Theoretical investigations on the pressure effects in spin-crossover materials: Reentrant phase transitions and other behavior

Pressure effects have been widely studied in spin-crossover (SCO) solids due to their immediate influence on the thermal dependence of the high-spin (HS) fraction. In most of the cooperative SCO materials, the applied pressure shifts the transition temperatures upward and decreases the thermal hysteresis widths to such an extent that it vanishes at some critical pressure. However, several other unexpected experimental features were found in the literature, showing that the applied pressure may (i) induce an increase of the thermal hysteresis width or even (ii) lead to a reentrant behavior on the thermal hysteresis whose width first increases for low applied pressures and then decreases at high-pressure values. These nonstandard behaviors have been classified as anomalous, even though the transition temperature always increases under pressure. In this theoretical contribution, we rationalize all these behaviors by describing the spin-crossover system under pressure with an elastic description accounting for the difference of lattice parameters between the low-spin (LS) and HS phases including the pressure effects. The analytical study of this elastic model in the homogeneous mechanical system demonstrated its isomorphism with an Ising-like model with infinitely long-range interactions, in which the pressure acts linearly on the ligand field and nonlinearly on the strength of the interactions. The resolution of this model brings to light the existence of a pressure-induced interplay between these two contributions allowing one to recover a wide range of normal and abnormal observed experimental thermal dependences of the spin transition under pressure.

Figure 1

(a) Thermal dependence of the HS fraction for different values of the ratio $\frac{J_0}{{\mathrm{\Delta }}_0}$, showing the presence of a critical point for ${\mathrm{\Delta }}_0=J_0.$ The spinodal curve, given by Eq. (16), is superimposed for clarity reasons. (b) $ln\left(\mathrm{\Delta }T\right)$ vs $ln\left(T_C-T_{eq}\right)$ showing a linear dependence, with a slope $\frac{3}{2}$, following the power law given by Eq. (17). In (b) $J_0$ ( ${\mathrm{\Delta }}_0$) is fixed to 120 K (300 K) when ${\mathrm{\Delta }}_0$ ($J_0$) is varied. The other parameters values used are $g=\sqrt{150}$.

Figure 2

(a) phase diagram exhibiting the pressure dependence of the spin transition $T_{eq}$ and the Curie temperature $T_C$. (b) The thermal dependence of the HS fraction under pressure showing the transformation from first-order to gradual spin transition as the pressure increases. The black curve is the spinodal curve, which delimits the bistable region. (c) The superlinear behavior of the thermal hysteresis width as a function of ${\left(T_C-T_{eq}\right)}^{3/2}$ obtained for the curves of (a). The parameter values are $\sqrt{g}=150, {\mathrm{\Delta }}_0=300.6$ K,$J_0=160$ K, $\alpha =40.1$ K/kbar, and $\gamma =4$ K/kbar.

Figure 3

(a) Phase diagram exhibiting the pressure dependence of the spin transition $T_{eq}$ and the Curie temperature $T_C$, showing the opposite situation with respect to that of Fig. 2. (b) Thermal dependence of the HS fraction under pressure showing a transformation from a gradual to first-order spin transition as the pressure increases. The black curve is the spinodal curve, which delimits the bistable region. (c) The superlinear behavior of the thermal hysteresis width as a function of ${\left(T_C-T_{eq}\right)}^{3/2}$ obtained for the curves of (a). The parameter values are $\sqrt{g}=150, {\mathrm{\Delta }}_0=400.1$ K, $J_0=120$ K, $\alpha =20.0$ K/kbar, and $\gamma =8$ K/kbar.

Figure 4

(a) Pressure dependence of the spin transition temperature $T_{eq}$ and the Curie temperature $T_C$, showing the existence of three regions: $T_{eq}>T_C, T_{eq}<T_C$, and $T_{eq}>T_C$, due to the nonmonotonous character of the Curie temperature with pressure. (b) Associated thermal dependence of the HS fraction for several applied pressure values, going from 0 to 20 kbar, exhibiting a reentrant behavior from gradual to first-order and then to gradual spin transition as $P$ increases. (c) Associated pressure dependence of the spin transition temperature (black curve) and upper (in red) and lower (in blue) switching temperatures of the thermal hysteresis, helping to visualize the observed reentrant phase transition. Parameter values used are $\sqrt{g}=150, {\mathrm{\Delta }}_0=250.5$ K,$J_0=23.9$ K, $\alpha =33.4$ K/kbar,$\gamma =35.8$ K/kbar, and $\delta =-2.78\mathrm{K}/\mathrm{kba}{\mathrm{r}}^2$.

Figure 5

(a) Pressure dependence of the spin transition temperature $T_{eq}$ and the Curie temperature $T_C$, showing the existence of three regions: $T_{eq}<T_C, T_{eq}>T_C$, and $T_{eq}<T_C$, due to the nonmonotonous pressure dependence of the Curie temperature with pressure. (b) Associated thermal dependence of the HS fraction for several applied pressure values, going from 0 to 21 kbar, exhibiting a reentrant behavior from first-order to gradual and then to first-order spin transition as $P$ increases. The black curves denote the spinodal lines. Parameter values used are $g=150, {\mathrm{\Delta }}_0=400.8$ K,$J_0=253$ K, $\alpha =33.4$ K/kbar,$\gamma =-13.2$ K/kbar, and $\delta =1.857\mathrm{K}/\mathrm{kba}{\mathrm{r}}^2$.