin S = 1/2 constituting the spin chains, which are dimerized due to the spin-Peierls

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Fig. 8. Reflectance changes at 3.0 eV induced by 532-nm and 1064-nm laser pulses, as a function of excitation density. Reprinted with permission from [45]. Copyright APS.

mechanism [36–38] and/or due to the modulation of the Coulomb interaction [40,39]. Thus, an electron–lattice interaction also plays an important role on the neutral-ionic transition of TTF-CA. From X-ray and neutron diffraction studies, the directions of dimerized molecular displacements are revealed to be three-dimensionally ordered, stabilizing the ferroelectric ground state [42,41]. The photoinduced phase transition in TTF-CA was reported first by Koshihara et al. [43]. Many molecules are converted from the ionic state to the neutral state by one photon. Both ionic-to-neutral and neutral-to-ionic transitions are photoinduced within the time scale of 100 ps [44]. In these studies, the excitation lights are set at 1.5–2.5 eV and polarized perpendicular to the stacking axis, which produce local intramolecular excitations at TTF sites. Note that the peak energy of the lowest charge-transfer excitation is 0.65 eV. 3.1.1. Initial-condition sensitivity Later, it is confirmed that the ionic-to-neutral transition is induced by intrachain charge-transfer photoexcitations with 1.2 eV, which falls on the high-energy side of the peak energy [45]. The excitation-density dependence of the yield of photoinduced neutral phases is studied for both 532-nm (2.3 eV) intramolecular excitations and 1064-nm (1.2 eV) chargetransfer excitations. As shown in Fig. 8, neutral domains are formed by the intramolecular excitations at TTF sites almost in proportion to the excitation density. In contrast, by the intrachain charge-transfer excitations, the ionic-to-neutral transition is induced only above a threshold excitation density [45]. Below the threshold, a macroscopic neutral domain cannot be generated by a single charge-transfer excitation. Thus, the photoinduced dynamics are very sensitive to the initial condition of the electronic state, from which the lattice relaxation starts. This study is extended to the excitation wavelengths from 800 to 2000 nm to demonstrate such a state-dependent feature [46]. The ionic-to-neutral transition is induced above the threshold excitation density in the whole range of chargetransfer excitations. The threshold increases rapidly with decreasing photon energy of the exciting laser light. The temporal evolution of reflectance changes is analyzed in detail with the help of rate equations [47]. 3.1.2. Adiabatic relaxation path The observation of such initial-condition sensitivity has motivated theoretical studies based on itinerant-electron models. Huai et al. have calculated the adiabatic relaxation path in a one-dimensional extended Hubbard model with alternating potentials and an electron–lattice coupling [39], H = −t0

X l,σ

+ Ď

Ď



cl,σ cl+1,σ + h.c. +

X X ∆X (−1)l nl + U nl,↑ nl,↓ + Vl (ql , ql+1 )δ nl δ nl+1 2

S1 X

S2 X

2

4

(ql − ql+1 )2 +

l

l

l

l

(ql − ql+1 )4 ,

(35)

l

Ď

where cl,σ (cl,σ ) is the creation (annihilation) operator of an electron with spin σ at site l, nl,σ = cl,σ cl,σ , nl = nl,↑ + nl,↓ , δ nl = nl − 2 for odd l, δ nl = nl for even l, ql is the dimensionless displacement of the lth molecule along the chain from its equidistant position. The distance between the lth and (l + 1)th molecules is given by dl,l+1 = d0 (1 + ql+1 − ql ), where d0 is the average intermolecular distance. The donor and acceptor molecules are located at odd and even sites, respectively. The total charge of the donor molecule at site l is −eδ nl = +e(2 − nl ), while that of the acceptor molecule is −eδ nl = −enl . In the neutral phase, the orbital of the donor molecule is almost doubly occupied, while that of the acceptor molecule is almost empty. In the ionic phase, both orbitals are almost singly occupied (Fig. 7).

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

13

Fig. 9. Schematic excited domains in ionic background.

The nearest-neighbor repulsion strength is assumed to depend nonlinearly on the intermolecular distance as Vl (ql , ql+1 ) = V0 + β1 (ql − ql+1 ) + β2 (ql − ql+1 )2 ,

(36)

where V0 is for the regular lattice, and β1 (β2 ) is the linear (quadratic) coefficient. The parameter t0 denotes the transfer integral, ∆ the level difference between the neighboring orbitals in the neutral limit, and U the on-site repulsion strength. The elastic energy is expanded up to the fourth order: the parameters S1 and S2 are the linear and nonlinear elastic constants. The values of the eight parameters are so determined that they reproduce the ab initio estimation of the transfer integral, the ionicity in the ionic phase, that in the neutral phase, the degree of dimerization in the ionic phase, the energies and relative strength of the charge-transfer absorption peaks in the ionic phase, and the charge-transfer absorption energy in the neutral phase. The lattice relaxation path starting from the Franck–Condon state above the ionic ground state with a single chargetransfer exciton is assumed to be described by the displacements, ql = (−1) q0 l





    l0 1 + ∆q tanh θ |l| − −1 , 2

(37)

where (−1)l q0 denotes the uniform dimerization in the ionic state, ∆q the amplitude of a local distortion in the excited domain, θ the width of the domain boundary, and l0 the domain size (Fig. 9). An interchain interaction between displacements is introduced to confine the domain in the ionic background: ph

Hinter =

X X l

Ki ql − (−1)l q0


2i

,

(38)

i=1,2,3

where ql denotes the displacement of a central chain surrounded by ionic environmental chains, and Ki the 2ith coefficient with respect to the distortion in the central chain. The adiabatic energy surface of the ground state and that of the first excited state are drawn as a function of the amplitude ∆q and the size l0 of the domain with the width of the domain boundary θ optimized [39]. The energy surface of the ground state has a plateau corresponding to a neutral domain separated from the ground state by a low potential barrier. This plateau makes the neutral domain long-lived, suppressing the decaying into the initial ionic ground state. In the energy surface of the first excited state, a local minimum corresponding to the charge-transfer exciton appears around the Franck–Condon state (around l0 ∼ 0 in Fig. 10). Another local minimum appears and corresponds to a neutral domain, which is separated from the Franck–Condon state by such a high potential barrier that the absorption of a single photon with 0.6 eV cannot overcome it (around l0 ∼ 40 in Fig. 10). The domain is stable only when its size exceeds a critical value because the energy required for creation of two domain boundaries makes a small domain unstable in the excited state. The ground and excited states with a neutral domain are very similar: their spin- and charge-density distributions are different only at the domain boundaries. Because a single charge-transfer exciton cannot trigger the formation of a macroscopic neutral domain, a large excess energy is needed at the very beginning of the relaxation (dash-dotted line in Fig. 10) to overcome the high barrier, for charge-transfer excitons to proliferate, and finally to form a macroscopic neutral domain. The process described above is only for the early nucleation stage of the photoinduced macroscopic transformation. 3.1.3. Coherent motion of a macroscopic domain To clarify the mechanism of the photoinduced transition in TTF-CA, it is important to observe the transition dynamics induced by a resonant excitation. By using femtosecond pump-probe reflection spectroscopy, the photoinduced ionic-toneutral transition is found to be driven in a picosecond time scale by the resonant excitation of the charge transfer band at 0.65 eV polarized along the stacking axis (i.e., from · · · D+ A− D+ A− D+ A− · · · to · · · D+ A− D0 A0 D+ A− · · ·), as shown in Fig. 11 [48]. This transient reflectivity probed at the peak energy of the intramolecular transition at TTF in the ionic phase (2.25 eV) shows that neutral strings are produced within 2 ps after the photoexcitation. The signals are accompanied with prominent oscillations. Their time characteristics for the resonant excitation at 77 K just below TNI show three kinds of coherent oscillations [49]. In a short time domain, an oscillation of frequency ω ∼ 56 cm−1 (period ∼ 0.60 ps) appears and

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Fig. 10. Energy along relaxation path, as a function of number of neutral sites in ionic background. From [39]. Reprinted with permission from JPS.

Fig. 11. Time evolution of transient reflectivity changes at 2.25 eV, (a) at 4 K and (b) at 77 K. Reprinted with permission from [48]. Copyright APS.

decays with a time constant of 3.5 ps. It is related with an optical mode of lattice phonon. In the formation process of a onedimensional neutral domain, the dimerized molecular displacements will transiently persist but finally be dissolved during the disappearance of spins in the molecules. In this dissolution process, the coherent oscillation does occur, modulating the degree of charge transfer ρ . As a consequence, the coherent oscillation is detected as the reflectivity change. The microscopic one-dimensional neutral domains are multiplied and then a semi-macroscopic stable neutral state is produced within 20 ps. After that, other coherent oscillations are observed. One with period ∼ 50 ps is due to a shock wave driven by the sudden volume change of about 0.6% at the ionic-to-neutral transition [46]. The other with period ∼85 ps (Fig. 11) is related with the valence instability because its intensity is considerably enhanced with increasing temperature up to TNI . This appears due to the modulation in the amount of molecules in the neutral states, indicating coherent motion of a neutral-ionic domain boundary over the macroscopic scale [48]. At 4 K, the time characteristics strongly depend on the excitation density (Fig. 11). For the lower excitation density (Nex = 0.02 × 1016 cm−2 ), the signal initially rises within the time resolution, sharply drops up to 2ps, and decays with a time constant about 300 ps. For the higher excitation density (Nex = 1.2 × 1016 cm−2 ), the initial rise is followed by the drastic increase in 20 ps, and the signal does not decay. A stable neutral state is suggested to be formed, although its lifetime is finite and it finally decays within 500 µs. The excitation-density dependence of the dynamics of the photoinduced neutral states is clearly shown by plotting the photoinduced changes in reflectivity at different delay times td as a function of Nex (Fig. 12). At 4 K, when Nex is less than 0.15 × 1016 cm−2 , the initial rise is proportional to Nex , but the signal almost disappears before td = 500 ps because the photoinduced neutral states decay with a time constant about 300 ps. When Nex exceeds 0.2 × 1016 cm−2 , the initial rise is saturated as a function of Nex , and the signal sharply increases by td = 500 ps because the neutral states multiply. Thus, the photoinduced ionic-to-neutral conversion by the high excitation density proceeds as shown in Fig. 13. At 77 K, the threshold excitation density is dramatically reduced. From the space-filling argument, the amount of initial D0 A0 pairs produced by one photon is estimated to be 8D0 A0 at 4 K and 24D0 A0 at 77 K. The energy of a one-dimensional neutral domain is theoretically shown to be lower than that of a single charge-transfer state when the neutral and ionic

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

15

Fig. 12. (a) Photoinduced reflectivity changes at 2.25 eV as a function of excitation density Nex at 4 and 77 K, 2 and 500 ps after the photoexcitation. (b) Schematic photoinduced I-to-N conversion by the low excitation density. Reprinted with permission from [48]. Copyright APS.

Fig. 13. Schematic evolution in photoinduced I-to-N transition.

phases are almost degenerate [50,36,51,52]. The fact that the size of the initial neutral domain at 77 K is larger than that at 4 K is reasonable because the former is closer to TNI and thus closer to the degeneracy. 3.1.4. Transition dynamics due to electron–lattice interactions Theoretically, real-time dynamics need to be calculated for further understanding. Then, the time-dependent Schrödinger equation is solved for an electron–lattice model without electron–electron interaction as a first step, mainly at half filling, H = −t0

X (l,l0 )σ

Ď



cl,σ cl0 ,σ + h.c. +

∆ X 2

1 − (−1)l nl − S

l



X l

ql (nl − 1) +

S X 2

l

q2l +

S 2ω

X 2

q˙ 2l ,

(39)

l

where two charge-density-wave states with opposite polarizations are almost degenerate [53,54] or exactly degenerate (∆ = 0) [55]. The lattice part is treated classically. The photoconversion is shown to be nonlinear. The threshold number of electron–hole pairs is one in the single-chain case, while it is about four in the four-chain case [53]. The aggregation of excited domains is discussed on the basis of the adiabatic potentials for the ground and excited states and demonstrated by the solution to the time-dependent Schrödinger equation. The photoexcited kink and antikink propagate with a constant velocity, dissipating energy to particular vibrational modes whose wave numbers are distributed around twice the Fermi wave number with width related to the kink width. This nonergodicity is suggested to promote the phase conversion processes. A precursor of transitions is also discussed from the stable phase to the metastable phase [54]. For one to three oneelectron excitations in the four-chain case, only locally relaxed structures are obtained as metastable states. Four oneelectron excitations are attracted to one another, forming a cluster of the one-dimensional nature. This one-dimensional domain structure is sandwiched between two domain walls. Its radiative transition probability is calculated as a function of the distance between the domain walls. The nonradiative decay probability is also calculated as a function of time. The one-dimensional domain is shown to be much more stable against these processes than the locally relaxed structure. In the two-chain case with exact degeneracy, the fraction of photoconversion is shown to be a quadratic function of the excitation density below the saturation at a half [55]. During the photoconversion process, macroscopic oscillations in the fraction are observed for high excitation density, which are essentially different from linear modes in equilibrium (Fig. 14). The peak frequency of its Fourier transform is much smaller than the bare phonon frequency. At low excitation density, the macroscopic oscillation is like a breather.

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Fig. 14. Sample-averaged dynamics of fraction of photoconversion for different excitation densities. Reprinted with permission from [55]. Copyright APS.

3.1.5. Photoinduced phase transition dynamics in TTF-CA In TTF-CA, electron–electron interactions are important to realize the ionic phase. Then, the kinetic energy of the P ˙ 2l , with the lth molecular mass ml , is added to the model (35): displacements, 12 l ml q H = −t0

X

Ď



cl,σ cl+1,σ + h.c. +

l,σ

X X ∆X (−1)l nl + U nl,↑ nl,↓ + Vl (ql , ql+1 )δ nl δ nl+1 2

l

l

S2 X 1X S1 X (ql − ql+1 )2 + (ql − ql+1 )4 + ml q˙ 2l . + 2

l

4

l

2

l

(40)

l

Miyashita et al. have calculated real-time dynamics of its ionicity and dimerization within the unrestricted Hartree–Fock approximation after the ionic phase is photoexcited [56]. Note that the time-dependent Hartree–Fock approximation for driven electron–lattice systems actually goes beyond the random phase approximation for equilibrium electron–lattice systems [57] in the sense that the former deals with interactions among collective modes and gives them finite lifetimes. In this study, the photoexcitation is introduced by changing the occupancy of orbitals in the initial state, as before [53– 55]. After the photoexcitation, the time-dependent Schrödinger equation is solved for the electronic part, and the classical equation of motion for the lattice part. As the number of orbitals whose occupancy is changed increases, i.e., with increasing photoexcitation density, more neutral domains are created. Above the threshold excitation density, the neutral phase is finally achieved. After the photoexcitation, ionic domains with wrong polarization generally appear, which reduce the correlation length of the staggered lattice displacement. As the degree of initial lattice disorder increases, more solitons appear between the ionic domains with opposite polarizations, which obstruct the growth of neutral domains and slow down the transition. Thus, the dynamics of different types of domain walls become complicated with increasing thermal lattice fluctuations. For the analysis of these complex dynamics, Fourier power spectra are useful for the ionicity in different time windows [56]. During the photoinduced transition, they are very broad, featureless and gradually decrease with frequency. The lowest-frequency components are due to very slow excitations accompanied with collective motion of neutral-ionic domain boundaries. Because this motion is sensitively affected by collisions with the solitons, the spectra are broad. After the transition, the broad background is suppressed and a peak emerges, which corresponds to the peak energy of a chargetransfer exciton in the random-phase-approximation spectra. The low-frequency part is still present due to slow chargetransfer excitations coupled with optical lattice vibrations. When the fraction of ionic-to-neutral conversion is plotted as a function of the number of excited electrons, the curve evolves with time as shown in Fig. 15. Immediately after the photoexcitation, it is close to a linear function. Above the threshold number, the fraction monotonically increases with time until the photoinduced phase transition is completed. Below the threshold, the fraction decreases and eventually becomes very small. This result is consistent with the experimental finding [48]. 3.1.6. Difference between I-to-N and N-to-I transitions More recently, experimental results have been summarized covering both the ionic-to-neutral and the neutral-toionic transitions with different excitation energies, excitation densities and temperatures [49]. From the excitation-energy dependence of the dynamics induced by very low photoexcitation density, it is found that the decay of the photoinduced neutral states becomes slower with increasing excitation energy. The very slowly decaying (with a time constant much longer than 500 ps and shorter than 500 µs) component of the signal, as a function of the excitation energy, is almost proportional to the excitation profile of the photoconductivity along the stacking axis. It suggests that the low-energy

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17

Fig. 15. Conversion from ionic to neutral phases, as a function of number of excited electrons in 100-site chain. The nonlinearity is strengthened with the passing of time after the photoexcitation, t = 1.5/ω, 15/ω, 100/ω and 250/ω, where ω denotes the bare phonon frequency. The initial lattice temperature is T = 10−2 t0 . From [56]. Reprinted with permission from JPS.

excitation near resonance generates only neutral domains consisting of D0 A0 pairs, and the higher-energy excitation generates also charged domains consisting of an odd number of neutral molecules, whose lifetime is much longer than the neutral domains (consisting of an even number of neutral molecules). The efficiency of the ionic-to-neutral transition is enhanced with increasing excitation energy, which is consistent with the dependence of the threshold excitation density on the excitation energy [46]. When a single crystal sample is irradiated by light, the amount of the photoexcited states exponentially decreases with increasing distance from the sample surface. In addition, in the pump-probe reflection spectroscopy, the absorption depth of the probe light depends on its energy and is different from that of the pump light. These facts affect the magnitude and spectral shape of transient reflectivity changes. Then, the transient reflectivity spectra are analyzed by a multilayer model [49]. There are two adjustable parameters in the model, which are the density of the photoinduced neutral states at the sample surface (x0 ) and the length characterizing its exponential decay from the sample surface along the normal direction to the surface (β −1 ). The total amount of the photoinduced neutral states Ntotal is proportional to x0 /β . At 4 K with high excitation density, x0 gradually increases, while β −1 is almost unchanged, for td < 20 ps. In this time region, the multiplication process of the neutral states generated just after the photoirradiation occurs two dimensionally in the plane parallel to the sample surface. For 20 ps < td < 40 ps, x0 continues to increase but its speed decreases, while β −1 increases. For td > 40 ps, x0 is almost unchanged, and β −1 continues to increase, though its speed decreases. Thus for td > 20 ps, the two-dimensional multiplication near the sample surface is almost saturated, and the multiplication occurs mainly in the direction perpendicular to the sample surface. In short, the ionic-to-neutral transition induced by the resonant excitation of the charge-transfer band proceeds with (1) initial formation of a confined one-dimensional neutral domain, (2) multiplication of such domains to semi-macroscopic neutral states by 20 ps, and (3) evolution in the direction normal to the sample surface (Fig. 13). The dynamics of the neutral-to-ionic transition induced by the resonant charge-transfer excitation are shown to be clearly different from the above [49]. For any excitation density Nex , the photoinduced reflectivity change initially shows a fast rise within the time resolution, due to the formation of ionic states in the neutral background, but it decays very rapidly within 20 ps even if Nex is large. The excitation-density dependence of the dynamics of the photoinduced ionic states is clearly shown by plotting the photoinduced changes in reflectivity at different delay times td as a function of Nex (Fig. 16). The signal at td = 0.3 ps is almost proportional to Nex up to 0.4 × 1016 cm−2 and then saturated. Thus, the initial rise is due to the formation of a one-dimensional ionic domain. In a short time domain, an oscillation of frequency ω ∼ 54 cm−1 (period ∼0.62 ps) appears, related with an optical mode of lattice phonon, and decays with a time constant of 7 ps. The signal becomes negative much before td = 500 ps, which is attributed to the thermal effect. Indeed, the observed reflectivity change is very similar to that caused by increased temperature. In short, although the one-dimensional ionic domains are initially produced by lights, they decay within 20 ps through nonradiative processes even if the density of the initial ionic domains is increased. Their energies are transferred to the lattice. There is no indication of multiplication of the ionic domains. These features are almost independent of the excitation energy from 0.65 to 1.55 eV. The photoinduced neutral-to-ionic conversion proceeds as shown in Fig. 17. A clear difference is thus demonstrated between the dynamics of the photoinduced ionic-to-neutral transition and those of the neutral-to-ionic transition.

3.1.7. Electronic-state-dependent transition dynamics To describe the dependence of the transition dynamics on the frequency, the amplitude and the duration of the laser pulse, we incorporate an oscillating classical electric field, E (t ) = Eext sin ωext t ,

(41)

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Fig. 16. Photoinduced reflectivity changes at 2.25 eV as a function of excitation density Nex at 90 K, 0.3 and 500 ps after the photoexcitation. The dashed line shows the linear relation. Reprinted with permission from [49]. Copyright APS.

Fig. 17. Schematic evolution in photoinduced N-to-I transition.

with frequency ωext and amplitude Eext for 0 < t < 2π Next /ωext with integer Next , into the Peierls phase of the transfer integral for an electron moving from the (l + 1)th site to the lth site,



t0 (t ) = t0 exp i

ed0 h¯ c



A(t ) ,

(42)

where e is the absolute value of the electronic charge, and c the light velocity. The time-dependent vector potential A(t ) is related to the electric field E (t ) by A(t ) = −c

Z

t

dt 0 E (t 0 ).

(43)

The time-dependent Schrödinger equation is solved again for the one-dimensional extended Hubbard model with alternating potentials and the electron–lattice coupling (40). The dynamics of the ionic-to-neutral transition are indeed qualitatively different from those of the neutral-to-ionic transition. When the dimerized ionic phase is photoexcited, the threshold behavior is observed again by plotting the final ionicity as a function of the increment of the total energy, i.e. as a function of the number of absorbed photons (Fig. 18) [58]. The ionicity PN is defined as ρ = 1 + (1/N ) l=1 (−1)l hnl i. The threshold photoexcitation density is higher (thus the photoexcitation is less effective) than the previous one obtained by changing the occupancy of orbitals in the initial state. More than 90% of the energy supplied by the oscillating electric field is not used for the transition even just above the threshold but absorbed into the electronic state, leading to deviation of the final state from the static self-consistent state in the metastable neutral phase. The threshold photoexcitation density is insensitive to the frequency (at and below the charge-transfer absorption peak), the amplitude or the duration of the pulse, though it becomes slightly higher for small amplitudes. Above the threshold photoexcitation density, the system finally enters a neutral state with equidistant molecules (Fig. 19). The growth of a neutral domain is spontaneous once triggered, even if the electric field is switched off during the growth. When compared with the threshold behavior derived from the rate equation [9], we do not observe a quantity corresponding to the threshold intensity Ith . Even if the amplitude of the pulse is small (i.e., even if the pulse is weak), a transition seems finally achieved by setting the pulse duration long enough. The deterministic dynamics are more efficient than the stochastic ones because of the restricted energy dissipation. For a given amount of increment of the total energy, the lattice dynamics relative to the charge dynamics in the ionic-toneutral transition depend on the character of the pulse [58]. When the pulse is strong and short, the charge transfer takes place on the same time scale with the disappearance of dimerization. When the pulse is weak and long, the dimerizationinduced ferroelectric polarization is disordered first to restore the inversion symmetry, and then the charge transfer takes place to bring the system to a neutral state (Fig. 20). The difference between the two time scales increases as the pulse is weakened. Between these time scales, a paraelectric ionic phase is realized. These dynamics are illustrated in Fig. 21.

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Fig. 18. Final ionicity as a function of number of absorbed photons in 100-site chain, when electric field of strength eEext d0 = 5ω and of frequency ωext = 28ω below linear-absorption peak at 32ω is applied to ionic phase. ω denotes the bare phonon frequency. The initial lattice temperature is T = 10−3 t0 . From [58]. Reprinted with permission from JPS.

Fig. 19. Evolution of ionicity with time, for different field strengths (eEext d0 /ω)2 and Next = 3. The electric field of frequency ωext = 28ω is applied (for 0 ≤ ωt ≤ 0.67) to the ionic phase at T = 10−3 t0 . From [58]. Reprinted with permission from JPS.

Fig. 20. Evolution of (a) ionicity and (b) staggered lattice displacement with time, after ionic phase at T = 10−3 t0 is photoexcited with ωext = 28ω, eEext d0 = 2ω and Next = 150 (for 0 ≤ ωt ≤ 33.7). From [58]. Reprinted with permission from JPS.

In the case of intramolecular excitations at 1.55 eV, such a new ionic phase with disordered polarizations is suggested to appear by time-resolved X-ray diffraction (Fig. 22) [59]. General reflections are modified after an incubation time of about 500 ps, whereas the intensity of the (030) reflection responsible for the ferroelectric order starts to decrease just after the laser pulse excitation. This similarity would be due to the fact that, when the pulse is weak, the supplied energy is not directly used to transfer charge along the chain.

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Fig. 21. Schematic evolution in photoinduced Iferro -Ipara -N transition.

Fig. 22. Relative intensity of Bragg reflection as a function of delay between laser pump and X-ray probe. This is associated with the photoinduced ionicto-neutral transition. General reflections are modified after an incubation time of about 500 ps (top), whereas the intensity of the (030) reflection starts to decrease just after the laser pulse excitation (bottom), indicating a two-step mechanism with an intermediate paraelectric (disordered) ionic phase. From [59]. Reprinted with permission from Elsevier.

When the neutral phase is photoexcited, the linear behavior is observed by plotting the final ionicity as a function of the increment of the total energy (Fig. 23) [60]. The linear coefficient is almost independent of the frequency of the field or the initial lattice temperature [61]. Thus, the neutral-to-ionic transition proceeds in an uncooperative manner. If the oscillating electric field is turned off before the transition is completed, the ionicity remains intermediate unless the energy dissipation is taken into account. The growth of a metastable domain is not spontaneous but forced by the external field. Therefore, the final state is determined merely by how many photons are absorbed to increase the ionicity. With dissipation, it would easily relax back to the initial neutral state. This result is consistent with the experimental finding in the neutral-to-ionic transition induced by intrachain charge-transfer photoexcitations (Fig. 16) [49]. Interchain coupling effects will be discussed later in Section 3.1.9. The above qualitative difference between the photoinduced ionic-to-neutral and neutral-to-ionic transition dynamics exists even if interchain couplings are taken into account. The photoinduced neutral-to-ionic transition has also been studied on the basis of time-resolved X-ray measurements [62]. They show that intramolecular photoexcitations at 1.55 eV can induce ferroelectrically ordered ionic states in the neutral phase. More details are discussed in Section 5.2. It takes a long time about 500 ps for the ionic state to be formed (Fig. 24). The decay time of the photogenerated ionic state is much longer than 1 ns. These results are clearly different from those observed by the femtosecond pump-probe spectroscopy [49]. Detailed infrared studies show that the neutral states are partially converted to the ionic states by applying a very small pressure [63]. If photoinduced sample heating and resultant extension of the irradiated area give rise to inhomogeneous stress in the surrounding neutral region, thermal effects may not be negligible [64] possibly in the photoinduced phenomenon on a 100 ps time scale observed in the time-resolved X-ray study. From the theoretical viewpoint, the qualitative difference described above between the two transitions is understood as due to the difference in the initial electronic states because the difference survives even if the electron–lattice coupling is turned off in the model. The ionic state is a Mott insulator caused by the Coulomb interaction, whereas the neutral state is a band insulator caused by the band structure. In a Mott insulator, all the electrons are so correlated that any one of them cannot easily make a first move. However, above the threshold excitation density, they cannot tolerate the increased total energy. Once some electrons move, they trigger the collective motion. In a band insulator, electrons move individually. The supplied energy is merely consumed to transfer electrons from the donor to acceptor molecules almost independently. They do not strongly influence the motion of other electrons. A similar collective charge-transport phenomenon is observed in

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Fig. 23. Final ionicity as a function of number of absorbed photons in 100-site chain, when electric field of strength eEext d0 = 2ω and of frequency ωext = 28ω below linear-absorption peak at 30ω is applied to neutral phase. ω denotes the bare phonon frequency. The initial lattice temperature is T = 10−2 t0 . From [60]. Reprinted with permission from JPS.

Fig. 24. Relative intensity of Bragg reflection as a function of delay between laser pump and X-ray probe. A large structural reorganization, associated with the neutral-to-ionic transition, follows the laser irradiation. The light-driven metastable ionic state is established after about 500 ps. From [62]. Reprinted with permission from AAAS.

field-effect transistors fabricated on organic single crystals of a quasi-one-dimensional Mott insulator [65], whose ambipolar characteristics are theoretically shown to be caused by balancing the correlation effect in the bulk with the Schottky barrier effect at interfaces [66]. Charge transport through metal-Mott insulator interfaces, where rectification is strongly suppressed even for large work-function differences, is qualitatively different from charge transport through metal-band-insulator interfaces, as shown both theoretically and experimentally [67]. A difference should appear also in coherence, which can be manifested by a double pulse [68]. When the dimerized ionic phase is photoexcited by a double pulse, the interference effect is numerically observed owing to the charge-lattice coherence. The two pulses constructively interfere with each other if the interval t2nd is a multiple of the period of the optical lattice vibration (Fig. 25). When the neutral phase is photoexcited by a double pulse, the interference effect is weaker in the calculation based on the one-dimensional model. Nevertheless, when the neutral phase is experimentally photoexcited, by changing the interval of the two laser pulses, the amplitude of the oscillation in the reflectivity of the intramolecular transition band sensitive to the degree of charge transfer has been shown to be periodically changed [69]. With increase of the pump intensity, the oscillation amplitude is dramatically enhanced. This indicates the importance of three-dimensional effects. In this ultrafast pump-probe experiment [69], it has been claimed that an ionic-phase domain in the neutral-phase background is very quickly formed even before the lattice started to move. A dynamical density-matrix renormalization group study proves this idea in terms of the extended Hubbard model with site–potential alternation [70]. Namely, the photoexcited states in the neutral phase near the neutral-ionic phase boundary possess ionic domains even without lattice dimerization. 3.1.8. Interchain coupling effects on the phase diagram Thus far, we have focused on the dynamics in purely one-dimensional systems. Although the short-time behavior may not suffer from interchain interactions, they are generally important. The coherent motion of the macroscopic neutral-ionic domain boundary [48] should be possible only when interchain interactions are present. Although the effect of interchain elastic couplings is considered in the context of confinement of a metastable domain [39], interchain electron–electron

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Fig. 25. Evolution of (a) ionicity and (b) staggered lattice displacement with time, for different intervals ωt2nd between two pulses of ωext = 28ω, eEext d0 = 8ω and Next = 3. The initial state is ionic at T = 10−3 t0 . From [68]. Reprinted with permission from JPS.

Fig. 26. Phase diagram obtained from neutron scattering, NQR, vibrational spectroscopy, and conductivity measurements. Npara , Ipara , and Iferro refer to the paraelectric neutral, paraelectric ionic, and ferroelectric ionic phases, respectively. C is the estimated critical point. Reprinted with permission from [41]. Copyright APS.

Fig. 27. Intermolecular electrostatic energy for each pair, with distance between their centers of mass in parenthesis. Reprinted with permission from [71]. Copyright APS.

interactions are much stronger. In fact, the long-range Coulomb interaction is responsible for stabilizing the ionic phase, which becomes paraelectric under high pressure (Fig. 26) [41]. The intramolecular charge distribution in the TTF and CA molecules is calculated by the ab initio quantum chemical method [71]. The interchain electrostatic energies between neighboring molecules are smaller than but comparable to the intrachain ones (Fig. 27). Because the molecules are tilted, the relative repulsion strengths are very different from the naive expectations based on the intermolecular distance measured in terms of the center of mass of each molecule. In the ionic phase, the interchain attractive interaction between the neighboring donor and acceptor molecules along the b0 axis is larger than any interchain repulsive one between the donors or between the acceptors.

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23

Fig. 28. Structural evidence for triple point in the pressure–temperature phase diagram. From [72]. Reprinted with permission from EDP Sciences.

The consequent interchain Coulomb attraction in the ionic phase is responsible for the discontinuous contraction along the b axis (Fig. 28) [72] and the pressure–temperature phase diagram containing the paraelectric ionic phase [41]. This fact is theoretically shown by employing the quasi-one-dimensional Blume–Emery–Griffiths model [73], H =−

X

Jk pl,j pl+1,j + Kk p2l,j p2l+1,j − ∆p2l,j −

l,j



X l,j

J⊥ pl,j pl,j+1 + K⊥ p2l,j p2l,j+1 + felst ,



(44)

where the classical variable on the lth site in the jth stack, pl,j , takes 0 for a neutral state and ±1 for an ionic state with polarization. It has a classical analogue of the charge-transfer energy (∆) and the intrachain (with subscript k) and interchain (with subscript ⊥), dipolar (J) and nonpolar (K ) interactions. An additional contribution from the interchain Coulomb attraction and the interchain elastic energy is written as, felst = −

e2 p2l,j p2l,j+1 b0 + y

1

+ M Ω⊥2 y2 ,

(45)

2

where b0 denotes the lattice constant along the b axis, y the distortion, M the reduced mass of donor and acceptor molecules, and Ω⊥ the frequency of the optical phonon along the b axis [74,75]. The transfer-matrix method that is exact in pure one dimension is combined with the interchain mean-field treatment. With increasing pressure, the ionic phase is relatively stabilized, which can be simulated by decreasing ∆. A typical phase diagram is shown in Fig. 29 [74], which is to be compared with Fig. 26. 3.1.9. Interchain coupling effects on transition dynamics In order to study the transition dynamics, we need to treat itinerant-electron models. Then, we add to the model (40) interchain electron–electron interactions [76,77], el Hinter =

X

Up δ nl,j δ nl,j+1 + Vp1 δ nl,j+1 δ nl+1,j ,




(46)

l,j

where j is the chain index, and the donor–acceptor coupling strength Vp1 is slightly larger than the donor–donor or acceptor–acceptor one Up (Fig. 27) [71]. Note that the interchain interactions are introduced between δ n’s, so that they

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2 Fig. 29. Phase diagram for Jk = 1, Kk = 0.4, K⊥ = 0.06, J⊥ = 0.03 and e4 /(2b40 M Ω⊥ Jk ) = 0.0095. TP and CP represent the triple point and the critical point, respectively. Reprinted with permission from [74]. Copyright APS.

Fig. 30. (a) Number of converted chains in ionic-to-neutral transition, and (b) final ionicity in neutral-to-ionic transition, as a function of number of absorbed photons per chain. The relation Up = 0.9Vp1 is used, where a strong-coupling Vp1 = 0.5 corresponds to TTF-CA. Reprinted with permission from [78]. Copyright APS.

are significant in the ionic phase, where Vp1 produces attraction. Their effects on the phase transition dynamics induced by intrachain charge-transfer photoexcitations are studied by solving the time-dependent Schrödinger equation for the quasione-dimensional model within the unrestricted Hartree–Fock approximation. Coupled ten 100-site chains are treated. When the ionic phase is photoexcited, the transition dynamics depend on the strengths of the interchain couplings. With weak interchain couplings, the interchain correlation is very weak during the transition. A first neutral domain is easily created by a low density of photons. By continuing the application of the electric field, next domains are created in the neighboring chains. Many photons are finally needed to complete the transition into the neutral phase. With strong interchain couplings comparable to those in TTF-CA, the interchain correlation is strong during the transition. To create a first neutral domain, more photons need to be absorbed. Once a domain is nucleated, nearby domains are almost simultaneously created in the neighboring chains to grow cooperatively. As a consequence, the density of absorbed photons needed to complete the transition is lower than in the weak-coupling case (Fig. 30(a)) [78]. Because of the strong interchain correlation, the growth of metastable domains coherently proceeds, which is consistent with the experimentally observed, coherent motion of the macroscopic neutral-ionic domain boundary [48]. Neutral domains in nearby chains simultaneously grow even if their nucleation is delayed by reducing the amplitude of the electric field. Thus, the coherent motion of the macroscopic neutral-ionic domain boundary is generally observed on a large time scale. Even if the interchain electron–electron interactions are weak, in the case of very large amplitude of the electric field, metastable domains appear quickly and almost simultaneously after the photoirradiation starts. Such coherence that is forced by the intense electric field survives only shortly and decays rather quickly. This situation may be related to the

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Fig. 31. Doublon and holon after photoirradiation of Hubbard model.

coherent oscillation observed in the photoexcited quarter-filled-band charge-ordered organic (EDO-TTF)2 PF6 salt [79]. Recently, its photoinduced oscillations are investigated both experimentally [80] and theoretically [81]. See also [82,84,83]. In addition, ultrafast pump-probe spectroscopy with 15 fs temporal resolution is applied to (TMTTF+ )2 dimers, as a model for the photoinduced phase transition of such charge-transfer compounds, providing a picture of structural relaxation following charge-transfer photoexcitation [85]. Here also, the vibrations are strongly coupled to the charge-transfer excitation. When the neutral phase is photoexcited, the converted fraction is still almost a linear function of the photoexcitation density. The supplied energy is merely consumed to transfer electrons almost independently. The growth of metastable ionic domains is not spontaneous but always forced by the external field. Interchain electron–electron interactions make the above function slightly nonlinear, increasing cooperativity a little (Fig. 30(b)) [78], but the above situation is almost unchanged. Finally, it should be noted that differences between I-to-N and N-to-I transitions exist even on the level of stochastic relaxations. Characteristic properties of relaxation processes in the neutral-ionic, paraelectric–ferroelectric phase transitions are discussed on the basis of a master equation in the spin-1 anisotropic Blume–Emery–Griffiths model [86]. Time-evolution equations are derived by an extension of the Saito–Kubo treatment [87] to the spin-1 anisotropic case. Numerical solutions show characteristic properties caused by the relation Jk > Kk > K⊥  J⊥ . The anisotropy becomes prominent during the neutral-to-ionic paraelectric-to-ferroelectric phase transition: the interchain ferroelectric ordering develops much more slowly than the intrachain one, owing to Jk  J⊥ . In contrast, the ionic-to-neutral phase transition proceeds rather isotropically on the level of the Saito–Kubo treatment. 3.2. Halogen-bridged metal-chain compounds Before going to halogen-bridged metal-chain compounds, we will treat closely related and generic problems on onedimensional Mott insulators by the one-band half-filled Hubbard model. Although there are many theoretical studies on the effects of chemical doping, those of photodoping are few and limited. For instance, photoinduced absorption spectra of an effective model in the limit of infinite on-site repulsion are calculated [88]. Then, we need to clarify whether photoexcited states are metallic and whether the photodoping is similar to the chemical doping. 3.2.1. Insulator-to-metal phase transition in one-band models Let us consider the one-dimensional extended Hubbard model with and without alternating potentials, H = −t0

X l,σ

Ď



cl,σ cl+1,σ + h.c. + ∆

X X X (−1)l nl + U nl,↑ nl,↓ + V nl nl+1 , l

l

(47)

l

where the notations are the same as before, although ∆ is used here instead of ∆/2. Here, we introduce the alternating potential to compare Mott and band insulators later. When the half-filled Hubbard model (∆ = 0) is irradiated by lights, electrons are excited as shown in Fig. 31, producing carriers consisting of doubly occupied sites (doublons) and those of empty sites (holons). The optical responses of photoexcited states in the Hubbard model (V = 0) are calculated [89]. The optical conductivity spectra and Drude weights of the ground and photoexcited states are obtained by the exact diagonalization method. The photoexcited state is shown to be metallic with a large Drude weight. A significantly large spectral weight is shifted to below the optical gap ∆opt from a holon–doublon continuum observed in the ground state. Furthermore, the optical conductivity of the photodoped system is shown to be very similar to that of the corresponding hole-doped system for V = 0. The optical conductivity of the ground state |ψ0 i is given by σ (ω) ≡ Dδ(ω) + σ reg (ω), where D is its Drude weight and reg σ (ω) the regular component [90]. The optical conductivity of the optically allowed, first excited state |ψ1opt i with energy E1opt is given by 0

σ1 (ω) ≡ D1 δ(ω) + σ1reg (ω) + σ1reg (ω),

(48)

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Fig. 32. (a) Integrated intensities and (b) their ratios in 1D Hubbard model as a function of U. (c) shows size dependence. From [89]. Reprinted with permission from JPS.

where D1 is the Drude weight of |ψ1opt i defined by D1 = −

π N

hψ1opt |Kˆ |ψ1opt i −

2π X |hψn |ˆj|ψ1opt i|2 N

n6=1opt

En − E1opt

,

(49)

with Kˆ being the kinetic term of Hamiltonian (47), ˆj the current operator defined by ˆj ≡ it0 reg0 1

reg

P

l,σ

(clĎ+1,σ cl,σ − clĎ,σ cl+1,σ ), |ψn i

the nth excited state, and En the corresponding energy. The quantities σ1 (ω) and σ (ω) are regular components [89]. For half-filled N-site chains, the periodic boundary condition is used for N = 4n and the antiperiodic boundary condition for N = 4n + 2. The Lanczos method is used for diagonalization. To show how the weight of the holon–doublon continuum 0 is reduced by photoexcitation, R ω ranging from U − 4t0 to RUω+ 4treg the following integrated intensities are calculated: I hd ≡ ω u σ reg (ω)dω, and I1hd ≡ ω u σ1 (ω)dω, where the lower limit l l ωl and the upper limit ωu are set to include only the contribution from the holon–doublon continuum above ∆opt . The total R R ∞ ∞ integrated intensities have also been calculated: I tot ≡ 0 σ (ω)dω, and I1tot ≡ 0 σ1 (ω)dω. These quantities are shown in Fig. 32. In contrast to σ (ω), to which the holon–doublon continuum above ∆opt mainly contributes, σ1 (ω) has a much smaller contribution from the continuum, although its total integrated intensity is larger than that of the ground state. As a consequence, I1hd is much smaller than I1tot , showing that low-energy components have a dominant contribution to σ1 (ω). The size dependence of I1hd /I hd comes from the density of photodoped carriers, 2/N. The large shift of the spectral weight from above ∆opt to low energies is the characteristic doping effect in the Mott insulator phase [91]. To see this in the photodoping case, σ (ω) and σ1 (ω) are compared in the Mott insulator phase (Fig. 33(a)) and in the band insulator phase (Fig. 33(b)). In σ1 (ω) of the band insulator phase, the contribution from the hole–electron excitations is similar to that in σ (ω) except for the absence of lowest peak at ∆opt . This single reg0

reg

difference between σ (ω) and σ1 (ω) comes from the addition of the negative σ1 (ω) to σ1 (ω) in Eq. (48). Indeed, the reg bare contribution from the hole–electron pair excitations σ1 (ω) is almost identical with σ (ω), demonstrating that the

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Fig. 33. Optical conductivity, (a) in Mott insulator phase of 1D Hubbard model (U = 12, ∆ = 0) and (b) in band insulator phase of that with alternating potentials (U = 0.1, ∆ = 4). From [89]. Reprinted with permission from JPS.

band structure is unchanged by photoirradiation. This is in contrast to the situation in the Mott insulator phase shown above. To compare photodoped and chemically doped systems, the optical conductivity is calculated in the ground state of the two-hole doped system. Both doped systems have a reduced holon–doublon continuum above ∆opt : the integrated intensities over the continuum are comparable and approach each other as U increases for V = 0 [89]. The Drude weight of the photodoped system is smaller than that of the chemically doped system roughly by the amount corresponding to the transition from the first excited state to a nearly degenerate holon–doublon state inherent in one-dimensional Mott insulators [92]. Next, we compare the effect of the nearest-neighbor repulsion V on the Mott insulator and that on the band insulator. Then, the optical responses are discussed in the model (47) with V [93]. The exact diagonalization method is used again to obtain optical conductivity spectra and Drude weights for the ground and photoexcited states. Different optical responses in these insulators are due to different behaviors against V of carriers: doublons and holons in the Mott insulator phase, and electrons and holes in the band insulator phase. The Drude weight of the lowest-energy photoexcited state D1 in the Mott insulator phase is shown in Fig. 34. In the half-filled extended Hubbard model, the energy of the photoexcited state is given by U − 4t0 if the doublon and the holon are unbound, and by U − V − 4(t02 /V ) if they are bound [94]. Because of the reduced effective transfer integral for the bound pair, 2t02 /V , the doublon and the holon gain energy by unbinding for small V . Thus, for V < 2t0 , they contribute to the metallic conductivity. The Drude weight D1 decreases very slowly with the system size N owing to the decreasing density of carriers. For V > 2t0 , the doublon and the holon are bound to form a neutral pair, which does not contribute to the conduction at zero energy. The Drude weight D1 decreases exponentially with N. The Drude weight D1 in the band insulator phase is shown in Fig. 35. It monotonically decreases with V for any N. The electron and the hole are bound to form an exciton with an infinitesimally small V [95]. Only at V = 0, D1 decreases very slowly with N, implying the metallic behavior. For any V > 0, D1 decreases exponentially with N, whose localization length becomes short with increasing V .

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Fig. 34. Drude weight of lowest-energy photoexcited state D1 , as a function of V , in Mott insulator phase with U /t0 = 20, ∆ = 0 and different N. From [93]. Reprinted with permission from IOP.

Fig. 35. Drude weight of lowest-energy photoexcited state D1 , as a function of V , in band insulator phase with U /t0 = 1, ∆/t0 = 3 and different N. From [93]. Reprinted with permission from IOP.

3.2.2. Insulator-to-metal phase transition in MX chains Halogen-bridged metal-chain compounds, often called MX chains, are intensively studied [96]. Their ground states depend mainly on the metal element. The nickel-chain compounds are Mott insulators due to large on-site Coulomb repulsion, while the palladium- and platinum-chain compounds are charge-density-wave (CDW) insulators due to relatively large electron–lattice interactions. In the latter, such nonlinear excitations as solitons and self-trapped excitons have attracted much attention of many scientists [97]. In the former, an enhancement of the third-order nonlinear optical susceptibility is reported [98]. A nickel-chain compound, [Ni(chxn)2 Br]Br2 (chxn = cyclohexanediamine), has nickel (Ni3+ )-bromine (I− ) chains as shown in Fig. 36. In this compound, four nitrogen atoms of two ligand (chxn) units coordinating a Ni ion produce such a strong ligand field that the Ni3+ ion is in a low-spin state (d7 : S = 1/2) with an unpaired electron in the dz 2 orbital. Half-filled one-dimensional electron systems are formed by the hybridization between Ni dz 2 and Br pz orbitals. The strong on-site Coulomb repulsion among 3d electrons opens a gap between the Ni 3d upper Hubbard band and the lower Hubbard band. The occupied Br 4p band is located inside this Mott–Hubbard gap, so that the lowest-energy electronic excitation is a charge-transfer excitation from the Br 4p band to the Ni 3d upper Hubbard band. Owing to the electron correlation, the ground state is a Mott insulator in a broad sense. When this compound is irradiated by lights, electrons are excited as shown in Fig. 36, producing carriers consisting of doubly occupied Ni sites and those of ‘‘empty’’ sites. These photogenerated carriers, when their density exceeds 0.1 per Ni site, enhance a Drude-like low-energy component in the optical conductivity immediately after photoirradiation (td = 0.1 ps), suggesting a Mott transition by photodoping (Fig. 37) [99]. For very low excitation density xph of about 10−3 photon/Ni site, a midgap absorption is observed around 0.4–0.5 eV. As xph increases, the low-energy part of the optical conductivity below 0.2 eV significantly grows. For xph above 0.1, the optical gap disappears. It is known that the chemical doping of Mott insulators enhances the Drude weight in the optical conductivity. Hole- or electron-doped two-dimensional copper oxides showing the high-Tc superconductivity are well-known examples [91]. For them also, photoirradiation is an effective method of producing carriers [100]. By integrating the optical conductivity from 0 to ω, one obtains the effective number of carriers, Neff (ω). The photodoping-density xph dependence of the photoinduced change in Neff (ω)[∆Neff (ω)] in the nickel-chain compound [99] is very similar to the hole-doping-density x dependence of Neff (ω) in the copper oxide La2−x Srx CuO4 [91], both for ω in the infrared region and for ω just below the charge-transfer

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Fig. 36. Schematic electronic structures of [Ni(chxn)2 Br]Br2 .

Fig. 37. (a) Reflectivity spectra and (b) spectra of imaginary part of dielectric permittivity, before (dashed lines) and at delay time td (solid lines) after photoexcitation at room temperature. The excitation energy is 1.55 eV (above the charge-transfer gap at 1.3 eV) and the excitation density xph is 0.5 photon/Ni site. Polarizations of both the pump and probe lights are parallel to the chain axis. Reprinted with permission from [99]. Copyright APS.

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Fig. 38. Schematic electronic and lattice structures of [Ni1−x Pdx (chxn)2X ]X2 (X = Cl, Br) [102].

gap. In other words, both the Drude weight and the total spectral weight transferred from the charge-transfer band in the photodoped system are similar to those in the chemically doped system. The photoinduced metallic state has an ultrashort lifetime. The temporal characteristics of the photoinduced reflectivity change observed at 0.12 eV agree with those of ∆Neff (0.5 eV) at different xph values. They are reproduced by the sum of three exponential functions, which depend on xph . The ultrafast decay component with a time constant τd = 0.5 ps, which is dominant for xph = 0.5, is characteristic of the photoinduced metallic states [99]. The decay dynamics of the photoexcited state in one-dimensional Mott insulators are also studied on Sr2 CuO3 [101], where the electron–electron scattering with emission of spin excitations is considered as a possible mechanism for the ultrafast relaxation. The electronic structures of halogen-bridged mixed-metal complexes, [Ni1−x Pdx (chxn)2X ]X2 (X = Cl, Br; 0 ≤ x ≤ 1), have been investigated through optical and magnetic measurements [102]. The system is shown to change with increasing x from charge-transfer insulator (often regarded as Mott insulator) to Mott–Hubbard insulator (in a narrow sense) at around x ∼ 0.3, and to Peierls insulator (also called CDW insulator) at x ∼ 0.9 (Fig. 38). Recent experiments on the photoinduced optical properties show a distinct difference between the Mott and CDW insulators. In the nickel-chain compound, the optical conductivity shows a Drude-like low-energy component just after the photoexcitation [99], as mentioned above. The palladium-chain compounds remain insulating with a finite optical gap even after the photoexcitation [103]. In the halogen-bridged metal-chain compounds, the effect of lattice fluctuations is also important. Because the palladium-chain compounds have the CDW order, they are much more sensitive to the lattice fluctuations than the nickel-chain compounds. The low-energy optical responses in the palladium-chain compounds are therefore easily suppressed. Recall that the behavior of carriers in the Mott insulator phase depends on whether V < 2t0 or V > 2t0 . The relations V < 2t0 and V > 2t0 are suggested to hold in the copper-oxide chains and the nickel-chain compounds, respectively, from the comparisons between their optical conductivity spectra and photoconductivity spectra [95]. The excitonic effect is significant in the nickel-chain compounds even in the Mott insulator phase. When the energy of the photoexcitation is large enough for the carriers to be separated, photoconductivity appears as well as the photoinduced metallic property [99]. It is also shown from luminescence measurements in [Ni(chxn)2X ]Y2 [(X, Y) = (Br, Br), (Cl, Cl), (Cl, NO3 )] and theoretical calculations in an extended Peierls–Hubbard model that the electron–lattice interaction is strongly suppressed in the photoexcited states of these one-dimensional Mott insulators [104]. For all the three compounds, luminescence bands have a very small Stokes shift. The adiabatic potentials for local lattice distortions calculated by the exact method for a small

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Fig. 39. Schematic electronic and lattice structures of [Pd(chxn)2 Br]Br2 .

system show no lattice relaxation unless the transfer integral is small. Larger systems are also studied in an effective model in the limit of infinite on-site repulsion. The binding energy of a self-trapped excitonic bound state decreases quickly to zero as a function of the transfer integral in both models. This is in contrast to that in the CDW phase, where the Stokes shift decreases gradually and never drops to zero with continuously changing the lattice deformation pattern. In short, the tendency toward localization is weak in the Mott insulator phase even from the viewpoint of lattice relaxation. Recently, fs pump-probe absorption spectroscopy is applied on a thin film sample of a Br-bridged Ni compound, [Ni(C14 − en)2 Br]Br2 /PMMA [105]. Ultrafast decay of excitons with 1.4 ps may be caused rapid energy transfer from charge to spin degrees of freedom through the emission of spin excitations in one-dimensional Mott insulators. Relatively slow relaxation of charge carriers with several tens of picoseconds is suggested to be caused by their random walk along chains through hopping processes affected by a polaronic effect. 3.2.3. CDW-to-Mott phase transition A photoinduced phase transition from the CDW insulator to the Mott insulator has been observed in a Br-bridged Pdchain compound, [Pd(chxn)2 Br]Br2 (Fig. 39) [106]. The Mott insulator is a metastable phase, which is usually hidden by the stable CDW phase. The lowest-energy optical excitation changes from the Pd2+ → Pd4+ charge transfer (0.7 eV) to the Pd3+ → Pd3+ charge transfer (0.55 eV) immediately after photoirradiation (within the time resolution of 0.14 ps) for excitation density xph = 0.025 photon/Pd site. The converted fraction was estimated at about 55% for td = 0.25 ps, indicating that approximately 22Pd3+ sites are produced by one photon. The photoinduced reflectivity changes at td = 0.15 ps and 10 ps are proportional to the excitation density up to xph = 2.4 × 10−3 photon/Pd site. The signals quickly decay through the geminate recombination process, i.e., the interaction among excited states is negligible, which is consistent with the linear behavior of the relaxation dynamics as a function of the excitation density. The effect of cooperativity is not observed. In the short time domain, an oscillation of frequency ω ∼ 90 cm−1 (period ∼ 0.36 ps) appears and decays with a time constant of 1.28 ps. In the Raman spectra, a peak is observed near 180 cm−1 but no counterpart is found near 90 cm−1 , which is rather close to 115 cm−1 observed by femtosecond time-resolved luminescence spectroscopy for [Pt(en)2 ][Pt(en)2 Br2 ](ClO4 )4 [107]. This transition is also theoretically studied by investigating possible relaxation paths [108]. Irrespective of whether the initial photoexcited state contains the lowest-energy exciton or a separate electron–hole pair, the macroscopic phase transformation is shown to be induced by light without energy barrier. The model used there takes account of both Pd 4dz 2 and Br 4pz orbitals, H = −

X l,σ

+

X l

Ď

t (l)(cl+1,σ cl,σ + h.c.) +

X

e(l)nl +

X l=odd

Vpp nl nl+2 +

X l=even

X

Up nl,↑ nl,↓ +

Vdd nl nl+2 +

Ud nl,↑ nl,↓

l=even

l=odd

l

V (l)nl nl+1 +

X

X Kl l

2

Ql2 ,

(50)

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Fig. 40. Adiabatic potentials as a function of Mott insulator domain size. (a) corresponds to the experiment, which is closer than (b) to the phase boundary between the Mott and CDW insulators. Reprinted with permission from [108]. Copyright APS. Ď

where cl,σ (cl,σ ) is the creation (annihilation) operator of a hole with spin σ at the lth site, and Ql is the displacement of the lth ion along the chain from its equidistant position. The number operators are for the holes. The other notations are standard. The elastic constant Kl is assumed to be infinite for the Pd sites (Ql = 0 for even l). The site-dependent parameters are defined as t (l) = tdp − β(Ql+1 − Ql ), e(l) = ep − 2α(Ql+1 − Ql−1 ) for odd l, e(l) = ed + α(Ql+1 − Ql−1 ) for even l, and V (l) = Vdp − α(Ql+1 − Ql ). The purely electronic, model parameters are determined from fitting to the experimental results [102], while some combinations of the electron–lattice coupling parameters α 2 /K and β 2 /K are chosen to reproduce the CDW gap. The electronic part is treated by the density-matrix renormalization group method, while the lattice part is treated classically. Relaxation paths from the lowest-energy exciton state are studied along the configuration ql ≡ Ql K /α for odd l (Br sites),



ql = q0 (−1)(l−1)/2 1 +

1 2

 tanh



l − l2

w



 − tanh

l − l1

w



,

(51)

where q0 is the CDW amplitude, l2 > l1 for the positions of the domain walls, and w is the half width of each domain wall. Adiabatic potentials for the ground state and the lowest excited singlet and triplet states are shown in Fig. 40 as a function of the domain size l0 ≡ l2 − l1 . In the realistic case (a), the lowest singlet exciton relaxes down to a Mott insulator state without energy barrier. In the case (b) that is located a little deeper in the CDW phase, the CDW gap is larger and a local minimum appears in the relaxation path. At the local minimum, however, the self-trapped exciton is spatially broad. Thus obtained relaxation path without energy barrier suggests linear processes, in that one photon makes a substantial size of converted Mott insulator domain. The photoinduced transition from the CDW to Mott insulator phases is experimentally observed, and its converted fraction shows a linear tendency as a function of the light intensity [106]. 3.3. Halogen-bridged binuclear metal complexes Halogen-bridged binuclear metal complexes have one-dimensional chains composed of repeating metal–metal–halogen (M–M–X) units, so they are called MMX-chain compounds. In the MMX chain, a one-dimensional electronic state is formed by dz 2 orbitals of M and pz orbitals of X. The average valence of M is 2.5, and three electrons exist per two dz 2 orbitals. Four charge-lattice-ordering states are theoretically expected in these systems [109–113]. In addition to the two states shown in Fig. 41, there are an average valence (AV) state (−X− − M2.5+ M2.5+ − X− − M2.5+ M2.5+ − X− −) and an alternating charge–polarization (ACP) state (−X− − M2+ M3+ − X− − M3+ M2+ − X− −). The charge-density-wave (CDW) and charge–polarization (CP) states are stabilized by the displacements of X− ions, while the ACP state is stabilized by the displacements of binuclear MM units. Two classes of MMX-chain compounds with different ligand molecules, pop (pop = P2 O5 H22− ), dta (dta = CH3 CS− 2 ) and its analogs, have been extensively studied, including R4 [Pt2 (pop)4 X]nH2 O (R = K, NH4 and X = Cl, Br) [114,115] and Pt2 (dta)4 I [117,116].

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Fig. 41. Schematic electronic and lattice structures of R4 [Pt2 (pop)4 I]nH2 O.

For Pt2 (dta)4 I, it has been suggested that the ACP is stabilized at low temperature [116]. Because it is composed of neutral chains without counterion, binuclear MM units are so easily displaced that the ACP state is favored, in contrast to the pop systems composed of charged chains and counterions. The ground state of R4 [Pt2 (pop)4 X]nH2 O (R = K, NH4 and X = Cl, Br) has been revealed to be the CDW state. The physical properties of these MMX-chain compounds are theoretically studied by applying the exact diagonalization and other methods to one-dimensional two- and three-band extended Peierls–Hubbard models [118]. For the pop systems, they are well explained by perturbation theories from the strong-coupling limit. As a general trend, the hybridization between the p orbital of X and the d orbital of M in the I-bridged compounds is larger than that in the Cl- and Br-bridged compounds. Then, I-bridged binuclear Pt complexes with pop are the most interesting in that a first-order phase transition between the nonmagnetic CDW state and the paramagnetic CP state (Fig. 41) is realized by modification of counterions located between chains, by increasing pressure, or by photoirradiation [119]. To control the electronic structures of the MMX-chain compounds, many I-bridged binuclear Pt complexes of R4 [Pt2 (pop)4 I]nH2 O and R02 [Pt2 (pop)4 I]nH2 O have been synthesized with counterions (R+ or R0 2+ ) of alkyl-ammonium [R = Cn H2n+1 NH3 , (Cn H2n+1 )2 NH2 , R0 = H3 N(Cn H2n )NH3 ] or alkali-metal (R = Na, K, NH4 , Rb, Cs) [114,119–121]. The substitution of the counterion greatly alters the distance between neighboring binuclear units (i.e., between two Pt ions bridged by the I ion) dMXM = dPtIPt . The change in dPtIPt indeed modifies various electronic parameters such as intersite electron–electron and electron–lattice interactions, realizing the nonmagnetic CDW state and the paramagnetic CP state (Fig. 42). In these compounds, two Pt ions are linked by four pop molecules, forming a binuclear Pt2 (pop)4 unit. The two neighboring Pt2 (pop)4 units are bridged by an I ion, forming the PtPtI chain structure. The iodine ion deviates from the midpoint between the two neighboring Pt ions toward Pt3+ . The CDW state is composed of Pt2+ –Pt2+ and Pt3+ –Pt3+ units and nonmagnetic because the neighboring Pt3+ ions form a singlet state. The CP state is composed of Pt2+ –Pt3+ units, where Pt3+ ions have a spin S = 1/2 coupled antiferromagnetically with each other, forming a one-dimensional paramagnetic spin chain. The optical gap energy ECT is larger in the CP phase than in the CDW state (Fig. 42), as theoretically expected [118]. The Pt–Pt stretching mode of the binuclear unit in the Raman spectra is split in the CDW phase, where two kinds of Pt–Pt units exist: Pt2+ –Pt2+ and Pt3+ –Pt3+ . The assignments of the electronic states are confirmed by measuring the spin susceptibility. From systematic optical, magnetic and X-ray studies, the phase diagram shown in Fig. 42 is obtained. The electronic states are controlled by dPtIPt , and the boundary between the CDW and CP phases is located at around dPtIPt = 6.1 Å. The CDW phase is observed for small dPtIPt , while the CP phase for large dPtIPt . The dependence of the electronic phase on dMXM is understood in a one-dimensional two-band three-quarter-filled Peierls–Hubbard model [123,122], H = −tMM

X Ď X Ď (c2l−1,σ c2l,σ + h.c.) − tMXM (c2l,σ c2l+1,σ + h.c.) l,σ

+β

X l

ql (n2l+1 − n2l ) + UM

l,σ

X l

nl,↑ nl,↓ + KMX

X l

q2l + (M /2)

X l

q˙ 2l ,

(52)

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Fig. 42. Magnitude of splitting in Pt–Pt stretching Raman band ∆ν (Pt–Pt) and optical gap energy ECT as a function of dPtIPt in various PtPtI chain compounds. Reprinted with permission from [119]. Copyright APS. Ď

Ď

where cl,σ creates an electron with spin σ at site l, nl,σ = cl,σ cl,σ , and nl = σ nl,σ . The binuclear unit contains two M (M = Pt) sites, 2l − 1 and 2l. The displacement of the X (X = I) ion between the two M sites 2l and 2l + 1, relative to that in the undistorted structure, is denoted by ql . The transfer integral within the unit is denoted by tMM , while that between the neighboring units through the X pz orbital by tMXM . The energy level of the M dz 2 orbital depends on q with linear coefficient β . The on-site repulsion strength is denoted by UM , the elastic constant for the MX bond by KMX , and the mass of the X ion by M. In the strong-coupling limit of this model, the two phases are energetically degenerate, although the inclusion of the intraunit Coulomb interaction VMM stabilizes the CP phase [118]. For simplicity, we ignore the intersite Coulomb interactions below, but the conclusion is unchanged. The degeneracy at VMM = 0 is lifted by the transfer integrals. In the CDW phase, the 2 intraunit transfer integral tMM lowers it by 2tMM /UM per binuclear unit because the excited state contains a doubly occupied 2 site. In the CP phase, tMM lowers it by tMM /(2β|u|) because the energy levels are split by the coupling β . The interunit 2 transfer integral tMXM lowers the energies of these phases by the same amount, tMXM /(2β|u|). As a consequence, a strong on-site repulsion UM favors the CP phase, while a strong electron–lattice coupling β favors the CDW phase, which have been confirmed by exact-diagonalization studies for reasonable values of transfer integrals [118]. With decreasing dMXM , the coupling β is strengthened because it originates from the electrostatic potential due to the X ion.

P

3.3.1. CDW-CP phase transition Because the phase diagram is a function of dMXM , a phase transition is expected to be driven by applying pressure to the compounds in the CP phase. Such a pressure-induced transition from the CP phase to the CDW phase has been observed with a large hysteresis loop (Fig. 43) [119]. The values of dPtIPt and ECT at the pressure-induced transition are indeed very close to those at the counterion-substitution-induced transition. At the points a and b in Fig. 43, the compound is in a metastable

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Fig. 43. Pressure dependence of optical gap ECT . The insets schematically show free-energy potential curves. Reprinted with permission from [119]. Copyright APS.

phase, as shown in the free-energy potential curves in the insets. Photoinduced transitions from the metastable phase to the stable phase are observed at these points. The photoinduced dynamics of charge density and lattice displacements are calculated in the model (52), after the addition of very small random numbers to the initial lattice variables q and q˙ , by solving the time-dependent Schrödinger equation for the electronic wave functions within the unrestricted Hartree–Fock approximation and the classical equation of motion for the lattice displacements. We first set the coupling β below the transition point within the hysteresis loop so that the CDW phase is metastable. When the metastable CDW phase is photoexcited, a transition to the stable CP phase is indeed realized above a threshold photoexcitation density [122]. The threshold excitation density sensitively depends on the relative stability of the two phases. With decreasing β (decreasing pressure), the energy difference of the two phases is enlarged, the energy barrier between the two is lowered, and the threshold excitation density is lowered. This dependence of the threshold excitation density is easily understood by drawing the diabatic potentials in the two phases, i.e., without considering the transition processes. However, the diabatic viewpoint breaks down when compared with a reverse transition from the CP phase to the CDW phase. We next set β above the transition point so that the CDW phase is stable. Even when the metastable CP phase is photoexcited, a transition to the stable CDW phase is hardly realized by any photoexcitation energy or density [122]. In order to understand the difference between the dynamics from the CDW phase and those from the CP phase, we need to take photoinduced charge-transfer processes into account. The optical excitations in the MMX chain in the CDW and CP ground states are schematically shown in Fig. 44. In the CDW phase, the interunit charge transfer needs a lower energy than the intraunit one because the latter costs large on-site repulsion UM . Through lattice relaxations with the halogen displacements, the 2 + 3 + 2 + 3+ valence state will be stabilized in a photoexcited state. Namely, in the CDW phase, an optical excitation will produce locally a CP state. Therefore, the transition from the CDW phase to the CP phase is easily induced by lights. In the CP phase, the opposite situation holds as long as tMM is larger than tMXM : the interunit charge transfer needs a higher energy even if long-range electron–electron interactions are present (as long as the intraunit nearest-neighbor repulsion VMM is larger than the interunit nearest-neighbor repulsion VMXM ) [118]. The intraunit charge transfer from the 2 + 3 + 2 + 3+ to 2 + 3 + 3 + 2+ valence states is the dominant optical excitation corresponding to the gap transition. Namely, in the CP phase, low-energy charge-transfer processes occur only within a binuclear unit. Therefore, the CDW state is never produced even locally. As a consequence, the CP-to-CDW transition is difficult to be achieved by the photoirradiation with energy near ECT . Even if the CP phase is irradiated by higher-energy photons, charge transfer among neighboring binuclear units from the 2 + 3 + 2 + 3+ to 2 + 2 + 3 + 3+ valence states takes place incoherently. Since the interunit charge transfer is energetically unfavorable, it is difficult for the CDW domains to proliferate. Its transition efficiency should be very low. Furthermore, the transition amplitude connecting the degenerate CDW states with opposite phases is much smaller than those connecting the degenerate CP states (Fig. 45). Suppose a CDW (CP) domain lies next to an anti-phase domain. The smallest unit in a CDW domain consists of two binuclear units. In order to reverse the phase of such a CDW domain, two electrons need to be transferred to the next binuclear unit bridged by the halogen ion. The smallest unit in a CP domain consists of only one binuclear unit. In order to reverse the phase of such a CP domain, only one electron needs to be transferred inside the binuclear unit. Therefore, a wrong phase is easily corrected in the CP phase, helping the growth of a single–phase CP domain. In other words, the coherence of the charge-lattice-order is easily recovered in the CP phase. In iodine-bridged binuclear platinum complexes, R4 [Pt2 (pop)4 I]nH2 O with cation R, phase transitions are indeed observed after the photoirradiation for a long time, 8 ms (for CDW-to-CP) and 30 s (for CP-to-CDW) [119]. To clarify whether

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Fig. 44. Schematic ground and photoexcited states. In the CDW phase, charge transfer among the binuclear units to form a CP domain needs a lower energy than that within the unit. In the CP phase, however, that to form a CDW domain needs a higher energy. Reprinted with permission from [122]. Copyright APS.

the transitions are induced by an optical process or a laser heating, the dependence of the converted fraction on the excitation photon density is investigated for two excitation energies Eex of 1.96 eV and 2.41 eV (Fig. 46) [124]. The converted fractions are estimated from the photoinduced changes in the integrated intensity of the Pt–Pt stretching Raman bands. At the point a in Fig. 43, the CDW-to-CP transition efficiency exhibits a clear threshold Nth , which strongly depends on Eex : Nth ∼ 1.4 × 1025 cm−3 at Eex = 1.96 eV and ∼ 3 × 1024 cm−3 at 2.41 eV. The magnitude of Nth is large, but the irradiated power of the lights is very small and the pulse duration is very long. The threshold behavior demonstrates that the observed transitions are driven not by a laser heating but by an optical process. At the point b, the CP-to-CDW transition could not be driven by the irradiation of 1.96-eV or 2.41-eV (nearly equal to ECT ) light, even if the intensity and duration of the light were changed. However, irradiation with 2.71-eV light for 30 s did result in a CP-to-CDW transition. Therefore, the efficiency of the photoinduced CP-to-CDW phase transition is much lower than that of the CDW-to-CP phase transition. The former is observed near the edge of the hysteresis loop [124]. 3.4. Spin chains coupled with phonons Toward realizing the switching of magnetism by light irradiation, materials near magnetic transition boundaries have been investigated from the expectation that their magnetic instability can be enhanced or suppressed by light. In view of the fact that light fields do not couple directly with spin states, a promising candidate for such optical control is the family of spin-Peierls systems. In these systems, one-dimensional paramagnetic spin states with S = 1/2 are converted to dimerized nonmagnetic states at low temperature in equilibrium. It is known that the introduction of carriers or nonmagnetic impurities can destabilize the nonmagnetic spin-Peierls state. The introduction of photocarriers also modifies the spin states, leading to a macroscopic change in magnetic properties. 3.4.1. Organic radical crystal, TTTA Among the systems possessing the spin-Peierls ground state, the organic radical crystal of 1,3,5-trithia-2,4,6triazapentalenyl (TTTA) has been investigated [125]. It shows a spin-Peierls-like first-order magnetic transition at around room temperature with an anomalously large hysteresis loop (230–305 K) [126]. The high-temperature and low-temperature phases have quite different crystal structures. At high temperature, the intercolumn intermolecular overlap cannot be neglected, as compared with the intracolumn one. In the low-temperature phase, the intracolumn molecular overlaps are considerably larger than those in the high-temperature phase, resulting in an enhancement of one dimensionality. The molecules are strongly dimerized along this direction. Thus, the transition includes a degree of structural character as well as the spin-Peierls character.

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Fig. 45. Schematic coherence-recovering processes. To shift the boundary between CDW domains with different phases, two electrons need to be transferred to the next-nearest-neighbor site beyond the halogen ion. To shift the boundary between CP domains with different phases, only one electron needs to be transferred to the nearest-neighbor site within the unit. Reprinted with permission from [122]. Copyright APS.

The magnetic transition of TTTA can be driven with a nanosecond laser pulse at room temperature and at a very low temperature of 11 K. Photocarrier doping suppresses the spin-Peierls instability [125]. At room temperature, a single-shot irradiation at 2.64 eV causes a permanent color change in the irradiated area from yellow-green of the low-temperature phase to red-purple of the high-temperature phase. The Raman spectra are accordingly altered. At 11 K outside the hysteresis loop also, the low-temperature-to-high-temperature phase conversion with a similar color change is observed. Although the photoinduced phase is essentially the same as the high-temperature phase, it is suggested to be different from the Raman spectra. Such a difference between the photoinduced phase and the thermally induced high-temperature phase has been reported in the spin-crossover complex, [Fe(2-pic)3 ]Cl2 EtOH (2-pic = 2-aminomethyl-pyridine) [14], and much more clearly in Cu(dieten)2 X2 (dieten = N, N-diethylethlenediamine, X = BF4 and ClO4 ) [127]. The conversion efficiency of the photoinduced phase transition shows a threshold behavior as a function of the absorbed photon density per unit volume. It suggests that a cooperative effect between photoexcited species plays an important role in the growth of the macroscopic phase change. (At a point where the material is in the metastable high-temperature phase within the hysteresis loop, however, the high-temperature-to-low-temperature transition could not be driven by irradiation at the same energy, even at very high excitation densities.) The conversion efficiency strongly depends on the excitation energy. The threshold photon density gradually increases as the excitation energy decreases. It suggests that the nature of the initial photoexcited states depends on the excitation energy. From the excitation profile of photoconductivity, the excitonic character is revealed at charge-transfer bands. The threshold photon density decreases as the photoconductivity increases, suggesting that photogenerated charge carriers promote the transition more effectively than excitons. Detailed relaxation processes from excited states with different orientations of charge transfers are studied more recently by femtosecond luminescence spectroscopy [128].

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Fig. 46. Excitation photon density dependence of converted fraction, (a) from CDW to CP at point a in Fig. 43 and (b) from CP to CDW at point b. The insets show the absorption spectra obtained from the polarized reflectivity spectra. The arrows indicate the energy positions of the excitation lights. From [124]. Reprinted with permission from World Scientific.

During these experimental studies, there had been a controversy over the origin of the lowest photoexcited state of TTTA. Then, the optical properties of one-dimensional dimerized Mott insulators are investigated in the one-dimensional dimerized extended Hubbard model [129]. Exact-diagonalization calculations and a perturbative analysis from the decoupled-dimer limit clarify that there are three relevant classes of charge-transfer states generated by photoexcitation: interdimer charge-transfer unbound states, interdimer charge-transfer exciton states, and intradimer charge-transfer exciton states. By fitting of the optical conductivity spectrum, it is concluded that the lowest photoexcited state of TTTA is the interdimer charge-transfer exciton state and the second lowest state is the intradimer charge-transfer exciton state. 3.4.2. Organic charge-transfer compound, alkali-TCNQ A photoinduced inverse spin-Peierls transition has been known to occur for long in potassium tetracyanoquinodimethane (K-TCNQ) crystals [130], which are prototypical one-dimensional Peierls–Hubbard systems. They are composed of onedimensional columns of TCNQ molecules which stand face to face along the a axis. Complete charge transfer between the TCNQ molecule and the cation causes a half-filled π -electron band. Electrons are nearly localized on TCNQ molecules due to the large on-site Coulomb energy (U ' 1.5 eV), which overwhelms the transfer integral (t0 ' 0.2 eV) [131]. Crystals of K-TCNQ undergo a weakly first-order phase transition accompanying the stack dimerization or spin-Peierls distortion at 395 K. The dimerization of molecular sites alternately modulates the electron-transfer energy. The absorption spectrum for the polarization along the columns in the low-temperature phase consists of a chargetransfer band with a maximum around 1.0 eV and a sideband around 1.4 eV. The splitting of the band has been ascribed to the coexistence of intradimer and interdimer charge-transfer excitations in the spin-Peierls phase [132]. Across the phase transition, the magnitude of the charge-transfer gap changes little because it is mostly governed by the on-site Coulomb energy U. The sideband disappears and the absorption profile is broadened above the transition temperature. Pulse- or cw-laser irradiation of a crystal with dimerized stacks produces mesoscopic-size domains with nearly regular stacks. The photoexcitation was done by an excimer-laser-driven dye laser (2.1 eV) with a pulse width of about 10 ns. The photoinduced absorption spectrum with polarization along the stacking axis is well reproduced by the calculated differential-absorption spectrum. It is consistent with the photoinduced change in the infrared vibrational spectra showing a decrease in the intensity of each ag band, which is optically inactive for the regular stack. Thus, photoexcitation in the spin-Peierls phase generates molecular states analogous to those of the high-temperature phase. The magnitude of the

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39

photoinduced reflectivity changes, or equivalently the converted fraction, is approximately proportional to the exciting light intensity. One absorbed photon is estimated to generate a regular domain of about 40 (TCNQ)− molecules. The photoreflectance signal decays according to t −1/2 , indicating that the photoconverted molecular domains are extinguished by recombination of a pair of domain walls walking randomly on the stack. The temperature dependence of the photocurrent exhibits quite a parallel behavior with that of the photoreflectance signal. It suggests that the photoconverted non-spinPeierls domains or their domain walls can carry an electric charge. 3.4.3. Ultrafast photoinduced inverse spin-Peierls transition Quite recently, ultrafast dynamics of the photoinduced melting of the spin-Peierls phase in K-TCNQ have been investigated by femtosecond (fs) pump-probe reflection spectroscopy [133]. Irradiation of a pump pulse of 1.55 eV with a width of 110 fs produces charge carriers forming nonmagnetic sites and melting the spin-Peierls phase within about 400 fs for a density of 0.095 photon/TCNQ. Instantly, the reflectivity decreases over a wide energy region (0.5–1.8 eV) due to the bleaching of the charge-transfer band. Below 0.5 eV, the reflectivity increases, showing a midgap absorption at about 0.25 eV. At delay time td = 1 ps, the photoinduced reflectivity becomes positive in the lower energy region of the chargetransfer band (0.5–0.9 eV). The spectral shape is very similar to the first derivative of the original reflectivity. In other words, photocarriers induce a redshift of the charge transfer band within 1 ps, indicating the photoinduced decrease of the dimerization. After this delayed increase of the photoreflectance, prominent oscillations are observed. The oscillatory component can be reproduced well by using the sum of three damped oscillators. The frequencies (the decay times) of these modes are 20 cm−1 (5.7 ps), 49 cm−1 (0.53 ps), and 90 cm−1 (0.57 ps). Phonon modes responsible for the dimerization are Raman active only in the spin-Peierls phase and show selection rules for light polarization depending on the directions of the displacements. The frequency (20 cm−1 ) of the long-lived oscillation is observed also in the corresponding Raman spectra, so that this is the lattice mode of the dimerized ground state. The long decay time is interpreted as due to the phonon mode of the ground state. This observation of the coherent oscillation with the same (i.e., not doubled) frequency as the Raman mode indicates that the dimerization is not completely lost by the photoirradiation. From the polarization dependence, this mode is assigned to the longitudinal optical (LO) mode corresponding to the dimerization along the stacking axis. For the oscillations of 49 and 90 cm−1 , no Raman bands are observed at the same frequencies. Their decay times are much shorter than that of the photocarriers ∼3 ps. Thus, they are attributed to local phonon modes in the photoexcited state. The transverse optical (TO) modes are possibly assigned to them. The TO modes do not play significant roles for the stabilization of the spin-Peierls phase, in contrast to the LO mode. More recently, photoinduced inverse spin-Peierls transitions in K-TCNQ and in Na-TCNQ are reported in detail [134]. It is clarified that, in Na-TCNQ, both the LO and TO modes play an important role in stabilizing the spin-Peierls phase. Such a difference between K-TCNQ and Na-TCNQ is caused by the difference of the crystal structures. 3.4.4. Adiabatic picture for melting of molecular dimerization Dynamical properties of such photoexcited states as above are theoretically studied in a one-dimensional Mott insulator with lattice dimerization [135]. Interdimer charge-transfer excitations indeed efficiently destabilize the dimerized phase, leading to the photoinduced inverse spin-Peierls transition. In addition, a mainly electronic origin of the midgap state is proposed. Used is the one-dimensional dimerized Hubbard model at half filling with elastic energy, H = −t0

X X N [1 + (−1)l δ](clĎ,σ cl+1,σ + clĎ+1,σ cl,σ ) + U nl,↑ nl,↓ + βδ 2 , l,σ

l

2

(53)

where N denotes the system size, β the elastic constant, and the other notations are standard. The exact diagonalization method is employed with the periodic boundary condition. To describe photoexcited states appearing in the optical conductivity spectra σ (ω), different approaches exist: odd and even charge-transfer states [132] and interdimer and intradimer charge-transfer states [135]. Both approaches are complementary to each other. Without dimerization, only odd states are optically active. For finite dimerization δ , a chargetransfer state with even parity becomes optically active acquiring a small spectral weight. This description is useful for small-δ systems. The interdimer charge-transfer state described below connects adiabatically to the lowest photoexcited state of the regular (i.e., undimerized) system (without nearest-neighbor repulsion), which is an odd charge-transfer state. Both the interdimer and intradimer charge-transfer states are linear combinations of the odd and even charge-transfer states. As δ increases, the degree of hybridization changes continuously. It is useful to introduce a perturbation theory that is justified in the decoupled-dimer limit, δ → 1. Both the interdimer and intradimer charge-transfer states below are found to have finite spectrum intensity even near the decoupled-dimer limit. The perturbation theory with respect to the interdimer transfer integral can treat the on-site repulsion and the intradimer correlation exactly [135]. Thus, it is appropriate in constructing a simple picture of photoexcited states. The zeroth-order ground state is the direct product of the ground states for dimers. One class of relevant photoexcited states in the decoupled-dimer limit consists of intradimer charge-transfer states, which are obtained by transferring an electron within a dimer. The other class of important photoexcited states consists of interdimer charge-transfer excitations, which

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Fig. 47. Schematic picture of ground state, intradimer and interdimer charge-transfer states. The ellipses show the singlet dimer states, and the up (down) arrows are electrons with up (down) spins. Reprinted with permission from [135]. Copyright APS.

Fig. 48. Dynamics of dimers in optically excited states. Effective transfer integrals for three- and one-electron dimers in |ψA i are much larger than that for a two-electron excited dimer in |ψB i. Reprinted with permission from [135]. Copyright APS.

are generated by transferring an electron between neighboring dimers (Fig. 47). For any positive U (and without nearestneighbor repulsion), it is easy to show that the excitation energies of the intradimer charge-transfer states are larger than those of the interdimer charge-transfer states, to the first order in the interdimer transfer integral. The energy differences are basically due to the kinetic energies of a doublon and a holon inside the respective dimers (i.e., due to the intradimer transfer integral). In the regular case (δ = 0), the optical conductivity spectrum σ (ω) has a band at U − 4t0 < ω < U + 4t0 corresponding to excitations from the lower to upper Hubbard bands. With finite δ introduced, new optically allowed states are generated, and some of the photoexcited states at δ = 0 get weakened spectral intensity or eventually disappear. However, the lowest photoexcited state survives up to the decoupled-dimer limit and is called |ψA i hereafter. It is an interdimer charge-transfer state. For δ > 0.4, a new optical state appears as a sharp peak slightly above ω = U and is called |ψB i hereafter. Its wave function clearly shows that it is an intradimer charge-transfer state (Fig. 48). The first-order degenerate perturbation theory shows that the three-electron dimer and the one-electron dimer propagate freely with effective transfer integrals proportional to t0 (1 − δ) in |ψA i. Meanwhile, the two-electron excited dimer in |ψB i has a much smaller effective transfer integral, which is proportional to t02 (1 − δ)2 . The perturbation theory explains the exact-diagonalization results of the excitation energies very well. The adiabatic potentials of the ground state |ψ0 i and the optically excited states |ψA i and |ψB i are shown in Fig. 49. That of |ψA i has a minimum at δ = 0, which demonstrates that the dimerized state is destabilized by the optical excitation from |ψ0 i to |ψA i. When the state |ψA i is photoexcited, the system can move toward the regular state (δ = 0) along its adiabatic potential unless the recombination of the doublon and the holon occurs. This is the realization of the inverse spin-Peierls transition. By contrast, the potential of |ψB i has a minimum at a finite value, suggesting that |ψB i does not annihilate the lattice dimerization although it can weaken it. Whether the lattice dimerization is stable or not in the charge-transfer states depends on how the spin singlets are destroyed by the charge-transfer excitations. Then, a quantity is calculated that measures the strength of the spin dimerization, D ≡ |hS0 · S1 i − hS1 · S2 i|. The results for the three states are shown in Fig. 50. The ground state has large spin dimerization even when δ is rather small. By contrast, that of |ψA i is strongly reduced, which indicates that a large part of the spin dimers is destroyed by the interdimer charge-transfer excitation. In |ψB i, D has a larger value than that of |ψA i. This result is consistent with the fact that the strength of |ψA i for destabilizing the dimer phase is much larger than that of |ψB i. Next, we discuss optical responses of the system after the photoexcitation to consider the appearance of the midgap state in the reflectivity spectrum of K-TCNQ immediately after the photoexcitation [133]. The photoexcitation in this experiment corresponds to |ψA i. Because the midgap state appears immediately after the photoexcitation, at first we do not consider the lattice relaxation. Then, the key quantity is the optical conductivity in the state |ψA i given by Eq. (48), where ψ1opt is

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41

Fig. 49. Adiabatic potentials of ground state |ψ0 i and optically excited states |ψA i and |ψB i, for N = 8. Reprinted with permission from [135]. Copyright APS.

Fig. 50. Spin dimerization, D ≡ |hS0 · S1 i − hS1 · S2 i|, in ground and optically excited states. Reprinted with permission from [135]. Copyright APS.

replaced by |ψA i. Its numerical-diagonalization results are shown in Fig. 51 for U /t0 = 15 and N = 8. There appear the large Drude peak and another pronounced low-energy peak. The presence of the Drude peak suggests the metallic property caused by photodoped carriers, a doublon and a holon [89,93]. The latter low-energy peak corresponds to the excitation from |ψA i to the nearly degenerate holon–doublon state inherent in one-dimensional Mott insulators [92]. As δ increases, these peaks lose their intensity and several new peaks appear. The high-energy peak around ω ∼ U corresponds to the intradimer charge-transfer excitation because |ψA i still includes a large number of dimers in the ground state of the two-site model. More interesting are midgap peaks around ω ∼ 4t0 , which develop with increasing δ and are closely related to the dimerization. To the zeroth order in the interdimer transfer integral, the excitation energy is given by 2t0 (1 +δ), as observed in Fig. 51. The first-order degenerate perturbation theory with respect to the interdimer transfer integral reproduces the exact-diagonalization results of the excitation energies and the spectral intensities very well [135]. Therefore, the observed midgap state in K-TCNQ might be due to the intradimer charge-transfer excitation from the photoexcited state |ψA i. However, this is inconsistent with the experimental fact that the Drude component that should be caused by the photoinduced carriers does not appear. An alternative mechanism is also discussed, where polaronic localized midgap states are generated by the photoexcitation [136]. Two types of phonon modes, intermolecular and intramolecular vibrations, were taken into account in the adiabatic approximation. Numerical results based on the density-matrix renormalization group demonstrate that the intermolecular lattice distortion is necessary to reproduce the photoinduced midgap absorption in KTCNQ. There are two types of midgap states. One is a usual polaronic state characterized by a localized elementary excitation. The other is superposition of two types of excitations, a doped-carrier state and a triplet-dimer state, which can be generally observed in one-dimensional dimerized Mott insulators.

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Fig. 51. Optical conductivity spectra in photoexcited state |ψA i for U /t0 = 15. Reprinted with permission from [135]. Copyright APS.

4. Photoinduced macroscopic parity violation and ferroelectricity In the previous sections, we have been concerned with various photoinduced structural phase transitions, which, after the complicated nonlinear lattice relaxation, finally result in false ground states. These states are, of course, no more luminescent. However, by recent experimental studies on SrTiO3 and KTaO3 , we have discovered a new-type photoinduced structural phase transition, which occurs only within the optical excited state, keeping this state still luminescent. In these materials, a photogenerated conduction electron results in a ferro- or super-para-electric domain with a macroscopic parity violation. 4.1. Large polaron, self-trapped polaron, linear and quadratic couplings An electron, excited from a valence band to a conduction one by a photon in an insulating solid, often forms a quasiparticle called polaron, being composed of the original electron and the phonon cloud around itself. This polaron effect, coming from the interaction of the electron with phonons or lattice vibrations, has been a matter of considerable interests for these 50 years, and many experimental and theoretical studies have already been devoted. It is one of the most basic themes related with various fields of the solid state physics, such as optics and electronic conductivities in many kinds of semiconductors and insulators [137–139]. At present, it is well-known, that this polaron can be clearly classified into two types, the large polaron and the self-trapped one, provided that the electron–phonon (e–p) interaction is short ranged in ordinary three dimensional crystals [138]. These two different types are brought about through the competition between the quantum itineracy of the electron and the strength of the e–p coupling. When the e–p coupling is weak as compared with the itineracy or the energy bandwidth of this conducting electron, we can get the large (or free) polaron, extending over a wide region of the crystal. In this case, the phonon cloud or the lattice distortion cloud around the electron has a large radius as compared with the lattice constant of this crystal. Meanwhile, this cloud is rather thin because of the weakness of this e–p coupling. These large polarons, once formed in a crystal by photoexcitations, can greatly contribute to increase the photoconductivity or the electronic conductivity. On the other hand, when the e–p coupling is strong as compared with the energy bandwidth, the electron self-induces a potential well of local lattice distortion only around a single lattice site of the crystal, and is trapped in it. This is the self-trapped (or small) polaron, which never contributes to the ordinary electronic conductivity. It is also well-established, that these two states, the large and self-trapped polaron states, are clearly separated by an energy barrier in the adiabatic potential surface, just like first-order phase transitions. Hence, the large polaron state can remain as a locally stable state in the adiabatic potential surface, even when the e–p coupling is strong enough to make the self-trapped state globally stable [138]. However, in this competition between the large polaron and the self-trapped one, the original e–p coupling is tacitly assumed to be linear and short ranged. That is, a local electron density at a certain position only linearly couples with a phonon field of the same position through a contact interaction. Consequently, if the original crystal lattice has space inversion symmetry, only the phonon mode with even parity can contribute to this coupling, and possible contributions coming from odd phonon modes are excluded. For this reason, in this section, we will see possible contributions coming from odd phonon modes under the condition that our starting crystal lattice has space inversion symmetry [140,141]. In this case, the conduction-band electron couples, not linearly, but quadratically with odd-mode phonons. The effects of this quadratic coupling have not been seriously taken

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43

Fig. 52. Schematic structure of SrTiO3 .

into account, or often been neglected, because this quadratic coupling can be easily merged into the original restoring force of this mode, if this force is very strong or hard. However, when this odd mode is quite soft and anharmonic, the quadratic e–p coupling, switched on by the photogeneration of an electron, will result in parity violating instabilities and ferroelectric phase transitions, which are absent in the cases of even modes. 4.2. Photoinduced phenomena in SrTiO3 This problem is closely related with recent optical experiments on the perovskite type SrTiO3 [142–145]. As shown in Fig. 52, the structural unit of this material is an O2− octahedron with a Ti4+ ion in its center [146]. The six apices of this octahedron are connected with each other three-dimensionally. The top of the valence band of this material is mainly composed of the 2p orbital of O, and the bottom of the conduction band is mainly composed of the 3d orbital of Ti [147]. In between, there is a wide indirect energy gap of about 3.2 eV [145,147]. Irradiating this wide gap material by ultraviolet light, Yasunaga [142], Takesada et al. [143], and Hasegawa et al. [144] have very recently found that the electronic conductivity and the quasi-static electric susceptibility are macroscopically enhanced. From the Hall coefficient measurements by Yasunaga [142], this increase of the conductivity has already been known to come, not from the holes, but from the electrons. Hence, a pair of the 3d electron and the 2p hole generated by the light is shown to separate into mutually independent, an electron and a hole, with no exciton effect in between. After this separation, only the electron remains as a mobile carrier, while the hole is assumed to be trapped and localized by some reason, which is unknown. On the other hand, the quasi-static electric susceptibility, or the real part of the quasi-static dielectric permittivity (≡ ε1 ), is shown to increase about 103 , between before and after this irradiation [143,144]. An extraordinarily long lived luminescence, arising from this irradiation, is also found by Hasegawa et al. [145] at 2.4 eV, and the electron and the hole are thus shown to recombine and disappear. Meanwhile, this luminescence has a large Stokes shift of about 0.8 eV. It tells us that the electron and the hole strongly couple with the phonon mode, which mainly corresponds to the breathing (A1g ) type motion of O2− s around Ti4+ , and has an energy of about 20 meV [145]. What is very interesting is that the aforementioned enhancement of ε1 disappears as this luminescence terminates. It clearly means that this enhancement is directly related to the presence of the electron or (and) the hole. While, quite recently, Nozawa et al. [148] have succeeded to observe the photoinduced macroscopic parity violation in SrTiO3 , using the x-ray inner core absorption spectroscopy. The 1s–3d inner core transition of the Ti ion, having an energy of about 4968 eV, is dipole forbidden, if this ion is exactly at the center of the octahedron (Fig. 52). However, it becomes dipole allowed partially, when the Ti ion moves to an off-center position. In this connection, they observed, not a microscopic, but a macroscopic increase of this transition intensity between before and after the UV light irradiation. 4.2.1. Quantum dielectric, soft-anharmonic T1u mode and quadratic coupling The perovskite type compound SrTiO3 is well-known as a quantum dielectric, and long before the aforementioned optical studies, many elaborate works have already been systematically devoted to the ground state properties of this material without photogenerated electron (hole) [149–154]. They are mainly concerned with low temperature properties of ε1 , in relation to soft and anharmonic natures of the T1u (TO) mode, which mainly corresponds to an off-center type displacement motion of a Ti4+ ion, from the central position of the O2− octahedron. In Fig. 53, we have schematically shown the pattern of this off-center type T1u mode, together with the aforementioned A1g mode. Qualitatively speaking, the Ti4+ ion is only loosely confined within this octahedron, and hence, it has a very large quantum fluctuation around the central position. Thus, we can expect to get various parity violating instabilities and ferroelectric phase transitions. However, this material SrTiO3 remains only in the super-paraelectric phase down to 0 K without global frozen parity violation, being called quantum dielectric with no ferroelectric phase transition. According to Müller et al. [150], this T1u mode has almost no, or even a negative restoring force, if it is described only in terms of this T1u mode coordinate. However, it has an effective positive frequency of about 1 meV, which comes from a

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K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

Fig. 53. Schematic nature of T1u and A1g modes. Reprinted with permission from [140]. Copyright APS.

quite specific anharmonicity, called ‘‘bi-quadratic mode–mode coupling’’ between this T1u mode and other acoustic phonon modes [150,151]. In many cases of soft-anharmonic phonons, we are often tempted to simply imagine an ordinary singlemode double-well type anharmonicity, composed of a positive quartic potential and a negative quadratic one, only in terms of this site localized T1u phonon coordinate. However, by Müller et al. [150] and also by Vogt [152], this type single-mode quartic anharmonicity is completely shown not to be realized in this material at low temperatures, while the single-mode sextic anharmonicity model is shown to be rather appropriate [152]. By the Raman scattering measurements, this soft T1u phonon is proved to couple not linearly but quadratically with electronic excitations [155], although its coupling strength seems to be one order of magnitude smaller as compared with the linear coupling of the breathing (A1g ) mode. Together with the aforementioned large quantum fluctuation, this quadratic coupling is expected to result in some macroscopic parity violation in the photoexcited states, although it is never realized in the ground state. 4.2.2. Possible scenario Let us now proceed to a possible scenario that describes the aforementioned enhancement of the conductivity and ε1 in the photoexcited state. By a UV photon irradiation, a conduction electron is generated in the 3d orbital of Ti, and this orbital makes an energy band with a width of about 2 eV, through its hybridization to the 2p orbital of O [147]. This electron linearly and strongly couples with the breathing (A1g ) mode of O around the Ti, and hence, as explained before, we have both the metastable large polaron state and the globally stable self-trapped state. Since the large polaron state is energetically higher than the self-trapped one, the photogenerated electron reaches this state first [156]. As explained before, this state can contribute to the electronic conductivity. On the other hand, this large polaron state quadratically couples with the T1u phonons, and the sign of this quadratic coupling constant is ‘‘negative’’ for the reason we see later in detail. Thus, the quadratic coupling makes this soft mode more ‘‘soft’’. This further softening due to the e–p coupling occurs at very large number of lattice sites, which are included in this large polaron radius. Since this mode is dipole active, this further softening consequently contributes to the enhancement of quasi-static ε1 . 4.2.3. Model Hamiltonian Keeping this scenario in mind, let us start from the following model Hamiltonian (≡ Hf ) that describes the electrons in the 3d conduction band of SrTiO3 , coupling linearly with the breathing mode and quadratically with the T1u mode, as, (h¯ = 1)

X

Hf = −Tf

`,`0 ,σ

+ Ď

(c`,σ

ωd X 2

`,i

Ď

c`0 ,σ

−

+ h.c.) − Sb ωb

X

n` A` +

`

D6`,i ∂2 + 2 3 m∂ D`,i

 ωb X 2

`

 ∂2 S d ωd X 2 − 2 + A` − n` D2`,i 2 ∂ A` `,i

! + Uf

X `

n`,α n`,β ,

Ď

n`,σ ≡ c`,σ c`,σ , n` ≡

X σ

n`,σ .

(54)

Here, c`,σ is the creation operator of an electron at lattice site ` with spin σ (=α, β) in a simple cubic crystal, and Tf is the transfer energy between neighboring two lattice sites ` and `0 . Sb is the dimensionless constant of the linear coupling between this electron and the site-localized breathing (A1g ) mode, whose energy is ωb and dimensionless coordinate is A` . On the other hand, Sd (> 0) is the dimensionless constant of the quadratic coupling between the electron and the site-localized T1u mode, whose energy is ωd and dimensionless coordinate in the direction i (=x, y, z ) is D`,i . Uf denotes the intra-site (intra-orbital) Coulomb repulsion. The dispersions of phonons are neglected, and only the long-wave characteristics of each mode are taken into account. The origin of a short range quadratic coupling between a site localized T1u mode and an electron around it, is already well-known to be an off-center effect [157,158]. It comes from the local mixing between the occupied atomic orbital of the electron, and unoccupied ones which are energetically higher than this occupied one but have opposite parities. This mixing

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is brought about by a linear change of the ligand crystal field coming from the off-center type T1u displacement of a central atom. Within the second-order perturbation theory, the resultant energy change (≡∆E) of the occupied state is given as

∆E = −D2

X ho|V |jihj|V |oi Ejo

j

,

(55)

where |oi and |ji denote occupied and unoccupied atomic orbitals, whose parities are opposite to each other, and Ejo is the energy difference between them. V is an electronic operator with T1u symmetry, and denotes the linear change of the ligand crystal field due to the T1u type displacement D, whose site and direction indices ` and i are omitted. We can see that ∆E is always negative as long as |oi is the highest occupied atomic orbital. Thus, we can get a negative quadratic interaction, and it is a driving force for the central atom to push off its original position [158]. We should note that this ‘‘off-center’’ effect is nothing but the local parity violation, resulting in a local and microscopic ferroelectricity. In the present case, the Wannier function of the conduction-band electron is mainly composed of the 3d orbital of Ti, hybridized with the 2p orbitals of O. This Wannier function mixes with the 4p orbital of Ti and the 3 s orbital of O due to the T1u type off-center displacement of the central Ti ion. In addition to these typical atomic orbitals, there are many other unoccupied higher electronic states, which can locally mix with the Wannier function. Thus, we can get a negative quadratic coupling as described by the fourth term of Eq. (54), where Sd only phenomenologically represents the effects of all aforementioned local mixings. The typical energy difference Ejo is about 10 eV, and hence this off-center effect is usually neglected. As mentioned before, if the original restoring force of this mode is very strong or hard, this quadratic coupling can be easily merged into this original hard force. However, in the cases of soft-anharmonic modes, the quadratic coupling becomes very important in connection with parity violating instabilities, even though it is not strong. It should be noted that, our purpose in the present section, is not to estimate this coupling energy Sd quantitatively. Our purpose is to clarify the possible parity violations coming from this quadratic e–p coupling, under the condition that this effect is one order of magnitude smaller than both the itineracy of the electron and the linear e–p coupling. As for the anharmonicity of the fifth term of Eq. (54), we have taken a sextic one, for the reason mentioned before in detail. While ‘‘m’’ in this fifth term denotes the dimensionless effective mass of this site-localized T1u mode. This mass will be determined so that the lowest vibronic excitation energy of this soft-anharmonic Hamiltonian becomes equal to ωd , which is obtained by the experiment [151]. We now rewrite Hf into the following dimensionless form as h ≡ Hf /ωb

X

= −t

`,`0 ,σ

+

Ď (c`,σ c`0 ,σ + h.c.) + u

γ sd X 6

`,i

d6`,i −

γ

X

n`,α n`,β − sb

X

n` a` +

`

`

sb X 2

`

a2` −

1 X ∂2 2sb

`

∂ a`

2

−

γ sd X 2

`,i

X ∂ , ∂ d2`,i `,i

n` d2`,i

2

1/3 2msd

(56)

where t ≡ Tf /ωb , −1/2

a` ≡ sb

u ≡ Uf /ωb ,

sb ≡ (Sb )2 ,

−1/6

A` ,

d`,i ≡ sd

γ ≡ ωd /ωb ,

sd ≡ (Sd )3/2 ,

(57)

D`,i .

Assuming t, u, sb , and sd are all greater than unity

(t , u, sb , sd )  1,

(58)

we can define an adiabatic Hamiltonian had as h → had ≡ −t

X `,`0 ,σ

+

Ď (c`,σ c`0 ,σ + h.c.) + u

γ sd X 6

`,i

X `

d6`,i .

n`,α n`,β − sb

X `

n` a` +

sb X 2

`

a2` −

γ sd X 2

`,i

n` d2`,i (59)

4.2.4. Variational method for polaron Let us now proceed to a variational calculation for a polaron. For this single polaron state |pi, we use a trial function [≡ φ(`)] as,

|pi ≡

X `

Ď φ(`)c`,α |0ii,

X `

|φ(`)|2 = 1,

(60)

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where |0ii is the true electron vacuum. After taking the expectation value of had with respect to this |pi, we get

*

+

hhad i = −t

X

(· · ·) − sb

X γ sd X γ sd X 6 sb X 2 hn`,α ia` + a` − hn`,α id2`,i + d`,i , 2

`

`,`0 ,σ

2

`

`,i

6

`,i

(61)

h· · ·i ≡ hp| · · · |pi. By the Hellmann–Feynman theorem, we obtain as

∂hhad i = 0, ∂ a`

∂hhad i = 0, ∂ d`,i

→ hn`,α i = a` ,

→ hn`,α i1/4 = d`,i .

(62)

Substituting this Eq. (62) into Eq. (61), we finally get

*

+

hhad i = −t

X

(· · ·) −

`,`0 ,σ

sb X

hn`,α i2 −

2

`

γ sd X hn`,α i3/2 . 3

(63)

`,i

4.2.5. Continuum approximation and super-paraelectric large polaron We now take the following Gaussian type trial function with a reciprocal localization length (≡ ∆p ) as

" φ(`) ∝ exp −

∆2p (` · `) 2

# ,

` = (`x , `y , `z ),

(64)

where `x , `y , and `z are the Cartesian components of `. Before we go in detail about the numerical results calculated by using these Eqs. (63) and (64), let us simply and qualitatively examine energetics of hhad i. For this reason, we, at first, use the continuum approximation and regard `i (i = x, y, z) as a continuous variable from −∞ to ∞. In this case, the first term of Eq. (63) is reduced as

* −t

+ X `,`0 ,σ

(· · ·) →

Z

∞

Z

∞

∞

dly

dlx −∞

Z

−∞

−∞

" dlz φ(`) −t

∂2 ∂2 ∂2 + + ∂`2x ∂`2y ∂`2z

!# φ(`),

(65)

where all the one-body energies are referenced from the bottom of the conduction band. The other two terms in Eq. (63) can also be easily calculated within this continuum approximation. The result is shown in Fig. 54 as a function of ∆p . When (γ sd ) = 0, the first term of hhad i, being the kinetic energy, is easily shown to increase in proportion to ∆2p , while the second term, being the energy gain due to the self-trapping by the breathing (A1g ) mode, decreases in proportion to −∆3p , as shown in Fig. 54(a). If these two competing quantities are of the same order, we get two minima of hhad i. The minimum at ∆p = 0 in Fig. 54(a) denotes the well-known large (or free) polaron, while the other minimum at large ∆p , not being written explicitly, is the self-trapped (small) polaron. It should be noted that all these states have even parities irrespective of ∆p . As mentioned before, these two states are separated by an adiabatic energy barrier, and hence, the large polaron state can be locally stable, even when the self-trapped polaron state is globally stable [137,138]. When (γ sd ) 6= 0, however, we can easily prove that the energy gain due to the T1u type parity violation, which is the 3/2 third term of Eq. (63), is proportional to −∆p , being the most dominant in the small limit of ∆p as shown in Fig. 54(b). Hence the previous minimum at ∆p = 0 now moves to a small but finite ∆p . Thus, we can get a very important conclusion within the adiabatic approximation. An infinitesimal γ sd is enough to change the even-parity large polaron into the parity violating one, although the energy gain due to this symmetry breaking is quite small. This parity violation occurs at a large number of lattice sites included in the large polaron radius, and hence it is a quasi-global parity violation, as schematically shown in Fig. 55. Meanwhile, the self-trapped state, not being written explicitly in Fig. 54(b), also has a parity violation, but it is quite local. Hence, we can call it the off-center type self-trapped state, where only the central atom will move off the original position, as shown in Fig. 55. This off-center effect will be rather slight, since the lattice distortion of this state is mainly dominated by the strong coupling of the breathing mode. It should be noted that this parity violation of the T1u mode is tacitly accompanied by the parity violation of the Wannier function itself, as mentioned in Eq. (55). Originally, this function has T2g or Eg symmetry of the Oh point group, since it is the 3d orbital of Ti. However, as the T1u type off-center displacement occurs, it mixes with unoccupied T1u type atomic orbitals, resulting in a finite dipole, and the parity violation in the electronic level too, although this change is hidden. It is essential that the phase of this local parity violation is quite random, and has no inter-site coherence, since phonons are site-localized ones, and the quadratic coupling is also independent of this phase. However, a weak external electric field may be enough to make it spatially coherent, as schematically shown in Fig. 55. In this sense, our new large polaron is super-paraelectric(SPE) one, which has a quasi-global parity violation. It is essentially the same as a charged and conductive ferroelectric domain.

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

(a) (γ sd ) = 0.

47

(b) (γ sd ) 6= 0.

Fig. 54. Schematic energetics of self-trapping, (a) without T1u mode and (b) with T1u mode. Reprinted with permission from [140]. Copyright APS.

Fig. 55. Schematic natures of the ground state, the super-paraelectric large polaron, and the off-center type self-trapped polaron. Reprinted with permission from [140]. Copyright APS.

Let us now examine the nature of the anharmonicity used in this theory. For the reason mentioned above in detail, the sextic anharmonicity is used in Eq. (54). In this sextic case, an infinitesimal γ sd is enough for the parity violation to occur. However, even if we change this sextic anharmonicity to other ones higher than quintic as,

ωd X D6`,i 2

`,i

3

→

ωd X |D2M `,i | 2

`,i

M

,

M > 2.5,

(66)

we can easily prove the occurrence of the same adiabatic instability for an infinitesimal γ sd . Meanwhile, in the case of the aforementioned single-mode quartic double-well, the instability already occurs in the ground state itself, and hence, the photogenerated electron may not induce further instabilities. 4.2.6. Numerical results Keeping these general characteristics in mind, let us see the results of further numerical calculations. The conduction bandwidth 12Tf is already known to be 2 eV from the non-relativistic augmented-plane-wave calculation [147]. According to the absorption spectra [145], the energy of the breathing mode ωb is also known to be about 20 meV, while, from Raman

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Fig. 56. Adiabatic energy per electron with respect to ∆p . The solid curve denotes the energy of the present model. The dashed curve denotes the energy without the T1u mode. Inset (a) gives the detailed information of the energy minima in the small ∆p region. A comparison between the single large polaron (solid line or dashed line) and the large bipolaron (dotted line) is also given. Inset (b) shows the contribution of each part of the Hamiltonian shown in Eq. (54). The solid, dot, and dash-dot curves denote the A1g part, the T1u part, and the (kinetic energy +6T) part, respectively. Parameters are Sb = 9.5 and Sd = 35.3. Reprinted with permission from [141]. Copyright APS.

scattering [151], the T1u mode ωd is about 1 meV. The Coulomb repulsion Uf of the 3d orbital is usually one order of magnitude greater than Tf , so we take Uf = 4 eV. On the other hand, m in Eq. (54) is assigned to be 1.294, so that the lowest transition energy between the ground state and the first excited one of the original sextic anharmonic oscillator fits the experimental observation [151]. The two coupling constants Sb and Sd can be determined, so that they, as a set, reproduce the observed large Stokes-shift (0.8 eV) of the luminescence [145], i.e., the global minimum point of the adiabatic surface has a 0.2 eV offset below the bottom of the conduction band. This value, 0.2 eV, being the binding energy of the self-trapped electron, is also confirmed from the new infrared absorption band induced by the ultraviolet light irradiation [159], and also from the photo-electron emission spectrum of SrTiO3 with an n-type low doping [160]. The quadratic coupling, being always subsidiary to the linear one, is assumed to be of the order of 10% of this offset. In contrast to the previous continuum approximation, we now apply a discrete lattice model with 1000 lattice sites each along x, y, and z directions. From Eqs. (63) and (64), we can thus obtain the adiabatic energy for a couple of typical values of Sb and Sd (Sb = 9.5, Sd = 35.3), in accordance with the aforementioned experiments [145,159,160]. The result is shown in Fig. 56, where the solid curve gives a typical diagram of the adiabatic surface of the one-electron system, which has two minima. At ∆p = 2.2, there is the global minimum, where the electron is tightly bound almost within one lattice site. This is the off-center type self-trapped polaron. In the region ∆p < 1.0, as we can see clearly from the solid curve in the inset (a) of Fig. 56, there is a rather shallow bound state, extended over about 1000 lattice sites. This state is equal to a charged quasi-macroscopic domain. Such a state is favorable for polarization, and can be called super-paraelectric (SPE) large polaron, as we have already schematically shown in Fig. 54(b) and Fig. 55. In the inset (b) of Fig. 56, each part of the energy of Eq. (63) is given. The solid, dot, and dashdot curves denote the energy of the A1g part, that of the T1u part, and the (electronic kinetic energy +6Tf ), respectively. In the region ∆p < 1.0, the T1u part is the most dominant. In fact, at ∆p = 1.0, the energy from the A1g part is almost zero, which indicates that the large polaron state is determined by a parity violation induced by the quadratic coupling between the photoexcited electron and the T1u phonons. Meanwhile, at the region ∆p > 1.5 the effect of the A1g mode rapidly increases, and at ∆p = 2.2 it becomes prevailing. So, the self-trapped state is dominated by the breathing mode. 4.2.7. Bipolaron Let us now proceed to a few but plural polaron cases. Even in these cases, we can calculate the total energy per electron by almost the same method as mentioned above. A two-electron system cannot form any strongly localized states, in contrast to the one-electron system, because of large Uf . However, two electrons can be loosely bound in an extended state, as shown by the minimum of the dotted curve in Fig. 56(a), since Uf does not work at all in this extended case. It is just an SPE large bipolaron. We can see that the SPE large bipolaron is also a charged and conductive macroscopic domain, and dipole active. These extended states as shown in Fig. 56(a) are very shallow in contrast to the self-trapped one, being metastable states relative to the latter. In addition, from the energy difference between the two adiabatic minima in Fig. 56(a), we can expect that two SPE large polarons will aggregate to form an SPE large bipolaron. Natures of various polaron clusters larger than the bipolaron are also discussed in detail by Qiu et al. [141].

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49

0 Fig. 57. Phonon softening and dielectric enhancement as a function of Sd . Square blocks are for phonon softening. ∆E10 is the energy difference between the ground state and the first excited state of the single sextic oscillator coupled with the electron quadratically. Filled circles are for dielectric ratio ∆ . Reprinted with permission from [141]. Copyright APS.

4.2.8. Phonon softening and dielectric enhancement The aforementioned e–p interactions, on the one hand, cause the electron to get into the previous two bound states, but, on the other hand, causes the frequency change of phonons. Next, we will proceed to this phonon frequency change due to e–p interactions. In the SPE large polaron state, Ti ions are correlated with the photoinduced electron, which is itinerant within the macroscopic domain. As a result, this polaron involves a great number of vibration modes. For brevity, we focus only on a single T1u mode at a typical lattice site, for example, the x direction of the central site of the SPE large polaron, and keep all coordinates of the other modes at their equilibrium positions given by the aforementioned adiabatic theory. So we can 0 ) only for this particular T1u mode, as get an effective Hamiltonian (≡ H1u 0

H1u

  ∂2 D6 = −Sd ωd |φ(0)| D + − + , 2 m∂ D2 3 2

2

ωd

(67)

where φ is the Gaussian type electronic state announced in Eq. (60). The square-block-curve in Fig. 57 gives the calculated 0 0 , as a function ) from the ground state to the first excited state of this single-mode Hamiltonian H1u transition energy (≡ ∆E10 of Sd . It is shown that the e–p interaction does cause softening of this mode. The stronger the quadratic coupling is, the greater the softening gets. Since the strength of the linear coupling Sb ωb is usually much larger than that of the quadratic coupling Sd ωd , we have taken the region Sd < 40 as a practical case. For the original T1u mode, its Hamiltonian (≡H1u ) is already given as H1u =

ωd

 −

2

∂2 D6 + m∂ D2 3



.

(68)

The static dielectric permittivity ε1 due to this original mode can be approximately calculated by

ε1 ∼

X |Dm0 |2 m

∆Em0

.

(69)

Here, Dm0 is the matrix element of the dipole transition from the vibrational ground state to the m-th excited state, and ∆Em0 is the energy difference between them. To reflect the variance of dielectric permittivity, we define a ratio

∆ε ≡ ε10 /ε1 and give its curve as a function of Sd by the filled circles in Fig. 57, where ε10 represents the static dielectric permittivity 0 calculated from H1u by the same method as Eq. (69). The static dielectric permittivity averagely increases by a half, and our calculations indicate that this increase mainly comes from the softening mentioned above. It should be noted that this enhancement of the static dielectric permittivity results only from one typical T1u mode, as we have mentioned in the previous discussion. Actually, the SPE large polaron is a macroscopic domain with hundreds and thousands of lattice sites involved. We have validated that, in each lattice site, this T1u mode will be softened by the e–p interaction, and as such results in huge dielectric enhancement respectively. Their corporative effect gives rise to a gigantic enhancement in the macroscopic static dielectric permittivity, which has been observed experimentally [143,144]. It should be mentioned that this photo-enhancement of the static dielectric permittivity occurs also in the SPE large bipolaron, or in other large polaron clusters.

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K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

As for the off-center type self-trapped polaron, the electron is tightly bound almost within one lattice site. Therefore, only few soft T1u modes are involved, where appears a hardening, instead of the softening. This agrees with the experimental fact that when the ultraviolet illumination is turned off, the phenomenon of dielectric enhancement disappears. The reason is that the SPE large polarons and bipolarons incline to transform into the self-trapped polarons by a quantum tunneling process [161], as schematically shown in Fig. 56 by the dot–dot–dashed arrows. The enhancement of the static dielectric permittivity comes mainly from the sextic anharmonic oscillator coupled with the photoinduced electron. This functional distinction results from the difference of parity. The T1u mode of odd parity is easily polarized and inclines to give rise to a conspicuous dipole. Meanwhile, the A1g mode has an even parity, and remains free from the polarization, even under the electric field. 4.2.9. Polaron transport In the absence of scatterings by phonons, the conduction electron moves freely under an external electric field. However, owing to e–p interactions, the conduction-band electron will distort its nearby lattice and results in lattice displacements around it. On the contrary, the electron might get heavier, because whenever it moves, there will be an induced lattice displacement dragged by it. This is just the mass enhancement of the polaron. In order to clarify this mass enhancement effect, we next study a one-site translational motion of the polaron by calculating the expectation value of the electronic P Ď transfer operator l al,σ al+1,σ between two polaron states with neighboring centers as schematically shown by the big dashed and big solid circles in Fig. 58, where r schematically denotes the large polaron radius. We define a ratio between the real polaron transfer and the hypothetical bare electron transfer as

 R= 

Ψ 0|

P

Ψele |

P

l

0

l

Ď

al,σ al+1,σ |Ψ



Ď al,σ al+1,σ |Ψele

,

(70)

where |Ψele i is the electronic state and |Ψ i is the polaron state encompassing both electronic and phonon configurations. The state with a prime distinguishes from that without a prime by the center of the polaron. Under the Hartree–Fock approximation, the polaron state can be reduced to the direct product of the electronic state and the phonon state as 0 | Ψph i. The state |Ψ i = |Ψele i · |Ψph i. The ratio R is then reduced to the inner product of two phonon states, R = hΨph |Ψph i encompasses many modes of T1u type as well as those of A1g type modulated by the e–p couplings. Namely,

hΨph 0

" #3 " # Y Y | Ψph i = hΨT1u (l) | ΨT1u (l + 1)i hΨA1g (l) | ΨA1g (l + 1)i , l

(71)

l

where |ΨT1u (l)i are the phonon ground states of the modulated single T1u modes with effective Hamiltonian

  ∂2 D6 H1u (l) = −Sd ωd |φ(l)| D + − + , 2 m∂ D2 3 0

2

2

ωd

(72)

and |ΨA1g (l)i are the phonon ground states of the modulated single A1g modes with effective Hamiltonian 0 H1g (l) = −Sb ωb |φ(l)|2 A +

ωb 2

 −

 ∂2 2 + A . ∂ A2

(73)

In Fig. 59, we give the result of R versus Sd for the self-trapped polaron as well as for the SPE large polaron. In the case of the SPE large polaron denoted by the square blocks, R has almost no deviation from 1, which indicates quite similar behavior to that of the free conduction-band electron. It suggests that the SPE large polaron has almost no mass enhancement. With the onset of UV irradiations, many charged SPE large polarons will be created. They move smoothly under an external electric field and contribute to the high electronic conductivity. This conductive property of the SPE large polaron is also applicable to the SPE large bipolaron, and can be employed to understand the observed large photoconduction in SrTiO3 . In the case of the self-trapped polaron as denoted by the triangles, R is almost 0. This indicates a gigantic mass enhancement, because the electron is tightly bound within a lattice potential well and entails a very strong external electric field to help it break away from the bondage. This also suggests a disappearance of the photocurrent in SrTiO3 when the UV irradiation is turned off. 4.2.10. Concluding remarks By a UV photon irradiation, a conduction electron is generated in the 3d orbital of Ti, and this electron linearly and strongly couples with the breathing (A1g ) mode of O around the Ti, and hence, as explained before, we have both the metastable large polaron state and the globally stable self-trapped state. Since the large polaron state is energetically higher than the self-trapped one, the photogenerated electron reaches this state first. As explained before, this state can contribute to the electronic conductivity. On the other hand, this large polaron state quadratically couples with the T1u phonons, and this quadratic coupling makes the soft mode more ‘‘soft’’. This further softening due to the e–p coupling occurs at a very

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

51

Fig. 58. Schematic diagram of polaron transport. Big circles denote the effective range of the polaron, with the radius r. Small filled circles denote bare phonon modes, and small unfilled circles denote the modes coupled with the electron. Reprinted with permission from [141]. Copyright APS.

Fig. 59. Mass enhancement, Sd -dependent ratio R. Triangles are for self-trapped polaron and square blocks are for SPE large polaron. Reprinted with permission from [141]. Copyright APS.

large number of lattice sites, which are included in this large polaron radius. Since this mode is dipole active, the further softening consequently contributes to the enhancement of quasi-static ε1 . As schematically shown in Figs. 55 and 56, this large polaron is well-known to decay into the self-trapped one through quantum tunneling process [161]. After that, it recombines to the hole with the aforementioned luminescence. Recently, the relaxation process in a three-dimensional SrTiO3 crystal is investigated by a nonadiabatic molecular-dynamics method [162]. The formation of ferroelectric domains after the ultraviolet illuminations is shown to be an ultrafast process. It is also noted that, without photodoping, the quantum paraelectric phase is achieved by quantum fluctuations of polarizations. The effect of photodoping on the quantum paraelectric SrTiO3 is also studied by using the one-dimensional quantum Ising model, where the Ising spin describes the effective lattice polarization of an optical phonon [163]. Two types of electron–phonon couplings were introduced through the linear and quadratic modulations of the transfer integral via lattice deformations. By the exact diagonalization and the perturbation theory, we find that photoinduced low-density carriers can drastically alter quantum fluctuations when the system locates near the quantum critical point between the quantum para- and ferro-electric phases. The enhancement of the static dielectric permittivity is accompanied with photocarriers with only little heavier mass than the bare electron mass.

5. Probing by time-resolved X-ray diffraction The understanding of photoinduced structural phase transitions is being developed by the emergence of fast and ultrafast X-ray science. Time-resolved X-ray diffraction directly accesses the dynamics of electronic, atomic and molecular motions in materials triggered by a pulsed laser irradiation. These laser pump and X-ray probe techniques now provide a great opportunity for direct observation of photoinduced structural phase transitions. The time-resolved photo-crystallography has given a strong impact on photoinduced chemical reactions and biochemical functions. There is a fundamental difference in nature between photoinduced phase transitions and these conventional homogeneous photoinduced processes where molecules are transformed independently. After introduction of basic facts on X-ray scattering, we will overview fast [165, 164] and ultrafast [166] X-ray scattering techniques and their recent results [167].

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5.1. Structural changes seen by X-ray scattering X-ray scattering experiments have contributed to revealing structural phase transitions in thermal equilibrium with or without a change in the symmetry. Data contain positions, intensities and shapes of peaks and spreading of diffuse ones. The occurrence of sharp peaks is the manifestation of a periodic or quasi-periodic long range structural order. The Bragg diffraction, however, only concerns the average structure. Below, the kinematical approximation is employed where the magnitude of the incident X-ray wave amplitude is assumed to be the same at all points in the sample. Then, the intensity is proportional to the square modulus of the coherent addition of the amplitudes scattered by all the electrons in the diffracting crystal. In the simplest case of a crystal with N 3 unit cells on a three-dimensional periodic lattice, the scattered intensity in the direction of a scattering vector Q = kd − ki , where kd and ki are respectively the diffracted and incident wave vectors, is given by

2 N X N N X X −iQ·Ruvw F (Q)e I (Q) = , u=1 v=1 w=1

(74)

where F (Q) is the scattered amplitude along Q for one unit cell. The summation leads to I (Q) =

sin2 (Nhπ ) sin2 (Nkπ ) sin2 (Nlπ ) sin2 (hπ ) sin2 (kπ ) sin2 (lπ )

|F (Q)|2 ,

(75)

where the scattering wave vector Q is decomposed on the reciprocal space basis, Q = ha∗ + kb∗ + lc∗ . The interference is constructive only for integer values of the hkl coordinates, giving rise to sharp peaks in the diffraction pattern. The sharpness of these Bragg peaks is governed by the inverse of N or imperfections in the crystal. The intensity of every Bragg peak is proportional to the square modulus of the amplitudes scattered by all the electrons in a unit cell, which is the Fourier transform of the electron density in the unit cell. For a spherical distribution of electrons around atoms at rest, the structure factor of the unit cell is given by F (Q) =

X

fj e−iQ·rj =

j

X

fj e−i2π(hxj +kyj +lzj ) ,

(76)

j

where fj is the atomic scattering factor or the form factor of the spherical atom j at position rj . Accurate determination of the electron density is, however, possible beyond the spherical approximation. Basically, the positions of Bragg peaks are related to the three-dimensional periodic lattice, their intensity to the atomic structure, and their shape to the size of diffraction volume. In real samples, each unit cell instantaneously has slightly different atomic positions from others owing to intrinsic thermal motions and/or disordering processes. The incoherent thermal vibrations attenuate the intensities of Bragg reflections according to the Debye-Waller factor. In a conventional X-ray diffraction experiment, the intensity of a Bragg reflection is integrated over a relatively long time, so that the derived structure is a time-averaged one in thermal equilibrium. In a pump-probe fast or ultrafast X-ray experiment, a quasi-instantaneous state is investigated because the time-averaging is limited by the duration of the X-ray probe pulse. In a photoinduced phase transition, the induced phase generally coexists with the stable phase, which is different from homogeneous random distribution of induced local states. The crystal structure is modified on a large scale, where photoinduced and stable domains diffract X-rays with their own intensities, Iphoto and Istable , incoherently with each other. If their lattice parameters are sufficiently different, large strains occur and the crystal often breaks by photoirradiation. If the lattice parameters are close, the Bragg peaks from the two coexisting phases are hardly separated. As a consequence, the measured intensity Imeasured becomes a weighted average with a fraction x for photoinduced domains, Imeasured = xIphoto + (1 − x)Istable . Generally, a refinement of both the structure of the photoinduced phase and its fraction x is difficult because they are highly correlated. An order parameter related to a symmetry breaking may appear or disappear depending on whether the symmetry of the photoinduced phase is lower or higher than that of the stable phase. If the symmetry is changed from the stable phase to the photoinduced phase, it is deduced from time-resolved X-ray diffraction. Some reciprocal nodes can be systematically absent in the diffraction pattern due to a structure factor set to zero by symmetry. The growth of the photoinduced phase can be followed either by the appearance of new Bragg reflections if a photoinduced long range order is being established or by a reduction in the intensities of specific reflections if disordering is photoinduced. In photoinduced dynamical processes, including precursor phenomena, the structure factor, F (Q)(t ) =

X j

fj e−iQ·(rj +1rj (t )) =

X

fj e−i2π[h(xj +∆xj (t ))+k(yj +∆yj (t ))+l(zj +∆zj (t ))] ,

(77)

j

is basically different from the Debye-Waller factor, which is due to the incoherent thermal excitation of phonons. Precursor nano-domains may be formed before a new three-dimensional structural order is established. For instance, lattice-relaxed charge-transfer strings are formed during the neutral-ionic transitions. Characteristic spatial fluctuations are introduced

K. Yonemitsu, K. Nasu / Physics Reports 465 (2008) 1–60

53

around the average structure, Fm = hF i + ∆Fm , and revealed by diffuse scatterings. In this case, the total X-ray intensity scattered by M unit cells has two components, I (Q) =

X m

|hF (Q)i|2 e−iQ·Rm + M

X h∆F0 (Q) · ∆Fm (Q)ie−iQ·Rm ,

(78)

m

where the first term corresponds to the Bragg diffraction by the average periodic structure and the second term to the additional diffuse scatterings spread in the reciprocal space depending on the nature of spatial fluctuations. The latter appear through the Fourier transform: e.g., a diffuse plane for a one-dimensional spatial correlation. Thus the time evolution of the shape and the size of the precursor nano-domains can be experimentally determined. When a three-dimensional order will be formed, these diffuse scatterings will condense in Bragg reflections. In general, photoinduced phase transitions proceed on different intrinsic spatial and time scales. Impulsive coherent atomic or molecular precursor short-range transformations take place first on a sub-picosecond time scale. A long-range transformation to a new photoinduced phase generally leads to a phase separation and is often governed by the motion of phase fronts or domain walls on the acoustic phonon time scale. If the latter motion is coherent on a large scale, domain wall oscillations can be generated and observed. One should care about differences between the volume probed by an X-ray and the part which is efficiently excited by a light pulse. When a material is exited with a laser pulse, the light can directly transform only a finite depth dL from the surface of the material. X-rays are also absorbed by the material, so that they can probe the sample only over a penetration depth dX , which is of the order of a few µm for conventional semiconductors with heavy elements or of the order of a few mm for molecular materials composed of light atoms. 5.2. Fast time-resolved X-ray diffraction for molecular materials Time-resolved X-ray diffraction is a useful technique to get key information on structural rearrangements associated with ultrafast phenomena. The available time resolution allows us to track very short-lived excited states. The time resolution is limited by the width of the X-ray pulse, which is generated by different techniques. One of them uses the electromagnetic radiation emitted by accelerated charged particles. Synchrotron sources take advantage of the time structure of electron bunches in the ring: each bunch emits its own X-ray pulse. The time width of an emitted X-ray pulse is typically between 30 and 150 ps [165]. This technique is used for photoinduced neutral-ionic phase transitions. The optical pump and X-ray probe techniques consist of exciting the sample with an optical pulse of typically 100-fs width and probing it with an X-ray pulse at a given delay dt during the relaxation of the photoinduced metastable state or during the transformation itself if the pulsed laser excitation triggers a photoinduced phase transition. Because one wants to probe the system in the transient state in a stroboscopic way, one has to wait for the complete relaxation of the sample before exciting it again. Then, the repetition rate is often set at about 1 kHz. Different X-ray pulses of 100-ps width, which are generated by different electron bunches inside the ring, are separated by a much shorter time. If one wants to probe the sample every ms, only one pulse per many has to be selected by a mechanical chopper and utilized for observation. It is technically difficult to synchronize the mechanical chopper, the synchrotron pulses and the laser pulses in order for the time of photoirradiation to be well defined. 5.2.1. Neutral-ionic transition in TTF-CA Here, we review 100-ps investigations performed by fast time-resolved X-ray diffraction on the neutral-to-ionic as well as the ionic-to-neutral photoinduced phase transitions in the TTF-CA compound. This illustrates that, thanks to a large number of Bragg spots measured, a synchrotron source allows one to determine the symmetry change and the nature of a photoinduced structural phase transition in relatively complex materials. Recall that the TTF-CA compound has a regular neutral state, · · · D0 A0 D0 A0 D0 A0 · · ·, and two degenerate, polar dimerized ionic states, · · · (D+ A− )(D+ A− )(D+ A− ) · · · and · · · (A− D+ )(A− D+ )(A− D+ ) · · ·. In thermal equilibrium, a transition is observed from the high-symmetry neutral phase to the low-symmetry ferroelectric ionic phase. In the neutral phase, the space group is P21 /n, manifested in the diffraction pattern by two types of systematic absences, (0k0) with k = 2n + 1 due to the presence of a two-fold axis [the two TTF (CA) molecules in a unit cell, related by the screw axis, are equivalent by the translation of b/2] and (h0l) with h + l = 2n + 1 due to the presence of a glide plane. In the ionic phase, the space group is Pn, where the screw-axis symmetry is lost with the center of inversion and (0k0) with k = 2n + 1 appear due to its symmetry breaking. The optical pump and X-ray probe experiments [62,59] have been performed with monochromatic X-ray pulses of 100±10 ps and ultra-short laser pulses at 1.55 eV (800 nm) with polarization parallel to the stacking axis. The excitation is off resonant because the charge-transfer band is centered on 0.65 eV. More than 6000 peaks were collected, which correspond to more than 800 or 1600 unique reflections in the high-symmetry or the low-symmetry phase. For the photoinduced neutral-to-ionic transition at 93 K > TNI = 81 K, data probed by X-rays are collected at 2 ns before and 1 ns after the laser excitation beyond a threshold density (Fig. 24) [62]. Before the excitation, the resolved structure is in agreement with the neutral structure (P21 /n) at 90 K in thermal equilibrium. After the excitation, a symmetry lowering is observed with the appearance of (0k0) reflections with k odd. The space group of the photoinduced phase is thus Pn, as for the ionic phase in thermal equilibrium, implying a ferroelectric long-range order.

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For the photoinduced ionic-to-neutral transition at 70 K, normal reflections except the (0k0) ones with k odd are modified after an incubation time of about 500 ps, whereas the intensity of the (030) reflection is modified on a shorter time scale (Fig. 22) [59]. Thus the intensity of the latter reflection characteristic of the change of the symmetry decreases immediately after the laser irradiation, which implies fast restoring the symmetry before the transition from the ionic state to the neutral state. It results from a first disordering process of ionic dimer strings after the photoexcitation in a similar way to the static pressure-induced transition from the ferroelectric ionic phase to the paraelectric ionic phase and finally to the paraelectric neutral phase in thermal equilibrium [41]. We discuss again the coexistence of stable and photoinduced phases. The neutral-to-ionic transition does not proceed via homogeneous random distribution of induced local states, but via a phase separation between macroscopic threedimensional domains of the stable and the photoinduced phases. Each domain incoherently diffracts X-rays with an associated intensity. The spatial resolution did not allow one to observe a significant shift of the spot position from the stable phase to the photoinduced phase. For each Bragg spot the measured intensity Imeasured was the sum of the contribution from the stable domains (Istable ) and that from the photoinduced domains (Iphoto ) weighted with the respective concentrations (1 − x) and x, Imeasured = xIphoto + (1 − x)Istable . Assuming that the structure of the photoinduced ferroelectric phase is the same as that of the ferroelectric ionic phase stable at low temperature, the converted fraction can be estimated from the variation of the diffracted intensities, Imeasured − Ineutral = x(Iionic − Ineutral ). For the photoinduced neutral-to-ionic transition, the converted fraction is estimated at around 35%.

5.3. Ultrafast time-resolved X-ray diffraction for hard materials To generate ultra-short X-ray pulses, laser-produced plasma sources can be used [166]. The X-ray radiation is emitted by the relaxation of electrons in plasma, which is excited by a 100 fs optical pulse. The X-ray pulse width is then limited by this width of the optical exciting pulse. However, the weak intensity of the X-ray beam produced by this technique limits its use to the measurement of few Bragg reflections. Nevertheless, laser-plasma based X-ray sources are table-top facilities in contrast to large facilities for synchrotron sources. At present their time resolution of about 100 fs is suitable for the time scale of small atomic displacements. In particular, this gives a possibility to directly observe the oscillation of an optical phonon by X-ray diffraction experiments. Another intrinsic property of this technique is that the time of photoirradiation is well defined since the same optical laser is used to pump the crystal and to generate the plasma X-ray pulse. Difficult synchronization problems are thus avoided. However, the X-ray pulse is divergently emitted through a solid angle of 2π , and only a small part of emitted X-ray photons can be collected by a monochromator. Thus, the number of photons reaching the sample is much smaller than that of emitted photons. Below, we review three sub-ps works utilizing a laser-produced plasma source: nonthermal surface melting in conventional semiconductors, insulator-to-metal transition in VO2 , and coherent optical phonons in bismuth. These hightemporal resolution experiments on conventional hard solids with only a few atoms in a unit cell are limited to the measurement of a few Bragg peaks. The samples were perfect single crystals, which lead to the fact that only the vicinity of the surface is probed by X-ray diffraction. The dynamical theory of diffraction has to be used instead of the kinematical approximation. The probed depth may be of the same order with the laser-light penetration depth.

5.3.1. Nonthermal surface melting in semiconductors Photoinduced nonthermal surface melting is observed in semiconductors. The term ‘‘nonthermal’’ means that the surface melting takes place on the sub-ps time scale before the typical electron–phonon thermalization time. It is characterized by the disappearance of the Bragg reflections, associated with the loss of the lattice periodicity. Such processes are ultrafast since the electronic excitation triggers impulsive coherent atomic motions on a time scale that can reach few hundred femtoseconds. The surface is transformed into a disordered quasi-liquid state before the thermal relaxation occurs. In the case of InSb, the pump-and-probe technique was used to excite the sample with 120-fs pulses of laser light and to detect the resulting structural changes [168]. 100-fs X-ray pulses were generated by focusing a second beam from the laser on a silicon target, and the pulses are then directed at the sample. The excitation pulse causes disorder in the sample, eliminating diffraction peaks so that the X-ray signal drops to zero with time constants as short as 350 fs. It is possible to tune the penetration depth of the X-ray from about 3500–1200 Å. The shorter the penetration depth is, the stronger the decrease of the Bragg reflection is. A loss of long-range order is established up to 900 Å inside the crystal. This is a direct signature of the formation of a liquid film at the surface of the sample. In general, transfer of energy form excited electrons to thermal motion of the lattice is expected to take many picoseconds, so thermal processes cannot account for the data. The sudden change of interatomic potential induced by the laser pulse immediately excites optical phonons. The longitudinal optical phonon (with a period of 160 fs) emission time from excited electrons is about 200 fs if screening is neglected, so that a high density of electrons can rapidly energize these vibrations appreciably. The Debye-Waller effect gives a decrease in diffracted intensity of only 7% close to the melting temperature, so the observed decreases of more than 50% require displacements far larger than the critical amplitude for thermal melting.

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5.3.2. Insulator-to-metal transition in vanadium oxide A photoinduced solid–solid phase transition is observed in VO2 [169,170]. It is known to undergo an insulator-to-metal transition at 340 K. Changes in the electronic band structure are associated with structural changes between a low-T monoclinic and a high-T rutile phase. A theory for this transition is proposed on the basis of cluster dynamical mean-field theory, in conjunction with the density functional scheme [171]. The interplay of strong electronic Coulomb interactions and structural distortions, in particular, the dimerization of vanadium atoms in the low-temperature phase, is claimed to play a crucial role. Structural probing was achieved by using an X-ray, which is generated by focusing laser pulses onto a moving copper wire, after the 50 fs, 800 nm, p-polarized optical pump pulse. The integrated X-ray reflectivity from the new phase, normalized to the signal from the low-temperature monoclinic crystal, shows the formation of the metallic rutile phase in two steps just after the laser excitation. The integrated diffraction from the rutile phase indicates that, for intermediate fluence range, a 250-nm thick layer of the material undergoes transformation in about 10 ps. The signal measured during the first few picoseconds shows that the first step in the structural phase transition occurs over a depth of 40–60 nm on the time scale within 500 fs at a high degree of electronic excitation (about 5 × 1021 cm−3 ). Such a time scale, shorter than the typical internal thermalization time of a nonequilibrium phonon distribution, means that this excitation regime cannot correspond to the conventional pathway for a first-order phase transition via nucleation and then growth of a new phase. It is suggested that excitation of a dense carrier population across the 600-meV band gap may significantly perturb the potential energy surface of the electronic ground state and depress the barrier separating the two phases. It is not clear whether the system becomes metallic from disruption of electronic correlations or from structural distortion. This issue is intimately related to the nature of the insulating phase of VO2 . It should be noted that the multi-THz conductivity of VO2 is traced during an insulator-metal transition triggered by a 12-fs 1.55-eV light pulse [172]. The femtosecond dynamics of lattice (40 meV < h¯ ω < 85 meV) and electronic (h¯ ω > 85 meV) degrees of freedom are spectrally discriminated. A coherent motion of V–V dimers at 6 THz modulates the lattice polarizability for approximately 1 ps. The electronic conductivity settles to a constant value already after one V–V oscillation cycle if the fluence is above a threshold. The mid-infrared electronic conductivity thus looks metallic, even though the lattice is far from equilibrium. Optical-pump terahertz-probe spectroscopy is also used to investigate the near-threshold behavior of the photoinduced phase transition [173]. Upon approaching the transition temperature, a reduction in the fluence required to drive the insulator-to-metal transition is observed, consistent with a softening of the insulating state due to an increasing metallic volume fraction. 5.3.3. Coherent optical phonons in bismuth The coherent atomic displacement of the lattice atoms in femtosecond-laser-excited bismuth is measured close to a phase transition [174]. Excitation of large-amplitude coherent optical phonons gives rise to a periodic modulation of the ultrafast X-ray diffraction efficiency. The X-ray pulses were generated by focusing femtosecond laser pulses from a titanium:sapphire laser system onto a thin titanium wire to produce a microplasma. The incoherent X-ray emission from the laser-produced plasma was collected and focused on the surface of the bismuth crystal. The optical phonons at the center of the Brillouin zone affect the X-ray diffraction intensity because of the changes in the structure factor through the coherent atomic motion. From the polarization of the A1g phonon involved [giving ∆xj , ∆yj and ∆zj in Eq. (77)], it is possible to calculate the effect of the atomic oscillation on the structure factor. For low laser fluence, clear oscillations of the (111) and (222) reflections are observed, and the modulated Bi–Bi distance affects the two reflections in an opposite way. It is thus possible to separate this coherent atomic motion inside the unit cell from the signal associated with incoherent dynamics due to melting. The frequency of the observed phonon is downshifted. The frequency shifts depend on the phonon oscillation amplitude. The effect is attributed to anharmonicity. For fluence higher than the melting threshold, the initial stage of large-amplitude optical phonon motion is followed by structural disordering of bismuth due to its melting. From the large change in the diffraction signal, it is estimated that the initial coherent atomic displacements exceed 10% of the nearest-neighbor distance at these high excitation levels. No oscillations of the diffraction signal occurred. 6. Summary We have reviewed the progress in research of photoinduced phase transitions mainly from the theoretical aspect, focusing on the dynamics of itinerant electron systems. First of all, before the introduction of itinerant-electron models, statistical models were treated. Cooperativity is somehow required for the initially local, photoinduced structural deformations to grow into macroscopic ones. Ising models could qualitatively explain relaxation processes and threshold behavior. Time evolutions are described with the help of transition probabilities, which are justified by coarse-graining the energy-exchange processes and are therefore appropriate on long time scales. They cannot describe dynamics depending on the details of the photoexcitation or the system itself. With increasing examples of photoinduced phase transitions and with improving time-resolutions of measurements, there have appeared a variety of photoinduced dynamics that cannot be described by stochastic processes. Now, it is possible to follow time evolutions of electronic states directly by different experimental techniques with resolution finer than phonon-related oscillations. Relevant models should include itinerant electrons, which are transferred by photoexcitations. The photoinduced dynamics sensitively depend on the energy and strength of pump light. Now, the materials that show photoinduced phase transitions range from one to three in electronic dimensions. See below for the latest progress. The

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time scales of the dynamics range widely for materials of any dimension. The variety also exists in the relative speed among the evolutions of spin, charge, and lattice degrees of freedom. We thus need to get down into specific transition dynamics in a specific group of materials. By accumulating the experimental and theoretical research works, we may be able to control the physical properties of selected materials intelligently with desired timing in future. It is noted that, between one- and two-dimensional strongly correlated electron systems, fundamental differences exist as shown in Section 6.1. In this article, we have mainly dealt with condensed molecular materials, where the electric conductivity is limited into low dimensions. They show a variety of electronic phases, depending on the characters of interactions, because the Fermi surface nesting tends to destabilize the metallic state and the formation of a long-range order tends to open a gap and to lower the total energy. It is also true that quantum fluctuations tend to destroy any long-range order in low dimensions. For example, low-dimensional organic conductors show spin-Peierls, antiferromagnetic or spin-densitywave, frustration-induced spin-liquid, charge-ordered or charge-density-wave, ferroelectric, superconducting, and other phases [3]. In photoinduced phase transitions in a narrow sense, stable and metastable phases are involved (i.e. thermal transitions are discontinuous) and the free-energy barrier between them contributes to nonlinear characters such as threshold behavior. Therefore, among the above electronic phases in low-dimensional organic conductors, only a few electronic phases are converted into another by photoirradiation. Nevertheless, many examples of photoinduced phase transitions are now known. Many one-dimensional materials have been studied so far for photoinduced phase transitions. At an early stage, A-B transitions were studied in polydiacetylenes, which are qualitatively described in statistical models. The most intensively studied is the neutral-ionic transitions in the mixed-stack charge-transfer complex, TTF-CA, which are presented in detail here. Its neutral-ionic transition originates from coupling of symmetry-preserving order parameter (i.e. ionicity) and (inversion-)symmetry-breaking order parameter (i.e. dimerization) leading to ferroelectricity in the ionic phase. The relation between the nonlinear character of the photoinduced dynamics and the electron correlation has been clarified. Related materials show quantum paraelectricity near the quantum critical point. The effect of quantum phonons on the photoinduced dynamics will become an important issue in near future. Halogen-bridged metal complexes show transitions among metal, Mott insulator, charge-density-wave, and charge– polarization phases. The latter two phases are both low-symmetry phases, where different symmetries are broken and their magnetic and dielectric properties are very different. The corresponding photoinduced phase transitions show quite different dynamics depending on the direction. The origin of the difference is clarified by investigating the microscopic charge-transfer processes. They are treated appropriately only in itinerant-electron models with relevant interactions. Now an increasing number of one- and two-dimensional organic conductors show photoinduced transitions from chargeordered to metallic phases [79,175]. For quasi-two-dimensional BEDT-TTF salts, see Section 6.2. Their time scales are widely distributed, depending on the effect of the relevant electron–phonon coupling. The studies of their dynamics sometimes urge us to reconsider the relevant interactions that stabilize the electronic states of interest. Roughly speaking, there is some (but not strict) systematics concerning the relations among nonlinearity, directional property, and stability of photoinduced phases. That is, high nonlinearity characterized by a threshold in photoinduced dynamics, discontinuous thermal transitions, photoinduced transitions in both directions, and rather long lifetimes of the photoinduced phases appear often together, as do low nonlinearity, continuous or weakly discontinuous thermal transitions, one-way photoinduced transitions, and short lifetimes of the induced phases. There are other combinations as well. The interrelations among these properties may sometimes be predictable, but they largely depend on the dimensionality, interaction type and range, order parameter(s) of the material. Further research works are necessary for really systematic understandings. When the understanding is deeper and the relations are clarified between the photoinduced response and the wavelength, strength, and shape of the laser pulse (including the interval of a double pulse when appropriate), the possibility of optical controlling or switching the electronic and structural properties will be widened. Coherent oscillations, which are often observed during the transition, may be a key to revealing the optimal timing for photoexcitations. It is because their observation implies that the time evolutions of electronic states are in phase in a macroscopic domain: a direct manifestation of cooperativity. Recently, coherent excitations are found in insulator-to-metal transitions in perovskite manganites also, as shown in Section 6.3. In this article, time-resolved X-ray diffraction is explained rather in detail. It should be added that femtosecond time-resolved photoemission is also used to investigate the time evolution of electronic structure in the Mott insulator 1T-TaS2 [176]. A collapse of the electronic gap is observed within 100 fs after optical excitation. The photoemission spectra and the spectral function calculated by dynamical mean-field theory show that this insulator-to-metal transition is driven solely by hot electrons. A coherently excited lattice displacement results in a periodic shift of the spectra lasting for 20 ps. Among important issues to be developed in future, are relations to quantum critical phenomena. So far, we have treated the lattice degrees of freedom as classical objects. By taking account of quantum tunneling between local minima of the lattice potential, photoinduced dynamics would be modified. Especially when the transition proceeds coherently, the quantum nature of phonons would play an important role. In this context, photoinduced large responses of quantum paraelectric systems would give useful information. In addition, further improvement of time resolution may allow the observation of coherence in electronic excitations. Even if photoinduced nonequilibrium dynamics are not directly related to any transition, understanding them will progress to other fields like visualization or light-driven pump in biological systems. Chains of hierarchical reactions would

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smoothly proceed if some condition on the time scales is satisfied. Similar conditions may appear for photoinduced melting of orders if phonons of different frequencies are involved [81]. The rapid progress in time-resolved spectroscopy and X-ray diffraction will promote the detailed probing of different time scales of photoinduced events. Below, we summarize the latest progress in higher-dimensional electron systems. 6.1. Differences between one- and two-dimensional systems Between one- and two-dimensional strongly correlated electron systems, fundamental differences exist in the dynamics of the photoexcited state, as shown in the Hubbard model [177,178] and in the extended Hubbard model [179], without electron–phonon coupling. In the one-dimensional case, spin-charge separation holds well even for the photoexcited state. Photogenerated doublons and holons move almost freely without friction from spin fluctuations, preserving the coherence. Thus, only the charge relaxation is observed. In the two-dimensional case, ultrafast relaxation originating from the transfer of photogenerated charges occurs in the antiferromagnetic background. It is followed by slower relaxation originating from the spin structure rearrangement with the new charge distribution. Consequently, the relaxation in the charge degrees of freedom is much faster than that in the spin degrees of freedom. These facts are manifested in the time dependence of the spin and charge correlation functions [177], the free induction decay, the transient four-wave mixing [178], in the light absorption spectra [179], and in the photoemission response [180]. Recently, charge dynamics in a one-dimensional Mott insulator has been experimentally investigated by femtosecond pump-probe reflection spectroscopy on an organic charge-transfer compound, (BEDT-TTF)(F2 TCNQ) [BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene, F2 TCNQ = 2, 5difluorotetracyanoquinodimethane] [181]. Low-energy spectral weight induced by photocarrier doping is concentrated on a Drude component irrespective of the doping density. A midgap state is never formed. These facts are explained by the concept of spin-charge separation characteristic of one-dimensional correlated electron systems. In general, the transition from a Mott insulator to a metal proceeds ultrafast because it is not accompanied with structural deformations. Thus, it is quite different from all the transitions involving lattice degrees of freedom. 6.2. Melting of charge order in quasi-two-dimensional organic conductors Photoinduced melting of charge order in quasi-two-dimensional BEDT-TTF salts have been investigated by femtosecond spectroscopy [175,182]. Comparative studies are performed on two polytypes exhibiting large [θ -(BEDT-TTF)2 RbZn(SCN)4 ] and small [α -(BEDT-TTF)2 I3 ] molecular rearrangements through the charge-ordering transition. Ultrafast melting of charge order in both compounds demonstrates the major contribution of the Coulomb interaction to the electronic instability. Their photoinduced dynamics are however qualitatively different: the θ -RbZn compound shows local melting of charge order and ultrafast recovery of charge order irrespective of temperature and excitation intensity, while the α -I3 compound shows critical slowing down. Thus, it is important to show theoretically how electron–phonon interactions are significant in the θ -RbZn compound. Then, charge ordering accompanied by lattice distortion in these organic conductors is studied in a two-dimensional 3/4-filled extended Hubbard model with Peierls-type electron–lattice couplings by the Hartree–Fock approximation at zero and finite temperatures [183,184], the exact-diagonalization method, and the thirdorder perturbation theory from the strong-coupling limit [185,186]. It is found that the horizontal-stripe charge-ordered state, which is experimentally observed in the θ -RbZn and α -I3 compounds, is stabilized by the self-consistently determined lattice distortion. Electron–phonon interactions including molecular rotations modifying transfer integrals are crucial to stabilize this charge order and to realize the low-symmetry crystal structure. The lattice effect on θ -RbZn is found be much stronger than that on α -I3 by any method, which is directly related to the crystal-symmetry difference. 6.3. Insulator-to-metal transitions in manganese oxides Coherent orbital waves are claimed to be observed by using sub-10-fs optical pulses in the photoinduced insulatormetal dynamics of a magnetoresistive manganite, Pr0.7 Ca0.3 MnO3 [187]. At room temperature, the time-dependent pathway towards the metallic phase is accompanied by coherent 31 THz oscillations of the optical reflectivity, significantly faster than all lattice vibrations. Furthermore, the electronic phase of the manganite Pr0.7 Ca0.3 MnO3 is controlled by a modeselective vibrational excitation [188]. Ultrafast switching from the stable insulating phase to a metastable metallic phase is observed via direct excitation of a phonon mode at 71 meV (17 THz). It is uniquely attributed to a large-amplitude Mn–O distortion that modifies the electronic bandwidth. The ultrafast spin and charge dynamics is investigated in the course of a photoinduced phase transition from an insulator with short-range charge order and orbital order to a ferromagnetic metal in perovskite-type Gd0.55 Sr0.45 MnO3 [189,190]. Transient reflectivity changes suggest that the metallic state is formed just after the photoirradiation [189] and decays within ∼1 ps. The magnetization, however, increases with the time constant of 0.5 ps and decays in ∼10 ps [190]. The relatively slow increase of the magnetization is attributable to the magneticfield-induced alignment of ferromagnetic domains in the initially produced metallic state and its slow decay to the partial recovery of the orbital order. Photoinduced switching is investigated on Pr0.55 (Ca1−y Sry )0.45 MnO3 [191], which locates in the critical region between the charge and orbital ordered insulator (CO/OOI) phase and the ferromagnetic metal (FM) phase. For a low excitation density, the charge and orbital order is partially melted and the microscopic quasi-FM domains are

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generated, which rapidly return to the CO/OOI state. When the excitation density exceeds a threshold, the CO/OOI phase transforms permanently to the FM one. Theoretical studies for photoinduced phase transitions in perovskite manganites have just started [192]. The transient optical absorption spectra of the one-dimensional extended double-exchange model are calculated by the density-matrix renormalization group method. A charge-ordered insulator becomes metallic just after photoirradiation, and the metallic state tends to recover the initial state. Spin and charge dynamics are highly correlated here. A theory of ultrafast photoinduced magnetization dynamics is already developed with the mean-field approximation for ferromagnetic semiconductors such as Ga(Mn)As [193–195]. Acknowledgments The authors are grateful to P. Huai, H. Inoue, K. Iwano, J. Kishine, M. Kuwabara, T. Luty, N. Maeshima, N. Miyashita, S. Miyashita, Y. Qiu, Y. Tanaka, C. Q. Wu, Y. Yamashita and H. Zheng for theoretical collaboration, and especially H. Cailleau, S. Iwai, S. Koshihara and H. Okamoto among many others for sharing their data prior to publication and for enlightening discussions. This work was supported by the Next-Generation Supercomputer Project (Integrated Nanoscience) and Grantsin-Aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

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