Heat capacity of mesoscopically inhomogeneous superconductors: theory and applications to MgB2

Electronic specific heat C as a function of temperature T was calculated for a mesoscopically disordered s-wave superconductor treated as a spatial ensemble of domains with continuously varying superconducting properties. Domains are supposed to have sizes L>ξ, where ξ is the coherence length. Specific calculations were performed for an exponential and Gaussian distributions of domain critical temperatures Tc0's. It is shown that the spatially averaged is proportional to T2 for low T, whereas the anomaly at Tc is substantially smeared even for small dispersions. For narrow gap distributions there exists an intermediate T range, where the curve can be relatively well approximated by an exponential Bardeen-Cooper-Schrieffer-like dependence with the effective gap smaller than the weak-coupling value. The results obtained successfully reproduce the salient features of the C(T) data for MgB2, where a wide superconducting gap distribution has been previously seen in tunnel, point-contact, photoemission and Raman spectra. The observed anomalously low ratio Δ(0)/Tc, where Δ(0) is the superconducting energy gap at T=0, can be also understood in the framework of our theory. After an analysis of the available experimental evidence a conclusion is made that the multiple-gap behavior of superconducting MgB2 is due to the spatial intrinsic non-homogeneities. Possible origins of the latter are discussed