Elastic-frustration-driven unusual magnetoelastic properties in a switchable core-shell spin-crossover nanostructure

Spin-crossover (SCO) solids have been studied for their fascinating properties, exhibiting first-order phase transitions and macroscopic bistabilities, accompanied by significant magnetic, structural, and optical changes. These exceptional properties make these materials promising for applications as high-density information storage and optical switches. Recently, the critical progress made in chemistry allowed the design of spin-crossover nanocomposites, combining the properties of two types of spin-crossover solids having different properties, like different lattice parameters, bulk moduli, transition temperatures, ligand fields, etc. In this paper, we consider a microscopic electroelastic description of a SCO nanostructure made of a SCO active core surrounded by a SCO active shell, for which we impose an unconventional elastic frustration at the core-shell interface. The detailed examination of the thermodynamic properties of such a nanocomposite, as a function of the lattice parameter misfit between the two constituents, revealed that the frustration causes unexpected behaviors on the thermal dependence of the average bond lengths, such as the emergence of two- or three-step spin transitions, with self-organization of the spin states in the plateau regions. These results highlight the nontrivial character of the magnetoelastic properties in switchable SCO nanoparticles.

Figure 1

(a) Schematic structure of the SCO nanocomposite. Blue and red dots are two spin-crossover sites belonging to different materials. (b) The configuration of the elastic interactions in the two-dimensional square model considered in this paper showing a central red ball connected by springs to gray and blue sites, representing the nearest and next-nearest neighbors, respectively. (c) The blue, green, and red solid lines denote $R_0^S\left(S_i,S_j\right),R_0^{\mathrm{int}}\left(S_i,S_j\right),$ and $R_0^C\left(S_i,S_j\right)$, respectively. The blue, green, and red dashed lines denote $d_0^S\left(S_i,S_j\right),d_0^{\mathrm{int}}\left(S_i,S_j\right)$, and $d_0^C\left(S_i,S_j\right)$, respectively. As a reference lattice, we take $R_0^S\left(-1,-1\right)=R_0^{\mathrm{int}}\left(-1,-1\right)=R_0^C\left(-1,-1\right)=1\left(R_{0,S}^{\mathrm{LL}}=R_{0,C}^{\mathrm{LL}}=R_{0,\mathrm{int}}^{\mathrm{LL}}=1\right)$.

Figure 2

Thermal variation of the HS fractions and lattice parameters of the uncoated 2D hollow (a) and core (b) nanoparticle (with square symmetry). The core size is $40\times 40$ and the hollow shell has five-layer thickness. The equilibrium NN distances in HS-HS, HS-LS, and LS-LS configurations are, respectively, equal to $1.05,1.025$, and $1.0$ nm for the core and $1.03,1.015$, and $1.0$ nm, for the shell. For both cases, the spatial distributions of the HS (red dots) and LS (blue dots) sites along the spin transition phenomenon are shown in the right panels.

Figure 3

Thermal variation of (a) the total HS fraction and (b) the average lattice parameter of the whole SCO nanocomposite for different equilibrium lattice parameters of the shell, which was varied from $R_{0S}^{\mathrm{HH}}=1.00\mathrm{to}1.05$ nm. The other lattice parameter values are given in Table 1. The used values of the elastic constants are $A_C=A_S=A_{\mathrm{int}}={10}^5\mathrm{K}\mathrm{n}{\mathrm{m}}^{-2}$ for NN interactions and $B_C=B_S=B_{\mathrm{int}}={10}^5\mathrm{K}\mathrm{n}{\mathrm{m}}^{-2}$ for the NNN interactions in the core (C), shell (S), and interface (int).

Figure 4

Thermal variation of (a) the shell HS fraction and (b) the average NN distance for different $R_{0S}^{\mathrm{HH}}$ values, varied from $R_{0S}^{\mathrm{HH}}=1.0\mathrm{to}1.05$ nm. (c, d) The temperature dependences of the core HS fraction and average NN distance for the same $R_{0S}^{\mathrm{HH}}$ values. The other lattice parameter values are given in Table 1. The other model parameters are given in the text.

Figure 5

High-spin shell lattice parameter dependence of the upper $\left(T^{+}\right)$ and lower $\left(T^{-}\right)$ transition temperatures of the core components, showing a clear vanishing of the thermal hysteresis for below the value, $R_{0,S}^{\mathrm{HH}}=1.01$ nm, as a result of pressure effects exerted by the shell on the core due to the lattice parameter misfit.

Figure 6

Selected snapshots depicting the spatial distribution of the HS fraction presented in Figs. 4 and 5 for the case $R_{0,S}^{\mathrm{HH}}=1.00,1.02,1.03,1.04,\mathrm{and}1.05\mathrm{nm}$ and temperature $T=229,174,144,115,$ and 87 K, respectively. The corresponding HS fraction (0.60, 0.52, 0.47, 0.43, 0.40) and lattice parameter (0.98, 0.99, 1.00, 1.01, 1.01) values can be easily read in Figs. 4 and 4 for the core lattice and Figs. 3 and 3 for the shell lattice. Note the emergence of labyrinth HS structures inside the core as a result of the spatial distribution of the strain. Here HS (in red) and LS (in blue) depict total HS occupation in the shell, and spatial distribution of the HS fraction in the core.

Figure 7

Lattice parameter profile along a horizontal line located at coordinate $j=\frac{N}{2}$, in the middle of the lattice, for several HS shell lattice parameters, and temperatures corresponding to the plateau regions of Figs. 4 and 4. The data of the HS shell part are represented with red squares and those of the LS core part are represented with blue filled circles. All curves correspond to a LS core surrounded by a HS shell. The horizontal dashed lines, corresponding to $d_{ij}=R_{0S}^{\mathrm{HH}}$ and $d_{ij}=1.0$ nm, are, respectively, associated with the equilibrium bond lengths of the HS shell and LS core.

Figure 8

Displacement fields for the different HS shell lattice parameter values (a) $R_{0,S}^{\mathrm{HH}}=1.00$ nm, (b) $R_{0,S}^{\mathrm{HH}}=1.02$ nm, (c) $R_{0,S}^{\mathrm{HH}}=1.03$ nm, (d) $R_{0,S}^{\mathrm{HH}}=1.04$ nm, and (e) $R_{0,S}^{\mathrm{HH}}=1.05$ nm, corresponding to the situations depicted in Figs. 7–7. The LS state, common to all figures, was used as a reference state for the calculations of the displacement field [see Eq. (9)].