Orbital Ordering in Correlated Perovskites

Orbital-ordering phenomena, the periodic arrangement of occupied orbitals in a lattice, play a fundamental role in determining the electronic and magnetic properties of many strongly correlated oxides. Since there are two competing mechanisms that lead to the same type of ordering, the origin of orbital order in real materials has been debated for years. The first mechanism, going back to the work of Kanamori [1], explains orbital ordering as the result of the interplay between electron-phonon coupling and orbital degeneracy: The Jahn-Teller effect leads to a cooperative lattice distortion, which, in turn, gives rise to a crystal-field splitting and orbital order. The second mechanism, conceived by Kugel and Khomskii [2], explains orbital order as the result of super-exchange in the presence of orbital degeneracy. Both mechanisms lead to the same type of order, so that the origin of orbital order has remained controversial.Using the LDA+DMFT approach, we are able to disentangle the effects of lattice distortions, electron localization, and orbital super-exchange. From the temperature- dependence of the orbital polarization for the paradigmatic compound KCuF3 we find that Kugel-Khomskii super-exchange, in the absence of crystal-field splitting, can drive the transition at about 350 K, a remarkably high temperature for a super-exchange driven transition. This means, nevertheless, that it cannot be the origin of orbital order in KCuF3, which is observed to temperatures well above 800 K [3].This apparently leaves the Kanamori mechanism as the cause of orbital order, for which a Landau-type phase transition is expected at high temperatures: 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 find, however, that for orbital ordering in KCuF3, 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. The new mechanism explains not only the absence of a phase transition in KCuF3, but even suggests the possibility of an inverted transition in closed-shell systems, where the ordered phase emerges only at high temperatures [4].[1] J. Kanamori, J. Appl. Phys. 31, 14S (1960)[2] K.I. Kugel and D.I. Khomskii, Zh. Eksp. Teor. Fiz. 64 1429 (1973)[3] E. Pavarini, E. Koch, and A.I. Lichtenstein, Phys. Rev. Lett. 101, 266405 (2008)[4] H. Sims, E. Pavarini, E. Koch, Phys. Rev. B 96, 054107 (2017