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) YuzbashyanAltshuler Urbina Richter Sangita Bose

2. L 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? Main goals

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

4. Relevant Scales  Mean level spacing Δ 0 Superconducting gap  F Fermi Energy L typical length l coherence length ξ Superconducting coherence length Conditions BCS  / Δ 0 << 1 Semiclassical 1/k F L << 1 Quantum coherence l >> L ξ >> L For Al the optimal region is L ~ 10nm

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

6. A little history 1959, Anderson: superconductor if  / Δ 0 > 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 T c 2008, Olofsson, estimation of fluctuations, no matrix elements!

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

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 f(L,  -  ’,  F ) is a simple function For l>>L ergodic theorems assures universality

10. Semiclassical (1/k F L >> 1) expression of the matrix elements valid for l >> L!! Technically is much more difficult because it involves the evaluation of all closed orbits not only periodic ω =  -  ’ A few months later 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 Universal function Importance of boundary conditions

12. 3d chaotic AL grain k F = 17.5 nm -1  = 7279/N mV  0 = 0.24mV L = 6nm, Dirichlet,  /Δ 0 =0.67 L= 6nm, Neumann,  /Δ 0, =0.67 L = 8nm, Dirichlet,  /Δ 0 =0.32 L = 10nm, Dirichlet,  /Δ 0,= 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 & analyticalCube & 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 Phonons Deviations from mean field Decoherence Geometrical deviations No Yes

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

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

17. submitted to Nature