Ab-initio study of the energy competition between Γand K valleys in bilayer transition metal dichalcogenides

TL;DR

This work uses ab-initio density-functional theory to determine whether the valence-band maximum in bilayer group VI TMDs lies at $\Gamma$ or $K$, focusing on the competition between interlayer tunneling, spin-orbit coupling, gate-induced fields, and $d$-orbital electronic correlations. By combining VASP structural relaxations for 2H and AA stackings with LAPW calculations in WIEN2k (LDA and local-mBJ) that include SOC, the authors map $\Delta E_{K-\Gamma}=E_K-E_\Gamma$ as a function of interlayer distance $h$ and external fields across homo- and heterobilayers of MX$_2$ (M=Mo,W; X=S,Se,Te). They find that increasing $h$ generally raises $E_K$ while lowering $E_\Gamma$, with local-mBJ correlations pushing the K valley higher and SOC strongly lifting degeneracy at K; gating effects are stronger at K, and field direction matters in heterobilayers. Pressure and gating emerge as practical knobs to tune the valley ordering, potentially enabling near-degenerate Γ and K valleys and the realization of moiré-derived, two-orbital correlated states in designed bilayer TMD systems. These results provide design rules for moiré materials by linking microscopic interactions to tunable valley energetics.

Abstract

Moiré engineering in two-dimensional van der Waals bilayer crystals has emerged as a flexible platform for controlling strongly correlated electron systems. The competition between valleys for the band extremum energy position in the parent layers is crucial in deciding the qualitative nature of the moiré Hamiltonian since it controls the physics of the moiré minibands. Here we use density functional theory to examine the competition between K and $Γ$ for the valence band maximum in homo- and hetero-bilayers formed from the transition metal dichalcogenides (TMD), MX\{_2} where M=Mo,W and X=S,Se,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 moiré materials.

Figures (9)

• Figure 1: (color online). Example of AA and 2H stacking in the case of the free standing $\textrm{MoTe}_\textrm{2}$/$\textrm{WTe}_\textrm{2}$ TMD heterobilayer. A vacuum of magnitude 20 $\text{\normalfont\AA}$ is added to the bulk c-axis lattice constant. $h$ is the interlayer separation, characterized by the vertical distance between metal atoms.
• Figure 2: (color online). The top four valence bands computed with the LmBJ functional and spin-orbit coupling, as a function of $h$ for 2H-stacked $\textrm{MoS}_\textrm{2}$/$\textrm{WS}_\textrm{2}$ TMD bilayers. A sketch of each crystal structure is set into each plot. The energies of the $\Gamma$ and K points are labelled as $E_\Gamma$ and $E_K$ respectively. These results demonstrate that $E_K$ increases relative to $E_\Gamma$ as $h$ increases, as observed in all cases considered.
• Figure 3: (color online). The energy difference between the local valence band maxima at $\Gamma$ and K points, $\Delta E_{K-\Gamma}\equiv E_K - E_\Gamma$, as a function of the interlayer distance $h$ calculated by LDA and local mBJ functionals for 2H stacked homobilayers. The value of $h$ at which $\Delta E_{K-\Gamma}$ crosses from negative ($\Gamma$-valley) to positive (K-valley) decreases when correlations are included by using the local mBJ potential. All calculations include spin-orbit coupling.
• Figure 4: (color online). The energy difference between the local band maxima $\Gamma$ and K points $\Delta E\equiv E_K - E_\Gamma$ as a function of the interlayer distance $h$ calculated by LDA and local mBJ functionals for heterobilayers. Analysis of the band characteristics shows that the metal atom at the K valley of the highest valence band is that of Tungsten for all heterobilayers.
• Figure 5: (color online). (top) The spin-orbit splitting $\Delta E_{SO}$ and (bottom) the bilayer splitting $\Delta E_{BL}$ as a function of the absolute interlayer separation $h$ for $\textrm{MoX}_2$ and $\textrm{WX}_2$ with $\textrm{X}=\textrm{S},\textrm{Se}$ and $\textrm{Te}$. The 2H stacked results using the local mBJ functional are presented.