Dynamical study of phase fluctuations and their critical slowing down in amorphous superconducting films

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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)
Sambandamurthy Ganapathy (UB)
Luke Bilbro (JHU)
Rolando Valdes Aguilar (JHU)
Minsoo Kim (UB)
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Outline

• Overview
• Broadband Corbino microwave spectrometer
• InOx thin films
• Results and discussion
• Conclusion

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Outline

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

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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

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Kosterlitz-Thouless - Berezinskii
Kosterlitz, Thouless J. Phys. C solid phys,
Vol. 6 1973 Berezinskii, Sov. Phys. JETP 32
(1971) 493
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Universal resistance curve
P. Minnhagen (1987)
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Non linear I-V characteristic
K. Epstein (1982)
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Universal Jump
McQueeny et al. (1984) He3-He4 mixtures of
different proportions DP proportional to
superfluid density - Measured via torsion
oscillator
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Frequency Dependent Superfluid Stiffness
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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.

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Outline

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

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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 ss1is2 over a broad microwave
  frequency range

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Corbino Spectrometer
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Outline

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

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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
4.4mm4.4mm Silicon substrate. ? High Tc at
high resistance 2.3K _at_ 7kW. Current films are
30nm thick morphologically homogeneous and
amorphous. Inherent disorder can be tuned by
thermal annealing slightly above room temperature
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Outline

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

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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
Tc0 an energy scale in 2D, but not a phase
transition
Y D(x,t) eif (x,t)
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Superconductor AC conductance
Real Conductivity Imaginary Conductivity
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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
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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.
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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.
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Frequency Dependent Superfluid Stiffness
Kosterlitz Thouless Berezenskii Transition
4TKTB T?
increasing ?
bare superfluid stiffness
Probing length set by diffusion relation.
Superfluid stiffnes
winf
?0
TKTB
Tm
Temperature
In 2D static superfluid stiffness survives at
finite frequency (amplitude is still well
defined). Finite frequency probes short length
scale. If wgt 1/t then system looks
superconducting. Approaches bare stiffness as
? gets big.
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Frequency Dependent Superfluid Stiffness
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Universal jump?
T?critical
T?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
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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
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Scaling in 2D superconductors Phase
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Scaling in 2D superconductors Phase
All temperature dependencies enter through
extracted ? and T?? from scaling
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Scaling in 2D superconductors Magnitude
t
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Scaling in 2D superconductors Magnitude
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Characteristic fluctuation rate
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Scaling in 2D superconductors
???/ 2?????1?GHz and T??????3?
???/ 2? ?????GHz and z? 1.58
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Vortex Activation?
a is the ratio of is the votex core energy µ , to
the votex core energy in the 2D XY model µXY
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)
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Vortex Activation?
We get 0.27K, which compares with estimate from
T?0 approximately 0.3 K Within BCS one expects
that ? T?0/8
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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.

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Scheme of sample
Scheffler et al.
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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. Hookes law U ½ kx2
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Kosterlitz Thouless Berzenskii Transition
increasing w
Superfluid stiffnes
bare superfluid density
winf
w0
sc phase q
TKTB
Tm
Temperature
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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 _at_ our
frequencies
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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
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Sigma2
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Superconductor AC Conductance
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