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Nanostructure morphology variation modeling and estimation for nanomanufacturing process yield improvement
h [electronic resource] /
by Gang Liu.
[Tampa, Fla] :
b University of South Florida,
Title from PDF of title page.
Document formatted into pages; contains 65 pages.
Thesis (M.S.I.E.)--University of South Florida, 2009.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
ABSTRACT: Nanomanufacturing is critical to the future growth of U.S. manufacturing. Yet the process yield of current nanodevices is typically 10% or less. Particularly in nanomaterials growth, there may exist large variability across the sites on a substrate, which could lead to variability in properties. Essential to the reduction of variability is to mathematically describe the spatial variation of nanostructure. This research therefore aims at a method of modeling and estimating nanostructure morphology variation for process yield improvement. This method consists of (1) morphology variation modeling based on Gaussian Markov random field (GMRF) theory, and (2) maximum likelihood estimation (MLE) of morphology variation model based on measurement data. The research challenge lies in the proper definition and estimation of the interactions among neighboring nanostructures. To model morphology variation, nanostructures on all sites are collectively described as a GMRF.The morphology variation model serves for the space-time growth model of nanostructures. The probability structure of the GMRF is specified by a so-called simultaneous autoregressive scheme, which defines the neighborhood systems for any site on a substrate. The neighborhood system characterizes the interactions among adjacent nanostructures by determining neighbors and their influence on a given site in terms of conditional auto-regression. The conditional auto-regression representation uniquely determines the precision matrix of the GMRF. Simulation of nanostructure morphology variation is conducted for various neighborhood structures. Considering the boundary effects, both finite lattice and infinite lattice models are discussed. The simultaneous autoregressive scheme of the GMRF is estimated via the maximum likelihood estimation (MLE) method. The MLE estimation of morphology variation requires the approximation of the determinant of the precision matrix in the GMRF.The absolute term in the double Fourier expansion of a determinant function is used to approximate the coefficients in the precision matrix. Since the conditional MLE estimates of the parameters are affected by coding the date, different coding schemes are considered in the estimation based on numerical simulation and the data collected from SEM images. The results show that the nanostructure morphology variation modeling and estimation method could provide tools for yield improvement in nanomanufacturing.
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Advisor: Qiang Huang, Ph.D.
Co-advisor: Jos Zayas-Castro, Ph.D.
Gaussian Markov random field
x Industrial and Management Systems Engineering
t USF Electronic Theses and Dissertations.
Nanostructure Morphology Variation Modeling and Estimation for Nanomanufacturing Process Yield Improvement by Gang Liu A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Enginee ring Department of Industrial and Management System s Engineering College of Engineering University of South Florida Co Major Professor: Qiang Huang Ph.D. Co Major Professor: Jos Zayas Castr o Ph.D. Michael Weng, Ph.D. Date of Approval: Oct ob er 3 0 200 9 Keywords : nanowire, Gaussian Markov r andom f ield, neighborhood structure, interaction, autoregressive scheme Copyright 200 9 Gang Liu
Acknowledgements First and foremost, I would like to express my sincere gratitude to Dr. Qiang Huang for his instantly inspiring guidance, hearty encouragement, valuable advices and constant support through the entire course of this study. I appreciate him so much for providing me this great opportunity to work in the Nanomanufacturing Quality Control Lab and introducing me to the new world. Secondly, special thanks for Dr. Jos Zayas Castro and Dr. Michael Weng for being my committee numbers and for the excellent suggestions they proposed in my defense and for my thesis. Next, thanks are given to the students from Advanced Materials Research Lab. Dr. Qiang Hu and Farah Alvi helped me to much exten t with the experimentation and scanning electron microscopy images collection. Thanks also go to my friends here, who give me sincere suggestions on the research work and help me with my study and life. The names es pecially need to be mentioned are: Xi Zhang, Hui Wang, Shaoqiang Chen, Yang Tan, Qingwei Li and Yu An from industrial and system management engineering department I should also say thanks to L. Xu who helped with the proof of absolute term. I gratefully acknowledge the financial support from NSF. Finally, I am heavily indebted to my parents, Q .W. Liu and X.Y. Xiang, who have been firmly supporting me and encouraging me from the behind. I would like to express my great appreciation and love to them.
i Table of Contents List of Tables iii List of Figures iv Abstract v Chapter 1 Introduction and Background 1 1.1 Introduction to Nanotechnology 1 1.1.1 Features at Nanoscale 2 1.1.2 Applications of Nanotechnology 3 1.2 Introduction to Nanowire 5 1.2.1 Features of Nanowire 6 1.2. 2 Quantum Confinement 7 1.2. 3 Classifications of Nanowire 8 1.2. 3 .1 Silicon Nanowire 8 1.2. 3 .2 ZnO Nanowire 1 0 1.2. 3 .3 Silica Nanowire 12 1.2. 3 .4 III IV Semiconducting N anowire 1 3 1.3 Nanowire Growth Process 1 4 1.3.1 Physical Vapor Deposition Method 1 4 1.3.2 Chemical Vapor Deposition 1 6 1.3.3 Solution B ased C hemical S ynthesis M ethods 1 7 1.4 Nanowire Growth Kinetics 1 7 1.5 Challenges of Current Nanomanufacturing Process 2 0 1. 6 Research Objective s 2 1 1. 7 Thesis St ructure 2 2 1. 8 Summary 2 2 Chapter 2 Experimental Methods and Characterization Techniques 2 3 2.1 Introduction to Experiment System 2 3 2.2 Characterization Techniques 2 4 2.2.1 Scanning Electron Microscopy 2 5 2.2.2 Nanowire Length Measuremen t Development 2 7 2.2.3 Results and Data Tables 3 0 2.3 Summary 3 3 Chapter 3 Nanostructure Morphology Variation Modeling 3 4 3.1 Space T ime R andom F ield M odel of N anostructure M orphology 3 4 3. 2 Motivation of Modeling via Gaussian Markov Random Fields 3 5 3. 3 Modeling of N anowire M orphology V ariation 3 6 3.4 Construction of Precision Matrix of GMRFs 38 3.4.1 Precision Matrix for Infinite Lattice 38
ii 18.104.22.168 Conditional Autoregression Scheme 38 22.214.171.124 Simultaneous Autoregressive Scheme 39 3.4.2 Precisi on Matrix for Finite Lattice 4 0 3. 5 The Simulation of Morphology Variation by Improper GMRF 4 3 3.6 Summary 4 5 Chapter 4 Parameters Estimation for Nanowire Morphology Variation Model 46 4.1 Autoregressive S cheme 46 4.2 Simultaneous Autoregressive Sche me and the Increments 47 4.3 Maximum Likelihood Estimation for Autoregressive Schemes 47 4. 4 Coding Methods on Lattice 5 0 4. 5 Parameter Estimation Method Using Simulation Data 5 2 4. 6 Model Estimation for Two Sets of Real Data 5 3 4.7 Summary 58 Chapter 5 Conclusions 5 9 References 6 1
iii List of Tables Table 1 Data of Lengths for the First Set of Nanowires 3 1 Table 2 Data of Lengths for the Second Set of Nanowires 3 2 Table 3 Data Simulated from the Model with Given Parameters 5 3 Table 4 Parameters Estimation for Data Set 3 under by MLE 5 3 Table 5 Parameters Estimation for Data Set 1 under by MLE 5 4 Table 6 Parameters Estimation for Data Set 1 under by MLE 5 5 Table 7 Parameters Estimation for Data Set 2 under by ML E 56 Table 8 Parameters Estimation for Data Set 2 under by MLE 5 6
iv List of Figures Fig .1 The Four Forms of Nanostruct ures 5 Fig .2 A Diagram for ZnO Nanowires Synthesis 2 4 Fig .3 Interactions of Electrons with the Sample 2 5 Fig .4 Electron Interactions with Specimen Scattering Events 2 6 Fig .5 SEM Image at the Magnification of 12K 27 Fig .6 ZnO Nanowires of the Same Regi on Observed from Different Angles 28 Fig .7 The Position Change of the Same Nanowire after the Rotation 28 Fig .8 The Two Sets of the Observed Nanowires 3 0 Fig .9 Morphology Variation from the Data Set 1 3 1 Fig .10 Morphology Variation from the Data Set 2 3 2 Fig .11 A 3 by 3 Lattice of Local Variability 4 1 Fig .12 Simulation for Central Increments 4 3 Fig .13 Simulation for F orward Increments with Interactions 4 4 Fig .14 Simulation for F orward Increments without Interactions 4 4 Fig .15 Simulation for Increm ents from Spatial Statistics 4 5 Fig .16 Coding Pattern for a First order Scheme 5 0 Fig .17 Coding Pattern for a Second order Scheme 5 1
v Nanostructure Morphology Variation Modeling and Estimation fo r Nanomanufacturing Process Yield Improvemen t Gang Liu ABSTRACT N anomanufacturing is critical to the future growth of U S manufacturing Y et the process yield of current nanodevices is typically 10% or less. Particularly in nanomaterials growth there may exist large variability across the sites on a substrate, which could lead to variab ility in properties. Essential to the reduction of variability is to mathematically describe the spatial variation of nanostructure. This research therefore aims at a method of modeling and estimating nanostructure morphology variation for process yield i mprovement. This method consists of (1) morphology variation modeling based on Gaussian Markov random field (GMRF) theory, and (2) maximum likelihood estimation (MLE) of morphology variation model based on measurement data. The research challenge lies in t he proper definition and estimation of the interactions among neighboring nanostructures. To model morphology variation, nanostructures on all sites are collectively describ ed as a GMRF. The morphology variation model serve s for the space time gr owth model of nanostructures. The probability structure of the GMRF is specified by a so called simultaneous autoregressive scheme, which defines the neighborhood systems for any site on a substrate. The neighborhood system characterizes the interactions a mong adjacent nanostructures by determining neighbors and their influence on a given site in terms of conditional auto regression. The conditional
vi auto regression representation uniquely determines the precision matrix of the GMRF. Simulation of nanostruct ure morphology variation is conducted for various neighborhood structures. Considering the boundary effects, both finite lattice and infinite lattice models are discussed. The simultaneous autoregressive scheme of the GMRF is estimated via the maximum lik elihood estimation (MLE) method. The MLE estimation of morphology variation requires the approximat ion of the determinant of the precision matrix in the GMRF. The absolute term in the double Fourier expansion of a determinant function is used to approximat e the coefficients in the precision matrix. Since the conditional MLE estimates of the parameters are affected by coding the date, different coding schemes are considered in the estimation based on numerical simulation and the data collected from SEM image s. The results show that the nanostructure morphology variation modeling and estimation method could provide tools for yield improvement in nanomanufacturing.
1 Chapter 1 Introduction and Background In this introductory chapter, the broad backgro und of the research work, nanotechnology and the relatively narrow b ackground of the research work, nanowire, will be discussed respectively and successively After the classification of n anowire, the growth process and the growth kinetics of nanowire wil l be introduced. Later sections deal with challenges of the current nanomanufacturing p rocess and the r obust design of nanowire synthesis to improve the yield of the nanomanufacturing process. The research work aims to improve the yields of the nanomanufac turing process by modeling the morph ology variation of nanowire. This motivation is directly from the common challenges in the nanomanufacturing process. The statement of research objective and the significance of the research follow the motivation. The ch apter concludes with the structure of the thesis. 1.1 Introduction to Nanotechnology The term "nanotechnology" has evolved over the years via terminology drift to mean "anything smaller than microtechnology," such as nano powders, and other things that are nan oscale in size, but not referring to mechanisms that have been purposefully built from nanoscale components. Nanotechnology actually refers to the science and engineering of materials, structures and devices in which at least one of the dimensions is 100 n m or less. In a broader perspective, nanotechnology also includes the fabrication techniques in which objects are designed and built by the specification and placement of individual atoms or molecules both in an additive and subtractive manner.
2 An accredit ed definition of nanotechnology by NNI (National Nanotechnology Initiative) is the understanding and control of matter at dimensions of roughly 1 to 100 nanometers, where unique phenomena enable novel applications. Encompassing nanoscale science, engineeri ng and technology, nanotechnology involves imaging, measuring, modeling, and manipulating matter at this length scale. 1.1. 1 Features at Nanoscale Because the nanostructures are very tiny in size, they have salient features at nanoscale dimensions as follo ws  Firstly, t he dimensions of nanostructures are comparable to the size of biomolecule s dynamics, and diffusion and transport mechanism s) Such an attribute makes them a s sensitive probes to unravel atomistic and molecular events with great accuracy. Secondly, the r atio of the surface area to total volume is a direct measure of reactivity for any system/material. When the volume is too l arge relative to surface area, diffusion cannot occur at sufficiently high rates as molecules/particles take time to reach the active surface. An increased surface area to volume ratio means increased exposure to the environment, which in turn translates a s a sensitive surface to minor surface perturbations. In addition, a high surface area to volume ratio provides the required acceleration to negotiate thermodynamic barriers thereby minimizing free energy. Also, nanostructures possess high surface to volu me ratio facilitating faster diffusion mechanism. Thirdly, o ne and two dimensional na nostructures are generally high aspect ratio (ratio of length to diameter) arrangements with directionality dependent (along va rious axes) material properties. For example, (a) the deflection or absorbance of incident light would vary with the texture of the nanostructures, (b) a mechanical
3 a magnetic property like hysteresis would change with orientation. Also, the nanostructures could be tuned to achieve desired material attributes leveraging size dependent characteristics. Active s urface is another key feature T he diminutive sizes of nanoscale materials enable (a) the entry of foreign m aterials through adsorption and diffusion, (b) specific chemical interaction, and (c) physisorption by acting as templates. Collectively coining t he above mentioned feature Fictionalizations nanostructures present an active surface for imp lementing various applications. Finally, t he advancements in semiconductor manufacturing in combination with precision metrology have enabled high density synthesis of nanostructures at a faster rate to achieve the real promise of deve loping and manufacturing new nanomaterials, devices, and products in a cost effective manner. 1.1.2 Applications of Nanotechnology Because nanotechnology is essentially a set of techniques that allow manipulation of properties at a very small scale, it can have many applications  The first application is in d rug delivery. Today, most harmful side effects of treatments such as chemotherapy are a result of drug delivery methods that don't pinpoint their int ended target cells accurately. Researchers at Har vard and MIT have been able to attach special RNA strands, measuring about 10 nm in diameter, to nanoparticles and fill the nanoparticles with a chemotherapy drug. These RNA strands are attracted to cancer cells. When the nanoparticle encounters a cancer c ell it adheres to it and releases the drug into the cancer cell. This directed method of drug delivery has great potential for treating cancer patients while producing fewer harmful side effects than those produced by conventional chemotherapy.
4 The second application relates to f abrics. The properties of familiar fabrics are being changed by manufacturers who are adding nano sized components to conventional materials to improve performance. For example, some clothing manufacturers are making water and stain repellent clothing using nano sized whiskers in the fabric that cause water to bead up on the surface. A third use of nanotechnology deals with the reactivity of m aterials. The properties of many conventional materials change when formed as nano sized par ticles (nanoparticles). This is generally because nanoparticles have a greater surface area per weight than larger particles; they are therefore more reactive to some other molecules. For example studies have show n that nanoparticles of iron can be effect ive in the cleanup of chemicals in groundwater because they react more efficiently to those chemicals than larger iron particles. N anotechnology has also been applied to improve the strength of m aterials. Nano sized particles of carbon (for example na notubes and bucky balls) are extremely strong. Nanotubes and bucky balls are composed of only carbon and their strength comes from special characteristics of the bonds between carbon atoms. One proposed application that illustrates the strength of nanosiz ed particles of carbon is the manufacture of t shirt weight bullet proof vests made out of carbon nanotubes. Another application of n anotechnology is in Micro/Nano ElectroMechanical Systems. The ability to create gears, mirrors, sensor elements, as well as electronic circuitry in silicon surfaces allows the manufacture of miniature sensors such as those used to activate the airbags in your car. This technique is called MEMS (Micro ElectroMechanical Systems). The MEMS technique results in close integration o f the mechanical mechanism with the necessary electronic circuit on a single silicon chip, similar to the method used to produce computer chips. Using
5 MEMS to produce a device reduces both the cost and size of the product, compared to similar devices made with conventional methods. MEMS is a stepping stone to NEMS or Nano ElectroMechanical Systems. NEMS products are being made by a few companies, and will take over as the standard once manufacturers make the investment in the equipment needed to produce nan o sized features. Molecular Manufacturing also involves nanotechnology now If you're a Star Trek fan, you remember the replicator, a device that could produce anything from a space age guitar to a cup of Earl Grey tea. Your favorite character just program med the replicator, and whatever he or she wanted appeared. Researchers are working on developing a method called molecular manufacturing that may someday make the Star Trek replicator a reality. The gadget these folks envision is called a molecular fabric ator; this device would use tiny manipulators to position atoms and molecules to build an object as complex as a desktop computer. Researchers believe in the future it may be possible to reproduce almost any inanimate object using this method. 1.2 Introduction to Nanowire Based on the structure and morpho logy of the nanostructures, the nanocomponents can be broadly classified as nanoparticles, nanowires/nanorods, nanotubes and nano thin films as shown on Fig.1  Fig 1 The Four Forms of N anostructures
6 Among the several nanocomponents, nanowires have been preferred for many functional applications due to their efficient transport of electrons, structural anisotropy and the possibility of quantum confinement. 1. 2 1 Features of N anowire A nanowire is a nanostructure, with the diameter of the order of a nanometer. Alternatively, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. The con ductivity of a nanowire is expected to be much less than that of the corresponding bulk material. This is due to a variety of reasons. First, there is scattering from the wire boundaries, when the wire width is below the free electron mean free path of the bulk material. In copper, for example, the mean free path is 40 nm. Nanowires less than 40 nm wide will shorten the mean free path to the wire width. Nanowires also show other peculiar electrical properties due to their size. Unlike carbon nanotubes, whos e motion of electrons can fall under the regime of ballistic transport (meaning the electrons can travel freely from one electrode to the other), nanowire conductivity is strongly influenced by edge effects. The edge effects come from atoms that lay at the nanowire surface and are not fully bonded to neighboring atoms like the atoms within the bulk of the nanowire. The unbonded atoms are often a source of defects within the nanowire, and may cause the nanowire to conduct electricity more poorly than the bul k material. As a nanowire shrinks in size, the surface atoms become more numerous compared to the atoms within the nanowire, and edge effects become more important. Due to their high Young's moduli, their use in mechanically enhancing composites is being investigated. Because nanowires appear in bundles, they may be
7 used as tribological additives to improve friction characteristics and reliability of electronic transducers and actuators. 1. 2 2 Quantum Confinement Quantum confinement is a unique phenomenon observed in nanowires which has been exploited to alter the electronic properties for application specific requirements. Quantum confinement describes the organization of energy levels into which electrons can transverse when excited and becomes more relev ant when the exciton (electron hole pair) has squeezed into a dimension that approaches a critical quantum measurement, called the exciton Bohr radius. Physical separation between an electron and hole constitutes the Bohr radius which varies with different substances. The confinement phenomenon is better understood taking into consideration the quantum mechanics behind it. Electrons and holes or in general particles are considered to possess wave like properties described by a spatial wave function. Constra ining them in a small space indicates only particular types of wave function are allowed. Each must be exactly the right length and shape to satisfy the Schrdinger wave equation while under the effect of the trap. The solution to the wave equation corresp onds to a particular characteristic length as well as a particular energy to form a "ladder", with higher rungs corresponding to higher energy. When the confining dimension is large compared to the wavelength of the particle, the particle behaves free ly w ith continuum energy levels. As the confining dimension decreases, the particle's energy increases discretely. This can be further explained by the photoluminescence phenomenon (process in which material absorbs photons and re emits photon s of characterist ic wavelength). The energy of the emitted light is governed by the material composition in the case of a bulk semiconductor. When the physical size of the semiconductor is considerably reduced to be much
8 smaller than the Bohr radius, additional energy is r equired to confine this excitation within the nanoscopic structure leading to a shift in the emission to shorter wavelengths. Since the electron is given a wave treatment, conventional law such as ld, Au is a better conductor than Pb; however, in atomic scale conduction Pb underpins Au by three orders of magnitude T he s ignificance of Quantum Confinement is reflect ed by the following two aspects: I) It i ncrease s the e ffici ency and p erformance of d evices C onventional lasers are created with thin film epitaxial of III V semiconductors. Though the bulk of modern lasers are highly functional and reliable, they are inefficient in terms of energy consumption and heat dissipation. Adopting 1 D nan owires or 0 D quantum dots enables higher efficiencies and bright er laser s with the possibility of on chip integration. II) External e n vironment can m anipulate the devices as well. T he discrete energy levels in a nanowire co uld easily be altered by changing the environment such as a c hanging the electric and magnetic field or the introduction of a dielectric interface. Quantum dots/particles/wires can be easily manipulated precisely by STM or AFM generated fields. 1. 2 3 Clas sifications of Nanowire Based on the electrical conductivity of the nanowires, they are broadly classified as metallic and semiconducting. Some of the reported configurations of metallic nanowires are that of Au, Ag, Ni, Pd and Co, while Si, ZnO, SnO 2 GaN and SiO 2 forms the repertoire of semiconducting nanowires. 1.2. 3 .1 Silicon Nanowire Silicon (Si) is the second most abundant element in the earth's crust, and its versatility in many applications is imparted by the semiconductor manufacturing
9 industry wh ich has developed the standardized infrastructure and the technical to process it. However, though Si is configured as a mechanical material for microstructures, its indirect band gap (~1.1eV), making it a poor light emitter. The development of efficient light emitting silicon devices or structures would enable optical interconnect on a new generation of chips. Hence, the research thrust was to engineer silicon to enhance its properties. One such effort led to the synthesis of Si nanowires. Many techniques have been applied to synthesize Si nanowires, ranging from laser ablation methods to solution techniques [ 4 5 ]. Routinely, ordered arrays of vertically aligned Si nanowires have been mass manufactured by Vapor Liquid Solid mechanism (VLS), invo lving silane as a source of silicon. McFarland et al [ 6 ] used Si nanowires as channels in Field Effect Transistors (FETs) to combat the size limitation encountered by the CMOS industry. In addition, sensitive biosensors were developed [ 7 ] by detecting th e voltage modulation on the silicon nanowires before and after immobilization of biomolecules. Some of the other reported applications of Si nanowires include (a) anode material lithium ion batteries, (b) field emitters, (c) thermoelectric materials and cu rrently in (d) photovoltaic applications. Silicon is an attractive anode material for lithium batteries because it has a low discharge potential and the highest known theoretical charge capacity. However, bulk volume changes by 400% upon insertion and extraction of lithium which results in pulverization and capacity fading. It has been shown that Si nanowire (Si NW) [ 8 ] battery electrodes circumvent these issues as they can accommodate large strain without pulv erization, provide good electronic contact and conduction, and display short lithium insertion distances. Recent investigations indicate the use of Si nanowires in photovoltaics [ 9 ]. Upon fabricating p n junctions conformally around the nanowire structure, the absorption of light can
10 be decoupled from minority carrier diffusion. Subsequently, minority carriers only have to diffuse tens to hundreds of nanometers to the charge separating junction, r ather than the tens to hundred s of microns typical of convent ional solar cells. Silicon nanowires as alternate material for carrier transport nanowires offer several performance and manufacturing benefits that may impact future PV applications. The key to mak ing silicon nanowire functional and leverage discernible e nhancement from their bulk counterpart is to introduce defects both at atomic scale (boron, phosphorous...etc ) and at high concentrations (erbium) in the case of nanophotonic applications. As far as nanowire growth is concerned, research investigations ai m to develop low temperature growth methodologies and silane free synthesis of wires. 1.2. 3 2 ZnO Nanowire An important class of functional nanomaterials arises from zinc oxide (ZnO) based nanostructures. Due to its wide band gap (3.37 eV), near band em ission, transparent conductivity, high exciton binding energy, and high breakdown strength, ZnO has been a promising nanomaterial for various applications. The inherent structure and a careful selection of process variables have enabled the nucleation of Z nO nanostructures with semiconducting, insulatin g and piezoelectric properties. Some of the potential applications [ 10 ] include: Gas s ensing is one application of ZnO nanostructures. ZnO is candidate material for gas sensing applications. ZnO shows oxygen deficiencies and surface defects in addition to high surface to volume ratio. Exposure to external ambient to ZnO nanorods induces a significant change in its conductivity/resistivity high selectivity and sensitivity.
11 Optical e mission by ZnO nanostructures is conducted. Zinc oxide is a very good light emitter. Electrically driven ZnO nanowire serves as high efficiency nanoscale light sources for optical data storage, imaging, and biological and chemical sensing. Energy h arvesting is a nother use of ZnO nanowires. Recently, ZnO nanowires have been used for energy harvesting applications. Piezoelectric ZnO NWs [ 11 ] have been used to convert nanoscale mechanical energy into electrical energy. In this investigation, the aligned NWs are def lected with a conductive atomic force microscope tip in contact mode. The coupling of piezoelectric and semiconducting properties in ZnO creates a strain field and charge separation across the NW as a result of its bending. The rectifying characteristic of the Schottky barrier formed between the metal tip and the NW leads to electrical current generation. The efficiency of the NW based piezoelectric power generator was estimated to be 17 to 30%. This approach has the potential of converting mechanical, vibr ational, and/or hydraulic energy into electricity for powering nanodevices. ZnO nanostructures are used to make Che m ical /Bio logical s ensors The piezoelectric property of ZnO nanowires has also been exploited for chemical and biological sensing. Th e transduction mechanism to detect the desired chemical/biological species originates from compensation of the polarization induced bound surface charge by interaction with polar molecules in the analyte. Also, the high aspect ratio and high isoelectric po int of ZnO nanowires lends for efficient immobilization of high concentration of acidic enzymes and the mediating effect by the redox reactions. The traditional challenge in growing aligned ZnO nanowires is the substrate compatibility due to lattice misma tch between ZnO and substrate. For example, ZnO
12 nanowires have a tendency to vertical ly align themselves on sapphire substrates where as in Si, process optimization is required. 1.2. 3 3 Silica Nanowire Silica is one of the omnipresent elements existing in nature, primarily in the form of sand. It is a common constituent of glass and its derivative like quartz. Undoubtedly, Silicon di oxide or silica has been the unsung hero of the semiconductor industry donning the role of an insulator. Bulk silica has b een used in a plethora of applications such as (a) an insulating medium in CMOS circuits, (b) a barrier to prevent interfacial reactions in high temperature processing, (c) dielectric matrices/medium for synthesis of new materials as well as plasmonic appli cations, (c) masking material for high energy processes while offering a chemically compatible surface for etching applications and (d) a cell culture medium and biocompatible surface for developing in vitro and in vivo biosensors. T he aforementioned prop erties are enhanced multi fold due to the high aspect ratio of the nanostructures and quantum confinement effects. Silica nanostructures have been manufactured in a variety of distinct topological forms (e.g. nanopores, nanospheres, nanowires, nanotubes, n optics, biosensing/clinical diagnostics, and catalysis and developing composites. In general, silica nanostructures are known to possess characteristic structural properties (e.g., two membered silicon diox ide rings [ 12 ], non bridging oxygen (NBO) defects [ 13 ]) quite unlike those of the bulk, which is often reflected in their physical/chemical properties. Recently, for example, silica nanowires and nanotubes have been shown to be intense blue/visible light e mitters [ 14 ], which have been linked to structurally non stoichiometric s group [ 15 ] fabricated long freestanding silica nanowires using a process with
13 diameters down to 50 nm that show atomic level surface s moothness and excellent diameter uniformity. The length of th e wires was estimated to be in tens of millimeters, giving them an aspect ratio larger than 50,000. It was shown that light can be launched along these wires by optical evanescent coupling. The w ires enable single mode operation and have very low optical losses within the visible to near infrared spectral range. Also, mechanical tests show that the wires have tensile strength in excess of 5 GPa. The wires were also resilient and flexible, easily b ending into microscopic loops. In contrast, silica nanotubes (SNTs) have proven to be a multifunctional nanostructure for biomedical applications [ 16 ] such as drug delivery and bioseparation. In a recent article, synthesis of well controlled shape coded SN Ts and their application to biosensing as a new dispersible nanoarray system have been described. It is mentioned that the shape coded SNTs can be easily identified by their different shapes (codes) using a conventional optical microscope. Because of the l ow density and high surface area of the hollow tubular structure compared to those of the spherical and rod structures, SNTs can be suspended and are stable in solution. 1.2. 3 4 III IV Semiconducting Nanowire III I V compou nd semiconductor nanowires form a n important sub set functional nanostructure for optoelectronic applications, due to the direct bandgap and high carrier mobility of these materials. Various nanowire based devices, including lasers and photodetectors, have already been demonstrated [ 17 1 8 ]. Compound semiconductor nanowires like GaAs, InAs and InP have been investigated. It has been reported that homogenous InAs nanowires [ 19 ] show a low resistance at room temperature as well as at liquid helium temperature. A positive gate voltage increas es the conductance of a wire significantly and is found to be very sensitive to the diameter of the nanowires. Alt ernatively, InP nanowires serve as electron super
14 elec tron attracting electrode and this scenario could boost thin film solar cell efficiency as recently reported [ 20 ]. The growth of these nanowires is often carried out in MOCVD (Metal Organic Chemical Vapor deposition) with poisonous gases. However, the nan owire growth is limited to certain substrates which offer lattice matching to stress free nucleation of wires. 1.3 Nanowire Growth Process During the last decade, one dimensional nanostructures have been extensively investigate d. Most of the studies have foc used on the synthesis and fabrication of the nanostructures. Typically, two categories of synthesis and fabrication techniques are chemical synthesis, self assembly and locatio n manipulations. The other is the cutting, etching, grinding etc. to fabricate nanoscale objects out of bulk materials. In the bottom up category, several approaches have bee n well established, which include an extensively explored vapor phase deposition method, including chemical vapor deposition (CVD) and physical vapor deposition (PVD), and liquid phase deposition (solution synthesis approach). 1. 3 1 Physical Vapor Depositi on Method The traditional physical vapor deposition method is dedicated to the thin film and coating technology, which generally rel ies on the physical vaporization and condensation processes of materials. Based on the source of energy, five typical method s have been developed which generally uses a resistor heater in a furnace to heat and facilitate the
15 vaporization of source materials and the nanostructure deposition process. This method has been fully developed as one of the most versatile methods for 1D nanostructure fabricatio n. The second type of PVD is the Electron beam PVD, which has been extensively used for fabrications of ceramic thin film and coa tings, but so far no reports ha ve been found for application in 1D nanostructure synthesis. The third atoms due to bombardment from a flux of impinging energetic particles. The consequences of the interaction between an incident particle and a solid surface are mostly dictated by the kinetic energy of the projectile particle, although its internal energy may also play a significant role. Surface diffusion is usually used for explanation of the nanoscale islands or rods g rowth during the sputtering process. Cathodic arc discharge is another important physical vapor deposition method. It is famous for synthesis of the first batch of carbon nanotubes in 1993. It has been very popular as a method for fabrication of carbon rel ated nanostructures, but there are very few applications in growth of oxide nanostructures. Pulse laser deposition (PLD) is a method for creation of nanostructures developed in the seminal work on YBa 2 Cu 3 O 7 x thin film growth by Venkatesan and co workers i n 1987 [ 21 ]. In t he process the source material is rapidly evaporated from a bulk target in a vacuum chamber by focusing a high power laser pulse on its surface. A large number of material types have been deposited as thin films using PLD. For some comple x and high T m (melting point) compounds, there is no doubt that using the PLD could be simpler and easier to receive a better quality of nanostructures. PLD has been used for synthesis of 1D nanostructure since early 90s in the last century. It has been su ccessfully used in carbon nanotube synthesis with high yield instead of using cathode arc discharge despite of a large amount of impurity. In the past decade, using
16 PLD method, a variety of 1D complex compound and core shell structured nanostructure have been synthesized. Outside the auspices of high performance semiconductor applications, PLD is as promising as any other technique for investigating new growth phenomena and systems, and it can offer some unique features. 1. 3 2 Chemical Vapor Deposition Ch emical vapor deposition (CVD) can be classified into six or more categories based on the difference of the vacuum control level, heating source and reactants gas types etc., as shown in Table I. Thermal CVD, LPCVD, LCVD, and MOCVD have been used to synthes ize several types of nanowires. Using a similar set up as the thermal evaporation method in PVD approach, thermal CVD just involves chemical reaction during the deposition process rather than a physical evaporation and deposition process. It has produced h igh quality carbon nanotub e and nanowires including Ga N [22 23 ] GaAs etc. Like thermal evaporation in PVD, in thermal CVD, a solid source could be placed upstream. The solid is vaporized through heating, which is transported by a carrier gas that feed s the catalyst and source material for nanowire growth. LPCVD is a technique for fabrication of a thermal oxide layer on semiconductor wafers in electronic industry. This approach uses a commercial oven or furnace as a main reaction chamber, which is gene rally composed of the reactant gas control and supply system and temperature/pressure control system. The pressure control is normally under 0.1 torr. Despite of an old technique for oxide film growth in electronic industry, recently the LPCVD has been a s uccessful method for synthesis of various nanostructures such as carbon nanotub e [ 24 25 ] and Si nanowir e [ 26 28 ] etc.
17 In Laser CVD, the CVD setup has been modified for laser ablation with controlled doses, leading to a control over nanowire lengths and diameters. Such a precision control made it possible to form superlattice s [ 29 ] or core shel l [ 30 ] structures with doping. MBE has been used to grow 1D nanostructure of III V group semiconductor compounds such as Ga N [ 31 ] GaA s [ 32 ] InGaA s [ 25 ] and Al GaA s [ 3 3 ] etc. ALD seem ing ly has been used for fabrication of templates and ordered core shell nanostructure s [ 3 4 37 ] due to its precise control on atomic layer by layer deposition. 1. 3 3 Solution B ased C hemical S ynthesis M ethods As a generic meth od for synthesis of nanostructures, chemical synthesis has been a powerful method for producing raw nanomaterials. Since the emergence of one dimensional nanomaterials, solution based chemical synthesis methods have been extensively explored for fabricatio n of one dimensional nanostructures. Many materials such as elemental noble metals Au, Pt, A g etc., tri or multi elemental compounds like Ba(Sr)TiO 3 [ 38 39 ], in nanowire like shape have been successfully synthesized via solution based chemical synthesis methods, which cannot be easily fabricated by using vapor phase methods. The 1D nanostructures synthesized by chemical synthesis can be easily achieved in a large yield, but the large amount of impurities usually contained in the product s always dra matically hinder their application, although complex post filtering and purification is useful for increasing its purity. From the cost point of view, a low cost chemical synthesis with high yield undoubtedly is ideal for future commercialization of 1D nan ostructure. 1.4 Nanowire Growth Kinetics The first most significant work on the mechanism of the unidirectional growth of semiconductor whiskers was published by Wagner and Ellis in 1964 [ 40 ]. They
18 produced Si whiskers on the Silicon substrate surface with A u as impurity by the Chemical Vapor Deposition growth technique and proposed the well known Vapor Liquid Solid (VLS) mechanism. It is believed that whiskers grow due to the diffusion of vapor atoms on the alloy drop surface and the deposition of supersatur ated liquid alloy on the surface under the drop [ 40 44 ]. In his published works, Givargizov discussed the four main steps during which nanowire was produced and he also drew some conclusions about the rate determining step [ 42 ]. In 1973, Givargizov and Che rnov put forward a theoretical model of whisker formation via VLS mechanism [ 41 ]. Since then, the VLS concept has been extended to include the nanometer scale whiskers and widely used for explanation of Nanowire growth and the phenomenological Givargizov C herno v model has been implemented [ 44 46 ]. The empirical formula from Givargizov and Chernov [ 41 ] for the whisker growth rate is shown by equation Here, is the vapor supersaturation, is the surface temperature, is the Boltzmann constant, is the unknown kinetic coefficient of growth for the solid liquid interface in the step of the fourth in [ 42 ], in whi ch is the specific surface energy of the interface between the nanowire and the vapor around it and represents the diameter. Givargizov and Chernov developed their equation based on the Gibbs Thomson effect o f elevation of chemical potential of cylindrically shaped crystal phase diameter D It is reasonable that the dependen t term appears in the equation. Kashchiev [ 47 48 ] also proposed his own formula for the growth rate of nanowires with diameter but he did not take Gibbs Thomson effect into account. He accounted for the transition from mono nuclei to poly nuclei mechanism of nucleation mediated growth of nanowires. This effect is combined with the finite size of the growing surfa ce on the substrate which affects the number of droplet s from which a nanowire is produced. I n
19 mono nuclei mechanism the growth rate is proportional to but in poly nuclei mechanism it is independent of so diameter should be taken into account when transition from mono nuclei to poly nuclei mechanism of nucleation mediated growth of nanowires is made Kashchiev s formula reads where coefficients and depend on while their dependence on can often be neglected. From the given expressions it follows that both models predict the increasing length diameter d ependence, thicker Nanowires thus growing faster than the thinner ones. Such length diameter dependence was obtained in many experiments on CVD growth of nanowires [ 41 43 ]. V.G. Dubrovskii and his co workers [ 44 46 ] made great efforts in growth kinetics of GaAs nanoscale whiskers grown on the GaAs surface activated by Au and they proposed a new kinetic model describing the growth of n anoscale whiskers based on VLS mechanism in 2004. They aimed at the development of a kinetic model on Nanowire growth describ ing this process in a rather general form, containing the Givargizov Chernov model as a particular case and accounted for both the Gibbs Thomson effect and for the finite size of growing facet simultaneously [ 46 ]. Although it was reported later by Givargiz ov that diffusion in the liquid phase could not be the time determining step [ 42 ], V. Ruth and J. P. Hirth introduced kinetics of diffusion controlled whisker growth in 1964 [ 49 ]. In 2005, J. Kikka and co workers [ 50 ] measured the growth rate of silicon na nowires that were grown at temperatures between 365 and 495C via VLS mechanism and they studied the functions between growth rate and Temperature, time and diameter of nanowires. They finally got formula where is the activation energy and is a constant.
20 1.5 Challenges of Current Nanomanufacturing Process Nanomanufacturing represents the future of U S manufacturing. Nanostructured meterials and processes have been predicted to increase their market impact to about $340 billion per year in the next 10 years [ 5 1]. It has also been predicted that within the next 10 years, at least half of the newly designed advanced materials and manufacturing processes will be built at the nano scale [ 52 ]. In the past decade, tremendous efforts have been devoted to basic nanoscience discovery, novel process development and concept proof of nano devices. Yet much less research activities have be en undertaken in nanomanufacturing to duplicate the s uccess of transforming quality and productivity performance of traditional manufacturing. The h igh cost of proces sing has been a major barrier to transferring the fast developing nanotechnology from laboratories to industry applications [ 53 ]. The process y ield of current nanodevices is typically 10% or less [ 54 55 ]. The main challenge here is to improve process repeatability; therefore there is an imperative need for process improvement methodologies for nanomanufacturing. Understanding the first principles of nanostructure synthesis is certainly critical to improving process yield. Yet the physical laws are not completely understood at nanoscale. Current growth kinetics models involve a large number of constants to be estimated at a high level of accuracy. These models provide understanding of growth behavior at coarse scale, lack of description of local variability cross different sites on the substrate. In traditional manufacturing, statistical quality control and engineering driven sta tistical analysis of manufacturing process have achieved great success in yield and productivity improvement. Yet the scale effects bring new challenges on quality control in nanomanufacturing. First, manufacturing of quality engineered
21 nanostructures dema nds on prediction, monitoring and control of process variations at multiple scales. Second, nanostructure growth kinetics discovered in nanophysics provides valuable insights for process improvement. However, deterministic kinetics models fail to address p rocess uncertainties. Third, there is a lack of in situ observation of most properties at the nanoscale during processing. The offline SEM (scanning electron microscope) or TEM (transmission electron microscopy) inspectation is time consuming and costly. T hese challenges call for systematic methodology to model and control nanomanufacturing processes. Efforts have been undertaken to develop robust design methods to synthesize desired nanostructures  and to study the reliability of nanoelectronics [59 61 ]. The work on nanomanufacturing process control has not been reported. Another big challenge lies in the characterization technology for nanoscale. Nanometrology is the science of measurement at the nanoscale level and it has a crucial role in order to pr oduce nanomaterials and devices with a high degree of accuracy and reliability Current characterizat ion techniques are slow in data acquisition especially in 3D measurement Researchers have been developing measurement methods such as p hotogrammetry [ 5 6] and 3D reconstruction [ 5 7]. 1.6 Research Objective s Huang (2009)  developed a nanowire growth process model at a fine scale for prediction. His model consists of two major components: nanowire morphology and local variability. The morphology componen t represents the overall t rend characterized by growth kinetics. The area specific variability, also named morphology variation, is less understood in nanophysics duo to complex interactions among neighboring nanowires. This thesis will mainly focus on n an ostructure m orphology v ariation m odeling over different regions on the same substrate and
22 parameters estimation for the model by specifying the interactions among nanowires in the same system of sites 1.7 Thesis Structure In this thesis, the first chapter, as stated above, is the introduction part, which shows the broad background of the research work. The second chapter will describe the experiment and characterization methods. A measurement method will be proposed to obtain the length information from 2 dim ensional SEM images. T wo sets of the data on ZnO nanowire length s has been collected. The nanostructure morphology variation model will be built in C hapter 3 using Gaussian Markov r andom f ield s (GMRFs) Several numerical examples will be conducted accordi ng to the types of increment which reflects the neighborhood structure s in a GMRF The parameters in the model stated in the third chapter will be estimated in the fourth chapter by maximum likelihood estimation. Before the estimation, simulation studies are conducted to validate the model estimation method The method then will be applied to the two sets of real data. The last chapter will be the conclusion of this research work. 1.8 Summary This chapter first introdu ces nanotechnology and nanowire, which provide s the board background of this research work. The growth process and the growth kinetics are then presented in details due to their importance in the research work. The research objective and the structure of the thesis are stated in the end.
23 C hapter 2 Experimental Methods and Characterization Techniques Over the past decades, research into ZnO nanostructures has been one of the most attractive topics in physics, chemistry and materials science due to their significance in both fundamen tal and technological fields. In this chapter, we grow ZnO nanowires by ch emical vapor deposition process and character ize ZnO nanowires from scanning electron microscopy (SEM) images. 2.1 Introduction to Experiment System The steps for ZnO nanowire synthes is in the experiment are briefly expressed by Fig.2 : 1. Clean both the outer and the inner circular tubes and the s ubstrate. This step is important because an airtight space to grow ZnO nanowires is necessary If the outer circular tube is contaminated i t is very likely that the system leaks gas. W hen clean ing the substrate, use methanol instead of water 2. Put 500 milligram ZnO powder and 500 milligram graphite in a slot container. In the replicat es of the experiment, the mass and the ratio will remain the same. P ut the gold deposited substrate in another slot container. 3. Put both of the containers into the inner circular tube and the ZnO powder is in front of the substrate. Notice that the therm ocouple should be in the middle of the powder for the accuracy of the temperature detection. 4. Close the vent valve, switch on the mechanical pump, connect the vacuum gauge and then open the vacuum valve slowly. After about 15 minutes, the pressure in the gauge should be around 30 mtorr or at least less than 100 mtorr
24 5. Open the hydrogen & Argon valve and then open t he H ydrogen & Argon cylinders ( H ydrogen should not be less than 200kPa or may be more, Argon is about 40psi). Switch on the mass flow contr oller. Channel 3 is for Argon and channel 4 is for hydrogen. Set up the gas flow rate of Argon and hydrogen P ressure in the gauge increases and will be stable around 500 mtorr. 6. Turn on the digital temperature gauge, set the temperature rate at 10C/min ( P appears on the top Then press the s R S appears at the bottom. Press the right button to go back to the temperature display). S et the final temperature. 7. Wh en the temperature reaches the final temperature, keep it at this high temperature for 30 minutes. Close the hydrogen valve set the final temperature to room temperature ( 25C ) and close argon valve when the temperature reached 250C. Fig .2 A D iag ram fo r ZnO N anowires S ynthesis 2.2 Characterization Technique s The main technique used to observe and characterize ZnO is scanning electron microscopy (SEM). But the SEM images are in fact purely two dimensional as the electron beam is scanned over the specimen surface No height information can be
25 extracted directly from the images. The length of the nanowire is computed from two images obtained by observing the sample from two different angles/directions. The measurement method will be described in this sectio n 2.2.1 Scanning Electron Microscopy An electron microscope is a type of microscope that uses electrons to illuminate the sample. Electron microscopes have much greater resolving power and can obtain higher magnifications than a light microscope. They can magnify specimens up to 2 million times, while the best light microscopes are limited to magnifications of 2000 times. Such a multi fold enhancement in the resolution is due to the smaller wavelength of the electrons. For example, the wavelength of an electron induced by a 10 kV machine would be around 0.0123 nm whereas the visible wavelength of light ranges from 400 700 nm. Before proceeding in detail, the interaction of electrons with the surface of the sample needs to be understood. Fig.3 Intera ctions of Electrons with the Sample [Courtesy: Geochemical Instrumentation and Analysis, Science Education Research Center, Carleton College]
26 When electrons bombard on to a surface, a number of interactions ( F ig .3 ) arise with the atoms of the target sample. Accelerated electrons can either : (a) pass through the sample without interaction; (b) undergo elastic scattering or (c) undergo inelastic scattering. The scattering events result in unique signatures that can be used for imaging and quantitative informat ion about the sample. Fig. 4 Electr on Interactions with Specimen Scattering Events [Courtesy: Geochemical Instrumentation and Analysis, Science Education Research Center, Carleton College] Typical signals include secondary electrons (SE), back scatter ed electrons (BSE), visible light, Auger electrons and characteristic X rays arising from different depths in the sample as shown on Fig.4 Secondary emission is produced by inelastic interactions of high energy electrons with valence electrons of atoms i n the specimen. After undergoing additional scattering events while traveling through the specimen, some of these ejected electrons emerge from the surface of the specimen. The secondary electrons emerge with energies less than 50 eV. Further, larger atoms (with a greater atomic number, Z) have a higher probability of producing an elastic collision
27 because of their greater cross sectional area. Consequently, the number of backscattered electrons (from a higher depth) reaching the detector is proportional to the mean atomic number of the sample. Thus, a "brighter" BSE intensity correlates with greater average Z in the sample, and "dark" areas have lower average Z. BSE images are very helpful for obtaining high resolution compositional maps of a sample thus e nabling the differentiation of heavy and light elements/compounds and the metal/dielectric nature of the sample. The solid state detectors for collecting the secondary and back scattered electrons are placed on the side and in line with the electron gun re spectively. Along with BSE image, the elemental and chemical composition is verified using energy dispersive spectroscopy. 2.2.2 Nanowire Length Measurement Development The SEM images are in fact purely two dimensional as the electron beam is scanned over the specimen surface No height information can be extracted directly from the images. Fig 5 is a SEM image of ZnO nanowires; the length shown in the picture is not the true length of the nanowire but the top view of the true length. Fig 5 SEM I mage a t the M agnification of 12K Based on the principles of photogrammetry [ 56 ], a general simplified measurement method for nanowire length is developed. The nanowires are observed
28 from two different angles/directions using scanning electron microscopy. B y the following computation method, the height information can be extracted from the images. Fig. 6 ZnO N anowires of the S ame R egion O bserved f rom D ifferent A ngles T he two SEM images of the same set of nanowire s are obtain ed from two different angles as s hown by Fig.6 Shown on the CAD figure (Fig.7) is the length of one nanowire observed at a certain angle and is the length of the same nanowire observed after t he rotation axis is tilted by an angle of 10 degr ee. Fig.7 T he P osition C hange of the S ame N anowire after the R otation
29 From Fig.7, and can be obtained directly from the SEM images, and is the distance at which the sample is moved to observe the same nanowire and thus can be easily calculated by subtraction between the horizontal line and the nanowire before and after the sample is tilted at an angle of 10 degree s can be computed by the equation ( 2 .1). ( 2 .1) When , are obtained the extended part of the line and can be calculated by equations ( 2 .2) and ( 2 .3) ( 2 .2) ( 2 .3) The length of is the left view of the whole length of the nanowire, the equation (3.4) for computation of the left view is ( 2 .4) is the top view of horizontal distance fro m the first position of the nanowire to the rotation axis; is the top view of horizontal distance from the second position of the nanowire to the rotation axis. The equation for is ( 2 .5) To get the value of which is the top view of the distance between the rotation axis and the imagined nanowire wh en it is on the substrate, and which is the left view of the angle in bet ween the rotation axis and the first position of the nanowire, we will have the nonlinear equations set, ( 2 .6)
30 For the nonlinear equations set, the optimal solution of the equation ( 2 .7) can be the right and ( 2 .7) When the optimal solution is obtained, the length of the nanowire could be calculated by the equation ( 2 .8). ( 2 .8) Note that two assump tions are taken in the computation method. The first assumption is that the electron beams are perpendicular to the surface of the substrate. In fact, the electrons are from a starting point; however, the assumption is acceptable because the distance from the starting point to the substrate surface is very long. The second assumption is that the nanowire being observed and the rotation axis are in the same plane. 2.2.3 Results and Data Tables By the above measurement method, the length of nanowires ca n be obtained. The data in this research are spatial data, so the locations of the data are criti cal in data collections. Fig. 8 shows the two sets of nanowires obtained from the scanning electron microscopy. Fig.8 T he T wo S ets of the O bserved N anowires
31 By the measurement method developed above, the length of the observed nanowires can be obtained. Data T able 1 (Data Set 1) shows the lengths of the nanowires from the first observation. The data are filled into a regular lattice for the convenien ce of modeling. Table 1: Data of Lengths for the First S et of Nanowires 5.01 2.27 3.07 2.58 3.24 3.61 3.43 2.41 1.97 3.52 4.32 3.65 1.71 1.54 3.17 2.49 2.38 3.34 2.91 3.26 1.88 2.46 2.63 2.92 2.66 3.79 1.94 3.15 2.79 4.48 3.91 3.25 1.89 2.51 4.27 3.74 2.89 2.65 3.03 1.87 4.56 2.32 2.18 2.27 2.14 4.19 4.77 3.04 2.65 3.07 2.73 2.33 3.33 2.59 4.82 2.48 3.67 4.57 4.46 4.65 2.57 2.53 2.36 2.68 3.58 2.60 3.92 3.13 3.69 3.97 2.96 2.31 3.51 2.57 2.56 3.62 2.76 2.75 2.83 3.31 2.91 3.54 2.85 2.79 1.80 2.08 2 .91 2.99 2.58 2.38 2.88 3.26 2.99 2.75 1.82 2.54 3.31 2.13 3.32 2.39 Fig.9 shows the morphology variation from the above data in lattice which is collected in one region on the substrate. Fig .9 Morphology V ariation from the Data Set 1
32 Data T able 2 (D ata Set 2) shows the lengths of the nanowires from the second observation. The size of this set of data is exactly the same as the first set and it is used for validation. Table 2 : Data of Lengths for the S econd S et of Nanowires 3.38 2.11 3.79 2.81 3.81 3 .21 2.88 2.36 2.97 2.45 1.86 2.01 2.83 2.71 2.48 3.13 1.75 2.27 3.41 2.89 2.33 2.03 2.24 1.86 1.92 2.17 2.48 2.86 2.23 1.98 2.13 2.12 3.06 2.42 2.63 2.49 2.91 1.95 2.73 2.07 3.09 2.53 2.59 2.01 1.67 2.35 1.75 2.26 2.79 2.10 2.42 2.65 2.34 2.18 2.54 2. 36 2.39 3.38 2.89 2.76 2.18 2.74 1.97 2.03 2.70 2.69 2.36 2.83 3.62 2.44 2.42 2.91 2.39 2.42 2.98 2.61 2.88 2.23 2.05 1.76 2.15 1.98 2.55 2.38 3.45 2.86 2.43 2.19 3.96 2.89 2.33 2.29 3.29 2.41 1.54 3.36 2.82 2.38 2.31 2.52 Fig.10 shows the morphology variation from the above data in lattice which is collected in a different region on the same substrate Fig.10 Morphology V ariation from the Data Set 2
33 2.3 Summary In this chapter, the experiment is described follow ed by the introduction of characteriza tion technique s The most popular and effective technique currently is sca nning electron microscopy T hrough a systematic investigation of the three dimensional sca nning techniques a simplified measurement method for ZnO nanowire length from two SEM image s obtained from two different angles has been developed. The measurement method follows the principles of Photogrammetry and the two SEM images are achieved by tilting the rotational axis at a given degree of angle. The computation code is developed due to the large amount of data and the difficulty to obtain the solution s for nonlinear equations set. Two set s of data, each of which is of size 10 by 10, are finally collected for the use of parameter estimation in the following chapters.
34 Chap ter 3 Nanostructure Morphology Variation Modeling Huang (2009) [6 2 ] point ed out that nanostructure growth is a dynamic process which is to be characterized by a space time random field model. His model consists of two majo r components: nanowire morphology and local variability. The morphology component represents the overall trend characterized by growth kinetics. The area specific variability, namely morphology variation, is less understood in nanophysics due to complex in teractions among neighboring nanowires. In this chapter, the nanostructure growth model will be introduced first and then we will focus on n anostructure m orphology v ariation m odeling on both infinite lattice and finite lattice by specifying the interaction s among nanowires in some neighborhood structure s 3.1 Space T ime R andom F ield M odel of N anostructure M orphology Let represent the quality features of nanowires on a substrate. At a certain scale and time the observed feature at site of the substrate is denoted as For simplicity of presentation we hereafter only show the case with All the features at sites are represented as (3.1) which is a space time random field on lattice
35 In the space time random field model, Huang [62 ] decompose d the nanostructure random fie ld into morphology or profile local variation and noise (3.2) where could be a plane or a surface with r elative complex profile evolving over the growth process. represents the local fluctuation riding on the morphology. A Gaussian Markov random field ( GMRF) is suggested to be applied to model which will be the ma in task in this chapter. The rationale of the morphology local decomposition is as follows. Currently there is better understanding of global behaviors of the growth kinetics but limited physical knowledge is available for area specific variability. The de composition therefore aims at engaging growth kinetics through and morphology variation modeling through Due to the lack of p hysical studies, we only consider the case that local variability at site is mainly determined by its interaction with neighbors and remains stable over time. 3.2 Motivation of Modeling via Gaussian Markov Random Fields It is suggested that a GMRF be applied to model the local variability in nanostructure growth. A ctual ly a n alternative method of modeling the nanostructure is to use the kriging model in spatial statistics [ 63 64 ]. would be expressed as where is a mean function and weak stationary Gaussian field(see a kriging example in computer experiments[ 65 ]). The kriging model also captures the trend and local variability, and there is a connection between Gaussian random field and Gaussian Markov random field [ 66 ]. We prefer a Markovian property for the local variability component and a noise term in
36 equation (3.2) for two reasons  : First, we intend to separate the modeling error and noise (see a general discussion in ). The modeling error in equa tion (3.2) dominates in due to the lack of understanding at fine scales. However, molecular dynamics simulation [ 67 ] often use, e.g. the Lennard Jones potential [ 68 ] to describe the interactions among nanowires or nanotubes. The sepa ration offers the opportunity to adopt result in molecular dynamics to model the covariance structure in at nanoscale in the future. Second, Gaussian Markov random field has sho wn the flexibility to specify the neighborhood structur e on a lattice and to improve the efficiency in computation [ 66 ]. Material properties such as anisotropy are easier to be modeled. Although the advantages of this modeling treatment may not be fully demonstrated until the multiscale modeling and control me thodologies are developed, it can be viewed as one viable strategy to model the nanostructure growth process with effective integration of nanophysics. 3.3 Modeling of N anowire M orphology V ariation As defined in equation (3.2), the area specific variability represents local fluctuation riding on the morphology. In addition, is expected to capture the dependence/interaction among neighboring sites. We adopt an intrinsic Gaussian Markov random field model with rank deficiency in a line for local variability It has density function [ 66 ] (3.3) where is precision matrix with rank and denotes the generalized determinant.
37 The local variability specified by equation (3.3) will mainly be determined by the structure of the precision matrix. For an IGMRF of first order ( ), the conditional mean of is simply a weighted average of its neighbors, not involving an overall mean, i.e., [ 66 ]. Notation denotes the sites s that are the nei ghbors of Let be modeled via a Gaussian Markov random field with precision matrix and the overall mean of be we have [ 66 ], (3.4) (3.5) (3.6) It is clear that to model nanostru cture morphology variation construct ing the precision matrix for a Gaussian Markov random fie ld is critical The process defined by equation (3.3) is called conditional autoregression Gaussian scheme  for which (3.7) It is worth indicating the distinction between this scheme and the process defined by simu ltaneous autoregressive scheme, typically (3. 8 ) where are independent Gaussian variants each with zero mean and variance In contrast to equation (3.3), the latter process has joint probability density function,
38 (3. 9 ) where is defined the same way as the in the conditional a utoregression Gaussian scheme, which will be constructed in the next section 3 4 Construction of Precision Matrix of GMRF s To construct the precision matrix for GMRF s the most critical issue is to define the neighborhood structure s which include two basic aspects: (i) labeled graph  and (ii) interactions among the nanostructure s. Labeled graph determines the related neighbors and the interactions are the weights of the related neighbors. 3. 4 .1 Precision Matrix for Infinite Lattice To construct precision matrix of GMRF s on infinite lattice s b oth of the above two aspect s in the neighborhood structures can be represented in conditional autoregression Gaussian scheme and the simultaneous autoregressive scheme. 3. 4 .1 .1 Co nditional Autoregression Scheme In the conditional autoregression Gaussian scheme ( 3.3 ), is the matrix whose diagonal elements are unity and whose off diagonal element is Clearly is symmetric and is nonsingular  For example, we have a conditional a utoregression Gaussian scheme as follows, (3.10) T he precision matrix for a 3 by 3 lattice would be, ( 3.11 )
39 From Rue and Held [66 ], for a conditional autoregression scheme, (3.12) the precision matrix has elements (3.13) It is assumed that the variance of is a constant and thus the precision matrix from equation (3.13) for a 3 by 3 la ttice is the multiplication of and equation (3.11). 3. 4 .1.2 Simultaneous Autoregressive Scheme For the simultaneous autoregressive scheme (3.9), the precision matrix Q will be (3.14) is defined the same as in the conditional autoregression scheme which is shown by Eq.(3.13). For example, we have a simultaneous autoregressive scheme as follows, ( 3.15 ) The precision matri x for a 3 by 3 lattice will be (3.16) This is calculated by equation (3.14), in which is the same as in equation (3.11).
40 3.4 2 Precision Matrix for Finite Lattice The precision matrices defined above are for the infinite lattice an d are definite. The case where is symmetric and positive semidefinite is of particular interest. The class is known under the name intrinsic Gaussian Markov random fields. We construct precision matrix of these random fields based on the concept of the order increments which are directly from the simultaneous autoregressive scheme. Like the simultaneo us autoregressive scheme, t he increments also contain both aspects in the neighborhood structures. The order increments define the related neighbors and the coefficients of the neighbors in the increments are the interactions namely the weights of the related neighbors. For example, the second order forward increments in two directions, Follow the computation method proposed by Rue and Held , to define the joint density of : = (3.17) T he precision matrix for a lattice with size of 3 by 3 shown by Fig. 11 is (3.18 )
41 The entries in pr ecision matrix are the interaction s of the corresponding combination of the sites. For example, the elements in the first row are the interaction s of the combinations from to when we study the site of By the above precision matrix, the edge effects can be obviously interpreted. The interactions of the combinations which contain are all zeroes because is o n the edge When we take the forward increments for constructing precision matrix for GMRFs only the interactions between and the coming sites, like and are not zeroes By comparing equatio ns (3.11), (3.16) with (3.18), we can figure out the differences between the precision matrices of GMRFs on infinite lattice and those on finite lattice. The precision matrices on infinite lattices are all non singular; the precision matrices on finite lat tices are singular and contain a large quantity of zero elements, which are called sparse matrices and will lead to efficiency in computation The edge effect is quite significant when the lattice is large and it is the reason for the rank deficiency of th e precision matrices for GMRFs. We have the conclusion s based on the above precision matrix, (3.1 9 ) (3. 20 ) Fig. 1 1 A 3 by 3 L attice of L ocal V ariability
42 Other examples of the increme nts are central increments (3.21 ) and the forward increme nts withou t diagonal terms (3.22 ). (3.2 1 ) (3.2 2 ) The fourth type of increments is from covariance functions and weights functions in Spatial Statistics [63 ] We know for two nodes and there would be a couple of covariance functions, such as: Exponential covariance function: Gaussian covariance function: Powered covariance function: where is Euclidean distance between two sites. To define a proper neighborhood system, it is obvious that two sites should be neighbors when their covariance is large In spatial statistics, we also know several types of weights functions, such as and Here, we choose exponential covariance function, the decision boundary is 0.001 and the weights function as an example Based on these ideas, we will have increments as follows: (3.2 3 ) In conclusion of this section the precision matrices are constructed from the conditional autoregression Gaussian scheme, the simultaneous autoregressive scheme and the increments. The intrinsic Gaussian Markov random field fro m the increments
43 plays a critical role in the model descri bed in the first section of C hapter 3 because the intrinsic Gaussian Markov random field is invariant to the addition of polynomials which is also menti oned in the paper by Huang [62 ]. When the prec ision is obtained, the morphology variation could be simulated from the model. 3 5 The Simulation of M orphology Variation by Improper GMRF With the precision matrix obtained in the above section we can simulate the morphology variation for improper GMRFs by the algorithm from Rue and Held [6 6 ]. We simulate the morphology variation for a lattice with 20 rows and 20 columns The simulation for the finite lattice from improper GMRF constructed based on the central increments is shown by Fig.12 The ce ntral increments are Fig 1 2 Simulation for C entral I ncrements The simulation for the finite lattice from improper GMRF constructed based on the forward increments is shown by Fig.13 The forward increments with interactions are
44 Fig .13 Simulation for F orward I ncrements with I nteractions The simulation for the finite lattice from improper GMRF constructed based on the forward increments without diagonal term s is shown by Fig.14 The forward increments without diagonal terms are Fig .14 Simulation for F orward I ncrements without I nteractions
45 The simulation for finite lattice from improper GMRF constructed based on the interactions in spatial statistics is shown by Fig.15 The increments are Fig .15 Simulation for I ncrements from S patial S tatistics 3. 6 Summary In this chapter the morphology decomposition model wa s introduced and we focus ed on the local variability by specifying th e interactions among the nanostructures. The Gaussian Markov random field was applied to model the morphology variation. The precision matrix play ed the central role in the random field. We construct ed a precision matrix based on the conditional autoregres sion Gaussian scheme, the simultaneous autoregressive scheme and the increments. Finally the morphology variation wa s simulated based on the order increments T his led to the intrinsic Gaussian Markov random field which is of practic al use in the research work.
46 Chapter 4 Parameters Estimation for Nanowire Morphology Variation Model The order increments representation of nei ghborhood structures captures the interactions among different nano structu res in the same region o n the substrate. The parameter estimat ion of the order increments model is therefore critical to the nanostructure growth model described in Chapter 3 The spatial interactions of lattices have been researched for several decades and some related schemes and estimation methods have been developed. In this chapter, spatial interactions among nanostructures are to be estimated by maximum likelihood estimation (MLE). 4.1 Autor egressive S cheme Assume we have an infi nite lattice of the local variability at the site of each site of which is normally distributed. For one site we have n : (4.1) W here, represents all the other sites except and are the mean value of and is 0 when and are local variabilit ies in different neighborhood structures. Equation (4.1) c an be written in the form of simultaneous autoregressive scheme (4.2) When the mean value o f the site over th e lattice is zero the above equation will become, (4.3)
47 The error item is the same as the increment stated in C hapter 3. Then the estimation will be the fitting of the unknown parameters in e quation (4.3), when the neighborhood structure is defined by the increment from the above equation 4.2 S imultaneous A utoregressive S cheme and the Increments As stated above the error term in Gaussian schemes are eventually the same as the increments whic h are fundamental in constructing a precision matrix for intrinsic Gaussian Markov Random Fields. For example, the second order increment in constructing a precision matrix for intrinsic Gaussian Markov Random Field about the local nanostructure morphology variation is (4.4) While the error term in the second order Gaussian simultaneous autoregressive scheme is (4.5) It is obvious tha t to estimate the coefficien t in the increments is to estimat e the parame ters in the error term in Gaussian simultaneous autoregressive scheme. 4.3 Maxim um L ikelihood Estimation for A utoregressive S chemes We begin by considering the estimation of the parameters in an auto normal scheme of the form (4. 1 ) but subject to the restr iction We assume that the dimensionality of the parameter space is reduced through having a particular structure and that is both unknown and independent of all the is the n x n matrix whose diagonal elements are unity and whose off diagonal (i ,j) element is For a given realization x, the corresponding likelihood function is then equal to (4.6)
48 It follows that the maximum likelihood estimate of will be given by  (4.7) Once the maximum likelihood estimate of has been fo und by s ubstituting (4.7) to (4.6), we find that c an be found by minimizing (4.8) Suppose then that we temporarily abandon the auto normal model above and decide instead to fit a simultane ous scheme of the form (4.2), again subject to and with having the same structure as in ( 4.6 ). Provided ( 4.6 ) is valid so is the present, but different, scheme. The likelihood function now becomes (4.9) and the new estimate of must be found by minimizing (4.10) The only real difficulty centers upon th e evaluation of the determinant we may use the semi analytical result of Whittle . To minimize (4.8), where is the absolute term in the power series expansion of (4.11) and where, neglecting boundary effects, (4.12) and denotes the empirica l autocovariance of lags and in and respectively Let us suppose we have a set of data on a regular lattice of the local variability of nanostructures namely ( ), so t he empirical
49 covariance of lag is (4.13) The absolute term in the power series expansion can easily be evaluated for given parameter s by appropriate numerical Fourier inversion. For the following scheme (4.14) T he absolute term in will be (4.15) Thus we will minimize (4.16) For another conditional autoregressive scheme (4.17) the following function will be minimized to get the parameters (4.18) T he absolute term in will be (4.1 9 ) T he second order conditional autoregression scheme is (4.20) For scheme (4.20), we will minimize (4.21)
50 fail would be due to the inconsistent likelihood approximation chosen and more due to his not accounting for the large scale variation (trend) in the data. While for the data in our model the trend part is excluded by the profile of the morphology. It is noticeable that the incr ements in d efining the intrinsic GMRFs in C hapter 3 are equivalent to the error terms in simultaneous autoregressive Gaussian schemes and the minimization of equation (4.8) is for auto normal schemes. It is also proved that the auto normal schemes and the simultaneous autoregressive Gaussian schemes are reciprocally transformable. For auto normal scheme (4.1 6 ), the simultaneous autoregressive Gaussian scheme when the global mean is equal to zero is (4. 22 ) The result from mi nimizing equation (4. 16 ) also matches equations (4 .22 ), which is equivalent to the increments in defining GMRFs when equation (4. 22 ) is rewritten as (4. 23 ) 4.4 Coding Methods on Lattice Besag (1974) introduced coding metho ds on the rectangular lattice in the context of binary data and the methods are effective for more general situations. In his methods, for e xample, to fit the first order scheme (4.14), the interior sites of the lattice is la beled alternatively by and + as shown by Fig.16. Fig.1 6 Coding P attern for a F irst order S cheme
51 Since the variables associated with the + sites, given all the other observations on the lattice, only depends on their immediate sites, they are reciprocally in dependent. This will result in the conditional likelihood, T he product is taken over all + sites. Conditional maximum likelihood estimates of the parameters in the scheme could be obtained following the procedures in the above section. The estimates of the same scheme could also be obtained by maximizing the likelihood function for the sites conditional on the immediate + sites. The results from the two procedures can be different to some extend and it is better to combine the two results appropriately. To fit a second order scheme, for example scheme (4.20) the interior sites are labeled by and + as shown in Fig. 17 . Fig. 17 Coding P attern for a S econd order S cheme Unlike Fig.1 6 which could interpre t the variables at both and + sites, Fig. 17 only interprets the joint distributions of the + site values given the site values. When the joint distribution of all the + sites is obtained, we can estimate the unknown parameters by the methods introduced above. By performing shifts of the entire coding framework over the lattice, four sets of estimates are available and these results should be combined appropriately.
52 Parameter estimation under coding patterns would provide us with convenience of selecti ng data regions. The combinations of the estimated par a meters under different coding patterns would approach the true interactions. We will use this method in future work. 4.5 Parameter Estimation Method Using Simulation Data To test the above estimation method, we will simulate a set of data from the given parameters and try to use that data to make estimations that will return to the original model. Supposing infinite lattice of the local morphology variations of nanostructures t he simulated data are from the following conditional autoregression scheme : where, meaning that the mean of local variability is zero. The mean will keep zero for the following cases, because the trend part is excluded by the profile of the global morphology determined by growth kinetics. By the s imulation method in C hapter 3, the simula ted data (Data Set 3) is shown in T able 3.
53 Table 3 : Data S imulated from the Model with G iven P arameters 0.397 0 .928 0.404 0.209 0.396 0.955 1.515 0.931 0.6047 0.277 1.107 1.297 2.939 1.409 1.546 1.166 2.503 1.758 0.3305 0.092 2.200 2.338 1.620 0.598 0.645 0.033 0.517 0.891 2.3252 0.699 1.457 0.784 0.605 0.410 1.402 2.263 1.967 1.420 0.8909 1.31 5 1.271 0.064 1.277 0.101 1.150 0.267 0.709 0.926 1.3963 0.365 1.594 0.643 1.016 1.544 1.514 1.066 2.61 1 1.098 1.8450 1.820 0.932 0.458 2.492 2.065 0.716 0.439 2.837 0.156 1.3642 0.954 2.281 1.834 2.197 1.940 0.426 1.616 1.803 0.836 0.6537 0.339 0.782 1.386 0.785 2.756 0.534 1.364 1.415 0.830 0.8775 0.936 0.602 0.969 0.598 1.120 3.341 1.536 1.531 0.125 0.1044 0.300 The model to be estimated should have the same form, By the maximum likelihood estimation, we can have the estimated results shown in Table 4 b y maximum likelihood estimation ( MLE ) method because the interactions to the left and to the right are assumed to be the same Table 4 : Parameters E stimation for Data Set 3 by MLE Parameters Values 0. 2 723 0. 2334
54 The above results are obtain ed by adding 1.5 global mean to the data i n Table 3 If we compare the estimated results with the given paramete rs, we have confidence that the estimation method is of relatively good accuracy. 4.6 Model Estimation for Two Sets of Real D ata From the observations of scanning electron microscopy, we collected two sets of data concerning the lengths of nanowires which are shown in the figures in Chapter 2. According to the two sets of data, we could have the parameter estimated by the maximum likelihood estimation method For the first set of data, we at the beginning fit it as the conditional autoregressive sche me of the first order, which is the same as scheme (4.14) In this case the interactions in the horizontal direction are the same, no matter whether the neighboring site is in the left or in the right. The interactions in the vertical direction are identified as well. The estimated parameters are shown in Table 5 by MLE method Table 5 : Parameters E stimation for Data Set 1 by MLE Pa rameters Values 0. 104 9 0. 11 19 From the results, we c an tell that the vertical interaction is larger tha n the interaction in the horizontal direction.
55 Then we fit the first set of data as another conditional au toregressive scheme of the first order, which is the same as scheme (4.17) In this case the interactions of the immediate nei ghbors in the horizontal direction could be different. The interactions in the vertical direction are fluctuant as well The estimated parameters are shown in Table 6 by MLE method Table 6 : Parameters E stimation for Data Set 1 by MLE Parameters Values 0.44 63 0. 3 924 0. 3196 0. 3964 From Table 6 we could have the conclusion that the interaction s of the coming sites are generally larger than those of the past sites. By comparing the results in Tabl e 6 with the results in Table 5 we could find the differences between the two results are relatively large, due to the different scheme used in the model. It should be concluded that the appropriate model, for instance, the right order and the accurate interactions, needs to be tried more times. For the second se t of data, to validate the parameters we obtained above, we also fit it as the conditional autoregressive scheme of the first order, which is the same as scheme (4.14)
56 It is the same case that the interactions in the horizontal direc tion are the same, no matter whether the neighboring site is in the left or in the right. The interactions in the vertical direction are identified as well. The estimated parameters are shown in Table 7 by MLE method Table 7 : Parameters E stimation for Data Set 2 by MLE Parameters Values 0.0 198 0. 10 67 From the results, we c an tell that the vertical interaction is much larger than the interaction in the horizontal direction. Then we fit the first set of data as another conditional autoregressive scheme of the first order, which is the same as scheme (4.17) It is obvious that the interactions of the immediate neighbors in the horizontal direction could be different. The interactions in the vertical direction are fluctuant a s well. The estimated parameters are shown in Table 8 by MLE method Table 8 : Parameters E stimation for Data Set 2 by MLE Parameters Values 0.4164 0.3989 0. 4989 0. 2 620
57 It is noticeable that the summation of the para meters would be around 1 which is an intrinsic requirement. Therefore it is needed to adjust the parameters to make their summation close to 1. By comparing the estimation results and the validation results, we could find that the interactions for differe nt regions could be significantly different. It is reasonable that the difference s between two close and similar regions are smaller, as shown in Table 8. The results from the first and the second analysis are similar while those from the third and the fou rth analysis are close. However, the differences between them are relatively large. The possible reason could be: a) the data are collected in the different regions of the same substrate and therefore it is very likely that the interactions are of great difference; b) the measurement method of the nan o wires contains simplification s an d assumption s for the s a k e of fast computation and the size of data is not big enough ; c) the first order model may not be the best for estimating the parameters for the interactions among the lengths of the nanowires, so complicated model will be conducted later in the resea r ch work. For a higher order scheme, the process is the same as the first order examples stated above. The computation might be time consuming and the coding pattern needs to be expanded by shifting the entire framework for more times. 4. 7 Summary In this chapter, p arameter estimation and validation for the nanostructu re morphology variation model have been accomplished using the maximum likelihood estimation. The conditional autoregress ion scheme and the simultaneous autoregressive scheme are introduced to construct the conditional and joint probability models A test is conducted to prove the feasibility of the method. Then parameter estimation is done on the two sets of the obtained da ta, the later one of which is for validation After the numerical examples, it is f oun d that the difference s between the
58 estimation results and the validation results are relatively large. The possible reason s are possibly due to the different regions from where the two sets of data are collected, the inaccuracy in the measurement and the improper neighborhood structure in the model selected
59 Chapter 5 Conclusion s In this research, a series study was carried out on nanostructure morphology variation modeling and estimation for nanomanufacturing process yield improvement. The following goals were achieved: a). A simplified method was proposed for measuring nanowire lengths from scanning electron microscopy images ; b). ZnO nanowire morphology variation modeling was built using Gaussian Markov Random Fields and various simulations were accomplished to form morphology library; c). Parameter estimation and validation for the model were accomplished f or the collected data. Firstly, throu gh a systematic investigation on the three dimensional scanning electron microscopy images, a simplified method for measuring ZnO nanowire length from two SEM images obtained from two different angles was developed. The measurement method follows the princ iples of Photogrammetry and the two SEM images are achieved by tilting the rotational axis at a given degre e of angle. Secondly, ZnO nanowire morphology variation model s were set up by Gaussian Markov Random Fields (GMRFs). Nanostructure morphology variat ion modeling is a typically spatial data analysis. In this thesis, the spatial data of ZnO length all over the substrate at a given time was studied by GMRFs. The sampling algorithm concentrates more on precision matrix, which is a key feature for GMRFs A library of morphology has been set up based o n various neighborhood structure Thirdly parameter estimation and validation for the nanostructure morphology variation model was accomplished based on the two sets of collected data using the
60 maximum likeli hood estimation method We estimated the parameters in two first order conditional autoregression schemes for both of the two sets. In t he first scheme the interactions in the horizontal direction are the same and the interactions in the vertical direction are identical as well, however the interactions in both directions could be different. In the other scheme, all the interactions between the four immediate neighbors and the central site could be fluctuant. From the results, we know both of the estimation results and the validation results show the same tendency of the interactions. The interactions in the horizontal direction are less than the interactions in the vertical direction. However, the differences bet ween them are relatively large possibly due to the different regions from where the two sets of data are collected, the inaccuracy in the measurement and the improper neighborhood structure in the model selected The higher order will be studied following the same procedures and better neighborhood structure for the nanostructures could be obtained at higher orders.
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