Orbital ordering in La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$ studied by model Hartree-Fock calculation

We have investigated orbital ordering in the half-doped manganite La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$, which displays spin, charge and orbital ordering, by means of unrestricted Hartree-Fock calculations on the multiband $p$-$d$ model. From recent experiment, it has become clear that La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$ exhibits a cross-type $(z^2-x^2\mathrm{/}{y}^2-z^2)$ orbital ordering rather than the widely believed rod-type $(3x^2-r^2\mathrm{/}3y^2-r^2)$ orbital ordering. The calculation reveals that cross-type $(z^2-x^2\mathrm{/}{y}^2-z^2)$ orbital ordering results from an effect of in-plane distortion as well as from the relatively long out-of-plane Mn-O distance. For the “Mn${}^{4+}$” site, it is shown that the elongation along the c-axis of the MnO${}_6$ octahedra leads to an anisotropic charge distribution rather than the isotropic one.