AC Transport in Really Really Dirty
Superconductors and near SuperconductorInsulator Quantum Phase Transitions
N. Peter Armitage
The Institute for Quantum Matter
Dept. of Physics and Astronomy
The Johns Hopkins University
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AC Transport in Really Really Dirty
Superconductors and near SuperconductorInsulator 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.
Mattis – Bardeen formalism: Electrodynamics of BCS
superconductor in the dirty limit
 ~ 0.D
Dissipation
T=0
Case II
(s-wave Supercond)
Case I
(SDW)
sn
sn
Case II
(s-wave Supercond)
1
Frequency /D
2
Case I
(SDW)
0.5
Frequency T/Tc
1
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 me, there is a sum
rule on the area of the real part of the conductivity (f-sum rule of
quantum mechanics).
2D Gap
2D Gap
2D Gap
2D Gap
Superconducting Fluctuations;
Thermal and Quantum
Temperature (Kelvin)
TKTB Tc0
Fluctuations can be enhanced in low
dimensionality, short coherence length,
and low sf density  dirty
- Transverse phase fluctuations
Vortices  x ei ≠ 0
– Longitudinal phase fluctuations;
“spin waves”;  . ei ≠ 0 (in neutral
superfluid)
Size set by phase `stifness’
Normal State
Amplitude Fluctuations
Phase Fluctuations
- Below Tc0 D>≠ 0
Superconductivity
Resistance W/ c
Thermal superconducting
fluctuations
Different T regimes of superconducting fluctuations
  De  Order parameter
- Amplitude (D) fluctuations; Ginzburg-Landau
theory; D ≠ 0
Amplitude Fluctuations
Superfluid (Phase) Stiffness …
Many of the different kinds of superconducting
fluctuations can be viewed as disturbance in phase field
  De  Order parameter
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
Superfluid (Phase) Stiffness …
Superfluid density can be parameterized as a phase stiffness:
Energy scale to twist superconducting phase 
q1
q2
q3
q4
q5
q6
 D eq
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.
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 
q1
q2
q3
q4
q5
q6
 D eq
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.
Kosterlitz-Thouless-Berezenskii Transition
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 Stifness s
Mermin-Wagner Theorem --> In 2D no true long-range ordered states with
continuous order parameters
TKTB
p/ s
bare superfluid
stiffness
s BCS
rs
TKTB
Temperature
Tc0
Superfluid stiffness falls discontinuously to zero at universal value of s/T
If r >> l2/d then charge superfluid effect should be minimal
Frequency Dependent Superfluid Stiffness …
Kosterlitz Thouless Berzenskii Transition
Superfluid stiffnes
TKTB = p/ s
increasing 
bare superfluid
density
=inf
Probing length set by
diffusion relation.
=0
TKTB
Temperature
Tm
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.
See W. Liu on Friday
Phase Stiffness(Kelvin)
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?
Superconductor-Insulator Transition
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
Phase Diagram for Homogeneous System?
Thermal
T0
Phase Dominated
Amplitude
Dominated
Transition:
“Dirty” Bosons
Transition
TKTB
Amplitude defined
Phase defined
 = D(x,t) ei (x,t)
Superconducting
Bc
Insulating
Quantum
“Bc2”
Can get it from s2 
Superfluid Stiffness @ 22 GHz
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?
Coupled 1D Josephson arrays, with two different JJs per unit cell 
(same as inhomogeneous superfluid density)
K
Considered extensively
in the context of the
bilayers cuprates
I
A new mode!  Oscillator
strength depends on
difference in JJ couplings
Super current depends on
weaker JJ coupling
L.N. Bulaevskii 1994
D. van der Marel and A. Tsvetkov, 1996
(probably many others)
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!
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 ?