Kohei Matsuura,Masaki Roppongi,Mingwei Qiu,Qi Sheng,Yipeng Cai,Kohtaro Yamakawa,Zurab Guguchia,Ryan P. Day,Kenji M. Kojima,Andrea Damascelli,Yuichi Sugimura,Mikihiko Saito,Takaaki Takenaka,Kota Ishihara,Yuta Mizukami,Kenichiro Hashimoto,Yilun Gu,Shengli Guo,Licheng Fu,Zheneng Zhang,Fanlong Ning,Guoqiang Zhao,Guangyang Dai,Changqing Jin,James W. Beare,Graeme M. Luke,Yasutomo J. Uemura,Takasada Shibauchi
Two superconducting states with broken time-reversal symmetry in FeSe1−xSx
iron-based superconductors,superconducting gap,muon spin relaxation,unconventional superconductivity,Bogoliubov Fermi surface
Significance When metal becomes superconducting, the Fermi surface becomes unstable, and a superconducting gap forms. In unconventional superconductors, the gap varies in momentum space and can become zero at points or lines on the original Fermi surface. More exotic superconducting states with time-reversal symmetry breaking (TRSB) have been theoretically proposed, but such states are rarely found. Our muon experiments on iron chalcogenide superconductors FeSe1−xSx in the orthorhombic and tetragonal phases provide evidence for TRSB below the transition temperatures. Moreover, the TRSB superconducting state in the tetragonal phase has a unique feature that not all electrons become superconducting. Our results reveal that this system exhibits two distinct superconducting states both with TRSB, which provides research directions for exotic superconductivity.
Two superconducting states with broken time-reversal symmetry in FeSe1−xSx

When metal becomes superconducting, the Fermi surface becomes unstable, and a superconducting gap forms. In unconventional superconductors, the gap varies in momentum space and can become zero at points or lines on the original Fermi surface. More exotic superconducting states with time-reversal symmetry breaking (TRSB) have been theoretically proposed, but such states are rarely found. Our muon experiments on iron chalcogenide superconductors FeSe1−xSx in the orthorhombic and tetragonal phases provide evidence for TRSB below the transition temperatures. Moreover, the TRSB superconducting state in the tetragonal phase has a unique feature that not all electrons become superconducting. Our results reveal that this system exhibits two distinct superconducting states both with TRSB, which provides research directions for exotic superconductivity.

Recent studies of FeSe1−xSx using high-quality single crystals up to x ∼ 0.25 have shown that electronic nematic order sets in at the structural transition temperature Ts, which can be completely suppressed by S substitution without inducing any magnetic ordering under ambient pressure (1, 2). The phase diagram of FeSe1−xSx exhibits a nematic quantum critical point at x ≈ 0.17 (3, 4), near which anomalous non-Fermi liquid transport properties have been reported (5, 6). The superconducting gap structure of FeSe1−xSx inside the nematic (orthorhombic) phase is very anisotropic, and it has been discussed that Δ(k) exhibits accidental line nodes or deep minima (7–12), indicating the unconventional nature of its superconductivity. In the nematic phase with C2 symmetry, the superconducting gap has twofold rotational symmetry, which can be described by the sum of two contributions Δs and Δd having s-wave and d-wave symmetries, respectively (13, 14). We should also consider the effect of nematic domains whose orientations are orthogonal to each other, and in neighboring domains, one of the s- and d-wave contributions has opposite signs, such as Δs ± Δd (13). Scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) studies on FeSe have shown that the nematic twin boundary affects the low-energy excitation spectrum, opening a gap near the boundary (8, 11). It has been suggested that across the twin boundary, Δs + Δd may be transformed into Δs − Δd through a time-reversal symmetry breaking (TRSB) state Δs + eiθΔd with 0 < θ < π, which is consistent with the gap opening near the boundary. However, direct evidence of TRSB has not been reported in superconducting FeSe. In the tetragonal phase of FeSe1−xSx (x > 0.17), the nematic order disappears, and thus, no twin boundary is observed in the STM measurements (15). It has been reported that although the electronic structure evolves smoothly across the critical concentration (15, 16), the superconducting properties change abruptly (15, 17, 18). In the tetragonal phase, the STM conductance spectra in the superconducting state have found a substantial zero-bias value, and the specific heat and thermal conductivity show large residuals at low temperatures. Recent theoretical studies have suggested that FeSe1−xSx may have an ultranodal superconducting state with Bogoliubov Fermi surfaces (BFSs), which can account for these unusual properties reported in the tetragonal phase. In conventional s-wave superconductors, the gap is almost isotropic in momentum space, and the quasiparticle density of state N(E) is essentially absent at low energies (Fig.1A). In unconventional superconducting states, the superconducting gap Δ(k) on the normal-state Fermi surfaces may have points or lines of zeros (nodes). Consequently, the low-energy quasiparticle excitations are governed by the nodal structure, as summarized in Fig.1 B and C. In contrast, the novel ultranodal superconducting state has extended two-dimensional surfaces of nodes (Δ(k)=0). These BFSs lead to extended zero energy states at zero field like a normal metal (Fig.1D). It has been shown theoretically by Agterberg et al. (19, 20) that such an exotic pairing state with BFSs can be energetically stable under certain conditions, if the system breaks TRS. A recent theoretical study considers a multiband system with intraband and interband interactions with broken TRS and suggests an ultranodal pair state with BFSs, which may apply to FeSe-based superconductors (21, 22). It is then extremely important to clarify whether TRS is broken or not in the FeSe1−xSx-pagination experimentally.

TRSB in the superconducting state can be studied by the zero-field muon spin relaxation (μSR) technique, which is a sensitive probe of the internal magnetic field inside the sample (23). When TRS is broken in the superconducting state, a small magnetic field can be induced near defects or TRSB domain boundaries (24), which can be detected by the change in the relaxation rate of μSR asymmetry. As shown in Fig.2A –C, we have measured zero-field μSR spectra at low temperatures for the collection of single crystals of FeSe (x = 0) in the nematic phase and FeSe1−xSx (x = 0.20 and 0.22) in the tetragonal phase. The comparisons of relaxation curves in the normal state above Tc (red symbols) and in the superconducting state below Tc (blue symbols) reveal a faster relaxation in the superconducting state for all samples. The relaxation rate Λ of the samples can be extracted through the fitting analysis considering background contributions (SI Appendix), and the obtained temperature dependence of Λ is shown in Fig.2D –F. The small and almost temperature-independent relaxation in the normal state is likely due to randomly oriented nuclear magnetic moments whose fluctuation speed is slower than the μSR time scale. For all the samples we measured, the relaxation rate increases below Tc, and the increase in Λ is observed in both non-spin-rotated (NSR) and spin-rotated (SR) configuration modes (Fig.2E). These results indicate that a finite magnetic field develops in the sample in the superconducting state. Moreover, for the samples tested under longitudinal magnetic fields of 200–2,000 G (x = 0 and 0.22), we find that the relaxation rate shows no enhancement below Tc (Fig.2 D and F), implying that the internal field below Tc observed in the zero-field μSR measurements is not fluctuating but static on the microsecond timescale. The increase of the relaxation rate is similar in magnitude both in the nematic and tetragonal phases, and the internal magnetic field at the lowest temperatures can be estimated as Bint ∼ 0.09, 0.12, and 0.07 G for x = 0, 0.20, and 0.22, respectively (Fig.2D–F). These values are significantly larger than the inevitable residual field (< 5 mG) in our shielded experimental setup. These results strongly indicate that TRS is broken in both nematic and tetragonal phases of superconducting FeSe1−xSx. In addition to the zero-field measurements, we have also performed the transverse-field μSR measurements under external fields of 330 G. From the relaxation rate analysis in the superconducting state (SI Appendix), we can extract the temperature dependence of the superfluid density, which is proportional to λ−2 ∝ ns/m*, where λ is the in-plane magnetic penetration depth and m* is the effective mass of quasiparticles. This gives information on the superconducting electron density ns, which can be compared with the total electron density n. The temperature dependence of λ−2 for x = 0 and 0.10 in the nematic phase and for x = 0.20 and 0.22 in the tetragonal phase is shown in Fig.3A. The data for all samples deviate from the conventional s-wave curve, consistent with the anisotropic gap in this system. We also note that the λ−2(T) data for FeSe (x = 0) are in good agreement with the previous μSR measurements for a 500-G c-axis field (25) (SI Appendix, Fig. S3). The absolute values of λ−2 in the zero-temperature limit become smaller in the tetragonal phase compared with those in the nematic phase (Fig.3A, Inset). Notably, this is opposite to the increasing trend of Fermi surface volumes as a function of x in FeSe1−xSx reported from the quantum oscillation measurements (16). More quantitatively, in Fig.4C, we compare the x-dependence of λ−2(T → 0) measured by μSR and that estimated from the Fermi surface structures based on quantum oscillations measurements (16) (SI Appendix). In FeSe (x = 0), the λ−2(T → 0) value is quantitatively consistent with the estimate from all measured orbits in quantum oscillations, indicating that all the carriers in the normal state are condensed into the superconducting pairing state. The comparison between measured and estimated λ−2 in FeSe1−xSx (Fig.4C) demonstrates that the superfluid density in the tetragonal phase is substantially suppressed from that expected from the normal-state electronic structure. Together with the reported phase diagram (Fig.4A) and the evolution of low-energy excitations with S composition x (Fig.4B), we can summarize the unusual transformation of the superconducting state from the SC1 phase to SC2 phase, driven by the disappearance of the nematic order in FeSe1−xSx, as follows. i) The superconducting transition temperature Tc shows an abrupt change between SC1 in the nematic phase and SC2 in the tetragonal phase (18). ii) The zero-bias tunneling conductance (15) and electronic specific heat (17, 18) at low temperatures are both largely enhanced in the SC2 phase, indicating the presence of low-lying quasiparticle excitations in the tetragonal phase. iii) The superfluid density shows a sizable suppression in the SC2 phase from that expected from the normal-state electronic structure. iv) TRS is broken in both SC1 and SC2 phases. These results indicate that FeSe1−xSx exhibits two different TRSB superconducting states SC1 and SC2, which are separated by the end point of the nematic order. The suppressed superfluid density in the tetragonal phase implies that not all electrons are condensed in the superconducting ground state, which is quite anomalous. The presence of unpaired electrons in the zero-temperature limit is not expected even in nodal superconductors but consistent with the presence of zero-energy excitations reported by the STM and specific heat measurements. These results in the SC2 phase are consistent with an ultranodal pair state with emergent BFSs suggested theoretically.

Here, we emphasize that the crystals used in this study are grown by the chemical vapor transport technique (3, 26), which is known as a method to obtain high-quality FeSe-based samples (1). The observation of quantum oscillations in a wide range indicates that the crystals are clean, and the STM results show that S atoms are distributed quite uniformly (15, 16). It is therefore highly unlikely that the low-lying excitations in SC2 originate from impurities or chemical inhomogeneities. This is reinforced by the STM measurements in the tetragonal phase, which have shown that the presence of zero-bias conductance is a robust feature insensitive to the positions (15). The fact that the onsets of these extraordinary excitations and the suppressions of Tc and λ−2 match the fate of the nematic order strongly suggests that the emergence of an ultranodal state is an intrinsic property triggered by the electronic change associated with the nematicity in this system. One may also consider the effect of multigap and the possibility of gapless Fermi surface in some bands, which could lead to the unpaired electrons. However, it has been widely discussed that the interband interactions are generally important for iron-based superconductors, and thus, there must be nonnegligible coupling between different bands. In such a case, it is natural to consider that all the bands have superconducting ground states even if they have different gap magnitudes. In principle, some of the bands may have very small gaps, but the temperature dependence of the superfluid density down to very low temperatures (Fig.3A) does not show any signatures of such an extremely small gap. Thus, we conclude that the multigap effect is unlikely the origin of the unpaired electrons. We should note that the theoretical model of ultranodal states for FeSe1−xSx does not provide microscopic origins at present (21). Moreover, it is not clear how the BFSs can relate to nematicity. Thus, a complete understanding of the observed behaviors requires further studies. However, in recent theoretical calculations of the diamagnetic current response, it has been shown that there is a limited range of degree of TRSB to have stable BFSs, in which the superfluid density becomes less than half of the original value (22). This appears to be in good correspondence with the case for SC2 in the tetragonal phase of FeSe1−xSx. The observed TRSB in the nematic phase of FeSe, in which no evidence of BFSs is found, also poses an intriguing question about the structure of the phase of the order parameter in the presence of the nematic twin boundary. As mentioned above, near the nematic twin boundary, we expect a large change in the phase θ in the complex order parameter Δs + eiθΔd. If the TRS is preserved deep inside the bulk, we expect θ = 0 or π away from the boundaries. Our zero-field μSR results show a similar change of relaxation rate below Tc in FeSe and tetragonal FeSe1−xSx with and without nematic twin boundaries, which imply that TRS is broken inside the bulk even for FeSe. In such a nematic TRSB superconducting state, θ is shifted from 0 or π in the bulk, but still, θ is different in the neighboring domains, and the phase can change near twin boundaries. This suggests that the degree of TRSB depends on the position near twin boundaries, which can explain the STM and ARPES results. Indeed, the original theory considering the orthorhombic domains by Sigrist et al. (13) has shown that the state with broken TRS only near twin boundaries exists in a limited temperature region above the bulk TRSB state. In FeSe, the superconducting gap structure is very anisotropic and twofold symmetric around the hole and electron bands (10, 11), which suggests that Δs and Δd have similar magnitudes. In such a case, it is plausible that the TRSB sets in very close to Tc, which seems consistent with the observed enhancement of the relaxation rate just below Tc. In addition to the TRSB and gap node topology, our present results for the superfluid density provide insights into phenomenologies of unconventional superconductors. Fig. 3B shows correlations between Tc and the effective Fermi temperature TF = EF/kB derived from the effective superfluid density ns/m* (27, 28). The points from FeSe1−xSx lie close to the nearly linear trend between Tc and TF reported in hole-doped and electron-doped cuprates, organic, and heavy-fermion superconductors (27, 28), which is a hallmark of strongly correlated superconductivity. This may suggest that the sudden decrease in Tc above the nematic end point is closely related to the reduction in the superfluid density due to the emergence of BFSs in the tetragonal phase (Fig.4 A and C). The doping evolution of FeSe1−xSx resembles the behaviors of overdoped cuprates, in the departure of superfluid density from the normal-state carrier density (28), existence of nematic quantum critical point (29), and orthorhombic to tetragonal transition. This analogy encourages high-precision experimental searches for TRSB in overdoped cuprates. Our μSR measurements reveal two different TRSB superconducting states in the phase diagram of FeSe1−xSx. The suppression of nematicity leads to the reduction in Tc as well as the suppressed superfluid density, which indicates that the superconductivity in the tetragonal phase is very exotic, having broken TRS and unpaired electrons. The results are consistent with the ultranodal state with BFSs proposed theoretically, but details of the BFSs including the size and the location in the momentum space require further studies. Together with the recent reports of Bose–Einstein condensation-like superconductivity in FeSe1−xSx (18, 30), our findings may open up a wide field of studies on the extended zero-energy excitations in fermionic bound states.

Single crystals of FeSe1−xSx (x = 0, 0.10, 0.20, and 0.22) used in this study were synthesized by the chemical vapor transport method (3, 26). For the μSR measurements, a sample mass density of about 200 mg/cm2 is required to stop the muon in the sample. For the pure FeSe samples, typical crystal dimensions are about 1 × 1 mm2 in the ab plane and several tens of μm in the c-axis direction and weigh about 1 to 5 mg. The sizes of the S-substituted samples are even smaller. Therefore, for each x, a large number of single-crystal samples (up to 100 pieces) were coaligned on a silver plate with Apiezon N grease (whose volume is around 10% of the sample mass) to cover an area of ∼1 cm2, so that the c axes were aligned perpendicular to the plate. For x = 0, 0.10, and 0.22, crystals were collected from a single batch each, but for x = 0.20, we used a few batches to extract enough crystals with a total mass of ∼170 mg. Energy-dispersive X-ray spectroscopy was used to determine the compositions of S-substituted samples by averaging the x values obtained for several samples taken from the batches. For x = 0.22, we also measured the temperature dependence of resistivity on a crystal taken from the same batch, and it was confirmed that there was no anomaly due to structural phase transition and that the superconducting transition temperature was around 4 K. In addition, the c-axis lattice parameters determined by X-ray diffraction experiments are consistent with the previous reports (3, 26).

μSR measurements were performed at the Centre for Molecular and Materials Science at TRIUMF in Vancouver, Canada. We conducted zero-field μSR measurements above 2 K with the M20D beamline, while the measurements below 2 K were conducted on the M15 surface muon channel with a dilution refrigerator. Stray magnetic fields at the sample position at M20D were measured by fluxgate magnetometer and found to be ∼2, 1, and 5 mG along three orthogonal directions. Stray fields were reduced to less than 5 mG in all directions in the dilution refrigerator using the method of ref. (31). The μSR analysis was performed in the time domain using the program MUSRFIT (32). More detailed information for the procedure of data analysis is described in SI Appendix.

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