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

Abstract

Muon spin rotation ($μ$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 $μ$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 $μ$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 $μ$SR as a powerful complementary technique to the established method of nuclear magnetic resonance for probing the spin susceptibility in superconductors.

Figures (3)

• Figure 1: Schematic illustration of the contributions to the muon Knight shift $K_{\mathrm {obs}}$ (thin red loop) and DC magnetic susceptibility measured by SQUID $\chi_{\mathrm {obs}}$ (thin blue loop). After correcting for geometry effects in both SQUID and $\mu$SR measurements, the remaining difference between the magnetic susceptibility $\chi$ (thick blue loop) probed by SQUID and the intrinsic muon Knight shift $K_{\mathrm {\mu}}$ (thick red loop) probed by $\mu$SR corresponds to the spin contact term $K_{\mathrm {spin\text{-}contact}}$, which directly reflects superconducting pairing symmetry.
• Figure 2: Schematic diagrams of (a) FLAME and (b) the tandem sample holder used for $\mu$SR. (a) FLAME consists of a pair of three muon (positron) detectors (left and right) plus forward and backward veto detectors. The muon detectors collect positron signals, and the veto detectors catch muons missing the sample. (b) The upper holder with one Sr$_2$RuO$_4$ crystal on a 25-$\mu$m-thick copper-foil and the lower holder with a reference Ag plate. The upper holder is capped with another copper-foil, shown on the right, ensuring good thermal anchoring. (c-d) The Fourier transforms of the muon time spectra at 2 K (red) and 0.05 K (blue) shown in SM supplement_2025. The line-shape remains symmetric within the fitting range. (c) One-crystal, (d) Six-crystals. (e) Muon Knight shift $K_{\mathrm {obs}}$ (left axis) and observed field shift $\Delta B_{\mathrm {obs}}$ vs. temperature for two distinct sample configurations. A magnetic field of 0.2 T is applied along [100] direction, indicated by green arrows. Red circles: six-crystals (top left); black circles: one-crystal(bottom); scale bars in photos are 1 mm; red and black squares at 2 K: Knight shift of the Ag reference. The anticipated diamagnetic shift appears only for one-crystal, whereas measurements with six-crystals show an increase in $K_{\mathrm {obs}}$ of 1100 ppm ($\approx$0.22 mT) at 0.05 K, attributable to additional stray-fields from neighboring diamagnetic crystals, as shown in the inset. A smaller difference in $K_{\mathrm {obs}}$ above $T_{\mathrm {c}}$ also arises from stray-fields of neighboring paramagnetic crystals.
• Figure 3: (a-b) Observed muon Knight shift $K_{\mathrm {obs}}$ and DC magnetic susceptibility $\chi$ of the same Sr$_2$RuO$_4$ crystal. (a) Temperature-dependence of $K_{\mathrm {obs}}$ in $B_{\mathrm {ext}}=0.4\text{-}0.85$ T along [100], calibrated using the Ag reference. (b) Temperature-dependence of $\chi=M/H$ in $B_{\mathrm {ext}}=0.4\text{-}0.85$ T along [100] measured on warming from 0.5 K to 2 K after field-cooling from 3 K. $B=\mu_0(M+H)$, where $M$ and $H$ are in A/m. (c-d) Derived intrinsic Knight shifts of Sr$_2$RuO$_4$ under various magnetic fields. (c) $K_{\mathrm {\mu}}$ obtained from $K_{\mathrm {obs}}$ by removing the contribution of Lorentz and demagnetizing field contributions. (d) Spin Knight shift represented by $K_{\mathrm {spin\text{-}contact}}$. The vertical scale is inverted in (d), reflecting the negative coupling between dipolar and contact fields of spin susceptibility. For comparison, the NMR Knight shift at 0.6916 T along [100] is also shown ishida_reduction_2020. O(1)$_\parallel$ and O(1)$_\perp$ represent two distinct planer O(1) sites. Excellent agreement is observed between NMR and $\mu$SR results at 0.7 T.