Electronic properties of the five principal stackings of boron nitride moir\'e bilayers

All theoretical calculations on boron nitride moir\'e bilayers report the properties of, at most, two possible stackings which preserve the monolayer hexagonal symmetry (i.e. the invariance upon rotations of 120$^\circ$). In this work, we demonstrate that, for a given moir\'e periodicity, the same symmetry is respected by five different stackings and not only two as always discussed in literature. We introduce some definitions and an appropriate nomenclature to identify unambiguously the twist angle and the stacking sequence of any BN bilayer with order-3 rotation symmetry. The nomenclature we introduce here and the method to calssify stacking sequences and the angles is completely general and can be applied to homobilayers of any hexagonal 2D materials. Moreover, we produce density functional theory predictions of the electronic structure of each of the five stacking sequences at six different twist angles and discuss the evolution of the gapwidth and band structure and as a function of these parameters. We show that the gap is indirect at any angle and in any stacking and we identify features that are conserved at any angle within the same stacking sequence.