A theory of finite size effects in BCS A theory of finite size effects in BCS superconductors: The making of a papersuperconductors: The making of a paper
Antonio M. García-Garcí[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
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
Talk to Emil
Quantum chaos, trace formula…
what?
Richardson equations, Anderson representation
…what?
Princeton 2005: A false start Superconductivity?, Umm,
semiclassical, fineSuperconductivity, spin, semiclassical
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?
Non oscillatory terms
Oscillatory terms in terms of classical quantities only
Semiclassical (1/kFL >> 1) expression of the spectral density,Gutzwiller, Berry
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
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
Mean level spacing
Δ0 Superconducting gap
F Fermi Energy
L typical length
l coherence length
ξ Superconducting coherence length
Conditions
BCS / Δ0 << 1
Semiclassical1/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
Fine but the matrix
elements?
I ~1/V?
In,n should admit a semiclassical expansion but how to proceed?
For the cube yes but for a chaotic grain I am not sure
With help we could achieve it
Winter 2006: From desperation to hope
),'()',(22 LfLk
B
Lk
AI F
FF
?
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
Semiclassical (1/kFL >> 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
Non oscillatory terms
Oscillatory terms in terms of classical quantities only
Semiclassical (1/kFL >> 1) expression of the spectral density,Gutzwiller, Berry
Expansion in powers of /0 and 1/kFL
2d chaotic and rectangular
3d chaotic and rectangular
Summer 2007
3d chaotic
The sum over g(0) is cut-off by the coherence length ξ
Universal function
Importance of boundary conditions
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!!
Universal function
Importance of boundary conditions
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
V = n/181 nm-3
Numerical & analytical Cube & parallelepiped
No role of matrix elementsVI /1)',( Similar results were known in the literature from the 60’s
Fall 2007, sent to arXiv!
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
Scars
N =4598
Anomalous enhancement of the quantum probability around certain unstable periodic orbits (Kaufman, Heller)
N =5490
Experimental detection possible (Yazdani)
No theory so trial and error
Is this real?
Real (small) Grains
Coulomb interactions
Phonons
Deviations from mean field
Decoherence
Geometrical deviations
No
No
Yes
Yes
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?
1. Disorder and finite size effects in superconductivity
2. AdS-CFT techniques in condensed matter physics
Control of superconductivity (Tc)
What?
Why?
Superconductivity
1. New high T1. New high Tcc superconducting superconducting
materialsmaterials2. Control of interactions and 2. Control of interactions and disorder in cold atoms disorder in cold atoms 3. New analytical tools3. New analytical tools
Why now?
4.Better exp control in condensed matter
arXiv:0904.0354v1
Test of localizationTest of localizationby Cold atomsby Cold atoms
Finite size/disorderFinite size/disorder effects in effects in
superconductivitysuperconductivity
GOALS
Comparison with Comparison with superconductingsuperconducting
grains exp.grains exp.
Numerical and theoretical analysis of experimental
speckle potentials
Comparison withComparison withexperiments experiments (cold atoms)(cold atoms)
Mean field regionMean field regionSemiclassical + knownSemiclassical + knownmany body techniquesmany body techniques
Comparison withComparison withexp. blackbodyexp. blackbody
Semiclassical techniques Semiclassical techniques plus Stat. Mech. resultsplus Stat. Mech. results
IDEA THEORY REALITY CHECK
Exp. verification Exp. verification of localizationof localization
BadBad GoodGood
Mesoscopic Mesoscopic statisticalstatistical
mechanicsmechanics
Great!Great!
Superconducting Superconducting circuits withcircuits with
higher criticalhigher criticaltemperaturetemperature
Qualitiy controlQualitiy controlmanufacturedmanufactured
cavitiescavities
Test of quantumTest of quantummechanicsmechanics
E. Yuzbashian, J. Urbina,B. Altshuler. D. Rodriguez
Wang Jiao
S. Sinha, E. Cuevas
0 53Time(years)
Easy Medium Difficult Milestone
Strong CouplingStrong CouplingAdS -CFT techniques AdS -CFT techniques
Great!Great!
Comparison cold Comparison cold atoms experimentsatoms experiments
Test Ergodic Hyphothesis Test Ergodic Hyphothesis Numerics + beyond Numerics + beyond semiclassical tech.semiclassical tech.
Novel states Novel states quantum matterquantum matter
Great!Great!
Comparison BEC-BCS physics
Theory of strongly interacting fermions
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