Double ellipsoidal Fermi surface model of the normal state of ferromagnetic superconductors

We model the normal state of ferromagnetic superconductors with two general ellipsoidal Fermi surfaces (FSs), one for each spin projection $\sigma =\{\uparrow ,\downarrow \}$, each with its ferromagnetically split chemical potential ${\mu }_{\sigma }$ and its three distinct single particle effective masses, $\{m_{i\sigma }\}$, the geometric mean of which is $m_{\sigma }$. We study this model in the presence of an arbitrarily oriented magnetic induction, $\mathbf{B}={\mu }_0\mathbf{H}+{\mathbf{M}}_{\mathbf{0}}$, where ${\mathbf{M}}_{\mathbf{0}}$ includes the Ising-like spontaneous ferromagnetic order, which for URhGe is in the $c$-axis direction above the superconducting transition temperature $T_c$. We assume the low-$T$ total particle density $\Sigma_{\sigma} n_{\sigma}({\bf B})$tobeindependentof${\bf B}$, and obtain a self-consistent asymptotic expansion for ${\sum }_{\sigma }{\mathrm{\Pi }}_{\sigma }^{3\mathrm{/}2}(\mathbf{B})$ in even powers of $\mathbf{B}$, where $\Pi_{\sigma}({\bf B})=m_{\sigma}({\bf B})\mu_{\sigma}({\bf B})$\mathrm{.}Weassumethatthe$\mu_{\sigma}({\bf B})$arelinearin${\bf B}$ for both spins due to the Zeeman interaction and that the remaining even $\mathbf{B}$ dependence in the ${\mathrm{\Pi }}_{\sigma }(\mathbf{B})$ arises only from $m_{\downarrow}({\bf B})$\mathrm{.}OuranalogousexpressionfortheSommerfeldconstant$\gamma({\bf B})$leadstogoodfitstothe$\gamma({\bf H})$ data of Aoki and Flouquet [J. Phys. Soc. Jpn. $\mathbf{\text{81}}$, 011003 (2012)] obtained for the ferromagnetic superconductor URhGe in the ferromagnetic, non-superconducting phase, with the applied magnetic field $\mathbf{H}$ along each of the three crystallographic directions. We discuss this model in terms of the reentrant superconducting properties of URhGe and UCoGe. This model can be generalized to an arbitrary number of ellipsoidal FSs.