Diffusion quantum Monte Carlo and GW study of the electronic properties of monolayer and bulk hexagonal boron nitride

We report diffusion quantum Monte Carlo (DMC) and many-body $GW$ calculations of the electronic band gaps of monolayer and bulk hexagonal boron nitride (hBN). We find the monolayer band gap to be in principle indirect; the valence-band maximum is at the $K$ point of the hexagonal Brillouin zone while the conduction-band minimum is at $\Gamma$; nevertheless, monolayer hBN is in effect a direct-gap material, owing to the delocalized nature of the electronic states at the conduction-band minimum at the $\Gamma$ point. $GW$ predicts much smaller quasiparticle gaps at both the single-shot $G_0W_0$ and the partially self-consistent $GW_0$ levels. In contrast, solving the Bethe-Salpeter equation on top of the $GW_0$ calculation yields an exciton binding energy for the direct exciton at the $K$ point in close agreement with the DMC value. Vibrational renormalization of the electronic band gap is found to be significant in both the monolayer and the bulk. Taking vibrational effects into account, DMC overestimates the band gap of bulk hBN, while $GW$ theory underestimates it.