We present a theoretical investigation of the electronic structure of rutile (metallic) and M$_1$ and M$_2$ monoclinic (insulating) phases of VO$_2$ employing a fully self-consistent combination of density functional theory and embedded dynamical mean field theory calculations. We describe the electronic structure of the metallic and both insulating phases of VO$_2$, and propose a distinct mechanism for the gap opening. We show that Mott physics plays an essential role in all phases of VO$_2$: undimerized vanadium atoms undergo classical Mott transition through local moment formation (in the M$_2$ phase), while strong superexchange within V-dimers adds significant dynamic intersite correlations, which remove the singularity of self-energy for dimerized V-atoms. The resulting transition from rutile to dimerized M$_1$ phase is adiabatically connected to Peierls-like transition, but is better characterized as the Mott transition in the presence of strong intersite exchange. As a consequence of Mott physics, the gap in the dimerized M$_1$ phase is temperature dependent. The sole increase of electronic temperature collapses the gap, reminiscent of recent experiments.
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