We investigate the influence of Landau Levels (LLs) and Zeeman energy, induced by an applied magnetic field B, on the critical temperature Tc for two-dimensional (2D) ultraclean metals using a fully quantum mechanical approach within the Bardeen-Cooper-Schrieffer (BCS) theory. In contrast to standard BCS theory, it allows for Cooper pair formation between electrons with opposite spins and momenta along the B direction, both on the same or on neighboring LLs. Our quantum mechanical treatment of LLs reveals that Tc (B) for electrons paired on the same LLs exhibits oscillations around the BCS critical temperature at lower magnetic fields, a phenomenon analogous to the de Haas-van Alphen effect. The Zeeman energy leads to a decrease in Tc (B) with increasing B for electrons paired both on the same and on neighboring LLs. Notably, as the g-factor increases, the amplitude of the B oscillations gradually diminishes until they vanish at higher magnetic fields. Conversely, for small g-factors, electron pairing on the same or on neighboring LLs can result in a re-entrant superconducting phase at very high magnetic fields.
Zhao et al. (Wed,) studied this question.