Computational Study of Confined States in Quantum Dots by an Efficient

Finite Difference Method

A THESIS

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTA

BY

Salman Butt

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

Dr. Jing Bai

January 2010

© Salman Butt 2010

i

Acknowledgements Page

Acknowledgements

First of all, I’d like to thank my adviser Dr. Jing Bai to give me an opportunity to

work on this thesis project and to guide me throughout the course of this research. I’d

also like to thank Professor Mohammed Hasan and Professor Jonathan Maps to be

my committee members. Also, I’d like to thank Professor Imran Hayee for the TA

support and advice to help me achieve my goals towards my degree. Next, I’d like to

thank Professor Jiann-Shiou Yang for his support and advice as a professor and the

department head, Professor Scott Norr for his understanding and support during my

final semester as GTA and GRA, Professor Stanley Burns for his support, Professor

Tom Ferguson for his advice and support as well as the rest of the ECE faculty. In

addition, I’d like to thank our ECE office staff, Shey Peterson and Kathy Bergh for

their great support and always being available to answer questions. I’d like to thank

Marvin for his technical support in setting up our workstation and providing the

software(s) needed for our research. Specially, I’d like to thank the financial support

from Grant-in-Aid research award (PI: Dr. Jing Bai) from the graduate school of UM

and Dr. Jing Bai’s start-up fund from UMD. Not to forget, thanks to my colleagues

for their friendship and support. Finally, I’d like to thank my family very much

whose everlasting support gave me great motivation towards completion of my thesis

and my degree.

ii

Dedication

This thesis is dedicated to the UMD ECE dept.

iii

Abstract

Semiconductor quantum dot systems have gained more attention in quantum computation

and optoelectronic applications due to the ease of bandstructure tailoring and three-

dimensional quantum confinement. Thus, an accurate solution of energy bandstructure

within the quantum dot is important for device design and performance evaluation. In this

paper, the solutions of bandstructures of quantum dot systems are presented by

implementing finite difference technique. To illustrate our analysis procedure, various

configurations of quantum dot systems were taken into account. In order to improve the

calculation efficiency of the finite difference solution in terms of time and memory

consumption, uneven divisions for the quantum dot confinement region were used. In

addition, we identified the optimum combination of divisions for each geometrical

configuration. Eventually, the eigenstate wavefunctions and eigenvalues were obtained

by directly solving the eigen-value problems. Overall, the generated results agreed

consistently with the published results obtained by other solution techniques.

Keywords: quantum dots, confinement states, finite difference method

iv

TABLE OF CONTENTS

List of Tables...………….…………………….………………………………..….……vi

List of Figures …………….…………………….………………………………..….…vii

Chapter 1: Introduction ….…………………….………………………………..….……1

1.1. Background ………..…………………………………………………1

1.2. Objective ………………….………………………………………… 2

1.3. Scope ……………….……………………………………………….. 3

Chapter 2: Literature Review…......…………………………………………..………… 5

2.1. Structure and Applications of Quantum Dots …...…………….……..5

2.2. Structure and Applications of Quantum Dot Superlattice System…...9

2.3. Modeling strategies of QDs and QD-SL ……………………..…..... 13

2.3.1. Modeling strategy for individual & coupled QDs …..….……...13

2.3.2. Modeling strategy for QD-SL …………………..………..…… 13

Chapter 3: Theoretical Background ………………….……………..………..…….......15

3.1. Schrödinger equation for quantum system….………………….…...15

3.2. Finite Difference method ………………..….……………………... 16

Chapter 4: Finite Difference Solution for Quantum Well Systems …………….……...18

4.1. Solution Procedure ……………………………………..………...…18

4.2. Results and Discussions ……………………………..………......… 19

Chapter 5: Finite Difference Solution Procedure for Quantum Dot System ……….… 23

5.1. Finite Difference formulation for quantum dot system……………..23

5.2. Quantum Dots in cubic shape ………………………..……………..28

5.2.1. Physical configuration and finite difference mesh …………….28

v

5.2.2. Skills Used …………………………..………………….…….. 29

5.2.3. Convergent Eigen values & Optimum Combination Processes..29

5.2.4. Results …………………………………………...………….....29

5.3. Quantum Dots in cuboid shape ………….….………..……………. 32

5.3.1. Physical configuration and finite difference mesh …………….32

5.3.2. Skills Used …………………………..………………….……...33

5.3.3. Convergent Eigen values & Optimum Combination Processes..33

5.3.4. Results ………………….…………………….......…………....33

5.4. Quantum Dots in pyramid shape …….…….………..……………...35

5.4.1. Physical configuration and finite difference mesh …………….35

5.4.2. Skills Used …………………………..………………….…….. 39

5.4.3. Convergent Eigen values & Optimum Combination Processes..40

5.4.4. Results ………………….……………………...……………....42

5.5. Discussion on CPU Usage: Strategies for Limited Memory……….49

Chapter 6: Quantum Dot Superlattice …...……………………………….…………... 50

6.1. Solution Procedure …………………….………………….………. 50

6.2. Results ….…………………………………………………………. 51

Chapter 7: Conclusions & Future Recommendations …………..….…………...….… 55

vi

LIST OF TABLES

Table 1: Analytical Comparison for Double Pyramidal QD Results …………………. 46

vii

LIST OF FIGURES

Figure 2.1. Schematic bandstructure of a quantum dot ……………….…….…….…….6

Figure 2.2. Electronic state transition in QD lasers ……………….…….…….………...7

Figure 2.3. Semiconductor Fluorescence …..…………..…………….…….…….…….. 7

Figure 2.4. Undoped QD-SL grown on substrate ……..………..….....…….…….……..9

Figure 2.5. Cross-section of undoped QD-SL grown on substrate ……....….….…...….10

Figure 4.2. 1st 3 Eigen States for Single QW with 100 meV………….…….………..... 19

Figure 4.3. 1st 3 Eigen States for Double QW with 100 meV.………….………………20

Figure 4.4. 1st 3 Eigen States for Double QW with 520 meV….……….….………...... 21

Figure 4.5. 1st 6 Eigen Functions for Double QW with asymmetric barrier …...………22

Figure 5.6. 3D Mesh for Single Cubic QD ………………………..………….………..28

Figure 5.7. 1st Eigen State Wavefunction for Double Cubic QDs ..…..………………..30

Figure 5.9. 3D Mesh for Single Cuboid QD ..……………………..………….………..32

Figure 5.10. 1st Eigen State Wavefunction for Double Cuboid QDs ..……………….....34

Figure 5.11. 3D Model for Pyramidal QD in cuboid matrix…...…………………......... 35

Figure 5.12. Mesh for Single Pyramidal QD and Barrier cross-section………..….........36

Figure 5.13. Enlarged View of Mesh for transition region Single Pyramidal QD….......37

Figure 5.14. Mesh for Double Pyramidal QDs and Barrier cross-section ………….......38

Figure 5.15. Convergent Test Results for Double QDs ………………………..….........41

Figure 5.16. 1st Eigen State Wavefunction (xz) Double Pyramidal QDs ……………… 43

Figure 5.17. 1st Eigen State Wavefunctions (xy) Double Pyramidal QDs …………..… 44

Figure 5.18. Graphical Comparison for Single & Double Pyramidal QD Results …......47

Figure 5.19. Graphical Comparison for Double Pyramidal QDs Results ………..……..48

viii

Figure 6.6. QD-SL: Electron miniband vs Cubic QDs (Interdot distance fixed)..……...52

Figure 6.7. QD-SL: Electron miniband vs Interdot distance (QD size fixed) .……..…. 53

Figure 6.8. QD-SL: Density of States vs Energy levels ………….…..……..………… 54

1

Chapter 1. Introduction

1.1 Background

Quantum dots (QDs) are commonly known as “artificial atoms”, since they provide the

opportunity to control the energy states of carriers by adjusting the confinements in all

three spatial dimensions. The QD superlattice (QD-SL) structure consists of multiple

arrays of QDs as more information is provided below. Quantum dot structures have

extensive applications in quantum computations and optoelectronic applications/devices

[1]. The QD-SL structure has been attractive in making the intermediate-band solar cells

(IBSCs) [2-4]. The maturity of crystal growth technologies, such as molecular beam

epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD), has facilitated

the growth of quantum dots with various geometries for a wide selection of material

systems.

In order to further explore the potential of QD structure and increase performance of QD

devices, understanding of confined energy states in QD structures is necessary. Thus,

accurate and efficient solutions for confined states in single-QD and double-QD systems

are pivotal in providing constructive details to facilitate the understanding of the more

complex structure of QD arrays. In this study, both the confinement of individual QD

structures and the miniband structure formed by QD-SL are investigated. The

confinement states for an individual QD and two coupled QDs are investigated through

the finite difference solution approach for the time-independent Schrödinger Equation.

The solution is generalized from the solution of one-dimensional confinement of

quantum well (QW) structures, which has been well reported in the literature [5-7].

2

Moreover, the miniband confinement of QD-SL structure is solved from Envelope

Function Approximation method.

1.2 Objective

The objective of this research is to solve for confined Eigen state energies as a result of

computations and simulations within QD structures to develop further know-how for the

more practical QD-SL systems. Among all the theoretical and numerical solutions of the

1D and 3D Schrödinger’s equations, the finite difference method (FDM) provides an

efficient and a convenient way in treating QDs with various complex geometries, which

can best approximate the practical shapes of QDs (pyramid or diamond) from fabrication.

In principle, the FDM ends up with the solution of Eigen-value problems. Here, the

eigen-values correspond to the Eigen energy levels of the confined states and the Eigen-

vectors correspond to the eigen state wavefunctions. Due to the spanning in 3D, large

memory usage in the computation brings challenge when deriving the solution of FDM

for QD systems. In our current work, we develop a mesh strategy for efficient memory

usage in solving QD system by using the 3D finite difference methodology. In principle,

besides directly solving the Eigen-value problems, we reduce the memory usage through

division optimization in 3D along with application of dissimilar and uneven divisions

between the dot and barrier matrix regions. Our simulator results agree well with results

presented in other sources. In summary, the technique presented here is applicable to

structures with limited number of QDs. For systems which constitute a larger number of

packed QDs i.e. the QD-SL structure, different solution procedures such as the 3D

Kronig-Penney model and the Envelope Function Approximation methods are applied.

3

1.3 Scope

The scope of this thesis is to present solutions based on results from implementation of an

efficient meshing strategy in 3D FDM solution for configurations comprising of

vertically and horizontally aligned coupled QDs. Following the first chapter of our paper

i.e., the introduction, the rest of the thesis is arranged as follows. In Chapter 2, we discuss

the literature which includes the review on the QD and QD-SL structures and their

practical applications. In addition, existing modeling strategies in the literature for the

similar QD systems are reviewed. Such information based on analysis of results derived

from other resources has proven to be useful in laying a basis for our current work. In

Chapter 3, we present the basic theories i.e. the theoretical formulations for the QW and

QD structures. The Finite Difference Method solution procedure for time-independent

Schrodinger equations is outlined. In Chapter 4, we present the QW structure by outlining

the solution procedures and the results derived from simulations for QW systems with

single and multiple barriers. Even though the solution of 1-D QW structure by FDM has

been very well studied in the literature, we take this as the starting-point of our model and

then expand it to 3D for QDs. In Chapter 5, we explain the solution procedures and the

generated results for various QD configurations of vertical and horizontal alignment of

QDs with cubic, cuboid and pyramidal geometries. Details for the CPU usage are also

discussed along with highlighting the various strategies implemented to utilize less space.

In Chapter 6, we focus on the solution of the QD-SL structure derived from the Envelope

Function Approximation method.

4

In Chapter 7, we conclude our thesis by summarizing the results obtained from the work

along with future recommendations to further expand the scope of our work.

5

Chapter 2. Literature Review

In this chapter, we introduce the structures for quantum dots and quantum dot

superlattice, as well as their applications in optics and electronics. Also, we survey the

current existing modeling strategy for the quantum dot systems.

2.1 Structure and Applications of Quantum Dots

The term "Quantum Dot" was coined by Mark Reed [8]. They were discovered by Louis

E. Brus [8]. Typically, quantum dots are small regions defined in a semiconductor

material with electrons confined to all three spatial dimensions and a size of order

10~100 nm [9]. Resultantly, the properties embedded within them are between the ones

for bulk semiconductors and discrete molecules [10-13]. Since the first studies in the late

1980’s, the physics of quantum dots has been a very active and fruitful research topic

[14].

The name “dot” suggests an exceedingly small region of space [14]. A semiconductor

quantum dot, however, is comprised of roughly a million atoms with an equivalent

number of electrons. Virtually all electrons are tightly bound to the nuclei of the material,

however, the number of free electrons in the dot can be very small; between one and a

few hundred. The deBroglie wavelength [15], or matter wavelength of these electrons is

comparable to the size of the dot and the electrons occupy discrete quantum levels (akin

to atomic orbitals in atoms) and have a discrete excitation spectrum. A quantum dot has

another characteristic, usually called the charging energy, which is analogous to the

ionization energy of an atom. This is the energy required to add or remove a single

electron from the dot [14]. The quantum confinement states [16] are states consisting of

6

energy confined to three dimensions inside a dot as the dot is within range of less than

100 nm scale, hence minimized to a unit of physical entity i.e. quantum.

Practically, quantum dots have proven to be useful systems to study a wide range of

electronic and optical applications. Semiconductor lasers are ubiquitous in modern

society and play a key role in technologies ranging from CD players to optical

telecommunications [17]. These QD lasers with an atomic like density of states are

expected to show ultra low-threshold current densities, ultrahigh temperature stability of

threshold current, ultrahigh differential gain increase, cutoff frequency and chirpfree

operation under direct current modulation. Potential device applications therefore range

from high-power semiconductor lasers to high-speed light sources for fiber-based data

transmission [18]. Below is a schematic (Fig.2.1) [18] illustrating how the QD lasers

work:

Based on Figure 2.1 above, a 3-D-array of dots vertically aligned along the growth

direction which is formed during the growth of multiple QD layers is illustrated

schematically. The ideal QD laser consists of a 3-D-array of dots with equal size and

Figure 2.1. Schematic bandstructure of a quantum dot

(laser

7

shape surrounded by a higher bandgap material which confines the injected carriers [18].

Figure 2.2 below further illustrates the electronic state transition among the QD confined

energy levels in QD lasers.

Also, semiconductor fluorescent quantum dots are nanometer-sized functionalized

particles that display unique physical properties making them particularly well suited for

visualizing and tracking molecular processes in cells using standard fluorescence

microscopy [19-21]. Below is Figure 2.3 explaining how the semiconductor fluorescence

works [22]:

Figure 2.3. Semiconductor Fluorescence

Figure 2.2. Electronic state transition in QD lasers

http://www.lce.hut.fi/publications/annual2001/a0451x.gif

.

http://www.fluorescence-foundation.org/lectures/chicago2009/lecture10.pdf.

8

In addition, semiconductor nanocrystal or quantum dot flourophores offer a more stable

and qualitative mode of fluorescence in situ hybridization for research and clinical

applications [23]. Although, semiconductor nanocrystals have been investigated for

several decades [23-25], compatibility of these minute crystals with interesting electro-

optical properties has only recently been demonstrated in biological experiments. Other

applications of quantum dots include nanomachines, neural networks, high-density

memory, LEDs and diode lasers. As far as nanomachines go, the electron transfer rates

are computed from the free energy change for a single electron transfer to or from a

quantum dot of size such that only charge quantization matters. For a small enough

dot,

the nanomachine device could operate at room temperature [26]. In neural networks, the

neural model system is that of a quantum dot molecule [27, 28] with five dots arranged as

the spots on a playing card [29]. Also, metal nano-dot (MND) films consist of a thin

oxide film that includes high density metal dots with nano-scale (quantum) [30]. In

addition, quantum-dot LEDs (QD-LEDs) are fabricated that contain only a single

monolayer of QDs, sandwiched between two thin films [31]. Lastly, InGaN QDs have

proven to be useful for localization of carriers in purple laser diodes [32].

9

2.2 Structure and Applications of Quantum Dot Superlattice system

QD superlattices are artificial crystals whose building block is a QD [33, 34]. In other

words, QD superlattices are multiple arrays or stacks of QDs [35] aligned vertically and

horizontally in all the three spatial dimensions. High quality QD-SLs are made of

InGaAs/GaAs where GaAs is used for barrier material. Other materials for QD-SL

include Ge/Si. Following are graphs (Figure 2.4 and Figure 2.5) [4] that illustrate the

structure of the QD-SL:

Figure 2.4. Image of an undoped InGaAs/GaAs (20 nm) QD SL grown

on GaAs substrate.

10

Basically, superlattices of nanometer- sized QDs can be generated either in solution as

the crystallization of a monodisperse colloid or at a solid or liquid interface as a thin,

ordered superlattice of dots [36]. Self-assembled and self-organized Ge/Si quantum dot

(QD) superlattices (SLs) have been grown by a solid-source MBE system with the

Stranski-Kranstanov (SK) growth mode [37]. Under the SK growth mode, the formation

of quantum dots is driven by the strain during epitaxy growth of InGaAs on a GaAs

substrate as the deposited layer exceeds a critical thickness [38]. In MBE, the growth

temperature is kept around 500oC with InGaAs or InAlAs used as the dot material while

GaAs and AlGaAs as the cladding (covering) layers [39]. Meanwhile, in the MOCVD, as

Figure 2.5. Cross-section image of an undoped InGaAs/GaAs (20 nm) QD

SL grown on {113}B GaAs substrate.

11

in other growth techniques that involve molecular species, surface reactions play an

important role [39]. Besides, various geometric structures for QDs have been

implemented within the QD-SL systems. Arrays of cubic and pyramidal QDs are

common for the typical QD-SL configuration. Such intricate configurations and

structures have recently been studied as their complexities have led to a wide array of

potential practical applications. Among them, high efficiency thin-layer solar cells are

very well-known. Solar cells have been the main focus recently for practical purposes of

energy conversion and as a means of providing renewable energy. Such novel

semiconductor structures have been studied and researched as they are also referred to as

the IBSC, a generic class of photovoltaic devices [2]. Basically, the way the QD-SL solar

cells work is that inside them, the QDs are modeled as a regularly spaced array of equally

sized dots in the respective matrix. Incorporating the effect of silicon’s anisotropic

effective mass is shown to reduce both the degeneracies of the isotropic solutions and the

energy separation between states [3]. These photovoltaic arrays have been in demand in

recent times due to interest in cost-effective solutions for renewable energy conversion.

In addition to the solar cells, collective electronic phenomena have been predicted for the

three-dimensional ordered superlattices of QDs [40]. Some of these collective electronic

phenomena include electrical transport and observation of macroscopic properties of QD-

SL solids. Also, far-infrared radiation can directly induce optical transitions between the

meV energy levels confined in the quantum dots. Known to be narrow band gap

semiconductor, QDs have been found to exhibit a well‐defined structure and have

near‐unity quantum yield with emission wavelength between 1.2 microns and 2 microns

determined by the particle size. Thus QDs, especially PbSe, have potential for application

12

as a monochromatic infrared light source [41]. To achieve a detectable signal requires dot

arrays with active sample areas on the order of 10 mm2 [42]. One of the requirements for

technological applications is a high spatial density of dots. Toward this end, several

groups have grown quantum-dot superlattices. For example, alternating growth of GaAs

and strained InAs yields layers of InAs dots embedded in GaAs. A fascinating feature of

such structures is that the dots in successive layers are spatially correlated [43-47]. Each

of the 6 citations mentioned here has a particular growth technology used. In the first one,

the growth of multilayer arrays of coherently strained quantum dots is investigated as a

simple model reproduces the observed vertical correlation between QDs in successive

layers. In the second case, the superlattices and thick QD layers grown on (100), (111),

and

(110) Si surfaces by MBE exhibit different growth

morphologies and defect

structures. The third citation mentions transmission electron microscopy for study of

microstructure of strained layers of InxGa1−xAs/GaAs grown by MBE. Based on the

fourth source, coherent InAs QDs separated by GaAs layers are shown to exhibit self-

organized growth along the vertical (i.e., growth) direction. Finally, the last citation

discusses multi-layer, vertically coupled, quantum dot structures investigated by using

layers composed of InAs QDs grown by MBE in the S-K growth mode. QD-SL structures

with distribution of density of states and discrete energy levels due to three-dimensional

quantum confinement provide the potential for better thermoelectric devices [48]. The

MBE growth of self-assembled QD-SL materials on planar substrates using the Stranski-

Krastanov growth mode yields improved thermoelectric (TE) figures of merit [49, 50].

Self-assembled quantum dot materials represent just one of a number of new approaches

[51] being investigated in order to enhance TE performance.

13

2.3 Modeling strategies of quantum dots and quantum dot superlattices

2.3.1 Modeling strategy for individual and coupled quantum dots

To develop a further practical understanding of the QD and the more complex QD-SL

structures, numerous modeling strategies and methodologies have been implemented to

solve for various significant factors including the confined eigenstate values. Such values

are integral as they are associated with energy values and eigenstate functions which help

in further understanding the properties imbedded within the material used for these

structures. For QD structures, finite difference method is used [52-54]. In order to save

memory in the solution procedure, the resulting eigen-value problem is transferred to the

solution a set of linear equations [52], provided that an appropriate perturbation value is

chosen. This perturbation has to be a non-zero value, otherwise the solution to this system

of linear equations would be uniformly zero. Another source implements the Jacobi-

Davidson [53] based method to solve for eigen-value problem after initially applying the

FDM [61]. Here, the Jacobi-Davidson method is used to compute the smallest possible

eigenvalues and the associated eigenvectors which involves the orthonormality and

orthogonality of the eigenvectors.

2.3.2 Modeling strategy for quantum dot superlattice

As far as the QD-SL systems go, similar modeling strategies are executed by

implementing the Envelope Function Approximation or Effective Mass Approach [3, 35].

This method involves assumption of electron potential to be a sum of three independent

14

periodic functions in orthogonal directions. Using this approach, the resulting potential is

separable into three independent components allowing carrier dispersion relation to be

obtained from the sum of solutions in each spatial direction. Here, multiple arrays of

cubic QDs are stacked together vertically and horizontally to form a QD-SL system

configuration. Models used are mathematically based upon the methods mentioned above

which are similar to the Kronig-Penney Model [55]. This model deals with single

electron moving in a one-dimensional crystal. It requires use of Bloch and hyperbolic

functions with product of the traveling wave solutions and a periodic function. That

function has the same periodicity as the potential. Overall, this model leads to both real

and imaginary solutions corresponding to the propagating and the forbidden electron

states. Also, as mentioned before, large memory consumption is the main problem for

computation of eigen values due to the spanning of the 3D structure. In our current work,

we present an efficient mesh strategy to save the memory in solving the eigen-value

problem resulting from 3D finite difference method. This method reduces the

computational memory in a different way compared to the method presented in

Harrison’s paper [52].

15

Chapter 3. Theoretical Background

3.1 Schrödinger equation for quantum system

The confinement states in a quantum system are analyzed through the Schrödinger

equation. The Schrödinger equation is as central to quantum mechanics as Newton's laws

are to classical mechanics. Solutions to Schrödinger's equation describe not only atomic

and subatomic systems, atoms and electrons, but also macroscopic systems [56].

The standard time-independent 1-D Schrödinger equation (3.1) [57-59] is as follows:

)()()(][2 2

22

xExxUdx

d

m

(3.1)

U corresponds to the quantum barrier potential, which is dependant on the spatial

dimensions, E represents the eigenenergy of the system energy, ψ symbolizes the

wavefunction of the corresponding eigenstate, ħ is the Planck’s constant and m is the

electron effective mass. The Schrödinger equation can be applied to solve the quantum

well system, where degree of electron freedom (Df) is 2 and degree of electron

confinement (Dc) is 1 as Df + Dc = 3 for all solid state systems [59].

For the electronic bandstructure of a quantum dot system, since the confinement is in 3D,

the confinement states can be described by the 3D time-independent Schrödinger’s

Equation (3.2) [58-60]:

),,(),,(),,(][2 2

2

2

2

2

22

zyxEzyxzyxUzyxm

(3.2)

16

Basically, different m was used for the QD and the matrix regions. For consistent

comparison with results derived from Harrison’s paper, same m was used for the QD and

matrix regions.

3.2 Finite Difference Method

The finite difference method approximates solutions to differential equations by replacing

derivative expressions with approximately equivalent difference quotients [63]. As far as

the derivation procedure for the finite difference method goes, the first derivative of is:

,)()(

lim)('0 x

xxxx

x

(3.3)

A reasonable approximation for that derivative would be:

,)()(

)('x

xxxx

(3.4)

for small value of Δx. This is known as the forward difference equation for the first

derivative. The approximation for the second-order derivative in FDM would be:

,)(

)()(2)()(''

2x

xxxxxx

(3.5)

which is important to solve the Schrödinger equation since the Schrödinger equation

involves the second-order derivative. The 1-D Schrödinger equation along the x-axis is as

follows, which can be used to solve for the 1-D quantum-well system:

)()()()(])(

)()(2)([

2 2

2

xxExxUx

xxxxx

m

(3.6)

17

Equation (3.6) can be expanded to 3-D in the (x, y, z) space as Eq. (3.7), which can be

used for solving the confinement states the quantum dot system:

)7.3(),,(),,(),,(),,(

])(

),,(),,(2),,(

)(

),,(),,(2),,(

)(

),,(),,(2),,([

2

2

2

2

2

zyxzyxEzyxzyxU

z

zzyxzyxzzyx

y

zyyxzyxzyyx

x

zyxxzyxzyxx

m

18

Chapter 4. Finite Difference Solution for Quantum Well Systems

4.1 Solution Procedure

The primary purpose for implementation of finite difference method on QWs is to

understand the finite different solution which helps in expanding the formulation to 3D

for more complex systems. In other words, solving for QW system helps explain the

behavior patterns of the electron along with the basic knowledge of the eigen state energy

pattern in 1D system.

Mathematically, the 1-D Schrödinger’s equation for FDM (3.6) as above is used for the

quantum well system. Matrix A (the term in brackets which is multiplied by –h2/2m in

Eq. (3.6)) along with rest of parameters for Eq. (3.6) are below:

ijjij

N

QW ExxUxAas

x

x

x

xA

)()()(

)(

)(

)(

,

21

...

121

121

12

)(

1 2

1

2 (4.1)

The barrier material used is AlInAs and the quantum well material is InGaAs.

Simulations were performed for both single and double QW along with position varying

electron effective mass and constant electron effective mass. For the single QW system,

the dimensions along x-axis for both the left and right barrier lengths were 25 nm and the

quantum well 50 nm. For the Double QW system, the dimensions for the left and right

barriers were kept at 25 nm while the middle barrier at 20 nm. Both the quantum wells

were restricted to 15 nm each.

19

4.2 Results and Discussions

In Figure 4.2 below, it is observed that eigenenergies are increasing based on n2 i.e. E2 ≈

22 E1 and E3 ≈ 3

2 E1. Also, it can be seen how the barrier widths affect the change in the

wavefunctions especially at the boundaries. Besides, the shapes of the wavefunctions

correspond correctly to the relative confined energy i.e. the 1st energy wavefunction has

extremum, the 2nd

has two extrema and the 3rd

has three. The extrema correspond to the

highest electron probability density. These characteristics of the wavefunctions

appropriately reflect the behavior of the particle(s) inside the quantum well.

n = 1

E2=4.326 meV

E1=1.087 meV

E3=9.680 meV

x (nm)

0

0

10 20 30 40 50 60 70 80 90 100

5

10

Ψ2(x)

Ψ3(x) n = 3

n = 2

Ψ1(x)

Figure 4.2. Results from Matlab for 1st 3 Eigen States for Constant Electron Effective

Mass in Single QW with potential of 100 meV

Ener

gy (

meV

)

20

In Figure 4.3 below, the first three relative eigen energies along with their respective

wavefunctions are displayed. The barrier potential is same as above but the well region is

reduced to 2 wells of 15 nm each with additional barrier region of 20 nm in the middle

which increases the eigen energies below. Also, the wavefunctions with even n are the

degenerate states for the respective odd n i.e. n=1&2 correspond to the same eigenstate.

Figure 4.3. Results for 1st 3 Eigen State Wavefunctions for Constant Electron

Effective Mass in Double QW with finite potential of 100 meV

10 20 30 40 50 60 70 80 90 100 0

0

2

4

6

8

10

12

x (nm)

n = 1 Ψ1(x)

n = 3 Ψ3(x)

n = 5 Ψ5(x)

E5=11.0436 meV

E3=4.9409 meV

E1=1.2401 meV

100 meV 100 meV 100 meV 0 meV 0 meV

Ener

gy (

meV

)

21

In Figure 4.4 below, similar behavior is observed for the wavefunctions except this time

the barrier potential has been increased to 520 meV.

Figure 4.4. Results for 1st 3 Eigen State Wavefunctions for Varying Electron Effective

Mass in Double QW with potential of 520 meV

E1=0.02256 eV

E3=0.09080 eV

E5=0.20511 eV 0.2

0.1

0

0 10 20 30 40 50 60 70 80 90 100

x (nm)

n=1 Ψ1(x)

n=3 Ψ3(x)

n=5 Ψ5(x)

B

(520 meV)

B

(520 meV)

B

(520 meV)

W

(0 meV)

W

(0 meV)

Ener

gy (

eV)

22

In Figure 4.5 below, the difference in the corresponding or relative wavefunctions can be

noticed due of the asymmetry of the barrier widths. The top two eigenfunctions represent

11st

eigenstate, the next two as 2nd

eigenstate and the bottom two for 3rd

eigenstate.

10 20 30 40 50

x (nm)

60 70 80 90 100 0

10 20 30 40 50

60 70 80 90 100 0

10 20 30 40 50

60 70 80 90 100 0

10 20 30 40 50

60 70 80 90 100 0

10 20 30 40 50

60 70 80 90 100 0

10 20 30 40 50

60 70 80 90 100 0

0.5

0

-0.5

0.5

0

-0.5

0.5

0

-0.5

0.5

0

-0.5

0.5

0

-0.5

0.5

0

-0.5

Ψ(x)

Figure 4.5. 1st 6 Eigenfunctions for Varying Electron Effective Mass in Double QW

23

Chapter 5. Finite Difference Solution Procedure for Quantum Dot

System

In this chapter, we present the finite difference formulation for the 3D quantum dot

system using even division lengths for cubic and cuboid geometries meanwhile using

uneven division lengths for the pyramidal geometry.

5.1 Finite Difference formulation of the quantum dot system

Equation (3.2) above is solved through the finite different method, in which the quantum

box of dimensions Lx, Ly and Lz can be discretized along each coordinate axis, with nx

divisions along x axis, ny along y axis and nz along z axis. For even division lengths, Δx =

Lx/nx, Δy = Ly/ny and Δz = Lz/nz were used. For uneven division lengths, the following

equations were used:

x1 n1: ii xwherexxfromrangingx , (5.1a)

y1 n1: jj ywhereyyfromrangingy , (5.1b)

z1 n1: kk zwherezzfromrangingz , (5.1c)

The indexes are defined in terms of i, j and k along x, y and z axes with ranges 1 ≤ i ≤ nx,

1 ≤ j ≤ ny and 1 ≤ k ≤ nz. Therefore, k-1 and k+1 represent the corresponding neighboring

points along z-axis, j-1 and j+1 along y-axis and i-1 and i+1 along x-axis.

Next, Equation (3.2) is transformed as follows:

24

.

111111

2

2

,,

,,,,

1

2

1

2

1

2*

2

1

1,,

2

1,,

1

,1,

2

,1,

1

,,1

2

,,1

*

2

kji

kjikji

kkkjjjiiie

kk

kji

k

kji

jj

kji

j

kji

ii

kji

i

kji

e

E

Uzzzyyyxxxm

zzzyyyxxxm

(5.2)

Here, *

em is effective mass of electron.

Equation (5.2) is then distributed in matrix A as follows:

AQD =

nznynxnznynx

ijkrkji

ijkpi

bjkbkj

ajkaj

kk

,,

111111

11

1111

111

22121

1111

000000

000000000

0000000

00000

000000

0

00000

000000

(5.3)

Here, α is varying, diagonally across the matrix A as

[

1

2

1

2

1

2*

2 111111

2 kkkjjjiiie zzzyyyxxxm

] since uneven

division lengths were used whereas for a uniform 3D mesh is ])(

1[6

2h where ∆h =

(nx * ny * nz) rows and columns

25

∆x = ∆y = ∆z. The rest of the corresponding non-zero column elements are represented

by neighboring points with values varying due to uneven division lengths i.e.

1

2

11

kkkzz

andz

fromranging (5.4a)

1

2

11

jjjyy

andy

fromrangingY (5.4b)

1

2

11

iiixx

andx

fromrangingX (5.4c)

relative to the z, y and x coordinates respectively.

Based on matrix A above, every diagonal element represents a particular index location

point of a 3D system. The rest of the column elements for each diagonal value represent

the relative neighboring point positions. For instance, the diagonal element from the 1st

row represents index location for a corner point in 3D. Corner points are recognized as

points with i = j = k = 1, i = j = 1 and k = nz, i = k = 1 and j = ny, i = 1 with j = ny and k =

nz, j = k = 1 and i = nx, j = 1 with i = nx and k = nz, i = z = 1 and j = ny and finally i = nx

with j = ny and k = nz. Therefore, there would be a total of 8 corner index location points

in a 3D barrier box. An example of a corner point would be (1,1,1). The corresponding

2nd

column element would represent the neighboring point relative to k+1 i.e. (1,1,2). The

(nz + 1)th

column element represents the neighboring point relative to j+1 i.e. (1,2,1).

Similarly, the (nz ny + 1)th

column element represents the neighboring point along x-axis

corresponding to i+1 i.e. (2,1,1). Hence, there would be 3 valid neighboring points for the

corner index location. In another case, the diagonal element of the (nz + 1)th

row would

be a representation of an index location point on the edge. Here, one of the edge points

26

would be when i = 1, 1 < j < ny and 1 ≤ k < nz e.g. (1,2,1). For such location point, the

neighboring points along y-axis would be j-1 and j+1 i.e. (1,1,1) and the (2nz + 1)th

column element i.e. (1,3,1). The neighboring point along x-axis would be i+1 i.e. (2,2,1)

which is the [(nz . ny) + nz + 1]th

column element. Also, the neighboring point along z-axis

corresponds to k+1 i.e. (1,2,2). Therefore, in the case of an edge index location, there will

be 4 valid neighboring points. In addition, an example of a surface index location point

would be the diagonal element of the (nz + 2)th

row. Here, a surface point is when i = 1, 1

< j < ny and 1 < k < nz e.g. (1,2,2). This time, there are 2 valid neighboring points with

respect to the z-axis i.e. k-1 and k+1 which would be (1,2,1) and (1,2,3) respectively. The

rest of the j+1, j-1 and i+1 neighboring coordinates will be one column element to the

right with respect to the (nz + 1)th

row i.e. (1,3,2), (1,1,2) and (2,2,2) respectively. Hence,

a total of 5 valid neighboring points in the case of surface index location point. Finally,

any inside location point will always be surrounded by 6 valid neighboring points. An

inside point would be when 1 < i < nx, 1 < j < ny and 1 < k < nz. An example would be

(2,2,2) which can be represented by the diagonal element of the (nz ny + nz + 2)th

row.

Here, the neighboring points along x-axis will be i-1 and i+1 i.e. (1,2,2) and (3,2,2).

These would be the (nz + 2)th

and [2 . (nz . ny) + nz + 2]th

column elements respectively.

The [(nz ny) + 2]th

and [(nz ny) + 2 nz + 2]th

column elements would represent the

neighboring points along y-axis i.e. (2,1,2) and (2,3,2). And, the location points along z-

axis would be z-1 and z+1 i.e. (2,2,1) and (2,2,3). These would be represented by the [(nz

ny) + nz + 1]th

and the [(nz ny) + nz + 3]th

column elements respectively.

Later, the potential values are distributed diagonally among elements of a separate matrix

with VD = 0 meV designated for QD index location points and height of VB = 276 meV

27

(used in the case of pyramidal QD) for Barrier index points. This matrix is significant as

it shows what potential is used for a particular barrier. Importantly, because of the QD

geometry, all the middle diagonal values are not VD but actually distributed based on the

barrier and quantum dot location or index points. To explain more clearly, all the index

location points starting from (1,1,1) till the index point before QD region were distributed

diagonally with value of VB. All the points after the last index location point of the QD

region till the last index location point of the Barrier box i.e. (nx , ny , nz) were also

diagonally assigned the value of VB. For the matrix elements representing the middle

region including both the QD and MBL (Middle Barrier Length) regions, all the diagonal

elements of the potential matrix were initially set to VB. Then, those middle diagonal

elements were systematically chosen by matching with the QD region index location

points and accordingly assigned with value of VD. The general potential matrix is as

follows:

B

B

BD

B

B

V

V

VorV

V

V

V

00

000

000

000

000

(5.5)

28

5.2 Quantum Dots in cubic shape

5.2.1 Physical configuration and finite difference mesh

For cubic quantum dot, the material for the dot region was selected as InGaAs. Because

of even symmetry in shape, the dimensions used for the dot were similar for all the three

coordinates. The 3D mesh structure as follows was generated to clearly explain and

illustrate the geometry for the cubic QD. The mesh shown below is not up to scale based

on the calculations below as a cross-section of the actual diagram is presented to show

the mesh lattice structure and the geographical placement of the cubic QD. Information

regarding the division lengths, memory and the calculation time are presented later.

Figure 5.6. 3D Mesh for a Single Cubic QD

35 35

25

15

30

20

10

5

35

30

25

20

15

10

5

0

z

25 15

30 20

10 5

x

y

29

5.2.2 Skills Used:

Even division lengths were used between dot and barrier region.

5.2.3 Convergent Eigen values and Optimum Combination Processes

Please see section 5.4.3 below.

5.2.4 Results

Following are results generated from simulations for coupled or double cubic QDs

aligned along x-axis. Due to similar and consistent geometrical shape along all three

axes, the results are similar mathematically for Double cubic QDs aligned along z-axis as

well. The parameters used for this computation were: divisions combination of 36x36x36

for nx ny nz, barrier box dimensions of 24x24x24 nm3, cubic QDs dimensions as 6x6x6

nm3, barrier potential of 520 meV, dot material as InGaAs i.e. effective mass constant of

0.043. Division lengths for the dot region were ≈ 0.35 nm/div and ≈ 0.7 nm/div for the

barrier region. Memory utilization was less than 11 MB and calculation time less than 1

hour.

30

Resultantly, the Code Output Eigenstate Results are as follows:

Figure 5.7. 3D Simulation Results for Double Cubic QDs: 1st Eigen State Function

Figure 5.7. above shows 2 sets of peaks which is expected for a small MBL between

double QDs. Also, the results provide a statistical comparison between the code-

generated ground state eigenvalue and the theoretical eigenvalue based on the analytical

equation (5.8) from Paul Harrison’s text [59]. The code generated ground state

eigenvalue is 335.8572 meV and the analytical eigenvalue as 335 meV:

][2

22

2

2

2

2

222

,,

z

z

y

y

x

x

zyxL

n

L

n

L

n

m

(5.8)

0

-0.005

-0.010

-0.015

-0.020

-0.025

-0.030

-0.035

-0.040

0 0 3 6 9 12 15 18 21 24

3 6 9 12 15 18 21 24

y z

Ψ

31

Based on the results above, a 0.25% discrepancy statistically verifies the accuracy and

authenticity of the results derived from our programming code for QDs with cubic

configuration. Overall, we can see how the confinement energies change between the

QW and QD as such additional effect of the 3D confinement makes an influence since

the number of eigen energies changes hence affecting the eigen states as well.

32

5.3 Quantum dots in cuboid shape

5.3.1 Physical configuration and finite difference mesh

For the cuboid quantum dot, the material for the dot region was also selected as InGaAs.

The 3D mesh structure as follows was generated to clearly explain and illustrate the

geometry for the cuboid QD. The mesh shown below is not up to scale based on the

calculations below as a cross-section of the actual diagram is presented to show the mesh

lattice structure and the geographical placement of the cuboid QD. Information regarding

the division lengths, memory and the calculation time are presented later.

Figure 5.9. 3D Mesh for Single Cuboid QD

25

20

15

0

0

5 10

15 20

30 25

35

x

5

10

15

20

30

25

35

y

z

33

5.3.2 Skills Used:

Even division lengths were used between dot and barrier region.

5.3.3 Convergent Eigen values and Optimum Combination Processes

Please see section 5.4.3 below.

5.3.4. Results

Following are results generated from simulations for coupled or double cuboid QDs

aligned along z-axis. Figure 5.10 show the eigen state wavefunctions generated. The

parameters used for this computation were: divisions combination of 36x36x36 for nx ny

nz, barrier box dimensions of 30x30x30 nm3, cuboid QDs dimensions as 12x12x6 nm

3,

barrier potential of 520 meV, dot material as InGaAs i.e. effective mass constant of

0.043. Division lengths for the dot region along x and y axis were ≈ 0.35 nm/div and ≈

0.2 nm/div along z axis. Division lengths for the barrier region were ≈ 0.8 nm/div.

Memory utilization was less than 11 MB and calculation time less than 1 hour.

34

Figure 5.10. 3D Simulation Results for Double Cuboid QDs: 1st Eigen State Function

z x

0

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06

0 4

8 12

16 20

24 28

0 4

8 12

16 20

24 28

Ψ

35

5.4 Quantum dots in pyramid shape

5.4.1 Physical configuration and finite difference mesh

As follows, the geometric structure and the mesh created for a pyramidal QD shows

clearly the uneven structure with square base. The QD region was divided into two parts

to achieve pyramidal geometry: square base points and remaining non-square base points

located above the square base region. Logically, the z coordinates are constant in the base

region whereas coordinates of all three axes vary inside the non-base part.

Figure 5.11. Matlab Simulation Generated Image displaying Pyramidal QD in cuboid

matrix structure

80

70

60

50

40

30

20

10

0

z

40 30

20 10

0

50

50

0 y x

5 10 20 15 25 30 35 40 45 50

Pyramidal QD in Cuboid Matrix Structure

L

H

W

36

The lattice illustrated in the mesh below clearly indicates the location points of the QD

with pyramidal structure within a grid of 520 Å x 520 Å x 650 Å using combinations of

46x46x60 divisions. Division size at convergence towards the dot region ranges from ≈

1.2 to 0.90 nm/div. The dimension size ranges from ≈ 0.70 to 0.90 nm/div which is

smaller than the converged division size.

Figure 5.12. Matlab Simulation Image displaying Mesh for Single Pyramidal QD

with cross section only of Barrier Mesh to focus mainly on the Dot Mesh

25

26

27

28

29

30

Lz (nm)

25 35

45

15

50 40

30 20

10 5 0

Ly (nm)

Lx (nm)

0

5

10

15

25

35

45

50

40

30

20

37

Due to symmetrical configuration, the concept for geometry of double cubic QDs was

similar to single cubic QD. In the case of double pyramidal QDs, it was very similar as

well to the single pyramidal QD which is also mentioned above. For specification

purposes, the widths used for Barrier Box were 52nm x 52nm x 65 nm or 520 Å x 520 Å

x 650 Å (x,y,z) whereas QD1 and QD2 pyramid dimensions were 14nm x 14nm base

with height 7nm and 12nm x 12nm base with height 6nm respectively. Division size at

convergence for larger dot ranges from ≈ 1.2 to 0.90 nm/div whereas for the smaller dot it

ranges from ≈ 1.1 to 0.80 nm/div. The dimension size for larger dot ranges from ≈ 0.70 to

0.90 nm/div whereas for the smaller dot ≈ 0.60 to 0.80 nm/div which is smaller than the

converged division size. Figure 5.14 as follows illustrates the geometry for Double QD

pyramids (14nm and 12nm) aligned vertically along the z-axis with MBL = 70 Å:

Figure 5.13. Enlarged view at the region between the dot and matrix region to show

the element size transition

38

Figure 5.14. Matlab Simulation Image displaying Mesh for Double Pyramidal QDs again

with cross section only of Barrier Mesh to focus mainly on the Dot Mesh

34

33

32

31

30

29

28

27

26

25

Lz (nm)

Ly (nm)

40 45 50

Lx (nm)

40

45

50

30

35

25

10

15

20

39

5.4.2 Skills Used

Uneven division lengths were used for the Barrier Box and the QD regions. The reasons

for applying uneven division lengths were primarily for achieving better accuracy in

computation using lesser divisions and for smoother transition of confined Eigen States

while satisfying boundary conditions. This was implemented by using approximately

twice the number and uneven divisions for quantum dot regions compared to that of the

barrier region. For instance, in the case of total of 46 x 46 x 60 divisions, there were

uneven division lengths for ∆x, ∆y and ∆z for the QD and barrier regions. For QD, LxD =

14 nm for the larger QD and LxD = 12 nm for the smaller QD were used. Hence, ∆x for

QD was kept within range of 0.70 nm/division and 0.90 nm/division whereas for the

barrier box, ∆x was close to 1.2-1.5 nm/division. Similarly for y-axis, LyD = 14 nm for the

larger QD and LxD = 12 nm for the smaller QD were used. Hence, ∆y for QD was kept

within similar range compared to ∆x. Also, uneven division lengths were applied for the

transition regions between the barrier and the QD regions. In the case of vertically

aligned QDs, the division lengths along the z-axis were not even also as uneven divisions

were distributed for the QDs while dealing with the MBL division lengths as well.

Overall, uneven division lengths were used along the three axes for the barrier regions

and the QD regions for attaining convergence and accuracy in Eigen state results while

using lesser divisions. Based on the uneven division sizes mentioned above, mesh

division sizes in Harrison’s paper were consistent and even ranging close to ≈ 0.98

nm/div. Also 53x53x53 elements were used according to his paper instead of our

46x46x60 elements. Our mesh is more efficient since it gives us flexibility and utilizes

smaller memory by using lesser divisions because of uneven division lengths.

40

5.4.3 Convergent Eigen values and Optimum Combination Processes

In general, by utilizing the latest MATLAB [62] software available (Version: 7.7.0. 471 –

R2008b), the process is initiated by executing code simulations using various

combinations of divisions. Numerous computations were necessary to accomplish

convergent Eigen values based on determining the optimum combination of divisions.

These combinations are a product of the number of divisions i.e. nx ny nz in all the three

spatial coordinates. Based on our CPU memory allocation, the product of these divisions

was kept less than or equal to the threshold number of 563 or 175616. This threshold

number was determined based on the capability of our machine with limited memory to

compute and generate Eigen state results without utilizing elongated time periods for

simulations. Also, various combinations were used systematically for each simulation to

achieve convergent results. For Double QDs aligned along z-axis, an assortment of

combinations was embedded and tested within our codes for each varying MBL (1-9 nm

or 10-90 Å). Such strategy is illustrated as follows (Figure 3) which shows Convergence

Tests for Double QDs with MBL = 1 nm. Combinations in nx ny nz format include

49x47x73, 49x48x73, 49x49x72 and 50x48x72 which correspond to 168119, 171696,

172872 and 172800 divisions respectively. These combinations were derived based on

the fact that the less significant nx and ny division values were chosen to be equivalently

low and while a greater value selected for the more pertinent nz. The overall product was

kept within the range of the threshold value mentioned above i.e. 175616. Importantly,

the number of divisions represents the number of Eigen states as well. Overall, applying

these systematic strategies enhances the performance of simulations hence generating the

desired results with better accuracy and consistency. As a result, a smooth convergence

41

trend, comprising of the last 3 combinations of divisions, is observed and circled below in

Figure 5.15:

Figure 5.15. Convergent Test Results for Double QDs with MBL = 1 nm

Statistically, the convergence pattern observed above is highlighted due to the reasonably

minute ground state eigen value differences among the last 3 systematic combinations.

Based on further observation, the % diff and the ground state eigenvalue difference

between the last 2 combinations (denoted by ∆ in meV) are minimal. Therefore, the

optimum combination is represented by 50x48x72 based on the threshold of the machine.

Based on observation, the latest division combination reflecting convergence after

implementing uneven division lengths was recorded as 46x46x60.

Overall, computations involving very large numbers of divisions are necessary to achieve

more accurate results. In addition, based on recurrent observations, the simulation

process for each combination close to the threshold value of 175616 divisions takes up to

150

145

140

135

130

125

120

Gro

und S

tate

Eig

en V

alu

e (m

eV)

Systematic Combinations vs Ground State Eigen Values

49*47*73 49*48*73 49*49*72 50*48*72

convergence region

137.7454

140.2578 140.5044 140.5054

42

2.5-3 hours on average. Based on the latest divisions using uneven divisions, the

simulation process takes about 1.5 hours on average hence increasing overall efficiency.

5.4.4 Results

The mc and the potential barrier height (Vo) parameters used for cubic and pyramidal

QDs were: mc for Cubic QDs = 0.043, mc for Pyramidal QDs = 0.0665 (same as

Harrison’s paper for comparison), barrier potential for Cubic case = 520 meV and barrier

potential for Pyramidal case = 276 meV.

Following are results generated from simulations for pyramidal QDs aligned along z-

axis:

43

Figure 5.16. 1st Eigen State energy wavefunction (xz) for Double QD pyramids at y = 26 nm with

MBL = 5 nm

44

Figure 5.17. 1st Eigen State energy wavefunctions for Double QD pyramids with MBL = 10 nm

at z = 68, 29 and 1 nm respectively

0

0

15

10

5

0 60

50 40 30 20

0 0 10

60 50 40 30 20 10 x y

50 40 30 20 0 0 10

60 50 40 30 20 10

x y

60

50 40

30 20

0 0 10

60 50

40 30

20 10

x y

Ψ(x, y, z = 68nm)

Ψ(x, y, z = 29nm)

Ψ(x, y, z = 1nm)

6

4

2

0

x 105

2

1

0 60

x 1013

60

60

45

As we can see from the wavefunction above, there is a peak in the center of x-y plane as

the geometry for the pyramidal QDs is symmetric along x and y planes. This was also

noticed in the mesh plots before. Also, the trend can be noticed for the wavefunctions as

there is a variation with the z axis i.e. when z = 1, 29 and 68 nm. The wavefunction has

the largest magnitude among the three when z = 1 nm. At z = 29 nm, the magnitude of

wavefunction decreased and it occupies greater area for the grid. Such variation trends

help understand better about the QD confinement as the behavior of the particles and

eigen states within the dot with confinement in all the three planes. Also, the pyramid

double-dot confinement with the single-dot pyramid case in the previous section shows

the effect from the additional dot i.e. how having the middle barrier length affects the

behavior of the eigen states.

Next, for the case of convergent ground state eigenvalues for Single QD pyramid and

Double QD pyramids along the vertical axis, an in-depth mathematical analysis is shown

below (Table 1). Significantly, this analysis highlights the close comparison between our

code simulation results using direct eigensolver and the results based on a simpler

computation procedure for eigen values using linear equation solutions. Consistently, the

initial formulations applied in both cases involved implementation of FDM. The code-

generated convergent Eigen value results along with the published Eigen values in meV

are recorded as follows:

46

MBL Code EVs

Single

QD pyr

Published

EVs Single

QD pyr

EV diff (∆)

and %

Discrepancy

(D)

Code EVs

Double

QD pyrs

Published

EVs

Double

QD pyrs

EV diff (∆)

and %

Discrepancy

(D)

0 nm 147.2614 147.59 ∆≈0.3286

D≈0.2226%

N/A

N/A

N/A

1 nm

N/A

N/A

N/A 139.2957 139.70 ∆≈0.4043

D≈0.2894%

2 nm

N/A

N/A

N/A 143.8438 144.30 ∆≈0.4562

D≈0.3161%

3 nm

N/A

N/A

N/A 146.0079 [144.30-

147.10]

∆<1

D<1%

4 nm

N/A

N/A

N/A 146.7599 147.10 ∆≈0.3401

D≈0.2312%

5 nm

N/A

N/A

N/A 146.9541 [147.10-

147.59]

∆<1

D<1%

7 nm

N/A

N/A

N/A 147.1112 [147.10-

147.59]

∆<1

D<1%

9 nm

N/A

N/A

N/A 147.1980 [147.10-

147.59]

∆<1

D<1%

10 nm

N/A

N/A

N/A 147.2584 147.59 ∆≈0.3316

D≈0.2247%

Table 1: Analysis [Comparison between Code (direct eigensolver) Results and the Published

Ground State Eigen Value (linear equation solutions)]

Importantly, consistency in analysis for the comparison above is valid due to application

of FDM in both cases initially.

Finally, plots for convergent ground state Eigen values for Single QD and Double QD

pyramids aligned along z-axis with Varying MBL = 1,2,3,4,5,7,9,10 nm along with

comparison between code Eigen values and published Eigen values are shown below in

Figures 5.18 and 5.19:

47

Figure 5.18. Smooth Curve showing graphical comparison between Convergent Energy Level

Results for Single and Double QD pyramids

150

149

148

147

146

145

144

143

142

141

140

139

138

Eig

en V

alu

e (m

eV)

1 2 3 4 5 6 7 8 9 10

Eigen Values (meV) vs MBL (nm)

Single QD pyramid

Double QD pyramids

147.2614

139.2957

143.8438

146.0079

146.7599 146.9541 147.1112 147.1980

147.2584

48

Figure 5.19. Smooth Curves showing graphical comparison between Convergent Energy Level

Code and Published Results (Double QD pyramids)

The smooth curves illustrated above demonstrate the convergence pattern for Double QD

pyramids with increasing MBL. Based on observation from the graph above, as MBL

approaches 9-10 nm, the ground state Eigen value for Double QD pyramids converges

approximately to the ground state Eigen Value for Single QD pyramid. Similar pattern is

also observed for Eigen values generated via implementation of the linear equation

method.

150

149

148

147

146

145

144

143

142

141

140

139

138

Published EVs & Matlab EVs (meV) vs MBL (nm)

0 1 2 3 4 5 6 7 8 9 10 11

Eig

en V

alu

e (m

eV)

150

149

148

147

146

145

144

143

142

141

140

139

138

139.2957

143.8438

146.0079

146.7599

146.9541

147.1112

147.1980

147.2584

Matlab EVs

Published EVs

139.70

144.30

146.40

147.10 147.35

147.50 147.56 147.59

Eig

en V

alue (m

eV)

MBL (nm)

49

5.5. CPU Usage: Strategies for Limited Memory

Although the available CPU memory allocation can easily support computations

involving matrices with respectable sizes, such memory was insufficient for

computations with large matrices with dimensions above 165000 x 165000. However, the

calculation memory can be largely reduced based on the fact that matrix A is a sparse

matrix. The computer performing this computation is equipped with a total of 4 Giga

(109) bytes or GB CPU RAM memory. With such a CPU RAM memory allocation,

simulations involving 563 or 175616 divisions ran for 2 to 3 hours. Meanwhile,

employing 603 or 216000 divisions potentially took several days for computation of

Eigen states. Hence, the number of divisions for simulations was kept less than 56x56x56

for computation of results without compromising the desired values. According to the

latest results, the simulations consumed 1.5 hours on average using 46x46x60 divisions

with uneven division lengths. Overall, by implementing techniques of optimum

combination and uneven divisions in our system, time and memory utilization were kept

under control. The memory consumption was restricted to the allocated limited machine

CPU memory meanwhile simulation time periods were lessened to hours from days

without compromising the overall results. In addition, such techniques paved the way for

more flexibility in selection for combination of divisions along with attaining better

accuracy and convergence for the desired Eigen state results.

50

Chapter 6. Quantum Dot Superlattice

6.1 Solution Procedure

The Schrödinger’s equation that describes the motion of a single hole in QD-SL system

can be written as follows:

The U(r) corresponds to an infinite sequence of quantum dots of size Lx, Ly and Lz

separated by barriers of thickness Hx, Hy and Hz. For our work, the dimensions for both

the quantum dot and barrier sizes were kept as 1-6 nm.

The 3D Schrödinger’s equation decouples into three identical (1D) quantum-well

superlattice equations. The 3D envelope wave function ψ(r) is represented as a product of

three 1D eigenfunctions χ as follows:

The total energy spectrum for this wave function is given by:

where En are eigenvalues of the one-dimensional Schrödinger’s equation.

Finally, the solution for Eq.6.1 is similar to:

)1.6()()(])()(

1

2[

*

2

rErrUrm

rr

)2.6()()()(),,(,,)( zyxzyxr nznynxnznynx

)3.6(nznynxnznynx EEEE

)4.6(,

)sin()sin()(2

1)cos()cos()cos(

0

*

*

*

*

aUEif

HkLkmk

mk

mk

mkHkLkqd WW

WW

BB

BW

WB

BW

51

where

Therefore, the envelope function approximation method is used as mentioned above,

which is also similar to the effective mass method. The Kronig-Penny method is useful

but as mentioned above, this method is limited only to the one dimensional system. The

numerical method based on the envelop function is useful in our case as it is used for the

3D case and it gives more flexibility when solving for superlattice systems.

6.2 Results

Following are results and discussions for the solution of GaAs/InGaAs cubic QDS

structure:

(a). 1st electron miniband vs QD size (1-6 nm) with Interdot distance fixed at 1 nm

Figure 6.6 shows the 1st electron miniband against QD size varying from 1x1x1 nm

3 to

6x6x6 nm3 with interdot distance fixed at 1 nm. Since the highest and lowest bands are

shown in the graph, one can also see the size of the bandgap. Based on observation, as the

QD size within the QDS structure increases, the miniband energy decreases.

)4.6(,0

)sinh()sin()(2

1)cosh()cos()cos(

0

*

*

*

*

bUEif

HkLkmk

mk

mk

mkHkLkqd WW

WW

BB

BW

WB

BW

)5.6(||2

,||2 *

0

*

Emk

UEmk

WWBB

52

Figure 6.6. 1st electron miniband vs Cubic QD with Interdot distance fixed (1 nm)

To explain the graph above, the higher and lower miniband energies are presented for the

miller index [111’]. The gap between the miniband energies represents the miniband

width. Also, as mentioned before, the miniband energy decreases with increase in dot

size.

2

1.5

0.5

1

0

-0.5

min

iban

d e

ner

gy

in

eV

1 2 3 4

Cubic QD size (nm)

5 6 7

111’

53

(b). 1st electron miniband vs Interdot distance (1-6 nm) with QD size fixed at 3 nm

Based on the figure below, the higher and the lower of the energy bands converge at

around 6 nm interdot distance again for the miller index [111’]. Also, the highest

miniband width occurs at interdot distance = 1nm.

Figure 6.7. 1st electron miniband vs Interdot distance with Cubic QD fixed

(c). Density of States vs energy levels

The density of states per unit energy and per unit volume is:

)8.6(|)(|)2(

2)(

3 qE

dSEg

q

E

2

1.5

1

0.5

0

-0.5

Ener

gy (

eV)

Interdot distance (nm)

1 2 3 4 5 6 7

111’

Lower

Higher

Electron Miniband Energy vs Interdot Distance with Cubic QDs sizes fixed

54

The graph generated is presented as follows:

Figure 6.9. Density of States vs energy levels

In the graph above, it is observed that the density of states is high after E = 0.4 eV and at

around E = 1.8 eV. The density of states decays in some region and stays consistent for

enegy values ranging from 1 to 1.6 eV. This pattern can also be observed from the

reference source.

55

Chapter 7. Conclusions & Future Recommendations

In conclusion, we employ the FDM technique to analyze the confined states in the single-

QD and double-QD system. For the single cubic quantum dot, our results agree well with

the analytical solution. In addition, our results also agree well with those published in the

literature for the system comprising of two vertically aligned pyramidal quantum dots.

The discrepancy in the comparison is within 1%. The major concern for using 3D FDM

to solve for quantum confinement structure is the memory usage in the calculation. In

order to compromise between calculation accuracy and memory spending, we used the

optimum divisions along the three spatial dimensions. Along each of the spatial

dimensions, we apply uneven divisions in the quantum dot and barrier matrix regions.

This technique works well systematically for a limited array of QDs. However, the

strategy of implementing FDM does not reflect as a completely effective approach for the

more complex multi-layered QD array structure, which could be solved through different

modeling strategies such as the Kronig-Penney model and the Envelope Function

Approximation.

As far as future recommendations go, structures with more than dual QDs could be tested

with our technique to check the capacity and accuracy of our system for such structures.

After extensive research, many efficient methodologies have been discussed above that

are suitable for multiple arrays of QDs (QD-SL) i.e. the Envelop Function and the

Kronig-Penny models. In addition, programming techniques can be tried from further

exploring the software toolboxes and help pages that might help to utilize less memory

space hence possibly improving the efficiency of the overall system.

56

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