Impurity bound states in fully gapped $d$-wave superconductors with subdominant order parameters

TL;DR

This paper investigates impurity-induced bound states in fully gapped $d$-wave superconductors with subdominant order parameters, revealing distinct behaviors for magnetic and potential impurities in $d+is$- and $d+id'$-wave states.

Contribution

It provides a detailed analytical and numerical study of impurity effects in fully gapped $d$-wave superconductors with subdominant order parameters, highlighting their fundamental differences.

Findings

Potential impurities do not produce bound states in $d+is$-wave superconductors.

Magnetic impurities induce bound states with zero-energy crossing at finite scattering in $d+is$-wave.

In $d+id'$-wave states, two pairs of bound states always form, with zero-energy crossing depending on particle-hole asymmetry.

Abstract

Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zero-energy modes and for probing bulk properties of the superconducting state. The high-temperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure $d$-wave superconductor only allows virtual bound states. Fully gapped $d$-wave superconducting states have however been proposed in several cuprate systems thanks to subdominant order parameters producing $d+is$- or $d+id'$-wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped $d$-wave superconductors. Using analytical T-matrix and complementary numerical tight-binding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in $d+is$- and $d+id'$-wave superconductors. In a $d+is$-wave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zero-energy level crossing at a finite scattering strength. On the other hand, a $d+id'$-wave symmetry always give rise to two pairs of bound states and only produce a reachable zero-energy level crossing if the normal state has a strong particle-hole asymmetry.