1. Benjamin Sacépé Institut Néel, CNRS & Université Joseph Fourier, Grenoble Localization of preformed Cooper-pairs in disordered superconductors Lorentz Center, Leiden 2011

2. Benjamin Sacépé Institut Néel, CNRS & Université Joseph Fourier, Grenoble Localization of preformed Cooper-pairs in disordered superconductors Thomas Dubouchet, Claude Chapelier, Marc Sanquer CEA Grenoble Maoz Ovadia, Dan Shahar Weizmann Institute of Science, Rehovot M. Feigel’man L.D. Landau Institut for Theoretical Phyiscs, Moscow L. Ioffe Rutgers University, Piscataway Lorentz Center, Leiden 2011

3. Superconductor-Insulator Transition (SIT) Main ingredients : 1.Disorderlocalization 2.Attractive pairing superconducting phase 3.Coulomb interactioncompetes with pairing 4.Reduced dimensionality affects 1,2 and 3 d = Quench condensed Bismuth D.B. Haviland, Y. Lui, A.M. Goldman, PRL (‘89)

4. Amorphous indium oxide G. Sambandamurthy, et al. PRL 94, 017003, (2005) Magnetic field-tuned SIT For similar results in TiN films see : T. Baturina, et al. PRL (2007) V. F. Gantmakher et al., JETP 82, 951 (1996)

5. 0.20K Superconductor Magnetic field-tuned SIT

6. 0.20K Insulator Positive magnetoresistance at low field : Superconducting correlations in insulators !? Superconductor Magnetic field-tuned SIT

7. 0.20K Insulator Positive magnetoresistance at low field : Superconducting correlations in insulators !? « Insulating correlations » in superconductors ??? Superconductor Magnetic field-tuned SIT

8. mK-STM setup : tunneling spectroscopy Cryostat : inverted dilution 50mK< T< 6K C. Chapelier’s setup, CEA Grenoble  Combined transport and spectroscopy measurements

9. Thickness : 15 nm (blue) and 30 nm (red)  3D regime Samples : e-gun evaporation onto Si/SiO 2 substrate of high purity In 2 O 3 under O 2 pressure 1 mm InO#1 InO#2 Amorphous Indium Oxyde (a-InO x )

10. Thickness : 15 nm (blue) and 30 nm (red)  3D regime 1 mm InO#1 InO#2 Amorphous Indium Oxyde (a-InO x ) V. F. Gantmakher et al., JETP 82, 951 (1996) Samples : e-gun evaporation onto Si/SiO 2 substrate of high purity In 2 O 3 under O 2 pressure

11. Fit : s-wave BCS density-of-states Typical spectrum measured at 50 mK InO#1 Tunneling spectroscopy of amorphous indium oxyde

12. Inhomogeneities of Δ(r) Map of the spectral gap Gaussian distribution For similar results in TiN and NbN films see : B. Sacépé, et al. PRL 101, 157006 (2008) M. Mondal, et al. PRL 106, 047001 (2011)

13. Spectra measured at different locations (T=50mK) G, Normalized Inhomogeneities of Δ(r) Gaussian distribution For similar results in TiN and NbN films see : B. Sacépé, et al. PRL 101, 157006 (2008) M. Mondal, et al. PRL 106, 047001 (2011)

14. BCS ratio Δ/T c =1.76 ? Fluctuations of Δ(r) and superconducting transition

15.  Definition of T c : zero-resistance state (macroscopic phase coherence)

16. G, Normalized Fluctuations of peak heights ! Fluctuations of the BCS peaks

20. Extreme case : « Insulating » gap Spectra measured at different locations (T=50mK)

21. G, Normalized High disorder sample Let’s approach the SIT Sample InO#2 : disorder  2 InO#1 InO#2

22. resistivity × 2 InO#1 T c ~ 1.7 K InO#2 T c ~ 1.3 K  Proliferation of gaps without peaks Let’s approach the SIT Increase of disorder InO#1 InO#2

23. Anderson model : Attractive interaction : Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor 65 A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001) Disorder 2D localization length : λ Superconductivity and disorder With increasing disorder :  Superconductivity becomes « granular-like »

24. λ Superconductivity and disorder With increasing disorder :  Superconductivity becomes « granular-like »  Spectral gap remains finite even at large disorder  Spectral gap is NOT anymore the SC order parameter λ Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor 65 A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001)

25. λ  Insulating gap induced by pairing interaction λ Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor 65 A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001) Superconductivity and disorder M. Ma, and P. A. Lee, PRB 32, 5658, (1985) M. Feigel’man, et al., PRL 98, 027001, (2007) M. Feigel’man, et al, Ann. Phys. 325, 1390 (2010) M. Feigel’man, et al, PRB (2010)

26. Insulating gap due to pairing P. W. Anderson, J. Phys. (Paris) Colloq. 37, C4-339 (1976) M. Ma, and P. A. Lee, PRB 32, 5658, (1985) K. A. Matveev and A. Larkin, PRL. 78, 3749, (1997) 65 A. Ghosal, et al. PRL 81, 3940, (1998) and PRB 65, 014501 (2001) M. Feigel’man, et al. PRL 98, 027001, (2007) M. Feigel’man, et al. Ann. Phys. 325, 1390 (2010) M. Feigel’man, et al. PRB 82, 184534 (2010) In the lowest order: Reduced BCS Hamiltonian built on eigenstates of the Anderson problem with In the high-disorder regime when

27. λ  Insulating gap induced by pairing interaction λ Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor 65 A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001) Superconductivity and disorder M. Ma, and P. A. Lee, PRB 32, 5658, (1985) M. Feigel’man, et al., PRL 98, 027001, (2007) M. Feigel’man, et al, Ann. Phys. 325, 1390 (2010) M. Feigel’man, et al, PRB (2010)

28. Superconductivity and disorder Recent QMC simulations K. Bouadim, Y. Loh, M. Randeria, N. Trivedi, arXiv:1011.3275 Disorder SC Insulator

29. Pairing gap in the insulator resistivity × 2  Proliferation of incoherent –localized— Cooper-pairs when approaching the SIT when approaching the SIT

30. Pairing gap in the insulator resistivity × 2  Proliferation of incoherent –localized— Cooper-pairs when approaching the SIT when approaching the SIT Simulations on the Bethe lattice Lev Ioffe, Misha Feigel’man M. Feigel’man, et al., PRL (2007) M. Feigel’man, et al, Ann. Phys. (2010) M. Feigel’man, et al, PRB (2010)

31. TcTc Pseudogap above T c : preformed pairs T-dependence of the local DOS Pseudogap in TiN and NbN films : B. Sacépé, et al. Nature Commun. 1:140 (2010) M. Mondal, et al. PRL 106, 047001 (2011)

32. Pseudogap in quasi-2D conventional superconductors A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983) B. Sacépé, et al. Nature Commun. 1:140 (2010) Superconducting fluctuations in quasi-2D TiN films ( thickness < 5nm )

33. Pseudogap above T c  The overall shape is the same

34. Local versus macroscopic phase coherence T peak : Temperature below which peaks start to grow  BCS peaks give a local signature of the superconducting phase coherence  BCS peaks give a local signature of the superconducting phase coherence BCS peaks appears at Tc independently of gap inhomogeneities

35. « Insulating » gap at T<<T c  Local pairing without phase coherence at T << Tc Formation of a pseudogap without BCS peaks at T>T c  Local pairing without phase coherence at T > T c Spectral signature of localized Cooper pairs Preformed Cooper-pairs Condensation versus localization of preformed Cooper pairs

36. Conclusions Localization of preformed Cooper pairs in disordered superconductors Nature Physics 7, 239 (2011) Preformed Cooper-Pairs above Tc Pseudogap in the DOS between Tc and ~ 3-4 Tc “Partial” condensation of pairs below Tc Rectangular spectra at 50mK = localized Cooper pairs SIT occurs through the localization of Cooper pairs Gap in the DOS remains & coherence peaks disappear

37. Spatial fluctuations Traces of spectra : fluctuations of the spectral properties (T=50mK)

38. Disorder-enhanced Coulomb interaction B. Altshuler, et al., Phys. Rev. Lett. 44, 1288, (1980) Coulomb interaction : Zero-Bias Anomaly

39. Soft coulomb gap Superconducting fluctuations We need a global theory We need a global theory ! TiN 1 Quantum corrections to the DOS

40. Superconducting fluctuations (2D Aslamasov-Larkin correction …) Disorder-enhanced Coulomb interaction (2D weak-localization/Aronov-Altshuler corrections) Quantum corrections to the conductivity

41. Dynamical Coulomb blockade P. Joyez and D. Estève, PRB 56, 1848, (1997)

42. Magnetic field-tuned SIT Huge magnetoresistance peak : superconductivity-related ? Amorphous indium oxyde d = 15 nm

43. Spectral gap map Coherence peak map Spectral fluctuations

44. Disorder-induced inhomogeneities in TiN B. Sacépé, et al. PRL 101, 157006 (2008) Titanium nitride T. Baturina (Novosibirsk) V. Vinokur (Argonne National Lab.) Gaped insulator made of localized Cooper pairs ? Disorder-tuned SIT in ultra-thin films of TiN