  We study the electronic properties of commensurate faulted bilayer graphene
by diagonalizing the one-particle Hamiltonian of the bilayer system in a
complete basis of Bloch states of the individual graphene layers. Our novel
approach is very general and can be easily extended to any commensurate
graphene-based heterostructure. Here, we consider three cases: i) twisted
bilayer graphene, ii) bilayer graphene where triaxial stress is applied to one
layer, and iii) bilayer graphene where uniaxial stress is applied to one layer.
We show that the resulting superstructures can be divided into distinct
classes, depending on the twist angle or the magnitude of the induced strain.
The different classes are distinguished from each other by the interlayer
coupling mechanism, resulting in fundamentally different low-energy physics.
For the cases of triaxial and uniaxial stress, the individual graphene layers
tend to decouple and we find significant charge transfer between the layers. In
addition, this piezoelectric effect can be tuned by applying a perpendicular
electric field. Finally, we show how our approach can be generalized to
multilayer systems.
