Effect of Spin-Orbit Interaction in Spin-Triplet Superconductor: Structure of ${\bf d}$-vector and Anomalous $^{17}$O-NQR Relaxation in Sr$_2$RuO$_4$

TL;DR

This paper investigates how spin-orbit coupling influences the structure of the d-vector and explains anomalous NQR relaxation in Sr₂RuO₄, revealing internal Josephson effects in its spin-triplet superconducting state.

Contribution

It extends Leggett's method to calculate spin-orbit coupling in Sr₂RuO₄, showing its role in aligning the d-vector and inducing internal Josephson effects in the superconductor.

Findings

Spin-orbit coupling aligns the pair angular momentum with the spin in equal-spin pairing.

The d-vector is directed into the ab-plane due to atomic-origin anisotropy.

The model explains the anomalous 17O-NQR relaxation observed in Sr₂RuO₄.

Abstract

Supposing the spin-triplet superconducting state of Sr$_2$RuO$_4$, the spin-orbit (SO) coupling associated with relative motion in Cooper pairs is calculated by extending the method for the dipole-dipole coupling given by Leggett in the superfluid $^{3}$He. It is shown that the SO coupling works only in the equal-spin pairing (ESP) state to make the pair angular momentum $\hbar{\vec L}$ and the pair spin angular momentum ${\rm i}{\vec d}\times{\vec d}^{*}$ parallel with each other. The SO coupling gives rise to the internal Josephson effect in a chiral ESP state as in superfluid A-phase of $^3$He with a help of an additional anisotropy arising from SO coupling of atomic origin which works to direct the {\bf d}-vector into $ab$-plane. This resolves the problem of the anomalous relaxation of $^{17}$O-NQR and the structure of {\bf d}-vector in Sr$_2$RuO$_4$.