Origins of spontaneous magnetic fields in Sr$_2$RuO$_4$

Abstract

The nature of the broken time reversal symmetry (BTRS) state in Sr$_2$RuO$_4$ remains elusive, and its relation to superconductivity remains controversial. There are various universal predictions for the BTRS state when it is associated with a multicomponent superconducting order parameter. In particular, in the BTRS superconducting state, spontaneous fields appear around crystalline defects, impurities, superconducting domain walls and sample surfaces. However, this phenomenon has not yet been experimentally demonstrated for any BTRS superconductor. Here, we aimed to verify these predictions for Sr$_2$RuO$_4$ by performing muon spin relaxation ($μ$SR) measurements on Sr$_{2-y}$La$_{y}$RuO$_4$ single crystals at ambient pressure and stoichiometric Sr$_2$RuO$_4$ under hydrostatic pressure. The study allowed us to conclude that spontaneous fields in the BTRS superconducting state of Sr$_2$RuO$_4$ appear around non-magnetic inhomogeneities and, at the same time, decrease with the suppression of $T_{\rm c}$. The observed behaviour is consistent with the prediction for multicomponent BTRS superconductivity in Sr$_2$RuO$_4$. The results of the work are relevant to understanding BTRS superconductivity in general, as they demonstrate, for the first time, the relationship among the superconducting order parameter, the BTRS transition, and crystal-structure inhomogeneities.

Figures (8)

• Figure 1: Zero-field $\mu$SR data for Sr$_{2-y}$La$_{y}$RuO$_4$ single crystals. (a) Time evolution of the muon spin asymmetry for $y=0.01$ in the normal (3 K) and superconducting (0.05 K) states. (b) Extracted ZF muon spin relaxation rate $\lambda$ vs. temperature for $y=0.01$. The solid curve represents the best fit to Eq. (\ref{['eq:lambda_T_fit']}) with the following parameters: $\lambda_0 = 0.0028(4)~\mu\mathrm{s}^{-1}$, $\Delta\lambda = 0.0110(6)~\mu\mathrm{s}^{-1}$, $n = 3.7(13)$, and $T_{\rm BTRS} = 1.08(5)$ K. The bulk superconducting transition temperature determined from the specific heat measurements is $T_c = 1.13(2)$ K and is consistent with the TF data shown in Fig. \ref{['fig:appendix_A1']}. The relaxation rate within the experimental errors is enhanced at $T_{\rm c}$. (c) Time evolution of the muon spin asymmetry for $y=0.04$ at 1.5 K and 0.05 K. (d) Extracted ZF relaxation rate $\lambda$ vs. temperature for $y=0.04$, which is essentially temperature independent in the measured temperature range. In the sample with $y=0.04$, bulk superconductivity is suppressed below 0.05 K as demonstrated in TF measurements (Fig. \ref{['fig:appendix_A1']}).
• Figure 2: Results of $\mu$SR measurements on Sr$_2$RuO$_4$ under hydrostatic pressure. (a) Zero-field (ZF) muon spin relaxation rate $\lambda$ versus temperature under Zero Pressure (0 GPa). The solid line represents the fit to the phenomenological model Eq. (\ref{['eq:lambda_T_fit']}) with parameters: $\lambda_0 = 0.0191(11)~\mu\mathrm{s}^{-1}$, $\Delta\lambda = 0.0142(26)~\mu\mathrm{s}^{-1}$, $n = 3.9(27)$, and $T_{\rm BTRS} = 1.01(7)$ K. The vertical dashed line indicates the superconducting transition $T_c = 1.24(1)$ K determined from TF measurements, showing a splitting between $T_{\rm BTRS}$ (dotted line) and $T_c$. (b) ZF relaxation rate $\lambda$ under High Pressure ($P = 1.37$ GPa). The solid line is the best fit with Eq. \ref{['eq:lambda_T_fit']}: $\lambda_0 = 0.0598(11)~\mu\mathrm{s}^{-1}$, $\Delta\lambda = 0.0118(15)~\mu\mathrm{s}^{-1}$, $n = 4$, and $T_{\rm BTRS} = 0.91(4)$ K. The vertical dashed line marks $T_c = 0.97(2)$ K. (c) and (d) Temperature dependence of the transverse-field (TF) Gaussian relaxation rate $\sigma$ at 0 GPa and 1.37 GPa, respectively. These measurements track the formation of the flux-line lattice and were used to determine the intrinsic bulk $T_c$ values indicated by the dashed lines in panels (a) and (b).
• Figure 3: Summary plot of the enhanced muon spin depolarization rate $\Delta\lambda(T=0)$ vs. $T_c$ in Sr$_2$RuO$_4$ and its derivatives. The plot summarizes the data from this study and from the literature. The overall behaviour supports the conclusion that $\Delta\lambda(T=0)\propto nJ_s$ with $n$ inhomogeneity or defect density and $Js$ the strength of spontaneous currents induced by non-magnetic inhomogeneities in the superconducting state with broken time reversal symmetry. For La-doped samples and for Sr$_2$RuO$_4$ under hydrostatic pressure, the density $n$ is nearly unchanged, resulting in a trend $\Delta\lambda \propto nJ_s \propto T_c^2$. In contrast, the samples with Ru inclusions and random disorder (presumably Ru vacancies) show enhanced $\Delta\lambda(T=0)$ dominated by the enhancement of $n$. For further explanations, see the text. The data from the literature PhysRevB.107.024508Nat.Phys.17grinenko2021natcommunJPS2014luke2000unconventionalluke1998PhysRevB.85.134527.
• Figure S1: TF $\mu$SR data for Sr$_{2-y}$La$_{y}$RuO$_4$. (a) Time evolution of the muon spin asymmetry for $y=0.01$ under $TF=50G$. (b) Gaussian decay rate $\sigma$ vs. temperature for $y=0.01$, showing an increase below $T_c = 1.13(2)$ K. (c) Time evolution of the muon spin asymmetry for $y=0.04$ under $TF=50G$. (d) $\sigma$ vs. temperature for $y=0.04$. The rate remains constant, demonstrating the suppression of bulk superconductivity.
• Figure S2: Temperature dependence of the specific heat, plotted as $C_p/T$, for Sr$_{2-y}$La$_{y}$RuO$_4$ single crystals. Superconductivity is suppressed homogeneously with La doping.