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Landau theory of metal-insulator transition in VO$_\text{2}$ doped with metal ions

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.