Numerical analysis of quantum dots nanostructures

The finite element model used consists of a pseudo thermal expansion of the buried quantum dot and the wetting layer so as to simulate the mismatch strain caused by the difference in the

NUMERICAL SIMULATION OF QUANTUM DOT

NANOSTRUCTURES

QUEK SIU SIN JERRY

(B.Eng (Hons.), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTORATE OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2003

ACKNOWLEDGEMENTS

1 A/Prof Liu Gui Rong (Supervisor) – for his supervision, guidance and many

valuable advice And for the many learning opportunities given throughout the years

2 Dr Han Xu (Manager, Centre for ACES) – for his support while working in

the Centre for ACES

3 Various staff and graduate students of Centre for ACES – for sharing their

ideas and knowledge

4 Staff of NUS IT Unit, CALES 1 – for their support in providing the computing

resources

CONTENTS

SUMMARY i

LIST OF SYMBOLS iii

LIST OF FIGURES v

LIST OF TABLES xi

1 INTRODUCTION 1

1.1 Literature Review 4

1.1.1 Development of Quantum Dots in Brief 4

1.1.2 Fabrication Techniques for Quantum Dots 8

1.1.3 Characterization of Quantum Dots 11

1.1.4 Self-organized Ordering of Quantum Dot Superlattices 13

1.1.5 Modeling and Simulation of Quantum Dot Heterostructure 15

2 FINITE ELEMENT FORMULATION FOR ANALYSIS OF QUANTUM DOT HETEROSTRUCTURES 20

2.1 The SK Growth Mode, Stress Relaxation and Vertical Correlation 20

2.2 Finite Element Modeling 22

2.2.1 2D Axi-symmetric Model 24

2.2.2 3D Model 32

2.3 Strain Energy Density Calculation 38

3 NUMERICAL FORMULATION OF QUANTUM DOT SURFACE EVOLUTION 40

3.1 Surface Roughening 40

3.1.1 Moving Least Squares (MLS) Interpolation 42

3.1.2 Surface Displacement Time Marching 44

4 FINITE ELEMENT NUMERICAL SIMULATIONS OF QUANTUM DOT HETEROSTRUCTURE 50

4.1 2D Axi-symmetric Finite Element Model 51

4.1.1 Model Parameters 51

4.1.2 Results and Discussions 52

4.2 3D Finite Element Model 62

4.2.1 Model Parameters 62

4.2.2 Results and Discussions 64

4.3 Effects of Elastic Anisotropy on the Self Organized Ordering of Quantum Dots 74

4.3.1 Model Parameters 74

4.3.2 Results and Discussions 77

5 NUMERICAL SIMULATION OF QUANTUM DOT SURFACE EVOLUTION 94 5.1 2D Quantum Dot Island Growth 94

5.2 Note on Sensitivity of Time Step Used 101

5.3 3D Quantum Dot Island Growth 103

5.3.1 Growth of CdSe Quantum Dots on ZnSe(001) 106

5.3.2 Growth of CdSe Quantum Dots on ZnSe(111) 117

6 CONCLUSION 123

REFERENCES 130

APPENDIX A A-1

APPENDIX B B-1

APPENDIX C C-1

SUMMARY

Nanotechonology is the key to future’s technology and while researchers build devices and novel materials at the nanoscale, the one major obstacle is the efficient mass-production of these nano-materials that have sizes of just a few atoms A particularly successful and interesting technique being used in recent years is by self-assembly Based on theories of lattice mismatch, coherent islands can be formed via self-assembly that are small enough for size-quantization effects to be noticed Such islands are often called quantum dots and they possess interesting opto-electronic properties with potential for novel device applications

The numerical simulation of quantum dot nanostructures is presented in this thesis The finite element method is used to analyze the stress and strain fields in the quantum dot heterostructure The stress fields play a very important role in the lattice arrangement of the heterostructures when quantum dot superlattice is fabricated Furthermore, the elastostatic fields in the quantum dot heterostructure are vital for determining the opto-electronic properties of these structures The finite element model used consists of a pseudo thermal expansion of the buried quantum dot and the wetting layer so as to simulate the mismatch strain caused by the difference in the lattice parameters of the dissimilar semiconductor materials A tied contact condition

is also modeled at the interface of the dissimilar materials, which basically ties the nodes belonging to the island to the matrix

Using results obtained from the finite element method, the strain energy density of a new wetting layer surface can be calculated The strain energy density is a major

contributor to the surface chemical potential and by analyzing the distribution, one can predict the preferential site whereby new quantum dot(s) will be formed on the surface It is shown that consideration of elastic anisotropy of the semiconductor materials has a significant effect on the vertical and lateral correlation of quantum dots in the heterostructure Various parameters like the variation in the thickness of the cap layer was also shown to affect the correlation The author hypothesized that at small cap layer thickness, quantum dot islands will tend to coalesce directly above the buried island When the thickness of the cap layer increases, the islands separate and move further away from the vertically correlated position

The author also developed a code to simulate the surface evolution of the quantum dot-growing layer A mesh-free approach is employed by using the moving least squares (MLS) interpolation method together with a forward marching finite difference method This method provides the approximation for solving the surface diffusion equation typical of most epitaxy processes The normal surface velocity and hence the displacements can be deduced from the surface flux The problem then becomes an initial-value problem and is then marched forward in time using a forward marching finite difference method This method will enable analysts to simulate real-time quantum dot formation and therefore able to predict the island shape, size and vertical and lateral ordering

µo Initial bulk chemical potential

εo Lattice mismatch strain

σo Lattice mismatch stress

αT Thermal expansion coefficient

A Anisotropic ratio

a i Lattice parameter of island material

a m Lattice parameter of matrix material

c ij Components of material elastic matrix, c

D m Dissipating function

fe Element force vector

H Cap layer thickness

h i Quantum dot island height

h w Wetting layer thickness

Jv

k Global stiffness matrix

ke Element stiffness matrix

N i Finite element shape functions

U h Approximation of global variables/generalized displacements

u i x displacement component of node i

U o Initial strain energy density

v i y displacement component of node i

v n Surface normal velocity

w(r) Weight function

w i z displacement component of node i

LIST OF FIGURES

Figure 1.1 Nature of electronic states in bulk material, quantum wells and quantum dots Top row: schematic morphology, bottom row: density of electronic states

4

Figure 1.2 Schematic representation of different nanostructures fabrication processes 7

Figure 2.1 Schematic diagram of epitaxial strain relaxation via island formation 21

Figure 2.2 Schematic representation of basic unit of quantum dot heterostructure 25

Figure 2.3 Geometry for 2D axi-symmetric finite element analysis 26

Figure 2.4 2D axi-symmetric finite element mesh 27

Figure 2.5 Tied contact in abaqus 31

Figure 2.6 3D finite element quarter model of heterostructure 33

Figure 2.7 3D finite element model of heterostructure with part of cap layer cut away to reveal the pyramidal shaped quantum dot 35

Figure 3.1 Schematic of surface re-construction by adding blocks to an initially planar surface 45

Figure 3.2 Flow chart of quantum dot surface evolution numerical procedure 49

Figure 4.1 2D stress (σxx ) distribution of matrix in the x direction (h i = 3nm, H = 5nm) 53

Figure 4.2 2D stress (σxx ) distribution of island in the x direction (h i = 3nm, H = 5nm) 53

Figure 4.3 Strain (εxx ) distribution through centre of heterostructure model (h i = 3nm, H = 5nm) 56

Figure 4.4 Strain (εzz ) distribution through centre of heterostructure model (h i = 3nm,

H = 5nm) 56

Figure 4.5 Stress (σxx ) distribution on top surface of heterostructure (h i = 3nm, H =

5nm) 57 Figure 4.6 Stress (σxx) distribution on top surface of heterostructure with different

island heights (H = 30nm) 59

Figure 4.7 Stress (σxx) distribution on top surface of heterostructure with different

cap layer thickness (h i = 3nm) 60 Figure 4.8 Strain energy density distribution of top surface wetting layer before

island formation 62 Figure 4.9 3D stress (σxx ) distribution of matrix in the y=0nm plane 64

Figure 4.10 3D stress (σxx ) distribution of island in the y=0nm plane 65

Figure 4.11 Stress (σxx ) distribution on top surface of heterostructure in the y=0nm

plane obtained using 2d and 3d models 66 Figure 4.12 3D stress (σxx ) distribution of matrix in the y=6nm plane 66

Figure 4.13 3D stress (σxx) distribution of matrix in the θ = 45 ° plane (where θ is

measured in counter clockwise from the y = 0nm plane) 67

Figure 4.14 3D stress (σxx) distribution of island in the θ = 45 ° plane (where θ is

measured in counter clockwise from the y = 0nm plane) 67

Figure 4.15 Hydrostatic strain on the top surface of cap layer using a 3D quarter

model with isotropic material properties 68 Figure 4.16 3D mesh of heterostructure with truncated pyramidal shaped island 70 Figure 4.17 Stress (σxx ) distribution of matrix in the plane of y = 0nm of

heterostructure with truncated pyramid island 71

Figure 4.18 Stress (σxx ) distribution of island in the plane of y = 0nm of

heterostructure with truncated pyramid island 72 Figure 4.19 Comparison of stress (σxx ) on top surface of cap layer (y = 0) of

heterostructures with truncated and full pyramid 73 Figure 4.20 Comparison of strain (εxx) through centre of truncated and full

pyramidal islands 73 Figure 4.21 Finite element model of basic unit of quantum dot heterostructure with part of the cap layer cut away to reveal the pyramidal shaped island 75 Figure 4.22 Strain energy density distribution of top surface of second CdSe wetting

layer with the dimensionless spacer thickness, H/h i = 4; island base is 20 x 20

nm; h i = 10 nm; crystal orientation of [001] for the normal of the top surface of the substrate and spacer-layer 78 Figure 4.23 Fringe plots of strain energy density of top surface of second CdSe

wetting layer with spacer-layer thickness of (a)H/h i = 2; (b) H/h i = 3; and (c)

H/h i = 4 79 Figure 4.24 Detailed fringe plot of the center region of top surface of second CdSe

wetting layer with spacer-layer thickness of H/h i = 3 80 Figure 4.25 Schematic diagram of spatial correlation of multiple layers of

CdSe/ZnSe(001) quantum dot structure in a tetragonal body centered

arrangement 82 Figure 4.26 Fringe plots of strain energy density of top surface of second InAs

wetting layer with spacer-layer thickness of (a)H/h i = 2; (b) H/h i = 3; and (c)

H/h i = 4 84 Figure 4.27 Strain energy density distribution of top surface of second CdSe wetting

layer with the dimensionless spacer thickness, H/h i = 4; island base is 20 x 20

nm; h i = 10 nm; crystal orientation of [111] for the normal of the top surface of the substrate and spacer-layer 85 Figure 4.28 Position and depth of minima with various spacer-layer thickness of

CdSe/ZnSe(111) 86 Figure 4.29 Schematic diagram of spatial correlation of multiple layers of

CdSe/ZnSe(111) quantum dot structure with the quantum dots displaced from the center in the [1-10] direction 87 Figure 4.30 Strain energy density distribution of top surface of second CdSe wetting

layer with the dimensionless spacer thickness, H/h i = 4 island base is 20 x 20

nm; h i = 10 nm; crystal orientation of [113] for the normal of the top surface of the substrate and spacer-layer 88 Figure 4.31 Schematic diagram of spatial correlation of multiple layers of

CdSe/ZnSe(113) quantum dot structure with the quantum dots displaced from the center in the [-3-32] direction 89 Figure 4.32 3D mesh of heterostructure with cuboid shaped island 90

Figure 4.33 Surface strain energy distribution of CdSe/ZnSe(001) of various H/h i

obtained using a (a) pyramidal island and (b) cuboidal island 91 Figure 4.34 Strain energy density distribution of top surface of second CdSe wetting

layer with the dimensionless spacer thickness, H/h i = 4; cuboid shaped island of dimensions 20 x 20 x 10 nm; crystal orientation of [001] for the normal of the top surface of the substrate and spacer-layer 92 Figure 4.35 Schematic illustration showing decay of strain field above the (a)

pyramidal island and (b) cuboidal island 93 Figure 5.1 Schematic model for the simulation of the kinetic growth of the quantum dot layer 95

Figure 5.2 2D growth of quantum dot wetting layer from t=0.01s to t=1s at time

steps of ∆t=0.01s 96

Figure 5.3 2D strain energy density distribution of wetting layer from t=0s to t=1s at time steps of ∆t=0.01s 98

Figure 5.4 2D chemical potential per unit volume (µ /Ω) distribution of wetting layer from t=0s to t=1s at time steps of ∆t=0.01s 99

Figure 5.5 2D quantum dot growth state at various cut-off times 100

Figure 5.6 Surface chemical potential at t = 2.0s 101

Figure 5.7 Unstable 2D simulation from t=0.02s to t=1s at time steps of ∆t=0.02s 103

Figure 5.8 Initial nodal distribution on surface of growing quantum dot layer 104

Figure 5.9 Gauss points distribution after applying mls interpolation 105

Figure 5.10 3D quantum dot growth state of CdSe/ZnSe(001) with H/h i = 2 at various cut-off times 107

Figure 5.11 Plot of island height with different growth times for CdSe/ZnSe(001); H = 20nm 108

Figure 5.12 Chemical potential distribution on surface of CdSe/ZnSe(001), H/h i = 2; (a) t = 0.5s; (b) t = 5.0s 109

Figure 5.13 Strain energy density distribution on surface of CdSe/ZnSe(001), H/h i = 2; (a) t = 0.5s; (b) t = 5.0s 110

Figure 5.14 3D quantum dot growth state of CdSe/ZnSe(001) with H/h i = 3 at various cut-off times 112

Figure 5.15 Contour plot of surface displacement of CdSe/ZnSe(001) with H/h i = 3 and t = 1s 113

Figure 5.16 Contour plot of surface displacement of CdSe/ZnSe(001) with H/h i = 3 and t = 5s 114

Figure 5.17 3D quantum dot growth state of CdSe/ZnSe(001) with H/h i = 4 at

various cut-off times 115

Figure 5.18 Contour plot of normal surface displacement at t = 1s and H = 40nm 116 Figure 5.19 Surface profile of CdSe/ZnSe(111) with H = 20nm at various cut-off

times 118 Figure 5.20 Plot of island height with different growth times for CdSe/ZnSe(111);

LIST OF TABLES

Table 4.1 Geometric parameters for 2D FE model 51

Table 4.2 Isotropic elastic properties of InAs and GaAs 52

Table 4.3 Lattice parameters of various semiconductor materials 54

Table 4.4 Anisotropic elastic constants for various semiconductor materials 63

Table 4.5 Distance of strain energy density minimas from center of wetting layer with various spacer-layer thicknesses and the crystallographic directions of where the minimas are formed 81

Table 5.1 Model parameters for growth simulation of quantum dot layer 96

1 INTRODUCTION

This project involves the numerical computation of various aspects of quantum dot nanostructures formation Quantum dots can be considered to be clusters of atoms in the nano-meter scale usually made up of semiconducting materials like Silicon (Si), Germanium (Ge), Indium Arsenide (InAs), Gallium Arsenide (GaAs) and so on Quantum dots are attracting much interest because of their unique electro-optic properties making them desirable for use in novel semiconductor and electronic devices

Being in the order of nanometer in size, quantum dots are significantly smaller than the micro-sized structures typical of current microelectronic circuits Hence they are difficult

to manufacture using standard lithographic techniques and therefore, various alternative methods are actually devised to produce them Recently, a particularly interesting and effective method is to grow the dots directly by depositing a thin film layer of material on

a substrate under appropriate growth conditions More often than not, the depositing material is different from the substrate material and hence, such structures are also known generally as heterostructures Due to the different lattice parameters between the depositing and substrate material, lattice mismatched strain occurs and the thin layer deposited on the substrate will break up into coherently strained islands Such a growth mode is typically known as the Stranski-Krastanow (SK) growth mode The driving force behind the SK growth mode is because of the strain energy stored in the lattice mismatched, epitaxial thin film layer, making it generally unstable Although considerable research has been put into understanding this self-assembly of quantum dot

structures, there are still many ambiguities in the actual growth process For example, there still exist a debate if this quantum dot growth should be explained from a kinetic or energetics point of view This leads to the motivation behind using computational simulation to analyze certain aspects of the quantum dot and thus gain a better understanding of the process

If one considers just the growth of a single epitaxial layer of quantum dots on the substrate, the positions of growth as well as the size of the quantum dots can be said to be highly statistical in nature Without a regular arrangement of uniformly sized quantum dots, the advantage of the discrete energy level the dots possessed cannot be utilized to its full potential To achieve better dot size and arrangement uniformity, the solution is to grow multiple layers of the heterostructure to form quantum dot superlattices It has been found that the elastic fields induced by quantum dots in a bottom layer will assert a strong influence on the ordering and possibly the size of quantum dots grown on a subsequent layer Therefore, this presents a tool for which one can tune the regular ordering and the size of quantum dot superlattices In this project, the author has used the finite element method to investigate this elastic interaction of the buried quantum dots and its subsequent quantum dots growth layer Anisotropic material properties as well as different crystal orientation of the materials were included in the analysis and interesting results were obtained

Other than using the finite element method to model the elastostatic effects of the quantum dot nanostructures, the morphology of the quantum dot island was also analyzed

Epitaxy can be considered as governed by surface diffusion of the adatoms involved Therefore, in analyzing the quantum dot island morphology, the surface diffusion equation is solved which basically describes the movement of atoms or material from a region of higher chemical potential to a region of lower potential The numerical approach employed is a meshless one utilizing the Moving Least Squares (MLS) interpolation method together with a forward-marching finite difference method By using this numerical method, the evolution of the quantum dot island with respect to time can be studied which further enhances the understanding of the spatial distribution, shape and size of quantum dots

This thesis consists of first a literature review of the various developments in the field of quantum dot nanostructures as well as a review of computational methods typically used

in the numerical simulation of such structures Following that, physical and mathematical theories of quantum dot island formation as well as the numerical approaches will be detailed This will be followed by a detailed description of the elastostatic analysis of quantum dot heterostructure The finite element models used consist of both two-dimensional (2D) and three-dimensional (3D) models Both isotropic material properties as well as anisotropic material properties will also be presented and compared Different parameters that affect the stress and strain fields will also be presented Then, a detailed presentation of the MLS approximation approach of solving for the quantum dot island morphology will be given

1.1 Literature Review

1.1.1 Development of Quantum Dots in Brief

Quantum dots are desirable for use in novel semiconductor and electronic devices because of their interesting electro-optic properties This is due to carrier localization in all three dimensions resulting in the density of electronic states of quantum dots becoming discrete as shown in Figure 1.1

Figure 1.1 Nature of electronic states in bulk material, quantum wells and quantum dots Top row: schematic morphology, bottom row: density of electronic states against energy

of electronic states

When the carrier motion in a solid is limited in layer of a thickness of the order of the carrier de Broglie wavelength (or the mean free pass if this number is smaller), the effects

of size quantization will be observed The de Broglie wavelength λ, depends on the

effective mass m eff of the carrier and on temperature T:

The mass of the charged carrier in Eq ( 1.1 ) is not the free electron mass but the effective mass of the electron (or hole) in the crystal As this mass can be much smaller than the free electron mass, size quantization effects can already be pronounced at a thickness ten to a hundred times larger than the lattice constant The introductory chapter

in the book by Bimberg et al (1999) gave a very concise historical development of

quantum dot heterostructures and a brief summary will be presented here By the late 1950s and early 1960s there is already the idea of using ultra-thin layers for studies of size quantization effects At that time, the research focus was on thin films of semimetals

on mica substrates obtained by vacuum deposition Nevertheless, thin films of semiconductors and metals were also studied At that time, the effect of increase in the effective bandgap with decreasing film thickness was demonstrated Though at that time, theoretical works were highly recognized, the experimental studies were limited by the inadequacy in technology

With the advancement in technology and the development of novel deposition techniques like molecular beam epitaxy and metal organic chemical vapor deposition in the late 1960s, the revolution started It became possible to insert coherent layers a few lattice constants thick, of a semiconductor of lower bandgap in a matrix with a larger bandgap, restricting carrier movement to only two dimensions In 1969-70, the use of multilayer

periodic semiconductor heterostructures (superlattices) for the creation of artificial materials with controlled width of minizones for carrier transport along the superlattice

grown at around 1971 A clear demonstration of size quantization effects became possible only after lattice-matched GaAs/AlGaAs structures with planar interfaces were realized by molecular beam epitaxy Researchers at that time also observed the effect of resonant tunneling, proving the application of quantum mechanics to describe transport phenomena in ultra-thin semiconductor heterostructures Interesting results were obtained from the optical studies of quantum wells and superlattices For example, the step-like characteristics of the absorption spectrum related to the two-dimensional character of the density of states in quantum wells (Figure 1.1) were observed

By the end of late 1980s, researchers have gained considerable understanding of quantum wells and superlattices and naturally, interests shift towards structures with further

reduced dimensionality – to quantum wires (Kapon et al, 1989) and quantum dots A

complete reduction of the remaining infinite extension of a quantum well in two dimensions to atomic values lead to carrier localization in all three dimensions and breakdown of classical band structure energy level model The resulting energy level structure of quantum dots is discrete (Figure 1.1), like in atomic physics, and many of the physical properties of quantum dots resemble that of an atom in a cage The study of single quantum dots and ensembles of quantum dots presents a new chapter in

atoms

Of course, having discovered and understood all these theories, to fabricate and make use

of quantum dots for device fabrication is not an easy task The first realization of quantum dots went as far back as the 1930s with nano-size semiconductor inclusions in glass as shown in the schematic diagram in Figure 1.2(a) It was not until 1984, when quantum confinement effects in such a system is confirmed experimentally Many approaches have been developed to fabricate quantum dots including artificial patterning

of thin layer structures into three-dimensional regions (Figure 1.2(b))

In the last decade however, greater success have been achieved using self-assembly effects like that during the growth of strained heterostructures as shown schematically in Figure 1.2(c) The best thing about this self-assembly process is that the fabrication process is compatible with present optoelectronic device technology With the discovery

of this process, the advancement in quantum dot nanotechnology took a huge leap ahead and much research work were, and still being carried out on various semiconductor systems Much of the research work in this dissertation is based on understanding and analyzing this self-assembly process using numerical simulation

1.1.2 Fabrication Techniques for Quantum Dots

Lithographic Techniques

By the end of the 1980s, the most straightforward method of fabricating quantum dots was by the patterning of quantum wells This technique has several advantages, which till today still attracts considerable attention Firstly, quantum dots of arbitrary lateral shape, size and arrangement can be realized depending on the resolution of the particular lithographic technique used Secondly, a variety of processing techniques are at the researcher’s disposal And finally, the technique is generally compatible with modern VLSI (very large scale integrated) semiconductor technology Lithographic techniques comprise of (Beaumount, 1991; Sotomayor Torres et al, 1994):

• optical lithography and holography,

• X-ray lithography,

• electron and focused ion beam lithography,

• scanning tunneling microscopy

To the best of the author’s knowledge, the resolution of optical lithography based methods is generally insufficient to fabricate structures of lateral dimensions of 20nm or less A maximum size of ≈20nm is required for the carriers to be sufficiently contained

at room temperature in typical semiconductors Having said that, optical lithography could nevertheless be used for fabrication of structures investigated at low temperatures whereby confinement is easily reached and also for the fabrication of patterned substrates that can be used subsequently by other techniques

X-ray lithography (Warren et al, 1986) has the advantage of much shorter wavelengths and can be used for nanostructures fabrication effectively However, an additional process is required to fabricate the mask Nevertheless, X-ray lithography has the potential for mass production of nanoscale structures

The most developed techniques for patterning quantum wells are electron beam lithography (Howard et al, 1985; Beaumount, 1991), focused ion beam lithography (Komuro et al, 1983), and also contact imprinting (Krauss, 1997) In electron beam lithography (EBL), the electron beam is usually emitted from a high-brightness cathode

or a field emission gun and focused on the substrate by a multilens system Resolution in the 10-20nm range was demonstrated at the beginnings of the 1980s (Craighead et al,

1983; Stern et al, 1984) Periodic nanostructures can also be fabricated using an electron beam interference technique (Fujita et al, 1995), which is particularly advantageous for large-scale fabrication of quantum wires and dots

Focused ion beam lithography (FIBL) has a beam diameter that is generally larger than that of an electron beam Nevertheless, FIBL has a number of properties advantageous for the fabrication of nanostructures:

• maskless etching (ion beam sputtering or stimulated chemical etching),

• maskless implantation of dopants,

• deposition of metallic structures,

• patterning of resists with strongly reduced proximity effect

Scanning tunneling microscopy (STM) also presents novel approaches for nanoscopic patterning and creating nanostructures (Snow et al, 1993; Snow and Campbell, 1994) Using the above lithography techniques, various methods of fabricating quantum dots are possible For example, free-standing quantum dots can be fabricated using dry or wet etching procedures; selective intermixing based on ion implantation for buried heterostructures (Werner et al, 1989); strain-induced lateral confinement (Kash, 1988, 1991); and growth of quantum dot nanostructures on patterned substrates (Kapon et al, 1987; Lebens et al, 1990; Madhukar,1993; Madhukar et al, 1993; Ishida et al, 1998)

Self-organized Quantum Dots

The physics behind self-organized quantum dots is actually a well-known phenomenon that involves the evolution of an initially two-dimensional growth into a three-dimensional corrugated growth front In 1937, Stranski and Krastanow proposed the possibility of island formation on an initially flat heteroepitaxial surface Eventually, the term “SK growth” was used in heteroepitaxy for such island formation More details regarding this growth process will be covered in Chapter 2 Goldstein et al (1985) observed for the first time, the formation of a regular pattern of islands in an InAs/GaAs superlattice The discovery of the possibility of exploiting this SK growth mode for the purpose of quantum dot fabrication stimulated much research interest In years to come, important breakthroughs were reported by various groups including Leonard et al (1993), Madhukar et al (1994), Moison et al (1994) and Bimberg et al (1995)

1.1.3 Characterization of Quantum Dots

Numerous methods are used for the physical characterization of quantum dots The methods can be divided into two main types namely: (i) direct imaging methods and; (ii) diffraction methods The direct imaging techniques include scanning tunneling microscopy (STM) (Guryanov et al, 1995), atomic force microscopy (AFM) (Kobayashi

et al, 1996; Tersoff et al, 1996) and transmission electron microscopy (TEM) (Yao et al, 1991; Nabetani et al, 1994; Christiansen et al 1994; Carlsson et al, 1998) Diffraction methods are methods such as reflective high-energy electron diffraction (RHEED)

(Nabetani et al, 1994) and X-ray diffraction (XRD) (Pal and Towe, 2001; Stangl et al, 2002)

STM has the advantage of being able to reveal directly the morphology of a surface on an atomic level and to enable manipulation of surface atoms Atomic force microscopy has

in principal atomic resolution The actual resolution of course depends on the specific size and shape of the tip Tip effects can actually modify the apparent height and to a larger extent the lateral size of the quantum dot However, one should note that STM and AFM plan-view measurements of uncovered dots are usually not performed at the growth temperature and therefore the surface morphology can thus be completely different from that directly after growth For example, physical measurements of quantum dots have been shown to vary with growth conditions like the deposition temperature (Ledentsov et

al, 1996) These problems are not apparent if STM cross-sectional experiments are performed on covered samples (Wu et al, 1997; Legrand et al, 1998)

Using high-resolution electron microscopy (HREM) or electron energy loss spectroscopy (EELS) provide information on quantum dot morphology which is frozen-in by direct coverage (Ruvimov et al, 1995; Kosogov et al; 1996), and therefore avoids the problems mentioned above for STM and AFM Nevertheless, these techniques are very time consuming and strain fields affect the apparent shape of the quantum dots

RHEED is highly surface-sensitive ultra-high vacuum technique used to monitor growth

in MBE systems Upon transformation of an initially two-dimensional ordered surface

with monolayer high islands into a corrugated structure, the RHEED pattern changes from streaky to spotty (Nebatani et al, 1994; Guryanov et al, 1996) Therefore, RHEED

is often used for in situ monitoring of quantum dot formation

X-ray diffraction techniques are useful for structural investigation after growth Krost (1996) reported results for single dot layers whereas Darhuber et al (1997) reported results for stacked layers for the InAs/GaAs and Ge/Si systems

In a nutshell, the above techniques generally reveal or at least give an idea of the shapes, sizes, island densities and other physical characteristics of quantum dots nanostructures For example, it was estimated by Nabetani et al (1994) using RHEED and TEM that indium arsenide (InAs) islands grown on gallium arsenide (GaAs) substrate have a pyramidal shape with a rectangular base It should be noted however that the precise geometry of the quantum dots is not clear, as the TEM images are usually difficult to interpret

1.1.4 Self-organized Ordering of Quantum Dot Superlattices

The positions of growth as well as the size of the quantum dots on an epitaxial layer by itself can be said to be highly statistical in nature Without a regular arrangement of uniformly sized quantum dots, the advantage of the discrete energy level the dots possessed cannot be fully utilized to its full potential To achieve better dot size and arrangement uniformity, the solution is to grow multiple layers of the heterostructure to

form quantum dot superlattices It is found that the elastic fields induced from the buried quantum dots will assert a strong influence on the subsequent quantum dot layer Xie et

al (1995) had shown experimentally that there is a strong vertical correlation in multiple layers of InAs quantum dots grown on GaAs substrate The InAs quantum dots tended to grow directly on top of the buried dots as a result of the tensile stress fields induced directly above the buried island They also provided an analytical description by deriving the vertical pairing probability, which deduced that the probability of vertical correlation drops with increasing spacer-layer thickness Tersoff et al (1996) also reported the observed vertical correlation as well as an increase in size uniformity of the dots via a simplified model based on continuum elasticity theory It should be noted however, that

in the above papers, the quantum dot structure is assumed to be an elastically isotropic medium

It was generally well accepted then that the observed vertical correlation is a direct result

of the stress or strain induced by the buried island, but Strassburg et al (1998) proved the contrary when anti-correlation is observed in CdSe quantum dots grown on ZnSe This finding resulted in a more detailed look into the spatial correlation between successive quantum dot layers by many researchers including Shchukin et al (1998, 2001) who studied the energetics of multi-layered quantum dot islands in two-dimension Elastic anisotropy is taken into account in their investigations and they deduced that the spatial correlation between successive quantum dot layers is dependent on the elastic interaction between layers, which exhibits an oscillatory decay with the separation between layers Thus, by varying the distance between successive layers, a transition occurs from vertical

correlation to an anti-correlation Springholz et al (1998, 2002) had also carried out an extensive study on the vertical and horizontal correlation between quantum dots in successive layers Their studies showed that the PbSe quantum dots grown on Pb1-

xEuxTe exhibit a trigonal lattice with face-centered cubic (FCC) like vertical stacking sequence

1.1.5 Modeling and Simulation of Quantum Dot Heterostructure

With the growing research interest in quantum dot nanostructures, numerical modeling and simulation of these structures also started to predict and analyze the properties and behaviors of such structures Important properties such as stress and strain, piezoelectric effects, quantum confinement effects, optical properties and so on are modeled using various mathematical models and techniques By understanding the physics behind quantum dot formation, the morphology of these nanostructures can also be mathematically simulated This review will briefly cover simulation of stress and strain

in quantum dots as well as the simulation of quantum dot island formation

Strain Distribution

The elastostatic distribution in and around a quantum dot structure is often of interest to many researchers because firstly, the induced stress fields of buried islands in multiple stacks affects the spatial correlation of islands in subsequent layers; secondly, the strain fields in the quantum dots affect the optical and electronic properties Jain et al (1996)

had reviewed the theory and experimental work on stresses in III-V semiconductor heterostructures.Many methods of modeling quantum dots heterostructures are developed

to analyze the stress or strain field The methods are generally categorized into three main categories:

• Analytical continuum approach

• Atomistic approach

• Numerical continuum approach

The analytical continuum approach is based on Eshelby's inclusion theory as shown by Downes et al (1997) This approach treats quantum dots as inclusions in an infinite matrix, and elastic isotropy and uniformity of elastic properties in both the quantum dots and matrix are assumed The atomistic approach often uses the valence force field (VFF) model with the Keating potential (Jiang and Singh, 1998; Kikuchi et al, 2001) And lastly, the numerical continuum approach basically utilizes the finite difference method (FDM) as shown by Grundmann et al (1995) and the finite element method (FEM) (Benabbas et al, 1999; Muralidharan, 2000, Liu and Quek, 2002, 2003) whereby geometrical symmetry of island shapes is assumed and the strain energy of the structure

is minimized In this project, the author uses the FEM to calculate the elastostatic fields

in a quantum dot heterostructure

Evolution of Quantum Dot Island

The surface evolution simulation is initially motivated by the need to understand thin film growth and defect formation in micro-electronic circuits Flat and defect-free surfaces are usually required for microelectronic semiconductor devices Therefore earlier research in numerical simulation or roughening thin film surfaces deals mainly with the motivation of preventing surface roughening in microelectronic devices Interestingly though, the mechanisms for forming the “unwanted” rough surface is actually the same for forming coherent nano-scale islands called quantum dots For typical conditions in the growth of quantum dots from heteroepitaxial films, the dominant mechanism of mass transport is by surface diffusion From this approach, several researchers have numerically simulated the morphology evolution of the quantum dot-growing layer

Gao (1991 a, b) formulated a first-order boundary perturbation method for the dimensional elasticity problem of an elastic inclusion embedded in an infinite dissimilar material His perturbation method sees applications in elastically induced morphological perturbations of surfaces, interfaces, voids, precipitates and inclusions in a stressed solid

two-He demonstrated that for a traction-free surface under a laterally applied bulk stress, the material surface becomes unstable and roughens His perturbation method was then applied in solving for stress-driven surface roughening in heteroepitaxial thin film structure (Gao, 1994)

Freund (1994, 1995) also performed numerical studies on the evolution of roughness on the surface of strained elastic material He solved the surface diffusion equation to obtain the normal velocity (and hence the normal displacement) of the evolving surface Freund also uses Gao’s (1991a, b) perturbation approach to approximate the change in surface strain on a wavy surface He demonstrated the phenomenon of instability of a flat, strained surface when it is perturbed

Zhang et al (1999) formulated a three-dimensional finite element method to predict surface evolution as a result of surface diffusion From the governing equation for surface diffusion, the weak form is derived and the finite element procedure is applied

In the weak form, the equation is stiff due to the curvature term and therefore they employed a semi-implicit Euler scheme, which allows larger time steps to be taken They had used a modified plate-bending element to interpolate the values of the change in displacement They demonstrated their method firstly on the evolution of voids in an interconnect of a microelectronic circuit which is induced by strain and electomigration Secondly they also applied their method to the island evolution in thin films Their studies showed good comparison with Gao’s (1991 a, b) and Freund’s (1994, 1995) perturbation approach From their studies using the finite element method, Zhang provided much insight into the formation of quantum dot islands in heteroepitaxial thin films (Zhang and Bower, 1999)

There are other researchers who did work in simulating the surface evolution of quantum dot surfaces Norris and Vemula (1998) used a coupled finite element and finite

difference method to study crevice formation in thin plates by stress driven diffusion The governing equations are similar to that of the heteroepitaxial surface The kinetic Monte Carlo atomistic model is also a common tool used to investigate the kinetics of thin film growth as shown by Gilmer et al (1998)

2 FINITE ELEMENT FORMULATION FOR ANALYSIS OF QUANTUM DOT HETEROSTRUCTURES

2.1 The SK Growth Mode, Stress Relaxation and Vertical

Correlation

There are generally three well-known hetero-epitaxial growth modes: a layer-by-layer growth called Frank-van der Merwe (FvdM) growth, a 3D island growth called Volmer-Weber (VW) growth, and lastly, the layer-by-layer together with 3D island growth called Stranski-Krastanow (SK) growth Detailed theories of these growth modes can be found

in most crystal and epitaxial growth textbooks (Example: Markov, 1995) This section will give a brief introduction to the concepts involved since it is of high relevance to the explanations of the results obtained

For lattice-matched systems, the growth mode is governed solely by the interface energy and the surface energy If the sum of the epilayer surface energy and the interface energy

is lower than the energy of the substrate surface, then the FvdM growth mode will occur This is commonly known as wetting Having high interface energy may cause a transition from the FvdM growth mode to the VW growth mode However, for lattice-mismatched systems with small interface energy, there is mismatch strain energy present

in the epilayer The initial growth may be a layer-by-layer growth when the substrate material has a higher surface energy than the epitaxial layer deposited But as the thickness of the layer increases above a critical thickness, the higher surface energy of the

substrate material is no longer felt Therefore, it no longer has any advantage in continuing to wet the substrate and as the strain energy exceeds the surface energy, islands will be formed to lower the energy in which the strain will be relaxed Thus, the

SK growth mode occurs

The strain relaxation by island formation without considering the surface energy can be visualised in a schematic way in Figure 2.1 Initially the depositing atom (A) will wet the substrate However, due to the difference in lattice parameter, there will be compressive strain induced on the deposited layer whereas at the same time, a tensile strain will occur

at the top of the substrate The next depositing atom (B) will choose to be deposited on top of A since depositing on the substrate will incur higher strain and hence higher energy This stacking of atoms results in the formation of an island

A

B

A

Figure 2.1 Schematic diagram of epitaxial strain relaxation via island formation

As alternating layers of the materials are grown, one will notice that under certain conditions, there will be a vertical correlation or anti-correlation in the growth of the islands That is to say that as subsequent wetting layers are grown, there will be a

tendency for the islands to either form directly above those at the bottom layer or to form

in other location(s) away from the buried islands This can be explained by the stress induced by the presence of the islands buried under the cap layer This induced stress modifies the original stress caused by the lattice mismatch to create potential wells on the quantum dot-growing surface In other words, the strain energy on the surface of the growing wetting layer will have minimas induced by the stress interaction of the buried islands

A typical deposition technique like the MBE is controlled by surface diffusion and it is driven by a gradient in the surface chemical potential The strain energy on the surface of the wetting layer is a contributor to the surface chemical potential and the resulting minimum potentials on the dot-growing surface implies that migrating adatoms will tend

to migrate to these minimum potential sites and thus form a spatial correlation with reference to the locations of the buried islands Hence, this provides a sound way of producing quantum dot superlattices, which gives interesting electronic and optical properties Further details of the above-mentioned correlation phenomenon can be found

in papers authored by Xie et al (1995), Benabbas et al (1999), Tersoff et al (1996), Strassburg et al (1998), Shchukin et al (1998, 2001) and Springholz et al (1998, 2002)

2.2 Finite Element Modeling

The finite element method is a numerical approach based on continuum material Continuum mechanics can be considered to be an examination of what is going on within

a body assumed to be a continuous matter with continuous properties without going into detail about the forces and motions of the atomic constituents However since quantum dots are typically in the nanometer scale that is very much the length scale of atoms, the question of the validity of using the finite element method or other continuum approach arises It is beyond the scope of the dissertation project to justify the use of the finite element method with experimental comparisons but nevertheless, in a paper by Carlsson

et al (1998), a detailed comparison between the strain state of uncovered quantum dots

obtained from electron microscopy and finite element calculations was carried out The results show that finite element calculations on free-standing quantum dots correspond closely to values measured by high resolution transmission electron microscopy (HRTEM) This paper therefore endorses the use of continuum material properties for nano-scaled systems, which before that, the validity is often just assumed

In this project, both two-dimensional (2D) and three-dimensional (3D) finite element models were used to carry out stress and strain analysis in and around the quantum dot nanostructures As mentioned, stress and strain analysis in and round the quantum dot is highly of interest to researchers due to the fact that it affects the spatial correlation of quantum dots when multiple hetero-epitaxial layers are grown and it is often required in the calculation and study of the opto-electronic properties of quantum dots

2.2.1 2D Axi-symmetric Model

Numerous studies using the FEM on quantum dot structures used 2-D axi-symmetric

model (Benabbas et al, 1999 and Muralidharan, 2000) Though it may not be fully

representative of the true shape of the island (for example that of a pyramidal shape for

InAs islands grown on GaAs as estimated by Nabetani et al, 1994), it has been found that

the general results do not differ very much Therefore, it may be sufficient for certain cases to just do a simpler, 2D model The concepts involved for the 2D as well as the 3D model are the same The main difference lies in the extra effort in modeling

Figure 2.2 shows a schematic diagram of a basic quantum dot unit that will be modeled The pyramidal shape for the island is used in the modeling The wetting layer together with the developed quantum dot island depicts a typical SK growth mode as mentioned in the previous section The wetting layer and the quantum dot are usually of a different material (hence different lattice parameter) from the substrate and the cap layer, therefore for convenience, the quantum dot together with the wetting layer will be collectively termed as the island, whereas the substrate together with the cap layer will be termed as the matrix