charge transport through a double quantum dot in the presence of dynamical disorder
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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