JOURNAL OF APPLIED PHYSICS
VOLUME 94, NUMBER 8
15 OCTOBER 2003
Simple one-dimensional model for electronic structure calculation
of unbiased and biased silicon quantum dots in Coulomb
blockade applications
Johann Sée,a) Philippe Dollfus, Sylvie Galdin, and Patrice Hesto
Institut d’Électronique Fondamentale (CNRS UMR 8622), Université Paris XI 91405 Orsay, France
共Received 27 May 2003; accepted 28 July 2003兲
This article presents a simple one-dimensional 共1D兲 model of electronic structure calculation able to
treat quantum dots 共QDs兲 under bias voltage. With a view to investigating complex Coulomb
blockade devices with multiple QDs, this model aims at providing accurate information on the QD
eigenstates within reasonable and optimized computation time. First, the electronic structure of an
unbiased QD is obtained from a self-consistent solution of the coupled Schrödinger/Poisson
equations as a function of the dot size and the charging state. By comparison with three-dimensional
共3D兲 calculations of total energy at given QD volume, we found that the 1D spherical approximation
appears to be very good for a wide range of dot shapes. We develop two techniques to include the
effect of external 3D bias potential that breaks the symmetry: 共i兲 a perturbation method and 共ii兲 an
expansion of the wave function on the eigenstates of the unbiased dot. If the validity of the first
technique is limited to small dots and/or low bias voltage, the latter gives excellent results over a
wide range of dot sizes and bias voltages. The results obtained for a single dot device using this 1D
model are carefully and successfully compared with a full 3D calculation. © 2003 American
Institute of Physics. 关DOI: 10.1063/1.1610803兴
Si nanocrystals by scanning tunneling spectroscopy5 and
used in memory device with self-aligned doubly-stacked dots
to improve the retention time.6 Finally, these effects have
been demonstrated at room temperature in single-electron
transistors.7,8 The high compatibility of such Si NC-based
devices with the mainstream Si technology makes them good
candidates for actual applications in combination with a conventional CMOS circuit.
In this context, the theoretical study of electronic structure and single-electron charging in NCs has become an active subject of research.9–13 The implementation of accurate
physical models aiming at describing and predicting the electrical behavior of QD-based devices appears as a strong need
for further development. To investigate such small structures,
a three-dimensional 共3D兲 description seems inescapable.
Nevertheless, in the case of a complex device with multiple
NCs, the use of a general 3D approach to calculating the
electronic structure of each NC is impractical from a computational point of view. Thus, considering that in many
cases Si and Ge NCs have a spherical shape,14,15 the development of a simple one-dimensional 共1D兲 approximation of
their electronic structure based on this spherical symmetry
seems particularly relevant to achieve reasonably fast computation. However, such a model seems to be intrinsically
unable to include the effect of 3D bias potential that breaks
the spherical symmetry.
In this article, we present a 1D model of electronic structure calculation capable of including the bias effect. The
electronic structure of an unbiased NC is obtained from a
self-consistent Schrödinger/Poisson solver. The Schrödinger
equation is solved within the frame of the Hartree method.
Two techniques have been developed to include the effect of
I. INTRODUCTION
Efforts to integrate sub-10-nm-scaled complementary
metal-oxide-semiconductor 共CMOS兲 devices will face significant technological and physical limitations in the near
future, so that the long-term capability of CMOS technology
to support further increase in density with low power consumption becomes questionable. Not to mention the technological issues, device and circuit designers will especially
have to consider 共i兲 the reduced and uncertain number of
electrons involved in the operation of conventional downscaled transistors and 共ii兲 the unavoidable quantum mechanical effects that tend to strongly influence or degrade the device electrical characteristics. That is why many studies are
currently focused on the development of novel devices turning the occurring wave mechanical effects to advantage. In
this field the concept of single-electron devices 共SEDs兲,
based on the Coulomb blockade effect in quantum-dot 共QD兲/
tunnel-junction systems, is very promising to achieve suitable low power operation.1 In the Coulomb blockade regime
the device size reduction becomes then a key-point to guarantee the control of single-electron transport at room temperature. Self-assembled semiconductor nanocrystals 共NCs兲
embedded in SiO2 appear as a good basic structure for the
realization of SEDs. The concept of multidot memories using
silicon or germanium NCs as a floating gate has been
demonstrated.2– 4 It could be extended and developed to design nonvolatile multibit memory cells of very small size,
highly reliable, and operating at low voltage. The Coulomb
blockade and quantization effects have been studied in single
a兲
Electronic mail: johann.see@ief.u-psud.fr
0021-8979/2003/94(8)/5053/11/$20.00
5053
© 2003 American Institute of Physics
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5054
Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
external bias. One consists in considering the bias potential
as a perturbation. The second method lies in the expansion of
the new wave functions on the basis of unbiased-dot eigenstates. The latter is proved to be an excellent approximation
on a wide range of dot sizes and bias voltages. The results
obtained for a single-dot device using this 1D model are
carefully compared with a full 3D calculation.
In the whole article, atomic units are used unless otherwise indicated. In these units, ប⫽1, 4 ␲ ␧ 0 ⫽1, elementary
electron charge 兩 e 兩 ⫽1, and free electron mass m e⫽1. The
unit of length is the Bohr radius (a 0 ⬇0.529 Å) and the unit
of energy is the Hartree (1 Hartree⬇27.212 eV).
II. GENERAL THEORETICAL FRAMEWORK
A. System under study
The basic idea of the study consists in characterizing the
Coulomb blockade phenomenon in a silicon QD via the influence on the electron gas energy of an additional electron
in the system. This leads us to solve a Schrödinger equation
describing the state of a many-electron system in a confining
potential V conf :
Ĥ ␺ 共 r1 , . . . ,rN 兲 ⫽E ␺ 共 r1 , . . . ,rN 兲 ,
共1兲
with Hamiltonian
N
Ĥ⫽
兺
i⫽1
⫹
1
2
冋
1
⫺ “ 共关 M 兴 ⫺1 “ 兲 ⫹V conf共 ri 兲
2
N
1
兺
j⫽1 ␧ 储 ri ⫺r j 储
j⫽i
册
FIG. 1. Schematic of a typical QD studied: we use a square-well potential of
3.1 eV depth, effective masses are taken equal to m Si⫽0.27 and m SiO2
⫽0.5 for Si and SiO2 , respectively. The dielectric constants are ␧ Si⫽11.7
and ␧ SiO2 ⫽3.8.
crystalline SiO2 . 17 We take ␧ Si⫽11.7 and ␧ SiO2 ⫽3.8 for the
relative dielectric constants of Si and SiO2 , respectively.
B. Hartree method
In the Hartree method, the wave function ␺ (r1 , . . . ,rN )
representing the whole electron gas is expressed as a product
of N wave functions representing each electron:
␺ 共 r1 , . . . ,rN 兲 ⫽⌽ 1 共 r1 兲 ⫻¯⫻⌽ N 共 rN 兲 .
共2兲
,
In this way, the interaction between electron i and the other
electrons can be calculated via the following Poisson equation, where ␧ is the relative dielectric permittivity:
“ 共 ␧“V couli 兲 ⫽4 ␲␳ i ,
where 关 M 兴 is generally the effective-mass tensor. In this
article, the tensor has been replaced by a scalar effective
mass to improve computation time. This choice is also dictated by the 1D approximation that we want to develop and
compare with a full 3D approach: the use of an effectivemass tensor would be a bar to a spherically symmetric approximation. Solving Eqs. 共1兲 and 共2兲 is a problem similar to
those met in atomic physics and the same theoretical methods can be employed. Among these methods, we have
adopted the Hartree method. In principle, this method does
not take into account the correlation between electrons, but
its efficiency in describing a many-electron system in a QD
has been demonstrated by comparison with a density functional approach, which makes it a good approximation.13 The
determination of the fundamental level of a QD containing N
electrons amounts then to solving a Schrödinger equation
coupled to a Poisson equation.
We are interested in a single silicon QD 共Fig. 1兲 surrounded by silicon dioxide, and we study the conductionband electrons in the model of finite square-well potential of
depth V 0 ⫽3.1 eV. The electron effective mass in silicon is
approximated by the harmonic mean of the transverse and
longitudinal effective masses; that is, m Si⫽3(1/m L
⫹2/m T) ⫺1 ⬇0.27. For electron effective mass in SiO2 , we
consider m SiO2 ⫽0.5 corresponding to the effective mass of
16
共3兲
共4兲
with the charge density ␳ i given by
␳ i ⫽⫺
冋兺
j⫽i
册
g j 兩 ⌽ j 兩 2 ⫹ 共 g i ⫺1 兲 兩 ⌽ i 兩 2 ,
共5兲
where g j represents the number of electrons on level i of
wave function ⌽ i .
Each level is twelvefold degenerate due to the spin and
the six-valleys’ degeneracies. The electron density n(r) of
the electron gas, for the fundamental level, may thus be written
n 共 r兲 ⫽
再
q
兺 12兩 ⌽ i兩 2 ⫹ p 兩 ⌽ q⫹1兩 2 ,
i⫽1
共6兲
N⫽12q⫹p
where q represents the number of fully occupied states and p
the number of remaining electrons in the last state.
The Schrödinger equation of a N-electron system can
then be decomposed in q⫹1 Schrödinger equations, one for
each level:
冋 冉 冊
册
1
1
“ ⫹V conf⫹V couli ⌽ i ⫽Ei ⌽ i ,
⫺ “
2
m
共7兲
and Ei is the energy of level i. The total energy of the system
is given by
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Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
兺 Ei ⫺ 2 冕 ⌽ *i V coul ⌽ i
i⫽1
N
E TOT⫽
1
i
5055
共8兲
d 3 r.
III. UNBIASED QUANTUM DOTS
A. Spherical symmetry approximation
The system defined by the Eqs. 共4兲, 共5兲, and 共7兲 can be
numerically solved for QD potential of any shape. Nevertheless, the computational resources needed for a full 3D resolution can be a bar to simulating a device as a whole. One
way to make these calculations faster lies in the use of the
symmetry properties of spherical QDs. Such approximation
has already been presented in a previous article.13 We first
propose to show that this approximation is not only excellent
for spherical QDs, but may be also used to correctly describe
QDs of any shape.
This model is based on the study of spherical silicon
QDs embedded in silicon dioxide so that permittivity, electron effective mass, and confinement potential have spherical
symmetry. Moreover, in order to factorize the wave function
⌽ i into a radial part u i (r)/r multiplied by a spherical
harmonic18 Y m
l ( ␪ , ␸ ),
u i共 r 兲
⌽ ml
i ⫽
r
B. Comparison with 3D resolution
Ym
l 共 ␪,␸ 兲,
m
Ym
l 共 ␪ , ␸ 兲 ⫽ 共 ⫺1 兲
冑
共9兲
2l⫹1 共 l⫺m 兲 ! im ␸ m
e P l 共 cos ␪ 兲 ,
4 ␲ 共 l⫹m 兲 !
共10兲
共with Legendre polynomials P m
l ), we use the central field
approximation which turns ␳ i (r) 共and consequently V couli )
into a function of r solely:
␳ i共 r 兲 ⫽
冕冕␳
i 共 r 兲 sin ␪
d␪d␸.
共11兲
The problem is then reduced to solving the 1D coupled
Poisson/Schrödinger system:
⫺
冋
册
l 共 l⫹1 兲
1 d2 ui
⫹ V conf⫹V couli ⫹
u
2m dr 2
2mr 2 i
⫺
dr
2
冉
冊
1 d 1/m d u i u i
⫽Ei u i ,
⫺
2 dr
dr
r
d 2 U couli
⫹
冉
共12兲
冊
1 d ␧ d U couli U couli
4␲
⫽
⫺
r ␳ i共 r 兲 ,
␧ dr
dr
r
␧
with V couli ⫽U couli /r and
␳ i 共 r 兲 ⫽⫺
FIG. 2. Total energy of the fundamental level of QDs as a function of the
number of confined electrons. Different shapes have been studied using 3D
calculation 共sphere, cube, pyramid, hemisphere兲 for comparison with the 1D
spherical model. All these dots have the same volume corresponding to a
30-Å-radius sphere.
1
4␲r2
冋兺
j⫽i
册
g j 兩 u j⫹1 兩 2 ⫹ 共 g i ⫺1 兲 兩 u i 兩 2 .
共13兲
共14兲
It should be noted that the central field approximation is
taken into account by the 1/4␲ factor in the definition of the
density 共14兲.
From now on, ‘‘1D model’’ refers to the approximation
just depicted.
In order to appreciate the accuracy of the 1D model, a
comparison with a full 3D resolution must be performed. The
computational methods underlying the resolution of the 3D
or 1D Poisson/Schrödinger equations use the finite difference
discretization and an Arnoldi algorithm19,20 to determine the
eigenvalues of large-scale sparse matrices. We plot in Fig. 2
the total energy of the fundamental level for various 3D QDs
of different shapes as a function of electron number N 共symbols兲. The solid line represents the result obtained from the
1D spherical approximation, in good agreement with 3D calculation. To make it relevant, the comparison of QD shapes
is made at a given volume. We can conclude that the 1D
model, far faster to compute than the 3D resolution, is thus
able to correctly describe the electronic structure of QDs of
any shape. Indeed, the error made using this model should be
smaller than the uncertainties related to the dot physical parameters 共radius, shape, electron effective mass, etc.兲. These
results are consistent with the comparison of cubic and
spherical QDs using the tight-binding calculation reported in
Ref. 9. Additionally, the shape of Si and Ge nanocrystals
embedded in SiO2 is proved to be quasi-spherical,14,15,21
which justifies all the better the use of a spherically symmetric 1D model.
Moreover, the 1D model not only gives relevant information on the energy of the system, but also on the electronic
density. The density calculated by the 1D model in a spherical QD 共containing either 1, 12, or 15 electrons兲 is plotted in
Fig. 3. It exhibits the same behavior as the 3D density shown
in Fig. 4 for a cubic dot: in both graphs, the maximum density for N⫽15 is significantly shifted due to the electrostatic
repulsive force between electrons, in addition to the transfer
in 2p orbitals.
Nevertheless, the study of unbiased QDs for Coulomb
blockade electronics devices is restrictive: a modeling taking
into account the effect of bias voltage must be investigated.
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J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
IV. BIASED QD DEVICE
A. Bias potential
For the structure in Fig. 1 under bias voltage ⌬V, the
Schrödinger Eq. 共7兲 simply turns into
冋 冉 冊
册
1
1
“ ⫹V conf⫹V couli ⫹V bias ⌽ i ⫽Ei ⌽ i ,
⫺ “
2
m
共15兲
where the additional term V bias is the local bias potential 共in
Hartree兲 along the x direction resulting from the bias voltage
⌬V applied between x 1 and x 0 . It may be extracted from the
resolution of the Poisson equation with the boundary conditions
FIG. 3. Electron density given by the 1D model in a spherical QD of radius
30 Å containing N electrons (N⫽1, 12, or 15兲.
In particular, with an eye to future realistic device investigation, an extension of the 1D model for a multiple QD system
under bias voltage should be of great interest because of the
relatively large amount of computation required by 3D calculation 共one 3D resolution of Poisson/Schrödinger equations for each voltage, each dot size, and each number of
electrons confined兲. The issue lies in the fact that applying a
bias potential breaks the spherical symmetry of the system;
that is why new approximations and techniques must be
found. This question is tackled in the next section.
再
“ 共 ␧“V bias兲 ⫽0,
V bias共 x 0 ,y,z 兲 ⫽0,
and
V bias共 x 1 ,y,z 兲 ⫽⫺⌬V.
共16兲
In this equation, ⌬V is the bias voltage 共in ‘‘atomic unit
volt’’兲 applied with respect to the reference potential
V bias(x 0 ,y,z). An example of the shape of the potential is
given in Fig. 5 for a 30-Å radius spherical QD surrounded by
a 24-Å silicon dioxide barrier: the effect of permittivity difference between Si and SiO2 is clearly observed.
Our goal is to extend the previous 1D model to the case
of biased structures, if possible by applying analytical corrections to include the effect of bias potential V bias . A first
way to simplify the problem consists in finding an analytical
model for the bias potential. A good choice for this analytical
model may be obtained if we solve the 3D Poisson Eq. 共16兲
by neglecting the derivative of V bias along y and z axes. This
equation then turns into a simple 1D equation
FIG. 4. Isodensity surface and density in the plane z⫽0 for a cubic QD containing N electrons (N⫽1, 12, or 15兲 whose volume corresponds to a 30-Å-radius
spherical QD. This graph illustrates that as the number of electrons increases, the distribution of electron density shifts from the center to the edges of the dot.
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Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
FIG. 5. 共a兲 Bias potential of a 30-Å-radius spherical QD surrounded by a
24-Å silicon dioxide barrier. The applied bias voltage is 0.6 V. 共b兲 Bias
potential profile along x axis at y⫽0 and z⫽0 for the same QD, the linear
approximation 共Eq. 18兲 is also presented.
再
⳵ V bias
⫽const,
⳵x
␧ 共 x,y,z 兲
V bias共 x 0 ,y,z 兲 ⫽0,
and
共17兲
V bias共 x 1 ,y,z 兲 ⫽⫺⌬V.
Solving this equation along the x axis gives us a linear piecewise potential slowly varying along y and z axes because of
the variation in dielectric constant due to the QD shape. To
make the model simpler, we take a linear approximation by
keeping in the whole device the same electrical field as in the
center of the dot. We thus find the final expression for V bias
independent of y and z:
V bias共 x,y,z 兲 ⫽⫺⌬V 共 Ax⫹B 兲 ,
with
冦
A⫽
B⫽
␧ SiO2 /␧ Si
L⫹2a 共 ␧ SiO2 /␧ Si⫺1 兲
⫺x 0 ⫹a 共 ␧ SiO2 /␧ Si⫺1 兲
,
共18兲
L⫹2a 共 ␧ SiO2 /␧ Si⫺1 兲
where a represents the QD radius and L the device length
x 1 ⫺x 0 . The comparison of this approximation with the potential given by the resolution of the 3D Poisson equation is
5057
FIG. 6. 共a兲 Energy levels obtained by 3D resolution of the coupled Poisson/
Schrödinger equations with either the exact bias potential 共symbols兲 or its
linear approximation. A spherical QD containing up to three electrons are
considered with a 30-Å radius and a 24-Å oxide barrier thickness. 共b兲 Electron density profile along x axis at z⫽0 and y⫽0 for the same QD. Linear
approximation and full 3D potential are compared.
presented in Fig. 5共b兲 and shows the good agreement of the
linear model with the actual potential inside the silicon QD.
Figure 6 shows, on the one hand, the fundamental energy
level resulting from 3D resolution of the coupled Poisson/
Schrödinger equations for 1, 2, or 3 electrons in the dot 关the
right electrode is taken as reference potential so that
V bias(x 0 )⫽0] and, on the other hand, the electron density for
one electron in the dot. The values obtained from the linear
approximation 共18兲 of the bias potential 共solid lines兲 match
very well with those obtained from the Poisson Eq. 共16兲
共symbols兲. This agreement is observed for various dot sizes
whatever the number of electrons in the dot. Additionally, the
fundamental energy level and the electron density is proved
to be weakly dependent on the barrier thickness 共result not
shown兲: the agreement between results obtained from Eqs.
共16兲 and 共18兲 remains good whatever the value of this parameter. Thus, the linear approximation 共18兲 is sufficiently
accurate to describe the effect of bias voltage on the electronic structure of these Si QDs. This approximation is used
in the rest of the article.
Moreover, some interesting information can be extracted
from the electron density cross section 共along symmetry
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5058
Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
FIG. 7. 共Color兲 Electron density cross section in the symmetry plane z⫽0 for a 60-Å spherical QD containing one electron for different bias voltages applied.
plane z⫽0) plotted in Fig. 7 for various bias voltages. Indeed, as expected, we remark that the electronic density concentrates on the region of lower potential. It implies two
important properties.
共1兲 First, this effect is opposed to the good functioning of
Coulomb blockade devices. Indeed, to take advantage of
the blockade property, the electrons should tunnel more
easily through the input barrier 共right barrier in Fig. 1兲
than through the output barrier 共left barrier in Fig. 1兲: an
electron can then stay a long enough time in the dot to
block the arrival of new electrons from the input junction. However, under the effect of bias potential, the
electron density concentrates near the output barrier
making easier the tunneling of electrons through this barrier.
共2兲 From a numerical point of view, the use of the central
field approximation to calculate the effect of bias potential is no longer valid because of the strongly nonspherical behavior of the wave function.
Both following subsections are then devoted to two possible
techniques to include the bias potential in the 1D model. The
first one uses the perturbation theory, while the second is
based on the expansion of the wave function on a good basis.
B. Perturbation theory
Let us consider the bias potential
Ĥ pert⫽⫺⌬V 共 Ax⫹B 兲
共19兲
as a perturbation compared with the unbiased dot Hamiltonian Ĥ 0 of Eq. 共7兲. Thanks to the 1D model, we can easily
determine the energies E n and wave functions ⌽ n characterizing the eigenstates of Ĥ 0 . According to the second-order
stationary perturbation theory, the corrected energy E ⬘0 of the
fundamental level can be written as a function of Ĥ pert , ⌽ n ,
and E n :
E 0⬘ ⫽E 0 ⫹ 具 ⌽ 0 兩 Ĥ pert兩 ⌽ 0 典 ⫹
兺
n⫽0
2
兩 具 ⌽ 0 兩 Ĥ pert兩 ⌽ ml
n 典兩
E 0 ⫺E n
, 共20兲
knowing that eigenstates 兩 ⌽ n 典 form an orthonormal basis.
Using the linear approximation of the bias potential 共18兲
and knowing that the unperturbed wave functions are functions of the spherical harmonics,
⌽ ml
n ⫽
u n共 r 兲 m
Y l 共 ␪,␸ 兲,
r
共21兲
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Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
FIG. 8. First- and second-order perturbation correction on the chemical
potential of two quantum dots of 30-Å and 60-Å radii containing 1, 2, or 3
electrons. The barrier thickness is equal to 24 Å.
we can semi-analytically derive the general matrix element
m l
具 ⌽ n ⬘⬘ ⬘ 兩 Ĥ pert兩 ⌽ ml
n 典 that can be rewritten
m
⬘
具 ⌽ n ⬘⬘ ⬘ 兩 Ĥ pert兩 ⌽ ml
n 典 ⫽ 具 ␮ m ⬘ Y l ⬘ 兩 ⫺xA⌬V 兩 ␮ m Y l 典
m l
m
⫹ 具 ⌽ n ⬘⬘ ⬘ 兩 ⌽ ml
n 典 关 ⫺B⌬V 兴 .
m l
共22兲
Substituting r sin(␪)cos(␸) for x in Eq. 共22兲 gives
具 ⌽ n ⬘⬘ ⬘ 兩 Ĥ pert兩 ⌽ ml
n 典 ⫽⫺A⌬V
m l
⫻
冕
⬁
0
冕 冕
␸ ⫽2 ␲
␸ ⫽0
␪⫽␲
␪ ⫽0
FIG. 9. Correction on the chemical potential using biased Hamiltonian diagonalization on the 17 first states (1s,2p,2s,3d,4f ,3p) of the unbiased
Hamiltonian for a quantum dot of 30-Å or 60-Å radius containing 1, 2, or 3
electrons. The barrier thickness is equal to 24 Å.
neglect their contribution to the corrected energy 共20兲. We
then obtain a second-order ‘‘Taylor’’ expansion of E 0⬘ as a
function of applied voltage ⌬V:
E 0⬘ ⫽E 0 ⫺ ␣ ⌬V⫹ ␤ ⌬V 2 ,
共24兲
␣ ⫽B,
共25兲
冋
u n ⬘ * ru n dr
2
A
␤⫽
E 0 ⫺E 1 冑6
2
Y l ⬘⬘ * Y m
l sin共 ␪ 兲
5059
冕
⫹⬁
0
册
2
u s * 共 r 兲 ru p 共 r 兲 dr .
共26兲
m
In this expression, the ten first orthonormal levels
(1s,2p,2s,3d)
⫻cos共 ␸ 兲 d ␸ d ␪
⫹ 关 ⫺B⌬V 兴 ␦ nn ⬘ ␦ ll ⬘ ␦ mm ⬘ .
共23兲
We remark that the angular part of the matrix element can be
derived analytically.
In fact, many of these elements are zero because of the
symmetry issue, as, for instance, the interaction term between s and d orbitals. Moreover, if we consider higher
eigenstates, the 1/(E 0 ⫺E n ) factor becomes small enough to
1s
再
u s 共 r 兲 Y 00 共 ␪ , ␸ 兲 /r
u p 共 r 兲 Y 11 共 ␪ , ␸ 兲 /r→p y
0
2 p u p 共 r 兲 Y 1 共 ␪ , ␸ 兲 /r→p z
u p 共 r 兲 Y ⫺1
1 共 ␪ , ␸ 兲 /r→p x
2s
u s * 共 r 兲 Y 00 共 ␪ , ␸ 兲 /r
共27兲
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5060
3d
Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
冦
u d 共 r 兲 Y 22 共 ␪ , ␸ 兲 /r→d x 2 ⫺y 2
u d 共 r 兲 Y 12 共 ␪ , ␸ 兲 /r→d zx
u d 共 r 兲 Y 02 共 ␪ , ␸ 兲 /r→d z 2
u d 共 r 兲 Y ⫺1
2 共 ␪ , ␸ 兲 /r→d zy
u d 共 r 兲 Y ⫺2
2 共 ␪ , ␸ 兲 /r→d xy
have been taken into account in the summation of Eq. 共20兲. It
should be noted that the second-order expansion of E ⬘0 is
proved to be a third-order approximation as well, the thirdorder term being zero thanks to symmetry arguments.
For two QDs of different sizes, the corrected energy is
plotted in Fig. 8 as a function of bias voltage for N⫽1,2,3.
We compare the full 3D calculation 共symbols兲 with 1D approximation using first-order 共dashed lines兲 or second-order
共solid line兲 correction. A correct agreement is only found at
low bias voltage or for small QDs in which the voltage drop
is small. This method is thus not suitable to include the bias
effect in the 1D model. In fact, even a fourth-order correction
is unable to describe correctly the energy structure of large
QDs: for these dots, the bias potential can no longer be considered as a perturbation.
C. Basis function decomposition
Another way to implement the bias voltage effect consists in a common method in quantum physics: the expan-
sion of the wave function on a good basis. Keeping the same
notation as in the previous section, it is obvious that the
eigenstates of the Hamiltonian Ĥ 0 without bias form the
most natural basis to diagonalize the full Hamiltonian:
Ĥ 0 ⫹Ĥ bias⫽Ĥ 0 ⫺ 共 Ax⫹B 兲 ⌬V.
共28兲
Once the eigenstates 兩 ⌽ n 典 of Ĥ 0 are calculated using the 1D
model, the determination of the matrix H 0 ⫽ 关 具 ⌽ i 兩 Ĥ 0 兩 ⌽ j 典 兴 i j
does not set any problem since this matrix is diagonal in the
basis of its eigenstates. The matrix H bias⫽ 关 具 ⌽ i 兩 Ĥ bias兩 ⌽ j 典 兴 i j
is directly deduced from Eqs. 共22兲 and 共23兲. The matrix
elements are computed semi-analytically using the angular
properties of spherical harmonics. The determination of
the eigenstates of the matrix H⫽H 0 ⫹H bias gives the
energies and wave functions of the biased QD structure containing N electrons. We present here the expression of H
taking into account the ten first orthonormal eigenstates of
Ĥ 0 共in basis order 1s,2s,2 p x ,2 p y ,2 p z ,3d xy ,3d zy ,3d zx ,
3d z 2 , 3d x 2 ⫺y 2 ):
H⫽H 0 ⫹H bias ,
共29兲
with
共30兲
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Sée et al.
J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
5061
FIG. 10. 共Color兲 Electron density cross section in the symmetry plane z⫽0 for a 60-Å spherical QD containing one electron for different bias voltages applied
calculated using the 1D model. This density exhibits the same properties as the full 3D density plotted in Fig. 7.
and knowing that the radial part u i of the wave function is
supposed to be real,
␣ ⫽⫺B⌬V,
W i j ⫽⫺A⌬V
共31兲
冕
⬁
0
u i ru j dr.
共32兲
For better accuracy, the basis is enlarged with 4 f and 3 p
eigenstates. In fact, this extension is only necessary for accurate determination of wave functions, but a 兵 1s,2s,2p,3d 其
basis is sufficient for the energy correction. The fundamental
energy level of a biased QD is shown in Fig. 9 for the same
type of QDs as in Fig. 8. In this graph, we compare a full 3D
modeling to the results given by the ‘‘extended 1D model’’
described earlier. We can notice the good results offered by
this extended 1D model whose main advantage lies in its
efficiency in term of computation time 共a quasi-immediate
result compared to about a one-day calculation for a full 3D
resolution to obtain the data plotted in Fig. 9兲.
Moreover, this simple 1D model not only gives information on the energy of the system but also on the wave function. Indeed, the knowledge of the eigenvectors of H offers
the opportunity of calculating the wave function in the device. Figure 10 presents the electronic density mapping in
the symmetry plane z⫽0 for a 60-Å spherical QD containing
one electron and for different bias voltages applied. This
graph is quasi-identical to the one given in Fig. 7 resulting
from a full 3D calculation. A more quantitative comparison
is shown in Fig. 11, which is a plot of the density as a
function of x at y⫽z⫽0. The 1D calculation 共solid lines兲
matches very well the full 3D resolution; the agreement is
still better for smaller QDs. This information on the wave
function is of first importance in view of simulating tunneling currents in Coulomb blockade devices.
V. CONCLUSION: APPLICATIONS FOR COULOMB
BLOCKADE DEVICES
Nowadays, the need for modeling tools in the field of
Coulomb blockade devices becomes more and more necessary, but, even using a powerful computer, the full 3D simulation of a realistic device containing many QDs of different
shapes and sizes seems impractical. It is strongly desirable to
develop new methods likely to yield fast simulation of complex devices while describing accurately the QDs under bias
voltage in terms of energy and wave functions. The 1D
model presented here achieves both objectives and can be
used as an elementary constituent of a wider 3D simulator of
semiconductor Coulomb blockade devices.
As an example, we can calculate the maximum number
N of electrons stored at 0 K in a spherical silicon QD as a
function of the bias voltage ⌬V. Indeed, this number is sim-
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5062
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J. Appl. Phys., Vol. 94, No. 8, 15 October 2003
FIG. 12. Fundamental energy level of a 60-Å-radius spherical silicon QD
surrounded by 24-Å silicon dioxide barriers, containing 1, 2, 3, or 4 electrons, as a function of the bias voltage applied between the two aluminum
electrodes. The maximum number of electrons in the dot is also represented.
The reference of energy is taken at the bottom of the silicon conduction
band in the dot. Dashed lines are plots of the Fermi level of both electrodes.
FIG. 11. Electron density profile along x axis at z⫽0 and y⫽0, for 30-Å
and 60-Å QDs containing one electron for different bias voltages applied.
3D and 1D results are compared.
ply the number of electrons giving the maximum energy
level in the dot E(N) below the right electrode Fermi potential E F right. In other words, we have
E F left⭐E 共 N 兲 ⭐E F right and E 共 N⫹1 兲 ⬎E F right.
共33兲
Figure 12 presents such results and shows that only a small
number of electrons can be stored in a relatively large QD of
60 Å. In this graph, the energy reference is taken at the
bottom of the silicon conduction band and the electrodes are
made of aluminum, whose work function is taken equal to
4.1 eV. This graph gives not only the maximum number of
electrons which can be stored simultaneously in the dot, but
also, and especially, the value of the different threshold voltages for which a new electron can transfer from the right
electrode to the dot.
Further investigation should consist in using the wave
function extracted from the 1D approximation of QDs to
implement a single-electron tunneling model. In the weakcoupling approximation, Bardeen’s tunnel formalism22 is a
good candidate to calculate the tunnel transition rates from
the wave functions given by our models. In this way, tunnel
rates are no more considered as fitting coefficients as usually
done in many calculations: they can be calculated as a function of the physical parameters of the system. Above all,
taking into account the influence of the bias voltage on the
tunnel transition rates should be of first importance considering the bias-induced shifting of the wave functions towards
the lower potential region. This work is currently under way.
A first application should be the study of charging effects
in a single QD device like the one experimentally investigated in Ref. 5. Such a comparison with experimental results
for a structure as simple as possible would allow us to validate our approach. It could then be used to quantitatively
determine the different physical parameters favorable to the
emergence of Coulomb blockade properties at room temperature. The model could be easily generalized to more
complex devices including several QDs, which is the strong
advantage of the 1D approximation presented in this work
over a full 3D calculation.
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