Transcript Anderson Localization (1957)

A theory of finite size effects in BCS
superconductors: The making of a paper
Antonio M. García-García
[email protected]
http://phy-ag3.princeton.edu
Princeton and ICTP
Phys. Rev. Lett. 100, 187001 (2008), AGG, Urbina, Yuzbashyan, Richter, Altshuler.
Yuzbashyan
Altshuler
Urbina
Richter
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
Princeton 2005: A false start
Superconductivity,
spin, semiclassical
Superconductivity?,
Umm, semiclassical,
fine
Talk to Emil
Quantum chaos,
trace
formula…what?
Richardson equations,
Anderson representation
…what?
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?
Semiclassical (1/kFL >> 1) expression
of the spectral density,Gutzwiller, Berry
Non oscillatory
terms
Oscillatory
terms in terms
of classical
quantities only
Maybe it is possible
Go ahead!
Corrections
to BCS
smaller or
larger?
Let’s think about this
This has not been done
before
It is possible but it is
relevant?
If so, in what range of
parameters?
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!
Relevant Scales
Δ0 Superconducting gap
L typical length
 Mean level spacing
l coherence length
F Fermi Energy
ξ Superconducting
coherence length
Conditions
BCS
/ Δ0 << 1
Semiclassical
1/kFL << 1
Quantum coherence
l >> L ξ >> L
For Al the optimal region is L ~ 10nm
Fall 06: Hitting a bump
3d cubic Al grain

In,n should admit a
semiclassical expansion
but how to proceed?
I ~1/V?
Fine but the
matrix
elements?
For the cube
yes but for a
chaotic grain
I am not sure
Winter 2006: From desperation to hope
?
A
B
I ( ,  ' ) 
 2 2  f (   ' ,  F L)
kF L kF L
With help we could
achieve it
Regensburg, we have got a problem!!!
Do not worry. It is not
an easy job but you are
in good hands
For l>>L ergodic
theorems assures
universality
Nice closed
results that do
not depend on
the chaotic
cavity
f(L,- ’, F) is a
simple function
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
Semiclassical (1/kFL >> 1) expression
of the spectral density,Gutzwiller, Berry
Non oscillatory
terms
Oscillatory
terms in terms
of classical
quantities only
Summer
2007
Expansion in
powers of /0
and 1/kFL
2d chaotic and
rectangular
3d chaotic and
rectangular
3d chaotic
The sum over
g(0) is cut-off by
the coherence
length ξ
Importance of
boundary
conditions
Universal function
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
2d chaotic
Importance of
Matrix elements!!
Importance of
boundary
conditions
Universal function
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
3d integrable
Fall 2007, sent to arXiv!
V = n/181 nm-3
Numerical & analytical
Cube & parallelepiped
I ( ,  ' )  1 / V
No role of matrix elements
Similar results were known in the literature from the 60’s
Spatial Dependence of the gap
The prefactor suppresses exponentially the
contribution of eigenstates with energy > Δ0
The average is only over a few
eigenstates around the Fermi surface
Maybe
some
structure is
preserved
N = 2998
Anomalous enhancement of the quantum
probability around certain unstable
periodic orbits (Kaufman, Heller)
Scars
Experimental detection possible (Yazdani)
No theory so trial and error
N =4598
N =5490
Is this real?
Real (small) Grains
Coulomb interactions
No
Phonons
No
Deviations from mean field
Yes
Decoherence
Yes
Geometrical deviations
Yes
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
Decoherence and
geometrical deformations
Decoherence effects and small geometrical
deformations in otherwise highly symmetric grains
weaken mesoscopic effects
How much?
To what extent are our
previous results robust?
Both effects can be accounted analytically by using an
effective cutoff in the semiclassical expressions
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
Fluctuations are robust provided that L >> l
Non oscillating
deviations present
even for L ~ l
The Future?
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
arXiv:0904.0354v1
THEORY
IDEA
S. Sinha, E. Cuevas
Test of localization
by Cold atoms
REALITY CHECK
Comparison with
experiments
(cold atoms)
Numerical and theoretical
analysis of experimental
speckle potentials
Bad
Finite size/disorder
effects in
superconductivity
E. Yuzbashian, J. Urbina,
B. Altshuler. D. Rodriguez
Mean field region
Semiclassical + known
many body techniques
Strong Coupling
AdS -CFT techniques
Semiclassical techniques
plus Stat. Mech. results
Wang Jiao
Test of quantum
mechanics
Great!
Good
Exp. verification
of localization
Superconducting
circuits with
higher critical
temperature
Comparison with
superconducting
grains exp.
Great!
Test Ergodic Hyphothesis
Numerics + beyond
semiclassical tech.
Mesoscopic
statistical
mechanics
GOALS
0
Theory of
strongly
interacting
fermions
Comparison
BEC-BCS
physics
Comparison cold
atoms experiments
Great!
Qualitiy control
manufactured
cavities
Comparison with
exp. blackbody
3
5
Time(years)
Easy
Medium
Difficult
Novel states
quantum matter
Milestone