Download - Wei Liu The Johns Hopkins University
![Page 1: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/1.jpg)
Dynamical study of phase fluctuations Dynamical study of phase fluctuations and their critical slowing down in and their critical slowing down in
amorphous superconducting filmsamorphous superconducting films
Wei LiuWei LiuThe Johns Hopkins UniversityThe Johns Hopkins University
Wei Liu, et al, Phys. Rev. B 84, 024511
(2011)
1
![Page 2: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/2.jpg)
Acknowledgement Acknowledgement
N. Peter Armitage (JHU)
Rolando Valdes Aguilar (JHU)
Luke Bilbro(JHU)
Sambandamurthy Ganapathy(UB)
Minsoo Kim(UB)
22
![Page 3: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/3.jpg)
OutlineOutline
Overview Broadband Corbino microwave
spectrometer InOx thin films Results and discussion Conclusion
33
![Page 4: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/4.jpg)
OutlineOutline
Overview Broadband Corbino microwave
spectrometer InOx thin film Results and discussions Conclusion
44
![Page 5: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/5.jpg)
Superconducting Superconducting fluctuationsfluctuations
Superconducting order Superconducting order parameter:parameter:
==eeii
• Amplitude Amplitude fluctuations: fluctuations: Ginzburg-Landau theoryGinzburg-Landau theory
• Phase fluctuations: Phase fluctuations: thermally generated free thermally generated free vorticesvortices
• Kosterlitz-Thouless-Kosterlitz-Thouless-Berezinskii phase Berezinskii phase transition: transverse transition: transverse phase fluctuations frozen phase fluctuations frozen out out
Ω/
Temperature (Kelvin)TKTB Tc0
Am
pli
tud
e F
luct
uat
ion
s
Ph
ase F
luctu
ati
on
s
Su
pe
rco
nd
uct
ivit
y
No
rmal
Sta
te
55
![Page 6: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/6.jpg)
Kosterlitz-Thouless - Kosterlitz-Thouless - Berezinskii Berezinskii
Kosterlitz, Thouless: J. Phys. C: solid phys, Vol. 6 1973
Berezinskii, Sov. Phys. JETP 32 (1971) 493
From V. Vinokur
Ω/
Temperature (Kelvin)TKTB Tc0
Am
pli
tud
e F
luct
uat
ion
s
Ph
ase F
luctu
ati
on
s
Su
pe
rco
nd
uct
ivit
y
No
rmal
Sta
te
6
![Page 7: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/7.jpg)
Universal resistance curveUniversal resistance curve
P. Minnhagen (1987)
77
![Page 8: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/8.jpg)
Non linear I-V characteristicNon linear I-V characteristic
K. Epstein (1982)
88
![Page 9: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/9.jpg)
Universal JumpUniversal Jump
McQueeny et al. (1984) He3-He4 mixtures of different proportions
Pproportional to superfluid density - Measured via torsion oscillator
99
![Page 10: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/10.jpg)
Frequency Dependent Superfluid StiffnessFrequency Dependent Superfluid Stiffness
1010
![Page 11: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/11.jpg)
ConclusionConclusion Unique system: continuous scan to measure Unique system: continuous scan to measure
complex conductivity down to 300 mK at complex conductivity down to 300 mK at microwave region; capable to perform finite microwave region; capable to perform finite frequency study on 2D quantum phase transition.frequency study on 2D quantum phase transition.
Superfluid stiffness acquires frequency Superfluid stiffness acquires frequency dependence at a transition temperature which is dependence at a transition temperature which is close to the universal jump value close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism.-consistent with Kosterlitz-Thouless-Berezinskii formalism.
Critical slowing down close to the phase transition Critical slowing down close to the phase transition and in general the applicability of a vortex and in general the applicability of a vortex plasma model above Tc. plasma model above Tc.
1111
![Page 12: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/12.jpg)
OutlineOutline
Motivation Broadband Corbino microwave Broadband Corbino microwave
spectrometerspectrometer InOx thin film Results and discussions Conclusion
1212
![Page 13: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/13.jpg)
Corbino Microwave SpectrometerCorbino 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
1313
![Page 14: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/14.jpg)
Corbino SpectrometerCorbino Spectrometer
1414
![Page 15: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/15.jpg)
OutlineOutline
Motivation Broadband Corbino microwave
spectrometer InOInOxx thin film thin film Results and discussion Conclusion
1515
![Page 16: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/16.jpg)
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
InOInOxx film growth film growth
amorphous granular
(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
1616
![Page 17: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/17.jpg)
OutlineOutline
Motivation Broadband Corbino microwave
spectrometer InOx thin film Results and discussionResults and discussion Conclusion
1717
![Page 18: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/18.jpg)
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
…
(x,t)ei(x,t)
Temperature (Kelvin)
Extracting TExtracting Tc0c0-The Cooper Paring -The Cooper Paring scalescale
1818
![Page 19: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/19.jpg)
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
Con
duct
ivity
806040200
Frequency
Real ConductivityImaginary Conductivity
Superconductor AC conductanceSuperconductor AC conductance
1919
![Page 20: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/20.jpg)
AC Response of a AC Response of a SuperconductorSuperconductor
Canonical response of a superconductor at low T
Real and imaginary part of conductance plotted as a function of frequency for different temperatures
2020
![Page 21: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/21.jpg)
Frequency Dependent Superfluid StiffnessFrequency Dependent Superfluid Stiffness
Superfluid density can be parameterized as a superfluid stiffness: Energy scale to twist superconducting phase e
1 23 4 5 6
Spin stiffness in discrete model.
2121
![Page 22: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/22.jpg)
Kosterlitz-Thouless-Berezenskii Transition
In 2D static superfluid stiffness falls discontinuously to zero at temperature set
by superfluid stiffness itself. Thermal vortex/anti-vortex proliferation at TKTB.
Sup
erflu
id s
tiffn
ess
TKTB
Temperature
Universal jump in Superfluid (Phase) Universal jump in Superfluid (Phase) StiffnessStiffness
2222
![Page 23: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/23.jpg)
Kosterlitz Thouless Berezenskii TransitionS
uper
fluid
stif
fnes
TKTB Tm
bare superfluid stiffness
=0
=inf
In 2D static superfluid stiffness survives at finite frequency (amplitude is still well defined). Finite frequency probes short length scale. If then system looks superconducting. Approaches ‘bare’ stiffness as gets big.
Temperature
increasing
Frequency Dependent Superfluid Stiffness Frequency Dependent Superfluid Stiffness ……
Probing length set by diffusion relation.
2323
![Page 24: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/24.jpg)
Frequency Dependent Superfluid Stiffness …Frequency Dependent Superfluid Stiffness …
2424
![Page 25: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/25.jpg)
Universal jump?
Tpredicted
Tcritical
Non-universaljump?
2525
![Page 26: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/26.jpg)
Superconductor AC Superconductor AC ConductanceConductance
2626
![Page 27: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/27.jpg)
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 AnalysisScaling Analysis
2727
![Page 28: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/28.jpg)
Close to transition scaling forms are expected.
Data collapse with characteristic relaxation frequency (T) = 1/
Important! Since pre-factors are real, phase of S is also phase of !With = tan-1(2/1). should collapse with one parameter scaling.
Scaling in superconductorsScaling in superconductors
Functional form may look unusual, but it is
not. Drude model obeys this form.
All temperature dependencies enter through extracted and T from scaling
2828
![Page 29: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/29.jpg)
Scaling in 2D superconductors: PhaseScaling in 2D superconductors: Phase
2929
![Page 30: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/30.jpg)
Scaling in 2D superconductors: PhaseScaling in 2D superconductors: Phase
All temperature dependencies enter through extracted and T from scaling
3030
![Page 31: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/31.jpg)
Scaling in 2D superconductors: Scaling in 2D superconductors: MagnitudeMagnitude
3131
t
![Page 32: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/32.jpg)
Scaling in 2D superconductors: Scaling in 2D superconductors: MagnitudeMagnitude
3232
![Page 33: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/33.jpg)
Characteristic fluctuation Characteristic fluctuation raterate
3333
![Page 34: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/34.jpg)
Scaling in 2D superconductorsScaling in 2D superconductors
GHz and z = 1.58GHz and T’
3434
![Page 35: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/35.jpg)
Vortex Activation?
our value of T’ is consistent with a reasonably small value of the vortex core energy
GHz and T’B. Halperin et al. J. Low Temp. Phys. 36, 599 (1979).
L. Benfatto et al. Phys. Rev. B 80,
3535
α is the ratio of is the votex core energy μ , to the votex core energy in the 2D XY model μXY
![Page 36: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/36.jpg)
Vortex Activation?
We get 0.27K, which compares with estimate from T
0 approximately0.3 K
Within BCS one expects that: ~ T
0/8
3636
T0/8
![Page 37: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/37.jpg)
ConclusionConclusion Unique system: continuous scan to measure Unique system: continuous scan to measure
complex conductivity down to 300 mK at complex conductivity down to 300 mK at microwave region; capable to perform finite microwave region; capable to perform finite frequency study on 2D quantum phase transition.frequency study on 2D quantum phase transition.
Superfluid stiffness acquires frequency Superfluid stiffness acquires frequency dependence at a transition temperature which is dependence at a transition temperature which is close to the universal jump value close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism.-consistent with Kosterlitz-Thouless-Berezinskii formalism.
Critical slowing down close to the phase transition Critical slowing down close to the phase transition and in general the applicability of a vortex and in general the applicability of a vortex plasma model above Tc. plasma model above Tc.
3737
![Page 38: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/38.jpg)
383838
Scheme of sampleScheme of sample
Scheffler et al.
![Page 39: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/39.jpg)
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
3939
![Page 40: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/40.jpg)
= sc phase q
Kosterlitz Thouless Berzenskii Kosterlitz Thouless Berzenskii TransitionTransition
Superfl
uid
sti
ffnes
TKTB Tm
bare superfluid density
w=0
w=inf
Temperature
increasing w
4040
![Page 41: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/41.jpg)
Q: What about ‘normal’ electrons?
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 …
A: Due to strong scattering ‘normal’ electrons give completely insignificant contribution @ our frequencies
0.001
0.01
0.1
1
Imag
inar
y C
ondu
ctiv
ity
12 3 4 5 6 7 8 9
102 3 4 5 6 7 8 9
100
Frequency
=32
=16
=8
=5
=3
=inf
0.001
0.01
0.1
1
Imag
inar
y C
ondu
ctiv
ity
12 3 4 5 6 7 8 9
102 3 4 5 6 7 8 9
100
Frequency
=32
=16
=8
=5
=3
=inf
4141
![Page 42: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/42.jpg)
Superconductor AC Superconductor AC ConductanceConductance
Close to transition scaling forms for the conductivity are
expected *.
Data collapse in terms of a characteristic relaxation
frequency (T) =
* Fisher, Fisher, Huse PRB, 1991
4242
![Page 43: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/43.jpg)
434343
Sigma2Sigma2
![Page 44: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/44.jpg)
Superconductor AC Superconductor AC ConductanceConductance
4444
![Page 45: Wei Liu The Johns Hopkins University](https://reader036.vdocument.in/reader036/viewer/2022062314/56814043550346895dabb052/html5/thumbnails/45.jpg)
454545
References:References:
1.1. Marc Scheffler, Broadband Microwave Spectroscopy on Correlated Electrons, Dissertation, Marc Scheffler, Broadband Microwave Spectroscopy on Correlated Electrons, Dissertation, Universität Stuttgart, Stuttgart,2004Universität Stuttgart, Stuttgart,2004
2.2. Riley Crane, Probing the Bose Solid: A finite frequency study of the magnetic field-tuned Riley Crane, Probing the Bose Solid: A finite frequency study of the magnetic field-tuned superconductor-insulator transition in two-dimensions, Dissertation, UCLA, CA, 2006superconductor-insulator transition in two-dimensions, Dissertation, UCLA, CA, 2006
3.3. James Clay Booth, Novel Measurements of the Frequency Dependent Microwave Surface James Clay Booth, Novel Measurements of the Frequency Dependent Microwave Surface Impedance of Cuprate Thin Film Superconductors, Dissertation, university of Maryland, 1996Impedance of Cuprate Thin Film Superconductors, Dissertation, university of Maryland, 1996
4.4. R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, SSurvival of superconducting correlations across the two-dimensional superconductor-insulator urvival of superconducting correlations across the two-dimensional superconductor-insulator transition: A finite-frequency study transition: A finite-frequency study ,, Phys. Rev. B 75, 184530 (2007) Phys. Rev. B 75, 184530 (2007)
5.5. R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, FFluctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional luctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional superconductorssuperconductors,, Phys. Rev. B 75, 094506 (2007) Phys. Rev. B 75, 094506 (2007)
6.6. Martin Dressel and George Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Martin Dressel and George Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge University Press, Cambridge, 2002).Matter (Cambridge University Press, Cambridge, 2002).
7.7. Marc Scheffler and Martin Dressel, Marc Scheffler and Martin Dressel, BBroadband microwave spectroscopy in Corbino geometry for roadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7 Ktemperatures down to 1.7 K,, Rev. Sci. Instrum. 76, 074702 (2005) Rev. Sci. Instrum. 76, 074702 (2005)
8.8. S. M. Girvin, Duality in Perspective, Science 25, Vol. 274. no. 5287, pp. 524 - 525 (1996)S. M. Girvin, Duality in Perspective, Science 25, Vol. 274. no. 5287, pp. 524 - 525 (1996)9.9. J. C. Booth, Dong Ho Wu, and Steven M. Anlage, J. C. Booth, Dong Ho Wu, and Steven M. Anlage, AA broadband method for the measurement of broadband method for the measurement of
the surface impedance of thin films at microwave frequenciesthe surface impedance of thin films at microwave frequencies,, Rev. Sci. Instrum. 65, 2082 Rev. Sci. Instrum. 65, 2082 (1994)(1994)
10.10. Marc Scheffler, Serife Kilic, and Martin Dressel, Strip-shaped samples in a microwave Corbino Marc Scheffler, Serife Kilic, and Martin Dressel, Strip-shaped samples in a microwave Corbino spectrometer, Rev. Sci. Instrum 78, 086106 (2007) spectrometer, Rev. Sci. Instrum 78, 086106 (2007)
11.11. James C. Booth, Dong-Ho Wu, and Steven M. Anlage, Measurements of the Frequency James C. Booth, Dong-Ho Wu, and Steven M. Anlage, Measurements of the Frequency Dependent Microwave Fluctuation Conductivity of Cuprate Thin Film Superconductors, Dependent Microwave Fluctuation Conductivity of Cuprate Thin Film Superconductors, Fluctuation Phenomena in High Temperature Superconductors, (Kluwer, Dordrecht, 1997), edited Fluctuation Phenomena in High Temperature Superconductors, (Kluwer, Dordrecht, 1997), edited by Marcel Ausloos and Andrei A. Varlamov, pp.151 - 178.by Marcel Ausloos and Andrei A. Varlamov, pp.151 - 178.
12.12. Haruhisa Kitano, Takeyoshi Ohashi and Atsutaka Maeda, Broadband method for precise Haruhisa Kitano, Takeyoshi Ohashi and Atsutaka Maeda, Broadband method for precise microwave spectroscopy of superconducting thin films near critical temperature, microwave spectroscopy of superconducting thin films near critical temperature, arxiv:0806.1421v1 (2008)arxiv:0806.1421v1 (2008)
13.13. V.F. Gantmakher and M.V. Golubkov, Width of the zero-field superconducting resistive transition 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)in the vicinity of the localization threshold, JETP LETTERS Vol. 73 (2001)
14.14. J. Corson, R. Mallozzi, J. Orenstein, J.N. Eckstein, I. Bozovic, Vanishing of phase coherence in J. Corson, R. Mallozzi, J. Orenstein, J.N. Eckstein, I. Bozovic, Vanishing of phase coherence in underdoped Biunderdoped Bi22SrSr22CaCuCaCu22OO8+8+δδ, ., .Nature, Vol. 398, Issue 6724, pp. 221-223 (1999)Nature, Vol. 398, Issue 6724, pp. 221-223 (1999)