Calculation by Eliashberg theory of critical current and critical electric field in thin superconducting films

Supercurrent field-effect transistors realized in thin metallic films hold a great promise for future microelectronic devices. In spite of intense research, a complete quantitative microscopic mechanism by which superconductivity in thin films is suppressed by an external DC electric field is missing. Here, for the case of NbN, we provide a quantitative description of superconductivity based on Eliashberg theory. This calculation is in the dirty limit and provides an estimate of the magnitude of the external electric field needed to suppress superconductivity in thick dirty NbN films of the order of 107 V/m, in agreement with experimental observations. We link this critical external electric field with the value of the critical density current and we provide a recipe for reducing the value of the critical electric field.