Upper critical field and critical supercurrent in twisted bilayer graphene

Superconductivity is observed in magic-angle twisted bilayer graphene over a wide range of moir\'e band fillings and electric gating conditions. Based on the optical-phonon-mediated mechanism assisted by adjustable Coulomb repulsion, we perform self-consistent Bogoliubov-de Gennes mean-field calculations to ascertain the characters of superconducting state generated at finite pairing momentum in twisted bilayer graphene. Our calculations show that the free energy of finite-momentum pairing state can be well described by the phenomenological Ginzburg-Landau theory of superconductivity with quadratic kinetic energy. The perpendicular upper critical field H_c2 possesses a sharp peak at the van Hove singularity of density of the states and decreases quickly away from it, with an averaged value comparable with experimental result. The critical supercurrent density exhibits a dome-like character in analogy to the mean-field superconducting transition temperature upon varying chemical potential. We discuss these results in connection with recent experimental observations, which may provide insights for understanding the exotic superconducting properties in twisted bilayer graphene.