Wei Liu The Johns Hopkins University

1. Dynamical study of phase fluctuations and their critical slowing down in amorphous superconducting films Wei Liu The Johns Hopkins University Wei Liu, et al, Phys. Rev. B 84, 024511 (2011)

2. Acknowledgement N. Peter Armitage (JHU) SambandamurthyGanapathy (UB) Luke Bilbro (JHU) Rolando Valdes Aguilar (JHU) Minsoo Kim (UB)

3. Outline Overview Broadband Corbino microwave spectrometer InOx thin films Results and discussion Conclusion

4. Outline Overview Broadband Corbino microwave spectrometer InOx thin film Results and discussions Conclusion

5. Superconducting fluctuations Superconducting order parameter: =ei Amplitude  fluctuations: Ginzburg-Landau theory Phase fluctuations: thermally generated free vortices Kosterlitz-Thouless-Berezinskii phase transition: transverse phase fluctuations frozen out c Ω/ Amplitude Fluctuations Superconductivity Phase Fluctuations Normal State Temperature (Kelvin) TKTB Tc0

6. Kosterlitz, Thouless: J. Phys. C: solid phys, Vol. 6 1973 Berezinskii, Sov. Phys. JETP 32 (1971) 493 Kosterlitz-Thouless - Berezinskii c Ω/ Amplitude Fluctuations Superconductivity Phase Fluctuations Normal State Temperature (Kelvin) TKTB Tc0 From V. Vinokur

7. Universal resistance curve P. Minnhagen (1987)

8. Non linear I-V characteristic K. Epstein (1982)

9. Universal Jump McQueeny et al. (1984) He3-He4 mixtures of different proportions DPproportional to superfluid density - Measured via torsion oscillator

10. Frequency Dependent Superfluid Stiffness

11. Conclusion Unique system: continuous scan to measure complex conductivity down to 300 mK at microwave region; capable to perform finite frequency study on 2D quantum phase transition. Superfluid stiffness acquires frequency dependence at a transition temperature which is close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism. Critical slowing down close to the phase transition and in general the applicability of a vortex plasma model above Tc.

12. Outline Motivation Broadband Corbino microwave spectrometer InOx thin film Results and discussions Conclusion

13. Corbino Microwave Spectrometer • Broadband microwave spectroscopy has traditionally been difficult • Most measurements with microwave cavities, but they are limited to some particular frequencies • Our broadband microwave Corbino spectrometer can scan from 10MHz to 40GHz with 1Hz resolution down to 300mK • Measure both component of complex ‘optical’ response σ=σ1+iσ2 over a broad microwave frequency range

14. Corbino Spectrometer

15. Outline Motivation Broadband Corbino microwave spectrometer InOx thin film Results and discussion Conclusion

16. InOx film growth (A) and (B) are AFM images of InOx samples grown at SUNY-Buffalo by varying growth conditions. (C) Transmission electron diffraction image of an amorphous, homogeneous sample showing the non-crystalline nature of the film granular amorphous Films prepared by e-gun evaporating high purity (99.999 %) In2O3 on clean 0.38mm thick 4.4mm*4.4mm Silicon substrate.  High Tc at high resistance – 2.3K @ 7kW. Current films are 30nm thick morphologically homogeneous and amorphous. Inherent disorder can be tuned by thermal annealing slightly above room temperature

17. Outline Motivation Broadband Corbino microwave spectrometer InOx thin film Results and discussion Conclusion

18. Extracting Tc0-The Cooper Paring scale Tc0 is extracted using the Aslamazov-Larkin theory for DC fluctuation superconductivity (amplitude fluctuations). The temperature scale at which Cooper pairs start to form c W/ Tc0 an energy scale in 2D, but not a phase transition … Temperature (Kelvin) Y = D(x,t)eif(x,t)

19. Superconductor AC conductance Real Conductivity Imaginary Conductivity

20. AC Response of a Superconductor Canonical response of a superconductor at low T Real and imaginary part of conductance plotted as a function of frequency for different temperatures

21. Frequency Dependent Superfluid Stiffness Superfluid density can be parameterized as a superfluid stiffness: Energy scale to twist superconducting phase Y = D eiq q3 q1 q4 q5 q2 q6 Spin stiffness in discrete model.

22. Universal jump in Superfluid (Phase) Stiffness Kosterlitz-Thouless-Berezenskii Transition 4TKTB = T Superfluid stiffness TKTB Temperature In 2D static superfluid stiffness falls discontinuously to zero at temperature set by superfluid stiffness itself. Thermal vortex/anti-vortex proliferation at TKTB.

23. Frequency Dependent Superfluid Stiffness … Kosterlitz Thouless Berezenskii Transition 4TKTB = T increasing  bare superfluid stiffness Probing length set by diffusion relation. Superfluid stiffnes w=inf =0 Tm TKTB Temperature In 2D static superfluid stiffness survives at finite frequency (amplitude is still well defined). Finite frequency probes short length scale. If w> 1/t then system looks superconducting. Approaches ‘bare’ stiffness as  gets big.

24. Frequency Dependent Superfluid Stiffness …

25. Universal jump? Tcritical Tpredicted Non-universal jump?

26. Superconductor AC Conductance

27. 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

28. Scaling in superconductors Close to transition scaling forms are expected. Data collapse with characteristic relaxation frequency (T) = 1/ Functional form may look unusual, but it is not. Drude model obeys this form. Important! Since pre-factors are real, phase of S is also phase of ! With = tan-1(2/1).  should collapse with one parameter scaling. All temperature dependencies enter through extracted  and T from scaling

29. Scaling in 2D superconductors: Phase

30. Scaling in 2D superconductors: Phase All temperature dependencies enter through extracted  and T from scaling

31. Scaling in 2D superconductors: Magnitude t

32. Scaling in 2D superconductors: Magnitude

33. Characteristic fluctuation rate

34. Scaling in 2D superconductors / 21GHz and T’3 / 2 GHz and z = 1.58

35. Vortex Activation? α is the ratio of is the votex core energy μ , to the votex core energy in the 2D XY model μXY = 0.38 our value of T’ is consistent with a reasonably small value of the vortex core energy / 21GHz and T’3 B. Halperin et al. J. Low Temp. Phys. 36, 599 (1979). L. Benfatto et al. Phys. Rev. B 80, 214506 (2009)

36. Vortex Activation? T0/8 We get 0.27K, which compares with estimate from T0 approximately 0.3 K Within BCS one expects that:  ~ T0/8

37. Conclusion Unique system: continuous scan to measure complex conductivity down to 300 mK at microwave region; capable to perform finite frequency study on 2D quantum phase transition. Superfluid stiffness acquires frequency dependence at a transition temperature which is close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism. Critical slowing down close to the phase transition and in general the applicability of a vortex plasma model above Tc.

38. Scheme of sample Scheffler et al. 38

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

40. Kosterlitz Thouless Berzenskii Transition increasing w Superfluid stiffnes bare superfluid density w=inf w=0 = sc phase q TKTB Tm Temperature

41. Q: What about ‘normal’ electrons? In principle there can be a contribution to s2 from thermally excited electrons and above gap excitations. Rough estimate, using Drude relations and approximate numbers … A: Due to strong scattering ‘normal’ electrons give completely insignificant contribution @ our frequencies

42. Superconductor AC Conductance Close to transition scaling forms for the conductivity are expected *. Data collapse in terms of a characteristic relaxation frequency (T) = 1/t * Fisher, Fisher, Huse PRB, 1991

43. Sigma2 43

44. Superconductor AC Conductance

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