Document [original]

Vol.:(0123456789)

Optical and Quantum Electronics (2020) 52:257

https://doi.org/10.1007/s11082-020-02356-y

1 3

Numerical simulation ofTEM images forIn(Ga)As/GaAs

quantum dots withvarious shapes

AniezaMaltsi1 · ToreNiermann2· TimoStreckenbach1· KarstenTabelow1 ·

ThomasKoprucki1

Received: 13 September 2019 / Accepted: 10 April 2020

Abstract

We present a mathematical model and a tool chain for the numerical simulation of TEM

images of semiconductor quantum dots (QDs). This includes elasticity theory to obtain the

strain profile coupled with the Darwin–Howie–Whelan equations, describing the propaga-

tion of the electron wave through the sample. We perform a simulation study on indium

gallium arsenide QDs with different shapes and compare the resulting TEM images to

experimental ones. This tool chain can be applied to generate a database of simulated TEM

images, which is a key element of a novel concept for model-based geometry reconstruc-

tion of semiconductor QDs, involving machine learning techniques.

Keywords Semiconductors· Quantum dots· TEM images· Electron wave propagation·

Strain

1 Introduction

The fabrication of semiconductor quantum dots (QDs) with specific electronic properties

would highly benefit from the assessment of QD geometry, distribution, and strain profile

in a feedback loop between epitaxial growth and analysis of their properties. To assist the

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under

Germany’s Excellence Strategy—MATH+: The Berlin Mathematics Research Center, EXC-2046/1—

Project ID: 390685689 (A.M) and within CRC 787 “Semiconductor Nanophotonics” under Project A4

(T.N).

This article is part of the Topical Collection on Numerical Simulation of Optoelectronic Devices.

Guest edited by Angela Thränhardt, Karin Hinzer, Weida Hu, Stefan Schulz, Slawomir Sujecki and

Yuhrenn Wu.

* Anieza Maltsi

[email protected]

1 Weierstrass Institute (WIAS), Mohrenstr. 39, 10117Berlin, Germany

2 Institut für Optik und Atomare Physik, Technische Universität Berlin, Straße des 17. Juni 135,

10623Berlin, Germany

A.Maltsi et al.

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257 Page 2 of 11

optimization of QDs transmission electron microscopy (TEM) can be used. However, a

direct 3D geometry reconstruction from TEM of bulk-like samples (thickness 100–200 nm)

by solving the tomography problem is not feasible due to its limited resolution (0.5–1nm),

the highly nonlinear behaviour of the dynamic electron scattering and strong stochastic

influences in the images.

Recently, we introduced a novel concept for 3D model-based geometry reconstruc-

tion (MBGR) of QDs from TEM imaging (Koprucki etal. 2018; Maltsi etal. 2018). The

approach is based on (a) an appropriate model for the QD configuration in real space

including a categorization of QD shapes (e.g. pyramidal or lens-shaped) and continuous

parameters (e.g. size, height), (b) a database of simulated TEM images covering a large

number of possible QD configurations and image acquisition parameters (e.g. bright field/

dark field, sample tilt), as well as (c) a statistical procedure for the estimation of QD prop-

erties and classification of QD types based on acquired TEM image data.

Here, we present as a first step towards MBGR: A mathematical model and a tool chain

for the numerical simulation of TEM images for semiconductor QDs. We perform a sim-

ulation study on lens-shaped and pyramidal indium gallium arsenide QDs embedded in

a gallium arsenide matrix and compare the resulting TEM simulations to experimental

derived images. It is known that TEM images are very sensitive to strain fields around QDs

and these fields are mostly responsible for the observed contrast. In order to link the con-

trasts in TEM images with shapes and concentration of these QDs it is crucial to combine

strain calculations with TEM image simulations. Previous examples of investigations with

combined finite element and image calculations include the explanation of surface relaxa-

tion contrasts along quantum wells by Jacob etal. (1998) and explanation of the typical

coffee-bean contrast around QDs by Benabbas etal. (1996). Previous observations of Liao

etal. (1998) showed that, for single excitation conditions, it is hard to distinguish between

effects of shape and crystal symmetries on the TEM images. By our systematic study of the

influence of the shape on the image contrast, we could identify excitation conditions that

allow to distinguish between lens-shaped and pyramidal QDs.

2 Imaging ofquantum dots bytransmission electron microscope

The principal ray path for conventional imaging of crystalline specimen within a TEM is

sketched in Fig.1. A parallel electron beam illuminates a specimen of typical thicknesses

below few hundred nanometers.

The periodic structure of the specimen diffracts the beam in discrete directions, which

fulfill the Laue condition for a certain reciprocal lattice point

𝐠

. Due to the finite extent

of the specimen in the beam direction, the Laue condition must not be exactly fulfilled.

Figure1c, d illustrates two possible cases, in the latter only two beams are exactly on the

Ewald sphere (two beam conditions).

The diffracted beams leaving the exit surface of the specimen are focussed again by the

objective into a magnified image of the specimen. The objective aperture allows to restrict

the set of transmitted beams forming this image. In general, the term bright field image is

used, when the undiffracted beam is included in the aperture (e.g. Fig.1a). Here we use the

term dark field for images, where only one diffracted beam forms the image (Fig.1b).

As further detailed in Sect.3 the obtained images in two beam conditions are mostly

sensitive to the displacement field components along the direction of the diffraction vector

𝐠

. Examples for TEM images of two QDs are shown in Fig.2. These are dark field images

Numerical simulation ofTEM images forIn(Ga)As/GaAs quantum…

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Page 3 of 11 257

of the excited beam obtained for two beam conditions, in two perpendicular directions.

Both images are showing a coffee-bean-like contrast along the direction of the respective

diffraction vector.

This sensitivity for certain components of the displacement field is also used for instance

in dislocation imaging. Depending on the dislocation type the displacement field has no

(a) (b)

Fig. 1 Ray path within TEM: a ray path for imaging a specimen, with the objective aperture selecting all

beams. b Ray path for a specimen in two beam conditions, where the objective aperture only selects the

strongly excited beam. c Ewald sphere for case a, d Ewald sphere for case b

Fig. 2 TEM images of InAs

QDs: a dark field of (040) beam,

sensitive to [010]-component of

strain, b dark field of (004) beam,

sensitive to [001]-component

of strain, adapted from Hartwig

(2011), both showing a coffee-

bean-like contrast.

2018 IEEE.

Reprinted, with permission, from

Koprucki etal. (2018)

A.Maltsi et al.

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257 Page 4 of 11

component in certain directions. These directions can be found by looking for diminishing

contrast under two-beam conditions for different diffraction vectors

𝐠

. Typically the contrast

vanishes for directions

𝐠

perpendicular to the Burger’s vector of the dislocation, i.e.

𝐠

⋅

𝐛=0

(DeGraef 2003).

3 Modeling ofTEM imaging ofsemiconductor nanostructures

The dynamical electron scattering in crystalline solids, e.g. semiconductor nanostructures, is

influenced by spatial variations in the material composition and by local deformations of the

lattice due to elastic strain. In order to model the TEM images we need to use elasticity theory

to obtain the strain profile and couple this with the equations describing the electron propaga-

tion through the sample.

The indium gallium arsenide QDs under consideration and the surrounding GaAs matrix

have different lattice constants. This induces mechanical stresses in the nanostructure. The

strain tensor is defined as

𝜺

(𝐮)=

∇𝐮+ (∇𝐮)T

)

, where the displacement

𝐮∶𝛺→ℝ3

and

the stress

𝝈

is linked to the strain by Hook’s law

𝝈=𝐂∶𝜺

. Here,

𝐂

is the elastic stiffness

tensor.

We model the elastic relaxation of the misfit-induced strain following the concept of Eshel-

by’s inclusion (Eshelby 1957). The approach was developed for the description of the elastic

relaxation within and around inclusions of a solid surrounded by a matrix. The GaAs matrix

provides a global reference for measuring displacements and strains. For simplicity we assume

that both matrix and inclusion (QD) exhibit cubic symmetry. If we remove the inclusion from

the matrix, it is stress-free, if the strain

𝜺

of the reference system equals the eigenstrain

𝜺∗

Assuming a stress-strain relation according to Hooke’s law, we can mimic the stress-free con-

dition for

𝜺=𝜺∗

in the reference system of the matrix by incorporating the eigenstress in the

momentum balance:

The boundary condition on the Neumann part of the boundary (stress-free relaxation) has

to be modified accordingly

The dynamical electron scattering is described by the Schrödinger equation,

where

𝐤𝟎

∈ℝ

describes the incoming electron beam and

U(𝐫)

the reduced electrostatic

lattice potential. We assume a propagation of the beam in x-direction. Writing the wave

function as a superposition of plane waves, for the directions given by the Laue condition:

we end up with the Darwin–Howie–Whelan equations (DeGraef 2003)

(1)

∇

⋅

(

𝝈

−

𝝈

∗)=0.

(2)

(

𝝈

−

𝝈

∗)

⋅

𝐧=0.

(3)

𝛥𝛹

+4𝜋

𝐤

𝛹=−4𝜋

U(𝐫)𝛹

(4)

𝛹

(𝐫)=

∑

𝐠

𝜓𝐠(x;y,z)e2𝜋i(𝐤0+𝐠)⋅𝐫,𝐫=(x,y,z)

(5)

d𝜓

𝐠

dx =i𝜋

[

2s𝐠𝜓𝐠+1

𝐧⋅(𝐤0+𝐠)

∑

𝐠

�

U𝐠−𝐠�𝜓𝐠�

]

,s𝐠=−𝐠⋅(2𝐤0+𝐠)

2𝐧⋅(𝐤0+𝐠)

Numerical simulation ofTEM images forIn(Ga)As/GaAs quantum…

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Page 5 of 11 257

The amplitudes

𝜓𝐠

depend only parametrically on the (y,z)-coordinates, which define a 2D

pixel array constituting the TEM image. The Fourier coefficients

U𝐠

of the reduced electro-

static potential

U(𝐫)

of the crystal introduce a coupling between the beams and the excita-

tion error

s𝐠

shows how well theLaue condition is satisfied. We consider only amplitudes

𝜓𝐠

with a small excitation error

s𝐠

. In our case the vector

𝐧

is the unit vector in x direction.

The influence of variations in the indium content

c(𝐫)

and of the lattice deformations

given by the displacement field

𝐮(𝐫)

can be approximated by the modification of the Fou-

rier coefficents according to

The projection of the displacement on the individual reciprocal lattice vector

𝐠

enters the

coupling as phase factor.

4 Numerical simulation ofTEM images fordifferent geometries

A simulated TEM image is generated by propagating the beams through the speci-

men for every pixel

(yi,zj)

i,j=1, …,N

. This is done numerically by solving the Dar-

win–Howie–Whelan equations (5) using displacements

𝐮

obtained from the elasticity prob-

lem (1) for a given geometry.

The excitation errors

s𝐠

entering the Darwin–Howie–Whelan equations depend on the

orientation of the crystallographic lattice with respect to the beam direction. The specimen

can be oriented such that only a small number of beams

𝐠

with amplitudes

𝜓𝐠

are excited.

For two-beam conditions, the Ewald sphere only cuts two reciprocal lattice points. Thereby,

most of the intensity is contained within the undiffracted

𝜓0

and a single diffracted beam

𝜓𝐠

while the other beams remain largely unexcited, see Fig.1d.

In imaging conditions, where the objective aperture selects only a single beam, the

image contrast is simply obtained by taking the square of the modulus of the beam’s ampli-

tude at the exit plane

x=xexit

For example for excitation of (040)-two beam conditions, the bright field is given by

I0(y,z)

while the dark field image is given by

I𝐠(y,z)

with

𝐠=(040)

4.1 Tool chain forsimulation ofTEM images

The simulation tool chain starts with a parametric geometry description of the QD shape,

e.g. using base length and height of the pyramidal QD. A representation of the geometry

by a triangulation is created using the 3D mesh generator TetGen (Si 2015), see Fig.5a.

The generated mesh enters a FEM-based elasticity solver of WIAS-pdelib (Fuhrmann etal.

2019) which computes the strain profile and the displacement field

𝐮

in the QD and in

the sample. Finally, the multi-beam solution is obtained by the Darwin–Howie–Whelan

solver pyTEM (Niermann 2019) depending on excitation conditions and the sample orien-

tation and results in the simulated TEM image. For the examples shown in the following

we assumed an acceleration voltage of 300

. The y and z range of the simulated TEM

images is -50nm to 50nm around the center of the quantum dot, sampled with 101 points.

The beams used are chosen from (020) systematic row for the (040) reflection and (002)

(6)

𝐠→U

�

𝐠

(𝐫)=

[

c(𝐫)U

InAs

𝐠

+(1−c(𝐫))U

GaAs

𝐠]

×exp(−2𝜋i𝐮(𝐫)⋅𝐠)

(7)

I𝐠

(y,z)=

𝜓

𝐠

exit

;y,z)

|2.

A.Maltsi et al.

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257 Page 6 of 11

systematic row for the (004) reflection. From these, the beams with the smallest excitation

error were used (6 beams in our case).

4.2 Spherical QD

As a first example, we consider a spherical InAs QD with radius 10nm embedded in a

GaAs matrix. Figure3b shows the dark field TEM image under (040)-two beam condi-

tions, revealing a coffee-bean like contrast which looks similar to the experimental results

shown in Fig.2. For comparison Fig.3a shows a simulated TEM image using in Eq. (6)

the effect of the material contrast only, neglecting strain effects. Even the qualitative behav-

ior differs strongly from the results of the fully-coupled simulation (Fig.3b) and from the

experimental observations (Fig.2). This demonstrates that the TEM contrast is dominated

by the strain profile around the QD, shown in Fig.3c, and the nonlocal behavior of the

mapping QD geometry (sphere) to TEM image (coffee-bean).

Fig. 3 Simulated TEM images: a dark field for material contrast only, b dark field also including influ-

ence of strain. The image contrast in a is strongly increased for better visualization. Due to the excitation

under (040)-two-beam conditions, the image contrast is sensitive to the [010]-component of the displace-

ment field, shown in c along a cross-section through the center of a spherical InAs QD as obtained by FEM.

2018 IEEE. Reprinted, with permission, from Koprucki etal. (2018)

Fig. 4 3D view of the geometry of a lens-shaped and a pyramidal quantum dot. The shadow indicates the

cutting plane for the cross sections shown in Fig.5

Numerical simulation ofTEM images forIn(Ga)As/GaAs quantum…

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Page 7 of 11 257

1.4

1.2

1.0

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

-1.4

3.5

3.0

2.5

2.0

1.5

1.0

0.5

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

-3.5

(a)

(b)

(c)

(d)

(e)

[010]

[001]

[100]

20nm

20nm g=(004)

g=(040)

Fig. 5 InGaAs (green) quantum dots (indium content 80%) embedded in a GaAs (red) matrix with differ-

ent shapes: lens-shaped QDs with circular base (two left columns) and pyramidal QDs (two right columns)

with different vertical aspect ratios, respectively. FEM simulation of the elastic relaxation of the misfit

induced strain: a geometry and FEM mesh, where we made a cut through the mesh by cell removal, so

the colour changes occur due to light reflections from the remaining tetrahedra. b

component and d

component of the displacement field along a yz-cross-section through the center of the QDs in Å. The strain

fields are not fully symmetric because we used an unstructured and coarse mesh. Simulated TEM images

for two different excitations: c dark field for (040) beam for excitation under (040) two beam conditions and

e dark field for (004) beam for excitation under (004) two beam conditions. The sample thickness is 150

nn and the acceleration voltage is

300 kV

. All images show the same 60 nm

60 nm field of view. (Color

figure online)

A.Maltsi et al.

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257 Page 8 of 11

4.3 Pyramidal andlens‑shaped QDs

For more realistic QD TEM images we study QDs with two different shapes: lens-shaped

QDs with circular base (diameter of 15nm) and (truncated) pyramidal QDs (baselength

of 15nm) with different vertical aspect ratios, respectively, and a thin wetting layer, see

Figs.4 and 5a. We assume an indium content of 80% for all cases.

For comparison with the experimental TEM images shown in Fig. 2, we simulated

TEM images for the same excitation conditions, namely (040)- and (004)-two beam condi-

tions, see Fig.5c, e, respectively. Consequently, the image contrast is sensitive to the y- and

z-component of the displacement field, shown in Fig.5b, d along a yz-cross section through

the center of the QDs, respectively.

For both, pyramidal and lens-shaped QDs, a coffee-bean like contrast can be found for

(040)-two beam conditions as it is also observed in the experimental data, see Fig.2. How-

ever, for (004)-two-beam conditions we observe a striking difference between the images:

The pyramidal QDs produce more a crescent-like contrast, whereas the contrast for the

lens-shaped QDs resembles the coffee-bean contrast again. This can be explained by the

different behavior of the elastic relaxation of the misfit induced strain: for lens-shaped

structures the relaxation takes place in the QD as well as below and above it, while for

pyramidal ones it is mainly concentrated in the QD itself and above, see Fig.5e.

Another observation we can make from the simulated images is a line of no contrast

across the center of lens-shaped QDs only. This characteristic feature is also observed

experimentally, see Fig.2. Therefore, even on the basis of the four different QD shapes we

studied, one can already conclude that a lens-shaped structure matches the experimental

(a) (c)

(b) (d)

Fig. 6 Experimental images a, b from Fig.2 compared to simulated images c, d of InGaAs quantum dots

(indium content 80%) embedded in a GaAs matrix from Fig.5. From top left to bottom right the shapes are

lens-shaped, flat lens, full pyramid and truncated pyramid. Dark field images are shown for the (040) beam

a, c and the (004) beam b, c. The yellow boxes indicate the same areas in both experimental and simulated

TEM images. Acceleration voltage is

300 kV

Numerical simulation ofTEM images forIn(Ga)As/GaAs quantum…

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Page 9 of 11 257

observations much better than pyramidal structures (truncated or not), see comparison in

Fig.6.

4.4 Database ofTEM images

In order to achieve an automatic classification of QD shapes from experimental TEM

images out next step is to use the tool chain to generate a database of TEM images. This

database will contain series of TEM images for QDs with various shapes, in addition to

pyramidal and lens-shaped, see Schliwa etal. (2007), and will depend on the size and

indium concentration as well as on the excitation conditions of the electron beam, e.g.

excited beam, excitation error, acceleration voltage. As an example we show a series of

(a)

(b)

(c)

(d)

20nm

Fig. 7 InGaAs lens-shaped quantum dots embedded in a GaAs matrix. Dark field images for (040) two

beam conditions: a thickness variation 120 nm, 130 nm, 140 nm, 150 nm, size of QD 15 nm, indium con-

tent 80%, position of QD at 75nm, b size variation 10 nm, 15 nm, 20 nm, 25 nm, sample thickness 150 nm,

indium content 80%, position of QD at 75 nm, c indium content variation 20%, 40%, 60%, 80%, sample

thickness 150 nm, QD size 15 nm, position of QD at 75 nm, d position variation along the beam direction

45 nm, 55nm, 65 nm, 75 nm, size of QD 15 nm, indium content 80%, sample thickness 150nm. All images

have the same gray scaling. Black corresponds to no intensity while white corresponds to the intensity of

the incoming beam. Acceleration voltage is

300 kV

A.Maltsi et al.

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257 Page 10 of 11

specimen parameter variations for a lens-shaped InGaAs quantum dot embedded in a GaAs

matrix: different values of the sample thickness in Fig.7a, the size of the QD in Fig.7b,

the indium content in Fig.7c and the position of the QD in the matrix relative to the top

surface in Fig.7d.

5 Conclusion

We presented a tool chain for the simulation of TEM images for QDs with realistic param-

eterized 3D geometries. Our computational tool chain is not restricted to QDs but can be

applied also to other epitaxial heterostructures or coherent precipitates. This includes the

surface relaxation of thin TEM lamellas or structures with spatially varying material com-

position, e.g. the QDs with varying indium content in our case.

Our study on four different QD shapes showed a strong sensitivity of image con-

trast to the characteristic profiles of the strain field. FEM investigations showed that the

strain field in growth direction allows to distinguish between pyramidal and lens-shaped

QDs. Therefore TEM images of QDs under strong beam conditions in the growth direc-

tion are suitable for the classification of QDs shapes. From comparison with experi-

mental data we can exclude a pyramidal structure (truncated or not) for InGaAs QDs,

assuming a constant homogenous indium content.

This tool chain will be used to create a database of simulated TEM images of QDs,

which will serve as a training set to apply deep learning techniques. Furthermore, statistical

procedures will be used to estimate continuous parameters such as height or base length of

the QD for the realization of our novel concept for model-based geometry reconstruction

(MBGR) of semiconductor QDs from TEM imaging. In addition, we plan to use methods

like elastic shape analysis, optimal transport, diffusion maps etc, in order to define a metric

on the space of TEM images and get a better understanding of its structure.

Funding Open Access funding enabled and organized by Projekt DEAL.

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References

Benabbas, T., François, P., Androussi, Y., Lefebvre, A.: Stress relaxation in highly strained InAs/GaAs struc-

tures as studied by finite element analysis and transmission electron microscopy. J. Appl. Phys. 80(5),

2763–2767 (1996). https:// doi. org/ 10. 1063/1. 363193

De Graef, M.: Introduction to Conventional Transmission Electron Microscopy. Cambridge University Press,

Cambridge (2003). https:// doi. org/ 10. 1017/ CBO97 80511 615092

Eshelby, J.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc.

Lond. A 241, 376–396 (1957). https:// doi. org/ 10. 1098/ rspa. 1957. 0133

Fuhrmann, J., Streckenbach, T., etal.: pdelib: a finite volume and finite element toolbox for PDEs. http:// pdelib.

org (2019). Accessed 13 September 2019

Numerical simulation ofTEM images forIn(Ga)As/GaAs quantum…

1 3

Page 11 of 11 257

Hartwig, M.: TEM-Untersuchungen der Spannungsfelder von In(Ga)As-Quantenpunkten. Bachelor’s thesis, TU

Berlin (2011)

Jacob, D., Androussi, Y., Benabbas, T., François, P., Lefebvre, A.: Surface relaxation of strained semiconduc-

tor heterostructures revealed by finite-element calculations and transmission electron microscopy. Philos.

Mag. A 78(4), 879–891 (1998). https:// doi. org/ 10. 1080/ 01418 61980 82399 62

Koprucki, T., Maltsi, A., Niermann, T., Streckenbach, T., Tabelow, K., Polzehl, J.: Towards model-based geom-

etry reconstruction of quantum dots from TEM. In: Djurišić, A., Piprek, J. (eds.) Proceedings of the 18th

International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018). IEEE, pp

115–116. https:// doi. org/ 10. 1109/ NUSOD. 2018. 85702 46 (2018)

Liao, X.Z., Zou, J., Duan, X.F., Cockayne, D.J.H., Leon, R., Lobo, C.: Transmission-electron microscopy study

of the shape of buried

InxGa1−xAs∕Gas

quantum dots. Phys. Rev. B 58, R4235–R4237 (1998). https:// doi.

org/ 10. 1103/ PhysR evB. 58. R4235

Maltsi, A., Koprucki, T., Niermann, T., Streckenbach, T., Tabelow, K., Polzehl, J.: Model-based geometry

reconstruction of quantum dots from TEM. Proc. Appl. Math. Mech. 18, e201800398 (2018). https:// doi.

org/ 10. 1002/ pamm. 20180 0398

Niermann, T.: pyTEM: a python-based TEM image simulation toolkit. Version 3.0.9 (2019)

Schliwa, A., Winkelnkemper, M., Bimberg, D.: Impact of size, shape, and composition on piezoelectric effects

and electronic properties of

In(Ga)As∕GaAs

quantum dots. Phys. Rev. B 76, 205324 (2007). https:// doi.

org/ 10. 1103/ PhysR evB. 76. 205324

Si, H.: TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Mathe. Softw. 41, 1–36

(2015). https:// doi. org/ 10. 1145/ 26296 97

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