Phonon-mediated unconventional $s$- and $f$-wave pairing superconductivity in rhombohedral stacked multilayer graphene

Understanding the origin of superconductivity in correlated two-dimensional materials is a key step in leveraging material engineering techniques for next-generation nanoscale devices. The recent demonstration of superconductivity in Bernal bilayer and rhombohedral trilayer graphene, as well as in a large family of graphene-based moir\'e systems, indicate a common superconducting mechanism across these platforms. Here we combine first principles simulations with effective low-energy theories to investigate the superconducting mechanism and pairing symmetry in rhombohedral stacked graphene multilayers. We find that a phonon-mediated attraction can quantitatively explain the main experimental findings, namely the displacement field and doping dependence of the critical temperature and the presence of two superconducting regions whose pairing symmetries depend on the parent normal state. In particular, we find that intra-valley phonon scattering favors a triplet $f$-wave pairing out of a spin and valley polarized normal state. We also propose a new and so far unexplored superconducting region at higher hole doping densities $n_h \approx 4 \times 10^{12}$ cm$^{-2}$, and demonstrate how this large hole-doped regime can be reached in heterostructures consisting of monolayer $\alpha$-RuCl$_3$ and rhombohedral trilayer graphene.