Momentum-resolved superconducting gap in the bulk of Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ from combined ARPES and $\mu$SR measurements

Here we present a calculation of the temperature-dependent London penetration depth, $\lambda(T)$, in Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ (BKFA) on the basis of the electronic band structure [1,2] and momentum-dependent superconducting gap [3] extracted from angle-resolved photoemission spectroscopy (ARPES) data. The results are compared to the direct measurements of $\lambda(T)$ by muon spin rotation ($\mu$SR) [4]. The value of $\lambda(T=0)$, calculated with \emph{no} adjustable parameters, equals 270 nm, while the directly measured one is 320 nm; the temperature dependence $\lambda(T)$ is also easily reproduced. Such agreement between the two completely different approaches allows us to conclude that ARPES studies of BKFA are bulk-representative. Our review of the available experimental studies of the superconducting gap in the new iron-based superconductors in general allows us to state that all hole-doped of them bear two nearly isotropic gaps with coupling constants $2\Delta/k_{\rm B}T_{\rm c}=2.5\pm1.5$ and $7\pm2$.


INTRODUCTION
The superconducting energy gap in the newly discovered iron-based superconductors naturally attracted much attention of physicists, and during one year of hard work, these materials were investigated by numerous experimental techniques [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. As the diversity of conclusions made about the symmetry and value of the gap is huge, which can be attributed to the various shortcomings of different methods, the situation seems to be far from clear. In this paper, based on the angle resolved photoemission spectroscopy (ARPES) and muon spin rotation (µSR) data taken from the same single crystals of Ba 1−x K x Fe 2 As 2 with T c = 32 K, we succeeded to reveal robust momentum dependence of the gap in this compound. The coupling constant, 2∆/k B T c , is 1 for the outer Γ-barrel and 6.8 for all other Fermi surface sheets. Furthermore, close inspection of many studies of different iron-based superconductors allows one to derive quite definitive conclusions about the gap in these materials.

THE LONDON PENETRATION DEPTH FROM THE ELECTRONIC BAND STRUCTURE
The London penetration depth, λ, can be expressed through the electronic band structure. For the quasi-twodimensional superconductor with equivalent a and b principal axes, the formula, relating in-plane penetration depth to the band dispersion, reads (in SI units) where v F is the Fermi velocity, ∆ k (T ) is the momentum-dependent superconducting gap, Σ is the scattering rate (in the following we assume clean limit, Σ = 0), dk is the element of the Fermi surface length, T is temperature, L c is the size of the elementary cell along the c axis, f T (ω) = [1+exp(ω/k B T )] −1 is the Fermi function, k B is the Boltzmann constant, h is the Planck's constant, ε 0 is the electric constant, c is the speed of light, and e is the elementary charge [31]. Formula (1) is consistent with results already presented in the literature [32], although the former accounts for a finite lifetime (see also Ref. 3), and for the four-fold symmetry of the problem (see also Ref. 33).

LOW-ENERGY ELECTRONIC BAND STRUCTURE OF Ba1−xKxFe2As2
The information, required to calculate λ(T ) via the formula (1), can be extracted directly from ARPES spectra. The temperature and momentum dependence of the superconducting gap were obtained in Ref. 3, and the band structure was qualitatively revealed in Refs. 1 and 2. The momentum dependence of the superconducting gap is quite easy to describe -the gap is large, ∆ k (T ) = ∆ large (T ), on the inner Γ-barrel and the propeller-like structure around the X point, and it is small, ∆ k (T ) = ∆ small (T ), on the outer Γ-barrel. The temperature dependence of the gap (see Fig. 1) is well fitted by the formula [34] ∆ large,small (T ) = ∆ large,small (0) · tanh with ∆ large (0) = 9.1 meV and ∆ small (0) < 4 meV.
Taking into account the mentioned momentum dependence of the gap, one can rewrite (1) in the following way: where I 1,2 are temperature-independent factors and D(∆, Σ , T ) is defined as See the Appendix for the evaluation of this integral. In Fig. 2 we present a quantitative investigation of the low-lying electronic band structure of Ba 1−x K x Fe 2 As 2 . The band dispersion is extracted from ARPES data taken at a temperature slightly above the superconducting transition. The Fermi velocities for the inner and outer Γ-barrels along the ΓX direction equal v ΓX iΓ = 0.51 eVÅ and v ΓX oΓ = 0.36 eVÅ respectively [see    (c), (h)], resulting in v iΓ = 0.52 eVÅ and v iΓ = 0.40 eVÅ on the average. Fermi momenta for the inner and outer Γ-barrels are k iΓ = 0.14Å −1 and k oΓ = 0.30Å −1 respectively [see Fig. 2(b-g), (j-k)]. Kinks in the dispersions of the inner and outer Γ-barrels around 25 meV have been described elsewhere [39]. For the electron-like X-pocket k e = 0.06Å −1 , and the depth of the band is ε e = 17 ± 3 meV [see Fig. 3, Fig. 4(b)], from where, assuming a parabolic band dispersion for this small pocket, we infer v e = 2ε e / k e = 0.57 eVÅ. For the hole-like blade pocket the average Fermi momentum equals k h = 0.06Å −1 and we estimate ε h as 5-15 meV [see Fig. 4(b)], thus the average Fermi velocity equals v h = 0.33 eVÅ. The c-axis lattice constant equals 13.3Å [35,37]. The low-energy band dispersion can be well approximated by the following formulas where L a is the in-plane lattice constant, which according to Refs. 35, 37 equals 3.90Å; 2. outer Γ-barrel 3. X-pocket 4. blades These dispersion relations are visualized in Fig. 4(c).

RESULTS
The penetration depth at T → 0 in the clean limit depends only on the band structure and does not depend on the value of the superconducting gap (provided it is not zero), and, therefore, can be calculated purely from ARPES without any additional assumptions: which results in λ(0) = 270 nm. This is in remarkable agreement with the value of 320 nm obtained by µSR [4], the more so when one takes into account the complementarity of the two methods. The temperature dependence of λ strongly depends on the values of the superconducting gap. Due to technical reasons, the small gap has not been determined precisely from ARPES measurements -only an upper limit of ∆ small < 4 meV was obtained [3]. Therefore, we use ∆ small as a fitting parameter when comparing λ(T ) calculated from ARPES to that determined from muon-spin depolarization rate in the µSR experiments (Fig. 5). The best fit of the normalized data corresponds to ∆ small = 1.1 meV. The good agreement between absolute values of λ at T = 0 from ARPES (270 nm) and µSR (320 nm) implies correct determination of the Fig. 5. The in-plane London penetration depth in single crystals of Ba1−xKxFe2As2 as calculated from ARPES with one adjustable parameter, ∆ small , and as measured directly by µSR. The temperature dependence of the normalized penetration depth is reproduced with the best accuracy for ∆ small = 1.1 meV, which is in agreement with our previous estimate ∆ small < 4 meV [3]. Contributions from different Fermi surface sheets are shown by different colors.
band dispersion in the vicinity of the Fermi level. The possibility to fit the normalized temperature dependence with only one fitting parameter implies (i) correct determination of the relative contributions from different Fermi surface sheets, (ii) perfect agreement between two independent experimental techniques concerning the value of ∆ large , and (iii) possibility to improve the estimate of ∆ small (now 2∆ small /k B T c 1) with respect to pure ARPES measurements (< 3) [3]. The general good agreement of ARPES and µSR studies of Ba 1−x K x Fe 2 As 2 allows us to state that ARPES experiments in this case are bulk-representative.

EXPERIMENTAL DETAILS
Single crystals of Ba 1−x K x Fe 2 As 2 were grown using Sn as flux in a zirconia crucible. The growth details are described in Ref. 37. The crystals were cleaved in situ and measured with Scienta SES R4000 analyzer at the base pressure of 5·10 −11 mBar. ARPES experiments were performed using the "1 3 ARPES" end station at BESSY. Details of the experimental geometry can be found in Ref. 38. µSR experiments were performed at the Swiss Muon Source (SµS), Paul Scherrer Institute (PSI, Switzerland).
ARPES measurements allow one not only to state that different bands bear different gaps, but also to reveal the complete momentum dependence of the gap magnitude -in Ba 1−x K x Fe 2 As 2 the large gap opens on the inner Γ-barrel and the propeller-like structure around the X point, while the small gap opens only on the outer Γ-barrel [3,13]. It is interesting to note that recent ARPES studies of the electron-doped compound BaFe 1.85 Co 0.15 As 2 have suggested that the smaller gap opens on the bands in the vicinity of X point, while the large one opens on the bands around In the figure, the points corresponding to the data taken on 122 systems are denoted by stars, points corresponding to 1111 systems are denoted by squares, and points corresponding to 011 (FeSe0.85) are denoted by spindlelike symbols. Stars, corresponding to the studies of Sr1−xKxFe2As2, are marked by "Sr", corresponding to BaFe2−2xCo2xAs2 are marked by "Co". For 1111 systems, the element Ln in the structural formula LnFeAsO1−xFx is given inside the squares. Critical temperature, Tc (K), is given as numbers above the symbols. Blue symbols correspond to the small gap, while maroon ones correspond to the large gap. Studies on the 122 crystals grown by Sn-flux method are shown as overturned stars. Points corresponding to the most comprehensive and quality studies are marked by an extra frame. There are two superconducting gaps in these systems -the "small" one and the "large" one, although some studies overlook one of the gaps [5,12,15,23,24,25,26].
Γ [17]. The anisotropy of the gap within one Fermi surface sheet has not been firmly established, although some evidence for small variations within the inner Γ-barrel (∼10%) was reported [3,14]. In addition, it is worthwhile noting that all of the above referred only to the magnitude (absolute value) of the gap. As suggested by NMR studies, the order parameter changes sign between different Fermi surface sheets [40].

CONCLUSIONS
In conclusion, we have derived low-energy electronic band structure of Ba 1−x K x Fe 2 As 2 from ARPES spectra.
Recently it was shown that ARPES allows one to explain and predict many tangible physical properties of the material, which depend on the low-lying electronic structure -transport properties [33,41,42], propensity of the system to form additional order [43], critical temperature of the superconducting transition [44]. In this paper we have presented a calculation of the London penetration depth from ARPES data (to the best of our knowledge,  it is the first calculation of such kind). A comparison of the obtained results to direct µSR measurements has shown good agreement, which allows us to state that we have determined the robust momentum dependence of the superconducting gap in the bulk of Ba 1−x K x Fe 2 As 2 . Namely, the gap distribution over the Fermi surface is consistent with those reported in our ARPES studies of this compound [3] -the gap is small (2∆ small /k B T c < 3) on the outer Γ-barrel, and large on the other parts of the Fermi surface (2∆ large /k B T c = 6.8). Furthermore, comparison to µSR measurements resulted in the improvement of the assessment of the small gap magnitude -its coupling constant turned out to be 1 instead of previous < 3.