BCS superconductivity in metallic nanograins: Finite-size corrections, low-energy excitations, and robustness of shell effects

Physical Review B - Condensed Matter and Materials Physics

Volumn 83, Issue 1, 2011, Pages

  García García, Antonio M ^a   Urbina, Juan Diego ^b   Yuzbashyan, Emil A ^c   Richter, Klaus ^b   Altshuler, Boris L ^d  

^a UNIVERSITY OF LISBON   (Portugal)

^b UNIVERSITY OF REGENSBURG   (Germany)

^c RUTGERS UNIVERSITY   (United States)

^d Department of Earth and Environmental Sciences   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79551574194     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.83.014510     Document Type: Article

References (62)

1. M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, New York, 1997).
2. R. Balian and C. Bloch, Ann. Phys. APNYA6 0003-4916 10.1016/0003-4916(70) 90497-5 60, 401 (1970).
3. R. Balian and C. Bloch, Ann. Phys. APNYA6 0003-4916 10.1016/0003-4916(71) 90286-7 64, 271 (1971).
4. P. W. Anderson, J. Phys. Chem. Solids JPCSAW 0022-3697 10.1016/0022-3697(59)90036-8 11, 26 (1959).
5. R. Parmenter, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.166.392 166, 392 (1968).
6. J. M. Blatt and C. J. Thompson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.10.332 10, 332 (1963);
7. C. J. Thompson and J. M. Blatt, Phys. Lett. PHLTAM 0031-9163 10.1016/S0375-9601(63)80003-1 5, 6 (1963).
8. B. Mühlschlegel, D. J. Scalapino, and R. Denton, Phys. Rev. B PHLTAM 1098-0121 10.1103/PhysRevB.6.1767 6, 1767 (1972).
9. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.108.1175 108, 1175 (1957).
10. D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.3241 74, 3241 (1995);
11. C. T. Black, D. C. Ralph, and M. Tinkham, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.76.688 76, 688 (1996).
12. Y. Guo, Y.-F. Zhang, X.-Y. Bao, Science SCIEAS 0036-8075 10.1126/science.1105130 306, 1915 (2004).
13. W. H. Li, C. C. Yang, F. C. Tsao, and K. C. Lee, Phys. Rev. B SCIEAS 1098-0121 10.1103/PhysRevB.68.184507 68, 184507 (2003);
14. W. H. Li, C. C. Yang, F. C. Tsao, S. Y. Wu, P. J. Huang, M. K. Chung, and Y. D. Yao, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.72.214516 72, 214516 (2005);
15. S. Bose, P. Raychaudhuri, R. Banerjee, and P. Ayyub, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.74.224502 74, 224502 (2006);
16. S. Reich, G. Leitus, R. Popovitz-Biro, and M. Schechter, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.147001 91, 147001 (2003).
17. A. Mastellone, G. Falci, and R. Fazio, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80.4542 80, 4542 (1998);
18. A. Di Lorenzo, R. Fazio, F. W. J. Hekking, G. Falci, A. Mastellone, and G. Giaquinta, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.550 84, 550 (2000).
19. H. Heiselberg, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.68.053616 68, 053616 (2003).
20. A. A. Shanenko, M. D. Croitoru, M. Zgirski, F. M. Peeters, and K. Arutyunov, Phys. Rev. B PLRAAN 1098-0121 10.1103/PhysRevB.74.052502 74, 052502 (2006);
21. M. D. Croitoru, A. A. Shanenko, and F. M. Peeters, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.76.024511 76, 024511 (2007).
22. V. N. Gladilin, V. M. Fomin, and J. T. Devreese, Solid State Commun. SSCOA4 0038-1098 10.1016/S0038-1098(01)00512-9 121, 519 (2002).
23. R. W. Richardson, Phys. Lett. PHLTAM 0031-9163 10.1016/0031-9163(63) 90259-2 3, 277 (1963);
24. R. W. Richardson and N. Sherman, Nucl. Phys. B NUPHA7 0029-5582 10.1016/0029-5582(64)90687-X 52, 221 (1964).
25. H. Boyaci, Z. Gedik, and I. Kulik, J. Supercond. 14, 133 (2001);
26. V. N. Gladilin, J. Tempere, I. F. Silvera, and J. T. Devreese, Phys. Rev. B NUPHA7 1098-0121 10.1103/PhysRevB.74.104512 74, 104512 (2006);
27. J. Tempere, V. N. Gladilin, I. F. Silvera, and J. T. Devreese, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.72.094506 72, 094506 (2005).
28. V. Z. Kresin and Y. N. Ovchinnikov, Phys. Rev. B NUPHA7 1098-0121 10.1103/PhysRevB.74.024514 74, 024514 (2006);
29. Y. N. Ovchinnikov and V. Z. Kresin, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2005-00158-7 45, 5 (2005).
30. V. Kresin, Y. N. Ovchinikov, and S. A. Wolf, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2006.05.006 431, 231 (2006);
31. B. Cao, C. M. Neal, A. K. Starace, Y. N. Ovchinnikov, V. Z. Kresin, and M. F. Jarrold, J. Supercond. Nov. Magn. PRPLCM 1557-1939 10.1007/s10948-008- 0314-3 21, 163 (2008).
32. H. Olofsson, S. Aberg, and P. Leboeuf, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.037005 100, 037005 (2008).
33. M. Farine, F. W. J. Hekking, P. Schuck, and X. Vinas, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.68.024507 68, 024507 (2003).
34. J. Dukelsky and G. Sierra, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.83.172 83, 172 (1999);
35. J. Dukelsky and G. Sierra, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.61.12302 61, 12302 (2000);
36. J. M. Roman, G. Sierra, and J. Dukelsky, Nucl. Phys. B NUPBBO 0550-3213 10.1016/S0550-3213(02)00317-6 634, 483 (2002);
37. R. A. Smith and V. Ambegaokar, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.77.4962 77, 4962 (1996).
38. K. A. Matveev and A. I. Larkin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.78.3749 78, 3749 (1997).
39. J. M. Gordon and A. M. Goldman, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.35.4909 35, 4909 (1987).
40. M. Ma and P. A. Lee, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.32. 5658 32, 5658 (1985).
41. M. Schechter, Y. Imry, Y. Levinson, and J. von Delft, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.63.214518 63, 214518 (2001).
42. E. A. Yuzbashyan, A. A. Baytin, and B. L. Altshuler, Phys. Rev. B PRLTAO 1098-0121 10.1103/PhysRevB.71.094505 71, 094505 (2005).
43. A. M. Garcia-Garcia, J. D. Urbina, E. A. Yuzbashyan, K. Richter, and B. L. Altshuler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.187001 100, 187001 (2008).
44. M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
45. See, for example, S. Tomsovic, D. Ullmo, and A. Bäcker, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.164101 100, 164101 (2008), and references therein.
46. M. Berry, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/10/12/016 10, 2083 (1977).
47. R. Moro, X. Xu, S. Yin, and W. de Heer, Science SCIEAS 0036-8075 10.1126/science.1083247 300, 1265 (2003).
48. J. Friedel, J. Phys. 10.1051/jp2:1992178 2, 959 (1992).
49. P. Meier, M. Brack, and S. C. Creagh, Z. Phys. D ZDACE2 0178-7683 10.1007/s004600050324 41, 281 (1997).
50. N. Pavloff and C. Schmit, Phys. Rev. B ZDACE2 1098-0121 10.1103/PhysRevB.58.4942 58, 4942 (1998).
51. P. Leboeuf, in Lecture Notes in Physics, edited by, J. M. Arias, and, M. Lozano, (Springer, New York, 2005).
52. R. Balian and B. Duplantier, Ann. Phys. APNYA6 0003-4916 10.1016/0003-4916(77)90334-7 104, 300 (1977).
53. R. Balian and C. Bloch, Ann. Phys. 69, 76 (1971).
54. Roughly speaking, the quantum ergodicity theorem 40 says that for classical systems with ergodic classical dynamics, the eigenfunctions converge to an uniformly distributed density in the semiclassical limit.
55. Y. Colin de Verdiere in Les Houches School Lecture Series LII, edited by, M. J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1989), p. 251.
56. The fact that quantum systems whose classical counterpart is chaotic have strong ergodic properties (in the sense that spatial averages for a single wavefunction are equivalent to energy averages over a set of eigenfunctions) was first conjectured by Berry. 31 The equivalent statement for disordered systems can be found, for example, in I. V. Gornyi and A. D. Mirlin, Phys. Rev. E APNYA6 1539-3755 10.1103/PhysRevE.65.025202 65, 025202 (R) (2002). There is strong evidence, both numerical and experimental, for this conjecture, see, e.g., Ref. 43.
57. E. B. Bogomolny, Physica (Amsterdam) 31D, 169 (1988).
58. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, UK, 1999).
59. J. D. Urbina and K. Richter, Phys. Rev. E APNYA6 1539-3755 10.1103/PhysRevE.70.015201 70, 015201 (R) (2004).
60. In certain situations, pairs of trajectories with arbitrarily small action differences can be constructed that describe important effects beyond the diagonal approximation; see, e.g., M. Sieber and K. Richter, Phys. Scr. 128 (2001).
61. A. M. Ozorio De Almeida, Hamiltonian Systems: Chaos and Quantization (Cambridge University Press, Cambridge, UK, 1988).
62. M. Sieber, J. Phys. A: Math. Gen. JPHAC5 0305-4470 10.1088/0305-4470/32/ 44/307 32, 7679 (1999).