Geometric Contribution to the Superfluid Density in Inversion Asymmetric Bilayer Graphene

We study superconductivity and exciton condensation in bilayer graphene-based two-dimensional crystal heterostructures in the presence of a broken $mathcal{C}_2$ symmetry.

The $mathcal{C}_2$ symmetry is broken by applying a displacement field perpendicular to the sample and results in a finite mass term $m$. The mass term leads to an enhancement of the superconductivity and exciton superfluid gap in bilayer graphene. We calculate the superfluid weight, which consists of two terms i) conventional contribution and ii) geometric contribution. The geometric contribution emerges from the quantum metric of the electron-hole bands. We find that the geometric contribution can be significantly enhanced by a finite mass term, and we show the effect of this enhancement on the Kosterlitz-Thouless transition temperature.