Department of Quantum Mechanics Seminar1

20 April 2004

Division of Open Systems Dynamics

CHARGE TRANSPORTTHROUGH A DOUBLE QUANTUM

DOT IN THE PRESENCE

OF DYNAMICAL DISORDER

CHARGE TRANSPORTTHROUGH A DOUBLE QUANTUM

DOT IN THE PRESENCE

OF DYNAMICAL DISORDER

Jan Iwaniszewskiwith

prof. Włodzimierz Jaskólski, prof. Colin Lambert, Lancaster, UK

Supported by The Royal Society, London

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

2

OutlineOutline

Charge transport in coupled semiconductor quantum dots• applications• description

Dynamical disorder• sources (defects)• two-level fluctuator

Charge transport in the presence of fluctuators• description• two-level system and fluctuator• modification of the current

Perspectives

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

3

Description of the systemDescription of the system

Model: single-level dots Coulomb blockade weak coupling to leads leads in thermal

equilibrium

ii,Xi,Xi,XX

ii,XXi,Xi,XSX

2

1RLS

leads

RL

ninteractio

SRSLS

aaH

ac)(VH

LRRLRRELLEH

HHHHHH

c.c.

L,T

R,T

ER

VR

VL

L R

J

EL

evolution of the total system

000

empty state

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

4

Reduction of the descriptionReduction of the description

reduced evolution

2nd order perturbation infinitely fast relaxations in leads

Born-Markov approximation

RRRRRRRRRRRRR

LLLLLLLLLLLLL

S

c,cc,cc,cc,c,cc

c,cc,cc,cc,c,cc

,Hdt

d

E

E

i

i

i

X

X

2

XX2

1X

X

2

XX2

1X

E1EVEg

EEVEg

E

f

f

1

XXX

X

1kT

EexpE

Eg

f

where

- density of states

- detuning

Fermi distribution

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

5

Rate equationsRate equations

)()][

)()][

)(22

)(22

22)(2

R,L,0M,N,MN

R,RL,L2L,RRLRLL,R

R,RL,L2R,LRLRLR,L

L,RR,L2R,RR0,0RR,R

L,RR,L2L,LL0,0LL,L

R,RRL,LL0,0RL0,0

M,N

i

i

i

i

i

i

EE

EE

RL

RL

E(E

E(E

IL

IR

R,RL,Le

R,RL,Lz

L,RR,Ly

L,RR,Lx

)(

i

10,0e0,0R,RL,L One electron only - Coulomb blockade

eze

ezyz

zyxy

yxx

)(

)(

RLRLRL ,, EE RL EE

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

6

Matrix calculusMatrix calculus

pσΑσ

00

0

0

00

,0

0

,

e

z

y

x

Apσ

pAσ 1s

stationary solution

eLLzLLL,LL0,0LLL )2(2)(2nΙ eee

L2R

22RL

2RL

L )(4)2(

2I

e

1

RLRL

RLL

1

2

1

2

2I

eeweak coupling to the leads

simplification for

T=0, αR=βL=0

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

7

Dynamical disorderDynamical disorder

Sources of dynamical disorder• phonon field• fluctuations of impurities• defects of the lattice

Model - two-level fluctuator (TLF)• the defect switches randomly

between two discrete states D

• its dynamics is governed by dichotomous Markov noise with correlation time τ=1/2γ

PPP

PPP

1)t(,0)t(,1)t( 2

x)t(2xbx)t(ax)t(

cx)t(bxax

cx)t(bax

dt

d

dt

d

dt

d

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

8

Perturbed systemPerturbed system

surrounding

L,T

R,T

ER

VR

VL

=

J

EL

LRRL))t((

000RRELLE)t(H

HHHH)t(HH

102

1

RLS

RLSRSLS

Assumption:Fluctuator varries slower thanrelaxation processes in the leads

pσΑσ

0000

000

000

0000

,

0

0

0

0

,

00

0

0

00

,0

0

,

1

111

e

z

y

x

1

0

000

e

z

y

x

0

Apσ

Apσ

IΑΑ

ΑΑΑ

p

pp

201

10

1

0

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

9

Current in the presence of TLFCurrent in the presence of TLF

slow fluctuator limit 0 LL

II)(I)(II2

110L10L2

10L

fast fluctuator limit )(II 0LL

L

0L0 II

4

4L D

NIstationary current (exact but cumbersome)

further simplifications•weak coupling to the leads•T=0

two cases:1. tuned =02. detuned ≠0

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

10

Stationary current for =0Stationary current for =0

L=0.2L=0.4L=0.6L=0.8 aL=0.01

L=0.2L=0.4L=0.6L=0.8 aL=0.001

aL=0.001aL=0.005aL=0.010aL=0.015

1=0.5

estimation of the position of minimum

111

2 0

21

20

Resonant decreasing of the current

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

11

Stationary current for ≠0Stationary current for ≠0

=0.00=0.05=0.10=0.151=0.5al.=0.01

=1.0=0.8=0.6=0.4=0.21=0.5al.=0.01

4

22

Resonant increasing of the current

estimation of the position of maximum

Div

isio

n o

f O

pen

Syste

ms D

yn

am

ics

20.0420.0420042004

12

What next?What next?

Characteristic of the current• coherency of the process• spectral properties of the current

Beyond the applied approximations• detailed (quantum) description of the fluctuator • detailed treatment of coupling to the leads• two electrons transport – weak Coulomb repulsion

Related problems• stochastic perturbation of energy levels • multilevel quantum dots• three- or multi- wells semiconductor structures

The aim of research

Optimal control of charge transportthrough semiconductor heterostructures

Department of Quantum Mechanics Seminar13

20 April 2004

Division of Open Systems Dynamics

The EndThe End