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Methods of Spatial Statistics for the Characterization of Dislocation Systems

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Methods of Spatial Statistics for the Characterization of Dislocation Systems Methods of Spatial Statistics for the Characterization of Dislocation Systems Von der Fakult¨atf¨urMathematik und Informatik der Technischen Universit¨atBergakademie Freiberg genehmigte DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium Dr. rer. nat., vorgelegt von M.Sc. Hamid Ghorbani geboren am 29. April 1977 in Isfahan, Iran Gutachter: Prof. Dr. Dietrich Stoyan, Freiberg Prof. Dr. Hans Joachim M¨oller, Freiberg Prof. Dr. Joachim Ohser, Darmstadt Tag der Verleihung: 21. Juli 2004 Table of Contents Acknowledgements . iii Summary . iv 1 Introduction 1 2 Methods of preprocessing of dislocation images 5 2.1 Description of raw dislocation images . 5 2.2 Automatic object recognition . 6 2.2.1 Shape ratios . 6 2.2.2 Methods for image analysis . 7 3 Statistical methods for the estimation of dislocation numbers 11 3.1 Rinio’s method . 11 3.2 Point process methods . 12 3.2.1 Classical intensity estimation . 12 3.2.2 Intensity estimation in the case of overlapping discs . 14 3.3 Random point processes . 16 3.3.1 Introduction . 16 3.3.2 The homogeneous Poisson point process . 16 3.3.3 Statistics for the homogeneous Poisson point process . 17 3.3.4 Poisson cluster processes . 18 3.4 Germ-grain models . 20 3.4.1 Fundamentals . 20 3.4.2 The Boolean model . 21 3.5 Intensity estimation for cluster processes . 24 3.5.1 The geometrical problem . 24 3.5.2 Mat´erncluster process . 26 3.5.3 Gauss-Poisson process . 26 3.5.4 Segment cluster process . 29 3.6 Line-based Poisson process . 36 i 3.6.1 The Poisson line process . 36 3.6.2 The spherical contact distribution . 38 3.7 Superposition . 39 3.8 Parameter estimation . 40 4 Modelling the probability distribution of the number of dislo- cations 51 4.1 Introduction . 51 4.2 Dislocation density and point density distribution function . 51 4.2.1 Empirical observations . 53 4.3 A physical model for dislocation growth . 55 4.3.1 Introduction . 55 4.3.2 Dislocation multiplication and exponential growth models 55 4.4 Three distributions . 66 4.4.1 Pareto distribution . 66 4.4.2 Weibull distribution . 68 4.4.3 Gumbel distribution . 69 4.5 Stochastic growth models leading to Pareto and Weibull distri- butions . 70 4.6 Fitting distributions to the empirical point density distribution function . 71 4.7 Summary of model fitting . 80 4.8 Estimating the number of dislocations . 81 4.9 A stochastic modification of the exponential growth model . 82 4.9.1 The double Pareto-lognormal distribution . 83 5 General growth laws and size or number distributions 90 5.1 Introduction . 90 5.2 Stochastic growth processes . 90 5.3 Examples . 93 5.3.1 Distribution of the number of galaxies . 93 5.3.2 Wildfire size distributions . 102 5.3.3 Particle size distribution in the fluidized bed . 103 Bibliography 106 ii Acknowledgments I wish to express my sincere gratitude to all those who helped me to write this thesis. I am specially grateful to my supervisor, Professor Dietrich Stoyan, for sug- gesting the PhD position in graduate school ‘Spatial Statistics’ and all his help and support. During the work I have benefitted from his mathematical sugges- tions and technical advices. I appreciate his efforts for reading the manuscript carefully and his suggestions to improve it. I am deeply indebted to my co-supervisor Professor Hans Joachim M¨ollerfor all his help, technical support and providing fruitful dialogues in good working atmosphere. Special thanks go to Professor Joachim Ohser for reading an earlier version of this thesis. I would like to thank Markus Rinio, Professor Vicent J. Mart´ınez, Professor Frederic P. Schoenberg and Professor Stefan Heinrich for kind permission to use their data. The valuable comments of Markus Rinio, Dr. Gunter D¨oge, Dr. Claudia Funke and Marco Menzel are appreciated. Acknowledgments are made to my colleagues, my friends, our coordinator Mrs. Gudrun Seifert and Mrs. Renate Ebert for their help in my first days in Freiberg. As always the greatest debt one owes is to the family. I would like to thank my parents who always encouraged me to go on. This thesis is dedicated to my family. The financial support of ‘Deutsche Forschungsgemeinschaft’ is appreciated. Hamid Ghorbani iii Summary Multicrystalline Silicon (mc-Si) is a solar graded raw material for industrial so- lar cell production. Compared to microelectronic-grade Silicon material, mc-Si material contains more impurities and a high density of extended crystal de- fects such as grain boundaries, dislocations and micropercipitate. Dislocations are line defects in crystals, which either begin and end on the crystal surface or form a closed curve inside the crystal. A volume of 1 mm3 Silicon may contain dislocations of a total length of 1 km. Dislocations largely control the electrical properties of the material and often impact the solar cell character- istics adversely. The main objective of the PhD thesis is to develop statistical methods for the analysis of systems of dislocations and similar structures. A problem of par- ticular importance is estimating the number of dislocations on Silicon wafers. After etching the surface of a Silicon wafer a system of overlapping circular etch pits is observed, which grew around the dislocation centers. In the sta- tistical analysis, dislocation objects are distinguished from other objects like grain boundaries, oxygen precipitates and artifacts on a given planar section of microcrystalline Silicons, by an image analysis procedure developed by physi- cists. Typically, the dislocation centers show a high degree of overlapping and chain-like clustering. In order to estimate the number n of dislocations on a given wafer, the etched dislocation pattern Z is considered as a germ-grain model, Z = X ⊕ b(o, R), as studied in stochastic geometry. The corresponding germ point pattern X consists of all dislocation centers and the grains are discs with fixed radius R, corresponding to the etch pits. Experimental observations showed that R = 1 µm. The problem of estimating n is equivalent to the problem of estimating the intensity λ (dislocation intensity) of the point process X. Statistical methods are developed to estimate the intensity λ of a germ-grain model with circular grains. While the aim is estimating the intensity of a point process X, the ob- servable is a random set, the union Z of overlapping discs with fixed radius R. Because of overlappings, intensity estimation in this setting is a complicated problem. In the classical case of a Boolean model (X is a homogeneous Poisson process) one of the successful methods to estimate the intensity of Poisson point process X uses the spherical contact distribution function Hs(·). In the case of Silicon wafers the assumption of a Poisson process X is unrealistic. Therefore, the following models for the point process X are considered: • Segment cluster process, iv • Line-based Poisson process, • Superposition of a segment cluster process and a Poisson process, • Superposition of a line-based Poisson process and a Poisson process. A segment cluster process is a Poisson cluster process where the cluster points are scattered on the segments, the centers of which are in the parent points. A line-based Poisson point process is a process where the points are scattered randomly on the lines of a Poisson line process. The approach for estimating the intensity of the Boolean models is developed in the thesis for non-Poissonian cases, where X is a segment cluster or line-based Poisson point process. Approximations for the spherical contact distribution X functions Hs (·) of the segment cluster process, line-based Poisson process and superposition of these two processes with Poisson point processes are derived, using numerical methods and simulation. This is useful for the statistical ap- Z proach, since the spherical contact distribution function of Z, Hs (·), can be X expressed in terms of Hs (·). By help of the ImageC software of Imtronic the ˆ Z empirical spherical contact distribution functions Hs (·) of dislocation patterns were estimated. The parameters of the different point process models are estimated using non- linear regression, by fitting the theoretical spherical spherical contact distribu- Z ˆ Z tion functions Hs (·) to their empirical counterparts Hs (·). The mean square error criterion is used for comparison between the models. Then the numbers of dislocations are estimated, using the model parameters of X. As expected, the estimated numbers which are obtained by the statistical methods used in the thesis are larger than estimates obtained by methods developed by physi- cists. An alternative method for estimating the numbers of dislocations starts from the intensity function. The dislocation pattern on the wafer is considered as an inhomogeneous planar point process with intensity function λ(x). The func- tion is summarized by the ‘point density distribution function’ D(·), which was introduced in this thesis. This function is estimated as follows area with estimated point density λˆ(x) ≤ t Dˆ(t) = , area of W where W is the window of observation. The point density distribution function is useful as a descriptor even for very irregular point processes. It can be parameterized and the corresponding pa- rameters can be estimated, which helps to come to interesting results for the v process of growth of number of dislocations. In the physical literature deterministic models are given which describe the so-called multiplication behavior of the number X(t) of movable dislocations at time t during crystallization. Since for the given samples of dislocations the dislocation density varies heavily, the deterministic physical models are refined so that X(t) is a random variable; the classical (deterministic) physical mod- els for dislocation growth process are replaced by stochastic growth processes, which are described as follows.
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