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A theory of finite size effects in BCS superconductors: The making of a paper

A theory of finite size effects in BCS superconductors: The making of a paper. Antonio M. Garc í a-Garc í a ag3@princeton.edu http://phy-ag3.princeton.edu Princeton and ICTP. Phys. Rev. Lett. 100, 187001 (2008), AGG, Urbina, Yuzbashyan, Richter, Altshuler. Urbina. Yuzbashyan. Altshuler.

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A theory of finite size effects in BCS superconductors: The making of a paper

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  1. A theory of finite size effects in BCS superconductors: The making of a paper Antonio M. García-García ag3@princeton.edu http://phy-ag3.princeton.edu Princeton and ICTP Phys. Rev. Lett. 100, 187001 (2008), AGG, Urbina, Yuzbashyan, Richter, Altshuler. Urbina Yuzbashyan Altshuler 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. Richardson equations, Anderson representation …what? Quantum chaos, trace formula…what? Princeton 2005: A false start Superconductivity?, Umm, semiclassical, fine Superconductivity, spin, semiclassical Talk to Emil

  4. Spring 2006: A glimmer of hope Semiclassical: To express quantum observables in terms of classical quantities. Only 1/kF L <<1, Berry, Gutzwiller, Balian, Bloch Gutzwiller trace formula Can I combine this? Is it already done?

  5. Semiclassical (1/kFL >> 1) expression of the spectral density,Gutzwiller, Berry Non oscillatory terms Oscillatory terms in terms of classical quantities only

  6. 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

  7. 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, fluctuations in Chaotic grains, no matrix elements!

  8. 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

  9. Fall 06: Hitting a bump 3d cubic Al grain  In,n should admit a semiclassical expansion but how to proceed? For the cube yes but for a chaotic grain I am not sure I ~1/V? Fine but the matrix elements?

  10. With help we could achieve it Winter 2006: From desperation to hope ?

  11. 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

  12. 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

  13. Semiclassical (1/kFL >> 1) expression of the spectral density,Gutzwiller, Berry Non oscillatory terms Oscillatory terms in terms of classical quantities only

  14. Summer 2007 2d chaotic and rectangular Expansion in powers of /0 and 1/kFL 3d chaotic and rectangular

  15. 3d chaotic The sum over g(0) is cut-off by the coherence length ξ Importance of boundary conditions Universal function

  16. 3d chaotic AL grain kF = 17.5 nm-1  = 7279/N mv 0 = 0.24mv From top to bottom: 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

  17. 2d chaotic Importance of Matrix elements!! Importance of boundary conditions Universal function

  18. 2d chaotic AL grain kF = 17.5 nm-1  = 7279/N mv 0 = 0.24mv From top to bottom: L = 6nm, Dirichlet, /Δ0=0.77 L= 6nm, Neumann, /Δ0,=0.77 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

  19. 3d integrable Fall 2007, sent to arXiv! 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

  20. Spatial Dependence of the gap The prefactor suppresses exponentially the contribution of eigenstates with energy > Δ0 Maybe some structure is preserved The average is only over a few eigenstates around the Fermi surface

  21. N = 2998

  22. Scars Anomalous enhancement of the quantum probability around certain unstable periodic orbits (Kaufman, Heller) Experimental detection possible (Yazdani) No theory so trial and error N =4598 N =5490

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

  24. Mesoscopic corrections versus corrections to mean field Finite size corrections to BCS mean field approximation Matveev-Larkin Pair breaking Janko,1994 The leading mesoscopic corrections contained in (0) are larger. The corrections to (0) proportional to  has different sign

  25. Decoherence and geometrical deformations Decoherence effects and small geometrical deformations in otherwise highly symmetric grains weaken mesoscopic effects To what extent are our previous results robust? How much? Both effects can be accounted analytically by using an effective cutoff in the semiclassical expressions

  26. D(Lp/l) The form of the cutoff depends on the mechanism at work Finite temperature,Leboeuf Random bumps, Schmit,Pavloff Multipolar corrections, Brack,Creagh

  27. Fluctuations are robust provided that L >> l Non oscillating deviations present even for L ~ l

  28. The Future?

  29. What? Superconductivity 1. Disorder and finite size effects in superconductivity 2. AdS-CFT techniques in condensed matter physics Why? Control of superconductivity (Tc) Why now? 1. New high Tc superconducting materials 2. Control of interactions and disorder in cold atoms 3. New analytical tools 4.Better exp control in condensed matter

  30. arXiv:0904.0354v1

  31. THEORY IDEA REALITY CHECK GOALS S. Sinha, E. Cuevas Comparison with experiments (cold atoms)‏ Test of quantum mechanics Numerical and theoretical analysis of experimental speckle potentials Test of localization by Cold atoms Great! Bad Good Exp. verification of localization Superconducting circuits with higher critical temperature Mean field region Semiclassical + known many body techniques Comparison with superconducting grains exp. Finite size/disorder effects in superconductivity Great! Theory of strongly interacting fermions Strong Coupling AdS -CFT techniques Comparison BEC-BCS physics E. Yuzbashian, J. Urbina, B. Altshuler. D. Rodriguez Comparison cold atoms experiments Great! Test Ergodic Hyphothesis Numerics + beyond semiclassical tech. Novel states quantum matter Mesoscopic statistical mechanics Qualitiy control manufactured cavities Semiclassical techniques plus Stat. Mech. results Comparison with exp. blackbody Wang Jiao 0 3 5 Time(years)‏ Easy Medium Difficult Milestone

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