Origin of the multiple charge density wave order in $1T-{\mathrm{VSe}}_2$

Transition-metal dichalcogenide $1T\text{−}{\mathrm{VSe}}_2$ 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 $1T\text{−}{\mathrm{VSe}}_2$ 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 $4\times 4\times 3$ CDW superstructure in bulk $1T\text{−}{\mathrm{VSe}}_2$, while the momentum-dependent electron-phonon coupling-induced $\sqrt{7}\times \sqrt{3}$ CDW superstructure is most stable in the freestanding monolayer $1T\text{−}{\mathrm{VSe}}_2$. For monolayer $1T\text{−}{\mathrm{VSe}}_2$, the substrate-induced compressive strain can turn the ground state into the $4\times 4$ CDW superstructure, while tensile strain preserves the $\sqrt{7}\times \sqrt{3}$ superstructure. Our results demonstrate the origin of the CDW orders in $1T\text{−}{\mathrm{VSe}}_2$ and shed light on the experimental observation of multiple CDW orders in monolayer $1T\text{−}{\mathrm{VSe}}_2$.

Figure 1

(a) Crystal structure of bulk $1T\text{−}{\mathrm{VSe}}_2$. The red (green) balls represent V (Se) atoms. (b) Fermi surface of bulk $1T\text{−}{\mathrm{VSe}}_2$. (c) Real and (d) imaginary parts of the electron susceptibility cross section in the plane of $q_z=\frac{1}{3}{c}^{*}$, and the corresponding maximum is marked by $Q_{//}$. The bottom panels in (c) and (d) are the real and imaginary parts of electron susceptibility along the $+k_z$ ($\mathrm{\Gamma }A$) direction, where $Q_{\mathrm{CDW}}$ is indicated by the crosses.

Figure 2

(a) Phonon dispersion curves of bulk $1T\text{−}{\mathrm{VSe}}_2$, where $Q_1$, $Q_1^{\prime }$, and $Q_2$ stand for the prominent imaginary frequencies. (b) Phonon linewidth of the lowest phonon mode of bulk $1T\text{−}{\mathrm{VSe}}_2$ in the $q_z=\frac{1}{3}{c}^{*}$ plane (top) and along the $+k_z$ ($\mathrm{\Gamma }A$) direction (bottom).

Figure 3

Superstructures of (a) $2\times \sqrt{3}$, (b)$\sqrt{7}\times \sqrt{3}$, and (c) $4\times 4$ CDW orders. (d) Evolution of the energy gain of various CDW superstructures under different strains.

Figure 4

Cross section of the (a) real and (b) imaginary parts of electron susceptibility of monolayer $1T\text{−}{\mathrm{VSe}}_2$ in the $q_z$ = 0 plane. (c) Phonon dispersion curves of monolayer $1T\text{−}{\mathrm{VSe}}_2$, where $Q_1$ and $Q_2$ stand for the two prominent imaginary frequencies. (d) Phonon linewidth of the lowest phonon mode of monolayer $1T\text{−}{\mathrm{VSe}}_2$ in the $q_z=0$ plane.

Figure 5

Phonon dispersion curves of monolayer $1T\text{−}{\mathrm{VSe}}_2$ under (a) and (b) compressive strain and (c) and (d) tensile strain.