Microscopic theory for electron-phonon coupling in twisted bilayer graphene

The origin of superconductivity in twisted bilayer graphene -- whether phonon-driven or electron-driven -- remains unresolved. The answer to this question is hindered by the absence of a quantitative and efficient model for electron-phonon coupling (EPC). In this work, we develop a first-principles-based microscopic theory to calculate EPC in twisted bilayer graphene for arbitrary twist angles without needing a periodic moiré supercell. We adopt a momentum-space model for the electronic and phonon structures and quantify the EPC using generalized Eliashberg-McMillan theory for superconductivity without an adiabatic approximation. Using this framework, we find that the EPC is significantly enhanced near the magic angle, and drops abruptly for both smaller and larger twist angles. In addition to a large electronic and phonon density of states, we find that a condition for large EPC is the resonance between flat bandwidth and the frequency of phonons that have the largest matrix elements. We also show that the EPC strength of a specific phonon corresponds to the moiré potential. In particular, we identify several Γ-phonon branches that contribute most significantly to the EPC, which are experimentally detectable via Raman spectroscopy.