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This is to certify that the
dissertation entitled

STRAIN ANALYSIS AT THE HETEROINTERFACES OF III-V
TERNARY ALLOYSIULTRA-THIN GaAsP/GaAs
SUPERLATTICES

presented by

WONJAE CHANG

has been accepted towards fulfillment
of the requirements for the

PHD. degree in ELECTRICAL ENGINEERING

 

 

 

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Date

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5/08 K:IProj/Aoc&Pres/ClRC/DateDue.indd

 

 

 

 

STRAIN ANALYSIS AT THE HETEROINTERFACES OF III-V TERNARY
ALLOYS: ULTRA-THIN GaAsP/GaAs SUPERLATTICES

By

WONJAE CHANG

A DISSERTATION
Submitted to
Michigan State University
In partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Electrical Engineering

2010

ABSTRACT

STRAIN ANALYSIS AT THE HETEROINTERFACES OF III-V TERNARY
ALLOYS: ULTRA-THIN GaAsP/GaAs SUPERLATTICES

By
WONJAE CHANG

The principal objective of this research was to characterize heterointerfaces of 111-
V ternary alloys consisting of ultra-thin GaAsP/GaAs superlattices grown by Molecular
Beam Epitaxial (MBE). The area of application for these materials is in optical devices,
operating under the visible and IR spectrum. The characterization of heterointerfaces
using high-resolution transmission electron microscopy (HRTEM) is divided into two
parts: 1) Strain analysis using advanced methodologies known as phase techniques, and
2) Evaluation of chemical contrast, along with minimization of artifacts on HRTEM
images, using the Composition Evaluation by Lattice Fringe Analysis (CELFA) method.

Phase techniques for stain analysis on HRTEM images can be divided into two
types: Geometric Phase Analysis (GPA) and Computational Fourier Transform Moire’
(CFTM) method. This research exploits the CFTM method and deals with demonstration
of the Computational Fourier Transform Moiré (CFTM) method for strain analysis with
monolayer accuracy in terms of theoretical implementation. In addition, experimental
results using ultra-thin GaAsP/GaAs superlattices are obtained. The CFTM method has
not been previously demonstrated for the strain analysis of ternary alloys using HRTEM.

Characterization of strain information on HRTEM images at heterointerfaces of
ternary alloys has limitations due to artifacts from the phase technique used for strain
analysis. In particular, small lattice-mismatched systems (f<3%) such as GaAsP/GaAs

superlattices pose additional challenges. From the methodology point of view, the phase

technique generates error at heterointerfaces due to the leakage effect, and it has been
shown that the error dominates for small-strained systems (f<3%) with low image
resolution. Hence, this research effort was pursued to overcome the leakage effect at
heterointerfaces and to minimize error from the phase technique, allowing quantitative
structural analysis. Mask-size optimization to suppress the leakage effect at
heterointerfaces was proposed and led to a quantitative value of measured strain at a
GaAso,36Po.14/GaAs superlattice. A simulation method using Matlab® for determining the
error of the strain values as we exploit the phase technique was proposed such that we
can evaluate the reliability of the phase technique with monolayer accuracy. Finally,
minimization of artifacts on HRTEM images was conducted and chemical contrast was

investigated using Composition Evaluation by Lattice Fringe Analysis (CELFA).

C0pyright by
WONJAE CHANG

2010

To my family. This could never have been written without your extreme sacriﬁces

and wonderful support.

ACKNOWLEDGMENTS

First of all, I give thanks to the Creator God for providing me with encouragement,
wisdom, and strength through my work on this dissertation. He is the Almighty and my
Comforter. I praise the God.

Words would not be sufficient to express my gratitude to Dr. Brown, chairman of my
dissertation committee, for his guidance and ﬁnancial support. Growth of Group III-
Nitride ﬁlms using Molecular Beam Epitaxy will be the most valuable asset in my life.

I owe a debt of gratitude to Dr. Reinhard and Dr. Hogan. Their guidance and comments
on my research were very helpful. I also appreciate Dr. Fan and Dr. Cramp for discussions
related with microscopy work. Without their support, I would not produce this outcome.

I would like to express my sincere appreciation to Dr. Hong, assistant professor in the
Mechanical Engineering Department at Michigan State University, for discussions of the
computational Fourier Transform moiré method. I also thank Dr. Goldman, professor in
Material Science and Engineering at the University of Michigan, for the opportunity of
growth of compound semiconductors using Molecular Beam Epitaxy.

I am deeply grateful to Karl Dersch for supporting clean room facilities in the Electrical

and Computer Engineering Department.

vi

TABLE OF CONTENTS

 

 

 

 

 

LIST OF TABLES .............................................................................................................. x
LIST OF FIGURES ........................................................................................................... xi
CHAPTER]
INTRODUCTION .............................................................................................................. l
CHAPTER2
MOLECUAR BEAM EPITAXY ....................................................................................... 5
2.1 Current Status of MBE growth for compound semiconductors ................................ 5
2.2 Veeco GEN-II MBE system ..................................................................................... 8
2.2.1 Transfer chamber ............................................................................................... 8
2.2.2 Growth chamber ................................................................................................. 9
2.3 Vacuum system ....................................................................................................... 12
2.3.1 Sorption pump .................................................................................................. 13
2.3.2 Ion pump .......................................................................................................... 14
2.3.3 Cryopump ........................................................................................................ 15
CHAPTER3
TRANSMISSION ELECTRON MICROSCOPY ............................................................ 16
3.1 Functions of Each Lens and Aperture ..................................................................... 16
3.1.1 Condenser Lens and Aperture .......................................................................... 16
3.1.2 Objective lens ................................................................................................... 19
3.1.3 Selected Area Aperture .................................................................................... 21
3.1.4 Objective aperture ............................................................................................ 22
CHAPTER4
ELECTRON ENERGY-LOSS SPECTROSCOPY AND ENERGY-FILTERED TEM . 25
4.1 Electron Scattering Theory ..................................................................................... 25
4.2 Electron Energy-Loss Spectroscopy (EELS) .......................................................... 27
4.2.1 Low energy loss range (<50eV) and Plasmons ............................................... 28
4.2.2 Thickness measurement of a sample ................................................................ 30
4.2.3 High Energy Loss Range (>50eV) ................................................................... 32
4.3 Energy Filtered TEM (EFTEM) ............................................................................. 33
CHAPTERS
HIGH-RESOLUTION TRANSMISSION ELECTRON MICROSCOPY ....................... 34
5.1 Fourier Transform ................................................................................................... 34

vii

5.2 Weak-Phase-Object Approximation ....................................................................... 35

 

 

 

 

 

 

5.3 Contrast Transfer Function ..................................................................................... 36

CHAPTER6

LITERATURE REVIEWS ............................................................................................... 38
6.1 Previous methods for strain analysis ....................................................................... 38
6.2 Regressional Analysis ............................................................................................. 39

CHAPTER7

PHASE TECHNIQUE ...................................................................................................... 42
7.1 Mathematical description for GPA and CFTM method ......................................... 43
7.2 Interpretation of phase image .................................................................................. 50
7.3 Previous applications with GPA method ................................................................ 52
7.4 Efforts to minimize error for strain analysis ........................................................... 54

CHAPTERS

EXPERIMENTAL RESULTS .......................................................................................... 56
8.1 MBE growth and TEM studies ............................................................................... 56
8.2 Ultra thin GaAso,g6Po,14 layer embedded in GaAs: Theoretical values of strain ..... 57
8.3 Strain analysis using the CF TM method ................................................................. 59

CHAPTER9

IMAGE SIMULATIONS ................................................................................................. 66
9.1 Leakage effect ......................................................................................................... 66
9.2 Impact of the leakage effect .................................................................................... 67
9.3 Reliability of phase technique ................................................................................. 71

9.3.1 Procedures of image simulations ..................................................................... 75
9.3.2 Error quantiﬁcation of the phase technique ..................................................... 81

CHAPTERIO

COMPOSITION EVALUATION BY LATTICE FRINGE ANALYSIS (CELFA) ....... 91

CHAPTERll

SUMMARY AND DISCUSSTION ................................................................................. 99

APPENDICES

APPENDIX A ................................................................................................................. 103
MATLAB CODE of an experimental image .............................................................. 103

APPENDIX B ................................................................................................................. 107
MATLAB CODE of image simulations ..................................................................... 107

APPENDIX C ................................................................................................................. 109
MATLAB CODE of cﬁm function used for image simulations ................................. 109

APPENDIX D ................................................................................................................. 112
STRUCTURE FACTOR FOR gallium arsenide phosphide ....................................... 112

viii

REFERENCES
LIST OF REFERENCES ................................................................................................ 114

ix

LIST OF TABLES

Table 4.1: Characteristics of the principal energy-loss processes ..................................... 30
Table 4.2: Plasmon loss data for 100eV electrons of several elements ............................. 31

Table 9.1: Estimate of (002) fringe spacing when a GaAs specimen is viewed along [110]
direction with Jeol 2200FS. Field of view is also provided. .................................... 77

Table 9.2: Measured strain at two monolayers of strained layers with g/3 Gaussian mask
in terms of resolution (number of pixels per ﬁ’inge spacing) and a strain value we
applied on an original image. Error within 25% is bounded with thick line ............ 90

Table D.1: Structure factors of GaAsl-xPx with major reﬂections ................................. 113

LIST OF FIGURES

Figure 1.1: Wavelength coverage by different LED materials from the visible spectrum to
the infrared. Shadowed and dashed lines are indirect Eg materials. .......................... 4

Figure 2.1: Gen II MBE system (a) Layout and (b) Gen II MBE system (CAR side) ...... 10
Figure 2.2: Transfer chamber (a) Inside transfer chamber (b) Trolley in transfer charnberll

Figure 2.3: Continuous Azimuthal Rotation (CAR) assembly. (a) Sample and Head
rotation feedthrough (b) Rotation-shielded head assembly (c) Heater assembly ..... 11

Figure 2.4: CAR side of Gen II MBE system. (a) Opening for CAR manipulator (b)
Installation of CAR into growth chamber ................................................................ 12

Figure 2.5: Shutters placed between effusion cells and a substrate. .................................. 12
Figure 2.6: The pumping system with a roughing pump and three sorption pumps. ........ 14
Figure 3.1: Schematic diagram of a TEM system .............................................................. 18

Figure 3.2: Ray diagram of condenser lens (a) A focused C2 lens illuminating a small
area of the specimen. (b) Effect of condenser aperture on the parallel nature of the
beam: a smaller aperture creates a more parallel beam. ........................................... 19

Figure 3.3: Two basic operations of the TEM imaging system. (a) Projected image on the
viewing screen (b) Diffraction pattern on the viewing screen. ................................ 20

Figure 3.4: Ray diagram of an objective lens. The distance from the object plane to the
lens (11), the distance from the lens to the image plane (v), and the distance from the
lens to the back focal plane (f). ................................................................................ 20

Figure 3.5: Ray diagram of SAD aperture. Insertion of an aperture in the image plane
results in the creation of a virtual aperture in the plane of the specimen. ................ 22

Figure 3.6: Optical ray diagram of bright-ﬁeld (BF) from a specimen to image plane. An
aperture is placed in the back focal plane of the objective lens which allows only
the direct beam to pass ............................................................................................. 23

xi

Figure 3.7: Ray diagram of a bright-ﬁeld image. Thicker or hi gher-Z area of the specimen
(gray area) will scatter more electrons off axis than thinner or lower-mass (white
area). Thus fewer electrons from the gray region fall on the equivalent area (black
area) of the image plane, which improves contrast relative to without the objective
aperture. .................................................................................................................... 24

Figure 4.1: Diagram for unscattered and scattered electrons ............................................. 25

Figure 4.2: Two mechanisms where an incident electron is scattered by an isolated atom.
(a) Elastic scattering (b) Inelastic scattering (inner-shell ionization) (c) Inelastic

scattering (outer-shell ionization) ............................................................................. 27
Figure 4.3: Inelastic scattering at phosphorous .................................................................. 29
Figure 4.4: Zero—loss and plasmon peaks in energy-loss spectrum. .................................. 30
Figure 4.5: Intensity under the zero-loss peak (10) and the total intensity. ........................ 32
Figure 5.1: An example of contrast transfer function, T(g), versus g ................................ 37

Figure 6.1: Schematic diagram showing a rigid lattice contraction across strained layers.41

Figure 7.1: Schematic diagram of displacement ﬁeld within a pair of atomic planes.
Position A and B indicates an initial position and its position after deformation,
which is represented by a displacement vector u(r). g is a reciprocal vector in
perfect crystal. .......................................................................................................... 52

Figure 8.1: TEM images viewed on [110] ZA. (a) A cross-sectional TEM image with low
magniﬁcation of a strained GaAso,g6P0.14/GaAs superlattice. (b) A cross-sectional
HRTEM image (XTEM) of a strained GaAsongQM /GaAs heterointerface taken
along [110] zone axis. Image size is 108 by 650 pixels and magniﬁcation is
0.031nm/pixel (Instrumental magniﬁcation of 600,000 x). ...................................... 60

Figure 8.2: A digital diffractograrn of the original HRTEM image in Figure 8-1(b). ....... 61

Figure 8.3: Strain proﬁle (exx) using the CF TM method of the experimental HRTEM
image with mask size (gm/3) ................................................................................... 62

Figure 8.4: (a) Measured strain at the GaAsogéPOM layers according to mask size. The x-
axis is the radius of the Gaussian mask. (b) A cross-sectional proﬁle of the (002)
reﬂection along the x-direction. The radius of the (002) reﬂection, along the x-
direction is around 15 pixels. ................................................................................... 65

Figure 9.1: Computer-generated lattice fringes using sinusoidal functions. Arrows
indicate coordinate system of x and y axes, corresponding to growth and interface
directions of Figure 1. A digital diffractogram is shown in the inset. The inset
shows the ﬁrst harmonic on the x-axis that can be used for strain analysis using the
phase technique. ....................................................................................................... 68

xii

Figure 9.2: Strain proﬁle, exx, of the simulations for (a) 2% and (b) 10% lattice
contraction at strained layers with two monolayers. Image resolution is 7 pixels per
lattice fringe spacing. ............................................................................................... 72

Figure 9.3: (a) Measured strain according to mask size and image resolution (number of
pixels per ﬁinge spacing) when applied strain at two monolayers of strained layers
is —2%. (b) A cross-sectional proﬁle of the ﬁrst harmonic used for phase technique.73

Figure 9.4: Measured strain according to mask size and image resolution (number of
pixels per fringe spacing) when applied strain at two monolayers of strained layers
is -10%. (b) A cross-sectional proﬁle of the ﬁrst harmonic used for phase
technique. ................................................................................................................. 74

Figure 9.5: Comparison of strain proﬁles on the GaAsP/GaAs heterostructure using
CFTM and GPA method. ......................................................................................... 75

Figure 9.6: (a) A computer-generated 2D lattice using Matlab software. Arrows indicate
coordinate system of x and y axes. A digital diffractogram is shown at inset. '
Rectangular region indicates the ﬁrst harmonic required for strain analysis along x-
direction. (b) Bragg- ﬁltered lattice fringe using a g/3 Gaussian mask centered on
the ﬁrst harmonic. (c) An unwrapped phase image by removing phase jump from
-1t to It. Notable change in phase at strained layers indicates displacement at
strained layers. .......................................................................................................... 78

Figure 9.7: Strain proﬁle on strained layers that is composed of 2 monolayers. An
askance step proﬁle was constructed for simulations, considering diffusion of
constituent materials at strained layers ..................................................................... 80

Figure 9.8: Strain proﬁle on strained layers composed of two monolayers with 4% lattice
contraction on strained layers. (a) Each proﬁle was plotted with different size of a
Gaussian function. Undershoot at strained layers becomes larger and larger with
increasing mask size of a Gaussian function (b) Strain was measured with mask
size of a Gaussian function where g/2 and g/3 correspond to 30 and 46 pixels
respectively at x-axis. ............................................................................................... 83

Figure 9.9: Measured strain in the case of -4% strain at strained layers we applied on an
original image. Mask size (Gaussian radius) was varied from 4 pixels to g/2 for
each resolution. Each graph represents resolution (number of pixels per fringe
spacing) where FS represents ﬁinge spacing in legend. .......................................... 84

Figure 9.10: Measured strain with different image resolutions (number of ﬁinge spacing)
with given strain values at strained layers. Image resolutions of (a) 7 pixels of
fringe spacing (b) 8 pixels of ﬁinge spacing (c) 9 pixels of fringe spacing (d) 10
pixels of fringe spacing (e) 12 pixels of fringe spacing (f) 14 pixels of fringe
spacing (g)16 pixels of fringe spacing (h) 18 pixels of fringe spacing (i) 20 pixels
of fringe spacing ....................................................................................................... 85

xiii

Figure 10.1: Selected area diffraction pattern with the GaAsongQM/GaAs superlattice,
showing only (000), (002), (004) beams are strong. ................................................ 95

Figure 10.2: Selected area diffraction pattern with the GaAso,g6Po_14/GaAs superlattice
after centering the (002) beam to optical axis from Fig. 10.1. ................................. 96

Figure 10.3: (a) A HRTEM image of the GaAso_g6Po,14/GaAs superlattice. Inset shows
symmetric SAD pattern for HRTEM imaging. (b) A low resolution image with
procedures of CELFA method. (c) A High resolution image with procedures of
CELFA method, showing lattice ﬁinge. .................................................................. 97

Figure 10.4: Elemental (phosphorus) map of the 5-period superlattice using energy-
ﬁltered TEM (EFTEM). ........................................................................................... 98

xiv

CHAPTER 1

INTRODUCTION

The growth of heterostructures by molecular beam epitaxy (MBE) provides
signiﬁcant advantages in band gap engineering. An important class of commercial
semiconductor materials that cover the visible and IR spectrum is the III-V ternary alloys

based on alloying GaAs and GaP, which are denoted as gallium arsenide phosphide

(GaA31 -xPx). When x<0.45, the alloy GaAsl_xPx is a direct bandgap semiconductor and

the wavelength ranges from about 630nm, red, for x=0.45 (GaAsO.5 5 P045 ) to 870 nm

for x=0, GaAs for the application of visible and IR detectors and emitters. When x>0.45,

the alloy GaAs]-xPx is an indirect bandgap semiconductor. However, if we add

isoelectronic impurities such as nitrogen (also group V element) into the semiconductor,

then some of these N atoms substitute for P atoms. Nitrogen doped indirect bandgap

GaAsl.xPx alloys are widely used in inexpensive green, yellow, and orange LEDs while

indium arsenide phosphide (InAsP) can be used within 1.3 - 3.0 pm. Recently, strain
balanced GaAsP/InGaAs quantum well solar cells on GaAs were reported by Ekins-
Daukes et a]. with increased cell efﬁciencies over standard GaAs single junction solar
cells [1]. Figure 1.1 summarizes some typical wavelengths for a few selected

semiconductor materials over the range from 0.4 to 1.7um.

The GaAsP and InAsP III-V ternary alloys are examples of mixed anion materials,
which are of current interest in the development of advanced devices. Mixed anion
materials composed of a few monolayers are used to form heterointerfaces and are
synthesized as strained layers on binary substrates such as GaAs and InP.

Mixed anion layers, having thicknesses of a few monolayers, exhibit fascinating
electrical and optical properties. The molecular beam epitaxy (MBE) growth technique
plays a vital role in making compositionally abrupt interfaces at mixed anion layers.
Therefore, the characterization of a few monolayers or heterointerfaces is essential to the
development of novel device structures. However, very few techniques have been
established to investigate strain with monolayer accuracy on High-Resolution
Transmission Electron Microscopy (HRTEM) images. Furthermore, the strain analysis of
heterointerfaces has been a challenge due to artifacts on HRTEM images.

The objective of this research includes four aspects. First, theoretical implementation
of the CF TM method on HRTEM images for strain analysis is performed, where the
CFTM method was originally developed on solder ball connect (SBC) assemblies [2].

Second, strain analysis is investigated at the heterointerfaces of small lattice-mismatched

systems (f<3%) such as GaAso.36P0.]4 /GaAs superlattices, where the theoretical strain
between GaAso.86P0.14 and GaAs is around one percent. For reliable strain analysis at

heterointerfaces of GaAso.g6P0.14/GaAs, this research addresses several challenges and

demonstrates the solutions to minimize errors resulting ﬁ'om the phase technique. Third,
the error of the CFTM method is quantiﬁed when two strained layers is analyzed for

strain information. Fourth, chemical contrast was investigated using the Composition

Evaluation by Lattice Fringe Analysis (CELFA) method and energy-ﬁltered TEM
(EFTEM). Another advantage of the CELFA method is that it can be used to minimize
errors resulting from the electron microscope.

In Chapter 2, a brief review of MBE research is provided, along with details of the
installation of an MBE at Michigan State University. The basic principles of TEM,
EELS, energy-ﬁltered TEM (EFTEM), and HRTEM are summarized from Chapter 2 to
Chapter 5. A literature review with previous techniques for strain analysis is introduced at
Chapter 6. Previously, there was a technique known as regressional analysis that was
demonstrated for a few monolayers with small lattice-mismatched materials. However, a
large error occurred for Si-doped GaAs and additional Gaussian smoothing was applied
to reduce this large error. Hence, strain analysis using the regressional techniques has
major challenges in dealing with a few monolayers.

The primary advantage of the phase technique is speed, the phase technique is much
faster than regressional analysis, because it can be implemented with digital image
processing from start to ﬁnish, and provides 2D strain mapping, while regressional
analysis provides only 1D mapping. The phase technique is summarized in Chapter 7.

Experimental results are addressed in Chapter 8. Simulations are described in Chapter
9 to understand the source of errors in the experimental results. The Composition
Evaluation by Lattice Fringe Analysis (CELFA) method is provided with experimental

results in Chapter 10. Conclusions are provided in Chapter 11.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 1.1: Wavelength coverage by different LED materials from the visible spectrum to
the infrared. Shadowed and dashed lines are indirect Eg materials [3].

CHAPTER 2

MOLECUAR BEAM EPITAXY

2.1 Current Status of MBE growth for compound semiconductors

Molecular Beam Epitaxy (MBE) is an epitaxial growth process that involves
chemical and physical reactions between atomic or molecular beams and a heated
substrate under ultra-high-vacuum (UHV). The primary advantage of MBE is the
capability of growing high-quality ﬁlms under ultra-high-vacuum (UHV) pressure with
atomic layer control over the epitaxial process. With the MBE process, ultrathin and
abrupt layers, as well as, continuous doping proﬁles can be obtained.

Thin-ﬁlm epitaxial crystal growth of compound semiconductors by MBE has been
explored in many attractive applications, including optoelectronics such as light-emitting
diodes (LEDs), Solar cells, and Laser Diodes; novel sensors; and transistors such as high
electron mobility transistors (HEMTs) and single electron transistors (SETS). This section
begins with a brief review of devices developed using the MBE technique and a
description of the GEN-II MBE system is provided in Chapter 2.2.

LEDs and Laser diodes convert electrical energy into incoherent or coherent optical
radiation by electrical injection into the junction region of the p-n diode. Recent research
in high-brightness blue and green LEDs [4] [5] and LDs [6] using wide bandgap
materials have brought rapid development of white-light LEDs and blue-violet LDs for

blue-ray discs. Signiﬁcant progress has been achieved even though the exact mechanism

responsible for radiative recombination in the presence of high threading dislocation
densities is under debate. Current research on LEDs and LDs is primarily concerned with
minimizing dislocations in order to achieve high intensity light output. For most devices,
the epitaxial layers have different crystal lattice parameters, leading to a lattice mismatch
between the two crystal structures. This causes dislocations, resulting in crystal defects.
Such crystal defects cause recombination of electron-hole pairs (EHPs). Progress in the
nitride research area has been achieved via insertion of a buffer layer grown at low
temperature between a GaN layer and a sapphire substrate, resulting in high quality
(0001)-oriented GaN layers [7] [8]. After that, the growth of high quality ternary alloys
InGaN [9] [10] and AlGaN [11] [12] on GaN were developed, leading to high quality
heterostructures, including quantum well (QW) structures [13] [14]. In order to increase
the intensity of light output for optical devices, a heterostructure device (HD) [15], e. g.
InGaN/GaN quantum well [16], or quantum dots (QDs) [17] have been used. Generally,
QDs inherently contain a lower density of structural defects due to a built-in strain ﬁeld;
however, QDs reduce radiative lifetime. Also, InGaN nanowire LEDs [18] and InGaN
dot-in-a-wire heterostructures [19] have been investigated. In addition, AlGaN/GaN
heterostructures have been studied for UV optoelectronics [20].

A photovoltaic device, also known as a solar cell, converts sunlight directly into
electricity by generating electron-hole pairs at a junction via internal photovoltaic effects.
For a given solar spectrum, conversion efﬁciency depends on the semiconductor
materials properties and the device structure [1] [21]. GaAs and Si solar cells have
comparable efﬁciencies, though theoretically GaAs with a higher bandgap is supposed to

have a better efﬁciency. The largest factor reducing the efﬁciency of a Si solar cell is the

unabsorbed photons with In) < Eg and short wavelength photons absorbed near the

surface. Both of these factors are improved if we use a tandem cell structure or
heterojunctions. The AlGaAs window layer overcomes the surface recombination
limitation and improves cell efﬁciency. Tandem or cascaded cells use two or more cells

in tandem or in cascade to increase the absorbed photons from the incident light. The ﬁrst

cell is made from a wider bandgap material and only absorbs photons with hp > Eg]. The

second cell absorbs photons that pass through the ﬁrst cell and have ho > Egz.

Progress in the grth technique for GaN materials has led to intense research of
AlGaN/GaN heterostructures for high electron mobility transistor (HEMT) applications,
particularly in the area of high power and high frequency devices [22]. Demonstration of
a two-dimensional electron gas (2DEG) in AlGaN/GaN heterostructures [23] was an
important result for achieving HEMT devices. Moreover, wide bandgap materials exhibit
high thermal stability and device operation at an elevated temperature (600°C) is
possible. Silicon based devices have typically been limited to 350°C. Also,
heterostructures with GaN materials have been focused on the area of sensor applications
such as gas, liquid, and pressure sensors due to the chemical inertness of the nitrides. For
example, the interaction of gaseous species with the Pt electrode leads to a change in the
electronic properties of the Pt-GaN [24] Schottky contact. Also, a liquid sensor [25] for
bio-applications, pressure sensors, and strain sensors [26] have been demonstrated using
AlGaN/GaN heterostructures.

Several emerging areas of research that utilize heterostructures include: SET (Single

Electron Transistors), single-electron memory devices using QDs (Quantum Dots) [27],

II-VI and III-V based dilute magnetic semiconductors for spin transistors and MRAM
(Magnetic Random Access Memory) that records information with electron spin [28].
One of the challenges that must be addressed for MRAM applications is the discovery of

materials that have acceptable room temperature ferromagnetism.

2.2 Veeco GEN-II MBE system

The GEN 11 MBE System at Michigan State University is conﬁgured with two
separate chambers, the main vacuum chamber for growth and an intro/transfer chamber.
The system layout is shown in Figure 2.1 (a) and (b). Each of these chambers is equipped
with an ultra-high-vacuum (U HV) pumping system and isolated by means of gate valve.
The grth Chamber's center axis for ﬁlm layer deposition is oriented horizontally and
the substrate is held facing the source ﬂange. The vacuum chambers are made of stainless
steel due to its low chemical reactivity. All system components should be able to
withstand baking at 200°C. The grth chamber contains a RHEED system and beam

ﬂux monitoring.

2.2.1 Transfer chamber

Substrates are mounted in a non-indium-bonded substrate holder, including UNI-

TM
Block Plus series for use in the 3" diameter sample and loaded on the cassette trolley as

shown in Figure 2.2. Once loaded onto the trolley, the transfer chamber is then pumped

down ﬁrst by the pumping station (to 10.6 Torr) and then by an ion pump. Substrates are

individually transferred from the trolley to the heated station of the CAR (Continuous

Azimuthal Rotation) manipulator using a transfer arm. Care should be taken to avoid

crashing or dropping of substrates since this is a very delicate process.

2.2.2 Growth chamber

The CAR assembly shown in Figure 2.3 is mounted horizontally in the growth
chamber (see Figure 2.4 (b)). Heater assembly should be faced down or up during
installation of the CAR assembly into the growth chamber since the opening in the
cryopanel is rectangular in shape (see Figure 2.4(a)). The substrate holder at the CAR
assembly can be rotated 300° about the main shaft. This rotational head movement allows
for aligning to the growth position, transfer, and ﬂux measurement position. The growth
position is also the correct position for Reﬂection High Energy Electron Diffraction
(RHEED) analysis.

The CAR assembly provides substrate heating by means of a wire wound heater (see
Figure 2.3(c)) with an integrated type C (W—26%Re/W-5%Re) thermocouple for
feedback. Desorption of native oxides can be carried out using the CAR assembly while
in the growth chamber. One advantage of desorption of the native oxide in the growth
chamber is that epitaxial grth can immediately start after the native oxide has been
eliminated. This helps to minimize surface contamination that might be introduced if
sample cleaning is done in a separate chamber.

Abrupt ﬁlm layers are achieved using shutters placed between the effusion cells and
the substrate as shown in Figure 2.5. Continuous changes in chemical composition via

ﬂux control are achieved by programmed variation of the temperature of effusion cells.

CAR . .
Ionrzatron

 

 

 

 

 

 

 

 

 

RHEED gun manipulator
\ gauge Ion pump
Effusron cells {—T } g Entry/Exit quick
- § q ‘— hatch port
~ I— Transfer chamber
\H’
A l

 

 

 

 

 

 

 

 

~ W 1 \ I
I: v / ) Transfer rod
/

l r
\

I Pumping cart

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cryo shrouds /
Fluorescence \An/gle valve
screen Rotating
substrate holder T Gate valve
CUOpMp
(a)
g : LNz house line
5 if: RHEED source
3 Solenoids &

Actuator stem to
control shutters

 

 

 

CAR

Manipulator Effusion cells
Growth S f -
chamber ource rrnge

 

(b)

Figure 2.1: Gen II MBE system (a) Layout and (b) Gen II MBE system (CAR side)

10

 

(b)

Figure 2.2: Transfer chamber (a) Inside transfer chamber (b) Trolley in transfer chamber

Head Rotation Beam Flux

Feedthrough Monitor Type C T/ C
sample ROtatlon (180° Opposite Sample)
F eedthrough

     

Heater Three pin
Assembly sample
mount
(a) (b) (c)

Figure 2.3: Continuous Azimuthal Rotation (CAR) assembly. (a) Sample and Head
rotation feedthrough (b) Rotation-shielded head assembly (c) Heater assembly

11

 

(b)

Figure 2.4: CAR side of Gen II MBE system. (a) Opening for CAR manipulator (b)
Installation of CAR into growth chamber

 

Figure 2.5: Shutters placed between effusion cells and a substrate.

2.3 Vacuum system

A very low (<10'lo Torr) or ultra high vacuum base pressure is essential for high
quality MBE growth. Under UHV conditions, the surface of a substrate can be
maintained clean and free of any contaminants. The beams of atomic particles for growth
do not suffer from momentum loss through collision with residual gas molecules. In order

to achieve the UHV condition, there is not a single pump that reaches ultra high vacuum

12

from atmosphere; instead several pumping stages are used including sorption pumps, ion

pumps, and a cryogenic pump.

2.3.1 Sorption pump

A triple system of sorption pumps linked with a scroll-type roughing pump is shown

in Figure 2.6. Before operating the sorption pumps, the roughing pump is used to bring

the growth and transfer chambers from atmosphere to ~10.3 Torr. Then an LN2 trap (not

shown in Figure 2.6) can further pump down ~5* 10'4 Torr. The sorption pumps with

liquid nitrogen for the GEN 11 MBE system are sequentially operated to bring the growth

and transfer chamber from ~5"‘10.4 Torr to 10.6 Torr. The sorption pumps remove

molecules inside the chamber by trapping them on the surface of the sorbent. The sorbent
is a solid porous material with large surface area and is typically made of a synthetic
zeolite material known as molecular sieve. When the sorption pump is cooled by liquid

nitrogen, the molecular sieve can absorb molecules until the pressure of the chamber

reaches ~10.6 Torr. After several pumping cycles, the pores of the molecular sieve

materials will become clogged with water vapor and the pumping efﬁciency will
deteriorate. Water vapor is removed by baking the pump up to 300°C for 2~4 hours while

pumping on the molecular sieve. The bakeout heater is shown in Figure 2.6.

13

A bakeout heater

 

Figure 2.6: The pumping system with a roughing pump and three sorption pumps.

2.3.2 Ion pump

Ion pumps do not use oil, in contrast to diffusion pumps, providing contaminant free
Ultra High Vacuum pumping (UHV). An ion pump ionizes gases and employs a strong
electrical potential, typically 3kV to 7kV between a stainless steel anode and a titanium
cathode. When a strong electrical potential is placed across the electrodes, a discharge is
initiated by ﬁeld emission. A magnetic ﬁeld is utilized to conﬁne the discharge within the
pump. Pumping action is achieved when gas molecules are ionized in the discharge, and
accelerated into the cathode with sufﬁcient kinetic energy that they are permanently
buried. In addition, this ion bombardment of the cathode, results in titanium being
sputtered off the cathode and deposited on the pump walls. Active gases are chemisorbed

by the titanium, contributing to the pumping action. Inert and lighter gases, such as He

and H2 do not effectively sputter and are absorbed by physisorption. Our system also

uses a Ti sublimation pump. The active titanium surface is produced by electrically
heating a ﬁlament to evaporate titanium, which in turn condenses on the walls of the

vacuum chamber.

14

2.3.3 Cryopump

The operation of a cryopump relies primarily on the principle that any surface will act
as a pump for a gas that condenses on a cold surface. The cryopump in our MBE system
incorporates a closed circuit helium refrigerator to cool its active surfaces. A compressor

is utilized to compress the helium refrigerant. The main advantage of cryopumps is that

they provide efﬁcient, oil-free pumping of all gases in the 10'3 to 10.IO Torr range,

achieving high pumping speed. One disadvantage is that the trapped gases remain in the
pump, hence, the pump needs to be regenerated periodically, by allowing it to warm to
room temperature while pulling vacuum on the cryopump using a separate vacuum
system in order to pump out all the adsorbed gases. Another disadvantage is that if the
cryopump fails unexpectedly, the adsorbed gases will be released into the vacuum
chamber, possibly damaging the MBE system. Hence, in the event of a power outage, the
high vacuum side gate valve should be closed immediately, before the pump starts

warming up.

15

CHAPTER 3

TRANSMISSION ELECTRON MICROSCOPY

Transmission electron microscopy (TEM) has been the technique most widely used
for the observation of dislocations and other crystal defects, such as stacking faults,
twins, grain boundaries, and voids. Static arrangements of defects are usually studied, but
in some cases, miniaturized tensile devices have been used to deform specimens (less
than 3mm in length) within a microscope, thus providing direct observation of dislocation
interactions and multiplication processes. TEM is applicable to a wide range of materials,
subject only to the condition that the specimen can be prepared in a very thin section, and
that they remain stable when exposed to a beam of high-energy electrons within a high
vacuum system. A schematic of a conventional transmission electron microscope is

shown in Figure 3.1.

3.1 Functions of Each Lens and Aperture

3.1.1 Condenser Lens and Aperture.

The condenser lens consists of two lenses (C1 and C2). The C1 and C2 lenses are
adjusted to illuminate the specimen with parallel beams of electrons from the gun as
shown in Figure 3.2. The C1 lens ﬁrst forms a de-magniﬁed image of the gun crossover

while the C2 lens controls illumination (converged or parallel beam) and the beam’s

16

intensity on a specimen. We call the C2 brightness. If we want to increase the intensity
of the beam on a speciﬁc area of a specimen, we focus the beam more. In other words,
when we intend to slightly minimize the area of the specimen that we are illuminating,
we simply change the C2 (brightness) lens so that it is focused. For normal image mode
we underfocus C2 (brightness) until the illuminated area on the specimen ﬁlls the
viewing screen. This is also the case for several alignments (i. e. Z-height correction,
condenser astigmatism, centering of current, and centering of voltage etc.). The Z-height
correction is to ﬁx the specimen’s height on the optic axis to allow us to obtain the same
objective lens current and thus a ﬁxed objective lens magniﬁcation. The centering of the
current involves the objective lens rotation alignment to ensure that the objective lens
ﬁeld is centered on the optic axis. As a result, direct electrons leaving from the specimen
see a symmetric ﬁeld as they pass through the lens. The voltage centering should be
preformed when a varying voltage is applied to the gun and the objective lens is aligned
to ensure the electrons remain on axis through the lens as their energy varies. Also, we
need to deliberately create a focused convergent beam on the specimen for the
convergent-beam mode. The convergent beam is a probe. We use such a probe to get a
signal from only a localized area of the specimen. For example, defect analyses with
EDX or EELS, scanning of the beam via STEM, or convergent-beam diffraction.

On the other hand, the condenser aperture excludes stray electrons. However, if we
use a smaller aperture, the angle of the beam convergence is decreased, which makes the

beam more parallel as shown in Figure 3.2 (b).

17

Gun_—’ U

   
    
   
 
 

Condenser Condenser
lens aperture
Objective .
lens Specrmen
Objective aperture
Diffraction lens Selected area aperture

Intermediate lens
Projector lens

Viewing screen

Camera/Film

Figure 3.1: Schematic diagram of a TEM system

18

 

   

""‘ C2 lens and
Condenser
aperture

 

 

  

<——— —>
(a) Specimen (b)

Figure 3.2: Ray diagram of condenser lens (a) A focused C2 lens illuminating a small
area of the specimen. (b) Effect of condenser aperture on the parallel nature of
the beam: a smaller aperture creates a more parallel beam[29].

3.1.2 Objective lens

The objective lens is the most important component of the TEM since it creates the
image on the image plane and diffraction patterns on the back focal plane. The ray
diagram from specimen to viewing screen is shown in Figure 3.3. Notice that switching
between diffraction pattern and image is conducted by the objective aperture and the
selected area aperture.

The objective lens takes the electrons leaving from the exit surface of the specimen
and disperses them to create a diffraction pattern in the back focal plane. It then
recombines them to form an image in the image plane (see Figure 3.4). If we divide
inﬁnitesimal points on the object, each inﬁnitesimal point is converged on the same point
of the image plane no matter how the scattering events occur in the specimen (see Figure

3.4).

19

     
     
   
    
   

[V V“ Specimen

‘ ‘7'. Objective lens

      

 

 

31‘ Back focal plane
7 r

l\ K Objective aperture/v
- ' Q Image Plane
\ SAD aperture

Intermediate lens
)))

 

 

 

 

 

_ _ Intermediate image

Projector lens

 

 

 

Viewing screen
Projected image Diffraction pattern

(a) (b)
Figure 3.3: Two basic operations of the TEM imaging system. (a) Projected image on the
viewing screen (b) Diffraction pattern on the viewing screen.

 

 

Back focal plane

 

Image plane
(conjugated plane of object)

 

A

 

Figure 3.4: Ray diagram of an objective lens. The distance from the object plane to the
lens (u), the distance from the lens to the image plane (v), and the distance
from the lens to the back focal plane (1).

20

From the Figure 3.4, any parallel beams from the object are converged at the same
point on the back focal plane. That is the principle behind the diffraction pattern. Another
process occurring between the object and image plane is magniﬁcation. From the
Newton’s lens equation, the relationship between the relative positions is:

l + l = .1.

u v f

And the magniﬁcation of a convex lens is given by:

M = £3: = 1:

AB u

Therefore, a high magniﬁcation can be realized by putting the object close to the lens,
making u small and M large. Otherwise, if we make the lens weaker, then the focal length
is increased. If f is enlarged but u is unchanged, then v must be correspondingly longer,
and the image magniﬁcation is larger. This magniﬁcation is the major role of the

intermediate lens and projector lens as shown in Figure 3.3.

3.1.3 Selected Area Aperture

The selected area aperture is an essential component for obtaining a diffraction
pattern. This operation is called selected-area diffraction (SAD). The position of the
selected area aperture is the conjugate plane of the specimen since we cannot insert the
selected area aperture in the specimen position. The conjugate plane is the image of the
objective lens as shown in Figure 3.4. The SAD aperture is inserted into the image plane
of objective lens. Then any electron that hits the specimen outside the area deﬁned by the
virtual aperture will hit the real diaphragm as shown in Figure 3.5. Thus, it will be

excluded from contributing to the diffraction pattern. By choosing a speciﬁc small area,

21

we can protect the camera from the high intensity of the electron beam. Aﬁer inserting
SAD aperture, we can obtain diffraction patterns by turning on the diffraction mode of
the diffraction lens of Figure 3.1, which switches the back focal plane of the objective
lens as its object. As shown in Figure 3.3, objective aperture should be taken out before
inserting the SAD aperture in order to select the back focal plane as an object. To
convert to image mode, we have to remove the SAD aperture and switch into image

mode of the diffraction lens.

Virtual aperture

  
 
  
   
    

Virtual diaphragm
Specimen —>

 

 

Diffraction Pattern (7 : ack Focal Plane

’ ‘ \ SAD diaphragm

‘ SAD aperture .

Figure 3.5: Ray diagram of SAD aperture. Insertion of an aperture in the image plane
results in the creation of a virtual aperture in the plane of the specimen.

3.1.4 Objective aperture

An additional principle of TEM operation is the bright-ﬁeld (BF) image and dark-
ﬁeld (DF) image. These two modes are created by the objective aperture. The bright-ﬁeld
(BF) image is generated by allowing only the forward beam to pass through the objective
aperture. On the contrary, the dark-ﬁeld (DF) image is generated by allowing any of the

diffracted beams to pass through the objective aperture. The contrast in BF and DF

22

images is called the diffraction contrast. Figure 3.6 shows the optical ray diagram of
bright-ﬁeld from a specimen to the image plane. The forward beam is converged at the

back focal plane on the optic—axis and forms the image on the image plane.

    
 

‘—— Specimen

 

 

‘_ Back focal plane

 

Optic axis

Image plane
(Bright ﬁeld image)

 

Figure 3.6: Optical ray diagram of bright-ﬁeld (BF) from a specimen to image plane. An
aperture is placed in the back focal plane of the objective lens which allows
only the direct beam to pass

When the electron beam encounters a specimen, many electrons are strongly scattered
due to mass-thickness or specimen orientation difference. By choosing only the direct
beam, we can improve contrast of the image. This is shown in Figure 3.7.

The other mode is the dark-ﬁeld (DF) image. The dark ﬁeld is formed by choosing a
diffracted beam of many diffracted beams. The DF image provides high contrast image

with low intensity while BF image provides low contrast image with high intensity.

23

Electron source

Hea element 1 l l
W \ ...--__-.

.. Lighter element
.. - /

Small angle
Large angle scattering
C

 

scattering

Blocked by
aperture

     
 

 

 

"“m ——*Obj ective aperture

Figure 3.7: Ray diagram of a bright-ﬁeld image. Thicker or higher-Z area of the specimen
(gray area) will scatter more electrons off axis than thinner or lower-mass
(white area). Thus fewer electrons from the gray region fall on the equivalent
area (black area) of the image plane, which improves contrast relative to

without the objective aperture.

24

CHAPTER 4

ELECTRON ENERGY-LOSS SPECTROSCOPY AND ENERGY-FILTERED TEM

4.1 Electron Scattering Theory

When a uniform intensity of electrons encounters a thin sample, the intensity is
modiﬁed and corresponds with the sample’s mass and thickness. In addition to the
intensity, scattering events change the angular distribution as shown in Figure 4.1. When
we consider scattering events, the simplest process is single scattering (i. e. an electron
undergoes either a single scattering event or no scattering) through the sample. If a
second scattering event happens, the electron could go back into the direct beam so that it
appears to have experienced no scattering. The single scattering assumption is often very

reasonable if the sample is thin.

 

' ' <— Sample

 

Scattered

Unscattered Electrons

electrons

Figure 4.1: Diagram for unscattered and scattered electrons

25

 

The scattering causes an angular distribution of electrons and can be mainly divided
into elastic and inelastic scattering (see Figure 4.2). The term elastic means that there is
no change in energy, but there is a change in the deﬂection angle. Inelastic scattering
refers to the beam of electrons that are scattered such that there is a change in energy
which involves the inner or outer shell atomic electrons. Inelastic interactions carry
information about the nature of the atoms, their electronic structure, and their bonding
with surrounding atoms.

Figure 4.2 (a) shows two mechanisms by which an incident electron is scattered by an
isolated atom. Coulombic interaction within the electron cloud results in low-angle (9)
elastic scattering, while Coulombic attraction by the nucleus results in backscattering.
Note that the potential within the electron cloud is always positive. Figure 4.2 (b) shows
inelastic scattering resulting from interaction between inner-shell electron (core electron)
and an incident electron beam. Figure 4.2 (c) shows inelastic scattering due to interaction
between outer-shell (valance electron) and an incident electron beam. Therefore, different
energies involved in inner and outer-shell scattering occupy different parts of the EELS

spectrum. This will be discussed further in Section 4.2.

26

 

 

 

(a) (b) (C)
Figure 4.2: Two mechanisms where an incident electron is scattered by an isolated atom.
(a) Elastic scattering (b) Inelastic scattering (inner-shell ionization) (c)
Inelastic scattering (outer-shell ionization)

4.2 Electron Energy-Loss Spectroscopy (EELS)

Electron energy loss spectroscopy (EELS) measures the energy loss of the electrons
that interact with the sample. It measures the energy transfer between the electrons in the
incident beam and atomic electrons. Information regarding the local environment of the
atomic electrons and nearest neighbor atoms can be obtained. The method used for
separating electrons of different energy involves a magnetic prism mounted above the
intermediate lens, since the magnetic prism is compact and provides sufﬁcient energy
resolution to distinguish all the elements in the periodic table.

When electrons pass through a uniform magnetic ﬁeld, they are deﬂected by the

magnetic ﬁeld. The amount of deﬂection is determined by the energy loss that the

27

 

electrons experienced. The greater the energy loss of the electron, the further it is
deﬂected. Therefore, electrons with different energies can be physically separated. This is
the operation principle of EELS.

The EELS can be mainly divided into three parts on the spectrum: zero-loss peak,
low-loss spectrum, and high-loss spectrum. First, the zero-loss peak is the most intense
spectrum and contains the elastic forward-scattered electrons that experienced very small
energy loss. Hence, these electrons have approximately the same energy as the incoming

electron beam. Second, the low-loss spectrum appears close to the zero-loss peak and is

 

much lower in intensity than the zero-loss peak. It contains electrons that interacted with b
the weakly-bound outer shell electrons, plasmons, or phonons of the sample, resulting in

low energy losses. The low-loss spectrum is generally located below 50eV. Third, the

high-loss spectrum contains electrons that interacted inelastically with tightly bound

inner-shell (core) electrons, and it appears above 50eV.

4.2.1 Low energy loss range (<50eV) and Plasmons

The plasmon process is the most common inelastic interaction (see Figure 4.3). The
plasmons are longitudinal wave-like oscillations of weakly bound electrons. When an
electron beam passes through the free electrons of the conduction band, plasmon (i. e.
collective oscillations of the free electrons of the conduction band) occurs. Hence, the
plasmons can happen in any material with weakly bound or free electrons and occurs

most often in metals.

28

I , r' " ‘ ‘ - ‘ Plasmon (~5-25eV)
\
\
z ’ / Interband transition (~5-25eV)

 
 
  
 

\ Free electron

  

 

\
j K shell
1

L Shell

I
I
v>\M shell
’ I

~~_—"

Figure 4.3: Inelastic scattering at phosphorous

The energy Ep lost by the beam of electrons as it generates a plasmon of resonance

frequency a) p is given by:

21
ne 2

 

 

h a) p h
E p = = ( [29]
27: 27: am
where a) p is the resonance frequency of the plasmon. The excited plasmon peak

absorbs the energy Ep for low loss EELS. We call the second spectrum a plasmon peak
(see Figure 4.4). Plasmon is the second most dominant peak of the energy-loss spectrum.
Note that the plasmon peak could include inter-intra band transition or lattice vibrations

(phonon). Generally, the plasmon peak includes all possible outer-shell interactions of

29

 

scattering events. However, there are times when we cannot distinguish the second
spectrum with plasmon, phonon, inner transition, or intra transition, since their energies
are close to each other. As a result, we generally call the second spectrum a plasmon

spectrum. The principal energy loss processes due to scattering are listed in Table 4.1.

Intensity
L
l Zero-loss

 

 

 

 

Plasmon
0 10 15 0 Energy Loss (eV)

Figure 4.4: Zero-loss and plasmon peaks in energy-loss spectrum.

Table 4.1: Characteristics of the principal energy-loss processes

 

 

 

 

 

 

 

 

Process EnergyLoss (eV)
Phonons ~0.02
Inter/lntra-band transitions 5-25
Plasmons ~5-25
Inner-shell ionization ~l 0-1000

 

4.2.2 Thickness measurement of a sample

The plasmon excitation is a useful way to measure the thickness of a TEM sample. A

typical value of the plasmon mean-free path 4,, is listed in Table 4.2. The l p is about a

few hundred nanometers. Hence, it is reasonable to expect at least one strong plasmon

peak in a thin sample. However, if the sample is thick, a number of individual losses

30

should occur. Hence, there is thickness information in the energy-loss spectrum since the

amount of all inelastic scattering increases with sample thickness.

Table 4.2: Plasmon loss data for 100eV electrons of several elements [29]

 

 

 

 

 

 

 

 

Material Ep(calc.) Ep(exp.) ,1. p (calc.)
(6V) (6V) (M)

Li 8.0 7.1 233

Be 18.4 18.7 102

A1 15.8 15.0 1 19

Si 16.6 16.5 1 15

K 4.3 3.7 402

 

 

 

 

 

In principle, the probability of a single inelastic scattering event is given by:

t
= :1— (t: thickness, ,1 : inelastic mean free path)

P

For multiple scattering of n events, the probability follows the Poisson distribution and is
expressed:

-t
In t n “Pf/I)

P :——: — ————9
” I, (,1) n!

where I n is the n-th scattered intensity, It is the total intensity.

—I
For n=0, ‘jo— = eXP(':1—) where 10 is the elastic scattered electrons. Therefore, we can
I

obtain a relationship between the thickness and the inelastic MFP (mean free path) versus

elastic scattered electrons and total intensity:

I
i = 1n(—’) .
2. 10

31

This gives us a relative thickness. Finally, the relative thickness multiplies the mean free

path and gives absolute thickness. The inelastic MFP is given by:

106FE
xi z 0

, ,1 is nm in unit, E is keV, ﬂ is collection semiangle, E is
{ l (ZﬂEO 0 m
Em n

Em )}

 

the average energy loss in eV and F is a relativistic correction factor. 5

The E m and F is given by:

 

 

E0 I
{1+——} :
F: 1022 j
E0 2 ,(F=0.618 at E0=200 keV)
1+ —
{ (511) }
Em = 7.6ZO°36

The 10 and I t are drawn in Figure 4.5.

/’0

 

 

 

Figure 4.5: Intensity under the zero-loss peak (10) and the total intensity.

4.2.3 High Energy Loss Range (>50eV)
The energy needed for ionization processes is transferred from the incoming electron
beam to a core-shell electron in the atom. Since the ionization process is characteristic of

the atom, the high-loss spectrum provides chemical information.

32

Electrons with the atoms can excite an empty state by accepting a speciﬁc value of
energy. The incident fast electrons which provide this energy suffer a loss of the same

amount. The minimum energy causing the ionization threshold is called the critical
ionization energy (EC). Of course, if the incident electron is greater than Ec, ionization
could be possible. However, the chances of ionization become less with increasing

energy above EC, since the value of the cross section decreases with increasing energy.

As a result, the ionization-loss electrons have an energy distribution that ideally shows a
sharp rise to maximum at E, followed by a slowly decreasing intensity above Ec back

toward the background coming from any previous peak. That is the reason we call it an
“edge”. The edge energy is determined where the first derivative is maximized or the

second derivative is zero. In the case of ionization energy of K-shell electron, we call it
EK and EL for L-shell ionization energy. The presence of certain elements can be

determined from the appearance of the edges in the spectrum.

4.3 Energy Filtered TEM (EFT EM)

If we consider a point in a sample, the point is transferred on a point in the TEM
image no matter what scattering events occur. That is how a lens works. However, we
can select speciﬁc electrons which have the same energy using EFTEM. The energy ﬁlter
component is installed between the intermediate lens and projector lens. This special
ﬁlter allows for selection of a very narrow window of energies in the EELS spectrum.

Using the corresponding electrons for imaging, EFTEM is performed.

33

. ‘o ‘ " 1' ' TWJL‘L'.‘

 

CHAPTER 5

HIGH-RESOLUTION TRANSMISSION ELECTRON MICROSCOPY

The HRTEM utilizes phase—contrast imaging and represents interference patterns
between the forward-scattered and diffracted electrons from a sample (see Figure 3.6).
The phase shift can be created by periodic potentials that the electron sees as it passes
through the sample, during image processing. Since a lens is not perfect, the image
processing modiﬁes the diffracted wave by the effects of defocus, energy spread,
spherical aberration, etc. The phase shiﬁ resulting from image processing is represented

with a contrast transfer function. This function is described in Section 5.3.

5.1 Fourier Transform

If we consider the wave function ‘I’(r) at the exit face of a sample, and ‘P(g) at the
back focal plane, then g is the reciprocal vector, and the diffraction pattern, ‘P(g) , is the

Fourier transform of the wave function at the exit face of the sample. Therefore,
ll’(g) = NO”)- T (8)

where T (g) is called the contrast transfer ﬁrnction and F is the Fourier Transform
operator. The intensity of the diffraction pattern in the back focal plane is I‘P(g)|2.

Consequently, the ﬁnal image wave function (D(r) is the inverse Fourier transform of

\I’(g). The ﬁnal image wave function will be<D(r) = ‘I’(r) ® T(r). The intensity of the

34

 

image is<D(r)-<D(r)* = |d>(r)|2.

5.2 Weak-Phase-Object Approximation

A general model of ‘I’(r) at the exit face of a sample can be described as below:
L1’0) = A(r)eXP(i¢t (x,y))
where A(r) is the amplitude and (I) t (x, y) is the phase, which depends on the thickness ff

of the sample. The A(r) can be set to unity since HRTEM utilizes phase-contrast

imaging. When the electron passes through the sample, its wavelength is modiﬁed from

 

A to 2' as given below:

 

 

A = —i— and 1'— h
VzmeE J2me(E + V(x, y,z))'

Hence, the phase change is given by:

 

E V ,.
d¢=3:d,_3zdz=adzd + 0w)-
1' ,1 ,1 JE

 

7! 7r
1) “ A—EVUayaZWZ = EV! (x,y) = 0V, (LY)

where the Vt (x, y) is the projected potential in the z-direction and 0' is the interaction

constant. As a result, the wave function at the exit face of the sample, ‘-P(r) , can be

written as T(r) = exp[i0'Vt (x,y)] .
If the sample is very thin (i. e. V t (x, y) << 1), we can expand the exponential ﬁrnction,
resulting in ‘P(r) = 1 + ith (x, y). The ﬁnal form of the wave function at the exit face of

the sample is called the weak-phase-object approximation, such that ‘P(r) is linearly
related to the potential of the sample when the sample is very thin. Generally, the TEM

sample is thin enough to use the weak-phase-object approximation.

35

5.3 Contrast Transfer Function

The contrast transfer function describes instrumental instability, defocus, imperfect
lens, etc. In real space, the contrast transfer ﬁrnction is a sinc function (i. e. point spread
function) and it is applied across the image by convolution theorem of the Fourier
transform. Hence, it produces a blurring effect limiting the resolution. In the back focal
plane, the contrast transfer function multiplies the diffracted wave and limits instrumental
resolution. The phase contrast function can be written as:

T(g) = “(3) - sin 10:)

where A(g) is the aperture function and 1(g)is the phase-distortion function describing
phase shift due to spherical aberration and defocus. The A(g) is unity when g’s are
allowed to pass the aperture and zero when the g’s are blocked. The 1(g) is affected by

spherical aberration and defocus given by:
1
x(g) = ”Aflgz + Ems/133:4

where C s is spherical aberration, 2. is the electron wavelength, g is magnitude of g , and

Af is defocus. Figure 5.1 shows a typical transfer function. The transfer function can be
optimized by adjusting Af and C s . The transfer function is oscillatory and there is a band

of good transmission. However the zeros do not contribute to the output signal.

The best transfer function can be obtained by three points of view. Firstly, the ﬁrst
crossover with g axis should be large so that a higher spatial resolution can be obtained.
The ﬁrst crossover is deﬁned as instrumental resolution limit. Secondly, the T(g) is large
so that information with a periodicity or spatial frequency corresponding to that value of

g will be strongly transmitted. Thirdly, the different values of g should have the same

36

contrasts, which means a ﬂat function at band region. As a result, all the atoms at atomic
column positions appear as either dark or bright spots. In summary, ﬂat response of the
contrast transfer function and beams as many as possible with identical phase transferred
through the optical system are ﬁrndamental principles of phase-contrast imaging in

HRTEM.

g(l/nm)

/\A
6 8V

 

Figure 5.1: An example of contrast transfer function, T(g), versus g.

37

CHAPTER 6

LITERATURE REVIEWS

6.1 Previous methods for strain analysis

Several methods have been demonstrated for measuring local deformations on
HRTEM images. The analysis of local deformations, i. e. strain analysis on HRTEM
images, can be broadly classiﬁed with two types: peak-ﬁnding methods based on
measuring the spacing of the image intensity between positions of maxima and minima
and phase techniques. A detailed description of phase technique is provided in Chapter 7.

The peak-ﬁnding method is to simply calculate spacing of the lattice fringes with sub-
pixel resolution in a given coordinate system by detecting pairs of intensity maxima. The
variation of lattice spacing can give local deformation in a speciﬁc direction. Examples of

the use of peak-ﬁnding methods include calculations of the cumulative sum (CUSUM)

[30] demonstrated for a coherently strained In0.32Alo.238b/InSb superlattice layer and

Lattice Fringe Spacing Measurements (LF SM) on the thick layers [31] and delta-doped
layers [32] using regressional analysis in a superlattice. An advanced technique using the
peak-ﬁnding method was demonstrated by Ourmazd et. a1. [33] [34] and Rosenauer et al.
[3 5]. They proposed a method that links both real and reciprocal space using real-space

vector pattern recognition, providing a 2D map of local displacement. This method was

38

demonstrated with a Stranski-Krastanov-island structure to analyze strain on an atomic

scale [36].

6.2 Regressional Analysis

Regressional analysis is one of the peak-ﬁnding methods and was developed to
overcome the artifacts at heterointerfaces on HRTEM images for strain analysis.
Regressional analysis utilizes rigid body displacement proposed by Dunin-Borkowski et
al. [32]. Rigid body displacement is deﬁned as the difference in lattice parameters
between a strained layer and a perfect crystal, where measurement of the difference in
lattice parameters is chosen far from the heterointerface. Hence, a major drawback of this
method is that one can not directly interpret strain information at the heterointerface.

This approach involves extrapolating the lattice fringes ﬁ'om regions on either side of
the layer towards it and then measuring the mismatch between them (See Figure 6.1). The
specimen thickness, defocus and imaging conditions must be unchanged between the two
regions of the image analyzed. Polynomial ﬁts were used to determine the position of
each peak. This method gives us rigid body displacement with monolayer accuracy.
Figure 6-1 describes an example of the rigid body displacement. If we assume that the
lattice constant of the reference regions is known and is 2A as shown in Figure 6-1 (b),
then the lattice constant of the strained layers (shadow region in Figure 6-1) is unknown
and it is a value we desire to calculate. In order to utilize regressional analysis we must
know the number of strained layers. In this example, we assume two monolayers of
strained layers are embedded as shown in Figure 6.1. Dashed lines in Figure 6.1 indicate

the positions of lattice planes in a perfect crystal that is extrapolated from the reference

39

regions toward the heterointerface. If measured mismatch between a dashed line and
original lattice position far from the heterointerface is 0.4A as illustrated in Figure 6.1,
then we can conclude that the lattice contraction per each strained layer is 0.2A because
two strained layers contribute the total rigid body displacement (0.4A), resulting in a
lattice constant of each strained layer of 1.8A.

Recently, groups [32] [3 7] have demonstrated local displacement at a few
monolayers with small lattice-mismatched materials (f<3%) using regressional analysis.
They calculated rigid lattice displacement. Dunin-Borkowski et al. [32] used silicon
doped GaAs for strain analysis. Based on SIMS (Secondary Ion Mass Analysis) data,
they used three different samples: 10.1 %, 4.8%, and 2.4% in silicon concentration. For
example, the theoretical strain at the silicon doped layer of 10.1% (SIMS concentration)
is around 0.74%. Hence, we can expect that strain analysis would not be able to reliably
detect the 0.74% strain. In reality, their experimental results were 50~100 times larger

than the predicted values. On the other hand, Liu et al. [3 7] investigated local

displacement at a few monolayers with an InAsxP]-x/InP superlattice where theoretical

strain at the InAsxP1-x layer was around 1.8%. However, a large error occurred and

additional Gaussian smoothing was applied to reduce this large error [3 7]. Hence, strain
analysis using regressional techniques has major challenges in dealing with a few

monolayers.

40

 

Lattice constant , _ ,
Strarned layers With two monolayers or heteromterface

J‘t
zAzlej Izlzx|2x|2xJ 2x]

 

.
é— ..'___ ..____ Q;— .I__

0.2A 0:4A 6.4/1 0.4A 0.4x

 

Position of lattice

2:151:18] in perfect Measured mismatch

Figure 6.1: Schematic diagram showing a rigid lattice contraction across strained layers.

41

CHAPTER 7

PHASE TECHNIQUE

Efforts to minimize artifacts at heterointerfaces in HRTEM images, along with digital
image processing, have provided the motivation to directly interpret strain information at
strained layers or heterointerfaces with monolayer accuracy. Efforts to minimize artifacts
in HRTEM images are addressed in Chapter 7.4 and Chapter 10. There have been two
methods that show promise for the analysis of strain at heterointerfaces with monolayer
accuracy. One is DALI (Digital Analysis of Lattice Images) software using peak-fmding
method proposed by Ourmazd et al. [33] [34] and Rosenauer et al. [36], and the other is
geometric phase analysis (GPA), using the phase technique proposed by Hytch [3 8]. The
advantage of these two methods is that they rapidly provide 2D mapping of strain
information through digital image processing, while regressional analysis can only
provide 1D mapping of strain information. In this research, the phase technique is used
exclusively to implement strain analysis, as this is a more direct method than using the

DALI software. In addition, the phase technique can be completely automated using

software capable of digital image processing, such as Matlab®.

The phase technique implemented here utilizes a moire’ pattern obtained via Fourier
ﬁltering. This approach provides highly sensitive measurement of local deformations on

2D images because the moire’ pattern effectively magniﬁes local deformations, allowing

42

analysis of deformations with sub pixel accuracy. The phase technique can be separated
into two methods: geometric phase analysis (GPA) [3 8] and Computational Fourier
Transform moiré (CFTM) [2].

Fundamental investigations using the GPA method were initially conducted on rubber
plates [39]. Subsequently, the GPA method was applied for HRTEM images by Hytch
[3 8]. The theoretical background for applying the GPA method to HRTEM images is
explained in Ref. [40], and several applications have been demonstrated on misﬁt
dislocations (MDs) [41] [42]. In addition, it has been used for the determination of
Burgers vectors [43], domain walls [40] [44], strained layers [44] [45] [46] [47],
antiphase boundaries [48], carbon nanotubes [48], high-angle annular dark-ﬁeld
(HAADF) imaging [49], and strain ﬁelds at dislocations [50].

The fundamental work for the CFTM method was conducted on solder ball connect
(SBC) assemblies [2]. The application of the CFTM method to HRTEM images has not
been previously demonstrated, even though the CFTM method requires less memory for
computation than that of the GPA method. This is achieved by avoiding unwrapping
algorithms, including line-by-line scanning to detect the 21: discontinuities on the phase
image in order to obtain strain distribution. Hence, we demonstrate the use of the CFTM
method in this work. Mathematical works are introduced at Section 8.1 for comparison

between GPA and CF TM.

7.1 Mathematical description for GPA and CFTM method

A HRTEM image is a discrete periodic image, and it can be expressed as a Fourier
series. Under the condition of a perfect crystal, the image intensity with position r can be

written as below [40]:

43

I(r>=ZHnexp(2m'gn ~r), (7.1)
n

where Hn is the Fourier coefﬁcients of the image and n represents the index of a

particular set of lattice fringes. The reciprocal vector, g, of a particular set of lattice
fringes has a length equal to the reciprocal of the lattice fringe spacing. The direction of
the reciprocal vector g is perpendicular to the lattice fringe. Subscript, g, represents each
reﬂection, i. e. a particular set of sinusoidal lattice fringes. The reciprocal vector, g, of the
particular set of lattice fringes has a length equal to the reciprocal of the lattice fringe
spacing. The direction of the reciprocal vector g is perpendicular to the lattice fringe.
When deformations are present in an original image due to a displacement u(r), the

following relationship exist [40]:
r = r - u(r) (7-2)

The expression in Equation (7-1) becomes:

1 (1‘) = 2 H n (1‘) explzmgn ° (1‘ — u(r))}, (7-3)
11

where the amplitude term, Hn(r), is also position-dependent.

The digital diffractograrn, I(k), of the deformed image I(r) can be obtained by Fourier

transform:

I(k) = [I(r)exp{—i2zzgn -r}dr, (7-4)
A

where A is the region of interest in the deformed image.

The expression in Eq. (7-4) can be expanded as below:

44

I (k) = EH n (r) exp {i27zgn - (r — u(r))}exp(—i272k - r)dr.

(7'5)
and then rewritten as follows:
I (k) = EH n (r) exp {i 272g” - r} exp {—i2ngn - u(r)}exp(—-i272k - r)dr. (7-6)
A n
I(k) = [211,1 (r) exp {—i272gn - u(r)}exp{—i27r(k — gn) - r)dr. (7-7)
A n

If a mask function M(k), centered at a reﬂection in the digital diffractogram I(k), is
introduced, then a particular set of lattice fringes can be extracted from the HRTEM
image, and displacement along a particular direction can be calculated. For instance, if a
Mask ﬁrnction M(k) is centered at a reﬂection located on the x-axis in the digital
diffractogram, lattice ﬁinges along the x-direction are extracted from the HRTEM image,

providing calculation of the displacement along the x-direction. If the reﬂection located
at the x-axis is referred to as gx, then the digital diffractogram, ng(k) is given by:
1gx(k)=1(k)M(k), (7-3)

where the mask function M(k) has a value of one within the mask area and a value of
zero outside of the mask area.

Equation 7-8 is equivalent to

ng(k)= AlHX(r)exp{—i272gx -uxIr)}epr—i27r(k—gx)-r}dr, (7-9)

where gx and ux(r) are the reciprocal vector and the displacement located along the x-

direction, respectively. H X(r) are the Fourier coefﬁcients inside of the mask area.

45

If we subtract the factor anx'r, the perfect lattice frequency, from the digital

diffractogram ng(k) by substituting

k=k—gx [2]. (7_10)
then the expression in Eq. (7-9) can be replaced with
1gx(i) = J11 X (r) exp {—iZIzgx -ux(r)}exp{—i2zi2 - r)dr, (7-1 1)

and local atomic displacement ux(r) remains in the form of anx ~ux (r) in the phase

term. The tilde (IE) indicates that the new digital diffractogram I gx (R) has a reﬂection

(i. e. the reciprocal vector gx) that is shifted to the origin with a mask centered at the

reﬂection in the digital diffractogram.

The expression in Eq. (7-11) is equivalent to

nga'l) = FlH X (r) exp {—2m'gx -ux<r>}I (m)

The inverse Fourier transform of both sides of Eq. (7-12) results in a complex image

given by
F"I1gx(i2)]=thr)exp{—2nigx -ux(r)} (m)
If we deﬁne the right side of Eq. (7-13) as Mg(r), then Mg(r) is called the conventional

moiré fringe [39] and can be expressed as

Mg (r) = HX(r) exp{—27n'gx -ux (r)} (7-14)

46

 

The phase image Pg(r) including values for local displacement ux(r) can be obtained by
argument of Eq. (7-13) [39].

Pg (r) = Phase [M g (r)} = —27zg x ~ux (r) (7-15)

_, ImIF‘tngtEm

:43“ ReIF‘IngdoI} (7'16)

 

The local displacement ux(r) can be deﬁned as

l
“MU-”mpgo‘) (7'17)

A one-dimensional strain distribution exx can be obtained from the phase image Pg(r) by

taking the derivative of the displacement, giving

8 =0ux(r)=_ 1 anO')
xx 6X 27r|gx| 6X

 

(7-18)

This method is called geometric phase analysis (GPA); where the derivative is calculated

numerically from the phase image (Pg(r)). Prior to taking the derivative, an unwrapping

process on the phase image, Pg(r), is required to remove discontinuities on the strain

distribution.

In contrast, for the CFTM method, the gradient is applied to Eq. (7-13), resulting in
F"Ii2nk1gx (k)] = —i2nV{gx 'ux(r)}H X (r) expr—zm'gx ~ux(r>} (749)
where the following differentiation theorem of Fourier transform was used:

F{M(x)'} = ianIMu?) (7-20)

47

 

Thus we obtain the expression:

—F“If£1gx(i)1
F"Ing<iE)I

 

Wax -ux(r)} = (7-21)

Equation 7-21 is equivalent to

— F"[K1gx(§)]
F"Ilgx(E)I ’

 

V{lgxlux(r)} = (7-22)

because the reciprocal vector gx , the perfect lattice frequency, is constant.

Therefore, equation 7-22 is equivalent to
F" IE1 g. (in
lgxlF"[1gx(k)]

 

vru. (r)} = (7-23)

The x—direction derivative of the displacement ux(r), using directional derivatives, is exx
and given by:

aux(r) _ — F"[E-nllgx(l7)]
6x nglF"Ing(E)I

 

 

exx = n] -V{ux(r)} = (7-24)

where n1 is a unit vector along the x-axis. Therefore, the CFTM method does not use the

phase image, Pg(r), and it is a direct method to calculate the strain distribution. Since the

CFTM method does not use the phase image, an unwrapping algorithm to remove phase
discontinuities on the phase image, Pg(r), is not required. (Compare Eq. (7-18) and (7-

24)). Similarly, strain tensor of CFTM method can easily be calculated as below:

48

 

‘3 ux (r) a ux (r)

e: exx ex); = 6x a);
eyx 9);); auy(r) auy(r) ,

 

 

 

 

 

 

(7-25)
6x 6y
where rest of each component is
6u (r) —F"If£-n21gx<i)1
= -V = x = ~ 7-26
ex), n2 {ux(r)} 6y ng|F_,ng(k)] ( )
au (r) —F"[E-n I (11)]
«aw—n1 V{uy(r)}- gx = I IF"[II 8:32)] (7-27)
gy gr
au (r) —F"[i2-n I 02)]
eyy =n2-V{uy(r)}= y - 2 gy (7-28)

6y lgylF“ [Ignién
where n; is a unit vector along the y-axis.

The digital diffractogram I gy (R) is obtained by a reﬂection whose an initial position is y-

axis before shifting to origin in the digital diffractogram. The “gy” represents a reciprocal

vector located in y-axis at the digital diffractogram.

For calculations with digital images the coordinates, r, refer to the coordinates of

pixels in the image. The reciprocal lattice vectors, gx and gy, appear in the equations, are

in pixels". This means that the coordinates of the reciprocal lattice vectors in the
numerical Fourier transform (with respect to the position k=0) must be divided by the
size of the original image. Strain tensor equations for digital image processing for

Matlab® are

49

-l kx ~
aux(r)__F [ngxm]

 

 

 

 

 

 

 

 

exx=n1-V{llx(r)}= ax — Bib—1U (11)] (729)
N gx
a, (r) —F“Ifllgx<ié)1
exy=n2-V{ux(r)}= gy = lgxl 1:4 ~ (730)
—N——F [ng(k)]
5., (r) —F"I'i‘-’-‘Igy(E)I i'l
eyx=nl'V{“y(r)}‘-‘ I; = lgyl N ,. (7-31)
-M~—F"I1gy(k)1
ex» = "2 -V{uy(r)} = all?) = .21" [”1900], (7-32) j

-1 ~
7 F [lgy (kn
where N and M are size of an original image along x-direction and y-direction

respectively with unit of pixel.

7.2 Interpretation of phase image

This section describes the meaning of phase ﬁinges in a phase image so that we can

interpret phase fringes in a phase image, Pg(r), created from GPA method. The phase
term, g . u(r) , in a phase image varies between 0 and 1, resulting in the term 272g - u(r)
varying between 0 and 27:. This is the reason a phase image Pg(r) has 21: discontinuities

on the phase image. Given a particular set of sinusoidal lattice fringes, i. e. a particular set
of atomic planes, a reciprocal vector (g) in a perfect crystal and a displacement vector

u(r) can be drawn within a pair of atomic planes as shown in Figure 7-1. The length of

50

the reciprocal vector g is equal to l/d, where d is the lattice fringe spacing in the
reference region (i.e. perfect crystal). By trigonometry, the projection component of u(r)

along a reciprocal vector g of the perfect crystal (i.e. reference lattice) is lu(r)| cosa as

shown in Figure 7.1 and it can be written with the dot product since this is the projection
of one vector on the other as:
1
|u(r)| cosa = IE g - u(r)
(7-33)
Since the length of the reciprocal vector g is equal to the reciprocal of the lattice ﬁinge
spacing, d, in the perfect crystal (i.e. reference lattice), Eq. 7-33 is

|u(r)l cosa = lilg - u(r) = —:—g - u(r) = dg - u(r).
g 2,“ (7-34)

In equation 7-34, the projection component of u(r) along a reciprocal vector g of the
perfect crystal, varies between 0 and d as below:

0 S dg - u(r) S d. (7-35)
By dividing by the lattice fringe spacing, d, and multiplying by 27: we obtain:

0 S 271g - u(r) S 27:, . (7-36)
such that a cycle of phase fringes, 2n discontinuity, in a phage image appears whenever
displacement on a HRTEM image is equal to the pitch of the reference lattice (i.e. lattice
fringe spacing, d), if the center of a Bragg reﬂection is exactly shifted to the origin in the

digital diffractogram.

51

 

 

 

|u(r)| cosa

 

 

d
Lattice ﬁinge spacing

A
V

Figure 7.1: Schematic diagram of displacement ﬁeld within a pair of atomic planes.
Position A and B indicates an initial position and its position after
deformation, which is represented by a displacement vector u(r). g is a
reciprocal vector in perfect crystal. Projection component of u(r) along a

reciprocal vector g of perfect crystal (i.e. reference lattice) is |u(r)| cosa.

7.3 Previous applications with GPA method

The reason that it is called Geometric Phase Analysis is that phase reconstructed from
the GPA method from HRTEM images does not correspond to the phase of the beams
emerging from the specimen, but corresponds to the information associated with the
positions of the fringes in the HRTEM images. In other words, the phase techniques such
as the GPA and CFTM methods do not rely on knowledge of the positions of atomic
columns in the crystalline specimen. It is based on the assumption of a constant spatial

relationship between the positions of fringes in the HRTEM images and the positions of

atomic columns in the crystalline specimen. The relationship between phase (Pg; and

52

 

sz) and displacement ﬁeld (ux and uy) using two non-colinear Bragg reﬂections (g1 and

g2) is [40]:
P g g u
g1 =_2 1x 1y x (7_37)
P g g u
g2 2x 2y y
.1 P
“x 1 glx gly g1
u ="2— g g P (738)
y 7’ 2x 2y g2

where glx and g] y are the x-component and y-component of a reciprocal vector (g1).
Similarly, g2x and g2y are the x-component and y-component of a reciprocal vector (g2).

The strain ﬁeld using the displacement ﬁeld is expressed [40]:

 

 

/ aux aux) ( anl anl \
e:[exx exy]: a? a? =_L[glx egT 6x 6y (7-39)
eyx 8” —_y -_y- 271' 82x 82y £9sz (3sz
\ 6x 6y 1 \ 6x 6y /

 

 

 

 

where the derivative is calculated numerically. Before taking the derivative, an
unwrapping process is required to obtain the strain ﬁeld.

Phase techniques can be used for various applications as well as determination of
local displacement at strained layers [44] [45] [46] [47]. Strain ﬁeld analysis on the
domain wall in PbTiO3 [40] demonstrated interchanging of lattice planes and rotation
angles between two domains. Asymmetric abruptness of the interfaces at a GaAs/AlAs
superlattice [45] was also demonstrated. It is also valuable to detect misﬁt dislocations,
since abrupt phase changes in a phase image appear due to an additional lattice plane
starting at the dislocations [50]. Other applications include calculating strain ﬁeld on

threading dislocations [42], edge dislocation [41], and boundaries of a quantum dot [51].

53

The dislocation core distribution tensor (5 ) [43] can be calculated by further

differentiation of the lattice distortion tensor ( ,8) as below:

(aux aux)

 

 

Ei=—curl ,B=—curl a? 6?
__X _X
K 33‘ 5y)

The tensor ((3?) vanishes in the whole region except inside of the dislocation core where

' :11?!

it forms characteristic peaks. Integration of the ail-(x, y) values gives us calculation of in-

 

plane Burgers vector, bi = - j aijds’ where Se is dislocation core surface. The
Sc

determination of the Burgers vector can be used to calculate interaction of dislocations
and information about the position of the maximum of the distortion at dislocations to

determine the character of the dislocation core [43].

7.4 Efforts to minimize error for strain analysis

The phase technique and peak-ﬁnding methods such as cumulative sum (CUSUM)
[30], Lattice Fringe Spacing Measurements (LFSM) [31], regressional analysis [32], and
DALI (Digital Analysis of Lattice Images) [36] extract strain information based on
positions of lattice fringes on a HRTEM image. However, the positions of ﬁinges in the
image do not have to correspond to atomic plane positions in a specimen. For instance,
image contrast on HRTEM images depend on specimen thickness, spherical aberration,
and lattice constant, etc. In order to minimize the difference between positions of atomic
planes in a specimen and lattice fringes in the image, several efforts are suggested. An

electron beam as large as possible is suggested to minimize electron-beam tilt [52]. A ﬂat

54

response of the contrast transfer function (CTF) is required to transfer the identical phase.
Achieving a ﬂat response for the contrast transfer function (CTF) will also minimize
inversion of contrast at interfaces where the structure varies rapidly. The best way to
minimize delocalization [53] is to use spherical aberration corrected (Cs-corrected)
transmission electron microscopy [54]. Contamination or amorphous surface layers at a
specimen must be minimized. It should be noted that precise alignment of the electron
microscope including current and voltage centering of the objective lens and correction of
astigmatism at high magniﬁcation should also be performed. The efforts to minimize
artifacts mentioned above will provide high-quality HRTEM images such that the atomic

column positions on HRTEM images correspond to those of the crystal at a specimen.

55

CHAPTER 8

EXPERIMENTAL RESULTS.

The primary objective of this chapter is strain analysis at the heterointerface of III-V
ternary alloys with ultra—thin GaAsP/GaAs superlattices, which can be used for optical
devices operating under the visible and IR spectrum. The motivation for strain analysis
on the GaAsP/GaAs heterointerface is that strain analysis for this system has not been
previously demonstrated due to the small strain value present. Such small lattice-
mismatches increase the critical thickness of the ﬁlms, which is the thickness above
which dislocations are likely to occur. In this research effort, we address issues
associated with strain measurement for small strain (f<3%) at heteroepitaxial ﬁlms using

the phase techniques known as the GPA and CFTM methods.

8.1 MBE growth and TEM studies

A superlattice with GaAsP/GaAs (190A) was grown on a GaAs (001) substrate by
MBE. The experimental details are provided in Ref. [55]. The results of HRXRD

dynamical simulations revealed that the composition of the ultra-thin GaAsP was

G3ASQ86POJ4. This sample was grown at 520°C with an As beam equivalent pressure

(BEP) of4><10'6 Torr and a Ga BEP of 1.5><1o'7 Torr. The GaAso.36P0.14 layer was

56

grown by exposing an As-stabilized GaAs surface to phosphorous for 50 seconds at a P2
BEP of 4><10-6 Torr, producing a few monolayers. This process was repeated, resulting

in a strained layer superlattice.

A TEM sample was prepared using the conventional cross-sectional method:
grinding, dimpling, and Ar ion milling. Low angle (3°) Ar ion milling was used to obtain
a comparatively uniform thickness over the superlattice. For the HRTEM imaging, a
JEOL 2200FS equipped with double-tilt specimen holder, operating at 200kV, was used

with Cs=0.5 mm, where the point resolution is 0.19nm.

8.2 Ultra thin GaAsoMPo,“ layer embedded in GaAs: Theoretical values of strain

The lattice constant parallel to the growth direction for the GaASO.86P0_]4 layer can

be calculated using the Vegard’s law approximation:

"(3(2713036100l4 = aGaAs ‘ xP(aGaAs ‘ aGaP)’ (84)
where xp is phosphorus composition and equal to 0.14. The lattice constants aGaAs and
aGap are 0.565nm 0.545nm, respectively. Hence, the lattice constant parallel to the
growth direction for the GaAsO_86P0_14 layer is 0.5622nm. Lattice distortion components

parallel to the grth direction, 1'. e. theoretical strain exx, can be deﬁned as below:

8 = “2 ‘a1 = aGa0.86A30.14 —aGaAs = “1 "down
x 9
“1 aGaAs aGaAs (3‘2)

57

where az and a1 are the lattice constants parallel to the growth direction for the GaAsO,36
P0314 layer and GaAs, respectively.

Under the assumption of pseudomorphic growth, the GaAsO.g6P0_14 layer is further

constrained to match the host lattice at GaAs in the plane parallel to the interface as

predicted by elasticity theory. Thus the lattice parameter of the GaAso336P0314 layer in a

_ J!

direction parallel to the growth direction is given by:

1+0

“1 = aGaAs ‘ —1 _ U (aGaAs ‘ aGaAso 86100 1 4), (8_3)

 

where u is Poisson’s ratio and (1+u)/(1~u) is a factor of contraction (or extension) of the
unit cell in a direction parallel to the growth direction when the unit cell parallel to the
interface direction is extended (or contracted) due to coherent strain[56]. The reason that
we include the factor, (1 +u)/(l - u) is that the biaxial in-plane tensile strain, due to the

GaP bonding, results in a uniaxial compressive stress along the growth direction at the

GaAso_36P0314 layer, such that we can presume that the superlattice is deformed

tetragonally [57]. Poisson’s ratio for both GaAs [58] and GaP [59] is 0.31. In Eq. (8-3),
we ignored surface relaxation of the strained layer for cross-sectional TEM because the

strained layer we investigated was one period of a 10-period superlattice grown near the

substrate and capped with a GaAs layer. Furthermore, the thickness of the GaAso.g6P0.14

layer is very small (< 6A) with respect to that of the GaAs superlattice. Consequently, the

lattice constant parallel to the growth direction of the GaAso.g6P0.14 layer is 0.559nm

58

when considering elasticity theory. Hence, the lattice distortion component parallel to the

growth direction exx is given by -1.0% using Eq. (8-2).

8.3 Strain analysis using the CFTM method

A low magniﬁcation image of a superlattice with GaAsP/GaAs (190A) grown on a
GaAs (001) substrate is shown in Figure 8-1 (a). The cross-sectional HRTEM image

taken along [110] zone axis is given in Figure 8-1 (b). The resolution is 7 pixels/ (002)

lattice fringe. The ultra-thin GaAsO.86P0.14 layer is located in the middle of this image.

By comparing the HRTEM image with electron microscopy software (EMS) [60], where
EMS is software for simulation and analysis of HRTEM images, the white spots in the
HRTEM image were determined to be columns of Ga and As pairs. Arrows on the
HRTEM image depict the growth direction (x-direction) and the interface direction (y-
direction). The heterointerface was grown coherently with strained layers because the
lattices are well aligned. Hence, we can assume that the strained layer is tetragonally
distorted, i. e. a pseudomorphic layer [57] [61]. In addition, we can expect that the proﬁle
of phosphorus will affect local deformations at the strained layer, since phosphorus will
be diffused into the GaAs layers.

As can be judged from Figure 8-1 (b), the HRTEM system at our disposal cannot

provide chemical contrast for GaAsO.g6P0_14 on GaAs because the difference in projected
potential between GaAsO_g6P0_14 and GaAs is small and structural variation does not take

place across GaAs0.36P0.14 layers. Since contrast in Figure 8-1 (h) does not directly relate

to the presence of phosphorus, the phase technique independent of the image contrast can

59

be used to detect GaAso.36P0314 and obtain strain distribution through the difference of

lattice constants across the layers.

Y‘l-“Ol
Interface direction

>

x=[001]
Growth direction

 

y=l-110l

 

Interface direction
§ (b) F1001]
‘g . Growth direction
*3 ‘.."I‘LlZQ‘.--Z
a . .. . . . .
8
3’ 100 200 300 400 500 600
g Positionalongx—dre-ction

Figure 8.1: TEM images viewed on [110] ZA. (a) A cross-sectional TEM image with low
magniﬁcation of a strained GaAso.86P0314/GaAs superlattice. (b) A cross-

sectional HRTEM image (XTEM) of a strained GaAso.36PO314 /GaAs
heterointerface taken along [110] zone axis. Image size is 108 by 650 pixels
and magniﬁcation is 0.031nm/pixel (Instrumental magniﬁcation of 600,000 X).

The CF TM method was applied to evaluate strain on the HRTEM image. All work
related with strain analysis in this research has been implemented using Matlab®. In the
ﬁrst stage, the original image was Fourier-transformed to obtain the digital diffractogram
I(k) as shown in Figure 8-2. Since in practice, local deformations along the growth
direction (x-direction) are the primary concern, an (002) reﬂection should be chosen, as it

is the ﬁrst harmonic along the x-direction and has the largest intensity.

6O

In the second stage, Fourier ﬁltering using a Gaussian mask centered at the (002)

reﬂection was performed. The Gaussian mask [40] we used is below,

2
M (k) = exp{—47r 52‘} (8-4)
g
and the mask size is g002/3.

(06-2) .

Y (Interface direction)

LX (Growth direction)

 

Figure 8.2: A digital difﬁactogram of the original HRTEM image in Figure 840)).

Figure 8-3 shows the strain proﬁle (exx) obtained using Eq. (7-24). This proﬁle was

vertically averaged along the direction parallel to the interface from a 2D strain image
(not shown here). Vertical averaging from a 2D strain image allows randomly distributed
noise on the image to be minimized. In Figure 8-3, we can clearly see two regions where
overshoot occurs. These overshoots are exhibited as a rippled appearance [46] and
provide evidence of the presence of a strained layer [46] because overshoot and
undershoot are typical phenomena that occur at strained layers. In addition, ﬂuctuations

are present in the reference regions (GaAs layers). It is not feasible to determine what

61

created the ﬂuctuations in the reference regions. However, we consider ﬂuctuations of

$0.7 % to be within an acceptable range for the reference region.

The results of Figure 8-3 show a lattice contraction of -4.9% at the strained
GaAsO.36P0.14 layers in the (001) direction. This is around ﬁve times larger than the

theoretical value (-1%) determined in the Chapter 8.2. To understand what causes this
large error from the initial estimate, we can assume that error comes from the phase
technique because accuracy of the phase technique at ultra-thin layers with small strain

has not been demonstrated. Error from the phase technique is address in Chapter 9.

 

 

    

 

' r ' r r 1 r r a I ' r ' ﬁ ' I ' T '
4 _ Ultra-thin GaAsP layer ..
2 - Overshoot Overshoot ‘
,
$3 0 -W\N/\\/\ W
(2
.§ -2 - _
1;,- I Measured strain
E _4 _ at straIned layers ‘
E , II oUndershoot
&> -6 _ -4.9 /o _
-3 _ _

exx

 

 

 

50 100 150 200 250 300 350 400 450 500 550
Position along x-direction [pixel]

Figure 8.3: Strain proﬁle (exx) using the CFTM method of the experimental HRTEM
image with mask size (g002/3).

62

Before investigating error from the phase technique, strain at the GaAsO336P0. 1 4

layers of the HRTEM image was measured as a function of mask size. During the Fourier
ﬁltering, mask size will determine the rippled appearance (overshoot and undershoot) of
the strain proﬁle because the mask size is related with the effective spatial and spectral
resolution. Therefore, smoother strain proﬁles can be obtained with reduced mask size;
thus we gain spatial information but at the cost of lost spectral resolution [46]. However,
it should be noted that we are primarily concerned with the strain at the center of the
strained layers instead of sharpness of the strain proﬁle across the strained layers. Hence,
we do not have to be concerned with the spectral resolution in this research. It is worth
restating that the strain at the center of the strained layers can be studied by reducing the
mask size in order to remove error in the form of rippled appearance such as overshoot
and undershoot. One weakness to this approach is that we may not detect asymmetry of
the strain proﬁle due to the small mask size, if it exists at the strained layers. Asymmetry
of the strain proﬁle is plausible due to asymmetry of the phosphorous composition along

the growth direction, which occurs when phosphorus is primarily diffused into the GaAs.

Characterization of error at GaASO_86PO.]4 strained layers has been investigated

according to mask size. Figure 8.4 (a) shows the measured strain at GaAso.36P0.14

strained layers according to mask size, where the radius of a Gaussian mask was varied
(See x-axis of Figure 8.4 (a)). A mask size of g/3 and g/2 corresponds to a radius of 30
and 45 pixels, respectively. The radius of the Gaussian mask is plotted from 3 to 40
pixels. It was observed that the magnitude of the measured strain is linearly increasing

with mask size in region-3. The discrepancy of measured strain at GaAsP strained layers

63

(i. e. -4.9% at g/3 mask size) is caused by undershoot (See Figure 8.3). We can assume
then that region-3 is primarily due to error in the form of a rippled appearance.

In addition, Figure 8.4 (a) shows that the measured strain gradually approaches -1.5%
at region-l and saturates for a while at region-2. The cause of saturation (region-2) in
Figure 8.4 (a) can be explained by the following observation: measured strain at strained
layers is saturated when the radius of the Gaussian mask is close to the actual radius of
the (002) reﬂection, i. e. around 12~15 pixels in this case. The cross-sectional proﬁles of
the (002) reﬂection are shown in Figure 8.4 (b) for estimate of size of the (002)
reﬂection. This implies that the strain information ranges over the width of the (002)
reﬂection. Therefore, the assumption that error occurred and contributed to the result of

Figure 8.3 is valid, and the error could be removed using a smaller mask size. The
resulting measured strain at GaAsO.36P0_14 strained layers would be around 1.5% lattice

contraction. We emphasize here that error in the form of rippled appearance, which is
linearly dependent on mask size, has not been previously reported. We suggest that this
error is related to the Fourier leakage effect, which is demonstrated through image

simulations at Chapter 9.

64

 

I ‘ l ' I ' l ' I ' l ' l ‘ l ‘ l

, Region-1 —I~— Measured strain d
-1 _ Region-2 _

I.

g§'-2_ 1“ q
:2: _4_ “I‘Regron-B d
h r 4
a -5- “I ~
g . ‘n .
2 -6‘ (a) t “\I‘ r

-7‘ g/3 " -

 

 

-8 . . . . . , . , . . . . T 1 . , . , .
0 4 8 12 16 20 24 28 32 36 40
Mask size (Radius of a Gaussian mask) [pixel]

 

800k ' l ' I ﬁ' T f ‘I f I ' I T I

 

 

700k~ ' ~

600k - -

(b)

500k - _

400k - e -
l \ .
300k - ..

1 ~15 pixels
200k I

Intensity [arb. units]

 

100k- . ., .,\
d I. . ./ \./ \ . . II
l-.;......’,. \I'.‘ / \I/ .' \ / ‘0"....., .0. , .4

0" ' I ' I ' I I ' I
-20 -15 -10 -5 0 5 1O 15 20

Cross-sectional proﬁle of (002) reﬂection [pixel]

 

 

 

Figure 8.4: (a) Measured strain at the GaAsO.36P0.14 layers according to mask size. The
x-axis is the radius of the Gaussian mask. (b) A cross-sectional proﬁle of the
(002) reﬂection along the x-direction. The radius of the (002) reﬂection, along
the x-direction is around 15 pixels.

65

CHAPTER 9

IMAGE SIMULATIONS

Our experimental results revealed that the measured strain linearly increases with
increasing mask size. Further details on error, when the phase technique is applied, can
be divided into two types: alias effect and leakage effect. The alias effect is related with
sampling. The phase technique starts with a digitized image that is taken using a Charge
Coupled Device (CCD) in the image plane of a microscope. Therefore, lattice planes in a
specimen are sampled with a CCD camera, resulting in lattice fringes on HRTEM
images. Typical resolution of the CCD camera is 1024 by 1024 pixels in the ﬁeld of
view. When the sampling frequency during digitization of a CCD camera is smaller than
the Nyquist frequency, the alias effect can occur. However, we found that the alias effect
does not occur if fringe spacing is larger than three pixels. Therefore, the reason for error
linearly increasing with increasing mask size can be related with the leakage effect, since

the leakage effect cannot be avoided when the phase technique is applied.

9.1 Leakage effect

The leakage effect is associated with the Fourier transform and cannot be avoided.
Therefore, the leakage effect limits accuracy of the phase technique since the phase

technique utilizes Fourier transform. The leakage effect occurs when the size of an image

66

is not an integer multiple of the pitch of the periodic crystal structure in the original
image. Since most HRTEM images are not commensurate, where commensurate means
an integer number of periods of the crystal structure ﬁt within the ﬁeld of view. The
leakage effect causes a spectral “smear” located about a center frequency, along with
spurious high frequency noise in the spectral space that results in rippled appearance such
as overshoot and undershoot, where the lattice constant abruptly changes in the spatial
domain. We demonstrate the rippled appearance caused by the leakage effect using
computer-generated images to investigate error from the phase technique. We have
simulated and measured the strain as a ﬁrnction of mask size for two cases of lattice

contraction: 2% and 10%.

9.2 Impact of the leakage effect

Figure 9.1 shows a one-dimensional lattice ﬁinge constructed using a sinusoidal
function, where the image resolution is the same as Figure 9-1 (i.e. 7 pixels per ﬁinge
spacing). In addition, the image size for all simulations in Chapter 9.2 is the same as the
experimental image in Figure 9-1 (108 by 650 pixels). The pitch along the x-direction is
different at unstrained and strained layers. The pitch of unstrained layers (reference
region) and strained layers are 7 pixels and 6.86 pixels, respectively. This represents a
2% lattice contraction at the strained layers along the x-direction. The strained layers

have two monolayers and are placed in the middle of the image as shown in Figure 9-1.

67

Strained layers with two monolayers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E
U) .5
g 2: 20 II
E c 40 Y=[-110]

.9 60 .
S U l I I
£- 2138 =[001lj
ID "" ,
DC: a 100 200 300 400 500 600

Postion along x-direction [pixel]

Figure 9.1: Computer-generated lattice fringes using sinusoidal functions. Arrows
indicate coordinate system of x and y axes, corresponding to grth and
interface directions of Figure 1. A digital diffractogram is shown in the inset.
The inset shows the ﬁrst harmonic on the x-axis that can be used for strain
analysis using the phase technique.

In implementing the phase technique for strain analysis, the same procedures have

been conducted using the ﬁrst harmonic shown in the inset of Figure 9.1. Figure 9.2

shows exx proﬁles using the CF TM method according to different mask sizes when the

lattice constants of strained layers positioned in the middle of an original image are 2%
and 10% smaller than the reference region. The mask size was varied up to g/2 for a
Gaussian mask. We observed a large discrepancy for the measured strain in the case of
2% lattice contraction as shown in Figure 9.2 (a) in the form of undershoot. In order to
clearly see the measured strain as a function of mask size, strain was measured with
different mask sizes for 2% and 10% lattice contraction.

Figure 9.3 and Figure 9.4 show the measured strain in the case of 2% and 10% lattice
contraction, along with the effects of different image resolutions. The mask size was
varied up to g/2 with a Gaussian mask. Different image resolutions (number of pixels per

ﬁinge spacing) were plotted together. It should be noted that as we increase

68 ’

 

magniﬁcation of a microscope, i.e. enlargement of the fringe spacing or improvement of
image resolution on HRTEM images, reﬂections will move towards the origin in
reciprocal space for images having the same size. Therefore, the magnitude of a
reciprocal vector g (distance between origin and a reﬂection in reciprocal space) is
smaller and smaller as we improve image resolution of the HRTEM images. In fact the
magnitude of a reciprocal vector g is equal to the reciprocal of the fringe spacing.
Therefore, the magnitude of g/2 gets smaller and smaller with increasing image
resolution (number of pixels per fringe spacing) as shown in Figure 9.3 and Figure 9.4, i

where the mask size (Gaussian radius) was varied from 4 pixels to g/2.

 

First, we observed that the linear dependence of error as a function of mask size
occurs only for the small strained (2% lattice contraction) systems as shown in Figure 9.3
(a). Error in the form of undershoot should be caused by phase technique because
artifacts from an electron microscope are not included in the simulated images. It veriﬁes
that the leakage effect during digital image processing creates error when the phase
technique is applied for strain analysis of small-strained systems at ultra-thin layers.

One way to minimize the leakage effects linear dependence on mask size is to
increase the image resolution. We can clearly see that the error is reduced with increasing
resolution of the image, as shown in Figure 9.3 (a). This demonstrates that the spurious
high frequency term in the spectral domain due to the leakage effect can be minimized
with an increase of image resolution.

Second, we can see that saturation occurs in region-2 when the mask size is close to
the size of the ﬁrst harmonic, i. e. 15 pixels, as shown in Figure 9.3 (a) and (b). The cross-

sectional proﬁles of the ﬁrst harmonic are shown in Figure 9.3 (b) for estimate of size of

69

the ﬁrst harmonic. This implies that the strain information is mainly positioned at or near
the center of the ﬁrst harmonic.

From comparative analysis of Figure 9.3 (a) and Figure 8.4 (a), it is observed that the

leakage effect at the GaAso_36P0.14 strained layers contributed to the linearly dependent

error for mask sizes in region-3 of Figure 8.4 (a). Clearly, the leakage effect dominates
for small-strained systems (f<3%) when the image resolution is 7 pixels per fringe
spacing. In order to reduce error at the strained layers, one can increase the image
resolution. Otherwise, only higher strained systems can use a resolution of 7 pixels per
fringe spacing for reliable analysis as shown in Figure 9.4 (a). At 10% lattice contraction
of the strained layers, reliable results were obtained with an image resolution of 7 pixels
per fringe spacing. An alternate way to interpret small strain values for HRTEM images
with 7 pixels per ﬁinge spacing is to reduce the mask size until mask size is close to the
size of the ﬁrst harmonic used for phase technique because the strain information is

mainly positioned at or near center of the ﬁrst harmonic. To sum up, we conclude that

strain of the GaAsO_36P0.14 strained layers would be around -1 .5% at Figure 8.4 (a) using

the alternate way (i.e. reducing mask size until it reaches size of (002) reﬂection) to
eliminate spurious high frequency term caused by leakage effect. However, more work is
needed to determine accuracy of the phase technique when mask size is reduced until size
of the ﬁrst harmonic in reciprocal space. This is the best way to increase image resolution
to determine small strain at ultra-thin layers when the phase technique is applied.

Now we compare strain analysis between the CF TM method and the GPA method on

the GaAso.86Po.14/GaAs heterostructure. In order to directly compare the CFTM and

70

GPA methods, strain proﬁles from both techniques are plotted together as shown in
Figure 9.5. From Figure 9.5, we can clearly see that the two proﬁles are exactly matched
such that the CFTM method is conﬁrmed experimentally by excellent agreement between

the CFTM and GPA methods.

9.3 Reliability of phase technique

Strain analysis at heterointerfaces that generally are composed of a few monolayers
requires methodology having capability to interpret monolayer accuracy. Therefore, after
the phase technique for HRTEM images was developed in 1993, the primary concern has
been the reliability of the phase technique in monolayer accuracy for strain analysis at
heterointerfaces. However, reliability of the phase technique in monolayer accuracy has
not been investigated yet. In this research, error of the phase technique is quantiﬁed for
monolayer accuracy. Two monolayers of strained layers were constructed using Matlab®
assuming that in-plane lattice constant (y-direction) is matched and lattice constant of
growth direction (x-direction) is strained such as coherent growth. Strained layers of two
monolayers are inserted in the middle of an image (See Figure 9.6). The procedures used

to produce the image simulations are provided in the following Section 9.3.1.

71

Figure 9.2:

 

 

 

 

 

Relative strain [%]

'3 ‘ Undershoot _. 9,3 ‘

 

(a) —> g/2

 

 

r

 

260 I 360 ' 460 ' 560 V 600

Position along x-direction [pixel]

o '160

 

 

 

Relative strain [%]

 

(b)
o '160'2601360'460'560'650

Position along x-direction [pixel]

 

 

 

Strain proﬁle, exx, of the simulations for (a) 2% and (b) 10% lattice
contraction at strained layers with two monolayers. Image resolution is 7
pixels per lattice fringe spacing.

72

 

  
  
  

Region-3

Measured strain [%]

' + 7 pixles/ fringe spacing
_4 _ —o— 8 pixels/ fringe spacing
—A— 9 pixels/ fringe spacing
I + 10 pixels/ fringe spacig
—o— 12 pixels/ fringe spacing

 

 

 

0'110'2'0'3'0'4'0'50'60
Mask size (Radius of a Gaussian mask) [pixel]

18.0k.,.........
16.0k3 ---------------------------------------------------------------------------------------- l
14.0k4' '
12.0k3
10.0k-‘
8.0k-I
6.0K:
4.0K;
2.0K:
0.0‘

'2.0k ' I r l ' I w r ' I ' I ' I '
-40 -30 -20 -10 0 10 20 30 40

Cross-sectional proﬁle of the ﬁrst harmonic [pixel]

 

 

 

 

 

Intensity [arb. units]

 

 

 

 

 

 

Figure 9.3: (a) Measured strain according to mask size and image resolution (number of
pixels per ﬁinge spacing) when applied strain at two monolayers of strained

layers is -2%. (b) A cross-sectional proﬁle of the ﬁrst harmonic used for phase
technique.

73

 

0 r I ' l ‘ I ' l ' T
d

+ 7 pixels/ fringe spacing

-2 - —o-— 8 pixels/ fringe spacing e
. —A— 9 pixels/ fringe spacing
_4 _ \ _ —v— 10 pixels/ fringe spacing _

—o— 12 pixels/ fringe spacing

Measured strain [%]
i:

 

 

 

-144,.,.,.,.,.
01020 3040 5060

 

18.0k
16.0kl
14.0kl
12.0kl
10.0k:
8.0k-‘
6.0k:
4.0K:
2.0k5
0.0‘
-2.0kJ....-......,...
40 -3o -20 -1o 0 10 20 30 4o
Cross-sectional proﬁle of the ﬁrst harmonic [pixel]

 

Intensity [arb. units]

 

 

 

 

 

 

Figure 9.4: Measured strain according to mask size and image resolution (number of
pixels per fringe spacing) when applied strain at two monolayers of strained

layers is -10%. (b) A cross-sectional proﬁle of the ﬁrst harmonic used for
phase technique.

74

 

2- a
a 0 -
.s '
g '2- -
(n 4
3 4- -
u
Lu .
(D
a: -Ge -

‘ —— exx using the CFTM method ‘
’8 _ + exx using the GPA method '-

 

 

‘10 ' I ' I ' I f I T ' I ' I ' I ' I .
50 100 150 200 250 300 350 400 450 500 550
Position along x-direction [pixel]

 

Figure 9.5: Comparison of strain proﬁles on the GaAsP/GaAs heterostructure using
CFTM and GPA method.

9.3.1 Procedures of image simulations

In this research, a simulation method using Matlab® for determining errors of the
strain values when the phase technique is applied has been proposed and the error of
phase technique was quantiﬁed. While the image size of all images for simulations was
maintained with around 620 by 620 pixels, two parameters are applied on a 2D lattice
image generated with Matlab® for simulations to determine errors of the strain values
with phase technique. One is a value of applied strain at strained layers with respect to
the reference regions on a 2D lattice image and the other is image resolution (pixels per
fringe spacing). In these simulations, the parameter of image resolution was applied

between 7 and 12 pixels because, in practice, the resolution (number of pixels per fringe

75

 

spacing) of a HRTEM image depends upon the lattice constant of the crystal structure
and is generally between 7 and 12 pixels per ﬁinge spacing (See Table 9.1). The
following procedures describe steps for the simulation method and coding of the
simulation method is provided at appendices B and C.

$3le An image with a 2D lattice was generated using a sinusoidal function along both
x and y directions, resulting in crossing lattice fringes displaying dot contrast (See Figure
9.6 (a)). Pitches along the y-direction are constant over the image. Pitches along the x-
direction are different at unstrained and strained layers. The pitch of unstrained layers
(reference region) and strained layers are 20 pixels and 18 pixels respectively. This
represents a 10% lattice contraction at strained layers along the x-direction. The strained
layers are placed in the middle of the image as shown in Figure 9.6 (a). The image size is
620 by 620 pixels.

M: Fourier transform was performed on the original image, resulting in a digital
diffractogram shown in the inset of Figure 9.6 (a).

Step 3: Fourier ﬁltering on the diffractogram was performed using a Gaussian mask

centered on the ﬁrst harmonic placed along x-direction for exx. The ﬁrst harmonic was

marked with rectangular box.
Step 4: The Bragg-ﬁltered lattice fringe, shown in Figure 9.6 (b), was obtained by

inverse Fourier transform using the ﬁrst harmonic without shifting the ﬁrst harmonic to

the origin in the diffractogram. An unwrapped phase image Pg (X) was obtained using

GPA method as shown in Figure 9.6 (c).

Step 5: Strain proﬁle exx shown in Figure 9.7 was obtained using Eq. 7.24 where g/3 was

76

selected for mask size of a Gaussian function in reciprocal space. A Gaussian mask
provides a smoothing effect in real space such that the strain proﬁle is not sharp
compared with a theoretical proﬁle (See Figure 9.7). The ringing effect or rippled
appearance that occurs in the spatial domain when we use the phase technique is caused
by abrupt spectral truncation in the frequency domain such as circular, rectangular or
diamond mask with sharp edges. It can be eliminated by use of a more gradual
attenuation such as a Gaussian mask, resulting in no rippled appearance in Figure 9.7. A
Gaussian mask loses high-frequency detail, leading to rounded edges; however, a
Gaussian mask is suggested to remove ring effect or rippled appearance for monolayer
accuracy.

S_te;L6_: Finally, values of measured strain were determined through various simulations
in terms of applied strain at strained layers and image resolution (i.e. number of pixels

per fringe spacing) and summarized in Figure 9.10 and Table 9.2.

Table 9.1: Estimate of (002) fringe spacing when a GaAs specimen is viewed along [110]
direction with Jeol 2200FS. Field of view is alscm'ovided.

 

 

 

 

 

 

 

 

Instrumental (002) fringe spacing Field of view with 1024 by
magniﬁcation (pixels) 1024 CCD (nm)

(Jeol 2200FS)

300,000 ~3 pixels 6lnm by 6lnm

400,000 ~5 pixels 47nm by 47nm

500,000 ~6 pixels 37nm by 37nm

600,000 ~7 pixels 3lnm by 3 lnm

800,000 ~10 pixels 24nm by24nm

1,000,000 ~11 pixels l9nm by l9nm

 

 

 

77

 

(a) Strained layers withl two monolayers

100IIIIIIIIIIIIIIIIQIIIIIIIZIILIII:

zooEE?EE?E§E?EEEEEEEEEE333533333???

300IIIIIZIZIIIiITTIZlIZIIIIIIIIIIII

  

- O o . I a o O I I I 0 O Q ~ I o O O 0 O O 0 0 O o o o O -

 
 

MEEEEEEEEEEEEEESEEE'W

sooEEEEEEEEEEEEEEFEEE 'D

ooo::::.:::::::::::;; X-dll'eCtloﬂ

 

 

 

1 00 200 300 400 500 600

Position along y-direction [pixel]

Position along x-direction [pixel]

Figure 9.6: (a) A computer-generated 2D lattice using Matlab software. Arrows indicate
coordinate system of x and y axes. A digital diffractogram is shown at inset.
Rectangular region indicates the ﬁrst harmonic required for strain analysis
along x-direction. (b) Bragg- ﬁltered lattice fringe using a g/3 Gaussian mask
centered on the ﬁrst harmonic. (c) An unwrapped phase image by removing
phase jump from —rt to 7r. Notable change in phase at strained layers indicates
displacement at strained layers.

78

Position along
y-direction [pixel]

Position along
y-direction [pixel]

(b)

100
200
300

400

 

500

600
100 200 300 400 500 600

Position along x-direction [pixel]

100
200
300
400

500

 

GOO

 

It ‘7
100 200 300 400 500 600

Position along x-direction [pixel]

Figure 9.6: Continued.

79

 

  

 

 

 

 

 

201) ' O
.___. 19.5- ’"2
22 4 f
E _ ‘E'
g 19.0" p-6 g
o co
8 a:
a 13.5 . Measured strain "8 E
.E . a)
u“. 13.04 "10 0:
Theoretical proﬁle
17 5 _ 0 Strain proﬁle using CFTM method "12

 

100 ' 200 360 ' 400 ' 560

Position along x-direction [pixel]

Figure 9.7: Strain proﬁle on strained layers that is composed of 2 monolayers. An
askance step proﬁle was constructed for simulations, considering diffusion of
constituent materials at strained layers.

80

9.3.2 Error quantiﬁcation of the phase technique

Figure 9.8 shows a result of simulations where two parameters are -4% (a value of

applied strain at strained layers with respect to reference regions on a 2D lattice image)

and 7 pixels per lattice fringe spacing (image resolution). Each exx proﬁle was plotted

with different radius of a Gaussian mask and the mask size was varied up to g/2 for a
Gaussian mask. Undershoot is observed due to the leakage effect similar to Figure 9.3
(a). In other words, linear dependence of error due to spurious high frequency noise from
the leakage effect on mask size occurs. This results in measured strains of -7.03% and -
9.33% at g/3 and g/2 Gaussian mask sizes, having errors of 76% and 133%, respectively.

The measured strain according to the mask size (Gaussian radius) is simulated using
2D lattice image (See Figure 9.6), where a strained value at strained layers is -4%, and
shown in Figure 9.9. The result is similar to previous simulations using 1D lattice image
as shown in Figure 9.3 (a). The simulation results shown in Figure 9.9 demonstrate again
that error due to leakage effect can be notably reduced with increasing image resolution
of an original image. For example, the measured strain was -3.76 with a g/3 Gaussian
mask in the case of image resolution of 12 pixels per fringe spacing while measured
strain was -7.03 with a g/3 Gaussian mask in the case of image resolution of 7 pixels per
fringe spacing. Therefore, we conclude that leakage effect causes undershoot at strained
layers, implying that resolution (number of pixels per ﬁinge spacing) has to be increased
to minimize error for reliability of phase technique as shown in Figure 9.3 (a) and Figure
9.9.

Finally, error has been quantiﬁed in terms of two parameters: the strained value we

applied in the middle of an original image and image resolution (number of pixels per

81

fringe spacing). Error quantiﬁcation is shown in Figure 9.10 and summarized in Table 9.2
where the procedures of image simulations are described in Chapter 9.3.1. All strains
were measured at a g/3 Gaussian mask. From Figure 9.10, error due to the leakage effect
is apt to become worse with small strain and poor resolution of an original image as
shown in Figure 9.10 (a), (b), and (c). The error can be minimized with increased fringe
spacing. From Table 9.2, we concluded that the phase technique can be exploited to
determine strain information with two monolayer accuracy and less than 25% error for
the values bounded with the thick line in Table 9.2. In the case of outside regions of the
bounded region of Table 9.2, values within 25% error and values exceeding 25% error

were sporadically positioned.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

0
-2 - -
g .
.g -4~ -
.3 - d
9 '6 ‘ ‘
g . > Undershoot
a; -8 . (a) -
J
9
-10 - XX -
o . 160 ' 260 ' 360 r460 T560 ' 600
Position along x-direction [pixel]
-1 - I ﬂ I ' T ' I -
. I
-2 _ \_ _
r—r . '1
°\.:. -3. \\ —-— Measured strain .
.5 « 3
E ‘4 r \\ -
'65 r '\
8 -5 d \-\I d
:9; -6 - (b) .\.‘.\. -I
2 -7 - l “is. *
\_\_\
-8 . g/3 ' .

 

 

Mask size [pixel]

Figure 9.8: Strain proﬁle on strained layers composed of two monolayers with 4% lattice
contraction on strained layers. (a) Each proﬁle was plotted with different size
of a Gaussian function. Undershoot at strained layers becomes larger and
larger with increasing mask size of a Gaussian function (b) Strain was
measured with mask size of a Gaussian function where g/2 and g/3 correspond
to 30 and 46 pixels respectively at x-axis.

83

 

—'I—- 21) pixels/ FS
0 18 pixels/ FS‘
A 16 pixels/ F3
* 14 pixels/ FS
316 12 pixels/ FS‘
X 10 pixels/ FS‘
—*—— 9 pixels/ FS ,
- 8 pixels/ FS
+ 7 pixels/ FS '

 

Measured strain [%]

 

 

 

'10 ' I ‘ I ' I ' I ' I
0 10 20 30 40 5O

Mask size [pixel]

Figure 9.9: Measured strain in the case of -4% strain at strained layers we applied on an
original image. Mask size (Gaussian radius) was varied from 4 pixels to g/2
for each resolution. Each graph represents resolution (number of pixels per
fringe spacing) where FS represents fringe spacing in legend.

84

Measured strain on various strain values
(At 7 pixels per fringe spacing )

'12 j I l T T I I ' T T T ‘ I ' T ‘/

 

Measured strain[%]
4
o
P

-2~ , ’ - - - Theorical value ‘
-1- , 0 Measured strain

A '4

 

 

 

o'4'-§Té'irle'lr-%Ve'iathI{42
Applied strain on an original image [%]

 

 

 

 

(3)
Measured strain on various strain values
13 (At 8 pixels per fringe spacing )
42-: o , ’ -
41: ,” -
40'" , ’ _‘
g -g- O O I p] -
c ' ‘. I ’
.§ -7: i ’ ..I
‘63 -6- ’ -
o , 0
E -5: I 9 '1
3 .4 S I
g -31 O ’ ’ 1
s 0,0 .
-21 , I - - - - Theoretical value .
-1j ’ ' 0 Measured strain 1
0 r ' I ' I r I ' I ' r ' I ' I ' I ' l ' Tﬁ *
0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11-12 -13
Applied strain on an original image [%]
(b)

Figure 9.10: Measured strain with different image resolutions (number of fringe spacing)
with given strain values at strained layers. Image resolutions of (a) 7 pixels of
fringe spacing (b) 8 pixels of fringe spacing (c) 9 pixels of ﬁinge spacing (d)
10 pixels of ﬁinge spacing (e) 12 pixels of fringe spacing (f) 14 pixels of
fringe spacing (g)l6 pixels of fringe spacing (h) 18 pixels of ﬁinge spacing (i)
20 pixels of ﬁinge spacing

85

Measured strain on various strain values

42 (At 9 pixels per fringe spacing )

 

Measured strain [%]
o
\

o . -
-Zj , ’ - - - - Theoretrcal value ,
-1 - I 0 Measured strain -

 

 

 

0t-I1'-'2'-8'-:4'-61-]6f-l7'-l81-I9'-1'0'-1l1-12
Applied strain on an original image [%]

 

 

 

 

(C)
Measured strain on various strain values
_12 (At 10 pixels per fringe spacing
-11 - I ’ I -
'10- ’0’ -‘
_ -9: o ’9 ’ -
°Z -8j , ’ 4
.E _ ’ j
e '7. O] ,0
‘6 '6‘ I "
.0 " I
a: -5- I I -
-4-: O 9’ O -
g '3‘- o , ’ I 7
'21 I 9 - - - - Theoretical value:
4: ’ 0 Measured strain 1
0 1 l ' I ' Iﬁ l l ' l ' I ' l ' I T l ' I
O -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11-12
Applied strain on an original image [%]
(d)

Figure 9.10: Continued.

86

Measured strain on various strain values

 

 

 

 

 

 

 

 

-12. . . . ,w12.nix2's.psrfrzneeﬁiasinag?
-11 .‘ I I ’ j
-10— 0 IO’ -

: g: ‘
- - D -.
E -7: 9 ’ _
Ia; ‘ , ’
-54 _
E -5. 9 ’ O ..
a 4 , ’ 4
to j o I 0 ‘
g '3'. , ’ i
‘2': O I oI - - - - Theoretical value 1
4: ’ 0 Measured strain 1
0 I I I I I I I I I I I I I I I 1 I I I I I I I
0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11-12
Applied strain on an original image [%]

(6)

Measured strain on various strain values
-121 . . . 4 6219905618 96! we? semi—w.
-11 .‘ o I , ’ _
'10-: I ’ ..

._ -9- , , -

i 431 ,O’ -

E '7: ,6 ~

2;; -6: 9’ -

E '5'. o ’6 ..

a '4: / I I ‘
i I 1
‘21 o , ’ - - - - Theoretical value 1
'11 ’ 0 Measured strain 1
0 ' I ' I ' I ' I I I ' I ' I ' I ' I ' Iﬁ I 1
O —1 -2 -3 -4 -5 -6 -7 -8 -9 -10-1l -12
Applied strain on an original image [%]
(t)

Figure 9.10: Continued.

87

 

Measured strain on various strain values

 

 

(At 16 pixels per fringe spacing )
'12 .. ' I f I ' I ' I ' I ' I ' I I T ' I ' I ' I ' I
-11- , , ’ _
-10- ,O’ _
l—D ” ’
§ '9: O ’0 1
c -3- , ’ a
'6 . I
*3 '7'. ,0 -
'O '6‘ 0’ 'l
9 -5- 9 ’ _
a I ,
8 '4‘ 9’ ‘
- I
2 -3: 0 l6 . _
-2- , ’ - - -Theoretical value _
-1 ; 9 ’ 0 Measured strain .'
O ' I f I ‘ Fr I ' I ' ' I T I ' I f

 

 

0'31 -2 -3 -4 -5 -6 3735 -9 -10-11-12
Applied strain on an original image [%]

Measured strain on various strain values
_1 2 (At 18 pixels per fringe spacing )

41-: ’-:

I
OI
I

 

Measured strain [%]
<1”
0
\

_2_j o, - - - - Theoretical value _I
1 ‘ o ’ 0 Measured strain

 

 

 

o 1'1'-'2'-'3'1351's'-'f-'a'-b'-1'o'-i(.12
Applied strain on an original image [%]

00

Figure 9.10: Continued.

88

 

Measured strain on various strain values

 

(At 20 pixels per fringe spacing )

 

 

'12‘ /
-111 /’/ -
-10‘ ,o’ -
_ -9: ’0’
E. -3.- 0’
E '7: x6 ‘
g '2? ,’o j
e " ‘ ’6
3 ~44 ,V i
g3 -3: ’O .
'21 I 9’ - - - - Theoretical value _
-1j O 0 Measured strain
0 ' I I I ' I ' I ' I ' I ' I 1 I fI r I ' I j
o -1 -2 -3 -4 -5 -6 -7 -8 -9 -1o-11-12
Applied strain on an original image [%]
(D

Figure 9.10: Continued.

 

89

 

Table 9.2: Measured strain at two monolayers of strained layers with g/3 Gaussian mask
in terms of resolution (number of pixels per fringe spacing) and a strain value
we applied on an original image. Error within 25% is bounded with thick line.

 

 

 

 

 

 

 

 

 

 

 

Given strain Fringe Spacing [pixel]
7 8 9 10 12 14 16 18 20
pixels pixels Jixels Pixels pixels pixels pixels . pixels pixels
-1% Strain -4.02 -3.38 -2.97 -2.55 -2 -1.64 -1.35 I -1.17 -1.02
Error (%) 302 238 197 155 g 100 64 35 17 2
-2% Strain -3.28 -2.72 -2.35 -2.08 I -l.84 -3.34 -2.73 I -2.36 -2.05
Error (%) 64 36 17.5 4 8 67 36.5 18 2.5
-3% Strain -3.02 -2.78 -5.44 -4.6 I -3.55 -3.04 -2.78 -3.67 -3.09
Error (%) 0.67 7.33 7 81.33 53.33 18.33 1.33 7.33 22.33 3
-4% Strain -7.03 -5.68 -4.85 -4.26 -3.76 -4.59 -3.99 -3.86 -4.19
Error (%) 75.75 42 21.25 6.5 6 14.76 0.25 3.5 4.75
-5% Strain -6.38 -5.39 -4.95 -4.68 -5.36 -4.81 -5.37 -S.02 —4.57
Error (%) 27.6 7.8 l 6.4 7.2 3.8 7.4 0.4 8.6
-6% Strain -6.29 -5.78 -7.5 -6.54 -5.73 -6.29 -5.75 -6.21 -5.58
Error (%) 4.83 3.67 25 9 4.5 4.83 4.17 3.5 7
-7% Strain -6.8 -8.44 -7.34 -6.9 -7.43 -6.81 -6.97 -7.55 -6.79
Error (%) 2.86 20.57 4.86 1.43 6.14 2.71 0.43 7.86 3
-8% Strain -9.93 -8.49 -7.95 -9.11 -7.95 -8.17 -8.48 -8.12 -7.83
Error (%) 24.13 6.13 0.63 13.88 0.63 2.13 6 1.5 2.13
-9% Strain -9.9 -9.05 -10.38 -9.46 -9.41 -9.93 -9.05 -9.28 -9.22
Error (%) 10 0.56 15.33 5.11 4.56 10.33 0.56 3.11 2.44
- Strain -10.47 -12 -10.79 -10.22 -10.16 -10.5 -10.3 -10.75 —9.9
10% 1
Error(%) J 4.7 20 7.9 2.2 1.6 5 3 7.5 1

 

 

9O

 

CHAPTER 10

COMPOSITION EVALUATION BY LATTICE FRINGE ANALYSIS (CELFA)

High-Resolution Transmission Electron Microscopy (HRTEM) involves the
interference of the unscattered and scattered beams and does not use the objective
aperture to obtain high resolution. These images usually have little contrast compared to
dark or bright ﬁeld images. However, the Composition Evaluation by Lattice Fringe
Analysis (CELFA) method is a composition evaluation procedure based upon a
chemically sensitive beam. For example, the (002) beam is the chemically sensitive beam
for zincblende structures. The chemically sensitive beam depends strongly on the
chemical composition. Hence, it is well suited for composition analysis and DALI
software [53] provides measurement of chemical composition based on amplitude of the
HRTEM images taken with the procedures for the CELFA method. In order to use the
CELFA method, HRTEM images should be formed using two-beam or three-beam
conditions [53], which generate a lattice ﬁ'inge image on the HRTEM image. Hence, it is
called Composition Evaluation by Lattice Fringe Analysis (CELFA) method.

However, accurate strain information on HRTEM images, especially at
heterointerfaces, is also suggested by the CELFA method. The reason is that HRTEM
images include artifacts inﬂuencing the positions of the atom columns in the HRTEM

images. The artifacts are mainly caused by delocalization [53] and the composition

91

dependence of the phases of the beams contributing to the image formation. When a
microscope is not equipped with Cs-corrector, the delocalization effect and chemical shift
of the phases are minimized using the (002) and (000) beams and a strong excitation of
the (004) beam [53], where the (002) beam is centered to the optical axis and the
objective aperture is centered around the (000) and the (002) beams. The reason that the
(002) beam is centered at the optical axis is to minimize aberrations and astigmatism.
This is called a two-beam centered dark-ﬁeld (CDF) image when two-beam conditions
are used, and one diffracted beam such as the (002) beam is centered to the optical axis.
The image with these two beams (i.e. (000) and (002) beams) interfering does not include
artifacts from a microscope such as delocalization and chemical shift, the lattice fringe
image will accurately correspond to the atomic columns in the crystalline specimen,
resulting in accurate determination of strain information. The accurate determination of
stain information using the CELFA method provides accurate composition evaluation
using DALI software. Details about optimized imaging conditions to minimize artifacts
from a microscope are discussed on Ref. [53].

However, the procedure of the CELFA method mentioned above is not very simple.
Firstly, the specimen should be tilted from ZA to make systematic row condition, which
shows only (00n) beams at selected area diffraction pattern. Secondly, the specimen
should be tilted again to make two-beam conditions to obtain good diffraction contrast,
where two-beam conditions is that only one diffracted beam and the (000) central beam
are strong. In this case of the CELFA method, the (002) chemical sensitive beam is used
for one diffracted beam in the case of the zincblende structure. Thirdly, the specimen

should be titled again to obtain a non-zero deviation parameter, S, producing a strong

92

.._,_.

excitation of (004) beam as well as the (000) and (002) beams. This is shown in Fig. 10.1

with the GaAso_g6P0.14/GaAs superlattice. The S is a vector and a measure of how far we

deviate from the exact Bragg condition. The value of S, amount of deviation from the
exact Bragg condition, is different in different material systems. Rosenauer et al. [53]
provides the value, S, because they developed software to determine the best value of S
for different material systems, and the software is not commercially released yet. Finally,
the (002) beam is centered to the optical axis as shown in Fig. 10.2, and the objective
aperture is centered around the (000) central beam and the (002) beam, resulting in
chemical sensitivity and minimized-artifacts in the lattice ﬁinge image as shown in F i g.
10.3 (c). Figure 10.3 (b) shows a low resolution image with the CELFA method, where
the procedure of the CELFA method is described above. We can clearly see that the
interface is clearer at Fig. 10.3 (b) than Fig. 8.1 (a) or energy-ﬁltered TEM (EFTEM)
shown in Figure 10.4. Figure 10.3 (c) shows a high resolution image with CELFA

method, which shows chemical lattice fringe at heterointerface of the

GaAso_36P0.14/GaAs superlattice due to the (002) chemical sensitive beams. As shown in

Figure 10.3 (c), we can see that the number of monolayers of GaAso_36P0_14 is around

2~3 with bright contrast.

In order to compare chemical contrast at the interface, energy-ﬁltered TEM (EFTEM)
was performed, and it is shown in Figure 10.4. Figure 10.4 shows a relatively rough
surface with bright contrast compared to the CELFA image; however, it is evident from

Figure 10.4 that phosphorus is mainly incorporated at the interface that is similar to

93

7'

 

Figure 10.3 (c). To sum up, the CELFA image shown in Fig. 10.3 (c) shows clearer and

sharper chemical contrast at the interface compared to energy-ﬁltered TEM (EFTEM).

However, in practice, the GaAsOO86POJ4/GaAs heterostructure is not well suited for

the CELFA method to determine strain information at the GaAsO_g6P0.14layers because

the lattice fringe is not clear. Hence, more work is needed to determine strain information

at the GaAsO.36P0.14/GaAs heterostructure when the CELFA method is applied. If we

can obtain nice and high-resolution images with the CELF A method, we can apply the
phase technique on the image also for strain analysis, thus it will be valuable to compare
results from the phase technique with results from the DALI software, since the DALI

software also provides strain analysis on HRTEM images. Finally, the structure factor of

GaAsl.xPx was calculated at Appendix D.

94

004 Kikuchi lines

(001))

 

Figure 10.1: Selected area diffraction pattern with the GaAso.36P0.14/GaAs superlattice,
showing only (000), (002), (004) beams are strong.

95

   
 

\

(002) (006)

Figure 10.2: Selected area diffraction pattern with the GaAso.36P0.14/GaAs superlattice
after centering the (002) beam to optical axis from Fig. 10.1.

96

 

(a) (b)

 

(C)

Figure 10.3: (a) A HRTEM image of the GaAsoog6P0114/GaAs superlattice. Inset shows
symmetric SAD pattern for HRTEM imaging. (b) A low resolution image
with procedures of CELFA method. (c) A High resolution image with
procedures of CELFA method, showing lattice fringe.

97

 

1-1101

 

Figure 10. 4. Elemental (phosphorus) map of the 5- period superlattice using energy-
ﬁltered TEM (EFTEM).

98

 

CHAPTER 1 1

SUMMARY AND DISCUSSTION

The primary research focus was strain analysis on heterointerfaces using the phase
techniques such as the GPA and the CFTM method. The CFTM method, developed on
solder ball connect (SBC) assemblies was not previously demonstrated on HRTEM
images. Here we demonstrated the use of the CF TM method for strain analysis of ultra-
thin layers from HRTEM images.

The advantage of the CFTM method with respect to the GPA method is that the
CFTM method requires less memory for computation than that of the GPA method. The
computational advantage is due to the fact that the CFTM method does not require a
phase unwrapping process to remove discontinuities. The unwrapping process involves
line-by-line scanning of the phase image to detect discontinuities; a process that requires
considerable memory for computation. Moreover, the unwrapping algorithm depends on
the conﬁguration of the phase fringes in the phase image. The process becomes very
complex if the phase fringes have a winding or irregular form. Therefore, the primary
advantage of the CF TM method is that it does not require the unwrapping process.

Strain using the phase technique has been performed on ultra-thin layers of GaAsP on
GaAs and the experimental results are introduced in Chapter 8. Our results revealed that

error, linearly dependent on mask size, exists in the HRTEM image. This error appears to

99

be caused primarily by the leakage effect at strained layers. We demonstrated that the
leakage effect dominates for small-strained systems (f<3%) with an image resolution of 7
pixels per fringe spacing, using computer-generated images. The image simulation
results were introduced in Chapter 9. Therefore, the resolution (number of pixels per

ﬁinge spacing) of the HRTEM image must be improved in order to remove the leakage

effect from the GaAsO.36P0_14 strained layers, regardless of quality of the HRTEM image.

Alternatively, we can reduce the mask size used to ﬁlter out spurious high frequency
terms due to the leakage effect at the cost of spectral resolution.

The leakage effect could be the principle limitation for interpreting strain using the
phase technique, where lattice constant abruptly changes such as at heterointerfaces or a
few monolayers. When the leakage effect exists at an interface with a few strained
monolayers and higher resolution (larger number of pixels per ﬁinge spacing) HRTEM
images are not available, then mask-size optimization provides an alternative way.

Our initial estimate of an experimental strain analysis using g/3 mask size of a
Gaussian mask revealed that the measured strain was around ﬁve times larger than the

theoretical value. However, based on mask-size optimization, we determined that a

quantitative value of measured strain at the GaAsoog6P014 /GaAs heterointerfaces would

be -1.5%. The calculated value (-1.5%) was reached by eliminating error from the phase
technique. Since the phase techniques such as GPA and CFTM methods do not rely on

knowledge of the positions of atomic columns in the crystalline specimen, accuracy

between the calculated value (-1.5%) and actual strain at the GaAso.36P0.14 /GaAs

100

heterointerfaces depends on quality of HRTEM imaging. The suggestions to minimize
artifacts from a microscope were summarized in Chapter 7.4.

A simulation method using Matlab® for determining error of the strain values when
the phase technique is applied has been proposed, and the error of the phase technique
was quantiﬁed for reliability of the phase technique with monolayer accuracy at Chapter
9.3. The image size of 2D lattice images generated with Matlab® are maintained at 620
by 620 pixels, and two monolayers of strained layers are placed in the middle of the 2D
lattice images. Two parameters were applied to the 2D lattice images for image
simulations. One is a value of strain at the strained layers and the other is the image
resolution of the original image. We observed that a Gaussian mask in reciprocal space
can remove the rippled appearance at the reference region with a computer-generated
image, and error due to the leakage effect is apt to become worse with small strain and
poor image resolution of an original image. The error was quantiﬁed with g/3 mask size
of a Gaussian ﬁmction in terms of two parameters in this research: a value of strain at
strained layers and image resolution. Based on image simulations, we concluded that the
phase technique can be exploited to determine strain information with monolayer
accuracy within 25% error and this region is bounded with a thick line in Table 9.2.

Efforts to minimize artifacts from a microscope and chemical contrast were

investigated using the CELFA method. Chemical contrast was compared with energy-

ﬁltered TEM. Two or three layers of GaAso.36P0.14 with bright contrast were identiﬁed
using CELF A method. However, in practice, the GaAs0.36P0_14 /GaAs superlattice is not

well suited for CELFA method to determine strain information at GaAsO,36P0_14 layers

101

because the lattice fringe is not clear. If nice and high-resolution images with CELFA
method can be obtained, we can apply the phase technique. Thus comparison of results
for strain analysis between the phase technique and DALI software could be future

research work.

102

APPENDIX A

MATLAB CODE OF AN EXPERIMENTAL IMAGE

close all
clear all

im=imread ('HR600K.tif‘); % Loading of image
[szl sz2 sz3]=size(im);

%Original image
ﬁgure(l);colorrnap(gray(256));imagesc(im); axis image

%Rotation of image to exactly align growth and interface direction to x-and y- direction
ro=imrotate(im,-52.3);

%Rotated image of the original image
ﬁgure(2); colormap(gray(256));imagesc(ro);axis image

%Crop of original image

a2=ro(689:796,543 :1 192);

a2=a2(1 : 108,1 :650);

%Display cropped image for original image

ﬁgure(3); colorrnap(gray(256));imagesc(a2);axis image

a2=double(a2);
[M,N]=size(a2);

%Fourier transform of an original image

f1 1 =fftshift(fﬁ2(a2));

a3=fftshift(abs(fft2(a2)));

% Display of digital diffractogram of an original image
ﬁguilt-14);

imagesc(log(a3));

% Length of (002) reciprocal vector
shiftu=94;

%Deﬁne mask function

103

[x y]=meshgrid(-N/2:N/2-l ,-M/2:M/2-1);
Fﬂipudm;

maska=zeros(M,N);

maskb=zeros(M,N);

%Meshgrid for mask ﬁmction
[X,Y]=meshgrid(1 :N,l :M);

for radius=90zl :90 %For loop for different mask sizes, radius=mask size

% Deﬁne mask function

a=4*pk

maska=exp(-a*((X- ((N/2+1)+shiftu) )."2+(Y-(M/2+1))."2)/(radius"2));
maskb=exp(-a*((X- ((N/2+1)-shiftu) )."2+(Y-(M/2+1))."2)/(radius"2));

%Display of Gaussian mask function

ﬁgure(5);

subplot(1,2,1);imagesc(maska); title(' maska (002)');
subplot(1,2,2);imagesc(maskb); title(' maskb (00-2)');axis tight;

uﬁ_1=maska.* ﬁl ;
uﬁ_2=maskb.*ﬁ1 ;

g=abs(uﬁ_1 );
h=abs(uﬁ_2);

%Display selected positive and negative harmonics
ﬁgure(6);

subplot(1 ,2,1);imagesc(g); title('FFT shift U');
subplot(1 ,2,2);imagesc(h);title('FF T shift -U');

uﬁ2_l=zeros(size(a2));
uﬁ2_l =uﬁ_l ;
uﬁ2_2=zeros(size(a2));
uﬁ2_2=uﬁ_2;
dofr=uﬁ2_1 +uﬁ2_2;

%Lattice Fringe image
cc=real(ifft2(fftshift(uﬁ2_1 )));
layer_02=real(ifft2(fftshift(dofr)));
% Shifting (002) reﬂection to origin
uﬁ3=circshift(uﬁ_1, [0, -94]);

uf_1=ifft2(fftshift(uﬁ3)); % Conventional Moire Fringe

104

 

[M,N]=size(uf_1);

%Phase map
=-angle(uf_1);

%Unwrapping process
WU=zeros (size(uf_1));

for i=1:10
for j=2zN
if u(i,j)-u(i,j-l)<-pi
WU (i,j)=WU (i,j-1)+1;
else
WU (iJ)=WU (id-1);
if u(i,j)-u(i,j-l)>pi
WU (i,j)=WU (i,j-l)-1;
else
WU (i,j)=WU (iJ-l);
end

end

end
end

for j=l:10
for i=11:M
if u(i,j)-u(i-1,j)<-pi
WU (i,j)=WU (i-1,j)+l;
else
WU (iJ)=WU (i-IJ);
if u(i,j)-u(i-l,j)>pi
WU (iJ)=WU (i-IJH;
else
WU (i,j)=WU (i-1,j);
end

end

end
end

for i=ll:M
forj=10+1 :N

105

 

if u(i,j)-u(i,j-1)<-pi

WU (i,j)=WU (i,j-1)+1;
else

WU (iJ)=WU (id-1);
if u(i,j)-u(i,j-1)>pi

WU (iJ)=WU(iJ-1)-l;
else

WU (i,j)=WU (i,j-l);
end

end

end
end

unwrap_u=u+WU*2* pi;

% Strain using numerical derivative
fx33=(6.91/(2*pi))*(1/6)* [-1,0,1; -1,0,1; -1,0,1];
fx3nurnber_u=conv2(u,fx33);
fx3number_unu=conv2(unwrap_u,fx33);

% Covstrain3: strain using numerical derivative on phase image
Covstrain3=-1 *fx3number_u';

% Covstrain3 unwrap: strain using numerical derivative on unwrapped phase image
Covstrain3unwrap(:,:,radius)=—1 *fx3number_unu';

% Display phase image

ﬁgure(7);

imagesc(u); xlabel('Position along x-direction [pixel]');ylabel(' Position along x-direction
[Pixel]'); colorbar

% Display unwrapped phase image

ﬁgure(8);

imagesc(unwrap_u); xlabel('Position along x-direction [pixel]');ylabel(' Position along x-
direction [Pixel]'); colorbar

[k1 ,k2]=meshgrid(-N/2 :N/2-1 ,-M/2 :M/2- l );
k2=-k2;

%Strain using CF TM method
dudx(:,:,radius)=real(ifft2(fftshift(uﬁ3.*k1/N))./(-shiftu/N* uf_1));

end

106

APPENDIX B
MATLAB CODE OF IMAGE SIMULATIONS

clear all
close all

% For loop for different mask sizes, B=mask size
for B=4:10:64

% PITCHES AT REFERENCE REGIONS [Unit:pixe1]

%p0=Pitch along y-direction, qO=Pitch along x-direction,

%m0=Number of periods along y—direction, n0=Number of periods along x-direction
p0=20; q0=20;

m0=32; n0=15;

% Deﬁne sinusodal function at reference regions
[x,y]=meshgrid(0:q0*n0-l,0:p0*m0—1);

r=4;

f0=zeros(size(X)); % x: size: 320 by 160
f0=sin(2*pi/q0*x)."r.*sin(2*pi/p0*y)."r;

% PITCHES AT STRAINED REGIONS (2 MONO LAYERS) [Unitzpixel]
%p3=Pitch along y-direction, q3=Pitch along x-direction,
%m3=Number of periods along y-direction, n3=Number of periods along x-direction

p3=20; q3=18; % In this example, 10% lattice contraction is applied.
m3=32; n3= 2;

% Deﬁne sinusodal function at reference regions
[x,y]=meshgrid(0:q3*n3-l,0:p3*m3-1);
d0=zeros(size(x)); % x: size: 320 by 34
d0=sin(2*pi/q3*x)."r.*sin(2*pi/p3*y)."r;

% Combine reference regions with strained layers in one image.
g=[ﬂ) d0 f0];

f=g(:,1:end);

[M N]=size(f);

% An Original image
ﬁgureﬂ);

107

 

 

imagesc(f);axis image; colormap gray

% Digital diffractogram of the image "f"
ﬁ1=fftshift(fft2(f));
a3=fftshift(abs(fft2(f)));
ﬁgure(2);
imagesc(a3);
[zl,22]=meshgrid(-N/2:N/2-1,-M/2:M/2-1);
22=-22;
ff=fft2(f);
affl =abs(ff(1,:));

%Find out lenght of a reciprocal vector of the ﬁrst harmonic.
peaku1=ﬁnd(affl (1 :round(end/2))==max(aff1 (2 :end)));
peaku=peakul -1 ;
peak__u1=-1 *peakul ;
peak_u=peak_u1 +1 ; '7

___._l'

 

% Function of Computational Fourier Transform Moire (CFTM)
[u,unwarp_u,dudx,uﬁ_1 ,uﬁ_2,dofr,uﬁ2_l ,cc,layer_02,k1,k2,ﬁ l ,uﬁ3]=cftm_Gau(B,f,pea
ku,peak_u);

g=abs(uﬁ_1); % Diffractogram showing only positive ﬁrst harmonic using an mask

=abs(uﬁ_2); % Diffractogram showing only negative ﬁrst harmonic using an mask
ﬁgure(3);

subplot(] ,2, 1 );colormap( gray(256));imshow( g)
subplot( 1 ,2,2);colorrnap(gray(256));imshow(h)

ﬁgure(4);
imshow(abs(dofr)); % Diffractogram for Bragg-ﬁltered lattice fringe

ﬁgure(5);
imshow(log(abs(uﬁ3))); % Diffractogram for CFTM method

ﬁgure(6); %Strain mapping using contour function
[C,h] = contour(k1,k2,dudx);
clabe1(C,h);title('dudx');colorbar

dudxmasksize(:,B)=dudx(M/2+7,:);
dudxmasksize_B=dudxmasksize';

 

%Plot strain using CFTM method

ﬁgure(7);plot(dudx(M/2+7,:),'.-'); xlabel('Pixel');ylabel('du/dx (Strain) ');title('Strain
proﬁle');axis tight;grid on;axis([1 643 -0.15 0.1 D;

end

108

APPENDIX C
MATLAB CODE OF CFTM FUNCTION USED FOR IMAGE SIMULATIONS

function
[u,unwarp_u,dudx,uﬁ__1 ,uﬁ_2,dofr,uﬁ2_1 ,cc,layer_02,k1,k2,ﬁ1 ,uﬁ3]=cftm_Gau(B,f,pea
ku,peak_u)

[M N]=size(f);

% Fourier transform of an original image (Digital diffractogram)
ﬁl =fftshift(fit2(f));
radius=B; %Mask radius

[X,Y]=meshgrid(1 :N,l :M);
% Deﬁne Gausian mask ﬁrnction
% maska=> Mask function for positive harmonic
% maskb=> Mask function for negative harmonic
=4*PU
maska=exp(-a* ((X- ((N/2+1)+peaku) )."2+(Y-(M/2+1))."2)/(radius"2));
maskb=exp(-a*((X- ((N/2+l)+peak_u) )."2+(Y-(M/2+l))."2)/(radius"2));

% Selecting the ﬁrst harmonic using masks
uﬁ_l =maska."‘ﬁ1 ;
uﬁ_2=maskb. *ﬁl ;

uﬁ2_1=zeros(size(f));
uﬁ2_1=uﬁ_l ;

uﬁ2_2=zeros(size(f));
uﬁ2_2=uﬁ_2;

dofr=uﬁ2_l +uﬁ2_2;

cc=real(ifft2(fttshift(uﬁ2_1)));
layer_02=real(ifft2(fftshift(dofr)));

% Shifting the ﬁrst harmonic to origin for CF TM method
uﬁ3=circshift(uﬁ_1 ,[O,peak_u]);

uf_1=ifft2(fftshift(uﬁ3)); % Conventional Moire Fringe
%Phase image

109

 

 

u=-angle(uf_1 );
% Unwrapping algorithm on phase image
WU=zeros (size(f));

for i=1:10
for j=2:N
if u(iJ)-u(i,j-1)<-pi
WU (i,j)=WU (iJ-1)+1;
else
WU (i,j)=WU (i,j-l);
if u(iJ)-u(iJ-l)>pi
WU (i,j)=WU (Li-1H;
else
WU (i,j)=WU (id-1);
end

end

end
end

for j=1 :10
for i=11:M
if u(iJ)-u(i-l J)<-pi
WU (i,j)=WU (i-1J)+1;
else
WU (iJ)=WU (i-1J);
if u(i,j)-u(i-l ,j)>pi
WU (i,j)=WU (i-1J)-1;
else
WU (iJ)=WU (i-IJ);
end

end

end
end

for i=11:M
for j=10+1 :N
if u(iJ)-u(i,j-1 )<-pi
WU (i,j)=WU (i,j-1)+1;
else
WU (iJ)=WU (iJ-l);
if u(i,j)-u(iJ-l)>pi
WU (iJ)=WU(iJ-1)-1;

110

 

 

else
WU (iJ)=WU (id-1);
end

end

end
end
unwarp_u=u+WU*2*pi;

[k1,k2]=meshgrid(-N/2:N/2-l,-M/2:M/2-l);

k2=-k2;

%Strain tensor of CF TM method
dudx=rea1(ifﬁ2(fftshiﬁ(uﬁ3 . *k 1 /N))./(-peaku/N *uf_1 ));
dudy=real(ifﬁ2(fﬁshift(uﬁ3.*k2/M))./(-peaku/N *uf_1));

111

APPENDIX D

STRUCTURE FACTOR FOR GALLIUM ARSENIDE PHOSPHIDE

In the kinematical approximation, the amplitude of a diffracted beam ghkl is
proportional to structure factor. The structure factor is to consider a unit cell of a crystal
(1')

containing N atoms with atomic scattering factors fe at positions ri (i=1, ....... N). The

structure factor of a beam for a binary material GaAs, given by
i7:
Fhk, = 4U?" + g“ exp<7<h+k+1»}. (13.1)

In the case of a ternary material G-aAsl_xPx, we assume a random dOistribution of As and

P on the metal sublattice. The structure factor of the ternary material is then obtained to a
good approximation by assuming, that the atomic scattering amplitude of the metal site is
a linear combination of the scattering amplitudes of the two types of metal atoms under

consideration:

Fhkl = 4{feGa +(1— x)feAS exp(127—r— (h + k + 1)) + xfeP exp(527£(h + k + l))}. (D-2)

structure factors of GaAsl_xPx with major reﬂections are shown in Table D. 1.

112

 

 

 

Table D. 1: Structure factors of GaAs] -xPx with major reﬂections.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Structure factors (100) (110) (200)

GaAs 0 0 4{fGa —fAS}

(FS_GaAs )

GaAsP 0 O 4{fGa—(l—x)fAS—xfp}

(FS_GaAsP )

1 = |F|2 0 0 16%., —<1 —x)fAS -fo}2

X=O O 0 4{f Ga - fAs}

FS(X) O O fGa_fAs+(fAs_fP)x

F fGa -fAs
5(0)

Structure factors (220) (400)

GaAs 4{fGa + f AS} 4{fGa + f AS}

(FS_GaAs)

GaAsP 4{fGa+(l—x)fAs+fo} 4{fGa+(l—x)fAs+fo}

(FS_GaAsP )

1 = |F|2 16{fGa +(l-x)fAS +xfp}2 16{fGa +(1-x)fAS +1ch2

X=0 4{fGa + f A S} 4{fGa + f A S}

F fGa+fAs+(fP_fAs)x fGa+fAs+(fP—fAs)x
SOC)

F fGa+fAs fGa+fAs
5(0)

 

 

113

 

 

 

 

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