Continuum model for relaxed twisted bilayer graphenes and moir\'{e} 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 moir\'{e} band states and the moir\'{e} phonon modes. By numerical calculation, we find that the electron-phonon coupling and phonon mediated electron-electron interaction are significantly enhanced in low twist angles due to the superlattice hybridization.