DYNAMICAL COULOMB BLOCKADE IN SHORT COHERENT CONDUCTORS
Frédéric Pierre
Carles Altimiras, Hélène le Sueur, Ulf Gennser, Antonella Cavanna, Dominique Mailly, F.P.
CNRS, Laboratory of Photonics and Nanostructures (LPN), Marcoussis
Ronald Cron, Michel Devoret, Daniel Estève, Philippe Joyez, Cristian Urbina
Quantronics, Service de Physique de l’État Condensé (SPEC), CEA-Saclay
LPN
PROBLEMATIC: IMPEDANCES COMPOSITION LAWS IN MESOSCOPIC CIRCUITS
Z1
V< Lφ
I1
V
<Lφ
I2
Z2
V / I1 = Z1 V / I2 = Z2
V>Lφ
I
200nm1µm
V / I = Z1 + Z2
HOW TO COMPOSE COHERENT CONDUCTORS IN A CIRCUIT?
MICROSCOPIC PICTURETUNNEL JUMP OF AN ELECTRON
TUNNEL EVENT CHARGE ON CAPACITANCE INCREASED BY et
HT = t ∑Tec+c + h.c.R L
CURRENT ACROSS A TUNNEL JUNCTIONELASTIC TUNNELING
EV Elastic tunneling
GT
G(V)
eV
GT
eV
I ∝ number of transitions allowed by Pauli principle
∝ V
Fermi golden rule
OHMIC BEHAVIOR
CURRENT ACROSS A TUNNEL JUNCTIONINELASTIC TUNNELING
V E
ε
Zenv(ν)
GT
eV
∫∫I(V)= GT/e dEdε P(ε) fL(E)(1-fR(E-ε)-fR(E)(1-fL(E-ε)))
G(V)
eV
GT
T=0K: G(V)=GT(1- dεP(ε))∫∞
Ve
G(V) SHOWS A DEPRESSION NEAR V=0
GTR
C
STATIC COULOMB BLOCKADE
Charge dynamics ignored if:
R >> RK = h/e2 ≈ 25.8kΩEC= e2/2C >> ∆E ≈ h/RC
Simple electrostatic problem:
P(ε)=δ(ε− EC)The charging energy EC=e2/2C has to be paid(T=0K)
STATIC COULOMB BLOCKADE
GTR
C
V eV>EC
eV
E
EC
eV<ECE
EC
eV
OK Forbidden by Pauli principle
G(V)
eV
GT
e2/2C
(kBT<< e2/2C)
P(ε)=δ(ε− EC)R >> RK = h/e2 ≈ 25.8kΩ
Coulomb blockade of G(V) at V< e/2C
DYNAMICAL COULOMB BLOCKADE
VQUANTUM DESCRIPTION OF Zenv
Zenv(ν)
G(V)
eV
ε=∑nihνi
Zenv=R//C<<h/e2
kBT
h/RC
GT
GT
Caldeira & Leggett
(T=0K)
P(ε)=Σ|<0,…,0|Te |n1, … nj, …> |2
≈ 2 θ(ε) Re Zenv(ε/h)/(εRK)Zenv<<h/e2
|n1, … nj, …>
See Ingold & Nazarov in "Single Charge Tunneling" (Ed. Grabert & Devoret, 1992)
CURRENT PULSES EXCITE THE MODES OF THE ENVIRONMENT
THE SCATTERING MATRIX DESCRIPTION OF COHERENT CONDUCTORS
COHERENT CONDUCTOR
tt’r r’
…SET OF INDEPENDENT
CONDUCTION CHANNELS
Landauer,Büttiker,Martin
MESOSCOPIC CODE: τi (Tunnel junction: τi <<1)
∑=
=N
iih
eG1
22 τ∑
∑=
=−
= N
ii
N
iii
I eIS1
1)1(
2τ
ττ FCS
Landauer formulaeFano factor
THE WAVE PACKET APPROACHsingle channel caseE
τ
1-τ
τ =1: finite G & noiseless current (T=0K)
δt=h/eV
I=eτ /δt=Vτ e2/h ; SI(ω≈0)=2eI(1-τ)GK Poisson
Fano
Martin & Landauer 1992
eV
f(E)
DYNAMICAL COULOMB BLOCKADE IN SHORT COHERENT CONDUCTORS
A. Levy Yeyati, A. Martin-Rodero, D. Esteve & C. Urbina, PRL 87, 46802 (2001)
D.S. Golubev & A.D. Zaikin, PRL 86, 4887 (2001)
A coherent conductor is NOT a perturbation
Zenv << h/e2≈25.8kΩ
Short coherent conductor
Same energy dependence as for tunnel junctionsBUT
Renormalized in amplitude by the same Fano factor as shot noise
DYNAMICAL COULOMB BLOCKADE IN SHORT COHERENT CONDUCTORS
HOW TO UNDERSTAND THE NOISE-DCB RELATIONSHIP?
P dissipated in environment (2 approaches):
∫∞
≈=0
)0()]/(Re[ hdShZP I εωε
∫∞
=0
)( εεε dPeIP
P(ε) = 2 θ(ε) F Re Zenv(ε/h)/(εRK)
2eI F
∑∑
=
=−
N
ii
N
iii
1
1)1(
τ
ττ
T=0KeV>>EC,h/RC
Integrants identification:
OK in perturbative regime at T=0K(no multi-photons processes)
EXPERIMENTAL TEST OF DCB THEORY IN COHERENT CONDUCTORS
Experimental requirements:
i) Known & tunable coherent conductorii) Known circuit of Re[Z(ω~GHz)]~1kΩ
Experiments described here:
A) Atomic contacts in on-chip resistive circuit (Quantronics-SPEC, 2001)
Pioneer experiment: DCB reduction at τ~1
B) QPC embedded in on-chip tunable 2DEG circuit (Phynano-LPN, 2007)
Test Fano reduction on τ1∈]0,1],τ2=0 & τ1=1,τ2∈[0,1]
DYNAMICAL COULOMB BLOCKADE IN ATOMIC CONTACTS
DCB SUPPRESSION IN WELL TRANSMITTED CHANNELS
Ronald Cron, Michel Devoret, Daniel Estève, Philippe Joyez, Cristian Urbina
Quantronics, Service de Physique de l’Etat Condensé (SPEC), CEA-Saclay
EXPERIMENTAL SETUP
Slide P. Joyez
MAKING ATOMIC CONTACTS
Slide P. Joyez
HOW TO EXTRACT THE MESOSCOPIC CODE?
Slide P. Joyez
0 1 2 3 40
1
2
3
4
I/G
0∆
eV/∆
Slide P. Joyez
0 1 2 3 40
1
2
3
4
I/G
0∆
eV/∆
HOW TO EXTRACT THE MESOSCOPIC CODE?
Slide P. Joyez
0 1 2 3 40
1
2
3
4
I/G
0∆
eV/∆
HOW TO EXTRACT THE MESOSCOPIC CODE?
Slide P. Joyez
DCB ON ATOMIC CONTACT
Fig. providedby P. Joyez
-2 -1 0 1 2
0.6
0.7
0.8
0.9
-2 -1 0 1 20.025
0.030
0.035
0.040
0.045C=0.45 fF; T=21 mK
τ=0.045
V (mV)
G/G
0
-30%
-20%
-10%
0%C=0.45 fF ; T= 23.5 mK
τ=0.845,0.07
V (mV)
-30%
-20%
-10%
0%
BCD Standard Levy-Yeyati et al.
δG
/Gto
t
Tunnel contactτ<<1
1 well transmittedchannel
RESULTS
DCB IN WELL TRANSMITTED CHANNELS REDUCED AS PREDICTED BY THEORY
R. Cron et al., Proceedings of the XXXVIth Rencontres de Moriond, Les Arcs, France, Jan. 20-27, 2001 (eds.: T. Martin, G. Montambaux, J. Trân Thanh Vân), p. 17
DYNAMICAL COULOMB BLOCKADE IN 2DEG QUANTUM POINT CONTACTS
DCB vs FANO FACTOR
Carles Altimiras, Hélène le Sueur, Ulf Gennser, Antonella Cavanna, Dominique Mailly, F.P.
ϕ Nano TeamLPN
Phynano, Laboratory of Photonics & Nanostructures (LPN), CNRS-Marcoussis
QUANTUM POINT CONTACTS IN 2DEGs
2DEG
IVDS
VQPC
van Wees; Wharam 1988
VQPC [V]
G [2
e2/h
]
τ1=0→1τ2=0 →1
τ3=0 → 1Landauer
G= = (N-1+τN)2e2
h ∑τi2e2
hτi=1 except 0<τN<1 for the last channel
AlGaAs
GaAs
EF≈10-20meV, nS≈1-5 1015m-2
le≈Lφ≈1-10µm, λF≈10-50nm
Orders of magnitude in 2DEGs:
QPC: A TEST-BED FOR COHERENT CONDUCTORS
QPC: A TEST-BED FOR COHERENT CONDUCTORS
Experimental test of quantum shot noise reduction in coherent conductors
Bru
it ∆I
2/ B
ruit
Sch
ottk
y
Kumar et al. (1996)
∑∑
=
=−
= N
ii
N
iii
IeIS
1
1)1(
2 τ
ττ
GQPC [2e2/h]
EXPERIMENT PRINCIPLE
τ1=0→1,τ2=0,τ1=1,τ2=0→1
VscQPC
R
I : Coulomb blockade
: no Coulomb blockade
R : several values
V
i) SET & MEASURE τi with no Coulomb blockade
ii) MEASURE COULOMB BLOCKADE amplitude at V≈0 for the same τi
iii) CHANGE COULOMB BLOCKADE by changing R for the same τi
SAMPLE MICROGRAPH
2DEG in GaAs/Ga(Al)As, nS=2.5 1015m-2, µ=55m2V-1s-1
200nm
VQPC [V]
GQ
PC[2
e2 / h]
Rseries=350Ω
τ1=GQPC,τ2=0
τ1=1,τ2=GQPC-1
VQPC
GQPC
G= (N-1+τN)2e2
h
-0.90 -0.85 -0.80 -0.75 -0.700
1
2
B=0.2TT=40mK
data fit saddle point(Büttiker model)
10µm
MEASURED QUANTUM POINT CONTACT
ACCURACY ON τi: ∆τi < 0.05
Estimate from max. ≠ with expected shape & between steps
TUNABLE CIRCUIT‘LARGE’ SERIES RESISTANCE
DVR
VSC
VQPC
VSC
VR
VSD
QPC
1.4kΩ
7kΩ
RSC
C
V
R=7kΩ
TUNABLE CIRCUIT‘SMALL’ SERIES RESISTANCE
D
VSC
VQPC
VSC
VSD
QPC
1.4kΩ
7kΩ
RSC
C
V
R=7kΩ//1.4kΩ=1.2kΩ
TUNABLE CIRCUIT‘SMALL’ SERIES RESISTANCE ‘SHORT CIRCUITED’ AT HIGH FREQUENCIES
DVQPC
VSD
QPC
1.4kΩ
7kΩ
RSC
C
V
R=1.2kΩ//RSC
COULOMB BLOCKADE MEASUREMENTS
-0.8 -0.6 -0.4 -0.2 0.0-2
-1
0
CAPACITIVE CROSS TALK
GQPC≅0.5 [2e2/h]
B=0.2TT=40mK
VSC [V]
δR[k
Ω]
R=7kΩR=1.2kΩ
no Coulomb blockade0.5 1.0 1.5 2.0
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
∆G
(GQ
PC,∆
V) [
2e2 /h
]
GQPC [2e2/h]
∆VQPC=0.31mV ∆VSC=0.33V
∆VQPC ∆VSC/1000
δRC
B
Coulomb blockade
δRCB: DYNAMICAL COULOMB BLOCKADE SIGNAL
T DEPENDENCE OF COULOMB BLOCKADE SIGNAL
0.05 0.1 0.15 0.20
1
2
3
4
5
R=7kΩ: data thyR=1.2kΩ: data thy
DYNAMICAL COULOMB BLOCKADE CALCULATIONS
i) Environment Renv//C- C=30fF (N.E.: [25-35]fF)- Coulomb blockade: Renv=R- "No" Coulomb blockade:
Renv=R // RSC=1kΩii) DCB reduction by F=1-τ=0.67T [K]
GQ
PCδR
CB
[%]
GQPC=0.33 [2e2/h]
EXPECTED AMPLITUDE & T DEPENDENCE
COULOMB BLOCKADE DEPENDENCE WITH MESOSCOPIC CODE τi
-0.8 -0.6 -0.4 -0.2 0.0
0
1
2
3
4
5
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
2.0
2.5
3.0
GQPC [2e2/h]
GQ
PCδR
CB
[%]
GQPC [2e2/h]0.5
11.5
GQ
PCδR
[%]
VSC [V]
R=1.2kΩ
COULOMB BLOCKADE DEPENDENCE WITH MESOSCOPIC CODE τi
-0.90 -0.85 -0.80 -0.75 -0.700
1
2
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
2.0
2.5
3.0
T=40mKB=0.2T
GQPC [2e2/h]
F=(1-τ1)τ2(1-τ2)(1+τ2)
R=1.2kΩ
GQ
PCδR
CB
[%]
VQPC [V]
F=
GQ
PC
[2e2 /h
] τ1=1τ2= GQPC-1
τ1=GQPCτ2=0
QUANTITATIVE AGREEMENT DATA/PREDICTION
SUMMARYKumar et al. 1996
GQPC [2e2/h]
Noi
se ∆
I2/S
chot
tky
nois
e
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
GQPC [2e2/h]
DC
B s
igna
l [re
lativ
e to
tunn
el ju
nctio
n]
(1-τ1)τ2(1-τ2)(1+τ2)
FOR A SHORT COHERENT CONDUCTOR:
COULOMB BLOCKADE CORRECTIONS ∝ SHOT NOISE (Renv<<h/e2)
R. Cron et al., Proceedings of the XXXVIth Rencontres de Moriond, Les Arcs (France), Jan. 20-27, 2001 (eds. T. Martin, G. Montambaux, J. Trân Thanh Vân), p. 17Altimiras, Gennser, Cavanna, Mailly, Pierre, PRL 99, 256805 (2007)
WHAT IS NEXT?
Large DCB corrections on short coherent conductorsKindermann et al. PRL 2003 (not quantitative)Safi & Saleur PRL 2004 (1 channel, purely resistive environment)
Mesoscopic environmentJoyez et al. PRL 1998 (high conductance tunnel junction)
GG nn
BC
1+∝δFCS:Kindermann et al. PRL 2003; PRB 2004
Finite size effectsNazarov PRB 1991Florens, Simon, Andergassen, Feinberg PRB 2007 (Kondo-DCB)
RonaldCron
MichelDevoret
DanielEstève
PhilippeJoyez
CristianUrbina
ϕ Nano Team
LPN
AntonellaCavanna
UlfGennser
Hélènele Sueur
CarlesAltimiras
DominiqueMailly