Experimental evidence for the elastic long-range character of the spin crossover transition in cooperative single crystals

We evidenced by means of optical microscopy the long-range character of the interactions acting in spin crossover (SCO) single crystals through a study of the optical density (OD) variation during the spin transition between high-spin (HS) and low-spin (LS) states. The time evolution of the local OD, measured in four single crystals, revealed the existence of precursor phenomena, which are manifest through the appearance of significant changes in the OD of the probed area well before the arrival of the front interface in the case of HS to LS transition and after the passage of the interface in the case of LS to HS transition. These two effects are attributed to the manifestation of long-range interactions that develop during the propagation of the HS/LS interface, which is accompanied by a volume change, generating elastic stresses in the material that deploy far from the sources. Interestingly, the experimental investigations indicate that the precursor and the after effect phenomena occur mostly in the HS phase, which is softer than the LS phase. These original experimental results are well reproduced using an electroelastic modeling in which we considered, in addition to the lattice misfit between HS and LS states, the existence of different elastic constants in the HS and LS phases.

Figure 1

(a) Temperature dependence of the HS fraction ($n_{\text{HS}}$), derived for OM data and adapted from Ref. [35], showing the occurrence of a thermal hysteresis loop, with alternating temperatures $T\simeq 153$ K and $T\simeq 163$ K for cooling and heating. The temperature sweep rate was $0.2\text{K}{\text{min}}^{-1}$. (b) OM images of a rectangular-shaped single crystal of compound (1a) including its dimensions in the high- and low-temperature phases [35]. Panels (c) and (e) display selected crystal snapshots with HS/LS front interface along the cooling (153 K) and heating processes (163 K), respectively. The blue and red arrows indicate the propagation direction of the interface. The blue and red squares correspond to the selected areas where the OD is probed. Panels (d) and (f) are the time dependence of the local green OD, measured in the previous defined probed area of the single crystal. Point $P$ appearing in the curves relates to the time at which the interface crosses the probed square region. Point $A$ and the dashed green circle highlight the region where the probe feels the strain field [35].

Figure 2

(a) Temperature dependence of the HS fraction ($n_{\text{HS}}$), derived for OM data and adapted from Ref. [35], showing the occurrence of a thermal hysteresis loop, with alternating temperatures $T\simeq 150$ K and $T\simeq 162$ K for cooling and heating. The temperature sweep rate was $0.2\text{K}{\text{min}}^{-1}$. (b) OM images of a triangular-shaped single crystal of compound (1b) including its dimensions in the high- and low-temperature phases [35]. Panels (c) and (e) display the crystal snapshots with HS/LS front interface along the cooling (150 K) and heating (162 K) regimes, respectively. The blue and red arrows indicate the propagation direction of the front. The blue and red squares correspond to the selected surface where the OD is measured. Panels (d) and (f) are the time dependence of the local green OD as a function of time, measured in the previous defined probed area of the single crystal. Point $P$ appearing in the curves relates to the time at which the interface crosses the probed square. Point $A$ and the dashed green circle highlight the region where the probe feels the strain field [35].

Figure 3

(a) Temperature dependence of the HS fraction ($n_{\text{HS}}$), derived for OM data and adapted from Ref. [36], showing the occurrence of a thermal hysteresis loop with alternating temperatures $T\simeq 143$ K and $T\simeq 153$ K for cooling and heating, respectively. The temperature sweep rate was $0.5\text{K}{\text{min}}^{-1}$. (b) OM images of the single crystal of compound (2) including its dimensions in the high- and low-temperature phases [36]. Panels (c) and (e) display the crystal snapshots with HS/LS front interface along the cooling and heating, respectively. The blue and red arrows indicate the propagation direction of the front. The blue and red squares correspond to the selected areas where the OD is probed. Panels (d) and (f) are the time dependence of the local green OD, measured in the previous defined probed area of the single crystal. Point $P$ appearing in the curves relates to the time at which the propagating interface crosses the square. Point $A$ and the dashed green circle highlight the region where the probe feels the strain field [36].

Figure 4

(a) Temperature dependence of the HS fraction ($n_{\text{HS}}$), derived for OM data and adapted from Ref. [37], showing the occurrence of a thermal hysteresis loop, with alternating temperatures $T\simeq 104$ K and $T\simeq 112$ K for cooling and heating. The temperature sweep rate is $0.2\text{K}{\text{min}}^{-1}$. (b) OM images of the single crystal of compound (3) including its dimensions in the high- and low-temperature phases [37]. Panels (c) and (e) display the crystal snapshots with the presence of a HS/LS front interface along the lower ($\sim 104$ K) and the upper ($\sim 112$ K) transition temperatures, respectively. The blue and red arrows indicate the propagation direction of the front. The blue and red squares correspond to the selected areas where the OD is probed. Panels (d) and (f) are the time dependence of the red OD, measured in the previous defined probed area of the single crystal. Point $P$ appearing in the curves relates to the time at which the propagating interface crosses the square. Point $A$ and the dashed green circle highlight the region where the probe feels the strain field.

Figure 5

(a) Snapshot of the relaxed lattice configuration with a fixed front interface position at $i=N_x/2$. The black square indicates the probed area. Blue and red regions represent the LS and HS phases, respectively. (b) Time dependence (MCS) of the elastic energy during the mechanical relaxation at fixed electronic configuration. The initial configuration is made of half spins $-1$ and $+1$ while all distances are set equal to those of the HS state. (c) Spatial distribution of the internal pressure field inside the lattice, corresponding to panel (a). The HS-LS interface is represented by the dashed white line. The lattice size is $N_x\times N_y=70\times 20$. The other parameter values are given in the text.

Figure 6

Time dependence of the local average distance $\langle d\rangle$ measured in the probed area, represented by the black square of Fig. 5, during the interface propagation along the (a) HS to LS and (b) LS to HS phase transitions. The simulations are performed at 10 K and the lattice size is $N_x\times N_y=70\times 20$.

Figure 7

Time dependence of the theoretical OD evaluated at the black square probed area (see the insets) along the (a) HS to LS and (b) LS to HS phase transitions. The curves are calculated using Eq. (6) and the average local distance $\langle d\rangle$ of Fig. 6. A very good qualitative agreement with the experimental results of compounds 1a, 1b, 2, and 3. The lattice size is $N_x\times N_y=70\times 20$.