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Superconducting Symmetry Phases and Dominant bands in (Ca-) Intercalated AA- Bilayer Graphene

Built on a realistic multiband tight-binding model, mirror symmetry is used to map a calcium-intercalated bilayer graphene Hamiltonian into two independent single layer graphene-like Hamiltonians with renormalized hopping. The quasiparticles exhibit two types of chirality. Here a quasi-particle consists of two electrons from opposing layers where possess an additional quantum number called "cone index" which can be regarded as the eigenvalue of mirror symmetry operations. To obtain tight-binding parameters, both effective monolayer Schrodinger equations are solved analytically and fitted to first-principles band structure results. Two quasi-particles (four electrons) can team up to build a Cooper pair with even or odd chirality. Treatment of the pairing Hamiltonian leads to two decoupled gap equations. The pairing of quasi-particles with different cone indexes is forbidden. The decoupled gap equations are solved analytically to obtain all the possible superconducting phases. Two nearly "flat bands" crossing the Fermi energy, each related to the graphene-like structures, are responsible for two distinct superconductivity gaps that emerge. Depending on how much these bands are affected by the intercalant and which is closer to the Fermi energy, distorted s-wave or d-wave superconductivity may become dominant. Numerical calculations reveal that d-wave superconductivity is dominant in both sectors. For these two dominant phases, within the range of 0-6 K which superconductivity has been observed, numerically the transition from single-gap to dual-gap superconductivity is possible. Adopting the two-gap viewpoint of superconductivity in C$_6$CaC$_6$, the dominant $d$-wave states should have the same critical temperature. Around $T_c=2K$ these two relations intersect, otherwise, superconductivity has been realized just in one of these two sectors and disappears in the other one.