Electron correlations and electron-lattice interactions in the metal-insulator, ferroelastic transition in VO2: A thermodynamical study

We present a model of the V${\mathrm{O}}_2$ phase transition which incorporates both electron-electron and electron-lattice interactions and accounts for the presence of two $d$ bands (${\pi }^{*}$ and $d_{\parallel }$) overlapping at the Fermi level of the metallic phase. Attention is given to the crystallographic symmetry change at the transition and to the properties of the associated order parameter. To take into account electron-electron interaction, we use the functional-integral treatment of the Hubbard Hamiltonian. Then, to account for the electron-lattice interaction, the center of gravity and the shape of the $d$ bands are chosen so as to depend explicitly on the lattice distortion. Finally spin-spin interactions are described in terms of an Ising-like model. This treatment leads to a free energy expression which, at a given temperature, depends on two variational parameters: the mean amplitude of the local moment $\mu$ and the amplitude of the lattice distortion $\eta$. Minimization of the free energy with respect to $\eta$ and $\mu$ leads to the temperature dependence of these quantities. The transition which is a first-order one, appears to be driven by electron correlations which, below the transition temperature, stabilize a distorted phase with paired local magnetic moments on the vanadium sites and a density of states gap between the lower Hubbard $d_{\parallel }$ and the ${\pi }^{*}$ band. Our model lends itself to discussion of the magnetic susceptibility which, in the insulating phase, is governed by the pairing of the local magnetic moments, induced by the electron-lattice interaction. Using accepted values of the electronic structure parameters, we find a fairly quantitative agreement with experiment. We account for all the features of the ferroelastic, metal-insulator transition of V${\mathrm{O}}_2$.