Superconductivity of disordered Dirac fermions

Abstract

We study the effect of disorder on massless, spinful Dirac fermions in two spatial dimensions with attractive interactions, and show that the combination of disorder and attractive interactions is deadly to the Dirac semimetal phase. First, we derive the zero temperature phase diagram of a clean Dirac fermion system with tunable doping level (μ) and attraction strength (g). We show that it contains two phases: a superconductor and a Dirac semimetal. Then we add disorder and show that arbitrarily weak disorder destroys the Dirac semimetal, turning it into a superconductor instead. Thus for Dirac fermions near charge neutrality, disorder actually assists superconductivity. We discuss the strength of the superconductivity for both long range and short range disorder. For long range disorder, the superconductivity is exponentially weak in the disorder strength. For short range disorder, a uniform mean field analysis predicts that superconductivity should be doubly exponentially weak in the disorder strength. However, a more careful treatment of mesoscopic fluctuations suggests that locally superconducting puddles should form at a much higher temperature, and should establish global phase coherence at a temperature that is only exponentially small in weak disorder. Thus, mesoscopic fluctuations exponentially enhance the superconducting critical temperature. We also discuss the effect of disorder on the quantum critical point of the clean system, building in the effect of disorder through a replica field theory. We show that disorder is a relevant perturbation to the supersymmetric quantum critical point. We expect that in the presence of attractive interactions, the flow away from the critical point ends up in the superconducting phase, although firm conclusions cannot be drawn since the renormalization group analysis flows to strong coupling. We argue that although we expect the quantum critical point to get buried under a superconducting phase, signatures of the critical point may be visible in the finite temperature quantum critical regime. Our results have implications for experiments on proximity induced superconductivity in Dirac fermion systems, where they imply an enormous disorder enhancement of the superconducting susceptibility. As a result, the proximity induced superconductivity in dirty systems is expected to be much stronger than that in clean systems at the Dirac point.