Emergence of h/e -period oscillations in the critical temperature of small superconducting rings threaded by magnetic flux

Physical Review B - Condensed Matter and Materials Physics

Volumn 77, Issue 22, 2008, Pages

  Wei, Tzu Chieh ^a   Goldbart, Paul M ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 46149087268     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.77.224512     Document Type: Article

References (32)

1. W. A. Little and R. D. Parks, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.9.9 9, 9 (1962).
2. M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1996), pp. 128-130.
3. See, e.g., J. R. Schrieffer, Theory of Superconductivity (Perseus Books, Reading, MA, 1983), Chap., pp. 240-244.
4. Y. Liu, Yu. Zadorozhny, M. M. Rosario, B. Y. Rock, P. T. Carrigan, and H. Wang, Science SCIEAS 0036-8075 10.1126/science.1066144 294, 2332 (2001)
5. H. Wang, M. M. Rosario, N. A. Kurz, B. Y. Rock, M. Tian, P. T. Carrigan, and Y. Liu, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.197003 95, 197003 (2005).
6. P. G. de Gennes, C. R. Acad. Sci. (Paris) Ser. II 0031-9007 292, 279 (1981).
7. R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.54.2696 54, 2696 (1985).
8. H. Bluhm, N. C. Koshnick, M. E. Huber, and K. A. Moler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.97.237002 97, 237002 (2006).
9. N. C. Koshnick, H. Bluhm, M. E. Huber, and K. A. Moler, Science SCIEAS 0036-8075 10.1126/science.1148758 318, 1440 (2007).
10. K. Czajka, M. M. Maśka, M. Mierzejewski, and Z. Śledź, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.72.035320 72, 035320 (2005).
11. F. Loder, A. P. Kampf, T. Kopp, J. Mannhart, C. W. Schneider, and Yu. S. Barash, Nat. Phys. ZZZZZZ 1745-2473 4, 112 (2008).
12. V. Juricic, I. F. Herbut, and Z. Tesanovic, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.187006 100, 187006 (2008).
13. Yu. S. Barash, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100. 177003 100, 177003 (2008).
14. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.106.162 106, 162 (1957).
15. L. P. Gor'kov, Sov. Phys. JETP 9, 1364 (1959) 0031-9007
16. L. P. Gor'kov, Sov. Phys. JETP 10, 998 (1960). 0031-9007
17. In c.g.s. unit, the single-particle flux quantum is hc/e. We shall set c=1 and =1. Even though in this unit the flux quantum becomes 2π/e, we shall refer to it by h/e instead.
18. See, e.g., A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971).
19. Although we take the ring to be of infinitesimally thin width for small but finite width compared to the radius and the coherence length, this set of equations can be obtained by averaging over the radial direction and ignoring the fluctuations (see Appendix). In our discussions we require that the mean average level spacing vF/d in the transverse direction is much greater than the superconducting gap Δ0, so that only the lowest level in the transverse direction is occupied. This leads to the condition that d ξ0, where ξ0 is the Cooper-pair size ξ0 = vF /π Δ0.
20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 259 and 556.
21. G. Sarma, J. Phys. Chem. Solids JPCSAW 0022-3697 10.1016/0022-3697(63) 90007-6 24, 1029 (1963).
22. K. Maki and T. Tsuneto, Prog. Theor. Phys. PTPKAV 0033-068X 10.1143/PTP.31.945 31, 945 (1964).
23. Note that here we assume the equilibrium normal state does not possess a persistent current. Ignoring the finiteness correction, the transition to normal follows the curve ABD in Fig. 3. If the normal state does possess a persistent current, the transition from superconducting to normal will follow the curve ABE instead. The inclusion of the finiteness correction only modifies the detail locations of these curves.
24. L. P. Gor'kov and A. A. Abrikosov, Sov. Phys. JETP 8, 1090 (1959). 0038-5646
25. A. V. Lopatin, N. Shah, and V. M. Vinokur, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.037003 94, 037003 (2005).
26. This can be seen from the fact that the function f (t) =lnt-ψ (1 2) + (1 2 + a t), is greater than zero for t≥0 when a>Γ/4π, whereas it crosses from negative values to positive at a certain value of t>0 when a<Γ/4π.
27. D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.3241 74, 3241 (1995).
28. A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975), Chap., Sec. 39.2.
29. R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1987), Chap..
30. J.-X. Zhu and Z. D. Wang, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.50.7207 50, 7207 (1994).
31. J.-X. Zhu, arXiv:0806.1084 (unpublished).
32. V. Vakaryuk, arXiv:0805.2626 (unpublished).