Anderson Localization (1957)

1
A theory of finite size effects in BCS
superconductors The making of a paper
Antonio M. García-García ag3_at_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
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 ltlt1,
Berry, Gutzwiller, Balian, Bloch
Gutzwiller trace formula
Can I combine this?
Is it already done?
5
Semiclassical (1/kFL gtgt 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?
Lets think about this
7
A little history
1959, Anderson superconductor if ?/ ?0 gt 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
?0 Superconducting gap
? Mean level spacing
l coherence length
? Superconducting coherence length
?F Fermi Energy
Conditions
BCS ?/
?0 ltlt 1
Semiclassical1/kFL ltlt 1
Quantum coherence l gtgt L ?
gtgt 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
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 lgtgtL ergodic theorems assures universality
f(L,?- ?, ?F) is a simple function
12
A few months later
Semiclassical (1/kFL gtgt 1) expression of the
matrix elements valid for l gtgt L!!
? ?-?
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 gtgt 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 ?
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, ?/?00.67 L
6nm, Neumann, ?/?0,0.67 L 8nm, Dirichlet,
?/?00.32 L 10nm, Dirichlet,
?/?0, 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, ?/?00.77 L
6nm, Neumann, ?/?0,0.77 L 8nm, Dirichlet,
?/?00.32 L 10nm, Dirichlet,
?/?0, 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 60s
20
Spatial Dependence of the gap
The prefactor suppresses exponentially the
contribution of eigenstates with energy gt ?0
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 gtgt 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
Control of superconductivity (Tc)
Why?
1. New high Tc superconducting materials
Why now?
2. Control of interactions and disorder in cold
atoms
3. New analytical tools
4.Better exp control in condensed matter
30
arXiv0904.0354v1
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
THEORY
GOALS
IDEA
REALITY CHECK
Comparison with experiments (cold atoms)?
Test of quantum mechanics
S. Sinha, E. Cuevas
Numerical and theoretical analysis of
experimental speckle potentials
Test of localization by Cold atoms
Great!
Bad
Exp. verification of localization
Good
Superconducting circuits with higher
critical temperature
Comparison with superconducting grains exp.
Mean field region Semiclassical known many body
techniques
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
5
3
Time(years)?
Easy
Medium
Difficult
Milestone