Phys. Rev. Lett. 100, 187001 (2008)

Yuzbashyan Rutgers

Altshuler Columbia

Urbina Regensburg

Richter Regensburg

Sangita Bose, Tata, Max Planck Stuttgart

Kern Ugeda, Brihuega

arXiv:0911.1559

Nature Materials2768, May 2010

Finite size effects in superconducting grains: from theory to experiments

Antonio M. García-García

L

1. Analytical description of a clean, finite-size non high Tc superconductor?

2. Are these results applicable to realistic grains?

Main goals

3. Is it possible to increase the critical temperature?

Can I combine this?

BCS superconductivity

Is it already done?

Finite size effects

V Δ~ De-1/

V finite Δ=?

Brute force?

i = eigenvalues 1-body problem

No practical for grains with no symmetry

Semiclassical techniques

1/kF L <<1 Analytical?

Quantum observables in terms of classical quantities Berry,

Gutzwiller, Balian, Bloch

Non oscillatory terms

Oscillatory terms in L,

Expansion 1/kFL << 1

Gutzwiller’s trace formula

Weyl’s expansion

Are these effects important?

Mean level spacing

Δ0 Superconducting gap

F Fermi Energy

L typical length

l coherence length

ξ SC 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

In what range of parameters?

Corrections to BCS smaller or larger?

Let’s think about this

Is it done already?

Is it realistic?

A little history

Parmenter, Blatt, Thompson (60’s) : BCS in a rectangular grain

Heiselberg (2002): BCS in harmonic potentials, cold atom appl.

Shanenko, Croitoru (2006): BCS in a wire

Devreese (2006): Richardson equations in a box

Kresin, Boyaci, Ovchinnikov (2007) Spherical grain, high Tc

Olofsson (2008): Estimation of fluctuations in BCS, no correlations

Superconductivity in particular geometries

Nature of superconductivity (?) in ultrasmall systems

Breaking of superconductivity for / Δ0 > 1? Anderson (1959)

Experiments Tinkham et al. (1995). Guo et al., Science 306, 1915, “Supercond. Modulated by quantum Size Effects.”

Even for / Δ0 ~ 1 there is “supercondutivity

T = 0 and / Δ0 > 1 (1995-)

Richardson, von Delft, Braun, Larkin, Sierra, Dukelsky, Yuzbashyan

Thermodynamic propertiesMuhlschlegel, Scalapino (1972)

Description beyond BCS

Estimation. No rigorous!

1.Richardson’s equations: Good but Coulomb, phonon spectrum?

2.BCS fine until / Δ0 ~ 1/2

/ Δ0 >> 1

We are in business!

No systematic BCS treatment of the dependence

of size and shape

?

Hitting a bump

Matrix elements?

I ~? Chaotic grains?

1-body eigenstates

I = (1 + A/kFL + ...?

Yes, with help, we can

From desperation to hope

),,'('22 LfLkB

LkAI F

FF

?

Semiclassical expansion for I

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 maybe we can use ergodic

theorems

Semiclassical (1/kFL >> 1) expansion for l !!

ω = -’/F

Relevant in any mean field approach with chaotic one body dynamics

Classical ergodicity of chaotic systemsSieber 99, Ozoiro Almeida, 98

Now it is easy

3d chaotic

ξ controls (small) fluctuations

Universal function

Boundary conditions

Enhancement of SC!

(i) (1/kFL)i

3d chaotic

Al grainkF = 17.5 nm-1

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

For L< 9nm leading correction comes from I(,’)

3d integrable

Numerical & analytical Cube & rectangle

From theory to experiments

Real (small) Grains

Coulomb interactions

Surface Phonons

Deviations from mean field

Decoherence

Fluctuations

No, but no strong effect expected

No, but screening should be effective

Yes

Yes

No

Is it taken into account?

L ~ 10 nm Sn, Al…

Mesoscopic corrections versus corrections to mean field

Finite size corrections to BCS

Matveev-Larkin Pair breaking Janko,1994

The leading mesoscopic corrections contained in (0) are larger

The correction to (0) proportional to has different sign

Experimentalists are coming

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

h= 4-30nm

Single isolated Pb, Sn

B closes gap

Tunneling conductance

Experimental output

Almost hemispherical

dI/dV )(T

Shell effects

Enhancement of fluctuations

Grain symmetry

Level degeneracy

More states around F

Larger gap

+

5.33 Å

0.00 Å

0 nm

7 nm

Pb

Do you want more fun? Why not

(T) > 0 for T > Tc

(0) for L < 10nm

Physics beyond

mean-field

Theoretical dI/dV

Fluctuations + BCS Finite size effects + Deviations

from mean field

dI/dV )(T

?Dynes formula

Beyond Dynes

Dynes fitting

>

no monotonic Breaking of mean field

Pb L < 10nm

Strongly coupled SC

Thermal fluctuations

/TcQuantum

fluctuations /,ED

Finite-size corrections

Eliashberg theory

Path integral

Richardson equations

Semiclassical

Scattering, recombination, phonon spectrum

Static path approach

Exact solution,

Previous part

Exact solution, BCS Hamiltonian

Thermal fluctuations

Path integral

0d grains homogenous

Static path approach

Hubbard-Stratonovich transformation

Scalapino et al.

Other deviations from

mean fieldPath integral?

Too difficult!

Richardson’s equationsEven worse!

BCS eigenvalues

But

OK expansion in /0 !

Richardson, Yuzbashyan, Altshuler

Pair breaking excitation

Pair breaking energy

D EDd

Blocking effect

Quantum fluctuations>>

Energy gap

Remove two levels closest to

EF

Only important ~~L<5nm

Putting everything together

Tunneling conductance

Energy gap Eliashberg

Thermal fluctuationsStatic Path Approach

BCS finite size effectsPart I

Deviations from BCSRichardson formalism

Finite T ~ Tc

(T), (T) from data(T~Tc)~ weak T dep

T=0

BCS finite size effectsPart I

Deviations from BCSRichardson formalism

No fluctuations!Not important h > 5nm

Dynes is fine h>5nm

(L) ~ bulk from data

What is next?

1. Why enhancement in average Sn gap?

2. High Tc superconductors

1 ½ . Strong interactionsHigh energy techniques

THANKS!

Strongly coupled

field theory

Applications in high Tc superconductivity

A solution looking for a problem

Powerful tool to deal with strong interactions

Transition from qualitative to quantitative

Hartnoll, Herzog

N=4 Super-Yang MillsCFT

Anti de Sitter spaceAdS

Holographic techniques in condensed matter

Phys. Rev. D 81, 041901 (2010)

JHEP 1004:092 (2010)

Weakly coupled

gravity dual

FrancoSanta Barbara

RodriguezPrinceton

AdS-CFT correspondence

Maldacena’s conjecture

QCD Quark gluon plasma

Condensed matter

Gubser, Son

2003

2008

Problems

1. Estimation of the validity of the AdS-CFT approach

2. Large N limitFor what condensed matter systems

are these problems minimized?Phase Transitions triggered by thermal

fluctuations

1. Microscopic Hamiltonian is not important 2. Large N approximation OK

Why?

1. d=2+1 and AdS4 geometry2. For c3 = c4 = 0 mean field results3. Gauge field A is U(1) and is a scalar4. The dual CFT (quiver SU(N) gauge theory) is known for some ƒ5. By tuning ƒ we can reproduce different phase transitions

Holographic approach to phase transitionsPhys. Rev. D 81, 041901 (2010)

How are results obtained?

1. Einstein equations for the scalar and electromagnetic field

2. Boundary conditions from the AdS-CFT dictionary

Boundary

Horizon

3. Scalar condensate of the dual CFT

Calculation of the conductivity

ikytixx erAA )(1. Introduce perturbation in the bulk

2. Solve the equation of motion

with boundary conditions HorizonBoundary

3. Find retarded Green Function

4. Compute conductivity

For c4 > 1 or c3 > 0 the transition becomes first order

A jump in the condensate at the critical temperature is clearly observed for c4 > 1

The discontinuity for c4 > 1 is a signature of a first order phase transition.

Results I

Second order phase transitions with non mean field critical exponents different are also accessible

1. For c3 < -1

2/112 cTTO

2. For 2/112

Condensate for c = -1 and c4 = ½. β = 1, 0.80, 0.65, 0.5 for = 3, 3.25, 3.5, 4, respectively

21

Results II

The spectroscopic gap becomes larger and the coherence peak narrower as c4

increases.

Results III

Future1. Extend results to β <1/2

2. Adapt holographic techniques to spin

3. Effect of phase fluctuations. Mermin-Wegner theorem?

4. Relevance in high temperature superconductors

E. Yuzbashyan, Rutgers

B. AltshulerColumbia

JD Urbina Regensburg

S. Bose Stuttgart

M. Tezuka Kyoto

S. Franco, Santa Barbara

K. Kern, StuttgartJ. Wang

SingaporeD. RodriguezQueen Mary

K. Richter Regensburg

Let’s do it!!

P. NaidonTokyo