Tuning the electronic structure of twisted transition metal dichalcogenides heterotrilayer with an applied electric field

Moiré superlattices generated by twisting two or more layers of semiconducting transition metal dichalcogenides (TMDs) have attracted great attention due to their ability to host strongly correlated states, such as Mott insulators at half-filling (one hole per moiré cell) and generalised Wigner crystals at fractional fillings. The top valence bands in these systems are derived either from the monolayer states in the K^-/K⁺ valley or alternatively from monolayer states in the Γ valley. Here, we demonstrate using density-functional theory calculations that the order of K^-/K⁺-derived and Γ-derived valence bands in a heterotrilayer consisting of a 2H-WSe₂ bilayer stacked on a MoSe₂ layer with a twist angle of 3.1° can be controlled by a perpendicular electric field. Interestingly, the wavefunctions of the Γ-derived bands are delocalized over the three layers and realize a honeycomb lattice at the moiré scale with inequivalent A and B sites, whereas K^-/K⁺- derived bands give rise to a triangular lattice and are fully localized on specific layers. From these ab initio insights, we construct an effective Hamiltonian to study the correlated hole states that arise when the system is doped. Our results show that twisted heterotrilayer TMDs are an ideal platform to investigate Hubbard models on both triangular and honeycomb lattices in the same system and are in agreement with experimental measurements.