Finite size effects in BCS: theory and experiments Antonio M. García-García [email protected]...
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Transcript of Finite size effects in BCS: theory and experiments Antonio M. García-García [email protected]...
Finite size effects in BCS: theory and Finite size effects in BCS: theory and experimentsexperiments
Antonio M. García-Garcí[email protected]
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 ?
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
Pb and Sn are very different because their coherence lengths are very different.
!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!
However in Sn is
very different