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Finite size effects in BCS: theory and experiments

Finite size effects in BCS: theory and experiments. Antonio M. Garc í a-Garc í a ag3@princeton.edu Princeton and IST(Lisbon). Phys. Rev. Lett. 100, 187001 (2008) (theory), submitted to Nature (experiments). Urbina. Yuzbashyan. Altshuler. Sangita Bose. Richter. Main goals.

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Finite size effects in BCS: theory and experiments

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  1. Finite size effects in BCS: theory and experiments Antonio M. García-García ag3@princeton.edu Princeton and IST(Lisbon) Phys. Rev. Lett. 100, 187001 (2008) (theory), submitted to Nature (experiments) Urbina Yuzbashyan Altshuler Sangita Bose Richter

  2. Main goals 1. How do the properties of a clean BCS superconductor depend on its size and shape? 2. To what extent are these results applicable to realistic grains? L

  3. How to tackle the problem λ Semiclassical: To express quantum observables in terms of classical quantities. Only 1/kF L <<1, Berry, Gutzwiller, Balian Gutzwiller trace formula Can I combine this? Is it already done?

  4. Relevant Scales L typical length Δ0 Superconducting gap  Mean level spacing l coherence length ξSuperconducting coherence length F Fermi Energy Conditions BCS / Δ0 << 1 Semiclassical1/kFL << 1 Quantum coherence l >> L ξ >> L For Al the optimal region is L ~ 10nm

  5. Maybe it is possible Go ahead! This has not been done before It is possible but it is relevant? Corrections to BCS smaller or larger? If so, in what range of parameters? Let’s think about this

  6. A little history 1959, Anderson: superconductor if / Δ0 > 1? 1962, 1963, Parmenter, Blatt Thompson. BCS in a cubic grain 1972, Muhlschlegel, thermodynamic properties 1995, Tinkham experiments with Al grains ~ 5nm 2003, Heiselberg, pairing in harmonic potentials 2006, Shanenko, Croitoru, BCS in a wire 2006 Devreese, Richardson equation in a box 2006, Kresin, Boyaci, Ovchinnikov, Spherical grain, high Tc 2008, Olofsson, estimation of fluctuations, no matrix elements!

  7. Hitting a bump λ/V ? For the cube yes but for a chaotic grain I am not sure In,n should admit a semiclassical expansion but how to proceed? I ~1/V? Fine but the matrix elements?

  8. Yes, with help, we can From desperation to hope ?

  9. Regensburg, we have got a problem!!! Do not worry. It is not an easy job but you are in good hands Nice closed results that do not depend on the chaotic cavity For l>>L ergodic theorems assures universality f(L,- ’, F) is a simple function

  10. A few months later Semiclassical (1/kFL >> 1) expression of the matrix elements valid for l >> L!! ω = -’ Technically is much more difficult because it involves the evaluation of all closed orbits not only periodic This result is relevant in virtually any mean field approach

  11. 3d chaotic The sum over g(0) is cut-off by the coherence length ξ Importance of boundary conditions Universal function

  12. 3d chaotic AL grain kF = 17.5 nm-1  = 7279/N mV 0 = 0.24mV L = 6nm, Dirichlet, /Δ0=0.67 L= 6nm, Neumann, /Δ0,=0.67 L = 8nm, Dirichlet, /Δ0=0.32 L = 10nm, Dirichlet, /Δ0,= 0.08 In this range of parameters the leading correction to the gap comes from of the matrix elements not the spectral density

  13. 3d integrable V = n/181 nm-3 Numerical & analytical Cube & parallelepiped No role of matrix elements Similar results were known in the literature from the 60’s

  14. Is this real? Real (small) Grains Coulomb interactions No Phonons No Deviations from mean field Yes Decoherence Yes Geometrical deviations Yes

  15. Is this really real? Sorry but in Pb only small fluctuations Are you 300% sure? arXiv:0904.0354v1

  16. However in Sn is very different !!!!!!!!!!!!!!!!!!!!!!!!!!!!! Pb and Sn are very different because their coherence lengths are very different.

  17. submitted to Nature

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