Impurity Effects on Caroli-de Gennes-Matricon Mode in a Vortex Core in Superconductors

TL;DR

This paper introduces an improved Green's function scheme to analyze impurity effects on the Caroli-de Gennes-Matricon modes in superconductors, applicable to various regimes and types of vortices, providing a more accurate understanding of impurity impacts.

Contribution

An improved scheme of Gor'kov Green's functions that accurately treats impurity effects on CdGM modes across different superconducting regimes and vortex types.

Findings

The scheme remains valid across various quasiclassical parameters.

Impurity effects depend on impurity strength and regime, affecting CdGM modes.

The method successfully describes impurity impacts in both s-wave and chiral p-wave superconductors.

Abstract

We develop a scheme of Gor'kov Green's functions to treat impurity effects on Caroli-de Gennes-Matricon (CdGM) mode in superconductors (SCs) by improving the Kopnin-Kravtsov scheme with respect to the coherence factors and applicability to various SCs. We can study the impurity effects keeping the discreteness of the energy spectrum in contrast to the quasiclassical theory. We can thus apply this scheme to the SCs with the small quasiclassical parameter $k_{\mathrm{F}}\xi_{0}$ (which is the product of the Fermi wavenumber $k_{\mathrm{F}}$ and the coherence length $\xi_{0}$ in pure SC at zero temperature) and/or superclean regime $\Delta_{\mathrm{mini}} \tau \gg1$ ($\Delta_{\mathrm{mini}}$ and $\tau$ denote, respectively, the level spacing of the CdGM mode called minigap and the relaxation time for CdGM mode and we take $\hbar =1$). We investigate the impurity effects as a white noise for a vortex in an s-wave SC and two types of vortices in a chiral p-wave SC, for various values of the quasiclassical parameters and impurity strengths (from moderately clean regime to superclean regime), and confirm the validity of this scheme.