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

Please visit

http://strongdisordersuperconductors.blogspot.com/

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CommentEtc.

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

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 ConductivityImaginary 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)

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

Mattis-Bardeen prediction for type II coherence

Klein PRB 1994

Thin films transmissionthrough Pb filmsPalmer 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

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

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 kx2

De Order parameter

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.

Superconductor AC Conductance @ T=0, D

Real ConductivityImaginary Conductivity

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.

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

If r >> l2/d then charge superfluid effect should be minimal

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 …

Pha

se S

tiffn

ess(

Kel

vin)

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 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

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

Superfluid Stiffness @ 22 GHzCan get it from s2

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 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

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 ?