Electronic structure of metallic and insulating phases of vanadium dioxide and its oxide alloys

${\mathrm{VO}}₂$ attracts much attention due to its metal-insulator transition. Alloying ${\mathrm{VO}}₂$ with MgO and ${\mathrm{GeO}}₂$ allows the band gap and the transition temperature to be varied. We find that the spin order plays a key role in creating the band gap in the low-temperature ${M}₁$ phase. For MgO alloying, the alloying fraction $n\phantom{\rule{0.28em}{0ex}}(\mathrm{M}{\mathrm{g}}_n{\mathrm{V}}_{1\ensuremath{-}n}{\mathrm{O}}_{2\ensuremath{-}n})$ is varied from 12.5 to 33.3%. The minimum band gap does not change without a structural rearrangement because both band edges of insulating ${\mathrm{VO}}₂$ consist of only V $3d$ states on sixfold-coordinated V sites. A crystal search finds that if the Mg fraction in the alloy is large enough ($>20%$), fivefold-coordinated V sites can have lower energy than the sixfold sites, and the band gaps are doubled. For ${\mathrm{GeO}}₂$ alloying, the insulating ${M}₁$ structure reverts to rutile because ${\mathrm{GeO}}₂$ has a rutile phase. The result matches the experimental observation and is very important in guiding ${\mathrm{VO}}₂$'s applications such as smart coating and nonlinear resistor.