Coexistence of itinerant ferromagnetism and a nonunitary superconducting state with line nodes: Possible application to ${\text{UGe}}_2$

We construct a mean-field theory for itinerant ferromagnetism coexisting with a nonunitary superconducting state, where only the majority-spin band is gapped and contains line nodes, while the minority-spin band is gapless at the Fermi level. Our study is motivated by recent experimental results, which indicate that this may be the physical situation realized in the heavy-fermion compound ${\text{UGe}}_2$. We investigate the stability of the mean-field solution of the magnetic and superconducting order parameters. Also, we provide theoretical predictions for experimentally measurable properties of such a nonunitary superconductor: the specific heat capacity, the Knight shift, and the tunneling conductance spectra. Our study should be useful for direct comparison with experimental results and also for further predictions of the physics that may be expected in ferromagnetic superconductors.

Figure 1

(Color online) The gap dependence on the ferromagnetic exchange interaction parameter $\tilde{I}=IN\left(0\right)$. The gap remains constant for $\tilde{I}\in \left[0,1\right]$, which correspond to a unitary phase. The gap ${\Delta }_0$ then starts growing with increasing $\tilde{I}$ for $\tilde{I}>1.0$, announcing the onset of a spontaneous magnetization. The analytical formula is based on Eq. (18).

Figure 2

(Color online) Self-consistently obtained solution for the superconducting gap ${\Delta }_0$ (red symbols) compared to the analytical expression Eq. (23) with $\gamma =1.70$ (blue lines), modeling a BCS-type temperature dependence.

Figure 3

(Color online) The phase diagram of our model in the $T\text{−}\tilde{I}$ plane. For $\tilde{I}>1.0$, a spontaneous magnetization arises and allows for the possible uniform coexistence of ferromagnetism and triplet superconductivity. Note that decreasing $\tilde{I}$ (going from left to right along the $x$ axis) corresponds to an increasing external pressure $p$. The abbrevations stand for nonunitary (NU), unitary (U), ferromagnetic (FM), and paramagnetic (PM).

Figure 4

(Color online) The left panel shows a plot of the specific heat capacity, using self-consistently obtained order parameters for three different values of $\tilde{I}$. The right panel shows relative jump (superconducting vs normal state) of the specific heat at the transition temperature as a function of the normalized exchange splitting between the spin bands, $\tilde{M}$. Numerically calculated values are shown in red; analytical results [Eq. (25)] using $\gamma \approx 1.70$ are shown in blue.

Figure 5

(Color online) Knight shift for a ferromagnetic superconductor, using self-consistently obtained order parameters, for three different values of $\tilde{I}$. Here, the field is applied $\mathbf{H}\parallel {\mathbf{d}}_{\mathbf{k}}$.

Figure 6

(Color online) Plot of the tunneling conductance of a normal/ferromagnetic superconductor junction for $\alpha =0$ and $\alpha =\pi /2$, using self-consistently obtained solutions at $T=0$. In the left column, we fix the tunneling barrier strength $Z=2{\text{mV}}_0/k_F=3$ and plot the conductance for several values of the Stoner interaction $\tilde{I}$. In the right column, we fix $\tilde{I}=1.01$ and plot the conductance for several values of $Z$.

Figure 7

(Color online) Illustration of the different symmetry states considered here and a qualitative sketch of the weighting factor ${\sigma }_N\text{cos}{\theta }_N$.

Figure 8

(Color online) Plot of the normalized conductance $G/G_0$ and bulk DOS ${\rho }_0$ for three different symmetries of the superconducting state in the tunneling limit. Only in the $p_y$-wave case is there a difference between these two quantities.