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The idea of combining different two-dimensional (2D) crystals in van derWaals heterostructures (vdWHs) has led to a new paradigm for band structureengineering with atomic precision. Due to the weak interlayer couplings, theband structures of the individual 2D crystals are largely preserved uponformation of the heterostructure. However, regardless of the details of theinterlayer hybridisation, the size of the 2D crystal band gaps are alwaysreduced due to the enhanced dielectric screening provided by the surroundinglayers. The effect can be on the order of electron volts, but its precisemagnitude is non-trivial to predict because of the non-local nature of thescreening in quasi-2D materials, and it is not captured by effectivesingle-particle methods such as density functional theory. Here we present anefficient and general method for calculating the band gap renormalization of a2D material embedded in an arbitrary vdWH. The method evaluates the change inthe GW self-energy of the 2D material from the change in the screened Coulombinteraction. The latter is obtained using the quantum-electrostaticheterostructure (QEH) model. We benchmark the G$\Delta$W method against fullfirst-principles GW calculations and use it to unravel the importance ofscreening-induced band structure renormalisation in various vdWHs. A mainresult is the observation that the size of the band gap reduction of a given 2Dmaterial when inserted into a heterostructure scales inversely with thepolarisability of the 2D material. Our work demonstrates that dielectricengineering \emph{via} van der Waals heterostructuring represents a promisingstrategy for tailoring the band structure of 2D materials.

Quasiparticle band structure engineering in van der Waals heterostructures via dielectric screening