[ [ File : Bardeen plaque uiuc.jpg|thumb|A commemorative plaque placed in the Bardeen Engineering Quad at the University of Illinois at Urbana-Champaign . It commemorates the Theory of Superconductivity developed here by John Bardeen and his students , for which they won a Nobel Prize for Physics in 1972 . ] ]

{ { short description|Microscopic theory of superconductivity } }

' '' 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| Heike Kamerlingh Onnes 's ] ] 1911 discovery . The theory describes superconductivity as a microscopic effect caused by a [ [ Bose–Einstein condensate|condensation ] ] of [ [ Cooper pair ] ] s. The theory is also used in [ [ nuclear physics ] ] to describe the pairing interaction between [ [ nucleon ] ] s 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 '' , < ref > { { cite journal|last=London|first=F.|title=On the Problem of the Molecular Theory of Superconductivity|journal=Physical Review|date=September 1948|volume=74|issue=5|pages=562–573|doi=10.1103/PhysRev.74.562|bibcode = 1948PhRv ... 74..562L } } < /ref > where [ [ Fritz London ] ] proposed that the [ [ Phenomenology ( particle physics ) |phenomenological ] ] [ [ London equations ] ] may be consequences of the [ [ quantum coherence|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 [ [ Superconducting coherence length|coherence length ] ] . John Bardeen then argued in the 1955 paper , `` Theory of the Meissner Effect in Superconductors '' , < ref > { { cite journal|last=Bardeen|first=J.|title=Theory of the Meissner Effect in Superconductors|journal=Physical Review|date=March 1955|volume=97|issue=6|pages=1724–1725|doi=10.1103/PhysRev.97.1724|bibcode = 1955PhRv ... 97.1724B } } < ! -- |access-date=May 3 , 2012 -- > < /ref > 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 '' . < ref > { { cite journal|last=Cooper|first=Leon|title=Bound Electron Pairs in a Degenerate Fermi Gas|journal=Physical Review|date=November 1956|volume=104|issue=4|pages=1189–1190|doi=10.1103/PhysRev.104.1189|issn=0031-899X|bibcode = 1956PhRv..104.1189C |doi-access=free } } < ! -- |access-date=May 3 , 2012 -- > < /ref >

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 '' . < ref > { { cite journal|last=Bardeen|first=J.|author2=Cooper , L. N.|author3=Schrieffer , J. R.|title=Microscopic Theory of Superconductivity|journal=Physical Review|date=April 1957|volume=106|issue=1|pages=162–164|doi=10.1103/PhysRev.106.162|bibcode = 1957PhRv..106..162B |doi-access=free } } < /ref > The demonstration that the phase transition is second order , that it reproduces the [ [ Meissner effect ] ] and the calculations of [ [ specific heat ] ] s and penetration depths appeared in the December 1957 article , `` Theory of superconductivity '' . < ref name=BCS_theory > { { cite journal|last=Bardeen|first=J.|author2=Cooper , L. N. |author3=Schrieffer , J. R. |title=Theory of Superconductivity|journal=Physical Review|date=December 1957|volume=108|issue=5|pages=1175–1204|doi=10.1103/PhysRev.108.1175|bibcode = 1957PhRv..108.1175B |doi-access=free } } < /ref > They received the [ [ Nobel Prize in Physics ] ] in 1972 for this theory .

In 1986 , [ [ high-temperature superconductivity ] ] was discovered in La-Ba-Cu-O , at temperatures up to 30 & nbsp ; K. < ref > { { cite journal|last=Bednorz|first=J . G.|author2=Müller , K. A.|s2cid=118314311|title=Possible highT c superconductivity in the Ba−La−Cu−O system|journal=Zeitschrift für Physik B : Condensed Matter|date=June 1986|volume=64|issue=2 |pages=189–193 |doi=10.1007/BF01303701|bibcode=1986ZPhyB..64..189B } } < /ref > Following experiments determined more materials with transition temperatures up to about 130 & nbsp ; K , considerably above the previous limit of about 30 & nbsp ; [ [ Kelvin|K ] ] . It is believed that BCS theory alone can not explain this phenomenon and that other effects are in play. < ref > { { cite journal|last=Mann|first=A.|title=High Temperature Superconductivity at 25 : Still In Suspense|journal=Nature|date=July 2011|volume=475|doi=10.1038/475280a|pmid=21776057|bibcode = 2011Natur.475..280M|issue=7356|pages=280–2|doi-access=free } } < /ref > 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 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 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 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 :

< blockquote > 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. < /blockquote >

===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 atom|ultracold gases ] ] of [ [ fermion ] ] s where a homogeneous [ [ magnetic field ] ] has been tuned to their [ [ Feshbach resonance ] ] . The original results of BCS ( discussed below ) described an [ [ Atomic orbital|s-wave ] ] superconducting state , which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the [ [ Atomic orbital|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 : < ref > { { cite web|url=http : //hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html|title=BCS Theory of Superconductivity|website=hyperphysics.phy-astr.gsu.edu|access-date=16 April 2018 } } < /ref >

* `` '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 [ [ electron ] ] s 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 [ [ boson ] ] s instead , which are bound by [ [ Bose–Einstein statistics|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 < sup > 202 < /sup > Hg with a different isotope < sup > 198 < /sup > Hg. < ref name=maxwell1950 > { { cite journal|last1=Maxwell|first1=Emanuel|title=Isotope Effect in the Superconductivity of Mercury|journal=Physical Review|volume=78|issue=4|pages=477|doi=10.1103/PhysRev.78.477|bibcode = 1950PhRv ... 78..477M |year=1950 } } < /ref >

* `` 'An [ [ Exponential growth|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 temperature increases toward the critical temperature . 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 < ref name= '' Giaever '' > Ivar Giaever - Nobel Lecture . Nobelprize.org . Retrieved 16 Dec 2010. http : //nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html < /ref > 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 `` T '' < sub > c < /sub > . 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 < ref name= '' Tinkham 1996 63 '' > { { Cite book

| first=Michael| last=Tinkham| year=1996

| title=Introduction to Superconductivity | pages=63

| publisher=Dover Publications

| isbn=978-0-486-43503-9 } } < /ref > $ \ Delta ( T =0) = 1.764 \, k _{\ r m B } T _{\ r m c }, $ independent of material . Near the critical temperature the relation asymptotes to < ref name= '' Tinkham 1996 63 '' / > $ \ Delta ( T \ t o T _{\ r m c })\approx 3.06 \, k _{\ r m B } T _{\ r m c }\sqrt{1-( T / T _{\ r m c })} $ which is of the form suggested the previous year by M. J. Buckingham < ref >

{ { cite journal

| last=Buckingham | first=M . J .

| title=Very High Frequency Absorption in Superconductors

|date=February 1956

| journal= [ [ Physical Review ] ]

| volume=101 | issue=4

| pages=1431–1432

| doi = 10.1103/PhysRev.101.1431

| bibcode = 1956PhRv..101.1431B } } < /ref > 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 ( [ [ exponential decay|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 .

* It also describes the variation of the [ [ upper critical field|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 `` T '' < sub > c < /sub > in terms of the electron-phonon coupling potential `` V '' and the [ [ Debye frequency|Debye ] ] cutoff energy `` E '' < sub > D < /sub > : < ref name=BCS_theory/ > $ k _{\ r m B }\, T _{\ r m c } = 1.134 E _{\ r m 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 ( element ) |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 ] ] , < ref > { { Cite journal|last=Maxwell|first=Emanuel|date=1950-05-15| title=Isotope Effect in the Superconductivity of Mercury|journal=Physical Review|volume=78|issue=4|pages=477| doi=10.1103/PhysRev.78.477|bibcode=1950PhRv ... 78..477M } } < /ref > and C. A. Reynolds , B. Serin , W. H. Wright , and L. B. Nesbitt. < ref > { { Cite journal|last1=Reynolds|first1=C . A.|last2=Serin|first2=B.|last3=Wright|first3=W . H.|last4=Nesbitt|first4=L . B.| date=1950-05-15|title=Superconductivity of Isotopes of Mercury|journal=Physical Review|volume=78|issue=4|pages=487|doi=10.1103/PhysRev.78.487|bibcode=1950PhRv ... 78..487R } } < /ref > 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 effect|Little–Parks experiment ] ] < ref name=Little > { { Cite journal | doi=10.1103/PhysRevLett.9.9| title=Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder| year=1962| last1=Little| first1=W . A.| last2=Parks| first2=R . D.| journal=Physical Review Letters| volume=9| issue=1| pages=9–12| bibcode=1962PhRvL ... 9 ... .9L } } < /ref > - One of the first { { citation needed|date=January 2018 } } 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 < ref > { { Cite journal|last1=Gurovich|first1=Doron|last2=Tikhonov|first2=Konstantin|last3=Mahalu|first3=Diana|last4=Shahar|first4=Dan|s2cid=119268649|date=2014-11-20|title=Little-Parks Oscillations in a Single Ring in the vicinity of the Superconductor-Insulator Transition|url=https : //www.researchgate.net/publication/268524767|journal=Physical Review B|volume=91|issue=17|pages=174505|doi=10.1103/PhysRevB.91.174505|arxiv=1411.5640|bibcode=2015PhRvB..91q4505G } } < /ref > indications of the importance of the [ [ Cooper pair ] ] ing principle .

==References==

{ { Reflist|30em } }

===Primary sources===

* { { cite journal |doi-access=free |doi=10.1103/PhysRev.104.1189 |title=Bound Electron Pairs in a Degenerate Fermi Gas |year=1956 |last1=Cooper |first1=Leon N. |journal=Physical Review |volume=104 |issue=4 |pages=1189–1190 |bibcode=1956PhRv..104.1189C } }

* { { cite journal |doi-access=free |doi=10.1103/PhysRev.106.162 |title=Microscopic Theory of Superconductivity |year=1957 |last1=Bardeen |first1=J . |last2=Cooper |first2=L . N. |last3=Schrieffer |first3=J . R. |journal=Physical Review |volume=106 |issue=1 |pages=162–164 |bibcode=1957PhRv..106..162B } }

* { { cite journal |doi-access=free |doi=10.1103/PhysRev.108.1175 |title=Theory of Superconductivity |year=1957 |last1=Bardeen |first1=J . |last2=Cooper |first2=L . N. |last3=Schrieffer |first3=J . R. |journal=Physical Review |volume=108 |issue=5 |pages=1175–1204 |bibcode=1957PhRv..108.1175B } }

==Further reading==

* John Robert Schrieffer , `` Theory of Superconductivity '' , ( 1964 ) , { { 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 } } .

* { { Cite book |editor=Cooper , Leon N |editor-link=Leon Cooper |editor2= Feldman , Dmitri |title=BCS : 50 Years ( book ) |publisher= [ [ World Scientific ] ] |year=2010 |isbn=978-981-4304-64-1 |title-link=BCS : 50 Years ( book ) } }

* Schmidt , Vadim Vasil'evich . The physics of superconductors : Introduction to fundamentals and applications . Springer Science & Business Media , 2013 .

==External links==

* ScienceDaily : [ https : //www.sciencedaily.com/releases/2006/08/060817101658.htm Physicist Discovers Exotic Superconductivity ] ( [ [ University of Arizona ] ] ) August 17 , 2006

* [ http : //hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html Hyperphysics page on BCS ]

* [ http : //ffden-2.phys.uaf.edu/212_fall2003.web.dir/T.J_Barry/bcstheory.html BCS History ]

* [ http : //www.aip.org/history/mod/superconductivity/03.html Dance analogy ] of BCS theory as explained by Bob Schrieffer ( audio recording )

* [ http : //www.cond-mat.de/events/correl16/manuscripts/koch.pdf Mean-Field Theory : Hartree-Fock and BCS ] in E. Pavarini , E. Koch , J. van den Brink , and G. Sawatzky : Quantum materials : Experiments and Theory , Jülich 2016 , { { ISBN|978-3-95806-159-0 } }

{ { four-fermion interactions } }

[ [ Category : Superconductivity ] ]