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, TTRSB, is generally found to match the critical temperature, Tc, within resolution. In combination with evidence for even parity, this result has led to consideration of a dxzâ±âidyz order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding TTRSBâ=âTc, but it has a hard-to-explain horizontal line node at kzâ=â0. Therefore, sâ±âid and dâ±âig order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain TTRSBâââTc. 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 Sr1.98La0.02RuO4 at zero pressure. Both hydrostatic pressure and La substitution alter Tc without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected, TTRSB should track changes in Tc, while if it is accidental, these transition temperatures should generally separate. We observe TTRSB to track Tc, supporting the hypothesis of dxzâ±âidyz order.
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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) superconductivity1,2,3,4,5,6,7,8,9,10,11.
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 pxâ±âipy or dxzâ±âidyz 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 Tc, the superconducting critical temperature, and the lower temperature TTRSB, 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 superconductors11,12, while both (U,Th)Be131,4 and UPt32,3,8 have split Tc and TTRSB.
Here, we study Sr2RuO4, an unconventional superconductor13,14, 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 superconductivity18,19,20. For two decades, the leading candidate state to explain these and other observations was the chiral p-wave state pxâ±âipy (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 fields21,22,23 and the recently discovered drop in the NMR Knight shift below Tc24,25. In combination with the above experimental support for TRSB superconductivity, this evidence compels consideration of dxzâ±âidyz order.
dxzâ±âidyz order would be a surprise because it has a line node at kzâ=â0, which under conventional understanding requires interlayer pairing, while in Sr2RuO4 interlayer coupling is very weak. It has been proposed that dxzâ±âidyz 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 URu2Si227,28. 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, sâ±âidxy30 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 dxzâ±âidyz, these require tuning to obtain TcâââTTRSB 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 Sr2âyLayRuO4. Both of these perturbations maintain the tetragonal symmetry of the lattice. If the order parameter has single-representation nature, TTRSB will track Tc. If the order parameter is of the composite-representation kind, with TTRSB matching Tc in clean, unstressed samples through an accidental fine tuning, then perturbations away from this point should in general split TTRSB and Tc, whether they preserve tetragonal lattice symmetry or not33. Here, we have observed a clear suppression of TTRSB at a rate matching the suppression of Tc. 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 \(\varnothing \sim 3\)âmm were affixed to oxygen-free copper foils, and assembled into an approximately cylindrical collection of total diameter \(\varnothing \sim 7\)âmm and total length l ~12âmm (see Fig. 1a). The c-axes of the separate crystals were aligned to within 3â.
The pressure cell used in the present study (refs. 34,35 and Fig. 1b) is a modification of a âclassic μSRâ clamped pressure cell35,36. 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 response34,35,36. 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 97âMeV/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 pâââ2.3âGPa37. The maximum pressure reached here was 0.95âGPa, 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 Tc was observed to decrease linearly with pressure, whereas in-plane uniaxial stress on a GPa scale causes a strong non-linear increase in Tc38.
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 Tc of 1.30(6) K (see Supplementary Fig. 1 in Supplementary Note 1), slightly below the limit of Tc of 1.50 K for a pure sample.
Tc and TTRSB 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, Pμ(t), can be determined; the time-evolution of this polarisation is determined by the magnetic fields in the sample40.
Tc is determined through transverse-field (TF) measurements. An external field Bext of 3âmT, as is generated by Helmholtz coils, was applied parallel to the crystalline c-axis and perpendicular to the initial muon spin polarisation Pμ(0). Measurements were performed in the field-cooled (FC) mode. Details of the method and analysis are given in the âMethodsâ section.
Example TF-μSR time spectra at pressure pâ=â0.95âGPa, and at a temperature above Tc and one below, are shown in Fig. 2a. Above Tc, the spins of muons stopped in both the sample and the pressure cell walls precess with frequency Ïâ=âγμBext (where γμâ=â2ÏâÃâ135.5âMHz/T is the muon gyromagnetic ratio). The muon spin polarisation is seen to relax substantially on a 10âμs 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 Tc, 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.
TF-μSR measurements were performed at 0, 0.25, 0.62, and 0.95âGPa. Data at 0 and 0.95âGPa 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 âMethodsâ section), respectively. From the sample contribution we extract a Gaussian relaxation rate, Ï, and the diamagnetic shift of the field inside the sample, BintâââBextâââMFC41 (MFC is the FC magnetisation). Figure 2b, c, respectively, shows the temperature dependence of Ï and BintâââBext. Ï is given by \({\sigma }^{2}={\sigma }_{\,\text{sc}}^{2}+{\sigma }_{\text{nm}\,}^{2}\), where Ïsc and Ïnm are the flux-line lattice and nuclear moment contributions, respectively. \({\sigma }_{\text{sc}}\propto {\lambda }_{ab}^{-2}\), where λab is the in-plane magnetic penetration depth; see ref. 42 and the âMethodsâ section. The onset of superconductivity can be seen in both Ï and BintâââBext, as a transition rounded on a scale of ~0.1 K. The heat capacity measurements show a similar distribution of Tcâs; see Supplementary Fig. 1 in Supplementary Note 1.
The pressure dependence of Tc 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 Tc. When fitting Ï(T) and Bint(T) with model functions, the statistical error on the Tcâs extracted is considerably smaller, meaning that the error on changes in Tc is low. A linear fit to Tc(p) yields a slope dTc/dpâ=ââ0.24(2)âK/GPa, which is in good agreement with literature data43,44,45. The unpressurised Tc 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.
TTRSB is determined through ZF measurements. The signature of TRSB is an enhancement in the muon spin relaxation rate below TTRSB, indicating the appearance of spontaneous magnetic fields. In these measurements, external fields were compensated to better than 2âμT, ruling out flux lines below Tc as the origin of this signal. An example of ZF-μSR time spectra above and below Tc, showing the faster relaxation below Tc, at pâ=â0.95âGPa 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 âMethodsâ 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 0.95âGPa, for independent measurements with the initial muon spin polarisation Pμ(0)â¥c and â¥ab, are shown in Fig. 2e, f; ZF data at 0.25 and 0.62âGPa are shown in Supplementary Figs. 3 and 4 in Supplementary Note 2.
To extract TTRSB, λ(T) is fitted with the following functional form:
λ0 is the relaxation rate above TTRSB, and Îλ is the enhancement due to spontaneous magnetic fields. Where data were obtained both for Pμ(0)â¥c and â¥ab, the exponent n is constrained to be the same for both polarisations. TTRSB, λ0, and Îλ were obtained independently for each pressure and muon spin polarisation. The resulting values of TTRSB are plotted in Fig. 2g.
Our ZF data yield the following three results:
-
(1)
Where data were taken both for Pμ(0)â¥c and â¥ab (that is, at 0 and 0.95âGPa), TTRSB and Îλ were found to be the same within resolution for both polarisations. [At 0âGPa, Îλâ=â0.027(4) and 0.033(3)âμsâ1, and at 0.95âGPa, 0.030(4) and 0.025(3)âμsâ1, for Pμ(0)â¥ab and Pμ(0)â¥c, respectively.] This agrees with the zero-pressure results of Luke et al.15. Because Îλ reflects fields perpendicular to Pμ(0), this result indicates that the spontaneous fields have no preferred orientation.
-
(2)
Îλ was found to be pressure-independent within resolution (including all pressures investigated: 0, 0.25, 0.62 and 0.95âGPa), having an average value of Îλâ=â0.026(2)âμsâ1. This value corresponds to a characteristic field strength BTRSBâ=âÎλ/γμâ=â0.031(2) mT. BTRSB 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).
-
(3)
A linear fit yields TTRSB(p)â=â1.27(3)âKâââpââ â0.29(5)âK/GPa. In other words, within resolution the rate of suppression of TTRSB under hydrostatic pressure matches that of Tc.
μSR on Sr1.98La0.02RuO4
Substitution of La for Sr adds electrons to the Fermi surfaces; in Sr2âyLayRuO4 this doping drives the largest Fermi surface through a Lifshitz transition from an electron-like to a hole-like geometry, at yâââ0.2050,51. At yâ=â0.02, the change in Fermi surface structure is minimal, and the main effect of the La-substitution is to suppress Tc, through the added disorder. Heat capacity data, measured on a small piece cut from the μSR sample, give Tcâ=â0.70(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 28âMeV/c, giving of ~0.2âmm implantation depth40. Representative TF-μSR time spectra above and below Tc, where the applied field is Bextâ=â2âmT parallel to the crystalline c-axis, are shown in Fig. 3a. Below Tc, the muon spin polarisation relaxes almost completely on a 10âμs time scale, showing that essentially the entire sample volume is superconducting. The TF Gaussian relaxation rate Ï is shown in Fig. 3b, and BintâââBext in Fig. 3c. These measurements yield Tcâ=â0.75(5) K. The heat capacity data are also shown in Fig. 3b.
ZF-μSR data are presented in Fig. 3d, e. Fitting with Eq. (1) returns Îλâ=â0.007(1)âμsâ1 and TTRSBâ=â0.8(1) K. This Îλ is noticeably smaller than that obtained from the undoped Sr2RuO4 sample, corresponding to an internal field BTRSBâââ0.01âmT. It is, however, within the range of previous results47. In qualitative agreement with data on a lower Tc Sr2RuO4, reported in ref. 46, though here with more data at Tâ>âTc to be certain of the base relaxation rate, this low value of Îλ shows that BTRSB is not straightforwardly related to defect density. At present, the origin of the sample-to-sample variation in BTRSB is unknown.
Longitudinal-field (LF) measurements can be employed to determine whether internal fields are static or fluctuating. If BTRSB is static, under an applied field parallel to Pμ(0) that is considerably larger than BTRSB, muon spin precession is greatly restricted and the spin polarisation does not relax (i.e. the muon spins decouple from BTRSB). In contrast, fluctuating BTRSB can still relax the muon spin polarisation40. Data shown in Fig. 3d, e indicate that Bextâ£â£Pμ(0)â=â3âmT fully suppresses the muon spin relaxation, and therefore that BTRSB 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 10âmT, considerably stronger than that for Sr2RuO4.
Heat capacity measurements
The specific heat measurements were performed at ambient pressure for several pieces of Sr2-yLayRuO4 single crystals. The results are presented in Fig. 3b, e for Sr1.98La0.02RuO4 (yâ=â0.02) and in Supplementary Fig. 1 in Supplementary Note 1 for Sr2RuO4 (yâ=â0.0), respectively. The specific heat jumps at Tc (ÎCel/γnTc, γn 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 Sr2-yLayRuO4 samples. Here \({T}_{{\rm{c}}}^{{\rm{SH}}}\) denotes the superconducting transition temperature determined from Cel/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 Tc. Remarkably, ÎCel/γnTc vs. Tc data points scale as \({T}_{{\rm{c}}}^{\alpha }\) with αâââ0.65, which is distinctly different from the BCS behaviour, where αâ=â0 (ÎCel/γnTcâ=âconst). Just a single point at Tcâââ3.5âK 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 âanisotropyâ 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 Tc.
Such scaling behaviour is rarely observed since the ratio ÎCel/γnTc is sensitive to a change of the superconducting gap structure and symmetry. Note that a similar scaling is reported for Fe-based superconductors, where ÎCel/γnTc follows approximately the BNC (Budâko-Ni-Canfield) scaling behaviour \({{\Delta }}{C}_{{\rm{el}}}/{\gamma }_{{\rm{n}}}{T}_{{\rm{c}}}\propto {T}_{{\rm{c}}}^{\alpha }\) with αâââ253, which is considered to be a consequence of the unconventional multiband sâ±âsuperconductivity. The change of the superconducting pairing state in the Ba1âxKxFe2As2 system results in abrupt change of the scaling behaviour leading to an intermediate sâ+âis state11. The monotonic ÎCel/γnTc vs. Tc 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 Tc and TTRSB47. Uniaxial pressure drives a strong increase in Tc, while TTRSB varies much more weakly, probably decreasing slightly with initial application of pressure. The microscopic mechanism yielding the signal observed at TTRSB, 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 experimentally54,55. 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 Tc 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 TTRSB, and it was further argued that this signal is extremely difficult to obtain from a purely magnetic mechanism. Nevertheless, the weak observed variation of TTRSB, while Tc 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 TTRSB with hydrostatic stress, at a rate matching the suppression of Tc. This result further strengthens the evidence that enhanced muon spin relaxation is an indicator of TRSB superconductivity: TTRSB tracks Tc 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 TTRSB versus Tc. The data reported here, on hydrostatically pressurised Sr2RuO4 and on unpressurised Sr1.98La0.02RuO4, fall on the TTRSBâ=âTc line, while the uniaxial pressure data from ref. 47 clearly deviate from this line.
Our central finding that TTRSB tracks Tc provides further support for the single-representation dxzâ±âidyz order parameter. Importantly, homogeneous dxzâ±âidyz is the only spin-singlet order parameter consistent with the selection rules imposed by ultrasound and Kerr effect data. Ultrasound data on Sr2RuO456,57 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 Tc for transverse modes. While these experimental results are not sensitive to the spin configuration, they impose other stringent conditions on the possible pairing symmetries58,59. 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 data24,25. 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 sâ+âidxy states are constructed to be compatible with the ultrasound measurements, but they are not chiral31,60. 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 âMethodsâ 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, TTRSB could be locked to Tc by hypothesising that superconductivity appears at the dislocations before the bulk, but tuning would then be required to obtain Tc and TTRSB that split under modest uniaxial stress.
Major challenges to dxzâ±âidyz order are the absence of a resolvable second heat capacity anomaly at TTRSB 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 coupling26,64,65.
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 dxzâ+âidzy-wave state, while all composite-representation states suffer from several deficiencies. We note, however, that there are also empirical challenges to a hypothesis of dxzâ±âidyz, 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
Sr2-yLayRuO4 single crystals
Single crystals of Sr2-yLayRuO4 were grown by means of a floating zone technique39. Samples for measurement under hydrostatic pressure (with yâ=â0) were cut from two rods, C140 and C171, that each grew along a ã100ã crystallographic direction. The rods have diameter \(\varnothing \simeq 3\)âmm. Two sections of length 8â12âmm 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 Sr2-yLayRuO4 crystal of length 8âmm. The La concentration was analysed by an electron-probe micro-analysis and was found to be yâââ0.02. Before the μSR measurements, this rod was then cleaved into two semi-cylindrical pieces, again with the flat faces â¥c.
The x-ray diffraction experiments performed on small powdered pieces cut from of each particular rod gave aâ=â0.3867ânm, câ=â1.273ânm for pure Sr2RuO4 and aâ=â0.3865ânm, câ=â1.274ânm for La-substituted sample.
Specific heat of Sr2-yLayRuO4 at ambient pressure
Specific heat measurements were performed at zero pressure for several pieces of Sr2-yLayRuO4 single crystals, cut from the rod used for μSR measurements.
For Sr2RuO4 used in hydrostatic pressure measurements, the electronic specific heat capacity Cel/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 Cel/T for a small piece cut from the Sr1.98La0.02RuO4μSR 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 ÏE1 beamlines, using the GPD35, and Dolly spectrometers (Paul Scherrer Institute, PSI Villigen, Switzerland). At the GPD instrument, experiments under pressure up to pâââ0.95 GPa on undoped Sr2RuO4 were performed. At the Dolly spectrometer, measurements of Sr1.98La0.02RuO4 at ambient pressure were conducted. At both instruments 4He cryostats equipped with the 3He insets (base temperature Tâââ0.25 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 Pμ(0) (TF-μSR) were performed. In two sets of ZF-μSR studies, Pμ(0) was set to be parallel to the c-axis and along the ab-plane, respectively. In TF-μSR measurements the small 3âmT magnetic field was applied parallel to the c-axis and perpendicular to Pμ(0).
At the Dolly instrument, in addition to ZF- and TF-μSR experiments, the LF measurements were performed. In these studies 3âmT magnetic field was applied parallel to the c-axis and to the initial muon spin polarisation Pμ(0).
The experimental data were analysed by separating the μSR signal on the sample (s) and the background (bg) contributions66:
Here A0 is the initial asymmetry of the muon spin ensemble, and As (Abg) and Ps(t) [Pbg(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:
Here Bint is the internal field in the sample, Ï is the initial phase of the muon spin ensemble, and γμâââ2ÏâÃâ135.5âMHz/T is the muon gyromagnetic ratio. The Gaussian relaxation rate Ï consists of the âsuperconductingâ, Ïsc, and nuclear moment, Ïnm, contributions and it is defined as: \({\sigma }^{2}={\sigma }_{{\rm{sc}}}^{2}+{\sigma }_{{\rm{nm}}}^{2}\). Here, Ïsc and Ïnm 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, Ïnm was assumed to be constant over the entire temperature range and was fixed to the value obtained above Tc, 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:
Here Ïpcâââ0.28âμsâ1 is the field and the temperature-independent relaxation rate of beryllium-copper35, and Bext 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 (Bextâ¥c), the superconductig contribution into the Gaussian relaxation rate Ïsc becomes proportional to the inverse squared in-plane magnetic penetration depth λab42. The proportionality coefficient between Ïsc 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 Ï and the diamagnetic shift BintâââBext are presented in Figs. 2b, c and 3b, c for Sr2RuO4 and Sr1.98La0.02RuO4 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:
Here GKT(t) is the Gaussian Kubo-Toyabe (GKT) relaxation function describing the magnetic field distribution created by the nuclear magnetic moments40,67:
ÏGKT 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:
with the temperature-independent relaxation rate ÏGKT,BeCuâââ0.35âμsâ135.
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 [Pμ(0)â¥ab and Pμ(0)â¥c] and pressure (pâ=â0.0, 0.25, 0.62 and 0.95âGPa) were fitted simultaneously with As, Apc, \({\sigma }_{{\rm{GKT,S{r}}_{2}Ru{O}_{4}}}\), ÏGKT,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 Sr2âyLayRuO4 leads to the corresponding Gaussian Kubo-Toyabe relaxation rate being nearly zero. Consequently, the analysis of ZF- and LF-μSR data for Sr1.98La0.02RuO4 was performed by using the simple-exponential decay function:
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 Sr1.98La0.02RuO4 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 D4h the even parity spin-singlet pairing states can be listed according to the irreducible representations of D4h, four one-dimensional ones A1g, A2g, B1g, B2g and a two-dimensional one Eu. The pair wave function ÏÎ(k) of the corresponding states are given by:
where Ï0(k) is a function of k invariant under all symmetry operations of the tetragonal lattice. We list here first the composite-representation TRSB states:
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:
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 results24,25. 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 ϵxyâ=âϵyx58,59. 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 B2g58,59,68. For the observed renormalisation of the speed of sound the superconducting order parameter has to couple linearly to ϵxy, thus, requiring that B2g is contained in the decomposition of \({\tilde{{{\Gamma }}}}_{j}\otimes {\tilde{{{\Gamma }}}}_{j}\). This only possible for \({\tilde{{{\Gamma }}}}_{3},{\tilde{{{\Gamma }}}}_{4}\) and \({\tilde{{{\Gamma }}}}_{7}\):
and
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, Bz which belongs to the representation A2g. 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:
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 data24,25 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 Eu 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.
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or in the Supplementary Information. Other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The work was performed at the Swiss Muon Source (SμS), Paul Scherrer Institute (PSI, Switzerland). We acknowledge fruitful discussions with Zurab Guguchia, Carsten Timm and Jing Xia. Matthias Elender is acknowledged for technical support. The work of R.G. is supported by the Swiss National Science Foundation (SNF Grant No. 200021-175935). The work of M.S and B.Z was financially supported by the Swiss National Science Foundation (SNSF) through Division II (Grant No. 184739). The work of V.G. was supported by DFG GR 4667/1. N.K. acknowledges the support from JSPS KAKENHI (Nos. JP18K04715, and JP21H01033) in Japan. Y.M. acknowledges funding by JPJS-CNR-SPIN Core-to-core programme (No. JPJSCCA20170002), and by JPJS KAKENHI Nos. JP15H05852, JP15K21717, and JP17H06136. The work of H.-H.K. was supported by DFG SFB 1143 and GRK 1621.
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R.K., V.G. and M.S. conceived the project. Data were taken by R.K., V.G., D.D. and R.G. R.K. and V.G. performed data analysis and interpreted the results together with M.S. B.Z. and M.S. provided the theoretical analysis. N.K. provided and characterised samples. R.K., V.G., M.S. and C.W.H. wrote the manuscript with inputs from all authors: D.D., R.G., B.Z., N.K., Y.M. and H.-H.K.
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Grinenko, V., Das, D., Gupta, R. et al. Unsplit superconducting and time reversal symmetry breaking transitions in Sr2RuO4 under hydrostatic pressure and disorder. Nat Commun 12, 3920 (2021). https://doi.org/10.1038/s41467-021-24176-8
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DOI: https://doi.org/10.1038/s41467-021-24176-8
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