arXiv:1001.1074v2 [cond-mat.supr-con] 18 May 2010 Specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ : evidence for multiband strong-coupling superconductivity P. Popovich, 1 A.V. Boris, 1,2 O.V. Dolgov, 1 A.A. Golubov, 3 D.L. Sun, 1 C.T. Lin, 1 R.K. Kremer, 1 and B. Keimer 1 $^{1}$Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany $^{2}$Department of Physics, Loughborough University, Loughborough, LE11 3TU, United Kingdom $^{3}$Faculty of Science and Technology, University of Twente, 7500 AE Enschede, The Netherlands (Dated: October 23, 2018) The specific heat of high-purity Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ single crystals with the highest reported superconducting T$_{c}$ = 38.5 K was studied. The electronic specific heat, C$_{p}$ , below T$_{c}$ shows two gap features, with Δ$_{1}$ ≈ 11 meV and Δ$_{2}$ ≈ 3 . 5 meV obtained from an α -model analysis. The reduced gap value, 2Δ $^{max}$/k$_{B}$ T$_{c}$ ≈ 6 . 6 , the magnitude of the specific heat jump, Δ C$_{p}$ ( T$_{c}$ ) /T$_{c}$ , and its slope below T$_{c}$ exhibit strong-coupling character. We also show that an Eliashberg model with two hole and two electron bands gives the correct values of T$_{c}$ , the superconducting gaps, and the temperature dependence of the free-energy difference. Figure 1a displays unprocessed specific heat data on the Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ and Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ single crys- tals. C$_{p}$ ( T ) /T of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ shows no signs of low- temperature upturns or Schottky anomalies at any magnetic field. In the low-temperature limit (inset in Fig. 1a), the data can be fitted by C ( T ) /T = γ (0) + βT $^{2}$, where γ (0) T rep- In this Letter we report a comprehensive study of the specific heat, C$_{p}$ , of an optimally hole-doped, high-purity The superconducting Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ and reference Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ single crystals were grown from self- flux in zirconia crucibles sealed in quartz ampoules under ar- gon atmosphere, as described earlier [9]. Their chemical com- positions were determined by energy-dispersive X-ray spec- trometry. The specific heat was measured in the temperature range between 2 and 200 K with a Physical Properties Mea- surement System (Quantum Design) using the thermal relax- ation technique. The samples were mounted on a standard puck using Apiezon N grease. The measurements showed no difference between several cleaved pieces from the same batches. Bulk superconductivity in Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ was confirmed by magnetic susceptibility, dc and infrared conduc- tivity measurements, whereas Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ did not show any signatures of superconducting or magnetic phase transitions. In the following, we present specific-heat data for a superconducting Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ crystal of weight 13.6 mg and for a Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ reference crystal of weight 9.9 mg. Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ single crystal with T$_{c}$ = 38.5 K (onset), the highest transition temperature reported to-date for this family of superconductors. The superconducting specific heat indicates two energy gaps with magnitudes 2Δ$_{1}$ = 6 . 6 k$_{B}$T$_{c}$ and 2Δ$_{2}$ = 2 . 2 k$_{B}$T$_{c}$ , in agreement with the ARPES, µ SR, and NMR results [5, 6]. We also calculated the specific heat from the Eliashberg spectral function in a four-band spin- fluctuation model. From the solution of the Eliashberg equa- tions we obtained the value of T$_{c}$ , the superconducting gaps, and the temperature dependence of the free-energy difference in good agreement with the experimental data. The averaged electron-boson coupling constant, λ av ≈ 1 . 9 , resulting from this analysis and consistent with the magnitude of the specific heat jump and its first derivative below T$_{c}$ , is distinctly larger than the estimated electron-phonon coupling [1]. The strong coupling suggests the interaction with low-energy ( ≲ 50 meV) interband spin-fluctuations as the pairing mechanism. The newly discovered iron pnictide superconductors present an unusual case of multiband superconductivity in the vicinity of a magnetic instability. In view of the weak electron-phonon coupling in this class of superconductors [1], magnetic excitations are the most promising candidates for the pairing boson. The analysis of experimental data and the theoretical modeling of these systems are, however, consid- erably complicated by their complex electronic structure that involves at least four energy bands crossing the Fermi surface. Despite intensive research, there is hence no consensus on the gap symmetry and on the nature of the pairing interaction. The thermodynamic properties are a key source of surface- insensitive information on the electronic interactions and on the structure of the superconducting gap function. For a su- perconducting order parameter with alternating sign, which appears to be present in at least some of the pnictides, the pairing interactions are strongly influenced by impurity scat- tering, so that sample quality and phase purity issues are cru- cial for calorimetric experiments. Presumably because of such issues, the current specific heat evidence of doped BaFe$_{2}$As$_{2}$ , a pnictide family for which sizable single crystals are avail- able, is still ambiguous. Recently two gaps were reported in electron-doped Ba(Fe$_{0}$$_{.}$$_{925}$Co$_{0}$$_{.}$$_{075}$)$_{2}$As$_{2}$ , supporting the re- sults of nuclear magnetic resonance (NMR) and muon spin relaxation ( µ SR) studies [2]. In hole-doped Ba$_{1}$$_{−}$$_{x}$K$_{x}$Fe$_{2}$As$_{2}$ , on the other hand, prior specific heat experiments have shown only one gap with values of 6 meV in a single crystal with T$_{c}$ = 36.5 K [3] and varying from 5.9 meV to 6.7 meV for 0.3 ≤ x ≤ 0.6 (36.0 K ≤ T$_{c}$ ≤ 37.3 K) in polycrystalline speci- mens [4], in contrast to the observation of two nearly isotropic gaps in angle-resolved photoemission (ARPES) experiments [5]. Alas, all of the specific heat data thus far reported suffer from a residual low-temperature non-superconducting elec- tronic contribution and show Schottky anomalies. In electron- doped BaFe$_{2}$As$_{2}$ , this complication is likely due to structural defects caused by Co substitution directly in the supercon- ducting FeAs planes. In hole-doped BaFe$_{2}$As$_{2}$ , the substi- tution of Ba by K does not disturb the structure of the FeAs planes, but gives rise to electronic phase separation in the un- derdoped regime [7, 8].