Editing Superconductivity (section)

=== Phase transition ===
[[File:Cvandrhovst.png|thumb|Behavior of heat capacity (''c''<sub>v</sub>, blue) and resistivity (''ρ'', green) at the superconducting phase transition]]
In superconducting materials, the characteristics of superconductivity appear when the temperature ''T'' is lowered below a critical temperature ''T''<sub>c</sub>. The value of this critical temperature varies from material to material. Conventional superconductors usually have critical temperatures ranging from around 20&nbsp;[[Kelvin|K]] to less than 1&nbsp;K. Solid [[Mercury (element)|mercury]], for example, has a critical temperature of 4.2&nbsp;K. As of 2015, the highest critical temperature found for a conventional superconductor is 203 K for H<sub>2</sub>S, although high pressures of approximately 90 gigapascals were required.<ref>{{cite journal |last1=Drozdov |first1=A. |last2=Eremets |first2=M. |last3=Troyan |first3=I. |last4=Ksenofontov |first4=V. |date=17 August 2015 |title=Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system |journal=Nature |volume=525 |issue=2–3 |pages=73–76 |arxiv=1506.08190 |bibcode=2015Natur.525...73D |doi=10.1038/nature14964 |pmid=11369082 |s2cid=4468914}}</ref> [[Cuprate superconductor|Cuprate superconductors]] can have much higher critical temperatures: [[YBCO|YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7</sub>]], one of the first cuprate superconductors to be discovered, has a critical temperature above 90&nbsp;K, and mercury-based cuprates have been found with critical temperatures in excess of 130&nbsp;K. The basic physical mechanism responsible for the high critical temperature is not yet clear. However, it is clear that a two-electron pairing is involved, although the nature of the pairing (<math>s</math> wave vs. <math>d</math> wave) remains controversial.<ref>{{cite book |last1=Tinkham |first1=Michael |title=Introduction to Superconductivity |date=1996 |publisher=Dover Publications, Inc. |isbn=0486435032 |location=Mineola, New York |page=16}}</ref>

Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external [[magnetic field]] is applied which is greater than the ''critical magnetic field''. This is because the [[Gibbs free energy]] of the superconducting phase increases quadratically with the magnetic field while the free energy of the normal phase is roughly independent of the magnetic field. If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field (proportional to the square root of the difference of the free energies at zero magnetic field) the two free energies will be equal and a phase transition to the normal phase will occur. More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of electrons that are superconducting and consequently to a longer [[London penetration depth]] of external magnetic fields and currents. The penetration depth becomes infinite at the phase transition.

The onset of superconductivity is accompanied by abrupt changes in various physical properties, which is the hallmark of a [[phase transition]]. For example, the electronic [[heat capacity]] is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear. At low temperatures, it varies instead as ''e''<sup>−''α''/''T''</sup> for some constant, ''α''. This exponential behavior is one of the pieces of evidence for the existence of the [[energy gap]].

The [[Second-order transition|order]] of the superconducting [[Phase transition#Order parameters|phase transition]] was long a matter of debate. Experiments indicate that the transition is second-order, meaning there is no [[latent heat]]. However, in the presence of an external magnetic field there is latent heat, because the superconducting phase has a lower entropy below the critical temperature than the normal phase. It has been experimentally demonstrated<ref>{{cite journal |author=Dolecek |first=R. L. |date=1954 |title=Adiabatic Magnetization of a Superconducting Sphere |journal=[[Physical Review]] |volume=96 |issue=1 |pages=25–28 |bibcode=1954PhRv...96...25D |doi=10.1103/PhysRev.96.25}}</ref> that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material.

Calculations in the 1970s suggested that it may actually be weakly first-order due to the effect of long-range fluctuations in the electromagnetic field. In the 1980s it was shown theoretically with the help of a [[Disorder field|disorder field theory]], in which the [[Vortex line|vortex lines]] of the superconductor play a major role, that the transition is of second order within the [[Type-II superconductor|type II]] regime and of first order (i.e., [[latent heat]]) within the [[Type-I superconductor|type I]] regime, and that the two regions are separated by a [[tricritical point]].<ref>{{cite journal |author=Kleinert |first=H. |date=1982 |title=Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition |url=http://www.physik.fu-berlin.de/~kleinert/97/97.pdf |journal=[[Lettere al Nuovo Cimento]] |volume=35 |issue=13 |pages=405–412 |doi=10.1007/BF02754760 |s2cid=121012850}}</ref> The results were strongly supported by Monte Carlo computer simulations.<ref>{{cite journal |author=Hove |first1=J. |last2=Mo |first2=S. |last3=Sudbo |first3=A. |date=2002 |title=Vortex interactions and thermally induced crossover from type-I to type-II superconductivity |url=http://www.physik.fu-berlin.de/~kleinert/papers/sudbotre064524.pdf |journal=[[Physical Review B]] |volume=66 |issue=6 |page=064524 |arxiv=cond-mat/0202215 |bibcode=2002PhRvB..66f4524H |doi=10.1103/PhysRevB.66.064524 |s2cid=13672575}}</ref>