Abstract
In this paper, we show that the scale of spatial variation of the order parameter ~ in extreme type II superconductors has a temperature dependence other than that of the temperature dependent Ginzburg Landau coherence length ~o( T). Furthermore for temperatures in the vicinity of the critical temperature Tc our results indicate that ~/~o( T) decreases with decreasing temperature. When an external magnetic field is applied to a type II superconductor, vortex lines form provided that the magnetic field is below the upper critical field H e2• 1 ) For temperatures just below the superconducting transition temperature T e, or for a magnetic field just below H e2, such vortices can be analysed within the Ginzburg Landau (GL) framework. 2 ) For temperatures further below Te the Eilenberger and Bogoliubov equations have been investigated, through interative or variational methods. 3 )-5) Kramer et a1. 3 ) and Gygi et a1. 4 ) find that the scale of spatial variation of the order parameter divided by the GL temperature dependent coherence length decreases with decreasing temperature. On the other hand Dorsey et a1. 5 ) claim that there is no appreciable temperature dependence in this quantity. Instead of formulating the problem in terms of Green's functions and the order parameter we adopt a path integral approach. Here the fermion fields associated with the electrons are integrated out, resulting in a formal expression for the free energy in terms of a functional determinant. The order parameter configuration is then given by minimisation of the free energy. In this paper we will be concerned with the quantity ?.=~/J[~o(T), where ~ is the scale of spatial variation of the order parameter and ~o( T) is the GL temperature dependent coherence length. Any temperature dependence of?. indicates that ~ has a temperature dependence that is different from the GL variation. This work aims to determine whether? has any temperature dependence. The vortex free energy, relative to that of the uniform configuration, at finite temperature, T=l//3, can be written as a ratio of two path integrals over Grassmann fields 6
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