PseudoGap Superconductivity and Superconductor-Insulator transition

1. PseudoGap Superconductivity and Superconductor-Insulator transition Mikhail Feigel’man L.D.Landau Institute, Moscow In collaboration with: Vladimir Kravtsov ICTP Trieste Emilio Cuevas University of Murcia Lev Ioffe Rutgers University Marc Mezard Orsay University 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 andno Coulomb in this talk !

3. Bosonic mechanism: Control parameter Ec = e2/2C

5. Plan of the talk • Motivation from experiments • BCS-like theory for critical eigenstates - transition temperature - local order parameter • 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.Lett99 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(T0/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 d = 5 nm . d = 20 nm I2: T0 = 0.38 K R0 = 20 kW T0 = 15 K R0 = 20 kW Kowal-Ovadyahu 1994 Baturina et al 2007 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 InOx and TiN

11. Major exp. data calling for a new theory • Activated resistivity in insulating a-InOx D.Shahar-Z.Ovadyahu 1992, V.Gantmakher et al 1996 T0 = 3 – 15 K • Local tunnelling data B.Sacepe et al 2007-8 • Nernst effect above Tc P.Spathis, H.Aubin et al 2008

12. Phase Diagram

13. Theoretical model Simplest BCS attraction model, butfor critical (or weakly) localized electrons H = H0 - g ∫ d3rΨ↑†Ψ↓†Ψ↓Ψ↑ Ψ= ΣcjΨj (r) Basis of localized eigenfunctions S-I transition atδL≈ Tc M. Ma and P. Lee (1985) :

14. Superconductivity at the Localization Threshold: δL→0 Consider Fermi energy very close to the mobility edge: single-electron states are extended butfractal 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 Tc

17. Fractality of wavefunctions 4 IPR: Mi = dr D2 ≈ 1.3 in 3D 3D Anderson model: γ = 0.57

19. Modified mean-field approximation for critical temperature Tc For small this Tc is higher than BCS value !

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

21. Tc from 3 different calculations Modified MFA equation leads to: BCS theory: Tc = ωD exp(-1/ λ)

22. Order parameter in real space for ξ = ξk

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

24. Tunnelling DoS Average DoS: Asymmetry in local 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 Tcis well-defined through the wholesystem 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 (Gid) 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 Local pairing energy ------- ------- EF --↑↓-- --  ↓-- K. Matveev and A. Larkin 1997 No many-body correlations Correlations between pairs of electrons localized in the same “orbital”

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

30. P(M) distribution

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

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

34. Correlation functionM(ω) No saturation at ω < δL : M(ω) ~ ln2 (δL /ω) (Cuevas & Kravtsov PRB,2007) Superconductivity with Tc < δL is possible This region was not found previously Here “local gap” exceeds SC gap :

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

36. Virial expansion results: Tcversus Pseudogap Transition exists even at δL >> Tc0

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

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 Tc

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

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

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 Tc 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 Tc 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 ?? Normally, Coulomb interaction is overscreened, with universal effective coupling constant ~ 1 Condition of universal screening: Example of a-InOx Effective Couloomb potential is weak: