Muon Knight shift as a precise probe of the superconducting symmetry of Sr$_2$RuO$_4$

Abstract

Muon spin rotation ($\mu$SR) measurements of internal magnetic field shifts, known as the muon Knight shift, is used for determining pairing symmetries in superconductors. While this technique has been especially effective for $f$-electron-based heavy-fermion superconductors, it remains challenging in $d$-electron-based superconductors such as Sr$_2$RuO$_4$, where the Knight shift is intrinsically small. Here, we report high-precision muon Knight shift measurements of superconducting Sr$_2$RuO$_4$. We observe that using multiple pieces of crystals, a common practice in $\mu$SR measurements, induces a substantial paramagnetic shift below the superconducting transition temperature, $T_c$, when a weak magnetic field is applied. We attribute such an unresolved paramagnetic shift to stray fields generated by neighboring diamagnetic crystals. To avoid this, one piece of crystal was used in this study. We experimentally determine the muon Knight shift of Sr$_2$RuO$_4$ in the normal state to be -116$\pm$7 ppm. By combining the observed muon Knight shift with independently determined bulk magnetization data from the same crystal used in $\mu$SR and carefully separating various contributions to the shift, we confirm a significant reduction in the spin Knight shift below $T_c$, consistent with spin-singlet-like pairing. This result constitutes the precise muon Knight shift measurement in a $d$-electron-based superconductor. Our results highlight the potential of $\mu$SR as a powerful complementary technique to the established method of nuclear magnetic resonance for probing the spin susceptibility in superconductors.