Ab-initio study of the energy competition between $Gamma$ and K valleys in bilayer transition metal dichalcogenides*

Moir'e engineering in two-dimensional van der Waals bilayer crystals has emerged as a flexible platform for controlling tunable strongly correlated electron systems. The competition between different valleys for the band extremum energy position in the parent layers is crucial in deciding the qualitative nature of the moir'e model since it controls the physics of the moir'e minibands. Here we use density functional theory to examine the competition between K and $Gamma$ for the valence band maximum in homo- and hetero-bilayers formed from the transition metal dichalcogenides (TMD), $ extrm{MX}_2$ where $ extrm{M}= extrm{Mo}, extrm{W}$ and $ extrm{X}= extrm{S}, extrm{Se}, extrm{Te}$. We shed light on how the competition is influenced by interlayer separation, which can be modified by applying pressure, by external gate-defined electric fields, and by transition metal atom $d$-orbital correlations. Our findings are related to several recent experiments, and contribute to the development of design rules for moir'e materials.