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Nanostructure interaction modeling and estimation for scalable nanomanufacturing

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Nanostructure interaction modeling and estimation for scalable nanomanufacturing

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Content NANOSTRUCTURE INTERACTION MODELING AND ESTIMATION FOR
SCALABLE NANOMANUFACTURING
by
Lijuan Xu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENGINEERING)
December 2013
Copyright 2013 Lijuan Xu
Dedication
To my Fianc´ e Fenye Bao.
To my parents and our big family!
ii
Acknowledgements
Pursuing a PhD in a foreign university is like an adventure. I feel very fortunate and
honored to have so many people helping me, encouraging me, guiding me, and sharing
with me joys and passions for over four years of my graduate study. Because of them, I
become more confident, independent, and mature. For them, I owe the greatest gratitude.
First of all, I would like to sincerely thank my PhD advisor Dr. Qiang Huang. He
has served as my advisor since the first date of my graduate study in University of
Southern California. He designed the curriculum based on my educational background,
encouraged me to learn more about Statistics, and guided me step by step to our research
field. He has strict requirements on our coursework and research and thus looks like a
tough mentor in the eyes of other graduate students. But we deeply understand that is
all because he cares about us. He expects us to be the best of ourselves and provides us
trainings and opportunities to grow better and stronger. He is always ready to cheer us
up when we feel frustrated and celebrate our progresses no matter how small they are.
Without him, my PhD study won’t be so fruitful.
I would also like to thank my doctoral committee members Dr. Yong Chen and Dr.
S. Joe Qin for their “challenging” questions and constructive suggestions to improve my
dissertation. Their unique thinking of my dissertation work has enhanced my under-
standing of the field and triggered deeper thoughts of future extensions.
iii
My qualifying exam committee members Dr. Sheldon M. Ross, Dr. Maged M.
Dessouky, Dr. Chongwu Zhou, and Dr. Jinchi Lv have also given me great supports.
Thank you so much for patiently listening to my research ideas and providing valu-
able inputs. Thank you for saying “excellent work” when greeting me and telling me
that I passed the qualifying exam. Your encouragements have lighten up so many dark
moments when I was struggling about my research problems.
My PhD study would not be such a pleasant experience without the support and
company of my friends and fellow graduate students. In particular, Yue Fu and Maoqing
Yao (PhD students in Dr. Chongwu Zhou’s group) and Kai Xu (PhD student in Dr. Yong
Chen’s group) have provided me tremendous help in collecting research data. Labmates
Li Wang, Jizhe Zhang and Jian Wu are always there for a research discussion and very
supportive whenever help is needed. And my friends in USC have brought me so many
joys. Because of them, I feel Los Angeles is like my second hometown.
My gratitude also goes to all faculty and staff in our department. Over the past four
years, they have provided me tremendous help in all aspects of my PhD life besides
research. Without them, it would not even be possible to live or study in USC. Thank
you so much for being so nice, patient, and supportive. I would like to thank the graduate
school and the university for providing me such a valuable learning experience as well.
Finally, I would like to give my deepest thanks to my parents, my grandparents, and
my late grandmother for their tireless love from the beginning of my life, to my younger
brother and my younger sister for enriching my life and serving as my best friends since
they were born, and to my Fianc´ e for trusting me, loving me, and supporting me.
iv
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures ix
Abstract xi
Chapter 1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Potentials of Nanotechnology . . . . . . . . . . . . . . . . . . 2
1.1.2 Importance of Scalable Nanomanufacturing . . . . . . . . . . . 3
1.1.3 Synthesis Variations and Nanostructure Interactions . . . . . . . 4
1.1.4 Research Challenges for Nanostructure Interaction Characteri-
zation and Control . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 State of the Art on Scale-up Nanomanufacturing Research . . . . . . . 8
1.2.1 Classification of Nanomanufacturing Research . . . . . . . . . 9
1.2.2 Nanostructure Metrology . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Nanostructure Feature Variation Quantification . . . . . . . . . 12
1.2.4 Nanostructure Synthesis Process Modeling . . . . . . . . . . . 13
1.2.5 Experimental Design for Robust Synthesis . . . . . . . . . . . 15
1.3 Research Tasks and Objectives . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 2 Nanostructure Interaction Modeling and Estimation for a Local
Region 19
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Modeling of Nanostructure Interactions . . . . . . . . . . . . . . . . . 23
2.3 Estimation of Nanostructure Interactions . . . . . . . . . . . . . . . . . 26
2.3.1 Maximum Likelihood Estimation for Stationary Interactions . . 27
2.3.2 Normality Tests on Transformed Residuals to Detect Defects . . 28
v
2.3.3 Local Likelihood Estimation for Regions with Defects . . . . . 30
2.4 Simulation Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Setups of Simulation Experiments . . . . . . . . . . . . . . . . 32
2.4.2 Interaction Extraction for Normal Growth Regions . . . . . . . 33
2.4.3 Interaction Extraction for Defects Detection . . . . . . . . . . . 35
2.5 Real Case Study: ZnO Nanowire Experiment Data Analysis . . . . . . 37
2.5.1 Data Collection: Nanowire Length Measurement . . . . . . . . 38
2.5.2 Preliminary Data Analysis . . . . . . . . . . . . . . . . . . . . 41
2.5.3 Interaction Extraction for Nanowire Bundle Detection . . . . . 42
2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 3 Nanostructure Interaction Modeling and Estimation for Whole
Substrate 48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Overall Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Modeling and Estimation of Nanostructure Interactions . . . . . . . . . 53
3.2.1 Extension of Local Region Stationary Interaction Modeling . . 54
3.2.2 EM Estimation of Interactions with Incomplete Measurement . 56
3.3 Tailored Space Filling Design for Site Selection in Each Region . . . . 62
3.4 Simulation Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Multiple Tiny Regions Versus One Large Region . . . . . . . . 66
3.4.2 EM Estimation of Interactions with Incomplete Measurement . 67
3.4.3 Tailored Space Filling Design for Each Region . . . . . . . . . 71
3.5 Real Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 4 Growth Process Modeling of III-V Nanowire Synthesis via SA-
MOCVD 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Growth Modeling of III-V Nanowires Synthesized via SA-MOCVD . . 79
4.2.1 Process Characteristics and Precursor Diffusion . . . . . . . . . 79
4.2.2 Notation Convention . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.3 Spatio-Temporal Growth Modeling . . . . . . . . . . . . . . . 82
4.2.4 Model Discussion . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Model Validation and Case Studies . . . . . . . . . . . . . . . . . . . . 88
4.3.1 Increased Nanowire Growth Rate . . . . . . . . . . . . . . . . 88
4.3.2 Dependence of Nanowire Height on Diameter . . . . . . . . . . 90
4.3.3 Spatial Profile of Nanowire Growth . . . . . . . . . . . . . . . 92
4.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 95
Chapter 5 Discussions and Future Extensions 96
vi
References 100
vii
List of Tables
1.1 Sub-categories of scale-up methodology and comparison . . . . . . . . 10
2.1 Parameter values set in simulation studies . . . . . . . . . . . . . . . . 32
2.2 Summary of MLEs for
; and
2
based on stationary datasets (95
samples) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 MLEs and 95% C.I.s of stationary GMRF parameters for NW length field 44
3.1 MLE estimation of nanostructure interactions based on simulated datasets 69
4.1 Height dependence on diameter . . . . . . . . . . . . . . . . . . . . . . 92
viii
List of Figures
1.1 Demonstration of the fabrication process of a nanowire-based solar cell 4
1.2 The decomposition of nanostructure synthesis variations . . . . . . . . 5
1.3 Four pillars of scale-up methodology research . . . . . . . . . . . . . . 9
1.4 Comparison of synthesis process modeling strategies . . . . . . . . . . 14
2.1 Illustration of nanostructure feature measurement in a local region of
interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Samples of candidate neighborhood structures . . . . . . . . . . . . . . 22
2.3 Overall strategies of modeling and estimating nanostructure interactions 26
2.4 Illustration of local likelihood estimation . . . . . . . . . . . . . . . . . 31
2.5 MLEs and 95% confidence bounds of
i
; i = 1;:::; 4 . . . . . . . . . . 34
2.6 Smoothing parameter selection by cross validation [22] . . . . . . . . . 36
2.7 Level plots of local likelihood estimated =f
1
;
2
;
3
;
4
g together
with No Growth sites (magenta squares) for one of simulated nonsta-
tionary datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 SEM images of ZnO Nanowires (a) top view and (b) tilted view . . . . 38
2.9 Geometric model for measuring NW length . . . . . . . . . . . . . . . 39
2.10 Level plot of real collected length field for ZnO nanowires . . . . . . . 42
2.11 Locations of nanowire bundles and no growth sites . . . . . . . . . . . 43
2.12 Nanowire (bundle) lengths VS. row indexes and column indexes of the
field excluding no growth sites where corresponds to lengths of nanowire
bundles and is for normally grown nanowires. . . . . . . . . . . . . . 43
2.13 Half-normal plot for transformed residuals ^ e, real case study . . . . . . 44
2.14 selection by cross validation, real case study . . . . . . . . . . . . . . 45
2.15 Level plots of =f
1
;
2
;
3
;
4
g together with top view SEM images. 46
3.1 Overall strategies of interaction analysis under metrology constraint. . . 50
3.2 Nanostructure sampling strategies. . . . . . . . . . . . . . . . . . . . . 51
3.3 Schematic illustration of non-measurement site classification . . . . . . 60
3.4 Sample maximin distance design that produces single conditionally depen-
dent group of non-measurement sites under diamond neighborhood struc-
ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Schematic illustration of simulated data structures . . . . . . . . . . . . 68
ix
3.6 EM estimation of interactions for simulated datasets. . . . . . . . . . . 69
3.7 EM estimation time with non-measurement ratios (simple random sam-
pling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8 EM estimation time with maximum cluster sizes (simple random sam-
pling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.9 Comparsion of maximum cluster sizes under tailored space filling sam-
pling and simple random sampling of non-measurement sites . . . . . . 72
3.10 Comparison of EM estimation time for tailored space filling sampling
and simple random sampling of non-measurement sites . . . . . . . . . 72
3.11 EM estimation of interactions for real nanowire length data. . . . . . . . 74
4.1 Schematic of SA-MOCVD fabrication (modified from [93]) . . . . . . 77
4.2 Schematic of precursor diffusion in SA-MOCVD synthesis . . . . . . . 80
4.3 Schematic of a patterned substrate and a synthesized nanowire in SA-
MOVPE process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Schematic illustration of the four precursor sources and their corre-
sponding contributions to nanowire growth based on mass conservation
principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Distribution of mask area among neighboring nanowires . . . . . . . . 84
4.6 Modeling and integration of skirt diffusion . . . . . . . . . . . . . . . . 86
4.7 Time-dependent growth rate in SA-MOCVD synthesis. . . . . . . . . . 89
4.8 Dependence of nanowire height on diameter . . . . . . . . . . . . . . . 91
4.9 Spatial profile of nanowire heights under different conditions . . . . . . 93
4.10 Our model’s fit to the InP nanowire height data . . . . . . . . . . . . . 93
x
Abstract
Nanotechnology serves the potential to address national and global needs in high-impact
opportunity areas such as energy, security, and public health. In the past decades,
numerous laboratory successes have been reported to apply functional nanomaterials
in promising applications that could potentially replace the current technology with
more advanced performance and novel functionalities. As the key enabler to scaling-
up laboratory successes, nanomanufacturing, however, has been the major bottleneck to
fulfill this promise because of its incapability of producing reliable and cost-effective
nanoscale materials, structures and devices. The current yield of nanodevices is 10%
or less. This low and unstable yield makes nanofabrication extremely expensive, hardly
predictable and performance of nanodevices highly uncertain.
To achieve scalable nanomanufacturing, a central issue is to control nanostructure
synthesis variations. Since the formation of nanostructures is sensitive to the interac-
tions among neighboring structures such as competition for source materials during
growth, it is therefore critical to develop metrics and methods to characterize nanos-
tructure interactions and link the interactions with structure variability. Fundamental
research challenges of developing such characterization and connection include three
aspects. (i) There is very limited physical knowledge of nanostructure interactions, let
alone a rigorous formulation. The only understanding is that nanostructures physically
xi
closer have more chances to have mutual impacts. But there are no concrete physi-
cal principles that we can rely on. Modeling nanostructure interactions are subject to
more uncertainties due to latent or unobserved factors. (ii) Nanostructure interaction
characterization is severely constrained by the scarcity of measurement data. Current
nanostructure inspection techniques such as Scanning Electron Microscope (SEM) and
Transmission Electron Microscope (TEM) are mainly two dimensional imaging. Taking
images and extracting information of nanostructure features e.g. nanowire length from
these 2D images are extremely time consuming and labor intensive. (iii) Although we
have relatively good understanding of growth kinetics to describe the general trend or
profile of nanostructure growth, the understanding is not conclusive. For most synthe-
sis processes, there is a lack of process models that could coherently explain all major
experimental observations. Uncertainty of this general trend modeling affects the esti-
mation of nanostructure interactions as well.
In this dissertation, we focus on the synthesis processes of nanowires and develop a
systematic modeling and estimation technique to characterize nanostructure interactions
for scale-up nanomanufacturing. We include three research tasks to deliver our charac-
terization methodologies and address the research challenges step by step. They are
(i) modeling nanostructure interactions statistically by Gaussian Markov random field
to capture possible interaction patterns and link with structure variability; (ii) estimat-
ing nanostructure interactions with limited measurement data to reduce the metrology
efforts of interaction mapping; and (iii) developing a growth process model for nanowire
synthesis to capture the spatial and temporal growth trend, which in turn assists the
investigation of nanostructure interactions. The three tasks together provide a complete
quantification of nanostructure growth variations.
xii
Both simulated and real experimental nanowire length data are used to demonstrate
and validate our strategies. Results show that our method indeed quantifies nanostruc-
ture feature variability accurately, requires reasonable amount of structure measurement,
and explains quantitatively all major experimental observations for III-V nanowire syn-
thesis via selective area metal organic chemical vapor deposition (SA-MOCVD) which
is a promising nano-fabrication technique for scalable nanomanufacturing.
Our work in this dissertation contributes to both nanotechnology research and nanoman-
ufacturing scale-up. For nanotechnology, our growth process model enhances our under-
standing of the synthesis mechanism and confirms the dominant role of precursor dif-
fusion in SA-MOCVD. Additionally, our interaction analysis reinforces the fact that
nanostructure interactions affect nanoscale feature variations. Our findings may trigger
further studies on molecular level and facilitate designs of novel synthesis processes that
provide better controlled nanostructures.
For nanomanufacturing, as our growth process model accurately captures the depen-
dence of nanowire geometries on process parameters, it guides process optimization and
guarantees the overall quality of synthesized nanowires. What’s more, our interaction
analysis is capable of assessing dense quantities of nanostructures and detecting possi-
ble structural defects within manufacturing relevant time spans. Our technique supports
systematic quality control for scale-up nanomanufacturing.
xiii
Chapter 1
Introduction
In this dissertation, we
1
systematically characterize nanostructure interactions to provide
a supporting tool for scalable nanomanufacturing. We focus on “bottom-up” synthesis
of nanowires and continuous quality features of nanowires e.g. length. By modeling
nanostructure interactions and linking them with structure variations, we are committed
to provide a benchmark metric of nanostructure quality, and assist diagnosis and control
of nanostructure variations.
In addition, we develop a growth process model for III-V nanowire synthesis via
selective area metal organic chemical vapor deposition (SA-MOCVD) technique which
is a very promising fabrication technique for scale-up nanomanufacturing. Our model
aims to capture the spatial and temporal growth trend of nanowires, coherently explain
various experimental observations quantitatively, and reinforce our understanding on
nanowire growth kinetics. Additionally, it is designed to facilitate the investigation of
nanostructure interactions.
In following part of this chapter, we will first discuss the general background and
motivation for scalable nanomanufacturing. We will then summarize the research chal-
lenges for characterizing nanostructure interactions and establishing a growth process
model. Following that we will review the existing literature on nanomanufacturing and
associated field from which we will define the research gap in more detail. At the end,
we will list our tasks and objectives of this dissertation.
1
All work written in this dissertation is primarily done by myself under the guidance of my aca-
demic advisor Professor Qiang Huang. Due to writing habit, “we” instead of “I” is used throughout this
manuscript to represent the single author.
1
1.1 Background and Motivation
1.1.1 Potentials of Nanotechnology
Nanotechnology, the branch of science and engineering that studies and manipulates
matter at nanoscale (1 to 100 nm, with 1 nm = 10
9
m) [68, 81], has the great potential
to address national and global needs in high-impact opportunity areas such as energy,
security, and public health [90]. In the past decades, numerous laboratory successes have
been reported to apply functional nanostructures in applications that could potentially
replace the current technology with more advanced performance and novel functionali-
ties [44, 98, 74, 117, 52, 105]. For example, high-density aligned nanotubes were fabri-
cated to build submicron nanotube transistors with superior properties [100] and CMOS
integrated circuits with great advantages [99]. Semiconductor nanowires were used to
build nanoscale optoelectronic devices with revolutionary advances [35, 102, 71]. Moti-
vated by laboratory successes, it has been expected that the impact of nanotechnology is
“at least as significant as the combined influences of antibiotics, the integrated circuit,
and human-made polymers” [79].
One of the contributing factors to the improved performances of nanomaterials is
the quantum effects which begin to dominate at nanoscale. As a consequence, proper-
ties such as melting point, fluorescence and electrical, magnetic and thermal behaviors
are size-dependent when materials reach nanoscale in one, two or three dimensions. For
instance, the thermal conductivity of Si nanowires can be two times smaller than that of
bulk silicon in the temperature range from 200K to 500 K [96]. This reduction of ther-
mal conductivity is desirable in applications such as thermoelectric cooling and power
generation [110]. In addition, at nanoscale the surface area is also greatly increased,
which improves significantly the chemical reactivity of materials. A notable example
is gold which is inert in bulk but can act as catalyst at room temperature in form of
2
nanoparticles. It is reported that catalysis by engineered nanomaterials have impacted
about 1/3 of the U.S. catalyst markets especially in the oil and chemical industries [20].
Unlike laboratory successes, we have seen relatively small number of industrial
applications of nanotechnology. Furthermore, most commercial products available on
market thus far are in the flavor of applying nanoscale materials in bulk applications.
Examples include incorporating nanoparticles in tennis balls for better durability and in
trousers and socks to keep people cool in the summer. A complete list could be found
on [88, 67]. Nanotechnology innovations have not generated the broad societal impact
as expected.
1.1.2 Importance of Scalable Nanomanufacturing
Scalable nanomanufacturing refers to the controlled, reliable and cost-effective man-
ufacturing of nanomaterials, structures and devices in commercial scale [42]. It is
the key enabler to transform discoveries of nano-science to real-world value-added
nanotechnology-enabled products [42]. At the same time, it is also the bottleneck to
fulfill the promise of nanotechnology and produce more advanced industry applications
of nanotechnology at the current stage.
The major problem with nanomanufacturing scale-up lies in the synthesis of basic
nanostructures such as nanowires which are the building blocks for more advanced func-
tional nanodevices. Take nanowire-based solar cells as an example. The fabrication
process can be coarsely divided into three steps as demonstrated in Figure 1.1 [27, 25].
Among them, the synthesis of nanowires and their axial or radial extended heteroge-
neous structures is the central step. To achieve optimized device efficiency, it is expected
that all nanostructures on the substrate have uniform shapes, diameters and lengths so
that when a transparent electrode is put on the top surface, all nanostructures get con-
nected and contribute to the energy transfer process congruously. On the other hand, the
3
variability in nanostructures and possible defective structures produced may severely
degrade device performance.
light
preparation of substrate synthesis of nanowires fabrication of solar cells
Figure 1.1: Demonstration of the fabrication process of a nanowire-based solar cell
Uniformity of synthesized nanowires directly determine the efficiency of
corresponding solar cells.
Due to synthesis variations, the current yield of nanodevice is only 10% or less [49].
This low and unstable yield makes nano-fabrication extremely expensive and hardly pre-
dictable. While in lab experiments intensive human involvement may help with ensuring
the quality of final products, an automatic and efficient quality assessment and variation
control strategy is urgently needed to achieve nanomanufacturing scale-up.
1.1.3 Synthesis Variations and Nanostructure Interactions
Nanostructure synthesis is normally a very complicated process which involves param-
eters and features at multiple scales. Take metal catalyzed VLS (vapor liquid solid) syn-
thesis of nanowires as an example. The growth uniformity is not only affected by tem-
perature and growth time but also impacted by the size uniformity of catalyst droplets
on the substrate [97]. To distinguish different levels of uncertainty, we may decompose
nanostructure synthesis variations into (i) variations on substrate level and (ii) variations
in a local region based on their scales and contributing factors [37].
4
As demonstrated in Figure 1.2, variations on the substrate level refer to the tem-
poral growth trend and spatial profile of nanostructure quality features e.g. length. The
magnitude of such variations indicate the overall quality of nanostructures for an experi-
mental run. In term of causes, variations on the substrate level are determined by process
parameters such as growth time and temperature that can also be controlled to the best
at substrate level. Most growth process models in literature focus on quantifying growth
variability at this scale [113].
On the other hand, variations in a local region represent the additional variability
among nanostructures besides the general growth profile. While local variations may
be negligible for traditional manufacturing, they are especially critical for nanodevice
performances as illustrated in Section 1.1.2 for nanowire based solar cells.
(a) Temporal trend
x location
y location
total diffusion
skirt diffusion, 150 um skirt
(b) Spatial trend (profile) (c) Local variations (ZnO sample)
Figure 1.2: The decomposition of nanostructure synthesis variations
ZnO nanowires are grown by Vapor-Liquid-Solid mechanism. The growth is under
atmosphere pressure and 950 °C temperature with 100 sccm Ar flow for 25 minutes
(excluding ramp-up and cooling down time)
It is known that nanostructure interactions such as competition for source materials
play important roles in determining the formation of nanostructures and thus explain
the local variability among nanostructures. An example is demonstrated in Figure 1.2
for VLS (Vapor-Liquid-Solid) growth of ZnO nanowires. In particular, due to the occu-
pancy of larger space and the absorption of more Zn vapor, large nanowire bundles can
5
inhibit the growth of neighboring nanowires [84]. As a result, stronger interactions act
as an indication for the existence of defective structures as well.
Additionally, when nanostructures are assembled into nanodevices, their interac-
tions also play important roles in determining the performance of nanodevices. Taking
Si nanowire solar cells as an example, not only the geometry but the arrangement of
nanowires affect efficiencies of light absorption and surface recombination [26]. Nanos-
tructure interaction is an important quality index for nanostructures and nanodevices.
In this work, our quantification of nanostructure variations thus targets on the inter-
action patterns among neighboring nanostructures. Characterizing nanostructure inter-
actions will not only provide a nanostructure quality metric but also promote our under-
standing at nanoscale. We hope our work can trigger deeper exploration of the process-
structure-property relation.
1.1.4 Research Challenges for Nanostructure Interaction Charac-
terization and Control
As stated previously, to achieve scaled-up, reliable and cost-effective manufacturing
of nano-products, a central issue is to characterize nanostructure synthesis variations.
Since nanostructure interactions play important roles in determining the formation of
nanostructures, we thus focus on nanostructure interaction characterization for impor-
tant quality features of nanostructures.
Fundamental research challenges of developing such interaction characterization
technique include three aspects.
(1) There is no quantitative physical principles of interactions that we can rely on.
To our best knowledge, there is no physical modeling of nanostructure interactions
for synthesis processes. The only understanding is that nanostructures physically
6
closer have higher chances to mutually impact on each other. For nanodevice
quality determination, the only approach seen in literature is through molecu-
lar dynamics simulation [53, 54, 36]. Due to the extremely heavy computation
involved, the simulation is only for several nanostructures and thus far from being
sufficient to be used in nanomanufacturing research. Characterizing nanostruc-
ture interactions are subject to more uncertainties because of latent or unobserved
factors at the lower end of nanoscale.
(2) Nanostructure interaction characterization is severely constrained by the scarcity
of measurement data.
Current techniques of inspecting nanostructures are mainly two dimensional
imaging such as Scanning Electron Microscopy (SEM) and Transmission Elec-
tron Microscopy (TEM). To measure nanostructure features such as lengths of
nanowires, we have to take SEM/TEM images first and then extract informa-
tion from the 2D images. Both processes are time consuming and labor intensive
[2, 72, 101, 111]. While characterization of local variations requires reasonable
amount of data, measuring nanostructures for a small area on the substrate would
mean a formidable metrology task.
(3) There is no conclusive understanding in nanostructure growth kinetics.
Although we have relatively good understanding of growth kinetics compared
to nanostructure interactions, the understanding is not conclusive. Taking SA-
MOCVD synthesis of III-V nanowires which is a very promising nano-fabrication
technique as an example, there are active debates whether the process is gov-
erned by self-catalyzed VLS/VS growth [11, 57, 62] or diffusion based selective
area epitaxy [16, 69, 40, 33, 15, 30]. In addition, although selective area epitaxy
is more widely accepted as the dominant mechanism, the diffusion sources and
7
their relative contributions are still controversial. As a result, there is no coherent
description for major experimental observations, needless to say a general model
that captures the spatial and temporal growth trend of nanowires quantitatively.
The lack of growth process modeling disables systematic process optimization. It
also puts more uncertainties on interaction analysis at finer scales.
To address the challenges, we propose to characterize nanostructure interactions step
by step. We first use Gaussian Markov random field (GMRF) to model nanostructure
interactions statistically to capture various patterns of interactions, and then develop
techniques to estimate nanostructure interactions with limited data to relax the metrol-
ogy constraint. We will also work on growth process modeling to reinforce our under-
standing of the growth kinetics and facilitate a more accurate description of nanostruc-
ture interactions.
1.2 State of the Art on Scale-up Nanomanufacturing
Research
Nanomanufacturing is a multidisciplinary research area. To achieve commercial appli-
cations of laboratory successes, nanotechnology community has been gradually shifting
their efforts from discovering new nanostructure morphologies to achieving controlla-
bility and reproducibility [106]. Researchers from statistics, quality engineering, and
reliability are also actively involved.
The importance of scalable nanomanufacturing has been well-recognized in recent
years. What constitutes the scale-up nanomanufacturing research, however, has not
been clearly defined in literature. In the following, we thus provide our understanding
of nanomanufacturing research first to set the basis of our literature review here. After
that, we will summarize the state of the art for each sub-field in detail.
8
1.2.1 Classification of Nanomanufacturing Research
Based on the research focuses, we may classify the scale-up nanomanufacturing
research into two categories:
C1. scale-up process research: identifying and developing nanomanufacturing pro-
cesses and processing techniques with the potential of economical production at
commercial scale, and
C2. scale-up methodology research: establishing modeling, simulation, and control
methodologies that enable and support economical production at commercial
scale.
The scale-up process research focuses on the process-level issues which are normally the
focuses of nanotechnology researchers. For instance, the discovery and experimentation
of SA-MOCVD synthesis are scale-up process research. Their work provides a plat-
form that gives the potential to economically fabricate well controlled III-V nanowires
in commercial scale. On the other hand, the second category - scale-up methodology
research devotes its attention to system-level methodological issues such as yield and
quality improvement to fulfill the promise of identified scale-up processes. Our work in
this dissertation falls into the scale-up methodology research category.
Quantity
Yield/Quality
Size
Throughput
Scale-up
Methodology
Figure 1.3: Four pillars of scale-up methodology research
We may further classify the scale-up methodology research into four sub-categories
based on their research objectives, nature of the problems, measurement of outcomes,
9
and methodology domains as in Table 1.1. For instance, we classify the objectives of the
scale up methodology as: (i) improving process repeatability (i.e. increasing quantity),
(ii) scaling up size, (iii) increasing production throughput, and (iv) reduce defects (i.e.
increasing yield). As demonstrated in Figure 1.3, quantity, size, throughput, and yield
consist of the four pillars of a scale-up methodology research, where cost is considered
as the implied outcome from the four pillars.
Table 1.1: Sub-categories of scale-up methodology and comparison
Quantity Size Throughput Yield
1!N S! L S! F L! H
Objective from one to
many
from small to
large size or area
from slow to fast
processing rate
from low to high
quality
Nature of
problem
improve process
repeatability
scale size up increase produc-
tion rate
reduce defects
Measure of
outcome
large number of
units with low
variations
full-scale geom-
etry at full-scale
system
short production
cycle
low fraction
nonconforming,
low defect rate
Potential
Methods
variation reduc-
tion
dimensional
analysis
process design process & qual-
ity control
In the current literature, most research work in scale-up methodology field focuses
on improving the process repeatability (i.e. quantity) or reducing process nonconfor-
mities (i.e. yield). Fewer work has yet been done to scale-up the size or increase the
production rate. We hope that our summary of scale-up methodology research here not
only presents an overall understanding of the field but also could trigger further explo-
rations in these less-investigated research areas.
In the following part of this section, we will focus our review on the scale-up
methodology research. For easier reading, we divide our reviews into subsections based
on research stages and approaches. Particularly, the sub-fields of scale-up methodology
research are (i) nanostructure metrology (ii) nanostructure feature variation quantifica-
tion, (iii) nanostructure synthesis process modeling, (iv) experimental design for robust
10
synthesis of nanostructures, and (v) reliability modeling and assessment of nanodevices.
We will skip the review on reliability modeling and assessment of nanodevices since it
is not closely related to our work in this dissertation. Interested readers may find a com-
plete review recently given by [43]. For other sub-fields, we will review them one by
one in detail and summarize the state of the art. We will also explain our contributions
in this dissertation that are related to each of them.
1.2.2 Nanostructure Metrology
Metrology is the basis for nanostructure quantification and improved nanomanufactur-
ing. However, based on current instrumentation, measuring nanostructures is a time-
consuming, expensive and complicated process. It is reported that revolutionary instead
of evolutionary advances are needed to characterize dense quantities of nano-elements
under manufacturing-relevant time spans, but “no known solutions” are available for
5-10 years down the road [73]. Current inspection techniques for nanostructures are
mainly two dimensional imaging such as Scanning Electron Microscopy (SEM) and
Transmission Electron Microscopy (TEM). Nanostructure features are then extracted
and reconstructed from their 2D projections on the images.
For nanoparticle measurement, the major problems are various degrees of overlap-
ping among particles and irregular shapes of particles [72]. [2] increases nanoparti-
cle recognition by modifying the image intensity and highlighting particle boundaries.
[24, 59, 29, 13] segregate overlapping particles based on elliptical shape assumptions.
[72] relaxed the elliptical shape constraints and proposes to firstly learn and construct the
shape statistics from those clearly identifiable particles and then use the shape statistics
to split overlapping particles in TEM images.
For 1D nanostructures such as nanowires and nanotubes, features of interest include
lengths, diameters, orientations and density. [104] estimates the distribution of nanotube
11
lengths by first isolating nanotubes and then measuring those uncensored nanotubes on
the AFM (Atomic Force Microscopy) images. Later on, [50] corrected the possible
bias in this estimation by a nonparametric model that incorporates censored nanotubes.
To the best of our knowledge, [101] is the only work that systematically studies the
measurement of diameters and density for 1D nanostructures. It is also the only notable
work besides our own that measures nanowire lengths and orientations. They use a
geometry model to recover length and orientation of a nanowire based on its projection
lengths on three SEM images taken from different angles. The disadvantage of their
approach is that it requires nanowires to be fully visible in all images which however
hardly happens in reality.
In this dissertation, we will develop a new geometry model for nanowire length mea-
surement that requires only nanowire tips to be visible. Additionally, we will create a
design for selecting which nanowires to measure and establish an interaction analysis
technique targeting this “incomplete feature measurement”. Our approaches will pro-
vide as accurate feature quantification with much reduced metrology efforts. Details
will be discussed in following chapters.
1.2.3 Nanostructure Feature Variation Quantification
Nanostructure feature variation quantification is the further distributional analysis of
nanostructure quality features based on their raw measurement data. In the current liter-
ature, not much work has been done in this field. What is more, most existing literature
is about dispersion modeling and quantification of nanoparticles which unfortunately
cannot be directly applied in our case.
In particular, [12] develops a novel inhomogeneous Poisson random field modeling
that incorporates both process parameters and nanoparticle interactions to describe the
dispersion of nanoparticles in polymer composites. [118] derives statistical indices to
12
quantify nanoparticle clustering in the matrix. And [116] studies a specific type of
particle clustering that is around the grain boundaries of metal matrix. While we may
borrow the ideas to model and quantify densities of 1D nanostructures, how to describe
the distributions of continuous nanostructure features such as length, orientation and
diameter remain to be investigated.
Furthermore, although nanostructure interactions have been widely recognized as
an essential factor to nanostructure variations, little work has ever investigated them
when modeling feature variations. The only notable exceptions are [12] and [37] which
however were either modeling number of nanoparticles [12] or focusing on multi-scale
modeling framework [37].
To fill the research gap, we investigate nanostructure interactions extensively for
continuous nanostructure features in this dissertation. We will use our extracted interac-
tions to quantify feature variations and link the variations with process characteristics to
assist defect detection and diagnosis. We will discuss the details in following chapters.
1.2.4 Nanostructure Synthesis Process Modeling
Modeling nanostructure synthesis process is to relate nano-product quality features with
process conditions such as temperature and gas flow rate. It is designed to capture the
general growth trend of nanostructures so that people can predict and control the overall
quality at substrate level. It is thus essential to scalable nanomanufacturing control and
sets the basis for accurate interaction analysis.
Majority of nano-fabrication research is related to synthesis process modeling. In
general, we can classify the literature into three categories (i) physical modeling e.g.
[80, 15], (ii) statistical modeling e.g. [18, 38], and (iii) physical-statistical modeling e.g.
[37, 19] based on their research strategies.
13
Amount of Data
Physical Knowledge

Statistical
Modeling

Physical Modeling

Physical-Statistical
Modeling
Cross-Domain
Modeling
Figure 1.4: Comparison of synthesis process modeling strategies
This figure first appears in [103] and is cited in the review paper [113] of our own.
As illustrated in Figure 1.4, these three modeling strategies have their own suitable
domains of application [103, 113]. Specifically, with sufficient physical knowledge,
physical modeling is readily a choice even when experimental data is less accessible.
As physical modeling comes from first principles of physics or material science, it has
more generality. But when growth mechanisms are debatable, uncertainties in the first
principles may invalidate the purely physical modeling approaches. In this case, model
validation based on experiment data is necessary.
On the other hand, statistical modeling mainly relies on the experiment data. It
is capable to capture the variation among nanostructures. At the same time, it faces
the issue of limited data in nanomanufacturing due to costly growth experiment and
structure characterization [112]. Moreover, the collected data may have large variations
because of the poor in situ control of process variables such as growth temperature.
As a result, a large pool of candidate models can statistically fit the data. The data
requirement for statistical modeling alone is rarely satisfied [103].
14
Physical-statistical modeling comes into play when physical knowledge or data
alone is insufficient. It is claimed that it can consolidate advantages of both physical
and statistical modeling approaches. But in reality, it normally relies on both of them.
Besides the above listed modeling approaches, Wang and Huang (2013) [103] made
the first attempt to devise the cross-domain model building and validation (CDMV)
approach. Their objective is to investigate a growth process when the uncertainties in
physical knowledge and data continue to increase and are beyond the application domain
of physical-statistical modeling.
Just as mentioned before, there is no definite superiority of one modeling approach
over another. What matters is just which approach we should apply in what situations.
For SA-MOCVD synthesis which will be discussed later in this dissertation, we choose
to model its growth process based on precursor diffusion theories. By validating our
developed model on real experimental data, we can not only coherently explain vari-
ous growth patterns but also confirm that the growth mechanism to be diffusion-driven
selective area epitaxy.
1.2.5 Experimental Design for Robust Synthesis
Both traditional techniques in design of experiments [109] and newly invented method-
ologies that are tailored for nanotechnology circumstances have been applied to improve
the yield of nanostructure synthesis. For instance, (fractional) factorial designs are
adopted to sequentially optimize the aspect ratios of ZnO nanowires [114], to investigate
the effect of reaction temperature and time on nanoparticle sizes [10] and to study how
AFM normal force, scanning speed and tip abrasion affect the output voltage of nano-
generators [85]. In addition, [115] proposed a statistical design augmentation method
called “level expansion” to address the dual needs of investigating nonlinear effects and
dealising experiment factors in the follow up experiments of nanostructure synthesis.
15
[46] created the sequential minimum energy design (SMED) based on electrostatics to
quickly carve out the large no growth regions in nanostructure experiments and sequen-
tially optimize the yield.
Besides model free designs discussed above, [119] has developed a locally D-
optimal design based on the exponential-linear modeling established in [38]. Their
design is used to optimize the experimental strategy so that model parameters can be
estimated with minimum runs. In this work, we won’t discuss synthesis design issues.
But developing robust experimental strategies based on our established process model-
ing would be probably more efficient than general model free designs.
1.3 Research Tasks and Objectives
In this dissertation, we would like to investigate “bottom-up” nanowire synthesis pro-
cesses and continuous quality features of nanowires. Based on reviews and discus-
sions presented previously, we decide to target nanostructure interaction analysis and
growth process modeling of III-V nanowire synthesis via SA-MOCVD. Our objective
is to establish systematic metrics and methods to quantify nanostructure variations and
link variations with process conditions so as to support scale-up nanomanufacturing.
Three major research tasks will be conducted to address the challenges, fill the research
gap, and achieve our goal step by step.
(i) Nanostructure interaction characterization for a local region on the substrate
This task aims to develop the fundamental framework of nanostructure interac-
tion modeling and estimation. It assumes a known structure for the general trend
of nanostructure features and focus on the interaction characterization of a local
region where nanostructure metrology is not a great concern. We adopt Gaussian
Markov random field (GMRF) to model nanostructure interactions statistically and
16
establish the connection between interaction and variability in terms of conditional
correlations. The benefits of using GMRF is that it has great flexibility of capturing
various types of interaction patterns which are well consistent with our intuitions
of physical interactions. This flexibility certainly helps at the initial stage when we
have limited knowledge about the interactions.
(ii) Mapping nanostructure interaction pattern across the whole substrate
The direct application of local region interaction characterization to the whole sub-
strate is constrained by the metrology. This task thus aims to reduce the metrology
efforts for interaction characterization. We optimize Expectation-Maximization
algorithm to our GMRF modeling and develop a tailored space filling design for
measurement site selection to estimate nanostructure interactions. With the devel-
oped technique, we are able to extract interaction patterns for any sample or com-
pare different samples with excellent accuracy. Our approach dramatically reduces
metrology efforts, and thus enables interaction analysis under manufacturing-
relevant time spans for commercial scale fabrication.
(iii) Growth process modeling for III-V nanowire synthesis via SA-MOCVD
This task aims to develop a process model that captures both spatial and tempo-
ral trend of nanowire growth so as to facilitate our investigation of nanostructure
interactions at finer scale. Due to superior properties of III-V nanowire synthesis
via SA-MOCVD, we will demonstrate our modeling approach based on this pro-
cess. Successful completion of this task will enhance our understanding on the
growth mechanisms in SA-MOCVD and coherently explain various experimental
observations. Our growth model contributes significantly to the nanotechnology
society as well.
17
The three tasks together will provide a complete quantification of nanostructure
growth and its quality. They support systematic process optimization and control and
thus set basis for scalable nanomanufacturing control. Detailed discussions of our
approaches will be presented in following chapters.
18
Chapter 2
Nanostructure Interaction Modeling
and Estimation for a Local Region
In this chapter, we discuss the interaction characterization for continuous nanostructure
features such as lengths, diameters etc. for any local region of interest on the substrate
(i.e. specimen or synthesis sample). We will establish a connection between nanostruc-
ture variations and nanostructure interactions under the framework of Gaussian Markov
random field (GMRF). This connection not only (i) provides a metric for assessing
nanostructure quality but also (ii) enables a method to automatically detect structural
defects and identify their patterns based on the underlying interaction patterns. Both
simulation and real case studies are conducted to demonstrate our developed methods.
The insights obtained from real case study agree with physical understanding. Extension
of the nanostructure interaction characterization to whole substrate will be discussed in
Chapter 3.
2.1 Problem Formulation
As introduced in Chapter 1, nanostructure interactions such as competition for source
materials contribute strongly to the formation of nanostructures and defects during the
growth process [84]. What is more, when fabricating functional nanostructures into
nanodevices or nanosystems, nanostructures also interactively determine the properties
19
of the nanoproducts [26]. Therefore, our quantification of nanostructure variations tar-
gets on the interaction patterns among nanostructures. For convenience, we call features
of nanostructures in a local region on the substrate as “(nanostructure) local features”.
Issues to be addressed in this chapter thus include (1) measuring nanostructure local
features, (2) modeling and estimating nanostructure interactions based on local fea-
ture measurement and (3) quantifying local feature variability and identifying structural
defects through interaction estimation.
• We adopt a discrete sampling approach to measure nanostructure features.
With certain abstraction, nanostructures can be viewed to disperse in a real space with
dimension 1, 2 or 3 depending on applications. Following steps are followed to measure
nanostructure features in each local region of interest.
Step 1. We divide the local region regularly into non-overlapping sites (grids) as illus-
trated in Figure 2.1. Denote the sites asr =fs
1
;s
2
;:::;s
n
g
Figure 2.1: Illustration of nanostructure feature measurement in a local region of interest
Here we adopt a discrete sampling approach. That is we first divide the local region
regularly into non-overlapping sites and then for each site we selectively measure one
of the nanostructures or average the feature of interest for all nanostructures.
20
Step 2. Measure and summarize nanostructures in site s
i
as X(s
i
). Altogether, we
obtain a random field of nanostructure features
X(r) =fX(s
1
);X(s
2
);:::;X(s
n
)g: (2.1)
In detail, if a site has only normally grown nanostructures, we measure a randomly
selected nanostructure or average all nanostructures in that site. Here, “normally grown”
means the nanostructure is a single crystal and has no structural defects. If a site has
defective structures, we measure the defective structures approximately by following the
same procedure as that for normally grown nanostructures (the strategy of measuring
nanowire bundles in real case study illustrates this procedure). And if there is no growth
in the site, we take the feature measurement as zero for that site.
Under current inspection techniques e.g. SEM and TEM, we need to extract and
reconstruct from the 2D projections of nanostructures to obtain their feature measure-
ment. Therefore, it’s usually difficult to measure defective structures due to their com-
plex shapes. We suggest measuring these defects in the same way as measuring normally
grown nanostructures because this approximation permits automatic measurement pro-
cessing but still allows identification of defect patterns (validated in real case study of
ZnO nanowire bundles). A method to calculate nanowire lengths from SEM images is
presented in real case study.
• To extract nanostructure interaction patterns, we model the feature field as a
Gaussian Markov random field (GMRF) and estimate its parameters based on
the feature measurementX(r).
GMRF is a random vector that follows multivariate Gaussian distribution and possesses
Markovian property in precision matrix. Through the modeling and estimation, we will
understand the interactions by finding for each sites
i
: (1) its neighbors i.e. those sites
21
that have interactions withs
i
and (2) the relation between sites
i
and its neighbors.
Based on our understanding of nanostructure interactions, we will follow two principles
when selecting the neighbors: (i) nanostructures physically closer have more chances to
have mutual impacts, and (ii) the parsimony of neighborhood structures.

(a)

(b)

(c)

(d)

Figure 2.2: Samples of candidate neighborhood structures
(a) first order (DS 1, i.e. diamond scheme of order 1); (b) second order (SS 1, i.e.
square scheme of order 1); (c) DS 2; and (d) SS 2 (up diagonal filled grids represent
neighbors of the solid filled grid)
Figure 2.2 demonstrates candidate neighborhood structures we will consider in this
work. They are of two forms: diamond schemes (DS) and square schemes (SS) of
different orders. In our analysis, we will select from the simplest structure (i.e. first
order) and gradually increase the complexity till our model well explains experimental
patterns. Through case studies we will show these two forms of neighborhood structures
can cover most commonly observed interaction schemes.
• Based on interaction estimation, we take a sequential strategy to quantify local
feature variability and detect structural defects.
We first model local features as stationary GMRF by assuming all nanostructures are
normally grown. We then develop normality tests to detect defects. If there is no defect,
the estimated stationary GMRF characterizes features’ local variability. Otherwise, we
relax the stationarity assumption and estimate interaction patterns for every site in the
field individually. Defect patterns are then identified by “abnormally” strong interac-
tions. Details are given in following sections.
22
Following this problem formulation, Sections 2.2 and 2.3 discuss the modeling and
estimation of nanostructure interactions respectively. Section 2.4 presents simulation
case studies that demonstrate and validate our proposed strategies. Section 2.5 provides
a real case study for ZnO nanowires. At the end, Section 2.6 will give a summary.
2.2 Modeling of Nanostructure Interactions
To separate different levels of uncertainty, we first decompose local nanostructure fea-
ture fieldX(r) into two parts: local trend or mean and local variability. Specifically we
take the following model
X(r) =Z
+ (r): (2.2)
HereZ
describes the local mean by a linear combination of explanatory variablesZ
that can be spatial location functions or concomitant data with each site [17]. (r) cap-
tures the local variability by describing interactions among neighboring nanostructures.
This decomposition is different from Huang (2011) [37] becauseZ
here extracts the
local trend for the region of interest and is not characterized by growth kinetics.
We adopt Gaussian Markov random field (GMRF) [5, 17] to model nanostructure
interactions and associate the interactions with local feature variability. Specifically, for
each sites
i
, we define its neighbors and specify the conditional distribution of (s
i
):
(s
i
)j(s
j
);j6=iN
0
@
X
s
j
s
i

ij
(s
j
);
2
(s
i
)
1
A
(2.3)
wheres
j
s
i
meanss
i
ands
j
are neighbors of each other,
2
(s
i
) is the conditional
variance and
ij
is a spatial coefficient that is zero unlesss
i
s
j
. We interpret interac-
tions among nanostructures as their mutual contributions on each other’s local variabil-
ity. Our modeling states, the conditional expectation of local feature variability in site
23
s
i
is a “weighted” sum of the local variabilities in its neighbors. Intuitively, the larger
the “weights” (
ij
), the stronger the interactions.
GMRF has been extensively used in areas including spatial statistics [61], image
analysis [28, 6] and disease mapping [8] etc. By varying the neighborhood structures
(illustrated in Figure 2.2) and model parameters therein (e.g.
2
(s
i
);
ij
), GMRF is
capable of modeling complex and diverse interaction structures. This property will cer-
tainly assist us to identify the patterns of nanostructure interactions and develop a proper
characterization of local feature variability.
DenoteD = diagf
2
(s
1
);:::;
2
(s
n
)g andB = (
ij
). From Brook’s Lemma [9],
when (IB) is invertible and (IB)
1
D is symmetric (i.e.
ij

2
(s
i
) =
ji

2
(s
j
))
and positive definite,X(r) follows a joint normal distribution as following:
X(r)N

Z
; (IB)
1
D

(2.4)
which establishes the spatial distribution of nanostructure features. In particular, the
covariance matrix = (IB)
1
D or equivalently the precision matrixQ =D
1
(I
B) clearly captures the local feature variability. Besides, it is easy to show the pairwise
conditional correlation for any two sitess
i
ands
j
is [17]:
corr (X(s
i
);X(s
j
)jx(s
k
);k6=i;j) =

ij
j
ij
j
p

ij

ji
: (2.5)
Quantitatively, the stronger the conditional correlations, the stronger the interactions
among neighboring nanostructures.
For a local region that has no defect (i.e. normal/uniform growth of nanostructures),
we may assume stationary interaction patterns for every site across the field. That is
24

2
(s
i
) =
2
and
ij
=
q
whereq =s
i
s
j
that only depends on the relative location
of the two sitess
i
ands
j
. The conditional characteristics are thus
(s
i
)j(s
j
);j6=iN
0
@
X
s
j
s
i

q
(s
j
);
2
1
A
(2.6)
for any sites
i
. Due to symmetry of the precision matrix, we have
ij
=
ji
8i;j and

q
=
q
for anyq > 0. Besides,D = diagf
2
(s
1
);:::;
2
(s
n
)g =
2
I. So the joint
distribution of nanostructure feature field turns to
X(r)N

Z
;
2
(IB)
1

: (2.7)
where matrix B with B
ij
=
s
i
s
j
is a block circulant matrix under
regular division of sites and periodic boundary conditions. Furthermore,
corr (X(s
i
);X(s
j
)jx(s
k
);k6=i;j) =
q
(Equation 2.5).
q
directly explains the inter-
actions between sites
i
and sites
j
.
We classify GMRF models (i.e. interaction modeling) into two categories depend-
ing on whether they assume stationary interaction patterns. Specifically, we call the
modeling in Equation (2.6) with stationarity assumptions “stationary GMRF model-
ing” or “stationary interaction modeling”. Correspondingly, GMRF of general forms in
Equation (2.3) is called “nonstationary” or “varying coefficient” GMRF (or interaction
modeling). Because nonstationary GMRF involves much more unknown parameters,
for model parsimony we always try stationary GMRF modeling first. Only for local
regions that have been detected to have defects, we then turn to nonstationary GMRF to
identify the detailed defect patterns.
25
2.3 Estimation of Nanostructure Interactions
As introduced in Section 2.2, we model nanostructure interactions in terms of condi-
tional correlations between sites. Under the framework of GMRF, nanostructure interac-
tions and local features are described by model parameters =f
ij
;
2
(s
i
);
g
i;j=1;:::;n
in Equation (2.4). Due to the computation of matrix determinantjIBj, maximum
likelihood estimation (MLE) approach is only feasible for stationary GMRF defined on
regular fields. Therefore, we follow strategies in Figure 2.3 to estimate nanostructure
interactions. Particularly, we apply MLE to estimate stationary interactions in normal
growth regions and local likelihood estimation (LLE) to estimate nonstationary interac-
tions in regions with defects. Normality tests on transformed residuals detect defects
initially based on stationary interaction estimation.
Figure 2.3: Overall strategies of modeling and estimating nanostructure interactions
Based on local feature measurement, we start from stationary interaction extraction and
then detect defects by testing the stationarity assumption of nanostructure interactions.
If there exist defects, we extract interaction patterns using nonstationary GMRF.
26
2.3.1 Maximum Likelihood Estimation for Stationary Interactions
If nanostructure interactions are modeled by stationary GMRF, the local features follow
a normal distributionN(Z
;
2
(IB)
1
). Since we divide regions of interest regu-
larly,IB is a block circulant matrix under periodic boundary conditions. We can thus
efficiently calculatejIBj through Fast Fourier Transformation (FFT). Consequently,
maximum likelihood estimation (MLE) is feasible and efficient in this case.
We summarize the MLE approach as following by referring to relevant literatures
[107, 5, 31, 7, 63, 17]. First, we can write the negative log likelihood as
2L(jx(r)) =2 logf(x(r)j)
=n log
2
log(jIBj) +
1

2
(x(r)Z
)
T
(IB)(x(r)Z
) +c
(2.8)
where =f
q
;
2
;
g
q>0
denotes the collection of unknown parameters,n is the total
number of random variables (sites) in the field and c is a known constant. Let ^ =
f
^

q
; ^
2
; ^
g be the MLE of. Then setting@L=@
2
= 0 we obtain:
^
2
=n
1
(x(r)Z
)
T
(IB)(x(r)Z
): (2.9)
Similarly setting@L=@
= 0 we can see ^
is the generalized LSE for linear regression:
x(r) =Z
+ withMVN(0; (IB)
1
), or equivalently,
^
= (Z
T
(IB)Z)
1
Z
T
(IB)x(r) (2.10)
Therefore,
^
= arg minL(jx(r)) with =f
q
g
q>0
and
L(jx(r)) = logfn
1
(x(r)Z ^
)
T
(IB)(x(r)Z ^
)gn
1
log(jIBj) (2.11)
27
Based on
^
, we can obtain ^
2
and ^
from Equations (2.9) and (2.10). In addition, if
we denote
2
as the unconditional variance for X(s
i
); 8i, then ^
2
= ^
2
(I
^
B)
1
11
(Equation 2.4).
^ is a consistent estimator of for stationary GMRF. Besides, ^ follows a joint
normal distribution with mean and precision matrix J() (the fisher information
matrix) asymptotically [58, 17]. Confidence intervals/elipsoid for ^ can be obtained
accordingly. Models of different neighborhood structures can also be compared through
asymptotic likelihood ratio tests (LRT) [107]. GMRF with the winning neighborhood
structure will be selected to describe the stationary interactions.
Estimation of stationary nanostructure interactions can be viewed as an overall esti-
mation of interactions for the local region. We use it as a benchmark to detect defects.
For normal growth regions, it also specifies the spatial distribution of nanostructure fea-
tures:N(Z ^
; ^
2
(I
^
B)
1
) and determines the interaction patterns for each site.
2.3.2 Normality Tests on Transformed Residuals to Detect Defects
Since normally grown nanostructures have stationary interaction patterns across sites,
deviation of our measurement data from the stationary interaction pattern indicates
potential defects. Therefore, we can formulate the defect detection problem as a prob-
lem of testing following hypotheses.
H
0
: measured nanostructure feature field follows stationary GMRF;
H
1
: measured nanostructure feature field follows nonstationary GMRF.
We define the transformed residuals as
e = (IB)
1=2
(X(r)Z
): (2.12)
28
Under the null hypothesisH
0
, we haveX(r) N(Z
;
2
(IB)
1
) by referring to
Equation (2.4) and (2.6). Therefore, the transformed residualse N(0;
2
I). Based
on stationary GMRF estimation, we obtain MLEs of transformed residuals as
^ e = (I
^
B)
1=2
(X(r)Z ^
): (2.13)
Taking into account the consistency property of ^ =f
^

q
; ^
2
; ^
g
q>0
, MLEs of trans-
formed residuals ^ e
i
; i = 1;:::;n are also i.i.d normal random variables asymptotically.
That is,
^ e
i
i.i.d.N(0;
2
) asymptotically fori = 1;:::;n (2.14)
if stationary GMRF really captures the interaction patterns among measurement data.
On the contrary, if there are some defects, due to the inaccurate interaction extraction
for defective sites, transformed residuals of defective sites would be larger comparing to
those of normal sites. Therefore, evaluating the deviation of ^ e
i
; i = 1;:::;n from i.i.d.
normal distributions will help detecting defects.
We perform normality tests on transformed residuals ^ e to detect defects. Normality
tests detect “outliers” among ^ e
i
; i = 1;:::;n. They are certainly the most efficient
detection method when the number of defects is not large. Both graphical methods
including normality plots and formal tests including Anderson-Darling test, Cramervon-
Mises test and Lilliefors test etc. can be used. Thode (2002) [89] has a good summary
of normality test methods. In this paper, we recommend half-normal plot for visual
judgment because it has a clearest indication of outliers. Anderson-Darling test is chosen
for formal detection because its calculation on departure of empirical distributions from
normality puts more weights on tails [86] and thus it is more sensitive to defects.
29
If normality tests indicate the existence of defects in our measurement data, local
likelihood estimation (LLE) technique is then used to estimate interactions across the
field. Patterns of defects will be identified by patterns of interactions.
2.3.3 Local Likelihood Estimation for Regions with Defects
For regions detected to have defects, we use varying coefficient GMRF to model the non-
stationary interactions among nanostructures. Since spatial parametersf
ij
g
i=1;:::;n;s
j
s
i
vary across sites, dimension of unknown parameters is larger than measurement data.
Traditional estimation methods (e.g. MLE, pseudo-likelihood, coding method etc.
[5, 7, 17]) are not directly applicable here.
We adopt local likelihood estimation (LLE) to estimate nanostructure interactions
[47]. Parametersf
ij
g
s
j
s
i
and
2
(s
i
) that define the interactions of any sites
i
with
its neighbors are estimated based on those sites physically nears
i
. As illustrated in
Figure 2.4, given window size = 4 we use sites within the corresponding circles to
estimate the interactions for colored target sites. DenoteR

(s
i
) (includings
i
) to be the
collection of sites included for estimatings
i
under. To estimate parametersf
ij
g
s
j
s
i
and
2
(s
i
), we assume stationary interaction patterns amongR

(s
i
). In addition, we
assign different weightW

k
(s
i
) to each sites
k
2R

(s
i
). The weightW

k
(s
i
) measures
how much we believe the interaction pattern ofs
k
is close to that ofs
i
.
For each site s
i
, pseudo likelihood [5, 17] is chosen to estimate its parameters
because of the total computation complexity. In this way,f
ij
;
2
(s
i
)g
s
j
s
i
is the
weighted least squares estimator for linear regression
X(s
k
) =
i0
+
X
s
j
s
i

ij
X(s
k
+s
j
s
i
) +
ik
;8s
k
2R

(s
i
) (2.15)
30
Figure 2.4: Illustration of local likelihood estimation
Feature measurement of sites within the red solid circle are used to estimate the
interaction pattern for red filled site,while feature measurement of sites within the blue
dashed circle are used to estimate the interaction pattern for blue filled site.
for any i = 1;:::;n under given . Here
i0
is the intercept and
ik
is the kth error
term. We select the smoothing parameter (i.e. the window size) by cross validations
[95]. Bias correction on raw estimation off
ij
g
i=1;:::;n;s
j
s
i
from Equation (2.15) is
then performed by referring to Equation (15) in [47]. Interested readers please refer to
[22, 47] and references therein for more details.
2.4 Simulation Case Studies
Simulation case studies are to illustrate and validate our proposed modeling and estima-
tion procedures for extracting nanostructure interactions and characterizing nanostruc-
ture local features.
31
2.4.1 Setups of Simulation Experiments
We simulated 100 i.i.d. datasets defined on 50 100 regular lattice r =
f(s;t)g
s=1;:::;50;t=1;:::;100
. These datasets mimic (replicates of) feature measurement
fields we may obtain from normal growth regions. Each dataset has structureX(r) =
Z
+ (r) as in Equation (2.2). The local trendEX(r) =Z
is defined as:
EX(s;t) =
0
+
1
s +
2
t: (2.16)
The nanostructure interaction field (r) follows stationary GMRF with second order
neighborhood structure. That is, for each sites
i
= (s;t)2r, we have
E(s;t)jf(s
0
;t
0
)g
(s
0
;t
0
)6=(s;t)
=
1
f(s 1;t 1) +(s + 1;t + 1)g
+
2
f(s 1;t) +(s + 1;t)g
+
3
f(s 1;t + 1) +(s + 1;t 1)g
+
4
f(s;t 1) +(s;t + 1)g (2.17)
and Var((s;t)jf(s
0
;t
0
)g
(s
0
;t
0
)6=(s;t)
=
2
. Values set for simulating the datasets are
summarized in Table 2.1. We include both positive and negative values for =
f
1
;
2
;
3
;
4
g to explore the most general patterns of nanostructure interactions.
Table 2.1: Parameter values set in simulation studies

0

1

2

1

2

3

4

2
Values 6.00 -0.03 -0.01 -0.1 0.25 0.05 0.15 1
32
2.4.2 Interaction Extraction for Normal Growth Regions
For each previously simulated dataset (referred as stationary datasets), we model it as
stationary GMRF with local trendEX(s;t) =
0
+
1
s +
2
t. We then estimate
parameters and interaction schemes for GMRF models by MLE.
Different neighborhood structures are compared based on asymptotic likelihood
ratio tests [107] at 95% confidence level. Among the 100 datasets, 95 datasets cor-
rectly select second order scheme to describe the interaction structure while the other 5
datasets select diamond structure (DS 2 in Figure 2.2).
As to parameter estimation, we summarize MLEs of
; and
2
in Table 2.2 based
on those 95 datasets that select second order scheme. From the results we can see
MLEs of parameters are quite stable among different datasets and are very close to their
corresponding true values. We also plot MLEs of
i
;i = 1;:::; 4 together with their
Table 2.2: Summary of MLEs for
; and
2
based on stationary datasets (95 samples)
Parameters True Val. Est. Mean 25% Qt. Median 75% Qt. Std. Dev.

0
6.00 6.0020 5.9633 6.0082 6.0514 0.0744

1
-0.03 -0.0299 -0.0310 -0.0298 -0.0287 0.0016

2
-0.01 -0.0101 -0.0107 -0.0101 -0.0095 0.0009

1
-0.10 -0.1013 -0.1088 -0.1020 -0.0904 0.0131

2
0.25 0.2512 0.2447 0.2526 0.2577 0.0102

3
0.05 0.0477 0.0413 0.0480 0.0539 0.0101

4
0.15 0.1506 0.1404 0.1495 0.1603 0.0140

2
1.00 0.9958 0.9837 0.9936 1.0078 0.0219
95% confidence bounds (theoretically estimated) in Figure 2.5. Vibrations of confidence
bounds are due to the randomness among different datasets (i.i.d. samples from the same
distribution). From the figure we can see the 95% confidence intervals of
i
;i = 1;:::; 4
contain corresponding true values almost all times.
Although we know there is no defect in our simulated data, we still do normality
tests to detect defects as described in Section 2.3. Among the 100 simulated datasets, 97
33
0 20 40 60 80 100
−0.16 −0.12 −0.08
MLE and 95% confidence intervals of beta_1
dataset index
estimates
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 95% C.B.
true values
MLE
0 20 40 60 80 100
0.20 0.22 0.24 0.26 0.28
MLE and 95% confidence intervals of beta_2
dataset index
estimates
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 95% C.B.
true values
MLE
0 20 40 60 80 100
0.00 0.02 0.04 0.06 0.08 0.10
MLE and 95% confidence intervals of beta_3
dataset index
estimates
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 95% C.B.
true values
MLE
0 20 40 60 80 100
0.10 0.12 0.14 0.16 0.18 0.20
MLE and 95% confidence intervals of beta_4
dataset index
estimates
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 95% C.B.
true values
MLE
Figure 2.5: MLEs and 95% confidence bounds of
i
; i = 1;:::; 4
The 95% confidence intervals of
i
contain corresponding true values almost all times.
Vibration of confidence bounds are due to the randomness among different datasets.
of them have an Anderson-Darlingp value greater than 5%. That is at 95% confidence
level only 3 out of 100 datasets are falsely detected to have defects. Our method for
detecting defects works well for normal growth regions.
The simulation study of stationary datasets shows our method can correctly extract
interaction patterns for normal growth regions. The estimated stationary GMRF accu-
rately describes features’ spatial variability. We also have high confidence that our sug-
gested normality tests don’t falsely indicate defects.
34
2.4.3 Interaction Extraction for Defects Detection
For each stationary dataset, we set randomly 50 250 sites in it as zeros (i.e. 1% 5%
zero ratio). They mimic those no growth sites caused by underlying deformities of
the substrate. By studying these nonstationary datasets, we hope our estimation of
nanostructure interactions and proposed normality tests can detect them and identify
their patterns.
Following the strategies summarized in Figure 2.3, we first model each nonstationary
dataset through stationary GMRF pretending all nanostructures are normally grown.
Then normality tests are performed to detect defects based on MLEs of the transformed
residuals (I
^
B)
1=2
(x(s)Z ^
). For the 100 nonstationary datasets, the Anderson-
Darling p values are all near zero (< 10
32
) indicating the existence of defects in all
datasets. Our defect detection result here is 100% consistent with reality.
Since all datasets are detected to have defects, we randomly select one of them to
illustrate the identification of detailed defect patterns. Based on the overall interaction
estimation for selected dataset, varying coefficient GMRF (Equation 2.3) with second
order neighborhood structure is adopted to model its nonstationary interactions. The
cross validation errors under different smoothing parameter are shown in Figure 2.6
for the selected dataset. Based on the calculation result, windows size is chosen as 12
finally.
We then estimate the nonstationary interactions by local likelihood technique under
= 12. Level plots of estimated
i
; i = 1;:::; 4 are in Figure 2.7. Locations of
designed no growth sites are depicted as magenta squares in Figure 2.7 to assist visual
comparison. In addition, we also add red arrows with the level plots to indicate cor-
responding directions of the interactions. Here
i
; 8i has the same meaning as that
explained in simulation setups. Therefore, values of
i
;8i determine the intensity of
interactions across the field.
35
5580 5600 5620 5640 5660
smoothing parameter lambda
total cross validation errors
9 12 15 18 21 24 27 30
Figure 2.6: Smoothing parameter selection by cross validation [22]
= 12 is chosen because it gives the smallest cross validation errors.
From the Figure 2.7 we can see local peaks of nanostructure interactions are always
related to those no growth sites. This observation is both consistent with the data: higher
correlation between zeros and their neighboring values and the physics: defects have
higher impact on neighboring sites due to competition of source materials etc.
Simulation studies presented above show that our proposed strategies to characterize
local feature variability and detect defects by extracting nanostructure interactions are
feasible and effective. Although no growth may be an extreme form of “defects”, our
results suggest it is feasible in general cases to identify defects by extracting interaction
patterns from feature measurement. The real case study of large bundle defects supports
this claim in the following section.
36
(a) Local likelihood estimation for
1
(b) Local likelihood estimation for
2
(c) Local likelihood estimation for
3
(d) Local likelihood estimation for
4
Figure 2.7: Level plots of local likelihood estimated =f
1
;
2
;
3
;
4
g together with
No Growth sites (magenta squares) for one of simulated nonstationary datasets
Local peaks of interactions (i.e. values of
i
) are related to those no growth sites.
2.5 Real Case Study: ZnO Nanowire Experiment Data
Analysis
In this section, we are going to study interactions among nanowires through length mea-
surement to automatically detect nanowire bundles (insets of Figure 2.8).
37
2.5.1 Data Collection: Nanowire Length Measurement
We measured a 27um 56:4um local region of a ZnO nanowire sample which was
synthesized through VLS (Vapor-Liquid-Solid) at 950

C, 1 atm. with 100 sccm Ar.
flow for 25 minutes. For the selected region, we first take SEM images from its top.
The stitched top view image from 80 SEM shots with resolution 12,000 (SEM images
similar to those shown in Figure 1.2 (c)) for the region is in Figure 2.8 (a). Then, we
tilt the ZnO nanowire sample by 8

and take SEM images similarly. The stitched tilted
view image is as Figure 2.8 (b). White stripes on the images are nanowire bundles. One
of them is enlarged in the insets of Figure 2.8 for a clearer view.
Figure 2.8: SEM images of ZnO Nanowires (a) top view and (b) tilted view
The images cover a 27um 56:4um local region on the substrate. White stripes on the
images are nanowire bundles. One of bundles is enlarged in the inserts.
38
As introduced in Section 2.1, we adopt a discrete sampling approach to measure
nanowires. Specifically, we divide the region (i.e. top view image) regularly into 4594
grids (sites) each of size 0:6um 0:6um. The grid size is selected such that for each
grid we roughly have two or three nanostructures. In this way, the sampling resolution
is good and we don’t artificially create any no growth sites.
For a site/grid that contains any normally grown nanowire, we would measure a
randomly selected nanowire in it. The algorithm of measuring nanowire length is a geo-
metric model which reconstructs from tip locations of the same nanowire on the two
images (Figure 2.8 (a) and (b)). We illustrate the geometric model in Figure 2.9. In
Figure 2.9: Geometric model for measuring NW length
the model,xyz represents the machine coordinates during SEM imaging. Among
them, axisx is the axis around which nanowire sample is tilted. X,Y andZ are pro-
jection distances between nanowire tip and the reference point along axis x, y and z
respectively. represents the projection angle of nanowire tip relative to the reference
point inyz plane. We useD to represent the actual distance between nanowire tip and
the reference point and useP to mean the projection length of nanowire on the top view
image. For distinction, we denote (:)
p
and (:)
d
as location parameters on top view and
39
tilted view images respectively. As illustrated in Figure 2.9, we then have the following
relations:
K
p
=
q
D
2
X
2
p
=Y
p
= cos(
p
) (2.18)
K
d
=
q
D
2
X
2
d
=Y
d
= cos(
d
) (2.19)
Z
p
=Y
p
tan(
p
) (2.20)
L =
q
Z
2
p
+P
2
(2.21)

p
=
d
+: (2.22)
whereL denotes the actual length of the nanowire. Among the parameters in Equations
(2.18) - (2.22), Y
p
and Y
d
are measurable as long as we can extract the tip locations
on both images. In addition, since axisx in Figure 2.9 corresponds to the axis around
which we tilt nanowire samples during SEM imaging, the difference
p

d
= is the
known tilting angle that is 8

in our case. Besides, tilting around axisx also means we
haveX
p
=X
d
that isX keeps unchanged during rotation. Therefore,
Y
p
= cos(
p
) =Y
d
= cos(
d
): (2.23)
The height of nanowire tip to the reference point can thus be solved as
Z
p
=
Y
p
cos()Y
d
sin()
(2.24)
SubstitutingZ
p
in Equation (2.21), we obtain the nanowire length
L =
q
P
2
+Z
2
p
=
s
P
2
+
(Y
p
cos()Y
d
)
2
sin
2
()
: (2.25)
40
For a site that has no growth, we take the length measurement as zero. And if a
site is occupied by a nanowire bundle, we measure the bundle approximately as mea-
suring straight nanowires. It means that we still follow the model designed for straight
nanowires and adopt Equations (2.18) - (2.25) to calculate the vertical distanceZ
p
and
nanowire bundle length L. But because of the curved shapes for nanowire bundles
(insets of Figure 2.8), the measurement of projection lengthP is not accurate and the
model for nanowire bundle lengthL (Equation 2.25) is an approximation.
Based on the strategy above, we obtain a 45 94 nanowire length field altogether.
The effect of measuring nanowire bundle length approximately on interaction estimation
and defect detection will be discussed in following subsections.
2.5.2 Preliminary Data Analysis
We first look at the data graphically. Figures 2.10 and 2.11 show level plot of the NW
length field (with contour lines) and locations of nanowire bundles and no growth sites
respectively. From the figure we can see, (1) neglecting no growth sites, nanowire
lengths generally decrease from top to bottom and from left to right; and (2) nanowire
bundles are normally surrounded by no growth sites indicating stronger impact of bun-
dles on their neighboring sites.
41
column
row
10
20
30
40
20 40 60 80
0
1
2
3
4
5
6
7
8
9
Figure 2.10: Level plot of real collected length field for ZnO nanowires
Nanowire lengths generally decrease from top to bottom and from left to right.
We further plot nanowire lengths with corresponding row and column indexes of
the field in Figure 2.12. Those no growth sites are neglected to give a clearer view
of the trend. Lengths of nanowire bundles (approximate) are indicated by blue circles.
Our findings are: (1) local trend of the lengths is generally linear with row and column
indexes; and (2) nanowire bundles cannot be identified directly by their lengths.
Therefore, we propose to detect bundles by extracting interaction patterns. Besides,
we take the local trend asEX(s;t) =
0
+
1
s +
2
t (same as Equation 2.16) with
s =f(s;t)g
s=1;2;:::;45;t=1;2;:::;94
in stationary GMRF modeling at first step.
2.5.3 Interaction Extraction for Nanowire Bundle Detection
We first model the length field as a stationary GMRF with its local trend as in Equation
(2.16) pretending we don’t know there are defects within the field. Our motivations are
(i) to get an overall estimate of local trend and nanowire interactions and (ii) to check
whether our method can automatically detect defects without looking at the images.
42
0 20 40 60 80
column
row
40 30 20 10 0
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Figure 2.11: Locations of nanowire bundles and no growth sites
represents nanowire bundles and denotes no growth sites. In the figure, nanowire
bundles are normally surrounded by no growth sites.
0 10 20 30 40
2 4 6 8
row index
NW lengths
0 20 40 60 80
2 4 6 8
col index
NW lengths
Figure 2.12: Nanowire (bundle) lengths VS. row indexes and column indexes of the
field excluding no growth sites where corresponds to lengths of nanowire bundles and
is for normally grown nanowires.
By stationary GMRF modeling, we select second order scheme at 95% confidence
level. Table 2.3 shows MLEs and 95% confidence intervals of the parameters. Based
on the estimation, we do normality tests on estimated transformed residuals ^ e = (I
^
B)
1=2
(x(s)Z ^
) to see whether there is defect in the field. Anderson-Darling test
gives a zerop value. Half-normal plot of ^ e is also depicted in Figure 2.13 to assist the
43
Table 2.3: MLEs and 95% C.I.s of stationary GMRF parameters for NW length field
Parameters
1

2

3

2
MLE 6.5648 -0.0295 -0.0254 1.0865
95% C.I. (6.4240, 6.7055) (-0.0335,-0.0256) (-0.0273, -0.0234) (1.0388,1.1342)
Parameters
1

2

3

4
MLE 0.0605 0.0810 0.0585 0.1384
95% C.I. (0.0322, 0.0888) (0.0510, 0.1111) (0.0302, 0.0869) (0.1100,0.1668)
detection. Both indexes strongly indicate the existence of defects. Our method indeed
detects defects automatically without observing the images.
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●● ●●●●● ● ●● ● ● ●●●● ●●● ● ● ●● ● ● ● ● ●●● ●● ● ●●●● ●● ● ●●●● ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●● ●●●●●●● ●● ● ●●●●●●● ● ●●●●● ●●●●●●●●●● ●● ●● ●●●●● ●●●●●●● ●●●●●●●●●●●●●●●● ● ●●●●●●●●●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 1 2 3 4
0 1 2 3 4 5 6
half−norm quantiles
sorted transformed residuals
Figure 2.13: Half-normal plot for transformed residuals ^ e, real case study
S-curve the transformed residuals indicates the violation of normality assumption.
Then we model the nanowire length field by second order varying coefficient GMRF
to identify detailed defect patterns. Model parameters are estimated by local likelihood
estimation (LLE) technique. Cross validation errors under different values are shown
in Figure 2.14. So = 12 is chosen. We plot estimated
i
; i = 1;:::; 4 across the
field on top of the top view SEM image in Figure 2.15. As expected, local peaks of
interactions always correspond to nanowire bundles. Although length measurement of
44
3800 4000 4200
smoothing parameter lambda
total cross validation errors
9 12 15 18 21 24 27 30
Figure 2.14: selection by cross validation, real case study
= 12 gives the lowest cross validation error.
nanowire bundles is an approximation, our extracted patterns of nanowire interactions
still identify the patterns of defects satisfactorily.
Nanowire bundles absorb more source materials during the growth process and
occupy additional space due to their curved shapes (insets of Figure 2.8). As a result,
nanowire bundles inhibit the growth of neighboring sites and may even cause no growth
areas around them (Figure 2.11). Consequently, their estimated interactions from the
length measurement data are stronger than other sites. That is why we may identify
defects by “abnormally” strong interactions.
By studying real collected nanowire length data, we can confirm although we take
approximate length measurement for nanowire bundles, our method to model and esti-
mate nanostructure interactions still automatically detects defects and identifies defect
patterns. Our studies of nanostructure interactions are consistent with known physical
understanding.
45
(a) Local likelihod estimation for
1
(b) Local likelihod estimation for
2
(c) Local likelihod estimation for
3
(d) Local likelihod estimation for
4
Figure 2.15: Level plots of =f
1
;
2
;
3
;
4
g together with top view SEM images.
Local peaks of interactions (dark color area) always correspond to nanowires bundles
(white stripes).
Remark 1: Our approach of modeling and estimating nanostructure interactions
enables not only the characterization of nanostructure features, but also the detection of
defects due to “abnormal” interactions among nanostructures.
Remark 2: Simulation study and real case study show the identification of no growth
sites caused by underlying deformities on the substrate and neighboring defective struc-
tures (large bundles) respectively. Further research can be done to distinguish these two
failure mechanisms.
46
2.6 Summary and Conclusion
In this chapter, we develop a technique to model and estimate interactions among neigh-
boring nanostructures for any local region of interest to assist the characterization of
local features and automatic diagnosis of structure defects.
We interpret interactions among nanostructures as the mutual contributions on neigh-
bors’ feature variability. Under the framework of Gaussian Markov random field, we
incorporate nanostructure interactions into the modeling of local features and establish
their connections with local feature variability by pairwise conditional correlations. Our
strategies of interaction estimation consist of three sequential steps: estimating the sta-
tionary interactions by MLE (maximum likelihood estimation) assuming no defects,
detecting defects by normality tests on transformed residuals based on stationary inter-
action estimation and estimating the nonstationary interactions by LLE (local likelihood
estimation) if the region has defects. Estimated nonstationary interaction patterns iden-
tify defect patterns.
Both simulation and real case studies of nanostructure interactions are consistent
with existing physical knowledge. Our approach indeed quantifies local feature vari-
ability and diagnoses defects accurately and automatically. Further research is needed
to distinguish different defect mechanisms. Deeper exploration of nanostructure interac-
tions will assist to generate new understandings of process-structure-property relation.
Our method of interaction analysis here requires complete nanostructure feature
measurement. Its direct extension to scalable nanomanufacturing systems is thus con-
strained by nanostructure metrology. In Chapter 3, we will discuss our strategies to relax
the metrology constraint and how we can extend our method here to quantify variations
of nanostructures on the whole substrate.
47
Chapter 3
Nanostructure Interaction Modeling
and Estimation for Whole Substrate
In Chapter 2 we have demonstrated a method to characterize nanostructure interactions
for a local region. It was shown that interaction analysis helps with quantifying nanos-
tructure growth quality and detecting structural defects. In this chapter, we extend our
interaction analysis to the whole substrate. The ultimate goal is to enable automatic
quality assessment of nanostructures within manufacturing relevant time spans and for
commercial scale so that it could be applied in scale-up nanomanufacturing systems.
As discussed previously, our original interaction analysis technique is built on complete
feature measurement. Therefore, the major bottleneck of directly extending our previ-
ous method is nanostructure metrology. Particularly, for current inspection techniques
such as scanning electron microscope (SEM), the key difficulties of measuring nanos-
tructures lie in two aspects: (i) taking and calibrating images for seamless coverage and
(ii) extracting and matching feature information from the images. In this chapter, we
will thus develop a tailored sampling strategy to relax the metrology constraint and cus-
tomize our interaction analysis to the corresponding “incomplete” feature measurement
to optimize our interaction analysis efficiency.
48
3.1 Introduction
3.1.1 Motivation
Automatic assessment of nanostructure quality within manufacturing relevant time
spans is essential for scale-up nanomanufacturing. Due to the critical roles played by
nanostructure interactions such as competing for source materials during growth pro-
cess, assessing nanostructures through interaction analysis has proved to be efficient in
Chapter 2. By extracting interactions, we could not only quantify nanostructure fea-
ture variability but also detect structural defects and identify their patterns. Interaction
analysis thus carries great potentials to facilitate scalable nanomanufacturing if we can
extend it from a local region to the whole substrate.
The major constraint for the direct extension of our previous approach is nanostruc-
ture metrology. On one hand, because the analysis algorithm was built on complete
feature measurement, mapping interaction patterns across the whole substrate would
demand complete information of all local regions. On the other hand, characterizing
nanostructure features is extremely labor intensive and time consuming under current
inspection techniques [101, 111]. To characterize dense quantities of nano-elements
under manufacturing-relevant time spans, it is necessary to have revolutionary instead
of evolutionary advances. But “no known solutions” are available for 5-10 years down
the road [73].
To better support nanomanufacturing scale up, a customized sampling strategy is
thus urgently needed to relax the metrology constraint but still maintain the desired
characterization resolution. At the same time, methods to extract nanostructure interac-
tions also have to be updated based on the corresponding sampled feature measurement
to ensure accurate structure assessment within desired time spans.
49
3.1.2 Overall Strategy
In view of the great potentials of interaction analysis and difficulties of nanostructure
metrology, we develop a two-step nanostructure sampling strategy as well as an interac-
tion estimation strategy to reduce feature measurement and speedup interaction analysis.
The overall framework is in Figure 3.1.
Objective: automatic and prompt
assessment of NS quality
Technique:
interaction analysis
importance of NS interactions
Constraint:
NS metrology
extracting feature
information from images
taking and calibrating
images
Interaction estimation strategy:
optimized EM estimation
Sampling strategy:
measuring selective
regions
Sampling strategy:
measuring selective sites
in each region
“incomplete” feature measurement
Figure 3.1: Overall strategies of interaction analysis under metrology constraint.
Modules in solid red correspond to research tasks in this chapter.
Sampling Strategy: For current inspection techniques such as scanning electron
microscope (SEM), the key difficulties of measuring nanostructures lie in two aspects:
(1) taking and calibrating images for seamless coverage and (ii) extracting and matching
50
feature information from the images. Our approach to measure nanowire lengths in
Chapter 2 well illustrates the point. For instance, due to limited coverage of SEM,
we may need to carefully take and calibrate over 200 images of neighboring areas to
cover a 0:1mm 0:1mm local region. Additionally, it is also computation intensive to
extract feature information from the two dimensional images. For example, to measure
nanowire lengths, we need to match SEM images taken from different angles and find
the same nanowire on each image among the nanowire forest [101, 111, 3].
Accordingly, to reduce metrology efforts but maintain the desired interaction analy-
sis resolution, we take following strategy to sample and measure nanostructures.
Step 1: We selectively measure multiple separated tiny regions spreading on the sub-
strate.
Step 2: For each selected region, we divide it regularly into non-overlapping sites
(grids) and only measure nanostructures in selected sites.
The above approach is explained in Figure 3.1 and illustrated in Figure 3.2.
(a) measurement regions across the substrate (b) non-measurement sites in each region
Figure 3.2: Nanostructure sampling strategies.
(a) measurement regions (blue squares) across the substrate (b) non-measurement sites
(patterned grids) in each measurement region
To better explore the whole growth region, standard space filling designs such as
maximin distance Latin hypercube designs are applied to select the measurement regions
51
(blue squares in Figure 3.2 (a)). And for each selected region, we will develop a tailored
space filling design in Section 3.3 to choose which sites to measure (e.g. white grids in
Figure 3.2 (b)). By selectively measuring partial of the sites, we can not only lighten
the burden of feature extraction but also maintain an appropriate interaction analysis
resolution. Moreover, our space filling design minimizes sampling biases and supports
subsequent interaction analysis (Figure 3.1).
We call those sites to be measured “measurement sites” and those not to measure
“non-measurement sites”. The resulting feature measurement is denoted as “incom-
plete” feature measurement in this work.
Please note, although we take a two-step procedure to sample nanostructures, our
strategy is different from the sequential or nested designs in literature [39, 77, 75, 76,
94, 78] for two reasons. First, regions and sites are all determined before the real mea-
surement takes place. There is no sequential evaluation. Second, our dataset consists
only feature measurement in the sites. There is no multiple levels of sampling or approx-
imation.
In addition, Latin hypercube based space filling designs [60] are not applicable to
select non-measurement sites for selected regions, because we have both pre-established
division of sites driven by desired resolution of interaction analysis and pre-determined
number of sites to measure due to metrology constraints.
Interaction Estimation Strategy: In accordance with the “incomplete” feature
measurement, we also update our interaction estimation method in Chapter 2 to ensure
accurate and prompt quality assessment of nanostructures. For the new estimation tech-
nique, we would like it to possess following key features. (i) It is capable to deal with
“missing values”. (ii) It can be easily fitted within our interaction modeling frame-
work. And (iii) it accurately estimates nanostructure interactions within desirable time
spans. In this work, we optimize Expectation-Maximization (EM) algorithm to estimate
52
nanostructure interactions. We will also customize the selection of measurement sites
to further enhance the interaction estimation efficiency.
By integrating our developed sampling strategy and interaction estimation strategy,
we are capable of characterizing nanostructures for scale-up nanomanufacturing sys-
tems. Details are discussed in following sections.
3.2 Modeling and Estimation of Nanostructure Interac-
tions
To facilitate our discussion, we introduce following notation for measurement regions,
sites and nanostructure features.
• R is the total number of measurement regions.
• l is the index of a measurement region withl2f1; 2;:::;Rg.
• n is the total number of sites in a measurement region.
• i orj is the index of a site in a measurement region withi; j2f1; 2;:::;ng.
• s
(l)
i
represents the spatial location of sitei in measurement regionl.
• r
l
=fs
(l)
1
;s
(l)
2
;:::;s
(l)
n
g denotes the complete collection of sites in regionl.
• m
l
represents the collection of measurement sites in regionl.
• o
l
=r
l
nm
l
is the collection of non-measurement (omitted) sites in regionl.
• A
l
denotes the matrix that maps the complete collection of sites to measured sites
in regionl, i.e.m
l
=A
l
r
l
.
• X(s
(l)
i
) represents nanostructure feature at sites
(l)
i
.
• X(r
l
) =fX(s
(l)
1
);X(s
(l)
2
);:::;X(s
(l)
n
)g is the collection of nanostructure fea-
tures at regionl.
53
• X(m
l
) is feature field for measurement sites in regionl. It is also called “mea-
sured feature field”.
• X(o
l
) =X(r
l
)nX(m
l
) is feature field for non-measurement sites in regionl.
Although here we assume different measurement regions are of the same size (i.e. every
region containsn sites), the modeling and estimation techniques developed in this chap-
ter apply in general cases. On the other hand, the equal size assumption is actually easy
to satisfy in practice since we normally determine the region sizes before doing the real
measurement.
3.2.1 Extension of Local Region Stationary Interaction Modeling
In Chapter 2 we have a complete discussion on interaction modeling for any local
region. Specifically, we adopted Gaussian Markov random field to describe the con-
ditional dependence among physically close (i.e. neighboring) nanostructures. We
started from stationary interaction modeling (assuming homogeneous interaction pat-
terns among nanostructures) and then turn to nonstationary interaction modeling for
detailed pattern recognition if defects are detected.
In this chapter, we focus on the first step: stationary interaction modeling and esti-
mation based on incomplete feature measurement. We will extend the stationary inter-
action modeling of one local region in Chapter 2 to the whole substrate by assembling
interaction relations from the multiple measured regions. We will also briefly men-
tion the defect detection in real case study but leave nonstationary interaction analysis
to interested readers. The extension of stationary interaction analysis to nonstationary
interaction analysis is easy by using local likelihood methods [47, 111].
We follow similar framework as in Chapter 2 to quantify nanostructure feature varia-
tion and detect structural defects. As the focus here is interaction analysis under metrol-
ogy constraints, we will discuss in detail the sampling strategy of nanostructures and the
54
stationary interaction estimation based on the incomplete feature measurement. We will
also briefly mention the detection of defects in real case study but leave non-stationary
interaction analysis to interested readers. The extension of stationary interaction analy-
sis to non-stationary interaction analysis will be easy by using local likelihood methods
[47, 111].
Similar to our previous modeling, we assume the feature measurement in each region
l can be decomposed into local trend and local variability as following:
X(r
l
) =Z(r
l
)
+ (r
l
): (3.1)
The local trend is modeled as a linear combination of explanatory variablesZ that can
be spatial location functions or concomitant data with each site [17]. For local variability
(r
l
), we use stationary Gaussian Markov random field to describe interactions among
neighboring sites in each region.
(s
(l)
i
)jf(s
(l)
j
)g
j6=i
N

X

ij
(s
(l)
j
);
2

(3.2)
Heref
ij
g are spatial coefficients that depend soly on the relative locations between
sitess
(l)
i
ands
(l)
j
and are zero unless the two sites are neighbors of each other.
2
is the
conditional variance of local variability at sites
(l)
i
given its neighbors.
We assume the local trend of each measurement region has the same functional form
and shares the same set of regression parameters
. It implies that the global trend for
the whole substrate is a smooth surface described byZ
. If measurement data objects
this assumption, it means we have a more complex trend profile due to large process
variations. In this case, one possible extension of the trend modeling here is to take a
different set of regression parameters
l
for each region and connect
l
with the process
55
conditions such as temperatureT . Interested readers may refer to Huang (2011) [37] for
the connection.
In addition, our model here also assumes the interaction parameter
ij
between any
two sitesi andj in regionl only depends on their relative locations. That is, we conjec-
ture homogeneous and nonisotropic interaction patterns across the substrate. When the
local trend is correctly captured, violation of the stationary interaction assumption here
indicates the existence of defects within measured regions.
By following similar derivations in Chapter 2, we can obtain the distribution of (mea-
sured) nanostructure feature field for each regionl. Specifically, we have
X(r
l
)N

Z(r
l
)
;
2
(IB)
1

(3.3)
whereB = (
ij
)
i;j=1;:::;n
is a block circulant matrix composed by interaction parameters

ij
. Additionally, asA
l
maps complete collection of sites in regionl to measured sites
i.e.m
l
=A
l
r
l
, the distribution of measured feature field is easily obtained as following:
X(m
l
)N

A
l
Z(r
l
)
;
2
A
l
(IB)
1
A
T
l

(3.4)
Since different regions are purposely selected to be tiny and fairly separated, we may
reasonably assume (measured) feature fields in different regions are independent.
3.2.2 EM Estimation of Interactions with Incomplete Measurement
Denote =f
;;
2
g as the collection of unknown parameters in the interaction mod-
eling. Estimating nanostructure interactions is then to estimate values of based on the
incomplete feature measurement from theR independent measurement regions. That is,
^ = arg max

logf

fx(m
l
)g
R
l=1
j

: (3.5)
56
The estimation difficulty mainly comes from the “incompleteness”. The non-square
mapping matrixA
l
in Equation (3.4) makes MLE (Maximum Likelihood Estimation)
[107, 5, 31, 17, 111] computationally infeasible.
We use EM (Expectation-Maximization) algorithm to estimate nanostructure inter-
actions. Those omitted sites (i.e. non-measurement sites) are treated as “missing at
random” observations, though here we omit them purposely. Comparing to other tech-
niques dealing with missing values such as imputation, interpolation and Bayesian data
analysis [55], EM algorithm takes advantages of the Markovian property in GMRF mod-
eling of nanostructure interactions in Equation (3.2) and the analytical property of like-
lihood function for complete feature field.
EM algorithm is an iterative method for finding the maximum likelihood estimates
of. Instead of maximizing directly the log likelihood for measured feature field (what
MLE does), EM algorithm iteratively calculates (E-Step) and maximizes (M-Step) the
conditional expectation of the log likelihood for complete feature field based on the
measured feature field and the current estimate for parameters at step p (i.e.
p
)
[21, 108]. In particular, we are to calculate and optimize the following:
Q(j
p
) =E

logf

fX(r
l
)g
R
l=1
j

jfx(m
l
)g
R
l=1
;
p

=
R
X
l=1
Eflogf (X(r
l
)j)jx(m
l
);
p
g (3.6)
where the decomposition is made possible because of the independence assumption for
different measurement regions.
Since the calculation in Equation (3.6) is similar for each measurement region, we
only demonstrate EM approach for one single region. Integrating feature measurement
from multiple regions can be easily derived.
57
To keep notation simpler, we denoteZ forZ(r
l
) and omit any subscript l in the
notation for following discussions. In addition, we introduce following notation for
more convenient presentation of the conditional statistics.
• E
cond
(:) =Ef:jx(m);
p
g represents the conditional expectation given measured
feature fieldx(m) and current parameter estimate
p
.
• Var
cond
(:) = Varf:jx(m);
p
g denotes the conditional variance.
• Cov
cond
(:;:) = Covf:;:jx(m);
p
g represents the conditional covariance.
Expectation Step of EM Estimation
The complete feature fieldX(r) follows a multivariate normal distribution as given in
Equation (3.3). We can thus easily obtain the conditional expectation of its log likeli-
hood function as following:
Q(j
p
) =Eflogf (X(r)j)jx(m);
p
g
=
E
cond
f(X(r)Z
)
T
(IB)(X(r)Z
)g
2
2

n
2
log (2
2
) +
1
2
log(jIBj): (3.7)
And the conditional expectation in Equation (3.7) can be expanded as:
E
cond

(X(r)Z
)
T
(IB)(X(r)Z
)

= (E
cond
X(r)Z
)
T
(IB)(E
cond
X(r)Z
)
+ Trf(IB)Var
cond
X(r)g: (3.8)
58
Therefore, to evaluate Q(:j:), keys are to compute the conditional expectation for
any non-measurement site and the conditional covariance between any two non-
measurement sites in Equation (3.8).
To speedup the calculation, we take advantages of the Markovian property in our
interaction modeling in Equation (3.2). Specifically, Markovian property allows us to
classify non-measurement sites into conditionally independent groups. In this way, the
conditional distribution of non-measurement feature field can be decomposed. Mathe-
matically, we have:
f(X(o)jx(m);
p
) =
Y
k
f(X(g
k
)jx(m);
p
)
=
Y
k
f(X(g
k
)jx(n
k
);
p
) (3.9)
wherefg
k
g represents conditionally independent groups of sites andn
k
denotes the
neighboring measurement sites for group k. Consequently, we may obtain the condi-
tional expectations and covariances by evaluating the conditional distribution for each
groupg
k
individually.
The algorithm for classifying non-measurement sites is as following. (i) If a non-
measurement site has all measured neighbors e.g. site A in Figure 3.3, we call it iso-
lated non-measurement site. Given measured feature fieldx(m), each “isolated non-
measurement site” is conditionally independent from any other non-measurement sites.
(ii) If a non-measurement site also has non-measurement sites as neighbors e.g. site B
in Figure 3.3, we call it clustered non-measurement site. We further group “clustered
non-measurement sites” into smallest clusters such that a site in one cluster is not the
neighbor of any site in other clusters. For instance, in Figure 3.3, sites B,C, and D form
a cluster and sites E - J form another cluster under second order neighborhood structure
where neighboring sites are the eight nearest sites such as sites 1-8 for site A.
59
It is easy to prove, groups constructed this way are conditionally independent. In
addition, the conditional distributions of each group will only depend on their neighbor-
ing measurement sites such as sites 1-8 for site A and sites 1” - 23” for the cluster of E
- J in Figure 3.3.
1 2 3 1’’ 2’’ 3’’
8
A
4 4’’ 5’’
H
6’’ 7’’
7 6 5 8’’
G
9’’
I
10’’ 11’’
1’ 2’ 3’ 4’ 5’ 12’’
F
13’’ 14’’
J
15’’
6’
B
7’
D
8’ 16’’ 17’’
E
18’’ 19’’ 20’’
9’ 10’
C
11’ 12’ 21’’ 22’’ 23’’
13’ 14’ 15’
isolated non-measurement sites clustered non-measurement sites
Figure 3.3: Schematic illustration of non-measurement site classification
Upon the classification of non-measurement sites, the calculation of conditional
expectations and covariances are quite routine for each group. In particular, for isolated
non-measurement sites, we obtain their conditional statistics directly from the interac-
tion modeling in Equation (3.2). That is:
E
cond
X(s
i
) = (Z
p
)
i
+
X

p
ij
(x(s
j
) (Z
p
)
j
)
Var
cond
X(s
i
) = (
2
)
p
(3.10)
60
And for each cluster of non-measurement sites, the calculation follows standard deriva-
tions for multivariate normal random variables. Specifically, we have:
E
cond
X(g
k
) =E(X(g
k
)jx(n
k
);
p
)
Var
cond
X(g
k
) = Var(X(g
k
)jx(n
k
);
p
) (3.11)
Equation (3.11) clearly shows that the interaction estimation is mostly affected by
those large clusters of non-measurement sites. To efficiently estimate nanostructure
interactions, it is thus desirable to have non-measurement sites in smaller groups. This
is exactly the motivation to develop tailored space filling designs in Section 3.3.
Maximization Step of EM Estimation
MaximizingQ(j
p
) based on its evaluation in Section 3.2.2 is a quite standard proce-
dure. Denote
p+1
=f
p+1
; (
2
)
p+1
;
p+1
g to be the EM estimates at stepp + 1. We
obtain (
2
)
p+1
and
p+1
by setting@Q=@
2
= 0 and@Q=@
= 0:
(
2
)
p+1
=
E
cond

(X(r)Z
)
T
(IB)(X(r)Z
)

n
(3.12)

p+1
=

Z
T
(IB)Z

1
Z
T
(IB)E
cond
X(r) (3.13)
Therefore,
p+1
= arg maxQ(j
p
) where

2
n
Q(j
p
) = log

(E
cond
X(r)Z
p+1
)
T
(IB)(E
cond
X(r)Z
p+1
)
n

+ log

P
s
i
2o
Var
cond
X(s
i
) 2
P
q>0;s
i

q
Cov
cond
(X(s
i
);X(s
i
+q))
n

log(jIBj)
n
+c (3.14)
61
with c being a constant. Here the second term is essentially log
Trf(IB)Var
cond
X(r)g
n
.
We obtain it by regrouping Tr[(IB)Var
cond
X(r)] in terms off
q
g
q>0
. To remark,
IB in Equation (3.14) is the precision matrix for complete dataX(r). Therefore,
its determinant can be easily computed by FFT (Fast Fourier Transform) through torus
approximation. Substituting
p+1
into Equations (3.12) and (3.13), we obtain EM esti-
mates (
2
)
p+1
and
p+1
at stepp + 1.
We iteratively perform the Expectation and Maximization steps described above
until the estimation
p
converges with certain tolerance. The converged values ~ of

p
will be our final parameter estimation.
3.3 Tailored Space Filling Design for Site Selection in
Each Region
This section aims to develop a sampling approach for selecting non-measurement
sites in each measurement region with given non-measurement ratio (number of non-
measurement sites to total number of sites). Our objective is to optimize both the accu-
racy and the efficiency of nanostructure interaction estimation.
A problem with simple random sampling is that with moderately large non-
measurement ratios it often produces clustering and poorly covered area. The associ-
ated consequences include: (1) nanostructures measured may not be representative over
the whole region, as a result of which the sampling brings extra biases to interaction
estimation; and (2) EM estimation of interactions are not computation efficient due to
those large clusters of dependent non-measurement sites. While classical space-filling
designs including distance-based designs [45, 64] and uniform designs [23] may well
solve problem (1), they have no control in clustering of dependent non-measurement
sites. Figure 3.4 shows a sample maximin distance design with nine points on a 5 5
62
griding. All non-measurement sites (patterned sites) are conditionally dependent under
diamond neighborhood structure (Figure 3.4(b)) which is usually evaluated for interac-
tion estimation. Consequently, all non-measurement sites form a single cluster making
EM algorithm inefficient.
(a) maximin space filling design (b) diamond neighborhood structure
Figure 3.4: Sample maximin distance design that produces single conditionally depen-
dent group of non-measurement sites under diamond neighborhood structure
To optimize interaction estimation efficiency, we develop a tailored space filling
design for selecting non-measurement sites. Design criteria are as following with
descending priorities.
C.1 size of the largest cluster of dependent non-measurement sites, smaller the better
C.2 number of isolated non-measurement sites, larger the better
C.3 number of conditionally independent groups, larger the better
It is intuitive that with smaller clusters, more isolated non-measurement sites and more
conditionally independent groups, we have more “space filling” non-measurement sites.
63
We use simulated annealing [51, 64] which is an iterative optimization method to
search the best space filling design with given non-measurement ratio. Four compo-
nents are contained in the algorithm which are (1) generation of initial designs, (2)
perturbation scheme, (3) updating criteria, and (4) termination conditions.
Initial Design Generation
When non-measurement ratio is low (e.g. <15%), we use simple random sam-
pling to generate the initial design. When non-measurement ratio is high, we
either inherit optimized designs of lower non-measurement ratios or use stratified
sampling to select sites from conditionally independent strata.
Perturbation Scheme
At each step, we perturb the current designD
current
by randomly replacing a non-
measurement site in the largest cluster with an existing measurement site. Denote
the new design asD
try
, the newly added non-measurement site as moveToSite
and the replaced design point as moveFromSite. Instead of summarizing the
new design D
try
from scratch, we re-classify only those non-measurement sites
affected by the movement. The re-classification algorithm is in Algorithm 1.
Updating Criteria
For each perturbation, we will either keep the current design unchanged or update
it with the newly generated one. The probability of updating is
p(D
try
;D
current
) = minf1; exp(
g(D
try
)g(D
current
)
T
)g (3.15)
where g(:) = a largestClusterSize +b numIsoSites +c numClusters
witha > b > c being positive constants. T is the temperature of annealing that
decreases along iterations. It is clear a better design based on criteria C. 1-3 has
64
larger chances to be accepted. Besides, we put most efforts to control the largest
cluster size.
Termination Conditions
The optimization process stops when either of the following two is met: (i) the
largest cluster size in current design is smaller than a pre-defined value; or (ii)
there is no design update in certain number of consequent iterations.
Algorithm 1 Re-classification of non-measurement sites
if moveToSite is an isolated non-measurement site then
isoSites inD
try
= isoSites inD
current
+ moveToSite
# The origCluster may decompose after the replacement
classiSites = otherSitesInOrigCluster inD
current
else
# Find non-measurement neighbors of moveToSite
nonMNeighs = non-measurement neighbors of
moveToSite inD
try
if ALL nonMNeighs belong to origCluster inD
current
then
classiSites = otherSitesInOrigCluster + moveToSite
else if NONE of nonMNeighs belong to origCluster then
nonNeigGroups = groups nonMNeighs belong to in
D
current
# moveToSite joins nonNeigGroups together
aNewCluster = moveToSite + sitesInNonNeigGroups
classiSites = otherSitesInOrigCluster
else
# Some belong to origCluster, some don’t
nonNeigGroups = groups nonMNeighs belong to in
D
current
otherSites = sites in nonNeigGroups but not in
origCluster
classiSites = otherSitesInOrigCluster + moveToSite
+ otherSites
end if
end if
Classify classiSites
UpdateD
try
accordingly
65
When non-measurement ratio increases, non-measurement sites form large clusters
more easily. As a consequence, more efforts are needed to obtain the space filling design.
However, as a design can be used for any dataset with compatible dimensions, time to
generate the sampling strategy is not a big concern.
3.4 Simulation Case Studies
This chapter is motivated to reduce the metrology effort for mapping nanostructure
interactions across the whole substrate of a specimen. Accordingly, we propose to (1)
measure nanostructures in multiple separated tiny regions instead of a large continuous
area and (2) measure partial of the sites in each measurement region. To efficiently
estimate nanostructures with incomplete feature measurement, we optimize EM algo-
rithm based on GMRF modeling and develop a tailored space filling design to select
non-measurement sites in each region. Through these approaches, we aim to estimate
interactions with highest accuracy, least computation effort and least feature measure-
ment requirement. In this section, we will demonstrate and validate these approaches
and their effectiveness by simulation case studies.
3.4.1 Multiple Tiny Regions Versus One Large Region
For the same amount of data collection, if we distribute it in multiple separated regions
instead of a single large region we can reduce SEM calibration since calibration is only
needed for neighboring images. In this subsection, we will validate the interaction esti-
mation accuracy is maintained by this multiple region approach as well.
We simulated 100 i.i.d. datasets of size 50 100 to mimic replicates of feature
measurement from large continuous regions. At the same time, we simulated 100 i.i.d.
data collections each of which contains 8 independent 25 25 datasets to represent
66
replicates of feature measurement obtained in 8 separated small regions. Structure of
simulated datasets is depicted in Figure 3.5. All datasets are GMRF with linear trend
and second order interaction patterns. Specifically for any sites
i
= (s;t) wheres andt
are row and column indexes in the datasets, we assume
EfX(s;t)g =
0
+
1
s +
2
t (3.16)
E (s;t)jf(s
0
;t
0
)g
(s
0
;t
0
)6=(s;t)
=
1
f(s 1;t 1) +(s + 1;t + 1)g
+
2
f(s 1;t) +(s + 1;t)g
+
3
f(s 1;t + 1) +(s + 1;t 1)g
+
4
f(s;t 1) +(s;t + 1)g (3.17)
Var (s;t)jf(s
0
;t
0
)g
(s
0
;t
0
)6=(s;t)
=
2
(3.18)
Values set for simulating the datasets are summarized in Table 3.1 under the column
True Val.
Parameter estimation results for multiple region approach and single large region
approach are summarized and compared in Table 3.1. Results show both approaches
give excellent quantification of feature variability. Sampling multiple regions indeed
reduces SEM calibration and achieves as good interaction estimation accuracy.
3.4.2 EM Estimation of Interactions with Incomplete Measurement
We use the 100 i.i.d 50100 datasets simulated in Section 3.4.1 as baseline and study the
EM estimation of interactions with different non-measurement ratios in this subsection.
In particular, we simulate 10 extra data collections by adding 3% 30% missing values
in the complete datasets. Non-measurement sites in these data collections are generated
by simple random sampling approach.
67
Single Region Approach
100 data collections
…
…

…
…
..

…
…

…
…
..

…
…

Collection 1
One 50 * 100 dataset
…
…

…
…
..

…
…

…
…
..

…
…

…
…
…
…

…
…
..

…
…

…
…
..

…
…

Collection 2
One 50 * 100 dataset
Collection 100
One 50 * 100 dataset
Multiple Region Approach
100 data collections
Collection 1
Eight 25* 25 datasets
…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…
Collection 2
Eight 25* 25 datasets
…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…
Collection 100
Eight 25* 25 datasets
…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…

..
…
…
..

…
…

…
…
…
…
…
…
…
…
Figure 3.5: Schematic illustration of simulated data structures
We summarize EM estimated means and 95% empirical confidence intervals for
interaction parameters
1
;:::;
4
in Figure 3.6. Results for each non-measurement ratio
are calculated based on 100 i.i.d datasets with corresponding amount of “missed” mea-
surement. MLEs off
i
g for complete datasets are also depicted for comparison. It
shows that we can reduce more than 30% nanostructure feature measurement but still
achieve comparable quantification of nanostructure variations.
68
Table 3.1: MLE estimation of nanostructure interactions based on simulated datasets
Multiple Region Single Region
True Val. Est. Mean 95% C.I. Est. Mean 95% C.I.

0
6.00 6.010 [5.888, 6.158] 6.001 [5.838, 6.142]

1
-0.03 -0.031 [-0.037, -0.026] -0.030 [-0.033, -0.027]

2
-0.01 -0.010 [-0.018, -0.003] -0.010 [-0.012, -0.008]

1
-0.10 -0.099 [-0.129, -0.078] -0.101 [-0.127, -0.080]

2
0.25 0.250 [0.223, 0.269] 0.251 [0.230, 0.270]

3
0.05 0.049 [0.027, 0.074] 0.048 [0.030, 0.069]

4
0.15 0.150 [0.125, 0.175] 0.150 [0.129, 0.182]

2
1.00 0.997 [0.956, 1.037] 0.996 [0.952, 1.038]
Estimation of beta1
Non−measurement Ratio %
Complete Data 6 9 12 15 18 21 24 27 30
−0.12 −0.1 −0.08
● ● ● ● ● ● ● ● ● ● ● _
_
_
_
_
_
_
_ _
_
_
_
_
_
_ _
_
_
_
_
_
_
Estimation of beta2
Non−measurement Ratio %
Complete Data 6 9 12 15 18 21 24 27 30
0.22 0.24 0.26
● ● ● ● ● ● ● ● ● ● ● _
_ _
_
_
_
_
_
_ _
_
_
_
_
_
_
_
_
_
_
_
_
Estimation of beta3
Non−measurement Ratio %
Complete Data 6 9 12 15 18 21 24 27 30
0.02 0.04 0.06 0.08
● ● ● ● ● ● ● ● ● ● ● _
_
_
_
_
_
_
_
_
_
_
_
_ _
_ _ _ _
_
_
_
_
Estimation of beta4
Non−measurement Ratio %
Complete Data 6 9 12 15 18 21 24 27 30
0.11 0.13 0.15 0.17
● ● ● ● ● ● ● ● ● ● ● _
_
_ _
_
_ _
_
_
_
_
_ _
_
_
_
_
_
_
_
_
_
Figure 3.6: EM estimation of interactions for simulated datasets.
: True Val.,: Est. Mean, j: 95% empirical C.I. based on 100 datasets.
69
Besides estimation accuracy, we also investigate the dependence of EM estimation
speed on non-measurement ratios. From Figure 3.7 we can see both averages and stan-
dard deviations of estimation time increase monotonically with non-measurement ratios.
This rapid increase roots in those large clusters of dependent non-measurement sites
generated by simple random sampling. Figure 3.8 validates our claim by depicting EM
estimation time with the largest cluster size under its highest evaluated interaction struc-
ture. Empirically, estimation time increases at least quadratically with maximum cluster
sizes since we have to evaluate the covariance matrix for each cluster. With simple
random sampling, there are even cases where all non-measurement sites are clustered
together e.g. the case with maximum cluster size 50 100 30% = 1500.
Non−measurement Ratio %
Estimation Time (Minutes)
3 6 9 12 15 18 21 24 27 30
20 60 100 140 180
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Avg. Est. Time
95% C.B.s of Est. Time
Figure 3.7: EM estimation time with non-measurement ratios (simple random sampling)
Therefore, to enhance interaction estimation efficiency so as to further reduce
metrology efforts, the key is to develop a new sampling approach that controls the
largest cluster size. This is exactly what we do in Section 3.3. We will examine the
effectiveness of our tailored space filling design in Section 3.4.3.
70
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 0 500 1000 1500
0 50 100 150
Max Cluster Sizes
Estimation Time (minutes)
Emp. Fitting: est.time = 2.2 + 3.7e−05 × max.cluster.size
2
Figure 3.8: EM estimation time with maximum cluster sizes (simple random sampling)
3.4.3 Tailored Space Filling Design for Each Region
We simulate 10 data collections with 3% to 30% non-measurement sites similar to those
in Section 3.4.2. The only difference is that non-measurement sites here are selected
by our tailored space filling design developed in Section 3.3 instead of simple random
sampling.
With our new approach, we obtain the largest cluster sizes in Figure 3.9 for each
dataset under their highest evaluated interaction structures. Comparing to simple ran-
dom sampling, our tailored space filling design successfully precludes large clusters.
As a result, both averages and standard deviations of EM estimation time are greatly
reduced (Figure 3.10).
Summarizing simulation case studies, we conclude our approaches of sampling
nanostructures and analyzing nanostructure interactions indeed greatly reduce the
metrology efforts and achieve excellent interaction estimation accuracy and efficiency.
71
Non−measurement Ratio %
Max Cluster Size
3 6 9 12 15 18 21 24 27 30
0 200 500 800 1100 1400
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● # non−measurement sites
simple random sampling
avg. simple random sampling
tailored space filling
avg. tailored space filling
Figure 3.9: Comparsion of maximum cluster sizes under tailored space filling sampling
and simple random sampling of non-measurement sites
Non−measurement Ratio %
Avg. Estimation Time (Minutes)
3 6 9 12 15 18 21 24 27 30
5 15 25 35 45 55
avg., tailored space filling design
avg., simple random sampling
Non−measurement Ratio %
Std.Dev Estimation Time (Minutes)
3 6 9 12 15 18 21 24 27 30
5 10 15 20 25
std., tailored space filling design
std., simple random sampling
Figure 3.10: Comparison of EM estimation time for tailored space filling sampling and
simple random sampling of non-measurement sites
3.5 Real Case Studies
In this section, we will study a 45 94 real collected ZnO nanowire length dataset that
includes measurement for both normally grown nanowires and nanowire bundles (one
72
type of structural defects). Interested readers may find more details in Chapter 2 and Xu
and Huang 2012 [111] about the dataset.
Our primary interest is to investigate whether and how much incomplete length mea-
surement affects interaction analysis and nanowire bundle detection. To this end, we
design 10 levels of incompleteness ranging from 3% - 30% by adding “missing obser-
vations” in the original length data. These “missing observations” are used to mimic the
sites we neglect during real measurement. In addition, under each level, we generate
100 replicates of datasets each of which has the same amount of “missing observations”
but different missing locations based on distinctive space filling designs.
We use optimized EM algorithm (Section 3.2.2) to extract interaction patterns from
each dataset. Since most of the datasets identify second order interaction structure with
four interaction parameters
1
;:::;
4
(Equation 3.17), we summarize estimation results
for these parameters in Figure 3.11.
Intuitively, we don’t see dramatic change in mean estimation of interaction parame-
ters with different levels of incompleteness except for
3
when non-measurement ratio
exceeds 21%. However, empirical confidence intervals are generally wider with larger
non-measurement ratios for all
i
; i = 1;:::; 4. Comparing to simulated stationary
datasets (Figure 3.6), unstable confidence intervals here indicate larger variations among
the 100 datasets with different missing observations, which may qualitatively implies the
existence of defects.
To quantitatively detect defects, we do normality tests on transformed residuals of
measured length field similarly as for complete data case [111]. Specifically, under the
null hypothesis that there are no defects within the field, interaction patterns are sta-
tionary across the field and thus measured feature field follows a normal distribution as
stated in Equation (3.4). The mapping matrixA
l
is known for each measurement region
l and the parameters are estimated as ~ by EM algorithm. Based on the consistency
73
Estimation of beta1
Non−measurement Ratio %
3 6 9 12 15 18 21 24 27 30
0.02 0.04 0.06 0.08
● ● ● ● ● ● ● ● ● ● _ _
_
_
_
_
_
_
_ _
_
_
_
_
_
_ _
_ _
_
Estimation of beta2
Non−measurement Ratio %
3 6 9 12 15 18 21 24 27 30
0.03 0.05 0.07 0.09 0.11
● ● ● ● ● ● ● ● ● ● _ _
_
_
_
_ _
_
_
_
_
_
_
_
_
_
_
_
_
_
Estimation of beta3
Non−measurement Ratio %
3 6 9 12 15 18 21 24 27 30
0.03 0.05 0.07 0.09 0.11
● ● ● ● ● ● ● ● ● ● _
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
Estimation of beta4
Non−measurement Ratio %
3 6 9 12 15 18 21 24 27 30
0.1 0.12 0.14 0.16 0.18
● ● ● ● ● ● ● ● ● ● _
_ _
_
_
_
_
_
_
_
_
_
_
_
_
_
_ _
_
_
Figure 3.11: EM estimation of interactions for real nanowire length data.
: Est. Mean, j: 95% empirical C.I. based on 100 datasets.
property of EM estimates, we then have the transformed residuals follows i.i.d. normal
distribution asymptotically. That is,

~ e
l
=
h
A
l
(I
~
B)A
T
l
i

1
2
[X(m
l
)A
l
Z(r
l
)~
)]

R
l=1
i.i.d.N(0;
2
) (3.19)
asymptotically. Therefore we may perform Anderson-Darling tests on transformed
residualsf~ e
l
g to determine whether there are structural defects [89, 111]. With the
normality test, all datasets here containing up to 30% missing observations are detected
to have defects. The defect detection result is 100% correct.
74
Real case studies in this section show that our new method of interaction analysis
guarantees to reduce metrology efforts more than 30% and at the same time provides
comparable interaction estimation accuracy and competitive defect detection precision.
It thus provides automatic and efficient quality assessment of nanostructures that enables
scale-up nanomanufacturing control.
3.6 Summary and Conclusion
In this chapter, we focus on the relaxation of metrology constraints for nanostructure
interaction analysis so that we can quantify nanostructure growth quality and detect
structural defects in scale-up nanomanufacturing systems.
Targeting on the limitations of current nanostructure inspection techniques, we
develop a two-step sampling strategy to reduce the metrology efforts. Specifically, we
sample multiple tiny regions over the substrate and develop tailored space filling designs
to selectively measure partial of the sites in each measurement region. This sampling
strategy not only maximizes the exploration of the growth region but also supports sub-
sequent interaction analysis.
In addition, we optimize Expectation-Maximization algorithm to estimate nanos-
tructure interactions based on corresponding “incomplete” feature measurement. By
classifying non-measurement sites into conditionally independent groups, we further
enhance the efficiency of interaction estimation.
Both simulation and real case studies show that our methods provide accurate and
prompt nanostructure assessment with significantly reduced metrology efforts. The
developed sampling and interaction estimation strategies facilitate online quality moni-
toring and control in scale-up nanomanufacturing systems. They serve as a supporting
tool for future nanomanufacturing.
75
Chapter 4
Growth Process Modeling of III-V
Nanowire Synthesis via SA-MOCVD
Our interests in the growth process modeling of III-V nanowire synthesis via SA-
MOCVD arise from two aspects. On one hand, we are motivated to facilitate the inter-
action analysis in previous chapters. As discussed in Introduction, growth process
modeling describes the overall growth trend on the substrate level which directly deter-
mines nanostructure quality. Therefore, we would like to develop a process model so as
to assist our interaction analysis for quality assessment and control at finer scales.
On the other hand, we have great interests in the growth process itself. In particular,
SA-MOCVD has been widely recognized as a promising nanowire fabrication technique
for scale-up nanomanufacturing, its growth mechanism however still remains unclear
in the literature. There are active debates on the effects of various process parame-
ters on nanowire growth. But there is a lack of growth process models that coherently
explain various growth patterns observed in experiments. Therefore, we are motivated
to develop a nanowire growth model to reinforce our understanding of the synthesis
mechanism and capture the spatial and temporal patterns of nanowire growth.
Our success will set basis for systematic process optimization and control and thus
support scalable nanomanufacturing control. In following part of this chapter, we will
focus on the synthesis process itself and make our contributions to fill the research gap.
76
4.1 Introduction
SA-MOCVD (Selective Area Metal-Organic Chemical Vapor Deposition) is a fabrica-
tion process for III-V nanowires that is promising for large scale nanomanufacturing.
On one hand, it inherits the advantages of MOCVD to be “economical”, flexible, and
robust, and produce high purity structures in high growth rates [87]. On the other hand, it
also possesses the strength of top-down lithography approach. Figure 4.1 demonstrates
the SA-MOCVD synthesis of GaAs nanowires. By defining well patterned openings in
the dielectric mask, SA-MOCVD could achieve precise position and shape control of
nanowires. Compared to conventional Au-catalyzed VLS (Vapor Liquid Solid) synthe-
sis, the catalyst-free property of SA-MOCVD also avoids any metal contamination.
Figure 4.1: Schematic of SA-MOCVD fabrication (modified from [93])
Despite of the great potentials of SA-MOCVD synthesis, the current fabrication has
limited capability for systematic process optimization and control. Studies have been
extensively conducted to investigate the dependence structure among nanowire geome-
tries (height and diameter), process conditions (e.g. temperature, growth time), and
patterning geometries (opening diameter and inter-distance i.e. pitch). Interesting exper-
imental observations include increased growth rate with time [41], reduced height with
diameter [92, 41, 70], and longer nanowires along substrate boundaries [15, 14]. While
77
these findings may provide valuable process guidance, there is unfortunately no coher-
ent description for all these phenomena, needless to say a general model that supports
quantitative process optimization.
A fundamental research gap for SA-MOCVD synthesis is a conclusive understand-
ing of the growth mechanism. In current literature, there are active debates whether the
process is governed by self-catalyzed VLS/VS growth [11, 57, 62] or diffusion based
selective area epitaxy [16, 69, 40, 33, 15, 30]. In addition, although selective area epi-
taxy is more widely accepted as the dominant mechanism, the diffusion sources and
their relative contributions are still controversial. Both empirical [92, 41] and physical
[15, 69, 82] models have been proposed with each capturing some aspects of the diffu-
sional effects. Besides being incomprehensive, these models often produce contradictive
conclusions. For instance, some argue that nanowire heighth should be proportional to
the inverse of diameter d i.e. h/ 1=d [32, 70, 41] while others believe h/ 1=d
2
[92, 41]. Conflicting statements from even the same research group put the growth
mechanism further into question.
In this dissertation, we attempt to develop a nanowire growth model for SA-MOCVD
synthesis based on selective area epitaxy mechanism. By quantifying contributions
from various diffusion sources, we will establish a functional relation between nanowire
height and process parameters, and examine the relation based on real experimental data.
Our objective is to confirm our understanding of the synthesis mechanism and explain
the spatial and temporal growth patterns observed in various experiments. Together with
our previous work on nanostructure interactions (ref. Chapters 2-3, and [111, 112]), our
model will provide a comprehensive description of nanostructure growth variations.
78
4.2 Growth Modeling of III-V Nanowires Synthesized
via SA-MOCVD
In this section, after briefly summarizing SA-MOCVD synthesis of III-V nanowires and
key growth process parameters, we classify precursor sources into different categories
and quantify their relative contributions mathematically. Our objective here is to estab-
lish a coherent functional relation between nanowire height and process parameters.
4.2.1 Process Characteristics and Precursor Diffusion
As demonstrated in Figure 4.1, SA-MOCVD synthesis begins by sputtering a layer of
dielectric mask onto the substrate. It then defines triangular patterned openings in the
mask by electron-beam [91] or nanosphere lithography [56]. As precursor elements can
only form bonds with the substrate but not the dielectric mask, nanowires preferentially
grow through the openings. In this way, those created openings directly determine the
location and diameter of the nanowires making SA-MOCVD robust and precisely con-
trollable.
The whole synthesis is conducted in a low-pressure (around 0.1 atm) horizon-
tal MOVPE reactor system. Because of faceting growth, final shapes of nanowires
are hexagonal pillars. At the same time, due to high surface energies of side facets,
nanowires preferentially grow vertically if the substrate surface has compatible crystal
orientation [91]. As a result, group-III and group-V atoms on mask and side surfaces
will diffuse to the nanowire top surface and drive the vertical growth.
Figure 4.2 depicts a schematic diagram of precursor diffusion and gas flow in SA-
MOCVD synthesis. Based on where atoms are first deposited, we may classify the
reactant supply into three sources which are (1) direct absorption on top surface, (2)
nanowire side surface diffusion, and (3) mask surface diffusion. In addition, while
79
Figure 4.2: Schematic of precursor diffusion in SA-MOCVD synthesis
neighboring nanowires closely share the mask area in central patterned region, the skirt
mask (left and right end in Figure 4.2) provide reactant almost exclusively for boundary
nanowires. To account their differences, we further classify mask surface diffusion into
(i) patterned mask surface diffusion and (ii) skirt diffusion. This classification will help
with explaining why nanowires are generally longer along the boundaries.
We focus on group-III atom diffusion modeling in this paper since group-V atoms
are sufficiently supplied in general. We will quantify contributions from the four sources
and establish a growth model of nanowire height. Our developed model will explain
various experimental observations quantitatively.
4.2.2 Notation Convention
Figure 4.3 depicts a patterned substrate and a synthesized nanowire in SA-MOCVD fab-
rication. We introduce following notation to describe the synthesis system and facilitate
the discussion of our growth model.
g: width of the patterned growth region.
w: width of the extra mask area which is also called “skirt area”.
80
p: pitch i.e. the distance between neighboring openings or nanowires.
d
0
: diameter of the lithographically defined openings.
d: diameter of synthesized nanowires.
h: height of nanowires.
t: growth time.
In addition, we useh(x;y;t) to represent the height of a nanowire at location (x;y) in
the patterned growth region at timet and take to represent an infinitesimal increment
of a parameter.
Figure 4.3: Schematic of a patterned substrate and a synthesized nanowire in SA-
MOVPE process
Typically, g and w are set in the range of 100 m to 1000 m (or larger) in lab
experiments, p is in the range of 500 nm to 2 m, and d
0
is in the range of 50 nm to
400 nm. Together with suitable temperature, precursor supply, and growth time (nor-
mally less than 30 minutes), nanowires dominantly grow vertically i.e. perpendicular to
the substrate surface in SA-MOCVD synthesis. Furthermore, they have quite uniform
81
diameters from base to top which are nearly identical to the designed opening diameter
d
0
[65, 66, 48]. In this work, we focus on III-V nanowire synthesis under these “nor-
mal process conditions” since they are the process settings enabling precise control of
nanowire growth.
4.2.3 Spatio-Temporal Growth Modeling
We develop our growth model based on selective area epitaxy mechanism. According to
mass conservation principle, nanowire growth is brought by the absorption of reactant
elements either through direct incorporation on top surface or by precursor diffusion
along nanowire side facets, patterned mask area, or skirt area.
Figure 4.4: Schematic illustration of the four precursor sources and their corresponding
contributions to nanowire growth based on mass conservation principle
82
We demonstrate our modeling idea in Figure 4.4. In particular, we can model the
volume increment of a nanowire at location (x;y) and timet as following.
p
3
2
d
2
h(x;y;t) =
(
C
1
p
3
2
d
2
+C
2
(2
p
3d)h(x;y;t)
)
t
+
(
C
3
p
3
2
(p
2
d
2
) +C
4
S(x;y)
)
t (4.1)
Here the left hand side represents the volume increment of the nanowire at timet and the
four terms in the right hand side correspond to precursor elements from (i) top surface
absorption, (ii) side facet diffusion, (iii) patterned mask surface diffusion, and (iv) skirt
diffusion respectively.
To reflect the fact that different sources may have varying precursor concentrations,
we use four distinctive coefficientsC
1
;:::;C
4
in our model. By definition, these coeffi-
cients may depend on process conditions such as temperature and gas flow rate, but are
independent of time and patterning geometries. They all have positive values.
In this study, we focus on the controlled nanowire growth before lateral overgrowth
stage [82, 48]. To this end, it is reasonable to assume a constant diameterd over time
for any single nanowire in our model. In addition, the lack of lateral overgrowth and
nanowires’ small sizes also allow us to reasonably assume a constant precursor concen-
tration on nanowire top surface, side facets, and patterned mask surface region respec-
tively based on diffusion theory [14]. Considering the diffusion length of precursors on
nanowire surfaces, we can thus well approximate precursor supplies from top surface
and side facets to be proportional to their surface areas.
Patterned mask diffusion In theory, precursor atoms from any part of the mask
surface region could diffuse to a particular nanowire and contribute to its growth. But in
reality, due to fully competition among nanowires, only atoms from neighboring mask
83
area may be captured. Figure 4.5 schemes how we divide the mask area among neigh-
boring nanowires. It is not to show the exact boundaries of contributing mask area but
to demonstrate the average “territory” for each nanowire. Furthermore, as the pattern-
ing pitchp is generally at least one-order smaller than the precursor diffusion length on
mask, we approximate the precursor supply from mask to be proportional to the assigned
area i.e. proportional to
p
3=2(p
2
d
2
) in our model (4.1).
Figure 4.5: Distribution of mask area among neighboring nanowires
Skirt diffusion To quantify the effect of skirt diffusion, we calculate the expected
amount of group-III atoms that diffuse to any particular nanowire from the whole skirt
area. The idea is similar to that for mask surface diffusion. But as demonstrated in
Figure 4.4, we cannot divide the skirt among nanowires as easily due to its location.
In addition, the skirt widthw is generally much longer than the diffusion lengthL
d
of
precursor atoms on mask surface. As a result, we have to differentiate atoms at varying
84
location. Specifically, as an atom has probability 1=(2rL
d
) expfr=L
d
g to diffuse to
a location that is distancer apart without any interruption [34], we could thus take
S(x;y)
Z
(u;v)2skirt area
1
2L
d
p
(xu)
2
+ (yv)
2
exp
(

p
(xu)
2
+ (yv)
2
L
d
)
dvdu (4.2)
for any nanowire at location (x;y) by neglecting boundary effects and possible interrup-
tions from nanowires closer to the boundaries.
In Equation (4.2), we prefer the cartesian representation because it fits well with
the geometric nature of skirt area which is demonstrated in Figure 4.6 (a). More for-
mally, we can expand the integration forS(x;y) in Equation (4.2) into four terms that
correspond to the four labelled regions in Figure 4.6 (a).
S(x;y) =
0
B
B
B
B
@
Z
w+g
w
Z
0
w
| {z }
1
+
Z
w+g
w
Z
w+g
g
| {z }
2
+
Z
0
w
Z
g
0
| {z }
3
+
Z
w+g
g
Z
g
0
| {z }
4
1
C
C
C
C
A

1
2L
d

1
p
(xu)
2
+ (yv)
2
exp
(

p
(xu)
2
+ (yv)
2
L
d
)!
dvdu (4.3)
Since the integration for each term is over a rectangle, function adaptIntegrate(.) in
‘cubature’ package in R could be used to easily integrate them numerically. This feature
earns us additional computation efficiency.
We plot in Figure 4.6 (b) a typical profile of skirt diffusion. It is obtained by setting
L
d
= 40um; g = 1000um, andw = 150um in Equation (4.3). From the figure we
can see a clear pattern that nanowires closer to the boundaries would benefit more from
the skirt diffusion. It indicates our model’s potential to capture the spatial pattern of
85
(a) Division of skirt area
x location
y location
total diffusion
skirt diffusion, 150 um skirt
(b) Profile of skirt diffusion
Figure 4.6: Modeling and integration of skirt diffusion
nanowire growth, which will be validated quantitatively by real experimental data in the
following section.
Divide each side by
p
3=2d
2
t in Equation (4.1) and consolidate constants if possi-
ble, and then we may simplify our model (4.1) as
h(x;y;t)
t
=C
1
+C
2
h(x;y;t)
d
+C
3
p
2
d
2
+C
4
S(x;y)
d
2
: (4.4)
Taking t! 0 and integrate over time, we obtain
h(x;y;t) =

exp(
C
2
d
t) 1

d
C
2

C
1
+C
3
p
2
d
2
+C
4
S(x;y)
d
2

(4.5)
by assumingh(x;y;t = 0) = 08(x;y). Concrete values of the coefficientsC
1
;:::;C
4
will be estimated based on experiment data.
86
4.2.4 Model Discussion
Our model (4.1) is originally motivated by the three-source modeling approach dis-
cussed in [41] and the investigation of nanowire height profiles on the substrate in [15].
Apart from being obviously different from [15], our growth model (4.5) is also clearly
distinctive from the height formula given in [41] although we share similar terms with
[41] in (4.1) before deriving the differential equation.
Firstly, our growth model claims an exponential growth of nanowire height before
lateral overgrowth. In contrast, authors in [41] concluded a linear growth rate due to
improper height approximation. Our model is consistent with the reported super-linear
growth in literature [82, 48]. It is also capable of explaining the observed growth rate
enhancement data [1] quantitatively.
Secondly, our modeled dependence between nanowire height h and diameter d is
rather different from that in [41]. Instead of compromising the 1=d and 1=d
2
relationship
[32, 70, 92] reported in literature, we integrate over the differential formula correspond-
ing to Equation (4.4). Studies in Section 4.3 will verify that our model better explains
the dependence structure.
What’s more, our growth model incorporates both growth time and location of the
nanowires. It thus has a unique capability of explaining both temporal and spatial pat-
terns in nanowire growth. In contrast to existing literature, our model provides an accu-
rate and direct functional equation between nanowire geometries and process parame-
ters, which will facilitate systematic process optimization.
87
4.3 Model Validation and Case Studies
As discussed in Section 4.1, following observations have been reported extensively in
literature by various studies [92, 41, 70, 4, 15, 14]. Particularly, for SA-MOCVD synthe-
sis of III-V nanowires people have seen (i) increased growth rate with time, (ii) reduced
height with diameter, and (iii) longer nanowires along substrate boundaries.
In this section, we will look into each of these observations. We are interested to
check our model’s conceptual consistence with these findings. More importantly, we
are excited to validate our model can precisely explain related experimental data.
4.3.1 Increased Nanowire Growth Rate
Nonlinear growth rate of III-V nanowires synthesized by SA-MOCVD was first reported
by Akabori et. al, 2003 [1] which investigated the height dependence on diameter and
growth time for InGaAs nanopillars. Figure 4.7 (a) shows the original figure from [1]
where black bullets are experimental data corresponding to nanowires grown for 7.5
minutes and gray diamonds correspond to those grown for 15 minutes. A linear model
of height with growth time was fitted in [1]. Results of their fitting in Figure 4.7 (a)
indicate that nanopillars grown for 15 minutes are more than twice longer of those with
same diameter but grown for half of the time i.e. 7.5 minutes. That is the growth rate of
nanopillars increases with growth time.
In [1], the authors believed that the enhanced growth rate was due to the re-
adsorption of group-III atoms desorbed from the SiO
2
mask. But the explanation is
far from being convincing. After [1], the only work that investigated this issue was [83]
which directly borrowed the three-source modeling approach in [41] and concluded a
88
linear relationship between h=t andh based on eight (8) data points. While the con-
clusion is ‘approximately’ correct when t is extremely small, a lack of a complete and
correct model of nanopillar height makes it incapable of explaining the data here.
(a) Nanopillar aspect ratio versus diameter.
A snapshot of Figure 3 in [1].
150 200 250 300 350
0 2 4 6 8 10 12
Akabori et al., 2003
Diameter,d (nm)
Aspect Ratio, h/d
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● 7.5 min
15 min
fitting
prediction
(b) Our model’s fit with the red line being our
prediction for 15 min. case.
Figure 4.7: Time-dependent growth rate in SA-MOCVD synthesis.
The data is extracted from [1].
Based on our growth model in Equation (4.5), we obtain the growth rate as
@h(x;y;t)
@t
= exp(
C
2
d
t)

C
1
+C
3
p
2
d
2
+C
4
S(x;y)
d
2

=C
2
h(x;y;t)
d
+C
1
+C
3
p
2
d
2
+C
4
S(x;y)
d
2
: (4.6)
It is obvious that growth rate
@h(x;y;t)
@t
increases with growth timet when other param-
eters are fixed. It is thus conceptually consistent with the experiment data in Figure
4.7(a). Furthermore, we can see from Equation (4.6) that the term that fosters growth
rate increase isC
2
h(x;y;t)
d
which corresponds to nanopillar side facet diffusion.
89
To quantitatively examine our model’s fit, we extract height, diameter and growth
time information from Figure 4.7 (a). As [1] doesn’t contain any information about
the nanopillars’ spatial location, we assume all nanopillars measured are taken from the
very center of the growth region. Therefore, we takeS(x;y) 0 for all nanopillars in
Equation (4.5). Our model structure is thus simplified as following for this case.
h(t) =

exp(
C
2
d
t) 1

d
C
2

C
1
+C
3
p
2
d
2

(4.7)
which gives the modeling equation for aspect ratio as
h(t)
d
=
1
C
2

exp(
C
2
d
t) 1

C
1
+C
3
p
2
d
2

: (4.8)
We fit our model in Equation (4.8) with the extracted data from Figure 4.7 (a). Results
are depicted in Figure 4.7 (b) where dash lines are individual fitting for either case. In
addition, we predict aspect ratios of nanopillars grown for 15 minutes based on model
fitting results from 7.5 min. data. The prediction is drawn as a red solid line in Figure
4.7 (b) which is as good as the fitting based on the 15 min. data itself.
This study validates that our model has accurately captured the time dependence of
nanopillar growth. Its success also indicates that side facet diffusion drives the growth
rate enhancement.
4.3.2 Dependence of Nanowire Height on Diameter
Our study in Section 4.3.1 has already covered the dependence of nanowire height on
diameter although it majorly focused on growth rate there. The prediction power has
demonstrated our model’s capability. In this section, we study another dataset in [1] to
further validate our model.
90
(a) Snapshot of Figure 3(a) in [1]
150 200 250 300 350
0 100 200 300 400 500 600 700
Refit to data in Fig. 3a of Akabori et al 2003
Diameter, d (nm)
Height, h (nm)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.36 nm/s, 7.5 min
0.18 nm/s, 15 min
0.09 nm/s, 30 min
model fitting
(b) Our model’s fit to the experiment data
Figure 4.8: Dependence of nanowire height on diameter
The data is extracted from [1].
Figure 4.8 (a) is a snapshot of Figure 3(a) in [1] which depicts the dependence of
nanopillar height on diameter. In the figure, legends “0.36 nm/s, 7.5 min.”, “0.18 nm/s,
15min.” and “0.09 nm/s, 30min.” represent different growth conditions [1]. It is clear
that nanowire height decreases with diameter in all conditions.
We extract height and diameter data from Figure 4.8 (a) for all conditions and fit
our growth model in Equation (4.7) to the extracted data. Fitting results are depicted in
Figure 4.8 (b). It confirms that our model accurately expresses the dependence structure
between nanowire height and diameter. Since designed opening diameterd
0
and pitchp
directly determine the distribution of nanowire diameterd [66, 70], our model supports
nanowire height control through patterning geometries.
Other than Figure 4.8 (b), we also summarize those estimated coefficient values in
Table 4.1. Corresponding standard errors are included in brackets to demonstrate our
91
Table 4.1: Height dependence on diameter
(coefficients are estimated by ‘nls’ function in R)
C
1
C
2
C
3
0.36 nm/s, 7.5 min. 7.728 (2.435) 19.386 (12.348) 7.142 (3.003)
0.18 nm/s, 15 min. 4.112 (0.871) 32.475 (11.352) 0.00 (0.372)
0.09 nm/s, 30 min. 1.533 (0.981) 14.412 (21.156) 0.00 (0.579)
estimation variation. Results tell us (i)C
1
is always statistically significant which con-
firms the universal importance of direct adsorption through top surface in all conditions
but (ii) the significance of side facet diffusion (C
2
) and mask diffusion (C
3
) may change
with growth conditions such as temperature. According to our study, mask diffusion
may contribute very little to nanowire vertical growth in slow-growth-rate conditions
(e.g. 0.18 nm/s and 0.09 nm/s in Table 4.1). This phenomena is possibly related to the
high desorption rates in those slow-growth-rate conditions [1].
4.3.3 Spatial Profile of Nanowire Growth
Another important contribution of our model is its capability to describe the spatial
profile of nanowire growth. Figure 4.9 (a) shows SEM images of InP nanowires grown
on different locations of the patterned region [15]. All images in the figure are in the
same scale. Nanowires near the boundary are obviously much longer than those in the
center area. Besides 350 m skirt width, authors in [15] also conducted experiments
with skirt width of 150m and 250m. Measured nanowire heights (with error bars)
in Figure 4.9 (b) show similar spatial patterns of nanowire growth in all cases.
Precursor diffusion from the skirt area is the major cause of this spatial pattern.
As a result, the wider the skirt width, the longer the nanowires. Observed patterns
in Figure 4.9 (b) are right consistent with our understanding and the proposed skirt
diffusion modeling in Equation (4.2).
92
(a) SEM images of InP nanowires
0 200 400 600 800 1000
1000 1500 2000 2500 3000

Position in the patterned region (um)
Height (nm)
● ● ● ● ● ● ● ● ● ● ● ● ● ● Skirt 350 um
Skirt 250 um
Skirt 150 um
(b) Measured nanowire heights
Figure 4.9: Spatial profile of nanowire heights under different conditions
Images and the data are extracted from [15]
0 200 400 600 800 1000
1000 1500 2000 2500 3000

Position in the patterned region (um)
Height (nm)
● ● ● ● ● ● ● ● ● ● ● ● ● ● Skirt 350 um
Skirt 250 um
Skirt 150 um
Figure 4.10: Our model’s fit to the InP nanowire height data
All data under different skirt widths are extracted from [15]
93
In this study, process parameters e.g. growth time and temperature and patterning
geometries are fixed. Therefore we may simplify our growth modeling in Equation (4.5)
as following to fit each set of data in Figure 4.9 (b).
h(x;y) =a +bS(x;y)
=a +b
Z
(u;v)2skirt area
1
2L
d
p
(xu)
2
+ (yv)
2
exp
(

p
(xu)
2
+ (yv)
2
L
d
)
dvdu
Here a;b and L
d
are parameters to be estimated based on experimental data. Fitted
results are depicted in Figure 4.10 as dash lines. Our simple modeling approach provides
surprising good performance. In addition, the estimated diffusion length parameterL
d
is 60m which is quite close to the reported diffusion length for InP in [14].
Our current model for skirt diffusion (Equation (4.3)) ignores any possible “blocking
effect” from nanowires closer to the boundary. Proper modeling of this “blocking effect”
may better explain those “lower-than-expected” central nanowires for 350 m case in
Figure 4.10 and thus further enhance our model performance.
In addition, we can notice in Figure 4.10 that height variations are generally larger
for nanowires closer to the boundary than those in the central substrate region and larger
for nanowires synthesized with wider skirt area. This phenomenon indicates that process
kinetics not only affect the mean part of general growth profile but also impact second
moment of nanowire height distribution. To fully capture nanowire growth profile, we
may need to extend our current “deterministic” growth process model into “stochastic”
models which incorporate randomness when deriving the incremental growth behavior
(Equation (4.4)) for each nanowire. Furthermore, integration of growth process mod-
els with our interaction analysis [111, 112] discussed previously may help to further
enhance our modeling power. We leave these interesting topics as future work.
94
4.4 Summary and Conclusion
In this chapter, we investigate SA-MOCVD synthesis of III-V nanowires which is
widely recognized as a promising fabrication technique for scale-up nanomanufactur-
ing. Based on our analysis of the epitaxial growth process, we decompose precursor
sources into four categories and establish their relationship to nanowire growth based
on mass conservation principle. A functional expression is successfully built between
nanowire height and process parameters including patterning diameter and pitch, spatial
location of nanowires, and growth time. Studies on real experimental data confirm that
our model accurately explains both spatial and temporal growth patterns of nanowires.
Our success provides extra support to the selective area epitaxy growth mechanism and
reinforces our understanding on precursor diffusion in SA-MOCVD synthesis.
In the future, we may investigate further the “blocking effect” in skirt diffusion and
incorporate variation modeling when deriving our growth process models to better cap-
ture the general growth profile of nanowires. Further integration of our growth process
model with interaction analysis will enhance our prediction skill and facilitate our under-
standing of nanowire growth behaviors at both substrate level and local scale.
95
Chapter 5
Discussions and Future Extensions
Scalable nanomanufacturing is the key enabler to transfer laboratory successes and ful-
fill the promise of nanotechnology. To achieve scale-up nanomanufacturing, a central
issue is to understand, quantify, and ultimately control nanostructure synthesis varia-
tions. In this dissertation, we are thus devoted to modeling nanostructure synthesis
process and capturing nanostructure variations at both micro-scale (the substrate level)
and nanoscale (variation among nanostructures). Our fundamental research goal is to
capacitate automatic nanostructure quality assessment and control for large-scale nano-
production systems.
To achieve our research goals step by step, we decompose our work in this disserta-
tion into three research tasks. They are (i) nanostructure interaction analysis for a local
region, (ii) nanostructure interaction analysis for the whole substrate and (iii) growth
process modeling for III-V nanwire synthesis via SA-MOCVD.
The first two tasks focus on the quantification of nanoscale quality feature variations.
Because nanostructure interactions play important roles in determining nanostructure
formation, our characterization of local variations targets on interaction patterns. Gaus-
sian Markov random field is adopted to model nanostructure interactions statistically.
Its definition of neighborhood structures is well consistent with our preliminary under-
standing of nanostructure interactions particularly the fact that nanostructures physi-
cally closer have more chances to mutually impact each other. With our methods, we
can find the interaction patterns among neighboring nanostructures, and quantify fea-
ture variations and identify structural defects based on extracted interaction schemes.
96
Together with our tailored nanostructure sampling strategies and customized interaction
analysis techniques established in task II, we can assess the growth at nanoscale for
dense-quantities of nano-elements within manufacturing relevant time spans.
In addition to local variation quantification, we also establish a growth process model
to explore synthesis variations at substrate level and facilitate our interaction analy-
sis at nanoscale. Different from the interaction analysis which generally applies to any
bottom-up fabrication process, this growth process model is designed for III-V nanowire
synthesis via SA-MOCVD. Our interests to this process are brought by the great poten-
tials of SA-MOCVD as a scalable nano-fabrication technique and the superior prop-
erties of III-V nanowires. Moreover, there are ongoing debates on the growth mecha-
nisms for III-V nanowire synthesis via SA-MOCVD. Our work facilitates fundamental
understandings of the nanowire growth behaviors. In this dissertation, we classify pre-
cursor sources in SA-MOCVD into four categories and establish their relationship to
nanowire growth based on mass conservation principle. Accordingly, we build a quanti-
tative model between nanowire height and process parameters. Our model successfully
captures the spatial and temporal trend of nanowire growth and well explains all major
experimental observations quantitatively. The success reinforces our understanding on
precursor diffusion in SA-MOCVD and confirms that the growth mechanism should be
selective area epitaxy.
To summarize, our work in this dissertation contributes to both nanotechnology
research and nanomanufacturing scale-up. For nanotechnology, our growth process
model enhances our understanding of the synthesis mechanism and confirms the dom-
inant role of precursor diffusion in SA-MOCVD. Additionally, our interaction analysis
reinforces the fact that nanostructure interactions affect nanoscale feature variations.
For nanomanufacturing, as our growth process model accurately captures the depen-
dence of nanowire geometries on process parameters, it guides process optimization
97
and guarantees the overall quality of synthesized nanowires. What’s more, our inter-
action analysis enables online quality assessment and defect detection at nanoscale for
scale-up nanomanufacturing systems. Our approaches thus support systematic process
monitoring and control.
In the future, we may extend the dissertation work to further support nanomanufac-
turing scale-up. Possible directions include but are not limited to the following.
• Nanostructure synthesis processes: First we need further research to better quan-
tify nanostructure synthesis variations and identify structural defects although we
have achieved satisfactory results in this dissertation. As summarized in pre-
vious chapters, further explorations are necessary to relate interaction patterns
with defective mechanisms so that we can distinguish structural defects of dif-
ferent kinds. Techniques in image analysis and pattern recognition field may be
helpful. For growth process modeling, we will have to incorporate the distribu-
tional variance when deriving general growth behaviors of nanostructures. Oth-
erwise, we may underestimate synthesis variations in general growth profiles and
fail to achieve satisfactory uniformity control. Seamless integration of nanostruc-
ture interaction analysis and growth process modeling will also be an interesting
research topic. It is expected that we will have enhanced prediction skills and
promoted understanding of nanostructure growth behaviors at both substrate level
and local scale when integrating these two together.
• Nanomanufacturing system: The fabrication of nanodevices or nano-systems is
a very complicated manufacturing process which involves multiscale process
parameters and quality features, and multiple sequential steps that are highly
correlated. In this dissertation, we majorly focus on micro-scale (i.e. the sub-
strate level) and nanoscale (i.e. local regions). In the future, we may investi-
gate structure behaviors at molecular level through simulations. In addition, we
98
may design experiments to explore the possibility of optimizing each processing
step. Our growth process models and interaction analysis techniques may pro-
vide extra guidance on the design. Device reliability and quality control strategies
are also worth investigating in the future for each stage. Furthermore, we may
study the nano-fabrication as a system and investigate the dependence (includ-
ing variation propagation) among sequential steps. By characterizing process-
structure-property relations, we will be able to systematically monitor, diagnose,
and control the manufacturing systems.
We hope our work in this dissertation will trigger further studies on scalable
nanomanufacturing especially in the field of nanomanufacturing scale-up methodology
research. With the efforts of our own and fellow researchers, we expect to fulfill the
promises of nanotechnology and realize well controlled nanomanufacturing in commer-
cial scale in the near future.
99
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Abstract (if available)

Abstract Nanotechnology serves the potential to address national and global needs in high-impact opportunity areas such as energy, security, and public health. In the past decades, numerous laboratory successes have been reported to apply functional nanomaterials in promising applications that could potentially replace the current technology with more advanced performance and novel functionalities. As the key enabler to scaling-up laboratory successes, nanomanufacturing, however, has been the major bottleneck to fulfill this promise because of its incapability of producing reliable and cost-effective nanoscale materials, structures and devices. The current yield of nanodevices is 10% or less. This low and unstable yield makes nanofabrication extremely expensive, hardly predictable and performance of nanodevices highly uncertain. ❧ To achieve scalable nanomanufacturing, a central issue is to control nanostructure synthesis variations. Since the formation of nanostructures is sensitive to the interactions among neighboring structures such as competition for source materials during growth, it is therefore critical to develop metrics and methods to characterize nanostructure interactions and link the interactions with structure variability. Fundamental research challenges of developing such characterization and connection include three aspects. (i) There is very limited physical knowledge of nanostructure interactions, let alone a rigorous formulation. The only understanding is that nanostructures physically closer have more chances to have mutual impacts. But there are no concrete physical principles that we can rely on. Modeling nanostructure interactions are subject to more uncertainties due to latent or unobserved factors. (ii) Nanostructure interaction characterization is severely constrained by the scarcity of measurement data. Current nanostructure inspection techniques such as Scanning Electron Microscope (SEM) and Transmission Electron Microscope (TEM) are mainly two-dimensional imaging. Taking images and extracting information of nanostructure features e.g. nanowire length from these 2D images are extremely time consuming and labor intensive. (iii) Although we have relatively good understanding of growth kinetics to describe the general trend or profile of nanostructure growth, the understanding is not conclusive. For most synthesis processes, there is a lack of process models that could coherently explain all major experimental observations. Uncertainty of this general trend modeling affects the estimation of nanostructure interactions as well. ❧ In this dissertation, we focus on the synthesis processes of nanowires and develop a systematic modeling and estimation technique to characterize nanostructure interactions for scale-up nanomanufacturing. We include three research tasks to deliver our characterization methodologies and address the research challenges step by step. They are: (i) modeling nanostructure interactions statistically by Gaussian Markov random field to capture possible interaction patterns and link with structure variability

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Asset Metadata

Creator Xu, Lijuan (author)

Core Title Nanostructure interaction modeling and estimation for scalable nanomanufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 11/21/2013

Defense Date 10/18/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag growth process modeling,nanomanufacturing,nanostructure interactions,nanowire,OAI-PMH Harvest,quality control,SA-MOCVD

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Huang, Qiang (committee chair), Chen, Yong (committee member), Qin, S. Joe (committee member)

Creator Email lijuanxu@usc.edu,sunshineiejulice@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-349129

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