Magnetic response of energy levels of superconducting nanoparticles with spin-orbit scattering

TL;DR

This paper investigates how energy levels in ultrasmall superconducting grains respond to magnetic fields, focusing on the effects of spin-orbit scattering and pairing correlations, and proposes level curvature as a probe for pairing.

Contribution

It introduces the use of level curvature distribution as a novel method to detect pairing correlations in superconducting nanoparticles.

Findings

g-factor distribution unaffected by pairing

Level curvature distribution sensitive to pairing

Level curvature can probe pairing correlations

Abstract

Discrete energy levels of ultrasmall metallic grains are extracted in single-electron-tunneling-spectroscopy experiments. We study the response of these energy levels to an external magnetic field in the presence of both spin-orbit scattering and pairing correlations. In particular, we investigate $g$-factors and level curvatures that parametrize, respectively, the linear and quadratic terms in the magnetic-field dependence of the many-particle energy levels of the grain. Both of these quantities exhibit level-to-level fluctuations in the presence of spin-orbit scattering. We show that the distribution of $g$-factors is not affected by the pairing interaction and that the distribution of level curvatures is sensitive to pairing correlations even in the smallest grains in which the pairing gap is smaller than the mean single-particle level spacing. We propose the level curvature in a magnetic field as a tool to probe pairing correlations in tunneling spectroscopy experiments.