Electron Dephasing and Energy Exchange in Diffusive Metal Wires:

Boulder Summer School 2005 – Lectures 2 & 3Norman Birge, Michigan State University

Collaborators:

F. Pierre, A. Gougam (MSU)S. Guéron, A. Anthore, B. Huard, H. Pothier,

D. Esteve, M. Devoret (CEA Saclay)

Work supported by NSF DMR

With special thanks to Hugues Pothier!

PrologueResistivity of metals

De Haas & de Boer, 1934

PtHigh T, phonons

Low T, impurities & disorder

dR/dT>0 always

But dR/dT<0 in some samples!

De Haas, de Boer, & van den Berg, 1934

Au

Suspect magnetic impurities

Fe in Cu:

J.P. Franck, Manchester, Martin (1961)

But how do they work?

The solution

∑ ⋅=i

i SsJHrr

The s-d exchange model:

⎟⎟⎠

⎞⎜⎜⎝

⎛−∝

KTTB logδρ

vJeETk FKB1−

≈Kondo temperature:

0<dTdρ

⇒Kondo’s result:

ν = density of states at EF

!!

Moral of the story: magnetic impurities dominate the low-temperature resistivity of metals, even

at concentrations as low as 0.01%

1960’s

Jump ahead 20 years … 1980’s

...111++=

− pheldisorder τττ

τρ 1

2nem

=

elastic

preserves quantum phase coherence

inelastic

destroys quantum phase coherence

WRONG!

disorder

phonons

R

T

ρ

Matthiessen’s rule:

Low T:elasticinelastic ττ

11<< Electrons maintain quantum phase

coherence over distance Lφ >> le

Electron transport in diffusive regime

1. Elastic scattering (film boundaries, impurities)

diffusive states

2. Inelastic scattering (phonons, other electrons, spins)

loss of phase coherence

energy exchange between electrons

el

φL

eFe vl τ=

eFlvD 31=

φφ τDL =

e-

Why Is the Phase Coherence Time Important?

• τφ limits quantum transport phenomena:– normal metals: weak localization, UCF, Aharonov-Bohm– superconductors: proximity & Josephson effects

• Localization theory assumes Lφ > ξ– no M-I transition if τφ saturates at low temperature

• example of quantum system coupled to environment

Predictions for τφ at low T(Altshuler & Aronov, 1979)

At low T, τφ limited by e-e interactions

Screening depends ondimensionality

τφ depends on dimensionality

transverse dimensionsL Dφ φτ= >« wires » (1d regime) :

( At energy E,

compare / withtransverse dimensions)

D Eh

( / rule the game )E φτ∼ h

( )...2, ,

...

x y zq q q Dq iω

⎛ ⎞⎜ ⎟⎜ ⎟+⎝ ⎠

∑

τφ(T) in wires

τφ

T

T-2/3

T-3

e-e

e-ph1 K

(Altshuler, Aronov, Khmelnitskii, 1982)

φτ −= +2 / 3 13( )A T B T

πν

⎛ ⎞= ⎜ ⎟

⎝ ⎠h

1/ 3214

B

F K

k RLwt R

A Screened Coulomb interaction at d=1

Temperature dependence of τφconfirmation of AAK theory in quasi-1D

Echternach, Gershenson, Bozler, Bogdanov & Nilsson, PRB 48, 11516 (1993)

(electron-electron)

T-3 (electron-phonon)

T-2/3

The Experimental ControversyMohanty, Jariwala and Webb, PRL 78, 3366 (1997)

T-2/3

T-3

Saturation of τφ:

Artifact of measurement ?If not, is it intrinsic ?

-0 .0 4 -0 .0 2 0 .0 0 0 .0 2 0 .0 4

3 .1 K

1 .1 7 K

2 1 0 m K

4 5 m K

1 0 -3

ΔR/R

B (T )

Ag

Interference of time-reversed paths ⇒ “weak-localization” correction to R

B reduces weak-loc. correction

I

V

B

L ~ 0.25 mm

Measuring τφ(T)

Lφ=(Dτφ)1/2e-

Measuring τφ(T) : raw data

-10 0 10

B (mT)

10-3

40 mK

165 mK

0.6 K

1.95 KAg(6N)c

ΔR/R

-15 0 15

5x10-4

40 mK

155 mK

0.8 K

2.5 KAg(5N)b

-15 0 15

10-3

37 mK

92 mK

0.7 K

1.6 K

ΔR

/R

Au(6N)

B (mT)-50 0 50

5x10-4

1.7 K

0.62 K

100 mK

37 mK

Cu(6N)d

Ag(6N) & Au(6N):ΔR grows as T decreases

Ag(5N) & Cu(6N):ΔR saturates below ~ 100mK

5N = 99.999 % source purity6N = 99.9999 % “ “

100 atoms ~ 25 nm

1 ppm ofimpurities :

τφ(T) in Ag, Au & Cu wires

Saturation of τφ is sample dependent

• Ag 6N, Au 6N→ agreement with AAK theory

• Ag 5N, Cu 6N→ saturation of τφ(T)

Low T behavior vs. Purity:

T-2/3 5N = 99.999 % source material purity6N = 99.9999 % “ “ “

0.1 1

1

10

Ag 6N Au 6N Ag 5N Cu 6N

τ φ (ns

)

T (K)

Quantitative comparison with AAK theoryfor clean samples

F. Pierre et al.,PRB 68, 0854213 (2003)

0.1 1

1

10

τ φ (ns)

T (K) πν

⎛ ⎞= ⎜ ⎟

⎝ ⎠h

1/ 3214

B

Fy

Kth

k RLwt R

A

φτ −= +2 / 3 13( )A T B T

1st method : τφ

2nd method : measure energy exchange rates

Investigation of inelastic processes

Distribution f(E)reflects the exchange rates

UU

R

BV

τφ(T)

T

T-2/3

T-3

e-e

e-ph

1 K

Another process dominates in not-so-pure samples?

?

Background: Shot noise in diffusive metal wiresSteinbach, Martinis and Devoret, PRL 76, 3806 (1996)

Theory:

Nagaev 1992,1995

Kozub & Rudin, 1995

∫∫ −= )),(1)(,(4 ExfExfdxdELR

SI

What does f(x,E) look like?

Distribution function -- textbook case(no shot noise)

Assumes complete thermalization -- tD >> telectron-phonon

Never true in mesoscopic metal samples at low T!

τD=L2/D

Distribution function for τD << τelectron-phonon

f(x,E) shaped by energy exchange

f(E)

1

E0-eU

E

x

1

E-eU 0

f(E)

E

xτD << τinteraction τD >> τinteractionτD=L2/D

free electrons “hot” electrons

Aside 1: Current through a tunnel junction

( )2

RLR

2

LF n ( eVE )n (E) f (E)2 dE M 1 f ( )eVE→ +

πΓ = − +

ν∫h

eV

V

( )I e → ←= Γ − ΓL R

( )

L

RL

RT

n ( eV

eV

E )1I dEeR

x

n (E)

f (E) f (E )

+

+

=

−

∫

NN junction:

T

n(E) 1 f(E)V

RI

= =

⇒ =

( )L Rf (E) f (E e1 V)←Γ = +−

Conductance of an N-X junction at T=0

V

N X( )

R

R

X

T

TeV

X

e

T

L

L

0

V

0

n (E )

f (E )

1I dEeR

1 dE

n (E)

f

eV

eV

eVn (E )

n

(E)

(E)

eR1 dE

eR

−

+

+

+

=

=

× −

=

∫

∫

∫

TX

d nI 1d RV

(e )V=

Spectroscopy of nX

eVdV

dI

How to measure f(E):tunnel spectroscopy using an N-S junction

V

N S

-0.4 -0.2 0.0 0.2 0.4

0

20

40

dI/d

V (µ

S)

V (mV)

eVΔ

Ag-Al 2 2

E

E − Δ

N out of equilibrium:spectroscopy of f(E)

V

N* S

eVΔ

-0.4 -0.2 0.0 0.2 0.4

0

20

40

dI/d

V (µ

S)

V (mV)

( )T

SNS f (E1I d eEe

V)n (E) (E)R

f= −−∫

ST

Nd n (E) eVI 1 d E )E

RVf

d(′ −

−= ∫

Experimental setup

UU

VS

tunnel junction

L=5 to 40 µm2

DLDiffusion time: 1 to 60 nsD

τ = =

( )dI VdV

numericaldeconvolution ( )f E

-0.6 -0.4 -0.210

20

V (

dI/d

V (µ

S)

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

V (mV)

dI/d

V (µ

S)

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

20

40

60

80

100

V (mV)

dI/d

V (µ

S)

-0.6 -0.4 -0.210

20

V (

dI/d

V (µ

S)

-0.2 0.00

1

E (meV)

f(E)

-0.2 0.00

1

E (meV)

f(E)

Summarize how to measure f(E):

UU

VS

U=0.2 mV

U=0 mV

f(E) dI/dV

-0.3 -0.2 -0.1 0.0 0.10

1

L=1.5 μmτD=0.36 nsf(E)

-0.3 -0.2 -0.1 0.0 0.1

L=5 μmτD=4.5 ns

Effect of the diffusion time τD on f(E)

E (meV) E (meV)

U=0.2 mV

longer interaction time ⇒ more rounding

H. Pothier et al., PRL 79, 3490 (1997)

0

1

τD=1.8 ns

Ag 6N (99.9999%)

f(E)

0

1

f(E)

τD=2.8 ns

Cu 5N(99.999%)

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.20

1

f(E)

E (meV)

τD=1.8 ns

Au 4N(99.99%)

Compare strength of interactions

effect of material ? effect of purity ?

Compare Dependence on U

0

1

U=0.1 mV

U=0.4 mV

f(E)

0

1

f(E)

-1.5 -1.0 -0.5 0.0 0.50

1

f(E)

E/eU

0

1L=20 µmτD=19 nsX=L/2

Ag 6N

f(E)

0

1

f(E)

L=5 µmτD=2.8 nsX=L/2

Cu 5N

-0.6 -0.4 -0.2 0.0 0.20

1

f(E)

E (meV)

L=1.5 µmτD=0.18 nsX=L/4

Au 4N

U=0.1, 0.2, 0.3 & 0.4 mV

Observe scaling law in Au 4N & Cu 5N but not in Ag 6N

Calculation of f(x,E)

{ }( ) { }( )in o2 ut

2f(E) f(E)t x

I x,E, f I x,E, fD∂ ∂+ = −

∂ ∂

Boltzmann equation in the diffusive regime (Nagaev, Phys. Lett. A, 1992):

xx-dx

x+dx

ε

E-ε

E

x 0 x Lf (E) f (E) Fermi function= == =

diffusion in real space

transfers of population in the energy space

Boundary conditions :

Calculation of f(x,E)

{ }( ) { }( )2 t

2

in ouI x,E, ff(E)x

I , , fD x E−∂

=∂

ε

E-ε

E E’

E’+ε

{ }( ) [ ] [ ]outI x,E, f dE'd f(E) 1 f(E ) f(E')K( 1 f(E') )= − − −ε ε ε + ε∫

e-e interactions :

fn3D

3 / 2κ

ε(Altshuler, Aronov, Khmelnitskii, 1982)

Boltzmann equation in the diffusive regime (Nagaev, Phys. Lett. A, 1992):

Theory of screened Coulomb interaction in the diffusive regime

(Altshuler & Aronov, 1979)

ingredients:polarisabilityoverlap ↑

↑

Prediction for 1D wire : 3 / 2( )K κε

ε =

( )κ π ν−

= h1

3/2e2 SFD

ε

2 4 2dq

D q ω⎛ ⎞∝⎜ ⎟+⎝ ⎠

∫

Experiment vs. Theory

-1 0

κ=0.425 ns-1meV-1/2

κthy=0.1 ns-1meV-1/2

0.2

Ag 6N0.5 mV0.4 mV0.3 mV0.2 mV0.1 mV

f(E)

E/eU

-1 0

κ=0.6 ns-1

Cu 5N

0.1 mV

0.2 mV

0.3 mV

0.4 mV

0.5 mV

E/eU

f(E)

κ=3.2 ns-1meV-1/2

κthy=0.1 ns-1meV-1/2

Cu 5N

0.1 mV

0.2 mV

0.3 mV

0.5 mV0.4 mV

K(ε)=κ ε-3/2

K(ε)=κ ε-2

Ag 6N: experiment agrees with theory

Cu 5N, Au 4N, Ag 5N: • energy exchange stronger than predicted• K(ε)=κ ε-2 fits data

f(E)τφ(T)

Ag5NCu6N,5N,4N Au4N

fast relaxation rates

Comparison of the results of the two methods

Ag6N

saturation

3 / 2K( ) κε

=ε

3 / 2

1K( ) ∝εε

2 / 3T−∝

f(E)

Comparison of the results of the two methods

τφ(T)

Ag5NCu6N,5N,4N Au4N

saturationfast relaxation rates

Ag6NCoulomb

interactions

Othermechanism

ROLE OF RESIDUAL IMPURITIES ?

3 / 2K( ) κε

=ε

3 / 2

1K( ) ∝εε

2 / 3T−∝

The two puzzlesτφ(T) measurements

0.1 1

1

10

Ag 6N Au 6N Ag 5N Cu 6N

τ φ (ns)

T (K)-0.3 0.0

0

1

0.3mV 0.2mV 0.1mV

τD=2.8 ns

Cu5

f(E)

E(meV)

Ag5N - Cu5N,4N - Au4N

f(E) measurements

Anomalous interactions in the less pure samples

The Kondo effect again

e

e

J .Sσur ur

Spin-flip scattering

increased resistivityreduction of τφ

Collective effect:Formation of a

singlet spin state

T

R

KT

Log(T)−β

sfγ

T

spin-flip

(total scattering rate)

phononsKondo

J1/νFKB eETk −≈

Nagaoka-Suhl expression of the spin-flip scattering rate near TK

1sf−τ

T

spin-flip

1 10 100

1

10

τ sf(T

)/(hv

F/2c im

p)

T/TK

Weak temperaturedependance near TK !!

Link to τφ(T) saturation?

From τsf to τφ

Another important timescale: τKLifetime of the spin state

of a magn. imp.

If τK < τsf Other electrons matter

Randomising effecte fph se e

1 1 1 1φ −

= + +τ τ τ τ

If τK > τsf The spin states of the mag. imp. seen by time-reversedelectrons are correlated

e fph se e

1 1 1 2φ −

= + +τ τ τ τ

( V.I. Fal’ko, JETP Lett. 53, 340 (1991) )

Comparison of τsf and τK

kBT EF BFk Tν concentration of electronsthat can spin-flip

cimp concentration of magnetic impurities

B impFIf k T c :>ν τK < τsf

impT 40 mK c (ppm)×>Numerically,

for Au, Ag, Cu, …

e fph se e

1 1 1 1φ −

= + +τ τ τ τ

Effect of magnetic impurities on τφ

0.1 1

1

10 Ag (6N)

Ag (5N) bare

1.0ppm Mn

0.3ppm Mn

e-e & e-ph scattering

spin-flipscattering

TK

τ φ (ns)

T (K) τφ(TK) =0.6 ns

cimp (ppm)

Spin-flip rate peaks at TK:

1 10 100

1

10

τ sf(T

)/(hν

F/2c im

p)

T/TK

ee p se h f

11 11φ −

+τ τ

+τ

=τ

Fit parameters:Ag(5N) bare: 0.13 ppm+ 0.3 ppm : 0.40 ppm+ 1 ppm : 0.96 ppm

F. Pierre et al.,PRB 68, 0854213 (2003)

Effect of magnetic impurities on τφ

0.1 1

1

10 Ag (6N)

Ag (5N) bare

1.0ppm Mn

0.3ppm Mn

e-e & e-ph scattering

spin-flipscattering

TK

τ φ (ns)

T (K)

Above TK : partial compensation of e-e and s-f

apparent saturation

Why can’t we just detect magnetic impurities with R(T) (the original Kondo effect)?

1 ppm of Mn is invisible in R(T)(hidden by e-e interactions)

1 2 3 4

0.2

0.4

0.6

0.8

1.0

1.2

δR/R

(‰)

T-1/2 (K-1/2)

Ag with 1 ppm Mn at B=30mT Fit: 2.4E-4T-1/2

Source material purity vs. sample purity: Cu samples

Magnetic impurities are invisible in R(T)

0.1 1 10

0.1

1

10

Ag 6N Cu 5N #1 Cu 5N #2 Cu 6N #1 Cu 6N #2 Cu 6N #3

τ φ (ns)

T (K)

1 2 3 4 5

1

2

3

4

δR/R

(‰)

1/T-1/2 (K)

• In all Cu samples τφ(T) saturates at low T• τφ(T) is strongly reduced but shows no dip

Measure τφ(B) from Aharonov-Bohm oscillations

I

V

BT=100 mK

Cu ring0.0 0.5 1.0 1.5

0

1

2

δG (

e2 /h)

B (T)

0.00 0.05-0.2

0.0

0.2

1.40 1.45-0.2

0.0

0.2

A.B. Oscillations vs. Magnetic Field

0 200 400 6000.0

0.5

1.0 B=0±0.1T B=1.2±0.1T

h/e

Pow

er s

pect

rum

AB

(a.u

.)

1/ΔB (1/T)

AB oscillations increase with B ⇒ presence of magnetic “impurities” !

Fourier Transform(T=100 mK)

40 mK noise floor

-20 -10 0 10 20

0.05

0.1

0.5

T=40 mK T=100 mK

ΔGh/

e * (k

BT/E

Th)1/

2 [e

2 /h]

2μBB/kBT

What about energy exchange ?

Energy exchange mediated by magnetic impuritiesKaminski and Glazman, PRL 86, 2400 (2001)

f(E)

EE-ε E’+ε

E’

E’+εE’E E-ε

Virtual state

* *Reinforced by Kondo effect

Energy exchange mediated by magnetic impuritiesvanishes when gμBB >> eU

f(E)

EE-ε E’+ε

E’

E’+εE’E E-ε

Virtual state

gµB

initial state:

final state:

Aside 2: Inelastic tunneling

eV

P(E)=probability to give energy E to environment

Ε

P(E) depends on environmental impedance

V

Zext(ω)

C,RT

V

RT

Z (ω)=

[ ] ( )

J(t

e

0t

iEt /

t

0

V

)

i

K

dI

J(t

P(E)

P(E

)

)

1 dEd R

1 e dt2

Re Z( )d2 e 1R

V

+

+∞ − ω

=

=π

ωω= −

ω

∫

∫

∫

h

h

At T=0, one obtains :

Resistive environmentV

R

C,RT

For K

2RRdI c tV

Vs .

d

⎛ ⎞∝ +⎜ ⎟⎜ ⎟

⎝ ⎠

-1 0 10.0

0.5

1.0

C=0.92fFT=40 mK R/RK=.08 (R=2.06kΩ)

RTdI

/dV

V(mV)

0.01 0.1 1

0.5

1.0

V(mV)kT eRC

h

RCeV h

<

UU

R

BV

Measure f(E) at B ≠ 0 using Dynamical Coulomb Blockade

-0.6 -0.3 0.0 0.3

18

21

dI/d

V (

µS)

-0 .6 -0.3 0.0 0.3

15

18

21

24

dI/d

V (

µS)

V (m V)

dI/d

V (

µS)

-0 .6 -0.3 0.0 0.3 0.6

15

18

21

24

V (m V)

V (m V)

-0.6 -0.3 0.0 0.3 0.6

18

21

V (m V)

dI/d

V (

µS)

U=0 mV

U=0.2 mVB=1.4 T

dI/dV → f(E) → electron-electron interactions

R=1 kΩ

A controlled experiment

Ag (99.9999%)

0.9 ppm Mn2+

implantation

Comparativeexperiments

bare

implanted

Effect of 1 ppm Mn on interactions ?

Left as is

25 nm

106 Ag1Mn

-0.2 0.0 0.2

0.9

1.0

-0.2 0.0 0.2

0.8

0.9

V (mV)

Rtd

I/dV

Experimental data at weak B

U = 0.1 mVB = 0.3 T

weak interaction

strong interaction

implanted

Rtd

I/dV

bare

U

( )

-0.5 0.0 0.5

0.9

1.0

V (mV)

Rtd

I/dV

( )

-0.5 0.0 0.5

0.9

1.0

V (mV)

Rtd

I/dV

-0.2 0.0 0.2

0.9

1.0

-0.2 0.0 0.2

0.8

0.9

V (mV)

Rtd

I/dV

Rtd

I/dV

U = 0.1 mVB = 0.3 TB = 2.1 T

Very weakinteraction

Experimental data at weak and at strong B

implanted

bare

0.1 1 10

0.1

1

10

τφ (ns)

T (K)

bare

implanted

Cbare =0.1 ppmCimplanted=0.9 ppm

Fits:

Coherence time measurementson the same 2 samples

Full U,B dependenceB

are

C=0

.1 p

pmIm

plan

ted

C=0

.9 p

pmU=0.1 mV

2.1 T1.8 T1.5 T1.2 T0.9 T0.6 T0.3 T

U=0.2 mV U=0.3 mV

-0.4 0.0 0.4

-0.2 0.0 0.2

0.8

0.9

1.0

1.1

1.2

V (mV)V (mV)V (mV)

RTd

I/dV

RTd

I/dV

-0.2 0.0 0.2

0.9

1.0

-0.4 0.0 0.4

-0.5 0.0 0.5

-0.5 0.0 0.5

Comparison with theoryB

are

C=0

.1 p

pmIm

plan

ted

C=0

.9 p

pm

2.1 T1.8 T1.5 T1.2 T0.9 T0.6 T0.3 T

U=0.1 mV U=0.2 mV U=0.3 mV

-0.4 0.0 0.4

-0.2 0.0 0.2

0.8

0.9

1.0

1.1

1.2

V (mV)V (mV)V (mV)

RTd

I/dV

RTd

I/dV

-0.2 0.0 0.2

0.9

1.0

-0.4 0.0 0.4

-0.5 0.0 0.5

-0.5 0.0 0.5

Goeppert, Galperin, Altshuler and Grabert, PRB 64, 033301 (2001)

1S2

⎛ ⎞=⎜ ⎟⎝ ⎠

Conclusions – Lectures 2 & 3Two methods to investigate interactions in wires

0.1 1

1

10

Ag 6N Au 6N Ag 5N Cu 6N

τ φ (ns

)

T (K)

τφ(T)

-0.4 -0.2 0.00

1

exp. U=0.1, 0.2, 0.3 mV theory

(κ3/2= 0.11 ns-1meV-1/2)f(E

)

E(meV)

f(E)

-0.2 0.0 0.2

0.8

0.9

1.0

1.1

1.2

RTd

I/dV

V (mV)

Moral of the story: even at concentrations as low as 1 ppm,magnetic impurities have a large influence on low-temperature electronic transport in metals.

Four consequences of electron-electron interactionsin quasi-1D diffusive wires (Altshuler & Aronov)

• loss of phase coherence: τφ ~ T-2/3 (AA+Khmelnitskii)• energy exchange between quasiparticles:

• correction to resistance: δR(T) ~ T-1/2

• correction to tunneling DOS, or dynamic Coulomb blockade: dI/dV ~ V2R/RQ R

∝ ε-3/2

References• Dephasing

– Gougam, Pierre, Pothier, Esteve, Birge, J. Low Temp. Phys. 118, 447 (2000).– Pierre & Birge, Phys. Rev. Lett. 89, 206804 (2002).– Pierre, Gougam, Anthore, Pothier, Esteve, Birge, Phys. Rev. B 68, 085413

(2003).

• Energy Exchange– Pothier, Gueron, Birge, Esteve, Devoret, Phys. Rev. Lett. 79, 3490 (1997).– Pothier, Gueron, Birge, Esteve, Devoret, Z. Physik. B 104, 178 (1997). – Pierre, Pothier, Esteve, Devoret, J. Low Temp. Phys. 118, 437 (2000).– Anthore, Pierre, Pothier, Esteve, Devoret, cond-mat/0109297 (2001).– Anthore, Pierre, Pothier, Esteve, Phys. Rev. Lett. 90, 076806 (2003).

• Both– Pierre, Pothier, Esteve, Devoret, Gougam, Birge, cond-mat/0012038 (2000).– Pierre, Ann. Phys. Fr. 26, N°4 (2001).– Huard, Anthore, Birge, Pothier, Esteve, to be published in PRL (2005).

Evidence for extremely dilute magnetic impuritieseven in purest samples

Sample Imp. TK(K) c (ppm)Ag(6N)a Mn 0.04 0.009

“ b “ “ 0.011“ c “ “ 0.0024“ d “ “ 0.012

Au(6N) Cr 0.01 0.02

100 atoms ~ 25 nm

1 ppm :

In the wire, 0.01 ppm = 3 impurities/µm

1

10

0.1 1

1

10

Ag(6N)c Ag(6N)b Ag(6N)a

Ag(6N)d Au(6N)

τ φ (ns)

T (K)

Compare τφ data with AAK and GZS theories

0.1 1 10 100 1000 10000 1000000.01

0.1

1

10MSU/Saclay: Ag6N; Au6NMaryland: Au (Mohanty et al.)Bell Labs: AuPd (Natelson et al.)

6

5

43

21

EC,D B

A

F

a bc

d

τ φmax

/τ eeAA

K (Tm

in)

τφ

max/τ0GZS