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)
1

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

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

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


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
5

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 6

P. Minnhagen
(1987)
7

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


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

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


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 microwave frequency range
13

14

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

x
amorphous granular
(A) and (B) are AFM images of InO x samples grown at SUNY-Buffalo by varying growth conditions.
(C) Transmission electron diffraction image of an amorphous, homogeneous sample showing the noncrystalline nature of the film
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 resistance – 2.3K @ 7k
W.
 High T c on at
Current films are 30nm thick morphologically homogeneous and amorphous.
Inherent disorder can be tuned by thermal annealing slightly above room temperature
16

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

c0
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

0.15
0.10
0.05
0
0
0.35
0.30
0.25
0.20
20
Real Conductivity
Imaginary Conductivity
80 40 60
Frequency
19

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.
21

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
.
22

Frequency Dependent Superfluid Stiffness …
Kosterlitz Thouless Berezenskii Transition
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.
23

Frequency Dependent Superfluid Stiffness …
24

T q critical
T q predicted
Non-universal jump?
25

26

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

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 is also phase of

!
With f
= tan -1 (

2
/

1
). f should collapse with one parameter scaling.
All temperature dependencies enter through extracted
W and T q
 from scaling
28

29

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

31

32

33

W

/ 2 11 GHz and T’ = 23K
W

/ 2  
GHz and z

= 1.58
34

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


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

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

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

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

44

7.
8.
9.
5.
6.
10.
11.
3.
4.
1.
2.
12.
13.
14.
Marc Scheffler, Broadband Microwave Spectroscopy on Correlated Electrons, Dissertation, Universität Stuttgart,
Stuttgart,2004
Riley Crane, Probing the Bose Solid: A finite frequency study of the magnetic field-tuned superconductor-insulator transition in two-dimensions, Dissertation, UCLA, CA, 2006
James Clay Booth, Novel Measurements of the Frequency Dependent Microwave Surface Impedance of Cuprate Thin
Film Superconductors, Dissertation, university of Maryland, 1996
R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, Survival of superconducting correlations across the two-dimensional superconductor-insulator transition: A finite-frequency study , Phys. Rev. B 75,
184530 (2007)
R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, Fluctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional superconductors, Phys. Rev. B 75, 094506 (2007)
Martin Dressel and George Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge
University Press, Cambridge, 2002).
Marc Scheffler and Martin Dressel, Broadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7
K, Rev. Sci. Instrum. 76, 074702 (2005)
S. M. Girvin, Duality in Perspective, Science 25, Vol. 274. no. 5287, pp. 524 - 525 (1996)
J. C. Booth, Dong Ho Wu, and Steven M. Anlage, A broadband method for the measurement of the surface impedance of thin films at microwave frequencies, Rev. Sci. Instrum. 65, 2082 (1994)
Marc Scheffler, Serife Kilic, and Martin Dressel, Strip-shaped samples in a microwave Corbino spectrometer, Rev. Sci.
Instrum 78, 086106 (2007)
James C. Booth, Dong-Ho Wu, and Steven M. Anlage, Measurements of the Frequency Dependent Microwave
Fluctuation Conductivity of Cuprate Thin Film Superconductors, Fluctuation Phenomena in High Temperature
Superconductors, (Kluwer, Dordrecht, 1997), edited by Marcel Ausloos and Andrei A. Varlamov, pp.151 - 178.
Haruhisa Kitano, Takeyoshi Ohashi and Atsutaka Maeda, Broadband method for precise microwave spectroscopy of superconducting thin films near critical temperature, arxiv:0806.1421v1 (2008)
V.F. Gantmakher and M.V. Golubkov, Width of the zero-field superconducting resistive transition in the vicinity of the localization threshold, JETP LETTERS Vol. 73 (2001)
J. Corson, R. Mallozzi, J. Orenstein, J.N. Eckstein, I. Bozovic, Vanishing of phase coherence in underdoped
Bi
2
Sr
2
CaCu
2
O
8+ δ , .
Nature, Vol. 398, Issue 6724, pp. 221-223 (1999)