Disordered Superconductors
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1 Cargese 2016 Disordered Superconductors Claude Chapelier, INAC-PHELIQS, CEA-Grenoble
2 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.
3 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, J.P. Burger la supraconductivité des métaux, des alliages et des films minces (Ed.Masson)
4 BCS theory for clean systems Bloch plane waves 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)
5 Theory of dirty superconductors 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) In superconducting grains, superconductivity disappears when the mean level spacing between different electronic states becomes greater than the superconducting gap Anderson criterion for superconductivity : ν Δ L 3 > 1
6 Localization in disordered metals P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958)
7 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958)
8 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Are Cooper pairs getting localized?
9 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Are Cooper pairs getting localized? Leggett s argument : every Bose system is superfluid at T=0 A.J. Legett, Topics in the theory of Helium Physica Fennica 8, 125 (1973)
10 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Are Cooper pairs getting localized? Leggett s argument : every Bose system is superfluid at T=0 A.J. Legett, Topics in the theory of Helium Physica Fennica 8, 125 (1973)
11 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Are Cooper pairs getting localized? Leggett s argument : every Bose system is superfluid at T=0 A.J. Legett, Topics in the theory of Helium Physica Fennica 8, 125 (1973) d = d = D.B. Haviland, Y. Lui, A.M. Goldman, PRL 62, 2180 (1989)
12 Localization in disordered metals P.A. Lee and T.V. Ramakrishnan Disordered electronic systems Rev. Mod. Phys. 57, 287(1985) P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958) Are Cooper pairs getting localized? Leggett s argument : every Bose system is superfluid at T=0 A.J. Legett, Topics in the theory of Helium Physica Fennica 8, 125 (1973) d = d = Strong localization, fluctuations and Coulomb interactions D.B. Haviland, Y. Lui, A.M. Goldman, PRL 62, 2180 (1989)
13 Localization in disordered superconductors 3D localization Insulator Metal Inhomogeneous superconducting state k F l 1 ν Δ ξ 3 loc ~ 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, (2001) M. Feigel man et al., Phys. Rev. Lett. 98, (2007) ; Ann.Phys. 325, 1390 (2010)
14 Localization in disordered superconductors Insulator Inhomogeneous superconducting state 3D localization Metal H 2D numerical simulation n n i.. int i i H t c c h c V n 0 i j i i, i, j, i, k F l 1 ν Δ ξ 3 loc ~ 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, (2001) M. Feigel man et al., Phys. Rev. Lett. 98, (2007) ; Ann.Phys. 325, 1390 (2010)
15 Localization in disordered superconductors Insulator Inhomogeneous superconducting state 3D localization Metal H 2D numerical simulation n n i.. int i i H t c c h c V n 0 i j i i, i, j, i, k F l 1 ν Δ ξ 3 loc ~ 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, (2001) M. Feigel man et al., Phys. Rev. Lett. 98, (2007) ; Ann.Phys. 325, 1390 (2010)
16 Localization in disordered superconductors Insulator Inhomogeneous superconducting state 3D localization Metal H 2D Monte Carlo calculation n n i.. int i i H t c c h c V n 0 i j i i, i, j, i, k F l 1 ν Δ ξ 3 loc ~ 1 k F l ~ 1 k F l 1 Cooper pairing beyond the mobility edge leads to an inhomogeneous superconductor The transition to an insulator requires quantum fluctuations 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, (2001) M. Feigel man et al., Phys. Rev. Lett. 98, (2007) ; Ann.Phys. 325, 1390 (2010)
17 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
18 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T T > T c Amplitude fluctuations ξ(t) τ = π ђ 8 k B (T T c ) A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
19 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T T > T c Amplitude fluctuations T > T c Amplitude and phase fluctuations ξ(t) ξ(t) τ = π ђ 8 k B (T T c ) A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
20 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T T > T c Amplitude fluctuations T > T c Amplitude and phase fluctuations 2D : Berezinskii Kosterlitz - Thouless ξ(t) ξ(t) τ = π ђ 8 k B (T T c ) A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
21 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T T > T c Amplitude fluctuations T > T c Amplitude and phase fluctuations 2D : Berezinskii Kosterlitz - Thouless ξ(t) ξ(t) τ = π ђ 8 k B (T T c ) δt c T c ~ G i(d) 1 ν k B T c ξ 0 d 2 4 d Clean 3D superconductor : G i(3) 80 k BT c E F 4 ~ Dirty 2D superconductor : 3D localized superconductor : G i(2) ν Δ ξ 3 loc ~ 1 e2 23 ђ R ξ 0 ~ξ loc G i ~1 A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
22 Superconducting fluctuations Thermal fluctuations ψ op = T eiφ T T > T c Amplitude fluctuations T > T c Amplitude and phase fluctuations 2D : Berezinskii Kosterlitz - Thouless ξ(t) ξ(t) τ = π ђ 8 k B (T T c ) δt c T c ~ G i(d) 1 ν k B T c ξ 0 d 2 4 d Clean 3D superconductor : G i(3) 80 k BT c E F 4 ~ Dirty 2D superconductor : 3D localized superconductor : e2 G i(2) 23 ђ R ν Δ ξ 3 loc ~ 1 ξ 0 ~ξ loc G i ~1 Disorder drastically enhances thermal fluctuations A. Larkin and A. Varlamov, Theory of fluctuations in superconductors, Oxford University Press (2006)
23 TIN Superconductor-Insulator transition TiN R [k ] ,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 T. I. Baturina, et al.prl 99, (2007) Sacépé et al., PRL 101, (2008)
24 TIN Superconductor-Insulator transition R [k ] ,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 Sacépé et al., PRL 101, (2008)
25 TIN Superconductor-Insulator transition Sacépé et al., PRL 101, (2008)
26 TIN Superconductor-Insulator transition Sacépé et al., PRL 101, (2008)
27 Superconducting fluctuations Quantum fluctuations ψ op = e iφ K.B. Efetov, Phase transition in granulated superconductors, JETP 51, 1016 (1980) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997)
28 Superconducting fluctuations Quantum fluctuations ψ op = e iφ D N φ > 1 2 N, φ E J = h π Δ 8 e 2 R t th( Δ 2T ) E c = e 2 4 π ϵ 0 D δ = 1 ν D 3 δ = 1 3 ν ξ loc K.B. Efetov, Phase transition in granulated superconductors, JETP 51, 1016 (1980) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997)
29 Superconducting fluctuations Quantum fluctuations ψ op = e iφ D N φ > 1 2 E J = h π Δ 8 e 2 R t th( Δ 2T ) N, φ Gallium E c = e 2 4 π ϵ 0 D δ = 1 ν D 3 δ = 1 3 ν ξ loc H. M. Jaeger, et al. Phys.Rev.B 34, 4920 (1986) K.B. Efetov, Phase transition in granulated superconductors, JETP 51, 1016 (1980) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997)
30 Superconducting fluctuations Quantum fluctuations N φ > 1 2 E J = h π Δ 8 e 2 R t th( Δ 2T ) E c = e 2 4 π ϵ 0 D δ = 1 ν D 3 δ = 1 3 ν ξ loc ψ op = e iφ N, φ D Gallium H 2D Monte Carlo calculation n n i.. int i i H t c c h c V n 0 i j i i, i, j, i, H. M. Jaeger, et al. Phys.Rev.B 34, 4920 (1986) K.B. Efetov, Phase transition in granulated superconductors, JETP 51, 1016 (1980) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997)
31 Superconducting fluctuations Quantum fluctuations N φ > 1 2 E J = h π Δ 8 e 2 R t th( Δ 2T ) E c = e 2 4 π ϵ 0 D δ = 1 ν D 3 δ = 1 3 ν ξ loc ψ op = e iφ N, φ D Gallium H 2D Monte Carlo calculation n n i.. int i i H t c c h c V n 0 i j i i, i, j, i, Parity gap H. M. Jaeger, et al. Phys.Rev.B 34, 4920 (1986) Δ p = δ 2 ln δ Δ = 1.76 k B T c K.B. Efetov, Phase transition in granulated superconductors, JETP 51, 1016 (1980) K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997) E g = + p K.A. Matveev and A.I. Larkin Parity effect in ground state energies of ultrasmall superconducting grains, Phys. Rev. Lett. 78, 3749(1997)
32 Coulomb depairing in disordered superconductors A.M Finkelstein Pis sma Zh. Esk. Theor. Fiz., 45, 46 (1987)
33 Coulomb depairing in disordered superconductors A.M Finkelstein Pis sma Zh. Esk. Theor. Fiz., 45, 46 (1987) R. A. Smith, M.Y. Reizer, and J. W. WIlkins Phys. Rev. B 51, 6470(1995)
34 Coulomb depairing in disordered superconductors A.M Finkelstein Pis sma Zh. Esk. Theor. Fiz., 45, 46 (1987) R. A. Smith, M.Y. Reizer, and J. W. WIlkins Phys. Rev. B 51, 6470(1995) Short range Coulomb interaction continuously decreases Tc and Δ in the same proportion
35 Superconductor-insulator transition : two scenarios Amorphous films Granular films Gallium Bismuth d = ψ op = T eiφ T d = 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
36 Superconductor-insulator transition : two scenarios Amorphous films Granular films Frydman, A., Physica C : Superconductivity 391, 189 (2003) Hsu, S.-Y., and Valles, J. M. Phys. Rev. B 48, 4164 (1993) Continuous decrease of T c Cooper pairing suppressed at the SIT SIT due to phase fluctuations Cooper pairs localized in grains
37 TIN Superconductor-Insulator transition TiN R [k ] ,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 Sacépé et al., PRL 101, (2008)
38 TIN Superconductor-Insulator transition TiN R [k ] ,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 Sacépé et al., PRL 101, (2008)
39 TIN Superconductor-Insulator transition TiN R [k ] ,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 2,0 2,0 2,0 G(V), normalized 1,5 1,0 0,5 = 154 µev T eff = 0,35 K G(V), normalized 1,5 1,0 0,5 = 225 µev T eff = 0,32 K G(V), normalized 1,5 1,0 0,5 = 260 µev T eff = 0,25 K 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] Increasing disorder Sacépé et al., PRL 101, (2008)
40 TIN Superconductor-Insulator transition TiN R [k ] ,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 2,0 2,0 2,0 G(V), normalized 1,5 1,0 0,5 = 154 µev T eff = 0,35 K G(V), normalized 1,5 1,0 0,5 = 225 µev T eff = 0,32 K G(V), normalized 1,5 1,0 0,5 = 260 µev T eff = 0,25 K 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] Increasing disorder Sacépé et al., PRL 101, (2008)
41 TIN Superconductor-Insulator transition TiN T c [K] Δ/T c ,0 2,0 2,0 G(V), normalized 1,5 1,0 0,5 = 154 µev T eff = 0,35 K G(V), normalized 1,5 1,0 0,5 = 225 µev T eff = 0,32 K G(V), normalized 1,5 1,0 0,5 = 260 µev T eff = 0,25 K 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] 0,0-1,0-0,5 0,0 0,5 1,0 V [mv] Increasing disorder Sacépé et al., PRL 101, (2008)
42 TIN Superconductor-Insulator transition T c [K] Δ/T c Sacépé et al., PRL 101, (2008)
![TIN Superconductor-Insulator transition T c [K] Δ/T c 4.7 1.](/docs-images/92/108402923/images/43-1.jpg "8 1.3 2.3 1 2.6 0.45 4 Sacépé et al.")
43 TIN Superconductor-Insulator transition T c [K] Δ/T c Sacépé et al., PRL 101, (2008)
44 Superconductivity and Coulomb interaction M. A. Skvortsov and M. V. Feigel man, Phys. Rev. Lett. 95, , (2005) Spatial fluctuations of Tc
45 Superconductivity and Coulomb interaction M. A. Skvortsov and M. V. Feigel man, Phys. Rev. Lett. 95, , (2005) Spatial fluctuations of Tc M.V. Feigelman and M.A. Skvortsov, Phys. Rev. Lett. 109, (2012) A.I. Larkin and Yu. N. Ovchinnikov, Sov. JETP 34, 1144 (1972)
46 Thermal dependence of the Density of States W. Escoffier, et al., PRL 93, , (2004)
47 Thermal dependence of the Density of States W. Escoffier, et al., PRL 93, , (2004)
48 Thermal dependence of the Density of States Pseudogap above Tc B. Sacépé et al., Nat. Comm., (2010)
49 Thermal dependence of the Density of States Pseudogap above Tc B. Sacépé et al., Nat. Comm., (2010)
50 Thermal dependence of the Density of States Pseudogap above Tc Pseudogap is due to pre-formed Cooper pairs B. Sacépé et al., Nat. Comm., (2010)
51 InOx Superconductor-Insulator transition InO x InO#2 V. F. Gantmakher et al., JETP 82, 951 (1996) InO#1 D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, (1992)
52 InOx Superconductor-Insulator transition InO x InO#2 V. F. Gantmakher et al., JETP 82, 951 (1996) InO#1 D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, (1992)
53 Thermal dependence of the Density of States InO x Pseudogap above Tc B. Sacépé et al., Nat. Phys. (2011)
54 Thermal dependence of the Density of States InO x Pseudogap above Tc Spectra without coherence peaks are the signature of localized pre-formed Cooper pairs B. Sacépé et al., Nat. Phys. (2011)
55 Point-Contact Andreev Spectroscopy Parity gap How to measure the order parameter? E g = + p T = Z 2 Normal metal Superconductor
56 Point-Contact Andreev Spectroscopy Parity gap How to measure the order parameter? E g = + p
57 Point-Contact Andreev Spectroscopy Parity gap How to measure the order parameter? E g = + p T = Z 2 Normal metal Superconductor Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, (1982) Contact regime Z value Z» 1 Z ~ 1 T = 300 mk Z «1 Tunnel regime
58 Point-Contact Andreev Spectroscopy Parity gap How to measure the order parameter? E g = + p T = Z 2 Normal metal Superconductor Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, (1982) Contact regime Tip Z value Z» 1 Sample Tip Z ~ 1 Sample T = 300 mk Z «1 Tunnel regime Tip Sample
59 Point-Contact Andreev Spectroscopy Parity gap How to measure the order parameter? E g = + p T = Z 2 Normal metal Superconductor Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, (1982) Contact regime Z value Z» 1 Tip Sample E g = + p Z ~ 1 Tip Sample T = 300 mk Z «1 Tunnel regime Tip Sample E g = + p
60 Point-Contact Andreev Spectroscopy InO x InOx film far from the Superconductor-Insulator Transition : Tc = 3.5K Homogeneous No pseudogap
61 Point-Contact Andreev Spectroscopy Close to the superconductor-insulator transition E g = + p Contact regime Z value Z» 1 Tip Sample E g = + p Z ~ 1 Tip Sample T = 300 mk Z «1 Tunnel regime Tip Sample E g = + p
62 Point-Contact Andreev Spectroscopy Close to the superconductor-insulator transition E g = + p Contact regime Z value Z» 1 Tip Sample E g = + p Z ~ 1 Tip Sample T = 300 mk Z «1 Tunnel regime Tip Sample E g = + p E g = + p
63 Superconductivity, disorder, Coulomb interaction and localization Disorder : Stong superconducting fluctuations above Tc Pseudogap due to preformed Cooper pairs Disorder & Coulomb interaction : Continuous decrease of Tc and with disorder Keeps /Tc ratio constant Spatial mesoscopic fluctuations of Tc Disorder & Localization : Tc decreases faster than with disorder : huge /Tc ratio Parity gap Strong spatial fluctuations of Localized Cooper pairs characterized by spectra without coherence peaks
64 Magnetic field studies through the SIT InO x G. Sambandamurthy et al., Phys. Rev. Lett. 92, , (2004) G. Kopnov et al., Phys. Rev. Lett. 109, , (2012)
65 Magnetic field studies through the SIT InO x Bi ϕ = h 2e G. Sambandamurthy et al., Phys. Rev. Lett. 92, , (2004) Stewart, Jr. et al., Science 318, 1273, (2007) H.Q. Nguyen et al., Phys. Rev. Lett. 103, (2009) G. Kopnov et al., Phys. Rev. Lett. 109, , (2012)
66 Magnetic field studies through the SIT InO x Bi ϕ = h 2e G. Sambandamurthy et al., Phys. Rev. Lett. 92, , (2004) Stewart, Jr. et al., Science 318, 1273, (2007) H.Q. Nguyen et al., Phys. Rev. Lett. 103, (2009) G. Kopnov et al., Phys. Rev. Lett. 109, , (2012) Cooper pair insulator?
67 Magnetic field studies through the SIT D. Sherman et al., Phys. Rev. Lett. 108, , (2012)
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