BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper,
and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh
Onnes's 1911 discovery. 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 Bardeen, Cooper, and Schrieffer
in 1957; they received the Nobel Prize in Physics for this theory in 1972.
== History == Rapid progress in the understanding of 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 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 (necessary for pairing)
is brought about indirectly by the interaction between the electrons and the vibrating crystal
lattice (the phonons). 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 (or more than just two electrons). Because the pairing
increases this energy barrier, kicks from oscillating atoms in the conductor (which
are small at sufficiently low temperatures) 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 (the current through the superconductor)
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 (in low temperature superconductors), this attraction is brought about indirectly
by the coupling of electrons to the crystal lattice (as explained above). 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 (discussed
below) 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 (attractively interacting) 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 (described as "a key piece in the puzzle")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 interactionsThe
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 superconductorsAn 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 temperaturethis 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 (where the state
of an electron can be changed by adding an arbitrarily small amount of energy). 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 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 (expressed in energy units) takes the universal value
Δ (
T =
0 )
= 1.764
k
B
T
c
,
{\displaystyle \Delta (T=0)=1.764\,k_{B}T_{c},}
independent of material. Near the critical temperature the relation asymptotes to
Δ (
T →
T
c
) ≈
3.06
k
B
T
c
1 −
( T
/
T
c
)
{\displaystyle \Delta (T\to T_{c})\approx 3.06\,k_{B}T_{c}{\sqrt {1-(T/T_{c})}}}
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 (exponentially) 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 (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 Supraleitfähigkeit.
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 level.
In its simplest form, BCS gives the superconducting transition temperature Tc in terms of the
electron-phonon coupling potential V and the Debye cutoff energy ED:
k
B
T
c
= 1.13
E
D
e
− 1
/
N (
0 )
V
,
{\displaystyle k_{B}\,T_{c}=1.13E_{D}\,{e^{-1/N(0)\,V}},}
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-pairing 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