Two-variable anharmonic model for spin-crossover solids: A like-spin domains interpretation
W. Nicolazzi, S. Pillet,* and C. LecomteLaboratoire de Cristallographie et Modlisation des Matriaux Minraux et Biologiques, UMR CNRS 7036, Institut Jean Barriol,
Nancy-Universit, BP 239, F-54506 Vandoeuvre-ls-Nancy, FranceReceived 9 June 2008; revised manuscript received 4 September 2008; published 4 November 2008
Spin-crossover SC complexes are one of the most fascinating examples of molecular bistability, whosesolid-state properties are tightly connected to cooperative interactions within the crystal lattice. A variety ofmacroscopic and microscopic models have been developed to explore the cooperative nature of the SC phe-nomenon. We present here a two-variable microscopic Ising-like model for SC solids, accounting for the elasticorigin of the cooperativity using coupled spin and translational lattice degrees of freedom. Within our model,the interaction between a pair of neighboring molecules in the crystal is dependent not only on their spin statesbut also on their separation distance, modeled by spin-dependent Lennard-Jones LJ potentials. This schemeleads explicitly to local variations in the interactions, associated to the local strain induced by the moleculeschanging their spin state. In essence, the LJ potentials provide the anharmonicity of the crystal lattice. Theequilibrium quasistatic properties of the proposed Hamiltonian are analyzed by Monte Carlo simulations ona regular deformable square lattice. We show that the spin dependence of the LJ potentials breaks the spin-statesymmetry in the free energy. The interplay between spin and lattice degrees of freedom shows itself in thetemperature evolution of the fraction of high-spin molecules and the mean lattice spacing, as a function of theintersite coupling. For strong coupling, like-spin domains nucleate and develop, evidenced by a double struc-ture in the distribution of lattice spacings; structural relaxation occurs at the domain walls. In the weaklycooperative situation, the mean lattice constant scales directly with the fraction of high-spin species; structuralrelaxation spans the entire system.
DOI: 10.1103/PhysRevB.78.174401 PACS numbers: 05.50.q, 64.60.De, 75.30.Wx, 75.60.d
The thermally induced spin-crossover SC phenomenonbetween the low-spin LS and the high-spin HS states ofFeII molecular complexes has been the subject of intenseresearch activities.1,2 The intrinsic bistability that these ma-terials exhibit originates from intramolecular vibronic cou-pling. The higher electronic and vibrational degeneracy inthe HS state leads to a LS to HS entropy increase, whichdrives the spin conversion. In addition to large variations inthe magnetic and optical properties upon the spin-statechange, drastic structural modifications occur, namely, atypical 0.2 Fe-N bond lengthening and a 2%5% unitcell expansion.3 Besides temperature or pressure changes, thespin conversion may alternatively be triggered by continuouslight excitation, albeit at very low temperature light-inducedexcited-spin-state trapping LIESST process,46 and also athigher temperature within the thermal hysteresis loop bymeans of an intense pulsed laser.7,8 Owing to theirtemperature- and light-induced switching potential, it is wellestablished that these molecular systems may have prospec-tive use as sensors or may be integrated in data storage de-vices.
In the solid state, strong electron-lattice coupling is at theorigin of the rich variety of behaviors SC materials exhibit,such as gradual spin crossover or abrupt spin conversionfirst-order character with often hysteresis in the case ofstrongly cooperative materials. Although the concept israther ill defined, cooperativity is often attributed to the largeHS to LS molecular volume change coupled to elastic inter-actions within the solid. In the framework of the continuumelasticity theory, it has been pictured as the buildup of an
internal pressure, proportional to the concentration of LSspecies.911 The local distortion strain of the crystal latticeaccompanying the spin transition has been traced back topoint defects at the iron site of the spin-changing molecule.Cooperativity shows itself in the abruptness of the thermaltransition and low-temperature sigmoidal HS to LS relax-ation curves, attributed to a self-accelerated process.12,13Polymeric SC materials with one-dimensional 1D chain ortwo-dimensional 2D layer structural architectures are em-blematic examples of highly cooperative systems, withsometimes extremely wide thermal hysteresis.14 In that case,the bridging ligands play the role of efficient spin-statepropagators and contribute to the anisotropic elasticity. Onthe other hand, for SC systems built from purely monomo-lecular entities, it is quite clear that the intermolecular con-tacts, through weak van der Waals, hydrogen bonds, or -stacking, transmit the interactions to long range and are thusresponsible for the elastic properties of the solid.15
Intensive theoretical investigations have been devoted tothe development of more or less sophisticated models of spintransition, treating cooperativity through various schemes.Among others, macroscopic models based on the thermody-namic theory of regular solutions were proposed. In the so-called Slichter and Drickamer SD model, cooperativity isdescribed in mean field through phenomenological enthalpyterms related to the intermolecular interactions.16 In the elas-tic model for SC systems introduced by Spiering andWillenbacher,911 SC molecules are considered as entitieswhose volume and shape depend on their spin state, insertedin an isotropic elastic medium. When a molecule undergoesthe spin transition, it creates a local strain which forces thenearest-neighbor molecules to undergo the spin transition aswell ferroelastic coupling. In its more elaborated form, the
PHYSICAL REVIEW B 78, 174401 2008
1098-0121/2008/7817/17440112 2008 The American Physical Society174401-1
Spiering scheme describes the crystal lattice within the De-bye approximation and allows for anharmonicity through theGrneisen approximation.11,17 A variety of microscopicIsing-like models were also designed, introducing a Hamil-tonian of interacting two-level units with different energiesand degeneracies, as proposed by Wajnflasz and Pick18,19 andBousseksou et al.20 These models can reproduce most of theequilibrium properties of SC materials as well as their dy-namic behavior, including the sigmoidal relaxation, the two-step transitions, and the photoinduced effects.2127 The iso-morphism between the macroscopic SD and microscopicIsing-like model in the mean field has been formulated.24,28A direct relation between the Ising coupling constant J andthe phenomenological cooperative factor in the SD schemehas been derived. The 1D version of the Ising-like model hasbeen further solved analytically.29 An alternative one-dimensional model of SC molecules coupled to a phononfield atom-phonon coupling model was established, the in-termolecular interactions being introduced as harmonicsprings.30,31 This 1D model was recently solvedanalytically32 and further extended to anharmoniccontributions.33 Various elastic models, for which the cou-pling constant depends on the intersite separation distance,have been proposed and investigated numerically throughmolecular dynamics MD methods34,35 and Monte CarloMC simulations.36,37
The mechanism by which the spin conversion proceeds inthe solid state is a fundamental question. In the case of co-operative materials, characterized by an abrupt thermal tran-sition and sigmoidal relaxation kinetics, it has been widelyargued that molecule clustering may play a key role. Thevery first experiments which have indirectly observed so-called like-spin domains LSDs are powder and single-crystal diffraction measurements.3841 These techniques areby no way sensitive to the molecular spin component but onthe contrary to lattice structural observables. In these ex-periments, LSD shows itself at the thermally and light-induced transitions by a clear separation of Bragg peaks cor-responding to ordered HS and LS phases with different cellconstants phase separation. Within this framework, a do-main consists of adjacent molecules, crystallographically or-dered to sufficiently long range i.e., comparable to the x-rayor neutron spatial coherence of the experiment, and withwell-defined intermolecular separation distances. The con-cept of LSD has emerged quite early in the macroscopic andmicroscopic model formulations, in which it is usually pic-tured as a set of correlated like-spin adjacent molecules. De-spite their success, Ising-like models are at present unable toexplain the Avrami kinetics of LSD nucleation and growthderived from diffraction techniques.38 This is well under-standable since under the assumption of pseudospins onrigid-lattice sites, the Ising-like models cannot describechanges in the positions of the molecules accompanying theelectronic configuration ordering and the associated local lat-tice distortion. Hence crystallographic LSD would not bedefined. The importance of going beyond a fixed-lattice Isingmodel has been stressed for investigating ordering process inCu-Au alloys42 or lipid bilayers,43 for instance. In the presentstudy, the strategy for introducing magnetoelastic couplingsin the Ising-like model of SC molecular solids is based on
the development of the intersite intermolecular interactionson pairwise potentials of the Lennard-Jones LJ form, con-sidering spin and lattice degrees of freedom through twocoupled on-site variables, which allows for continuous mo-lecular displacements. As a general matter of fact, these po-sition changes are expected to highly affect the characteris-tics and even the nature of the transition since, even at thelevel of global elastic deformations coupled to