Superconducting fluctuations, interactions and disorder : a subtle alchemy

Les défis actuels de la supraconductivité Dautreppe 2011 Superconducting fluctuations, interactions and disorder : a subtle alchemy Claude Chapelier, Benjamin Sacépé, Thomas Dubouchet INAC-SPSMS-LaTEQS, CEA-Grenoble

Superconductivity in pure metals Kamerlingh Onnes, H., "Further experiments with liquid helium. C. On the change of electric resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures." Comm. Phys. Lab. Univ. Leiden; No. 120b, 1911.

Superconductivity in pure metals and alloys Kamerlingh Onnes, H., "Further experiments with liquid helium. C. On the change of electric resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures." Comm. Phys. Lab. Univ. Leiden; No. 120b, 1911. J.P. Burger la supraconductivité des métaux, des alliages et des films minces (Ed.Masson)

BCS theory for clean systems k k ξ 0 Bloch plane waves φ k,σ = 1 V e ikr φ k,σ = 1 V e ikr k k J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. B. 108, 1175, (1957) J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106, 162, (1957)

Theory of dirty superconductors ξ0 l H t =H 0 +H dis φ n,σ = n <n k>φ k,σ φ n,σ= n <n k> φ k, σ E n Same energy if the scatterers are non-magnetic (time-reversed symmetry) P.W. Anderson, J. Phys. Chem. Solids. 11, 26, (1959) A.A. Abrikosov & I.P. Gorkov, Sov. Phys. JETP 8, 1090, (1959) Nota : if the particle size become small, the mean level spacing between different would become greater than the superconducting gap and superconductivity will be forbidden

Localization and superconductivity P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Insulator Metal Inhomogeneous superconducting state k F l 1 k F l 1 k F l 1 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) M. Feigel man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)

Superconductor-insulator transition : two scenarios Bosonic Ψ(r,T)= (r,t)e iϕ(r,t) M.P.A Fisher and G. Grinstein Anderson, Presence of quantum diffusion in twodimensions : universal resistance at the superconductor-insulator transition Phys. Rev. Lett 64, 587(1990)

Superconductor-insulator transition : two scenarios Bosonic Ψ(r,T)= (r,t)e iϕ(r,t) Fermionic M.P.A Fisher and G. Grinstein Anderson, Presence of quantum diffusion in twodimensions : universal resistance at the superconductor-insulator transition Phys. Rev. Lett 64, 587(1990)

Superconductor-insulator transition : the role of Coulomb interaction Amorphous films Bismuth Granular films Gallium d = Bosonic Ψ(r,T)= (r,t)e iϕ(r,t) d = Fermionic D.B. Haviland, Y. Lui, A.M. Goldman, PRL 62, 2180 (1989) H. M. Jaeger, et al. Phys.Rev.B 34, 4920 (1986) Continuous decrease of T c Cooper pairing suppressed at the SIT Competition between E C and E J Cooper pairs localized in grains

Superconductor-insulator transition in highly disordered TiN films 8 7 6 5 k F l 1 R [kω] 4 3 2 1 0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 T [K] TiN 1 TiN 2 TiN 3 B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008) B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)

Superconductor-insulator transition in highly disordered TiN films H = n n int λ i i i + (..) ( µ ) H = t c c + h c + V n 0 iσ jσ i i, σ < i, j>, σ i, σ A. Ghosal et al., Phys. Rev. Lett 81, 3940 (1998); Phys. Rev. B 65, 0145001 (2001)

Beyond the mean-field BCS theory : superconducting fluctuations Order parameter Ψ(r,T)= (r,t)e iϕ(r,t) Ginzburg-Landau functional (zero magnetic field) F[Ψ(r)] = F n + dv{a Ψ(r) 2 + b 2 Ψ(r) 4 + 4mi Ψ(r) 2 } a = α(t T c ) ξ 2 = 2 4mαT c τ GL = π 8k B (T T c ) ξ(t) = ξ 0 Tc T T c G i e2 23 R G i ǫ=ln( T T c ) 1 A. Larkin & A. Varlamov, in theory of superconducting fluctuations (Oxford)

superconducting fluctuations in highly disordered TiN films 8 7 6 5 R [kω] 4 3 2 1 0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 T [K] TiN 1 TiN 2 TiN 3

superconducting fluctuations in highly disordered TiN films W. Escoffier et al., Phys. Rev. Lett. 93, 217005, (2004) B. Sacépé et al., Nat. Comm., (2010)

superconducting fluctuations in highly disordered TiN films B. Sacépé et al., Nat. Comm., (2010)

Superconductor-insulator transition in highly disordered InO films High disorder (red) and low disorder (blue) k F l<1 InO#1 InO#2 InO#3 InO#1 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

Superconductor-insulator transition in highly disordered InO films Map of the spectral gap Spectra measured at different locations (T=50mK) G, Normalized G, Normalized Gaussian distribution

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

Coherence peaks Spatial fluctuations of the coherence peaks height Statistical study G, Normalized G, Normalized R = G( ) G( ev > ) G( ev > )

Incoherent spectra Full spectral gap without coherence peaks Statistical study R = G( ) G( ev > ) G( ev > )

Incoherent spectra Full spectral gap without coherence peaks Statistical study G, Normalized R = G( ) G( ev > ) G( ev > ) G, Normalized High disorder

Proliferation of incoherent spectra at the superconductor-insulator transition Increase of disorder σ ~ 8% InO#3 T c ~ 1.7 K resistivity 2 σ ~ 16% InO#1 T c ~ 1.2 K Proliferation of spectra without coherence peaks B. Sacépé et al., Nat. Phys. (2011)

Signature of localized Cooper pairs Numerical calculations: superconductivity with disorder A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001) Superconducting Insulating With increasing disorder: Superconductivity becomes «granular-like» Spectral gap is not the SC order parameter

Pseudogap state above Tc

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

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

Fractal superconductivity near the mobility edge p parity gap : pairing of 2 electrons in localized wave functions E gap = p + BCS BCS BCS gap : long-range SC order between localized pairs p BCS gap Tunneling spectroscopy (single-particle DOS) Tunnel barrier parity gap E gap = p + BCS BCS gap Point-contact spectroscopy (Andreev reflection = transfer of pairs) Transparent interface E gap = p + BCS

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

Point-Contact Andreev Spectroscopy From tunnel to contact regime Contact regime Control parameter: Contact resistance R c = V Bias / I t Tunnel regime Agrait, N. et al., Phys. Rev. B 46, 9 5814 (1992)

Point-Contact Andreev Spectroscopy From tunnel to contact in a-inox R c =6k Contact regime T=50mK R c =65k Tunnel regime

Distinct energy scales for pairing and coherence From tunnel to contact in a-inox BCS Contact regime E gap = p + BCS. Contact: additional peaks at ev ± 200 µev. Energy scale BCS independent of R c E gap Tunnel regime

Distinct energy scales for pairing and coherence Two distinct energy scales in a-inox E gap (r)= p (r)+ BCS 11 evolutions on 3 samples BCS [10-4 ev] BCS =E gap BCS probed by AR remains uniform E gap probed by STS fluctuates E gap [10-4 ev] Distinct energy scales for pairing and coherence in disordered a-ino x

Conclusion Interactions : Coulomb Tc decreases continuously in homogeneously disordered systems Inhomogeneous superconducting state Spatial fluctuations of the gap : a precursor to Cooper pair localization Fluctuations : 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 Distinct energy scales for pairing and coherence STS measures E gap and Andreev reflection measures BCS