Editing High-temperature superconductivity (section)

== Theoretical models ==
Multiple hypotheses attempt to account for HTS. 

[[Resonating valence bond theory|Resonating-valence-bond theory]] 

Spin fluctuation hypothesis<ref>{{cite journal |author=Mann |first=Adam |date=July 20, 2011 |title=High-temperature superconductivity at 25: Still in suspense |journal=Nature |volume=475 |issue=7356 |pages=280–2 |bibcode=2011Natur.475..280M |doi=10.1038/475280a |pmid=21776057 |s2cid=205066154 |doi-access=}}</ref> proposed that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as [[Paramagnon|paramagnons]].<ref>{{citation |last=Pines |first=D. |title=The Gap Symmetry and Fluctuations in High-Tc Superconductors |date=2002 |volume=371 |pages=111–142 |series=NATO Science Series: B |contribution=The Spin Fluctuation Model for High Temperature Superconductivity: Progress and Prospects |place=New York |publisher=Kluwer Academic |doi=10.1007/0-306-47081-0_7 |isbn=978-0-306-45934-4}}</ref><ref>{{cite journal |author=Monthoux |first1=P. |last2=Balatsky |first2=A. V. |last3=Pines |first3=D. |name-list-style=amp |date=1991 |title=Toward a theory of high-temperature superconductivity in the antiferromagnetically correlated cuprate oxides |journal=Physical Review Letters |volume=67 |issue=24 |pages=3448–3451 |bibcode=1991PhRvL..67.3448M |doi=10.1103/PhysRevLett.67.3448 |pmid=10044736 |doi-access=free}}</ref>{{dubious|date=August 2016}} 

Gubser, Hartnoll, Herzog, and Horowitz proposed holographic superconductivity, which uses holographic duality or [[AdS/CFT correspondence]] theory as a possible explanation of high-temperature superconductivity in certain materials.<ref>Jan Zaanen, Yan Liu, Ya Sun K.Schalm (2015). ''Holographic Duality in Condensed Matter Physics''. Cambridge University Press, Cambridge.</ref> 

Weak coupling theory suggests superconductivity emerges from antiferromagnetic spin fluctuations in a doped system.<ref name="Mont1992">{{cite journal |last1=Monthoux |first1=P. |last2=Balatsky |first2=A. |last3=Pines |first3=D. |year=1992 |title=Weak-coupling theory of high-temperature superconductivity in the antiferromagnetically correlated copper oxides |journal=[[Physical Review B]] |volume=46  |issue=22  |pages=14803–14817 |doi=10.1103/PhysRevB.46.14803 |bibcode=1992PhRvB..4614803M |pmid=10003579}}</ref> It predicts that the pairing wave function of cuprate HTS should have a ''d''<sub>x</sub><sup>2</sup>-y<sup>2</sup> symmetry. Thus, determining whether the pairing wave function has ''d''-wave symmetry is essential to test the spin fluctuation mechanism. That is, if the HTS [[order parameter]] (a pairing wave function as in [[Ginzburg–Landau theory]]) does not have ''d''-wave symmetry, then a pairing mechanism related to spin fluctuations can be ruled out. (Similar arguments can be made for iron-based superconductors but the different material properties allow a different pairing symmetry.)

Interlayer coupling theory proposes that a layered structure consisting of BCS-type (''s''-wave symmetry) superconductors can explain superconductivity by itself.<ref name="Chak1993">
{{cite journal
 |last1=Chakravarty |first1=S.
 |last2=Sudbø |first2=A.
 |last3=Anderson |first3=P. W.
 |last4=Strong |first4=S.
 |year=1993
 |title=Interlayer Tunneling and Gap Anisotropy in High-Temperature Superconductors
 |journal=[[Science (journal)|Science]]
 |volume=261  |issue=5119  |pages=337–340
 |doi=10.1126/science.261.5119.337 |pmid=17836845
 |bibcode=1993Sci...261..337C |s2cid=41404478
}}
</ref> By introducing an additional tunnelling interaction between layers, this model explained the anisotropic symmetry of the order parameter as well as the emergence of HTS.  

In order to resolve this question, experiments such as [[photoemission spectroscopy]], [[Nuclear magnetic resonance|NMR]], [[specific heat capacity|specific heat]] measurements, were conducted. The results remain ambiguous, with some reports supporting ''d'' symmetry, with others supporting ''s'' symmetry. 

Such explanations assume that superconductive properties can be treated by [[mean-field theory]]. It also does not consider that in addition to the superconductive gap, the [[pseudogap]] must be explained. The cuprate layers are insulating, and the superconductors are doped with interlayer impurities to make them metallic. 

The transition temperature can be maximized by varying the [[dopant]] concentration. The simplest example is La<sub>2</sub>CuO<sub>4</sub>, which consists of alternating CuO<sub>2</sub> and LaO layers that are insulating when pure. When 8% of the La is replaced by Sr, the latter acts as a dopant, contributing holes to the CuO<sub>2</sub> layers, and making the sample metallic. The Sr impurities also act as electronic bridges, enabling interlayer coupling. Proceeding from this picture, some theories argue that the pairing interaction is with [[phonons]], as in conventional superconductors with [[Cooper pairs]]. While the undoped materials are antiferromagnetic, even a few percent of impurity dopants introduce a smaller pseudogap in the CuO<sub>2</sub> planes that is also caused by phonons. The gap decreases with increasing charge carriers, and as it nears the superconductive gap, the latter reaches its maximum. The transition temperature is then argued to be due to the percolating behaviour of the carriers, which follow zig-zag percolative paths, largely in metallic domains in the CuO<sub>2</sub> planes, until blocked by charge density wave [[domain walls]], where they use dopant bridges to cross over to a metallic domain of an adjacent CuO<sub>2</sub> plane. The transition temperature maxima are reached when the host lattice has weak bond-bending forces, which produce strong electron–phonon interactions at the interlayer dopants.<ref name="pnas">
{{cite journal
 |last=Phillips |first=J.
 |date=2010
 |title=Percolative theories of strongly disordered ceramic high-temperature superconductors
 |journal=[[Proceedings of the National Academy of Sciences of the United States of America]]
 |volume=43 |issue=4 |pages=1307–10
 |pmid=20080578 |pmc=2824359 |doi=10.1073/pnas.0913002107
 |bibcode=2010PNAS..107.1307P
|doi-access=free
 }}
</ref>

=== D symmetry in YBCO ===
[[File:Meissner effect p1390048.jpg|thumb|Small magnet levitating above a high-temperature superconductor cooled by [[liquid nitrogen]]: this is a case of [[Meissner effect]].]]

An experiment based on flux quantization of a three-grain ring of [[Yttrium barium copper oxide|YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7</sub> (YBCO)]] was proposed to test the symmetry of the order parameter in the HTS. The symmetry of the order parameter could best be probed at the junction interface as the Cooper pairs tunnel across a [[Josephson junction]] or weak link.<ref name=Gesh1987>{{cite journal |last1=Geshkenbein |first1=V. |last2=Larkin |first2=A. |last3=Barone |first3=A. |year=1987 |title=Vortices with half magnetic flux quanta in ''heavy-fermion'' superconductors |journal=[[Physical Review B]] |volume=36  |issue=1  |pages=235–238 |doi=10.1103/PhysRevB.36.235 |bibcode=1987PhRvB..36..235G |pmid=9942041}}</ref> It was expected that a half-integer flux, that is, a spontaneous magnetization could only occur for a junction of ''d'' symmetry superconductors. But, even if the junction experiment is the strongest method to determine the symmetry of the HTS order parameter, the results have been ambiguous. [[John R. Kirtley]] and C. C. Tsuei thought that the ambiguous results came from the defects inside the HTS, leading them to an experiment where both clean limit (no defects) and dirty limit (maximal defects) were considered simultaneously.<ref name=Kirt1995>
{{cite journal
 |last1=Kirtley |first1=J. R.
 |last2=Tsuei |first2=C. C.
 |last3=Sun |first3=J. Z.
 |last4=Chi |first4=C. C.
 |last5=Yu-Jahnes |first5=Lock See
 |last6=Gupta |first6=A.
 |last7=Rupp |first7=M.
 |last8=Ketchen |first8=M. B.
 |year=1995
 |title=Symmetry of the order parameter in the high-{{mvar|T}}<sub>c</sub> superconductor YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7−δ</sub>
 |journal=[[Nature (journal)|Nature]]
 |volume=373  |issue=6511  |pages=225–228
 |doi=10.1038/373225a0 |bibcode=1995Natur.373..225K |s2cid=4237450
}}
</ref> Spontaneous magnetization was clearly observed in YBCO, which supported the ''d'' symmetry of the order parameter in YBCO. But, since YBCO is [[orthorhombic]], it might inherently have an admixture of ''s'' symmetry. By tuning their technique, they found an admixture of ''s'' symmetry in YBCO within about 3%.<ref name=Kirt2006>{{cite journal |last1=Kirtley |first1=J. R. |last2=Tsuei |first2=C. C. |last3=Ariando |first3=A. |last4=Verwijs |first4=C. J. M. |last5=Harkema |first5=S. |last6=Hilgenkamp |first6=H.
 |year=2006 |title=Angle-resolved phase-sensitive determination of the in-plane gap symmetry in YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7−δ</sub>
 |journal=[[Nature Physics]]
 |volume=2  |issue=3   |pages=190–194
 |doi=10.1038/nphys215 |bibcode=2006NatPh...2..190K |s2cid=118447968
|url=https://ris.utwente.nl/ws/files/6613933/angle.pdf }}
</ref> Also, they found a pure ''d''<sub>x<sup>2</sup>−y<sup>2</sup></sub> order parameter symmetry in tetragonal Tl<sub>2</sub>Ba<sub>2</sub>CuO<sub>6</sub>.<ref name=Tsue1997>
{{cite journal |last1=Tsuei   |first1=C. C. |last2=Kirtley |first2=J. R.
 |last3=Ren     |first3=Z. F.
 |last4=Wang    |first4=J. H.
 |last5=Raffy   |first5=H.
 |last6=Li      |first6=Z. Z.
 |year=1997
 |title=Pure ''d''<sub>x<sup>2</sup>−y<sup>2</sup></sub> order-parameter symmetry in the tetragonal superconductor Tl<sub>2</sub>Ba<sub>2</sub>CuO<sub>6+δ</sub>
 |journal=[[Nature (journal)|Nature]]
 |volume=387  |issue=6632  |pages=481–483
 |doi=10.1038/387481a0 |bibcode=1997Natur.387..481T |s2cid=4314494
}}
</ref>

=== Spin-fluctuation mechanism ===
The lack of exact theoretical computations on such strongly interacting electron systems has complicated attempts to validate spin-fluctuation. However, most theoretical calculations, including phenomenological and diagrammatic approaches, converge on magnetic fluctuations as the pairing mechanism. 

==== Qualitative explanation ====
In a superconductor, the flow of electrons cannot be resolved into individual electrons, but instead consists of pairs of bound electrons, called Cooper pairs. In conventional superconductors, these pairs are formed when an electron moving through the material distorts the surrounding crystal lattice, which attracts another electron and forms a bound pair. This is sometimes called the "water bed" effect. Each Cooper pair requires a certain minimum energy to be displaced, and if the thermal fluctuations in the crystal lattice are smaller than this energy the pair can flow without dissipating energy. Electron flow without resistance is superconductivity.

In a high-{{mvar|T}}<sub>c</sub> superconductor, the mechanism is extremely similar to a conventional superconductor, except that phonons play virtually no role, replaced by spin-density waves. Just as all known conventional superconductors are strong phonon systems, all known high-{{mvar|T}}<sub>c</sub> superconductors are strong spin-density wave systems, within close vicinity of a magnetic transition to, for example, an antiferromagnet. When an electron moves in a high-{{mvar|T}}<sub>c</sub> superconductor, its spin creates a spin-density wave around it. This spin-density wave in turn causes a nearby electron to fall into the spin depression created by the first electron (water-bed). When the system temperature is lowered, more spin density waves and Cooper pairs are created, eventually leading to superconductivity. High-{{mvar|T}}<sub>c</sub> systems are magnetic systems due to the Coulomb interaction, creating a strong Coulomb repulsion between electrons. This repulsion prevents pairing of the Cooper pairs on the same lattice site. Instead, pairing occurs at near-neighbor lattice sites. This is the so-called ''d''-wave pairing, where the pairing state has a node (zero) at the origin.