SUPERCONDUCTIVITY IN FLAT BANDS: NUMERICAL ANALYSIS OF THE BCS GAP AND SUPERCONDUCTING CROSSOVER

Abstract

Superconductivity is characterized by zero electrical resistance and the expulsion of magnetic fields below a critical temperature, phenomena first identified by Onnes and later clarified through the Meissner–Ochsenfeld effect. In conventional metals, these properties arise from the formation of Cooper pairs and the opening of an energy gap in the electronic spectrum, as described by BCS theory. The BCS framework predicts that both the superconducting gap and the critical temperature depend exponentially on the pairing interaction and the density of states, which limits the achievable superconducting transition temperatures in dispersive electronic bands. Recent interest has shifted towards flat-band systems, materials whose electronic bands have extremely low dispersion and correspondingly high density of states. In such systems, even weak interactions can lead to strong pairing, and the gap can scale linearly with interaction strength rather than exponentially. This makes flat bands a promising route toward enhanced or unconventional superconductivity, as recently shown experimentally in magic angle twisted bilayer graphene and theoretically analysed for systems like the Lieb lattice. In this work, the self-consistent BCS gap equation is numerically solved to investigate how the superconducting gap behaves across a range of model parameters. Specifically, I study the dependence of the gap on bandwidth, density of states, coupling strength, and temperature. The results demonstrate clear distinctions between flat-band and conventional regimes, including strong gap enhancement at small bandwidth and unusually large ∆ 𝑘𝐵 𝑇𝑐 ratios. A validation test in the perfectly flat-band limit confirms agreement with known analytical predictions. These findings provide insight into the crossover between dispersive and flat band superconductivity and highlight the conditions under which flat band driven pairing can emerge.