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

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 $m_1, m_2$, and $m_3$ in the presence of an arbitrarily directed magnetic induction ${\bm B}=B(\sinθ\cosϕ,\sinθ\sinϕ,\cosθ)$ onto those of a spherical FS. We then obtain the transformed gap equation for a transformed pairing interaction $\tilde{V}(\hat{\tilde{\bm k}},\hat{\tilde{\bm k}}')$ appropriate for any orbital order parameter symmetry. We use these results to calculate the upper critical induction $B_{c2}(θ,ϕ)$ for an orthorhombic ferromagnetic superconductor with transition temperatures $T_{\rm Curie}>T_c$. We assume the FS is split by strong spin-orbit coupling, with a single parallel-spin ($\uparrow\uparrow$) pairing interaction of the \textit{p}-wave polar state form locked onto the $\hat{\bm e}_3$ crystal axis normal to the spontaneous magnetization ${\bm M}_0\perp\hat{\bm e}_3$ due to the ferromagnetism. The orbital harmonic oscillator eigenvalues are modified according to $B\rightarrow Bα$, where $α(θ,ϕ)=\sqrt{m_3/m}\sqrt{\cos^2θ+γ^{-2}(ϕ)\sin^2θ}$, $γ^2(ϕ)=m_3/(m_1\cos^2ϕ+m_2\sin^2ϕ)$ and $m=(m_1m_2m_3)^{1/3}$. At fixed $ϕ$, the order parameter anisotropy causes $B_{c2}$ to exhibit a novel $θ$-dependence, which for $γ^2(ϕ)>3$ becomes a double peak at $0^{\circ}<θ^{*}<90^{\circ}$ and at $180^{\circ}-θ^{*}$, 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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