Electronic band structure of high-symmetry homobilayers of transition metal dichalcogenides

High-symmetric homobilayer transition metal dichalcogenides (TMDs) are important members of the bilayer (BL) van der Waals material family. Here we present a systematic study of the electronic band structure in low-energy regime in homo-BL TMD structures by using the standard $k·p$ method. Six types of BL TMD stacking configurations, which satisfy the $C_3$ symmetry are considered and they are ${\mathrm{H}}_{\mathrm{M}}^{\mathrm{M}}, {\mathrm{H}}_{\mathrm{X}}^{\mathrm{M}}, {\mathrm{H}}_{\mathrm{X}}^{\mathrm{X}}, {\mathrm{R}}_{\mathrm{M}}^{\mathrm{M}}, {\mathrm{R}}_{\mathrm{X}}^{\mathrm{M}}$, and ${\mathrm{R}}_{\mathrm{M}}^{\mathrm{X}}$. The intrinsic spin-orbit coupling (SOC) in the conduction and valence bands and the phase of interlayer hopping matrix elements are included in our investigation. Taking BL ${\mathrm{MoS}}_2$ as an example, we examine the electronic energy spectra, the electron density of states, and the Fermi energies in these BL structures. We find that the electron energy dispersions in high-symmetric BL TMDs are not parabolic-like, where the band parameters (such as the energy gap, the effective electron band mass and the fourth-order correction coefficient in different subbands) depend markedly on the stacking configurations. Interestingly, the spin splitting in $\mathrm{H}$-stacked BL TMDs is suppressed because of center-inversion symmetry and time-reversal symmetry. Importantly, the phase of the interlayer hopping matrix element affects significantly the electronic properties of ${\mathrm{H}}_{\mathrm{X}}^{\mathrm{X}}$ and ${\mathrm{R}}_{\mathrm{M}}^{\mathrm{M}}$ stacked BL TMDs. The methodology and the results presented in this study can foster further exploration of the basic physical properties of BL TMDs for potential applications in electronics and optoelectronics.