The electronic structure of the Mott insulator VO$_{\mathrm{2}}$: the strongly correlated metal state is screened by impurity band.

A Mott insulator VO$_{\mathrm{2}}$ (3$\mathrm{d}^{1})$ has a direct gap ($\Delta_{direct}\propto V_{direct})$ of 0.6 eV and an indirect gap of $\Delta_{act}\propto V_{direct}\approx $0.15 \textit{eV} coming from impurity indirect band. At $\mathrm{T}_{c}$, $\Delta_{direct}=\Delta_{act}=$ O is satisfied and the insulator-to-metal transition (IMT) occurs. The metallic carriers near core region can be trapped when a critical onsite Coulomb $U_{c}$ exists. Then, a potential energy is defined as \[ V_{g}=\left( V_{direct}+U_{c} \right)+V_{indirect} \] \begin{equation} \label{eq1} \thinspace \thinspace \thinspace \thinspace \thinspace =-(2 \mathord{\left/ {\vphantom {2 {3)E_{F}(1+e(N_{tot} \mathord{\left/ {\vphantom {N_{tot} {n_{tot})(1-\mathrm{exp}({-\Delta }_{act} \mathord{\left/ {\vphantom {{-\Delta }_{act} {k_{B}T)))+U_{c}}}} \right. \kern-\nulldelimiterspace} {k_{B}T)))+U_{c}}}}} \right. \kern-\nulldelimiterspace} {n_{tot})(1-\mathrm{exp}({-\Delta }_{act} \mathord{\left/ {\vphantom {{-\Delta }_{act} {k_{B}T)))+U_{c}}}} \right. \kern-\nulldelimiterspace} {k_{B}T)))+U_{c}}}}}} \right. \kern-\nulldelimiterspace} {3)E_{F}(1+e(N_{tot} \mathord{\left/ {\vphantom {N_{tot} {n_{tot})(1-\mathrm{exp}({-\Delta }_{act} \mathord{\left/ {\vphantom {{-\Delta }_{act} {k_{B}T)))+U_{c}}}} \right. \kern-\nulldelimiterspace} {k_{B}T)))+U_{c}}}}} \right. \kern-\nulldelimiterspace} {n_{tot})(1-\mathrm{exp}({-\Delta }_{act} \mathord{\left/ {\vphantom {{-\Delta }_{act} {k_{B}T)))+U_{c}}}} \right. \kern-\nulldelimiterspace} {k_{B}T)))+U_{c}}}}, \end{equation} where $V_{direct}=-(2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3)E_{F}$ is the screened Coulomb pseudopotential at K $=$ 0. $\mathrm{\Delta \rho =}N_{tot} \mathord{\left/ {\vphantom {N_{tot} {n_{tot}\approx 0.018\% }}} \right. \kern-\nulldelimiterspace} {n_{tot}\approx 0.018\% }$ [\underline {1}] is defined as the critical doping quantity, where $n_{tot}$ is the carrier density in the direct band and $N_{tot}$ is the carrier density in the impurity band. In $U_{c}<(2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3)E_{F}$ case, it sustains the insulator state. However, when both $U_{c}>(2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3)E_{F}$ and $\Delta_{act}\mathrm{=}$ 0 by excitation are satisfied, the IMT occurs in V$_{\mathrm{g}}\ge $ 0. This indicates that the excitation ($\Delta_{act}=$ 0) breaks the Coulomb equilibrium (V$_{\mathrm{g}}$\textless 0 and insulator sustaining $U_{c})$ in \textit{Eq.} ($\mathrm{1})$; the Coulomb energy changes from $U_{c}$ to a $U{