Angular dependence of the upper critical induction of clean $s$- and $d_{x^2-y^2}$-wave superconductors with self-consistent ellipsoidal effective mass and Zeeman anisotropies

TL;DR

This study models the angular dependence of the upper critical magnetic field in anisotropic superconductors with different pairing symmetries, revealing distinct patterns that can help identify the pairing type in experiments.

Contribution

It introduces a self-consistent method incorporating effective mass anisotropy and Zeeman effects to analyze the upper critical field in unconventional superconductors.

Findings

For anisotropic s-wave, $B_{c2}$ aligns with the lowest effective mass direction.

d-wave pairing shows four-fold or two-fold symmetry patterns in $B_{c2}$ depending on anisotropy.

The patterns of $B_{c2}$ can distinguish pairing symmetries in clean superconductors.

Abstract

We employ the Schr{\"o}dinger-Dirac method generalized to an ellipsoidal effective mass anisotropy in order to treat the spin and orbital effective mass anisotropies self consistently, which is important when Pauli-limiting effects on the upper critical field characteristic of singlet superconductivity are present. By employing the Klemm-Clem transformations to map the equations of motion into isotropic form, we then calculate the upper critical magnetic induction $B_{c2}(\theta, \phi, T)$ at arbitrary directions and temperatures $T$ for isotropic $s$-wave and for anisotropic $d_{x^2-y^2}$-wave superconducting order parameters. As for anisotropic $s$-wave superconductors, the reduced upper critical field $b_{c2}$ is largest in the direction of the lowest effective mass, and is proportional to the universal orientation factor $\alpha(\theta,\phi)$. However, for $d_{x^2-y^2}$-wave pairing, ${\bm B}_{c2}(\pi/2,\phi,T)$ exhibits either a four-fold pattern with $C_4$ symmetry just below the transition temperature $T_c$ that rotates by $\pi/4$ as $T$ is lowered, or a two-fold pattern with $C_2$ symmetry, depending upon the planar effective mass anisotropy. This provides a new method to distinguish these pairing symmetries in clean unconventional superconductors.