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

TL;DR

This study uses Hartree-Fock calculations on a multiband model to analyze the orbital ordering in La$_{0.5}$Sr$_{1.5}$MnO$_4$, revealing a cross-type orbital pattern driven by lattice distortions.

Contribution

It demonstrates that cross-type orbital ordering in La$_{0.5}$Sr$_{1.5}$MnO$_4$ arises from in-plane distortions and Mn-O distance effects, challenging previous assumptions of rod-type ordering.

Findings

Cross-type orbital ordering is confirmed by calculations.

In-plane distortions influence orbital patterns significantly.

MnO$_6$ octahedral elongation causes anisotropic charge distribution.

Abstract

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 $(z^2-x^2/y^2-z^2)$ orbital ordering rather than the widely believed rod-type $(3x^2-r^2/3y^2-r^2)$ orbital ordering. The calculation reveals that cross-type $(z^2-x^2/y^2-z^2)$ 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.