Quantum size effects on the perpendicular upper critical field in ultra-thin lead films

We report the thickness-dependent (in terms of atomic layers) oscillation behavior of the perpendicular upper critical field $H_{c2\perp}$ in the ultra-thin lead films at the reduced temperature ($t=T/T_c$). Distinct oscillations of the normal-state resistivity as a function of film thickness have also been observed. Compared with the $T_c$ oscillation, the $H_{c2\perp}$ shows a considerable large oscillation amplitude and a $\pi$ phase shift. The oscillatory mean free path caused by quantum size effect plays a role in $H_{c2\perp}$ oscillation.