BCS theory
Encyclopedia
BCS theory — proposed by Bardeen
, Cooper, and Schrieffer
(BCS) in 1957 — 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 pairs of electrons
into a boson
-like state. The theory is also used in Nuclear Physics
to describe the pairing interaction between nucleons
in an atomic nucleus.
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 announced in February 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 July 1957 article, Theory of superconductivity. They received the Nobel Prize in Physics
in 1972 for this theory. The 1950 Landau-Ginzburg theory of superconductivity is not cited in either of the BCS papers.
In 1986, "high-temperature superconductivity
" was discovered (i.e. superconductivity at temperatures considerably above the previous limit of about 30 K
; up to about 130 K). It is believed that BCS theory alone cannot explain this phenomenon and that other effects are at play. These effects are still not yet fully understood; it is even possible that they also control superconductivity even at low temperatures for some materials.
become unstable against the formation of Cooper pair
s. 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 "condensation" of Cooper pairs. These have some bosonic properties, while bosons, at sufficiently low temperature, can form a large Bose-Einstein condensate. Superconductivity was simultaneously explained by Nikolay Bogoliubov, by means of the so-called Bogoliubov transformation
s.
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 phonon
s). 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. There are a lot of such electron pairs in a superconductor, so that they overlap very strongly, forming a highly collective "condensate". Breaking of one pair results in changing of energies of remained macroscopic number of pairs. If the required energy is higher than the energy provided by kicks from oscillating atoms in the conductor (which is true at low temperatures), then the electrons will stay paired and resist all kicks, thus not experiencing resistance. Thus, the collective behaviour of "condensate" is a crucial ingredient of superconductivity.
" 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.
John Bardeen
John Bardeen was an American physicist and electrical engineer, the only person to have won the Nobel Prize in Physics twice: first in 1956 with William Shockley and Walter Brattain for the invention of the transistor; and again in 1972 with Leon Neil Cooper and John Robert Schrieffer for a...
, Cooper, and Schrieffer
John Robert Schrieffer
John Robert Schrieffer is an American physicist and, with John Bardeen and Leon N Cooper, recipient of the 1972 Nobel Prize for Physics for developing the BCS theory, the first successful microscopic theory of superconductivity.-Biography:...
(BCS) in 1957 — is the first microscopic theory
Microscopic theory
A microscopic theory is one that contains an explanation at the atomic or subatomic level in contrast to a higher level or classical macroscopic or phenomenological theory. eg in superconductivity BCS theory is a microscopic theory....
of superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
since its discovery in 1911. The theory describes superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
as a microscopic effect caused by a "condensation" of pairs of electrons
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...
into a boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
-like state. The theory is also used in Nuclear Physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
to describe the pairing interaction between nucleons
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
in an atomic nucleus.
History
The mid-1950s saw rapid progress in the understanding of superconductivity. It began in the 1948 paper, On the Problem of the Molecular Theory of Superconductivity where Fritz LondonFritz London
Fritz Wolfgang London was a German theoretical physicist. His fundamental contributions to the theories of chemical bonding and of intermolecular forces are today considered classic and are discussed in standard textbooks of physical chemistry.With his brother Heinz, he made a significant...
proposed that the phenomenological
Phenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
London equations
London equations
The London equations, developed by brothers Fritz and Heinz London in 1935,relate current to electromagnetic fields in and around a superconductor...
may be consequences of the coherence of a quantum state. In 1953, Brian Pippard
Brian Pippard
Sir Alfred Brian Pippard, ScD, FRS , was a British physicist. He was Cavendish Professor of Physics from 1971 until 1984 and an Honorary Fellow of Clare Hall, University of Cambridge, of which he was the first President...
, motivated by penetration experiments, proposed that this would modify the London equations via a new scale parameter called the coherence length
Coherence length
In physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. The significance is that interference will be strong within a coherence length of the source, but not beyond it...
. John Bardeen
John Bardeen
John Bardeen was an American physicist and electrical engineer, the only person to have won the Nobel Prize in Physics twice: first in 1956 with William Shockley and Walter Brattain for the invention of the transistor; and again in 1972 with Leon Neil Cooper and John Robert Schrieffer for a...
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
John Robert Schrieffer
John Robert Schrieffer is an American physicist and, with John Bardeen and Leon N Cooper, recipient of the 1972 Nobel Prize for Physics for developing the BCS theory, the first successful microscopic theory of superconductivity.-Biography:...
. The theory was first announced in February 1957 in the letter, Microscopic theory of superconductivity. The demonstration that the phase transition is second order, that it reproduces the Meissner effect
Meissner effect
The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...
and the calculations of specific heats and penetration depths appeared in the July 1957 article, Theory of superconductivity. They received the Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...
in 1972 for this theory. The 1950 Landau-Ginzburg theory of superconductivity is not cited in either of the BCS papers.
In 1986, "high-temperature superconductivity
High-temperature superconductivity
High-temperature superconductors are materials that have a superconducting transition temperature above . From 1960 to 1980, 30 K was thought to be the highest theoretically possible Tc...
" was discovered (i.e. superconductivity at temperatures considerably above the previous limit of about 30 K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
; up to about 130 K). It is believed that BCS theory alone cannot explain this phenomenon and that other effects are at play. These effects are still not yet fully understood; it is even possible that they also control superconductivity even at low temperatures for some materials.
Overview
At sufficiently low temperatures, electrons near the Fermi surfaceFermi surface
In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline...
become unstable against the formation of Cooper pair
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...
s. 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 "condensation" of Cooper pairs. These have some bosonic properties, while bosons, at sufficiently low temperature, can form a large Bose-Einstein condensate. Superconductivity was simultaneously explained by Nikolay Bogoliubov, by means of the so-called Bogoliubov transformation
Bogoliubov transformation
In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an...
s.
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 phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
s). 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. There are a lot of such electron pairs in a superconductor, so that they overlap very strongly, forming a highly collective "condensate". Breaking of one pair results in changing of energies of remained macroscopic number of pairs. If the required energy is higher than the energy provided by kicks from oscillating atoms in the conductor (which is true at low temperatures), then the electrons will stay paired and resist all kicks, thus not experiencing resistance. Thus, the collective behaviour of "condensate" is a crucial ingredient of superconductivity.
More 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. The original results of BCS (discussed below) described an "s-waveS-wave
A type of seismic wave, the S-wave, secondary wave, or shear wave is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves....
" superconducting state, which is the rule among low-temperature superconductors but is not realized in many "unconventional superconductors", such as the "d-wave
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
" 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
High-temperature superconductivity
High-temperature superconductors are materials that have a superconducting transition temperature above . From 1960 to 1980, 30 K was thought to be the highest theoretically possible Tc...
.
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
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
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.
Successes of the BCS theory
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 principlePauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
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 statesDensity of statesIn solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
at the Fermi energyFermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....
. 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 experiments and in reflection of microwaves from superconductors.
- BCS theory predicts the dependence of the value of the energy gap E 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 of 3.5, independent of material. Near the critical temperature the relation asymptotes to
-
- which is of the form suggested the previous year by M. J. Buckingham in Very High Frequency Absorption in Superconductors 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 tinTinTin is a chemical element with the symbol Sn and atomic number 50. It is a main group metal in group 14 of the periodic table. Tin shows chemical similarity to both neighboring group 14 elements, germanium and lead and has two possible oxidation states, +2 and the slightly more stable +4...
.
- 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 effectMeissner effectThe Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...
, i.e. the expulsion of a magnetic fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
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 OchsenfeldRobert OchsenfeldRobert Ochsenfeld was a German physicist born on May 18, 1901 in Hilchenbach . In 1933 he discovered with Walter Meissner the Meißner-Ochsenfeld effect.He died on December 5, 1993 in Hilchenbach....
in their 1933 article Ein neuer Effekt bei Eintritt der Supraleitfähigkeit.
- It also describes the variation of the critical magnetic fieldUpper critical field-Upper critical field:The upper critical field is the magnetic field which completely suppresses superconductivity in a Type II superconductor at 0K ....
(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 DebyeDebye frequencyThe Debye frequency of a crystal is a theoretical maximum frequency of vibration for the atoms that make up the crystal . It was proposed by Peter Debye as part of the Debye model. It is generally not the same as the actual maximum oscillation frequency for the crystal, due to approximations in the...
cutoff energy:
- Here N(0) is the electronic density of statesDensity of statesIn solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
at the Fermi energyFermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....
. 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 isotopeIsotopeIsotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
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 mercuryMercury (element)Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...
isotopes, although a few days before publication they learned of each other's results at the ONR conference in Atlanta, GeorgiaAtlanta, GeorgiaAtlanta is the capital and most populous city in the U.S. state of Georgia. According to the 2010 census, Atlanta's population is 420,003. Atlanta is the cultural and economic center of the Atlanta metropolitan area, which is home to 5,268,860 people and is the ninth largest metropolitan area in...
. 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 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.
Further reading
- John Robert SchriefferJohn Robert SchriefferJohn Robert Schrieffer is an American physicist and, with John Bardeen and Leon N Cooper, recipient of the 1972 Nobel Prize for Physics for developing the BCS theory, the first successful microscopic theory of superconductivity.-Biography:...
, Theory of Superconductivity, (1964), ISBN 0-7382-0120-0 - Michael TinkhamMichael TinkhamMichael Tinkham was an American physicist.-Professional life:Michael Tinkham was Rumford Research Professor of Physics and Gordon McKay Research Professor of Applied Physics at Harvard University...
, Introduction to Superconductivity, ISBN 0-4864-3503-2 - Pierre-Gilles de GennesPierre-Gilles de GennesPierre-Gilles de Gennes was a French physicist and the Nobel Prize laureate in physics in 1991.-Biography:...
, Superconductivity of Metals and Alloys, ISBN 0-7382-0101-4.
External links
- ScienceDaily: Physicist Discovers Exotic Superconductivity (University of ArizonaUniversity of ArizonaThe University of Arizona is a land-grant and space-grant public institution of higher education and research located in Tucson, Arizona, United States. The University of Arizona was the first university in the state of Arizona, founded in 1885...
) August 17, 2006 - Hyperphysics page on BCS
- BCS History
- Dance Analogy of BCS theory as explained by Bob Schrieffer (audio recording)