Energy States of Two Electrons in a Parabolic Quantum
Dot in a Magnetic Field
Mohammad El-Said
To cite this version:
Mohammad El-Said. Energy States of Two Electrons in a Parabolic Quantum Dot in
a Magnetic Field. Journal de Physique I, EDP Sciences, 1995, 5 (8), pp.1027-1036.
<10.1051/jp1:1995181>. <jpa-00247113>
HAL Id: jpa-00247113
https://hal.archives-ouvertes.fr/jpa-00247113
Submitted on 1 Jan 1995
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Phys.
J.
I
IFance
(1995)
5
1027-1036
1995,
AUGUST
1027
PAGE
Classification
Physics
Abstracts
78.50Ge
78.55Cr
Energy States of
Magnetic Field
a
Mohammad
Electrons
Two
in
Quantum
Parabolic
a
Dot
in
El-Said
Department
Physics,
of
University,
Mediterranean
Eastern
Majusa (N. Cyprus),
Gazi
Mersin
10, Turkey
(Jleceived
1994,
March
8
Abstract.
Trie
energy
in
potential
1/N expansion
parabolic
a
shifted
revised
December
27
of
spectra
an
method
magnetic
used
is
accepted
interacting
two
applied
1994,
field
solve
to
3
electrons
interaction
electron-electron
with
different
in
effective-mass
the
obtained.
are
Hamiltonian.
and
the
on
shown.
is
states
energies
are
The
influence
effect
dependence
Trie
confinement
dot
Trie
significant
its
by
confined
dot
quantum
a
strength
arbitrary
of
ground-state energy
level-crossings in states with different angular
momenta
energy
ground-state energy on trie magnetic field strength for various
Comparisons show that our
calculated
spectra of the quantum
of the
May 1995)
the
on
of the
presented.
is
good
agreement
confine
electrons
m
works.
Introduction
1.
With
ali
in
such
recent
small
levels.
far
the
infrared
reported
bave
technology,
nanofabrication
dimensions
structures
The
structures
in
progress
spatial
three
in
electrons
semiconductor
are
fully quantized
gated
on
(FIR) experiments
discrete
states
has
it
in
been
these
possible
called
structures
into
and
a
etched
microstructures
to
quantum dots (QDS). In
discrete
spectrum of energy
GaAs/AlGaAs
and
[1-3]. Trie growing
Insb
interest
motivated
by trie physical elfects and trie potential device applications of the
in this field is
QDS, to which many
theoretical
expenmental [2, 3] and
[4-18] works have been devoted. Trie
The effects
magnetic field dependence plays a useful role in identifying the absorption features.
of the magnetic field on the state of the impurity (8] and
excitons il1-15] confined in QD have
been extensively
studied.
Kumar, Laux and Stern [4] have self-consistently solved the Poisson
GaAs/AlGaAs for both cases:
and Schrôdinger equations and
obtained
the electron
states
in
fields
heterojunctions.
The
and
for
magnetic
applied
perpendicular
the
results of
to
zero
in
their
work [4]
indicated
that the
confinement
potential can be approximated by a simple oneparameter adjustable parabolic potential. Merkt, Huser and Wagner [Si have presented a study
of
quantum
were
taken
dots
magretc-optical
deviations
©
Les
in
from
Editions
which
account.
into
both
the
Pfannkuche
magnetic
and
field
radiation
response to far-infrared
the parabolic
confinement.
More
de
Physique
1995
and
the
electron-electron
Gerhardts
[6] bave devoted a
(FIR) of quantum-dot
recently,
De
Groote,
interaction
terms
study to
helium, accounting
Hornos and Chaplik
theoretical
the
for
I?1
JOURNAL
1028
PHYSIQUE
DE
I
N°8
investigated the thermodynamic properties of quantum
electron-electron
spin effect, in addition to the
interaction
effect
show
trie
of
this
study
of
trie
electron-electron
is
to
purpose
have
dots
the
and
the
dot
quantum
ground
the
in
of the
state
work,
this
In
we
shall
of two
including the
for trie
non-vanishing
with
states
as
the
shifted
use
confined
spectra
interaction
numbers
quantum
magnetic field strength
the
system
azimuthal
1IN expansion
electrons
in
quantum
a
method
taking into
consideration
The
magnetic field terms.
obtain
to
magnetic field. The
given in
are
a
1IN expansion
shifted
final
the
conclusions
method
described
is
an
transitions
energy
expression
effective-mass
the
following terms: trie
electron-electron
interaction,
parabolic confinement potential. The work is organized as follows. In
the
Hamiltonian
of the two interacting electrons, parabolically
confined
tonian
of
spectra
the
increases.
to
by solving
dot
trie
on
and
Section
in
2,
the
present
we
QD, subjected
the
Section
in
Hamil-
applied field and
Results
3.
and
section.
Theory
2.
approximation (EMA), trie Harniltonian for an interacting pair of
confined in a quantum dot by parabolic potential of the form m*wor~ /2 in a magnetic
electrons
field applied parallel to the z-axis land perpendicular to trie plane where trie
electrons
are
follows,
restricted
to move) in trie syrnrnetric
gauge is written
as
Within
effective-mass
the
~~ÎÎ
~
where
twc-dimensional
the
electron
in
momentum
mass
the
Î~~"~~~
~
vectors
and r2
ri
~
describe
positions of trie first and the
the
(x,y) plane, respectively. L] stands for the
eB/m*c, m* and E
for each electron and toc
of the medium, respectively.
dielectric
constant
the
magnetic field
B
and
frequency
confinement
trie
~
The
natural
and
effective
units
of
length
and
to
energy
be
used
second
angular
orbital
cyclotron frequency,
frequency w depends
given by
the
are
Trie
effective
both
on
j
~°Î+(
~°"
of the
z-component
and is
Mo
Il)
r~l'
E
=
and
rÎ~
ÎÎ~~~Î
~
(2)
trie
are
effective
Bohr
radius
a*
~(~
=
m
effective
an
magnetic
Upon introducing
Hamiltonian
the
~~
Rydberg
I?i in
center-of-mass
R*
field
trie
=
Trie
2m*a*
the
relative
~.
R
~~ ~ ~~
và
=
equation (1) can be written
Hamiltonian,
motion
motion
constant
strength.
center-of-mass
Î* ~~
~~
and
~~
dimensionless
~
as
and the
a
relative
Î~"~~~
~
=
2R*
coordinates
~~
r
separable parts
of two
sum
j
plays trie role
ÎÎ~ ~~
~~
và
=
that
of
the
,
represent
~~~
Hamiltonian,
~~
Î* ~~
~
Î~"~~~
~
~Î~~~
~
~~~
N°8
STATES
ENERGY
Equation (3)
ergies,
describes
the
OF
Hamiltonian
Encm,~ncm
by
labelled
numbers.
the
Hamiltonian.
Trie
make
the
1IN
hW +
~j~
(mc~n
Hamiltonian
The
>.
the
are
of the
coexistence
with
solution
Expansion
1IN expansion
shifted
The
analytic
exact
i)
azimuthal
total
with
oscillator
obtaining eigenenergies
to
of the
n~m
and
QUANTUM
IN
harmonic
12nc~n + lmc~nl +
=
reduced
states
ELECTRONS
of the
o, 1, 2, ...)
=
is
energy
Shifted
The
3.
(nc~n
numbers, (nc~nmc~n,
quantum
terms
radial
problem
The
TWO
=
the
1029
eigenen-
well-known
là)
mc~n
0, +1, +2, +3, ...)
quantum
relative
En~~~n of the
labelled by the CM and
electron-electron
special
present
DOT
and
functions
motion
relative
oscillator
the
possible.
not
Method
method,
being
the spatial dimensions, is
perturbation
pararneter that
N
pseudoperturbative
a
directly related
proposes
a
[20-22]. Trie aspect of this rnethod has been dearly stated by Irnbo
relevant to this
method.
Following
[20,21] who had displayed step-by-step
calculations
work, we present here only the analytic expressions which are required to determine trie
technique in the
to trie coupling
et ai.
their
that
sense
it
is
not
constant
states.
energy
Trie
rnethod
where
k
trie
resulting
large
barrier
momentum
will
Schrôdinger equation, for
radial
N-dirnensional
an
1-j
useful
get
to
parameter,
This
in
())
~~
+
space
~~
arbitrary
an
cylindrically
as,
Vlrlj Wlrl
+
=
E~wlrl
(GI
N + 2m.
=
order
In
by writing the
starts
potential,
syrnrnetric
give
in
a
rise
to
Î~
like
at
where
large k,
potential
which
zeroth-order
dassical
result.
k
=
suitably
be
must
effective
an
sensible
potential
of trie
behaves
term
1/k expansion,
from
results
k-limit
trie
so
does
and
a
vary
should
with
equation
a
[16].
potential
not
Hence,
k
defined
Î
(6)
at
in
is
a
suitable
Since
trie
behave
shift
angular
similarly.
large
values
of k
terms
of trie
shift
becomes,
parameter
'~~~~~~
vjr)
and
Q
is
method
1là.
a
scaling
constant
in
solving
consists
Trie
leading
+
=
ro is trie
contribution
minimum
to
term
trie
energy
of trie
comes
ÎÎ ~Î
to
shift
the
origin
~~Î~~~
ro
x
~~~
potential, given by
effective
to
expansion
parameter
from
~
=
convenient
mi
be
2r(V'(ro)
It is
+
specified from equation (10). Trie shifted 1IN
equation ii) systematically in terms of trie expansion
to
~~~~~~~
where
~w~r~
by
=
trie
ki jr
Q
(10)
definition
ro) /ro
l~~~
JOURNAL
1030
and
expanding
of
in the
x
equation ii)
with
series
one-dimensional
about
energy
Trie
anharmonic
and
0 in
=
scaling
the
frequency
of trie
+
vu j~
~~
3
the
eigenvalues
energy
in
explicit
The
introduces
~ ~~°~ ~
of ii
k to
(n~
to
and
monic
potentials.
Coulomb
1là
between
agreement
l12)
third
order)
~Î~
~~
chosen
expansion
and
equation (14)
From
'~~Î
~
the
shirt
the
~~~~
ÎÎ(
parameter
first
which
a,
the
in
term
energy
(14)
0.
=
~
as
Trie
~~
~~
a
read
make
to
as
so
ro
an
1/2
Appendix.
trie
in
is
))
+
By requiring
derivatives.
o
~~
~
additional
an
~7j
of1Ii (up
powers
and j2 are given
degree of freedom,
vanish, namely,
forms
of order
series
"
ÎO
~"~"~
potential
the
and
ro
powers
is
°
and
k, Q>
of
Schrôdinger equation for
oscillator frequency, the
anharmonic
the
of
terms
coefficients
trie
order in the
sonne
in
N°8
Comparing
x.
determine
we
constant
parameter
of
powers
ones
oscillator,
anharmonic
eigenvalue
x
corresponding
the
PHYSIQUE I
DE
analytic
exact
results
for
the
a=2-(2n~+1)a
where
For
radial
by trie
the
n~
n-
=
related
number
quantum
relation
and a*,
of R*
units
in
trie
n~ is
numbers
(m( -1.
Energies
twc-dimensional
N
case,
=
ro
(for
Results
4.
particular
a
the
energy
and
principal n and magnetic
lengths in equations (6-15)
the
and
2, equation (10) takes
=
computing
(là)
quantum
m
expressed
are
respectively.
h
Once
to
har-
obtain
we
is
quantum
and
state
2 + 2m
a
=
the
following form,
(16)
Q~~~
confining frequency)
is
determined,
trie
of
task
relatively simple.
Conclusions
Figures 1-à and Tables I and II. The relative ground-state energy
for zero-magnetic field case, against the
confinement
length is
displayed in Figure 1. The present results (dotted line) clearly show an excellent agreement
with the
nurnerical
results of reference [Si (solid fine). In Figure 2, the first low-energy levels,
Dur
(00
results
>
(00 >, (10
>
17.88,
The
is
the
effective
increases
w
linear
consistent
w,
the
relationship
with
reference
relative
using
obviously
levels
frequency
thus
and (20 > of the
electron
energy
in
motion,
relative
frequency
confinement
E=
presented
are
of trie
mass
show
Hamiltonian
as
a
function
of the
"
=
a
linear
confining
between
dependence
on
the
effective
frequency. As
the
effective
constant
rnev
Iii.
effective
dominates
the
and
interaction
term
terni
energy
energy
result
the energy
and the frequency is
maintained.
This
Iii.
electron-electron
investigate trie elfiect of the
quantum dot, we have plotted in Figure 3 the total
TO
presented
are
appropriate to Insb, where the dielectric
parameters
confinement
m*
0.014me and
1-à
energy hwo
interaction
ground-state
on
trie
energy
energy
(00; où
spectra
>
of trie
of the
full
ENERGY
N°8
STATES
ELECTRONS
TWO
OF
QUANTUM
IN
DOT
1031
+3
~
0
0>- SiAiE
8
.
f
=
Ù-O
~~
10
uJ
ib~
5
io
i~/2
Fig.
The
l.
length fo
confinement
of
ground
relative
state
~m i
=
(00
energy
1/2
o~
for
for
>
trie
magnetic
zero
electrons
in
a
dot
quantum
field:
reference
as
[Si;
a
(...)
function
present
calculations.
~
*-*
~
*-*
~
~lJ
0
5
10
20
15
25
30
35
40
two
electrons
l4~
Fig.
of
Insb
Harniltonian
the
ratio
Iow-lying
The
2.
made
as
for
wc/wo.
electron-electron
increases,
the
electron-electron
The
a
energy
relative
function
of
independent
The figure shows,
coulombic
electrons
are
coulombic
level-crossings
(00 >, (10
states
confinement
and
as
interaction
further
energy,
are
>
frequency
terni
>
for
interacting
significant
a
m
quantum
a
dot
electrons
energy
as
a
enhancement
function
when
of
the
Furtherrnore, as trie magnetic field
on.
QD, resulting in an increase of the repulsive
tumed
is
in
the
and in effect
in
(20
w.
expect,
we
squeezed
shown
and
Figure
the
4.
levels.
energy
We bave displayed
trie
eigenenergies of
JOURNAL
1032
PHYSIQUE
DE
I
N°8
(0050>-STATE
05
0
15
25
2
3
35
45
4
5
uJc/~
Fig.
The
3.
ratio
of trie
total
w~/wo.
For
ground-state
independent
(00; 00
energy
>
two-electrons
for
in
a
interacting
and
dot
quantum
as
a
function
electrons.
m
0
05
15
25
2
35
3
45
4
5
tJc/v~~
Fig.
Trie
4.
trie
in
of trie
eigenenergies
confined in
the
states
quantum
o, -1, -2,
(oo; om >; m
3a*
quantum dot of size la
states
trie
total
parabolically
electrons
-5, for
=
the
strength
increases
energy
non-vanishing quantum
with
"
the
number
of
(00j 0m >, m
dot of size fo
a
as
state
m
=
=
of trie
o
decreases,
ratio
increases
thus
5, for
interacting
w~/wo.
two
..,
3a*
interacting
two
function
m
0, -1, -2,
=
as
a
toc /wo.
while
As
the
to
ratio
parabolically
electrons
leading
of the
function
a
trie
energy
sequence
confined
magnetic
of the
of
field
states
different
N°8
Table I.
The
ENERGY
STATES
roots
determined
ro
ELECTRONS
TWO
QUANTUM
IN
by equation (16) for quantum
(m) against
numbers
quantum
azimuthal
OF
the
ratio
dot
DOT
1033
non-vanishing
with
states
toc /wo.
m
to/to~
ground
-3
-4
-5
4.262
4.827
5.457
6.075
6.659
7.209
2
3.692
3.682
4.212
4.719
5.193
5.635
3
3.188
2.943
3.398
3.825
4.229
4.586
states,
as
the
of
structure
spatial
reported in reference
higher trie angular
relative
the
extent
QD for
the
toc /wo,
azimuthal
electron-electron
is the
Table I trie
roots
ro
Î-e jr)
=
momentum.
2/ro,
in the
interaction
This
motion.
between
of the
the
system,
angular
the
(m( increases,
number
quantum
interaction
larger
relative
distance
angular
different
with
states
trie
as
list in
we
of the
The
:
larger
trie
interacting
[16]. In trie
momentum
function
wave
therefore,
and
numerically,
this
in
-2
lower
trie
is
-1
is
trie
momentum
the
iii].
electrons
leading
root
term
ro
also
of the
energy
electrons
the
and
increases
serres
trie
confirm
To
potential for the interacting
At particular
values of
the
energy
the
by
larger
caused
ratio
thus
trie
expression,
decreases.
In Figure 5, we have shown the dependence of the ground-state
energy on the magnetic field
and
strength for confinement energies: hwo
mev.
For
value of trie magnetic
12
6
constant
field, trie larger trie confinement
the
of
the
electrons in
trie
interacting
greater
energy,
energy
Hamiltonian
the QD. The spin effect can be included in trie
equation il) added to the centresolvable
of-mass
term, and equation (3) is still an analytically
part as a space-independent
=
harmonic
We
of
the
In
a
m
recent
i/w
IA
x
for
two
increases
10~~
1/w
at
Quantum
for the ground state energies (00 >
frequencies
results of Taut [18].
Hamiltonian
confining
with the
at
work, Taut has reported a particular analytical solution of the Schrôdinger
Trie Table
shows
electrons
externat
harmonic
potential.
interacting
in
an
difference
trie
between
results
noticeably
until it
becomes
both
decreases
in
dots
with
=
calculated
our
results
1419.47.
than
more
interacting electrons, provided
relative
II,
Table
dilfierent
relative
very
iii.
Hamiltonian
compared,
have
equation
as
oscillator
coordinates
electrons
also
cari
electron-electron
trie
electrons
between
studied.
be
interaction
~2
V((ri
rj ()
and
=
~ Î~'31
Trie
term
Hamiltonian
for
ne-
depends only
on
trie
parabolically
confined
in the
,
Hamiltonians.
potential
Trie parabolic
leads
separable
only
potential
which
1,2,3,
to a
Hamiltonian, equation
Hamiltonian.
Trie CM
by trie one-partide
motion
part is
(3), with trie electron
replaced by trie total mass M
nom* and trie electron charge
mass
total
charge
Q
replaced by trie
Trie relative
Hamiltonian
part, which involves only the
nee.
and
cylindrically
relative
coordinates
symmetric potential and con be handled
momenta, bas a
by trie 1/N-expansion technique. When trie confining potential is quadratic, FIR spectroscopy
quantum
form
dot,
V(ri)
is
separable
two
that
miw(r)/2,1
=
into
a
CM
and
=
,
a
relative
ne is trie
described
=
=
JOURNAL
1034
PHYSIQUE
DE
(00>~
5
15
10
20
N°8
I
STATE
25
30
35
40
45
Î~
Fig.
The
5.
confinement
insensitive
is
ground-state
relative
trie
to
contains
all trie
between
trie
i
QR, being
=
electron-electron
of trie
states
eigenenergies for
expression
energy
effects
interaction
~jeiri
dipole operator
Trie
(00
magnetic
>
against
the
because
of trie
CM
energy
strength
field
for
different
two
energies.
does
equation
~~
namely,
same,
Mc
and
relative
then
it
=
couple
not
then
to
states
of
the
là),
does
not
change
transitions
relative
Consequently,
radiation
H~ which
induces
the
~~
Trie
motions.
does
dipole operator
Trie
affect
not
Harniltonian,
the
remains
but
variable,
CM
pure
interactions.
CM
CM
the
a
Hamiltonian.
because
the
toc
in
the
absorption
FIR
m*c
experiments see only the feature of the single-electron energies. There are only
+1) and the FIR
dipole transitions (lhm
at frequencies
resonance
occurs
two
allowed
=
w~=~+i~
Many
experiments
different
that
and
trie
observed
In
of
conclusion,
confinement
with
trie
numerical
also
shown
its
numbers.
on
the
in
trie
energy
electron
an
and
thus
review
recent
very
a
obtained
results
trie
significance
The
interactions
shifted
Hamiltonian
effect
on
the
1IN
of the
of
Merkt
of the
energy
et ai.
system
trie
actual
article
[19].
spectra of two
The
level-crossings
rnethod
interaction
in
states
yields quick
validity of Kohn's
theorem
parabolic
potential
is
in
a
nurnber
interacting
method
[Si, Taut [18] and Wagner
electron-electron
expansion
system.
proved the
have
of
energies and rnagnetic field strength.
have
and
bave
we
frequency
resonance
electron-electron
independent of
well, as reported by Wixforth
dots
quantum
on
il?)
terni
with
results
electrons
electrons
shown
bas
et ai.
on
of
as
without
the
function
good agreement
[16]. Dur
ground
trie
different
a
in
azimuthal
putting
calculations
state
energy
quantum
restrictions
STATES
ENERGY
N°8
The
II.
Table
by 1IN expansion
OF
ELECTRONS
TWO
IN
QUANTUM
DOT
ground-state energies (in atomic nuits) of the relative
ut dijferent frequencies, compared with the results of
1035
Hamiltonian
Taut
i/to
i/N Expansion
Taut
4
o.4220
o.6250
20
0.1305
0.1750
54.7386
0.0635
0.0822
lis.299
o.0375
o.0477
523.102
o.0131
o.o162
1054.54
0.0081
O.ÙIOÙ
1419.47
0.0067
0.0081
calculated
f18j.
Acknowledgments
would
I
hke
to
thank
colleagues
my
Prof.
M.
Halilsoy and
The
discussions.
and helpful
encouragements
and Re§at Ako(lu is also acknowledged.
Akilhoca
technical
for
Appendix
The
explicit
used
as
unit
Assoc.
Prof.
assistance
Dr.
received
E.
from
Aydiro(lu,
Betül
B.
A
forms
of
of the
parameters
and
energy
ii
and
ii
j2
given
are
following.
in the
Here
R*
and a*
are
length, respectively.
=
a~~ [e(
3c2e4
ci e2 +
6cieie3
+
+
c4e()
IA.1)
and
'Î2
T7
"
+
T12
+
T16
T3
+
T4
(~.~)
'~~~~~~
T7
=
°~~ ÎT2
Ti
T12
~
"
ld
T16
"
O
+
+
ÎTS + T9 + T10 +
~
T5
+
~Î
T3
cid2
=
=
+
2eidi
3c2d4
+
+
~~'~~
ÎT13 + Ti4 + T151
2c5e(
c3d6
T2
T4
=
ciel
+12c2e2e4
=
6cieid3
~~
(~.~)
l~lll
with
Ti
~~
+
30c2eids
JOURNAL
1036
T5
T8
Tio
6cid3ei
=
4e(e2
=
24ce(e4
"
T13
8eie3
=
2c4e3d3
+
+
c's, d's
ci
c2
c3
"
1 +
2n~
=
1 +
=
3 +
2n~
8n~
e's
and
8c7eie3e4
Tii
108cieie3
+
c4
+
+
2n)
6n)
4n)
+
"
c6
30n)
sIn)
+
42n)
+
~~~
~Z
40n~
13 +
=
j
where
~~
=
1, 2, 3, 4
=
a)
j2
=
ô~
63
=
=
31 +
78n)
"
57 +
=
31 +
189n~
109n~
78)
+
+
225n)
+141n)
1, 2, 3, 4, 5, 6.
e~
W
Q
-i
=
titi
~_
2(2
Q
2
c9
=
2
a)
64
Î~
=
ô~-7 +ÉÉ
=-3-É%
ô~
cs
Q~/2
=
2
Q
~@~
62
and
w
~
c7
34n)
28n)
-MW
~~
5
4
é~=
+
QÎ/2
~3
48c4eie3
=
as,
+
30n~
+
12cse(e4
"
T14
21+ 59n~ +
II
=
c5
8c4e2e(
=
30c9e3
=
given
N°8
10c6e3d5
=
T9
+
parameters
are
T6
I
36cieie2e3
T15
Where
PHYSIQUE
DE
Q
4
References
iii
Sikorski
[2]
Demel
[3]
Lorke
Kumar
[SI
Merkt
and
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A.
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[12]
Halonen
[13]
Einevoll
788.
+
150n)
+ 94 +
n)