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.

Theoretical prediction of a strongly correlated Dirac metal

“…3). It is worth noting that the disorder introduced by Zn doping, while weak, might even, counterintuitively, support superconductivity in this limit 29 . Finally, a subtlety could arise for significant V: In order to avoid onsite and nearest-neighbour repulsion, this could drive a transition from a nodeless B 1u f-wave to a nodal B 2u f-wave 30 , but for realistic values of V this is highly unlikely.…”

Superconductivity in two-dimensional disordered Dirac semimetals

“…Previous theoretical and numerical studies [18,21,22,30] have found that, different from ordinary metals, the Cooper theorem is no longer valid in undoped Dirac semimetals. Because the fermion density of states (DOS) vanishes linearly near the band-touching points, an infinitely weak attraction is not sufficient to bind Dirac fermions together to form Cooper pairs.…”

“…Recently, Nandkishore et al [30] and Potirniche et al [31] have studied the effects of random chemical potential on superconductivity. The main conclusion reached in the mean-field analysis [30] is that superconductivity is enhanced. In this paper, we will consider the impact of all the three types of disorder.…”

“…The Cooper pairing of Dirac fermions and the resultant superconducting transition are two subjects of considerable interest [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The superconductivity might be mediated by various bosonic modes, such as phonons or plasmons, in graphene [18].…”

“…Because the fermion density of states (DOS) vanishes linearly near the band-touching points, an infinitely weak attraction is not sufficient to bind Dirac fermions together to form Cooper pairs. Cooper pairing can be triggered only when the attraction strength exceeds certain threshold [18,21,22,30], i.e., |u| > |u c | with u c being a finite critical value. Hence, there is a quantum phase transition between the semimetallic and superconducting phases, and the critical value u c defines the quantum critical point (QCP).…”