electron dephasing and energy exchange in...
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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
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