Origin of Jahn-Teller Distortion and Orbital Order in ${\mathrm{LaMnO}}_3$

The origin of the cooperative Jahn-Teller distortion and orbital order in ${\mathrm{LaMnO}}_3$ is central to the physics of the manganites. The question is complicated by the simultaneous presence of tetragonal and ${\mathrm{GdFeO}}_3$-type distortions and the strong Hund’s rule coupling between $e_g$ and $t_{2g}$ electrons. To clarify the situation we calculate the transition temperature for the Kugel-Khomskii superexchange mechanism by using the local density $\mathrm{\text{approximation}}+\mathrm{\text{dynamical}}$ mean-field method, and disentangle the effects of superexchange from those of lattice distortions. We find that superexchange alone would yield $T_{\mathrm{KK}}\sim 650\mathrm{K}$. The tetragonal and ${\mathrm{GdFeO}}_3$-type distortions, however, reduce $T_{\mathrm{KK}}$ to $\sim 550\mathrm{K}$. Thus electron-phonon coupling is essential to explain the persistence of local Jahn-Teller distortions to $\gtrsim 1150\mathrm{K}$ and to reproduce the occupied orbital deduced from neutron scattering.

Figure 1

Structure of ${\mathrm{LaMnO}}_3$ at 300 K [8]. The conventional cell is orthorhombic with axes $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, and contains 4 formula units. The pseudocubic axes (left corner) are defined via $\mathbf{a}=(\mathbf{x}-\mathbf{y})(1+\alpha )$, $\mathbf{b}=(\mathbf{x}+\mathbf{y})(1+\beta )$, and $\mathbf{c}=2\mathbf{z}(1+\gamma )$, with $\alpha$, $\beta$, $\gamma$ small numbers. For sites 1 and 3 the long (short) bond $l$ ($s$) is $\sim$ along $\mathbf{y}$ ($\mathbf{x}$), vice versa for sites 2 and 4 ($d$-type pattern). All Mn sites are equivalent. The symmetries that transform them into a site of type 1 are $x\leftrightarrow y$ (site 2) $z\to -z$ (site 3), $x\leftrightarrow y$, $z\to -z$ (site 4).

Figure 2

Right: $\mathrm{LDA}+\mathrm{DMFT}$ spectral function for the room temperature structure $R_{11}$ for different $U$. $\Uparrow$ ($\Downarrow$) indicates states with $e_g$ spins parallel (antiparallel) to ${\mathbf{S}}_{t_{2g}}$. Left: $\mathbf{k}$-resolved spectral function for $U=5\mathrm{eV}$.

Figure 3

Orbital polarization $p$ (left) and (right) occupied state $|\theta ⟩$ ($|-\theta ⟩$) for sites 1 and 3 (2 and 4) as a function of temperature. Solid line: 300 K ($R_{11}$) and 800 K ($R_{2.4}^{800\mathrm{K}}$) structures. Dots: orthorhombic structures with half ($R_6$) or no ($R_0$) Jahn-Teller distortion. Pentagons: 2 (full) and 4 (empty) sites CDMFT. Dashes: ideal cubic structure ($I_0$). Circles: $U=5\mathrm{eV}$. Diamonds: $U=5.5\mathrm{eV}$. Triangles: $U=6\mathrm{eV}$. Squares: $U=7\mathrm{eV}$. Crystal-field splitting (meV): 840 ($R_{11}$), 495 ($R_6$), 219 ($R_0$), 168 ($R_{2.4}^{800\mathrm{K}}$), and 0 ($I_0$).