Unified picture of lattice instabilities in metallic transition metal dichalcogenides

Transition metal dichalcogenides (TMDs) in the $1T$ polymorph are subject to a rich variety of periodic lattice distortions, often referred to as charge-density waves (CDWs) when not too strong. We study from first principles the fermiology and phonon dispersion of three representative single-layer transition metal disulfides with different occupation of the $t_{2g}$ subshell: ${\mathrm{TaS}}_2\left(t_{2g}^1\right),{\mathrm{WS}}_2\left(t_{2g}^2\right)$, and ${\mathrm{ReS}}_2\left(t_{2g}^3\right)$ across a broad range of doping levels. While strong electron-phonon interactions are at the heart of these instabilities, we argue that away from half-filling of the $t_{2g}$ subshell, the doping dependence of the calculated CDW wave vector can be explained from simple fermiology arguments, so that a weak-coupling nesting picture is a useful starting point for understanding. On the other hand, when the $t_{2g}$ subshell is closer to half-filling, we show that nesting is irrelevant, while a real-space strong-coupling picture of bonding Wannier functions is more appropriate and simple bond-counting arguments apply. Our study thus provides a unifying picture of lattice distortions in $1T$ TMDs that bridges the two regimes, while the crossover between these regimes can be attained by tuning the filling of the $t_{2g}$ orbitals.

Figure 1

Band structures calculated from first principles for monolayers of (a) $1T\text{−}{\mathrm{TaS}}_2$, (b) $1T\text{−}{\mathrm{WS}}_2$, and (c) $1T\text{−}{\mathrm{ReS}}_2$. The Fermi level is set to zero. Calculated bare static susceptibility along $\mathrm{\Gamma }M$ for (d) $1T\text{−}{\mathrm{TaS}}_2$, (e) $1T\text{−}{\mathrm{WS}}_2$, and (f) $1T\text{−}{\mathrm{ReS}}_2$. Calculated dispersion for the lowest acoustic phonon branch along $\mathrm{\Gamma }M$ for (g) $1T\text{−}{\mathrm{TaS}}_2$, (h) $1T\text{−}{\mathrm{WS}}_2$, and (i) $1T\text{−}{\mathrm{ReS}}_2$.

Figure 2

Fermi surface of monolayer $1T\text{−}{\mathrm{TaS}}_2$ (undoped and hole doped). The shaded area delimits the Brillouin zone. Nesting vectors for the undoped case have been drawn.

Figure 3

(a) Ball-and-stick representation of the $1T^{\prime }$ phase of ${\mathrm{WS}}_2$ with an isovalue plot of one of the two equivalent bonding $t_{2g}$ Wannier functions (WFs). W-W bonds have been drawn to facilitate visualization. Each bond accommodates a bonding $t_{2g}$ WF centered on it. (b) Aligned ligand field and modified ligand field energy diagrams for the $1T$ and $1T^{\prime }$ phases. The bonding (b), nonbonding (nb), and antibonding (ab) $t_{2g}$ states are labeled. (c) Calculated band structure along high-symmetry directions for $1T^{\prime }\text{−}{\mathrm{WS}}_2$. The orbital weights of the bonding and antibonding $t_{2g}$ WFs are color-coded. The Fermi level is set to zero. (d)–(f) Corresponding plots for the $1T^{″}$ phase of ${\mathrm{ReS}}_2$.