Review of U-based ferromagnetic superconductors: Comparison between UGE₂, URHGE, and UCOGE

The discovery in 2000 that the ferromagnetic (FM) compound UGe₂ (T_Curie = 52 K at ambient pressure) becomes superconducting under a pressure of P = 1.1 GPa until it enters the paramagnetic (PM) phase above P_c = 1.6 GPa was a surprise, despite the fact that such a possibility was emphasized in theory four decades ago. Successive searches for new materials (URhGe and UCoGe) led to the discovery of the coexistence of superconductivity (SC) and ferromagnetism at ambient pressure. Furthermore in UCoGe, it was found that SC survives in the PM regime from P_c = 1.1 to 4 GPa. The novelty is that SC also emerges deep inside the FM regime but with strong FM fluctuations. Focus has been on low-temperature experiments under extreme conditions of magnetic field (H), pressure, and uniaxial stress. NQR and NMR experiments are unique tools to understand the interplay between the spin dynamics and the Cooper pairing. We choose to present the SC properties from the knowledge of quasiparticle dressing in the normal phase (renormalized band mass, m_B plus the extra dressing originating from FM fluctuations, m⁺⁺). In UGe₂, strong interplay exists between Fermi surface (FS) reconstructions in the cascade of different FM and PM ground states and their magnetic fluctuations. Similar phenomena occur in URhGe and UCoGe but, at first glance, the SC seems to be driven by the FM fluctuations. The weakness of the FM interaction in these two compounds gives the opportunity to observe singular features in magnetic field scans depending on their field orientation with respect to the FM sublattice magnetization (M₀). We will show that for UCoGe, which has the smallest ordered moment, a longitudinal field scan (H ∥ M₀) leads to a drastic decrease in the FM fluctuations with direct consequences on SC properties such as the upward curvature of the upper critical field. A transverse field scan (H M₀) leads to suppression of the Curie temperature, T_Curie; the consequence is a boost in FM fluctuations, which leads to a reinforcement of SC. Contrary to the two examples of Ising FM UGe₂ and UCoGe, the singularity in URhGe is the weakness of the magnetocrystalline term between the choice of ferromagnetism along the c- or b-axis; the most noteworthy feature is the detection of reentrant SC on each side of the H switch at H_R from the c easy axis of magnetization to the b-axis. All the experimental results give evidence that the SC in these three materials originates from the FM fluctuations, which are amplitude modes of magnetic excitations in the FM state. Spin-triplet pairing has been anticipated in the FM superconductors and was actually observed by Knight-shift measurements in the SC state of UCoGe. Their fascinating ð P; T; H Þ phase diagrams are now well established. Of course, a new generation of experiments will elucidate subtle effects by obtaining their SC order parameters. While the FSs of UGe₂ have been experimentally well determined, those of URhGe and UCoGe have been poorly determined, and thus a quantitative comparison with band structure calculations cannot be achieved. Up to now, Angle-resolved photoemission spectroscopy (ARPES) measurements have only given the flavor of the electronic bands at the Fermi level. Discussion is presented on how different theoretical approaches can describe the various phenomena discovered by experimentalists. Following the new hot subject of topological superconductors, proposals have been made for UCoGe, which is a great challenge for ambitious researchers!.