Simple Two-Dimensional Model for the Elastic Origin of Cooperativity among Spin States of Spin-Crossover Complexes
Masamichi Nishino, Kamel Boukheddaden, Yusuké Konishi, Seiji Miyashita
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Simple Two-Dimensional Model for the Elastic Origin of Cooperativity among SpinStates of Spin-Crossover Complexes
Masamichi Nishino , , ∗ Kamel Boukheddaden , Yusuk´e Konishi , , and Seiji Miyashita , Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan Groupe d’Etudes de la Mati`ere Condens´ee, CNRS-Universit´e de Versailles/St.Quentin en Yvelines 45 Avenue des Etats Unis, F78035 Versailles Cedex, France Department of Physics, Graduate School of Science,The University of Tokyo, Bunkyo-Ku, Tokyo, Japan CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan (Dated: November 26, 2018)We study the origin of the cooperative nature of spin crossover (SC) between low spin (LS) andhigh spin (HS) states from the view point of elastic interactions among molecules. As the size of eachmolecule changes depending on its spin state, the elastic interaction among the lattice distortionsprovides the cooperative interaction of the spin states. We develop a simple model of SC with intraand intermolecular potentials which accounts for the elastic interaction including the effect of theinhomogeneity of the spin states, and apply constant temperature molecular dynamics based on theNos´e-Hoover formalism. We demonstrate that, with increase of the strength of the intermolecularinteractions, the temperature dependence of the HS component changes from a gradual crossoverto a first-order transition.
PACS numbers: 75.30.Wx 75.50.Xx 75.60.-d 64.60.-i
The discovery of LIESST (light-induced excited spinstate trapping) phenomena has accelerated studies offunctional spin-crossover (SC) molecular solids. SC com-pounds have been studied intensively not only becauseof their potential applicability to novel optical devices,e.g., optical data storage and optical sensors, etc, butalso because of the fundamental scientific interest in themechanism of the phase transition and the accompaniednon-linear relaxation processes .To control electronic and magnetic states of SC com-pounds, it is important to understand the bistable natureof these compounds. The SC transition between the low-spin (LS) and high-spin (HS) states can be induced bychange of temperature, pressure, magnetic field, light-irradiation, etc. It has been clarified that the interactionbetween spin states causes various types of cooperativephenomena between the LS and HS phases .In order to take into account the cooperativity in theSC transition, Wajnflasz and Pick (WP) proposed anIsing model , in which the spin state is described by afictitious spin ( σ = − J between the spin statesare introduced in the form H = − J P h i,j i σ i σ j + P i (∆ − k B T ln g ) σ i , where g is the degeneracy ratio between theHS and LS states. Using this model, the change betweena gradual crossover and a first-order transition has beenwell explained as a function of the parameters J , ∆ and g .So far the WP model and its extensions called “Ising-likemodels” have been widely used for the description of theSC transition and related relaxation phenomena includ-ing photoinduced effects. Although the Ising-like modelshave captured several important features , theorigin of the parameters remains unclear due to the dras-tic simplifications involved.The importance of the elastic interaction in the SC transition has been investigated andthe elastic energy of the system with the density distri-bution of the LS and HS sites has been phenomenolog-ically analyzed . The dependences of elastic constantson the spin state have been also investigated in a one-dimensional (1D) two-level model and in a 1D vibroniccoupling model . However, local degrees of freedom(change of lattice) can be traced out in one dimension.Thus, no phase transition occurs in one dimension.Through the electron-distortion interaction, that is thevibronic coupling, the size of the molecule changes withthe spin state. The distance (relative coordinate) be-tween the central transition metal and the surround-ing ligands changes. This distortion causes interactionsamong the spin states of molecules as depicted in Figs. 1(a), (b), and (c). In the present study, we focus on thelattice distortions in higher dimensions caused by the dif-ference of the molecular sizes due to the different spinstates. These local distortions interact with one anotherelastically which causes a long range effective interactionbetween the spin states.We perform molecular dynamics (MD) simulations ona 2D system with a simple square lattice. The in-tramolecular potential energy depending on the molec-ular size is given by a double-well adiabatic potential V intra i ( r i ), which is a function of the radius r i of the i thmolecule. Let p i be the corresponding relative momen-tum and let m be the reduced mass.We set an intermolecular binding interaction be-tween SC molecules (the i th and j th molecules) as V inter ij ( X i X j , r i , r j ), where X i = ( X i , Y i ) is the co-ordinate of the center of the i th molecule (Fig. 2 (a)).The corresponding momentum is P i = ( P X i , P Y i ) andthe mass of the molecule is M .To model this scenario we apply the following Hamil- LS molecule HS molecule
FeFe (a)(b) (c)
FIG. 1: (a) a schematic picture of the LS molecule and the HSmolecule. The HS molecule is bigger than the LS molecule.(b) ((c)) shows a schematic picture of the lattice distortionfor a HS (LS) molecule and surrounding LS (HS) molecules. tonian: H system = X i P i M + X i p i m + X i V intra i ( r i ) (1)+ X h i,j i V inter ij ( X i , X j , r i , r j ) . For simplicity we here consider only one symmetricvibration mode (isotropic volume expansion of molecules)as an active dominant mode .In order to clarify the effect of distortion, we adoptintermolecular binding potentials independent of themolecular states, although it is expected that the inter-molecular binding is looser for HS molecules than for LSmolecules.As the intramolecular potential, we adopt a double wellparabolic function V ( x ), where x is defined as the differ-ence of the radius from that of the ideal LS state. V ( x )has minima at x = 0 (ideal LS) and x = 1 (ideal HS).Setting r LS = 9 and r HS = 10 for the ideal radius of theLS molecule and that of the HS molecule, respectively,the radius of the molecule is r = r LS + x .When a parabolic potential for the LS state ( y = ax )and that for the HS state ( y = b ( x − c ) + d ) are mixedby off-diagonal element J , the lowest potential functionwith coefficient A is given by V ( x ) = A { d + b ( c − x ) + ax (2) − p J + ( d + b ( c − x ) − ax ) } . Because the entropy of the harmonic oscillator ( H = p m + Kx ) is S = N k B (cid:16) − ln ¯ hk B T q Km (cid:17) , the entropydifference between the LS and HS states is given by∆ S = S HS − S LS = N k B ln r K LS K HS . (3) i r j r molecule ( ) ii YX , ( ) kk YX , k r (a) ( ) jj YX , -1012345-1 -0.5 0 0.5 1 1.5 2 V ( x ) x (b) x T (c) FIG. 2: (a) a schematic picture of the model. (
X, Y ) isthe coordinate of the center of each molecule and r is itsradius. (b) Intramolecular potential, where x is the growthof r from r LS . (c) Temperature dependence of h x i withoutintermolecular interactions. Thus the ratio of the degeneracy between the HS andLS states is g = q K LS K HS = p ab , which corresponds to g in the WP-model. We take A = 10, a = 10, b = 0 . c = 1 . d = 0 . J = 0 .
04, which gives g = 10. In real-istic materials there are several sources of the differenceof the entropy. However, in order to focus on only thelattice effect, we adopt a large value of q K LS K HS (= 10). InFig.2 (b), the intramolecular potential V ( x ) is depicted.The energy difference between the LS stable point andthe HS stable point is ∆ E LS − HS = 1 .
075 and the energydifference between the LS stable point and the unstablepoint ( x c = 0 .
19) is ∆ E act . = 1 . x , i.e., h x i = Tr x exp( − β H system )Tr exp( − β H system ) (4)for V inter ij = 0, calculated by a numerical integration, isgiven in Fig. 2 (c).Next, we consider the intermolecular potential: V inter ij ( X i , X j , r i , r j ) between the nearest neighbors( i th and j th molecules). We take V inter ij ( X i , X j , r i , r j ) = f ( d ij ) , (5)where d ij = | X i − X j |− ( r i + r j ) . We treat phenomenain which the lattice distortion is not so large and it doesnot break the lattice structure. In this case, the quali-tative features of the phenomena do not depend on thedetails of the potential form, and thus we adopt one ofthe simplest forms f ( u ) = D (cid:16) e a ′ ( u − u ) + e − b ′ ( u − u ) (cid:17) , (6)where a ′ = 0 . b ′ = 1 . u is a constant such that f ( u ) has the minimum at u = 0. When molecules (circlesin Fig. 2 (a)) i and j contact each other ( d ij = 0), thefunction has a minimum value.In order to maintain the crystal structure (the coor-dination number), we introduce a potential between thenext-nearest neighbors ( i and k , see Fig. 2 (a)) V inter ik ( X i , X k , r i , r k ) = f ( d ik − ∆ r ) (7)with a ′ = 0 . b ′ = 0 .
2, which is much smaller thanthat of the nearest neighbors. Next-nearest neighborsdo not contact each other as depicted in Fig. 2 (a), andthere is a spatial gap between them. For simplicity, weassume here that next-nearest neighbors are most stabi-lized when the gap is ∆ r = 2( √ − r , where we take¯ r = ( r LS + r HS ) / r can be temperature de-pendent. We focus on the dependence of the spin state onthe strength of the intermolecular interaction, and thuswe study the dependence on D . Common D is used forboth Eqs. (5) and (7).To study the temperature dependence, we adopt theNos´e-Hoover method to generate the canonical en-semble for a given temperature T . The Hamiltonian ofthe thermal reservoir is given by H therm = P s Q + 3 N k B T ln s, (8)where s is a scaling factor, P s is the conjugated momen-tum of s and Q is an effective mass associated with s .Therefore, the total Hamiltonian including the effect ofthermal reservoir is given by H total = H system + H therm .Applying the Nos´e-Hoover formalism to the presentsystem, the time evolution of the system is realized ac-cording to the following equations of motion. dr i dt = p i m , (9) dp i dt = − ∂V intra ∂r i − ∂V inter ∂r i − ξp i , (10) d X i dt = P i M , (11) d P i dt = − ∂V inter ∂ X i − ξ P i , (12) dsdt = sξ, (13) dξdt = 1 Q "X i p i m + X i P i M − N k B T , (14)where V inter stands for the summation of the intermolec-ular potentials for the nearest and next-nearest pairs, and ξ ≡ P s Q . x T (a) x T (b) x T (c)(c) x T (d) FIG. 3: Temperature dependences of h x i for the values of (a) D = 10, (b) D = 20, (c) D = 28, and (d) D = 42. The openred circles (blue squares) denote h x i in the warming (cooling)process. We adopt x as a parameter to characterize the spinstate. We study the open-boundary system of L = 26 ×
26 molecules. We warm up the system from T = 0 . . T =0 .
1. At each temperature, 40000 MD steps are discardedas transient time and subsequent 20000 MD steps areused to measure x with the time step ∆ t = 0 .
01. Weemploy an operator decomposition method in which thenumerical error is of the order O (∆ t ). We set m = 1 . M = 1 .
0, and Q = 1 .
0. ( h x i does not depend on m , M ,and Q in the equilibrium state.)In Figs. 3 (a), (b), (c) and (d), the temperature depen-dences of h x i are shown. When D = 10 (Fig. 3 (a)), h x i in the warming process and that in the cooling processoverlap, indicating that a smooth (gradual) SC crossoveris realized. When the interaction parameter becomeslarger: D = 20 (Fig. 3 (b)), variation of h x i becomessharper, which implies that the SC transition becomesmore cooperative.When D = 28 (Fig. 3 (c)), a clear hysteresis loop of h x i is found. As the interaction parameter increases fur-ther: D = 42 (Fig. 3 (d)), the hysteresis width becomeslarger. Here, we found that when the interaction betweenmolecules becomes large, the SC transition changes froma gradual crossover to a first order transition. The criti-cal value of D is D critical ≃ (a) (b)(c) (d) FIG. 4: Snapshots of configurations. HS molecules (red cir-cles) are allocated when r is larger than r = r LS + x c . LSmolecules are drawn by blue circles. (a) Complete LS phase.(b) Complete HS phase. (c) A snapshot of configuration at T = 0 . D = 10 ( L = 16 ), where 80molecules are in the HS state. (d) A snapshot of configura-tion at T = 1 . D = 42 ( L = 16 ), where79 molecules are in the HS state. a snapshot of configuration at T = 0 . D = 10 and that at T = 1 . D = 42 are given, wherethe concentration of HS molecules is about 30% in both configurations. Although the number of HS molecules isalmost the same, the average cluster size of HS in (d)is bigger than in (c). This indicates that there is highercorrelation between spin states of molecules in the case ofstrong intermolecular interaction (case (d)), which pro-motes first-order transition.In this study, we investigated the cooperativity of spin-crossover phenomena induced by the elastic interactionamong lattice distortions which are triggered by the dif-ference of molecular sizes caused by the different spinstates. This effect is inherent to the high dimensional-ity (2D and 3D). The present 2D model can be appliedstraightforwardly to the 3D case. Although the latticerelaxation through a change of molecular sizes has beenstudied phenomenologically by a mean-field treatment ,as far as we know, this is the first attempt to investigatethe cooperativity attributed to the effect of local distor-tions (fluctuation) and that of the propagation to theoverall lattice.The present work was supported by Grant-in-Aid forScientific Research on Priority Areas and for Young Sci-entists (B) from MEXT of Japan, and supported byGrant-in-Aid from Minist`ere de l’Education Nationaleand CNRS (PICS France-Japan Program) of France. Thepresent work was also supported by the MST Founda-tion. The numerical calculations were supported by thesupercomputer center of ISSP of Tokyo University. ∗ Corresponding author. Email address:[email protected] P. G¨utlich, A. Hauser and H. Spiering, Angew. Chem. Int.Ed. Engl. , 2024 (1994) and references therein. S. Decurtins, P. G¨utlich, C.P.K¨ohler, H. Spiering, and A.Hauser, Chem. Phys. Lett. , 1 (1984). O. Kahn and C. Jay Martinez, Science , 44 (1998). J. F. L´etard, P. Guionneau, L. Rabardel, J. A. K. Howard,A. E. Goeta, D. Chasseau, and O. Kahn, Inorg. Chem., , 4432 (1998). A. Hauser, J. Jefti´c, H. Romstedt, R. Hinek and H. Spier-ing, Coord. Chem. Rev. , 471 (1999). J. A. Real, H. Bolvin, A. Bousseksou, A. Dworkin, O.Kahn, F. Varret, and J. Zarembowitch, J. Am. Chem. Soc. , 4650 (1992). N. Shimamoto, S. Ohkoshi, O. Sato, and K. Hashimoto,Inorg. Chem. , 678 (2002). M. Sorai and S. Seki, J. Phys. Chem. Solids, , 555 (1974). J. Wajnflasz and R. Pick, J. Phys. (Paris). Colloq , C1-91 (1971). A. Bousseksou, J. Nasser, J. Linares, K. Boukheddaden,and F. Varret, J. Phys. I France , 1381 (1992); A.Bousseksou, F. Varret, and J. Nasser, ibid. , 1463 (1993);A. Bousseksou, J. Constant-Machado, and F. Varret, ibid. , 747 (1995). K. Boukheddaden, I. Shteto, B. Hˆoo, and F. Varret., Phys.Rev. B , 14796 (2000); ibid. 14806 (2000). M. Nishino, S. Miyashita, and K. Boukheddaden, J. Chem.Phys. , 4594 (2003). M. Nishino, K. Boukheddaden, S. Miyashita, andF.Varret., Phys. Rev. B , 064452 (2005). S. Miyashita, Y. Konishi, H. Tokoro, M. Nishino, K.Boukheddaden, and F.Varret, Prog. Theor. Phys. , 719(2005). R. Zimmermann and E. K¨onig, J. Phys. Chem. Solids ,779 (1977). T. Kambara, J. Chem. Phys. , 4199 (1979). P. Adler, L. Wiehl, E. Meißner, C.P. K¨ohler, H. Spiering,and P. G¨utlich, J. Phys. Chem. Solids
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