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AC Transport in Really Really Dirty Superconductors and near Superconductor-Insulator Quantum Phase Transitions N. Peter Armitage The Institute for Quantum Matter Dept. of Physics and Astronomy The Johns Hopkins University. Please visit http://strongdisordersuperconductors.blogspot.com/ - PowerPoint PPT PresentationTRANSCRIPT
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AC Transport in Really Really Dirty Superconductors and near Superconductor-
Insulator Quantum Phase Transitions
N. Peter Armitage
The Institute for Quantum MatterDept. of Physics and AstronomyThe Johns Hopkins University
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Please visit
http://strongdisordersuperconductors.blogspot.com/
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CommentEtc.
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AC Transport in Really Really Dirty Superconductors and near Superconductor-
Insulator Quantum Phase Transitions
N. Peter Armitage
The Institute for Quantum MatterDept. of Physics and AstronomyThe Johns Hopkins University
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Effects of disorder on electrodynamics of superconductors?
“Low” levels of disorder captured by BCS based Mattis-Bardeen; Dirty limit (1/t >> D).
Higher levels of disorder one must progressively consider…
Fluctuating superconductivity (thermal fluctuations)
Quantum transition to insulating state? Quantum fluctuations?;Character of insulating state?
Effects of inhomogeneity self-generated granularity
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Superconductor AC Conductance @ T=0, D
Real ConductivityImaginary Conductivity
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Mattis – Bardeen formalism: Electrodynamics of BCS superconductor in the dirty limit
Sign depends on whether perturbation is even or odd under time reversal. Dipole matrix element is odd, so Case II coherence factors.
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Case I(SDW)
Case II(s-wave Supercond)
Diss
ipati
on
Case I(SDW)
Case II(s-wave Supercond)
Frequency /D Frequency T/Tc
1 2 0.5 1
sn
sn
Mattis – Bardeen formalism: Electrodynamics of BCS superconductor in the dirty limit
T = 0 ~ 0.D
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Mattis-Bardeen prediction for type II coherence
Klein PRB 1994
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Thin films transmissionthrough Pb filmsPalmer and Tinkham 1968(earlier Glover and Tinkham 1957)
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Cavity perturbation of Nb samples; Klein PRB 1994
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Mattis-Bardeen prediction for type II coherence
Klein PRB 1994
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For a collection of particles of density n of mass me, there is a sum rule on the area of the real part of the conductivity (f-sum rule of
quantum mechanics).
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2D Gap
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2D Gap
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2D Gap
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2D Gap
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Superconducting Fluctuations;Thermal and Quantum
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Different T regimes of superconducting fluctuations De Order parameter
- Amplitude (D) fluctuations; Ginzburg-Landau theory; D ≠ 0
- Below Tc0 D>≠ 0
- Transverse phase fluctuationsVortices x ei ≠ 0
– Longitudinal phase fluctuations; “spin waves”; . ei ≠ 0 (in neutral
superfluid)
Temperature (Kelvin)
TKTB Tc0
Am
plitu
de F
luct
uatio
ns
Phas
e Fl
uctu
ation
s
Sup
erco
nduc
tivity
Nor
mal
Sta
te
Thermal superconducting fluctuations
Res
ista
nce
W/c
Siz
e se
t by
phas
e `s
tifne
ss’
Fluctuations can be enhanced in low dimensionality, short coherence length, and low sf density dirty
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Amplitude Fluctuations
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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, concrete, etc.) has a form that goes like square of generalized coordinate
e.g. Hooke’s law
U = ½ kx2
De Order parameter
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Superfluid (Phase) Stiffness …Superfluid density can be parameterized as a phase stiffness: Energy scale to twist superconducting phase D eq
q1 q2 q3 q4 q5 q6 Uij = - T cos Dqij
(Spin stiffness in discrete model. Proportional to Josephson coupling)
Energy for deformation has this form in any continuous elastic medium.T is a “stiffness”, a spring constant.
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Superconductor AC Conductance @ T=0, D
Real ConductivityImaginary Conductivity
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Superfluid (Phase) Stiffness …Superfluid density can be parameterized as a phase stiffness: Energy scale to twist superconducting phase D eq
q1 q2 q3 q4 q5 q6 Uij = - T cos Dqij
(Spin stiffness in discrete model. Proportional to Josephson coupling)
Energy for deformation has this form in any continuous elastic medium.T is a “stiffness”, a spring constant.
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Supe
rflui
d St
ifnes
s s
TKTB Tc0
bare superfluid stiffness
s BCS
TKTB p/ s
Temperature
rs
Kosterlitz-Thouless-Berezenskii TransitionMermin-Wagner Theorem --> In 2D no true long-range ordered states with
continuous order parametersKTB showed that one can have
topological power-law ordered phase at low T
<(0) (r)> ~ 1/r
Since high T phase is exponentially correlated <(0) (r)> ~ e -r/ a finite temperature transition exists
Transition happens by proliferation (unbinding) of topological defects (vortex - antivortex) Coulomb gas
Superfluid stiffness falls discontinuously to zero at universal value of s/T
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If r >> l2/d then charge superfluid effect should be minimal
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Kosterlitz Thouless Berzenskii TransitionSu
perfl
uid
stiff
nes
TKTB Tm
bare superfluid density
=0
=inf
TKTB = p/ s
In 2D static superfluid density falls discontinuously to zero at temperature set by superfluid density itself. Vortex proliferation at TKTB.
Superfluid stiffness survives at finite frequency (amplitude is still well defined). Approaches ‘bare’ stiffness as w gets big.
Temperature
increasing Probing length set
by diffusion relation.
Frequency Dependent Superfluid Stiffness …
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Pha
se S
tiffn
ess(
Kel
vin)
See W. Liu on Friday
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Time scales?
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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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Characteristic fluctuation rate of 2D superconductor
See W. Liu on Friday
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And what about at higher disorders?
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Left:Bi film grown onto amorphous Ge underlayer on Al2O3 substrate. Data suggests a QCP [Haviland, et al., 1989]
Right:Ga film deposited directly onto Al2O3 substrate. [Jaeger, et al., 1989]
Thickness tuning tunes disorder; dominant scattering is surface scattering
Superconductor-Insulator Transition
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Amplitude Dominated Transition
T0
Bc “Bc2”
TKTB
Ther
mal
Quantum
= D(x,t) ei (x,t)
Superconducting
Phase defined
Phase Diagram for Homogeneous System?
Insulating
Amplitude defined
Phase Dominated Transition: “Dirty” Bosons
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Superfluid Stiffness @ 22 GHzCan get it from s2
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By Kramers-Kronig considerations, to get large imaginary conductivity one must have a narrow peak in the real part.(Stay tuned for Liu et al. 2012. Full EM response through the SIT. Preview on Friday W. Liu.)
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Effects of inhomogeneities?
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L.N. Bulaevskii 1994D. van der Marel and A. Tsvetkov, 1996(probably many others)
Coupled 1D Josephson arrays, with two different JJs per unit cell (same as inhomogeneous superfluid density)
Considered extensively in the context of the bilayers cuprates
K
I
A new mode! Oscillator strength depends on difference in JJ couplings
Super current depends on weaker JJ coupling
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EF
In random system, the supercurrent response will be governed by weakest link (strength of delta function is set by weakest link). Spectral weight (set by average of links) has to go somewhere by spectral weight conservation. (Remember coupling is density and there is a sum rule on conductivity set by density).
Finite frequency absorptions set by spatial average of superfluid density!
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Many models addressing these general ideas.
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Much newer work… (sorry Nandini…)
How to discriminate the ballistic response of a Cooper pair that crosses a scing patch in time t from a homogeneously fluctuating superconductor on times t ?
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Phase fluctuation effects important
Evidence for non-trivial electrodynamic response on insulating side of SIT
Inhomogeneous superfluid density gives dissipation
How can we discriminate the ballistic response of a Cooper pair that crosses a scing patch in time t from a homogeneously fluctuating superconductor on times t ?