Atomistic substrate relaxation effects in the band gaps of graphene on hexagonal boron nitride

We assess the impact of atomistic substrate lattice relaxation schemes in the primary band gap at charge neutrality and the secondary valence band gap of graphene on hexagonal boron nitride (G/h-BN) as a function of twist angle. For zero twist angle, the primary gap decreases from $\sim 30$~meV in fully relaxed suspended G/h-BN bilayers, to $\sim 9$~meV when the remote h-BN substrate layer is kept rigid, and down to $\sim 3$~meV in completely rigid structures. In the presence of relaxations, the primary gap shows a maximum near $\sim 0.6^{\circ}$ coinciding with energetic stabilization due to alignment between the moiré pattern and the graphene lattice vectors, while the secondary valence band gap drops from $\sim 12$~meV down to zero beyond twist angles of $\sim 1^{\circ}$. A small but finite primary gap on the order of $\sim 1$~meV, with a mass sign favoring electronic occupation of carbon atop boron, persists across twist angles from $0^{\circ}$ to $30^{\circ}$ for all sliding configurations, and switches sign for twist angles between $30^{\circ}$ and $60^{\circ}$.