Superconductor to insulator transition: a short overview on recent ideas. C.Castellani

Transcription

1 Superconductor to insulator transition: a short overview on recent ideas C.Castellani Collaborations L.Benfatto and J.Lorenzana (Roma), G.Seibold (Cottbus) G.Lemarié (Toulouse),D.Bucheli(PhD,Roma) References Seibold et al. PRL 108, (2012) G. Lemarié et al. PhysRevB 87, (2013).

2 Final fate of dirty superconductors by increasing disorder Two energy scales pairing energy Δ gap : gap in tunneling (DOS) and optics superfluid stiffness D s : phase coherence and superfluid response

3 Disappearing of SC by increasing disorder Fermionic vs bosonic mechanism Superconductor-Insulator Transition (SIT) Fermionic mechanism (Finkelstein): disorder enhances Coulomb repulsion, pairing strength decreases, both Tc and Δ go to zero FM or FI Bosonic mechanism (Fisher, Ma & Lee, etc.) direct localization of Cooper pairs, finite pairing in the insulating phase

4 Jim Valles, Leiden 2011 Thickness tuned SIT: amorphous Bi films (50nm) Δ gap 0 as T Tc Insulator : Weakly localized electrons Small positive MR Δ gap 0 at T>T c Localized Cooper Pairs with activated transport R=R 0 e To/T T 0 0 at SIT Giant positive MR Bosonic mechanism at work

5 Jim Valles, Leiden 2011 Thickness tuned SIT: amorphous Bi films (50nm) Δ gap 0 as T Tc Insulator : Weakly localized electrons Small positive MR Δ gap 0 at T>T c Localized Cooper Pairs with activated transport R=R 0 e To/T T 0 0 at SIT Giant positive MR Bosonic mechanism at work from granular

6 Jim Valles, Leiden 2011 Thickness tuned SIT: amorphous Bi films (50nm) Δ gap 0 as T Tc Insulator : Weakly localized electrons Small positive MR Δ gap 0 at T>T c Localized Cooper Pairs with activated transport R=R 0 e To/T T 0 0 at T c Giant positive MR Trivial consequence of artificial nanostructure

7 Superconductor to insulator transition: bosonic mechanism at work Homogeneous InO, TiN, NbN films share the same features : Δ gap stays 0 above T c activated transport R=R 0 e To/T, giant positive MR Are they inhomogeneous? Effectively inhomogeneous? General mechanism for inhomogeneity?

8 Ioffe and Mezard proposal for disordered superconductors low temperature glassy phase (with one-step replica symmetry breaking) self-generated inhomogeneity on a mesoscopic scale (much) larger than the scale of disorder and of pairing (ξ sc ) Ioffe and Mezard, PRL 105, (2010); Feigelman, Ioffe and Mezard, PRB 82, (2010)

9 Ioffe and Mezard proposal for disordered superconductors Ising model in a transverse random field (strong coupling pairing with <σ x i> = superconducting order parameter) on a Cayley tree with K-branching # z " i i! i $ g K# < ij> H = $ /!! Mean field- cavity method: recursion formula for the Weiss field acting on σ x i x i x j B i = g /K# < " x j > $ g/k# tanh& % 2 2 j j % 2 2 j + B j j + B j at low T only few paths contribute to pairing susceptibility from boundary to root χ 0L = <σ x 0>/ h x L (by mapping to directed polymer problem) B j root boundary

10 Ioffe and Mezard proposal for disordered superconductors Ising model in a transverse random field (strong coupling pairing with <σ x i> = superconducting order parameter) on a Cayley tree with K-branching # z " i i! i $ g K# < ij> H = $ /!! Mean field- cavity method: recursion formula for the Weiss field acting on σ x i x i x j B i = g /K# < " x j > $ g/k# tanh& % 2 2 j j % 2 2 j + B j j + B j B j Superconductivity on filaments: self-generated inhomogeneity on a mesoscopic scale root boundary

11 Ioffe and Mezard proposal for disordered superconductors Ising model in a transverse random field (strong coupling pairing with <σ x i> = superconducting order parameter) on a Cayley tree with K-branching # z " i i! i $ g K# < ij> H = $ /!! Mean field- cavity method: recursion formula for the Weiss field acting on σ x i x i x j B i = g /K# < " x j > $ g/k# tanh& % 2 2 j j % 2 2 j + B j j + B j Anomalous distribution of the local order parameter s=<σ x i> : P(s) s -α and non self-averaging properties B j root boundary

12 Ioffe and Mezard proposal for disordered superconductors Various questions: Quasi-1d, non self-averaging and anomalous distribution will survive from Cayley tree to real lattices (2d)? (The replica symmetry breaking issue in finite d) Intrinsically strong coupling (bosonic) model: what for a fermionic model of superconductivity from weak to strong coupling? attractive Hubbard model Price to pay: from cavity methods to mean field, however MF can be a reasonable description of the ordered phase

13 The attractive Hubbard model we consider the attractive Hubbard model with on site disorder H # # $ # + = $ t a!!! " ij i a j $ U n n i i i i! in < >, & % i! and solve the mean field Bogoliubov-de Gennes eqs on a 2d finite cluster at T=0 with site dependent SC order parameter Δ i = U <a + ia + i >, and-v 0 <ξ i <V 0 We can study weak coupling with sizeable non locality ξ sc 10 lattice spacing and more parameters (e.g. filling)

14 The attractive Hubbard model Parameters: t=1, U=1.5 10, V 0 =0.1 3,<n>=0.1 1 Various known results from previous BdG and Monte Carlo. Huge literature. See: Ghosal, Randeria and Trivedi, PRL(1998) and PRB (2001) Even for not too large U (with V 0 1) superconductivity is established by coherence of local pairs. Cfr: Feigelman et al Ann.Phys (2010) Strong variations of local SC order parameter Δ i ( Δ gap spectral gap from local Density of States) superconducting islands (inhomogeneity)

15 The attractive Hubbard model Distribution of the local order parameter s i =2Δ i /U 2<a + ia + i > <σ x i>. Note: Δ i is not the DOS gap, it is a measure of coherence predictions of IM in the ordered phase (g critical value): P(s) s m typ/s 1+m for large s, with m=m(k,t/v 0 )<1. Unbounded distribution but for the physical constraint s 1. Averaged <s> >> s typ with s typ =exp<lns> We find a qualitative agreement (broad P(s)), but a different distribution

16 Distribution of the order parameter Very crowded: P(s) Hubbard U=5, g=t 2 /UV 0 =0.08, n=0.85 U=5, g=0.2,n=1 s i =2Δ i /U 2<a + i a+ i > <σx i > XX-Z 2DCMF (Monthus and Garel 2012), g=0.4 MF, g=0.2 (dasheddotted) Cayley, K=3, g=0.2 (dotted)

17 Distribution of the order parameter Universal by rescaling R=(lns-lns typ )/σ s lns typ =<lns> σ s 2 =<(lns- lns typ ) 2 > Hubbard U=5,g=t 2 /UV 0 =0.08, n=0.85 U=5, g=0.2,n=1 XX-Z 2DCMF (Monthus and Garel 2012), g=0.4 MF, g=0.2 (dasheddotted) Cayley, K=3, g=0.2 (dotted)

18 Distribution of the order parameter Universal distribution: within our MF approach the rescaling seems to work independently of parameters in a wide range, with the variance σ s -lns typ increasing for g 0 In the disordered phase analogy with directed polymer problem physics in 2D: lns L = -c L+L ω u with ω 1/3>0 (ω=0 on the Cayley tree) L=distance from ordered boundary, u=random variable with Tracy-Widom distribution

19 magenta dashed dotted line, BdG with U = 1.5, n = 0.875, L = 25, g = 0.2, i.e., V 0 = 3.33.

20 Distribution of the order parameter Comparison with experiments Hypothesis: Δ(r) correlates with relative height of coherent peaks in local density of states in STM (Sacépé et al, Nature Phys. 7, 239, (2011): STM in InO films) Here we compare with data on NbN films from Tata Institute group (Pratap Raychaudhuri and collaborators)

21 Sacépé et al, Nature Phys. 7, 239, (2011): STM in InO films Define peak height h=(g peak -G min )/G min SC OP

22 Lemarié et al. PRB 87, (2013) STM in NbN films

23 Lemarie et al. PRB 87, (2013) STM in NbN films Distribution of the local peak height h SC order parameter s

24 Lemarie et al. PRB 87, (2013) STM in NbN films Rescaled distribution of the local peak height h s=sc OP R s =(lns-lns typ )/σ s

25 Inhomogeneity and glassy physics NbN films M. Chand et al PRB 2012 Inhomogeneity Increasing disorder The emergent mesoscopic inhomogeneity is a signature of glassy superconductivity? Can we make predictions on currents and superfluid density?

26 Current response in a disordered SC It is a tricky job, even within MF: in a clean system J q =χ BCS (q)a q and χ BCS (q 0) gives the superfluid density D S with χ BCS given by the bubble expression with no vertex corrections. χ BCS is not gauge invariant, but it is enough since a transverse A does not change the phase of the SC order parameter Δ In the presence of disorder this is wrong! We solve the BdG eqs in the presence of A and evaluate the related local current density J(r)

27 Current patterns Current in the presence of a finite transverse A, by allowing for the local phases θ i of the BdG solutions to relax to the applied field A A A U =5t, V 0 =2t, n=0.85 size=20x20 map: local lines: constant phase "# i $ 2A arrows: local current without phase relaxation G. Seibold, L.Benfatto, J. Lorenzana and C. Castellani PRL 108, (2012) with phase relaxation Unidimensional patterns for the current: glassy-like behavior

28 Current patterns Current in the presence of a finite transverse A, by allowing for the local phases θ i of the BdG solutions to relax to the applied field A A A without phase relaxation G. Seibold, L.Benfatto, J. Lorenzana and C. Castellani PRL 108, (2012) with phase relaxation D s BCS >>D s = =1/L 2 2 E(A)/ A 2 Missing superfluid density D s BCS - D s = spectral weigth of intragap σ(ω) from collective modes Unidimensional patterns for the current: glassy-like behavior

29 Sub-gap contribution of phase fluctuations Optical conductivity with vertex corrections: in-gap spectral weight due to phase fluctuations Increasing disorder Increasing SC coupling K full = j J J: collective modes (phase, amplitude, charge)

30 Sub-gap contribution of phase fluctuations Optical conductivity with vertex corrections: in-gap spectral weight due to phase fluctuations Increasing disorder Increasing SC coupling

31 Sub-gap contribution of phase fluctuations Optical conductivity with vertex corrections: in-gap spectral weight due to phase fluctuations Increasing disorder Increasing SC coupling Collective-modes contribution at low energies Smaller optical gap than STM gap? Anomalies in G-THz Spectroscopy?

32 Conclusions Disordered superconductors (near SIT): glassy physics? Anomalous P(Δ) distribution Quasi-1D current paths Physical view: coupled SC islands with large variation of Josephson couplings, competition between localization of Cooper pair and coherence Open problems: dynamics (optics), insulating phase and critical behavior at SIT

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34 Thickness tuned SIT: amorphous Bi films Jim Valles, Leiden 2011

35 Thickness tuned SIT: amorphous Bi films Jim Valles, Leiden 2011

36 Thickness tuned SIT: amorphous Bi films Jim Valles Lorentz Center, Leiden 2011

37 Thickness tuned SIT: amorphous Bi films Stewart et al. Science 318, 1273 (2007)

38 Superconductor-Insulator Transition Activated behaviour in amorphous indium oxyde films T 0 Δ = 1.76 T c T c Gantmakher et al. D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, (1992) JETP82, 951 (1996) Shahar and Ovadyahu PRB46, (1992) V. F. Gantmakher et al., JETP 82, 951 (1996) T o gaped insulator? superconductivity related?

39 Superconductor-Insulator Transition Titanium nitride thin films T. I. C. Strunk, M. R. Baklanov, and A. Satta T. I. Baturina, et al. PRL 99, (2007) PRL 98, (2007) Huge magnetoresistance peak : superconductivity-related?

40 Superconductor-Insulator Transition Titanium nitride thin films I1 : T 0 = 0.25 K I2 : T 0 = 0.38 K I3 : T 0 = 0.61 K 1/T [K] T. I. Baturina, et al. PRL 99, (2007) Activated behaviour in insulating films : superconductivity-related?

41 Superconductor-Insulator Transition T 0 Δ = 1.76 T c Activated regime : gap in the density of states? T c Non-monotonous magneto-resistance in insulators : superconducting correlations? T. I. Baturina, et al. PRL 99, (2007)

42 Current response in a disordered SC U =5t, V 0 =2t, n=0.85 size=20x20 small A x (linear regime) color map for Δ i lines=constant phase arrows=strength of current on links Almost 1d path

43 Current response in a disordered SC Current from BCS response It does not obey continuity equation D s BCS >>D s =1/L 2 2 E(A)/ A 2 Missing superfluid density D s BCS - D s = spectral weigth of intragap σ(ω) from collective modes

44 Current response in a disordered SC Superfluid density from D s =1/L 2 2 E(A)/ A 2 U =5t, n=0.85 vs V 0 Average over disorder configurations Q/D s measure of anharmonic terms

45 Current response in a disordered SC Distributions of the superfluid density with increasing lattice size U =5t, V 0 =2t n=0.85

46 Current response in a disordered SC U =5t, V 0 =2t, n=0.1 size=20x20 small A x (linear regime) color map for Δ i arrows=strength of current on links Almost 1d path

47 Current response in a disordered SC U =5t, V 0 =2t, n=0.1 size=20x20 small A x (linear regime) color map for charge n i arrows=strength of current on links Almost 1d path

48 Current response in a disordered SC Gap - charge correlation

49 Current response in a disordered SC

50 Current response in a disordered SC

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52 Lemarie et al. PRB 87, (2013) STM in NbN films Define peak height h=(g peak -G min )/G min SC order parameter s