Excitonic response of AA′ and AB stacked hBN bilayers

In this paper, we discuss the optical response due to the excitonic effect of two types of hBN bilayers: AB and ${\mathrm{AA}}^{\ensuremath{'}}$. Understanding the properties of these bilayers is of great utility to the study of twisted bilayers at arbitrary angles since these two configurations correspond to the limit cases of ${0}^{\ensuremath{\circ}}$ and ${60}^{\ensuremath{\circ}}$ rotation. To obtain the excitonic response, we present a method to solve a four-band Bethe-Salpeter equation by casting it into a one-dimensional problem, thus greatly reducing the numerical burden of the calculation when compared with strictly two-dimensional methods. We find results in good agreement with ab initio calculations already published in the literature for the ${\mathrm{AA}}^{\ensuremath{'}}$ bilayer, and predict the excitonic conductivity of the AB bilayer, which remains largely unstudied. The main difference in the conductivity of these two types of bilayers is the appearance of a small, yet well-resolved resonance between two larger ones in the AB configuration. This resonance is due to a mainly interlayer exciton, and is absent in the ${\mathrm{AA}}^{\ensuremath{'}}$ bilayer. Also, the conductivity of the AB bilayer is due to both intralayer and interlayer excitons and is dominated by $p$-states, while intralayer $s$-states are the relevant ones for the ${\mathrm{AA}}^{\ensuremath{'}}$ configuration, like in a monolayer. The effect of introducing a bias in the ${\mathrm{AA}}^{\ensuremath{'}}$ bilayer is also discussed.