A central issue in unconventional superconductors is the symmetry and structure of the superconducting gap, which is intimately related to the superconducting (SC) mechanism. In the superconductors reported so far, the gap can be classified into three types: full-gap, in which the gap completely opens; point-node type, in which the gap disappears in zero-dimension (0D); and line-node type, in which the gap disappears in 1D-line, on the Fermi surface (FS). In conventional phonon-mediated superconductors with attractive pairing interaction, a fully-gapped s-wave SC gap appears. On the other hand, in unconventional superconductors mediated by repulsive interactions such as spin fluctuations, anisotropic non-s-wave states with 0D-point or 1D-line nodes may appear. Very recently, however, a new class of SC gap structure has been proposed theoretically, in which the gap nodes appear in a wider region of the Fermi surface forming a 2D nodal plane. This plane, coined as the Bogoliubov Fermi surface (BFS) (1), may appear in multi-band superconductors in the absence of time-reversal symmetry. The SC state with BFSs is suggested to be a fascinating superconducting state, providing several exotic superconducting phenomena, including the spontaneous Volovik effect (1), additional instabilities of the BFS in the SC state (2, 3), and impurity-induced odd-frequency pairing (4). However, several requirements, such as time-reversal symmetry breaking, make the BFS state extremely difficult to achieve. Furthermore, direct observation of this state is challenging because it requires high-energy-resolution experiments in momentum space.
FeSe is one of the most intriguing systems among Fe-based superconductors (FeSCs) (5). It has the simplest crystal structure consisting solely of SC Fe-Se layers (6) and shows electronic nematic order without long-range magnetic ordering under ambient pressure, in contrast to the other FeSCs (7). The nematicity is suppressed by isoelectronic sulphur substitution and a nematic critical point (NCP) appears at x ~ 0.17 in FeSe1-xSx (8). FeSe is a compensated semimetal, whose Fermi surface consists of a very small hole pocket around the G-point and very small electron pockets near the zone corner. Remarkably, the ratio of Fermi energy eF to kBTc has been estimated as 0.3 for the hole pocket and ~1.0 for the electron pocket (9), indicating that FeSe is located in the crossover region from Bardeen–Cooper–Schrieffer (BCS)-type to Bose-Einstein–condensation (BEC)-type superconductivity. It has been shown that the superconducting properties dramatically change when crossing the NCP. In fact, very recent high-energy-resolution laser-based angle-resolved photoemission spectroscopy (laser ARPES) reports that BEC SC feature is pronounced for x = 0.21 in the tetragonal phase beyond the NCP (10). In addition, the scanning tunnelling spectroscopy (STS), thermal conductivity, and specific heat measurements reveal two distinct superconducting pairing states separated by the NCP (11–13). The most mysterious feature in the tetragonal phase beyond the NCP is the presence of an unusually large residual density of states (DOS), despite being a very clean system as revealed by the observation of the quantum oscillations. Based on these results, along with possible time-reversal symmetry breaking reported by mSR measurements (14, 15), it has been theoretically proposed that the observed anomalous large residual DOS may be attributed to the emergence of the BFS (16).
In this study, the detailed SC gap structure of 22 % S-substituted FeSe is determined by the ultralow-temperature and high-resolution laser ARPES measurements. We show that the observed gap structure provides evidence for the emergence of the BFS.
The hole FS around the G point is determined, and the positions of Fermi momenta (kF) are resolved. The symbols in Figs. 1a-c indicate the kF positions, where the energy distribution curves (EDCs) are measured below and above Tc to evaluate the size of the SC gap, obtained from the peak positions of the momentum distribution curves (MDCs) at EF. We stress that the oval FSs are reported in the nematic phase (17). Therefore, the circular-shaped FS, which reflects the four-fold rotational symmetry of the crystal structure, demonstrates that the system is in the tetragonal phase. This is further confirmed by the orbital components of the hole FS. While for the nematic phase of FeSe, the overall hole FS can be observed by either s- or p-polarized light depending on the direction of major and minor axes of the oval FS, a limited part of the FS can be observed by both polarizations for FeSe0.78S0.22. This indicates that comparable contributions from Fe 3dzx and 3dyz orbitals to the hole FS, while in the nematic phase of FeSe, the hole FS is contributed from almost a single component of the Fe 3dzx orbital. We note that although the quantum oscillation measurements detected two hole pockets, only the outer hole pocket shown as a red circle in Fig. 1d can be observed. This is due to the kz dispersion, and the inner pocket sinks below EF around the kz region near the Z point probed by 7-eV laser.
Next, we evaluate the SC-gap size from the detailed analysis of temperature-dependent EDCs. For the EDCs around the FS angle j = 90° (Fig. 2b), the SC-gap opening below Tc can be demonstrated by the observation of a distinct kink as indicated by the arrow in Fig. 2b. The appearance of the kink rather than a coherence peak is consistent with the previous ARPES and STS results in the tetragonal samples of FeSe1-xSx (10–12). In stark contrast, the EDCs around j = 0° (Fig. 2a) cross at the same point on EF below and above Tc, indicating the absence of the SC gap. We note that for j = 90° the gap opens below 25 K well above Tc (Fig. 2d). This pseudogap formation is consistent with the previous study reporting that BEC superconductivity is induced by the disappearance of the nematic ordering in FeSe1-xSx (10).
The EDCs at kF positions shown in Figs. 1a-c have been measured at 2 K below Tc and 25 K in the normal state. The magnitude of the SC gap at 2 K, which is evaluated by fitting the EDCs to the Dynes function (19), is plotted as a function of j in Figs. 3a and 3b. The green and red symbols represent the results measured with s- and p-polarized light, respectively. The error bars of the evaluated SC-gap sizes are ±200 meV as indicated by the solid bars. Because the lowest limit of the evaluable gap size is also 200 meV, for the points where the gap size was evaluated as a lower value by the fitting, only error bars are shown without plots of markers. There are two remarkable features in the angular dependence of the SC gap. First, SC gap disappears in a wide FS-angle range between j ~150° and 210°. Second, the SC gap exhibits a two-fold symmetry despite the four-fold tetragonal crystal structure. We found that the analyses of the gap anisotropy by a simple two-fold symmetric gap function cannot reproduce the observed gap structure, and strong deviations have been found around j ~0° and 180° (see Fig. S2). These results are not consistent with 0D-point and 1D-line nodes, implying the emergence of 2D nodal planes.
The 2D nodal planes observed by the present measurements are consistent with the ultranodal behavior showing large residual density of states reported in specific heat, thermal conductivity, and STS measurements. Indeed, the zero-bias conductance in the STS data for x~0.25 reaches about 1/4 of the normal-state value, which is in reasonable agreement with the estimated volume of the possible BFSs (Fig. 3c) suggested from the momentum dependence of the gap (Fig. S3). These results, together with the broken time-reversal symmetry revealed by mSR (14, 15), provide evidence for the emergence of the BFS.
An unexpected observation is that the extended nodal planes are found around j = 0° and 180°, demonstrating the two-fold symmetry of the superconducting gap, which breaks the four-fold symmetry of the normal-state Fermi surface in FeSe0.78S0.22 with the tetragonal crystal structure (Fig. 1). We argue possible origins of this two-fold symmetry. First is an additional instability of the BFS, which may lead to rotational symmetry breaking at low temperatures (2, 3). Indeed, inversion-symmetry breaking in the BFS state has been proposed theoretically owing to such instability. However, we point out that such a possibility is unlikely because the observed temperature dependence of the EDCs (Fig. 2) shows that the two-fold symmetry seems to be already present in the pseudogap state above Tc, which is associated with the BEC superconductivity. Second is the interplay between the inter- and intra-band pairing interactions. In iron-based superconductors, it is widely discussed that interband interactions between hole and electron pockets can lead to the s± pairing symmetry. In addition, the intraband interactions within the electron pockets promote anisotropic gap instability with dx2-y2 symmetry (20). Although there is no microscopic theory, an interplay between such two different symmetry channels may be responsible for the rotational symmetry breaking in the superconducting state, giving rise to twofold SC gap anisotropy. Indeed, in a theoretical model with anisotropic intraband gaps, there is a parameter range showing twofold BFSs which appear to be consistent with our results (16). The third is the effect of strong spin-orbit interaction, which has been considered to be crucial for the formation of nonmagnetic nematic ordering in FeSe-based superconductors (21). A nematic superconducting state that breaks the normal-state rotational symmetry has been discussed in the topological superconductor candidate, doped Bi2Se3, in which strong spin-orbit interaction is also important (22). It has been found that nematic fluctuations are largely enhanced near the nematic critical point (8), and such fluctuations with the spin-orbit interactions might lead to nematic superconductivity outside the NCP. At present, however, it remains a challenging issue to clarify the origin of the intriguing nematic BFSs.
Our discovery of the very unusual superconducting states with nematic BFSs in the tetragonal phase of FeSe1-xSx opens new research directions of nontrivial electron pairing. There are several tantalizing issues such as the intertwined relationship between BFS, nematic order, and BCS-BEC crossover.