Phys. Rev. Lett. 100, 187001 (2008) Yuzbashyan Rutgers Altshuler Columbia Urbina Regensburg Richter Regensbur g 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
Finite size effects in superconducting grains: from theory to experiments. Antonio M. Garc í a- Garc í a. Phys. Rev. Lett. 100, 187001 (2008) . Sangita Bose, Tata, Max Planck Stuttgart. arXiv:0911.1559 . Yuzbashyan Rutgers. Altshuler Columbia. Nature Materials 2768, May 2010. - PowerPoint PPT Presentation
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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
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?