Effects of surface roughness on the paramagnetic response of small unconventional superconductors

TL;DR

This study investigates how surface roughness affects the magnetic response of small unconventional superconductors, revealing that $d$-wave disks are sensitive to roughness while $p$-wave disks are robust, with implications for experimental detection.

Contribution

It provides a self-consistent theoretical analysis of surface roughness effects on paramagnetic responses in small unconventional superconductors, highlighting the role of orbital symmetry.

Findings

Surface roughness suppresses paramagnetic phase in $d$-wave disks.

$p$-wave disks maintain paramagnetic response despite surface roughness.

Paramagnetic state is more stable than the normal state, suggesting experimental detectability.

Abstract

We theoretically study effects of surface roughness on the magnetic response of small unconventional superconductors by solving the Eilenberger equation for the quassiclassical Green function and the Maxwell equation for the vector potential simultaneously and self-consistently. The paramagnetic phase of spin-singlet $d$-wave superconducting disks is drastically suppressed by the surface roughness, whereas that of spin-triplet $p$-wave disks is robust even in the presence of the roughness. Such difference derives from the orbital symmetry of paramagnetic odd-frequency Cooper pairs appearing at the surface of disks. The orbital part of the paramagnetic pairing correlation is $p$-wave symmetry in the $d$-wave disks, whereas it is $s$-wave symmetry in the $p$-wave ones. Calculating the free-energy, we also confirm that the paramagnetic state is more stable than the normal state, which indicates a possibility of detecting the paramagnetic effect in experiments. Indeed our results are consistent with an experimental finding on high-$T_c$ thin films.