Thermally assisted ordering in Mott insulators

Landau theory describes phase transitions as the competition between energy and entropy: The ordered phase has lower energy, while the disordered phase has larger entropy. When heating the system, ordering is reduced entropically until it vanishes at the critical temperature. This picture implicitly assumes that the energy difference between the ordered and disordered phases does not change with temperature. We show that for orbital ordering in the Mott insulator ${\mathrm{KCuF}}_3$, this assumption fails qualitatively: entropy plays a negligible role, while thermal expansion energetically stabilizes the orbitally ordered phase to such an extent that no phase transition is observed. To understand this strong dependence on the lattice constant, we need to take into account the Born-Mayer repulsion between the ions. It is the latter, and not the Jahn-Teller elastic energy, which determines the magnitude of the distortion. This effect will be seen in all materials where the distortion expected from the Jahn-Teller mechanism is so large that the ions would touch. Our mechanism explains not only the absence of a phase transition in ${\mathrm{KCuF}}_3$, but even suggests the possibility of an inverted transition in closed-shell systems, where the ordered phase emerges only at high temperatures.

Figure 1

Crystal structure of ${\mathrm{KCuF}}_3$. Inequivalent atoms inside the $I4/mcm$ unit cell (thick black lines) are shown in color (Cu: brown; F: green; K: violet). The additional atoms in gray show the pseudocubic setting in which the network of corner-sharing octahedra becomes apparent. The pseudocubic axes are defined as $\mathbf{x}=(\mathbf{a}+\mathbf{b})/2, \mathbf{y}=(-\mathbf{a}+\mathbf{b})/2$, and $\mathbf{z}=\mathbf{c}/2$. For clarity, lattice distortions are exaggerated twofold.

Figure 2

Distortion parameter $\delta$ as a function of lattice constant $a$ in thermally expanding ${\mathrm{KCuF}}_3$ [12] and under hydrostatic pressure [13], for ${\mathrm{RbCuF}}_3$ [14], and (${\mathrm{NH}}_4$)${\mathrm{CuF}}_3$ [15], compared to our calculations and the values obtained for constant short Cu-F distance $s_{\min }$.

Figure 3

DFT+$U$ gain in energy per formula unit as a function of the distortion parameter $\delta$ for experimental unit cells at different temperatures. With thermal expansion, the minimum of the energy curve moves to larger distortions ${\delta }_{\min }$ and deepens. Lines are fits to guide the eye.

Figure 4

Change in DFT+$U$ energy as a function of Cu-F distance $s$ for different lattice constants $a$. For $s$ smaller than $s_{\min }$, the energy curves are practically independent of the actual lattice. For $s$ larger than $s_{\min }$, each curve reaches a maximum at the undistorted position $s=a/2\sqrt{2}$.

Figure 5

Dependence of the calculated elastic constant $m{\omega }_0^2$ for the distortion about $\mathrm{\Delta }=0$ on the lattice parameter $a$. For the open-shell systems, the values are calculated by DFT+$U$ with $U=7$ eV and $J=0.9$ eV for ${\mathrm{KCuF}}_3$ and ${\mathrm{RbCuF}}_3$, and $U=6$ eV and $J=0.9$ eV for ${\mathrm{KCrF}}_3$, while the closed-shell ${\mathrm{CsZnF}}_3$ is already gapped without a Hubbard $U$. Large filled circles indicate the experimental lattice constant at room temperature; open circles indicate the DFT+$U$ or DFT relaxed lattice constant, which are between 1–2 % larger than the experimental values. For all compounds, the elastic constant changes sign when the lattice constant gets large enough. The larger the cation, the larger the critical lattice constant $a_c$. For the smallest, Zn, the relaxed structure is tantalizingly close to the value required for an inverted Landau transition.