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, in part due to 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 requiring a periodic moir\'e supercell. Our approach combines a momentum-space continuum model for both electronic and phononic structures with a generalized Eliashberg-McMillan theory beyond the adiabatic approximation. Using this framework, we find that the EPC is strongly enhanced near the magic angle. The superconducting transition temperature induced by low-energy phonons peaks at $1.1^\circ$ around 1 K, and remains finite for a range of angles both below and above the magic angles. We predict that superconductivity persists up to $\sim 1.4^\circ$, where superconductivity has been recently observed despite the dispersive electronic bands. Beyond a large density of states, we identify a key condition for strong EPC: resonance between the electronic bandwidth and the dominant phonon frequencies. We also show that the EPC strength of a specific phonon corresponds to the modification of the moir\'e potential. In particular, we identify several $\Gamma$-phonon branches that contribute most significantly to the EPC, which are experimentally detectable via Raman spectroscopy.