Anomalous angular dependence of the upper critical induction of orthorhombic ferromagnetic superconductors with completely broken p-wave symmetry

Abstract

We employ the Klemm-Clem transformations to map the equations of motion for the Green functions of a clean superconductor with a general ellipsoidal Fermi surface (FS) characterized by the effective masses m1,m2, and m3 in the presence of an arbitrarily directed magnetic induction B=B(sinθcosϕ,sinθsinϕ,cosθ) onto those of a spherical FS. We then obtain the transformed gap equation for a transformed pairing interaction Ṽ(k̃̂,k̃̂′) appropriate for any orbital order parameter symmetry. We use these results to calculate the upper critical induction Bc2(θ,ϕ) for an orthorhombic ferromagnetic superconductor with transition temperatures TCurie>Tc. We assume the FS is split by strong spin-orbit coupling, with a single parallel-spin (↑↑) pairing interaction of the p-wave polar state form locked onto the ê3 crystal axis normal to the spontaneous magnetization M0⊥ê3 due to the ferromagnetism. The orbital harmonic oscillator eigenvalues are modified according to B→Bα, where α(θ,ϕ)=m3/mcos2θ+γ−2(ϕ)sin2θ, γ2(ϕ)=m3/(m1cos2ϕ+m2sin2ϕ) and m=(m1m2m3)1/3. At fixed ϕ, the order parameter anisotropy causes Bc2 to exhibit a novel θ dependence, which for γ2(ϕ)>3 becomes a double peak at 0∘<θ*<90∘ and at 180∘−θ*, providing a sensitive bulk test of the order parameter orbital symmetry in both phases of URhGe and in similar compounds still to be discovered.

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