Giant enhancement of superconductivity in ultrathin $a$-Pb films by a parallel magnetic field: effect of film thickness

We have observed a pronounced enhancement of superconductivity in ultrathin homogeneously disordered amorphous Pb films in the presence of a parallel magnetic field. Our experiments are carried out in a modified dilution refrigerator capable of \textit{in situ} film growth and sample rotation, thus allowing for a detailed examination of the effects of perpendicular and parallel magnetic field on the same film at incrementally larger thicknesses. With increasing thickness, the film is tuned from an insulator to a superconductor where its zero-field T$_{c}$ increases with the thickness. For these films we observe enhancement of T$_{c}$ by a parallel magnetic field, while a perpendicular magnetic field of any strength causes a degradation of T$_{c}$. The magnitude of the T$_{c}$ enhancement appears to exhibit a non-monotonic dependence on the film thickness (zero-field T$_{c})$. The enhancement is as large as 13{\%} and persists in fields as high as 8 T for certain thicknesses. We will discuss the possible origin of the field-enhancement of superconductivity.