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Specific heat of Ba$_{0.59}$K$_{0.41}$Fe$_{2}$As$_{2}$, and a new method for identifying the electron contribution: two electron bands with different energy gaps in the superconducting state

We report measurements of the specific heat of Ba$_{0.59}$K$_{0.41}$Fe$_{2}$As$_{2}$, an Fe-pnictide superconductor with $T_c$ = 36.9 K, for which there are suggestions of an unusual electron pairing mechanism. We use a new method of analysis of the data to derive the parameters characteristic of the electron contribution. It is based on comparisons of $α$-model expressions for the electron contribution with the total measured specific heat, which give the electron contribution directly. It obviates the need in the conventional analyses for an independent, necessarily approximate, determination of the lattice contribution, which is subtracted from the total specific heat to obtain the electron contribution. It eliminates the uncertainties and errors in the electron contribution that follow from the approximations in the determination of the lattice contribution. Our values of the parameters characteristic of the electron contribution differ significantly from those obtained in conventional analyses of specific-heat data for five similar hole-doped BaFe$_{2}$As$_{2}$ superconductors, which also differ significantly among themselves. They show that the electron density of states is comprised of contributions from two electron bands with superconducting-state energy gaps that differ by a factor 3.8, with 77$\%$ coming from the band with the larger gap. The variation of the specific heat with magnetic field is consistent with extended $s$-wave pairing, one of the theoretical predictions. The relation between the densities of states and the energy gaps in the two bands is not consistent with a theoretical model based on interband interactions alone. Comparison of the normal-state density of states with band-structure calculations shows an extraordinarily large effective mass enhancement, for which there is no precedent in similar materials and no theoretical explanation.