Free Energy and Magnetic Penetration Depth of a $d$-Wave Superconductor in the Meissner State

TL;DR

This paper analyzes how thermal, nonlocal, and nonlinear effects influence the free energy and magnetic penetration depth in a d-wave superconductor under weak magnetic fields, providing insights into experimental observations.

Contribution

It introduces a comprehensive calculation of free energy and penetration depth considering thermal, nonlocal, and nonlinear effects in d-wave superconductors.

Findings

Nonlocal or nonlinear effects cut off linear-T dependence of free energy and penetration depth.

Nonlocal effects modify the nonlinear Meissner effect at very low temperatures.

Predictions align with experimental data on cuprate superconductors.

Abstract

We investigate the free energy and the penetration depth of a quasi-two-dimensional d-wave superconductor in the presence of a weak magnetic field by taking account of thermal, nonlocal and nonlinear effects. In an approximation in which the superfluid velocity $v_s$ is assumed to be slowly varying, the free energy is calculated and compared with available results in several limiting cases. It is shown that either nonlocal or nonlinear effects may cut off the linear-$T$ dependence of both the free energy and the penetration depth in all the experimental geometries. At extremely low $T$, the nonlocal effects will also generically modify the linear $H$ dependence of the penetration depth ("nonlinear Meissner effect") in most experimental geometries, but for supercurrents oriented along the nodal directions, the effect may be recovered. We compare our predictions with existing experiments on the cuprate superconductors.