[Mahan] Many-Particle Physics, Gerald D. Mahan, 2nd ed., Plenum, 1990.
[TIN] Introduction to Superconductivity, Michael Tinkham, 2nd ed., McGraw-Hill Internation Ed, 1996.
[BCS] Theory of Superconductivity, J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys Rev, 108, 1175 (1957). (the famous BCS Theory)
Wikipedia references: J. Bardeen, L. Cooper, and J.R. Schrieffer.
1911
[BCS] Superconductivity was discovered by Onnes.
1928
[BCS] The Sommerfeld-Bloch individual-particle model gives a fairly good description of normal metals, but fails to account for superconductivity. In this theory, it is assumed that in first approximation one may neglect correlations between the positions of the electrons and assume that each electron moves independently in some sort of self-consistent field determined by the other conduction electrons and the ions. Wave functions of the metal as a whole are designated by occupation of Bloch individual-particle states of energy ε(k) defined by wave vector k and spin σ; in the ground state all levels with energies below the Fermi energy, εF, are occupied; those above are unoccupied.
1933
[BCS] A major advance was the discovery of the Meissner effect, which showed that a superconductor is a perfect diamagnet; magnetic flux is excluded from all but a thin penetration region near the surface.
1934
[BCS] Of the various two-fluid models used to describe the thermal properties, the first and best known is that of Gorter and Casimir, which yields a parabolic critical field curve and an electronic specific heat varying as T3.
1935
[BCS] London and London proposed a phenomenological theory of electromagnetic properties in which diamagnetic aspects were assumed basic. F. London suggested a quantum-theoretic approach to a theory in which it was assumed that there is somehow a coherence or rigidity in the superconducting state such that the wave functions are not modified very much when a magnetic field is applied.
1950
[Mahan] Fröhlich was the first to realize that electrons could interact by exchanging phonons and that this interaction could be attractive. He was the first to suggest that superconductivity was caused by the electron-phonon interaction.
[Mahan] The isotope effect verified the Fröhlich hypothesis that the electron-phonon interaction caused superconductivity. The isotope effect was discovered for the metal Hg by Maxwell and Reynolds et al.
[BCS] Fröhlich's theory, which makes use of a perturbation-theoretical approach, does give the correct isotopic mass dependence for H0, the critical field at T = 0K, but does not yield a phase with superconducting properties and further, the energy difference between what is supposed to correspond to normal and superconducting phases is far too large.
[BCS] A variational approach by J. Bardeen (1950, 1951) ran into similar difficulties. Both theories are based primarily on the self-energy of the electrons in the phonon field rather than on the true interaction between electrons, although it was recognized that the latter might be important (1951).
1952
[Mahan] The electron-phonon interaction gives a scattering from a Bloch state defined by the wave vector k to k' = k ± x by absorption or emission of a phonon of wave vector x. It is this interaction which is responsible for thermal scattering. Its contribution to the energy can be estimated by making a canonical transformation which eliminates the linear electron-phonon interaction terms from the Hamiltonian. In second order, there is one term which gives a renormalization of the phonon frequencies, and another, H2, which gives a true interaction between electrons, independent of the vibrational amplitudes. A transformation of this sort was given first by Fröhlich in a formulation in which Coulomb interactions between electrons were disregarded.
1953
[BCS] The concept of coherence has been emphasized by A.B. Pippard, who, on the basis of experiments on penetration phenomena, proposed a nonlocal modification of the London equations in which a coherence distance ξ0 is introduced. The current density at a point is given by an integral of the vector potential over a region surrounding the point. ξ0 is of the order of 10-4 cm in a pure metal. For a very slowly varying A, the Pippard expression reduces to the London form.
[BCS] Nakajima showed how such interactions could be included. Particular for the long-wavelength part of the interaction, it is important to take into account the screening of the Coulomb field of any one electron by other conduction electrons.
[BCS] Two-fluid models which yield an energy gap and an exponential specific heat curve at low temperatures have been discussed by Ginsburg and Bernardes (1957).
1955
[Mahan] Schafroth showed that a charged boson gas, when undergoing a Bose-Einstein condensation, would exhibit many of the superconducting properties - but not those known now, such as the energy gap in the excitation spectrum.
[BCS] Blatt, Butler, and Schafroth, have introduced the concept of a "correlation length," roughly the distance over which the momenta of a pair of particles are correlated (not to be confused with Pippard's coherence distance). M.R. Schafroth has argued that there is a true Meissner effect only if the correlation length is effectively infinite. In BCS theory (1957), the correlation length is most reasonably interpreted as the distance over which the momentum of virtual pairs is the same. The authors of BCS believed that in this sense, the correlation length is effectively infinite. The value of q is exactly zero everywhere in a simply connected body in external field. When there is current flow, as in a torus, there is unique distribution of q values for minimum free energy.
[BCS] The average velocity, v0, of electrons at the Fermi surface, is required for penetration phenomena. As pointed out by Faber and Pippard, this parameter is most conveniently determined from measurements of the anomalous skin effect in normal metals in the high-frequency limit. Faber and Pippard suggested that if ξ0 is written as ξ0 = a h v0 / kTc , the dimensionless constant a has approximately the same value for all superconductors and they find it equal to about 0.15 for Sn and Al.
[BCS] J. Bardeen pointed out that an energy-gap model would most likely lead to the Pippard version, and we found this is to be true of the present theory. Our theory of the diamagnetic aspects thus follows along the general lines suggested by London and by Pippard.
[BCS] Such effects (screening effect of above) are included in a more complete analysis by Bardeen and Pines, based on the Bohm-Pines collective model, in which plasma modes are introduced for long wavelengths.
1956
[Mahan] The first inkling of the BCS theory was a letter by Cooper who pointed out that the ground state of a normal metal was unstable at zero temperature. We define a normal state as one which is neither superconducting nor magnetic. The instability is an indication that the metal prefers to be in another state, in this case the superconducting.
[TIN] The basic idea that even a weak attraction can bind pairs of electrons into a bound state was presented by Cooper in 1956. He showed that the Fermi sea of electrons is unstable against the formation of at least one bound pair, regardless of how weak the interaction is, so long as it is attractive. This result is a consequence of the Fermi statistics and of the existence of the Fermi-sea background, since it is well known that binding does not ordinarily occur in the two-body problem in the three dimensions until the strength of the potential exceeds a finite threshold value.
[BCS] From analysis of data on transmission of microwave and far infrared radiation through superconducting films of tin and lead, Glover and Tinkham find a = 0.27.
[BCS] To obtain the ground state function, we observe that the interaction Hamiltonian connects a large number of nearly degenerate occupation number configurations with each other via nonzero matrix elements. If the matrix elements were all negative in sign, one could obtain a state with low energy by forming a linear combination of the basis functions with expansion coefficients of the same sign. The magnitude of the interaction energy obtained in this manner would be approximately given by the number of configurations which connect to a given typical configurations which connect to a given typical configuration times an average matrix element. This was demonstrated by one of our authors (L.N. Cooper, Phys. Rev. 104, 1189 (1956)) by solving a problem in which two electrons with zero total momentum interact via constant negative matrix elements in a small shell above the Fermi surface. It was shown that the ground state of this system is separated from the continuum by a volume independent energy. This type of coherent mixing of Bloch states produces a state with qualitatively different properties from the original states.
1957
[Mahan] BCS Theory successfully describes the superconducting properties of weak superconductors such as aluminum.
[BCS] In a preliminary communication (Phys Rev 106, 162 (1957)), we gave as a criterion for the occurrence of a superconducting phase that for transitions such as Δε < hω, the attractive H2 dominate the repulsive short-range screened Coulomb interaction between electrons, so as to give a net attraction. We showed that how an attractive interaction of this sort can give rise to a cooperative many-particle state which is lower in energy than the normal state by an amount proportional to (hω)2, consistent with the isotope effect.
[BCS] In the theory, the normal state is described by the Bloch individual-particle model. The ground-state wave function of a superconductor is formed by taking a linear combination of many low-lying normal state configurations in which the Bloch states are virtually occupied in pairs of opposite spin and momentum. If the state k↑ is occupied in any configuration, -k↓ is also occupied. The average excitation energy of the virtual pairs above the Fermi sea is of the order of kTc.
[BCS] Excited states of the superconductor are formed by specifying occupation of certain Bloch states and by using all the rest to form a linear combination of virtual pair configurations. There is thus a one-ton-one correspondence between excited states of the normal and superconducting phases.
[BCS] Our theory also accounts in a qualitative way for those aspect of superconductivity associated with infinite conductivity and a persistent current flowing in a ring. When there is a net current flow, the paired states ( k1↑, k2↓) have a net momentum k1+ k2 = q, where q is the same for all virtual pair states, and so can only produce fluctuations about the current determined by q. Nearly all fluctuations will increase the free energy; only those which involves a majority of the electrons so as to change the common q can decrease the free energy. These latter are presumably extremely rare, so that the metastable current carrying state can persist indefinitely.
[BCS] Our picture differs from that of Schafroth, Butler, and Blatt, who suggest that pseudo-molecules of pairs of electrons of opposite spin are formed. They show if the size of the pseudo-molecules is less than the average distance between them, and if other conditions are fulfilled, the system has properties similar to that of a charged Bose-Einstein gas, including a Meissner effect and a critical temperature of condensation. Our pairs are not localized in this sense, and our transition is not analogous to Bose-Einstein condensation.
[BCS] Our theory is based on a rather idealized model in which anisotropic effects are neglected. It contains three parameters, two corresponding to N(εF) and v0, and one dependent on the electron-phonon interaction which determines Tc.
[BCS] We neglect the effects of the momentum dependent cutoff on the expression for the current density; as shown explicitly by P.W. Anderson (private communication) errors introduced are negligible in the weak coupling limit. ... We assume the collective excitations make a negligible contribution to the current form A0 (J. Bardeen, Nuovo Cimento 5, 1766 (1957)).
[BCS] Another problem, not yet solved, is the calculation of the paramagnetic susceptibility of the electrons in a superconductor, such as is required to account for Reif's (Phys. Rev. 106, 208 (1957)) data on the Knight shift in the nuclear paramagnetic resonance of colloidal mercury. Our ground state is for total spin S = 0. It is possible to construct states analogous to those used in spin-wave theory in which each virtual pair has a small net spin, and for which the energy varies continuously with S.
[BCS] We would like to mention particularly discussion with C.P. Slitchter and L.C. Hebel on calculation of matrix elements, with D. Pines on the criterion for superconductivity, and with K.A. Brueckner on the exactness of the solution for the ground state.
1960
[Mahan] Further refinements of the theory had led to the strong coupling theory of Eliashberg which describes well the properties of strong superconductors such as lead. The paired electrons behave, in some respects, as bosons.
[Mahan] The observation of the energy gap in the excitation spectrum by electron tunneling done by Giaever provided a dramatic verification of the BCS theory.
1962
[Mahan] Josephson predicted the coherent tunneling of pairs, which was also quickly observed. These experiments provide a detailed verification of the BCS theory.
[Mahan] Cohen et al introduced the concept of the tunneling Hamiltonian, which became universally adopted for the discussion of tunneling in superconductors.
1963
[Mahan] Balian and Werthamer solved the BCS equations for S=1, showed that the triplet state had smaller binding energy and was therefore less favored. (However, the recent of superfluidity in He3 are based on the premise that the pairing occurs in the triplet state. Thus triplet pairing is possible and may exist in heavy fermion solids such as UPt3 and UBe13. )
1964 - 1965
[Mahan] An extensive comparison between the theory and experiment is provided in two volumes of Superconductivity edited by R.D. Parks (1965) and in the books by Rickayzen and Schrieffer (1964).
1968
[Mahan] The distinction between weak and strong is roughly given by the value of the electron-phonon mass enhancement factor λ, as shown by McMillan.
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