Tutorial on elastic interaction models for multistep spin-crossover transitions

Spin-crossover (SC) compounds exhibit a variety of phase transitions induced by temperature, pressure, light irradiation, etc.^1–21 Their switchable magnetic, optical, and structural properties have drawn much attention in their potential applications to memory devices and sensors. In SC phenomena, transitions between the low-spin (LS) and high-spin (HS) states occur, and a wide variety of transition patterns are realized.

With increasing and decreasing temperature, various SC compounds show one-step SC transitions between the HS and LS phases. Their temperature dependence of the HS fraction, $f HS$⁠, shows a smooth single, discontinuous, or hysteresis curve. In the smooth single curve, the transition is a second-order transition or a mere change (not phase transition). In the discontinuous or hysteresis curve, a first-order transition occurs. In some SC compounds, besides the LS and HS phases, another phase with some pattern in alignment of LS and HS molecules is realized as a thermodynamics phase. An antiferromagnetic (AF)-like phase is a typical one, in which LS and HS molecules align alternately, and this is realized as an intermediate-temperature phase in two-step SC transitions.^22–31 

In recent experiments, more complex SC transition patterns were observed in three-step transitions^32–35 and more than three-step transitions.^36–40 A variety of ordered phases such as a LS–LS–HS–HS layered phase⁴¹ and commensurate and incommensurate stripe-ordered phases¹⁴ were found. With the development of nanotechnologies, SC nanoparticles and nanocomposites have also attracted much attention because of controllable finite-size effects.^42–47 Among them, recently, a core-shell nanocomposite made up of two different SC compounds⁴⁸ has been synthesized, which shows SC transitions with steps. Their synergy or competing effects owing to the combination of different SC compounds with finite-size effects are fascinating. Multistep SC phenomena with complex orderings are a current topic, and metastability or multistability is a key to understand these phenomena. In this paper, we present a tutorial study for modeling and simulation of SC materials and show that various patterns of SC transitions with steps are realized.

From the viewpoint of theoretical studies of SC phenomena, Ising-like models have been intensively studied, and several important properties of SC phenomena have been derived.^{28–30,49–52} However, the Ising interaction was introduced phenomenologically for simplicity, and the origin of the cooperative interactions has been ignored. On the atomistic scale, the molecular volume (or size) varies between the LS and HS states (Fig. 1) through the vibronic coupling between the electronic state and the structure, and the difference in the molecular size induces lattice distortion, which causes an elastic interaction between SC molecules (Fig. 2). Thus, volume expansion or contraction of SC crystals is accompanied with SC transitions. This elastic interaction is an essential ingredient in the cooperative character of SC materials.

In recent years, modeling of the cooperative interaction from the microscopic viewpoint has been focused on, and a new class of SC models based on cooperative elastic interactions, in which variation of the molecular size is taken into account, has been developed.^53–82 For the metal center whose spin switches between $S=0$ and $S=2$ in the case of Fe(II), for example, the molecular size (metal–ligand distance) changes depending on the electronic state as shown in Fig. 1. The cooperative nature of the elastic interaction has been intensively studied using this class of models, and important features of the cooperativity in SC phenomena have been found. Unlike the Ising-like models, in this modeling, the elastic interaction generates a long-range (LR) interaction. This LR interaction is important for the cooperativity as well as short-range (SR) interactions, and the interplay between the SR and LR interactions, which leads to frustration and synergy effect in SC ordering, is also significant especially for metastable and multistable states, leading to two-step and multi-step SC transitions.^{68,79,81,83–86} Interestingly, the importance of the LR interaction depends on the type of SC transitions. The LR interaction of elastic origin is relevant in first-order transitions^79,81 and a one-step second-order (continuous) transition,^55,65,68 while it is not relevant for second-order (continuous) transitions through the AF-like⁷⁹ or ferrimagnetic (FR)-like phases.⁸¹ The dynamics of phase transitions in the elastic interaction models is also characteristic and different from that of Ising-like models. In one-step transitions between the LS and HS phases, clustering in the bulk is suppressed, while clustering occurs from the corners (less probability from the sides) and does not occur from the inside.^61,66 On the other hand, in the Ising-like models, clustering occurs from the inside as well as the corners and sides. It should be noted that clustering from the inside is not always suppressed in the elastic interaction models. For example, in the second-order transitions through the AF-like or FR-like phases, clustering occurs also from the bulk,^68,79 in which the elastic distortion energy is relatively small.

Lattice parameter misfit causes an elastic frustration leading to unexpected behavior in SC transitions with steps.⁷⁶ SC models based on elastic interactions have been investigated for SC nanocomposites by modifying the surface environment,^64,87,88 and they have shown hysteresis curves with size dependence, which are consistent with corresponding experiments. The effect of lattice parameter misfit is important especially around the interface. Very recently, elastic interaction models have also been applied in a SC core-SC shell nanocomposite, and the thermodynamic properties of the nanocomposite have been studied as a function of the lattice parameter misfit between the two components around the interface.^89,90

The effect of the elastic interaction has been taken into account also in macroscopic approaches. SC transitions with steps have recently been studied in a Landau theory considering that non-symmetry-breaking and symmetry-breaking order parameters are coupled to the volume change.^91,92

In the present paper, we study elastic interaction models for SC systems in which LS, HS, and AF-like phases are realized as thermodynamics phases (upper panels in Fig. 3) and also those in which the LS, HS, LS-rich FR-like, and HS-rich FR-like phases are realized as thermodynamics phases (lower panels in Fig. 3). We analyze several phase diagrams for those systems, and making use of the phase diagrams, we show SC transition patterns with one-step, two-step, and three-step SC transitions. We also investigate an elastic interaction model for core-shell SC nanocomposites and show the occurrence of several two-step SC transitions originating from the difference in the model parameters between the core and shell.

The rest of the paper is organized as follows. In Sec. II, the model and method are presented. In Sec. III, we show the phase diagrams for square and triangular-lattice SC systems with metastable phases. Using the phase diagrams, we discuss thermodynamic properties of possible phases and SC transition patterns with steps. In Sec. IV, we model SC core-shell nanocomposites and give SC transition patterns with steps. We discuss characteristic features of the SC core-shell nanocomposites with an analysis of clustering and local pressure profiles. Section V is devoted to summary.

In simulation of elastic interaction models for SC transitions, Monte Carlo (MC) and molecular dynamics (MD) methods have often been used. In many cases, MC methods are easier to handle to simulate SC models than MD methods,^53,57 and many studies have been performed by MC methods. A combination between MD and MC methods is effective to study dynamics with different time scales between spin and lattice motions.⁶⁹ First, we give a brief instruction for the methods.

We can use a combination between the MD and MC methods, in which the change of the spin state is performed by the MC method and the deformation of the lattice is done by the MD method. The merit of this method is to control the relative ratio of the time scales of the spin-state change and lattice relaxation, which was applied to the property of the roughness of the interface growth depending on the two time scales.⁶⁹ 

Here, we use periodic boundary conditions in the MC method to study bulk properties in the $NPT$ ensemble (Sec. III) and open boundary conditions (OBCs) to investigate surface nucleation and clustering in finite size systems and properties of nanocomposite (Sec. IV) . The pressure is set to $P=0$⁠. In the $NPT$ MC method, we choose a molecule (site $i$⁠), update $σ i$⁠, $R i$⁠, and $X → i$⁠, and then update the volume of the total system under $P=0$⁠. In OBCs, the volume change is automatically realized. One MC step (MCS) is defined as $N$ repetitions of the update of a molecule. We apply $10 4$– $10 6$ MCS for measurement of physical quantities after the same number of MCS for equilibration.

We found that in spite of drastic simplification, the model (13) with an AF-like coupling, $J<0$⁠, on the square lattice catches essential features of the phase diagram for the elastic interaction model (5) with $J<0$ in both weak and strong elastic interaction cases. Owing to competing SR and LR interactions, unusual “horn structures” appear in the phase diagram of the model (13) with large $A(>0)$ as realized in that of the model (5) in the strong elastic interaction case (see also in Sec. III B).^79,83,84

We study one-step SC transitions between the LS and HS spin phases (upper left and middle configurations in Fig. 3). The transition is theoretically classified to three types: first-order transition, second-order transition, and gradual change. As we see later, the second-order transition is a special case, and practically, there are two types of transitions.

We can get a good outlook about the classification using the field–temperature (⁠ $h−T$⁠) phase diagram of the model (5) with $J=0$ or $J>0$ (ferromagnetic-like case) illustrated in Fig. 5. The feature of the phase diagram is independent of the space dimension and the lattice symmetry of the system. The pink dashed line is the coexistence line of the LS and HS phases. The right edge of the coexistence line is the critical point (⁠ $T c$⁠). This point has the mean-field universality,^55,65,68 which indicates a long-range interaction nature. The upper (lower) region is the HS (LS) phase, and the orange lines are spinodal lines. Therefore, the metastable LS (HS) phase exists between the upper (lower) spinodal line and the coexistence line. The region at high temperatures is a disordered phase, which connects the HS and LS phases continuously.

If the elastic interaction is stronger, $T c$ is higher. If $T SC< T c$⁠, the SC transition is of first order and is accompanied with a thermal hysteresis whose width is $T 2− T 1$ in Fig. 5. If $T SC= T c$⁠, the SC transition is a second-order (continuous) transition. If $T SC> T c$⁠, the SC transition is a gradual change. Here, we give an example of the first-order transition in Fig. 6 for $k 1=80000 K / nm 2$⁠, $D=1000 K$⁠, $J=0$⁠, $g=200$⁠, and $R HS/ R LS=1.1$ (⁠ $R LS=1$ nm) on the square lattice for $L=50$⁠, where $L= N 1 / 2$⁠.

Next, we study two-step SC transitions through an intermediate AF-like phase (upper right configuration in Fig. 3) in which LS and HS molecules align alternately. Here, we consider the AF-like SR (Ising) interaction, i.e., $J<0$⁠, on the square lattice.

In the same manner as the one-step SC transition, we perform a phase diagram analysis to classify the type of the two-step SC transitions.⁷⁹ There are basically three patterns for the two-step SC transition.

• double second-order (continuous) transitions.

• first-order and second-order (continuous) transitions.

• double first-order transitions.

We depict the $h−T$ phase diagram in Fig. 7 when the elastic interaction is not so strong. The green line denotes a critical line. This line has the Ising universality, in which the SR interaction is relevant.^68,79 The upper pink dashed line denotes the coexistence line of the HS and AF-like phases and the lower pink dashed line that of the AF-like and LS phases. The upper and lower orange lines are the spinodal lines of the AF-like phase, while the upper and lower violet lines are those of the HS and LS phases. The metastable AF phase exists between the upper orange solid and upper pink dashed lines and also between the lower pink dashed and lower orange solid lines. The metastable HS phase exists between the upper pink dashed and upper violet solid lines, while the metastable LS phase exists between the lower pink dashed and lower violet solid lines.

In Fig. 7(a), we show a path (blue thin line) for a temperature change (case II), and it crosses the critical line (green) (⁠ $T= T 3$⁠), the spinodal line of the LS phase (⁠ $T= T 2$⁠), and the spinodal line of the AF phase (⁠ $T= T 1$⁠). In the heating process, the LS phase changes to the AF-like phase at $T= T 2$ and the AF-like phase changes to the HS phase at $T= T 3$⁠, while in the cooling process, the HS phase changes to the AF-like phase at $T= T 3$ and the AF-like phase changes to the LS phase at $T= T 1$⁠. We plot the temperature dependences of $f HS$ and $V n$ in Fig. 8(a) for case II, where $V n=1.21$ in the perfect HS phase. Here, we used $J=−100$ K, $k 1=40000 K/ nm 2$⁠, $D=1030$ K, $g=1000$⁠, and $R HS/ R LS=1.1$ (⁠ $R LS=1$ nm) for $L=40$⁠. The corresponding $m sg$ is given in Fig. 8(b). We find that the feature of the volume change is similar to that of $f HS$ and observe volume jumps in the hysteresis loop.

In Fig. 7(b), we give a path for a two-step SC transition with double first-order transitions (case III), in which in the heating process, the LS phase changes to the AF-like phase at $T 2$ and the AF-like phase changes to the HS phase at $T 4$⁠, while in the cooling process, the HS phase changes to the AF-like phase at $T 3$ and the AF-like phase changes to the LS phase at $T 1$⁠. In addition to the two-step transition, the one-step transition between the AF-like and HS phases can be realized with another parameter set.⁷⁹ 

If the elastic interaction is strong, the feature of the phase diagram changes. Their unusual “horn structures,” each of which is surrounded by F-like spinodal lines, disorder (D) spinodal lines, and a critical line, appear at high temperatures. These structures cause a different kind of a two-step transition, in which a transition from the LS to AF to D to HS phase in the heating process and from the HS to D to AF to LS phase in the cooling process occurs.⁷⁹ 

We investigate spin-crossover transitions on triangular-lattice systems.^73,81 If a SR AF-like interaction is introduced into a triangular-lattice (nonbipartite) SC system such as $[ Fe ( bbtr ) 3] ( ClO 4 ) 2$⁠,⁹⁵ a frustration of the spin-state configuration, which accompanies an elastic frustration, is induced. Here, in addition to the NN SR (Ising) interaction, a small NNN SR (Ising) interaction, $J 2$⁠, is also considered. If $J 1≥0$ (F-like) and/or $J 2≥0$ (F-like), the phase diagram is similar to that of Fig. 5. Here, we take into account $J 1<0$ (AF-like) and $J 2≥0$⁠, which causes FR-like thermodynamic phases (lower left and right configurations in Fig. 3).

For the case of relatively weak elastic interactions, a typical phase diagram is given in Fig. 9. The two black solid lines in the phase diagram are second-order phase-transition lines with the three-state Potts universality.^81,96,97 The pink dashed line at $H=0$ is the coexistence line of the HS-rich and LS-rich FR-like phases. The yellow line at $H=0$ is a Berezinskii–Kosterlitz–Thouless (BKT) critical line, in which the right end point (right black circle) is a BKT transition point,^98–103 while the left end point is a second-order phase-transition point (left black circle).⁷³ The upper (lower) pink dashed line is the coexistence line of HS (LS) and HS-rich (LS-rich) FR-like phases. The violet lines are spinodal lines for F-like (HS and LS) phases. The upper (lower) orange line is the spinodal line for the HS-rich (LS-rich) FR-like phase against the stable F-like phase. The upper (lower) green line is the spinodal line for the LS-rich (HS-rich) FR-like phase against the stable HS-rich (LS-rich) FR-like phase.

We give a path in temperature variation depicted by a thin blue line in Fig. 9. This line crosses one of the critical lines at $T 4$⁠, the BKT line at $T 3$⁠, the spinodal line for the LS phase at $T 2$⁠, and the spinodal line for the LS-rich FR-like phase at $T 1$⁠. We set the parameter values as $k 1=40000 K/ nm 2$⁠, $R HS/ R LS=1.03$ (⁠ $R LS=1$ nm), $J 1=−100$ K, and $J 2=20$ K. This parameter set gives a phase diagram like Fig. 9. For $D=1520$ K and $ln⁡g=11.692$⁠, the path of the temperature change is like the blue line. The temperature dependences of $f HS$ and $V n$ for $L=36$ for this path are given in Fig. 10(a) and that of $M 3$ is shown in Fig. 10(b). The feature of $V n$ is similar to that of $f HS$⁠, where $V n=1.06$ in the perfect HS phase. We find a three-step SC transition with first-order, BKT, and second-order phase transitions at low, middle, and high temperatures, respectively, and observe that the LS-rich and HS-rich FR-like phases appear in the middle temperature region. In the relatively weak elastic interaction case, five typical SC transition patterns for one-step to four-step SC transitions can be realized.⁸¹ 

In the relatively strong elastic interaction with $J 1<0$ and $J 2>0$⁠, the feature of the phase diagram changes.⁸¹ The BKT phase disappears, and alternatively, a first-order transition point appears as the end point of the coexistence line between the HS-rich and LS-rich FR-like phases, at which the two critical lines terminate. The three coexistence lines and six spinodal lines are extended to the higher-temperature side. When $J 1<0$ and $J 2=0$⁠, i.e., only NN SR interaction, the topology of that is similar to the phase diagram with $J 1<0$ and $J 2>0$ for the relatively strong elastic interaction, but the regions of the coexistence lines are very small.⁸¹ Theoretically, there can be a variety of transition patterns in the triangular SC system: three, four, three, and two typical patterns for one-step, two-step, three-step, and four-step SC transitions, respectively.⁸¹ 

In the properties of SC nanocomposites, finite size effects are important. Before investigating SC nanocomposites, we study clustering features in a SC nanosystem. Here, we use the elastic interaction model (5) with open boundary conditions for $L=40$⁠. The parameters are set as $k 1=90000 K/ nm 2$⁠, $D=1200$ K, $J=0$⁠, and $g=200$⁠.

The temperature dependence of $f HS$ is plotted in Fig. 11. Corresponding snapshots of the configuration are given in Fig. 12. It is a characteristic that clustering of LS (HS) domains in the HS (LS) phase occurs from the corners during the transition from the HS (LS) to LS (HS) phase, and the clustering of LS molecules is sharper than that of HS molecules because of lower temperature. Clustering from the sides has a much lower probability owing to lower stability of the clusters, and clustering from the inner part is unrealistic owing to high distortion energy. This clustering feature is different from that of the Ising-like model, in which clustering occurs not only from the corners but also from the sides and the inner part. $J>0$ enforces the cluster formation, and the clustering feature is similar to the case of $J=0$⁠.

Here, the elastic interaction model (5) is applied to core-shell nanosystems. We use a core-shell nanostructure illustrated in Fig. 13.^89,90 The core has a square shape composed of $L C× L C$ molecules, and the shell has a shape of a thick frame of width $L S$ surrounding the core. The numbers of molecules in the core and the shell are $L C 2$ and $4 L S( L S+ L C)$⁠, respectively, and the total number is (⁠ $L C + 2 L S ) 2$⁠.

We use the same elastic constant in the core, shell, and interface, i.e., $k 1 C= k 1 S= k 1 I$⁠, and the same HS/LS degeneracy ratio $g$⁠, i.e., $g C= g S=200$ for simplicity. The Ising interaction is not considered: $J C= J S=0$⁠. We set $L C=40$ and $L S=5$⁠.

Variations of $R HS C$ and $R HS S$ cause changes of the strength of the elastic interactions in the core and shell, respectively, which influences the interaction between the core and shell. These variations lead to different transition features. In the present paper, we consider the following two cases.

• the shell SC transition temperature is higher: $T SC C< T SC S$⁠.

• the core SC transition temperature is higher: $T SC S< T SC C$⁠.

As reference data for the core-shell system, we depict the temperature dependences of $f HS$ of the uncoated core with $R HS C=1.02$⁠, 1.06, and 1.1 nm and of the hollow shell in Fig. 14. For larger $R HS C$⁠, the first-order transition nature is stronger in the uncoated core.

In Fig. 15, we show the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$ for $R HS C=1.02$ nm in the core-shell system. We find a smooth two-step transition with the first step at $T∼ T SC C$ and the second step at $T∼ T SC S$⁠. Both $f HS C$ and $f HS S$ show smooth curves without hysteresis. The size of the core HS molecule (⁠ $R HS C=1.02$ nm) is close to the core LS molecule, and the elastic interaction in the core is small. Above the lower transition temperature, the shell LS molecules around the interface are stabilized around $T∼ T SC S$ since $R HS C$ is close to $R LS S$⁠, and the transition point in the shell slightly shifts to the high-temperature side. This causes a smooth transition in the shell, in which the small hysteresis observed in the hollow shell in Fig. 14 disappears. The size of the shell HS molecule (⁠ $R HS S=1.1$⁠) is not close to that of the shell LS molecule, and the elastic interaction in the shell is larger than that in the core, but the shell has a large surface (open boundary) and the elastic interaction around the surface is suppressed.

Next, we consider a case of a larger size of the core HS molecule. In Fig. 16, we show the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$ for $R HS C=1.06$⁠. We find a hysteresis loop in $f HS$⁠, which reflects a hysteresis loop of $f HS C$⁠. The SC transition point in the core is located at a little higher temperature than $T= T SC C$⁠. Compared to the case $R HS C=1.02$ nm (Fig. 15), the shell HS fraction, $f HS S$⁠, changes at a lower temperature, and the transition becomes shaper. A large elastic interaction caused by a large difference in the molecular size between the LS and HS states in the core is the origin of the hysteresis. The core HS molecule is bigger than the shell LS molecule. Therefore, the core HS molecules around the interface have a positive pressure before the transition to the HS state in the core, which shifts the transition point in the core to a higher temperature, while the shell LS molecules around the interface have a negative pressure before the transition to the HS state in the shell, which accelerates a change to the HS state in the shell.

We study a case of a further large size of the core HS molecule. In Fig. 17, we show the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$ for $R HS C=1.1$⁠. We find that $f HS$ shows a large hysteresis at low temperatures. It should be noted that in this hysteresis loop, there are two contributions, namely, hysteresis loops of $f HS C$ and $f HS S$⁠. In the heating process, some of the shell LS molecules become the HS state before the transition to the HS state in the core is completed, and in the cooling process, some of the core HS molecules become the LS state before the transition to the LS state in the shell is completed. Specifically, the LS–HS transition in the core and that in the shell occur concertedly. Here, the change of $f HS$ cannot be expected from the values of $T SC C$ and $T SC S$⁠.

The larger size of the core HS molecule causes a strong elastic interaction and a large hysteresis of $f HS C$⁠. In the heating process, the metastability in the core is very strong because the transition to the HS state in the core occurs against a positive pressure from the shell LS molecules around the interface, and thus, the right branch of the hysteresis loop shifts to the high-temperature side. On the other hand, in the cooling process, the metastability in the core is not so strong because the transition to the LS state in the core is driven by a positive pressure caused by the shell LS molecules around the interface.

At low temperatures, e.g., 30 K, the local pressure is almost uniform with relatively small positive or negative values. It should be noted that at around $T=80$ K in the heating process, only small HS domains are generated in the core to reduce elastic distortion. The LS and HS molecules in the core have relatively low and high pressures, respectively. With further heating, the HS molecules stay in the HS, but many LS molecules switch to the HS state because of entropy gain. At 90 K in the heating process, the shell LS molecules at the sides (not corners) change to HS ones, and the transition to the HS state in the shell around the corners is delayed. There, the local pressure of the side molecules in the shell is low, and the HS domains are stable. At high temperatures (⁠ $150$ K), the state is close to the saturated HS state, but the fluctuation of the local pressure is found. At $T=80$ K in the cooling process, clustering of the LS molecules occurs from the corners in the shell. With further cooling to $T=30$ K, the rest shell HS molecules change to LS molecules, and simultaneously, clustering of the core LS molecule occurs from the corners of the core, around which the local pressure of the core HS molecules is high and this accelerates the transition to the LS state. Accompanying the clustering, small domains of the LS molecule appear inside the core to minimize the elastic distortion.

As reference data for the core-shell system, we illustrate the temperature dependences of $f HS$ of the hollow shell with $R HS S=1.01$⁠, 1.08, and 1.1 nm and of the uncoated core in Fig. 20. For larger $R HS S$⁠, the first-order transition nature in $f HS$ is stronger in the hollow shell.

In Fig. 21, we show the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$ for $R HS S=1.01$ nm. $f HS$ shows a two-step transition with a low-temperature smooth transition and a high-temperature transition with a hysteresis loop. In the heating (cooling) process, after the transition in the shell (core) is completed, the transition in the core (shell) occurs. The size of the HS molecule in the shell is small; thus, the elastic interaction in the shell is small, and a continuous transition between the LS and HS states in the shell takes place at $T∼ T SC S$⁠.

It should be noted that a hysteresis does not appear at the lower transition temperature but appears at the higher transition temperature (175–190 K), although the hysteresis width is smaller than that of the uncoated core in Fig. 20. In the case of the two-step transition through the AF-like phase studied in Sec. III B, a hysteresis appears at the lower transition temperature and a stronger metastable state is realized at low temperatures. In this core-shell system, however, a hysteresis at the higher transition temperature is easier to realize. The shell HS molecular size, $1.01$ nm, is small, and the molecules in the core change to the HS state under a large positive pressure. This effect reinforces the stability of the LS state in the core and causes a higher transition temperature, $T∼185$ K, in the core than $T= T SC C$⁠. This also makes the location of the left branch of the hysteresis loop at a higher-temperature side, and as a result, the hysteresis width becomes smaller than that of the uncoated core in Fig. 20.

We consider a larger size of the HS molecule in the shell, $R HS S=1.08$⁠. In Fig. 22, we show the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$⁠. $f HS$ shows a two-step transition with a small hysteresis loop at a little higher temperature than $T SC S$ and a large hysteresis loop at a high temperature, which is lower than $T= T SC C$⁠. It should be noted that in the heating process, before the transition to the HS states is completed in the shell, the transition to the HS state starts in the core, and in the cooling process, before the transition to the LS state is completed in the core, the transition to the LS state starts in the shell.

We study this case (Fig. 22) microscopically by analyzing snapshots of molecular configuration in Fig. 23 and of the local pressure in Fig. 24. In the heating process, clustering of HS molecules in the shell occurs from the corners around $T=50$ K. On further heating to $T=140$ K, the HS domains expand in the shell and simultaneously clustering of HS molecules in the core occurs from the corners, in which the core LS molecules have a low pressure and this helps them to easily change to the HS state. However, this clustering is not so clear, and small HS domains appear inside the core.

In the cooling process, small LS domains appear inside the core at $T∼110$ K to reduce the distortion due to the negative pressure appearing in the bulk. On further cooling to 50 K, the rest core HS molecules, which have high pressure, change to the LS state, and clustering of shell LS molecules from the sides of the shell occurs. The corner shell molecules change to the LS state in the end.

For a further large size of the shell HS molecule, $R HS C=1.1$⁠, we give in Fig. 25 the temperature dependences of $f HS$⁠, $f HS C$⁠, and $f HS S$⁠. $f HS$ shows a large hysteresis with a neck. In the heating process, a part of the shell LS molecules changes fast to the HS ones at $T∼60$ K. The rest of the shell LS molecules gradually change to the HS ones up to $T∼150$ K, and simultaneously, the core LS molecules change to the HS ones. In the cooling process, the shell HS molecules convert to the LS ones following the change of the core HS molecules to the LS ones, and in the ending of the transition to the LS state, $f HS$ and $f HS S$ change slowly.

The elastic interaction caused by lattice distortion owing to molecular size difference between the low-spin (LS) and high-spin (HS) molecules plays an essential role in cooperativity of SC phenomena. In this paper, we presented a tutorial study on elastic interaction models for spin-crossover (SC) compounds. Using the models, we studied two- and multi-step behavior in SC transitions in several systems. The competition and/or synergy effects between short-range interactions and long-range interactions of elastic origin cause a variety of thermodynamic and metastable phases. We analyzed several typical phase diagrams with metastable phases. Considering the path corresponding to temperature variation in the phase diagrams, we can classify what kind of transitions and what transition patterns can be realized with increasing and decreasing temperature.

Firstly, we showed basic one-step SC transitions between the LS and HS phases. Secondly, we presented two-step SC transitions between the LS, antiferromagnetic (AF)-like, HS phases on the square lattice, and possible patterns of two-step transitions. The qualitative feature of the phase diagram for the one-step SC transition and that for the two-step transition with relatively small elastic interactions do not change in three dimensions. Next, we investigated three-step SC transitions between the LS, LS-rich ferrimagnetic (FR)-like, HS-rich FR-like, and HS phases on the triangular lattice, and showed a three-step SC transition. In the systems introduced in this study, many patterns of SC transitions with one to four steps are theoretically plausible. Applying this analysis to recently discovered multi-step SC materials in experiments, details in transition mechanisms would be found.

We also investigated SC transitions with steps in a SC core-SC shell nanocomposite model. We analyzed the two cases: (I) the core has a lower transition temperature than the shell (⁠ $T SC C< T SC S$⁠), and (II) the core has a higher transition temperature (⁠ $T SC C> T SC S$⁠). We presented and discussed the temperature dependences of the HS fractions of the total system, core, and shell with an analysis of snapshots of the molecular configuration and the local pressure. We focus on lattice parameter misfit leading to an elastic frustration, which is an important ingredient for complex SC transitions in core-shell systems. The molecular size difference between the LS and HS state in the core and that in the shell affect the interaction between the core and shell, as well as the elastic interaction at the interface. Tuning the differences, a variety of SC transitions with steps are realized. For stronger interaction between the core and shell, the LS–HS transition in the core and that in the shell occur concertedly. In two-step transitions through the AF-like phase, the lower-temperature transition has a larger hysteresis. In two-step transitions in case II, however, the higher-temperature transition with a larger hysteresis is possible.

We hope that our study contributes to fully understanding of multistep SC phenomena and further development of SC studies.

This work was supported by Grants-in-Aid for Scientific Research C (Nos. 18K03444 and 20K03809) from the MEXT of Japan. Y.S. and K.B. acknowledge the ANR (Agence Nationale de la Recherche) for the funding received from project Mol-CoSM (No. ANR-20-CE07-0028-02), and all the authors acknowledge the LIA (International Associate French Japan Laboratory) for the financial support.

The data that support the findings of this study are available from the corresponding author upon reasonable request.