Oscillatory magnetic field-dependent critical temperatures of ultraclean Type-I superconductors.

The influence of the Zeeman energy and the Landau levels (LLs) arising from an applied magnetic fieldBupon the critical temperatureT_cis studied using a fully quantum mechanical method within the framework of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity that forms from an ultraclean metal. As in semiclassical treatments, we found that two electrons can form Cooper pairs with opposite spins and momenta in theBdirection while either in the same or in neighboring LLs. However, the fully quantum mechanical treatment of the LLs causesT_c(B) for electrons paired on the same LL to oscillate about the critical temperature of the BCS theory, similar to that of the de Haas-van Alphen effect. The Zeeman energy causesT_c(B) to decrease in an oscillatory fashion with increasingBfor electrons paired either on the same or on neighboring LLs. For the Zeemang > 1, pairing on neighboring LLs results in the highestT_c(B). Forg < 1, pairing on the same LLs gives the highestT_c(B). In addition,T_c(B) for electrons paired on neighboring LLs exhibits an apparent symmetry aroundg = 2, as the oscillatory critical temperature behaviors are nearly identical forg=2±δ.