Intrinsic effects of the boundary condition on switching processes in spin-crossover solids
Masamichi Nishino, Cristian Enachescu, Seiji Miyashita, Kamel Boukheddaden, Francois Varret
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Intrinsic effects of the boundary condition on switching processes in spin-crossoversolids
Masamichi Nishino , , , ∗ Cristian Enachescu , Seiji Miyashita , , Kamel Boukheddaden , and Fran¸cois Varret Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Japan Department of Physics, Alexandru Ioan Cuza University, Iasi, Romania Department of Physics, Graduate School of Science,The University of Tokyo, Bunkyo-Ku, Tokyo, 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 CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan (Dated: November 8, 2018)We investigated domain growth in switching processes between the low-spin and high-spin phasesin thermally induced hysteresis loops of spin-crossover (SC) solids. Elastic interactions amongthe molecules induce effective long-range interactions, and thus the boundary condition plays asignificant role in the dynamics. In contrast to SC systems with periodic boundary conditions, whereuniform configurations are maintained during the switching process, we found that domain structuresappear with open boundary conditions. Unlike Ising-like models with short-range interactions,domains always grow from the corners of the system. The present clustering mechanism providesan insight into the switching dynamics of SC solids, in particular, in nano-scale systems.
PACS numbers: 75.30.Wx 75.50.Xx 75.60.-d 64.60.-i
Spin-crossover (SC) compounds have been studied in-tensively because of their peculiar physical properties dueto competition between the low energy of the low-spin(LS) state and the high entropy of the high-spin (HS)state [1, 2, 3, 4]. SC transitions are induced by changes intemperature, pressure, etc. The LS state can be excitedby photo-irradiation to a long-lived HS state at low tem-peratures, which is called LIESST (light induced excitedspin state trapping) [5], and reverse LIESST (HS to LS)can also be obtained at a different wavelength [6]. Thesecontrollable and functional properties [1, 7, 8, 9, 10, 11]would bring potential applicability to novel optical de-vices, e.g., optical data storage and optical sensors.The LS and HS states couple through a vibronic mech-anism and the size of the SC molecule changes withthe spin state. The distortion caused by the change ofmolecular size induces a kind of elastic interaction amongthe spin states of molecules. The importance of theelastic interaction has been reported for SC transitions[1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. The mechanismof the phase transition induced by the elastic interactionhas been studied and various new aspects have been re-vealed. For example, using periodic boundary conditions,it has been shown that effective long-range interactionssuppress domain growth, and uniform configurations aremaintained even near the critical temperature [16]. Inthe process of switching the configuration uniformity isalso maintained, which is considered an intrinsic propertyof systems with the effective long-range interactions. Inthese systems, we do not expect the critical opalescencedue to growth of large clusters.Nowadays SC compounds are a focus of nano-scienceand technology [22, 23, 24, 25, 26]. On the nano-scale, such as powder or thin film samples, the boundary effectis important. In particular, in systems with long-rangeinteractions the concept of the thermodynamic limit maynot be well defined and the effect of the boundary mustbe considered carefully.In this Letter we investigate how a SC system witheffective long-range interactions switches between thebistable states, using open boundary conditions. We an-alyze characteristic features of the heating and coolingprocesses in thermal hysteresis loops with open bound-ary conditions (OBC), and compare to periodic boundaryconditions (PBC).We adopt a simple SC model for the square lattice,which represents general characteristics. In the model,both intramolecular and intermolecular interactions aretaken into account [20], H = N X i =1 P i M + N X i =1 p i m + N X i =1 V intra i ( r i ) (1)+ X h i,j i V inter ij ( X i , X j , r i , r j ) . Here, X i and P i represent the coordinate and its con-jugate momentum of the center of mass for the i thmolecule. Conjugate variables r i and p i are definedfor the totally symmetric mode for the i th molecule,which is the most important intramolecular motion [27].We also define the variable x as x = r − r LS , where r LS (= 9) is the ideal radius of the LS molecule. Thatof the HS molecule is r HS = r LS + 1. The intramolec-ular potential energy V intra i ( x i ) is shown by the solidcurve in Fig. 1 (a). We adopt the intermolecular poten-tial V inter ij ( X i , X j , r i , r j ) between the nearest neighbors V ( x ) x (a) LS HS (b) FIG. 1: (color online) (a) Intramolecular potential energy V ( x ) shown by the solid (blue) curve. A realistic value ω LS ω HS = 2 is adopted [27]. The dotted curves are LS and HSpotential energies without quantum mixing. A LS molecule(blue small circle) and HS molecule (red large circle) are inset.(b) Thermal hysteresis loop for D = 20 and L = 100 with theopen boundary. Open circles denote HS fraction. and next-nearest neighbors, where D is the strength ofthe intermolecular interaction [28].We study the present model by a molecular dynamicsmethod, in which we introduce a mechanism to controlthe large entropy difference between the HS and LS states[20]. When the intermolecular interaction is stronger, thesystem exhibits a thermal hysteresis. Using PBC, uni-form configurations are maintained during the transitionbetween the HS and LS phases. Using OBC, however, amacroscopic inhomogeneity is produced.Here we focus on the dynamics of a relatively largehysteresis loop ( D =20). In Fig. 1 (b) the temperaturedependence of the HS fraction [20] is given for a sys-tem of N = L = 100 × T = 0 . T = 0 .
1. At each temperature,first 400,00 MD steps were discarded as transient time,and then 200,00 MD steps were used to measure physicalquantities with the MD time step ∆ t = 0 .
01. The tran-sition from the LS to HS state (LS → HS) occurs around T = 1 . → LS) around T = 0 . T < . (a) (b)(c) (d) FIG. 2: (color online) Snapshots of configurations in thecooling process from the HS to LS phase. Red (blue) circlesdenote HS (LS) molecules. (a) T = 0 . t = 33108, (b) T =0 . t = 33506, (c) T = 0 . t = 33692, (d) T = 0 . t = 33804. actions, where the nucleation occurs from the inner partor the sides in large systems [29]. In order to clarify thisdifference, we study energy dependence on the clusterpattern, and we find that the growth from sides is notacceptable in the elastic model.As the initial state we set a round LS domain (closedquadrant) at a corner in the complete HS phase. Then,we move all molecules slowly by reducing the total poten-tial energy of the system. We define N ILS as the numberof LS molecules in the LS domain in the initial state, and N SLS as that in the stationary state. R ILS is the numberof LS molecules in the horizontal ( X ) (or vertical ( Y ))direction for the domain with N ILS .Figure 3 (a) illustrates the configuration in the station-ary state when we set R ILS = 7 and N ILS = 39. In thiscase, the quadrant shape was maintained and the numberof LS molecules in the domain did not change during thesimulation, i.e., N ILS = N SLS , although a small shift of theposition and radius for each LS molecule was observed,which means that the LS domain in the stationary stateis at least locally stable.We investigate ∆ E , defined as the difference of energiesbetween the stationary state and the complete HS phase.The dependence of ∆ E on N ILS is plotted by circles inFig. 4. We find that ∆ E becomes lower as N ILS increases,which indicates that the larger LS domain at the corneris energetically favorable. We checked that it holds truefor larger system size ( L ). Thus, the domain will grow ifsome small noise (thermal fluctuation) assists the system (a) (b)(c) (d) FIG. 3: (color online) (a) Stationary configuration of a roundLS domain at a corner in HS phase. (b) Initial configura-tion of a round LS domain at a side (c) A configuration inthe intermediate state. Green color denotes unstable state ofmolecules. (d) Configuration in the stationary state. Unsta-ble LS molecules changed to HS molecules and the initiallyround LS domain does not exist any more. -100-80-60-40-200200 10 20 30 40 50 60 70 80cornerside ∆ E N ILS (22) (26) (26)
FIG. 4: (color online) ∆ E vs. N ILS for LS domains at thecorner and the side. For LS domains at the corner, N ILS is thesame as N SLS , unlike LS domains at the side. The number ina parenthesis is the value of N SLS when the initial LS domainshape is not maintained and N SLS varies from N ILS . to relax.Next, we study the stability of LS domains (closedsemicircle) at a side. Figure 3 (b) shows the initialstate of the configuration of a LS domain ( R ILS = 7, N ILS = 78) at a side. In this case, the initial config-uration is found to be unstable due to high distortion.Several LS molecules change back to HS molecules andthe number of LS molecules changes ( N ILS > N
SLS ).Figure 3 (c) is a configuration in the intermediate stateand the final stable configuration is shown in Fig. 3 (d).The molecules colored green (in gray in black-and-whiteprint) in Fig. 3 (c) have intermediate radii (0.3 < r < E for LS domains atthe side as a function of N ILS . Unlike the case of thecorner, ∆ E is not a simple decreasing function of N ILS for larger N ILS . It is worth noting that N SLS is no moreequal to N ILS for 40 ≤ N ILS , and takes a value for theconfiguration in the stationary state, which is given in aparenthesis in Fig. 4. LS domains which are bigger thana critical size are unstable at the side.If we set a round LS domain in the center of the HSphase, due to a huge distortion, the domain becomes veryunstable and it collapses easily to reduce the number ofLS molecules.We checked the qualitatively same tendency for thestability of LS domains when the system size ( L ) is larger.These observations lead to a major conclusion; LS do-mains cannot grow from the sides or the inner part ofthe system, which is different from the results of short-range interaction models.This conclusion suggests that different nucleation pro-cesses exist between short-range interaction models andthe elastic model. We define P (corner), P (side), and P (inner part) as the nucleation rate from a corner, aside, and the inner part, respectively. Generally, they arefunctions of the size and temperature, and P > P > P because the surface energy increases in this order. How-ever, if we take into account the possible location of nucle-ation, the probabilities to observe nucleation at a corner,a side, and the inner part of the system are P , L × P ,and L × P , respectively. Then, in short-range models,the relation P < L × P < L × P will hold for larger L (linear dimension), and nucleation occurs in the innerpart in large systems. Thus, so called multi-nucleationprocess takes place [29].In the elastic model, however, P and P are essen-tially zero and P (corner) is the only probability fornucleation. Therefore, even if the system size is large,nucleation (clustering) always starts from corners.We next investigate the process in heating. Snapshotsof transient states from the LS to HS phase in the heat-ing are given in Fig. 5 (a) − (d). Here, we also find localclusters of HS molecules around the corners, but in con-trast to the case of the process from the HS to LS phase(left branch of the hysteresis loop), a large homogeneousregion appears as is observed in periodic boundary con-ditions.We consider the reason for the difference of the chang-ing pattern between the heating (LS to HS) and cooling(HS to LS) processes. In the SC system, the HS andLS states are not equivalent and we may expect differenttypes of relaxations for the cooling and heating processes.In the cooling process at low temperatures, the energystability is more important than the entropy gain andthe nucleation from a corner is the most favorable. Inthe heating process at high temperatures, however, theentropy gain becomes more important, and the configu- (a) (b)(c) (d) FIG. 5: (color online) Snapshots of configurations in theheating process from the LS to HS phase. Red (blue) circlesdenote HS (LS) molecules. (a) T = 1 . t = 5452, (b) T = 1 . t = 5522, (c) T = 1 . t = 5672, (d) T = 1 . t = 5748. ration may change uniformly, which can be seen in theinner part of the system.In summary, we have studied effects of the boundarycondition in a SC model with effective long-range inter-actions. We found that domains always grow from cor-ners, which exhibits a striking contrast to the cases ofshort-range interaction models. In the heating process,an entropy-driven mechanism causes a smearing of clus-ters, and the configuration is close to that with the peri-odic boundary condition.The existence of macroscopic domains in SC com-pounds has been suggested in experimental studies of X-ray diffraction [30, 31, 32]. The present study could givean insight into that suggestion. Dynamical propertiesof SC materials with OBC are important for studies ofnano-scale systems, where the boundary plays a crucialrole.The present work was supported by Grant-in-Aid forScientific Research on Priority Areas (17071011) and forScientific Research C (20550133), and by the Next Gen-eration Super Computer Project, Nanoscience Programfrom MEXT of Japan. CE thanks to PNII 1994 Roma-nian CNCSIS Ideas Grant. The numerical calculationswere supported by the supercomputer center of ISSP ofTokyo University. ∗ Corresponding author. Email address:[email protected][1] P. G¨utlich and H. A. Goodwin (ed), Spin Crossover inTransition Metal Compounds I, II, III. (Springer, Berlin,2004).[2] E. K¨onig, Struct. Bonding (Berlin) , 51 (1991).[3] A. Hauser, J. Jefti´c, H. Romstedt, R. Hinek and H. Spier-ing, Coord. Chem. Rev. , 471 (1999).[4] M. Sorai, M. Nakano, and Y. Miyazaki, Chem. Rev. ,976 (2006).[5] S. Decurtins, P. G¨utlich, K. M. Hasselbach, A. Hauser,and H. Spiering, Inorg. Chem. , 2174 (1985).[6] A. Hauser, J. Chem. Phys. , 2741 (1991).[7] T. Tayagaki and K. Tanaka, Phys. Rev. Lett. , 2886(2001).[8] J.F. L´etard, J. Mater. Chem., , 2550 (2006).[9] W. Gawelda et al ., Phys. Rev. Lett. , 057401 (2007)[10] M. Lorenc et al ., Phys. Rev. Lett. , 028301 (2009).[11] O. Fouch´e, J. Degert, G. Jonusauskas, C. Bald´e, C. Des-planche, J.F. L´etard, E. Freysz, Chem. Phys. Lett. ,274 (2009)[12] R. Zimmermann and E. K¨onig, J. Phys. Chem. Solids ,779 (1977).[13] P. Adler, L. Wiehl, E. Meißner, C.P. K¨ohler, H. Spiering,and P. G¨utlich, J. Phys. Chem. Solids
517 (1987).[14] N. Willenbacher and H. Spiering. J. Phys. C: Solid StatePhys. , 1423 (1988).[15] M. Nishino, K. Boukheddaden, Y. Konishi, and S.Miyashita, Phys. Rev. Lett. 98, 247203 (2007).[16] S. Miyashita, Y. Konishi, M. Nishino, H. Tokoro, and P.A. Rikvold, Phys. Rev. B , 014105 (2008).[17] Y. Konishi, H. Tokoro, M. Nishino, and S. Miyashita,Phys. Rev. Lett. 100, 067206 (2008).[18] K. Boukheddaden, M. Nishino, and S. Miyashita, Phys.Rev. Lett. 100, 177206 (2008).[19] W. Nicolazzi, S. Pillet, and C. Lecomte, Phys. Rev. B , 174401 (2008).[20] M. Nishino, K. Boukheddaden, and S. Miyashita, Phys.Rev. B , 012409 (2009).[21] C. Enachescu, L. Stoleriu, A. Stancu, and A. HauserPhys. Rev. Lett. , 257204 (2009).[22] S. Cobo, G. Moln´ar, J. A. Real, and A. Bousseksou,Angew. Chem. Int. Ed. , 5786 (2006).[23] E. Coronado, J. R. Gal´an-Mascar´os, M. Monrabal-Capilla, J. Garc´ıa-Mart´ınez, and P. Pardo-Iba˜nez, Adv.Mater. , 1359 (2007).[24] G. Moln´ar, S. Cobo, J. A. Real, F. Carcenac, E. Daran,C. Vieu, and A. Bousseksou, Adv. Mater. , 2163(2007).[25] F. Volatron, L. Catala, E. Rivi`ere, A. Gloter, O. St´ephan,and T. Mallah, Inorg. Chem. , 6584 (2008).[26] I. Boldog, A. B. Gaspar, V. Mart´ınez, P. Pardo-Iba˜nez,V. Ksenofontov, A. Bhattacharjee, P. G¨utlich, J. A. Real,Angew. Chem. Int. Ed. ,85 (2004) and references therein.[28] The intermolecular potential [15] is defined as V inter ij ( X i , X j , r i , r j ) = f ( d ij − ∆ r ), where f ( u ) = D (cid:16) e a ′ ( u − u ) + e − b ′ ( u − u ) (cid:17) . The variable u is a con- stant such that f ( u ) has the minimum at u = 0 and d ij = | X i − X j | − ( r i + r j ). For the nearest neigh-bors, ∆ r = 0, a ′ = 0 . b ′ = 1 . r = 2( √ − r with¯ r = ( r LS + r HS ) / a ′ = 0 . b ′ = 0 . , 5080 (1994). [30] N. Huby et al ., Phys. Rev. B , 020101(R) (2004).[31] S. Pillet , J Hubsch and Lecomte , Eur. Phys. J. B, ,541 (2004).[32] K. Ichiyanagi, J. Hebert, L. Toupet, H. Cailleau, P.Guionneau, J.-F. L´etard, and E. Collet, Phys. Rev. B73