Please visit strongdisordersuperconductors.blogspot/ Read Post Comment Etc.

1. AC Transport in Really Really Dirty Superconductors and near Superconductor-Insulator Quantum Phase Transitions N. Peter Armitage The Institute for Quantum Matter Dept. of Physics and Astronomy The Johns Hopkins University

3. Please visit http://strongdisordersuperconductors.blogspot.com/ Read Post Comment Etc.

4. AC Transport in Really Really Dirty Superconductors and near Superconductor-Insulator Quantum Phase Transitions N. Peter Armitage The Institute for Quantum Matter Dept. of Physics and Astronomy The Johns Hopkins University

5. Effects of disorder on electrodynamics of superconductors? “Low” levels of disorder  captured by BCS based Mattis-Bardeen; Dirty limit (1/t >> D). Higher levels of disorder one must progressively consider… Fluctuating superconductivity (thermal fluctuations) Quantum transition to insulating state? Quantum fluctuations?; Character of insulating state? Effects of inhomogeneity self-generated granularity

6. Superconductor AC Conductance @ T=0,  Real Conductivity Imaginary Conductivity

7. Mattis – Bardeen formalism: Electrodynamics of BCS superconductor in the dirty limit Sign depends on whether perturbation is even or odd under time reversal. Dipole matrix element is odd, so Case II coherence factors.

8. Mattis – Bardeen formalism: Electrodynamics of BCS superconductor in the dirty limit w ~ 0.2D T = 0 Case II (s-wave Supercond) Case I (SDW) sn Dissipation sn Case II (s-wave Supercond) Case I (SDW) 1 2 0.5 1 Frequency  Frequency T/Tc

9. Mattis-Bardeen prediction for type II coherence Klein PRB 1994

10. Thin films transmission through Pbfilms Palmer and Tinkham 1968 (earlier Glover and Tinkham 1957)

11. Cavity perturbation of Nbsamples; Klein PRB 1994

12. Mattis-Bardeen prediction for type II coherence Klein PRB 1994

13. For a collection of particles of density n of mass me, there is a sum rule on the area of the real part of the conductivity (f-sum rule of quantum mechanics).

16. 2D Gap

17. 2D Gap

18. 2D Gap

19. 2D Gap

20. Superconducting Fluctuations; Thermal and Quantum

21. Different T regimes of superconducting fluctuations • e Order parameter • Amplitude ( fluctuations; Ginzburg-Landau theory; ≠ 0 • - Below Tc0<2>≠ 0 • - Transverse phase fluctuations • Vortices xei≠ 0 • Longitudinal phase fluctuations; • “spin waves”; .ei≠ 0 (in neutral superfluid) Thermal superconducting fluctuations c ResistanceW/ Amplitude Fluctuations Phase Fluctuations Superconductivity Normal State Size set by phase `stifness’ Temperature (Kelvin) TKTB Tc0 Fluctuations can be enhanced in low dimensionality, short coherence length, and low sf density  dirty

22. Amplitude Fluctuations

24. Superfluid (Phase) Stiffness … Many of the different kinds of superconducting fluctuations can be viewed as disturbance in phase field e Order parameter Energy for deformation of any continuous elastic medium (spring, rubber, concrete, etc.) has a form that goes like square of generalized coordinate e.g. Hooke’s law U = ½ kx2

25. Superfluid (Phase) Stiffness … Superfluid density can be parameterized as a phase stiffness: Energy scale to twist superconducting phase Y = D eiq q3 q1 q4 q5 q2 q6 Uij = - T cosDqij (Spin stiffness in discrete model. Proportional to Josephson coupling) Energy for deformation has this form in any continuous elastic medium. T is a “stiffness”, a spring constant.

26. Superconductor AC Conductance @ T=0,  Real Conductivity Imaginary Conductivity

27. Superfluid (Phase) Stiffness … Superfluid density can be parameterized as a phase stiffness: Energy scale to twist superconducting phase Y = D eiq q3 q1 q4 q5 q2 q6 Uij = - T cosDqij (Spin stiffness in discrete model. Proportional to Josephson coupling) Energy for deformation has this form in any continuous elastic medium. T is a “stiffness”, a spring constant.

28. Kosterlitz-Thouless-Berezenskii Transition Mermin-Wagner Theorem --> In 2D no true long-range ordered states with continuous order parameters KTB showed that one can have topological power-law ordered phase at low T < rr Since high T phase is exponentially correlated < (r)> ~ e -r/ a finite temperature transition exists Transition happens by proliferation (unbinding) of topological defects (vortex - antivortex)  Coulomb gas TKTB p/2 rs bare superfluid stiffness Superfluid Stifness s rsBCS rs  Tc0 TKTB Temperature Superfluid stiffness falls discontinuously to zero at universal value of s/T

29. If r >> l2/d then charge superfluid effect should be minimal

30. Frequency Dependent Superfluid Stiffness … Kosterlitz Thouless Berzenskii Transition TKTB = p/2 rs increasing w bare superfluid density Probing length set by diffusion relation. Superfluid stiffnes w=inf w=0 Tm TKTB Temperature In 2D static superfluid density falls discontinuously to zero at temperature set by superfluid density itself. Vortex proliferation at TKTB. Superfluid stiffness survives at finite frequency (amplitude is still well defined). Approaches ‘bare’ stiffness as w gets big.

31. Phase Stiffness(Kelvin) See W. Liu on Friday

32. Time scales?

33. Fisher-Widom Scaling Hypothesis “Close to continuous transition, diverging length and time scales dominate response functions. All other lengths should be compared to these” Scaling Analysis

34. Characteristic fluctuation rate of 2D superconductor See W. Liu on Friday

35. And what about at higher disorders?

36. Superconductor-Insulator Transition Left: Bi film grown onto amorphous Geunderlayer on Al2O3 substrate. Data suggests a QCP [Haviland, et al., 1989] Right: Ga film deposited directly onto Al2O3 substrate. [Jaeger, et al., 1989] Thickness tuning tunes disorder; dominant scattering is surface scattering

37. Phase Diagram for Homogeneous System? T0 Phase Dominated Transition: “Dirty” Bosons Amplitude Dominated Transition TKTB Thermal Amplitude defined Phase defined Y = D(x,t)eif(x,t) Bc Superconducting “Bc2” Insulating Quantum

38. Can get it from s2  Superfluid Stiffness @ 22 GHz

39. By Kramers-Kronig considerations, to get large imaginary conductivity one must have a narrow peak in the real part. (Stay tuned for Liu et al. 2012. Full EM response through the SIT. Preview on Friday W. Liu.)

40. Effects of inhomogeneities?

41. Coupled 1D Josephson arrays, with two different JJs per unit cell  (same as inhomogeneous superfluid density) Considered extensively in the context of the bilayerscuprates w2K w2I A new mode! Oscillator strength depends on difference in JJ couplings Super current depends on weaker JJ coupling L.N. Bulaevskii 1994 D. van derMarel and A. Tsvetkov, 1996 (probably many others)

42. EF In random system, the supercurrent response will be governed by weakest link (strength of delta function is set by weakest link). Spectral weight (set by average of links) has to go somewhere by spectral weight conservation. (Remember coupling is density and there is a sum rule on conductivity set by density).  Finite frequency absorptions set by spatial average of superfluid density!

43. Many models addressing these general ideas.

46. Much newer work… (sorry Nandini…) How to discriminate the ballistic response of a Cooper pair that crosses a scing patch in time tfrom a homogeneously fluctuating superconductor on times t?

47. Phase fluctuation effects important Evidence for non-trivial electrodynamic response on insulating side of SIT Inhomogeneous superfluid density gives dissipation How can we discriminate the ballistic response of a Cooper pair that crosses a scing patch in time tfrom a homogeneously fluctuating superconductor on times t?