1. PseudoGap Superconductivity and Superconductor-Insulator transition In collaboration with: Vladimir Kravtsov ICTP Trieste Emilio Cuevas University of Murcia Lev Ioffe Rutgers University Marc Mezard Orsay University Mikhail Feigel’man L.D.Landau Institute, Moscow Short publication: Phys Rev Lett. 98, 027001 (2007)

2. Superconductivity v/s Localization Granular systems with Coulomb interaction K.Efetov 1980 et al “Bosonic mechanism” Coulomb-induced suppression of Tc in uniform films “Fermionic mechanism” A.Finkelstein 1987 et al Competition of Cooper pairing and localization (no Coulomb) Imry-Strongin, Ma-Lee, Kotliar-Kapitulnik, Bulaevsky-Sadovsky(mid-80’s) Ghosal, Randeria, Trivedi 1998-2001 There will be no grains and no Coulomb in this talk !

3. Bosonic mechanism: Control parameter E c = e 2 /2C

5. Plan of the talk 1.Motivation from experiments 2.BCS-like theory for critical eigenstates - transition temperature - local order parameter 3.Superconductivity with pseudogap - transition temperature v/s pseudogap 4. Quantum phase transition: Cayley tree 5. Conclusions and open problems

6. Example: Disorder-driven S-I transition in TiN thin films T.I.Baturina et al Phys.Rev.Lett 99 257003 2007 Specific Features of Direct SIT: Insulating behaviour of the R(T) separatrix On insulating side of SIT, low-temperature resistivity is activated: R(T) ~ exp(T 0 /T) Crossover to VRH at higher temperatures Seen in TiN, InO, Be (extra thin) – all are amorphous, with low electron density There are other types of SC suppression by disorder !

7. Strongly insulating InO and nearly-critical TiN. Kowal-Ovadyahu 1994Baturina et al 2007 I2: T 0 = 0.38 K R 0 = 20 k  d = 5 nm d = 20 nm T 0 = 15 K R 0 = 20 k  What is the charge quantum ? Is it the same on left and on right?

8. Giant magnetoresistance near SIT (Samdanmurthy et al, PRL 92, 107005 (2004)

9. Experimental puzzle: Localized Cooper pairs. D.Shahar & Z.Ovadyahu amorphous InO 1992 V.Gantmakher et al InO D.Shahar et al InO T.Baturina et al TiN

10. Bosonic v/s Fermionic scenario ? None of them is able to describe data on InO x and TiN

11. Major exp. data calling for a new theory Activated resistivity in insulating a-InO x D.Shahar-Z.Ovadyahu 1992, V.Gantmakher et al 1996 T 0 = 3 – 15 K Local tunnelling data B.Sacepe et al 2007-8 Nernst effect above T c P.Spathis, H.Aubin et al 2008

12. Phase Diagram

13. Theoretical model Simplest BCS attraction model, but for critical (or weakly) localized electrons H = H 0 - g ∫ d 3 r Ψ ↑ † Ψ ↓ † Ψ ↓ Ψ ↑ Ψ = Σ c j Ψ j (r) Basis of localized eigenfunctions M. Ma and P. Lee (1985) : S-I transition at δ L ≈ T c

14. Superconductivity at the Localization Threshold: δ L → 0 Consider Fermi energy very close to the mobility edge: single-electron states are extended but fractal and populate small fraction of the whole volume How BCS theory should be modified to account for eigenstate’s fractality ? Method: combination of analitic theory and numerical data for Anderson mobility edge model

15. Mean-Field Eq. for T c

17. 3D Anderson model: γ = 0.57 D 2 ≈ 1.3 in 3D Fractality of wavefunctions IPR: M i = 4 drdr

19. Modified mean-field approximation for critical temperature T c For smallthis T c is higher than BCS value !

20. Alternative method to find Tc: Virial expansion (A.Larkin & D.Khmelnitsky 1970)

21. T c from 3 different calculations Modified MFA equation leads to: BCS theory: T c = ω D exp(-1/ λ)

22. Order parameter in real space for ξ = ξ k

23. Fluctuations of SC order parameter SC fraction = Higher moments: prefactor ≈ 1.7 for γ = 0.57 With Prob = p << 1 Δ(r) = Δ, otherwise Δ(r) =0

24. Tunnelling DoS Asymmetry in local DoS: Average DoS:

25. Neglected : off-diagonal terms Non-pair-wise terms with 3 or 4 different eigenstates were omitted To estimate the accuracy we derived effective Ginzburg- Landau functional taking these terms into account

26. Superconductivity at the Mobility Edge: major features -Critical temperature T c is well-defined through the whole system in spite of strong Δ(r) fluctuations -Local DoS strongly fluctuates in real space; it results in asymmetric tunnel conductance G(V,r) ≠ G(-V,r) -Both thermal (Gi) and mesoscopic (Gi d ) fluctuational parameters of the GL functional are of order unity

27. Superconductivity with Pseudogap Now we move Fermi- level into the range of localized eigenstates Local pairing in addition to collective pairing

28. 1. Parity gap in ultrasmall grains K. Matveev and A. Larkin 1997 No many-body correlations Local pairing energy Correlations between pairs of electrons localized in the same “orbital” ------- ------- E F --↑↓-- --  ↓--

29. 2. Parity gap for Anderson- localized eigenstates Energy of two single-particle excitations after depairing:

30. P(M) distribution

31. Activation energy T I from Shahar- Ovadyahu exp. and fit to theory The fit was obtained with single fitting parameter = 0.05 = 400 K Example of consistent choice:

33. Critical temperature in the pseudogap regime Here we use M(ω) specific for localized states MFA is OK as long as MFA: is large

34. Correlation function M(ω) No saturation at ω < δ L : M(ω) ~ ln 2 (δ L / ω) (Cuevas & Kravtsov PRB,2007) Superconductivity with Tc < δ L is possible This region was not found previously Here “local gap” exceeds SC gap :

35. Critical temperature in the pseudogap regime We need to estimate MFA: It is nearly constant in a very broad range of

36. T c versus Pseudogap Transition exists even at δ L >> T c0 Virial expansion results:

37. Single-electron states suppressed by pseudogap Effective number of interacting neighbours “Pseudospin” approximation

38. Third Scenario Bosonic mechanism: preformed Cooper pairs + competition Josephson v/s Coulomb – S I T in arrays Fermionic mechanism: suppressed Cooper attraction, no paring – S M T Pseudospin mechanism: individually localized pairs - S I T in amorphous media SIT occurs at small Z and lead to paired insulator How to describe this quantum phase transition ? Cayley tree model is solved (L.Ioffe & M.Mezard)

39. Qualitative features of “Pseudogaped Superconductivity”: STM DoS evolution with T Double-peak structure in point-contact conuctance Nonconservation of full spectral weight across T c

40. Superconductor-Insulator Transition Simplified model of competition between random local energies (ξ i S i z term) and XY coupling

46. Phase diagram Superconductor Hopping insulator g Temperature Energy RSB state Full localization: Insulator with Discrete levels MFA line gcgc

48. Fixed activation energy is due to the absence of thermal bath at low ω

49. Conclusions Pairing on nearly-critical states produces fractal superconductivity with relatively high T c but very small superconductive density Pairing of electrons on localized states leads to hard gap and Arrhenius resistivity for 1e transport Pseudogap behaviour is generic near S-I transition, with “insulating gap” above T c New type of S-I phase transition is described (on Cayley tree, at least). On insulating side activation of pair transport is due to ManyBodyLocalization threshold

50. Coulomb enchancement near mobility edge ?? Condition of universal screening: Normally, Coulomb interaction is overscreened, with universal effective coupling constant ~ 1 Example of a-InO x Effective Couloomb potential is weak:

51. Class of relevant materials Amorphously disordered (no structural grains) Low carrier density ( around 10 21 cm -3 at low temp.) Examples: InO x NbN x thick films or bulk (+ B-doped Diamond?) TiN thin films Be, Bi ( ultra thin films)