Dichotomy of Electron-Phonon Coupling in Graphene Moir\'e Flat Bands

Graphene moire superlattices are outstanding platforms to study correlated electron physics and superconductivity with exceptional tunability. However, robust superconductivity has been measured only in magic-angle twisted bilayer graphene (MA-TBG) and magic-angle twisted trilayer graphene (MA-TTG). The absence of a superconducting phase in certain moire flat bands raises a question on the superconducting mechanism. In this work, we investigate electronic structure and electron-phonon coupling in graphene moire superlattices based on atomistic calculations. We show that electron-phonon coupling strength lambda is dramatically different among graphene moire flat bands. The total strength lambda is very large (lambda>1) for MA-TBG and MA-TTG, both of which display robust superconductivity in experiments. However, lambda is an order of magnitude smaller in twisted double bilayer graphene (TDBG) and twisted monolayer-bilayer graphene (TMBG) where superconductivity is reportedly rather weak or absent. We find that the Bernal-stacked layers in TDBG and TMBG induce sublattice polarization in the flat-band states, suppressing intersublattice electron-phonon matrix elements. We also obtain the nonadiabatic superconducting Tc that matches well with the experimental results. Our results clearly show a correlation between strong electron-phonon coupling and experimental observations of robust superconductivity.