  Metal-ion doping can effectively regulate the metal-insulator transition
temperature in $\mathrm{VO}_2$. Experiments found that the pentavalent and
hexavalent ion doping dramatically reduces the transition temperature while the
trivalent ion doping increases the transition temperature and induces
intermediate phases. Based on the phase-field model of the metal-insulator
transition in $\mathrm{VO}_2$ we developed previously, we formulate a Landau
potential of the metal-ion-doped $\mathrm{VO}_2$ taking account of the effects
of doping on the electron correlation and lattice structure. The effect of
metal-ion doping on the lattice structure is accounted for in a
phenomenological way. Using the Landau potential, we calculate the
temperature-dopant-concentration phase diagrams of $\mathrm{VO}_2$ doped with
various metal ions consistent with the experiments and provide explanation to
the different behaviors of different metal-ion doping. The phenomenological
theory can provide estimations of phase diagrams of $\mathrm{VO}_2$ doped with
other metal ions.
