IIE Transactions (2011) 43, 1–11Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408171003795335

Physics-driven Bayesian hierarchical modeling of thenanowire growth process at each scale

QIANG HUANG

Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California,Los Angeles, CA 90089-0193, USAE-mail: qiang.huang@usc.edu

Received September 2008 and accepted February 2010

Despite significant advances in nanoscience, current physical models are unable to predict nanomanufacturing processes underuncertainties. This research work aims to model the nanowire (NW) growth process at any scale of interest. The main idea is tointegrate available data and physical knowledge through a Bayesian hierarchical framework with consideration of scale effects. Ateach scale the NW growth model describes the time–space evolution of NWs at different sites on a substrate. The model consists of twomajor components: NW morphology and local variability. The morphology component represents the overall trend characterized bygrowth kinetics. The area-specific variability is less understood in nanophysics due to complex interactions among neighboring NWs.The local variability is therefore modeled by an intrinsic Gaussian Markov random field to separate it from the growth kinetics in themorphology component. Case studies are provided to illustrate the NW growth process model at coarse and fine scales, respectively.

Keywords: Nanomanufacturing, nanostructure growth process modeling, morphology and local variability, growth kinetics, intrinsicGaussian Markov random field, Bayesian hierarchical model estimation, Markov chain Monte Carlo simulation

1. Introduction

Nanomanufacturing represents the future of U.S. manu-facturing in that it has been estimated that nanostruc-tured materials and processes will increase their mar-ket impact to about $340 billion per year in the next10 years (National Science Foundation, 2001). In thepast decade, tremendous efforts have been devoted tobasic nanoscience discovery, novel process development,and concept proof of nanodevices. Yet much less re-search activity has been undertaken in nanomanufactur-ing to duplicate the success of transforming quality andproductivity performance of traditional manufacturing.High processing costs are a major barrier to transferringlaboratory-based nanotechnology to industry applications(National Center for Manufacturing Science, 2006). Theprocess yield of current nanodevices is typically 10% orless (The National Academy, 2002; Kuo, 2006). Hence,there is a need for process improvement methodologies fornanomanufacturing.

Understanding the first principles of nanostructure syn-thesis is certainly critical to improve process yield. How-ever, the physical laws are not completely understood atnanoscale levels. Current growth kinetics models are gen-erally deterministic (Ruth and Hirth, 1964; Wagner andEllis, 1964; Kikkawa et al., 2005; Dubrovskii et al., 2006)

and involve a large number of constants that have to beestimated to a high level of accuracy. These models providean understanding of growth behavior at a coarse scale, butthey lack a description of local variability across the dif-ferent sites on a substrate. Due to processing uncertainties,existing models are unable to predict realistic nanoman-ufacturing processes (National Nanotechnology InstituteGroup, 2005).

In traditional manufacturing, statistical quality controland engineering-driven statistical analysis of manufactur-ing processes have achieved great success in yield and pro-ductivity improvement (Wu and Hamada, 2000; Shi, 2006;Montgomery, 2009). However, scale effects result in newchallenges for quality control in nanomanufacturing. First,manufacturing of quality-engineered nanostructures de-mands prediction, monitoring, and control of process vari-ations at multiple scales; e.g., the overall nanowire (NW)growth rate versus the growth rates at different sites on asubstrate. Second, the growth kinetics of the nanostructureprovides valuable insights for process improvement. How-ever, deterministic kinetic models fail to address processuncertainties. Third, there is a lack of in situ observation ofmost properties at the nanoscale during processing. OfflineScanning Electron Microscopy (SEM) or TransmissionElectron Microscopy (TEM) inspection is time consum-ing and costly. These challenges call for a systematic

0740-817X C© 2011 “IIE”

2 Huang

methodology to model and control nanomanufacturingprocesses.

Efforts have been undertaken to develop robust designmethods to synthesize desired nanostructures (Dasguptaet al., 2008) and to study the reliability of nanoelectronics(Luo et al., 2004, 2006; Bae et al., 2007; see reviews by Jenget al. (2007) and Lu et al. (2009)). There has been limitedwork on nanomanufacturing process control. This articlefocuses on developing a physics-driven statistical growthmodel at any scale of interest to enable subsequent researchwork on multiscale model integration, process monitoring,and control. Following the introduction, Section 2 intro-duces the outline of a physics-driven hierarchical modelingstrategy using the NW growth process as an illustrative ex-ample. Details of the modeling approach and procedureare presented in Section 3. Section 4 provides two casestudies to demonstrate the developed NW growth processmodel at both coarse and fine scales. A summary is given inSection 5.

2. NW growth process and a hierarchical processmodeling strategy

Nanomaterials such as nanotubes or NWs are expectedto have wide applications in nanoelectronics and optoelec-tronic devices (Lu and Lieber, 2006). However, the pro-cess of synthesizing nanomaterials is complex. As shownin Fig. 1, a metal catalyst such as gold is first depositedonto the {111} surface of a Si substrate. The substrate isthen heated above the eutectic temperature to form a liquidAu-Si alloy. Using a Chemical Vapor Deposition (CVD)process, the liquid becomes the preferred site for absorb-ing Si atoms from the vapor, causing the liquid to becomesupersaturated with Si. The deposited particles travel tothe liquid–solid interface and vertically grow layer by layermediated by nucleation. The driving force for this so-calledVapor–Liquid–Solid (VLS) growth mechanism is the super-saturation of semiconductor atoms (Si) in the liquid alloy(Wagner and Ellis, 1964).

In addition, nanostructure growth normally involvesmultiple correlated processing steps; e.g., catalyst depo-sition and VLS growth. Processing uncertainties duringdeposition can cause variations in the shape and diame-ter of the metal catalyst from site to site, as illustrated byregions A and B in Fig. 2(a). During the VLS growth pro-cess, the NWs in region A can preferentially absorb speciesfrom the vapor and therefore inhibit the growth in regionB. This phenomenon can aggrivate the growth variabilityon a substrate (see Fig. 2(b)).

Effective control of such a complex NW growth processdemands a process model integrated with nanophysics. Thechallenge lies in knowledge disparity at different scales;i.e., limited understanding of growth kinetics at fine scales(greater uncertainties) versus the relatively established workat coarse scales (lower uncertainties). The following three

Fig. 1. Vapor–liquid–solid mechanism of NW growth (from Wag-ner and Ellis (1964)).

principles are essential when developing our modelingframework.

1. Openness: In light of the rapid advances in nanoscience,the modeling framework should be open enough to in-tegrate the future knowledge at fine scales without dra-matic changes to the model structures.

2. Separation: The model components with different lev-els of uncertainty should be separable so that duringmodel estimation the model structures with higher un-certainties will have no or little impact on those withlow uncertainties due to better physical understandingof these processes.

3. Structured simplicity: Due to limited measurement data,it is preferable to have a simple process model (contain-ing fewer unknown parameters) with a physical structureembedded with growth knowledge.

Fig. 2. Growth variability within a silicon substrate: (a) depositionvariability; (b) VLS process.

Nanomanufacturing quality control 3

Fig. 3. Physics-driven hierarchical modeling of growth process atmicro/nanoscale.

Specifically, the existing kinetics models that depict theNW growth at coarse scale (Ruth and Hirth, 1964; Kikkawaet al., 2005; Dubrovskii et al., 2006) and their model struc-ture will be adopted here to describe NW morphology.Based on the separation principle, the local variability com-ponent, for which no physical models are available to de-scribe growth variability across sites, is constructed to beinvariant to morphology. This morphology-local variabilityseparation is cast into a Bayesian hierarchical framework.As shown in Fig. 3, at a fine scale, the quality features X(s, t)of NWs on a substrate consist of morphology ηk−1(s, t),local variability φ(s), and noise ε, where t and s denotetime and the collection of sites on a substrate, respectively.Model parameters ξ of X(s, t) are determined by processvariables from growth kinetics. The classical Bayesian hi-erarchical modeling approach (Rubin, 1980), illustrated inthe right panel of Fig. 3, has long been applied to integrateavailable knowledge for model building (see engineeringapplications in Kennedy and O’Hagan (2001), Reese et al.(2004), and Bayarri et al. (2007)). The main contributionof this article is to provide a nanomanufacturing processmodeling methodology that addresses the issue of knowl-edge disparity at different scales through morphology-localvariability separation under a Bayesian hierarchical mod-eling framework. Similar work has not been reported in theliterature.

3. Hierarchical modeling of the NW growth process

The nanostructure growth is a dynamic process that is char-acterized by a space–time random field model in this work.The random field model has the hierarchical structure de-fined in Fig. 3.

3.1. Space–time random field representationof nanostructures

Let X represent the quality features of the NWs on a sub-strate. At a certain scale and time t, the observed featureat site si of the substrate can be denoted as X(si , t), si ∈

Fig. 4. Vertically aligned NWs with a mean length of 400 nm(after Chik et al., 2004).

Rp, p ∈ {1, 2, 3}, i = 1, 2, . . . , n. For simplicity of presen-

tation we hereafter only show the case with p = 1. As anexample, X(s1, t) could be 150 nm representing the aver-age length of all NWs at site A of the substrate shown inFig. 2(b). All the features at sites s = (s1, s2, . . . , sn)T arerepresented as X(s,t):

X(s, t) = [X(s1, t), X(s2, t), . . . , X(si , t), . . . , X(sn, t)]T,

(1)which is a space–time random field on lattice s. A realexample is shown in Fig. 4 (Chik et al., 2004) where theNW lengths vary from site to site and a random field offersa better characterization of the nanostructures. A similaridea has been developed for the stochastic modeling ofmicrostructures (Sobczyk and Kirkner, 2001). However, itshould be noted that that work mainly focuses on (offline)mathematical characterization of material structures. Wego beyond that by linking the mathematical model withprocess physics to set the foundation for in situ processcontrol.

3.2. Morphology–local decomposition of nanostructurerandom field

We propose to decompose the nanostructure random fieldX(s, t) into morphology or profile ηk−1(s, t), local variationφ(s), and noise ε:

X(s, t) = ηk−1(s, t) + φ(s) + ε, (2)

where ηk−1(s, t) could be a plane or a surface with a relativecomplex profile that evolves over the growth process. Thesubscript k − 1 represents the order of a polynomial, whichwill be introduced later in Equation (3). φ(s) represents lo-cal fluctuations on top of the morphology. We suggest an In-trinsic Gaussian Markov Random Field (IGMRF) modelfor φ(s). Figure 5 illustrates the idea in a two-dimensionalgraph for simplicity.

4 Huang

X(s)

s1 s2 sn... ...sj

X(sj)

(s)

(sj) +

NWs

Fig. 5. Morphology-local variability separation of the NW lengthfield.

The rationale of the morphology-local variability sepa-ration is as follows. Currently there is a good understandingof the global behavior of the growth kinetics. However, onlylimited physical knowledge is available on area-specific vari-ability because of latent and unobserved factors. The sepa-ration therefore aims at engaging growth kinetics throughηk−1(s, t) and local variation modeling through φ(s).

Due to the lack of physical studies, we only consider thecase where the local variability at site si is mainly deter-mined by its interaction with neighbors and remains stableover time. At the initial stage of modeling work, this time-invariant assumption simplifies the model structure in thehope that the dynamics are be captured by the morphologyηk−1(s, t). The model in Equation (2), however, could beextended to consider φ(s, t).

Remark 1: An alternative method of modeling the nano-structure is to use the kriging model in spatial statistics(Ripley, 1981; Cressie, 1993). X(si , t) would be expressed asX(si , t) = µ(si , t) + Z(si ), where µ(si , t) is a mean functionand Z(si ) is weak stationary Gaussian field (see a krigingexample in Joseph et al. (2008)). The kriging model alsocaptures the trend and local variability, and actually thereis a connection between Gaussian random field and GMRF(Rue and Held, 2005). We prefer a Markovian property forlocal variability component and a noise term ε in Equa-tion (2) for three reasons.

1. We intend to separate the modeling error and noise (seea general discussion in Kennedy and O’Hagan (2001)).The modeling error in Equation (2) dominates in φ(s)due to the lack of understanding at fine scales. However,molecular dynamics simulations (Rafii-Tabar, 2008) of-ten use, e.g., the Lennard–Jones potential (Lennard-Jones, 1924) to describe the interactions among NWsor nanotubes. The separation gives the opportunity tomodel the structure of φ(s) at the nanoscale.

2. The IGMRF has shown the flexibility to specify theneighboring structure on a lattice and has low compu-

tational requirements (Rue and Held, 2005). Materialproperties such as anisotropy are easier to model. It is aviable strategy to model the nanostructure growth pro-cess with effective integration of nanphysics.

3. In the hierarchical model estimation through MarkovChain Monte Carlo (MCMC) simulation, IGMRF suchas a conditional autoregressive model has shown bet-ter computational efficiency (Banerjee et al., 2004). Oursimulation studies also support the claim.

3.2.1. Modeling of nanowire morphologyThe morphology is devised to have order k − 1 polynomialrepresentation on s:

ηk−1(s, t) =

⎛⎜⎜⎜⎜⎝

η(s1, t)η(s2, t)

...η(sn, t)

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 s1 · · · 1(k − 1)!

sk−11

1 s2 · · · 1(k − 1)!

sk−12

...... · · · ...

1 sn · · · 1(k − 1)!

sk−1n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

β0(t)β1

...βk−1

⎞⎟⎟⎟⎟⎠

= Sk−1βk−1 (3)

where βk−1 = [β0(t), β1, . . . , βk−1]T are coefficients withspecific model structures. βk−1 is in the middle layer of thehierarchical framework given in Fig. 3. The detailed struc-ture of βk−1 is derived as follows to connect with processvariables θ.

The literature on growth kinetics does not report an in-tegrated model relating the NW length with time and tem-perature; see, for example, Ruth and Hirth (1964), Kikkawaet al. (2005) and Dubrovskii et al. (2006). We propose theintercept β0(t) to be

β0(t) = α1 exp[− Ea

kBT0− α2

t

]if β0(t) < Lf , or t < t0,

(4a)

β0(t) = α3 exp[− Ea

kBT0

]t + b if β0(t) ≥ Lf , or t ≥ t0.

(4b)

where t0 is the unknown transition point; Ea, kB, T0, andLf are the activation energy (kJ/mol), Boltzmann con-stant (J/K), average temperature (K), and diffusion length,respectively. Definitions of these quantities can be foundin Ruth and Hirth (1964), Kikkawa et al. (2005), and

Nanomanufacturing quality control 5

Dubrovskii et al. (2006). A continuity constraint is imposedat the transition point t0; that is,

α1 exp[− Ea

kBT0− α2

t0

]= α3 exp

[− Ea

kBT0

]t0 + b.

The unknown coefficients α1, α2, and α3 are adopted tocombine a set of physical constants such as degree of su-persaturation, atomic volume, diffusion coefficient, adatomconcentration, etc. Combining multiple unknowns into oneunknown means that much less data is required in the modelestimation. The drawback is that the model has to be esti-mated again if any of those physical constants are subject tochange. Since our ultimate goal is to control and monitorgrowth conditions, data under the same process conditionwill be collected anyway and the drawback is therefore nota concern.

The structure of the rest coefficients in βk−1 or[β1, . . . , βk−1]T are determined by properties of the NW’srandom field. Little work has reported in the literature onthe kinetics of NW morphology except for the overall trendor the intercept term β0(t). By the definition of ηk−1(s, t) inEquation (3), we can easily show that

η(si ) − η(s j ) =[

si − s j ,12

(s2

i − s2j

), . . . ,

1(k − 1)!(

sk−1i − sk−1

j

)](β1, . . . , βk−1)T. (5)

Equation (4) suggests that if the temperature has a space–time profile, i.e., being T(s, t) instead of a constant T0, thegrowth rate variability may lead to a complex morphol-

ogy. By defining q def= [si − s j ,12 (s2

i − s2j ), . . . , 1

(k−1)! (sk−1i −

sk−1j )]T, the following model is postulated for the structure

of [β1, . . . , βk−1]T:

η(si ) − η(s j ) = qT[β1, . . . , βk−1]T = α4 exp[− Ea

kBT(q, t)

],

(6)

where T(q, t) represents the temperature gradient betweentwo sites at time t. If the gradient does not vary with time,we can drop the time index in the model. Since temperatureis probably one of the key growth variables, here we chooseit to illustrate the modeling strategy. If other key processvariables are considered in Equation (6), those variablesmay modify the constant α4 by imposing a certain structure.To further consider the uncertainty of estimating α1–α4with limited data, we can introduce prior distributions forα1–α4 and update them with new observations.

3.2.2. Modeling of the NW’s area-specific variabilityAs defined in Equation (2), the area-specific variabil-ity φ(s) represents local fluctuations on top of themorphology. In addition, φ(s) is expected to capturethe dependence/interaction among neighboring sites.Based on Remark 1, we adopt an IGMRF modelwith rank deficiency k for local variability φ(s) =

[φ(s1), φ(s2), . . . , φ(si ), . . . , φ(sk−1)]T. It has density func-tion (Rue and Held, 2004):

f [φ(s)] = (2π)−n−k

2 |Q|∗ 12 exp

[12

[φ(s) − µ∗]TQ[φ(s) − µ∗]]

(7)

where Q is precision matrix with rank n − k and | ◦ |∗ de-notes the generalized determinant. Note that the rank de-ficiency k of precision matrix Q corresponds to an orderk − 1 polynomial in Equation (3). For an IGMRF of firstorder (k = 1), the conditional mean of φ(si ) is simply aweighted average of its neighbors, not involving an overallmean; that is (Rue and Held, 2004):

E[φ(si )|φ(−si )] = − 1Qii

∑j : j∼i

φ(s j ).

Notation j : j ∼ i denotes the sites si that are neighbors ofsi .

The unique property of an IGMRF of an order k isthat it is invariant to an order k − 1 polynomial (see de-tails in Rue and Held (2004)). That is another major reasonfor adopting an IGMRF in this study. This property fa-cilitates the separation of morphology from local variabil-ity, which gives the freedom to integrate physical laws intomorphology modeling without compromising the flexibil-ity of statistical modeling of area-specific variability anddependence.

Hence, the local variability φ(s) specified by Equation (7)will be mainly determined by the structure of the precisionmatrix Q. One key factor is the order of the IGMRF orthe rank deficiency of Q. A low-order k(≤ 5) is preferred inour model for two reasons. First, a large k would result in ahigh-order polynomial ηk−1, resulting in more fluctuationinto morphology. This may cause difficulties in distinguish-ing between morphology and local variability. Second, it ishard to properly connect a higher order morphology withprocess variables because it requres a better understand-ing of growth kinetics at finer scales. For similar levels ofthe goodness of fit, we tend to choose the model with thelower order. Hence, the morphology-local variability sepa-ration is a compromise between limited process knowledgeat fine scales and the desire to model and control localvariability.

It should be noted that the morphology-local variabilityseparation model can take a slightly different form fromEquation (2); for instance,

X(s, t) = µ(s, t) + ε, (8a)log[µ(s, t)] = log[ηk−1(s, t)] + φ(s). (8b)

The logarithmic transformation of the mean µ(s, t), andnot X(s, t), is based on the observation that the growthkinetics models usually take an exponential form (Equa-tions (4) and (6)). The terms ηk−1(s, t) and φ(s) fol-low the model structures defined in Equations (4) to (7).

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The transformation enables the morphology-local mode toaccommodate a wide range of growth conditions and pro-vides more flexibility in model fitting.

3.2.3. Nanostructure growth model estimationThe nanostructure growth model of Equations (2) to (7)integrates growth kinetics with statistical modeling of un-certainties through the Bayesian hierarchical frameworkshown in Fig. 3. The model inputs for estimation and pre-diction are from two sources; i.e., (limited) process/productdata and prior physical knowledge for the hyper-parameterλ (Fig. 3). The process/product data to be collected include,for instance, the growth temperature and its temporal–spatial distribution in the CVD process, carrying gas flowrate, NW length, and distribution on a substrate.

At fine scales, a number of issues make the model esti-mation process challenging. The first is a “missing data”problem. Since in situ nanoscale sensing technology is stillunder development (Howe et al., 2008), nanoscale mea-surements on the process or product during processing arescarce. Offline SEM or TEM inspection is extremely time-consuming and may only provide information on a fewtiny areas on a substrate at the time of decision making.Therefore, in situ process/product information and offlineproduct data at the nanoscale are either missing or incom-plete. Second, the number of unknown parameters is stillrelatively large even though we consolidated the unknownphysical constants (Equation (4)). Third, there is normallyno analytical solution for the hierarchical model set up byEquations (2) to (7). While maximum likelihood estima-tion is generally difficult to apply, the MCMC simulationapproach (Tanner and Wong, 1987; Liu, 2001) is fre-quently used to address the aforementioned issues in modelestimation.

Taking the NW growth as an example, let the NW lengthfield X = (Xobs, Xmiss), where Xobs is the observed NWlengths at some sites and Xmiss at the other sites is unob-servable due to resource/time constraints. If two or morereplicates are available, sites with missing measurement canvary from substrate to substrate. Given the in situ mea-surement of temperature T and prior distributions f0(θ)for process variables θ such as α1 ∼ α3 in Equation (4),the conditional distribution of θ under available observa-tion can be obtained. By drawing samples via the MCMCmethod, we can estimate θ and ξ in the hierarchical model(Fig. (3)). More details about MCMC can be found in Liu(2001). In the case studies we will implement Gibbs sam-pling using the WinBUGS software (Spiegelhalter et al.,2008) for model estimation.

4. Case studies

The case studies will demonstrate the procedure of hierar-chical modeling and estimation of the NW growth pro-

Fig. 6. Growth rate versus time, temperature (from Kikkawaet al. (2007)).

cess under uncertainties. Figure 6 shows how the NWlength varies with temperature and time (Kikkawa et al.,2005). The data therein were collected over time (t =15 s, 30 s, 180 s, 900 s) under six growth conditions(T = 365◦C, 380◦C, 400◦C, 420◦C, 430◦C, 440◦C). The firstfour conditions are used for model building. We do not con-sider the two high-temperature conditions because thereare no observations at times 180 s and 900 s. The growthin Kikkawa et al. (2004) may not last that long for highertemperature settings.

In the first example, we will focus on modeling andBayesian estimation of the NW growth process at the coars-est scale; i.e., the overall growth a cross the entire sub-strate. Compared to the existing NW kinetics models, wenot only consider the uncertainties but also estimate thetransition times under different process conditions withmissing values. In the second example, we will simulate lo-cal variation based on the data in Kikkawa et al. (2007).Since we were unable to identify any work that studiedthe growth variations over a substrate as a function oftemperature distribution, the second example only usesa first-order IGMRF model to describe the local varia-tions. It should be noted from Fig. 6 that the amount ofdata is quite limited compared to the number of modelparameters that need to be estimated. Results suggest thatthe MCMC method works very well in this missing valuescenario.

The three-level hierarchical modeling procedure is sum-marized in Fig. 7, which will be detailed in the twocases.

4.1. Case 1: dynamic NW length model at the coarsest scale

At the coarsest scale we treat the whole substrate as onesite. NW length X(s, t) = X(t) with s = 0. The hierarchical

Nanomanufacturing quality control 7

X(s,t) = (s,t)

Prior distributions for α1 ~ α4 , Ea, κ,

(s,t) = ηk-1(s,t) + ϕ(s)

ε

),(~ 2I0ε εσMN

111 --),(- kkk t =s

+= tTk

E

dt

d

B

a2

01

0 -exp ααβ

=− )(-exp],...,[ 411 ,tTk

E

B

aTk

T

qq αββ

IGMRF f[φ (s)] Eq.(7)

Eq.(3)

Eq.(4)

Eq.(6)

2εσIn situ measurement of

growth temperature T

Level 1

Level 2

Level 3

Fig. 7. The hierarchical modeling procedure.

model for this growth process can be set up as follows:

Level 1 : X(t) ∼ N(µ(t), σ 2ε ).

Level 2 : µ(t) = β0(t) = It<t0

[α1 exp

(− Ea

kBT− α2

t

)]

+It≥t0

[α3 exp

(− Ea

kBT

)t + b

]

Level 3 : Non-informative priors : σε ∼ uniform(0, 20),t0(T) ∼ uniform(1, 40), α2(T) ∼ uniform(1, 20),α1, α3 ∼ uniform(0, 1030).

where

b = α1 exp(

− Ea

kBT0− α2

t0

)− α3 exp

(− Ea

kBT0

)t0,

and

IA(t) ={

1 if t ∈ A,

0 otherwise.

Since model parameters such as diffusion length Lf , tran-sition time t0, and α3 defined in Equation (4) vary with tem-perature T, four prior distributions were assigned to eachparameter under four temperature conditions. For simplic-ity of presentation, we denote the parameter as a functionof temperature when assigning priors. The activation en-ergy Ea is 230 kJ/mol (Kikkawa et al., 2004). If needed,this parameter can be easily introduced as a prior distribu-tion under the current framework.

The computational issue for this model setup was thatthe MCMC procedure often crashed because it was sam-pling from such a wide range, particularly for the priors ofα1 and α3. Using a piece-wise nonlinear regression (Seberand Wild, 1989) we could obtain a rough estimate for α1and α3, which were of the order of 1019 and 1017, respec-tively. To avoid the problem of sampling large numbers, we

reparameterized µ(t) to create the following model:

Level 1 : X(t) ∼ N(µ(t), σ 2

ε

).

Level 2 : µ(t) = It<t0

[a1 exp

(−α2

t

)]

+ It≥t0

[a3t + a1 exp

(−α2

t0

)− a3t0

].

Level 3 : σε ∼ uniform(0, 20), t0(T) ∼ uniform(1, 40), α2(T) ∼ uniform(1, 20),a1 ∼ N(µa1, σ

2a1

), a3 ∼ N(µa3, σ2a3

).

log(µa1 ) = log(α1) − Ea

kBT, log(µa3 )

= log(α3) − Ea

kBT,

log(α1) ∼ N(0, 0.001), log(α3) ∼ N(0, 0.001),σa1, σa3 ∼ uniform(0, 10).

During WinBUGS simulation, we ran three Markovchains at the same time with initial values far apart fromeach other. The first 50 000 iterations were the burn-in pe-riod, while the following 50 000 iterations were used forcomputing posterior statistics. The computation time wasof the order of a few minutes for 50 000 iterations us-ing a 64-bit desktop workstation. Gelman-Rubin statisticsprovided by WinBUGS were around one and confirm theconvergency.

The numerical results are summarized in Table 1 and thefitted model based on posterior means is shown in Fig. 8.The model seems to fit the data well. The 95% confidenceinterval for the parameters is generally wide due to thelimited amount of data. The findings from the results areas follows.

1. As defined in Equation (4), α1 exp[−Ea/kBT0] changesthe scale of the curve, while α2 fine tunes the curve shape.This makes sense because a higher temperature may in-crease the growth level, but the growth dynamics that are

8 Huang

0 10 20 30 40 50 60

050

100

150

Time(s)

Leng

th(n

m)

Fitted Model

365oC380oC400oC420oC

0 200 400 600 800

020

040

060

080

010

00

Time(s)

Leng

th(n

m)

Fitted Model

365oC380oC400oC420oC

Fig. 8. Fitted model β̂0(t) for case 1.

reflected by the shape of length curve are expected tobe the same. As confirmed by the experimental data inFig. 6, the analysis to some degree justifies the proposedstructure for β0(t). The shape parameter α2 can be usefulto fine-tune the growth curve.

2. The most sensitive parameter is α1 or its intermediateparameter a1(T). It is correlated with α2 by the natureof the model structure. When more data are available,the introduction of a correlation structure between α1and α2 will be of use.

3. The parameter α3 has a wide interval. However, the slopeof the linear phase or a3(T) is more robust, as suggestedby its intervals. The main reason for this behavior isthat the linear structure requires relatively less data com-pared to the exponential phase. Therefore, from the datacollection point of view, more data are required duringthe exponential growth period when a high confidencefor α1 is needed.

4. This naturally brings up the issue of the exponential–linear transition point t0(T). The posterior estimates

Table 1. Bayesian estimates via MCMC for case 1

Parameter Mean Standard deviation Median 95% C.I.

α1 1.32 × 1019 6.8 × 1013 4.44 × 1019 [0, 2.72 × 1046]a1 (365◦C) 0.59 7.76 0.33 [−15.17, 16.62]a1 (380◦C) 1.63 7.85 1.08 [−14.19, 17.95]a1 (400◦C) 9.20 11.47 7.93 [−10.9, 32.28]a1 (420◦C) 16.49 15.59 17.53 [−9.10, 43.92]

α2 (365◦C) 15.76 8.31 15.88 [1.82, 29.29]α2 (380◦C) 15.65 8.34 15.70 [1.77, 29.29]α2 (400◦C) 10.89 8.72 8.09 [1.15, 28.73]α2 (420◦C) 7.34 8.60 2.30 [1.03, 28.24]

α3 2.83 × 1017 7.29 × 1013 2.74 × 1017 [0, 3.27 × 1044]a3 (365◦C) 0.11 0.02 0.11 [0.07, 0.16]a3 (380◦C) 0.15 0.02 0.15 [0.10, 0.19]a3 (400◦C) 0.51 0.02 0.51 [0.47, 0.56]a3 (420◦C) 1.25 0.03 1.25 [1.20, 1.30]

σε 19.81 0.19 19.86 [19.3, 19.99]t0 (365◦C) 20.14 11.29 19.96 [1.89, 39.01]t0 (380◦C) 19.52 11.24 18.99 [1.86, 38.86]t0 (400◦C) 10.20 8.15 7.88 [1.25, 32.37]t0 (420◦C) 4.79 3.19 4.07 [1.09, 12.56]

Nanomanufacturing quality control 9

Table 2. Bayesian estimates via MCMC for case 2

Parameter Mean Standard deviation Median 95% C.I.

α1 1.01 × 1019 5.62 × 1013 0.87 × 1019 [0, 8.20 × 1045]α2 (365◦C) 15.75 8.32 15.86 [1.78, 29.31]α2 (380◦C) 15.66 8.33 15.74 [1.79, 29.28]α2 (400◦C) 10.63 8.65 7.65 [1.15, 28.64]α2 (420◦C) 6.96 8.40 2.18 [1.03, 28.1]

α3 2.52 × 1017 4.69 × 1013 2.09 × 1017 [0, 2.20 × 1044]κ−2 12.71 4.15 12.83 [4.76, 19.56]t0 (365◦C) 20.21 11.29 20.09 [1.90, 38.99]t0 (380◦C) 19.51 11.26 18.97 [1.85, 38.85]t0 (400◦C) 10.57 8.36 8.25 [1.26, 33.32]t0 (420◦C) 5.04 3.31 4.34 [1.10, 13.11]σε 19.81 0.19 19.87 [19.31, 20.0]

of t0(T) are surprisingly good considering the fact thatthere are missing observations before the transition timefor T = 400◦C and 420◦C. The estimation is in agree-ment with the physics of the system in that the transitiontime decreases with temperature and the balancing pointis reached faster at higher energies. This agreement canbe attributed to the fact that the physical model was em-bedded into the model at the earliest opportunity. Theembedded “prior” knowledge to certain degree compen-sates for the lack of data.

4.2. Case 2: dynamic NW length model at a finer scale

Suppose at a finer scale that we are interested in investigat-ing both the overall growth and the final growth variabilityacross the substrate. For the case of presentation, we studythree equally divided grids aligned on the substrate. Whenthe morphology is relatively uniform, we can use the first-order IGMRF to approximate the local variability; i.e.,the conditional mean of φ(si ) is simply a weighted aver-age of its neighbors. X(s, t) = [X(1, t), X(2, t), X(3, t)]T,where s = [1, 2, 3]T. Using the morphology-local vari-ability seperation model or X(s, t) = η0(s, t) + φ(s) + ε,we have η0(s, t) = S0β0 = (1, 1, 1)Tβ0(t). The density andprecision matrix for φ(s) are (Rue and Held, 2005):

f [φ(s)] ∝ κ (n−3)/2 exp[−1

2φ(s)TQφ(s)

]

with Qi j = κ

⎧⎪⎨⎪⎩

ni , if i = j,−1, if i ∼ j,

0, otherwise,(9)

where ni represents the number of neighbors of site i , andi ∼ j means i and j are neighbors.

For each temperature setting, we simulated the three-gridIGMRF with κ = 1/50. The simulated data were added tothe growth data at t = 900 s in Kikkawa et al. (2004). The

hierarchical model in case 1 was extended as follows:

Level 1 : X(t, s) ∼ N(µ(t, s), σ 2ε ).

Level 2 : µ(t, s) = It<t0

[a1 exp

(−α2

t

)]

+ It≥t0

[a3t + a1 exp

(−α2

t0

)−a3t0

]+It=900sφ(s).

Level 3 : φ(s) ∼ CAR(κ), κ−2∼ uniform(0, 20)

Other non-informative priors are the same asthose in case 1

where CAR represents the conditional autoregressive rep-resentation of the IGMRF. The WinBUGS-defined func-tion car.normal for IGMRF was used in this MCMCsimulation.

The numerical results are given in Table 2. In general, theprecision of parameter estimation is slightly better (α1, α3,a3(T) omitted) or similar (α2(T), t0, σε, a1(T) omitted) incomparison with case 1. The posterior means of the modelparameters are very close to those of case 1. The estimatedκ̂−2 is larger that the true value. This can be attributed tothe small number of grids prescribed in the simulation. Alarger number of grids provides more data and confidenceto assess their correlations.

Clearly, the procedure is directly applicable to finer scaleswith measurement at each observation time; e.g., t = 15 s,30 s, 180 s, 900 s, if data are available. A more challengingissue is to correctly estimate the order of the local variabil-ity φ(s). Initial work on this problem has been presentedin Liu (2009) and interested readers are referred to thatpublication.

5. Summary

To date only limited work has been conducted to modelnanomanufacturing processes under uncertainties. This ar-ticle has developed a physics-driven process model and

10 Huang

modeling methodology to enable the prediction of the newgrowth process at each scale.

To take advantage of existing growth kinetics models,we conducted morphology-local variability seperation ofthe nanostructure field, in which the extended growth ki-netics model was incorporated to reflect the morphologyand local variability was represented by an IGMRF. Themorphology model describes the general time–space evo-lution of the nanostructure growth specified by key pro-cess variables. To take into account the large uncertaintydue to the limited number of measurements and physicalknowledge at fine scales, the morphology-local variabil-ity model was recast into a Bayesian hierarchical frame-work. Two case studies were provided to demonstrate themodeling and estimation procedure at both coarse and finescales.

This article reports initial attempts to address thechallenging issues in a new area with vast researchopportunities.

Acknowledgements

The work is supported by National Science Foundationunder grant CMMI-1002580. We appreciate the commentsfrom anonymous reviewers that helped to improve the qual-ity of the manuscript. S. Chen is thanked for providingFig. 2.

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Biographies

Qiang Huang is currently an Assistant Professor in the Daniel J. EpsteinDepartment of Industrial and Systems Engineering, University of South-ern California, Los Angeles. He was an Assistant Professor and then anAssociate Professor in the Department of Industrial and ManagementSystems Engineering at the University of South Florida from 2003 to

2009. Funded by the National Science Foundation, his research focuseson modeling and analysis of complex systems for quality and produc-tivity improvement, with a special interest in nanomanufacturing andnanoinformatics. He has served as an Associate Editor (Quality, Microand Nanomanufacturing Systems) for SME Journal of ManufacturingSystems since 2008. He is one of the editors for a special issue of theIIE Transactions on Quality and Reliability Engineering/Manufacturingand Design: “Quality, Sensing and Prognostics Issues in Nanomanufac-turing.” He is a member of scientific committee (Editorial Board) forthe North American Manufacturing Research Institution (NAMRI) ofSME, 2009–2011. He has been an Associate Editor (Automation in Meso,Micro and Nano-Scale) for 2009 and 2010 IEEE Conference on Automa-tion Science and Engineering. He is a member of IIE, INFORMS, SME,and ASME.