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

Abstract

Abstract We employ the Schrödinger-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 c 2 ( θ , ϕ , 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, B c 2 is largest in the direction of the lowest effective mass, and is proportional to the universal orientation factor α ( θ , ϕ ) . However, for d x 2 − y 2 -wave pairing and vanishing planar effective mass anisotropy, B c 2 ( π / 2 , ϕ , T ) exhibits a four-fold azimuthal pattern with C 4 symmetry the maxima of which are along the crystal axes just below the transition temperature T c , but these maxima are rotated by π / 4 about the z axis as T is lowered to 0. However for d x 2 − y 2 -wave pairing with weak planar effective mass anisotropy, B c 2 ( π / 2 , ϕ , T ) exhibits a two-fold pattern with C 2 symmetry for all T   ⩽   T c , which also rotates by π / 4 about the z axis as T is lowered to 0. These low planar effective mass anisotropy cases provide a new method to distinguish s-wave and d x 2 − y 2 -wave pairing symmetries in clean unconventional superconductors.