Orbital Superstructures in Spinels

Orbital degrees of freedom often lead to specific types of orbital and spin ordering. Complicated and interesting superstructures are observed in B-sublattice of spinels. This is connected with the geometric frustration of this lattice and with the interconnection of edge-sharing MO$_{6}$ octahedra, which is especially important for transition metals with partially-filled t$_{2g}$ levels. In some such systems (MgTi$_{2}$O4, CuIr$_{2}$S$_{4}$, AlV$_{2}$O$_{4})$ there appears strange superstructures with the formation of spin gap states. In other cases (ZnV$_{2}$O$_{4})$ structural transitions, apparently connected with orbital ordering, are followed by long-range magnetic ordering. Last but not least, the famous Verwey transition in magnetite Fe$_{3}$O$_{4}$ leads to a very complicated structural pattern, accompanied by the appearance of ferroelectricity. In this talk I will discuss all these examples, paying main attention to an interplay of charge, spin and orbital degrees of freedom. In particular, for MgTi$_{2}$O4, and CuIr$_{2}$S$_{4}$ we proposed the picture of orbitally-driven Peierls state [1]. Similar phenomenon can also explain situation in ZnV$_{2}$O$_{4 }$[2], although the corresponding superstructure has not yet been observed experimentally. Finally, I propose the model of charge and orbital ordering in magnetite [3], which uses the idea of an interplay of site- and bond-centered ordering [4] and which seems to explain both the structural data and the presence of ferroelectricity in Fe$_{3}$O$_{4 }$ below Verwey transition. \newline \newline [1] D.I.Khomskii and T.Mizokawa, Phys.Rev.Lett. \textbf{94}, 156402 (2005); \newline [2] Hua Wu, T.Mizokawa and D.I.Khomskii, unpublished; \newline [3] D.I.Khomskii, unpublished; \newline [4] D.V.Efremov, J.van den Brink and D.I.Khomskii, Nature Mater. \textbf{3}, 853 (2004)