Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

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Failed Theories of Superconductivity 51

Apparently unaware of the fact that Bloch’s ﬁrst theorem is valid even in

the presence of the ionic potential, they claimed, without giving further

details, to have found asymmetric occupations with a net momentum and

therefore a spontaneous current. Born and Cheng were correct to point out

that interactions aﬀect the distribution in momentum space, and become

important for systems with large portions of the Fermi surface close to the

Brillouin zone boundary. Yet, a correct analysis of their own formalism

should have led them to the conclusion that asymmetric occupations do not

correspond to the lowest energy state.

Fritz London strongly disagreed with Heisenberg and Born. Being chal-

lenged by those ideas, he formulated his own microscopic theory of supercon-

ductivity.

In a 1948 paper, he ﬁrst demonstrated that Heisenberg’s theory

cannot yield superconductivity as a matter of principle. He then pointed out

that Heisenberg missed the leading term in his treatment of the Coulomb

interaction, which, in London’s view, is the exchange interaction, discovered

by Heisenberg himself many years before.

London then proposed that

this exchange interaction, when it favors ferromagnetism, is responsible for

superconductivity as it can lead to an “attraction in momentum space.”In

materials known then, superconductivity is not caused by this eﬀect, but

rather by vibrations of the crystal lattice, i.e. sound excitations. On the

other hand, the superﬂuid state of

He,

–

and superconductivity in a

number of strongly correlated electron systems have been discussed in terms

of an exchange of collective degrees of freedoms, such as ferromagnetic, an-

tiferromagnetic or current ﬂuctuations,

–

that bear some resemblance to

what London had in mind. Of course, our understanding of these materials

is based on the BCS theory. London’s ideas, while interesting, remained

vague. The most remarkable aspect of this 1948 paper by Fritz London was

however that it emphasized clearly that a superconductor is a macroscopic

object in a coherent quantum state: “The superconductor as a pure quan-

tum mechanism of macroscopic scale.” This view was also one of the key

elements of the elegant phenomenological theory by Ginzburg and Landau

that was proposed soon thereafter.

If we talk about the creators of unsuccessful theories of superconductivity

we should not ignore that Bardeen himself belonged, at least temporarily, to

this honorable club. One attempt to solve the problem, brieﬂy mentioned

in an abstract in Physical Review, goes back to 1941 and is based on the

idea that electrons couple to small periodic distortions of the lattice.

his scenario, the electron-lattice interaction would cause small lattice dis-

tortions with large unit cells containing approximately 10

atoms, leading

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52 J. Schmalian

to 10

bands in a tiny Brillouin zone. Bardeen estimated that approxi-

mately 10

of those bands were located in the vicinity of ±k

of the Fermi

energy. Since some bands would have a very small mass, he expected them

to contribute to a large diamagnetic response. He further estimated that

a fraction of approximately 10

−6

of the electrons would participate in this

diamagnetic shielding. As the reasoning is based upon lattice distortions,

Bardeen concluded that the electron-lattice coupling of good superconduc-

tors should be large, a conclusion that also followed from the BCS theory,

yet for very diﬀerent reasons.

Another key experiment that elucidated the origin of superconductiv-

ity was the measurement of the isotope eﬀect at Rutgers University in the

group of Bernard Serin

and by Emanuel Maxwell

at the National Bureau

of Standards, Washington, D.C. in 1950. These experiments found a change

in the transition temperature upon changing the ion mass via isotope sub-

stitution, and demonstrated that vibrations of the crystalline lattice were

closely tied to the emergence of superconductivity. In fact, Herbert Fr¨ohlich

predicted the isotope eﬀect theoretically.

In 1950 Bardeen and, indepen-

dently, Fr¨ohlich worked on the problem and came to the conclusion that

vibrations of the crystalline lattice lead to a net attraction between electrons

and are a likely cause of superconductivity.

48,49

This attraction is present

even if one includes the electron–electron Coulomb interaction as shown by

Bardeen and Pines.

For an historical account of this period of time, see

Ref. 51. Starting in 1950 one should not refer to Bardeen’s or Fr¨ohlich’s ideas

as failed, rather as incomplete. The link between the established attraction

between electrons due to sound excitations and superconductivity was still

unresolved. For example, in 1953 Fr¨ohlich developed a beautiful theory for

one-dimensional system with electron lattice coupling

and obtained in his

calculation a gap in the electron spectrum, as was in fact seen in a number of

experiments. The theory is a perfectly adequate and very interesting start-

ing point for the description of sliding charge density waves. We know now

that the one-dimensionality makes the model somewhat special and that the

gap is incomplete in three dimensions. Most importantly, the theory did not

explain superconductivity and did not account for the Meissner eﬀect. Still,

there is strong evidence that sliding charge density waves, as proposed by

Fr¨ohlich to explain superconductivity, have in fact been observed in quasi-

one-dimensional conductors; see, for example, Refs. 53 and 54.

Richard Feynman is another scientist who battled with the problem

of superconductivity. When he was asked in 1966 about his activities in

the ’50s he answered: “There is a big vacuum at that time, which is my

September 13, 2010 15:44 World Scientiﬁc Review Volume - 9.75in x 6.5in ch4

Failed Theories of Superconductivity 53

attempt to solve the superconductivity problem — which I failed to do.”

55,56

Feynman’s views, shortly before the BCS theory appeared, are summarized

in the proceedings of a meeting that took place the fall of 1956.

He stated

that the Fr¨ohlich–Bardeen model was, in his view, the correct approach;

it only needed to be solved adequately. Even by performing a very accu-

rate analysis using the method of perturbation theory, i.e. by including the

vibrations of the crystal without fundamentally changing the properties of

the electrons, Feynman was still unable to obtain a superconducting state.

From this calculation, he concluded that the solution must be beyond the

scope of perturbation theory, which turned out to be correct. The formula-

tion of such a theory was accomplished by John Bardeen, Leon N. Cooper

and J. R. Schrieﬀer.

After one goes over these unsuccessful attempts to understand supercon-

ductivity, there seem to be a few natural conclusions that one can draw:

ﬁrst, the development of the BCS theory is among the most outstanding

intellectual achievements in theoretical physics. Second, theories emerge

quite diﬀerently from how they are taught in the classroom or presented in

textbooks, and, ﬁnally, an inapplicable theory, even a wrong theory, can be

interesting, inspiring and may turn out to be useful in another setting.

Acknowledgments

The author acknowledges Paul C. Canﬁeld, Rafael M. Fernandes, Jani Geyer

and Vladimir G. Kogan for a critical reading of the manuscript.

References

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6. L. Kelvin, Philos. Mag. 3, 257 (1902).

7. P. H. E. Mejier, American J. Phys. 62, 1105 (1994).

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9. E. Schr¨odinger, Annalen der Physik 79, 361 (1926); ibid. 79, 489 (1926); ibid.

81, 109 (1926).

10. A. Einstein, Theoretische Bemerkungen zur Supraleitung der Metalle, in

Het Natuurkundig Laboratorium der Rijksuniversiteit te Leiden,inde Jaren,

Vol. 429, 1904–1922 (Eduard Ijdo, Leiden) (1922).

11. W. P. Su, J. R. Schrieﬀer and A. J. Heeger, Phys. Rev. B 22, 2099 (1980).

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12. A. J. Heeger, S. Kivelson, J. R. Schrieﬀer and W.-P. Su, Rev. Mod. Phys. 60,

781 (1988).

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14. M. Cardona, Albert Einstein as the father of solid state physics, in 100 anys

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16. J. J. Thompson, Philos. Mag. 30, 192 (1915).

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(1921).

18. W. Heisenberg, Zeitschrift f¨ur Physik 49, 619 (1928).

19. F. Bloch, Zeitschrift f¨ur Physik 52, 555 (1928).

20. L. Hoddeson, G. Baym and M. Eckert, Rev. Mod. Phys. 59, 287 (1987).

21. L. D. Landau, Physikalische Zeitschrift der Sowjetunion 4, 43 (1933).

22. R. Kronig, Zeitschrift f¨ur Physik 78, 744 (1932).

23. R. Kronig, Zeitschrift f¨ur Physik 80, 203 (1932).

24. P. Drude, Annalen der Physik 306, 566 (1900); ibid. 308, 369 (1900).

25. W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933).

26. L. D. Landau, Physikalische Zeitschrift der Sowjetunion 4, 675 (1933).

27. L. D. Landau, Sov. Phys. JETP 7, 19 (1937); ibid. 7, 627 (1937).

28. D. Bohm, Phys. Rev. 75, 502 (1949).

29. S. Goudsmit and G. E. Uhlenbeck, Physica 6, 273 (1926); G. E. Uhlenbeck and

S. Goudsmit, Naturwissenschaften 47, 953 (1925).

30. F. A. Lindemann, Philos. Mag. 29, 127 (1915).

31. J. J. Thompson, Philos. Mag. 44, 657 (1922).

32. E. P. Wigner, Phys. Rev. 46, 1002 (1934).

33. L. Brillouin, Comptes Rendus Hebdomadaires des Seances de L Academie des

Sciences 196, 1088 (1933); J. Phys. Radium 4, 333 (1933); ibid. 4, 677 (1933).

34. W. Heisenberg, Zeitschrift f. Naturforschung 2a, 185 (1947); ibid. 3a, 65 (1948).

35. The Electron Theory of Superconductivity, in Two Lectures by W. Heisenberg

(Cambridge Univ. Press, 1949).

36. M. Born and K. C. Chen, Nature 161, 968 (1948); ibid. 161, 1017 (1948).

37. F. London, Phys. Rev. 74, 562 (1948).

38. A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).

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40. D. Vollhardt and P. W¨olﬂe, The Superﬂuid Phases of Helium 3 (Taylor &

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42. T. Moriya and K. Ueda, Rep. Progr. Phys. 66, 1299 (2003).

43. A. V. Chubukov, D. Pines and J. Schmalian, in Physics of Superconductors 1,

eds. K. H. Bennemann and J. B. Ketterson (Springer, 2003).

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46. C.A.Reynolds,B.Serin,W.H.WrightandL.B.Nesbitt,Phys. Rev. 78, 487

(1950).

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Failed Theories of Superconductivity 55

47. E. Maxwell, Phys. Rev. 78, 477 (1950); ibid. 79, 173 (1950).

48. H. Fr¨ohlich, Proc. Phys. Soc. A 63, 778 (1950); Phys. Rev. 79, 845 (1950);

Proc. Phys. Soc. A 64, 129 (1951).

49. J. Bardeen, Phys. Rev. 79, 167 (1950); ibid. 80, 567 (1950); ibid. 81, 829 (1951).

50. J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955).

51. L. Hoddeson, J. Statist. Phys. 103, 625 (2001).

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Condensed Matter 13, L389 (2001).

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and S. Uchida, Science 297, 584 (2002).

55. Interview of Richard F. Feynman, by Charles Weiner, June 28, 1966, p. 168,

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September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

NUCLEAR MAGNETIC RESONANCE AND

THE BCS THEORY

Charles P. Slichter

Department of Physics, University of Illinois,

1110 West Green Street, Urbana, IL 61801, USA

The author describes the inspiration for the experiment by Hebel and

Slichter to measure the nuclear spin–lattice relaxation time in super-

conductors, the design considerations for the experiment, the surprising

experimental results, their theoretical treatment using the Bardeen–

Cooper–Schrieﬀer theory, and how comparing the nuclear relaxation results

with those for ultrasound absorption conﬁrmed the central idea of the BCS

theory, the BCS pair wave function.

The year 2007 was the 50th anniversary of the famous Bardeen–Cooper–

Schrieﬀer (BCS) theory of superconductivity. To celebrate it, we held

a four-day symposium in Urbana. I was one of the speakers. I talked

about the experiment my remarkable student Chuck Hebel and I did,

inspired by a talk given by John Bardeen. It gave me an opportunity

to set our experiment in an historical context, describe a bit about how

John’s thinking was evolving, and present some of the thoughts and

motivation behind what we did that are not in our published papers. That

talk and all the others of the Symposium are available on the web at:

http://www.conferences.uiuc.edu/bcs50/video.html.

In 1951, John Bardeen joined the faculty of the University of Illinois

with an appointment as Professor, half in the Department of Electrical

Engineering (EE), half in the Department of Physics. As an undergrad-

uate at the University of Wisconsin, he had majored in EE, but had built

a strong background in other areas with teachers such as Warren Weaver in

mathematics and J. H. Van Vleck in physics. His PhD at Princeton had been

in Mathematics, but his thesis advisor was the physicist Eugene Wigner. He

was in fact Wigner’s second PhD student, and the second one to work on

a problem in solid-state physics. Fred Seitz, Wigner’s ﬁrst PhD student

September 17, 2010 9:51 World Scientiﬁc Review Volume - 9.75in x 6.5in ch5

58 C. P. Slichter

Fig. 1. John Bardeen in his Physics Department oﬃce in Urbana, early 1950s.

Fig. 2. Fred Seitz, early 1950s.

had been one of John’s ﬁrst friends at Princeton. Through this friendship,

Illinois had learned that John was unhappy at Bell Laboratories and was

able to persuade John to come to Illinois with this joint appointment.

John established an experimental program in semiconductors in EE, with

post docs supervising the laboratory activity. In physics, he set up a pro-

gram in theoretical physics, focusing on the subject of his long time interest,

superconductivity.

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Nuclear Magnetic Resonance and the BCS Theory 59

Fig. 3. A happy J. H. Van Vleck receiving early morning congratulations on

receiving the 1977 Nobel Prize in Physics. In his typical self-deprecating way, he

joked with friends saying, “I thought I was probably ruled out by the Statute of

Limitations.”

I had come to Illinois right out of graduate school in September 1949.

My undergraduate advisor was J. H. Van Vleck, who had moved from the

University of Wisconsin to Harvard in the early 1930s. He had recommended

to me to do graduate work at Harvard and then later suggested I ask Purcell

to be my thesis advisor to study paramagnetic salts, a long time interest of

Van Vleck, by electron spin resonance (ESR). I was Purcell’s third student

after Nico Bloembergen and George Pake and the only one doing electron

spin resonance. I built an x-band rig to measure crystal ﬁeld splittings of

paramagnetic salts, and a second one with high power to measure their spin

lattice relaxation times.

Erwin Hahn had just completed his PhD in Urbana that June do-

ing pulsed nuclear magnetic resonance (NMR), and over the summer had

discovered spin echoes. We had a wonderful year together before he left for

Stanford as an NRC Postdoctoral Fellow.

By the time John arrived, my students and I had been doing experi-

mental NMR studies of metals (Dick Norberg, Don Holcomb),

ESR studies

of conduction electrons (Tom Carver, Bob Schumacher)

and double reso-

nance studies that conﬁrmed Overhauser’s theory

of dynamic polarization

of nuclei in metals (Tom Carver).

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60 C. P. Slichter

Fig. 4. Photo of Erwin Hahn in his naval uniform, submitted as part of his appli-

cation to graduate school in physics at the University of Illinois. During the war,

Erwin was a radar technician.

The year 1950 had been a big year for superconductivity with the dis-

covery of the isotope eﬀect, the publication of the Ginsberg–Landau theory,

and Pippard’s proposal of his non-local version of the London equations.

In 1954, evidence was found that the heat capacity of superconductors de-

pended exponentially on inverse temperature.

Bardeen

proposed that a

simple model of a superconductor, the conventional free electron theory of

metals with a gap inserted in the density of states in energy, would lead to

the Meissner eﬀect and a non-local version of the London equations.

In 1954 he gave a talk at Illinois in which he tried to show how an electron-

lattice interaction would lead to an energy gap (this was not the BCS the-

ory). I was in the audience. I did not really understand most of what John

was saying, but as a result of all the work my students had been doing, I

immediately realized that a gap in the density of states at the Fermi en-

ergy would have a big eﬀect on the nuclear spin-lattice relaxation time, T

Nuclear relaxation arises by a process in which the nuclear magnetic mo-

ment scatters a conduction electron to absorb or emit the energy to ﬂip the

nuclear spin. The small size of the nuclear magnetic moment means that

only a small amount of energy is exchanged in such a scattering event. The