Сильно неупорядоченные сверхпроводники и квантовые фазовые переходы

1. М.В. ФейгельманИнститут теоретической физики им. Л. Д. ЛандауМосковский физико-технический институт Сильно неупорядоченные сверхпроводники и квантовые фазовые переходы сверхпроводник-изолятор Школа МИФИ, 25.09.2010

2. Phys Rev B 40 182 (1989) Почему и как происходит трансформация сверхпроводящего состояния в диэлектрическое ? Есть много разных ответов, часть из них будет представлена на этих двух лекциях В.Ф.Гантмахер и В.Т. Долгополов УФН 180, 3 (2010)

3. Lecture 1 • Disordered superconductors (basics) • Suppression of superconductivity by disorder: 3 types of materials and 3 major mechanisms • Granular superconductors and artificial arrays 1). Superconductivity in a single grain 2) Granular superconductors: experiments 3) Theories of SIT. Which parameter drives the S-I transition ? 4) BKT transitions in 2D JJ arrays 5) 2D JJ arrays with magnetic field: Bose metal ?? • Homogeneously disordered thin films 1) Suppression of Tc by disorder-enhanced Coulomb 2) Mesoscopic fluctuations of Tc near the QPT 3) Quantum S-M transition: SC islands on top of a poor metal film Experimental realization: graphen

4. Disordered superconductors (classical results) • Potential disorder does not affect superconductive transition temperature (for s-wave) – A.A.Abrikosov & L.P.Gor’kov 1958 P.W.Anderson 1959 • In the “dirty limit” l << ξ0 coherence length decreases asξ ~ (l ξ0 )1/2 whereas London length grows asλ ~ l -1/2 Accuracy limit: semi-classical approx. kFl >> 1 or (the same in another form) G = σ (h/e2) ξd-2 >> 1 What happens if G ~ 1 ?

5. “Anderson theorem” leads to BCS gap equation const Approx. 1 = g ∫ N(0) (dξ/ξ) th(ξ/2T)

6. Major types of disordered superconductive materials • Very strongly disordered metallic alloys with usual carrier density (~1022/cm3) bulk conductivity near Mott limit, G ~ 1 • Granular superconductive metals or artificially prepared arrays of islands inter-granular conductance Gt ~ 1 • Homogeneously disordered “poor metals” with low (~< 1021/cm3) carrier density (Second lecture)

7. Superconductivity v/s Localization • Granular systems with Coulomb interaction “Bosonic mechanism” • Coulomb-induced suppression of Tc in uniform films “Fermionic mechanism” • Competition of Cooper pairing and localization (no Coulomb) (2nd lecture)

8. I. Granular arrays Reviews: I.Beloborodov et al, Rev. Mod.Phys.79, 469 (2007) R.Fazio and H. van der Zant, Phys. Rep. 355, 235 (2001)

9. Grain radius a >> λF g ≈ const Examples: Phys Rev B 1981 Phys Rev B 1987

10. Basic energy scales Grain radius a is large on atomic scale: kF a >> 1 Coulomb energyEc= e2/a Level spacing inside grainδ = 1/(4a3 ) Intra-grain Thouless energy Eth = ћD0/4a2 δ << Eth << Ec g0 = Eth/ δ = 2500 Example: Al grains with a=20 nm δ = 0.02 K Eth = 50 K Ec = 1000 K Typical temperature range 4 K < T < 300 K Γ ~ 1K for gT = 50

11. Intergrain coupling Low transmission, but large number of transmission modes σT = (4πe2/ћ) 2 <|tpk|2> gT = σT h/e2 - dimensionless inter-grain conductance Effective diffusion constant Deff = Γ a2/ћ << D0 A) Granular metal gT ≥ 1 Narrow coherent band Γ =gT δ<< Eth Γ~ 1K for gT = 50 B) Granular insulator gT ≤ 1 Nearest-neighbors coupling only !

12. 1) Superconductivity in a single grain • What is the critical size of the grain ac? • What happens if a < ac ? • Assuming ξ0 >>a >> ac , what is the critical magnetic field ?

13. Critical grain size Mean-field theory gap equation: Δ = (g/2) Σi Δ/[εi2 + Δ2]1/2 Level spacing δ << Δallows to replace sum by the integral and get back usual BCS equation Grain radius a >> ac = (1/ Δν )1/3

14. Ultra-small grains a<<ac • No off-diagonal correlations • Parity effect ------- ------- EF --↑↓-- --  ↓-- K. Matveev and A. Larkin PRL 78, 3749 (1997) Perturbation theory w.r.t. Cooper attraction: Take into account higher-order terms (virtual transitions to higher levels):

15. Critical magnetic field for small grain A. Larkin 1965 ac << R << ξ0 Orbital critical field for the grain Local transition temperature Tc is determined by equation: Which follows from Deparing parameter (orbital) Zeeman term alone leads to >> ac Orbital deparing prevails at R > Ro-z~

16. 2) Granular superconductors: experiments • Very thin granular films • 3D granular materials • E-beam - produced regular JJ arrays

17. Thin quenched-condensed films A. Frydman, O. Naaman, R. Dynes 2002 Sn grains Pb grains

18. Granular v/s Amorphous films A.Frydman Physica C 391, 189 (2003)

19. Phys Rev B 40 182 (1989) Conclusion in this paper: control parameter is the normal resistance R. Its critical value is RQ = h/4e2 = 6.5 kOhm

20. Bulk granular superconductors Sample thickness 200 nm

21. Bulk granular superconductors

22. Artificial regular JJ arrays

25. What is the parameter that controls SIT in granular superconductors ? • Ratio EC/EJ ? • Dimensionless conductance g = (h/4e2) R-1 ? (for 2D case)

26. 3) Theoretical approaches to SIT • K.Efetov ZhETF 78, 2017 (1980) [Sov.Phys.-JETP 52, 568 (1980)] Hamiltonian for charge-phase variables • M.P.A.Fisher, Phys.Rev.Lett. 65, 923 (1990) General “duality” Cooper pairs – Vortices in 2D • R.Fazio and G.Schön, Phys. Rev. B43, 5307 (1991) Effective action for 2D arrays

27. K.Efetov’s microscopic Hamiltonian Control parameter Ec = e2/2C Artificial arrays: major term in capacitance matrix is n-n capacitance C qi and φi are canonically conjugated

28. Logarithmic Coulomb interaction Artificial arrays with dominating capacitance of junctions: C/C0 > 100 Coulomb interaction of elementary charges U(R) = For Cooper pairs,xby factor 4

29. M.P.A.Fisher’s duality arguments Insulator is a superfluid of vortices R In favor of this idea: usually SIT in films occurs at R near RQ Insulator RQ • Problems: i) how to derive that duality ? • What about capacitance matrix • in granular films ? • iii) Critical R(T) is not flat usually T • Competition between Coulomb repulsion and Cooper pair hopping: Duality charge-vortex: both charge-charge and vortex-vortex interaction are Log(R) in 2D. Vortex motion generates voltage: V=φ0 jV Charge motion generates current: I=2e jc At the self-dual point the currents are equal → RQ=V/I=h/(2e)2=6.5kΩ.

30. Can we reconcile Efetov’s theory and result of “duality approach” ? We need to account for capacitance renormalization to due to virtual electron tunneling via AES [PRB 30, 6419 (1984)] action functional Virtual tunneling Charging Sts = g Josephson (if dφ/dt << Δ) Sts = (3g/32 Δ) Cind = (3/16) ge2/ Δ

31. Mean-field estimate with renormalized action SC transition at T=0: J = gΔ/2 Strong renormalization of C:

32. Can one disentangle “g” and “EC/EJ” effects ? JETP Lett. 85(10), 513 (2007) This model allows exact duality transformation Control parameter Experimentally, it allows study of SIT in a broad range of g and/or EJ/EC

33. 4) Charge BKT transition in 2D JJ arrays Logarithmic interaction of Cooper pairs 2e R.Fazio and G.Schön, Phys. Rev. B43, 5307 (1991) U(R) = 8 EC Temperature of BKT transition is T2 = EC/π Not observed ! The reason: usually T2 is above parity temperature Interaction of pairs is screened by quasiparticles T1 = EC/4π Charge BKT is at (unless T* is above T2 )

34. 5) “Bose metal” in JJ array ? At non-zero field simple Josephson arrays show temperature-independent resistance with values that change by orders of magnitude.

35. Dice array (E.Serret and B.Pannetier 2002; E.Serret thesis, CNRS-Grenoble) Foto from arxiv:0811.4675 At non-zero field Josephson arrays of more complex (dice) geometry show temperature independent resistance in a wide range of EJ/Ec

36. The origin of “Bose metal” is unknown Hypothesis: it might be related to charge offset noise

37. Josephson Arrays Elementary building block

38. Josephson Arrays Elementary building block Al Al2O3 Al - + + - +

39. Conclusion for granular superconductive arrays:1) basic features of SIT are understood2) really quantitative SIT theory is not constructed yet

40. II. Homogeneously disordered SC films 1) Suppression of Tc by disorder-enhanced Coulomb 2) Mesoscopic fluctuations of Tc near the QPT 3) Quantum S-M transition: SC islands on top of a poor metal film 4) Experimental realization: graphen

41. 1) Suppression of Tc in amorphous thin films by disorder-enhanced Coulomb interaction Theory Experiment Review: Generalization to quasi-1D stripes: Yu. Oreg and A. M. Finkel'stein Phys. Rev. Lett. 83, 191 (1999) Similar approach for 3D poor conductor near Anderson transition: P. W. Anderson, K. A. Muttalib, and T. V. Ramakrishnan, Phys. Rev. B 28, 117 (1983)

42. Amorphous v/s Granular films

43. Suppression of Tc in amorphous thin films: experimental dataand fits to theory J.Graybeal and M.Beasley Pb films

44. Suppression of Tc in amorphous thin films: qualitative picture • Disorder increases Coulomb interaction and thus decreases the pairing interaction (sum of Coulomb and phonon attraction). In perturbation theory: g = σ h/e2 Return probability in 2D Roughly, But in fact BCS-type problem with energy-dependent coupling must be solved The origin of the correction term ??????? It is “revival” of strong Coulomb repulsion, due to slow diffusion at g ~ 1

45. Why Coulomb repulsion does not kill phonon attraction (sometimes)? Coulomb pseudo-potential μ(ω) = μ0 / [1+ μ0 ln (EF/ω)] Normally, μ0 ~ 1 and ln (EF/ω) >> 1 “Tolmachev Logarithm” ω ~ ωD ~ 0.05 eV whereas EF ~ 5 eV At the energy scale ωDrelevant for phonon process, full amplitude in the Cooper channel can be attractive at smallλ λ = λ0 - 1/ln (EF/ω) > 0

46. Why disorder leads to “Coulomb revival” ? Matrix element of Coulomb repulsion: Uij = ∫Ψi2(r)Ψj2(r’) U(r-r’) dr dr’ U(r) is short-range <Ψi2(r) Ψj2(r)> = V-1 [1 + (1/g) ln (1/τωij ) ] Uniform term subject to Tolmachev Log Enhancement due to disorder

47. Diagrams for Coulomb suppression of the Cooperon

49. Fluctuations of the local Density of States Review: A.Mirlin, F.Evers, Rev. Mod. Phys. (2008) β= 1 (T-invariant) or = 2 (no T-invariance)

50. Renormalization Group: summation of major diagrams In the form developed in M.F., A.Larkin & M.Skvortsov (Phys.Rev.B 2000)