Finite size effects in BCS: theory and Finite size effects in BCS: theory and experimentsexperiments

Antonio M. García-Garcíaag3@princeton.edu

Princeton and IST(Lisbon)Phys. Rev. Lett. 100, 187001 (2008) (theory), submitted to Nature (experiments)

Yuzbashyan AltshulerUrbina

Richter

Sangita Bose

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

How to tackle the problem

Semiclassical: To express quantum observables in terms of classical quantities. Only 1/kF L <<1, Berry, Gutzwiller, Balian

Gutzwiller trace formula

Can I combine this?

Is it already done?

λ

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

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, estimation of fluctuations, no matrix elements!

Hitting a bump

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

λ/V ?

Yes, with help, we can

From desperation to hope

),,'()',(22 LfLk

B

Lk

AIV 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

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

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

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

Is this real?

Real (small) Grains

Coulomb interactions

Phonons

Deviations from mean field

Decoherence

Geometrical deviations

No

No

Yes

Yes

Yes

Is this really real?

arXiv:0904.0354v1

Sorry but in Pb only small

fluctuations

Are you 300% sure?

Pb and Sn are very different because their coherence lengths are very different.

!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!

However in Sn is

very different

10 15 20 25 300.4

0.6

0.8

1.0

1.2

1.4

1.6

Expt. data Theory

No

rma

lize

d G

ap

(m

eV

)

Particle height (nm)

submitted to Nature