The spinel-structured lithium manganese oxide (LiMn$_2$O$_4$) is a material currently used as cathode for secondary lithium-ion batteries, but whose properties are not yet fully understood. Here, we report a computational investigation of the inversion thermodynamics and electronic behaviour of LiMn$_2$O$_4$ derived from spin-polarised density functional theory calculations with a Hubbard Hamiltonian and long-range dispersion corrections (DFT+$U-$D3). Based on the analysis of the configurational free energy, we have elucidated a partially inverse equilibrium cation distribution for the LiMn$_2$O$_4$ spinel. This equilibrium degree of inversion is rationalised in terms of the crystal field stabilisation effects and the difference between the size of the cations. We compare the atomic charges with the oxidation numbers for each degree of inversion. We found segregation of the Mn charge once these ions occupy the tetrahedral and octahedral sites of the spinel. We have obtained the atomic projections of the electronic band structure and density of states, showing that the normal LiMn$_2$O$_4$ has half-metallic properties, while the fully inverse spinel is an insulator. This material is in the ferrimagnetic state for the inverse and partially inverse cation arrangement. The optimised lattice and oxygen parameters, as well as the equilibrium degree of inversion, are in agreement with the available experimental data. The partially inverse equilibrium degree of inversion is important in the interpretation of the lithium ion migration and surface properties of the LiMn$_2$O$_4$ spinel.
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