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