The External Field Dependence of the BCS Critical Temperature

Abstract

Abstract:We consider the Bardeen-Cooper-Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg-Landau equation.

The BCS Critical Temperature in a Weak External Electric Field via a Linear Two-Body Operator

“…This subleading correction is determined by an effective linear Ginzburg-Landau functional which emerges on the macroscopic scale 1/h determined by the external potential. We hereby recover a similar result for the critical temperature as in the full non-linear BCS theory in [4]. This is of course not unexpected since we deal with the second derivative around the normal state of the BCS functional.…”

“…Let us compare our results here with those in [4] where we also computed the shift of the critical temperature. The results of [4] concern a definition of the critical temperature in the non-linear BCS functional, whereas here we base our definition of critical temperature on a quadratic approximation to the BCS functional around the normal state. Both notions lead to the same result to order h 2 .…”

“…The work [4] relied on [3] where the Ginzburg-Landau functional was derived from the BCS functional close to the critical temperature by means of a rather intricate proof. In view of this, the goal of the present paper is twofold.…”

“…First, we explain the strategy from [1] in a simpler setting, and second, we derive the linearized Ginzburg-Landau equation in a simpler way as in the full non-linear case [3]. One difference compared to the work [3,4] is the fact that we do not restrict ourselves to a finite box and therefore omit the periodicity assumptions. Further, we work in relative and center-of-mass coordinates which is natural in terms of the before mentioned two-scale structure.…”

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We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in [4] using the strategy introduced in [1], where we considered the case of an external constant magnetic field.

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