Moir\'{e} phonons in the twisted bilayer graphene

We study the in-plane acoustic phonons in twisted bilayer graphenes using the effective continuum approach. We calculate the phonon modes by solving the continuum equation of motion for infinitesimal vibration around the static relaxed state with triangular domain structure. We find that the moir\'{e} interlayer potential only affects the in-plane asymmetric modes, where the original linear dispersion is broken down into miniphonon bands separated by gaps, while the in-plane symmetric modes with their linear dispersion are hardly affected. The phonon wave functions of asymmetric modes are regarded as collective vibrations of the domain-wall network, and the low-energy phonon band structure can be qualitatively described by an effective moir\'{e}-scale lattice model.