Details of the thermodynamical derivation of the Ginzburg–Landau equations

Abstract

We examine the procedure of thermodynamical derivation of the Ginsburg-Landau equation for current, which is given unclear and contradictory interpretations in existing textbooks. We clarify all steps of this procedure and find as a consequence a limitation on the validity range of the thermodynamic Ginsburg-Landau theory, which does not seem to be explicitely stated up to now: we conclude that the thermodynamic theory is applicable only to a superconducting specimen that is not a part of an external current-carrying loop.

Use of the modified Ginzburg–Landau equations in high temperature superconductors

“…However, such a problem needs careful study as it is shown in ref. [22]. It was shown that the free energy variation depends on whether the variation is going with respect to the vector potential A or with respect to the magnetic field itself.…”

Ginzburg–Landau equations and boundary conditions for superconductors in static magnetic fields

“…The second term in (5) is the energy difference between the normal state and the superconducting phase which is expanded in even powers in the complex 'order-parameter wave function' Ψ( r),…”

“…The usual expression for the coupling between condensate and vector potential involves the absolute square of the gauge-invariant gradient [1,[3][4][5][6][7][8]; the original Ginzburg-Landau functional and the corresponding boundary conditions are treated with mathematical rigor in [9].…”

“…The Ginzburg-Landau energy functional in (5) and its four components are invariant under local gauge transformations,…”

“…Here, δ dG is the de Gennes length. Because this length scale did not appear in the original Ginzburg-Landau treatment it is sometimes argued that δ dG is of 'microscopic origin' and thus beyond the reach of a phenomenological theory [4,5].…”