Phys. Rev. Lett. 100, (2008) Yuzbashyan Rutgers Altshuler Columbia Urbina Regensburg Richter Regensburg Sangita Bose, Tata, Max Planck Stuttgart Kern Ugeda, Brihuega arXiv: 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 T c 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~  D e -1/ V finite Δ=?

Brute force?  i = eigenvalues 1-body problem No practical for grains with no symmetry Semiclassical techniques 1/k F L <<1Analytical? Quantum observables in terms of classical quantities Berry, Gutzwiller, Balian, Bloch

Non oscillatory terms Oscillatory terms in L,  Expansion 1/k F L << 1 Gutzwiller’s trace formula Weyl’s expansion

Are these effects important?  Mean level spacing Δ 0 Superconducting gap  F Fermi Energy L typical length l coherence length ξ SC coherence length Conditions BCS  / Δ 0 << 1 Semiclassical 1/k F L << 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 T c Olofsson (2008): Estimation of fluctuations in BCS, no correlations Superconductivity in particular geometries

Nature of superconductivity (?) in ultrasmall systems Breaking of superconductivity for  / Δ 0 > 1? Anderson (1959) Experiments Tinkham et al. (1995). Guo et al., Science 306, 1915, “Supercond. Modulated by quantum Size Effects.” Even for  / Δ 0 ~ 1 there is “supercondutivity T = 0 and  / Δ 0 > 1 (1995-) Richardson, von Delft, Braun, Larkin, Sierra, Dukelsky, Yuzbashyan Thermodynamic properties Muhlschlegel, Scalapino (1972) Description beyond BCS Estimation. No rigorous! 1.Richardson’s equations: Good but Coulomb, phonon spectrum? 2.BCS fine until  / Δ 0 ~ 1/2

 / Δ 0 >> 1 We are in business! systematic 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/k F L +...?

Yes, with help, we can From desperation to hope ? 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/k F L >> 1) expansion for l !! ω =  -  ’/  F Relevant in any mean field approach with chaotic one body dynamics Classical ergodicity of chaotic systems Sieber 99, Ozoiro Almeida, 98

Now it is easy

3d chaotic ξ controls (small) fluctuations Universal function Boundary conditions Enhancement of SC! (i)  (1/k F L) i

3d chaotic Al grain k F = 17.5 nm -1  0 = 0.24mV L = 6nm, Dirichlet,  /Δ 0 =0.67 L= 6nm, Neumann,  /Δ 0, =0.67 L = 8nm, Dirichlet,  /Δ 0 =0.32 L = 10nm, Dirichlet,  /Δ 0,= 0.08 For L< 9nm leading correction comes from I( ,  ’)

3d integrable Numerical & analyticalCube & 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 No Is it taken into account? L ~ 10 nm Sn, Al…

Mesoscopic corrections versus corrections to mean field Finite size corrections to BCS Matveev-LarkinPair 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: v1 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

Shell effects Enhancement of fluctuations Grain symmetry Level degeneracy More states around  F Larger gap

+

0 nm 7 nm

Pb Do you want more fun? Why not  (T) > 0 for T > T c  (0)  for L < 10nm Physics beyond mean-field

Theoretical dI/dV Fluctuations + BCS Finite size effects + Deviations from mean field dI/dV ? Dynes formula Beyond Dynes

Dynes fitting  >   no monotonic Breaking of mean field

Pb L < 10nm Strongly coupled SC Thermal fluctuations  /T c Quantum fluctuations  / ,E D 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 field Path integral? Too difficult! Richardson’s equations Even worse! BCS eigenvalues But OK expansion in  /  0 ! Richardson, Yuzbashyan, Altshuler Pair breaking excitation

Pair breaking energy D  E D d   Blocking effect Quantum fluctuations >> Energy gap  Remove two levels closest to E F  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 ~ T c (T),  (T) from data  (T~T c )~ 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 T c superconductors 1 ½. Strong interactions High energy techniques

THANKS!

Strongly coupled field theory Applications in high Tc superconductivity Powerful tool to deal with strong interactions Transition from qualitative to quantitative Hartnoll, Herzog N=4 Super-Yang Mills CFT Anti de Sitter space AdS Holographic techniques in condensed matter Phys. Rev. D 81, (2010) JHEP 1004:092 (2010) Weakly coupled gravity dual Franco Santa Barbara Rodriguez Princeton AdS-CFT correspondence Maldacena’s conjecture QCD Quark gluon plasma Condensed matter Gubser, Son

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 AdS 4 geometry 2. For c 3 = c 4 = 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 transitions Phys. Rev. D 81, (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 1. Introduce perturbation in the bulk 2. Solve the equation of motion with boundary conditions Horizon Boundary 3. Find retarded Green Function 4. Compute conductivity

For c 4 > 1 or c 3 > 0 the transition becomes first order A jump in the condensate at the critical temperature is clearly observed for c 4 > 1 The discontinuity for c 4 > 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 c 3 < For Condensate for c  = -1 and c 4 = ½. β = 1, 0.80, 0.65, 0.5 for  = 3, 3.25, 3.5, 4, respectively Results II

The spectroscopic gap becomes larger and the coherence peak narrower as c 4 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. Altshuler Columbia JD Urbina Regensburg S. Bose Stuttgart M. Tezuka Kyoto S. Franco, Santa Barbara K. Kern, Stuttgart J. Wang Singapore D. Rodriguez Queen Mary K. Richter Regensburg Let’s do it!! P. Naidon Tokyo