Spin–orbit coupling and superconductivity

Scope

Spin–orbit coupling links electronic motion to spin and changes the normal-state band structure and magnetic response. It does not, by itself, specify the attractive interaction responsible for superconductivity. In a nonmagnetic system without inversion symmetry, antisymmetric spin–orbit coupling can split bands and allow pairing components that inversion would separate. Permission by symmetry is not a measurement of their relative weights. Gor'kov and Rashba's two-dimensional model explicitly leaves the pairing mechanism unspecified and assumes its spin–orbit splitting remains small compared with the Fermi energy. R1 R2

In an ideal Rashba model, the spin texture lies in the plane and winds perpendicular to momentum. In the Ising description used for monolayer NbSe2, opposite valleys have predominantly opposite out-of-plane spin orientations. These are different textures and give different directional responses. A spin–orbit energy expressed as an equivalent field is not an applied laboratory field. Strong spin–orbit coupling and a large upper critical field do not independently certify triplet pairing or topological superconductivity. R1 R6 R8

Mathematical assumptions

For a two-spin effective model, write $h(\mathbf k)=\xi(\mathbf k)I+\mathbf g(\mathbf k)\cdot\boldsymbol\sigma$, with even $\xi$ and odd $\mathbf g$ under momentum reversal. Its normal energies are $\xi\pm|\mathbf g|$. An even-frequency gap may be written $\widehat\Delta=[\psi+\mathbf d\cdot\boldsymbol\sigma]i\sigma_y$, with even singlet component $\psi$ and odd triplet vector $\mathbf d$. The chosen spin or pseudospin basis matters; this two-component model is not a complete multiorbital classification. Neither the normal splitting nor the algebraic gap decomposition determines a self-consistent interaction. R1 R2

An independent calculation imposes a real, purely singlet gap $\widehat\Delta=\psi i\sigma_y$. Its Bogoliubov excitation energies are $E_\pm=\sqrt{(\xi\pm g)^2+\psi^2}$, where $g=|\mathbf g|$. For a nonzero Matsubara frequency, define $f_\pm=-\psi/(\omega^2+E_\pm^2)$. The anomalous propagator decomposes as $F=[f_s+\mathbf f_t\cdot\boldsymbol\sigma]i\sigma_y$, with $f_s=(f_++f_-)/2$ and $\mathbf f_t=(f_+-f_-)\hat{\mathbf g}/2$. Thus a momentum-resolved triplet pair amplitude can be nonzero although the imposed gap has no triplet component. These quantities should not both be labeled an experimental “triplet fraction.” R1

One hundred independently assembled matrices reproduce the spectra and fermionic relation $F(\mathbf k,i\omega)=-F^T(-\mathbf k,-i\omega)$; one hundred decompositions match the analytic amplitudes. Three controls check cancellation at zero spin–orbit coupling or at $\xi=0$, and a nonzero amplitude at generic parameters. The result distinguishes a propagator from a gap interaction. It does not compute a momentum-integrated mixing ratio, susceptibility, transition temperature or topological invariant. R1

Frigeri and colleagues' weak-coupling treatment neglects small particle-hole-asymmetry mixing terms and studies pairing channels separately. Alignment of $\mathbf d$ with $\mathbf g$ avoids a leading suppression term; it does not select the stable material state. Their erratum adds an omitted linear contribution to the MnSi spin–orbit vector and rescales the original Figure 2 parameter by dividing by $\sqrt{3/2}$. Claims about which p-wave component is suppressed therefore require the corrected vector and its relative coefficients. R2 R3

Experimental criteria

Nishiyama and colleagues report a nearly unchanged Knight shift and approximately cubic temperature dependence of the relaxation rate in Li2Pt3B, interpreted as evidence for triplet pairing with nodes. Their crushed polycrystalline sample has a zero-field susceptibility transition at 2.68 K. For boron NMR, 0.26 T gives a transition near 2.1 K, while 0.35 T gives approximately 1.8 K. The 0.1 K spectrum is a measurement temperature, not a transition. The Li2Pd3B comparison uses a different field, 1.46 T, and the Knight-shift interpretation includes an orbital contribution and a fitted gap model. R4

Häfliger and colleagues instead find low-temperature saturation in μSR-derived penetration depth and activated specific heat in Li2Pt3B, favoring a fully gapped interpretation. Their final text nevertheless notes that a simple single-gap fit to the penetration-depth data is unsatisfactory and that triplet states cannot be excluded. The low-field temperature scans extend to 30 mK, while the field-dependent comparison is made at 1 K; calorimetry reaches 0.1 K and uses a 4 T normal-state reference. These different probes and protocols must remain separately identifiable. R5

Ising-protected superconductivity provides another route to an enhanced field scale. In the NbSe2 study, the approximately 35 T scale is fitted, whereas contact-conductance data extend to 31.5 T. In the TaS2 study, 34.5 T is an experimental ceiling and 65.6 T is a fit parameter. Orbital geometry, alignment, gap scale and the resistance criterion accompany any comparison with a weak-coupling Pauli estimate. A larger ratio alone does not determine the spin structure of the gap. R6 R7

Limits and disputes

The Li2Pt3B disagreement cannot be settled by labeling one result “bulk” and another “surface”: NMR also probes bulk material. The μSR/calorimetry paper discusses sample inhomogeneity and metallic inclusions, but those possibilities are not independently demonstrated explanations for every contrasting measurement. Retain both experimental interpretations and their assumptions. Absence of a coherence peak is not, by itself, a unique nodal or triplet diagnostic. Complete scientific acceptance remains open, including raw-data comparisons, final-version checks and the wider subsequent literature. R4 R5

A topological assignment requires a specified superconducting Hamiltonian, symmetry class and appropriate invariant, together with evidence that the physical system realizes the assumed state. Normal-state spin splitting, absence of inversion and a superconducting transition do not supply those steps. Likewise, a crystal's effective spin–orbit texture cannot be inferred quantitatively from a simple atomic-number scaling rule. Use a material-specific band model and experimentally constrained parameters before proposing a pairing mechanism or device advantage. R1 R8

Discovery implications

Separate normal-state texture, superconducting gap, pair correlations and measured magnetic response in both language and data records. Compare field directions and sample conditions; retain corrections to theoretical papers and competing experimental interpretations. Useful spin–orbit engineering should predict an observable change under controlled gating, thickness or symmetry changes, then test it against alternatives such as orbital depairing and multiband response. An adjustable model parameter is a research hypothesis until the same model explains independent observations. R2 R3 R6 R7

Related topics

Singlet and triplet pairing, Pauli limit, and Majorana zero modes.

Material cases

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