1. Superconductivity Introduction Disorder & superconductivity : milestones BCS theory Anderson localization Abrikosov, Gorkov Anderson theorem 191119571958 - 195919871990 Time 2000 - 2010 Bosonic scenario Fermionic scenario Localization vs.Superconductivity A. Kapitulnik, G. Kotliar, Phys. Rev. Lett. 54, 473, (1985) M. Ma, P.A. Lee, Phys. Rev. B 32, 5658, (1985) G. Kotliar, A. Kapitulnik, Phys. Rev. B 33, 3146 (1986) M.V. Sadowskii, Phys. Rep., 282, 225 (1997) A.Ghosal et al., PRL 81, 3940 (1998) ; PRB 65, 014501 (2001) B.K. Bouadim, Y. L. Loh, M. Randeria, N. Trivedi, Nature Physics (2011) M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010) A.M. Finkel’stein JETP Lett. 45, 46 (1987) M.P.A. Fisher, PRL 65, 923 (1990) Kamerlingh Onnes, H., "On the change of electric resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures."

2. TIN Superconductor-Insulator transition Increasing disorder Sacépé et al., PRL 101, 157006 (2008)

3. TIN Superconductor-Insulator transition Increasing disorder Sacépé et al., PRL 101, 157006 (2008)

4. Superconducting fluctuations correction to the DOS Preformed Cooper pairs above Tc A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983) Pseudogap B. Sacépé, et al., Nature Communications (2010)

5. Localization of preformed Cooper pairs in highly disordered superconducting films Meso2012 Thomas Dubouchet, Benjamin Sacépé, Marc Sanquer, Claude Chapelier INAC-SPSMS-LaTEQS, CEA-Grenoble T. Baturina, Institute of semiconductor Physics - Novosibirsk V. Vinokur, Material Science Division, Argonne National Laboratory - Argonne M. Baklanov, A Satta, IMEC - Leuven Maoz Ovadia, Dan Shahar, The Weizmann Institute of Science Mikhail Feigel'man, L. D. Landau Institute for Theoretical Physics Lev Ioffe, Serin Physics laboratory, Rutgers University

6. Samples : e-gun evaporation of high purity In 2 O 3 onto Si/SiO 2 substrate under O 2 pressure D. Shahar, Weizmann Institute of Science InOx Amorphous Indium Oxide Thickness : 15 nm (red & grey) and 30 nm (bleu) – 3D regime InO#1 InO#3 1 mm

7. Nearly critical samples High disorder (red) and low disorder (bleu) k F l e ~ 0.4-0.5 < 1  localized regime (Ioffe-Regel criterion) T c comprised between 1K and 2K Carrier density : N = 3.5 x 10 21 cm -3 InO#1 InO#2 InO#1 InO#3 D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, 10917 (1992) InOx V. F. Gantmakher et al., JETP 82, 951 (1996)

8. Fit : s-wave BCS density of states Typical spectrum measured at 50 mK InO#3 STM and transport measurements  Absence of quasi-particle excitations at low energies

9. Spatial fluctuations of the spectral gap Δ (r) II.1 Superconducting inhomogeneitiesSuperconducting inhomogeneities Map of the spectral gap Gaussian distribution InO#3

10. II.1 Superconducting inhomogeneitiesSuperconducting inhomogeneities Spectra measured at different locations (T=50mK) Spatial fluctuations of the spectral gap Δ (r) G, Normalized Gaussian distribution InO#3

11. II.1 Superconducting inhomogeneitiesSuperconducting inhomogeneities Fluctuations of Δ (r) and superconducting transition Spatial fluctuations of ∆ / (1.76k B )

12. Spatial fluctuations of the coherence peaks height Coherence peaks G, Normalized

13. Spatial fluctuations of the coherence peaks height Coherence peaks Statistical study G, Normalized InO#3 G, Normalized

14. Incoherent spectra Statistical study Full spectral gap without coherence peaks G, Normalized InO#3

15. Incoherent spectra Statistical study G, Normalized High disorder Full spectral gap without coherence peaks InO#1

16. λ λ Superconducting Insulating A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001) Signature of localized Cooper pairs

17. λ « Insulating » gap E gap  Localized Cooper pairs Superconducting gap Δ  delocalized Cooper pairs G, Normalized Signature of localized Cooper pairs

18. resistivity × 2 InO#3 T c ~ 1.7 K InO#1 T c ~ 1.2 K  Proliferation of spectra without coherence peaks Proliferation of incoherent spectra at the superconductor-insulator transition ~ ~ B. Sacépé, et al., Nature Physics (2011)

19. resistivity × 2 InO#3 T c ~ 1.7 K InO#1 T c ~ 1.2 K  Proliferation of spectra without coherence peaks Proliferation of incoherent spectra at the superconductor-insulator transition ~ ~ High disorder Low disorder B. Sacépé, et al., Nature Physics (2011) M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)

20. Pseudogap state above Tc TcTc

21. TcTc

23.  BCS peaks appear along with superconducting phase coherence Macroscopic quantum phase coherence probed at a local scale T peak ~ T c Definition of Tc

24.  Phase coherence signaled at T c independently of ∆(r)  Justification of T c Definition of Tc Macroscopic quantum phase coherence probed at a local scale

25. Signatures of the localization of Cooper pairs A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001) K. Bouadim, Y. L. Loh, M. Randeria, N. Trivedi, Nature Physics (2011) M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010) Spatial fluctuations of the spectral gap rectangular-shaped” gap at T«T c Statistic distribution of coherence peaks height Pseudogap regime above T c Anomalously large spectral gap

26. Superconductivity beyond the mobility edge M. Feigel’man et al., Phys. Rev. Lett. 98, 027001, (2007) M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)  E gap = Δ p + Δ BCS ∆ p “parity gap”: pairing of 2 electrons in localized wave functions ∆ BCS “BCS gap”: long-range SC order between localized pairs BCS model built on fractal eigenfunctions of the Anderson problem Two contributions to the spectral gap

27. Superconductivity beyond the mobility edge M. Feigel’man et al., Phys. Rev. Lett. 98, 027001, (2007) M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)  E gap = Δ p + Δ BCS ∆ p “parity gap”: pairing of 2 electrons in localized wave functions ∆ BCS “BCS gap”: long-range SC order between localized pairs BCS model built on fractal eigenfunctions of the Anderson problem Two contributions to the spectral gap E gap = Δ p + Δ BCS “parity gap” “BCS gap” Tunneling spectroscopy (single-particle DOS) E gap = Δ p + Δ BCS “BCS gap” Point-contact spectroscopy (Andreev reflection = transfer of pairs) How to measure the SC order parameter ? Tunnel barrier Transparent interface

28. Point-Contact Andreev Spectroscopy Conductance of a N/S contact Sample Tip Sample Tip Sample Tip Normal metal Superconductor Barrier : parameter Z Transmission : T = 1 / (1+ Z 2 ) Z » 1 Z ~ 1 Z « 1 T = 300 mK Z value Tunnel regime Contact regime Single-particle transfer ~ T Two-particles transfer ~ T 2 Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, 7 4515 (1982)

29. Point-Contact Andreev Spectroscopy From tunnel to contact in a-InOx Tunnel regime Contact regime T=50mK R c =65kΩ R c =6kΩ E gap ∆ BCS E gap = Δ p + Δ BCS

30. TcTc Andreev signal : evolution with T E gap (T) = Δ p + Δ BCS (T)  E gap and ∆ BCS evolve on distinct temperature range  ∆ BCS : local signature of SC phase coherence. ∆ BCS evolves between 0 and ~ T c. E gap evolves between 0 and ~ 3-4 T c T-evolution of Andreev signal

31. Distinct energy scales for pairing and coherence Two distinct energy scales in a-InOx  Distinct energy scales for pairing and coherence in disordered a-InO x E gap (r) = Δ p (r) + Δ BCS ∆ BCS probed by AR remains uniform E gap probed by STS fluctuates ∆ BCS [10 -4 eV] E gap [10 -4 eV] 11 evolutions on 3 samples ∆ BCS = E gap

32. Conclusion : Fluctuation and localization of preformed Cooper pairs Fluctuating Cooper-Pairs above Tc “Partial” condensation of these preformed pairs below Tc SIT occurs through the localization of Cooper-pairs But inhomogeneous superconducting state at T<Tc Homogeneously disordered system with a continuous decrease of Tc Distinct energy scales for pairing and coherence