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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.

Figure 1: Band dispersion of superconducting Ba0.6K0.4Fe2As2.
figure 1

a, ARPES spectral intensity measured at 10 K plotted on a false-colour scale as a function of the in-plane momentum () and binding energy along Γ–X using 46-eV photons, which corresponds to kz=0. Two hole-like bands (α (inner) and β (outer)) are observed. b, Second derivative of the spectral intensity plot as shown in a. c, A set of EDCs within the E–k range indicated by the red rectangle in a. The red EDC is at kF of the α band. d, Second derivative plot of the dispersion along Z–R (kz=π) measured using 32-eV photons. Three hole-like bands (α (inner), α′ (middle) and β (outer)) are observed.

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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.

Figure 2: kz dispersion of quasiparticles and Fermi surface warping.
figure 2

a, Comparison of the α band dispersions at kz=0 (red dots) and kz=π (black dots) extracted from the peak position of momentum distribution curves along Γ–X. The solid lines are linear fits to the corresponding dispersions. The green dots denote the binding energies of the quasiparticle peak of the α band as shown in b measured at a fixed (slightly below kF) using different photon energies (or different kz). b, Corresponding EDCs to the green dots shown in a. The black vertical bars indicate the binding energies of the quasiparticle (QP) peaks. The red (blue) EDC corresponds to kz=0 (kz=π). Note that the ‘shoulder’ at the higher binding energy (∼20 meV) of the EDC measured at h ν=58 eV is from the α′ band. c, kz dispersion of the α band extracted from a and b. The solid curve is the fit using coskz. d, kz dependence of the in-plane kF of the α-FS parallel to Γ–X. The solid curve is the coskz fit. e,f, Same as d, but for the β-FS and electron-like (γ/δ)-FS. The error bars in a and c are standard deviations of the Lorentzian fit to the coherent peaks. The error bars of kF in d,e and f are standard deviations of the Lorentzian fit to MDC peaks at the Fermi energy. The error bars of kz in c,d,e and f are obtained from the estimated uncertainty of the inner potential (∼±1 eV) according to .

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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.

Figure 3: kz dependence of the superconducting gaps.
figure 3

a, Photon-energy-dependent EDCs measured at the left kF on the α-FS along Γ–X or its parallel directions with different kz. Red (blue) EDCs correspond to kz=0 (kz=π). The black blocks indicate the binding energies of the coherent peaks. b, Corresponding symmetrized EDCs of the ones shown in a. The dashed line at 12 meV is a guide to the eyes for viewing the variation of the SC gap at different h ν. c, Extracted values of the SC gap (defined as the half value of peak-to-peak positions in the symmetrized EDCs) on the α-FS at different photon energies. The dots (squares) are obtained from the left (right) side of kF on the α-FS. d–f, Same as a–c but on the electron-like FSs (γ/δ) along Γ–M or its parallel directions. The dashed line in e is at 11.5 meV as a guide to the eyes for viewing the SC gap at different h ν. The error bars are twice the derivation of the Lorentzian fit to coherent peaks.

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Figure 4: 3D superconducting gap function Δ (kx, ky, kz).
figure 4

a, The SC gap values on the α-FS (red dots), β-FS (blue dots), γ/δ-FS (green dots) and α′-FS (brown dots) as functions of |coskxcosky|. The black solid line is the gap function |Δ|=Δ0|coskxcosky|, with Δ0=12.3 meV. b, Same as a but as functions of coskz. The solid lines are independent linear fits to the SC gaps on the different FS sheets using a generic 3D gap function |Δ(kx,ky,kz)|=|Δ2D(kx,ky)(1+ηcoskz)|. c, Same as b but using a single 3D gap function |Δ(kx,ky,kz)|=|Δ1coskxcosky+(Δ2/2)(coskx+cosky)coskz|, with Δ1=12.3 meV and Δ2=2.07 meV, to fit all the SC gaps. The error bars are standard deviations of the measured SC gaps.

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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.