Phys. Rev. Lett. 100, 187001 (2008)
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Transcript of Phys. Rev. Lett. 100, 187001 (2008)
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 Materials
2768, 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 ~1/V? Chaotic
grains?
1-body eigenstates
I = 1/V + ...?
Yes, with help, we can
From desperation to hope
),,'('22 LfLk
B
Lk
AI 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 = -’
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!
3d chaotic
Al grain
kF = 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 /Tc
Quantum 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 ED
d
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 fluctuations
Static Path Approach
BCS finite size effects
Part I
Deviations from BCS
Richardson formalism
Finite T ~ Tc
(T), (T) from data
(T~Tc)~ weak T dep
T=0
BCS finite size effects
Part I
Deviations from BCS
Richardson 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 limit
For 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 geometry
2. For c3 = c4 = 0 mean field results
3. Gauge field A is U(1) and is a scalar
4. The dual CFT (quiver SU(N) gauge theory) is known for some ƒ
5. 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 conditionsHorizon
Boundary
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
2
1
Results II
The spectroscopic gap becomes larger and the coherence peak narrower as c4
increases.
Results III
Future
1. 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
Singapore
D. RodriguezQueen Mary
K. Richter Regensburg
Let’s do it!!
P. NaidonTokyo