Origin of the multiple charge density wave order in 1T−VSe2

Transition-metal dichalcogenide $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$ experimentally exhibits multiple charge density wave (CDW) orders, but its origin is still under debate. Using first-principles calculations, we investigate the origin of CDW orders in $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$ and clarify the ground state of CDW in the freestanding monolayer. Our results show that both Fermi-surface nesting and electron-phonon coupling account for the $4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}3$ CDW superstructure in bulk $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$, while the momentum-dependent electron-phonon coupling-induced $\sqrt{7}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ CDW superstructure is most stable in the freestanding monolayer $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$. For monolayer $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$, the substrate-induced compressive strain can turn the ground state into the $4\ifmmode\times\else\texttimes\fi{}4$ CDW superstructure, while tensile strain preserves the $\sqrt{7}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ superstructure. Our results demonstrate the origin of the CDW orders in $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$ and shed light on the experimental observation of multiple CDW orders in monolayer $1T\text{\ensuremath{-}}{\mathrm{VSe}}_{2}$.