Controllable phase transitions between multiple charge density waves in monolayer 1T-VSe$_2$ via doping and strain engineering

Two-dimensional (2D) materials are known to possess emergent properties that are not found in their bulk counterparts. Recent experiments have shown a $\sqrt7 \times \sqrt3$ charge density wave (CDW) in monolayer 1T-VSe$_2$, in contrast to the $4\times 4\times 3$ phase in bulk. Here, via first-principles calculations, we show that multiple CDW phases compete in monolayer VSe$_2$, the ground state of which can be tuned by charge doping and in-plane biaxial strain. With doping, the $\sqrt7 \times \sqrt3$ CDW of the pristine VSe$_2$ transfers to a $3 \times \sqrt3$ and $4\times 4$ phase, the latter of which is a projection of the bulk counterpart, at critical doping concentrations of around 0.2 holes per formula unit and 0.25 electrons per formula unit, respectively. The $4\times 4$ CDW phase can also be stabilized under compressive strain. Although electron-phonon coupling is prevailing in the CDW formation, we show that Fermi surface nesting is a good starting point to explain most of these transitions in monolayer 1T-VSe$_2$. These results make VSe$_2$ an appealing material for electronic devices based on controllable CDW phase transitions.