As presented in their original form, the above results pertain almost
exclusively to type-I superconductors. The main topic of interest
here however, is the magnetic properties of type-II
superconductors, whose magnetization curve is depicted in Fig. 2.4.
Below *T*_{c}, in the presence of an externally applied magnetic
field *H*, three distinct phases are recognizable, dependent on the
strength of the applied field. Below a
*lower critical field*
*H*_{c1} (*T*)the superconductor is in the Meissner state with full expulsion of
magnetic flux from its interior. For an applied
field above an *upper critical field*
*H*_{c2} (*T*), magnetic flux fully penetrates the type-II material and
returns it to its normal state. Within Ginzburg-Landau theory, the lower
and upper critical fields may be written as:

(20) |

and,

(21) |

respectively.

If the applied field lies between
*H*_{c1} (*T*)and
*H*_{c2} (*T*), there is a partial penetration of flux into the sample leading
to regions in the interior which are superconducting and others which
are in the normal state; this is often referred to as the *mixed state*.
Ginzburg and Landau defined type-II superconductors as those with
.
The high-temperature superconductors are extreme
type-II, with large GL parameters
(*i.e.*
),
large upper critical fields
*H*_{c2} (*T*), and small lower critcal fields
*H*_{c1} (*T*) [32,33,34].

In 1957, the Russian physicist Alexei A. Abrikosov predicted the existence
of type-II superconductors by considering the solution of the GL equations
for
[35]. In particular, he considered
the case where the externally applied magnetic field H is only slightly
below
*H*_{c2} (*T*); for in this region one obtains approximate solutions
resembling those of the linearized GL equations [13]. The
solutions revealed the presence of a periodic microscopic magnetic
field distribution, transverse to the applied field. More precisely his
efforts predicted a periodic square array of thin filaments of magnetic
flux in the mixed state. Consequently, literature sometimes refers to
the mixed state as the *Abrikosov regime* or for reasons to soon
be made apparent, the *vortex state*.

In the core region of a filament, the magnetic field is high and the
material does not superconduct here;
that is these regions are in the normal state. The
magnetic field is screened from the rest of the sample by supercurrents
which circulate around each filament. It is common to refer
to the filaments as *vortex lines* and the
array of filaments as a *vortex lattice*. At the vortex center where
the magnitude of the local field is largest, the density of superconducting
electrons *n*_{s} and hence the order parameter
is zero.
As one moves radially out from the center of a vortex core, *n*_{s}increases and the supercurrents screen a greater amount of flux. At a
radius on the order of the coherence length ,
*n*_{s} approaches the value in the bulk of the sample. It follows
that the density of the supercurrents is greatest
near the edge of the vortex core where the
screening of the local magnetic field is at a maximum. Outside of the vortex
core, magnetic field leaks into the superconducting regions of the sample
( see Fig. 2.5), in much the same
way magnetic field penetrates the surface
of the semi-infinite
superconductor in Figure 1.2. The superconducting order parameter is, for all
intents and purposes, constant beyond regions of the order
around
the vortex cores [36].
It may be noted here that
Abrikosov's prediction of the vortex state was remarkable in that it
precluded any concrete experimenital proof of its existence. He derived
the following relations between *H*_{c1}, *H*_{c2}, and the
thermodynamic critical field *H*_{c}:

(22) |

and,

(23) |

At a field

In the dirty limit (
), one can write the upper critcal field
as [14]:

(24) |

Thus it is clear that increasing the impurity content of the superconductor and thus shortening the mean free path

One may wonder how London's magnetic penetration depth
could be
incorporated into a theory for type-II superconductors. Although London's
original theory is indeed valid for
,
it makes no
mention of an upper critical field
*H*_{c2} (*T*). Nevertheless, one
could anticipate some extension of the London model applicable to
type-II superconductors by consideration of specific features. To start
with, F. London theoretically predicted that magnetic flux is quantized
in a superconducting annulus. An analogous treatment of the supercurrents
that shield the vortex cores, leads one to deduce in several different
ways [13,17], that the flux associated with each vortex is
quantized and defined by:

where is called the

For an isolated vortex line, the field as determined in the extreme type-II
limit
is [35]:

where r is the distance from the axis of the vortex line, and denotes a modified Bessel function of the second kind. For small distances ( ), from the vortex-line axis, Eq. (2.26) reduces to [13]:

(27) |

while further away ( ):

(28) |

The interaction between vortex lines becomes significant when their
separation is
[38]. In a type-II superconductor,
the local magnetic field outside a vortex core decays in an exponential
manner so that one
is justified in describing the field in this region by the London
equations. In a crude sense, the mixed state of a type-II
superconductor may be visualized as a type-I superconductor with a
regular array of normal state regions (*i.e.* the vortex cores)
embedded in its interior. The magnetic penetration depth
is then
a measure of the length over which magnetic flux leaks into the
superconducting regions that encompass the vortex cores.

The existence of the mixed state can be argued through free energy
considerations. Consider a sample in an applied magnetic field
*H*_{applied} with a normal phase and a superconducting phase separated by
a single interface. Realistically, the changeover between the two
phases is not abrupt. Magnetic field leaks into the
superconducting region a distance
,
resulting in a
contribution to the free surface energy of the interface that is negative.
In addition, the order parameter
decreases to zero
over a distance ,
thereby decreasing the actual volume of
the sample which superconducts. This leads to a positive-energy
contribution at the interface. The sum of both phenomena give the
net surface energy per unit length at the interface as [39]:

(29) |

A type-I superconductor where , has positive surface-energy; while type-II superconductors with , have negative surface-energy. A more precise calculation in the context of GL-theory predicts that the sign of the surface energy changes at [17]. For a type-I superconductor in an applied field, the presence of normal-superconducting interfaces would increase the total free energy density because of positive surface-energy contributions. This situation is of course most unfavourable. For this reason, in the intermediate state of a type-I superconductor large normal regions will be formed which contain many flux quanta [13]. On the other hand, in the case of a type-II superconductor, negative surface-energy contributions from normal-superconducting interfaces would reduce the free energy density. Consequently, the total energy is minimized by introducing as many of these interfaces as possible. This is exactly what happens in the mixed state of a type-II superconductor.

It has been argued that the vortex model borne out of Abrikosov's solution of the Ginzburg-Landau equations is the most energetically favourable form for the magnetic flux to assume inside the superconductor [37]. Maximizing the surface-to-volume ratio of the normal regions leads to the lowest energy situation in the mixed state, because this provides the smallest positive surface-energy contribution. To achieve this, one can form normal regions which constitute either lamina of very small thickness () or alternatively, filaments of small diameter (). Theoretical calculations indicate that the latter formation is lowest in energy for [14].

Since Abrikosov's original prediction of a square vortex lattice, subsequent
solutions of the GL equations for magnetic fields just below the upper
critical field
*H*_{c2} (*T*), have convincingly shown that a periodic
triangular array of vortex lines has the lowest free energy of all
possible periodic solutions, and hence is the most stable
configuration [13,40,41]. Decoration experiments which
utilize small iron particles to make the vortices visible through an
electron microscope have not only provided the best confirmation of
the triangular arrangement but have also verified the flux quantum of
Eq. (2.25) [42]. It is clear from such
experiments which verify
the flux quantum that it is energetically more favourable to increase
the number of vortices as one increases the magnetic field, rather than
increasing the amount of flux in each vortex [37].
Thus in an isotropic type-II superconductor,
the vortex lines repel one another as a result of the magnetic force
between them, to form an equilateral triangular lattice in accordance
with energy minimization.

2001-09-28