Hc2and the κ2-Parameter of a Superconductor with Cubic Symmetry

Abstract

The upper critical field Hc2 and the K 2-parameter of a pure superconductor with the cubic crystal symmetry are calculated near the transition temperature. The result obtained for I-Ic 2 is in good agreement with the experimental result of Farrell et al. It is found that the direc­ tion of the induced magnetic field is not parallel to that of the external magnetic field applied in an arbitrary direction. Therefore, the t.:: 2-parameter becomes a tensorial quantity. The ratio MxiMz is found to be about 0.015 at the reduced temperature t=0.7, where iY.lx and Mz are respectively magnetizations perpendicular and parallel to the applied external field. § l. Introduc11:im:Ji Reed et al. have experimentally shown that the upper critical field Hc2 has an anisotropic nature in several cubic type II superconductors and that the re­ lative anisotropy of Hc2 decreases as the reduced temperature t increases, and approaches zero as t-> 1. 1 ) They have attributed this nature to the anisotropy of the Fermi surface and to the non-locality of the Ginzburg-Landau equation. Hohenberg and W erthamer derived expressions for Hc2 in the nearly local limit and also in the limit of small anisotropy. 2 ) They have successfully shown that the anisotropy of Hc2 comes from the non-local correction to the original G-L equation as well as from the cubic symmetry of the Fermi surface. Farrell et al. have recently made precise measurements of the upper critical field of niobium crystal and analyzed their result assuming a four-term Kubic harmonic expression. 3 ) The calculation of Hohenberg and W erthamer in the nearly local limit contains only three-term Kubic harmonics. In this paper, we derive a theoretical expression for 1-Ic 2 involving four-term Kubic harmonics, extending the work of Hohcnberg and Werthamer to a higher order term in the non-locality. Further, we investigate a possible coupled effect from the non-locality and the anisotropic nature of the cubic crystal on the tc 2parameter. We restrict our attention to the study of 1-Ic 2 and tc 2 for a pure superconductor only near the transition temperature region (G-L region). In § 2, we will give the basic G-L equations for our system, and an expres­ sion for 1-Ic2 will be derived in § 3. In § 4 is contained the discussion of the effect of anisotropy on the induced magnetic field and also of the tc 2-parameter.