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** Up:** Superconductivity
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The Ginzburg-Landau theory revolves around the concept of a complex order
parameter
,
a pseudowavefunction describing the
centre of mass motion of the Cooper pairs [1]. The
distribution
is directly proportional to the
superelectron density
*n*_{s}(**r**) [19]. This theory assumes
local electrodynamics and is strictly valid only at temperatures near the
transition temperature *T*_{c} [1].
The order parameter must be small and vary slowly to derive the
Ginzburg-Landau differential equations.

The Ginzburg-Landau differential equations proceed from minimising the
Helmholtz free energy *f* of the superconducting
state [1][19]. The free energy *f* is
expanded in powers of the order parameter
and its
spatial derivative
,
and the
effect of a magnetic field on a particle of charge *e*^{*} and mass *m*^{*} is
included. Minimising the resulting free energy expression with respect to
the order parameter
and the internal field
*B*(**r**)gives the Ginzburg-Landau equations

where
**A**(**r**) is the vector potential associated with the
internal field
.
The quantities
and
are
expansion coefficients which depend on the penetration
depth
and the thermodynamic critical field *H*_{c}.
Relations (2.12) and (2.13) form coupled differential
equations for the order parameter
and the vector
potential
**A**(**r**). The first equation resembles the
Schrödinger equation for a free particle plus a nonlinear term.
This nonlinear term
encourages the order parameter
to spread evenly
throughout space. The second equation quantum mechanically describes a
current of particles of charge *e*^{*} and mass *m*^{*}. Calculations based on
the microscopic BCS theory show that the effective charge *e*^{*} is twice
the usual electronic charge *e*. Equation (2.12) also hints at another
important length scale for superconductivity, the temperature dependent
coherence length
.

The temperature dependent coherence length
characterizes the
distance over which changes in the order parameter
occur [1]. For this reason a finite coherence length implies a gradual spatial evolution between superconducting and normal regions.
Unlike the temperature independent BCS coherence length ,
the
Ginzburg-Landau coherence length
grows with temperature *T* in a
manner similar to the penetration depth
.

The ratio
of the penetration depth of a
superconductor to its coherence length determines whether this material
exhibits a vortex lattice. Figure 2.2 depicts the scenarios
associated with the two extremes of the Ginzburg-Landau parameter .

In both cases the order parameter
rises from zero to its
maximum value, and the internal field
**B** drops from its maximum to
zero, across the border from normal to superconducting domains. In the
situation
a zone exists where the field
**B** has been
substantially expelled and the order parameter
is not
yet maximal. Here the condensation energy is too small to completely
compensate for the Gibbs free energy increase caused by the negative
magnetisation. This leads to a positive surface energy in connection with
the domain wall between the superconducting and the normal material.
Conversely, in the case
,
a boundary region arises where the
positive diamagnetic energy is not enough to counteract the Gibbs free
energy reduction stemming from the growing number of superelectrons. The
surface energy is now negative. Calculations reveal that the surface energy
of a domain wall becomes zero for a Ginzburg-Landau parameter
of
.
Type I labels superconductors with a
smaller
than this, while type II refers to those with a greater .
In type II
superconductors the negative surface energy generates a regular array of
flux tubes, each associated with a single flux quantum
.
A flux line,
also known as a vortex, is quasinormal at its centre, where the order
parameter
,
the supercurrent density
**J**(**r**)and the BCS energy gap
all vanish. The detailed structure of a vortex
forms the subject of the next section.

** Next:** The Vortex Core
** Up:** Superconductivity
** Previous:** BCS Theory
*Jess H. Brewer *

2001-10-31