
Equation (3) follows if /pS ¼ 0 even in the presence
of a magnetic field (described in the London gauge).
In a normal metal /pS changes in such a way that
/p eAS is very small, giving just the small Landau
diamagnetism. Particularly in his book, published in
1950, London emphasized that the rigidity of the
wave functions in a magnetic field could arise from a
long-range order in the momentum distribution of
the electrons.
According to the London theory, the shielding
currents near the surface of a superconductor in a
magnetic field flow in a layer of average thickness
determined by a penetration depth l ¼ l
L
¼ L
1=2
,
determined by n
s
, the density of superconducting
electrons. In the late 1940s, Pippard, working at
Cambridge, found that the penetration depth in-
creases markedly with added impurities. Impurities
decrease the scattering mean free path of electrons in
the normal state but have little effect on their density.
For this and other reasons, he proposed a nonlocal
form of the London relation in Eqn. (3) in which
J
s
ðrÞ is given by an integral of AðrÞ over a region
surrounding the point r of size determined by a co-
herence distance x.Ifl is the electron mean free path,
x
1
¼ x
1
0
þ l
1
ð4Þ
where x
0
is the coherence distance in the pure metal,
typically of order 10
7
m.
About the same time (1950), Ginzburg and Landau
proposed a different sort of modification of the Lon-
don equations. They were interested in trying to un-
derstand the energy of the boundary between
superconducting and normal domains in the inter-
mediate state. The intermediate state consists of a
series of normal and superconducting domains in the
form of slabs parallel to the applied field. The field B
is equal to the critical field in the normal domains and
vanishes in the interior of the superconducting do-
mains. The change from n
s
¼0 in the normal domains
to the equilibrium value in the superconducting do-
mains occurs gradually across the boundary over a
distance determined by the two lengths in the prob-
lem, x and l.
Thus they needed to have a theory that describes
changes of n
s
(r) in space. Further, in accord with
Landau’s theory of second-order phase transitions,
they described the ordered low-temperature phase by
an order parameter that vanishes as T-T
c
. Ginzburg
and Landau showed remarkable insight in proposing
on phenomenological grounds that the order para-
meter be a complex function CðrÞ¼jCðrÞjexp½i jðrÞ
with amplitude and phase. The order parameter
describes both the density n
s
and the velocity v
s
of
superconducting condensate:
n
s
¼jCðrÞj
2
m
v
s
¼ _rjðrÞe
AðrÞ
)
ð5Þ
Because of pairing, it is now known that m
¼ 2 m,
e
¼ 2e and n
s
¼ n
s
=2.
Since CðrÞ must be single valued, the line integral
of rj around a closed loop must be a multiple of 2p.
If the loop is in the interior of a superconductor
where v
s
¼0 (such as around the interior of a torus
containing a persistent current), the enclosed flux F ¼
H
A:dl ¼ðh=e
Þ
H
rjdl must be a multiple of a flux
unit F
0
¼ h=e
.
By expanding the free energy in powers of C(r),
they derived a nonlinear Schro
¨
dinger-like equation to
determine C(r). This equation was coupled with
Maxwell’s equations to determine C(r) and thus the
supercurrent flow subject to appropriate boundary
conditions. The expansion is presumed to be valid
just below T
c
where C(r) is small. They used the
equations to determine the boundary energy as well
as other properties of superconductors.
To have a Meissner effect, the boundary energy
must be larger than a critical value. Otherwise the
superconductor could break up into a structure with
very thin normal regions that would allow the flux
to penetrate a distance l on either side. The decrease
in magnetic energy from the flux penetration would
more than compensate for the higher free energy of
the normal regions. Ginzburg and Landau found
that the criterion could be described by a parameter
k ¼ l=x.Ifko1=O2, the superconductor is stable
against normal domain formation for all fields less
than H
c
and there is a perfect Meissner effect.
If k41=O2, the flux begins to penetrate when the
field is greater than a lower critical field H
c1
oH
c
, but
the metal remains superconducting up to a higher
critical field, H
c2
4H
c
. Since magnetic flux can
penetrate for H
c1
oHoH
c2
, the increase in magne-
tic energy is less than that for a complete Meissner
effect, so that the transition to the normal state does
not occur until the higher critical field is reached. The
upper critical field H
c2
may be as large as 20 T
or greater for some intermetallic compounds. High-
field superconducting magnet wire is made with such
materials.
Superconductors with ko1=O2 that exhibit a per-
fect Meissner effect are called type I, those with
k41=O2 are called type II. Type-II superconductors
were first investigated by Shubnikov at Kharkov in
the Soviet Union in 1937 and 1938. His work stopped
when he fell victim to a purge.
It was first thought that flux penetrates type-II su-
perconductors through thin layers of normal domains
as it does in the intermediate state. Then Abrikosov
derived another solution, published in 1957, of the
Ginzburg–Landau equations that he interpreted as a
periodic array of quantized vortex lines. The order
parameter C(r) goes to zero on the axis of the lines.
Supercurrent circulating around the axis gives rise
to one unit of flux F
0
. With this theory, Abrikosov
was able to account in a quantitative way for the
magnetization curves observed by Shubnikov.
1148
Superconducting Materials: BCS and Phenomenological Theories