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Supercurrent field-effect transistors made from thin metallic films are a promising option for next-generation high-performance computing platforms. Despite extensive research, there is still no complete quantitative microscopic explanation for how an external dc electric field suppresses superconductivity in thin films. This study aims to provide a quantitative description of superconductivity as a function of film thickness based on Eliashberg's theory. The calculation considers the electrostatics of the electric field, its realistic penetration depth in the film, and its effect on the Cooper pair, which is described as a standard s-wave bound state according to BCS theory. The estimation suggests that an external electric field of approximately 10^8 V/m is required to suppress superconductivity in films 10--30 nm thick, which aligns with experimental observations. Ultimately, the study offers ``materials-by-design'' guidelines for suppressing supercurrent when an external electric field is applied to the film surface. Furthermore, the proposed framework can be easily extended to investigate the same effects for ultrathin films.
Zaccone et al. (Tue,) studied this question.