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BCS-to-BEC evolution details

Again an article in PHYSICS TODAY (by Carlos Sá de Melo, October 2008, page 45) has incorrectly implied that Anthony Leggett was the first person to study the crossover from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation. On page 47 of the article, it states that “a clear picture of the BCS-to-BEC evolution at zero temperature didn’t emerge until 1980, when Anthony Leggett realized that the physics could be captured by a simple description in real space of paired fermions with opposite spins.” Although the model I considered in my 1969 paper¹ is slightly different from Leggett’s, figure 4 in my paper clearly shows regions where pairing without superconductivity occurs and where superconductivity is limited by the Bose- condensation temperature of pairs, and on page 458 I discuss a limit at which the diameter of pairs is small compared with the distance between them.

I also disagree with a statement in the box on page 47 of Sá de Melo’s article that “the evolution from a Bardeen-Cooper-Schrieffer superfluid to a Bose–Einstein condensation superfluid cannot be studied in . . . superconductors.” At least in ceramic samples of SrTiO₃ with 3% of the titanium replaced by zirconium, the transition has been studied by varying the carrier concentration via differing heat treatments to produce different concentrations of oxygen vacancies.² It is possible that such a transition may be found in other superconducting semiconductors when people start to search for suitable materials. However, in three dimensions the pairing strength has to be above some threshold value to obtain the possibility of reaching the Bose-gas regime. Also, many authors think that the BEC regime occurs in underdoped cuprates,^3,4 while the consensus is that overdoped samples are BCS-like.

References

1. 1. D. M. Eagles, Phys. Rev. 186, 456 (1969) [SPIN].
2. 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) ; D. M. Eagles, Solid State Commun. 60, 521 (1986) [INSPEC]; D. M. Eagles, R. J. Tainsh, C. Andrikidis, Physica C 157, 48 (1989) [INSPEC].
3. 3. Q. Chen, J. Stajic, S. Tan, K. Levin, Phys. Rep. 412, 1 (2005) [INSPEC].
4. 4. A. S. Alexandrov, J. Supercond. Nov. Magn. 20, 481 (2007) .

Sá de Melo replies: I thank D. M. Eagles for his comments. My state-ment concerning the evolution from Bardeen-Cooper-Schrieffer to Bose–Einstein condensation superfluids was about clarity and not who was the first to propose the idea. Although I appreciate Eagles’s work, I still think that Anthony Leggett’s papers¹ are the clearest presentation on the topic up to 1980.

In his very interesting book written in 1964, John Blatt describes the BEC theory of superconductivity and its relation to the BCS theory.² As he recounts, the possibility of pairing without superconductivity and Bose condensation of electron pairs at a lower temperature was suggested as early as 1946 by Richard Ogg Jr. In 1954 and subsequent years, Max Schafroth developed a firm theoretical framework for such pairing, but it was not supported by experimental evidence: No preformed pairs were found, and the BEC temperature was much too high for known superconductors.

Blatt’s book thoroughly documents a clear connection between a variational, fixed-number ground state comprising antisymmetrized products of pair wavefunctions and the BCS equations that Eagles used in his 1969 paper. Eagles noticed that the BCS and number equations at zero temperature could be used to describe the evolution from BCS to BEC superconductivity in low-carrier-density thin films or bulk materials for fixed (and sufficiently strong) interaction and changing carrier density. However, Blatt’s main interest was in metals with high carrier densities.

If one reads the full statement in the box entitled “Feshbach resonances” on page 47 of my article, it is clear that I am talking about changing interactions for fixed density and not changing density for fixed interactions. I do not think it is currently possible in any superconductor to control the interactions between fermions. Thus my statement is accurate.

In many instances the change in carrier concentration in thin films or three-dimensional (bulk) systems is made via chemical doping, which may introduce disorder or distortions in the crystal structure and thus change the system in more than the desired way. I have serious doubts that ceramic samples of zirconium-doped strontium titanate described in Eagles’s letter indeed show the BCS-to-BEC evolution; the references provided do not offer clear evidence of such a phenomenon.

Since cuprate superconductors are known to be quasi-2D and have d-wave pairing symmetry, the evolution from BCS to BEC superconductivity for sufficiently strong fixed interactions and changing carrier density would be accompanied by a topological quantum phase transition³ and would not be a crossover as in the s-wave case. However, as much as I might want such a transition to be found for the cuprates, current experimental evidence shows that their normal state and superconducting properties are more complicated than simple BCS-to-BEC theories would predict.

The beauty of the observation of the BCS-to-BEC evolution in ultracold atoms is that many of the materials difficulties encountered in standard condensed matter can be avoided and the tuning of the interaction for fixed low density can be controlled. However, one should keep an open mind that a similar realization may appear elsewhere. For instance, if carrier density could be controlled and changed continuously via electrostatic doping in good-quality thin films or at the surface of bulk low-carrier-density superconductors like some titanates or cuprates, then there is a good chance of observing the BCS-to-BEC evolution as a function of carrier density as envisioned by Eagles in 1969.

References

1. 1. A. J. Leggett, in Modern Trends in the Theory of Condensed Matter: Proceedings of the XVI Karpacz Winter School of Theoretical Physics, February 19–March 3, 1979, Karpacz, Poland, A. Pekalski, J. Przystawa, eds., Springer, New York (1980); A. J. Leggett, J. Phys. Colloq. 41, C7-19 (1980).
2. 2. J.M. Blatt, Theory of Superconductivity, Academic Press, New York (1964).
3. 3. L.S. Borkowski, C.A.R. Sá de Melo, [LINK]; R.D.Duncan, C.A.R. Sá de Melo, Phys. Rev. B 62, 9675 (2000) [INSPEC].