Mechanism for Orbital Ordering in ${\mathrm{KCuF}}_3$

The Mott insulating perovskite ${\mathrm{KCuF}}_3$ is considered the archetype of an orbitally ordered system. By using the local-density $\mathrm{\text{approximation+dynamical}}$ mean-field theory method, we investigate the mechanism for orbital ordering in this material. We show that the purely electronic Kugel-Khomskii superexchange mechanism alone leads to a remarkably large transition temperature of $T_{\mathrm{KK}}\sim 350\mathrm{K}$. However, orbital order is experimentally believed to persist to at least 800 K. Thus, Jahn-Teller distortions are essential for stabilizing orbital order at such high temperatures.

Figure 1

Left: Crystal structure and orbital order in $a$-type [12] ${\mathrm{KCuF}}_3$. Cu is at the center of F octhaedra enclosed in a K cage. The conventional cell is tetragonal with axes $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, where $a=b$, $c=0.95a\sqrt{2}$. The pseudocubic axes are defined as $\mathbf{x}=(\mathbf{a}+\mathbf{b})/2$, $\mathbf{y}=(-\mathbf{a}+\mathbf{b})/2$, and $\mathbf{z}=\mathbf{c}/2$. All Cu sites are equivalent. For sites 1, the long (short) bond $l$ ($s$) is along $\mathbf{y}$ ($\mathbf{x}$). Vice versa for sites 2. Orbital $|2⟩$ (see Table ), occupied by one hole, is shown for each site. Right: Jahn-Teller distortions at sites 1, measured by $\delta =(l-s)/(l+s)/2$ and $\gamma =c/a\sqrt{2}$. $R$ is the experimental structure, $R_{\delta }$ and $I_{\delta }$ two ideal structures with reduced distortions, and $I_0$ is cubic.

Figure 2

LDA $3d$-bands (in eV) of ${\mathrm{KCuF}}_3$ for the real crystal (R) and less distorted structures (see Fig. 1). Lines: $e_g$-bands. Dashes: $t_{2g}$-bands. The Fermi level is at zero.

Figure 3

Orbital polarization $p$ as a function of temperature calculated with $\mathrm{LDA}+\mathrm{DMFT}$ (R, circles, triangles). R: $U=7\mathrm{eV}$, experimental structure. Circles: $U=7\mathrm{eV}$, idealized structures $R_{\delta }$ and $I_{\delta }$ (see Fig. 1) with decreasing crystal-field. Triangles: $U=9\mathrm{eV}$, $I_0$ only. Squares: cluster DMFT for the experimental structure and $U=7\mathrm{eV}$.