Effective tight-binding model for $M{X}_2$ under electric and magnetic fields

We present a systematic method for developing a five-band Hamiltonian for the metal $d$ orbitals that can be used to study the effect of electric and magnetic fields on multilayer $M{X}_2$ ($M=\mathrm{Mo}$,W and $X=\mathrm{S}$,Se) systems. On a hexagonal lattice of $d$ orbitals, the broken inversion symmetry of the monolayers is incorporated via fictitious $s$ orbitals at the chalcogenide sites. A tight-binding Hamiltonian is constructed and then downfolded to get effective $d$-orbital overlap parameters using quasidegenerate perturbation theory. The steps to incorporate the effects of multiple layers, external electric and magnetic fields, are also detailed. We find that an electric field produces a linear-$k$ Rashba splitting around the $\mathrm{\Gamma }$ point, while a magnetic field removes the valley pseudospin degeneracy at the $\pm K$ points. Our model provides a simple tool to understand the recent experiments on electric and magnetic control of valley pseudospin in monolayer dichalcogendies.

Figure 1

The $2H_b$-type crystallographic unit cell of ${\mathrm{MoS}}_2$ contains two ${\mathrm{MoS}}_2$ layers shifted by ($a/3,-a/3,c/2$) from each other. A monolayer of ${\mathrm{MoS}}_2$ contains a mirror plane whereas a double layer has an inversion point in the middle as shown.

Figure 2

The electronic band structures of monolayers (1L) and doublelayers (2L) of ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$ with spin-orbit coupling calculated using DFT without external electric field. The blue, red, and green symbols correspond to orbitals characters of $d_{z^2}, d_{x^2-y^2}+d_{xy}$, and $d_{xz}+d_{yz}$, respectively. The size of the symbols is proportional to the strength of the character.

Figure 3

(a) Valence and conduction bands for ${\mathrm{WS}}_2$ monolayer from DFT calculations with applied electric field of $E=0.8$ eV/Å. The bands show linear Rashba spin splitting around the $\mathrm{\Gamma }$ point. (b) Change in charge density for ${\mathrm{WS}}_2$ monolayer when electric field is applied ($\mathrm{\Delta }\rho ={\rho }_0-{\rho }_E$). The red and blue surfaces are for isovalues of $0.0006/\text{Å}{}^3$ and $-0.0006/\text{Å}{}^3$, respectively. The downward electric force on the electrons moves charge from the top S to the bottom one.

Figure 4

The variation of the (a) band gap and (b) Rashba parameter (${\alpha }_R$) in the valence bands near the $\mathrm{\Gamma }$ point as a function of electric field from first-principles calculations.

Figure 5

Comparison of TB and DFT bands without SOC and external potential for the monolayer and bilayer ${\mathrm{WS}}_2$. Parameter values for the TB calculation are given in Table 1.

Figure 6

Band structure calculated using the tight-binding model in the presence of external electric and magnetic fields for ${\mathrm{WS}}_2$ monolayers and bilayers with the parameters from Table 1. For the finite electric field ($E\ne 0$) case (in electron volts) ${\gamma }_1=0.2, {\gamma }_2=0.05$, and $\mathrm{\Delta }=0.12$ are used, while for the finite magnetic field ($B\ne 0$) case $\mu =-0.1$ is used.

Figure 7

The conduction and valence bands of monolayer ${\mathrm{WS}}_2$ calculated using tight-binding model in the 2D hexagonal Brillouin zone under finite electric field with ${\gamma }_1=0.2,{\gamma }_2=0.05$, and $\mathrm{\Delta }=0.12$ (middle panel) and finite magnetic field with $\mu =-0.1$ (bottom panel) are shown. Cross-sectional plots of the valence bands around $\mathrm{\Gamma }$ are shown in the right. Spin-orbit parameter is taken to be $2{\lambda }_{\text{W}}=0.36$ eV to show the effects more clearly.

Figure 8

Tight-binding model with second-neighbor coupling compared to DFT calculations of ${\mathrm{WS}}_2$ monolayer. The full list of parameters are (in eV) ${\varepsilon }_1=0.4, {\varepsilon }_2=2.3, {\varepsilon }_3=3.7, t_1=-0.01, t_2=0, t_3=0.03, t_4=-0.4, t_5=0.05, t_6=0.8, t_7=0.75, t_8=0.83, t_9=0.8, V_{dd\sigma }^{\prime }=0.07, V_{dd\pi }^{\prime }=0.18$, and $V_{dd\delta }^{\prime }=-0.09$.