BCS theory

Last updated July 29, 2024 • 9 min readFrom Wikipedia, The Free Encyclopedia

In physics, theBardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

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",^[1] where Fritz London proposed that the phenomenological London equations may be consequences of the 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 coherence length. John Bardeen then argued in the 1955 paper, "Theory of the Meissner Effect in Superconductors",^[2] 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".^[3]

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".^[4] The demonstration that the phase transition is second order, that it reproduces the Meissner effect and the calculations of specific heats and penetration depths appeared in the December 1957 article, "Theory of superconductivity".^[5] 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 K.^[6] Following experiments determined more materials with transition temperatures up to about 130 K, considerably above the previous limit of about 30 K. 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.^[7] 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 pairs. 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 transformations.

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 phonons). Roughly speaking the picture is the following:

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.

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 gases of fermions where a homogeneous magnetic field has been tuned to their Feshbach resonance. The original results of BCS (discussed below) described an s-wave superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the 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:^[8]

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 electrons 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 bosons instead, which are bound by 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 the most abundant natural mercury isotope, ²⁰²Hg, with a different isotope, ¹⁹⁸Hg.^[9]

An 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 greatly 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^[10] 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_c. 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^[11] $\Delta (T=0)=1.764\,k_{\rm {B}}T_{\rm {c}},$ independent of material. Near the critical temperature the relation asymptotes to^[11] $\Delta (T\to T_{\rm {c}})\approx 3.06\,k_{\rm {B}}T_{\rm {c}}{\sqrt {1-(T/T_{\rm {c}})}}$ which is of the form suggested the previous year by M. J. Buckingham^[12] 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 (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 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_c in terms of the electron-phonon coupling potential V and the Debye cutoff energy E_D:^[5] $k_{\rm {B}}\,T_{\rm {c}}=1.134E_{\rm {D}}\,{e^{-1/N(0)\,V}},$ 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 square-root of 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 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,^[13] and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt.^[14] 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 experiment ^[15] - One of the first^{[ citation needed ]} indications to the importance of the Cooper-pairing principle.

Related Research Articles

Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered, even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source.

Unconventional superconductors are materials that display superconductivity which does not conform to conventional BCS theory or its extensions.

High-temperature superconductors are defined as materials with critical temperature above 77 K, the boiling point of liquid nitrogen. They are only "high-temperature" relative to previously known superconductors, which function at even colder temperatures, close to absolute zero. The "high temperatures" are still far below ambient, and therefore require cooling. The first breakthrough of high-temperature superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller. Although the critical temperature is around 35.1 K, this new type of superconductor was readily modified by Ching-Wu Chu to make the first high-temperature superconductor with critical temperature 93 K. Bednorz and Müller were awarded the Nobel Prize in Physics in 1987 "for their important break-through in the discovery of superconductivity in ceramic materials". Most high- $T$ _c materials are type-II superconductors.

In condensed matter physics, a Cooper pair or BCS pair is a pair of electrons bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper.

Leon N. Cooper is an American physicist and Nobel Prize laureate who, with John Bardeen and John Robert Schrieffer, developed the BCS theory of superconductivity. His name is also associated with the Cooper pair and the BCM theory of synaptic plasticity.

<span class="mw-page-title-main">Fermionic condensate</span> State of matter

A fermionic condensate is a superfluid phase formed by fermionic particles at low temperatures. It is closely related to the Bose–Einstein condensate, a superfluid phase formed by bosonic atoms under similar conditions. The earliest recognized fermionic condensate described the state of electrons in a superconductor; the physics of other examples including recent work with fermionic atoms is analogous. The first atomic fermionic condensate was created by a team led by Deborah S. Jin using potassium-40 atoms at the University of Colorado Boulder in 2003.

The Little–Parks effect was discovered in 1962 by William A. Little and Ronald D. Parks in experiments with empty and thin-walled superconducting cylinders subjected to a parallel magnetic field. It was one of the first experiments to indicate the importance of Cooper-pairing principle in BCS theory.

<span class="mw-page-title-main">History of superconductivity</span>

Superconductivity is the phenomenon of certain materials exhibiting zero electrical resistance and the expulsion of magnetic fields below a characteristic temperature. The history of superconductivity began with Dutch physicist Heike Kamerlingh Onnes's discovery of superconductivity in mercury in 1911. Since then, many other superconducting materials have been discovered and the theory of superconductivity has been developed. These subjects remain active areas of study in the field of condensed matter physics.

<span class="mw-page-title-main">Pseudogap</span> State at which a Fermi surface has a partial energy gap in condensed matter physics

In condensed matter physics, a pseudogap describes a state where the Fermi surface of a material possesses a partial energy gap, for example, a band structure state where the Fermi surface is gapped only at certain points.

A charge density wave (CDW) is an ordered quantum fluid of electrons in a linear chain compound or layered crystal. The electrons within a CDW form a standing wave pattern and sometimes collectively carry an electric current. The electrons in such a CDW, like those in a superconductor, can flow through a linear chain compound en masse, in a highly correlated fashion. Unlike a superconductor, however, the electric CDW current often flows in a jerky fashion, much like water dripping from a faucet due to its electrostatic properties. In a CDW, the combined effects of pinning and electrostatic interactions likely play critical roles in the CDW current's jerky behavior, as discussed in sections 4 & 5 below.

In materials science, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, they are sometimes referred to as heavy electrons. Heavy fermion materials have a low-temperature specific heat whose linear term is up to 1000 times larger than the value expected from the free electron model. The properties of the heavy fermion compounds often derive from the partly filled f-orbitals of rare-earth or actinide ions, which behave like localized magnetic moments.

A Peierls transition or Peierls distortion is a distortion of the periodic lattice of a one-dimensional crystal. Atomic positions oscillate, so that the perfect order of the 1-D crystal is broken.

The Mattis–Bardeen theory is a theory that describes the electrodynamic properties of superconductivity. It is commonly applied in the research field of optical spectroscopy on superconductors.

Superstripes is a generic name for a phase with spatial broken symmetry that favors the onset of superconducting or superfluid quantum order. This scenario emerged in the 1990s when non-homogeneous metallic heterostructures at the atomic limit with a broken spatial symmetry have been found to favor superconductivity. Before a broken spatial symmetry was expected to compete and suppress the superconducting order. The driving mechanism for the amplification of the superconductivity critical temperature in superstripes matter has been proposed to be the shape resonance in the energy gap parameters ∆n that is a type of Fano resonance for coexisting condensates.

The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase can arise in a superconductor in large magnetic field. Among its characteristics are Cooper pairs with nonzero total momentum and a spatially non-uniform order parameter, leading to normal conducting areas in the superconductor.

<i>BCS: 50 Years</i> Book by Leon Cooper

BCS: 50 Years is a review volume on the topic of superconductivity edited by Leon Cooper, a 1972 Nobel Laureate in Physics, and Dmitri Feldman of Brown University, first published in 2010.

Several hundred metals, compounds, alloys and ceramics possess the property of superconductivity at low temperatures. The SU(2) color quark matter adjoins the list of superconducting systems. Although it is a mathematical abstraction, its properties are believed to be closely related to the SU(3) color quark matter, which exists in nature when ordinary matter is compressed at supranuclear densities above ~ 0.5 10³⁹ nucleon/cm³.

The condensate of electron quadruplets is a proposed state of matter in which Cooper pairs are formed but do not exhibit long-range order, but electron quadruplets do. One example of the proposed electron quadruplet condensates is charge-4e superconductivity. Another example is "quartic metal" phase is related to but distinct from those superconductors explained by the standard BCS theory; rather than expelling magnetic field lines as in the Meissner effect, it generates them, a spontaneous Nernst effect that indicates the breaking of time-reversal symmetry. After the theoretical possibility was raised, observations consistent with electron quadrupling were published using hole-doped Ba_1-xK_xFe₂As₂ in 2021.

In physics, the Matthias rules refers to a historical set of empirical guidelines on how to find superconductors. These rules were authored Bernd T. Matthias who discovered hundreds of superconductors using these principles in the 1950s and 1960s. Deviations from these rules have been found since the end of the 1970s with the discovery of unconventional superconductors.

Kohn–Luttinger superconductivity is a theoretical mechanism for unconventional superconductivity proposed by Walter Kohn and Joaquin Mazdak Luttinger based on Friedel oscillations. In contrast to BCS theory, in which Cooper pairs are formed due to electron–phonon interaction, Kohn–Luttinger mechanism is based on fact that screened Coulomb interaction oscillates as $and can create Cooper instability for non-zero angular momentum .$

References

↑ London, F. (September 1948). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562.
↑ Bardeen, J. (March 1955). "Theory of the Meissner Effect in Superconductors". Physical Review. 97 (6): 1724–1725. Bibcode:1955PhRv...97.1724B. doi:10.1103/PhysRev.97.1724.
↑ Cooper, Leon (November 1956). "Bound Electron Pairs in a Degenerate Fermi Gas". Physical Review. 104 (4): 1189–1190. Bibcode:1956PhRv..104.1189C. doi: 10.1103/PhysRev.104.1189 . ISSN 0031-899X.
↑ Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (April 1957). "Microscopic Theory of Superconductivity". Physical Review. 106 (1): 162–164. Bibcode:1957PhRv..106..162B. doi: 10.1103/PhysRev.106.162 .
1 2 Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (December 1957). "Theory of Superconductivity". Physical Review. 108 (5): 1175–1204. Bibcode:1957PhRv..108.1175B. doi: 10.1103/PhysRev.108.1175 .
↑ Bednorz, J. G.; Müller, K. A. (June 1986). "Possible high T_c superconductivity in the Ba−La−Cu−O system". Zeitschrift für Physik B: Condensed Matter. 64 (2): 189–193. Bibcode:1986ZPhyB..64..189B. doi:10.1007/BF01303701. S2CID 118314311.
↑ Mann, A. (July 2011). "High Temperature Superconductivity at 25: Still In Suspense". Nature. 475 (7356): 280–2. Bibcode:2011Natur.475..280M. doi:10.1038/475280a. PMID 21776057.
↑ "BCS Theory of Superconductivity". hyperphysics.phy-astr.gsu.edu. Retrieved 16 April 2018.
↑ Maxwell, Emanuel (1950). "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477. Bibcode:1950PhRv...78..477M. doi:10.1103/PhysRev.78.477.
↑ Ivar Giaever - Nobel Lecture. Nobelprize.org. Retrieved 16 Dec 2010. http://nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html
1 2 Tinkham, Michael (1996). Introduction to Superconductivity. Dover Publications. p. 63. ISBN 978-0-486-43503-9.
↑ Buckingham, M. J. (February 1956). "Very High Frequency Absorption in Superconductors". Physical Review . 101 (4): 1431–1432. Bibcode:1956PhRv..101.1431B. doi:10.1103/PhysRev.101.1431.
↑ Maxwell, Emanuel (1950-05-15). "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477. Bibcode:1950PhRv...78..477M. doi:10.1103/PhysRev.78.477.
↑ Reynolds, C. A.; Serin, B.; Wright, W. H.; Nesbitt, L. B. (1950-05-15). "Superconductivity of Isotopes of Mercury". Physical Review. 78 (4): 487. Bibcode:1950PhRv...78..487R. doi:10.1103/PhysRev.78.487.
↑ Little, W. A.; Parks, R. D. (1962). "Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder". Physical Review Letters. 9 (1): 9–12. Bibcode:1962PhRvL...9....9L. doi:10.1103/PhysRevLett.9.9.
↑ Gurovich, Doron; Tikhonov, Konstantin; Mahalu, Diana; Shahar, Dan (2014-11-20). "Little-Parks Oscillations in a Single Ring in the vicinity of the Superconductor-Insulator Transition". Physical Review B. 91 (17): 174505. arXiv: 1411.5640 . Bibcode:2015PhRvB..91q4505G. doi:10.1103/PhysRevB.91.174505. S2CID 119268649.

Primary sources

Cooper, Leon N. (1956). "Bound Electron Pairs in a Degenerate Fermi Gas". Physical Review. 104 (4): 1189–1190. Bibcode:1956PhRv..104.1189C. doi: 10.1103/PhysRev.104.1189 .
Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (1957). "Microscopic Theory of Superconductivity". Physical Review. 106 (1): 162–164. Bibcode:1957PhRv..106..162B. doi: 10.1103/PhysRev.106.162 .
Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (1957). "Theory of Superconductivity". Physical Review. 108 (5): 1175–1204. Bibcode:1957PhRv..108.1175B. doi: 10.1103/PhysRev.108.1175 .

External links

Hyperphysics page on BCS
Dance analogy Archived 2011-06-29 at the Wayback Machine of BCS theory as explained by Bob Schrieffer (audio recording)
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

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[1] London, F. (September 1948). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562.

[2] Bardeen, J. (March 1955). "Theory of the Meissner Effect in Superconductors". Physical Review. 97 (6): 1724–1725. Bibcode:1955PhRv...97.1724B. doi:10.1103/PhysRev.97.1724.

[3] Cooper, Leon (November 1956). "Bound Electron Pairs in a Degenerate Fermi Gas". Physical Review. 104 (4): 1189–1190. Bibcode:1956PhRv..104.1189C. doi: 10.1103/PhysRev.104.1189 . ISSN 0031-899X.

[4] Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (April 1957). "Microscopic Theory of Superconductivity". Physical Review. 106 (1): 162–164. Bibcode:1957PhRv..106..162B. doi: 10.1103/PhysRev.106.162 .

[BCS_theory-5] 1 2 Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (December 1957). "Theory of Superconductivity". Physical Review. 108 (5): 1175–1204. Bibcode:1957PhRv..108.1175B. doi: 10.1103/PhysRev.108.1175 .

[6] Bednorz, J. G.; Müller, K. A. (June 1986). "Possible high T_c superconductivity in the Ba−La−Cu−O system". Zeitschrift für Physik B: Condensed Matter. 64 (2): 189–193. Bibcode:1986ZPhyB..64..189B. doi:10.1007/BF01303701. S2CID 118314311.

[7] Mann, A. (July 2011). "High Temperature Superconductivity at 25: Still In Suspense". Nature. 475 (7356): 280–2. Bibcode:2011Natur.475..280M. doi:10.1038/475280a. PMID 21776057.

[8] "BCS Theory of Superconductivity". hyperphysics.phy-astr.gsu.edu. Retrieved 16 April 2018.

[maxwell1950-9] Maxwell, Emanuel (1950). "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477. Bibcode:1950PhRv...78..477M. doi:10.1103/PhysRev.78.477.

[Giaever-10] Ivar Giaever - Nobel Lecture. Nobelprize.org. Retrieved 16 Dec 2010. http://nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html

[Tinkham_1996_63-11] 1 2 Tinkham, Michael (1996). Introduction to Superconductivity. Dover Publications. p. 63. ISBN 978-0-486-43503-9.

[12] Buckingham, M. J. (February 1956). "Very High Frequency Absorption in Superconductors". Physical Review . 101 (4): 1431–1432. Bibcode:1956PhRv..101.1431B. doi:10.1103/PhysRev.101.1431.

[13] Maxwell, Emanuel (1950-05-15). "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477. Bibcode:1950PhRv...78..477M. doi:10.1103/PhysRev.78.477.

[14] Reynolds, C. A.; Serin, B.; Wright, W. H.; Nesbitt, L. B. (1950-05-15). "Superconductivity of Isotopes of Mercury". Physical Review. 78 (4): 487. Bibcode:1950PhRv...78..487R. doi:10.1103/PhysRev.78.487.

[Little-15] Little, W. A.; Parks, R. D. (1962). "Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder". Physical Review Letters. 9 (1): 9–12. Bibcode:1962PhRvL...9....9L. doi:10.1103/PhysRevLett.9.9.

[16] Gurovich, Doron; Tikhonov, Konstantin; Mahalu, Diana; Shahar, Dan (2014-11-20). "Little-Parks Oscillations in a Single Ring in the vicinity of the Superconductor-Insulator Transition". Physical Review B. 91 (17): 174505. arXiv: 1411.5640 . Bibcode:2015PhRvB..91q4505G. doi:10.1103/PhysRevB.91.174505. S2CID 119268649.

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