Fate of superconductivity in disordered Dirac and semi-Dirac semimetals

TL;DR

This paper investigates how weak disorder and vertex corrections influence s-wave superconductivity in Dirac and semi-Dirac semimetals, revealing that disorder can promote or suppress superconductivity depending on the system.

Contribution

The study incorporates vertex corrections into the superconducting gap equations for Dirac and semi-Dirac semimetals, providing a generalized approach to understand disorder effects.

Findings

Weak disorder reduces the critical attraction in 3D Dirac semimetals.

Superconductivity can be induced by arbitrarily weak attraction in 2D and semi-Dirac semimetals.

Vertex corrections generally promote superconductivity, with minimal impact in bilayer graphene.

Abstract

The influence of weak disorder on the superconductivity in ordinary metals can be formally described by the Abrikosov-Gorkov diagrammatic approach. The vertex correction is ignored in this approach because an inequality $k_F l \gg 1$, where $k_F$ is the Fermi momentum and $l$ mean free path, is satisfied in ordinary metals with a large Fermi surface. In a Dirac semimetal that has discrete Fermi points, this inequality may break down even for arbitrarily weak disorder since $k_F \rightarrow 0$, and thus the vertex correction could be important. We incorporate the vertex correction into the self-consistent equations of the superconducting gap and the disorder scattering rate, and then apply the generalized approach to study how $s$-wave superconductivity is affected by random chemical potential in two- and three-dimensional Dirac semimetals, as well as two-dimensional semi-Dirac semimetal. In the clean limit, superconductivity is formed only when the pairing interaction strength is greater than some critical value in these materials. Adding random chemical potential to the system promotes superconductivity by generating a finite fermionic density of states at the Fermi level. In three-dimensional Dirac semimetal, the critical attraction strength is reduced by weak disorder, but remains finite. In the other two cases, superconductivity is induced by arbitrarily weak attraction. Including the vertex correction does not change these qualitative results, and actually could further promote superconductivity in the weak-attraction regime. Bilayer graphene is quite special in that its zero-energy density of states is nonzero despite the existence of Fermi points. Due to this peculiar property, superconductivity is always slightly suppressed by random chemical potential, and the impact of vertex correction is nearly negligible.