Quantum size effect on the paramagnetic critical field in free-standing superconducting nanofilms

The quantum size effect on the in-plane paramagnetic critical field in Pb(1 1 1) free-standing nanofilms is investigated with the use of the spin-generalized Bogoliubov-de Gennes equations. It is shown that the critical field oscillates as a function of the nanofilm thickness with the period ∼ 2 ML (even-odd oscillations), modulated by the beating effect. The calculated values of the critical field for different nanofilm thicknesses are analyzed in the context of the Clogston-Chandrasekhar limit. It is found that the critical field for superconducting nanofilms differs from this limit. This phenomena is explained in terms of quantization of the electron energy caused by the confinement of electron motion in a direction perpendicular to the film. The thermal effect and thickness-dependence of electron-phonon coupling on the value of the critical magnetic field are also studied.