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-Garcíaag3@princeton.edu

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