Editing Superconductivity (section)

== Elementary properties ==
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Several physical properties of superconductors vary from material to material, such as the critical temperature, the value of the [[Energy gap#Superconductors|superconducting gap]], the critical magnetic field, and the critical current density at which superconductivity is destroyed. On the other hand, there is a class of properties that are independent of the underlying material. The Meissner effect, the quantization of the [[Magnetic flux quantum|magnetic flux]] or permanent currents, i.e. the state of zero resistance are the most important examples. The existence of these "universal" properties is rooted in the nature of the [[Symmetry breaking|broken symmetry]] of the superconductor and the emergence of [[off-diagonal long range order]]. Superconductivity is a [[Phase (matter)|thermodynamic phase]], and thus possesses certain distinguishing properties which are largely independent of microscopic details. Off diagonal long range order is closely connected to the formation of [[Cooper pair|Cooper pairs]].

=== Zero electrical DC resistance ===
[[File:CERN-cables-p1030764.jpg|thumb|Electric cables for accelerators at [[CERN]]. Both the massive and slim cables are rated for 12,500 [[Amperes|A]]. ''Top:'' regular cables for [[Large Electron–Positron Collider|LEP]]; ''bottom:'' superconductor-based cables for the [[Large Hadron Collider|LHC]]]]
[[File:Cross_section_of_preform_superconductor_cable.jpg|thumb|Cross section of a preformed superconductor rod from the abandoned [[Superconducting Super Collider|Texas Superconducting Super Collider (SSC)]]]]
The simplest method to measure the [[electrical resistance]] of a sample of some material is to place it in an [[electrical circuit]] in series with a [[current source]] ''I'' and measure the resulting [[voltage]] ''V'' across the sample. The resistance of the sample is given by [[Ohm's law]] as ''R&nbsp;=&nbsp;V&nbsp;/&nbsp;I''. If the voltage is zero, this means that the resistance is zero.

Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in [[Superconducting magnet|superconducting electromagnets]] such as those found in [[Magnetic resonance imaging|MRI]] machines. Experiments have demonstrated that currents in superconducting coils can persist for years without any measurable degradation. Experimental evidence points to a lifetime of at least 100,000 years. Theoretical estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature.<ref name="Gallop2" /> In practice, currents injected in superconducting coils persisted for 28 years, 7 months, 27 days in a superconducting gravimeter in Belgium, from August 4, 1995 until March 31, 2024.<ref>{{cite journal |last1=Van Camp |first1=Michel |last2=Francis |first2=Olivier |last3=Lecocq |first3=Thomas |year=2017 |title=Recording Belgium's Gravitational History |journal=Eos |language=en-US |volume=98 |doi=10.1029/2017eo089743 |doi-access=free}}</ref><ref>{{cite journal |last1=Van Camp |first1=Michel |last2=de Viron |first2=Olivier |last3=Watlet |first3=Arnaud |last4=Meurers |first4=Bruno |last5=Francis |first5=Olivier |last6=Caudron |first6=Corentin |year=2017 |title=Geophysics From Terrestrial Time-Variable Gravity Measurements |url=https://hal.archives-ouvertes.fr/hal-01631032/file/Reviews%20of%20Geophysics%20-%202017%20-%20Van%20Camp%20-%20Geophysics%20From%20Terrestrial%20Time%25u2010Variable%20Gravity%20Measurements.pdf |journal=[[Reviews of Geophysics]] |language=en |volume=55 |issue=4 |pages=2017RG000566 |bibcode=2017RvGeo..55..938V |doi=10.1002/2017rg000566 |issn=1944-9208 |s2cid=134876430}}</ref> In such instruments, the measurement is based on the monitoring of the levitation of a superconducting niobium sphere with a mass of four grams.

In a normal conductor, an electric current may be visualized as a fluid of [[Electron|electrons]] moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into [[heat]], which is essentially the vibrational [[kinetic energy]] of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance and [[Joule heating]].

The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound ''pairs'' of electrons known as [[Cooper pair|Cooper pairs]]. This pairing is caused by an attractive force between electrons from the exchange of [[Phonon|phonons]]. This pairing is very weak, and small thermal vibrations can fracture the bond. Due to [[quantum mechanics]], the [[energy spectrum]] of this Cooper pair fluid possesses an ''[[energy gap]]'', meaning there is a minimum amount of energy Δ''E'' that must be supplied in order to excite the fluid. Therefore, if Δ''E'' is larger than the [[thermal energy]] of the lattice, given by ''kT'', where ''k'' is the [[Boltzmann constant]] and ''T'' is the [[temperature]], the fluid will not be scattered by the lattice.<ref>{{cite book |last1=Tinkham |first1=Michael |title=Introduction to Superconductivity |date=1996 |publisher=Dover Publications, Inc. |isbn=0486435032 |location=Mineola, New York |page=8}}</ref> The Cooper pair fluid is thus a [[superfluid]], meaning it can flow without energy dissipation.

In the class of superconductors known as [[Type II superconductor|type II superconductors]], including all known [[High-temperature superconductor|high-temperature superconductors]], an extremely low but non-zero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of [[Abrikosov vortex|magnetic vortices]] in the electronic superfluid, which dissipates some of the energy carried by the current. If the current is sufficiently small, the vortices are stationary, and the resistivity vanishes. The resistance due to this effect is minuscule compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen into a disordered but stationary phase known as a "vortex glass". Below this vortex glass transition temperature, the resistance of the material becomes truly zero.

=== 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>

=== Meissner effect ===
{{Main|Meissner effect}}
[[File:Meissner_effect.ogv|thumb|Meissner effect in a high-temperature superconductor (black pellet) with a NdFeB magnet (metallic)]]
When a superconductor is placed in a weak external magnetic field ''H'', and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected but instead, the field penetrates the superconductor but only to a very small distance, characterized by a parameter&nbsp;''λ'', called the [[London penetration depth]], decaying exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100&nbsp;nm.

The Meissner effect is sometimes confused with the kind of [[diamagnetism]] one would expect in a perfect electrical conductor: according to [[Lenz's law]], when a ''changing'' magnetic field is applied to a conductor, it will induce an electric current in the conductor that creates an opposing magnetic field. In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field.

The Meissner effect is distinct from this{{snd}}it is the spontaneous expulsion that occurs during transition to superconductivity. Suppose we have a material in its normal state, containing a constant internal magnetic field. When the material is cooled below the critical temperature, we would observe the abrupt expulsion of the internal magnetic field, which we would not expect based on Lenz's law.

The Meissner effect was given a phenomenological explanation by the brothers [[Fritz London|Fritz]] and [[Heinz London]], who showed that the electromagnetic [[Thermodynamic free energy|free energy]] in a superconductor is minimized provided <math display="block"> \nabla^2\mathbf{H} = \lambda^{-2} \mathbf{H}\, </math> where ''H'' is the magnetic field and ''λ'' is the London penetration depth.

This equation, which is known as the [[London equation]], predicts that the magnetic field in a superconductor [[Exponential decay|decays exponentially]] from whatever value it possesses at the surface.

A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value ''H''<sub>c</sub>. Depending on the geometry of the sample, one may obtain an intermediate state<ref>{{cite book |author=Landau |first1=Lev D. |title=Electrodynamics of Continuous Media |last2=Lifschitz |first2=Evgeny M. |date=1984 |publisher=Butterworth-Heinemann |isbn=978-0-7506-2634-7 |series=[[Course of Theoretical Physics]] |volume=8 |location=Oxford, England |language=en-uk}}</ref> consisting of a baroque pattern<ref>{{cite journal |author=Callaway |first=David J. E. |date=1990 |title=On the remarkable structure of the superconducting intermediate state |journal=[[Nuclear Physics B]] |volume=344 |issue=3 |pages=627–645 |bibcode=1990NuPhB.344..627C |doi=10.1016/0550-3213(90)90672-Z}}</ref> of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value ''H''<sub>c1</sub> leads to a mixed state (also known as the vortex state) in which an increasing amount of [[magnetic flux]] penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength ''H''<sub>c2</sub>, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called [[Fluxon|fluxons]] because the flux carried by these [[Abrikosov vortex|vortices]] is [[Quantum|quantized]]. Most pure [[Chemical element|elemental]] superconductors, except [[niobium]] and [[Carbon nanotube|carbon nanotubes]], are Type I, while almost all impure and compound superconductors are Type&nbsp;II.

=== London moment ===
{{Main|London moment}}
Conversely, a spinning superconductor generates a magnetic field, precisely aligned with the spin axis. The effect, the London moment, was put to good use in [[Gravity Probe B]]. This experiment measured the magnetic fields of four superconducting gyroscopes to determine their spin axes. This was critical to the experiment since it is one of the few ways to accurately determine the spin axis of an otherwise featureless sphere.