Orbital ordering transition in the single-layer manganites near half doping: a weak-coupling approach

The roles of crystal-field splitting and Jahn-Teller distortions on the orbital ordering transition are investigated in the single-layer manganites near half doping. Crystal-field splitting of energy levels favoring the $d_{3z^2-r^2}$ occupancy provides not only the correct Fermi surface topology for La$_{0.5}$Sr$_{1.5}$MnO$_4$ having a circular electron pocket around the $\Gamma$ point, but also enhances the flatness of the hole pocket around the M point thereby improving the nesting. In the presence of the circular electron pocket, Jahn-Teller distortions are found to be crucial for the transition to the transverse orbital ordering with ordering wavevector ($0.5\pi, 0.5\pi$). In the hole-doping regime $0.5 \leq x \leq 0.7$, the orbital ordering wavevector shows a linear dependence on the hole concentration in accordance with the experiments.