Success of BCS theory
Shibnath Samanta (PH12D055)
PhD Scholar, Dept. of Physics, IIT Madras.
ABSTRACT
The most successful theoretical model that gives the first correct explanation to superconductivity is known as
BCS theory. In 1957, more than 40 years after the discovery of superconductivity, three physicists, Bardeen,
Cooper and Schrieffer, finally found the correct explanation to superconductivity in metals. Hence the model
has been named as BCS theory after their initials. They proposed that the electrons form pairs, called cooper
pairs before forming a collective quantum wave. Almost all superconducting properties of metals and behavior
of characteristic length and other parameters have been explained well by BCS theory. The theory is also used
in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.
Superconductivity was discovered by Heike
Kamerlingh Onnes on April 8, 1911 in Leiden.
Superconductivity is a quantum mechanical
phenomenon. It is characterized by the Meissner
effect. A number of theory have been developed by
many scientists to explain different properties of
superconductor such as Zero electrical DC
resistance, Superconducting phase transition,
critical thermodynamic parameters, Meissner effect,
London moment, penetration depth, coherence
length etc. The first phenomenological theory was
London theory. Then in 1950 the phenomenological
Gingburg-Landau theory was devised by Landau
and ginzburg. It was successful in many aspects
including the classification of Type-I and Type-II
superconductor. Finally the most successful theory
comes in 1957, proposed by Bardeen, Cooper and
Schrieffer.
The main idea of the BCS theory relies on
the quantum nature of electrons. In a metal,
electrons are waves. Each of these electrons is
relatively independent and follows its own path
independent of other electrons. In a superconductor,
the majority of these electrons merge in order to
form a large collective wave. In quantum physics,
we call it macroscopic quantum wave function or
condensate. When the collective wave is formed, it
requires each member to move at same speed. In a
metal, an individual electron is easily diverted by a
flaw or an atom that is too big. In a superconductor,
this same electron can be diverted only if, at the
same time all other electrons of the collective wave
are diverted in the exactly same manner. The flaw
in a single atom surely cannot do that, the wave will
not be diverted and thus it will not be slowed
down
1-2
.
BCS derived several important theoretical
predictions that are independent of the details of the
interaction, since the quantitative predictions
mentioned below hold for any sufficiently weak
attraction between the electrons and this last
condition is fulfilled for many low temperature
superconductors - the so-called weak-coupling
case
3
. These have been confirmed in numerous
experiments:
According to BCS theory Specific heat(1), flux
quantization(2), critical fields(3,4,5), oscillation of
DC Josephson current(6) are as follows
T
c
s
e
T T
c
/ ) 0 (
2 / 3
) 0 (
34 . 1
A
|
.
|
\
| A
=
..1
0
2
u = = u n
e
nhc
2
2
0
1
t
u
~
c
H ...3
2
0
2
tc
u
~
c
H 4
c c c
H H H ~
2 1
.5
eV 2
= e .6
The electrons are bound into Cooper pairs, and
these pairs are correlated due to the Paulis
exclusion principle for the electrons, from which
they are constructed. Therefore, in order to break a
pair, one has to change energies of all other pairs.
This means there is an energy gap for single-particle
excitation
4
. This energy gap is highest at low
temperatures but vanishes at the transition
temperature when superconductivity ceases to exist.
The BCS theory gives an expression that shows
how the gap grows with the strength of the
attractive interaction and the (normal phase) single
particle density of states at the Fermi energy.
Furthermore, it describes how the density of states
is changed on entering the superconducting state,
where there are no electronic states any more at the
Fermi energy. The energy gap is most directly
observed in tunneling
5
experiments and in reflection
of microwaves from superconductors.
BCS theory predicts the dependence of the
value of the energy gap E
g
at temperature T on the
critical temperature T
c
. The ratio between the value
of the energy gap at zero temperature and the value
of the superconducting transition temperature
(expressed in energy units) takes the universal value
of 3.52. It is independent of material. Near the
critical temperature the relation
6
asymptotes to
|
|
.
|
\
|
=
c
c B g
T
T
T K E 1 52 . 3
7
This is based on the fact that the superconducting
phase transition is second order, that the
superconducting phase has a mass gap.
Due to the energy gap, the specific heat of
the superconductor is suppressed strongly
(exponentially) at low temperatures. There is no
thermal excitations left. However, before reaching
the transition temperature, the specific heat of the
superconductor becomes even higher than that of
the normal conductor (measured immediately above
the transition) and the ratio of these two values is
found to be universally given by 2.5.
BCS theory correctly predicts the Meissner
effect, i.e. the expulsion of a magnetic field from
the superconductor and the variation of the
penetration depth (the extent of the screening
currents flowing below the metal's surface) with
temperature. This had been demonstrated
experimentally by Walther Meissner and Robert
Ochsenfeld in their 1933 article Ein neuer Effekt bei
Eintritt der Supraleitfhigkeit.
It also describes the variation of the critical
magnetic field (above which the superconductor can
no longer expel the field but becomes normal
conducting) with temperature. BCS theory relates
the value of the critical field at zero temperature to
the value of the transition temperature and the
density of states at the Fermi energy.
In its simplest form, BCS gives the
superconducting transition temperature in terms of
the electron-phonon coupling potential and
the Debye cutoff energy
8
:
V N
D c B
e E T k
) 0 ( / 1
14 . 1
=
..8
Here N(0) is the electronic density of states at the
Fermi energy.
The BCS theory reproduces the isotope
effect, which is the experimental observation that
for a given superconducting material, the critical
temperature is inversely proportional to the mass of
the isotope used in the material. The isotope effect
was reported by two groups on the 24th of March
1950, who discovered it independently working
with different mercury isotopes, although a few
days before publication they learned of each other's
results at the ONR conference in Atlanta, Georgia.
The two groups are Emanuel Maxwell, who
published his results in Isotope Effect in the
Superconductivity of Mercury and C. A. Reynolds,
B. Serin, W. H. Wright, and L. B. Nesbitt who
published their results 10 pages later
in Superconductivity of Isotopes of Mercury. The
choice of isotope ordinarily has little effect on the
electrical properties of a material. But affect the
frequency of lattice vibrations, this effect suggested
that superconductivity be related to vibrations of the
lattice. This is incorporated into the BCS theory,
where lattice vibrations yield the binding energy of
electrons in a Cooper pair.
Although BCS theory seems to fail in the
case of some superconductors such as cuprates or
pnictides, at present it is the most successful theory
to explain superconductivity in metals.
References:
1. Bardeen, J.; Cooper, L. N., Schrieffer, J. R.
"Microscopic Theory of
Superconductivity". Physical
Review 106 (1): 162164, (April 1957).
2. Bardeen, J.; Cooper, L. N.; Schrieffer, J. R.
"Theory of Superconductivity". Physical
Review 108 (5): 11751204 (December
1957).
3. Leon N. Cooper and Damitri; BCS:50
Years World Scientific Publishing Co. Pvt.
Ltd. ISBN: 978-9814304641
4. Aruldhas & Rajagopal; Modern Physics
PHI Learning Pvt. Ltd
5. Page- 433, Charles P. Poole, Jr., Horacio A.
Farach, Richard J. Creswick;
Superconductivity; Academic Press,
ISBN: 987-0-12-088761-3
6. Page-203, Ajoy Saxena; High Temperature
Superconductor, Springer, ISBN: 987-3-
642-00711-8
7. M. J. Buckingham, Very High Frequency
Absorption in Superconductors Physical
Review-101, 1431-1432 (1956)
8. Page-8, Ran Yang; Time Resolved
Magnetic flux and Ac-current Distribution
in Superconducting YBCO thin films and
Multifilaments
