B. Huard& Quantronics group
Interactions between electrons,mesoscopic Josephson effect and
asymmetric current fluctuations
Quantum electronics
DC AMPS
DC AMPS
L L/2
I 2 I
R L
Macroscopic conductors
Mesoscopic conductors
R L
Quantum mechanicschanges the rules
important for L < Lphase coherence length
Overview of the thesis1) Phase coherence and interactions between electrons in a disordered metal
2) Mesoscopic Josephson effects 3) Measuring high order current noise
150 nm
VI
Tool for measuring the asymmetry of I(t) ?
IB
superconductor
I() for elementary conductor
t
Overview of the thesis1) Phase coherence and interactions between electrons in a disordered metal
2) Mesoscopic Josephson effects 3) Measuring high order current noise
150 nm
VI
Tool for measuring the asymmetry of I(t) ?
IB
superconductor
I() for elementary conductor
t
Electron dynamics in metallic thin films
+ +++
+++
++
le
L
Grain boundariesFilm edgesImpurities
Elastic scattering
- Diffusion- Limit conductance
Coulomb interactionPhononsMagnetic moments
Inelastic scattering
- Limit coherence (L)- Exchange energy
Typically, F le L ≤ L
150 nm
How to access e-e interactions ?
1st method : weak localization
B
R(B) measures L
In a wire
Pie
rre e
t al., P
RB
(20
03
)
First measurement: Wind et al. (1986)
B (mT)
Diffusion time : (20 ns for 20 µm)D
E
? eU
U
U=0
f(E)
Occupiedstates
How to access e-e interactions ?
2nd method : energy relaxation
U
E
eU
f(E)
Distribution function and energy exchange rates
« weak interactions »
D int.
U
E
eU
f(E)
« strong interactions »
Distribution function and energy exchange rates
D int.
f(E) interactions
E
f(E)
« weak interactions »
E
f(E)
« strong interactions »
Distribution function and energy exchange rates
D int.D int.
Understanding of inelastic scattering
Coulomb interaction
Magnetic moments
Interaction1st method
Weak localization2nd method
Energy relaxation
OK
Wind et al. (1986)
OK
stronger thanexpected
Pierre et al. (2003)
dependence on B as expected
Anthore et al. (2003)
Pierre et al. (2000)
µeV0.01 0.1 1 10 100
Probed energies:
Understanding of inelastic scattering
Coulomb interaction
Magnetic moments
Interaction1st method
Weak localization2nd method
Energy relaxation
OK
Wind et al. (1986)
OK
stronger thanexpected
Pierre et al. (2003) Anthore et al. (2003)
Pierre et al. (2000)
several explanations dismissed (Huard et al., Sol. State Comm. 2004)
Quantitative experiment(Huard et al., PRL 2005)
dependence on B as expected
UU
R
BVR
I
U=0 mV
Access e-e interactions : measurement of f(E)
Dynamical Coulombblockade (ZBA)
UU
R
BVR
Istrong interactionweak interaction
U=0.2 mVU=0 mV
Measurement of f(E)
Dynamical Coulombblockade (ZBA)
Quantitative investigation of the effects of magnetic impurities
Ag (99.9999%)
0.65 ppm Mn implantation
bare
implanted
Left as is
Comparative experimentsusing methods 1 and 2
Huard et al., PRL 2005
1st method : weak localization
Best fit of L(T) for 0.65 ppm consistent with implantation
0.03 ppm compatible with < 1ppm dirt
0.10.02 TK 1 10TK1
10
30
3
L µm
Coulomb
spin-flipphonons
0.65 ppm Mn
0.1 0 0.1 0.2 0.3VmV
0.9
0.92
0.94
0.96
0.98
RtIdVd
0.1 0 0.1 0.2 0.3VmV
0.75
0.8
0.85
0.9
RtIdVd
2nd method : energy relaxation
U = 0.1 mVB = 0.3 TT= 20 mK
weak interaction
strong interaction
bare
implanted0.65 ppm Mn
0.1 0 0.1 0.2 0.3VmV
0.9
0.92
0.94
0.96
0.98
RtIdVd
0.1 0 0.1 0.2 0.3VmV
0.75
0.8
0.85
0.9
RtIdVd
Spin-flip scattering on a magnetic impurity
energy
f(E)
E
E E
E
- dephasing- no change of energy
*rate maximal at Kondo temperature
At B=0
Interaction between electrons mediated by a magnetic impurity
f(E)
E
E- E’+
E’
E’+E’E E-
Virtual state
Kaminski and Glazman, PRL (2001)
* *Enhanced by Kondo effect
Interaction mediated by a magnetic impurity :effect of a low magnetic field (gµBeU)
f(E)
E
E- E’+
E’
E’+E’E E-
Virtual state
EZ=gµB
Modified rate* *
EZ
E-EZ
Spin-flip scattering on a magnetic impurity :effect of a high magnetic field (gµB eU)
f(E)
E
E-EZ
EZ
Reduction of the energy exchange rate
eU E-
EZModified rate
E’+
E’
Virtual state
0.1 0 0.1 0.2 0.3VmV
0.9
0.92
0.94
0.96
0.98
RtIdVd
0.1 0 0.1 0.2 0.3VmV
0.75
0.8
0.85
0.9
RtIdVd
0.1 0 0.1 0.2 0.3VmV
0.9
0.92
0.94
0.96
0.98
RtIdVd
0.1 0 0.1 0.2 0.3VmV
0.75
0.8
0.85
0.9
RtIdVd
B = 0.3 T(gµBB = 0.35 eU)
B = 2.1 T(gµBB = 2.4 eU)
Very weakinteraction
Experimental data at low and at high B
implanted0.65 ppm Mn
bare
U = 0.1 mV
U = 0.1 mV
T= 20 mK
B0.3 TB0.6 TB0.9 TB1.2 TB1.5 TB1.8 TB2.1 T
0 0.60.3VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.3 mV
Various B and U
0 0.40.2VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.2 mV
0 0.20.1VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.1 mV
0 0.60.3VmV0.9
1
1.1
RtIdVd
bareU0.3 mV
0 0.40.2VmV0.9
1
1.1
RtIdVd
bareU0.2 mV
0 0.20.1VmV0.9
1
1.1
RtIdVd
bareU0.1 mV
T= 20 mK
Comparison with theoryUsing theory of Goeppert, Galperin, Altshuler and Grabert PRB (2001)
0 0.60.3VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.3 mV
0 0.40.2VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.2 mV
0 0.20.1VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.1 mV
B0.3 TB0.6 TB0.9 TB1.2 TB1.5 TB1.8 TB2.1 T
0 0.60.3VmV0.9
1
1.1
RtIdVd
bareU0.3 mV
0 0.40.2VmV0.9
1
1.1
RtIdVd
bareU0.2 mV
0 0.20.1VmV0.9
1
1.1
RtIdVd
bareU0.1 mV
Only 1 fit parameter for all curves : e-e=0.05 ns-1.meV-1/2 (Coulomb interaction intensity)
Coulomb interaction intensity e-e
0.02 0.1 1
0.02
0.1
1
energy relaxation weak localization
best
fit
for
e-e
(ns -
1 m
eV
-1
/2
)
expected for e-e (ns -1 meV
-1/2 )
Unexplained discrepancy
µeV
0.01
0.1
1
10
100
1/ 232
2
cross section area2
resistance / lengthF
e e e
Experiments on 15 different wires:
0 0.20.1VmV
0.9
1
1.1
RtIdVd
bareU0.1 mV
0 0.40.2VmV
0.9
1
1.1
RtIdVd
bareU0.2 mV
0 0.60.3VmV
0.9
1
1.1
RtIdVd
bareU0.3 mV
0 0.20.1VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.1 mV
0 0.40.2VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.2 mV
0 0.60.3VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.3 mV
Conclusions on interactions
Quantitative understanding of the role played by magnetic impurities
but Coulomb interaction stronger than expected
0.10.02 TK 1 10TK1
10
30
3
L µm
Coulomb
spin-flipphonons
Overview of the thesis1) Phase coherence and interactions between electrons in a disordered metal
2) Mesoscopic Josephson effects 3) Measuring high order current noise
150 nm
VI
Tool for measuring the asymmetry of I(t) ?
IB
superconductor
I() for elementary conductor
t
Case of superconducting electrodes
L R
L R
I
Supercurrent through a weak link ?
Unified theory of the Josephson effect
Furusaki et al. PRL 1991, …
B
VI
Coherent Conductor (L«L)
Transmission probability 2 2't t
Landauer
Collection of independent channels
r r’t
t’
Conduction channels
a(E)e-i
"e"
"h"
"h"
"e"
| | ieN S
Andreev reflection
probability amplitude
Andreev reflection (1964)
a(E)ei
a(E)e-i
E()
2
+
-
0
cos 2E
2 current carrying bound states
LR
"e"
"h"
"h"
"e"
= 1Andreev bound states
arg ( ) arg ( ) 0 mod 2R Li ia E e a E e
in a short ballistic channel ( < )
E→
E←
2arg mod 2a E
a(E)e-iR
a(E)eiL
←→
L R
a(E, L) a(E, R)"e"
"h"
"h"
"e"
< 1
-
0
E-
2
E()
+ E+
2 1
21 sin 2E
Central prediction of the mesoscopic theory
of the Josephson effect A. Furusaki, M. Tsukada (1991)
Andreev bound statesin a short ballistic channel ( < )
L R
a(E, L) a(E, R)"e"
"h"
"h"
"e"
< 1
-
02
E()
21 sin 2E
Central prediction of the mesoscopic theory
of the Josephson effect
A. Furusaki, M. Tsukada (1991)
Andreev bound statesin a short ballistic channel ( < )
0.5
0.99
0
),
1 ( ,I
E
CURRENT
0.5
0.99
I()
20
IS SV
{ 1 … N }
A few independent conduction channelsof measurable and tunable transmissions
J.C. Cuevas et al. (1998)E. Scheer et al. (1998)
Atomic orbitals
Quantitative test using atomic contacts .
,II
Quantitative test
I-V { 1 … N }
insulating layer
counter-support
Flexiblesubstrate
metallic film pushing rods
pushing rods
counter-supportwith shielded coil
sample
3 cm
Atomic contact
2 µm
Al
Metallic bridge(atomic contact)
Tunnel junction
max03 0.7 µ0 nA A tII
Ib
( )I
It
VHow to test I() theory
Strategy :
1)Measure {1,…,M}
2)Measure I()
V>0
V=0
Ib
V0
I
2/e
Switching of a tunnel junction .
It
V
Ib
circuit breaker : Ib>I V>0 stable
(circuit breaker)
open circuit : 2/e >V>0
0 0.1 0.2 0.3VmV0
15
30
45
60
IAn22e
23e
24e
AC1
AC2
AC3
0 0.1 0.2 0.3VmV0
15
30
45
60
IAn22e
23e
24e
Ib
I( )
It
VMeasure {1,…,M}
Measure I(V)
method: Scheer et al. 1997
0.957 ± 0.01
0.992 ± 0.003
AC3
AC2
AC1
Transmissions
0.089±0.06
0.185±0.05
0.12±0.015
0.115±0.01
0.088±0.06
0.11±0.01
0.11±0.01
0.62 ± 0.01
Ib
I( )
It
V
Measure I()
(circuit breaker)
max
03 0.7 µ0 nA A tII
swI ( ) I( )
bareswI ( ) ( )I I
V0
2/e
I Ibare
Ib
I( )
0
40
20
0
20
40
0
40
20
0
20
40
0
40
20
0
20
40
sw swI I
nA
Measure I()
0.957 ± 0.01 0.992 ± 0.0030.62 ± 0.01
0
40
20
0
20
40
0
40
20
0
20
40
0
40
20
0
20
40
sw swI I
nA
Theory : I() + switching at T0
Comparison with theory I()
0.957 ± 0.01 0.992 ± 0.0030.62 ± 0.01
0
40
20
0
20
40
0
40
20
0
20
40
0
40
20
0
20
40
sw swI I
nA
Comparison with theory I()
0.957 ± 0.01 0.992 ± 0.0030.62 ± 0.01
Overall good agreement
but with a slight deviation at 1
Theory : I() + switching at T0
Overview of the thesis1) Phase coherence and interactions between electrons in a disordered metal
2) Mesoscopic Josephson effects 3) Measuring high order current noise
150 nm
VI
Tool for measuring the asymmetry of I(t) ?
IB
superconductor
I() for elementary conductor
t
0 n
Full counting statistics
P(n) characterizes
Average current
during
Vm
ne/=I
I
t
pioneer: Levitov et al. (1993)
Need a new tool to measure it
10 0 10 20 30 40 50
1010
107
104
101
Independent tunnel events Poisson distribution
P(n) is asymmetric
P(n)
n
n
Well known case : tunnel junction
Simple distribution detector calibration
Log scale
Which charge counter ?
Vm
I
I
t
Tunnel junction
Charge counter: Josephson junction
t
VmIm
Im
Im
Im
I
Rlarge
Clarge
RlargeClarge 20 µs
-I
Switching rates
Proposal : Tobiska & Nazarov PRL (2004)
Charge counter: Josephson junction
t
VmIm
Im
Im+Ib
0
Im -Ib
Ib
t
Im
I
-I
I
-I
Im +Ib
Ib
0.25 0.5 0.75 1 1.25 1.5 1.75ImµA
0
0.02
0.04
0.06
0.08
0.1
0.12
R
Asymmetric current fluctuations
Im (µA)
cste (30 kHz)
There is an asymmetry
|Ib| so that
Gaussian noise
0 0.25 0.5 0.75 1 1.25 1.5 1.75ImµA
0
0.02
0.04
0.06
0.08
0.1
0.12
R
Asymmetric current fluctuations
Im (µA)
Disagreement with existing theory
Ankerhold (2006)
cste (30 kHz)
|Ib| so that
0
40
20
0
20
40
0 0.20.1VmV
0.9
1
1.1R
tIdVdbare
U0.1 mV
0 0.40.2VmV
0.9
1
1.1
RtIdVd
bareU0.2 mV
0 0.60.3VmV
0.9
1
1.1
RtIdVd
bareU0.3 mV
0 0.20.1VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.1 mV
0 0.40.2VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.2 mV
0 0.60.3VmV
0.8
0.9
1
1.1
RtIdVd
implantedU0.3 mV
0 0.25 0.5 0.75 1 1.25 1.5 1.75ImµA
0
0.02
0.04
0.06
0.08
0.1
0.12
R
Conclusions
Decoherence and interactions indisordered metals
Quantitative experimentsOpen : Coulomb intensity
Unified theory ofJosephson effect
Quantitative agreementwith fundamental relationPersp. : spectro and manip.
of Andreev states
Tool for measuringhigh ordercurrent noise
Tool sensitive to high order noise OKOpen : Interpretation ?
I (nA
)
Coulomb interaction discrepancy explanations- Extrinsic energy exchange processes ?
- Quasi-1D model inappropriate ?
- Diffusive approximation invalid ?
- Hartree term stronger than expected ?
- Theory valid at equilibrium only ?
Experiment near equilibrium
E
f(E)
0
1
Magnetic impurities and 2 level systems cannotexplain the discrepancy (bad fits)
Slight error at the lowest probed energieswould furthermore reduce the intensity e-e
Never been investigated
Strong enough if Ag very close to ferromagnetic instability
Yes, same result close to equilibrium