A Review of Statistical Methods for Quality Improvement and Control

A Review of Statistical Methodsfor Quality Improvement and

Control in Nanotechnology

JYE-CHYI LU

Georgia Institute of Technology, Atlanta, GA 30332-0205, USA

SHUEN-LIN JENG

National Cheng Kung University, Tainan 701, Taiwan

KAIBO WANG

Tsinghua University, Beijing 100084, P. R. China

Nanotechnology has received a considerable amount of attention from various fields and has become

a multidisciplinary subject, where several research ventures have taken place in recent years. This field is

expected to affect every sector of our economy and daily life in the near future. Besides advances in physics,

chemistry, biology, and other science-based technologies, the use of statistical methods has also helped the

rapid development of nanotechnology in terms of data collection, treatment-effect estimation, hypothesis

testing, and quality control. This paper reviews some instances where statistical methods have been used

in nanoscale applications. Topics include experimental design, uncertainty modeling, process optimization

and monitoring, and areas for future research efforts.

Key Words: Automatic Control; Experimental Design; Nanomanufacturing; Physical-Statistical Modeling;

Statistical Quality Control; Stochastic Modeling.

“N

ANOTECHNOLOGY is the understanding and con-trol of matter at dimensions of roughly 1 to

100 nanometers (nms, [a nanometer equals 10−9 me-ters]), where unique phenomena enable novel appli-cations. Encompassing nanoscale science, engineer-ing, and technology, nanotechnology involves imag-ing, measuring, modeling, and manipulating matterat this length scale. At the nanoscale, the physical,chemical, and biological properties of materials differin fundamental and valuable ways from the proper-

Dr. Lu is a Professor in the School of Industrial and Sys-

tems Engineering. His email address is [email protected].

Dr. Jeng is an Associate Professor in the Department of

Statistics. His email address is [email protected].

Dr. Wang is an Assistant Professor in the Department of

Industrial Engineering. His email address is kbwang@tsinghua.

edu.cn.

ties of individual atoms and molecules or bulk mat-ter. Nanotechnology R&D is directed toward under-standing and creating improved materials, devices,and systems that exploit these new properties.” (Na-tional Nanotechnology Initiative (2008)).

In the near future, it is expected that nanotechnol-ogy will impact every sector of our economy and dailylife. Roco (2004) characterized the development ofnanotechnology into four generations, each with dif-ferent featured products. The first generation, start-ing from about 2001, is characterized by passivenanostructures; the major focus of this generationis on nanostructured materials and tools for mea-surement and control of nanoscale processes. Popularexamples include nanoparticle synthesis and process-ing, nanocoating, various catalysis, etc. The secondgeneration is characterized by active nanostructures,according to Roco (2004). The research focus movesfrom nanostructured materials to novel devices and

Journal of Quality Technology 148 Vol. 41, No. 2, April 2009

STATISTICAL METHODS FOR QUALITY IMPROVEMENT AND CONTROL IN NANOTECHNOLOGY 149

FIGURE 1. Four Generations of Nanotechnology Appli-

cations.

device system architectures. Popular research topicsinclude nanobiosensors and nanodevices, nanoscaletools, nanoscale instrumentation, and nanomanufac-turing. Modeling and simulation of nanoprocesses arealso important topics in this generation. The thirdgeneration, featuring 3-D nanosystems and systemsof nanosystems, is expected to shift toward hetero-geneous nanostructures and supermolecular systemengineering. As an example, research on nanoscaleelectromechanical systems will be greatly promotedin this stage. Roco (2004) selected heterogeneousmolecular nanosystems as the feature products of thefourth generation.

Figure 1 presents the overlapping generations ofnanotechnology products and their respective man-ufacturing methods and research focus identified bythe International Risk Governance Council (IRGC)(see Roco (2004) for details). We will use this sum-mary to lead the discussion of where statistical meth-ods can contribute (or have contributed) in nanotech-nology development in the remainder of this paper.

Various government agencies, private corpora-tions, and venture capitalists have created programsto support R&D in nanotechnology. According toLux Research, the R&D investments in nanotech-nology worldwide amounted to $12.4 billion in 2006.The United States Department of Energy started ananotechnology program to develop new materialsfor improving fuel efficiency and providing efficientlighting. Likewise, the Department of Defense haslaid out a research plan to develop new approachesand processes for manufacturing novel, reliable, lowercost, higher performance, and more flexible elec-

tronic, magnetic, optical, and mechanical devices.Even the research topics from the National Institutesof Health include new nanomaterials for interfacingwith living tissues and molecular and cellular sensingfor gathering diagnostic data inside human bodieswhere traditional instruments cannot reach.

Many challenges confronting nanotechnology re-search call for solutions from a multidisciplinary ap-proach. Because statistical techniques have made siz-able impacts in many technology fields in the past,statistics are expected to play an important role intackling these challenges and boosting the develop-ment of nanotechnology. The purpose of this paperis to present examples of applications of statisticalmethods in nanotechnology research and to iden-tify possible new statistical approaches for solvingemerging problems in nanotechnology. Specifically,we highlight the following challenges that provide op-portunities for statistical applications:

First, statistical procedures help us learn moreabout the formation processes of nanocomposites. Re-cently, researchers have reported new progress in de-veloping nanocomposites. Forming processes dictateproperties of nanocomposites. These processes usu-ally involve complex chemical and mechanical reac-tions, where even minor changes of environmentalfactors or process settings may result in unexpectedoutcomes. Because the theory of the nanocomposite-forming process has not matured yet, many develop-ments rely on experimental studies. Due to the highcosts associated with experimental runs and sam-ple measurements and the complex relationship be-tween process outcomes and controllable/noise fac-tors, new statistical experimental design methods areneeded. Section 2 reviews recent work on applyingexperimental designs in nanocomposites applicationsand categorizes them based on the types of designsadopted in each study.

Second, statistical modeling helps deal with spe-cial data types and processes. Data collected fromprocesses for developing nanoscale materials some-times exhibit forms that are different from the usualpatterns seen in conventional manufacturing situa-tions. Such characteristics may include experimen-tal spaces with many isolated regions of low or zeroyields and high-frequency signals shown in spatial do-mains. Therefore, extensions of commonly used sta-tistical techniques are required to draw useful infor-mation from such types of data. Beyond these exten-sions, stochastic modeling of the forming processesfor characterizing multilevel or multiscale uncertain-

Vol. 41, No. 2, April 2009 www.asq.org

150 JYE-CHYI LU, SHUEN-LIN JENG, AND KAIBO WANG

ties is another valuable topic. Section 3 summarizesrecent work on data collection, statistical analysis,and uncertainty modeling in nano research and pro-vides examples for researchers that are facing similarproblems.

Third, statistics help improve low-quality andhigh-defect processes. Because nanosyntheses andnanofabrications are not well controlled thus far,defect rates of nanocomposites remain rather high.Mass and high-yield production is the key stepconfronting nanomanufacturing research. Statisticalquality control and productivity improvement tech-niques, which proved to be tremendously helpful intraditional manufacturing, are needed in nanomanu-facturing to enhance product quality and improveproduction efficiency. Extension of the traditionalquality monitoring and improvement tools to accom-modate the potential flood of sensor information isan interesting statistics research topic. Section 4 re-views the use of statistical process control (SPC)and automatic process control (APC) techniques innanoapplications. The feedback control section pro-vides examples of using stochastic partial differentialequations (SPDEs) to describe thin-film depositionprocesses. Based on these SPDEs and results in ki-netic Monte-Carlo simulations, multiscale automaticprocess controllers (APCs) are developed.

Last, statistics help improve low and unpredictableproduct reliability. Reliability models for nanocom-posites are not well developed. However, through in-vestigation of nanomaterials interfacing with envi-ronmental and stress factors, the reliability charac-teristics of nanodevices can be better understood andpredicted. Jeng et al. (2007) provide a comprehensivereview on recent reliability studies in nanoapplica-

tions. To keep this article brief, reliability issues willnot be discussed in this review.

The remainder of this paper is organized as fol-lows. Section 2 reviews successful applications of de-sign of experiments (DOEs) techniques in nanotech-nology research, which covers regular, nonregular,and robust parameter designs. Section 3 reviews var-ious data collection, statistical analysis, and mod-eling methods in the literature. Specifically, workson probability distribution and variation modelingof nanostructure characteristics, treatment of high-frequency signal and spatial data as well as stochasticmodeling techniques are presented. Section 4 reviewsrecent development of SPC and quality-control tech-niques. Examples of process monitoring and feedbackcontrol, especially multiscale modeling and controltechniques, are illustrated in this section. Section 5concludes this review with suggestions for future re-search on quality improvement and control in nan-otechnology.

Design of Experiments

Due to the applications in nanoelectronics, pho-tonics, data storage, and sensing, synthesizing nanos-tructures is a research topic of foremost importancein nanotechnology (Dasgupta et al. (2008)). How-ever, the synthesis process is extremely sensitive tocontrol settings and environmental noises. For ex-ample, to generate the nanostructures of nanosaws,nanowires, and nanobelts shown in Figure 2, Das-gupta et al. (2008) shows that the control factors,such as temperature and pressure, have a heavy im-pact on the final output of a synthesis process. More-over, situations occur in which multiple responsesare studied with functional relationships containing

FIGURE 2. Nanosaws, Nanowires, and Nanobelts (from Dasgupta et al. (2008)).

Journal of Quality Technology Vol. 41, No. 2, April 2009


many potential factors of interest. In order to ro-bustly optimize several properties of nanoproducts,the impact of each factor, plus their interactions, onprocess outputs should be studied using advancedstatistical techniques. In the literature, DOE hasbeen employed as the major tool for exploring therelationship between controllable/noise factors andprocess responses. Regular designs, including full fac-torial and fractional factorial designs, response sur-face methodologies, and nonregular designs (e.g., D-optimal designs) have been successfully utilized tooptimize synthesis processes and investigate materialproperties. This section reviews the use of DOE tech-niques and robust parameter designs in nanotechnol-ogy and gives examples of how these techniques canbe utilized in the nanomanufacturing processes. Thisreview covers studies in the following journals: AAPSPharmSciTech; Carbon; Chemical Engineering Jour-nal, Colloids and Surfaces A: Physicochemical En-gineering Aspects; International Journal of Pharma-ceutics; Journal of Physical Chemistry B; Journal ofthe American Statistical Association; Materials andDesign; Microelectronics Reliability; Nanotechnology;and Powder Technology.

Regular Designs

Basumallick et al. (2003) investigated the synthe-sis processes for Ni–SiO2 and Co–SiO2 nanocompos-ites, of which the physical properties are very sen-sitive to process parameters. This attribute makesthe response surface very rugged. Because process-ing conditions have significant impact on the proper-ties of nanocomposites, the authors considered threefactors with three levels each in their experimenta-tion. Instead of using a conventional three-level frac-tional factorial design, the authors designed eightruns using a two-level full factorial design and addedthree runs setting all factors at the middle level. Us-ing regression equations fitted by the 11-run exper-iment outcomes, the fractional conversion values ofthe nanocomposites were modeled as a function ofheating rate, composite concentration, and the start-ing temperature of the reaction. Becaise the responsesurface is very rugged, in order to improve the datacollection and analysis methods used in this paper,we recommend the experimental designs (and theirextensions) commonly used in computer experiments(e.g., Dasgupta et al. (2008)) for efficiently collectingcostly data. Moreover, robust process-optimizationideas (e.g., Taguchi (1986)) could be used so thatthe physical properties of the nanocomposites are lesssensitive to the noise factors.

Lin et al. (2003) studied the surface and grainstructure of silver-plated film on a copper lead frame.The surface roughness was measured by atomic forcemicroscopy (AFM), while the surface thickness wasobtained by nanoindentation measurement using aUMIS-2000 nanoindenter. Additionally, the grainstructure was measured by a transmission electronmicroscope (TEM). The impact of the silver-platedfilm-surface characteristics and grain structures tothe quality of wedge bonding between gold wire andsilver-plated lead frame were investigated througha 24 factorial design. Four design parameters werebond time, bond force, electrical current, and the twolead frame types. The experimental results showedthat the surface characteristics and grain structureshave a great impact on the quality of bonding.

To systematically obtain a model of factors lead-ing to an optimum response, many researchers usethe response surface methodology (RSM) for datacollection and process optimization. RSM should alsobe considered when complex models are needed forcharacterizing a nanosynthesis or nanomanufactur-ing process. Prakobvaitayakit and Nimmannit (2003)presented a process to prepare polylactic-co-glycolicacid (PLGA) nanoparticles by interfacial depositionfollowing the solvent-displacement technique. Theyinvestigated the process through a 23 factorial de-sign experiment. Multiple responses—particle size,amount of encapsulated material, and encapsulationefficiency—were considered in the experiments. Tooptimize multiple responses, the authors used a si-multaneous optimization technique with a desirabil-ity function. The desirability function converts eachresponse into an individual function that varies overthe range [0, 1] and takes the value one when the re-sponse is at its target value and less than one if not(see page 425 in Montgomery (2005) for details). Bydefining the overall desirability as the product of allindividual functions, optimizing multiple responsesis achieved by maximizing the overall desirability.With this method, the experimental design helpedchoose the optimal formulation ingredients for thenanoparticles. Now, the composites formed by PLGAnanoparticles are being commercially used for drug-delivery systems.

Barglik-Chory et al. (2004) used a second-orderresponse surface to study the main effects and in-teractions of three controllable factors in synthesiz-ing a type of semiconductor nanoparticles, colloidalCdS. The quantum effects in nanoparticles provideddiscrete energy levels, and the semiconductor bandgap exhibited strong size dependence. Through con-



trolled experiments, the influence of environmentalfactors on nanoparticles could be investigated andquantified, potentially leading to an improvement inthe nanofabrication efficiency.

After identifying the significant parameters byusing Taguchi’s parameter design method (Taguchi(1986)), Hou et al. (2007) used RSM to build a rela-tionship between five process parameters and averagegrain size of nanoparticles. They found that the rela-tionship between process parameters and grain sizecan be modeled with a second-order equation. Usingthe integrated genetic algorithm and RSM approach,the optimal settings of these five parameters in thenanoparticle milling process were determined.

Yordem et al. (2008) used RSM to investigateeffects of material choices and process parame-ters on the diameter of electrospun polyacrylonitrilenanofibers. Three explanatory factors—voltage, solu-tion concentration, and collector distance—and tworesponse variables—fiber diameter and coefficient ofvariation—were studied. At each level of the collectordistance, the other two factors were varied and theresultant fiber diameter was recorded. For the pur-pose of predicting fiber diameter and coefficient ofvariation, polynomial regression models were fittedto the experimental data at each level of the collec-tor distance. Interactions among the factors were alsoidentified. The authors suggested a narrower windowof factor space for future nanofiber production.

Nazzal et al. (2002) used a three-level Box–Behnken design to evaluate the effect of formula-tion ingredients on the release rate of ubiquinonefrom its adsorbing solid compact and to obtain anoptimized self-nanoemulsified tablet formulation. Tostudy the effects of three independent variables onsix responses, 15 runs of experiments were con-ducted. The formulation ingredients—copolyvidone,maltodextrin, and microcrystalline cellulose—wereshown to have a significant effect on the emulsionrelease rate. However, this article did not show howto allocate these ingredients to optimize the multipleobjectives.

Using a radio-frequency plasma reactor, Cota-Sanchez et al. (2005) used a 24 factorial designto study fullerenes synthesis. The response variablewas C60 yield and the four operating parameterswere reactor pressure, raw-material feed rate, car-bon black–catalyst ratio, and generator-plate power.Using ultraviolet spectrometric analysis to measure

yield, they found that the significant factors thataffect the C60 synthesis are the reactor pressure,plate power, and feed rate. Under the optimal op-erating condition, the yields were synthesized up toabout 7.7 wt.%; moreover, nanotubes were success-fully produced. Further optimization of other param-eters, such as the particle injection and quenchingconditions, through the use of RSM might lead toenhanced yields of these carbon nanostructures.

In order to study the flexural properties and dis-persions of a ceramic material, SiC, Yong and Hahn(2005) employed a two-level full-factorial design toinvestigate interactions between coupling agent anddispersant. A central composite design was usedto determine the optimal dosages of chosen fac-tors to achieve the maximum flexural strength andmaximum particle dispersion. Experimental resultsshowed that two objectives could be optimized si-multaneously at the same factor levels. When an op-timal dosage is employed, the nanoparticle reinforce-ment can enhance the mechanical properties of thecomposites.

Compared with factorial designs, split-plot de-signs are suitable for situations in which physicalrestrictions on a process exist and certain combina-tions of factor levels are difficult to reach. Nembhardet al. (2006) presented a split-plot design for inves-tigating nanoscale milling of submicron channels ona gold layer. Similar to a synthesis process that in-volves complex chemical and mechanical reactions,such a milling process was sensitive to various con-trollable and noise factors. By treating the experi-ment as a two-stage process, whole-plot and subplotfactors were identified. Results showed that split-plotexperiments in nanomanufacturing reduce labor andcosts and were often effective at detecting the effectsof subplot factors.

In order to achieve high yield and reproducibilityof a synthesis process, Dasgupta et al. (2008) con-ducted a full-factorial experiment. Two parameters,temperature and pressure, with five and nine levels,respectively, were varied during the experiment andthree response variables, the numbers of nanosaws,nanowires, and nanobelts, were recorded. Becausethe total number of the nanostructures was a con-stant, the authors proposed simultaneously modelingthe probability of generating different nanostructuresusing a multinomial generalized linear model (GLM).The optimal settings obtained from this research ledto a significant improvement in the process yield.



Nonregular Designs

When a regular factorial design is not feasibledue to possible limitations in experimental run-size and factor-level selections, some nonregular de-signs may be considered for nano process modelingand optimization. Among others, successful applica-tions of D-optimal designs in nanotechnology havebeen demonstrated by several researchers. Comparedwith the regular designs, D-optimal designs may usenonorthogonal design matrices to reduce experimen-tal runs and minimize the variances of coefficientsassociated with a specific model setting.

Fasulo et al. (2004) studied the extrusion pro-cessing of thermoplastic olefin (TPO) nanocompos-ites. Properties of TPO composites are influencedby the forming process. The authors chose differentcombinations of processing factors, including melttemperature, feed rate, and extruder screw-rotationspeed, and investigated both surface appearance andphysical/mechanical properties of the nanocompos-ites. A D-optimal design was adopted in the researchto characterize the relationship between the qualitymeasures and explanatory factors. Useful suggestionsfor optimizing process output were obtained.

Dasgupta (2007) developed a sequential minimumenergy design (SMED) to deal with complex responsesurface with multiple optima for the yield of nanoma-terial synthesis by using fewer design points. Figure3 illustrates the response surface of the yield and theselected design points. Compared with traditionaldesigns, the SMED design can probe high-yield re-gions and avoid nonyield points more effectively.

Robust Parameter Design

One major challenge confronting nanosynthe-sis/manufacturing processes is the high variation inexperimental results. Most synthesis processes arevery sensitive to environmental or noise factors.Therefore, robust parameter design (e.g., Taguchi(1986)) has been considered by several researchersto reduce experimental variation and enhance pro-cess yield and production efficiency. Figure 4 illus-trates one of the key ideas of the robust parameterdesign. The level setting of control factors will bedifferent by including the noise factors in the exper-iment when there are interactions between controland noise factors.

Nanoparticles have been widely utilized in manyindustrial applications, such as carbon nanotube,nanoceramics, and nanocompound materials. The

wet-type milling machine is a recently developed toolto produce nanoparticles and to avoid aggregationeffect. Because of its simplicity and applicability toall classes of materials, this machine is becomingpopular. Hou et al. (2007) applied the robust pa-rameter design method to optimize a nanoparticlemilling process. They considered the following fiveprocess factors, each with three levels, to improvethe nanoparticle milling process: the milling time,flow velocity of circulation systems, rotation velocityof agitator shaft, solute-to-solvent weight ratio, andfilling ratio of grinding media. The response variablewas the nanoparticle grain size, which is measuredby the Coulter multisizer equipment. To save experi-mental cost, the L27 orthogonal array was used. Theexperimental results showed that all five process fac-tors significantly affect the grain size.

Kim et al. (2004) implemented the robust designmethod with an L9 orthogonal array to optimize therecipe for preparing nanosize silver particles. The sil-ver nanoparticles have been widely used in chemicaland medical industry due to their unique propertiesof conductivity and resistance to oxidation. The ob-jective was to determine the experimental conditionswhere the size of nanoparticle is small and has lessvariability. The following three factors were consid-ered in the experiment: molar concentration ratio,dispersant concentration, and feed rate. The responsevariables are the average size and the size distribu-tion of silver nanoparticles. The concentration of dis-persant was identified as the most influencing factoron the average particle size and the size distribution.Using the derived optimal conditions, silver nanopar-ticles can now be prepared with small size varianceusing the derived optimal condition.

In another paper about the synthesis of nanopar-ticles, Kim et al. (2005) used Taguchi’s method tooptimize a new microemulsion method for prepar-ing TiO2 nanoparticles. An L8 orthogonal array wasused as the design of experiment for the five factors:H2O surfactant value, H2O/TEOT value, ammoniaconcentration, feed rate, and reaction temperature.The derived optimal condition led to the least sizevariability in nanoparticles.

Data Collection, Statistical Analysis,and Physical–Chemical–Statistical

Modeling

Because measurements with complicated datapatterns are frequently seen in experiments in nan-otechnology, the collection, analysis, and modeling



FIGURE 3. Response Surface of the Yield of a Nanostructure and the Selected Design Points (from Dasgupta (2007)).

of data in nanoengineering are vital topics. This sec-tion focuses on high-frequency and/or spatial datasignals seen in nanotechnology studies. In addition,probability and stochastic models of nanoscale mea-surements and process variations are reviewed. Theprimary journals reviewed here include: Combus-tion Theory and Modeling, IEEE Transactions onNanotechnology, IEEE Transactions on Reliability,IEEE Transactions on Semiconductor Manufactur-ing, IEEE Transactions on Very Large Scale Integra-

tion (VLSI) Systems, IEICE Transactions on Elec-tronics, International Journal of Engineering Sci-ence, International Journal of Plasticity, Journal ofthe Mechanics and Physics of Solids, Physical Re-view Letters, Solid-State Electronics, and Surface &Coatings Technology.

Sampling Plans

Effective sampling plans are critical to model thenanofabrication processes using fewer data. The fol-



FIGURE 4. One of the Key Ideas of the Robust Parameter

Design.

lowing sampling techniques can be used to select rep-resentative data for analysis: simple random sam-pling, stratified sampling, cluster sampling, system-atic sampling, importance sampling, and their hy-brids. However, new sampling techniques have beendeveloped for the nanofabrication process.

Zhao et al. (2004) developed a new double sam-pling technique to test interconnects and buses invery large scale integration (VLSI) circuits. Due tothe rapid size scaling and high-speed operation of in-tegrated circuits (ICs) using nanometer technologies,electromagnetic noise sources and their effects on in-terconnects have become extremely significant. Thebasic idea for this sampling technique was to sam-ple test data for loading into two flip-flops at a fixedtime interval and check whether the resultant datastreams were consistent with each other. Thus, in-consistent and noisy signals with spikes could be cap-tured. This technique was capable of detecting errorscaused by electromagnetic noise effects in nanofabri-cation.

Chang et al. (2005) proposed a new method todetect the read failure in a static random accessmemory (SRAM) cell using a critical-point samplingtechnique. The idea was similar to the importancesampling. In sub–100-nm technologies, the analyticalmodel previously used fails to accurately match real-istic simulation results due to various short channeleffects and different leakage components. It is prefer-able to employ a full-scale transient Monte Carlo(MC) simulation; however, using the MC methodusually takes a large number of iterative runs to ob-tain an accurate read failure probability. Thus, ratherthan deriving the entire voltage-transfer characteris-

FIGURE 5. Idea Illustration of Importance Sampling.

tic curve, the authors measured the SRAM cell sta-bility at certain representative points on the curvefor a specified voltage value. The experimental resultshowed that their model achieves high accuracy andwas 20 times faster in computational speed.

Proper use of importance sampling will providea more accurate inference or save the sampling costwith smaller sample size. This is a very helpful sta-tistical technique for collecting expensive samples inmany nanofabrication processes. Figure 5 illustratesthe general idea. Suppose the target of the inferenceis the mean of a statistic m(X), i.e. E(m(X)), wherem(x) only depends on the sample values x greaterthan a constant c. Then a proper choice of a newdensity h(x) of X will make the sample mean ofm(X)f(X)/h(X) an unbiased estimator of the tar-get with smaller variance. The main character of thenew density h(x) is to produce more samples on thearea (greater than c) that affects the values of m(x).See page 87 in Davison (2003) for more details ofimportance sampling.

Probability Distribution and VariationModeling

As technology shrinks to the nanoscale region,probability distributions for material and structureperformance may change to different forms. More-over, variability in device performance becomes amajor issue in the circuit-design stage. The followingparagraphs review several papers studying probabil-ity distributions of nanoparticle measurements andprocess variations in IC manufacturing.

In ordnance technologies, the reproducibility of a



thermite’s burning behavior is critical for both safetyand performance evaluation. As a particle’s diameterapproaches the nanoscale, the burn-rate calculationbecomes increasingly sensitive to variations in theparticle diameter. In the study by Granier and Pan-toya (2004), the burn-rate estimates for nanoscalethermites were statistically evaluated with a proba-bility density function (pdf) of the particle-size dis-tribution and a diameter-dependent burn-rate equa-tion. Based on a series of scanning electron mi-croscopy (SEM) images, a model of mass fractal ag-gregates was used to interpret the scattering data.A volume-weighted particle-size distribution was ob-tained. Both single mode and bimodal particle-sizedistributions were studied using Gaussian and amixture of Gaussian distributions, respectively. Theanalysis showed that, as the particle size reduced tothe nanoscale, the size distribution, rather than theaverage particle size alone, became increasingly im-portant. Large variability in the burn rate was as-sociated with a large standard deviation in particlesizes.

Bazant and Pang (2007) studied the pdf ofstrength of nanostructures based on a nanoscaleatomic lattice. To predict a failure event with an ex-tremely low probability, the cumulative distributionfunction (cdf) of strength of quasi-brittle structureswas modeled as a chain of representative volume el-ements (RVE). Each of the RVEs was statisticallycharacterized by a hierarchical model consisting ofbundles (via a parallel coupling) of two long sub-chains. Each of the subchains consisted of subbundlesof two or three long sub-subchains of sub-subbundlesand so on. Eventually, the nanoscale atomic latticewas reached. They gave physical reasons that thefailure of interatomic bonds follows a thermally ac-tivated process governed by stress-dependent activa-tion energy barriers. The tail of the cdf of the RVEstrength followed a power-law model. Finally, theyconcluded that the distribution of strength of thequasi-brittle nanostructure is a Weibull.

The physical process of surface formation innanocomposites has several stochastic components.Chen et al. (2005) studied barrier effects impactingsurface formation using oxidized porous silicon. Ac-cording to their experimental results, a Gaussian dis-tribution fit the data of oxidized porous silicon sam-ples well. The barrier height turned out to have aGaussion distribution, and the photon energy wasshown to be a function of the barrier heights.

Hydrogenated amorphous silicon (a-Si:H), the

prototypical disordered semiconductor, is an impor-tant material for use in nanoscale optoelectronic ap-plications including sensors, flat-panel displays, 2Dmedical imaging, and photovoltaics. Belich et al.(2003) reported that the noise distribution in a-Si:Hwas non-Gaussian in nature, a surprising feature inmacroscopic samples at and above room tempera-ture. Traditionally, the noise arose from an ensembleof statistically independent fluctuators. This led toGaussian-distributed noises. One possible explana-tion of the non-Gaussian noises was due to (nonlin-ear) interactions between fluctuators.

The stored charge in each logical node of a VLSIcircuit decreases with decreased dimensions and de-creased voltages in nanotechnology. Weak radiationcan cause disturbance in the circuit signals, lead-ing to increased soft-error failures. The fault/error-detection probability can serve as a measure ofsoft-error failure, which is an increased threat inthe nanodomain logic node. In order to model thefault/error-detection probability, Rejimon and Bah-nja (2005) presented a Bayesian network for error-sensitivity analysis in VLSI circuits. Their procedureprovided a framework to obtain the joint pdfs of allvariables in the network for calculating the error-detection probability.

As the technology node goes down to 90 nm andbelow, variability in device performance becomes acrucial problem in the design of ICs. In the past,die-to-die variability, which was well managed by theworst-case design technique, dominated the within-die variability. Onodera (2006) pointed out that thestatistical nature of the variability has been changedsuch that the within-die variability is growing. Thischaracteristic presents a challenge in circuit-designmethodology. His paper provided measurable resultsof variability in 0.35-, 0.18-, and 0.13-μm processesand explained the trend of variability. It also showedthat a circuit that was designed optimally underthe assumption of deterministic delay is now mostsusceptible to random fluctuation. This result indi-cates the need of applying statistical thinking in thecircuit-design methodology.

For solving process-variation problems in nano-scale-IC manufacturing, Li et al. (2005) pro-posed a novel projection-based extraction approach,PROBE, to efficiently create quadratic response sur-face models and capture both interdie and intradievariations with affordable computation cost. Insteadof fitting a full-rank quadratic model, PROBE ap-plied a projection operator and found an optimal



low-rank model by minimizing approximation er-rors, which were defined by the Frobenius norm. InPROBE, the modeling accuracy and parsimony canbe tuned by increasing or decreasing the dimensionof the projection space. Several examples from digi-tal and analog circuit-modeling applications demon-strated that PROBE can generate accurate responsesurface models while achieving up to 12 times fasterspeed as compared with traditional process-variationsimulation and modeling methods.

Stochastic Modeling

Due to the fact that a large proportion of vari-ability in nano synthesis/growth processes are notwell explained by known physical models, stochasticmodeling techniques have served as an effective wayin characterizing such processes.

Miranda and Jimenez (2004) proposed a stochas-tic logistic model based on a Wiener process for char-acterizing the breakdown dynamics of ultrathin gate

oxides (at 2-nm level) and for understanding the ef-fect of voltage stress on the leakage current. Themodel had two components: a deterministic termwith a logistic function describing the mean failurebehavior and a random term with a Wiener processrepresenting uncertainties and noises. Throughoutthe model, the nano-material–degradation dynamicswere captured using a small set of parameters.

To characterize degradation of leakage currents of3-nm gate oxides, Hsieh and Jeng (2007) considered anonhomogeneous Weibull compound Poisson modelwith accelerated stresses. The oxides were in squaremetal-oxide semiconductor (MOS) capacitors grownon p-Si. One hundred twenty capacitors were irra-diated at three levels of ion density, and the leak-age current versus gate voltage was measured beforeand after each irradiation step. Figure 6 shows thesketch of the leakage current of the gate oxides atthree ion levels and two accelerated voltages. Theyprovided maximum-likelihood estimates of model pa-rameters and derived the breakdown-time distribu-

FIGURE 6. Leakage Currents of 3-nm Gate Oxides Under Accelerated Stresses (Voltages; from Hsieh and Jeng (2007)).



tion. To check the proposed models for the degra-dation measurements and the rate of breakdown-event occurrence, goodness-of-fit tests were consid-ered. The estimated nanodevice reliabilities were cal-culated at lower stress conditions.

Time series of nanosystem measurements exhibitintermittency. At random times, the system switchesfrom state on (or up) to state off (or down) and viceversa. Margolin and Barkai (2005) investigated thenonergodic properties of blinking nanocrystals mod-eled by a Levy-walk stochastic process. The processwas characterized based on the sequence of on andoff sojourn times. The times of the on–off events weremutually independent and were drawn at randomfrom a pdf that followed a power-law distributiongenerated by a fractional Poisson process.

It has been widely recognized that novel nanoelec-tronic devices, such as carbon nanotubes and molec-ular switches, have high manufacturing defects dueto the stochastic nature of the bottom-up physicaland chemical self-assembly processes. Qi et al. (2005)reported that it was very challenging to manufac-ture nanocomputers with a device density of 1012

chips due to faulty components pervasive in the de-vice. The authors studied the behavior of a NAND-multiplexing system with a Markov chain of distribu-tions that could be unimodal or bimodal, dependingon whether the probability of NAND gates was largeror smaller than a threshold value.

High-Frequency Signal and Spatial DataAnalysis

Due to the use of advanced data-collection equip-ment, spatial data and/or high-frequency signals arecollected in many nano applications. Because suchdata signals contain rich information about processstatus or product quality, effective (and efficient)statistical analysis methods are greatly importantfor extracting knowledge. The following paragraphspresent a few examples.

Spatial statistical models have been used in mod-eling nanoscale structures. Chen and Lee (2004) usedlattice points to represent atomic bonding units ina crystal. The structure of the unit together withthe network of lattice points determined the crystalstructure and the physical properties of the nano-material. In their experiments, polycrystalline solidsconsist of randomly distributed grains and grainboundaries. The size of grains was usually in thenano/microscale. Each grain was modeled as crystal-lized solid by micromorphic theory, while the grain

boundaries were modeled by classical continuum the-ory. Within each grain, the atomic motion led to thecontinuum lattice deformation from nanoscale to mi-croscale. A multiscale spatial model was then used tocharacterize the material behavior of polycrystallinesolids.

In order to understand the macroscopic prop-erties of heterogeneous materials, modeling spatialcorrelations for microstructures is needed. Jeffersonet al. (2005) analyzed microstructures of polymernanocomposites and discovered that the materialmeasurements did not exhibit randomness from a sin-gle homogeneous distribution. The traditional empir-ical model became inappropriate. A mixture of twodistributions that defined the transition probabilitiesbetween two phases of the composites was introducedas a tool to examine the randomness and periodicityin the microstructure.

For manufacturing high-performance films witha thickness of only a few nanometers, nondestruc-tive testing methods are required to ensure pro-duction quality. Schneider et al. (2002) developeda laser-acoustic technique based on surface acous-tic waves for testing nanometer film’s hard-coatingquality. A nitrogen-pulse laser was used to generate awide-band surface acoustic wave. When propagatingthrough the nanofilm surface, the wave signals wererecorded using a digitizing oscilloscope. Applying theFourier transformation to the high-frequency laser-acoustic signals, the authors obtained empirical esti-mates of thickness and hardness of the films. Becausethe recent popular tool for modeling high-frequencysignals is wavelets, it might be interesting to see theapplicability of wavelets and the results they lead toin analyzing laser-acoustic signals. See papers in thesubsection of process monitoring of functional datafor examples.

Statistical Process Control (SPC)and Quality Control

As a traditionally popular technique in manufac-turing, SPC is expected to serve the role of identify-ing assignable causes and thus reducing variations innano-manufacturing processes. However, in this earlystage of research in nanotechnology, nano-productionsystems have not matured yet. Hence, publicationson SPC application to nano-technology research arescarce. This section highlights two important issuesappearing in nano technology: functional data moni-toring and feedback control. Although most of thefeedback control studies deal with control-theory,



i.e., they are not statistically oriented, this reviewdoes dig out a few sensor-data–based control studies.In addition, several multiscale modeling and controlpublications illustrate the potential of future statis-tics research in the important nanotechnology de-velopment areas (see the description in the secondgeneration of Figure 1). This review covers the lat-est progress on nano-SPC published in the follow-ing journals: Computers & Chemical Engineering,Control Engineering Practice, IEEE Transactions onPlasma Science, IEEE Transactions on Semiconduc-tor Manufacturing, Journal of Process Control, Jour-nal of Quality Technology, Nanotechnology, Surface& Coatings Technology, and Wear.

Process Monitoring of Functional Data

Functional data characterize quality or reliabilityperformance of many manufacturing processes. Theyare very informative in process monitoring and con-trolling for nanomachining, ultrathin semiconductorfabrication, and several other manufacturing pro-cesses seen in the literature. In particular, waveletanalysis has been popular in modeling and monitor-ing functional data.

Ganesan et al. (2003) combined online detectionwith offline modeling strategies in the nanodevicewafer-polishing process. Through wavelet-based mul-tiresolution analysis methods, the delamination de-fects were identified by analyzing the nonstationarysensor signals based on acoustic emission and co-efficient of friction. Jiang and Blunt (2004) intro-duced a wavelet model to extract linear and curve-like features from complicated nonstationary surface-texture patterns in nanometer morphological struc-tures. Through wavelet decompositions and recon-structions, Das et al. (2005) removed noises from thecoefficient of friction signals in their chemical me-chanical polishing of nanodevices and used a sequen-tial probability ratio test to set up a control chart formonitoring process conditions.

Feedback Control

Sensor-Data–Based Control

Due to the ultrasmall size of objects and very finemanufacturing processes, using data collected fromvarious sensors for developing precise automatic con-trol rules is critical in nanotechnology studies. For in-stance, Ohshima (2003) pointed out that most tradi-tional feedback controllers using existing sensors formicro and larger systems were not effective for deal-ing with nanomachines, where physical and chem-

ical phenomena occur in very short time windows.The author considered in-situ feedback schemes tocontrol physical–chemical reactions in their materialprocessing courses. Klepper et al. (2005) developedan in-situ feedback–control tool to improve the re-producibility of a nanostructure deposition process.In order to reduce variability in the coating pro-cess, the investigators needed to find ways to measureprocess variations. Experiments were set up to testthe assumption that plasma discharge was sensitiveto the surface-roughness variation. The experimentsresulted in the identification of the correlation be-tween hydrogen atomic emission and formation ofmetal-carbide coatings. Through the control of theemission, variability in the nanostructure composi-tion and the wear performance of the coating wascontrolled by a closed-loop feedback algorithm.

Lin et al. (2003) studied precise positioning issuesin a nanoscale drive system. To guarantee the desiredprecision, the authors used an integral type of con-troller in the positioning. The study demonstratedthe possibility of achieving a very precise position-ing at the resolution level of sensor measurements.Lantz et al. (2005) introduced a novel micromachinedsilicon-displacement sensor with displacement resolu-tion of less than 1 nm for providing accurate position-ing information. This new technique can detect thedisplacement by measuring the difference betweenthe resistances of the two sensors.

Multiscale Modeling and Control

A few factors, such as the following, motivate thecurrent active research on control of multiscale pro-cesses (see the preface of the special issue on controlof multiscale and distributed process systems in 2005,volume 29, Computers and Chemical Engineering fordetails).

(a) key technological needs in semiconductor man-ufacturing, microtechnology, nano technology,and biotechnology; and

(b) recent developments in actuator and sensortechnology that make control of material mi-crostructures, spatial profiles, and product-sizedistributions feasible and practical.

Using the thin-film growth process as an example,Christofides and Armaou (2006) provided a review ofrecently developed methods for controlling multiscaleprocesses. The typical thin-film growth process hasbeen widely used in microelectronic devices, nano-materials, and nanocomposites (see, e.g., Ng et al.(2003), Sneh et al. (2002), Mae and Honda (2000)).



To achieve better control of multiscale processes, pro-cess modeling and prediction by using physical, sta-tistical, or simulation methods to characterize com-plicated process outputs becomes a critical issue. Thefollowing provide a few examples.

Precise control of film properties in nanoscalesemiconductor manufacturing requires models thatpredict how the film state (in the microscopic scale) isaffected by changes in controllable parameters (in themacroscopic scale). Lou and Christofides (2003) de-veloped a multiscale model that involves coupled par-tial differential equations (PDEs) for modeling thegas phrase of the material deposition process. Sub-sequently, the authors used a kinetic Monte-Carlo(kMC) simulator to model the atom adsorption, des-orption, and surface-migration processes for shapingthin-film microstructures. There are strong interac-tions between the macro- and microscale phenomena.For example, the concentration of the precursor inthe inlet gas governs the rate of adsorption of atomson the surface, which, in turn, influences the sur-face roughness. On the other hand, the density of theatoms on the surface affects the rate of desorption ofatoms from surface to the gas phrase, which, in turn,influences the gas concentration of the precursor.

Lou and Christofides (2003) constructed an effi-cient estimator for the surface roughness at the time-scale comparable to the real-time evolution of theprocess using discrete on-line measurements. Then,the estimated roughness was fed into a proportional-integral (PI) controller. Application of the esti-mator/controller to the multiscale process modeldemonstrated successful regulation of the surfaceroughness at the desired value.

In nanoscale semiconductor manufacturing, thekMC simulation methods have been used for

1. predicting microscopic properties of thin films(e.g., surface roughness);

2. studying the dynamics of complex material de-position processes including multiple chemicalspecies with both short-range and long-rangeinteractions; and

3. performing predictive control design to controlfinal surface roughness.

kMC is not available in closed form, thus makingit difficult for use in system-level analysis and designand implementation of model-based feedback-controlsystems.

Instead of using the kMC approach in develop-

ing feedback-control schemes, for many depositionand sputtering processes, a system of stochastic lin-ear/nonlinear PDEs is derivable based on micro-scopic rules corresponding to the so-called masterequation, which describes the stochastic nature ofthe thin-film growth process (Van Kampen (1992)).Lou and Christofides (2006) proposed using nonlin-ear stochastic ordinary differential equations (ODEs)(an approximation of the stochastic PDEs) for devel-oping computationally efficient feedback controllers.Their objective was to control the expected rough-ness of the surface. The solution strategy is to controlthe covariance of the thin-film growth states in thenonlinear stochastic PDE system for various spatiallocations. Based on various simulation runs, it is clearthat the proposed nonlinear feedback-control methodcan reduce the surface roughness to the desired level,while also effectively reducing the variance of the fi-nal surface roughness.

Different control strategies have been used to con-trol multiscale processes. Both proportional-integral(PI) controllers (Lou and Christofides (2003), Louand Christofides (2006), Gallivan (2005)) and model-based predictive controllers (Ni and Christofides(2005a, b), Christofides and Armaou (2006)) havebeen implemented in practice to control thin-filmgrowth surface roughness and other parameters ofinterest. Figure 7 shows a general framework of pre-dictive controllers. The difference between the realand predicted output are compared; such informa-tion is then fed into the optimizer to generate an op-timal set point for the next step. Multiscale models

FIGURE 7. A Framework of Predictive Controllers with

Multiscale Models.



can be employed in building the predictive model. Innano applications, a multiscale model can be used tocharacterize both macroscale and microscale behav-ior of a process. Usually, manipulated variables areapplied to only the macroscale behavior of a process.Formulating an objective function is a key task in de-riving the optimal set point. Most objectives in themultiscale control publications focused on the upperscale process performance. See Fenner et al. (2005)for a study of incorporating objectives from multi-ple process stages in deriving an automatic-controlpolicy. The following gives another example.

The use of PI or predictive controllers has success-fully improved process stability and product qual-ity. However, in controlling a nanopositioner device,Salapaka et al. (2002) noted that the conventional PIcontroller does not meet the requirements for posi-tioning. Instead, the authors considered an H-infinitycontroller to incorporate both performance and ro-bustness in the objective function and found the op-timum based on the H-infinity norm. As the objectiveis defined on multiple scales, the H-infinity norm iscalculated as the supremum of the objective func-tion on all scales. The optimization method basedon H-infinity is a feasible way to solve multiscaleproblems and has substantially improved positioningspeed and precision in the research.

Conclusion and Future Research

This article has reviewed applications of statisti-cal techniques in nano technology. As the physicaland chemical properties of nanoscale materials dif-fer fundamentally from the properties of individualatoms and molecules or bulk matter, various chal-lenges have occurred in designing, analyzing, andmanufacturing nanodevices. Therefore, both conven-tional and new statistical techniques are expected toplay important roles in nanotechnology research.

This paper first investigates the experimental de-sign issues of nanodevice fabrication processes. Di-verse application examples include fractional facto-rial designs, response surface designs, and robust pa-rameter designs. For future nanotechnology researchto make efficient use of these methods, we feel thatthe following issues are worthy of consideration:

(a) Choice of experimental-design methods. In someapplications, simply full factorial or fractionalfactorial designs may be sufficient for modelbuilding and process understanding. However,as shown in several examples, nanofabricationprocesses can be very sensitive to small changes

in controllable and noise factors. Moreover, theprocess outcomes may not be continuous ran-dom variables. In this case, irregular designsor new experimental design methods might beconsidered. Finally, special experimental con-straints due to physical limitations may limitthe use of certain designs.

(b) Analysis of experimental data. Due to theuse of advanced measurement tools, spatialdata, high-frequency signals, and/or qualitativemeasures are frequently observed in nano re-search. Combined with the multistage natureof the nanodevice–fabrication processes, statis-tical analysis of such complicated and/or largesize data with many explanatory and noise fac-tors provides many challenges. New algorithms,such as those developed by Jeong et al. (2006),Yuan and Kuo (2006), and Wang and Tsung(2007), might be useful for dealing with thesenew data types. Variable selection tools, such asYuan et al. (2007), could be effective for dealingwith large size factors constrained with exper-imental design structures. However, more re-search is needed in this field, especially for mul-tistage, multiscale, and multilevel processes.

(c) Use of computer simulations. When the physi-cal experimentation is costly and time consum-ing, the use of computers for process and devicesimulations is expected to become more preva-lent in nano research. Sometimes, processes canbe very complicated, leading to lengthy com-puter experiments. Thus, the design and anal-ysis of computer experiments, as well as theintegration between computer and physical ex-periments, are topics that deserve more futureresearch efforts.

(d) Interaction between statisticians and experts inmaterial science, physics, and other disciplines.Nanotechnology is a multidisciplinary subject.When statisticians are more involved in under-standing the issues in nanoresearch, more effec-tive procedures can be developed to deal withchallenging application problems. Statisticiansshould be important players in novel scientificdiscoveries.

Quality issues have been emphasized in early pro-duction stages of nanodevices. The challenges con-fronted when applying SPC and APC techniques innano applications are confounded with the new datatypes discussed above. Moreover, when online sensorinformation is available, it is interesting to see the of-



fline robust parameter designs extended (e.g., Joseph(2003)) to take advantage of the new information use-ful for process adjustments. See Edgar et al. (2000)and Del Castillo (2006) for a review in the topics ofautomatic control and statistical process adjustment.

Research on reliability is also critical to nanode-vices and their fabrication processes. The intrinsicmechanism of failures for nanoproducts and the mod-eling of their lifetimes are areas of future researchinterest. Readers may refer to Jeng et al. (2007) fora detailed review in nanoreliability studies and seeother papers in the same issue of the journal forspecific topics. A different, but related, direction ofnew research is the “reliability” of measurement andpositioning systems in nanomanufacturing. Becausenanodevices are so small and the nanomanufacturingprocess requires speedy data collection, how can oneknow if the measurements are accurate and, moreimportant, if the nanosubsystems are placed in thecorrect location within a very small-sized nanosys-tem? See Xia et al. (2007) and Qian and Wu (2007)for examples of new statistical methods developedfor integrating information with different accuracyfor statistical inference.

With the fast development of nanotechnology andthe introduction of mass production of nanodevices,statistics is expected to play an increasingly impor-tant role in both academic and industrial fields. Thisreview summarizes successful applications of statis-tical methods in nanotechnology seen in the litera-ture, along with a few interesting topics for futureresearch.

Acknowledgments

The authors would like to thank the editor and theanonymous referees for their insightful comments andsuggestions, which have helped improve the qualityof this review greatly. Dr. Jeng’s research was partlysupported by a grant from the Landmark Project ofNational Cheng Kung University. Dr. Wang’s workwas supported by the National Natural Science Foun-dation of China (NSFC) under grant 70802034. Dr.Lu’s research was partially supported by the US Na-tional Science Foundation under the award 0400071.

References

General ReferencesJeng, S.-L.; Lu, J.-C.; and Wang, K. (2007). “A Review

of Reliability Research on Nanotechnology”. IEEE Transac-tions on Reliability 56(3), pp. 401–410.

National Nanotechnology Initiative. (2008). “NanotechFacts”. Available at http://www.nano.gov/html/facts/homefacts.html.

Roco, M. C. (2004). “Nanoscale Science and Engineering:Unifying and Transforming Tools”. AIChE Journal 50(5),pp. 890–897.

Design of Experiments

Barglik-Chory, C.; Strohm, C. R. H.; and Muller, G.

(2004). “Adjustment of the Band Gap Eenergies of Biosta-bilized CdS Nanoparticles by Aapplication of Statistical De-sign of Experiments”. Journal of Physical Chemistry, SeriesB 108(23), pp. 7637–7640.

Basumallick, A.; Das, G. C.; and Mukherjee, S. (2003).“Design of Experiments for Synthesizing In Situ Ni-SiO2and CO-SiO2 Nanocomposites by Non-isothermal ReductionTreatment”. Nanotechnology 14(8), pp. 903–906.

Cota-Sanchez, G.; Soucy, G.; Huczko, A.; and Lange,

H. (2005). “Induction Plasma Synthesis of Fullerenes andNanotubes Using Carbon Black-Nickel Particles”. Carbon43(15), pp. 3153–3166.

Dasgupta, T. (2007). “Robust Parameter Design for Auto-matically Controlled Ssystems and Nanostructure Synthe-sis”. Ph.D. Dissertation, Georgia Institute of Technology,Atlanta, GA.

Dasgupta, T.; Ma, C.; Joseph, V. R.; Wang, Z. L.; and Wu,

C. F. J. (2008). “Statistical Modeling and Analysis for Ro-bust Synthesis of Nanostructures”. Journal of the AmericanStatistical Association 103, pp. 594–603.

Fasulo, P. D.; Rodgers, W. R.; Ottaviani, R. A.; andHunter, D. L. (2004). “Extrusion Processing of TPONanocomposites”. Polymer Engineering and Science 44(6),pp. 1036–1045.

Hou, T.-H.; Su, C.-H.; and Liu, W.-L. (2007). “ParametersOptimization of a Nanoparticle Wet Milling Process Usingthe Taguchi Method, Response Surface Method and GeneticAlgorithm”. Powder Technology 173(3), pp. 153–162.

Kim, K. D.; Han, D. N.; and Kim, H. T. (2004). “Optimiza-tion of Experimental Conditions Based on the Taguchi Ro-bust Design for the Formation of Nano-Sized Silver Parti-cles by Chemical Reduction Method”. Chemical EngineeringJournal 104(1–3), pp. 55–61.

Kim, K. D.; Kim, S. H.; and Kim, H. T. (2005). “Applyingthe Taguchi Method to the Optimization for the Synthesisof TiO2 Nanoparticles by Hydrolysis of TEOT in Micelles”.Colloids and Surfaces: A—Physicochemical and EngineeringAspects 254(1–3), pp. 99–105.

Lin, T. Y.; Davison, K. L.; Leong, W. S.; Chua, S.; Yao,

Y. F.; Pan, J. S.; Chai, J. W.; Toh, K. C.; and Tjiu,

W. C. (2003). “Surface Topographical Characterization ofSilver-Plated Film on the Wedge Bondability of Leaded ICPackages”. Microelectronics Reliability 43, pp. 803–809.

Montgomery, D. C. (2005). Design and Analysis of Experi-ments. New York, NY: Wiley.

Nazzal, S.; Nutan, M.; Palamakula, A.; Shah, R.; Za-

ghloul, A. A.; and Khan, M. A. (2002). “Optimizationof a Self-Nanoemulsified Tablet Dosage Form of UbiquinoneUsing Response Surface Methodology: Effect of FormulationIngredients”. International Journal of Pharmaceutics 240(1–2), pp. 103–114.

Nembhard, H. B.; Acharya, N.; Aktan, M.; and Kim,

S. (2006). “Design Issues and Analysis of Experiments in



Nanomanufacturing”. In Handbook of Industrial and Sys-tems Engineering, Badiru, A. B. ed., pp. 17.1–17.24. BocaRaton, FL: CRC Taylor & Francis.

Prakobvaitayakit, M. and Nimmannit, U. (2003). “Opti-mization of Polylactic-Co-Glycolic Acid Nanoparticles Con-taining Itraconazole Using 2ˆ3 Factorial Design”. AAPSPharmSciTech 4(4), pp. 1–9.

Taguchi, G. (1986). Introduction to Quality Engineering: De-signing Quality into Products and Processes. Tokyo: TheAsian Productivity Organization.

Yong, V. and Hahn, H. T. (2005). “Dispersant OptimizationUsing Fesign of Rxperiments for SiC/Vinyl Ester Nanocom-posites”. Nanotechnology 16(4), pp. 354–360.

Yordem, O. S.; Papila, M.; and Menceloglu, Y. Z. (2008).“Effects of Electrospinning Parameters on PolyacrylonitrileNanofiber Diameter: An Investigation by Response SurfaceMethodology”. Materials and Design 29(1), pp. 34–44.

Data Collection, Analysis, and Modeling

Bazant, Z. P. and Pang, S.-D. (2007). “Activation EnergyBased Extreme Value Statistics and Size Effect in Brittleand Quasibrittle Fracture”. Journal of the Mechanics andPhysics of Solids 55, pp. 91–131.

Belich, T. J.; Shen, Z.; Campbell, S. A.; and Kakalios, J.

(2003). “Non-Gaussian 1/f Noise as a Probe of Long-RangeSstructural and Electronic Disorder in Amorphous Silicon”.In Proceedings of SPIE, vol. 5112, pp. 67–77.

Chang, I. J.; Kang, K.; Mukhopadhyay, S.; Kim, C. H.;

and Roy, K. (2005). “Fast and Accurate Estimation ofNanoscaled SRAM Read Failure Probability Using CriticalPoint Sampling”. In Proceedings of the 2005 IEEE CustomIntegrated Circuits Conference, pp. 439–442.

Chen, S. Y.; Huang, Y. H.; and Cai, B. N. (2005). “Nor-mal Distribution of Confinement Energy from a Photolumi-nescence Line Shape Aanalysis in Oxidized Porous Silicon”.Solid-State Electronics 49(6), pp. 940–944.

Chen, Y. P. and Lee, J. D. (2004). “Multiscale Modeling ofPolycrystalline Silicon”. International Journal of Engineer-ing Science 42(10), pp. 987–1000.

Granier, J. J. and Pantoya, M. L. (2004). “The Effect ofSize Distribution on Burn Rate in Nanocomposite Ther-mites: A Probability Density Function Study”. CombustionTheory and Modeling 8, pp. 555–565.

Hsieh, M.-S. and Jeng, S.-L. (2007). “Accelerated DiscreteDegradation Models for Leakage Current of Ultra-Thin GateOxides”. IEEE Transactions on Reliability 56(3), pp. 369–380.

Jefferson, G.; Garmestani, H.; Tannenbaum, R.; Gok-

hale, A.; and Tadd, E. (2005). “Two-Point Probabil-ity Distribution Function Aanalysis of Co-Polymer Nano-Composites”. International Journal of Plasticity 21, pp.185–198.

Li, X.; Le, J.; Pileggi, L. T.; and Strojwas, A. (2005).“Projection-Based Performance Modeling for Inter/Intra-Die Variations”. In Proceedings of the 2005 IEEE/ACM In-ternational Conference on Computer-Aided Design, pp. 721–727.

Margolin, G. and Barkai, E. (2005). “Nonergodicity ofBlinking Nanocrystals and Other Levy-Walk Processes”.Physical Review Letters 94(8), 080601(1–4).

Miranda, E. and Jimenez, D. (2004). “A New Model for theBreakdown Dynamics of Ultra-Thin Gate Oxides Based on

the Stochastic Logistic Differential Equation”. In Proceed-ings of the 24th International Conference on Microelectron-ics, pp. 625–628.

Onodera, H. (2006). “Variability: Modeling and Its Impacton Design”. IEICE Transactions on Electronics E89-C(3),pp. 342–348.

Qi, Y.; Gao, J.; and Fortes, J. (2005). “Markov Chains andProbabilistic Computation—A General Framework for Mul-tiplexed Nanoelectronic Systems”. IEEE Transactions onNanotechnology 4(2), pp. 194–205.

Rejimon, T. and Bahnja, S. (2005). “Time and Space Effi-cient Method for Accurate Computation of Error DetectionProbabilities in VLSI Circuits”. Proceedings of the 2005 IEEComputers and Digital Techniques 152(5), pp. 679–685.

Schneider, D.; Siemroth, P.; Schulke, T.; Berthold, J.;

Schultrich, B.; Schneider, H. H.; Ohr, R.; and Petereit,

B. (2002). “Quality Control of Ultra-Thin and Super-HardCoatings by Laser-Acoustics”. Surface and Coatings Tech-nology 153(2–3), pp. 252–260.

Zhao, Y.; Dey, S.; and Chen, L. (2004). “Double SamplingData Checking Technique: An Online Testing Solution forMultisource Noise-Induced Errors on On-Chip Interconnectsand Buses”. IEEE Transactions on Very Large Scale Inte-gration (VLSI) Systems 12(7), pp. 746–755.

Statistical Process Control and QualityControl

Christofides, P. D. and Armaou, A. (2006). “Control andOptimization of Multiscale Process Systems”. Computers &Chemical Engineering 30(10–12), pp. 1670–1686.

Das, T. K.; Ganesan, R.; Sikder, A. K.; and Kumar, A.

(2005). “Online End Point Detection in CMP Using SPRTof Wavelet Decomposed Sensor Data”. IEEE Transactionson Semiconductor Manufacturing 18(3), pp. 440–447.

Fenner, J. S.; Jeong, M. K.; and Lu, J.-C. (2005), “Opti-mal Automatic Control of Multistage Production Processes”.IEEE Transactions on Semiconductor Manufacturing 18(1),pp. 94–103.

Gallivan, M. A. (2005). “An Estimation Study for Controlof a Lattice Model of Thin Film Deposition”. Computers &Chemical Engineering 29(4), pp. 761–769.

Ganesan, R.; Das, T. K.; Sikder, A. K.; and Kumar,

A. (2003). “Wavelet-Based Identification of DelaminationDefect in CMP (Cu-Low k) Using Nonstationary Acous-tic Emission Signal”. IEEE Transactions on SemiconductorManufacturing 16(4), pp. 677–685.

Jiang, X. and Blunt, L. (2004). “Third Generation Waveletfor the Extraction of Morphological Features from Micro andNano Scalar Surfaces”. Wear 257, pp. 1235–1240.

Klepper, C. C.; Carlson, E. P.; Hazelton, R. C.; Yad-

lowsky, E. J.; Feng, B.; Taher, M. A.; and Meyer, H. M.

(2005). “H-Alpha Emission as a Feedback Control Sensor forReactive Sputter Deposition of Nano-Structured, Diamond-Like Carbon Coatings”. IEEE Transactions on Plasma Sci-ence 33(2), pp. 799–807.

Lantz, M. A.; Binnig, G. K.; Despont, M.; and Drechsler,

U. (2005). “A Micromechanical Thermal Displacement Sen-sor with Nanometre Resolution”. Nanotechnology 16(8), pp.1089–1094.

Lin, T.; Pan, Y.; and Hsieh, C. (2003). “Precision-Limit Po-sitioning of Direct Drive Systems with the Existence of Ffric-tion”. Control Engineering Practice 11, pp. 233–244.



Lou, Y. and Christofides, P. D. (2003), “Estimation andControl of Surface Roughness in Thin Film Growth Using Ki-netic Monte-Carlo Models”. Chemical Engineering Science58, pp. 3115–3129.

Lou, Y. M. and Christofides, P. D. (2006). “Nonlinear Feed-back Control of Surface Roughness Using a Stochastic PDE:Design and Application to a Sputtering Process”. Industrial& Engineering Chemistry Research 45(21), pp. 7177–7189.

Mae, K. and Honda, T. (2000). “Growth Mode Variations ofThin Films on Nano-Faceted Substrates”. Thin Solid Films373(1–2), pp. 199–202.

Ng, H. T.; Li, J.; Smith, M. K.; Nguyen, P.; Cassell,

A.; Han, J.; and Meyyappan, M. (2003). “Growth of Epi-taxial Nanowires at the Junctions of Nanowalls”. Science300(5623), pp. 1249–1249.

Ni, D. and Christofides, P. D. (2005a). “Dynamics and Con-trol of Thin Ffilm Surface Microstructure in a Complex De-position Process”. Chemical Engineering Science 60(6), pp.1603–1617.

Ni, D. and Christofides, P. D. (2005b). “Multivariable Pre-dictive Control of Thin Film Deposition Using a StochasticPDE model”. Industrial & Engineering Chemistry Research44(8), pp. 2416–2427.

Ohshima, M. (2003). “Control and Design Problems in Mate-rial Processing—How Can Process Systems Engineers Con-tribute to Material Processing?” Journal of Process Control13, pp. 599–605.

Salapaka, S.; Sebastian, A.; Cleveland, J. P.; and Sala-

paka, M. V. (2002). “High Bandwidth Nano-Positioner: ARobust Control Approach”. Review of Scientific Instruments73(9), pp. 3232–3241.

Sneh, O.; Clark-Phelps, R. B.; Londergan, A. R.; Win-

kler, J.; and Seidel, T. E. (2002). “Thin Film AtomicLayer Deposition Equipment for Semiconductor Processing”.Thin Solid Films 402(1–2), pp. 248–261.

Van Kampen, N. G. (1992). Stochastic Processes in Physicsand Chemistry. Amsterdam, The Netherlands: North-Holland.

Future Research

Edgar, T. F.; Butler, S. W.; Campbell, W. J.; Pfeif-

fer, C.; Bode, C.; Hwang, S. B.;, Balakrishnan, K. S.;

and Hahn, J. (2000). “Automatic Control in Microelectron-ics Manufacturing: Practices, Challenges, and Possibilities”.Automatica 36(11), pp. 1567–1603.

Del Castillo, E. (2006). “Statistical Process Adjustment:A Brief Retrospective, Current Status, and Some Opportu-nities for Further Work”. Statistica Neerlandica 60(3), pp.309–326.

Jeong, M. K.; Lu, J.-C.; and Wang, N. (2006). “Wavelet-Based SPC Procedure for Complicated Functional Data”. In-ternational Journal of Production Research 44(4), pp. 729–744.

Qian, Z. and Wu, C. F. J. (2007). “Bayesian HierarchicalModeling for Integrating Low-Accuracy and High-AccuracyExperiments”. Technometrics in press.

Joseph, V. R. (2003). “Robust Parameter Design with Feed-Forward Control”. Technometrics 45, pp. 284–292.

Sung, H. J.; Su, J.; Berglund, J. D.; Russ, B. V.;

Meredith, J. C.; and Galis, Z. S. (2005). “The Use ofTemperature-Composition Combinatorial Libraries to Studythe Effects of Biodegradable Polymer Blend Surfaces on Vas-cular Cells”. Biomaterials 26, pp. 4557–4567.

Wang, K. and Tsung, F. (2007). “Run-to-Run Process Ad-justment Using Categorical Observations”. Journal of Qual-ity Technology 39(4), pp. 312–325.

Wang, N.; Lu, J.-C.; and Kvam, P. (2007). “Multi-Level Spa-tial Modeling and Decision-Making with Application in Lo-gistics Systems”. Technical report, The School of Industrialand Systems Engineering, Georgia Institute of Technology,Atlanta, GA.

Xia, H. F.; Ding, Y.; and Mallick, B. K. (2007). “BayesianHierarchical Model for Integrating Multi-Resolution Metrol-ogy Data”. Paper presented in the best student paper com-petition, INFORMS Conference—Quality, Statistics and Re-liability Section, Seattle, WA.

Yuan, M.; Joseph, V. R.; and Lin, Y. (2007). “An EfficientVariable Selection Approach for Analyzing Designed Exper-iments”. Technometrics in press.

Yuan, T. and Kuo, W. (2006). “Defect Pattern Recognitionin Semiconductor Fabrication Using Model-Based Clusteringand Bayesian Inference”. Paper presented in the best studentpaper competition, INFORMS Conference—Quality, Statis-tics and Reliability Section, Pittsburgh, PA.

∼


Documents

A Review of Statistical Methods for Quality Improvement and Control