Quantum size effects in thermodynamic superconducting properties of ultra thin films

By using multi-bands BCS theory, we have calculated the superconductivity energy gap and the critical temperature of a thin-film metallic superconductor. The thermodynamic superconducting characteristics such as critical magnetic field, specific heat, as well as the tunneling conductance are investigated for varying film thickness and temperature. We find the oscillation of thermodynamic superconducting properties with the film thickness, including the thermodynamic critical field $H_c$, the specific heat of normal, superconducting state, and position of the differential tunneling conductance peak. The two universal constants the nth sub-band energy gap $\Delta_n$ at temperature $T=0K$ over $k_BT_c$, and the specific heat jump at $T_c$ over normal state specific heat at $T_c$ are independent of the film thickness. Their values are the same as in the bulk superconductor.