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)

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Acknowledgement

N. Peter Armitage (JHU) Rolando Valdes Aguilar (JHU) Luke Bilbro (JHU) Sambandamurthy Ganapathy (UB) Minsoo Kim (UB)

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Outline

     Overview Broadband Corbino microwave spectrometer InO x thin films Results and discussion Conclusion 3

Outline

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

Superconducting fluctuations

 Superconducting order parameter:  = 

e i

 • Amplitude  fluctuations: Ginzburg-Landau theory • Phase fluctuations: thermally generated free vortices • Kosterlitz-Thouless Berezinskii phase transition: transverse phase fluctuations frozen out

Temperature (Kelvin) T KTB T c0

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Kosterlitz-Thouless Berezinskii

Kosterlitz, Thouless: J. Phys. C: solid phys, Vol. 6 1973 Berezinskii, Sov. Phys. JETP 32 (1971) 493 From V. Vinokur Temperature (Kelvin) T KTB T c0

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Universal resistance curve

P. Minnhagen (1987) 7

Non linear I-V characteristic

K. Epstein (1982) 8

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

Frequency Dependent Superfluid Stiffness 10



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. 11

Outline

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

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=σ 1 +iσ 2 over a broad 13

Corbino Spectrometer

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Outline

     Motivation Broadband Corbino microwave spectrometer InO x thin film Results and discussion Conclusion 15

InO

x

film growth

amorphous granular

(A) and (B) are AFM images of InO conditions. amorphous, film x samples grown at SUNY-Buffalo by varying growth (C) Transmission electron diffraction image of an homogeneous sample showing the non crystalline nature of the Films prepared by e-gun evaporating high purity (99.999 %) In 2 O 3 clean 0.38mm thick 4.4mm*4.4mm Silicon substrate.  High T c high resistance – 2.3K @ 7k W.

on at Current films are 30nm thick morphologically homogeneous and amorphous. Inherent disorder can be tuned by thermal annealing slightly above room temperature 16

Outline

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

Extracting T

c0

-The Cooper Paring scale

T c0 is extracted using the Aslamazov-Larkin theory for DC fluctuation superconductivity (amplitude fluctuations).

Temperature (Kelvin) The temperature scale at which Cooper pairs start to form T c0 an energy scale in 2D, but not a phase transition …  =  (x,t) e i f (x,t) 18

Superconductor AC conductance

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0 0 20 Real Conductivity Imaginary Conductivity 40 60 Frequency 80 19

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 20

Frequency Dependent Superfluid Stiffness Superfluid density can be parameterized as a superfluid stiffness:

Energy scale to twist superconducting phase

 = 

e

iq q 1 q 2 q 3 q 4 q 5 q 6

Spin stiffness in discrete model.

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

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Frequency Dependent Superfluid Kosterlitz Thouless Berezenskii Transition Stiffness … 4T KTB = T  increasing  bare superfluid stiffness Probing length set by diffusion relation.

 =inf  =0 T KTB Temperature T m In 2D static superfluid stiffness survives at finite frequency (amplitude is still well defined). Finite frequency probes short length scale. If > 1/t then system looks superconducting. Approaches ‘bare’ stiffness as  gets big.

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Frequency Dependent Superfluid Stiffness … 24

Universal jump?

T q critical T q predicted Non-universal jump?

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Superconductor AC Conductance

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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 27

Scaling in superconductors

Close to transition scaling forms are expected.

Data collapse with characteristic relaxation frequency W (T) = 1/ t Functional form may look unusual, but it is not. Drude model obeys this form.

Important! Since pre-factors are real, phase of

S

With f = tan -1 (  2 /  1 ). f is also phase of  should collapse with one parameter scaling.

!

All temperature dependencies enter through extracted W and T q  from scaling 28

Scaling in 2D superconductors: Phase

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Scaling in 2D superconductors: Phase

All temperature dependencies enter through extracted W and T q  from scaling 30

Scaling in 2D superconductors: Magnitude

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Scaling in 2D superconductors: Magnitude

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Characteristic fluctuation rate

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Scaling in 2D superconductors

W  / 2 11 GHz and T’ = 23K W  / 2   GHz and z  = 1.58

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Vortex Activation?

= 3 W  / 2 11 GHz and T’ = 23K α is the ratio of is the votex core energy μ , to the votex core energy in the 2D XY model μ XY our value of T’ is consistent with a reasonably small value of the vortex core energy B. Halperin et al. J. Low Temp. Phys. 36, 599 (1979).

L. Benfatto et al. Phys. Rev. B 80

,

21456 (2) 35

Vortex Activation?

T

q

0 /8

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



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. 37

Scheme of sample

Scheffler et al.

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 kx 2 39

Kosterlitz Thouless Berzenskii Transition

bare superfluid density w =0 increasing w w =inf T KTB Temperature T m = sc phase q 40

Q: What about ‘normal’ electrons?

1 0.1

0.01

1/t =32 1/t =16 1/t =8 1/t =5 1/t =3 1/t =inf In principle there can be a contribution to  2 from thermally excited electrons and above gap excitations.

Rough estimate, using Drude relations and approximate numbers … 0.001

1 2 3 4 5 6 7 8 9 10 Frequency 2 3 4 5 6 7 8 9 100 A: Due to strong scattering ‘normal’ electrons give completely insignificant contribution @ our frequencies 41

Superconductor AC Conductance

Close to transition scaling forms for the conductivity are expected * .

Data collapse in terms of a characteristic relaxation frequency W (T) = 1/t * Fisher, Fisher, Huse PRB, 1991 42

Sigma2

Superconductor AC Conductance

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References:

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