In order to understand some of the superconducting mechanisms involving external electric fields at nanometric scales, a Lorentz-covariant extension of the phenomenological Ginzburg-Landau theory has been developed by analogy with the Higgs model of particle physics. Among the specific properties of this model, it has been shown that the phase diagram of some particular geometry submitted to crossed electric and magnetic fields in a stationary situation provides a criterion involving the applied electric field, which could discriminate between the usual Ginzburg-Landau theory and its covariant extension. A sub-microscopic device has been manufactured using microelectronics lithography techniques and was used to perform transport measurements at very low temperatures. However, the experimental measurements of the phase diagram do not reproduce the expectations based whether on the usual or the extended model, suggesting a screening of the electric field by some mechanism which is not accounted for by these phenomenological approaches.
A microscopic approach has therefore been developed to extend the s-wave channel of the BCS theory in a relativistic framework, using the functional integral formalism of Finite Temperature Field Theory. In particular, the effective action related to the Ginzburg-Landau free energy was obtained up to second order in the fluctuations of the electromagnetic field and of the superconducting condensate density. This allowed for the identification of the electric and magnetic penetration lengths, inclusive of their dependences on temperature and the chemical potential, which fully explain the experimental results. Several analytic expressions have also been provided for the effective potential in the full range of temperatures between 0 K and the critical temperature, among which the Ginzburg-Landau potential was shown to reproduce this effective potential within the limited range of temperatures where it is expected to be valid.