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Anisotropic but nodeless superconducting gap in the presence of spin density wave in iron-pnictide superconductor NaFe1-xCoxAs

The coexisting regime of spin density wave (SDW) and superconductivity in the iron pnictides represents a novel ground state. We have performed high resolution angle-resolved photoemission measurements on NaFe1-xCoxAs (x = 0.0175) in this regime and revealed its distinctive electronic structure, which provides some microscopic understandings of its behavior. The SDW signature and the superconducting gap are observed on the same bands, illustrating the intrinsic nature of the coexistence. However, because the SDW and superconductivity are manifested in different parts of the band structure, their competition is non-exclusive. Particularly, we found that the gap distribution is anisotropic and nodeless, in contrast to the isotropic superconducting gap observed in an SDW-free NaFe1-xCoxAs (x=0.045), which puts strong constraints on theory.

Most unconventional superconductors appear in the vicinity of a certain magnetically ordered phase [1]. Magnetism is suggested to play a critical role in the pairing mechanisms of the cuprates [2], heavy Fermion superconductors [2,3], and even organic superconductors [4]. For iron-pnictide superconductors, a spin density wave (SDW) phase appears next to the superconducting (SC) phase [5][6][7], and in some cases, they even coexist [8][9][10][11][12][13], which gives a unique SC ground state. While the coexisting SDW and SC phases may have significant impact on the SC mechanism [9], much is not clear about the subtle interacting nature between magnetism and superconductivity [14]. In fact, theories based on s ++ pairing symmetry suggest that there must be nodes in the SC gap in this regime [15] and the coexisting SDW and SC phases cannot be microscopic [9]. On the other hand, theories based on s +− pairing symmetry suggest nodeless SC gap in the presence of weak magnetic order; moreover, the coexistence may cause angular variation of the SC gap, and even give rise to nodes in the limit of strong antiferromagnetic (AFM) ordering [15,16], as indicated in a thermal conductivity study on Ba 1−x K x Fe 2 As 2 [17].
The coexistence of SDW and superconductivity in various iron pnictides has been illustrated by neutron scattering [8][9][10][11][12], nuclear magnetic resonance [18,19], and angle-resolved photoemission spectroscopy (ARPES) experiments [13]. Recent scanning tunneling microscope (STM) studies show the real-space coexistence and competition of SDW and superconductivity in NaFe 1−x Co x As [20,21] . However so far, little is known regarding the electronic structure of the coexisting phase in the momentum space, such as its SC gap distribution, and how the two orders coexist and compete on the same electronic structure. In this paper, we report ARPES studies on NaFe 0.9825 Co 0.0175 As in this coexisting regime. The band structure reconstruction corresponding to the SDW for-mation and the SC gap could be observed on the same bands, which provides a direct evidence for the intrinsic coexistence of the two orders. We found that SDW formation does not cause much depletion of the states near the Fermi energy (E F ), therefore, it allows the superconductivity to occur. Moreover, the SC gap distribution is found nodeless on all Fermi surface sheets: it is isotropic on the hole pocket, but it is highly anisotropic on the electron pockets. Our results reveal the distinct electronic properties of the coexisting phase and provide explicit constraints on theory.
High-quality NaFe 0.9825 Co 0.0175 As single crystals were synthesized by the self-flux method described elsewhere [22]. The SC transition temperature (T c ) is determined by the magnetic susceptibility measurements with a SQUID magnetometer [ Fig. 1(a)], which shows an onset drop at 20.5 K. Resistivity measured by PPMS indicates zero resistivity below 18 K, and a structural transition at T S = 36 K. Our neutron scattering data show that the SDW transition temperature (T N ) is 28 K [ Fig. 1(b)]. ARPES data were taken with various photon energies in circular polarization at the 1-Cubed beamline of BESSY II, other photoemission measurements were performed either with 21 eV photons at beamline 5-4 of the Stanford Synchrotron Radiation Laboratory (SSRL), or with randomly polarized 21.2 eV light from an in-house SPECS UVLS helium discharging lamp at Fudan University. All the data were taken with SCIENTA R4000 electron analyzers; the overall resolution is set to 6 meV or better and the typical angular resolution is 0.3 • . The samples were cleaved in situ, and measured under ultra-high vacuum, so that the aging effects are negligible in the data.
The general electronic structure of NaFe 0.9825 Co 0.0175 As is rather similar to the well studied NaFeAs [22][23][24].  The gap size is estimated through an empirical fit as described in detail in Ref. [28]. The inset on the top right corner shows the Fermi surface of NaFe 0.9825 Co 0.0175 As. The two solid lines mark cut #1 and cut #2 along which the data in panels (d) and (e) are located, respectively. The two dashed lines on the bottom plane are their projections. The photoemission data in panels (c) and (e) were acquired in-house, and others were collected at SSRL.
patch-like feature point around Γ (0, 0), and two orthogonal elliptical pockets around the zone corner. The photoemission intensity along cut #1 across Γ is plotted in Fig. 1(d), where three bands, α, β and γ could be resolved, but only γ crosses E F and gives the hole Fermi surface. The band top of α is just below E F , and contributes to the small patch in the zone center. Figure 1(e) plots the photoemission intensities at the zone corner, where two electron-like bands, δ and η, could be observed. As previous photon energy dependent study has revealed the negligible k z dispersion of NaFeAs [23], the overall Fermi surface topology of NaFe 0.9825 Co 0.0175 As is summarized in the inset on the top right corner of Fig. 1.
The signature of SDW on the electronic structure has been extensively studied before [22,[24][25][26], which is mainly manifested as a remarkable band reconstruction. As shown in Fig. 1(d), β shifts significantly with decreased temperature. To illustrate the subtle band reconstruction of γ, Fig. 1(f) plots the momentum distribution curves (MDCs) near the Fermi crossing of γ at several binding energies near E F at 45 and 10 K, and Fig. 1(g) plots the MDC at E F − 15 meV as a function of temperature. It is clear that γ first shifts in one direction due to the SDW [24], and then splits into two at low temperatures. Our recent ARPES study on the mechanically de-twinned NaFeAs has shown that the β and γ bands disperse differently along the ferromagnetic (FM) and AFM directions, which gives an appearance of band splitting in the twinned sample here as noted by the subscripts in Fig. 1 [24]. Similar reconstruction effects can be observed in the energy distribution curves (EDCs) as well in Fig. 1(h). As shown by the temperature dependence of the EDC peak positions summarized in Fig. 1(i), the electronic structure reconstruction occurs above the structural transition due to the fluctuations of the SDW and electronic structure nematicity [24,27]. It evolves smoothly across the structural and Neel transitions, and saturates below 20 K, with the separation of β AFM and β FM reaching 32 meV and the shift of γ reaching 3 meV. The reconstruction of δ and η is subtle, nevertheless in Fig. 1(e), their features in the MDCs at E F clearly show finite shifts as well [24]. On the other hand, SC gap opens just below T c , as illustrated by the symmetrized EDCs of the γ band with respect to E F in Fig. 1(j) and the fitted SC gap in Fig. 1(k). The fact that the signatures of both the superconductivity and SDW emerge in the same band structure confirms their intrinsic coexistence. Furthermore, the band reconstruction due to SDW mainly occurs over a large energy and momentum scales for β below E F , and it leaves the states on all the Fermi surfaces  largely intact in this doping regime, therefore superconductivity could occur in the presence of SDW here. The SC gap is mapped out extensively over the entire Brillouin zone. Figure 2(a) shows the symmetrized photoemission intensity along four momentum cuts across the γ hole Fermi surface in the k z = 6π plane. The suppression of the spectral weight around E F indicates the opening of the SC gap. In Fig. 2(b1), the symmetrized EDCs along the γ pocket clearly show sharp coherent peaks, and SC gaps of similar amplitude. Data from other k z planes in Figs. 2(b2)-2(b5), and data from another sample taken with more photon energies in Fig. 2(c) show that the gap is isotropically 5 meV on the γ pocket, as also summarized in Fig. 4(a). Now we turn to the SC gap on the electron Fermi surfaces around the zone corner. Figure 3(a) shows symmetrized photoemission intensity for six momentum cuts across the δ/η pockets in the k z = 6π plane, where the SC gap opens on both Fermi surfaces. Collecting the symmetrized EDCs at various k F 's along the δ pocket, Fig. 3(b1) demonstrates an anisotropic gap distribution, where the gap is about 7 meV in the flat part of the ellipse, and significantly drops to 4 meV near θ = 0 • , 180 • . Moreover, such a behavior is observed for all five sampled k z 's as shown in Figs. 3(b1)-3(b5). Similarly, such an anisotropic gap distribution is observed for η but rotated by 90 • [Figs. 3(c1)-3(c5)]. The weak k z dependence is further illustrated with more data taken at k z = 5.5π, 6.3π, and 6.5π with 21, 28, and 30 eV photons respectively in the supplementary material [Fig. S1].
The gap distribution of NaFe 0.9825 Co 0.0175 As is summarized in Figs. 4(a)-4(c). The gaps along the γ hole Fermi surface show isotropic distribution, while the gaps on the δ and η pockets vary significantly from 4 to 7 meV. As a comparison, Figures 4(d)-4(e) show the isotropic in-plane gap distribution on individual Fermi surfaces for an SDWfree NaFe 0.955 Co 0.045 As sample (T c = 20 K), which are retrieved from the symmetrized EDCs provided in the supplementary material [Fig. S2]. The gap is about 5 meV on the hole pocket, and 5.4 meV on the electron pockets. Such an isotropic in-plane gap distribution has been observed before in NaFe 0.95 Co 0.05 As as well [29]. Furthermore, Fig. 4(f) compares both the Fermi surfaces and the SC gap distributions of NaFe 0.9825 Co 0.0175 As and NaFe 0.955 Co 0.045 As. The hole pocket of NaFe 0.955 Co 0.045 As is slightly smaller as expected from cobalt doping, and the ellipticity of its electron pockets is smaller as well.
So far in ARPES experiments, the in-plane anisotropy of SC gap has been observed only for LiFeAs [30,31], Fe(Te,Se) [32], KFe 2 As 2 [33], and Ba 1−x K x Fe 2 As 2 [34] among all the iron-based superconductors, but none of them is in the coexisting regime. The small gap anisotropy on one of the hole pockets of Ba 1−x K x Fe 2 As 2 is within the experimental error that less than 0.6 meV difference over the 9∼10 meV gap amplitude is observed [34]. The moderately anisotropic gap on a hole Fermi surface of LiFeAs might be a mere consequence of the Fermi surface topology, since it is qualitatively consistent with the gap function △(k) = △ 0 cosk x cosk y predicted based on the s +− pairing symmetry [30,31]. For NaFe 0.9825 Co 0.0175 As, the large ellipticity gives a variation of |cosk x cosk y | from ∼ 0.98 in the flat region to ∼ 0.91 on the tip, which could not explain the over 40% change of the gap based on the Fermi surface topology. We note that an anisotropic gap distribution around the zone corner has also been revealed in LiFeAs, which deviates from the canonical s +− -wave gap function and was explained in terms of the band hybridization [31]. Consistently, the diviation there is most prominent around θ = 45 • where the hybridization is the strongest. However, the anisotropic behavior in NaFe 0.9825 Co 0.0175 As deviates the gap function remarkably around θ = 0 and 90 • , which is away from Fermi surface region of mixed orbital character. For Fe(Te,Se), the anisotropy of the SC gap on the hole pocket was suggested to be a consequence of sizable second-nearest-neighbor interactions, while the anisotropic and nodal gap on a hole pocket of KFe 2 As 2 may be related to strong intra-pocket scattering [35], or specific orbital characters near Z [28]. Alternatively, the angular variation in the d xy orbital content of the γ Fermi surface was predicted to cause anisotropic gap distribution on the electron pockets [35]. However, since NaFe 0.9825 Co 0.0175 As and NaFe 0.955 Co 0.045 As have similar Fermi surface, orbital characters and interaction parameters, NaFe 0.955 Co 0.045 As would have exhibited anisotropic gap if these had been the causes here. Therefore, the highly anisotropic gap distribution on the electron pockets of NaFe 0.9825 Co 0.0175 As is most likely a direct consequence of the coexisting SDW. The typical spectra at the Fermi crossings of the γ and δ bands taken at 1 K in BESSY-II. The intensity ratio of the residual spectral weight at E F is referred to the coherence peak height. Two Gaussians with 6 meV full-width-half-maximum are overlaid.
Theories based on the s +− paring symmetry have suggested the nodeless and anisotropic gap distribution in the presence of weak SDW [15,16]. Consistently, compared with NaFeAs [24], much weaker SDW order is present in NaFe 0.9825 Co 0.0175 As: the band folding due to the SDW order is negligible, and no SDW gap induced by the hybridization with the folded bands is observed here. In a recent theoretical study, it was predicted that even weak SDW order will cause appreciable gap anisotropy [16]. Particularly, it was found that the gap at the tip region of the electron Fermi surface is smaller than that at the flat region, in good agreement with our observation. Futhermore, the observed nodeless SC gap disallows the paring mechanism based on the s ++ pairing symmetry that predicts SC gap nodes in the SDW state [9,15].
The prominent band reconstruction of β observed here with a 32 meV separation between the dispersions along the AFM and FM directions is smaller than the 46 meV observed in NaFeAs [22]. Such a band reconstruction energy scale is distinct at a specific doping, and is correlated with the SDW transition temperature as observed in Sr 1−x K x Fe 2 As 2 [13]. Therefore, the sharp band dispersion with a single set of band reconstruction energy scale, plus the resolution limited width of the superconducting coherent peak [ Fig. 4(g)], highlight the homogeneous nature of the electronic state in the momentum space. Moreover, although the shielding fraction of the bulk sample is 75% based on our susceptibility measurements, the ARPES data are taken on a small region (0.05 mm × 0.2 mm) of the cleaved surface. As shown in Fig. 4(g), the photoemission intensity at E F in the superconducting state is negligible, which suggests the absence of non-superconducting region. That is, there is no phase separation of superconducting regions and non-superconducting SDW regions in the coexisting phase. Our results thus rule out the appearance of macroscopic phase separation and further support the intrinsic coexistence. These are consistent with a recent STM study on the coexisting phase of NaFe 1−x Co x As (x=0.014) [21], where the coexistence was found to occur microscopically in an anti-correlated but non-exclusive way between the two orders. Such a non-exclusive coexistence can be understood based on our observation of the indirect competition between SDW and superconductivity in the electronic structure.
Note that, the energy scales observed in STM for both the "SDW gap" feature (∼ 17 meV, and it should be a momentumintegrated effect of the band reconstruction) and SC coherence peak (∼ 5 meV) are quite independent of space. This is further consistent with the single set of SDW/SC energy scales observed here by ARPES.
Our neutron scattering data on the same sample reveals that static antiferromagnetic long-range order coexists with superconductivity, similar to the static antiferromagnetic order/superconductivity coexisting BaFe 2−x Ni x As 2 samples [14]. The intensity of the SDW diffraction peak decreases upon entering the SC state, suggesting a competition between the two orders [ Fig. 1(b)]. The magnitude of the SDW order could be monitored directly from the energy scale of the band reconstruction. However, we did not observe any remarkable change of band reconstruction below T c , which suggests that the competition between the two orders does not affect the magnitude of the local SDW order at the fast time scale of photoemission (∼ 1 f s). Alternatively, since the itinerant electrons near E F could play an important role in stabilizing the long-range SDW order [36], when the SC gap opens, the coherence of SDW order could be suppressed. Consequently, the enhanced fluctuation of the local SDW order could be responsible for the observed suppression of the effective (or timeaveraged) moment at the quasi-elastic neutron scattering time scale (≫ 1 ps) [37].
To summarize, we have revealed detailed electronic structure in the superconductivity/SDW coexisting regime of NaFe 1−x Co x As (x=0.0175), and signature in the momentum space for the intrinsic microscopic coexistence. We found that SDW does not cause a noticeable depletion of the states at the Fermi energy, which allows the superconductivity to emerge. Therefore, it explains why the two orders could coexist in a non-exclusive way. Moreover, we show that the anisotropy of the SC gap on the electron pockets is likely a distinct consequence of the coexisting SDW order, while the absence of gap node puts strong constraints on the pairing symmetry in theory of iron-based superconductors.
We  respectively. The superconducting gap magnitudes were determined by fitting the symmetrized EDCs with a typical superconducting-state spectral function [1]. The superconducting gaps on the hole and electron Fermi surfaces are nodeless and isotropic for the SDW-free NaFe 0.955 Co 0.045 As. Note that, since the Fermi crossings of δ and η are very close in NaFe 0.955 Co 0.045 As, we did not plot the symmetrized EDCs separately here for these two Fermi surfaces. * Electronic address: dlfeng@fudan.edu.cn