Emergent Odd-Frequency Superconducting Order Parameter near Boundaries in Unconventional Superconductors

TL;DR

This paper develops a self-consistent theory showing that odd-frequency superconducting order parameters can emerge near boundaries in p-wave superconductors, stabilized by electron-phonon interactions and linked to mid-gap bound states.

Contribution

It introduces a novel self-consistent framework for odd-frequency order parameters in unconventional superconductors, extending quasi-classical theory to include frequency dependence.

Findings

Odd-frequency order parameter is stabilized near boundaries with electron-phonon interactions.

Mid-gap bound states induce the emergent odd-frequency order parameter.

Coexistence of odd- and even-frequency order parameters near boundaries.

Abstract

It was previously suggested that an odd-frequency pair amplitude exists in the vicinity of boundaries in unconventional superconductors. We develop this idea and quest for a novel superconducting order parameter with an odd-frequency dependence. For this purpose, we focus on p-wave superconductors and extend the quasi-classical theory to include the odd-frequency dependence in the order parameter. Both of the frequency and spacial dependences of the order parameter are determined self-consistently. Under a finite electron-phonon interaction, it is found that an odd-frequency order parameter is stabilized near the boundary and coexists with the even-frequency one. By analyzing the induced odd-frequency pair amplitude in terms of the superconducting quasi-particle wavefunction, it is found that the mid-gap bound state generates the emergent odd-frequency order parameter.