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We describe here a theory of a quantum dot in an electrically tunable MoSe$_2$/WSe$_2$ heterostructure. Van der Waals heterostructures allow for tuning their electronic properties beyond their monolayer counterparts. We start by determining their electronic structure using density functional theory. We obtain the type-II band alignment and close in energy conduction band minima (valleys) at the $K$ and $Q$ points in the Brillouin zone. The valence band maxima, also energetically close, are located at the $K$ and $\Gamma$ points. By analyzing the Kohn-Sham wavefunctions, we describe the layer, spin, and orbital contributions. Next, we construct an \textit{ab initio}-based tight-binding model, which helps us to better understand the complexity of the interlayer interactions. We determine the effect of a vertical electric field, showing that vertical gating enables control of valleys extrema and their occupancy. Finally, we employ the tight-binding model to investigate laterally gated quantum dots and analyze the influence of a perpendicular electric field on their energy spectrum. Our results demonstrate that tuning the electric field enables control over the valley character of the quantum dot states, selectively localizing them in either the $K$ or $Q$ valleys, as evidenced by their characteristic degeneracies and wavefunctions. |
