BCS theory is the first microscopic theory of superconductivity since its
discovery in 1911. The theory describes superconductivity as a microscopic
effect caused by a condensation of Cooper pairs into a boson-like state.
The theory is also used in nuclear physics to describe the pairing
interaction between nucleons in an atomic nucleus. It was proposed by John
Bardeen, Leon Cooper, and John Robert Schrieffer in 1957; they received the
Nobel Prize in Physics for this theory in 1972.
History Rapid progress in understanding
superconductivity gained momentum in the mid-1950s. It began with the 1948 paper,
"On the Problem of the Molecular Theory of Superconductivity", where Fritz
London proposed that the phenomenological London equations may be
consequences of the coherence of a quantum state. In 1953, Brian Pippard,
motivated by penetration experiments, proposed that this would modify the
London equations via a new scale parameter called the coherence length.
John Bardeen then argued in the 1955 paper, "Theory of the Meissner Effect in
Superconductors", that such a modification naturally occurs in a
theory with an energy gap. The key ingredient was Leon Neil Cooper's
calculation of the bound states of electrons subject to an attractive force
in his 1956 paper, "Bound Electron Pairs in a Degenerate Fermi Gas".
In 1957 Bardeen and Cooper assembled these ingredients and constructed such a
theory, the BCS theory, with Robert Schrieffer. The theory was first
published in April 1957 in the letter, "Microscopic theory of
superconductivity". The demonstration that the phase transition is second
order, that it reproduces the Meissner effect and the calculations of specific
heats and penetration depths appeared in the December 1957 article, "Theory of
superconductivity". They received the Nobel Prize in Physics in 1972 for this
theory. In 1986, high-temperature
superconductivity was discovered in some materials at temperatures up to about
130 K, considerably above the previous limit of about 30 K. It is believed that
BCS theory alone cannot explain this phenomenon and that other effects are in
play. These effects are still not yet fully understood; it is possible that
they even control superconductivity at low temperatures for some materials.
Overview At sufficiently low temperatures,
electrons near the Fermi surface become unstable against the formation of Cooper
pairs. Cooper showed such binding will occur in the presence of an attractive
potential, no matter how weak. In conventional superconductors, an
attraction is generally attributed to an electron-lattice interaction. The BCS
theory, however, requires only that the potential be attractive, regardless of
its origin. In the BCS framework, superconductivity is a macroscopic
effect which results from the condensation of Cooper pairs. These have
some bosonic properties, and bosons, at sufficiently low temperature, can form a
large Bose–Einstein condensate. Superconductivity was simultaneously
explained by Nikolay Bogolyubov, by means of the Bogoliubov transformations.
In many superconductors, the attractive interaction between electrons is brought
about indirectly by the interaction between the electrons and the vibrating
crystal lattice. Roughly speaking the picture is the following:
An electron moving through a conductor will attract nearby positive charges in
the lattice. This deformation of the lattice causes another electron, with
opposite spin, to move into the region of higher positive charge density. The
two electrons then become correlated. Because there are a lot of such electron
pairs in a superconductor, these pairs overlap very strongly and form a highly
collective condensate. In this "condensed" state, the breaking of one
pair will change the energy of the entire condensate - not just a single
electron, or a single pair. Thus, the energy required to break any single pair
is related to the energy required to break all of the pairs. Because the
pairing increases this energy barrier, kicks from oscillating atoms in the
conductor are not enough to affect the condensate as a whole, or any individual
"member pair" within the condensate. Thus the electrons stay paired together
and resist all kicks, and the electron flow as a whole will not experience
resistance. Thus, the collective behavior of the condensate is a crucial
ingredient necessary for superconductivity.
= Details = BCS theory starts from the assumption
that there is some attraction between electrons, which can overcome the
Coulomb repulsion. In most materials, this attraction is brought about
indirectly by the coupling of electrons to the crystal lattice. However, the
results of BCS theory do not depend on the origin of the attractive
interaction. For instance, Cooper pairs have been observed in ultracold gases of
fermions where a homogeneous magnetic field has been tuned to their Feshbach
resonance. The original results of BCS described an s-wave superconducting
state, which is the rule among low-temperature superconductors but is
not realized in many unconventional superconductors such as the d-wave
high-temperature superconductors. Extensions of BCS theory exist to
describe these other cases, although they are insufficient to completely
describe the observed features of high-temperature superconductivity.
BCS is able to give an approximation for the quantum-mechanical many-body state
of the system of electrons inside the metal. This state is now known as the
BCS state. In the normal state of a metal, electrons move independently,
whereas in the BCS state, they are bound into Cooper pairs by the attractive
interaction. The BCS formalism is based on the reduced potential for the
electrons attraction. Within this potential, a variational ansatz for the
wave function is proposed. This ansatz was later shown to be exact in the dense
limit of pairs. Note that the continuous crossover between the dilute and dense
regimes of attracting pairs of fermions is still an open problem, which now
attracts a lot of attention within the field of ultracold gases.
= Underlying evidence = The hyperphysics website pages at
Georgia State University summarize some key background to BCS theory as follows:
Evidence of a band gap at the Fermi level
the existence of a critical temperature and critical magnetic field implied a
band gap, and suggested a phase transition, but single electrons are
forbidden from condensing to the same energy level by the Pauli exclusion
principle. The site comments that "a drastic change in conductivity demanded
a drastic change in electron behavior". Conceivably, pairs of electrons might
perhaps act like bosons instead, which are bound by different condensate rules
and do not have the same limitation. Isotope effect on the critical
temperature, suggesting lattice interactions
The Debye frequency of phonons in a lattice is proportional to the inverse
of the square root of the mass of lattice ions. It was shown that the
superconducting transition temperature of mercury indeed showed the same
dependence, by substituting natural mercury 202Hg with a different isotope
198Hg. An exponential rise in heat capacity
near the critical temperature for some superconductors
An exponential increase in heat capacity near the critical temperature also
suggests an energy bandgap for the superconducting material. As
superconducting vanadium is warmed toward its critical temperature, its
heat capacity increases massively in a very few degrees; this suggests an
energy gap being bridged by thermal energy.
The lessening of the measured energy gap towards the critical temperature
this suggests a type of situation where some kind of binding energy exists but
it is gradually weakened as the critical temperature is approached. A binding
energy suggests two or more particles or other entities that are bound together
in the superconducting state. This helped to support the idea of bound
particles - specifically electron pairs - and together with the above helped to
paint a general picture of paired electrons and their lattice
interactions. Implications
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. These have been
confirmed in numerous experiments: The electrons are bound into Cooper
pairs, and these pairs are correlated due to the Pauli 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, unlike in the normal metal.
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 single particle density of
states at the Fermi level. 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 level. The energy gap is most
directly observed in tunneling experiments and in reflection of
microwaves from superconductors. BCS theory predicts the dependence of
the value of the energy gap Δ at temperature T on the critical
temperature Tc. The ratio between the value of the energy gap at zero
temperature and the value of the superconducting transition temperature
takes the universal value independent of material. Near the
critical temperature the relation asymptotes to
which is of the form suggested the previous year by M. J. Buckingham based
on the fact that the superconducting phase transition is second order, that
the superconducting phase has a mass gap and on Blevins, Gordy and Fairbank's
experimental results the previous year on the absorption of millimeter waves by
superconducting tin. Due to the energy gap, the specific heat
of the superconductor is suppressed strongly at low temperatures, there
being 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 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 with temperature. This had been
demonstrated experimentally by Walther Meissner and Robert Ochsenfeld in their
1933 article Ein neuer Effekt bei Eintritt der Supraleitfähigkeit.
It also describes the variation of the critical magnetic field 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 level. In its simplest form, BCS gives the
superconducting transition temperature Tcin terms of the electron-phonon
coupling potential V and the Debye cutoff energy ED:
where N(0) is the electronic density of states at the Fermi level. For more
details, see Cooper pairs. 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 24 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. 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
does affect the frequency of lattice vibrations. This effect suggests that
superconductivity is related to vibrations of the lattice. This is
incorporated into BCS theory, where lattice vibrations yield the binding
energy of electrons in a Cooper pair. Little-Parks experiment - One of the
ﬁrst indications to the importance of the Cooper-paring principle.
See also Magnesium diboride, considered a BCS
superconductor Quasiparticle
Little–Parks effect, one of the first indications of the importance of the
Cooper pairing principle. References
= Primary sources = L. N. Cooper, "Bound Electron Pairs in a
Degenerate Fermi Gas", Phys. Rev 104, 1189 - 1190.
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Microscopic Theory of
Superconductivity", Phys. Rev. 106, 162 - 164.
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of
Superconductivity", Phys. Rev. 108, 1175.
Further reading John Robert Schrieffer, Theory of
Superconductivity,, ISBN 0-7382-0120-0 Michael Tinkham, Introduction to
Superconductivity, ISBN 0-486-43503-2 Pierre-Gilles de Gennes,
Superconductivity of Metals and Alloys, ISBN 0-7382-0101-4.
Cooper, Leon N ; Feldman, Dmitri. BCS: 50 Years. World Scientific. ISBN
978-981-4304-64-1. External links
ScienceDaily: Physicist Discovers Exotic Superconductivity August 17, 2006
Hyperphysics page on BCS BCS History
Dance Analogy of BCS theory as explained by Bob Schrieffer
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