We report peculiar momentum-dependent anisotropy in the superconducting gap observed by angle-resolved photoemission spectroscopy in BaFe2(As1-xPx)2 (x = 0.30, Tc = 30 K). Strongly anisotropic gap has been found only in the electron Fermi surface while the gap on the entire hole Fermi surfaces are nearly isotropic. These results are inconsistent with horizontal nodes but are consistent with modified s± gap with nodal loops. We have shown that the complicated gap modulation can be theoretically reproduced by considering both spin and orbital fluctuations.
In the iron-based superconductors, most of the experimental studies have so far indicated that the superconducting (SC) gap is nodeless and opens on all Fermi surfaces (FSs) in contrast to the d-wave SC gap of the high-Tc cuprate superconductors1. However, it is unclear whether the SC order parameter is a conventional s wave without sign change (s++ wave) or its sign changes between FSs (s± wave). According to theories of spin-fluctuation-mediated superconductivity, various SC gap structures, such as nodeless s±-wave, nodal s-wave, and nodal d-wave, are realized depending on the FS geometry2,3,4. On the other hand, observation of orbital-polarized states5 and the strong softening of the shear modulus C666,7 stimulate theoretical studies of the orbital-fluctuation-mediated superconductivity, which favor the s++ state without sign change8,9,10.
In contrast to the full-gap superconductivity in most of the iron-based superconductors, some systems show signatures of line nodes in the SC gap. While most of the nodal SC compounds such as KFe2As211,12 show low Tc < 10 K, BaFe2(As1-xPx)213,14,15 has a relatively high Tc (~30 K) and is a suitable system for this purpose. In an angle-resolved thermal conductivity measurement16, four-fold symmetry as a function of magnetic field direction has been observed in the vortex state, suggestive of loop-like line nodes on the electron FS17,18.
Angle-resolved photoemission spectroscopy (ARPES) is the most direct and powerful technique to unambiguously determine the location of line nodes in momentum space. For example, “octet line nodes” in KFe2As2 has been recently reported12. In BaFe2(As1-xPx)2, laser ARPES studies19,20 have revealed nearly isotropic and FS-independent superconducting gaps on the hole FSs around the Z point, indicating the importance of orbital fluctuations in the pairing mechanism8,9,10. However, in a recent study using synchrotron radiation by Zhang et al.21, a “horizontal line node”22,23 has been observed on the outer hole FS around the Z point. Because no in-plane anisotropy was found in both ARPES results19,21, the fourfold symmetry of the thermal conductivity16 remains to be explained.
In this study, in order to resolve this controversy, we have performed an ARPES study of BaFe2(As1-xPx)2 and clarified that there is no horizontal node on the hole FSs and the SC gaps are isotropic for all kz's, consistent with the laser ARPES study19,20. Furthermore, we have revealed that gap nodes or at least strong anisotropy exists on the inner electron FS, which enables us to understand the fourfold symmetry in the thermal conductivity16. We have shown that a theoretical model including not only spin fluctuations but also orbital fluctuations well reproduces the observed SC gap structure, suggestive of a pairing mechanism beyond spin fluctuations.
Energy gaps of the hole Fermi surfaces
Energy gaps of the hole FSs for BaFe2(As1-xPx)2 (x = 0.30, Tc = 30 K) around the Z point20,21 are presented in Fig. 1. We observed at least two hole dispersions, implying that two out of the three bands are nearly degenerated as in the previous studies19,21,24. The spectra at T = 15 K show a dip at EF for both bands, indicating a clear gap opening. The temperature dependence of the energy distribution curves (EDCs) at the Fermi momentum kF [Figs. 1(c1), 1(c2)] shows a clear gap opening even slightly above Tc, unlike the previous result21.
We have also investigated the gap anisotropy of the hole FSs [Fig. 1(e)]. In Figs. 1(d1)–1(d3), we show an example of the EDC at kF below and above Tc. The symmetrized EDC below Tc have been divided by those above Tc and we obtain peak structure at ~6 meV [Fig. 1(d3)]. Since this structure evolves at low temperature as a shoulder in the EDCs [Fig. 1(d1)], we attributed this peak to a SC peak. Similar shoulders are observed at various kF points on the outer hole FS [Fig. 1(e)] and their energies are nearly isotropic, consistent with the laser ARPES result19. Thus, the four-fold oscillation in the angle-resolved thermal conductivity measurements16 is hardly explained by the isotropic character of the outer hole band.
Next, we present ARPES spectra of the hole FSs with various photon energies and hence for various kz's as shown in Fig. 2. The photon energies hν ~ 23 eV and ~35 eV correspond to the kz of the Γ and Z points, respectively [Fig. 2(d)]. Three hole dispersions are clearly resolved near the Γ point [Fig. 2(e)]. For the β and γ FSs [Figs. 2(b) and 2(c)], the gap energies are ~5–8 meV and are nearly independent of photon energy, i.e., independent of kz. For the α and β bands [Figs. 2(a) and 2(b)], the EDCs around the Z point (~35 eV) show a hump (at ~12–18 meV) with a shoulder (at ~7 meV). While the shoulder signifies a superconducting gap as in Fig. 1, the humps for the α and β FSs indicate a pseudogap25,26. The SC gap is nearly independent of hν [panel (f)] and, hence, the hole FSs do not have an horizontal line node at any kz.
Energy gaps of the electron Fermi surfaces
In order to investigate the possible existence of line nodes on the electron FSs, a photon energy of hν = 40 eV with a circularly polarized light was used as shown in Fig. 3, because with this polarization signals from the inner and outer electron FSs can be distinguished due to matrix element effects as shown in panel (a). In the inner electron FS [Fig. 3(b)], while the EDC at the Fermi surface angle θFS = 80° shows a clear shift (~5 meV) from EF, the shift of the EDC at θFS = 170° is smaller (<~2 meV), indicating a strong anisotropy of the superconducting gap.
As in panel (b), the leading edge of the spectrum is useful to detect the gap anisotropy. Therefore, the crossing point between the EDCs below and above Tc [Figs. 3(c) and 3(d)] will reflect the magnitude of the SC gap. Also, we have determined the energy of SC peak in the same manner as in Fig. 1(d1)–(d3). In Figs. 3(e) and 3(f), the SC gap magnitudes estimated from the crossing point and the peak energy are plotted as a function of θFS. These plots indicate that the inner FS has a gap minimum near the edge of the FS (θFS ~ 0 or 180°). We have also confirmed this strongly anisotropic SC gap for the inner electron FS with several experimental conditions. (See Supplementary Information, Figs. S2 and S3.) Although the minimum gap value in the inner FS appears to be finite, it should be remembered that ARPES has finite kz resolution (inverse of the photoelectron mean-free-path ~0.5 nm). If the line nodes are distributed in a three dimensional shape such as a loop-like node17,19,27, kF's with finite gaps inevitably contribute to EDCs and the gap would look finite.
Here, we illustrate a possible momentum dependence of the SC gap function from the present ARPES study in Fig. 4(a). The outer hole FS near the Z point does not show a signature of vertical nor horizontal line nodes. In contrast to the isotropic gap on the hole FSs, the gap minimum has been observed on the inner electron FS at the longer edge of the ellipsoid-shaped FS cross-section [Fig. 3 and Fig. 4]. Khodas and Chubukov31 have shown the folded model of the electron pockets and deduced vertical loop-like nodes in the electron FSs. Even if this is the case, it is still difficult to reproduce the loop-like node if there are three hole FSs as in the present results28. Thus, the observed momentum dependence of the SC gap magnitude is difficult to understand from the spin-fluctuation-mediated mechanism alone. The comparable sizes of the SC gaps of the three hole FSs, that is, for FSs of different orbital character indicate that the inter-orbital scattering is required for the Cooper pair formation8,9,10. Also, the disappearance of the xy hole FS is required in the spin-fluctuation-mediated mechanism to achieve a strongly anisotropic gap or line nodes in the electron FSs2,27, while we have clearly observed three hole bands including the xy hole FS around the Γ point (Fig. 2).
The three-dimensional gap structure in ref. 28 obtained by a model calculation including orbital fluctuations, shown in Fig. 4(b), shows common features with the experimental results of the hole and electron FSs. In this model, when the inter-orbital scattering is moderately strong, the highly anisotropic gap of the electron FS is expected to have nodes in the crossover region between the s± and s++-wave superconductivity. According to the calculated results28, when a loop-like node appears at the longer edge of the electron FSs, a large part of the FS has an order parameter of opposite sign to that of the hole FSs. Thus, the present observation of the gap anisotropy in the electron FS indicates the realization of nearly s±-wave SC state.
The present result on the SC gap of the hole FSs for the x = 0.30 sample described above have been qualitatively reproduced for a slightly overdoped x = 0.38 sample as shown in Supplementary Fig. S4 and, therefore, can be considered as robust features. Also, in the present study, while the two electron FSs have gap values ~8–10 meV, the hole FSs show relatively small values ~5–8 meV. This can be contrasted with the previous ARPES study of Ref. 21, which shows similar gap values between electron and hole FSs except around the Z point. Here, we discuss possible origins of the discrepancy in the SC gap anisotropy with the previous study by Zhang et al.21. We should note that the samples were grown by the self-flux method in our study while a flux-free method was applied in Ref. 21. In the process of the sample growth, starting compounds are different between these methods. Therefore, we speculate that such a difference in the sample growth may give differences in the local structure and/or unknown defects, hence, the detailed electronic structure.
In the present study, the strongly anisotropic SC gap and the absence of a horizontal node are suggestive of the mechanism beyond spin fluctuations such as orbital fluctuations. As a possible origin, inter-orbital interaction, electron-phonon interaction8, and vertex correction of the Coulomb interaction29 have been discussed so far. Since these interactions may depend on the doping level and material families, the pairing mechanism beyond spin fluctuations should be revealed in future systematic studies of the SC gap in various kinds of the iron pnictide superconductors.
Single crystals of BaFe2(As1-xPx)2 with x = 0.30 (Tc = 30 K) were grown by a self-flux method30. The details of the sample growth, characterization are described in Supplementary S1. ARPES experiments were carried out at BL 9A of Hiroshima Synchrotron Radiation Center (HiSOR) and BL-28A of Photon Factory (PF). At HiSOR BL 9A, a Scienta SES-R4000 analyzer and a circularly-polarized light were used with the total energy resolution of ~7–8 meV. At PF BL-28A, a Scienta SES-2002 analyzer and a circularly- and linearly-polarized light were used with the total energy resolution of ~8–10 meV. The crystals were cleaved in situ at T = 8–13 K in an ultra-high vacuum of ~5 × 10−11 Torr. Calibration of the Fermi level (EF) of the samples was achieved by referring to that of gold. In-plane (kx, ky) and out-of-plane electron momenta (kz) are expressed in units of π/a and 2π/c, respectively, where a = 3.92 Å and c = 12.8 Å are the in-plane and the out-of-plane lattice constants. Here, the x and y axes point towards nearest neighbor Fe atoms. The correspondence between kz and hν has been determined in our previous ARPES studies20,24.
We are grateful to K. Kuroki and D.L. Feng for enlightening discussions. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Area “Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics” from MEXT, the Japan-China-Korea A3 Foresight Program and a Grant-in-Aid for Young Scientist (B) (22740221) from the Japan Society for the Promotion of Science. ARPES experiments were carried out at HiSOR (Proposal No. 10-B-27 and 11-B-1) and Photon Factory (Proposal No. 2009S2-005, No. 2012S2-001, and No. 2012G075).
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Decoupling of critical temperature and superconducting gaps in irradiated films of a Fe-based superconductor
Superconductor Science and Technology (2018)