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

We employ the Schr{ö}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 $B_{c2}(\theta ,\varphi ,T)$ 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 $\alpha (\theta ,\varphi )$. However, for $d_{x^2-y^2}$-wave pairing, ${\mathbf{B}}_{c2}(\pi /2,\varphi ,T)$ 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.

arXiv comment12 pages, 8 figures

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• Persistent Josephson tunneling between Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$ flakes twisted by 45${}^{\circ }$ across the superconducting dome