Orbital and spin ordering physics of the Mn₃O₄ spinel

Transition metal (TM) oxides with orbital, spin and lattice degrees of freedom have always been an interesting playground for physicists. Geometrical frustration adds more exotic phenomena in these systems. The Mn₃O₄ spinel is one such system in which orbital-spin-lattice couplings are very active in deciding the ground state. Indeed, this material is known to prossess a complex phenomenology of structure-, orbital-, and spin-ordering transitions. There are two cations in this system: Mn²⁺ (3d⁵) with spin S=5/2 and no orbital degrees of freedom, and Mn³⁺ (3d⁴) with spin S=2 and active orbital degrees of freedom (ODF). The Mn³⁺ ODF changes lattice symmetry (cubic to tetragonal) via a Jahn-Teller distortion. In this presentation, I will show how ODF in Mn³⁺ strongly affects spin superexchange via a 90^o TM-Oxygen-TM bond angle. The derived spin-orbital Hamiltonian obtains orbital ordering, and thence spin ordering. Interaction among Mn³⁺ spins as well as between Mn²⁺ and Mn³⁺ spins then give the well-known triangular Yafet-Kittel type spin ordering of this spinel. The remaining lattice degeneracy (a-b axis) is finally lifted via spin-lattice interaction, and can be tuned via external magnetic fields and applied pressure.