Anomalous angular dependence of the upper critical induction of orthorhombic ferromagnetic superconductors with completely brokenp-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 $m_1, m_2$, and $m_3$ in the presence of an arbitrarily directed magnetic induction ${\bm B}=B(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ 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 $B_{c2}(\theta,\phi)$ 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 $B\rightarrow B\alpha$, where $\alpha(\theta,\phi)=\sqrt{m_3/m}\sqrt{\cos^2\theta+\gamma^{-2}(\phi)\sin^2\theta}$, $\gamma^2(\phi)=m_3/(m_1\cos^2\phi+m_2\sin^2\phi)$ and $m=(m_1m_2m_3)^{1/3}$. At fixed $\phi$, the order parameter anisotropy causes $B_{c2}$ to exhibit a novel $\theta$-dependence, which for $\gamma^2(\phi)>3$ becomes a double peak at $0^{\circ}<\theta^{*}<90^{\circ}$ and at $180^{\circ}-\theta^{*}$, providing a sensitive bulk test of the order parameter orbital symmetry in both phases of URhGe and in similar compounds still to be discovered.