Effective lattice Hamiltonian for monolayer MoS${}_2$: Tailoring electronic structure with perpendicular electric and magnetic fields

We propose an effective lattice Hamiltonian for monolayer MoS${}_2$ in order to describe the low-energy band structure and investigate the effect of perpendicular electric and magnetic fields on its electronic structure. We derive a tight-binding model based on the hybridization of the $d$ orbitals of molybdenum and $p$ orbitals of sulfur atoms and then introduce a modified two-band continuum model of monolayer MoS${}_2$ by exploiting the quasidegenerate partitioning method. Our theory proves that the low-energy excitations of the system are no longer massive Dirac fermions. It reveals a difference between electron and hole masses and provides trigonal warping effects. Furthermore, we predict a valley-degeneracy-breaking effect in the Landau levels. In addition, we also show that applying a gate voltage perpendicular to the monolayer modifies the electronic structure, including the band gap and effective masses.

Figure 1

(a) Band structure of ML-MDS consisting of five bands in which two are spin split in the valence band. The dot-dashed line refers to one spin and the dashed line denotes another spin component. Solid lines refer to the spin-degenerate band. (b) A comparison between the band structure calculated by the present theory (solid lines) and the results calculated in Ref. 19 (dashed lines) based on density functional theory. Notice that our theory works quite well around the $\mathbf{K}$ point for the particle (hole or electron) density less than ${10}^{14}$ cm${}^{-2}$ (for instance, $E_F-E_{\mathrm{CBM}}\simeq 0.2$ eV). Here, $a_0=a\cos \theta$ where $a$ is the length of the Mo-S bond and $\theta$ is the angle between the bond and the $xy$ plane. (c) Side and top views of the lattice structure where the Mo atom (larger green sphere) is surrounded by six S atoms.

Figure 2

Contour plots of the conduction (top panel) and valence (bottom panel) bands in momentum space for spin-up components together with isoenergy lines to guide the eye. While the conduction band shows almost isotropic dispersion, trigonal warping occurs in the valence band around the $K$ points, due to the difference of the orbital structures of the conduction and valance bands.

Figure 3

Spin-up conduction and valence energy bands, referred to the energy $E_v=E_{\text{VBM}}+U/2$, for different values of the vertical bias.

Figure 4

Variation of the gap versus induced potential which is almost linear in the chosen range of $U$. Insets show the changes in the effective hopping integral and the effective masses with $U$. Using the form of $U$ dependence of the parameters, we can also estimate the linear change of $\delta \alpha \approx -0.15U/\text{eV}$ and $\delta \beta \approx -1.95U/\text{eV}$.