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APS/123-QED
Self–consistent simulation of quantum wires defined by local oxidation of Ga[Al]As
heterostructures
Christian P. May∗ and Matthias Troyer†
Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland
Klaus Ensslin‡
Solid State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland
(Dated: February 2, 2008)
We calculate the electronic width of quantum wires as a function of their lithographic width
in analogy to experiments performed on nanostructures defined by local oxidation of Ga[Al]As
heterostructures. Two–dimensional simulations of two parallel oxide lines on top of a Ga[Al]As
heterostructure defining a quantum wire are carried out in the framework of Density Functional
Theory in the Local Density Approximation and are found to be in agreement with measurements.
Quantitative assessment of the influence of various experimental uncertainties is given. The most
influential parameter turns out to be the oxide line depth, followed by its exact shape and the effect
of background doping (in decreasing order).
PACS numbers: 73.61.Ey, 71.55.Eq
I. INTRODUCTION
There have been several semiconductor quantum dot
calculations assuming simplifications such as parabolic
confining potentials1,2,3 and a number of self–consistent
solutions of coupled Schro¨dinger and Poisson’s equations
in III/V semiconductor nanostructures.4,5,6,7,8,9,10,11,12,13
However in particular the depletion mechanism in
two–dimensional electron gases (2DEG) caused by local
oxidation of the surface of AlGaAs/GaAs heterostruc-
tures with an atomic force microscope (AFM) has so far
only been studied in simplified models assuming a non–
realistic geometry.14 By AFM induced oxidation,15,16
confining walls can be defined with high accuracy en-
abling long quantum wires with only a few modes. Previ-
ous studies on quantum wires have either used analytical
approaches,17 lacked self–consistency18 or applied simple
geometrical assumptions.19
In this work, we perform a self–consistent numeri-
cal simulation of the effect of oxide lines on top of an
AlGaAs/GaAs heterostructure assuming a realistic ox-
ide line profile. In particular, a structure with two ox-
ide lines defining a quantum wire is studied within the
framework of Density Functional Theory (DFT)20 assum-
ing the Local Density Approximation (LDA)21 and the
results thereof are compared to available experimental
data. The main point of this paper is to treat this specific
kind of semiconductor nanostructure on a quantitative
level, which has not been done before. Moreover, var-
ious uncertainties like background charges are assessed
quantitatively in order to rank them by importance and
judge their respective influences. Obtaining results for
this type of quantum wires which agree with experiments
represents a major step towards a full self–consistent sim-
ulation of AFM lithography defined III/V semiconductor
nanostructure devices such as quantum dots and quan-
tum point contacts with realistic potentials.
II. METHODOLOGY
In order to solve the electrostatic problem, it is neces-
sary to solve Poisson’s equation. In our case of a two–
dimensional calculation, it reads
∇ (ǫ(x, y)∇)Φ(x, y) = −ρ(x, y)ǫ0
, (1)
where ǫ represents the space–dependent dielectricity ten-
sor and ρ denotes the charge density, given by the ion-
ized donor/acceptor concentrations Nn, Np as well as the
electron density n:
ρ(x, y) = q(Nn(x, y)−Np(x, y)− n(x, y)). (2)
Furthermore, we solve a one–particle Schro¨dinger equa-
tion
[
−~
2
2
∇
(
1
m∗(x, y)∇
)
+ V (x, y)
]
Ψ(x, y) = E Ψ(x, y).
(3)
In Eq. (3), m∗ stands for the space–dependent effective
mass, while the potential V consists of the band edge off-
sets ∆Ec at heterostructure interfaces, the electrostatic
potential Φ and exchange and correlation terms, which
are taken into account through the explicit parameteri-
zation Vxc given by Hedin and Lundqvist.22
V (x, y) = −qΦ(x, y) + ∆Ec(x, y) + Vxc(ρ(x, y)). (4)
The exchange–correlation potential explicitly reads
Vxc(ρ) =
−2Ry∗
π 23 (49 )
1
3 rs
(
1 + 0.7734 rs
21
log
(
1 +
21
rs
))
, (5)
where
rs(x, y) =
(
4
3
πa∗3ρ(x, y)
)− 13
(6)

2while a∗ and Ry∗ are the effective Bohr radius and Ryd-
berg constant in the respective material, which are given
by
a∗(x, y) = 4πǫ0ǫ(x, y)
~2
m∗(x, y)q2 (7)
Ry∗(x, y) = q
2
8πǫ0ǫ(x, y)a∗(x, y)
. (8)
Material parameters have been taken from well es-
tablished sources23 (the permittivity ǫ of GaAs and
Al0.3Ga0.7As is taken as 13.18 and 12.24 respectively, the
effective masses of Γ valley electrons m∗/m0 in GaAs and
Al0.3Ga0.7As are assumed to be 0.067 and 0.092 respec-
tively, while the conduction band offset between those
two materials amounts to ∆Ec=0.23 eV in our simula-
tions).
We discretize both Eq. (3) and Eq. (1) in a finite–
difference approach, allowing for different lattice spacings
in orthogonal directions.
Eq. (3) represents an eigenvalue problem, which we
have to solve only in a restricted domain as the elec-
tron density equals zero for practical purposes far away
from the interface. For computational efficiency rea-
sons, the domain has been chosen as small as possible
such that it does not change the numerical result com-
pared to the solution within the whole heterostructure
domain. Since only the lowest states are needed, the
Lanczos algorithm24 is a suitable method for obtaining
the eigenstates and their respective energies. We have
chosen the freely available IETL30 implementation for
this purpose. Having obtained a set of kmax eigenstates
Ψk allows us to calculate the electron density n by inte-
grating the Fermi distribution multiplied by the density
of states of the orthogonal directions over energy. In the
case of a two–dimensional simulation, it reads
n =
kmax
∑
k=1
|Ψk|2
∫ ∞
Ek
√
2m∗
π~
√
E − Ek
1
1 + exp
(
E−Ef
kbT
) dE,
(9)
where Ef and Ek denote the Fermi energy and k–th
eigenvalue respectively, while T remains at liquid Helium
temperatures of 4K throughout the investigation.
For the solution of Eq. (1), we rely on the Biconjugate
Gradient Stabilized Method.25 As boundary conditions
for the system as depicted in Fig. 1, we demand a van-
ishing electric field in the bulk (bottom of Fig. 1) and
Fermi level pinning on the semiconductor surface.26 The
left and right side of Fig. 1 are connected by periodic
boundary conditions, carefully paying attention to choose
the lateral extension large enough to prevent images of
the oxide line potential to have a considerable effect.
As both the Lanczos algorithm and the Biconjugate
Gradient Stabilized Method mainly consist of matrix–
vector products, a parallel matrix–vector class has been
implemented in order to achieve fast computation and
ensure extensibility to three dimension which will impose
significantly higher computational demands.
FIG. 1: (Color online) Schematic drawing of heterostructure
layout: A: 5nm GaAs, B: 8nm Al0.3Ga0.7As, C: 7nm GaAs,
D: 2nm Al0.3Ga0.7As n-doped, E: 15nm Al0.3Ga0.7As, F: bulk
GaAs.
Starting with a trial potential, equations (1) and
(3) are iteratively solved until a self–consistent solu-
tion is found. To ensure convergence, suitable damping
schemes27 have to be applied in order to reach the desired
equilibrium solution.
The heterostructure we want to investigate is schemati-
cally depicted in Fig. 1, following a design which has been
realized and characterized experimentally.15,28,29 Layer
D is n–doped in order to generate a 2DEG at the het-
erostructure interface. However, due to the fabrication
process, it is known that the bulk GaAs (denoted F) is
usually unintentionally p–doped. In our simulations, this
results in an additional degree of freedom since for ev-
ery background doping Np a modulation doping Nn can
be found that yields the experimentally measured sheet
density Ns = 4.5 × 1011cm−2 of the 2DEG 37nm below
the surface (see later discussion and Fig. 2). This re-
lationship has been established using a one–dimensional
self–consistent simulation (without oxide lines) along the
growth direction. We then apply the afore–mentioned
numerical method to the structure as defined in Fig. 1,
performing two–dimensional simulations in the plane per-
pendicular to the two parallel oxide lines in order to study
the influence of parameter variations on the electronic
width of the quantum wire. The latter forms in between
the projections of the oxide lines on the 2DEG plane. In
the experiment it has been established that the dimen-
sions of the oxide line below and above the semiconductor
surface are approximately the same. It was also demon-
strated that the electronic properties of a device confined
by oxide lines is not changed if the oxide lines are removed
e.g. by HCl etching. Therefore we use in the following
the term ”oxide line” to describe the shape of the groove
in the semiconductor surface which leads to a depletion
of the electron gas below.
The precise shape and size of the oxide growth under
the semiconductor surface cannot easily be probed ex-
perimentally. For the exact shape of the oxide line, we
therefore choose a Gaussian–like form similar to what has
been observed in the experiment.16 Expressed in terms

3of the width w1 as a function of depth d, it reads
w1(d) = 2b
√
− ln
[
1 +
(
1− dd0
)
(
e−(
w0
b )
2
− 1
)
]
, (10)
where d0 and w0 represent the maximum depth and
width, respectively, while the parameter b characterizes
the width at half depth.
The main aim of this work is now to assess the influ-
ence of these unknown parameters on the electronic wire
width quantitatively. In order to obtain independent pa-
rameters to vary we choose a definition of the lithographic
width that does not depend on the exact oxide line shape.
Following the experiment we therefore define the litho-
graphic width wli as the horizontal distance of the oxide
lines at a depth d0/2 and investigate the influence of rela-
tive width b/w0, maximum depth d0 and the background
doping Np respectively in the plane defined by the elec-
tronic and the lithographic width. The maximum width
is kept constant at w0 = 200nm throughout the simula-
tions.
III. RESULTS
Experimental electronic width values wel28,29 as a
function of lithographic width wli are plotted in Figures
3–5 together with simulations for various parameter set-
tings. In our simulation results, we define the electronic
width wel as the spatial distance over which the self–
consistent potential is below the Fermi energy. A linear
interpolation of the experimental values yields a slope
of 1.13± 0.07, while fittings for all simulations result in
slopes in the range of 1.00±0.02. The experimental error
bars have been estimated based on available AFM scans
and typical results for the wire width obtained from a fit-
ting of the positions of the minima of the low-field mag-
netoresistance. The simulation results are within these
error bounds.
Since the background doping Np, the maximum oxida-
tion depth d0 and the relative oxide line width b/w0 are
only known within certain ranges, we now investigate all
three of them within their physically meaningful ranges
and discuss their respective influences on the electronic
width.
In order to assess the influence of the background dop-
ing uncertainty, two possible doping concentration com-
binations (corresponding to the two points marked in Fig.
2) are simulated and their effect on the electronic width
is shown in Fig. 3. The effect of the background doping
turns out to be of minor importance.
In the following, we choose two extreme cases of possi-
ble oxide line shapes, b = 20w0 corresponding to a close
to rectangular profile and b = w0/2 which corresponds
to the situation where the oxide depth decreases quickly
from the center to the edge of the oxide line. All other
parameters are kept constant. In Fig. 4, we show the
configuration at which the influence of b is largest. It
33
33.5
34
34.5
35
35.5
36
1013 1014 1015
N n

[10
18

cm
-
3 ]
Np [cm-3]
Ns=4.5 × 10
11
cm-2
FIG. 2: (Color online) Required modulation n–doping Nn for
a given background p–doping Np yielding a constant sheet
density Ns = 4.5× 1011 cm−2 of the 2DEG.
0
50
100
150
200
250
300
350
400
450
50 100 150 200 250 300 350
w
el

[nm
]
wli [nm]
experimental data
Np=10
14
cm-3, b=w0/2, d0=12nm
Np=10
15
cm-3, b=w0/2, d0=12nm
FIG. 3: (Color online) Influence of background doping uncer-
tainty on electronic width: Electronic width wel as a function
of lithographic width wli for two simulated cases at constant
b = w0/2 and oxide line depth d0 = 12nm for different back-
ground doping levels Np (1014 cm−3 and 1015 cm−3) as well
as experimental values. The symbols denote specifically cal-
culated or measured values, the lines are linear fits to the
corresponding data points.
turns out that the choice of b is more relevant for the
comparison of experimental and simulated data than the
background doping uncertainty.
Finally, all simulations have been carried out at two
maximum oxide lines depths (d0 = 12nm and d0 = 13nm)
which are most likely to represent the physical reality.
Again, Fig. 5 displays the line in configuration space at
which the maximum depth exerts the most significant
influence leaving all other parameters constant. This pa-
rameter turns out to be the crucial one affecting the elec-
tronic width as the 2DEG is depleted at larger lateral
distances for larger values of d0. Figure 5 clearly shows
that the minimum lithographic width for the population

4 0
50
100
150
200
250
300
350
400
450
50 100 150 200 250 300 350
w
el

[nm
]
wli [nm]
experimental data
Np=10
15
, b=w0/2, d0=12nm
Np=10
15
, b=20w0, d0=12nm
-12
-10
-8
-6
-4
-2
0
-100 -50 0 50 100
FIG. 4: (Color online) Influence of oxide line shape uncer-
tainty on electronic width: Electronic width wel as a function
of lithographic width wli for two simulated cases at constant
background doping Np = 1015 cm−3 and oxide line depth
d0 = 12nm for different width parameters b (w0/2 and 20w0)
as well as experimental values. The symbols denote specifi-
cally calculated or measured values, the lines are linear fits
to the corresponding data points. Inset: geometrical shape of
a single oxide line. The same color code as in the electronic
width graph has been used to identify the different parameter
sets (all axis units are nm).
of the wire, i.e. the value of wli where wel goes to zero,
increases with increasing oxide line depth d0. For this
special case a situation can be realized where the elec-
tronic width wel is larger than zero for a negative value
of the lithographic width wli. This peculiar configuration
arises because of the definition of wli. It means that the
electron gas in the wire can laterally extend significantly
under the oxidized areas, an effect which can be con-
firmed experimentally and is also known from split–gate
defined quantum point contacts.
We note that our exploration of the parameter space
given by experimental uncertainties does not affect the
slope of the electronic width vs. lithographic width curve,
but only adds constant offsets. This fact and the slope
of unity we discovered agree with intuition in the case of
a lithographic width exceeding all other physical length
scales involved. However, on a smaller scale, there should
be a higher slope due to screening as well as exchange and
interaction effects, which cannot be observed using this
oxide line shape because the lithographic width at half
depth is still too wide for these phenomena to have a
significant effect. In order to further investigate the be-
havior at smaller lithographic widths, we therefore imple-
mented another shape allowing for extremely steep oxide
line walls, which reads
w2(d) = 2
√
c23 −
2c2c3
d + c1
. (11)
We achieve steep walls with the desired oxide
distance by choosing the constants c1=−1053/62,
0
50
100
150
200
250
300
350
400
450
50 100 150 200 250 300 350
w
el

[nm
]
wli [nm]
experimental data
Np=10
14
, b=20w0, d0=12nm
Np=10
14
, b=20w0, d0=13nm
-12
-10
-8
-6
-4
-2
0
-100 -50 0 50 100
FIG. 5: (Color online) Influence of oxide line depth uncer-
tainty on electronic width: Electronic width wel as a function
of lithographic width wli for two simulated cases at constant
background doping Np = 1014 cm−3 and width parameter
b = 20w0 for different oxide line depths d0 (12nm and 13nm)
as well as experimental values. The symbols denote specifi-
cally calculated or measured values, the lines are linear fits
to the corresponding data points. Inset: geometrical shape of
a single oxide line. The same color code as in the electronic
width graph has been used to identify the different parameter
sets (all axis units are nm).
0
10
20
30
40
50
60
70
20 25 30 35 40 45 50
w
el

[nm
]
wli [nm]
Np=10
14
, d0=12nm, steep profile
-12
-10
-8
-6
-4
-2
0
-100 -50 0 50 100
FIG. 6: (Color online) Electronic width wel vs. lithographic
width wli curve in the small lithographic width limit at back-
ground doping Np = 1014 cm−3 and oxide line depth d0 = 12
nm with a steep profile (see text). Inset: geometrical shape
of a single oxide line (all axis units are nm).
c2=−55575
√
62/1922, c3=450
√
62/31. Fig. 6 shows the
resulting slope of 3 ± 0.6 for very close oxide lines. This
extreme choice of parameters is used to test the theoret-
ical limit rather than representing the actual geometry.
Extrapolation down to the intersection of the elec-
tronic width vs. lithographic width curve with the x–
axis (lithographic width axis) allows one to read off the
physically interesting depletion length. It turns out that

5moderate parameter variations can easily account for the
difference between a positive depletion length (i.e. the
quantum wire gets cut off at a finite oxide line distance)
and a negative depletion length (it remains conducting).
IV. CONCLUSIONS
Given that experimental error bars are not explicitly
taken into account, the simulation data are in reasonable
agreement with experiments. Of all parameters investi-
gated, the oxidation depth turns out to be the most influ-
ential one. However, also the exact shape of the oxide line
as well as the background doping uncertainty contribute
to the overall error bounds (in decreasing order).
Given the exact physical setup of a nanostructure, the
result of electronic width calculations is precisely deter-
mined by the well–known laws of quantum mechanics and
electrodynamics. Since in experiments the exact shape
is never exactly known and also differs from sample to
sample, it is not straightforward to calculate the poten-
tial profile of more complex geometries.16 We envision,
however, that simulations using the methods presented in
this paper will be useful to design novel structures and to
obtain a better understanding on the electrostatic action
of in-plane and top gates. Most importantly, simulations
enable us to point out which parameters have the most
significant effect. Better experimental control is therefore
desirable for these parameters, in this particular case the
oxide line depth.
Acknowledgments
We thank Jean-David Picon for discussions. All sim-
ulations were run on the Beowulf cluster Gonzales oper-
ated by ETH Zurich and facilities at the Swiss National
Supercomputing Center CSCS. This work was supported
by ETH Research Grant TH-12 06-3 and the Studien-
stiftung des deutschen Volkes.
∗ Electronic address: cmay@itp.phys.ethz.ch
† Electronic address: troyer@phys.ethz.ch
‡ Electronic address: ensslin@phys.ethz.ch
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