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 moiré 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 [Finney et al., Proc. Natl. Acad. Sci. USA 119, e2118482119 (2022); Gao et al., arXiv:2412.01578]. 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 moiré potential. In particular, we identify several $\mathrm{\Gamma }$-phonon branches that contribute most significantly to the EPC, which are experimentally detectable via Raman spectroscopy.

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• Electron-phonon coupling
• Superconductivity
• Graphene