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The setup: The superconductive film is extended in the x and y direction and has thickness L along the z-direction. The electric field Ez pointing along the direction of the small black arrows is applied to both sides of the film, by means of two charge distributions A and B, taken to be equal. The penetration length is taken to be approximately 1 nm.
Superconductors are fascinating states of matter where charge can flow without resistance. In principle, once you set a current flowing in a superconductor it will never stop. This leads to many unique phenomena, such as the Meissner effect where superconductors repel magnetic fields. However, when a magnetic field is strong enough it does penetrate the superconductor, leading to vortices, and eventually destroying superconductivity.
A natural question might be to ask - what about electric fields? Can they lead to the same phenomena (a phase transition to the normal state)? The general lore among researchers in this field was that the answer is no. Essentially, this is because of screening effects in metals. Unless the electric field was ridiculously large (so large the material is destroyed), it was not possible to use an external field to force the superconductor to become a normal metal.
Yet, surprising results from an experimental group in Rome led to this lore being challenged. They observed that in very thin films of superconducting material, something that seemed like a phase transition happened at large (but not ridiculous) electric fields. In particular, as the electric field was tuned up, the superconducting current decreased.
In our paper, we sought to explain this phenomenon in terms of a phase transition to the normal state. We modified the standard Ginzburg-Landau description of superconductivity to allow for the electric field to modify the standard couplings. The net result was that we reproduced, in a smoothed out manner, the experiments observed by the Rome group.
The results: a): Numerical computation of the averaged critical current Ic normalized against its values at T = 5 mK, I¯c , as a function of the applied electric field E0 (see (1)) normalized against the 5 mK critical electric field Ēc . The curves are the numerical simulations and the dots are the experimental data. The values of I¯c at different temperatures are taken from the experiments. b): The suppression of the electric field effect as a function of the thickness of the film L for T = 5mK and E=Ec. Red dots are experimental data.
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