On Landau quantization of Cooper pairs in a heat bath

The standard Bethe-Salpeter equation, incorporating the BCS model interaction (V) as its kernel, is generalized to include temperature (T), and an external magnetic field (H) by invoking the Matsubara recipe and the Landau quantization (LQ) scheme. The resulting equation for Hc(T) is solved for some selected elements to compare the superconducting parameters in these at T=Tc, H=0 and T=0, H=Hc. Variation of these parameters with T is studied by assuming the empirical law that relates Hc(T) with T, whence it follows that this law can be understood in terms of a linear dependence of V on T. Earlier work invoking LQ, and its connection with the de Haas-van Alphen oscillations in the context of superconductivity, is briefly reviewed. It is shown that these oscillations follow the variations in the number of Landau levels as the field is varied; they are not dependent on whether the superconductor is of type I or II. The possibility of realizing Hc's exceeding Hc(T=0) of an element is investigated, leading to a suggestion about how it may be achieved at T=2.05 K (for Sn). Generalization of our equation for Tc(H) to cater to composite superconductors is discussed.