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
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http://strongdisordersuperconductors.blogspot.com/
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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
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
Superconductor AC Conductance
@
T=0,
D
Real Conductivity
Imaginary Conductivity
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.
Case I
(SDW)
Case II
(s

wave Supercond)
Dissipation
Case I
(SDW)
Case II
(
s

wave
Supercond
)
Frequency
/D
Frequency T/T
c
1
2
0.5
1
s
n
s
n
Mattis
–
Bardeen formalism: Electrodynamics of BCS
superconductor in the dirty limit
T = 0
~ 0.D
Mattis

Bardeen prediction for type II coherence
Klein PRB 1994
Thin films transmission
through
Pb
films
Palmer
and
Tinkham
1968
(earlier Glover and
Tinkham
1957)
Cavity perturbation of
Nb
samples; Klein PRB 1994
Mattis

Bardeen prediction for type II coherence
Klein PRB 1994
For a collection of particles of density
n
of mass m
e
, there is a sum
rule on the area of the real part of the conductivity (
f

sum rule of
quantum mechanics).
2
D
Gap
2
D
Gap
2
D
Gap
2
D
Gap
Superconducting Fluctuations;
Thermal and Quantum
Different T regimes of superconducting fluctuations
D
e
Order
parameter

Amplitude (
D)
fluctuations;
Ginzburg

Landau
theory;
D
≠ 0

Below T
c0
D
>
≠ 0

Transverse
phase fluctuations
Vortices
x
e
i
≠ 0
–
Longitudinal phase fluctuations;
“spin waves”;
.
e
i
≠ 0 (in neutral
superfluid
)
Temperature (Kelvin)
T
KTB
T
c0
Amplitude Fluctuations
Phase Fluctuations
Superconductivity
Normal State
Thermal superconducting
fluctuations
Resistance
W
/
c
Size set by phase `stifness’
Fluctuations can be enhanced in low
dimensionality, short coherence length,
and low
sf
density
dirty
Amplitude Fluctuations
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 kx
2
D
e
Order parameter
Superfluid (Phase) Stiffness …
Superfluid
density can be parameterized as a phase stiffness
:
Energy scale to twist superconducting phase
D
e
q
q
1
q
2
q
3
q
4
q
5
q
6
U
ij
=

T cos
Dq
ij
(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.
Superconductor AC Conductance
@
T=0,
D
Real Conductivity
Imaginary Conductivity
Superfluid (Phase) Stiffness …
Superfluid
density can be parameterized as a phase stiffness
:
Energy scale to twist superconducting phase
D
e
q
q
1
q
2
q
3
q
4
q
5
q
6
U
ij
=

T cos
Dq
ij
(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.
Superfluid Stifness
s
T
KTB
T
c0
bare superfluid
stiffness
s
BCS
T
KTB
p/
s
Temperature
r
s
Kosterlitz

Thouless

Berezenskii Transition
Mermin

Wagner Theorem

> In 2D no true long

range ordered states with
continuous order parameters
KTB 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
If
r
>>
l
2
/d then charge
superfluid
effect should be minimal
Kosterlitz Thouless Berzenskii Transition
Superfluid stiffnes
T
KTB
T
m
bare superfluid
density
=0
=
inf
T
KTB
=
p/
s
In 2D static superfluid density falls discontinuously to zero at temperature
set by superfluid density itself. Vortex proliferation at T
KTB
.
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 …
Phase Stiffness(Kelvin)
See W. Liu on Friday
Time scales?
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
Characteristic fluctuation rate of 2D
superconductor
See W. Liu on Friday
And what about at higher disorders?
Left:
Bi film grown onto amorphous
Ge
underlayer
on Al
2
O
3
substrate. Data
suggests a QCP [
Haviland
, et al.,
1989]
Right:
Ga
film deposited directly onto
Al
2
O
3
substrate. [Jaeger, et al.,
1989]
Thickness tuning tunes disorder; dominant scattering is surface scattering
Superconductor

Insulator
Transition
Amplitude Dominated
Transition
T
0
B
c
“B
c2
”
T
KTB
Thermal
Quantum
=
D
(x,t
)
e
i
(
x,t
)
Superconducting
Phase defined
Phase Diagram for Homogeneous System?
Insulating
Amplitude defined
Phase Dominated
Transition: “Dirty” Bosons
Superfluid
Stiffness
@ 22
GHz
Can get it from
s
2
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.)
Effects of
inhomogeneities
?
L.N.
Bulaevskii
1994
D. 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
E
F
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!
Many models addressing these general ideas.
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
?
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
?
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