Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction

We construct an analytic continuum model to describe the electronic structure and the electron-phonon interaction in twisted bilayer graphenes with arbitrary lattice deformation. Starting from the tight-binding model, we derive the interlayer Hamiltonian in the presence of general lattice displacement and obtain a long-wavelength continuum expression for smooth deformation. We show that the continuum model correctly describes the band structures of the lattice-relaxed twisted bilayer graphenes. We apply the formula to the phonon vibration and derive an explicit expression of the electron-phonon matrix elements between the moiré band states and the moiré phonon modes. By numerical calculation, we find that the electron-phonon coupling is significantly enhanced in low twist angles due to the superlattice hybridization. At the magic angle, in particular, the phonon-mediated electron-electron interaction is found to be comparable to the order of the inverse density of states, suggesting that the system is in the strong coupling regime.