Analytical Eliashberg−BKT theory of superfluid stiffness in graphene superconductors

Abstract

Graphene-based superconductors offer a natural setting for studying the competition between microscopic pair formation and macroscopic phase coherence in two-dimensional Dirac materials. Using the Nambu−Gor'kov formulation of Eliashberg theory, we derive the superfluid stiffness from the gauge-invariant electromagnetic response in the static limit. For monolayer graphene we obtain K_xx^mono = e²E_F/[π(1+λ)] and the phase stiffness D_s^mono(0) = E_F/[4π(1+λ)], where E_F is the Fermi energy and λ is the electron−phonon coupling; the Nelson−Kosterlitz criterion then gives the upper-bound BKT temperature T _BKT ≤ E_F/[8(1+λ)] and identifies a pseudogap-like regime where phase fluctuations dominate. Extending the framework to a bilayer in the weak-interlayer-coupling limit yields an additive stiffness, suggesting enhanced phase coherence when interlayer phase locking is effective.