Ab initio tight-binding Hamiltonian for transition metal dichalcogenides

We present an accurate ab initio tight-binding Hamiltonian for the transition metal dichalcogenides, ${\mathrm{MoS}}_2, {\mathrm{MoSe}}_2, {\mathrm{WS}}_2, {\mathrm{WSe}}_2$, with a minimal basis (the $d$ orbitals for the metal atoms and $p$ orbitals for the chalcogen atoms) based on a transformation of the Kohn-Sham density functional theory Hamiltonian to a basis of maximally localized Wannier functions. The truncated tight-binding Hamiltonian, with only on-site, first, and partial second neighbor interactions, including spin-orbit coupling, provides a simple physical picture and the symmetry of the main band-structure features. Interlayer interactions between adjacent layers are modeled by transferable hopping terms between the chalcogen $p$ orbitals. The full-range tight-binding Hamiltonian can be reduced to hybrid-orbital $\mathbf{k}·\mathbf{p}$ effective Hamiltonians near the band extrema that capture important low-energy excitations. These ab initio Hamiltonians can serve as the starting point for applications to interacting many-body physics including optical transitions and Berry curvature of bands, of which we give some examples.

Figure 1

(a) Top view of a single-layer TMDC crystal structure with hexagonal primitive vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$. The smaller yellow circles represent the two chalcogen atoms separated by $d_{X-X}$ in the $z$ direction and the larger brown circles the metal atoms. (b) Perspective side view for the TMDC $2H$ crystal structure composed of $X^A-M-X^B$ monolayers with $a$ the in-plane lattice constant and $c/2$ the separation between two units along the $\widehat{z}$ axis.

Figure 2

Schematic diagram for all hopping terms in the TBH: $M-M$ coupling (red) with six first-neighbor pairs; $X-X$ coupling (green) with six first-neighbor pairs; $X-M$ coupling (blue) with three first-neighbor pairs. Additional couplings between three second-neighbor $X-M$ pairs (gray dashed) are used to improve the accuracy. The top bar indicates the labels of parameters that correspond to the different types of hopping terms, $t^{\left(n\right)}, n=1,...,6$.

Figure 3

TBH (red lines) band structure along $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$ and the comparison with DFT results (blue circles) for monolayer spinless (a) ${\mathrm{MoS}}_2$, (b) ${\mathrm{WSe}}_2$.

Figure 4

Spin splitting for the highest pair of valence bands in ${\mathrm{MoS}}_2$: (a) along the $\mathrm{\Gamma }-\mathrm{K}$ direction as obtained from TBH+SOC (red lines) and DFT (blue dots) calculations. (b) The angular dependence with fixed $k_{\parallel }=\mathrm{K}/2$ and $k_{\parallel }=\mathrm{K}$ of TBH+SOC (red dots); the black lines are the fitted $cos\left(3\theta \right)$ functions.

Figure 5

$V_{pp,\sigma }\left(r\right)$ (upper) and $V_{pp,\pi }\left(r\right)$ (lower) for interlayer pairs as function of the pair distance $r$ in a ${\mathrm{MoS}}_2$ bilayer unit. Black, red, green, blue, cyan, brown are data points from 1st, 2nd, 3rd, 4th, 5th, 6th nearest distances. The solid black lines going through the points are the fits as obtained from Eq. (17). The inset is the perspective view for the interlayer coupling between one of the S atom on the top unit (orange) to all other S atoms on the bottom unit (yellow).

Figure 6

(a) TBH band structure (red lines) along $\mathrm{\Gamma }$-M-$\mathrm{K}-\mathrm{\Gamma }$ and the comparison with DFT results (blue circles) for a ${\mathrm{MoS}}_2$ bilayer in the $2H$ structure. (b) The overlap matrix element strength between two layers with the interlayer coupling Hamiltonian that leads to splitting of single-layer states, on a scale from red (strongest) to white (weakest).

Figure 7

Band structure of the bulk, 3D crystal of ${\mathrm{MoS}}_2$, in the $2H$ structure, as obtained from DFT calculations (blue circles) and from the TBH (red lines) derived from the bilayer system (including interlayer coupling). The symbols $\mathrm{\Gamma }$-M-$\mathrm{K}-\mathrm{\Gamma }$ correspond to $k_z=0$, while those with primes correspond to $k_z=\pi /c$.

Figure 8

(a) Comparison of the $\mathbf{k}·\mathbf{p}$ Hamiltonian (red lines) and the DFT results (blue dots) for the ${\mathrm{MoS}}_2$ monolayer with spin-orbit coupling at ${\mathrm{K}}_{+}$. (b) Energy contours in eV for the highest valence band centered at ${\mathrm{K}}_{+}$. The deviation from spherical symmetry is due to the $f_4$ term.

Figure 9

${\mathrm{MoS}}_2$ TBH: (a) ${\sigma }_{+}$ absorption and comparison with joint density of states (JDOS); the inset shows the $k$-resolved degree of polarization $\eta \left(\mathit{k}\right)$. (b) Berry curvature in units of Å${}^2$.

Figure 10

Rescaled TBH Hamiltonian (red lines) and its comparison with $\mathit{GW}$ (blue circles) band structure results for a ${\mathrm{MoS}}_2$ single-layer unit.