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BCS-to-BEC evolution details

Carlos A. R. Sá de Melo

Sá de Melo replies: I thank D. M. Eagles for his comments. My statement 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 papers 1 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 transition 3 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.REFERENCESSection:ChooseTop of pageREFERENCES << REFERENCES 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) Google Scholar A. J. Leggett, J. Phys. Colloq. 41, C7–19 (1980). https://doi.org/10.1051/jphyscol:1980704 , Google ScholarCrossref2. J. M. Blatt, Theory of Superconductivity, Academic Press, New York (1964). Google Scholar3. L. S. Borkowski, C. A. R. Sá de Melo, http://arxiv.org/abs/cond-mat/9810370v1 , Google Scholar R. D. Duncan, C. A. R. Sá de Melo, Phys. Rev. B 62, 9675 (2000). https://doi.org/10.1103/PhysRevB.62.9675 , Google ScholarCrossref, ISI© 2009 American Institute of Physics.

Physics TodayAug 1, 20090 citations

Historical Notes on Feshbach and Shape Resonances

D. M. Eagles

Daniel Kleppner’s Reference Frame piece implies incorrectly that the crossover between Bardeen-Cooper-Schrieffer and Bose–Einstein condensation theory was first studied by Anthony Leggett in 1980. Actually, my early work on this topic was published 11 years before then. 1 1. D. M. Eagles, Phys. Rev. 186, 456 (1969) https://doi.org/10.1103/PhysRev.186.456. In the mid-1980s, I and colleagues in Australia also found the first example 2 2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1. of a system that lay on the Bose-gas side of the BCS–Bose gas transition, in a ceramic sample of 3% zirconium-doped strontium titanate (SrTiO3), although with a pairing temperature much lower than that predicted for the 1969 model used for this material. In three dimensions there is a threshold in the coupling strength for the existence of Bose-gas superconductivity at very low carrier concentrations, and for strengths slightly above the threshold, the pair-binding energy is proportional to the square of the difference of the coupling from the threshold value. Thus, for example, a calculated coupling error that gives a value 5% above threshold instead of 1% above would make a factor of 25 error in the pair-binding energy.A later attempt 3 3. C. Andrikidis, R. J. Tainsh, D. M. Eagles, Physica B 165, 1517 (1990) https://doi.org/10.1016/S0921-4526(09)80344-X. to reach the Bose-gas regime in macrocrystalline samples of Zr-doped SrTiO3 with similar carrier concentrations to those in the ceramic sample mentioned indicated that some nonuniform state prevented the Bose-gas regime being reached. However, there is still much interesting physics awaiting study in this material.REFERENCESSection:ChooseTop of pageREFERENCES <<1. D. M. Eagles, Phys. Rev. 186, 456 (1969) https://doi.org/10.1103/PhysRev.186.456. Google ScholarCrossref, ISI, CAS2. R. J. Tainsh, C. Andrikidis, Solid State Commun. 60, 517 (1986) https://doi.org/10.1016/0038-1098(86)90729-5 , Google ScholarCrossref, ISI, CAS D. M. Eagles, Solid State Commun. 60, 521 (1986) https://doi.org/10.1016/0038-1098(86)90730-1. , Google ScholarCrossref, ISI, CAS3. C. Andrikidis, R. J. Tainsh, D. M. Eagles, Physica B 165, 1517 (1990) https://doi.org/10.1016/S0921-4526(09)80344-X. Google ScholarCrossref© 2005 American Institute of Physics.

Physics TodayFeb 1, 20050 citations

Aspects of BCS to BEC “Crossover” as illustrated by fulleride superconductivity

Richard H. Squire, Norman Henry March

Abstract The pioneering works by Eagles, Leggett, and Nozieres/Schmitt‐Rink [Eagles, Phys Rev, 1969, 186, 456; Leggett, In: Modern Trends in the Theory of Condensed Matter, Springer‐Verlag: Berlin, 1980; Nozieres and Schmitt‐Rink, Low Temp Phys, 1985, 59, 980] (reviewed and augmented by Randeria [Randeria, Bose‐Einstein Condensation, Cambridge University Press, Cambridge, 1995, p 255]) emphasize that in the limits of the models studied both at T = 0 and T ≠ 0, the “cross‐over” from a BCS‐type to a BEC‐type superconductor is continuous. The BCS and BEC “end points” seem to be well‐established. However, in the intermediate region—home to fulleride and high temperature superconductors—considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more “singular‐type phenomena” such as quantum critical points, “stripe” formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with “cold” atom fermion‐boson crossover theory [Chen, Stajic, Levin, 2005, e‐print cond‐mat/0508603], we can establish that the resonance previously discussed [Squire and March, Int J Quant Chem, 2006, 106, 3343] is a result of the crossing of the fermion band by the boson band. While the ground state appears to remain continuous, the paired energy gap becomes transformed. We discuss features of the resonance and the experimentally observed pre‐formed “BEC Cooper pair” formation, essential to the boson‐fermion resonance theory. In addition some of the various singular phenomena discussed above can be put more into perspective. Finally, in both limits the relation of characteristic lengths to the inverse Fermi momentum is strongly emphasized, as is the role of the chemical potential near the pseudogap regime. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

International Journal of Quantum ChemistryJan 1, 20075 citations

History of the BCS to Bose-Einstein Transition

D. M. Eagles

Attention is drawn to the fact that some publications state or imply incorrectly that the BCS-Bose gas transition was first studied by Leggett in 1980, whereas I published a paper on this subject in 1969. Attention is also drawn to evidence from colleagues and myself in the 1980's that a ceramic sample of 3% Zr-doped SrTiO3 was on the Bose-gas side of the transition. This was, I think, the first example of a sample of any material which reached the Bose-gas regime, and also gave a world record by a factor of over one hundred for the lowest carrier concentration (~10^15cm^{-3}) at which superconductivity (or strictly the beginning of a transition) has been observed to occur.

arXiv:cond-mat/0501598Jan 25, 20050 citations

When superconductivity crosses over: From BCS to BEC

Qijin Chen, Zhiqiang Wang, Rufus Boyack +2 more

The theory of unconventional superconductors continues to provide profound puzzles. The crossover between the weakly coupled Bardeen-Cooper-Schrieffer (BCS) state and the strong-pairing Bose-Einstein condensate (BEC) provides a useful perspective on how to address these questions. This paper describes a self-consistent framework for thinking about the crossover regime in between these two limits. The review discusses to what extent this BCS-BEC theory applies to a range of classes of superconducting materials including the cuprates, iron pnictides, twisted bilayer graphene, and interfacial superconductivity among others.

arXiv:2208.01774Reviews of Modern PhysicsMay 23, 202464 citations

Superconductor superfluid density from the Bardeen–Cooper–Schrieffer/Bose crossover theory

Israel Chávez, M. Grether, MANUEL DE LLANO

Abstract The superfluid density $$n_{s}(T)$$ n s ( T ) of a superconductor is calculated based on the generalized Bose–Einstein condensation (GBEC) theory that addresses a fully-interacting ternary boson-fermion gas mixture of free electrons as fermions, plus two-electron Cooper pairs (2eCPs) and also, explicitly, two-hole Cooper pairs (2hCPs), both as bosons. Here we consider two special cases (i) 100%–0% (i.e., with no condensed 2hCPs) and (ii) 0%–100% (i.e., with no condensed 2eCPs). Subsumed in GBEC are the Bardeen–Cooper–Schrieffer (BCS) and Bose–Einstein condensation (BEC) theories along with the BCS-BEC crossover theory extended with 2hCPs. We find that in the weak-coupling regime $$n_{s}(0)$$ n s ( 0 ) agrees with data from the Uemura et al. (2004) graph for several elemental SCs by taking in 3D with a quadratic energy-dispersion relation while in 2D with a linear relation are much too far below the data. In the strong-coupling regime the linear behavior of critical temperature $$T_{c}$$ T c vs $$n_{s}(0)$$ n s ( 0 ) obtained here is just as Božović et al. (2016) found. However, in 2D with a linear relation accounting for 0%–100%, $$n_{s}(T)/n_{s}(0)$$ n s ( T ) / n s ( 0 ) compares well with some high- $$T_{c}$$ T c -cuprate SC data between the two coupling regimes.

SN Applied SciencesJun 13, 20220 citations

BCS-to-BEC evolution details

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BCS-to-BEC evolution details

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