Magnetic-field oscillations of the critical temperature in ultraclean, two-dimensional Type-I superconductor

We investigate the influence of Landau Levels (LLs) and Zeeman energy, induced by an applied magnetic field $\mathbf{B}$, on the critical temperature $T_c$ 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 $\mathbf{B}$ direction, both on the same or on neighboring LLs. Our quantum mechanical treatment of LLs reveals that $T_c(\mathbf{B})$ 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 $T_c(\mathbf{B})$ with increasing $\mathbf{B}$ for electrons paired both on the same and on neighboring LLs. Notably, as the $g$-factor increases, the amplitude of the $\mathbf{B}$ 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.