Angular dependence of the upper critical induction of clean s- and dx2- y2-wave superconductors with self-consistent ellipsoidal effective mass and Zeeman anisotropies

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 inductionBc2(θ,ϕ,T)at arbitrary directions and temperaturesTfor isotropics-wave and for anisotropicdx2-y2-wave superconducting order parameters. As for anisotropics-wave superconductors,Bc2is largest in the direction of the lowest effective mass, and is proportional to the universal orientation factorα(θ,ϕ). However, fordx2-y2-wave pairing and vanishing planar effective mass anisotropy,Bc2(π/2,ϕ,T)exhibits a four-fold azimuthal pattern withC₄symmetry the maxima of which are along the crystal axes just below the transition temperatureT_c, but these maxima are rotated byπ/4about thezaxis asTis lowered to 0. However fordx2-y2-wave pairing with weak planar effective mass anisotropy,Bc2(π/2,ϕ,T)exhibits a two-fold pattern withC₂symmetry for allT ⩽ Tc, which also rotates byπ/4about thezaxis asTis lowered to 0. These low planar effective mass anisotropy cases provide a new method to distinguishs-wave anddx2-y2-wave pairing symmetries in clean unconventional superconductors.

Keywords: Zeeman energy; angle dependence; anisotropic Fermi Surface; the upper critical field.