Atomistic Simulation of Realistically Sized Nanodevices Using NEMO 3D: Part II
−
Applications
Gerhard Klimeck
1,2
, Shaikh Ahmed
1
, Neerav Kharche
1
, Marek Korkusinski
3
, Muhammad Usman
1
, Marta Prada
1
, and Timothy B. Boykin
4
Abstract
―
In Part
I, development and deployment of a general Nanoelectronic Modeling tool (NEMO 3D) has been discussed. Based on the atomistic valenceforce field (VFF) and the
sp
3
d
5
s
*
nearestneighbor tightbinding models, NEMO 3D enables the computation of strain and electronic structure in nanostructures consisting of over 64 and 52 million atoms, corresponding to volumes of (110nm)
3
and (101nm)
3
, respectively. In this part, successful applications of NEMO 3D are demonstrated in the atomistic calculation of singleparticle electronic states of realisticallysized (1) selfassembled quantum dots (QDs) including longrange strain and piezoelectricity, (2) stacked quantum dot system as used in quantum cascade lasers, (3) SiGe quantum wells (QWs) for quantum computation, and (4) SiGe nanowires. These examples demostrate the broad NEMO 3D capabilities and indicate the necessity of multimillion atomistic electronic structure modeling.
Index Terms
―
Atomistic simulation, NEMO 3D, Nanostructures, Strain, Piezoelectricity, Valley splitting, Quantum computation, Tight binding, Keating model, Quantum dot, Quantum well, Nanowire.
I. I
NTRODUCTION
HIS article describes NEMO 3D capabilities in the simulation of 3 (three) different classes of nanodevices of carrier confinement in 3, 2, and 1 dimensions in the GaAs/InAs and SiGe materials systems.
Single and Stacked Quantum Dots
(
confinement in 3 dimensions
). Quantum dots (QDs) are solidstate semiconducting nanostructures that provide confinement of charge carriers (electrons, holes, excitons) in all three spatial dimensions resulting in strongly localized wave functions, discrete energy eigenvalues and subsequent interesting physical and novel device properties [1][2][3][4][5]. Existing nanofabrication techniques tailor QDs in a variety of types, shapes and sizes. Within bottomup approaches, QDs can be realized by colloidal synthesis at benchtop conditions.
1
School of Electrical and Computer Engineering and Network for Computational Nanotechnology, Purdue University, West Lafayette, IN 47907, USA. Tel: (765) 494 9212, Fax: (765) 494 6441, Email: gekco@purdue.edu
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109.
3
Institute for Microstructural Sciences, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario K1A 0R6.
4
Electrical and Computer Engineering Dept., The University of Alabama in Huntsville, Huntsville, AL 35899.
Quantum dots thus created have dimensions ranging from 2– 10 nanometers corresponding to 100–100,000 atoms. On the other hand, selfassembled quantum dots (SAQDs), in the coherent StranskiKrastanov heteroepitaxial growth mode, nucleate spontaneously within a lattice mismatched material system (for example, InAs grown on GaAs substrate) under the influence of strain in certain physical conditions during molecular beam epitaxy (MBE) and metalorganic vapor phase epitaxy (MOVPE) [1][6]. The strain produces coherently strained quantumsized islands on top of a twodimensional wettinglayer. The islands can be subsequently buried to form the quantum dot. Semiconducting QDs grown by selfassembly are of particular importance in quantum optics [7][8], since they can be used as detectors of infrared radiation, optical memories, and in laser applications. The deltafunctionlike energy dependence of density of states and the strong overlap of spatially confined electron and hole wavefunctions provide ultralow threshold current densities, high temperature stability of the threshold current and high material and differential quantum gain/yield. Strong oscillator strength and nonlinearity in the optical properties have also been observed [1][8]. Selfassembled quantum dots also have potential for applications in quantum cryptography as single photon sources and quantum computation [9][10]
.
In electronic applications QDs have been used to operate like a singleelectron transistor and demonstrate pronounced Coulomb blockade effect. Selfassembled QDs, with an average height of 1–5 nm, are typically of size (base length/diameter) 5–50 nm and consist of 5,000–2,000,000 atoms. Arrays of quantummechanically coupled (stacked) selfassembled quantum dots can be used as optically active regions in highefficiency, roomtemperature lasers. Typical QD stacks consist of 3–7 QDs with typical lateral extension of 10–50 nm and dot height of 1–3 nm. Such dots contain 5–50 million atoms in total, where atomistic details of interfaces are indeed important [11].
Quantum Wires
(
confinement in 2 dimensions
)
.
For quite some time, nanowires have been considered a promising candidate for future building block in computers and information processing machines [12][13][14][15][16]. Nanowires are fabricated from different materials (metal, semiconductor, insulator and molecular) and assume different crosssectional shapes, dimensions and diameters. Electrical conductivity of nanowires is greatly influenced by edge effects
T
on the surface of the nanowire and is determined by quantum mechanical conductance quanta. In the nanometer regime, the impact of surface roughness or alloy disorder on electronic bandstructure need be atomistically studied to further gauge the transport properties of nanowires.
Quantum Wells
(
confinement in 1 dimension
)
.
QW devices are already a defacto standard technology in MOS devices and QW lasers. They continue to be examined carefully for ultrascaled devices where interfacial details turn out to be critical. Composite channel materials with GaAs, InAs, InSb, GaSb, and Si are being considered [17][18], which effectively constitute QWs. Si QWs buffered/strained by SiGe are considered for Quantum Computing (QC) devices where valleysplitting (VS) is an important issue [19]. Si is desirable for QC due to its long spindecoherence times, scaling potential and integrability within the present microelectronic infrastructure. In strained Si 6fold valleydegeneracy of Si is broken into lower 2fold and raised 4fold valleydegeneracies. The presence of 2fold valleydegeneracy is a potential source of decoherence which leads to leakage of quantum information outside qubit Hilbert space. Therefore, it is of great interest to study the lifting of remaining 2fold valley degeneracy in strained Si due to sharp confinement potentials in recently proposed [19] SiGe/Si/SiGe quantum well (QW) heterostructures based quantum computing architectures.
Fig. 1. Simulated InAs/GaAs quantum dots with dome and pyramidal shape. Two simulation domains are shown.
D
elec
: central domain for electronic structure calculation, and
D
strain
: larger/outer domain for strain calculation. In the figure:
s
is the substrate height,
c
is the cap layer thickness,
h
is the dot height,
d
is the dot diameter and
b
is the dot base length.
II.
S
IMULATION
R
ESULTS
(A) Strain and Piezoelectricity in InAs/GaAs Single QDs
The dome and pyramid shaped InAs QDs that are studied first in this work are embedded in a GaAs barrier material (schematic shown in Figure 1) and have diameter (base length) and height of 11.3 nm and 5.65 nm respectively, and are positioned on a 0.6nmthick wetting layer [20][21]. The simulation of strain is carried out in the larger computational box (width
D
strain
and height
H
), while the electronic structure computation is usually restricted to the smaller domain (width
D
elec
and height
H
elec
). All the strain simulations in this category fix the atom positions on the bottom plane to the GaAs lattice constant, assume periodic boundary conditions in the lateral dimensions, and open boundary conditions on the top surface. The inner electronic box assumes closed boundary conditions with passivated dangling bonds [22]. The strain domain contains ~3 M atoms while the electronic structure domain contains ~0.3M atoms.
Fig. 2. Atomistic
diagonal
strain profile along the [001],
z
direction. (a) Dome shaped dot with Diameter,
d
= 11.3 nm and Height,
h
= 5.65 nm. (b) Pyramidal dot with Base,
b
= 11.3 nm and Height,
h
= 5.65 nm. Strain is seen to penetrate deep inside the substrate and the cap layer. Also, noticeable is the gradient in the trace of the hydrostatic strain curve (
Tr
) inside the dot region that results in optical polarization anisotropy and nondegeneracy in the electronic conduction band
P
. Atomistic strain thus lowers the symmetry of the quantum dot. Fig. 3. Conduction band wavefunctions and spectra (eV) for first eight energy levels in the (a) Dome shape and (b) Pyramidal quantum dot structures. Atomistic strain is included in the calculation. Note the optical anisotropy and nondegeneracy in the
P
energy level. The first state is oriented along [110] direction and the second state along [110] direction.
Impact of strain
. Strain modifies the effective confinement volume in the device, distorts the atom bonds in length and angles, and hence modulates the local Bandstructure and the confined states. Figure 2 show the diagonal (biaxial)
[100] [010] [001] 100 [010] 00
0.120.080.0400.0401020304050
DISTANCE,
z
[nm]
S T R A I N
0.10.1000.04
01020304050
ε
xx
&
ε
yy
ε
zz
Tr
substrate cap layer QD
(b)
0.120.080.0400.0401020304050
DISTANCE,
z
[nm]
S T R A I N
0.10.1000.04
01020304050
ε
xx
&
ε
yy
ε
zz
Tr
substrate cap layer QD
(a)
components of strain distribution along the [001] direction in both the quantum dots (cut through the center of the dot). There are two salient features in both these plots: (a) The atomistic strain is longranged and penetrates deep into both the substrate and the cap layers, and (b) all the components of biaxial stress has a nonzero slope inside the quantum dot region. The presence of the gradient in the trace of the hydrostatic strain introduces unequal stress in the zincblende lattice structure along the depth, breaks the equivalence of the [110] and [110] directions, and finally breaks the degeneracy of the first excited electronic state (the so called
P
level). Figure 3 shows the wavefunction distribution for the first 8 (eight) conduction band electronic states within the device
Fig. 4. Atomistic
offdiagonal
strain profile along the
z
(vertical) direction which in effect induces polarization in the quantum dot structure. (a) Dome shape dot with Diameter,
d
= 11.3 nm and Height,
h
= 5.65 nm. (b) Pyramidal dot with Base,
b
= 11.3 nm and Height,
h
= 5.65 nm. Fig. 5. Potential surface plot of a (a) dome shape (b) pyramidal quantum dot in the
XY
plane at
z
= 1 nm from the base of the dot.
region for both the dots (in a 2D projection). Note the optical anisotropy and nondegeneracy in the first excited (
P
) energy level. The first
P
state is oriented along the [110] direction and the second
P
state along the [110] direction. The individual energy spectrum is also depicted in this figure which reveals the value of the
P
level splitting/nondegeneracy (defined as
E
110
– E
110
) to be about 5.73 meV and 10.85 meV for the dome shaped and pyramidal quantum dots, respectively. Although both the two dots have the same qualitative trend in diagonal strain profiles and similar wavefunction distribution, the reason for a larger split and hence pronounced anisotropy of
P
level in the pyramidal quantum dot is due to the presence of larger gradient of the hydrostatic strain, as can be seen in Figure 2, inside the dot region. In other words, as far as crystal symmetry lowering is concerned, atomistic strain has stronger impact in the pyramidal dot than it has in the dome shaped dot.
Fig. 6. Potential along [110] and [110] directions at z = 1 nm from the base of the dot. Notice the induced polarization in the potential profile and the unequal values of potential along the [110] and [110] directions. Also, dome shape dot induces stronger potential (
d/b
= 11.3 nm and
h
= 5.65 nm).
Impact of piezoelectric field
.
The presence of nonzero offdiagonal strain tensor elements leads to the generation of a piezoelectric field in the quantum dot structure, which is incorporated in the simulations as an external potential by solving the Poisson equation on the zincblende lattice. Figure 4 shows the atomistic offdiagonal strain profiles in both the quantum dots with heights,
h
of 5.65 nm and diameter (base length) of 11.3 nm. The offdiagonal strain tensors are found to be larger in the dome shaped dot. The offdiagonal strain tensors are used to calculate the firstorder polarization in the underlying crystal (please see Ref. [20] for the governing equations) which gives rise to a piezoelectric charge distribution throughout the device region and then used to calculate the potential by solving the Poisson equation. The relevant parameters for the piezoelectric calculation are taken from Ref. [20]. Experimentally measured polarization constants of GaAs and InAs materials (on unstrained bulk) values of 0.16 C/m
2
and 0.045 C/m
2
are used. The second
0.20.150.10.0500.050.125303540
DISTANCE,
z
[nm]
S T R A I N ( % )
ε
xy
ε
xz
&
ε
yz
substrate cap layer QD WL
0.20.150.10.0500.050.125 30 35 40
DISTANCE,
z
[nm]
S T R A I N ( % )
substr cQ
ab
604020020400 10 20 30 40 50
DISTANCE,
xy
[nm]
P I E Z O P O T E N T I A L [ m V ]
[110][110]
Dome Pyramid
order piezoelectric effect [23] is neglected here because of unavailability of reliable relevant polarization constants for an InAs/GaAs quantum dot structures.
Fig. 7. Conduction band wavefunctions for first three energy levels in the dome shaped quantum dot structure with diameter,
b
= 11.3 nm and height,
h
= 5.65 nm (a) without strain and piezoelectricity,
E
[110]

E
[110]
= 1.69meV (b) with atomistic strain,
E
[110]

E
[110]
= 5.73 meV and (c) with strain and piezoelectricity,
E
[110]

E
[110]
= 2.84 meV. Piezoelectricity
flips
the wavefunctions. Fig. 8. Conduction band wavefunctions for first three energy levels in the pyramidal quantum dot structure with base,
b
= 11.3 nm and height,
h
= 5.65 nm (a) without strain and piezoelectricity,
E
[110]

E
[110]
= 2.02meV (b) with atomistic strain,
E
[110]

E
[110]
= 10.85 meV and (c) with strain and piezoelectricity,
E
[110]

E
[110]
= 0.74 meV. Piezoelectricity does
not
flip the wavefunctions.
The calculated piezoelectric potential contour plots in the
XY
plane are shown in Figure 5 revealing a pronounced polarization effect induced in the structure. It is found that in both the dots piezoelectric field alone favors the [110] orientation of the
P
level. Shown in Figure 6 is the asymmetry in potential profile due to
atomistic
strain and inequivalence in the piezoelectric potential along [110] and [110] directions at a certain height
z
= 1 nm from the base of the dots. Figures 7 and 8 show the conduction band wavefunctions for the ground and first three excited energy states in the dome and pyramidal quantum dot structures with diameter (base length) of 11.3 nm and height,
h
of 5.65 nm, respectively. In Figures 7a and 8a strain and piezoelectricity are
not
included in the calculation. The weak anisotropy in the
P
level is due to the atomistic interface and material discontinuity. Material discontinuity mildly favors the [110] direction in both the dots. In Figures 7b and 8b atomistic strain and relaxation is included resulting in a 5.73 meV (dome) and 10.85 meV (pyramidal) splits in the
P
energy levels. Strain favors the [110] direction in both the dots. In Figures 7c and 8c piezoelectricity is included on top of strain inducing a split of 2.84 meV (dome) and 9.59 meV (pyramid) in the
P
energy level. There is a noticeable difference in Figures 7c and 8c. In the case of a dome shaped dot (Figure 7c), the first
P
state is oriented along [110] direction and the second state along [110] direction; piezoelectricity thereby has not only introduced a global shift in the energy spectrum but also
flipped
the orientation of the
P
states. In the case of a pyramidal dot (Figure 8c) the energetic sequence of the
P
states remains unchanged. The underlying reason behind this difference in orientation polarization due to piezoelectricity can be explained by the unequal potential induced as depicted in the 1D potential plot in Figure 6, which really is induced by the offdiagonal crystal distortion depicted in Figure 4. The pyramidal dot does not buildup as much offdiagonal strain due to the alignment of its facets with the crystal. As a result the piezoelectric fields are reduced.
Fig. 9. Electron state energies in the quantum dot molecule as a function of interdot separation. The strain simulation domain contains 8–13 M atoms and the electronic structure domain contains 0.5–1.1M atoms.
(B) Stacked Quantum Dot System
Selfassembled quantum dots can be grown as stacks where the QD distance can be controlled with atomic layer control. This distance determines the interaction of the artificial atom states to form artificial molecules. The design of QD stacks becomes complicated since the structures are subject to inhomogeneous, longrange strain and growth imperfections
1.341.351.361.371.381.391.41.412 4 6 8 10 12 14 16
DOT SEPARATION [nm]
E N E R G Y [ e V ]
E1E2E3E4
v
E5E6
25nm5nmddStrain domainElectronic domain10nm20nm20nm20nm
25nm5nmddStrain domainElectronic domain10nm20nm20nm20nm
such as nonidentical dots and interdiffused interfaces. Quantum dot stacks consisting of three QD layers are simulated next (see inset of Figure 9). The InAs quantum dots are disk shaped with diameter 10 nm and height 1.5 nm positioned on a 0.6 nm thick wetting layer. The substrate thickness under the first wetting layer is kept constant at 30nm and the cap layer on top of the topmost dot is kept at 10 nm for all simulations. The strain simulation domain contains 8–13 M atoms and the electronic structure domain contains 0.5–1.1M atoms.
Fig. 10. First five electron states wavefunction magnitudes (columns) with QD 2, 3, 4, 6, 10 and 12 nm separation (rows).
Figure 9 shows the electron state energy as a function of interdot separation. In a system without inhomogeneous strain one would expect the identical dots to have degenerate eigenstate energies for large dot separations. Strain breaks the degeneracy even for large separations. The strain field clearly extends over the distance of 15 nm quantum dot separation (which is why they physically do not grow on top of each other). As the dot separation is narrowed the dots interact with each other mechanically through the strain field as well as quantum mechanically through wavefunction overlaps. The set of lowest states E02 clearly show the state repulsion of bonding and antibonding molecular states for short interdot distances. Figure 10 shows crosssectional cuts in the growth direction and one lateral direction through the middle of the 3D wavefunctions. The wavefunctions are quite clearly separated into the individual dots with little overlap across the dots for dot separations of 15 nm and 10 nm. For 26 nm separation, wavefunction overlap can be observed. The reduction of E2 energy with decreasing distance for 2–4 nm can be associated with a crossover of psymmetry states. Whether or not the coupled dot system favors the topmost or bottommost QD to peak the ground state wavefunction is a complicated interplay of strain, QD size, and wavefunction overlap. Only a detailed simulation can reveal that interplay.
Fig. 11. (a) Schematic of a SiGe/Si/SiGe QW heterostructure grown on [001] substrate. The crystal symmetry directions are along
x
and
z
. (b) Schematic of a quantum well unit cell grown on
]01[
n z
′
miscut substrate. The unit cell is periodic along
]01[
n x
′
and
y
′
directions and confined in
]01[
n z
′
direction. Miscut angle is
( )
n
T
1tan
1
−
=
θ
. The step height is one atomic layer
)4/(
a
, where
a
is lattice constant. (c) Band structure of 5.26 nm thick flat QW along x and
0
2
miscut QW along
x
′
direction. Flat QW shows the presence of two nondegenerate valleys separated by an energy know as VS. Miscut QW shows the presence of two degenerate valleys centered at
0
x
k
′
±
. Interaction between these valleys at
0
=±
′
x
k
causes a minigap (
∆
m
) as shown in the inset. Lowest valleys are degenerate. Here,
Si L
naa
=
and
n=28
for
0
2
miscut. SiGe buffer layers are not included in electronic structure calculation domain for these plots.
(C) SiGe Quantum Well
Miscut (vicinal) surfaces (Figure 11b) as opposed to flat surfaces (Figure 11a) are often used to ensure uniform growth of Si/SiGe heterostructures. Miscut has a dramatic effect on bandstructure of Si QW. Bandstructure of a flat Si QW has two valleys centered at
0
=±
x
k
and separated by an energy known as valleysplitting (VS) [24][25]. VS in a flat QW is a result of interaction among states in bulk zvalleys centered at
m z
k k
=
, where
m
k
is position of the valleyminimum in strained Si. In a miscut QW lowest lying valleys are
(a) Flat
SiGe Si SiGe
]001[
z
]100[
x
]01[
n z
′
]01[
n x
′
(b) Miscut
SiGe Si SiGe
(c)