Phase separation in doped systems with spin-state transitions

Spin-state transitions, observed in many transition-metal compounds containing ${\text{Co}}^{3+}$ and ${\text{Fe}}^{2+}$, may occur with the change in temperature and pressure but also with doping, in which case the competition of single-site effects and kinetic energy of doped carriers can favor a change in the spin state. We consider this situation in a simple model, formally resembling that used for manganites in Kugel, Rakhmanov, and Sboychakov, Phys. Rev. Lett. 95, 267210 (2005). Based on such a model, we predict the possibility of a jumplike change in the number of ${\text{Co}}^{3+}$ ions undergoing spin-state transition caused by hole doping. A tendency to the electronic phase separation within a wide doping range is demonstrated. Phase diagrams with the regions of phase separation are constructed at different values of the characteristic parameters of the model.

Figure 1

(Color online) The energies of type 1 and 2 states as function of doping $x$. The solid curve (red) corresponds to the more favorable in energy homogeneous state, whereas the states with higher energies are shown by dashed lines (blue), see the text. The dot-dashed line (green) corresponds to the energy of an inhomogeneous state, obtained by Maxwell construction. ${\Delta }_1/zt=0.2$.

Figure 2

(Color online) The densities of ${\text{Co}}^{3+}$ ions in intermediate- $\left(n^e=n_{\text{IS},{\text{Co}}^{3+}}\right)$ and low-spin $\left(1-n^h=n_{\text{LS},{\text{Co}}^{3+}}\right)$ homogeneous states as a function of $x$ at ${\Delta }_1/zt=0.2$. At $x_1<x<x_2$, both spin states of ${\text{Co}}^{3+}$ coexist, whereas at $x=x_2$, there occurs a jumplike transition to the purely IS state of ${\text{Co}}^{3+}$ ions. Dashed lines illustrate the possible behavior of $n_{\text{IS},{\text{Co}}^{3+}}$ and $n_{\text{LS},{\text{Co}}^{3+}}$ for the type 1 state similar to that described in Ref. 29.

Figure 3

(Color online) Magnetic moment per one ${\text{Co}}^{4+}$ ion versus doping $x$ at different values of the hopping integral $t$: (a) $t/J_H=2$ and (b) $t/J_H=4$. Solid and dot curves (red) correspond to the homogeneous states. The behavior $M$ in the phase-separated state is shown by the dashed line (blue).

Figure 4

(Color online) Possible homogeneous states of the system under study at different values of the hopping integral $t$: (a) $t/J_H=1$ and (b) $t/J_H=1.5$. The boundaries between analogous phases in the upper and lower parts of the phase diagram are shown by the same lines (solid, dashed, or dot-dashed).

Figure 5

(Color online) Phase diagrams including the phase-separated states of the system under study at different values of hopping integral $t$: (a) $t/J_H=1$; (b) $t/J_H=1.3$; and (c) $t/J_H=1.5$. PS I is the phase-separated state including the regions without itinerant charge carriers, corresponding to LS ${\text{Co}}^{3+}$, and those with completely delocalized charge carriers promoted to IS ${\text{Co}}^{3+}$. PS II is the similar phase-separated state where the regions with and without itinerant charge carriers correspond to IS ${\text{Co}}^{4+}$ and HS ${\text{Co}}^{4+}$, respectively. Regions 1 at panel (b) correspond to the charge carriers located at HS ${\text{Co}}^{4+}$ and LS ${\text{Co}}^{3+}$, where the charge transfer between Co sites is suppressed due to the spin blockade.