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Relaxation dynamics of two-step spin-crossover
Polyhedron 24 (2005) 2852–2856 www.elsevier.com/locate/poly
Relaxation dynamics of two-step spin-crossover Masamichi Nishino
a,*
, Kamel Boukheddaden b, Seiji Miyashita c, Franc¸ois Varret
b
a
b
Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan Laboratoire de Magne´tisme et d’Optique, CNRS-Universite´ de Versailles/St. Quentin en Yvelines, 45 Avenue des Etats Unis, F78035 Versailles Cedex, France c Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan Received 8 October 2004; accepted 1 December 2004 Available online 11 July 2005
Abstract We study static and dynamical properties of the two-step spin-crossover solids. A Monte Carlo method with the Arrhenius transition probability is applied to a two-sublattice model with an intrasublattice interaction and an intersublattice interaction. The intrasublattice interaction favors the uniform order of the low spin state or high spin state in each sublattice. The intersublattice interaction favors the opposite orders in the two sublattices. The present model describes well the two-step phase transition. There, an intermediate-temperature ÔantiferroÕ phase, where the sublattices have different states, appears between low-temperature phase (LS phase) and high-temperature phase (HS phase), as observed in the experiments. We focus on the relaxation from the saturated HS state which is realized by photo excitation. Two-step relaxation is found: a fast relaxation with sigmoidal shape and following slow relaxation. The former is due to the Arrhenius transition rate and the latter is a result of stochastic relaxation from the metastable antiferro state, which reproduces well the relaxation observed in the experiments. 2005 Elsevier Ltd. All rights reserved. Keywords: Spin-crossover; Dynamics; Two-step
1. Introduction Spin-crossover transition induced by a change in temperature, pressure, etc. has been studied extensively [1–3]. Among spin-crossover compounds, the iron (Fe(II)) complexes are typical ones. The low spin (LS) state (S = 0) described by ðt62g Þ in low temperatures is converted to the high spin (HS) state (S = 2) described by ðt42g e2g Þ and vice versa. This transition between the LS and HS states is found to be a cooperative transition of correlated molecules, in which changes of molecular vibration and structure are accompanied. The spin crossover compounds have potential applications for a new type of information storage because of their bistability. So far various types of spin-crossover transitions *
Corresponding author. Tel.: +81298592472; fax: +81298604706. E-mail address: [email protected] (M. Nishino).
0277-5387/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.poly.2005.03.165
induced by temperature change have been observed, e.g., gradual transition, abrupt transition, two-step transition, etc. [1–12], where the elastic interaction plays an important role. Here, we focus on the spin-crossover transition with two steps, which has been found in compounds made of either di-iron [7–9] (called binuclear) or mono-iron systems [10–12]. In order to realize the intermediate-temperature phase, competing interactions are necessary and indeed the two-step spin-crossover transitions were modeled by introducing antagonist interactions [7–9,13,14]. Thus, we adopt a model with an intrasublattice interaction which favors neighboring molecules of identical electronic (or spin) states (HS–HS or LS–LS) and an intersublattice interaction which favors the opposite configuration (i.e., HS–LS or LS–HS). We call the latter interaction Ôantiferromagnetic-likeÕ or Ôantiferro-elasticÕ coupling.
M. Nishino et al. / Polyhedron 24 (2005) 2852–2856
After the discovery of light induced excited spin state trapping (LIESST) [15], a number of investigations of the metastable photo-excited HS states have been studied. There the relaxation curves of HS fraction show a typical sigmoidal shape, associated with ‘‘self-accelerated’’ relaxation [16–18]. A dynamical mean-field treatment [21,22] of Wajnflasz and Pick [19,20] with the Arrhenius transition probability reproduced well this behavior, which gives the microscopic origin of the HauserÕs formula [16–18]. Relaxation curve of the HS fraction from the photoexcited HS states with the two-step relaxation was observed recently [6]. The first fast relaxation and next slow relaxation, which cannot be explained by the mean-field (MF) theory. We reproduced this two-step relaxation by a Monte Carlo (MC) method [23]. In this paper, we present the equilibrium properties with a variety of two-step phase transition and the non-equilibrium properties with the two-step relaxation of the HS fraction using a MC method.
2. Models and methods We consider the model introduced by Bousseksou et al. [8], which is an extension of the Wajnflasz and Pick (WP) [19–22,24] model to the case of the two-equivalent sublattices coupled ÔantiferromagneticallyÕ X X A BB ^ ¼ J AA H SA S Bi S Bj i Sj J hi2A;j2Ai
J
AB
X
hi2B;j2Bi
B SA i Sj
þD
X
hi2A;j2Bi
S i ¼ 1; . . . ; 1; 1; . . . 1 ; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}u |fflfflffl{zfflfflffl}r
B SA i þ Si ;
tion and the phonon degrees of freedom. The parameter D is the ligand-field energy. Here, we assume that D is the same for the two sublattices. The ÔNaClÕ lattice structure (fcc) for the two sublattices A and B is considered (Fig. 1(b)). Thus, the coordination numbers between and within sublattices are given by zAB = 6 for A–B bonds and zAA = zBB = z = 12 for A–A and B–B bonds. We consider the case of equivalent sublattices, then we take JAA = JBB = J. Using a temperature dependent field, this model can be expressed formally as follows [21,22]: X X X A B ^ ¼ J H SA S Bi S Bj J AB SA i Sj J i Sj hi2A;j2Ai
ð1Þ
where the states S = 1 and 1 denote the LS and HS states, respectively. JAA and JBB are the ÔferromagneticÕ (JAA > 0, JBB > 0) intrasublattice interactions and JAB is the AF (JAB < 0) intersublattice interaction. The potential energy of the LS and HS states is shown schematically in Fig. 1(a). The degeneracy u and r for the LS and HS state comes from the angular momentum, the spin momentum, the molecular vibra-
hi2B;j2Bi
ln g X A þ D S i þ S Bi ; 2b i r with g ¼ ; u
b¼
hi2A;j2Bi
1 ; kBT
ð2Þ
where the degeneracy effect is renormalized into the parameter g. The degeneracy ratio g increases up to 100–1000 and becomes temperature-dependent [21,22]. However, for simplicity, we take g = 100. The parameter D is set as D = 400 K, which gives a realistic order of magnitude for the transition temperature. In order to study the dynamical properties of the system, we use the general master equation d P ðS 1 S k S N ; tÞ dt 1X X W k ðS k ! S 0k ÞP ðS 1 S k S N ; tÞ ¼ s k S 6¼S 0 k
i
2853
k
1X X W k ðS 0k ! S k ÞP ðS 1 S 0k S N ; tÞ; þ s k S 6¼S 0 k
ð3Þ
k
where P(S1 Sk SN, t) is the probability of the state {S1 Sk SN} at time t and W k ðS k ! S 0k Þ is the transition probability at the kth site. We adopt the Arrhenius dynamics, which is suitable to describe experimental data for the relaxation process from the HS state [21,22]. The transition probability is given by 1 1 W k ðS k ! S 0k Þ ¼ expðbðE0 þ S k yÞÞ; s 2s0
ð4Þ
0 state where Sk is one of thePstates (±1) and P SBk is the other ln g 0 A AB ðS k 6¼ S k Þ and y ¼ J A S þ J S D þ . Here, B 2b E0 denotes an intramolecular energy barrier, and s0 is a quantity related to the time scale. We take s0 = 1. The system has two intrinsic order parameters. One is the fraction of HS molecules defined by
Fig. 1. (a) Schematic diagram of the potential energy of the LS and HS states. The degeneracy of LS state and HS state is taken into account. The axis denotes metal–ligand distance. (b) The model (fcc lattice) with coupling J(>0) and JAB(<0).
fHS ¼
mF þ 1 . 2
ð5Þ
B Here, we define theP quantity mF ¼ mA þm , where 2 P the magA 1 netization mA ¼ N i2A hS i i and mB ¼ N1 i2B hS Bi i are
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M. Nishino et al. / Polyhedron 24 (2005) 2852–2856
the net ÔmagnetizationÕ per site in the sublattices A and B, respectively. N is the number of all A and B sites, and Æ æ denotes the thermodynamical average. The magnetization mF expresses the degree of ÔferromagneticÕ order (i.e., HS–HS, or LS–LS) and fHS takes a value between 0 and 1. The other order parameter is given by mA mB mAF ¼ . ð6Þ 2 This order parameter expresses the degree of the ÔantiferromagneticÕ (i.e., LS–HS or HS–LS) order.
3. Static and dynamical properties Two-step phase transitions by the MC method are shown in Fig. 2. The temperature dependence of the HS fraction is shown in Fig. 2(a) and the temperature dependence of AF order parameter is shown in Fig. 2(b). Here, 10 000–50 000 MC steps are taken as waiting time to make the system reach the equilibrium state, and then we perform 10 000–50 000 MC steps for measuring the thermodynamical quantities (fHS). The parameter JAB is set as JAB = 12 K and J is changed as J = 40 K (), 16 K (e), and 12 K (·).
The system size is L3 = 203. In the low temperatures the system is in the LS phase (all down state) as depicted in Fig. 2 (c)-(I). As the temperature is increased, a transition occurs and fHS reaches the value fHS = 1/2. Then a plateau region appears and the other transition occurs leading to the HS phase (Fig. 2(c)-(III)). This plateau region shows the presence of the phase of the AF order as depicted in Fig. 2(c)-(II). Here, we find three typical two-step transitions: (i) first-order transition at the lower transition temperature and first-order transition at the higher transition temperature as the case J = 40 K; (ii) first-order transition at the lower transition temperature and gradual transition at the higher transition temperature as the case J = 16 K; (iii) gradual transition at the lower transition temperature and gradual transition at the higher transition temperature as the case J = 12 K. In the following, we discuss the condition of these three cases [6,23]. We define TC as the critical temperature of 3D fcc Ising model, which is given by TC = 9.79524 J [25]. Here, we also define TC1 and TC2 as 2D 2zAB jJ A j and k B ln g 2D þ 2zAB jJ AB j . ¼ k B ln g
T C1 ¼ T C2
ð7Þ
Fig. 2. (a) Temperature dependence of fHS of two-step transitions. JAB = 12 K. J = 40 K (s), 16 K (e), and 12 K (·). The point I is the initial point of the relaxation dynamics. (b) Temperature dependence of the mAF of two-step transitions. (c) Ordered states on the x–y (or y–z) plane of the lattice. (I) LS phase, (II) AF phase, and (III) HS phase.
M. Nishino et al. / Polyhedron 24 (2005) 2852–2856
The temperatures TC1 and TC2 correspond to the transition temperatures of sharp double first-order transitions [6,23]. Which type of transition occurs depends on values of TC, TC1 and TC2. When TC1 < TC2 < TC, first-order transition occurs twice. When TC1 < TC < TC2, a first-order transition occurs at TC1, while a gradual transition occurs around TC2. When TC < TC1 < TC2, two successive gradual transitions occur. Next, we study the relaxation dynamics from the HS phase. We investigate the case of J = 40 K, in which a strong metastable AF phase exists in the low-temperature region near the lower transition temperature. We take the saturated HS state (fHS = 1) at T = 105 K as the initial state depicted by I in Fig. 2(a) and observe the relaxation to the LS state. Relaxation curves of the two order parameters fHS and mAF are observed, respectively, in Fig. 3(a) and (b). The bold line is the average relaxation curve obtained by sampling of 10 independent processes with different sequences of random numbers. We found that the relaxation process consists of two distinct regimes. Fast relaxation with a sigmoidal (convex) shape takes place from the saturated HS fraction (fHS = 1) to half of it (fHS . 0.5), and slow concave relaxation to the LS phase (fHS = 0) follows. The former relaxation with a sigmoidal shape occurs from the HS to AF state and this is due to the Arrhenius dynamics. On the other hand the latter relaxation is that from the AF to LS state. Because the AF state is a strong metastable state, this relaxation is governed by the nucleation process [23]. From the view point of the most probable path (MPP) on the mean-filed free energy, a potential barrier exists between the AF and LS states as we found in the contour diagram before [23]. There the relaxation from the metastable AF phase to LS phase depends on the energy barrier height and temperature. When the thermal fluctuation overcomes this potential energy barrier, relaxation to the LS state occurs. In Fig. 3(c), an example of nucleation of the LS state in the AF phase is shown. The nucleation process occurs
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stochastically and the time of the jump is distributed widely as the Poisson process [26]. As a result, the sample-averaged relaxation shows an exponential (concave) relaxation. This shape agrees with the experimental result very well [6]. The mean field theory [6] does not reproduce concave relaxation but convex relaxation where the thermal fluctuation is not taken into account. The introduction of short-range correlations by different analytical methods [27] leads to obtain long tail behavior [28,29] In Fig. 3(a), the thin lines are examples of relaxation with different sequences of random numbers. The upper thin solid line shows a case where the state stays long in the metastable AF state. Here, the concave relaxation curve is observed, which is similar to that obtained by the mean-filed results. On the other hand, the AF state lives in a very short period in the case of the lower thin line.
4. Summary and discussion We have investigated the static and dynamical properties of the two-step spin-crossover compounds. Using an extended WP-model which consists of two-equivalent sublattices with an intra- (ÔferroÕ) interaction and an inter- (ÔantiferroÕ) sublattice interaction, we present several types of two-step transitions. It was found with this model that basically there exists three types of two-step transitions. In the first case, a first-order transition occurs twice. In the second case, a first-order transition occurs at the lower transition temperature and a gradual transition occurs at the higher transition temperature. In the third case two successive gradual transitions occur. The condition for three types of the two-step transition was also given. We have studied the relaxation from the fully saturated HS state, which simulates the relaxation from the photo excited saturated high spin state. It was found that the relaxation of the HS fraction consists of two parts. First, fast relation from the HS to AF state occurs. A sigmoidal shape is reproduced in the initial stage
Fig. 3. (a) Relaxation of fHS at T = 105 K. The bold line is the average of 10 samples. Thin solid lines are examples of one randon number sequence. (b) Relaxation of mAF at T = 105 K. The bold line is the average of 10 samples. Thin solid lines are examples of one randon number sequence. (c) Droplet of the LS state () in the AF phase.
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M. Nishino et al. / Polyhedron 24 (2005) 2852–2856
of the relaxation, which is typical in relaxations of spincrossover compounds in experiments. This sigmoidal shape is due to the Arrhenius transition probability in the Monte Carlo simulation as has been pointed out in the mean-field theory [6]. Following the fast relaxation, the slow relaxation from the AF to LS state occurs. When a droplet of the LS state grows in the AF phase and the size of the droplet becomes over the critical droplet, nucleation occurs, which is a stochastic process. If the nucleation rate is low, the stochastic single-nucleation process appears where the AF state relaxes to the LS state suddenly as a Poisson process. On the other hand, if the nucleation rate is high the system shows the multi-nucleation (Avrami) relaxation. They give concave slow relaxation as the average process. This two-step relaxation, i.e., fast sigmoidal and slow exponential relaxation, agree with that found in the experiments [6] very well. We discussed here the dynamics only focusing on the relaxation from the HS state. Elucidation of various dynamical aspects by photoinduced phenomena [30,31] would be an important problem in the future. In the case of simple spin models as the present treatment, the stable state relaxes rather fast through the local nucleation. More realistic treatment of the elasticity would be a key to investigate a large scale separation of the phases which was observed in the experiment [32].
References [1] E. Ko¨nig, Struct. Bond. 76 (1991) 51. [2] P. Gu¨tlich, A. Hauser, H. Spiering, Angew. Chem. 106 (1994) 2109. [3] P. Gu¨tlich, A. Hauser, H. Spiering, Angew. Chem., Int. Ed. Engl. 33 (1994) 2024, and references therein. [4] C.P. Ko¨hler, R. Jakobi, E. Meissner, L. Wiehl, H. Spiering, P. Gu¨tlich, J. Phys. Chem. Solids 51 (1990) 239. [5] R. Jakobi, H. Spiering, P. Gu¨tlich, J. Phys. Chem. Solids 53 (1992) 267.
[6] K. Boukheddaden, J. Linares, E. Codjovi, F. Varret., V. Niel, J.A. Real, J. Appl. Phys. 93 (2003) 1. [7] J.A. Real, H. Bolvin, A. Bousseksou, A. Dworkin, O. Kahn, F. Varret, J. Zarembowitch, J. Am. Chem. Soc. 114 (1992) 4650. [8] A. Bousseksou, F. Varret, J. Nasser, J. Phys. I 3 (1993) 1463. [9] J.A. Real, J. Castro, A. Bousseksou, M. Verdaguer, R. Burriel, M. Castro, J. Linares, F. Varret, Inorg. Chem. 36 (1997) 455. [10] H. Ko¨ppen, E.W. Mu¨ller, C.P. Ko¨hler, H. Spiering, E. Meissner, P. Gu¨tlich, Chem. Phys. Lett. 91 (1982) 348. [11] V. Petrouleas, J.P. Tuchagues, Chem. Phys. Lett. 135 (1987) 21. [12] D. Boinnard, A. Bousseksou, A. Dworkin, J.M. Savariault, F. Varret, J.P. Tuchagues, Inorg. Chem. 33 (1994) 271. [13] A. Bousseksou, J. Nasser, J. Linares, K. Boukheddaden, F. Varret, J. Phys. I 2 (1992) 1381. [14] A. Bousseksou, J. Nasser, F. Varret, J. Magn. Magn. Mater. 140 (1995) 1511. [15] S. Decurtins, P. Gu¨tlich, C.P. Ko¨hler, H. Spiering, A. Hauser, Chem. Phys. Lett. 1 (1984) 139. [16] A. Hauser, Coord. Chem. Rev. 11 (1991) 275. [17] A. Hauser, P. Gu¨tlich, H. Spiering, Inorg. Chem. 25 (1986) 4345. [18] A. Hauser, J. Chem. Phys. 94 (1991) 2741. [19] J. Wajnflasz, R. Pick, J. Phys. Status Solidi 40 (1970) 537. [20] J. Wajnflasz, R. Pick, J. Phys. (Paris). Colloq 32 (1971) C1. [21] K. Boukheddaden, I. Shteto, B. Hoˆo, F. Varret, Phys. Rev. B 62 (2000) 14796. [22] K. Boukheddaden, I. Shteto, B. Hoˆo, F. Varret, Phys. Rev. B (2000) 14806. [23] M. Nishino, K. Boukheddaden, S. Miyashita, F. Varret., Phys. Rev. B 68 (2003) 224402. [24] M. Nishino, S. Miyashita, K. Boukheddaden, J. Chem. Phys. 118 (2003) 4594. [25] C. Domb, M.S. Green (Eds.), Phase Transition and Critical Phenomena, vol. 3. [26] P.A. Rikvold, H. Tomita, S. Miyashita, S.W. Sides, Phys. Rev. E 49 (1994) 5080, and references therein. [27] H. Mamada, S. Takano, J. Phys. Soc. Jpn. 25 (1968) 675. [28] K. Boukheddaden, F. Varret, S. Salunke, J. Linares, E. Codjovi, Phase Transit. 75 (2002) 733. [29] B. Hoˆo, K. Boukheddaden, F. Varret, Eur. Phys. J. B 17 (2000) 449. [30] M. Nishino, K. Yamaguchi, S. Miyashita, Phys. Rev. B 58 (1998) 9303. [31] M. Nishino, S. Miyashita, Phys. Rev. B 63 (2001) 174404. [32] Y. Ogawa, S. Koshihara, K. Koshino, T. Ogawa, C. Urano, H. Takagi, Phys. Rev. Lett. 84 (2000) 3181.
Relaxation dynamics of two-step spin-crossover Masamichi Nishino
a,*
, Kamel Boukheddaden b, Seiji Miyashita c, Franc¸ois Varret
b
a
b
Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan Laboratoire de Magne´tisme et d’Optique, CNRS-Universite´ de Versailles/St. Quentin en Yvelines, 45 Avenue des Etats Unis, F78035 Versailles Cedex, France c Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan Received 8 October 2004; accepted 1 December 2004 Available online 11 July 2005
Abstract We study static and dynamical properties of the two-step spin-crossover solids. A Monte Carlo method with the Arrhenius transition probability is applied to a two-sublattice model with an intrasublattice interaction and an intersublattice interaction. The intrasublattice interaction favors the uniform order of the low spin state or high spin state in each sublattice. The intersublattice interaction favors the opposite orders in the two sublattices. The present model describes well the two-step phase transition. There, an intermediate-temperature ÔantiferroÕ phase, where the sublattices have different states, appears between low-temperature phase (LS phase) and high-temperature phase (HS phase), as observed in the experiments. We focus on the relaxation from the saturated HS state which is realized by photo excitation. Two-step relaxation is found: a fast relaxation with sigmoidal shape and following slow relaxation. The former is due to the Arrhenius transition rate and the latter is a result of stochastic relaxation from the metastable antiferro state, which reproduces well the relaxation observed in the experiments. 2005 Elsevier Ltd. All rights reserved. Keywords: Spin-crossover; Dynamics; Two-step
1. Introduction Spin-crossover transition induced by a change in temperature, pressure, etc. has been studied extensively [1–3]. Among spin-crossover compounds, the iron (Fe(II)) complexes are typical ones. The low spin (LS) state (S = 0) described by ðt62g Þ in low temperatures is converted to the high spin (HS) state (S = 2) described by ðt42g e2g Þ and vice versa. This transition between the LS and HS states is found to be a cooperative transition of correlated molecules, in which changes of molecular vibration and structure are accompanied. The spin crossover compounds have potential applications for a new type of information storage because of their bistability. So far various types of spin-crossover transitions *
Corresponding author. Tel.: +81298592472; fax: +81298604706. E-mail address: [email protected] (M. Nishino).
0277-5387/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.poly.2005.03.165
induced by temperature change have been observed, e.g., gradual transition, abrupt transition, two-step transition, etc. [1–12], where the elastic interaction plays an important role. Here, we focus on the spin-crossover transition with two steps, which has been found in compounds made of either di-iron [7–9] (called binuclear) or mono-iron systems [10–12]. In order to realize the intermediate-temperature phase, competing interactions are necessary and indeed the two-step spin-crossover transitions were modeled by introducing antagonist interactions [7–9,13,14]. Thus, we adopt a model with an intrasublattice interaction which favors neighboring molecules of identical electronic (or spin) states (HS–HS or LS–LS) and an intersublattice interaction which favors the opposite configuration (i.e., HS–LS or LS–HS). We call the latter interaction Ôantiferromagnetic-likeÕ or Ôantiferro-elasticÕ coupling.
M. Nishino et al. / Polyhedron 24 (2005) 2852–2856
After the discovery of light induced excited spin state trapping (LIESST) [15], a number of investigations of the metastable photo-excited HS states have been studied. There the relaxation curves of HS fraction show a typical sigmoidal shape, associated with ‘‘self-accelerated’’ relaxation [16–18]. A dynamical mean-field treatment [21,22] of Wajnflasz and Pick [19,20] with the Arrhenius transition probability reproduced well this behavior, which gives the microscopic origin of the HauserÕs formula [16–18]. Relaxation curve of the HS fraction from the photoexcited HS states with the two-step relaxation was observed recently [6]. The first fast relaxation and next slow relaxation, which cannot be explained by the mean-field (MF) theory. We reproduced this two-step relaxation by a Monte Carlo (MC) method [23]. In this paper, we present the equilibrium properties with a variety of two-step phase transition and the non-equilibrium properties with the two-step relaxation of the HS fraction using a MC method.
2. Models and methods We consider the model introduced by Bousseksou et al. [8], which is an extension of the Wajnflasz and Pick (WP) [19–22,24] model to the case of the two-equivalent sublattices coupled ÔantiferromagneticallyÕ X X A BB ^ ¼ J AA H SA S Bi S Bj i Sj J hi2A;j2Ai
J
AB
X
hi2B;j2Bi
B SA i Sj
þD
X
hi2A;j2Bi
S i ¼ 1; . . . ; 1; 1; . . . 1 ; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}u |fflfflffl{zfflfflffl}r
B SA i þ Si ;
tion and the phonon degrees of freedom. The parameter D is the ligand-field energy. Here, we assume that D is the same for the two sublattices. The ÔNaClÕ lattice structure (fcc) for the two sublattices A and B is considered (Fig. 1(b)). Thus, the coordination numbers between and within sublattices are given by zAB = 6 for A–B bonds and zAA = zBB = z = 12 for A–A and B–B bonds. We consider the case of equivalent sublattices, then we take JAA = JBB = J. Using a temperature dependent field, this model can be expressed formally as follows [21,22]: X X X A B ^ ¼ J H SA S Bi S Bj J AB SA i Sj J i Sj hi2A;j2Ai
ð1Þ
where the states S = 1 and 1 denote the LS and HS states, respectively. JAA and JBB are the ÔferromagneticÕ (JAA > 0, JBB > 0) intrasublattice interactions and JAB is the AF (JAB < 0) intersublattice interaction. The potential energy of the LS and HS states is shown schematically in Fig. 1(a). The degeneracy u and r for the LS and HS state comes from the angular momentum, the spin momentum, the molecular vibra-
hi2B;j2Bi
ln g X A þ D S i þ S Bi ; 2b i r with g ¼ ; u
b¼
hi2A;j2Bi
1 ; kBT
ð2Þ
where the degeneracy effect is renormalized into the parameter g. The degeneracy ratio g increases up to 100–1000 and becomes temperature-dependent [21,22]. However, for simplicity, we take g = 100. The parameter D is set as D = 400 K, which gives a realistic order of magnitude for the transition temperature. In order to study the dynamical properties of the system, we use the general master equation d P ðS 1 S k S N ; tÞ dt 1X X W k ðS k ! S 0k ÞP ðS 1 S k S N ; tÞ ¼ s k S 6¼S 0 k
i
2853
k
1X X W k ðS 0k ! S k ÞP ðS 1 S 0k S N ; tÞ; þ s k S 6¼S 0 k
ð3Þ
k
where P(S1 Sk SN, t) is the probability of the state {S1 Sk SN} at time t and W k ðS k ! S 0k Þ is the transition probability at the kth site. We adopt the Arrhenius dynamics, which is suitable to describe experimental data for the relaxation process from the HS state [21,22]. The transition probability is given by 1 1 W k ðS k ! S 0k Þ ¼ expðbðE0 þ S k yÞÞ; s 2s0
ð4Þ
0 state where Sk is one of thePstates (±1) and P SBk is the other ln g 0 A AB ðS k 6¼ S k Þ and y ¼ J A S þ J S D þ . Here, B 2b E0 denotes an intramolecular energy barrier, and s0 is a quantity related to the time scale. We take s0 = 1. The system has two intrinsic order parameters. One is the fraction of HS molecules defined by
Fig. 1. (a) Schematic diagram of the potential energy of the LS and HS states. The degeneracy of LS state and HS state is taken into account. The axis denotes metal–ligand distance. (b) The model (fcc lattice) with coupling J(>0) and JAB(<0).
fHS ¼
mF þ 1 . 2
ð5Þ
B Here, we define theP quantity mF ¼ mA þm , where 2 P the magA 1 netization mA ¼ N i2A hS i i and mB ¼ N1 i2B hS Bi i are
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the net ÔmagnetizationÕ per site in the sublattices A and B, respectively. N is the number of all A and B sites, and Æ æ denotes the thermodynamical average. The magnetization mF expresses the degree of ÔferromagneticÕ order (i.e., HS–HS, or LS–LS) and fHS takes a value between 0 and 1. The other order parameter is given by mA mB mAF ¼ . ð6Þ 2 This order parameter expresses the degree of the ÔantiferromagneticÕ (i.e., LS–HS or HS–LS) order.
3. Static and dynamical properties Two-step phase transitions by the MC method are shown in Fig. 2. The temperature dependence of the HS fraction is shown in Fig. 2(a) and the temperature dependence of AF order parameter is shown in Fig. 2(b). Here, 10 000–50 000 MC steps are taken as waiting time to make the system reach the equilibrium state, and then we perform 10 000–50 000 MC steps for measuring the thermodynamical quantities (fHS). The parameter JAB is set as JAB = 12 K and J is changed as J = 40 K (), 16 K (e), and 12 K (·).
The system size is L3 = 203. In the low temperatures the system is in the LS phase (all down state) as depicted in Fig. 2 (c)-(I). As the temperature is increased, a transition occurs and fHS reaches the value fHS = 1/2. Then a plateau region appears and the other transition occurs leading to the HS phase (Fig. 2(c)-(III)). This plateau region shows the presence of the phase of the AF order as depicted in Fig. 2(c)-(II). Here, we find three typical two-step transitions: (i) first-order transition at the lower transition temperature and first-order transition at the higher transition temperature as the case J = 40 K; (ii) first-order transition at the lower transition temperature and gradual transition at the higher transition temperature as the case J = 16 K; (iii) gradual transition at the lower transition temperature and gradual transition at the higher transition temperature as the case J = 12 K. In the following, we discuss the condition of these three cases [6,23]. We define TC as the critical temperature of 3D fcc Ising model, which is given by TC = 9.79524 J [25]. Here, we also define TC1 and TC2 as 2D 2zAB jJ A j and k B ln g 2D þ 2zAB jJ AB j . ¼ k B ln g
T C1 ¼ T C2
ð7Þ
Fig. 2. (a) Temperature dependence of fHS of two-step transitions. JAB = 12 K. J = 40 K (s), 16 K (e), and 12 K (·). The point I is the initial point of the relaxation dynamics. (b) Temperature dependence of the mAF of two-step transitions. (c) Ordered states on the x–y (or y–z) plane of the lattice. (I) LS phase, (II) AF phase, and (III) HS phase.
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The temperatures TC1 and TC2 correspond to the transition temperatures of sharp double first-order transitions [6,23]. Which type of transition occurs depends on values of TC, TC1 and TC2. When TC1 < TC2 < TC, first-order transition occurs twice. When TC1 < TC < TC2, a first-order transition occurs at TC1, while a gradual transition occurs around TC2. When TC < TC1 < TC2, two successive gradual transitions occur. Next, we study the relaxation dynamics from the HS phase. We investigate the case of J = 40 K, in which a strong metastable AF phase exists in the low-temperature region near the lower transition temperature. We take the saturated HS state (fHS = 1) at T = 105 K as the initial state depicted by I in Fig. 2(a) and observe the relaxation to the LS state. Relaxation curves of the two order parameters fHS and mAF are observed, respectively, in Fig. 3(a) and (b). The bold line is the average relaxation curve obtained by sampling of 10 independent processes with different sequences of random numbers. We found that the relaxation process consists of two distinct regimes. Fast relaxation with a sigmoidal (convex) shape takes place from the saturated HS fraction (fHS = 1) to half of it (fHS . 0.5), and slow concave relaxation to the LS phase (fHS = 0) follows. The former relaxation with a sigmoidal shape occurs from the HS to AF state and this is due to the Arrhenius dynamics. On the other hand the latter relaxation is that from the AF to LS state. Because the AF state is a strong metastable state, this relaxation is governed by the nucleation process [23]. From the view point of the most probable path (MPP) on the mean-filed free energy, a potential barrier exists between the AF and LS states as we found in the contour diagram before [23]. There the relaxation from the metastable AF phase to LS phase depends on the energy barrier height and temperature. When the thermal fluctuation overcomes this potential energy barrier, relaxation to the LS state occurs. In Fig. 3(c), an example of nucleation of the LS state in the AF phase is shown. The nucleation process occurs
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stochastically and the time of the jump is distributed widely as the Poisson process [26]. As a result, the sample-averaged relaxation shows an exponential (concave) relaxation. This shape agrees with the experimental result very well [6]. The mean field theory [6] does not reproduce concave relaxation but convex relaxation where the thermal fluctuation is not taken into account. The introduction of short-range correlations by different analytical methods [27] leads to obtain long tail behavior [28,29] In Fig. 3(a), the thin lines are examples of relaxation with different sequences of random numbers. The upper thin solid line shows a case where the state stays long in the metastable AF state. Here, the concave relaxation curve is observed, which is similar to that obtained by the mean-filed results. On the other hand, the AF state lives in a very short period in the case of the lower thin line.
4. Summary and discussion We have investigated the static and dynamical properties of the two-step spin-crossover compounds. Using an extended WP-model which consists of two-equivalent sublattices with an intra- (ÔferroÕ) interaction and an inter- (ÔantiferroÕ) sublattice interaction, we present several types of two-step transitions. It was found with this model that basically there exists three types of two-step transitions. In the first case, a first-order transition occurs twice. In the second case, a first-order transition occurs at the lower transition temperature and a gradual transition occurs at the higher transition temperature. In the third case two successive gradual transitions occur. The condition for three types of the two-step transition was also given. We have studied the relaxation from the fully saturated HS state, which simulates the relaxation from the photo excited saturated high spin state. It was found that the relaxation of the HS fraction consists of two parts. First, fast relation from the HS to AF state occurs. A sigmoidal shape is reproduced in the initial stage
Fig. 3. (a) Relaxation of fHS at T = 105 K. The bold line is the average of 10 samples. Thin solid lines are examples of one randon number sequence. (b) Relaxation of mAF at T = 105 K. The bold line is the average of 10 samples. Thin solid lines are examples of one randon number sequence. (c) Droplet of the LS state () in the AF phase.
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of the relaxation, which is typical in relaxations of spincrossover compounds in experiments. This sigmoidal shape is due to the Arrhenius transition probability in the Monte Carlo simulation as has been pointed out in the mean-field theory [6]. Following the fast relaxation, the slow relaxation from the AF to LS state occurs. When a droplet of the LS state grows in the AF phase and the size of the droplet becomes over the critical droplet, nucleation occurs, which is a stochastic process. If the nucleation rate is low, the stochastic single-nucleation process appears where the AF state relaxes to the LS state suddenly as a Poisson process. On the other hand, if the nucleation rate is high the system shows the multi-nucleation (Avrami) relaxation. They give concave slow relaxation as the average process. This two-step relaxation, i.e., fast sigmoidal and slow exponential relaxation, agree with that found in the experiments [6] very well. We discussed here the dynamics only focusing on the relaxation from the HS state. Elucidation of various dynamical aspects by photoinduced phenomena [30,31] would be an important problem in the future. In the case of simple spin models as the present treatment, the stable state relaxes rather fast through the local nucleation. More realistic treatment of the elasticity would be a key to investigate a large scale separation of the phases which was observed in the experiment [32].
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