177:. 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
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781:, and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt. 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.
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161:. It is experimentally very well known that the transition temperature strongly depends on pressure. In general, 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.
133:. 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".
202:
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
611:) 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.
393:
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
263:
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
258:
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
201:
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
392:
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
373:
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 â specifically electron pairs â
595:
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
136:
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
206:
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
401:. 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.
383:
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
207:
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.
625:(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.
20:
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
328:
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 the most abundant natural
593:
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was discovered in La-Ba-Cu-O, at temperatures up to 30 K. Following experiments determined more materials with transition temperatures up to about 130 K, considerably above the previous limit of about
482:
192:
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
618:, 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.
487:
106:
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
415:. 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
255:
Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of high-temperature superconductivity.
765:, which is the experimental observation that for a given superconducting material, the critical temperature is inversely proportional to the square-root of the mass of the
2748:
259:
Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational
304:. The site comments that "a drastic change in conductivity demanded a drastic change in electron behavior". Conceivably, pairs of electrons might perhaps act like
653:
357:
is warmed toward its critical temperature, its heat capacity increases greatly in a very few degrees; this suggests an energy gap being bridged by thermal energy.
353:
An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting
1616:
Schmidt, Vadim Vasil'evich. The physics of superconductors: Introduction to fundamentals and applications. Springer
Science & Business Media, 2013.
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superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the
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used in the material. The isotope effect was reported by two groups on 24 March 1950, who discovered it independently working with different
220:. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the
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1994:
1953:
2346:
1634:
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Little, W. A.; Parks, R. D. (1962). "Observation of
Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder".
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129:, motivated by penetration experiments, proposed that this would modify the London equations via a new scale parameter called the
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in E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky: Quantum materials: Experiments and Theory, JĂźlich 2016,
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Reynolds, C. A.; Serin, B.; Wright, W. H.; Nesbitt, L. B. (1950-05-15). "Superconductivity of
Isotopes of Mercury".
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and together with the above helped to paint a general picture of paired electrons and their lattice interactions.
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1948:
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isotopes, although a few days before publication they learned of each other's results at the ONR conference in
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and penetration depths appeared in the
December 1957 article, "Theory of superconductivity". They received the
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BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the
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2010:
588:{\displaystyle \Delta (T\to T_{\rm {c}})\approx 3.06\,k_{\rm {B}}T_{\rm {c}}{\sqrt {1-(T/T_{\rm {c}})}}}
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the existence of a critical temperature and critical magnetic field implied a band gap, and suggested a
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1370:"Little-Parks Oscillations in a Single Ring in the vicinity of the Superconductor-Insulator Transition"
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superconductors - the so-called weak-coupling case. These have been confirmed in numerous experiments:
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1958:
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depend on the origin of the attractive interaction. For instance, Cooper pairs have been observed in
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London, F. (September 1948). "On the
Problem of the Molecular Theory of Superconductivity".
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1911 discovery. The theory describes superconductivity as a microscopic effect caused by a
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Buckingham, M. J. (February 1956). "Very High
Frequency Absorption in Superconductors".
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Maxwell, Emanuel (1950-05-15). "Isotope Effect in the Superconductivity of Mercury".
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Mann, A. (July 2011). "High Temperature Superconductivity at 25: Still In Suspense".
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The electrons are bound into Cooper pairs, and these pairs are correlated due to the
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BCS theory predicts the dependence of the value of the energy gap Î at temperature
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754:(0) is the electronic density of states at the Fermi level. For more details, see
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independent of material. Near the critical temperature the relation asymptotes to
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787:- One of the first indications to the importance of the Cooper-pairing principle.
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Gurovich, Doron; Tikhonov, Konstantin; Mahalu, Diana; Shahar, Dan (2014-11-20).
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Maxwell, Emanuel (1950). "Isotope Effect in the Superconductivity of Mercury".
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http://nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html
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Bardeen, J. (March 1955). "Theory of the Meissner Effect in Superconductors".
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It was proposed by Bardeen, Cooper, and Schrieffer in 1957; they received the
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743:{\displaystyle k_{\rm {B}}\,T_{\rm {c}}=1.134E_{\rm {D}}\,{e^{-1/N(0)\,V}},}
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In its simplest form, BCS gives the superconducting transition temperature
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Isotope effect on the critical temperature, suggesting lattice interactions
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The lessening of the measured energy gap towards the critical temperature
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Ivar Giaever - Nobel Lecture. Nobelprize.org. Retrieved 16 Dec 2010.
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Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (December 1957).
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of BCS theory as explained by Bob Schrieffer (audio recording)
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are forbidden from condensing to the same energy level by the
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2773:
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2651:
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305:
244:. The original results of BCS (discussed below) described an
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Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (April 1957).
224:(as explained above). However, the results of BCS theory do
2710:
1036:
Bednorz, J. G.; MĂźller, K. A. (June 1986). "Possible high T
477:{\displaystyle \Delta (T=0)=1.764\,k_{\rm {B}}T_{\rm {c}},}
1297:
2783:
597:
811:, one of the first indications of the importance of the
276:
summarize some key background to BCS theory as follows:
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1462:
1498:
Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (1957).
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Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (1957).
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near the critical temperature for some superconductors
656:
490:
421:
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This suggests a type of situation where some kind of
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At sufficiently low temperatures, electrons near the
1035:
181:. Superconductivity was simultaneously explained by
635:in terms of the electron-phonon coupling potential
1196:
1194:
742:
587:
476:
197:). Roughly speaking the picture is the following:
3316:
1430:"Bound Electron Pairs in a Degenerate Fermi Gas"
904:"Bound Electron Pairs in a Degenerate Fermi Gas"
1191:
607:of the superconductor is suppressed strongly (
2742:
2277:
1673:
1040:superconductivity in the BaâLaâCuâO system".
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990:
988:
83:to describe the pairing interaction between
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2270:
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1144:
1042:Zeitschrift fĂźr Physik B: Condensed Matter
287:(described as "a key piece in the puzzle")
1523:
1488:
1465:"Microscopic Theory of Superconductivity"
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1020:
985:
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953:"Microscopic Theory of Superconductivity"
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173:become unstable against the formation of
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1645:Mean-Field Theory: Hartree-Fock and BCS
1262:
1200:
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1150:
866:
621:It also describes the variation of the
25:Microscopic theory of superconductivity
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1758:Two-dimensional conformal field theory
1575:Superconductivity of Metals and Alloys
1427:
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901:
831:
267:
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312:and do not have the same limitation.
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333:, Hg, with a different isotope, Hg.
13:
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1205:. Dover Publications. p. 63.
694:
676:
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614:BCS theory correctly predicts the
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543:
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491:
465:
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272:The hyperphysics website pages at
252:high-temperature superconductors.
154:high-temperature superconductivity
14:
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2253:Template:Quantum mechanics topics
1620:
1558:Introduction to Superconductivity
1203:Introduction to Superconductivity
1127:"BCS Theory of Superconductivity"
800:, considered a BCS superconductor
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3284:
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2248:
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1592:; Feldman, Dmitri, eds. (2010).
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902:Cooper, Leon (November 1956).
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761:The BCS theory reproduces the
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1:
2859:Spontaneous symmetry breaking
1500:"Theory of Superconductivity"
1131:hyperphysics.phy-astr.gsu.edu
997:"Theory of Superconductivity"
819:
79:. The theory is also used in
408:on the critical temperature
308:instead, which are bound by
7:
2217:Quantum information science
1541:Theory of Superconductivity
791:
603:Due to the energy gap, the
164:
117:may be consequences of the
10:
3341:
3039:Spin gapless semiconductor
2948:Nearly free electron model
2614:Technological applications
1404:10.1103/PhysRevB.91.174505
310:different condensate rules
211:
187:Bogoliubov transformations
101:
98:for this theory in 1972.
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2003:
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1941:
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1695:
1627:Hyperphysics page on BCS
1201:Tinkham, Michael (1996).
596:waves by superconducting
390:Pauli exclusion principle
302:Pauli exclusion principle
149:in 1972 for this theory.
37:BardeenâCooperâSchrieffer
3158:Bogoliubov quasiparticle
2902:Quantum spin Hall effect
2794:BoseâEinstein condensate
2758:Condensed matter physics
2373:London penetration depth
1539:John Robert Schrieffer,
1525:10.1103/PhysRev.108.1175
1455:10.1103/PhysRev.104.1189
1428:Cooper, Leon N. (1956).
1250:10.1103/PhysRev.101.1431
1022:10.1103/PhysRev.108.1175
929:10.1103/PhysRev.104.1189
274:Georgia State University
240:has been tuned to their
179:BoseâEinstein condensate
141:and the calculations of
69:Heike Kamerlingh Onnes's
2666:List of superconductors
2544:By critical temperature
1913:2D free massless scalar
1806:Quantum electrodynamics
1733:QFT in curved spacetime
1571:Pierre-Gilles de Gennes
1490:10.1103/PhysRev.106.162
1355:10.1103/PhysRevLett.9.9
1335:Physical Review Letters
978:10.1103/PhysRev.106.162
889:10.1103/PhysRev.97.1724
785:LittleâParks experiment
623:critical magnetic field
2234:Quantum thermodynamics
2158:On shell and off shell
2153:Loop quantum cosmology
1995:N = 4 super YangâMills
1954:N = 1 super YangâMills
1821:Scalar electrodynamics
1811:Quantum chromodynamics
1713:Conformal field theory
1689:Quantum field theories
1320:10.1103/PhysRev.78.487
1285:10.1103/PhysRev.78.477
1173:10.1103/PhysRev.78.477
854:10.1103/PhysRev.74.562
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589:
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147:Nobel Prize in Physics
96:Nobel Prize in Physics
57:John Robert Schrieffer
22:
3034:Topological insulator
2968:Anderson localization
2312:Bean's critical state
2207:Quantum hydrodynamics
2202:Quantum hadrodynamics
1826:Scalar chromodynamics
777:. The two groups are
745:
590:
479:
199:
19:
2912:AharonovâBohm effect
2799:Fermionic condensate
2487:By magnetic response
2178:Quantum fluctuations
2148:Loop quantum gravity
1718:Lattice field theory
1595:BCS: 50 Years (book)
654:
488:
419:
236:where a homogeneous
3303:Physics WikiProject
2978:tight binding model
2958:Fermi liquid theory
2943:Free electron model
2892:Quantum Hall effect
2873:Electrons in solids
2439:persistent currents
2424:LittleâParks effect
2212:Quantum information
1816:Quartic interaction
1516:1957PhRv..108.1175B
1481:1957PhRv..106..162B
1446:1956PhRv..104.1189C
1396:2015PhRvB..91q4505G
1347:1962PhRvL...9....9L
1312:1950PhRv...78..487R
1277:1950PhRv...78..477M
1242:1956PhRv..101.1431B
1165:1950PhRv...78..477M
1097:2011Natur.475..280M
1054:1986ZPhyB..64..189B
1013:1957PhRv..108.1175B
969:1957PhRv..106..162B
920:1956PhRv..104.1189C
881:1955PhRv...97.1724B
846:1948PhRv...74..562L
809:LittleâParks effect
268:Underlying evidence
2864:Critical phenomena
2399:Andreev reflection
2394:Abrikosov vortices
2098:NambuâJona-Lasinio
2026:Higher dimensional
1933:WessâZuminoâWitten
1723:Noncommutative QFT
1637:2011-06-29 at the
1062:10.1007/BF01303701
798:Magnesium diboride
740:
585:
474:
285:at the Fermi level
242:Feshbach resonance
185:, by means of the
183:Nikolay Bogolyubov
110:proposed that the
61:microscopic theory
23:
3325:Superconductivity
3312:
3311:
3198:Exciton-polariton
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3055:Thermoelectricity
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2642:quantum computing
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2464:superdiamagnetism
2293:Superconductivity
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2258:
2121:
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1653:978-3-95806-159-0
1609:978-981-4304-64-1
1374:Physical Review B
1212:978-0-486-43503-9
583:
395:density of states
218:Coulomb repulsion
65:superconductivity
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3203:Phonon polariton
3095:Amorphous magnet
3075:Electrostriction
3070:Flexoelectricity
3065:Ferroelectricity
3060:Piezoelectricity
2917:Josephson effect
2897:Spin Hall effect
2877:
2876:
2854:Phase transition
2836:
2819:Luttinger liquid
2766:States of matter
2751:
2744:
2737:
2728:
2727:
2673:bilayer graphene
2647:Rutherford cable
2559:room temperature
2554:high temperature
2484:
2483:
2444:proximity effect
2419:Josephson effect
2363:coherence length
2286:
2279:
2272:
2263:
2262:
2251:
2250:
2168:Quantum dynamics
1841:YangâMillsâHiggs
1796:Non-linear sigma
1786:EulerâHeisenberg
1771:
1770:
1682:
1675:
1668:
1659:
1658:
1613:
1600:World Scientific
1529:
1527:
1510:(5): 1175â1204.
1494:
1492:
1459:
1457:
1440:(4): 1189â1190.
1416:
1415:
1389:
1365:
1359:
1358:
1330:
1324:
1323:
1295:
1289:
1288:
1260:
1254:
1253:
1236:(4): 1431â1432.
1223:
1217:
1216:
1198:
1189:
1183:
1177:
1176:
1148:
1142:
1141:
1139:
1137:
1123:
1117:
1116:
1080:
1074:
1073:
1033:
1027:
1026:
1024:
1007:(5): 1175â1204.
992:
983:
982:
980:
948:
942:
941:
931:
914:(4): 1189â1190.
899:
893:
892:
875:(6): 1724â1725.
864:
858:
857:
829:
749:
747:
746:
741:
736:
735:
734:
717:
699:
698:
697:
681:
680:
679:
668:
667:
666:
594:
592:
591:
586:
584:
579:
578:
577:
567:
550:
548:
547:
546:
536:
535:
534:
514:
513:
512:
483:
481:
480:
475:
470:
469:
468:
458:
457:
456:
341:exponential rise
294:phase transition
131:coherence length
115:London equations
112:phenomenological
3340:
3339:
3335:
3334:
3333:
3331:
3330:
3329:
3315:
3314:
3313:
3308:
3252:
3233:Granular matter
3228:Amorphous solid
3214:
3139:
3125:Antiferromagnet
3115:Superparamagnet
3088:Magnetic phases
3079:
3043:
2992:
2953:Bloch's theorem
2926:
2868:
2849:Order parameter
2842:Phase phenomena
2837:
2828:
2760:
2755:
2725:
2720:
2691:
2661:
2604:
2563:
2550:low temperature
2539:
2518:
2473:
2429:Meissner effect
2382:
2378:Silsbee current
2351:
2317:GinzburgâLandau
2295:
2290:
2260:
2255:
2238:
2190:Quantum gravity
2117:
2076:Particle theory
2071:
2050:
1999:
1973:
1937:
1901:
1855:Low dimensional
1850:
1791:GinzburgâLandau
1762:
1753:Topological QFT
1691:
1686:
1639:Wayback Machine
1623:
1610:
1554:Michael Tinkham
1536:
1534:Further reading
1504:Physical Review
1469:Physical Review
1434:Physical Review
1424:
1422:Primary sources
1419:
1366:
1362:
1331:
1327:
1300:Physical Review
1296:
1292:
1265:Physical Review
1261:
1257:
1229:Physical Review
1224:
1220:
1213:
1199:
1192:
1184:
1180:
1153:Physical Review
1149:
1145:
1135:
1133:
1125:
1124:
1120:
1105:10.1038/475280a
1091:(7356): 280â2.
1081:
1077:
1039:
1034:
1030:
1001:Physical Review
993:
986:
957:Physical Review
949:
945:
908:Physical Review
900:
896:
869:Physical Review
865:
861:
834:Physical Review
830:
826:
822:
794:
779:Emanuel Maxwell
713:
706:
702:
701:
693:
692:
688:
675:
674:
670:
662:
661:
657:
655:
652:
651:
649:
634:
616:Meissner effect
573:
572:
568:
563:
549:
542:
541:
537:
530:
529:
525:
508:
507:
503:
489:
486:
485:
464:
463:
459:
452:
451:
447:
420:
417:
416:
414:
381:
331:mercury isotope
326:Debye frequency
270:
230:ultracold gases
222:crystal lattice
214:
167:
139:Meissner effect
137:reproduces the
104:
81:nuclear physics
59:) is the first
26:
12:
11:
5:
3338:
3328:
3327:
3310:
3309:
3307:
3306:
3294:
3291:Physics Portal
3282:
3270:
3257:
3254:
3253:
3251:
3250:
3245:
3240:
3238:Liquid crystal
3235:
3230:
3224:
3222:
3216:
3215:
3213:
3212:
3207:
3206:
3205:
3200:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3149:
3147:
3145:Quasiparticles
3141:
3140:
3138:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3105:Superdiamagnet
3102:
3097:
3091:
3089:
3085:
3084:
3081:
3080:
3078:
3077:
3072:
3067:
3062:
3057:
3051:
3049:
3045:
3044:
3042:
3041:
3036:
3031:
3029:Superconductor
3026:
3021:
3016:
3011:
3009:Mott insulator
3006:
3000:
2998:
2994:
2993:
2991:
2990:
2985:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2940:
2934:
2932:
2928:
2927:
2925:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2883:
2881:
2874:
2870:
2869:
2867:
2866:
2861:
2856:
2851:
2845:
2843:
2839:
2838:
2831:
2829:
2827:
2826:
2821:
2816:
2811:
2806:
2801:
2796:
2791:
2786:
2781:
2776:
2770:
2768:
2762:
2761:
2754:
2753:
2746:
2739:
2731:
2722:
2721:
2719:
2718:
2713:
2708:
2703:
2698:
2693:
2689:
2685:
2680:
2675:
2669:
2667:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2639:
2634:
2629:
2627:electromagnets
2624:
2618:
2616:
2610:
2609:
2606:
2605:
2603:
2602:
2597:
2592:
2587:
2582:
2577:
2571:
2569:
2568:By composition
2565:
2564:
2562:
2561:
2556:
2551:
2547:
2545:
2541:
2540:
2538:
2537:
2535:unconventional
2532:
2526:
2524:
2523:By explanation
2520:
2519:
2517:
2516:
2511:
2510:
2509:
2504:
2499:
2490:
2488:
2481:
2479:Classification
2475:
2474:
2472:
2471:
2466:
2461:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2406:
2401:
2396:
2390:
2388:
2384:
2383:
2381:
2380:
2375:
2370:
2368:critical field
2365:
2359:
2357:
2353:
2352:
2350:
2349:
2344:
2339:
2337:MattisâBardeen
2334:
2329:
2324:
2322:KohnâLuttinger
2319:
2314:
2309:
2303:
2301:
2297:
2296:
2289:
2288:
2281:
2274:
2266:
2257:
2256:
2243:
2240:
2239:
2237:
2236:
2231:
2226:
2225:
2224:
2214:
2209:
2204:
2199:
2198:
2197:
2187:
2186:
2185:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2133:Casimir effect
2129:
2127:
2123:
2122:
2119:
2118:
2116:
2115:
2110:
2108:Standard Model
2105:
2100:
2095:
2090:
2085:
2079:
2077:
2073:
2072:
2070:
2069:
2064:
2058:
2056:
2052:
2051:
2049:
2048:
2043:
2038:
2033:
2028:
2023:
2018:
2013:
2007:
2005:
2001:
2000:
1998:
1997:
1992:
1987:
1981:
1979:
1978:Superconformal
1975:
1974:
1972:
1971:
1966:
1961:
1959:SeibergâWitten
1956:
1951:
1945:
1943:
1942:Supersymmetric
1939:
1938:
1936:
1935:
1930:
1925:
1920:
1915:
1909:
1907:
1903:
1902:
1900:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1858:
1856:
1852:
1851:
1849:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1813:
1808:
1803:
1798:
1793:
1788:
1783:
1777:
1775:
1768:
1764:
1763:
1761:
1760:
1755:
1750:
1745:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1705:
1699:
1697:
1693:
1692:
1685:
1684:
1677:
1670:
1662:
1656:
1655:
1642:
1629:
1622:
1621:External links
1619:
1618:
1617:
1614:
1608:
1590:Cooper, Leon N
1586:
1568:
1551:
1535:
1532:
1531:
1530:
1495:
1475:(1): 162â164.
1460:
1423:
1420:
1418:
1417:
1380:(17): 174505.
1360:
1325:
1290:
1255:
1218:
1211:
1190:
1178:
1143:
1118:
1075:
1048:(2): 189â193.
1037:
1028:
984:
963:(1): 162â164.
943:
894:
859:
840:(5): 562â573.
823:
821:
818:
817:
816:
813:Cooper pairing
806:
801:
793:
790:
789:
788:
782:
763:isotope effect
759:
739:
733:
729:
726:
723:
720:
716:
712:
709:
705:
696:
691:
687:
684:
678:
673:
665:
660:
647:
643:cutoff energy
632:
626:
619:
612:
601:
582:
576:
571:
566:
562:
559:
556:
553:
545:
540:
533:
528:
523:
520:
517:
511:
506:
502:
499:
496:
493:
473:
467:
462:
455:
450:
445:
442:
439:
436:
433:
430:
427:
424:
412:
402:
380:
377:
376:
375:
371:binding energy
366:
365:
359:
358:
350:
349:
335:
334:
321:
320:
314:
313:
289:
288:
281:Evidence of a
269:
266:
238:magnetic field
213:
210:
166:
163:
143:specific heats
103:
100:
89:atomic nucleus
24:
9:
6:
4:
3:
2:
3337:
3326:
3323:
3322:
3320:
3305:
3304:
3295:
3293:
3292:
3287:
3283:
3281:
3280:
3271:
3269:
3268:
3259:
3258:
3255:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3225:
3223:
3221:
3217:
3211:
3208:
3204:
3201:
3199:
3196:
3195:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3150:
3148:
3146:
3142:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3092:
3090:
3086:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3052:
3050:
3046:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3014:Semiconductor
3012:
3010:
3007:
3005:
3002:
3001:
2999:
2995:
2989:
2986:
2984:
2983:Hubbard model
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2935:
2933:
2929:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2884:
2882:
2878:
2875:
2871:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2846:
2844:
2840:
2835:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2771:
2769:
2767:
2763:
2759:
2752:
2747:
2745:
2740:
2738:
2733:
2732:
2729:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2668:
2664:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2619:
2617:
2615:
2611:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2585:heavy fermion
2583:
2581:
2578:
2576:
2573:
2572:
2570:
2566:
2560:
2557:
2555:
2552:
2549:
2548:
2546:
2542:
2536:
2533:
2531:
2528:
2527:
2525:
2521:
2515:
2514:ferromagnetic
2512:
2508:
2505:
2503:
2500:
2498:
2495:
2494:
2492:
2491:
2489:
2485:
2482:
2480:
2476:
2470:
2467:
2465:
2462:
2460:
2459:supercurrents
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2391:
2389:
2385:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2360:
2358:
2354:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2304:
2302:
2298:
2294:
2287:
2282:
2280:
2275:
2273:
2268:
2267:
2264:
2254:
2246:
2241:
2235:
2232:
2230:
2229:Quantum logic
2227:
2223:
2220:
2219:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2196:
2193:
2192:
2191:
2188:
2184:
2181:
2180:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2163:Quantum chaos
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2138:Cosmic string
2136:
2134:
2131:
2130:
2128:
2124:
2114:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2080:
2078:
2074:
2068:
2065:
2063:
2060:
2059:
2057:
2053:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2011:Pure 4D N = 1
2009:
2008:
2006:
2002:
1996:
1993:
1991:
1988:
1986:
1983:
1982:
1980:
1976:
1970:
1967:
1965:
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1946:
1944:
1940:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1908:
1904:
1898:
1895:
1893:
1892:ThirringâWess
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1867:BulloughâDodd
1865:
1863:
1862:2D YangâMills
1860:
1859:
1857:
1853:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1778:
1776:
1772:
1769:
1765:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1738:String theory
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1708:Axiomatic QFT
1706:
1704:
1703:Algebraic QFT
1701:
1700:
1698:
1694:
1690:
1683:
1678:
1676:
1671:
1669:
1664:
1663:
1660:
1654:
1650:
1646:
1643:
1640:
1636:
1633:
1632:Dance analogy
1630:
1628:
1625:
1624:
1615:
1611:
1605:
1601:
1597:
1596:
1591:
1587:
1584:
1583:0-7382-0101-4
1580:
1576:
1572:
1569:
1567:
1566:0-486-43503-2
1563:
1559:
1555:
1552:
1550:
1549:0-7382-0120-0
1546:
1542:
1538:
1537:
1526:
1521:
1517:
1513:
1509:
1505:
1501:
1496:
1491:
1486:
1482:
1478:
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171:Fermi surface
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127:Brian Pippard
124:
123:quantum state
120:
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2469:superstripes
2414:flux pumping
2409:flux pinning
2404:Cooper pairs
2306:
2244:
2173:Quantum foam
2113:Stueckelberg
2067:ChernâSimons
2004:Supergravity
1743:Supergravity
1728:Gauge theory
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3120:Ferromagnet
2938:Drude model
2907:Berry phase
2887:Hall effect
2454:SU(2) color
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2055:Topological
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1882:Sine-Gordon
1872:GrossâNeveu
1781:BornâInfeld
1748:Thermal QFT
1341:(1): 9â12.
399:Fermi level
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3135:Spin glass
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2997:Conduction
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2449:reentrance
1836:YangâMills
1543:, (1964),
1306:(4): 487.
1271:(4): 477.
1159:(4): 477.
820:References
815:principle.
3193:Polariton
3100:Diamagnet
3048:Couplings
3024:Conductor
3019:Semimetal
3004:Insulator
2880:Phenomena
2804:Fermi gas
2387:Phenomena
2245:See also:
1964:Super QCD
1918:Liouville
1906:Conformal
1877:Schwinger
1412:119268649
1387:1411.5640
1070:118314311
938:0031-899X
708:−
555:−
519:≈
501:→
492:Δ
423:Δ
298:electrons
152:In 1986,
119:coherence
3319:Category
3267:Category
3248:Colloids
2622:cryotron
2580:cuprates
2575:covalent
2332:Matthias
2300:Theories
2041:Type IIB
2036:Type IIA
2021:4D N = 8
2016:4D N = 1
1985:6D (2,0)
1949:4D N = 1
1928:Polyakov
1887:Thirring
1696:Theories
1635:Archived
1136:16 April
1113:21776057
792:See also
639:and the
355:vanadium
283:band gap
234:fermions
165:Overview
157:30
85:nucleons
3279:Commons
3243:Polymer
3210:Polaron
3188:Plasmon
3168:Exciton
2716:more...
2600:organic
2143:History
2126:Related
1923:Minimal
1774:Regular
1512:Bibcode
1477:Bibcode
1442:Bibcode
1392:Bibcode
1343:Bibcode
1308:Bibcode
1273:Bibcode
1238:Bibcode
1161:Bibcode
1093:Bibcode
1050:Bibcode
1009:Bibcode
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877:Bibcode
842:Bibcode
775:Atlanta
771:mercury
767:isotope
397:at the
264:gases.
212:Details
195:phonons
102:History
30:physics
3178:Phonon
3173:Magnon
2931:Theory
2789:Plasma
2779:Liquid
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2327:London
2083:Chiral
2031:Type I
1846:Yukawa
1767:Models
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306:bosons
261:ansatz
250:d-wave
246:s-wave
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67:since
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2657:wires
2652:SQUID
2222:links
2195:links
2183:links
2103:NMSSM
2088:Fermi
1831:Soler
1801:Proca
1408:S2CID
1382:arXiv
1066:S2CID
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641:Debye
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1897:Toda
1649:ISBN
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1579:ISBN
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1545:ISBN
1207:ISBN
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1109:PMID
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2637:NMR
2632:MRI
2507:1.5
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2342:RVB
2307:BCS
2046:11D
1520:doi
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