Elaboration of the alpha-Model Derived from the BCS Theory of Superconductivity
The single-band alpha-model of superconductivity [H. Padamsee, et al., J. Low Temp. Phys. {\bf 12}, 387 (1973)] is a popular model that was adapted from the single-band Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity mainly to allow fits to electronic heat capacity versus temperature T data that deviate from the BCS prediction. The model assumes that the normalized superconducting order parameter Delta(T)/Delta(0) and therefore the normalized London penetration depth lambda_L(T)/lambda_L(0) are the same as in BCS theory, calculated using the BCS value alpha_BCS = 1.764 of alpha = Delta(0)/(kB Tc), where kB is Boltzmann's constant and Tc is the superconducting transition temperature. On the other hand, to calculate the electronic free energy, entropy, heat capacity and thermodynamic critical field versus T, the alpha-model takes alpha to be an adjustable parameter. Here we write the BCS equations and limiting behaviors for the superconducting state thermodynamic properties explicitly in terms of alpha, as needed for calculations within the alpha-model, and present plots of the results versus T and alpha that are compared with the respective BCS predictions. Mechanisms such as gap anisotropy and strong coupling that can cause deviations of the thermodynamics from the BCS predictions, especially the heat capacity jump at Tc, are considered. Extensions of the alpha-model that have appeared in the literature such as the two-band model are also discussed. Tables of values of Delta(T)/Delta(0) and lambda_L(T)/lambda_L(0) calculated from the BCS theory using alpha = alpha_BCS are provided, which are the same in the alpha-model by assumption. Tables of values of the entropy, heat capacity and thermodynamic critical field versus T for seven values of alpha including alpha_BCS are also presented.