High pressure induced secondary and tertiary gaps in relaxed graphene on hexagonal boron nitride

We study the electronic phase diagram of fully relaxed moire superlattices in the monolayer graphene on hexagonal boron nitrogen (MLG/BN) with varying twist angle (theta) and high pressure. In systems with small theta, the secondary gap (Delta((e))(S)) on the electron side and the tertiary gap (Delta((h))(T)) on the hole side are opened by the high pressure up to 8 GPa, and the pressure also greatly enhances the primary band gap (Delta(0)) at the charge neutrality point and the secondary gap (Delta((h))(S)) on the hole side. Under a given high pressure, the gaps Delta((h))(T), Delta((e))(S), and Delta((h))(S) become closed in succession with varying theta from 0 degrees to 2 degrees. In contrast to the direct Delta(0) and Delta((h))(S) at small theta, Delta((e))(S) only becomes direct for a range of medium pressure, and Delta((h))(T) remains indirect. Compared with the local continuum Hamiltonian ((H) over cap (c)), the effects of nonlocal terms and those beyond the first harmonics in the full Hamiltonian become important for the quantitative values of the gaps under high pressure, while the qualitative behavior of the gaps with increasing pressure can still be understood by the variations of parameters in (H) over cap (c). The band isolation due to the pressure induced gaps leads to emergence or evolution of the van Hove singularities, the ballistic resistance peaks, and the peaks of the density of orbital magnetic moments, which may be observed experimentally.