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Coulomb Blockade Theory: Double Quantum Dots & Activated Conduction in Semiconductors, Study notes of Electronics

The theory of coulomb blockade in semiconductor devices, focusing on recent experiments with double quantum dots and the regime of activated conduction. The periodic dependence of conductance on the voltage applied to a gate electrode, caused by electrostatic energy arising from a change in charge of the grain. The document also presents the results of theoretical studies on the effects of temperature and asymmetry on the coulomb blockade peaks.

What you will learn

  • What is the role of asymmetry in the Coulomb blockade peaks observed in experiments with double quantum dots?
  • How does temperature affect the Coulomb blockade peaks in experiments with double quantum dots?
  • What causes the periodic dependence of conductance on the voltage applied to a gate electrode in semiconductor devices?

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bg1
Chapter
2.
Theory
of
Coulomb Blockade
Chapter
2.
Theory
of
Coulomb
Blockade
in
Semiconductor
Devices
Academic
and
Research
Staff
Professor
Patrick
A.
Lee,
Dr.
Konstantin
Matveev
Technical
and
Support
Staff
Margaret
E.
O'Meara
2.1
Project
Description
Sponsor
Joint
Services Electronics
Program
Grant
DAAH04-95-1-0038
Project
Staff
Professor
Patrick
A.
Lee,
Dr.
Konstantin
Matveev
The
conductance
through
a
small
grain
weakly
coupled
to
metallic
leads
shows
periodic
depend-
ence
on
the
voltage
applied
to
a
gate
electrode.
This phenomenon
is
observed
in
both
metallic'
and
semiconductor
2
devices,
and
is
commonly
referred
to
as the
Coulomb
blockade.
This
behavior
of
con-
ductance
is
caused
by
electrostatic
energy
arising
due to
a
change
in
the
charge
of
the
grain
(quantum
dot)
by
an
electron
tunneling
through
it.
The
Coulomb
blockade
is
typically
observed
in
structures
with
a
well
defined
quantum
dot,
which
is
separated
from the leads by
tunneling
barriers.
Recently,
the
experimental
observation
of
Coulomb
blockade
has
become
possible
in
more
sophisti-
cated
devices.
An
experimental
group
at
Harvard
has
studied
the
Coulomb
blockade
in
double
quantum
dots.
3
In
a
number
of
other
experiments,
4
the
phenomenon
of
Coulomb
blockade
was
observed
at
unusually
high
temperatures
T-
100
K,
where
the
transport
may
be
due
to the
activation
of
electrons over
the
barriers.
In
the past
year,
our
research group worked
on
the
theoretical
problems
raised
by
these
experiments.
2.1.1
Transport
through
Double
Quantum
Dots
In
these experiments,
3
the Coulomb
blockade
oscil-
lations
of
conductance
through
a
system
of
two
quantum
dots (figure
1)
were
studied.
The
contacts
between
the
quantum
dots
and
the
leads were
in
the
weak tunneling
regime, with
conductances
G,r
<<
e2
/h.
On
the
other
hand,
the
contact
between
the dots
was
tuned
by
adjusting
the
gate
voltage
Vo
in
such
a
way
that
the
corresponding
transmission
coefficient
To
scanned
the
whole
region from
0
to
1.
As
a
result,
a
series
of
conduc-
tance
dependences
on
the
gate
voltage,
at
different
values
of
To,
were measured.
5
The
most
basic
fea-
tures
of
the
experimental
data,
including
the
split-
ting
of
the
Coulomb
blockade
peaks
when
To
increases
from
0
to
1,
were
explained
in
our
pre-
vious
work.
6
We
have
now
extended
the
theoretical
description
to
include
the
effects
of
temperature
and
the small
asymmetry
of
the
double
dot
system
on
the
heights
and
shapes
of
the
Coulomb
blockade
peaks
observed
in
the
experiments.
3
The
experiments
have
demonstrated
that
small
tun-
neling
between
the dots
gives
rise to
a
small
split-
ting
of
the
Coulomb
blockade
peaks.
At
non-zero
1
H.
Grabert
and
M.H.
Devoret,
Single
Charge
Tunneling,
(New
York: Plenum Press,
1992).
2
M.A.
Kastner,
Rev.
Mod. Phys.
64:
849 (1992).
3
F.R.
Waugh
et
al.,
Phys.
Rev. Lett.
75:
705
(1995);
F.R.
Waugh
et
al.,
Phys.
Rev.
B
53:
1413
(1996).
4
K.
Murase
et
al.,
Microelectr.
Eng.
28:
399
(1995);
E.
Leobandung
et
al.,
Appl.
Phys.
Lett.
67:
2338
(1995);
E.
Leobandung
et
al.,
Appl.
Phys.
Lett.
67:
938
(1995).
5
K.A.
Matveev,
L.I.
Glazman,
and
H.U.
Baranger,
Phys.
Rev.
B
54:
5637
(1996).
6
K.A.
Matveev,
L.I.
Glazman,
and
H.U.
Baranger,
Phys.
Rev.
B
53:
1034
(1996).
pf3
pf4

Partial preview of the text

Download Coulomb Blockade Theory: Double Quantum Dots & Activated Conduction in Semiconductors and more Study notes Electronics in PDF only on Docsity!

Chapter 2. Theory of Coulomb Blockade^ in^ Semiconductor

Devices

Academic and^ Research^ Staff

Professor Patrick A. Lee, Dr. Konstantin Matveev

Technical and Support Staff

Margaret E.^ O'Meara

2.1 Project Description

Sponsor

Joint Services Electronics Program Grant DAAH04-95-1-

Project Staff

Professor Patrick A. Lee, Dr. Konstantin Matveev

The conductance through a small grain weakly coupled to^ metallic^ leads^ shows^ periodic^ depend- ence on the voltage^ applied^ to^ a^ gate^ electrode. This phenomenon is observed in^ both^ metallic'^ and semiconductor 2 devices, and is commonly^ referred to as the Coulomb blockade. This behavior^ of^ con- ductance is caused by electrostatic energy^ arising due to a change in the charge^ of^ the^ grain (quantum dot) by an electron tunneling through^ it.

The Coulomb blockade is typically observed in structures with a well defined quantum dot, which^ is separated from the leads by tunneling barriers. Recently, the experimental observation of^ Coulomb blockade has^ become^ possible^ in^ more^ sophisti- cated devices. An experimental group^ at^ Harvard has studied the Coulomb blockade in^ double quantum dots.^3 In a number^ of^ other^ experiments,^4 the phenomenon of Coulomb^ blockade^ was observed at unusually high^ temperatures^ T-^^100 K, where the transport may^ be^ due^ to the^ activation

of electrons over the barriers. In the past year, our research group worked on the theoretical problems raised by^ these^ experiments.

2.1.1 Transport through^ Double^ Quantum Dots

In these experiments, 3 the Coulomb blockade oscil- lations of conductance through a system of two quantum dots (figure^ 1)^ were^ studied.^ The^ contacts between the^ quantum^ dots^ and^ the^ leads^ were^ in the weak tunneling regime,^ with^ conductances G,r << e 2 /h. On the other^ hand,^ the^ contact between the dots was tuned^ by^ adjusting^ the^ gate voltage Vo in such a way that the corresponding transmission coefficient To scanned the whole region from 0 to 1. As a result, a series of conduc- tance dependences on the gate voltage, at^ different values of To, were measured.^5 The most basic^ fea- tures of the experimental data, including^ the^ split- ting of the Coulomb blockade peaks^ when^ To increases from 0 to 1, were explained in^ our^ pre- vious work. 6 We have now extended the^ theoretical description to include the effects of temperature and the small asymmetry of the double dot system on the heights and shapes of the Coulomb^ blockade peaks observed^ in^ the^ experiments.^ 3

The experiments have demonstrated^ that^ small^ tun- neling between the dots gives rise^ to^ a^ small^ split- ting of the Coulomb blockade peaks.^ At^ non-zero

1 H. Grabert and M.H. Devoret, Single Charge Tunneling, (New York: Plenum Press,^ 1992).

2 M.A. Kastner, Rev. Mod. Phys. 64: 849 (1992).

3 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh et^ al.,^ Phys.^ Rev.^ B^ 53:^^1413 (1996).

4 K. Murase et al., Microelectr. Eng. 28: 399 (1995); E. Leobandung et^ al.,^ Appl.^ Phys.^ Lett.^ 67:^^2338 (1995);^ E.^ Leobandung^ et^ al., Appl. Phys. Lett. 67: 938 (1995).

5 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 54: 5637 (1996).

6 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 53: 1034 (1996).

Figure 1. Schematic view of the double quantum dot system. The dots are formed by applying negative voltage to the gates (shaded); the solid line shows the boundary of the 2D electron gas (2DEG). V! and V, create tunnel barriers between the dots and the leads while Vo controls the transmission coefficient through the constriction connecting the dots.

temperatures, the^ splitting^ may^ or^ may^ not^ be resolved depending on^ the^ relation^ between^ the temperature T and the charging energy Ec. We have studied the whole temperature dependence of the split peaks analytically; the results are pre- sented graphically in figure 2.

In the experiments with double dots, one usually tries to make the two quantum dots symmetric. However, a small asymmetry is unavoidable, and is reflected in the actual experimental data. For example, when the gate voltage is chosen in such a way that an electron can tunnel to the first dot without changing its energy, tunneling^ to^ the^ other dot can still cost some^ electrostatic^ energy.^ This energy itself depends on^ the^ gate^ voltage^ X,^ and leads to a periodic in X suppression of the peak heights. Our (^) results are summarized in figure 3, which strongly resembles the actual experimental data.^7

0.

10 -8 -6 -4 -2 0 2 4 Gate voltage X

6 8 10

Figure 2. The evolution of the split peaks with temper- ature. The gate voltage X is plotted in units of the dis- tance between splitted peaks. The peak splitting is

observable at a sufficiently low temperature, /3 Ec/T > 2.

7 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh (^) et al., Phys. Rev. 6 53: 1413 (1996).

54 RLE Progress Report Number 139

  • = 0. --- =^ 1. / - 2.

/ \ .......... = 5

/.

/\

--^ /^ - /^ .: .\ \

metry of the barrier, -1 <^ A^ <^ 1.^ For^ symmetric^ bar- riers A = 0, and^ the^ peaks^ are^ similar^ to^ those^1 o^ of the blockade of tunneling conductance. It^ is^ inter- esting, however, that the activation energy EcX^ for the conductance (1) at off-peak values of the gate voltage is smaller than in the tunneling case 10 by^ a factor of two.

For an asymmetric barrier **A ***^ 0,^ and^ the^ peaks^ (1) have an unusual asymmetric shapes. This effect is specific to^ the^ activated^ regime.^ Thus,^ by^ mea- suring the^ Coulomb^ blockade^ peaks,^ one^ may^ be able to identify the mechanism of^ transport through the quantum dot.

2.3 Publications

Matveev, K.A., L.I. Glazman, and H.U. Baranger. "Coulomb Blockade of Tunneling Through a Double Quantum Dot." Phys. Rev. B 54: 5637 (1996).

Matveev, K.A., and L.I. Glazman,^ "Coulomb Blockade of Activated Conduction." Phys. Rev. B 54: 10339 (1996).

Matveev, K.A., L.I.^ Glazman,^ and^ H.U.^ Baranger. "Tunneling Spectroscopy of^ Quantum^ Charge Fluctuations in the Coulomb Blockade." Phys. Rev. B 53: 1034 (1996).

10 L.I. Glazman and R.I. Shekhter, J. Phys. CM 1: 5811 (1989).

56 RLE Progress Report Number^139