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

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.

💡 Research Summary

This paper presents a comprehensive first‑principles investigation of the competition between the Γ and K valleys for the valence‑band maximum (VBM) in both homo‑ and hetero‑bilayers of group‑VI transition‑metal dichalcogenides (TMDs) MX₂ (M = Mo, W; X = S, Se, Te). Using density‑functional theory (DFT) with several exchange‑correlation functionals (LDA‑SO, PBE‑SO, PBE‑SO‑D3, and the modified Becke‑Johnson (mBJ) potential) the authors systematically explore how structural and electronic parameters control whether the VBM resides at Γ or K.

Two high‑symmetry stackings are considered: the energetically favored 2H configuration (180° rotation between layers) and the AA configuration. Structural relaxations are performed with VASP, followed by all‑electron band‑structure calculations with WIEN2k. The interlayer distance h, which can be tuned experimentally by pressure, encapsulation, or an external gate‑defined electric field, is identified as the primary knob governing the Γ–K competition.

The key findings are: (1) Reducing h enhances interlayer tunnelling, which strongly raises the Γ‑point valence band due to a large bilayer splitting, thereby favoring a Γ‑VBM. Conversely, increasing h diminishes this tunnelling, allowing the spin‑orbit‑split K‑point states (which are relatively insensitive to h) to dominate, leading to a K‑VBM. (2) The choice of metal atom strongly influences the outcome. Tungsten‑based compounds exhibit larger intrinsic spin‑orbit coupling (ΔSO) than molybdenum analogues, shifting the Γ–K crossover toward larger h and making K‑VBM more common. (3) The chalcogen atom follows a monotonic trend S → Se → Te: heavier chalcogens increase the spatial extent of p‑orbitals, strengthen d‑p hybridisation across the layers, and increase the bilayer splitting at Γ, thus promoting a Γ‑VBM at a given h. (4) Electronic correlation effects, captured by the mBJ functional, raise the energy of the more localized K‑valley states relative to the more delocalised Γ states, effectively moving the Γ–K crossover to smaller h compared with plain LDA. (5) In heterobilayers the broken symmetry between the two layers introduces a non‑negligible bilayer splitting at K, which generally stabilises a K‑VBM, especially when the constituent monolayers individually favour Γ. However, for S‑based heterostructures a sufficiently large increase of h can still drive a Γ‑VBM, suggesting a route to electrically or mechanically switch the valley character. (6) An external perpendicular electric field creates a potential difference between the layers, modestly shifting ΔE_K‑Γ by 10–30 meV for realistic field strengths (≈ 0.5 V/Å), offering an additional, experimentally accessible tuning parameter.

From these systematic trends the authors formulate practical design rules for moiré‑engineered TMD bilayers: (i) Control the interlayer spacing via pressure, substrate encapsulation, or gate‑induced electrostatic pressure to toggle between Γ‑ and K‑VBM. (ii) Select W‑based compounds and heavier chalcogens to bias toward K‑VBM, while using Mo‑based and lighter chalcogens to favour Γ‑VBM. (iii) Employ the AA stacking only when a Γ‑VBM is desired at larger h, recognizing that AA is energetically less stable than 2H. (iv) Use heterobilayers to stabilise K‑VBM, but exploit the sensitivity of S‑based heterostructures to h for valley switching. (v) Apply modest out‑of‑plane electric fields for fine‑tuning of the valley hierarchy.

These insights provide a quantitative foundation for engineering the valley landscape in twisted or untwisted TMD moiré superlattices, thereby guiding the realization of correlated phases such as Mott insulators, superconductivity, and topological states that depend critically on whether the low‑energy moiré bands inherit Γ‑ or K‑valley character.