Unsplit superconducting and time reversal symmetry breaking transitions in Sr₂RuO₄ under hydrostatic pressure and disorder

Abstract

There is considerable evidence that the superconducting state of Sr₂RuO₄ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking, its onset temperature, T_TRSB, is generally found to match the critical temperature, T_c, within resolution. In combination with evidence for even parity, this result has led to consideration of a d_xz ± id_yz order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding T_TRSB = T_c, but it has a hard-to-explain horizontal line node at k_z = 0. Therefore, s ± id and d ± ig order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain T_TRSB ≈ T_c. To obtain evidence distinguishing these two possible scenarios (of symmetry-protected versus accidental degeneracy), we employ zero-field muon spin rotation/relaxation to study pure Sr₂RuO₄ under hydrostatic pressure, and Sr_1.98La_0.02RuO₄ at zero pressure. Both hydrostatic pressure and La substitution alter T_c without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected, T_TRSB should track changes in T_c, while if it is accidental, these transition temperatures should generally separate. We observe T_TRSB to track T_c, supporting the hypothesis of d_xz ± id_yz order.

Introduction

For unconventional superconductors identifying the symmetry of the order parameter is crucial to pinpoint the origin of the superconductivity. Unconventional pairing states are distinguished from conventional ones by a non-trivial intrinsic phase structure which causes additional spontaneous symmetry breaking at the superconducting phase transition. This can lead, for instance, to a reduction of the crystal symmetry or the loss of time reversal symmetry. Indeed, several superconductors are known, which show experimental responses consistent with time reversal symmetry breaking (TRSB) superconductivity^{1,2,3,4,5,6,7,8,9,10,11}.

TRSB superconducting states are formed by combining two or more order parameter components with complex coefficients. These components may be degenerate by symmetry, belonging to a single irreducible representation of the crystalline point group (as in the case of p_x ± ip_y or d_xz ± id_yz superconductivity on a tetragonal lattice), or they may come from different representations (for example, ${d}_{xy}\pm i{d}_{{x}^{2}-{y}^{2}}$ superconductivity on a tetragonal lattice). In the following, we refer to the former as single-representation and the latter as composite-representation order parameters. For composite-representation order parameters, the two components will generally onset at different temperatures. The higher transition temperature becomes T_c, the superconducting critical temperature, and the lower temperature T_TRSB, the temperature where TRSB onsets. The possibility of composite order parameters is usually dismissed out of hand, because it is unusual for two components that are not related by symmetry to be close enough in energy. However, there are a few known examples: s and ${d}_{{x}^{2}-{y}^{2}}$ are relatively close in energy in iron-based superconductors^11,12, while both (U,Th)Be₁₃^1,4 and UPt₃^2,3,8 have split T_c and T_TRSB.

Here, we study Sr₂RuO₄, an unconventional superconductor^13,14, in which the origin of the superconductivity remains a mystery. Evidence that this superconductor breaks time reversal symmetry comes from zero-field muon spin rotation/relaxation (ZF-μSR) experiments¹⁵ and polar Kerr effect measurements¹⁶. Phase-sensitive probes using a corner SQUID device give further support¹⁷. Moreover, the Josephson effect between a conventional superconductor and Sr₂RuO₄ reveal features compatible with the presence of superconducting domains, as expected for TRSB superconductivity^18,19,20. For two decades, the leading candidate state to explain these and other observations was the chiral p-wave state p_x ± ip_y (the lattice symmetry of Sr₂RuO₄ is tetragonal), which has odd parity and therefore equal spin pairing. However, there is compelling evidence against an order parameter with such spin structure. This evidence includes paramagnetic limiting for in-plane magnetic fields^21,22,23 and the recently discovered drop in the NMR Knight shift below T_c^24,25. In combination with the above experimental support for TRSB superconductivity, this evidence compels consideration of d_xz ± id_yz order.

d_xz ± id_yz order would be a surprise because it has a line node at k_z = 0, which under conventional understanding requires interlayer pairing, while in Sr₂RuO₄ interlayer coupling is very weak. It has been proposed that d_xz ± id_yz order might be obtained through multi-orbital degrees of freedom; in this model the order parameter symmetry is encoded in orbital degrees of freedom, so interlayer pairing is not required²⁶. This form of pairing is also under consideration for URu₂Si₂^27,28. However, so far it has not been unambiguously confirmed in any material. To avoid horizontal line nodes, the composite-representation order parameters $s\pm i{d}_{{x}^{2}-{y}^{2}}$²⁹, s ± id_xy³⁰ and ${d}_{{x}^{2}-{y}^{2}}\pm i{g}_{xy({x}^{2}-{y}^{2})}$^31,32 have also recently been proposed for Sr₂RuO₄. In contrast to d_xz ± id_yz, these require tuning to obtain T_c ≈ T_TRSB on a tetragonal lattice.

In this work, to test whether the order parameter of Sr₂RuO₄ is of single- or composite-representation type we perform ZF-μSR measurements on hydrostatically pressurised Sr₂RuO₄ and on La-doped Sr_2−yLa_yRuO₄. Both of these perturbations maintain the tetragonal symmetry of the lattice. If the order parameter has single-representation nature, T_TRSB will track T_c. If the order parameter is of the composite-representation kind, with T_TRSB matching T_c in clean, unstressed samples through an accidental fine tuning, then perturbations away from this point should in general split T_TRSB and T_c, whether they preserve tetragonal lattice symmetry or not³³. Here, we have observed a clear suppression of T_TRSB at a rate matching the suppression of T_c. Our experimental results provide evidence in favour of single-representation nature of the order parameter in Sr₂RuO₄.

Results

μSR on Sr₂RuO₄ under hydrostatic pressure

The hydrostatic pressure measurement setup is shown schematically in Fig. 1. Sr₂RuO₄ crystals of diameter $\varnothing \sim 3$ mm were affixed to oxygen-free copper foils, and assembled into an approximately cylindrical collection of total diameter $\varnothing \sim 7$ mm and total length l ~12 mm (see Fig. 1a). The c-axes of the separate crystals were aligned to within 3^∘.

Fig. 1: Setup for hydrostatic pressure experiments.

a Sr₂RuO₄ sample, consisting of semi-cylindrical pieces glued on oxygen-free copper foils. The top and the bottom panels are the front and the side view, respectively. The crossed circle and the arrow indicate the orientation of the c-axis. b Construction of the pressure cell³⁴. The sample and the pressure medium are surrounded only by beryllium-copper (the pressure cell body and the teflon cap support). The parts of the cell with strong μSR response (teflon cap and tungsten carbide piston) are far from the sample and outside of the muon beam. The initial muon spin polarisation P_μ(0) and the external field B_ext in TF-μSR measurements are aligned along the x- and y-axes, respectively. By rotating the cell about the z-axis, the angle between P_μ(0) and the sample c-axis can be varied.

The pressure cell used in the present study (refs. ^34,35 and Fig. 1b) is a modification of a “classic μSR” clamped pressure cell^35,36. It consists of a main body that encloses the sample and pressure medium, a teflon cap with a metallic support, a tungsten carbide piston, a pressing pad and a clamping bolt (not shown) that holds the piston in place. All the metallic parts of the cell apart from the piston are made from a nonmagnetic beryllium-copper alloy, which is known to have a temperature-independent μSR response^34,35,36. The main feature of this cell is that the only materials placed in the muon beam are the sample, the pressure medium and this CuBe alloy. The muons had a typical momentum of 97 MeV/c, sufficient to penetrate the walls of the pressure cell. The pressure medium was 7373 Daphne oil, which at room temperature solidifies at a pressure p ≈ 2.3 GPa³⁷. The maximum pressure reached here was 0.95 GPa, and therefore hydrostatic conditions are expected. The pressure was determined by monitoring the critical temperature of a small piece of indium (the pressure indicator) placed inside the cell with the Sr₂RuO₄ sample. Confirmation that essentially hydrostatic conditions were attained is provided by the fact that T_c was observed to decrease linearly with pressure, whereas in-plane uniaxial stress on a GPa scale causes a strong non-linear increase in T_c³⁸.

The samples used here were grown by the standard floating zone method³⁹. Measurements of heat capacity of pieces cut from the ends of the rods used here revealed an average T_c of 1.30(6) K (see Supplementary Fig. 1 in Supplementary Note 1), slightly below the limit of T_c of 1.50 K for a pure sample.

T_c and T_TRSB were both obtained by means of μSR, ensuring that both quantities were measured for precisely the same sample volume. In the μSR method, spin-polarised muons are implanted, and their spins then precess in the local magnetic field. By collecting statistics of decay positrons in selected direction(s), the muon polarisation as a function of time after implantation, P_μ(t), can be determined; the time-evolution of this polarisation is determined by the magnetic fields in the sample⁴⁰.

T_c is determined through transverse-field (TF) measurements. An external field B_ext of 3 mT, as is generated by Helmholtz coils, was applied parallel to the crystalline c-axis and perpendicular to the initial muon spin polarisation P_μ(0). Measurements were performed in the field-cooled (FC) mode. Details of the method and analysis are given in the “Methods” section.

Example TF-μSR time spectra at pressure p = 0.95 GPa, and at a temperature above T_c and one below, are shown in Fig. 2a. Above T_c, the spins of muons stopped in both the sample and the pressure cell walls precess with frequency ω = γ_μB_ext (where γ_μ = 2π × 135.5 MHz/T is the muon gyromagnetic ratio). The muon spin polarisation is seen to relax substantially on a 10 μs time scale. This is because ~50% of muons are implanted into the CuBe, where the nuclear magnetic moments of Cu rapidly relax their polarisation. Below T_c, the internal field in the sample becomes highly inhomogeneous due to the appearance of a flux-line lattice, and so the polarisation of the muons that implanted in the sample also relaxes quickly.

Fig. 2: Effect of pressure on T_c and T_TRSB in Sr₂RuO₄.

a TF-μSR time-spectra above and below T_c measured at p = 0.95 GPa and B_ext = 3 mT, with B_ext∥c. The plotted quantity is the detection asymmetry between two positron detectors, which is proportional to the muon spin polarisation P_μ(t). The solid lines are fits of Eq. (2), with the sample and the pressure cell contributions described by Eqs. (3) and (4), respectively. b, c Temperature dependencies of the Gaussian relaxation rate σ and the diamagnetic shift B_int − B_ext ∝ M_FC at p = 0.0 and 0.95 GPa. Arrows indicate the position of the superconducting transition temperature T_c at p = 0.0 GPa. d ZF-μSR time-spectra above and below T_c, measured at p = 0.95 GPa and with initial muon spin polarisation P_μ(0)∥c. The solid lines are fits of Eq. (2), with the sample and the pressure cell parts described by Eqs. (5) and (7). e, f Temperature dependencies of the ZF exponential muon spin relaxation rate λ at p = 0.0 and 0.95 GPa. In e, P_μ(0)∥ab, and in f, P_μ(0)∥c. The solid lines are fits of Eq. (1) to the data. Arrows indicate the position of T_TRSB at p = 0.0 GPa. g Dependence of T_c and T_TRSB on pressure. Open circle correspond to an average T_c of 1.30(6) K determined from specific heat data (see Supplementary Fig. 1 in Supplementary Note 1). The displayed error bars for μSR data correspond to one standard deviation from the χ² fit⁷¹. The displayed error bars for T_c indicate the rounding of the transition on a scale of approximately 0.1 K. The error bars for μSR data and T_TRSB correspond to one standard deviation from the χ² fit⁷¹.

TF-μSR measurements were performed at 0, 0.25, 0.62, and 0.95 GPa. Data at 0 and 0.95 GPa are shown in Fig. 2, and at the other two pressures in Supplementary Figs. 3 and 4 in Supplementary Note 2. Data are analysed as a sum of background and sample contributions, given by Eqs. (3) and (4) (in the “Methods” section), respectively. From the sample contribution we extract a Gaussian relaxation rate, σ, and the diamagnetic shift of the field inside the sample, B_int − B_ext ∝ M_FC⁴¹ (M_FC is the FC magnetisation). Figure 2b, c, respectively, shows the temperature dependence of σ and B_int − B_ext. σ is given by ${\sigma }^{2}={\sigma }_{\,\text{sc}}^{2}+{\sigma }_{\text{nm}\,}^{2}$, where σ_sc and σ_nm are the flux-line lattice and nuclear moment contributions, respectively. ${\sigma }_{\text{sc}}\propto {\lambda }_{ab}^{-2}$, where λ_ab is the in-plane magnetic penetration depth; see ref. ⁴² and the “Methods” section. The onset of superconductivity can be seen in both σ and B_int − B_ext, as a transition rounded on a scale of ~0.1 K. The heat capacity measurements show a similar distribution of T_c’s; see Supplementary Fig. 1 in Supplementary Note 1.

The pressure dependence of T_c is shown in Fig. 2g. The error bars in the figure are the rounding on the transition, and can be taken as an absolute error on T_c. When fitting σ(T) and B_int(T) with model functions, the statistical error on the T_c’s extracted is considerably smaller, meaning that the error on changes in T_c is low. A linear fit to T_c(p) yields a slope dT_c/dp = −0.24(2) K/GPa, which is in good agreement with literature data^43,44,45. The unpressurised T_c is found to be 1.26(5) K, in good agreement with 1.30(6) K found in the heat capacity measurements, see Supplementary Note 1.

T_TRSB is determined through ZF measurements. The signature of TRSB is an enhancement in the muon spin relaxation rate below T_TRSB, indicating the appearance of spontaneous magnetic fields. In these measurements, external fields were compensated to better than 2 μT, ruling out flux lines below T_c as the origin of this signal. An example of ZF-μSR time spectra above and below T_c, showing the faster relaxation below T_c, at p = 0.95 GPa is presented in Fig. 2d. The pressure cell background is T-independent, so the increased signal decay comes from the sample. The sample contribution was modelled by a two-component relaxation function: ${\rm{GKT}}(t)\cdot \exp (-\lambda t)$, in accordance with the results of refs. ^{5,6,9,15,46,47}; see also the “Methods” section. Here, GKT(t) is the Gaussian Kubo-Toyabe function describing the relaxation of muon spin polarisation in the random magnetic field distribution created by nuclear magnetic moments, and $\exp (-\lambda t)$ is a Lorentzian decay function accounting for appearance of spontaneous magnetic fields. Temperature dependencies of the exponential relaxation rate, λ, at 0 and 0.95 GPa, for independent measurements with the initial muon spin polarisation P_μ(0)∥c and ∥ab, are shown in Fig. 2e, f; ZF data at 0.25 and 0.62 GPa are shown in Supplementary Figs. 3 and 4 in Supplementary Note 2.

To extract T_TRSB, λ(T) is fitted with the following functional form:

$$\lambda (T)=\left\{\begin{array}{ll}{\lambda }_{0},\hfill&T\;> \;{T}_{{\rm{TRSB}}} \\ {\lambda }_{0}+{{\Delta }}\lambda \left[1-{\left(\frac{T}{{T}_{{\rm{TRSB}}}}\right)}^{n}\right],&T\;<\;{T}_{{\rm{TRSB.}}}\end{array}\right.$$

(1)

λ₀ is the relaxation rate above T_TRSB, and Δλ is the enhancement due to spontaneous magnetic fields. Where data were obtained both for P_μ(0)∥c and ∥ab, the exponent n is constrained to be the same for both polarisations. T_TRSB, λ₀, and Δλ were obtained independently for each pressure and muon spin polarisation. The resulting values of T_TRSB are plotted in Fig. 2g.

Our ZF data yield the following three results:

• (1) Where data were taken both for P_μ(0)∥c and ∥ab (that is, at 0 and 0.95 GPa), T_TRSB and Δλ were found to be the same within resolution for both polarisations. [At 0 GPa, Δλ = 0.027(4) and 0.033(3) μs⁻¹, and at 0.95 GPa, 0.030(4) and 0.025(3) μs⁻¹, for P_μ(0)∥ab and P_μ(0)∥c, respectively.] This agrees with the zero-pressure results of Luke et al.¹⁵. Because Δλ reflects fields perpendicular to P_μ(0), this result indicates that the spontaneous fields have no preferred orientation.

• (2) Δλ was found to be pressure-independent within resolution (including all pressures investigated: 0, 0.25, 0.62 and 0.95 GPa), having an average value of Δλ = 0.026(2) μs⁻¹. This value corresponds to a characteristic field strength B_TRSB = Δλ/γ_μ = 0.031(2) mT. B_TRSB has been found to vary from sample to sample⁴⁷, and this value is in line with previous reports (see refs. ^15,46,48,49 and Table 1).

  Table 1 Enhancement of the exponential relaxation rates Δλ and corresponding values of the spontaneous magnetic fields B_TRSB = Δλ/γ_μ caused by formation of TRSB state in Sr₂RuO₄ and related compounds.

• (3) A linear fit yields T_TRSB(p) = 1.27(3) K − p ⋅ 0.29(5) K/GPa. In other words, within resolution the rate of suppression of T_TRSB under hydrostatic pressure matches that of T_c.

μSR on Sr_1.98La_0.02RuO₄

Substitution of La for Sr adds electrons to the Fermi surfaces; in Sr_2−yLa_yRuO₄ this doping drives the largest Fermi surface through a Lifshitz transition from an electron-like to a hole-like geometry, at y ≈ 0.20^50,51. At y = 0.02, the change in Fermi surface structure is minimal, and the main effect of the La-substitution is to suppress T_c, through the added disorder. Heat capacity data, measured on a small piece cut from the μSR sample, give T_c = 0.70(5) K, where the error reflects the width of the transition (see Supplementary Fig. 2 in Supplementary Note 1).

This sample was studied at zero pressure. With no pressure cell material in the beam, the background is much smaller. The typical muon momentum was 28 MeV/c, giving of ~0.2 mm implantation depth⁴⁰. Representative TF-μSR time spectra above and below T_c, where the applied field is B_ext = 2 mT parallel to the crystalline c-axis, are shown in Fig. 3a. Below T_c, the muon spin polarisation relaxes almost completely on a 10 μs time scale, showing that essentially the entire sample volume is superconducting. The TF Gaussian relaxation rate σ is shown in Fig. 3b, and B_int − B_ext in Fig. 3c. These measurements yield T_c = 0.75(5) K. The heat capacity data are also shown in Fig. 3b.

Fig. 3: TRSB in Sr_1.98La_0.02RuO₄.

a TF-μSR time-spectra above and below T_c measured at B_ext = 2 mT with B_ext∥c. The solid lines are fits of Eq. (3) to the data. b, c Temperature dependencies of the Gaussian relaxation rate σ and the diamagnetic shift B_int − B_ext, respectively. Arrows indicates the superconducting transition temperature T_c, determined from the TF-μSR data. The blue curve in b is the electronic specific heat C_el/T, measured on a small piece cut from the μSR sample. d ZF- and LF-μSR time-spectra. ZF data from above and below T_c, measured with P_μ(0)∥c, are shown. The LF data are from T well below T_c, and with B_ext = 3 mT ∥P_μ(0). The solid lines are fits of Eq. (8). e Temperature dependence of the ZF and LF exponential relaxation rate λ. The solid red line is the fit of Eq. (1) to ZF λ(T) data. The blue curve is, again, C_el/T. Arrows indicates positions of T_c and T_TRSB. f Double logarithmic plot of the normalised specific heat jump ${{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}^{{\rm{SH}}}$ versus ${T}_{{\rm{c}}}^{{\rm{SH}}}$ [γ_n is the Sommerfeld coefficient and ${T}_{{\rm{c}}}^{{\rm{SH}}}$ is the transition temperature determined from C_el/T(T) by means of equal-entropy construction, see Supplementary Fig. 1 in Supplementary Note 1]. Values of ${{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}^{{\rm{SH}}}$ are determined in a way presented in Supplementary Fig. 2 in Supplementary Note 1. Filled symbols: data from this work; open symbols: data taken from refs. ^47,52. The displayed error bars for μSR data correspond to one standard deviation from the χ² fit⁷¹. The error bars for ${{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}^{{\rm{SH}}}$ and ${T}_{{\rm{c}}}^{{\rm{SH}}}$ indicate uncertainty in selecting the temperature range for linear fit below T_c.

ZF-μSR data are presented in Fig. 3d, e. Fitting with Eq. (1) returns Δλ = 0.007(1) μs⁻¹ and T_TRSB = 0.8(1) K. This Δλ is noticeably smaller than that obtained from the undoped Sr₂RuO₄ sample, corresponding to an internal field B_TRSB ≈ 0.01 mT. It is, however, within the range of previous results⁴⁷. In qualitative agreement with data on a lower T_c Sr₂RuO₄, reported in ref. ⁴⁶, though here with more data at T > T_c to be certain of the base relaxation rate, this low value of Δλ shows that B_TRSB is not straightforwardly related to defect density. At present, the origin of the sample-to-sample variation in B_TRSB is unknown.

Longitudinal-field (LF) measurements can be employed to determine whether internal fields are static or fluctuating. If B_TRSB is static, under an applied field parallel to P_μ(0) that is considerably larger than B_TRSB, muon spin precession is greatly restricted and the spin polarisation does not relax (i.e. the muon spins decouple from B_TRSB). In contrast, fluctuating B_TRSB can still relax the muon spin polarisation⁴⁰. Data shown in Fig. 3d, e indicate that B_ext∣∣P_μ(0) = 3 mT fully suppresses the muon spin relaxation, and therefore that B_TRSB is static on a microsecond time scale, in agreement with data on clean Sr₂RuO₄ reported in ref. ¹⁵. We note that LF measurements were not performed on the hydrostatically pressurised sample because the decoupling field for the Cu background is of the order of 10 mT, considerably stronger than that for Sr₂RuO₄.

Heat capacity measurements

The specific heat measurements were performed at ambient pressure for several pieces of Sr_2-yLa_yRuO₄ single crystals. The results are presented in Fig. 3b, e for Sr_1.98La_0.02RuO₄ (y = 0.02) and in Supplementary Fig. 1 in Supplementary Note 1 for Sr₂RuO₄ (y = 0.0), respectively. The specific heat jumps at T_c (ΔC_el/γ_nT_c, γ_n is the Sommerfeld coefficient) were further obtained in a way presented in Supplementary Fig. 2 in Supplementary Note 1.

Figure 3f summarises the ${{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}^{{\rm{SH}}}$ vs. ${T}_{{\rm{c}}}^{{\rm{SH}}}$ data for our Sr_2-yLa_yRuO₄ samples. Here ${T}_{{\rm{c}}}^{{\rm{SH}}}$ denotes the superconducting transition temperature determined from C_el/T vs. T measurement curves by means of equal-entropy construction algorithm, see Supplementary Fig. 1a in Supplementary Note 1. In addition, we have also included some literature data for Sr₂RuO₄ with different amount of disorder⁴⁷, and for Sr₂RuO₄ under uniaxial strain⁵². In total, Fig. 3f compares Sr₂RuO₄ samples with a factor of five variation in T_c. Remarkably, ΔC_el/γ_nT_c vs. T_c data points scale as ${T}_{{\rm{c}}}^{\alpha }$ with α ≈ 0.65, which is distinctly different from the BCS behaviour, where α = 0 (ΔC_el/γ_nT_c = const). Just a single point at T_c ≃ 3.5 K deviates from the scaling behaviour, which might be associated with tuning the electronic structure of Sr₂RuO₄ close to a van Hove singularity⁵². The results presented in Fig. 3f indicate, therefore, that the perturbation changes the gap structure on the Fermi surface, i.e. its “anisotropy” or the distribution among the three different bands which can lead to a renormalisation of the specific heat jump being not simply proportional to the normal-state-specific heat above T_c.

Such scaling behaviour is rarely observed since the ratio ΔC_el/γ_nT_c is sensitive to a change of the superconducting gap structure and symmetry. Note that a similar scaling is reported for Fe-based superconductors, where ΔC_el/γ_nT_c follows approximately the BNC (Bud’ko-Ni-Canfield) scaling behaviour ${{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}\propto {T}_{{\rm{c}}}^{\alpha }$ with α ≈ 2⁵³, which is considered to be a consequence of the unconventional multiband s ± superconductivity. The change of the superconducting pairing state in the Ba_1−xK_xFe₂As₂ system results in abrupt change of the scaling behaviour leading to an intermediate s + is state¹¹. The monotonic ΔC_el/γ_nT_c vs. T_c behaviour obtained in the present study suggests, therefore, that La-substitution do not yield a change of the superconducting gap symmetry. Consequently, the superconducting gap structure does not undergo a significant change due to effects of disorder and it remains the same as in bare Sr₂RuO₄ compound.

Discussion

In a previous ZF-μSR experiment, in-plane uniaxial pressure, which does lift the tetragonal symmetry of the unpressurised lattice, was found to induce a strong splitting between T_c and T_TRSB⁴⁷. Uniaxial pressure drives a strong increase in T_c, while T_TRSB varies much more weakly, probably decreasing slightly with initial application of pressure. The microscopic mechanism yielding the signal observed at T_TRSB, a weak enhancement in muon spin relaxation rate, remains unclear: the main proposed mechanism, magnetism induced at defects and domain walls by a TRSB superconducting order, is unproved experimentally^54,55. At present, the link between enhanced muon spin relaxation and TRSB superconductivity is, therefore, mainly empirical, based on: (1) the facts that it is a signal seen in only a small fraction of known superconductors, (2) it generally appears at T_c and (3) the general notion that TRSB superconductivity can in principle generate magnetic fields, while muons detect magnetic fields. In ref. ⁴⁷, careful checks were performed to rule out instrumentation artefact as the origin of the signal at T_TRSB, and it was further argued that this signal is extremely difficult to obtain from a purely magnetic mechanism. Nevertheless, the weak observed variation of T_TRSB, while T_c varied strongly, raised some doubt as to whether this signal is in fact associated with the superconductivity.

Here, we have observed a clear suppression of T_TRSB with hydrostatic stress, at a rate matching the suppression of T_c. This result further strengthens the evidence that enhanced muon spin relaxation is an indicator of TRSB superconductivity: T_TRSB tracks T_c when tetragonal lattice symmetry is preserved, while the splitting induced by uniaxial pressure shows unambiguously that it is a distinct transition, and not an artefact through some unidentified mechanism of the superconducting transition itself. Figure 4 shows T_TRSB versus T_c. The data reported here, on hydrostatically pressurised Sr₂RuO₄ and on unpressurised Sr_1.98La_0.02RuO₄, fall on the T_TRSB = T_c line, while the uniaxial pressure data from ref. ⁴⁷ clearly deviate from this line.

Fig. 4: Relation between T_TRSB and T_c.

Dependence of the time reversal symmetry breaking temperature T_TRSB on the superconducting transition temperature T_c. The closed symbols correspond to the results obtained in present studies under hydrostatic pressure up to 0.95 GPa in pure Sr₂RuO₄ (diamonds) and in the La-doped Sr_2−yLa_yRuO₄ with T_c = 0.75(5) K (square). The open squares are the uniaxial pressure data for undoped Sr₂RuO₄ from ref. ⁴⁷. The dashed line corresponds to T_TRSB = T_c. The minus signs at the pressure values denote the effect of ‘compression' of the sample volume. The error bars are the same as defined in Figs. 2 and 3 and in ref. ⁴⁷.

Our central finding that T_TRSB tracks T_c provides further support for the single-representation d_xz ± id_yz order parameter. Importantly, homogeneous d_xz ± id_yz is the only spin-singlet order parameter consistent with the selection rules imposed by ultrasound and Kerr effect data. Ultrasound data on Sr₂RuO₄^56,57 show a type of renormalisation that is not possible for a single-component order parameter on a tetragonal lattice: a jump in ultrasound velocity at T_c for transverse modes. While these experimental results are not sensitive to the spin configuration, they impose other stringent conditions on the possible pairing symmetries^58,59. The polar Kerr effect mentioned above is a second experiment which provides symmetry-related constraints, being compatible only with chiral pairing states¹⁶. These two selection rules are obeyed by both the chiral p-wave and chiral d-wave state, though as noted in the Introduction, p-wave order appears to be ruled out by NMR Knight shift data^24,25. In contrast, the composite-representation states do not satisfy the requirements for both selection rules. The ${d}_{{x}^{2}-{y}^{2}}+i{g}_{xy({x}^{2}-{y}^{2})}$ and s + id_xy states are constructed to be compatible with the ultrasound measurements, but they are not chiral^31,60. The $s+i{d}_{{x}^{2}-{y}^{2}}$ state violates both selection rules²⁹. It can be generally stated that any composite-representation pairing states in a tetragonal crystal, composed of components of two one-dimensional representations, would satisfy at most one of the two selection rules (see the “Methods” section).

We note that there is a recent proposal for inhomogeneous superconductivity in Sr₂RuO₄: single-component (${d}_{{x}^{2}-{y}^{2}}$) in the bulk, but two-component (${d}_{{x}^{2}-{y}^{2}}+i{g}_{xy({x}^{2}-{y}^{2})}$) in the strain fields around dislocations⁶¹. The combination of phase locking between adjacent dislocations and a preferred orientation to the dislocations would result in a bulk chirality. In this proposal, T_TRSB could be locked to T_c by hypothesising that superconductivity appears at the dislocations before the bulk, but tuning would then be required to obtain T_c and T_TRSB that split under modest uniaxial stress.

Major challenges to d_xz ± id_yz order are the absence of a resolvable second heat capacity anomaly at T_TRSB in measurements on uniaxially pressurised Sr₂RuO₄⁵², and, as already noted, the theoretical challenges in obtaining a horizontal line node in a highly two-dimensional metal⁶². We note in addition that an analysis of low-temperature thermal conductivity data indicated vertical, rather than horizontal, line nodes in Sr₂RuO₄⁶³. The theoretical objection to horizontal line nodes might be overcome through the complex nature of the multi-orbital band structure, including sizable spin-orbit coupling^26,64,65.

So we may conclude that our ZF-μSR data combined with the selection rules for ultrasound and polar Kerr effect and the NMR Knight shift behaviour are consistent with the single-representation chiral d_xz + id_zy-wave state, while all composite-representation states suffer from several deficiencies. We note, however, that there are also empirical challenges to a hypothesis of d_xz ± id_yz, and that the difficulty in reconciling apparently contradictory experimental results in Sr₂RuO₄ may mean that one or more major, apparently solid results will in time be found to be incorrect, either for a technical reason or in interpretation. Further experiments are therefore necessary.

Methods

Sr_2-yLa_yRuO₄ single crystals

Single crystals of Sr_2-yLa_yRuO₄ were grown by means of a floating zone technique³⁹. Samples for measurement under hydrostatic pressure (with y = 0) were cut from two rods, C140 and C171, that each grew along a 〈100〉 crystallographic direction. The rods have diameter $\varnothing \simeq 3$ mm. Two sections of length 8–12 mm were taken from each rod. These were then cleaved, forming semi-cylindrical samples with flat surfaces perpendicular to the c-axis (see Fig. 1a).

The effect of La doping on the TRSB transition was studied on a single original Sr_2-yLa_yRuO₄ crystal of length 8 mm. The La concentration was analysed by an electron-probe micro-analysis and was found to be y ≃ 0.02. Before the μSR measurements, this rod was then cleaved into two semi-cylindrical pieces, again with the flat faces ⊥c.

The x-ray diffraction experiments performed on small powdered pieces cut from of each particular rod gave a = 0.3867 nm, c = 1.273 nm for pure Sr₂RuO₄ and a = 0.3865 nm, c = 1.274 nm for La-substituted sample.

Specific heat of Sr_2-yLa_yRuO₄ at ambient pressure

Specific heat measurements were performed at zero pressure for several pieces of Sr_2-yLa_yRuO₄ single crystals, cut from the rod used for μSR measurements.

For Sr₂RuO₄ used in hydrostatic pressure measurements, the electronic specific heat capacity C_el/T was measured for four samples: one sample cut from each end of both the C140 and C171 sections. Results are presented in Supplementary Fig. 1 in Supplementary Note 1.

The temperature dependence of C_el/T for a small piece cut from the Sr_1.98La_0.02RuO₄μSR sample is presented in Fig. 3b, e and Supplementary Fig. 2 in Supplementary Note 1.

μSR experiments and μSR data analysis procedure

The muon spin rotation/relaxation (μSR) experiments were performed at the μE1 and πE1 beamlines, using the GPD³⁵, and Dolly spectrometers (Paul Scherrer Institute, PSI Villigen, Switzerland). At the GPD instrument, experiments under pressure up to p ≃ 0.95 GPa on undoped Sr₂RuO₄ were performed. At the Dolly spectrometer, measurements of Sr_1.98La_0.02RuO₄ at ambient pressure were conducted. At both instruments ⁴He cryostats equipped with the ³He insets (base temperature T ≃ 0.25 K) were used.

At the GPD instrument, measurements in zero-field (ZF-μSR) and with the field applied transverse to the initial muon spin polarisation P_μ(0) (TF-μSR) were performed. In two sets of ZF-μSR studies, P_μ(0) was set to be parallel to the c-axis and along the ab-plane, respectively. In TF-μSR measurements the small 3 mT magnetic field was applied parallel to the c-axis and perpendicular to P_μ(0).

At the Dolly instrument, in addition to ZF- and TF-μSR experiments, the LF measurements were performed. In these studies 3 mT magnetic field was applied parallel to the c-axis and to the initial muon spin polarisation P_μ(0).

The experimental data were analysed by separating the μSR signal on the sample (s) and the background (bg) contributions⁶⁶:

$${A}_{0}P(t)={A}_{{\rm{s}}}{P}_{{\rm{s}}}(t)+{A}_{{\rm{bg}}}{P}_{{\rm{bg}}}(t).$$

(2)

Here A₀ is the initial asymmetry of the muon spin ensemble, and A_s (A_bg) and P_s(t) [P_bg(t)] are the asymmetry and the time evolution of the muon spin polarisation for muons stopped inside the sample (outside of the sample), respectively.

In a case of μSR under pressure studies, the background contribution (~50% of total μSR response) is determined by the muons stopped in the pressure cell body. At ambient pressure experiment the small background contribution (of the order of 5%) is caused by muons stopped in the sample holder and the cryostat windows.

In TF-μSR experiments, the sample contribution was analysed by using the following functional form:

$${P}_{{\rm{s}}}^{{\rm{TF}}}(t)=\exp \left[-\frac{{\sigma }^{2}{t}^{2}}{2}\right]\cos ({\gamma }_{\mu }{B}_{{\rm{int}}}t+\phi).$$

(3)

Here B_int is the internal field in the sample, ϕ is the initial phase of the muon spin ensemble, and γ_μ ≃ 2π × 135.5 MHz/T is the muon gyromagnetic ratio. The Gaussian relaxation rate σ consists of the “superconducting”, σ_sc, and nuclear moment, σ_nm, contributions and it is defined as: ${\sigma }^{2}={\sigma }_{{\rm{sc}}}^{2}+{\sigma }_{{\rm{nm}}}^{2}$. Here, σ_sc and σ_nm characterise the damping due to the formation of the flux-line lattice in the superconducting state and of the nuclear magnetic dipolar contribution, respectively. In the analysis, σ_nm was assumed to be constant over the entire temperature range and was fixed to the value obtained above T_c, where only nuclear magnetic moments contribute to the muon depolarisation rate (see Supplementary Fig. 3a in Supplementary Note 2).

The pressure cell contribution was described by using the following equation:

$${P}_{{\rm{pc}}}^{{\rm{TF}}}(t)=\exp \left[-\frac{{\sigma }_{{\rm{pc}}}^{2}{t}^{2}}{2}\right]\cos ({\gamma }_{\mu }{B}_{{\rm{ext}}}t+\phi).$$

(4)

Here σ_pc ≃ 0.28 μs⁻¹ is the field and the temperature-independent relaxation rate of beryllium-copper³⁵, and B_ext is the externally applied field.

The solid lines in Fig. 2a correspond to the fit of TF-μSR data by using Eq. (2) with the sample and the background parts described by Eqs. (3) and (4). For the data presented in Fig. 3a the background contribution was described by non-relaxing function ${P}_{{\rm{bg}}}^{{\rm{TF}}}(t)=\cos ({\gamma }_{\mu }{B}_{{\rm{ext}}}t+\phi)$. The good agreement between the fits and the data demonstrates that the above model describes the experimental data rather well.

With the external magnetic field applied along the crystallographic c-axis (B_ext∥c), the superconductig contribution into the Gaussian relaxation rate σ_sc becomes proportional to the inverse squared in-plane magnetic penetration depth λ_ab⁴². The proportionality coefficient between σ_sc and ${\lambda }_{ab}^{-2}$ depends on the value of the applied field, the symmetry of the flux-line lattice and the angular dependence of the superconducting order parameter.

The temperature dependencies of the Gaussian relaxation rate σ and the diamagnetic shift B_int − B_ext are presented in Figs. 2b, c and 3b, c for Sr₂RuO₄ and Sr_1.98La_0.02RuO₄ samples, respectively.

In ZF and LF-μSR experiments the sample contribution includes both, the nuclear moment relaxation and an additional exponential relaxation λ caused by appearance of spontaneous magnetic fields¹⁵:

$${P}_{{\rm{s}}}^{{\rm{ZF}}}(t)={{\rm{GKT}}}_{{\rm{s}}}(t)\ {e}^{-\lambda t}.$$

(5)

Here GKT(t) is the Gaussian Kubo-Toyabe (GKT) relaxation function describing the magnetic field distribution created by the nuclear magnetic moments^40,67:

$${\rm{GKT}}(t)=\frac{1}{3}+\frac{2}{3}(1-{\sigma }_{{\rm{GKT}}}^{2}{t}^{2})\ {e}^{-{\sigma }_{{\rm{GKT}}}^{2}{t}^{2}/2}.$$

(6)

σ_GKT is the GKT relaxation rate.

Muons implanted in beryllium-copper pressure cell body sense solely the magnetic field distribution created by copper nuclear magnetic moments and described as:

$${P}_{{\rm{pc}}}^{{\rm{ZF}}}(t)={{\rm{GKT}}}_{{\rm{pc}}}(t)$$

(7)

with the temperature-independent relaxation rate σ_GKT,BeCu ≃ 0.35 μs⁻¹³⁵.

Fits of Eq. (2), with the sample and pressure cell parts described by Eqs. (5) and (7), to the ZF-μSR data were performed globally. The ZF-μSR time-spectra taken at each particular muon spin polarisation [P_μ(0)∥ab and P_μ(0)∥c] and pressure (p = 0.0, 0.25, 0.62 and 0.95 GPa) were fitted simultaneously with A_s, A_pc, ${\sigma }_{{\rm{GKT,S{r}}_{2}Ru{O}_{4}}}$, σ_GKT,BeCu, and λ₀ as common parameters, and λ as individual parameter for each particular data set. The solid green and purple lines in Fig. 2d correspond to the fit of ZF-μSR data by using Eq. (2) with the sample and the background parts described by Eqs. (5) and (7).

Note that the absence of strong nuclear magnetic moments in Sr_2−yLa_yRuO₄ leads to the corresponding Gaussian Kubo-Toyabe relaxation rate being nearly zero. Consequently, the analysis of ZF- and LF-μSR data for Sr_1.98La_0.02RuO₄ was performed by using the simple-exponential decay function:

$${P}_{{\rm{s}}}^{{\rm{ZF,LF}}}(t)={e}^{-\lambda t}.$$

(8)

The solid lines in Fig. 3d correspond to the fit of ZF-μSR data by using Eq. (2) with the sample part described by Eq. (8) and the non-relaxing background ${P}_{{\rm{bg}}}^{{\rm{ZF,LF}}}(t)=1$.

The temperature dependencies of the exponential relaxation rate λ are presented in Figs. 2e, f and 3e for Sr₂RuO₄ and Sr_1.98La_0.02RuO₄ samples, respectively.

Symmetry properties of the order parameters

Several order parameters have been proposed for the time reversal symmetry breaking superconducting state of Sr₂RuO₄. We would like here to give a brief overview on the different options and the symmetry requirements to satisfy the selection rules for two experiments: ultrasound velocity renormalisation for the transverse c₆₆-mode and the polar Kerr effect. For tetragonal crystal symmetry with the point group D_4h the even parity spin-singlet pairing states can be listed according to the irreducible representations of D_4h, four one-dimensional ones A_1g, A_2g, B_1g, B_2g and a two-dimensional one E_u. The pair wave function ψ_Γ(k) of the corresponding states are given by:

$$\begin{array}{ll}{\hskip -61pt}{\psi }_{{A}_{1g}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})&s\,\hbox{-}\text{wave}\,\\ {\psi }_{{A}_{2g}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}}){k}_{x}{k}_{y}({k}_{x}^{2}-{k}_{y}^{2})\quad &{g}_{xy({x}^{2}-{y}^{2})}\,\hbox{-}\text{wave}\,\\ {\hskip -24pt}{\psi }_{{B}_{1g}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})({k}_{x}^{2}-{k}_{y}^{2})&{d}_{{x}^{2}-{y}^{2}}\,\hbox{-}\text{wave}\,\\ {\hskip -44pt}{\psi }_{{B}_{2g}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}}){k}_{x}{k}_{y}&{d}_{xy}\,\hbox{-}\text{wave}\,\\ {\hskip 10pt}{\psi }_{{E}_{g}}({\boldsymbol{k}})=\{{\psi }_{0}({\boldsymbol{k}}){k}_{x}{k}_{z},{\psi }_{0}({\boldsymbol{k}}){k}_{y}{k}_{z}\}&\{{d}_{xz},{d}_{yz}\}\,\hbox{-}\text{wave}\,\end{array}$$

(9)

where ψ₀(k) is a function of k invariant under all symmetry operations of the tetragonal lattice. We list here first the composite-representation TRSB states:

$$\begin{array}{ll}{\tilde{{{\Gamma }}}}_{1}={A}_{1g}\oplus {A}_{2g}:&s+ig\,\hbox{-}\text{wave}\,\\ {\tilde{{{\Gamma }}}}_{2}={A}_{1g}\oplus {B}_{1g}:&s+id\,\hbox{-}\text{wave}\,\\ {\tilde{{{\Gamma }}}}_{3}={A}_{1g}\oplus {B}_{2g}:&s+id^{\prime} \,\hbox{-}\text{wave}\,\\ {\tilde{{{\Gamma }}}}_{4}={B}_{1g}\oplus {A}_{2g}:&d+ig\,\hbox{-}\text{wave}\,\\ {\tilde{{{\Gamma }}}}_{5}={B}_{2g}\oplus {A}_{2g}:&d^{\prime} +ig\,\hbox{-}\text{wave}\,\\ {\tilde{{{\Gamma }}}}_{6}={B}_{1g}\oplus {B}_{2g}:&d+id^{\prime} \,\hbox{-}\text{wave}\,\\ \end{array}$$

(10)

Note that in general different representations correspond to different critical temperature. Thus, to obtain a single superconducting phase transition for the composite states an accidential degeneracy of two representations is necessary. The two states proposed so far are ${\tilde{{{\Gamma }}}}_{2}$^29,30 and ${\tilde{{{\Gamma }}}}_{4}$^31,32. The two-dimensional representation allows for the combination:

$${\tilde{{{\Gamma }}}}_{7}={E}_{g}:\,{\text{chiral}}\,\ d\,\hbox{-}\text{wave}\,$$

(11)

with a pair wave function ${\psi }_{{E}_{g}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}}){k}_{z}({k}_{x}\pm i{k}_{y})$ as proposed in refs. ^26,62. All composite states, ${\tilde{{{\Gamma }}}}_{1-6}$, can be constructed by electron pairing within the RuO₂ planes, while the state ${\tilde{{{\Gamma }}}}_{7}$ requires interlayer pairing. Due to the spin-singlet nature all states are compatible with the new NMR Knight shift results^24,25. All TRSB state are expected to generate internal spontaneous currents around defects, such as surfaces and domain walls and, consequently, under present understanding are compatible with the μSR experiments¹⁵.

Next we consider the two selection rules. For the coupling to the lattice we restrict consideration to the mode which corresponds to the elastic constant c₆₆, which is connected with the strain tensor element ϵ_xy = ϵ_yx^58,59. This is active for transverse modes with a wave vector in the plane, e.g. [100] and a polarisation perpendicular also within the plane. This strain tensor component belongs by symmetry to the representation B_2g^58,59,68. For the observed renormalisation of the speed of sound the superconducting order parameter has to couple linearly to ϵ_xy, thus, requiring that B_2g is contained in the decomposition of ${\tilde{{{\Gamma }}}}_{j}\otimes {\tilde{{{\Gamma }}}}_{j}$. This only possible for ${\tilde{{{\Gamma }}}}_{3},{\tilde{{{\Gamma }}}}_{4}$ and ${\tilde{{{\Gamma }}}}_{7}$:

$${\tilde{{{\Gamma }}}}_{3}\otimes {\tilde{{{\Gamma }}}}_{3}={\tilde{{{\Gamma }}}}_{4}\otimes {\tilde{{{\Gamma }}}}_{4}=2{A}_{1g}\oplus 2{B}_{2g}$$

(12)

and

$${\tilde{{{\Gamma }}}}_{7}\otimes {\tilde{{{\Gamma }}}}_{7}={A}_{1g}\oplus {A}_{2g}\oplus {B}_{1g}\oplus {B}_{2g}.$$

(13)

The selection rule resulting in the polar Kerr effect requires the order parameter to couple by symmetry to the z-component of the magnetic field, B_z which belongs to the representation A_2g. Again we consider the decomposition of the corresponding representations of the different pairing states. We find that only ${\tilde{{{\Gamma }}}}_{1},{\tilde{{{\Gamma }}}}_{6}$ and ${\tilde{{{\Gamma }}}}_{7}$ satisfy the condition. The only pairing state which appears to obey both selection rules is the chiral d-wave state. None of the composite pairing states can satisfy both conditions. Among them there are the states ${\tilde{{{\Gamma }}}}_{2}$ and ${\tilde{{{\Gamma }}}}_{5}$ which are in conflict with both selection rules.

Turning to the odd parity states the analogous picture arises with:

$$\begin{array}{ll}{{\boldsymbol{d}}}_{{A}_{1u}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})(\hat{{\boldsymbol{x}}}{k}_{x}+\hat{{\boldsymbol{y}}}{k}_{y})\\ {{\boldsymbol{d}}}_{{A}_{2u}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})(\hat{{\boldsymbol{x}}}{k}_{y}-\hat{{\boldsymbol{y}}}{k}_{x})\\ {{\boldsymbol{d}}}_{{B}_{1u}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})(\hat{{\boldsymbol{x}}}{k}_{x}-\hat{{\boldsymbol{y}}}{k}_{y})\\ {{\boldsymbol{d}}}_{{B}_{2u}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})(\hat{{\boldsymbol{x}}}{k}_{y}+\hat{{\boldsymbol{y}}}{k}_{x})\\ {\hskip -2pt}{{\boldsymbol{d}}}_{{E}_{u}}({\boldsymbol{k}})={\psi }_{0}({\boldsymbol{k}})\{\hat{{\boldsymbol{z}}}{k}_{x},\hat{{\boldsymbol{z}}}{k}_{y}\}.\end{array}$$

(14)

here listed in the convenient d-vector notation for spin-triplet pairing states (see ref. ⁶⁸). It is important to note that all composite phases from combination of two pairing states of one-dimensional representation are c-axis equal spin state and would be in agreement with present time NMR Knight data^24,25 and had been proposed as possible states in refs. ^69,70. These states are also called helical state in literature, as they are topologically non-trivial with helical surface states. The Knight shift experiments disagree with expectations of the state in representation E_u which yields the chiral p-wave state.

Again we have to make composite states of the one-dimensional representation to obtain TRSB phases. Analogous to the even parity case we do not find any composite state which satisfies both selection rules, in contrast to the chiral p-wave state which behaves the same way as the chiral d-wave state in this respect.