Superconductors and near Superconductor-

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Nov 15, 2013 (3 years and 8 months ago)

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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/


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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

?