Ferromagnetism and superconductivity: coexistence and pairing evidence

Scope

Intrinsic ferromagnetism and superconductivity coexist in the uranium compounds UGe2, URhGe and UCoGe under specific conditions. Establishing two orders in one material, establishing their spatial overlap, and determining the pairing wavefunction are separate tasks. This page concerns bulk materials; superconducting correlations induced across an engineered ferromagnet interface are a different problem. Pressure, composition, field direction and the experimental transition criterion belong with each result. R3 R4 R6 R7

Ferromagnetic fluctuations can provide an equal-spin pairing interaction in a model. Fay and Appel's longitudinal-fluctuation calculation predicts a nonmonotonic superconducting transition as the ferromagnetic boundary is approached. It does not imply that increasing magnetization always improves superconductivity. Static ordered moment, dynamic magnetic susceptibility and the electronic exchange splitting are different quantities. Their abstract supplies model motivation, not a measured pairing interaction for every uranium compound. R1

Mathematical assumptions

Consider an inversion-symmetric spin model with normal Hamiltonian $H_0=\xi_{\mathbf k}I-h\sigma_z$. Here $h$ is an energy, not a magnetic field in tesla. With an imposed singlet gap $\Delta_0$, the four Bogoliubov energies are $\pm\sqrt{\xi_{\mathbf k}^2+|\Delta_0|^2}\pm h$. The gap held fixed closes spectrally at $|h|=|\Delta_0|$ when $\xi_{\mathbf k}=0$. This spectral condition does not locate the thermodynamic transition of a self-consistent superconductor. R2

At zero temperature, a separate weak-coupling comparison with constant single-spin density of states $N(0)$ gives $F_s-F_n=N(0)(-\Delta_0^2/2+h^2)$. Its energy crossing occurs at $|h|=\Delta_0/\sqrt2$, before that fixed-gap spectral closure. This reference assumes a homogeneous singlet state and a spin-only field. Orbital effects, finite-momentum pairing, strong coupling and spin-orbit coupling are outside it. The original Clogston paper's identity is verified; this explicit energy example is independently checked. R2

For diagonal equal-spin gaps, the positive branches instead read $E_\uparrow=\sqrt{(\xi-h)^2+|\Delta_\uparrow|^2}$ and $E_\downarrow=\sqrt{(\xi+h)^2+|\Delta_\downarrow|^2}$. The imposed gaps pair states within their respective spin bands. Their spectra do not establish immunity to magnetic fields: the band populations, interaction, orbital motion and gap amplitudes can change. All 150 singlet and equal-spin matrix examples satisfy Hermiticity, the analytic spectra and particle-hole symmetry between opposite momenta. The triplet momentum factor is odd. R1 R10

Write $\Delta=(\mathbf d\cdot\boldsymbol\sigma)i\sigma_y$. Then $\Delta\Delta^\dagger=|\mathbf d|^2I+\mathbf q\cdot\boldsymbol\sigma$, with $\mathbf q=i\mathbf d\times\mathbf d^*$. A unitary gap has $\mathbf q=0$. Equal-spin pairing can be unitary: $\mathbf d=(0,-i,0)$ gives $\Delta=I$ at the selected momentum. By contrast, $\mathbf d=(-1/2,-i/2,0)$ gives $\Delta=\mathrm{diag}(1,0)$, a nonunitary example with one ungapped spin band. One hundred complex-vector checks verify the identity and nonnegative squared gaps. This spin-basis illustration is not a determination of total sample magnetization or a realistic spin-orbit-coupled band structure. R10

Experimental criteria

Saxena and colleagues' UGe2 discovery uses resistivity and ac susceptibility under pressure. The published phase-diagram caption defines the superconducting temperature by a 50-percent resistivity drop; it scales that axis by ten for display. Its susceptibility caption uses approximately 1.3 GPa for the main magnetic curve and 1.5 GPa for the superconducting inset. These are distinct conditions. The paper reports magnetic survival below the superconducting transition, but defers the detailed neutron analysis. Selected methods and captions are checked here; the original plots are not visually audited. R3

Aoki and colleagues' URhGe discovery adds a thermodynamic superconducting signature at ambient pressure. Its abstract explicitly leaves the pairing symmetry and dominant mechanism unidentified. Levy and colleagues subsequently report a second superconducting field pocket around a spin reorientation near 12 T for a field along the crystal b axis, above the field that suppresses the first pocket. This ties superconductivity to magnetic evolution without uniquely identifying a gap symmetry. The complete discovery curves and reentrant phase boundaries remain outside this selected review. R4 R5

For Huy and colleagues' UCoGe sample 3, the resistive onset is 0.82 K, while heat capacity and thermal expansion begin their broad anomalies near 0.66 K. An idealized equal-entropy construction places a sharp transition near 0.45 K. Those values represent different criteria, not three competing measurements of one exact universal constant. The approximately 3 K Curie temperature is magnetic and must not populate a superconducting-temperature filter. The thermodynamic anomalies support bulk superconductivity; the suggested triplet mechanism remains an inference. R6

Ohta and colleagues' separate polycrystalline UCoGe sample has a susceptibility onset near 0.7 K and midpoint near 0.55 K. At 95 mK, similar nuclear recovery at peaks with and without internal magnetic fields supports superconducting response in both regions. The slow relaxation component accounts for about 30 percent of the measured recovery, not a demonstrated uniform superconducting response at every site. Pulse-condition sensitivity and model assumptions matter. A self-induced vortex state is proposed to explain the components; this is not a direct vortex image. R7

Limits and disputes

UTe2 should not be inserted into the established ambient-pressure ferromagnetic-coexistence list simply because its discovery paper invokes nearly ferromagnetic fluctuations. That paper provides superconductivity and unconventional-pairing clues, without establishing static long-range ferromagnetism. Later neutron work reports antiferromagnetic fluctuations and a resonance; its absence of detected ferromagnetic scattering is restricted to the measured window. Neither result by itself fixes the full pairing interaction. R8 R9

Complete scientific acceptance remains open. Selected source passages, local-probe figures and independent matrix checks support the discussion. Detailed magnetic and superconducting volume calibration, raw-data analysis, complete version comparisons and material-specific spin-orbit models remain unresolved. A large critical field, a small net moment or coexistence alone cannot replace a consistent set of spin, thermodynamic and phase-sensitive constraints. R4 R6 R7 R10

Discovery implications

Keep magnetic ordering temperature separate from superconducting onset, midpoint and bulk-anomaly criteria. Preserve specimen identity, field orientation and pressure for every observation. Test whether a pairing model accounts jointly for the measured magnetic response and superconducting evidence; do not turn one suggestive signature into a unique pairing assignment. R1 R3 R6 R7

Related topics

Ferromagnetic fluctuations, singlet and triplet pairing, heavy-fermion superconductors, and antiferromagnetism and superconductivity.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.