Abstract
The iron-pnictide superconductors have a layered structureformed by stacks of FeAs planes from which the superconductivity originates. Given the multiband and quasi three-dimensional1 (3D) electronic structure of these high-temperature superconductors, knowledge of the quasi-3D superconducting (SC) gap is essential for understanding the superconducting mechanism. By using the kz capability of angle-resolved photoemission, we completely determined the SC gap on all five Fermi surfaces (FSs) in three dimensions on Ba0.6K0.4Fe2As2 samples. We found a marked kz dispersion of the SC gap, which can derive only from interlayer pairing. Remarkably, the SC energy gaps can be described by a single 3D gap function with two energy scales characterizing the strengths of intralayer Δ1 and interlayer Δ2 pairing. The anisotropy ratio Δ1/Δ2, determined from the gap function, is close to the c-axis anisotropy ratio of the magnetic exchange coupling Jc/Ja b in the parent compound2. The ubiquitous gap function for all the 3D FSs reveals that pairing is short-ranged and strongly constrains the possible pairing force in the pnictides. A suitable candidate could arise from short-range antiferromagnetic fluctuations.
Similar content being viewed by others
Main
Angle-resolved photoemission spectroscopy (ARPES) has played an important role in revealing the electronic structure of the pnictides. These measurements have typically been carried out at a fixed incident photon energy (h ν) and varying incident angles that map out the planar band dispersion as a function of kx and ky. Thus far, four FS sheets have been observed with two hole pockets centred around the Γ (0, 0) point and two electron pockets around the M (π,0) point in the unfolded 2D Brillouin zone. Below the superconducting transition temperature Tc, nodeless SC gaps open everywhere on the FS sheets3,4,5,6, pointing to a pairing order parameter with an s-wave symmetry in the a–b plane, in agreement with a number of theoretical results7,8,9,10,11. However, there are other experiments that have indicated possible nodes in the superconducting gap of some pnictides, either line nodes in the a–b plane or nodes along the c axis12,13,14. It is well known that on tuning the incident photon energy h ν, the allowed direct transitions will shift in energy and, consequently, in the momentum perpendicular to the a–b plane (kz), which enables the determination of the electronic dispersion along the c axis. In the free-electron final-state approximation, the conversion is given by , where V0 is an experimentally determined inner potential15. Several earlier ARPES studies have used this kz-resolving capability to probe the 3D dispersion of the normal-state electronic structure in BaFe2As2-based pnictides, and found a large kz band dispersion in the orthorhombic phase in the vicinity of the parent compound16,17,18,19,20. The 3D nature of the superconducting gap, which is critically important to the understanding of pnictide superconductivity, has yet to be studied.
Figure 1a shows the spectral intensity measured at 10 K at a photon energy of 46 eV, which corresponds to the reduced kz=0 (see discussion below), plotted as a function of the binding energy and in-plane momentum along the Γ–X direction. Two hole-like bands are clearly observed, corresponding to the α (inner) and β (outer) bands respectively, in both the spectral intensity plot and the second derivative intensity plot (Fig. 1b). The energy distribution curves (EDCs) show that the α band disperses towards the Fermi level (EF) and bends back (Fig. 1c) as a result of the opening of the superconducting gap of ∼12 meV, consistent with earlier ARPES experiments3,6. When the photon energy is tuned to 32 eV (kz=π), in addition to the α and β bands observed around the zone centre (the Z point), a third hole-like band (Fig. 1d) emerges between the α and β bands. Although the band calculations21,22,23 have predicted the existence of the third hole-like band in the pnictides, mostly for the 1111 pnictides, there is no accurate prediction of the observed kz dispersion of this band for the 122 compound. We label this new band as the α′ band, the full dispersion and Fermi surface of which will be described in a separate paper.
Comparing Fig. 1b and d quantitatively, it is clear that the α band dispersion undergoes a parallel shift in binding energy under different photon energies, as shown in Fig. 2a. Moreover, this modulation exhibits a well-defined periodicity in the photon energy, indicating that the excess energy is carried by the coherent interlayer quasiparticle tunnelling with well-defined momentum kz. Converting the photon energy into momentum kz with an inner potential of 15 eV (ref. 16), we find that the periodicity in kz is remarkably close to that expected from the lattice spacing between the adjacent Fe layers, that is, 2π/c′=0.951 Å−1, with c′=c/2=6.6 Å (due to bilayer FeAs in Ba0.6K0.4Fe2As2; ref. 24). To determine the kz dispersion of the α band, we stay at a fixed in-plane momentum such that the binding energy varies in a region sufficiently away from the SC gap, and plot the quasiparticle dispersion (Fig. 2b) as a function of kz, as shown in Fig. 2c. The dispersion can be described remarkably well by
with an interlayer hopping amplitude in the binding energy range around 20 meV (along the Γ–X direction). Here and hereafter, for notational simplicity, kx,ky,kz carries the units of 1/a,1/b,1/c′, respectively. The kz dispersion results in the warping of the α-FS sheet along kz. One important implication of the observed kz dispersion is that this band cannot be a surface state, which would have no kz dispersion. To describe the underlying FS quantitatively, we extrapolate the fitted dispersion given in equation (1) to the Fermi level and display the in-plane Fermi wave vector (kF) along Γ–X for the α, β and the electron-like (γ/δ) bands as a function of kz in Fig. 2d–f, respectively. The FS area variation (defined as δ in (1±δ)AFS) in the a–b plane is ∼10%, 4% and 1% for the α, β and γ/δ bands, respectively, with the same periodicity along kz.
Having established the kz dispersion of the quasi-3D electronic structure, we turn to the kz dependence of the SC gaps on different FS sheets obtained using many photon energies. Figure 3a shows a collection of the EDCs at the Fermi crossings of the α band for different photon energies h ν=30–60 eV. Appreciable gap variations are clearly visible in the symmetrized EDCs in Fig. 3b. The extracted SC gap values (defined as Δ3Dα) at both left and right Fermi crossings are plotted in Fig. 3c as a function of the photon energy (left axis) and c-axis momentum kz (right axis). Remarkably, Δ3Dα shows rather large periodic variations from ∼9 to ∼13 meV, then back to ∼9 meV as kz moves from Z to Γ and back to Z. A similar kz dependence of the smaller SC gap on the β band is also observed, varying from ∼5 to ∼7 meV, as shown later in Fig. 4. However, the kz variation of the SC gap on the electron-like (γ/δ)-FS sheets is much smaller, as indicated in Fig. 3d–f, where the Bogoliubov quasiparticle peak is shown to remain at a nearly constant energy of ∼11.5 meV as kz varies from Γ to Z. We note that an appreciable kz dependence of pairing strength associated with the 3D band structure has been predicted in this material25, although the predicted certain in-plane anisotropy and gap nodes have not been observed in our experiments.
It is interesting to note that the observed gap values (Fig. 3c,f) and the FS warping (Fig. 2d,f) along kz are anti-correlated; that is, when the in-plane FS area at a fixed kz becomes larger (smaller) the SC gap becomes smaller (larger) on the same plane. At first glance, one might conclude that the gap variation along kz originates predominantly from the tunnelling-induced FS warping, that is, the kz dependence of the in-plane Fermi vector kF, as is implied by the simplest form of an in-plane s± gap function Δs±=Δ0coskxcosky. However, this turns out not to be the case. The <10% change in the in-plane Fermi vector is too small to account for the large, near 40%, gap variations because of the small ‘gap velocity’ in the pnictides. This is clearly seen from Fig. 1, where the near doubling of the Fermi vector in going from the α-FS to the β-FS results only in a gap change from 12 to 6 meV. To illustrate this point further, we plot in Fig. 4a the measured gap values as a function of |coskxcosky| at the Fermi points for the α, α′, β, and the electron (γ/δ) bands. Notice that although the average gap value follows this 2D form, the marked deviations induced by the kz dispersion could indicate that the pairing is not purely 2D and that there is a further driving force, namely, the pairing between the layers, that is predominantly responsible for the gap dispersion with kz.
For an anisotropic layered superconductor with interlayer coupling, we adopt a simple form of the gap function,
This is a direct generalization of the expression for Bardeen–Cooper–Schrieffer superconductors with an isotropic in-plane gap function26. As the FS warping is rather small, Δ2D(kx,ky) is approximately a constant. We expect this expression to be a good approximation, where η is a measure of the ratio of the interlayer to in-plane pairing strength. In Fig. 4b, we plot the measured Δ3D on different FS sheets as a function of coskz. To increase the accuracy, we used only the lower photon energy part of the gap dispersion and averaged over the left and right crossings. Figure 4b shows that equation (2) fits the data rather well, with the anisotropy ratios η∼0.17, 0.13 and −0.01 for the α, β and γ/δ bands, respectively (the gap values of the α′ band are resolvable only near kz=π). It is interesting to note that the values of η for the α and β bands are consistent with the ratio of the exchange coupling Jc/Ja b in the magnetically ordered parent compound, as extracted from the spin wave dispersion measured by neutron scattering2, where Jc∼5 meV is the interlayer coupling and Ja b∼30 meV is the next-nearest-neighbour coupling in the Fe plane. Note that k-averaged Jc would have most contributions from the bands with appreciable kz dispersion, such as α, α′ and β hole-like bands.
The observation of the cosine dependence of the SC gap on kx, ky and kz indicates that the gap function reflects the lattice symmetry and that the predominant pairing could be short-ranged in real space. Under such an assumption, and taking into account the lattice symmetry, the leading terms of a generalized s-wave gap function can be written as Δ1coskxcosky+(Δ2/2)(coskx+cosky)coskz+Δ3(coskxcosky)coskz. It is possible that the gap parameters of Δ1, Δ2 and Δ3 have some band/orbital dependence. For simplicity, we choose band/orbital-independent parameters to fit the data. We found that Δ3≪Δ2 as a result of the vanishingly small η of the γ/δ electron-like FS around (π,0). Remarkably, the remaining terms Δ1coskxcosky+(Δ2/2)(coskx+cosky)coskz fit all the gap values on the different FS sheets fairly well, with Δ1=12.3 meV and Δ2=2.07 meV, as shown in Fig. 4c. The ratio of Δ2/Δ1 is nearly the same as the ratio of Jc/Ja b. We have also checked that the SC gap on each observed FS sheet along different in-plane directions fits well to this 3D gap function.
Our finding of a single 3D superconducting gap function for all five different FS pockets indicates that there are only two dominant pairing energy scales, one in-plane and one out-of-plane, in pnictide superconductors. It points to a common origin for the pairing strengths on all the observed FS sheets, independent of their different density of states. Moreover, because this gap function can be obtained by decoupling the 3D next-nearest-neighbour antiferromagnetic exchange couplings within the pairing channel, our results are consistent with pairing induced by short-ranged antiferromagnetic fluctuations in the iron-pnictide superconductor9,10,27.
Methods
We have carried out systematic photon-energy-dependent ARPES measurements in the superconducting state of the optimally hole-doped Ba0.6K0.4Fe2As2 superconductors (Tc=37 K). The high-quality single crystals used in our study were grown by the flux method28. High-resolution ARPES measurements were conducted at the SIS beamline of the Swiss Light Source. The photon energy used in the experiments is between 20 and 110 eV, with different circular and linear polarization. The energy resolution is 8–20 meV depending on the photon energy, and the momentum resolution is below 0.02 Å−1. Samples were cleaved in situ and measured at 10 K in a working vacuum better than 5×10−11 torr. The mirror-like sample surface was found to be stable, without obvious degradation during a typical measurement period of 24 h. Many samples have been measured and reproducible results have been obtained in these samples.
References
Yuan, H. Q. et al. Nearly isotropic superconductivity in (Ba,K)Fe2As2 . Nature 457, 565–568 (2009).
Zhao, J. et al. Low energy spin waves and magnetic interactions in SrFe2As2 . Phys. Rev. Lett. 101, 167203 (2008).
Ding, H. et al. Observation of Fermi-surface-dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2 . Europhys. Lett. 83, 47001 (2008).
Zhao, L. et al. Multiple nodeless superconducting gaps in (Ba0.6K0.4)Fe2As2 superconductor from angle-resolved photoemission spectroscopy. Chin. Phys. Lett. 25, 4402–4405 (2008).
Wray, L. et al. Momentum-dependence of superconducting gap, strong-coupling dispersion kink, and tightly bound Cooper pairs in the high-Tc (Sr,Ba)1−x(K,Na)xFe2As2 superconductors. Phys. Rev. B 78, 184508 (2008).
Nakayama, K. et al. Superconducting gap symmetry of Ba0.6K0.4Fe2As2 studies by angle-resolved photoemission spectroscopy. Europhys. Lett. 85, 67002 (2009).
Mazin, I. I., Singh, D. J., Johannes, M. D. & Du, M. H. Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1−xFx . Phys. Rev. Lett. 101, 057003 (2008).
Kuroki, K. et al. Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO1−xFx . Phys. Rev. Lett. 101, 087004 (2008).
Seo, K., Bernevig, B. A. & Hu, J-P. Pairing symmetry in a two-orbital exchange coupling model of oxypnictides. Phys. Rev. Lett. 101, 206404 (2008).
Wang, F. et al. Functional renormalization-group study of the pairing symmetry and pairing mechanism of the FeAs-based high-temperature superconductor. Phys. Rev. Lett. 102, 047005 (2009).
Cvetkovic, V. & Tesanovic, Z. Multiband magnetism and superconductivity in Fe-based compounds. Europhys. Lett. 85, 37002 (2009).
Fletcher, J. D. et al. Evidence for a nodal-line superconducting state in LaFePO. Phys. Rev. Lett. 102, 147001 (2009).
Nakai, Y. et al. 31P and 75As NMR evidence for a residual density of states at zero energy in superconducting BaFe2(As0.67P0.33)2 . Phys. Rev. B 81, 020503(R) (2010).
Reid, J-Ph. et al. Nodes in the gap structure of the iron arsenide superconductor Ba(Fe1−xCox)2As2 from c-axis heat transport measurements. Phys. Rev. B 82, 064501 (2010).
Hüfner, S. Photoelectron Spectrsoscopy (Springer, 1995).
Vilmercati, P. et al. Evidence for three-dimensional Fermi-surface topology of the layered electron-doped iron superconductor Ba(Fe1−xCox)2As2 . Phys. Rev. B 79, 220503(R) (2009).
Liu, C. et al. Three- to two-dimensional transition of the electronic structure in CaFe2As2: A parent compound for an iron arsenic high-temperature superconductor. Phys. Rev. Lett. 102, 167004 (2009).
Malaeb, W. et al. Three-dimensional electronic structure of superconducting iron pnictides observed by angle-resolved photoemission spectroscopy. J. Phys. Soc. Jpn. 78, 123706 (2009).
Brouet, V. et al. Nesting between hole and electron pockets in Ba(Fe1−xCox)2As2 (x=0–0.3) observed with angle-resolved photoemission. Phys. Rev. B 80, 165115 (2009).
Thirupathaiah, S. et al. Orbital character variation of the Fermi surface and doping dependent changes of the dimensionality in BaFe2−xCoxAs2 from angle-resolved photoemission spectroscopy. Phys. Rev. B 81, 104512 (2010).
Singh, D. J. & Du, M. H. Density functional study of LaFeAsO1−xFx: A low carrier density superconductor near itinerant magnetism. Phys. Rev. Lett. 100, 237003 (2008).
Ma, F. & Lu, Z-Y. Iron-based layered compound LaFeAsO is an antiferromagnetic semimetal. Phys. Rev. B 78, 033111 (2008).
Xu, G., Zhang, H., Dai, X. & Fang, Z. Electron–hole asymmetry and quantum critical point in hole-doped BaFe2As2 . Europhys. Lett. 84, 67015 (2008).
Rotter, M., Tegel, M. & Johrendt, D. Superconductivity at 38 K in the iron arsenide (Ba1−xKx)Fe2As2 . Phys. Rev. Lett. 101, 107006 (2008).
Graser, S. et al. Spin fluctuations and superconductivity in a three-dimensional tight-binding model for BaFe2As2 . Phys. Rev. B 81, 214503 (2010).
Bulaeveskii, L. N. & Zyskin, M. V. Energy gap in layered superconductors. Phys. Rev. B 42, 10230 (1990).
Si, Q. & Abrahams, E. Strong correlations and magnetic frustration in the high Tc iron pnictides. Phys. Rev. Lett. 101, 076401 (2008).
Chen, G. F. et al. Transport and anisotropy in single-crystalline SrFe2As2 and A0.6K0.4Fe2As2 (A=Sr,Ba) superconductors. Phys. Rev. B 78, 224512 (2008).
Acknowledgements
We thank X. Dai, B. A. Bernevig and Z. Fang for valuable discussions. This work was supported by grants from the Chinese Academy of Sciences, NSF, the Ministry of Science and Technology of China, NSF, DOE of US, and the Sino-Swiss Science and Technology Cooperation.
Author information
Authors and Affiliations
Contributions
Y-M.X., Y-B.H., X-Y.C., E.R. and M.R. carried out the experiments. Y-M.X. and Y-B.H. analysed the data. H.D., Y-M.X., J-P.H. and Z.W. designed the experiments. Z.W., H.D., Y-M.X. and J-P.H. wrote the paper. G-F.C., P.Z., N-L.W., C-L.Z. and P-C.D. synthesized materials. All authors discussed the results and commented on the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Xu, YM., Huang, YB., Cui, XY. et al. Observation of a ubiquitous three-dimensional superconducting gap function in optimally doped Ba0.6K0.4Fe2As2. Nature Phys 7, 198–202 (2011). https://doi.org/10.1038/nphys1879
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys1879
This article is cited by
-
Deep-ultraviolet nonlinear optical crystals: concept development and materials discovery
Light: Science & Applications (2022)
-
Revealing the intrinsic superconducting gap anisotropy in surface-neutralized BaFe2(As0.7P0.3)2
Communications Physics (2021)
-
Angle-resolved photoemission spectroscopy and its application to topological materials
Nature Reviews Physics (2019)
-
Two-gap superconductivity in CaFe0.88Co0.12AsF revealed by temperature dependence of the lower critical field Hc1c (T)
npj Quantum Materials (2019)
-
Iron arsenides with three-dimensional FeAs layer networks: Can(n+1)/2(Fe1−xPtx)(2+3n)Ptn(n−1)/2As(n+1)(n+2)/2 (n = 2, 3)
Scientific Reports (2016)