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 publications: Phys Rev Lett. 98, 027001 (2007) arXiv:0909.2263

2. 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

3. Superconductivity v/s Localization • Coulomb-induced suppression of Tc in uniform films “Fermionic mechanism” A.Finkelstein (1987) et al • Granular systems with Coulomb interaction K.Efetov (1980) M.P.A.Fisher et al (1990)“Bosonic mechanism” • 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 in this talk !

4. We consider amorphous systems with direct S-I transition

5. Superconductor-Insulator: experimental evidence Direct evidence for the gap above the transition (Chapelier, Sacepe). Activation behavior does not show gap suppression at the critical point as a function of the disorder (Sahar, Ovaduyahu, 1992)! First look: critical behavior as predicted by boson duality (Haviland, Liu, Goldman 1989, 1991)

6. Class of relevant materials • Amorphously disordered (no structural grains) • Low carrier density ( around 1021 cm-3 at low temp.) Examples: InOxNbNxthick films or bulk(+ B-doped Diamond?) TiN thin filmsBe (ultra thin films) Special example: nanostructured Bi films (J.Valles et al)

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

8. Bose model (preformed Cooper pairs) R Insulator In superconducting films have cores → friction → vortex motion is not similar to charges → duality is much less likely in the films → intermediate normal metal is less likely in Josephson arrays (no vortex cores) RQ T • Competition between Coulomb repulsion and Cooper pair hopping: Duality charge-vortex: both charge-charge and vortex-vortex interaction are Log(R) in 2D. Vortex motion generates voltage: V=φ0 jV Charge motion generates current: I=2e jc At the self-dual point the currents are equal → RQ=V/I=h/(2e)2=6.5kΩ. M.P.A.Fisher 1990

9. Duality approach to SIT in films Duality transformation (approximate) Insulating state of Josepshon array Model of Josephson array (SC state) ????? Duality transformation (non-existing) Insulating state of disordered film Continous amorphous disordered film (SC state) ????????

10. Superconductor-Insulator: experimental evidence If duality arguments are correct, the same behavior should be observed in Josephson arrays… At zero field simple Josephson arrays show roughly the critical behavior. However, the critical R is not universal. (Zant and Mooji, 1996) and critical value of EJ/Ec differs. First look: critical behavior as predicted by boson duality (Haviland, Liu, Goldman 1989, 1991)

11. Superconductor-Insulator: experimental evidence If Josephson/Coulomb model is correct, the same behavior should be observed in Josephson arrays… At non-zero field simple Josephson arrays show temperature independent resistance with values that change by orders of magnitude. (H. van der Zant et al, 1996)

12. Superconductor-Insulator: experimental evidence If Josephson/Coulomb model is correct, the same behavior should be observed in Josephson arrays… At non-zero field simple Josephson arrays show temperature independent resistance with values that change by orders of magnitude. (H. van der Zant et al, 1996)

13. Superconductor-Insulator: experimental evidence If Josephson/Coulomb model is correct, the same behavior should be observed in Josephson arrays… BUT IT IS NOT At non-zero field Josephson arrays of more complex (dice) geometry show temperature independent resistance in a wide range of EJ/Ec. (B.Pannetier and E.Serret 2002)

14. Bosonic v/s Fermionic scenario ? None of them is able to describe data on InOx and TiN 3-d scenario: competition between Cooper pairing and localization (without any role of Coulomb interaction)

15. 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) :

16. 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

17. Mean-Field Eq. for Tc

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

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

22. Order parameter in real space for ξ = ξk SC fraction =

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

24. 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”

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

26. P(M) distribution

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

28. 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 :

29. Critical temperature in the pseudogap regime MFA: 3 ) ( ~ We need to estimate It is nearly constant in a very broad range of

30. Tcversus Pseudogap Transition exists even at δL >> Tc0

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

32. Qualitative features of “Pseudogaped Superconductivity”: • STM DoS evolution with T • Double-peak structure in point-contact conductance • Nonconservation of full spectral weight across Tc V Ktot(T) Tc Δp T

33. 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)

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

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

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

43. 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

44. 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:

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

46. 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

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

48. 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

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

50. 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