Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Alam, Jennifer Sherry
Master of Applied Science - MASc
University of British Columbia
Disordered semiconductors, including poly-crystalline and amorphous semiconductors, have unique properties that can be attributed to the disorder present within their atomic structure. While the distribution of electronic states associated with a defect-free crystalline semiconductor terminates abruptly at the band edges, in a disordered semiconductor, electronic states encroach into this otherwise empty gap region. This encroachment of the electronic states into the band gap region of a disordered semiconductor can be credited for many of its unique properties. Unfortunately, due to its random nature, the quantitative characterization of the disorder that is present within a disordered semiconductor has proven a difficult task to grapple with. The distribution of potential fluctuations found within such a material has a direct impact on the distribution of electronic states within a disordered semiconductor. The aim of this thesis is to characterize this distribution of potential fluctuations. In particular, I use a Monte Carlo approach, where simulations, replicating the random nature of the defects found within a disordered semiconductor, are performed a large number of times in order to obtain the distribution of potential fluctuations associated with such a semiconductor. With the distribution of potential fluctuations evaluated, the impact on the corresponding distribution of electronic states will be probed.
Monte Carlo analysis for theevaluation of the distribution ofpotential fluctuations within adisordered semiconductorbyJennifer Sherry AlamB.Sc., BRAC University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2018c© Jennifer Sherry Alam, 2018The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis entitled:Monte Carlo analysis for the evaluation of the distribution of potential fluc-tuations within a disordered semiconductorsubmitted by Jennifer Sherry Alam in partial fulfillment of the require-ments of the degree of Master of Applied ScienceDr. Stephen K. O’leary, School of EngineeringSupervisorDr. Jahangir Hossain, School of EngineeringSupervisory Committee MemberDr. Wilson Eberle, School of EngineeringSupervisory Committee MemberDr. Yang Cao, School of EngineeringUniversity ExamineriiAbstractDisordered semiconductors, including poly-crystalline and amorphoussemiconductors, have unique properties that can be attributed to the dis-order present within their atomic structure. While the distribution of elec-tronic states associated with a defect-free crystalline semiconductor termi-nates abruptly at the band edges, in a disordered semiconductor, electronicstates encroach into this otherwise empty gap region. This encroachment ofthe electronic states into the band gap region of a disordered semiconduc-tor can be credited for many of its unique properties. Unfortunately, dueto its random nature, the quantitative characterization of the disorder thatis present within a disordered semiconductor has proven a difficult task tograpple with. The distribution of potential fluctuations found within such amaterial has a direct impact on the distribution of electronic states withina disordered semiconductor. The aim of this thesis is to characterize thisdistribution of potential fluctuations. In particular, I use a Monte Carloapproach, where simulations, replicating the random nature of the defectsfound within a disordered semiconductor, are performed a large number oftimes in order to obtain the distribution of potential fluctuations associatedwith such a semiconductor. With the distribution of potential fluctuationsevaluated, the impact on the corresponding distribution of electronic statesiiiAbstractwill be probed.ivLay SummaryMost electronic devices are made of materials known as semiconductorswhose properties depend on how the atoms within are distributed. Disor-dered semiconductors, on the other hand, do not have a repeating patternwithin their atomic structure. Due to the presence of disorder, it becomesdifficult to predict the properties of this material. Hence, it can be help-ful to use statistics for the prediction of these properties. In my thesis, Iaim to model the problem in a Monte Carlo simulation, where a model forthe defects within the atomic structure that causes the random nature ofdisordered semiconductors is replicated and is then simulated over a largenumber of times. Finally, these results are observed for the probability ofthese defects being present and their affect on the properties of disorderedsemiconductors.vPrefaceThis thesis is an original and unpublished work by the author, JenniferSherry Alam, under the supervision of Dr. Stephen K. O’Leary.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . 92.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9viiTABLE OF CONTENTS2.2 The atomic distribution for the various forms of condensedmatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The distribution of electronic states . . . . . . . . . . . . . . 132.4 Forms of disorder . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Approaches employed for DOS evaluations of disordered semi-conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Free electron evaluation of the DOS function . . . . . . . . . 202.7 Introduction to probability . . . . . . . . . . . . . . . . . . . 222.8 Sample-space . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.9 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.10 The axioms of probability . . . . . . . . . . . . . . . . . . . . 262.11 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . 282.12 The probability density function . . . . . . . . . . . . . . . . 292.13 Distributions of continuous random variables . . . . . . . . . 332.14 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . 38Chapter 3: The Monte Carlo method . . . . . . . . . . . . . . . 423.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 The random number generator . . . . . . . . . . . . . . . . . 433.3 Empirical distribution function . . . . . . . . . . . . . . . . . 483.4 Random number generators of other distributions . . . . . . . 493.5 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 53Chapter 4: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 614.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61viiiTABLE OF CONTENTS4.2 The DOS functions for a single electron in a random potentialprofile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Modeling the random potential . . . . . . . . . . . . . . . . . 654.4 Monte Carlo approach . . . . . . . . . . . . . . . . . . . . . . 704.5 Monte Carlo results . . . . . . . . . . . . . . . . . . . . . . . 734.5.1 The power-law potential . . . . . . . . . . . . . . . . . 754.5.2 Exponential potential . . . . . . . . . . . . . . . . . . 854.5.3 Step function potential . . . . . . . . . . . . . . . . . . 1004.6 Closed-form analytical results . . . . . . . . . . . . . . . . . . 1024.6.1 On the distribution of points . . . . . . . . . . . . . . 1064.6.2 The Poisson point process . . . . . . . . . . . . . . . . 1074.6.3 Analytical result for the PDF . . . . . . . . . . . . . . 1174.7 DOS evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 124Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 133References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136ixList of TablesTable 2.1 Energy gaps of semiconductors. . . . . . . . . . . . . . 17Table 4.1 Simulation variables for power-law potential. . . . . . 77Table 4.2 Simulation variables for exponential potential. . . . . . 86Table 4.3 Simulation variables for step function potential. . . . . 100xList of FiguresFigure 1.1 Transistors on an integrated circuit. . . . . . . . . . . 2Figure 1.2 Examples of the applications of large area electronicdevices. . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.3 DOS functions of semiconductors . . . . . . . . . . . 7Figure 2.1 Atomic distribution of semiconductors. . . . . . . . . 12Figure 2.2 Energy levels of an atom. . . . . . . . . . . . . . . . . 14Figure 2.3 Energy levels splitting. . . . . . . . . . . . . . . . . . 16Figure 2.4 Formation of energy bands. . . . . . . . . . . . . . . . 16Figure 2.5 Venn diagrams of set operations. . . . . . . . . . . . . 27Figure 2.6 Sample space and random variables. . . . . . . . . . . 30Figure 2.7 PMF of flipping a coin. . . . . . . . . . . . . . . . . . 31Figure 2.8 Uniform distribution PDF and CDF. . . . . . . . . . 35Figure 2.9 Exponential distribution PDF and CDF. . . . . . . . 37Figure 2.10 Gaussian distribution PDF and CDF. . . . . . . . . . 39Figure 2.11 Cauchy distribution PDF and CDF. . . . . . . . . . . 40Figure 3.1 Histogram of outputs from a random number generator. 47Figure 3.2 CDF of a random number generator. . . . . . . . . . 50Figure 3.3 Transformation of CDF. . . . . . . . . . . . . . . . . 54xiLIST OF FIGURESFigure 3.4 CDF of a transformed exponential random numbergenerator . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.5 Circle within a square . . . . . . . . . . . . . . . . . . 57Figure 3.6 Monte Carlo simulations for the value of pi . . . . . . 58Figure 3.7 Monte Carlo evaluation of pi . . . . . . . . . . . . . . 59Figure 3.8 Monte Carlo integration . . . . . . . . . . . . . . . . . 60Figure 4.1 Potentials in a disordered semiconductor. . . . . . . . 63Figure 4.2 Uniformly distributed points. . . . . . . . . . . . . . . 68Figure 4.3 Uniformly distributed points with point of evaluation. 71Figure 4.4 Potential point in a cube. . . . . . . . . . . . . . . . . 74Figure 4.5 Power-law potential. . . . . . . . . . . . . . . . . . . . 76Figure 4.6 CDF of power-law potential for γ = 1. . . . . . . . . 78Figure 4.7 Monte Carlo simulations of power-law potential withγ = 4 and M = 1, 000 . . . . . . . . . . . . . . . . . . 81Figure 4.8 Monte Carlo simulations of power-law potential withγ = 6 and M = 1, 000 . . . . . . . . . . . . . . . . . . 82Figure 4.9 Monte Carlo simulations of power-law potential withγ = 8 and M = 1, 000 . . . . . . . . . . . . . . . . . . 83Figure 4.10 Stability boundary lengths for power-law potentialprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.11 Monte Carlo simulations for power-law potential forγ = 4 at L = 5. . . . . . . . . . . . . . . . . . . . . . 87Figure 4.12 Monte Carlo simulations for power-law potential forγ = 6 at L = 1. . . . . . . . . . . . . . . . . . . . . . 88xiiLIST OF FIGURESFigure 4.13 Monte Carlo simulations for power-law potential forγ = 8 at L = 0.5. . . . . . . . . . . . . . . . . . . . . 89Figure 4.14 Exponential potential. . . . . . . . . . . . . . . . . . . 90Figure 4.15 Monte Carlo simulations of exponential potential withk = 1 and M = 1, 000 . . . . . . . . . . . . . . . . . . 92Figure 4.16 Monte Carlo simulations of exponential potential withk = 5 and M = 1, 000 . . . . . . . . . . . . . . . . . . 93Figure 4.17 Monte Carlo simulations of exponential potential withk = 10 and M = 1, 000 . . . . . . . . . . . . . . . . . 94Figure 4.18 Stability boundary lengths for exponential potentialprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.19 Monte Carlo simulations for exponential potential fork = 1 at L = 15. . . . . . . . . . . . . . . . . . . . . . 97Figure 4.20 Monte Carlo simulations for exponential potential fork = 5 at L = 5. . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.21 Monte Carlo simulations for exponential potential fork = 10 at L = 2. . . . . . . . . . . . . . . . . . . . . . 99Figure 4.22 Step function potential. . . . . . . . . . . . . . . . . . 101Figure 4.23 Monte Carlo simulations for step potential for a = 0.5at L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 4.24 Monte Carlo simulations for step potential for a = 1at L = 3. . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 4.25 Monte Carlo simulations for step potential for a = 2at L = 4. . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 4.26 Uniformly distributed points on a line. . . . . . . . . 108xiiiLIST OF FIGURESFigure 4.27 PMF of a Poisson point process. . . . . . . . . . . . . 113Figure 4.28 CDF of a Poisson point process. . . . . . . . . . . . . 114Figure 4.29 CDF obtained analytically and from a Monte Carlosimulation. . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 4.30 DOS of Gaussian distribution in linear scale. . . . . . 128Figure 4.31 DOS of Gaussian distribution in semi-log scale. . . . . 129Figure 4.32 DOS of power-law distribution for γ = 6 in linear scale.131Figure 4.33 DOS of power-law distribution for γ = 6 in semi-logscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132xivList of Acronymsa-Si:H Hydrogenated amorphous siliconc-Si Crystalline siliconC.B. Conduction bandDOS Density of statesPDF Probability denstiy functionPMF Probability mass functionV.B. Valence bandxvAcknowledgementsI would like to express my gratitude to my supervisor, Dr. Stephen K.O’Leary, for giving me the opportunity for my journey in graduate studies.I thank him for the support and mentoring he provided during the processof completing this thesis.I would also like to mention the unwavering support provided to meby my family and friends, near and far, that kept me going and motivatedthroughout the whole process. This would not have been possible withoutyou all.xviDedicationTo Mommy, Jess, Jena and Arnab.xviiChapter 1IntroductionMicroelectronic circuits and systems have shaped the course of humandevelopment since the genesis of the transistor in 1947 [1]. Progress inconventional microelectronics is usually achieved through reductions in thedimensions of the constituent electronic devices which comprise a micro-electronic circuit or system. The microelectronics industry achieves im-provements through ever-smaller integrated circuits with ever-increased ca-pabilities. In 1965, Gordon Moore, the co-founder of Intel, predicted thatthe number of transistors on an integrated circuit would double every twoyears for about a decade, i.e., until 1975 [2]. This prediction, which is nowfamously referred to as Moore’s Law, has held true for over half a centurynow, and experts within the industry expect it to remain valid for at leastanother decade [3]; see Figure 1.1. Moore’s law provides the microelectronicsindustry with a benchmark against which progress within the industry maybe gauged. Accordingly, smaller, faster, and more economical integrated-circuits are expected to be developed and deployed over the next decade [6].With consumers interested in obtaining slimmer phones, lighter laptops, andsmarter watches, Moore’s law remains a driving force for progress within themicroelectronics field and this seems likely to remain true for many years tocome.1Chapter 1. Introduction1971 1980 1990 2000 2010 2017Year10310410510610710810910101011Number of transistorsFigure 1.1: The number of transistors on an integrated circuit between 1971and 2017. The data in this figure is from Roser and Ritchie [4] and Cavinet al. [5]. The online version of this figure is depicted in color.2Chapter 1. IntroductionWhile conventional microelectronics is focused on achieving progressthrough a reduction in device dimensions, there exists another branch ofelectronics that requires size in order to properly function. Solar panels andelectronic displays, for example, must be able to be spread out over largeareas. Scanners and digital X-ray imagers also must be of sufficient sizein order to be effective. These types of electronic devices are collectivelyreferred to as examples of large area electronic devices. Large area electron-ics, while constituting a small fraction of the overall electronics industry,is playing an increasingly important role in today’s technological offerings,with modern applications ranging the gamut from bio-sensors to image de-tectors [7]. Some applications for large area electronic devices are depictedin Figure 1.2.The semiconducting materials prepared for use in conventional micro-electronic circuits and systems are grown from a melt under pristine con-ditions. In large area electronics, however, the emphasis is on depositingthin-films uniformly and inexpensively over large areas, i.e., of the order of asquare-meter [6, 7]. Accordingly, the philosophy underlying the preparationof materials for large area electronic devices differs from that employed forthe case of conventional microelectronics. Most materials employed for largearea electronic device applications are fabricated from the gaseous phase,and the conditions employed are not nearly as exacting as those requiredfor the preparation of the materials employed for conventional microelec-tronics. Thin-films prepared from the gaseous phase are poly-crystalline oramorphous in form1 [8]. The formation of purely crystalline forms of these1The distribution of atoms within poly-crystalline and amorphous semiconductors ex-3Chapter 1. Introduction(a) Flexible displays.Army to display flexible technologyc©U.S. Army RDECOM, CC BY 2.0.(b) Information displays.IMGP5045 c©Matt Buck, CC BY-SA2.0.(c) OLED lighting systems.EL-Light (OLED) 2.jpg c©Thirteen-fri, CC BY-SA 3.0(d) Radio-Frequency (RF) tags.ATH.ENA Ticket RFID c©DuncanHull, CC BY 2.0(e) Solar panels.NPS Climate Change Response, Pub-lic Domain Mark 1.0Figure 1.2: Examples of the applications of large area electronic devices.The online version of this figure is depicted in color.4Chapter 1. Introductionmaterials from these approaches to deposition is simply not possible.The properties of a given material are determined, in large measure, bythe nature of the atoms and by how these atoms are distributed. Disorder inthe distribution of atoms, such as that present within a poly-crystalline oramorphous semiconductor, plays a key role in shaping the physical propertiesof the material [9]. Although a crystalline semiconductor can also exhibitforms of disorders within it, such as dopants and atomic displacements owingto the presence of lattice vibrations, overall, it still exhibits a long-rangeform of order. Long-range order is completely absent for the case of poly-crystalline and amorphous semiconductors [10, 11].Disorder within semiconductors takes on many forms. There are vari-ations in the bond angles and bond lengths. There are foreign atoms, in-troduced for doping or alloying purposes. There are also vacancies anddangling bonds that are present [12]. Understanding what role the variousforms of disorder play in shaping the properties of these materials has beena focus of intensive investigation for many years. In order to understand therelationship between the material properties and the device performance,an understanding of the role that disorder plays in shaping these materialproperties must first be acquired.The density of states (DOS) function, N(E), where N(E)∆E providesfor the number of electron states between [E,E + ∆E], per unit volume, isan important theoretical construct whereby the optical and electronic prop-erties of a given material may be understood, there being a DOS functionhibits an element of randomness not present within an ideal crystalline semiconductor.This matter is further discussed in Chapter 2.5Chapter 1. Introductionassociated with the conduction band and another associated with the va-lence band. In a defect-free crystalline semiconductor, the DOS functionsterminate abruptly at the band edges of the conduction band and the va-lence band. In contrast, in a disordered semiconductor, the DOS functionsencroach into the otherwise empty gap region, the amount of encroachmentbeing related to the amount of disorder [10, 13]. The DOS functions asso-ciated with a representative defect-free crystalline semiconductor are con-trasted with those associated with a disordered semiconductor in Figure 1.3.The development of an understanding as to how disorder shapes the formof the DOS functions is the ultimate goal of this analysis.In the characterization of a disordered semiconductor, it is often in-structive to consider the distribution of one-electron states within a randompotential profile. In order to provide a solution for this problem, the under-lying distribution of potential fluctuations must be known. In this thesis, Idevelop a model wherein the distribution of potential fluctuations may beexplicitly evaluated and then examine what impact this distribution has onthe corresponding DOS functions. The entire analysis is cast within theframework of the effective mass approximation, each band, i.e., the conduc-tion band and the valence band, having its own random potential profile.This thesis is organized in the following manner. In Chapter 2, thebackground material, relevant for this thesis, is presented. Then, in Chap-ter 3, I provide some perspective on the use of a Monte Carlo simulationapproach, such an approach being employed for the purposes of this anal-ysis. In Chapter 4, a detailed examination of the distribution of potentialfluctuations, and its role in shaping the form of the corresponding DOS6Chapter 1. IntroductionValence band Conduction bandDOSEGap(a) Energy bands in a defect-free crystalline semiconductor.Valence band Conduction bandDOSE(b) Energy bands in a disordered semiconductor.Figure 1.3: The conduction band and valence band DOS functions for arepresentative defect-free crystalline semiconductor and a disordered semi-conductor.7Chapter 1. Introductionfunctions, is presented. Finally, in Chapter 5, the conclusions of this thesisand recommendations for further study are provided.8Chapter 2Background2.1 OverviewIn large area electronics, disordered semiconductors, which include semi-conductors of the poly-crystalline and amorphous forms, are employed [7].The material properties of these disordered semiconductors play an impor-tant role in determining the performance of the large area electronic devicesthat are fabricated from these materials. The disorder that is present withinthese materials plays an important role in shaping the corresponding mate-rial properties [13]. Accordingly, understanding the fundamental role thatdisorder plays in shaping the material properties of disordered semiconduc-tors has been a focus of research within the large area electronics field formany years. This continues to be so today and seems likely to remain so formany years to come.In this chapter, the background material necessary in order to properlytreat disordered semiconductors is provided. As the disorder that is presentwithin a disordered semiconductor is related to the distribution of atomswithin it, I start by providing some insights into the distribution of atomsthat is found within disordered semiconductors. Given that probability the-ory underlies much of this analysis, a brief primer on this subject is also92.2. The atomic distribution for the various forms of condensed matterprovided within the scope of this chapter. At the conclusion of this chapter,all of the details, related to my subsequent analysis, should be in the reader’sgrasp.This chapter is organized in the following manner. In Section 2.2, thedistribution of atoms within the various forms of condensed matter are laidout. Then, in Section 2.3, the distribution of electronic states is introduced.The various forms of disorder, which are present within a disordered semi-conductor, are then introduced in Section 2.4. In Section 2.5, the variousapproaches that have been employed in the evaluation of the DOS functionare discussed, the form of the free electron DOS function being introducedin Section 2.6. Then, I discuss key aspects of probability theory, Section 2.7introducing the subject matter. Sample-space is then presented in Section2.8, set-theory being discussed in Section 2.9. The axioms of probability arepresented in Section 2.10. In Section 2.11, random variables are introduced,the probability density function being presented in Section 2.12. Finally, anumber of key continuous random variable probability density functions arepresented in Section 2.13.2.2 The atomic distribution for the various formsof condensed matterThere are three commonly encountered forms of condensed matter: crys-talline, poly-crystalline, and amorphous. Semiconductors may be preparedin all three forms of condensed matter2 [14]. Crystalline semiconductors are2A liquid form can exist as well.102.2. The atomic distribution for the various forms of condensed matterthose in which the constituent atoms are arranged in a periodic and orderedfashion. Poly-crystalline semiconductors, by way of contrast, are found toexhibit small crystallites, i.e., crystallized regions, that are randomly ori-ented with respect to each other, each crystalline region being separatedfrom its neighbouring crystalline region with grain boundaries. In amor-phous semiconductors, however, there is a complete absence of crystallinity,the bond lengths and bond angles exhibiting random variations from bond-to-bond [10, 11, 14]. Two-dimensional representations of the atomic distri-butions corresponding to crystalline, poly-crystalline, and amorphous semi-conductors are depicted in Figure 2.1.Semiconductors in their crystalline form may be characterized in termsof the long-range order that they exhibit. Through a periodic repetitionof a basic unit cell, an entire crystal structure may be generated. Disor-dered semiconductors, by way of contrast, lack this long-range order. Eachlocal region within a disordered semiconductor is distinct from every otherlocal region. Accordingly, disordered semiconductors lack the symmetriesthat are present within their crystalline counterparts. In actual fact, whilethey may lack long-range order, most disordered semiconductors, such ashydrogenated amorphous silicon (a-Si:H), exhibit a form of short-range or-der [15]. That is, many of the atoms within such a disordered semiconductorengage in the same type of bonding that is present within the correspondingcrystalline form, the nearest-neighbor bonding lengths and bonding angleswithin crystalline silicon (c-Si), for example, being very similar to thosefound within a-Si:H. As the nature of the chemical bonding that occurswithin a disordered semiconductor is similar to that which occurs within its112.2. The atomic distribution for the various forms of condensed matter(a) Crystalline semiconductor.(b) Poly-crystalline semiconductor.(c) Amorphous semiconductor.Figure 2.1: Two-dimensional representations of the atomic distributions cor-responding to crystalline, poly-crystalline, and amorphous semiconductors.The online version of this figure is depicted in color.122.3. The distribution of electronic statescrystalline counterpart, many of the electronic properties are expected tobe similar; the nature of the chemical bonds is one of the principal factorsthat determines the characteristics of a given material [16]. For the pur-poses of this analysis, I primarily focus on properties that are unique to thedisordered state.2.3 The distribution of electronic statesIn order to quantitatively characterize the energy levels that are presentat the atomic level, a quantum mechanical approach must be employed.Quantum mechanics tells us that an isolated atom exhibits a discrete spec-trum of energy levels [17]; see Figure 2.2. This model of the atom, and itscorresponding electronic states, acts as a cornerstone of modern chemistry,equipping the scientific community with a framework within which muchcan be explained.If two isolated atoms are brought together, as seen in Figure 2.3, thewave-functions associated with the individual atoms will overlap and thiswill split the energy levels associated with the individual atoms [17]. Thissplitting will increase as the atoms are brought together, i.e., as the wave-functional overlap increases. If N atoms are brought together, the wave-functions associated with the individual atoms will overlap and this willsplit the energy levels associated with the individual atoms into N levels.Condensed matter may be viewed as the bringing together of N atoms,where N approaches infinity. In such a case, the splitting of energy levelscorresponds to the introduction of bands of allowed energy levels, i.e., a132.3. The distribution of electronic statesNucleusEnergy levelsFigure 2.2: Energy levels of an isolated atom. The online version of thisfigure is depicted in color.142.3. The distribution of electronic statescontinuum, separated by energy levels within which there are no electronicstates, i.e., gaps [18].Consider, for example, the case of N silicon atoms. If the atoms areseparated through an infinite separation, the energy levels will assume thoseof the isolated silicon atom. There will thus be 6N valence electron statesassociated with the highest energy states; for each silicon atom, there are sixvalence electron states associated with the outer p sub-shell, and two valenceelectron states associated with the outer s sub-shell (this being at a lowerenergy than the p sub-shell), these electron states being occupied by fourvalence electrons. As the distance between the atoms is reduced, however,the wave-functions associated with the individual silicon atoms will overlap,and a splitting in the energy levels will occur. As the number of atomsbecome infinite, i.e., as N →∞, the N discrete levels associated with eachisolated atomic energy level becomes a continuum, i.e., they form a band.At the equilibrium inter-atomic distance, which is determined through aminimization of the free energy, the highest energy band is comprised of4N electronic states, the next highest energy band also being comprised of4N electronic states. At zero temperature, all of the 4N electronic statesassociated with the highest energy band are empty, this band being referredto as the conduction band, all of the 4N electronic states associated withthe next highest energy band being occupied, this band being referred to asthe valence band. The difference in energy between the lowest part of theconduction band and the highest part of the valence band is referred to asthe energy gap, E [17, 19]. Figure 2.4 shows an example of the formationof bands and the energy gap. Many of the properties of a semiconductor152.3. The distribution of electronic statesV(r)Figure 2.3: Energy levels splitting when two atoms are brought together.psInteratomic distanceEnergyaConduction bandValence bandBand gapEcEV6N States2N States4N states6N States2N States8N States4N statesFigure 2.4: Formation of the energy bands as the distance between the atomsreduces to its equilibrium separation.162.3. The distribution of electronic statesTable 2.1: Table of energy gaps associated with various semiconductors.The data in this table is from Kittel [20].Material Symbol Bandgap at 300 K (eV)Silicon Si 1.11Germanium Ge 0.66Gallium arsenide GaAs 1.43Gallium phosphide GaP 2.25Indium phosphide InP 1.27are directly related to this gap. A list of energy gaps, for a variety ofsemiconductor materials, is depicted in Table 2.1.In semiconductor physics, the distribution of electronic states, i.e., howthe states are distributed with respect to energy, is widely recognized asplaying an important role in determining the material properties. Typically,the distribution of electronic states is characterized in terms of a DOS func-tion, N(E), N(E)∆E denoting the number of electronic states, betweenenergies [E,E + ∆E], per unit volume, there being one DOS function as-sociated with the conduction band, another DOS function being associatedwith the valence band. In a defect-free crystalline semiconductor, these DOSfunctions terminate abruptly at the valence band and conduction band bandedges, there being no electronic states in the gap region between these bandedges. In contrast, in a disordered semiconductor, these DOS functions en-croach into the otherwise empty gap regions, the amount of encroachmentincreasing in response to increases in the amount of disorder [10]. Under-standing how disorder influences the shape of these DOS functions will bethe primary goal of this analysis.172.4. Forms of disorder2.4 Forms of disorderDisorder takes on many forms in a disordered semiconductor. Withinamorphous semiconductors themselves, there are variations in the bondlengths and bond angles, and in the geometry as to how the bonds them-selves are arranged. There are vacancies and voids that are distributedthroughout the material. There is the presence of foreign atoms and impu-rities, these sometimes being deliberately introduced for doping and alloyingpurposes [15]. Within poly-crystalline semiconductors, there are also grainboundaries that are present between the randomly oriented crystallites thatare distributed throughout the material. While each type of disorder hasits own impact on the amount of disorder that is present, for the purposesof this analysis, I will treat disorder as being a fundamental property of thematerial, and present all forms of disorder as contributing to some overallmeasure of disorder. This follows in the tradition of the analysis of Cody[21], in which it was demonstrated that the thermal and structural forms ofdisorder are equivalent in their effect on the material properties.2.5 Approaches employed for DOS evaluations ofdisordered semiconductorsThe role that disorder plays in shaping the form of the DOS functionshas been the subject of considerable analysis over the years. In this sec-tion, a variety of approaches, which have been employed in the past, areconsidered for the evaluation of the DOS functions associated with disor-182.5. Approaches employed for DOS evaluations of disordered semiconductorsdered semiconductors. The strengths and weaknesses of each approach arediscussed.Initial DOS determinations corresponding to disordered semiconductorsfocused on determining the distribution of electronic states from first prin-ciple evaluations, involving treating the individual atoms within such asemiconductor and their quantum mechanically determined wave-functions.Typically, a model was employed for the distribution of atoms, which rep-resented a sub-set of the overall structure. Disorder was introduced, eitherthrough the presence of vacancies or through the placement of foreign atoms.Examples of such calculations include the treatment of a-Si:H through theuse of the coherent potential approximation [22, 23].Unfortunately, the computational resources demanded of such an anal-ysis have rendered them inherently limited. State-of-the-art models arepresently at most 1,000 atoms in size. Characterizing the disorder thatis present within a disordered semiconductor in terms of the probabilities ofevents, such as how the atoms are distributed, will limit the resolution ofsuch a computation to events that have probabilities that are at least 10−3or greater. Unfortunately, many of the most interesting aspects related tothe role that disorder plays in shaping the electronic properties occur owingto statistical fluctuations or events that occur with probabilities that are assmall as 10−10 and below. Therefore, such an approach to the treatment ofdisordered semiconductors is inherently limited.The treatment of disorder within the framework of the single electroneffective mass perspective has been demonstrated to provide a more effectiveapproach. A variety of analyzes, wherein the interaction of an individual192.6. Free electron evaluation of the DOS functionelectron, endowed with an effective mass, with a random potential, havebeen presented. Examples of such an analysis include the analysis of John etal. [24] and others [25, 26], in which the distribution of potential fluctuationsis treated in concert with the potential well treatment of the electron states.The semi-classical treatment of O’Leary et al. [13] provides yet anotherexample. In most cases, a Gaussian distribution of potential fluctuations issimply assumed. It is often argued that in a disordered semiconductor thereis a sum over a large number of sources of disorder and, through the centrallimit theorem, this means that the distribution of potential fluctuations isGaussian in form.O’Leary and Lim [27], within the framework of an effective mass treat-ment, view the various sources of disorder within a disordered semiconductoras corresponding to individual “point” defects. The total potential at anygiven position is thus viewed as being the sum of potentials associated withthe individual “point” defects. This perspective allows one to go beyondthe elementary Gaussian distribution of potential fluctuations assumption.O’Leary and Lim [27] demonstrate that for most amorphous semiconductors,the central limit theorem is only marginally valid.2.6 Free electron evaluation of the DOS functionIt is instructive to consider the form of the conduction band and valenceband DOS functions within the framework of a free electron treatment. Thatis, it is useful to evaluate the form of the DOS functions corresponding to“constant” conduction band and valence band potential profiles. In order to202.6. Free electron evaluation of the DOS functionperform such analysis, I pursue the course of analysis outlined by Streetmanand Banarjee [17].Boundary conditions are placed on the electrons, where the electrons areconsidered to be in a cube, with side lengths of L. The three-dimensionalwave function for the electrons is thusψ = A exp[j2piL(nxx+ nyy + nzz)]. (2.1)Considering that Schro¨dinger equation for the free-electron may be expressedas,− }22m∇2ψ = Eψ, (2.2)i.e., the potential is set to zero, substituting the formal wave function intoEq. (2.2), it is seen thatE =}22m[(2piLnx)2+(2piLny)2+(2piLnz)2]. (2.3)The DOS corresponds to the number of electronic states that are allowedper unit volume with respect to energy. In three-dimensional space, the ~k-vectors may be expressed askx = 2pinxL,ky = 2pinyL,kz = 2pinzL,and the three-dimensional volume, V = L3. The number of states in the212.7. Introduction to probabilityvolume of a ~k-space is thusL3(2pi)3∆k× 2, (2.4)where the factor of 2 corresponds to the two possible spin states. The equa-tion can then be written in terms of E-space using the following relationship,N(E)∆E =2(2pi)3(∆k). (2.5)The band structure is thenE(k) =}2k22m∗. (2.6)Therefore, the relationship between ~k and E can be written as,dk =[√m∗21}]1√EdE. (2.7)Using this relationship, the DOS for a three-dimensional structure is asfollows,N(E)dE =2(2pi)24pik2dk=√2pi2(m∗}2)3/2E1/2dE.(2.8)2.7 Introduction to probabilityProbability, although steeped in tradition, has been an active field inresearch for many years. Probability theory has found applications in a222.7. Introduction to probabilitydiverse range of fields, ranging the gamut from the physical sciences tothe humanities. Researchers have been able to use probabilistic tools inorder to make improved predictions about the outcome of seemingly randomexperiments3 and allowing one to quantitatively predict the frequency withwhich one obtains a given outcome. By definition, probability provides ameasure of the chance of an event occurring. It is a tool that provides aquantitative measure of the frequency of a given event and allows for theprediction of future events, i.e., it provides a basis for probabilistic reasoning[28].In a given experiment with many possible outcomes, such as rolling a diceor flipping a coin, probability determines the likelihood of getting a certainoutcome, such as rolling a six for the case of rolling a dice or obtaining atail for the case of flipping a coin. When presented with a set of possibleevents, probability serves as a tool that determines the chances of theseoutcomes occurring in different situations [29]. As such, it is possible tobuild up a probabilistic model for different situations, which includes allpossible outcomes, each such event having a certain likelihood of occurring.In the following sections, a brief review on the theoretical foundationsof probability theory is provided. In particular, I explore the definitionsunderlying probability theory, starting with its axioms, and examine therules that must be followed for an effective course of probabilistic reasoningto be valid. I also aim to provide the background required in order toprobabilistically model a given problem at hand. In the interests of brevity,3By random experiment, I am referring to an experiment with a number of possibleoutcomes, where the outcome of a given experimental realization is not known prior tothe experiment actually being performed.232.8. Sample-spaceonly topics that are the most relevant to the subject matter at hand withinthe scope of this analysis are discussed.2.8 Sample-spaceA probabilistic model provides a mathematical analog of an experiment,consistent with the real conditions underlying such an experiment. In or-der to build a probabilistic model out of an experiment, all of the possibleoutcomes of the experiment must be considered. The complete set of ex-perimental outcomes within a probabilistic experiment is referred to as thesample-space [29]. Each element in sample-space, S, corresponds to a pos-sible outcome of the experiment. Consider, for example, the experimentof rolling a six-sided dice, where the outcome will be any number, rang-ing from one to six. For this case, the sample-space would be comprisedof six elements, ranging from one through six, inclusive. Similarly, in theexperiment of flipping a coin, the sample-space would be comprised of twoelements, heads and tail, these corresponding to all possible outcomes of thisparticular experiment.The experiments provided thus far correspond to sample-spaces whichare comprised of a finite number of points. The concept of sample-spacemaybe more broadly applied to cases in which there exists an infinite numberof points. For example, the experiment of selecting a positive integer wouldlead to a sample-space comprised of an infinite number of possible outcomes,i.e., 1, 2, 3,. . . . Another possible experiment that would correspond to asample-space with an infinite number of elements would be the selection of242.8. Sample-spacenumbers between 0 and 1. The sample-space corresponding to this particularexperiment would lead to a tabulation of all numbers between 0 and 1.The two previously mentioned examples that have been provided thusfar correspond to sample-spaces with infinite number of elements. There is,however, a clear distinction between these cases. In the former example, eachelement in the sample-space can be sequentially tabulated. Thus, assumingthat one had a sufficient amount of time, one could eventually “count” thenumber of elements up to and including a certain finite value. In contrast,the latter example presents a more challenging type of infinity. In particular,each value between 0 and 1 corresponds to an infinite number of other valuesbetween 0 and 1. One therefore can not even start to tabulate all of thevalues between 0 and 1. Thus, the former sample-space is referred to ashaving a “countable” infinity whilst the latter sample-space is referred to ashaving an “uncountable” infinity [30]. Further discussions on these mattersmay be found in the scientific literature corresponding to number theory.Collectively, a given set of sample points from sample-space is consideredto be an event, which is denoted using capital letters; for example, event A.The probability of each of such an event, expressed as P[A] for event A, forexample, provides for the probability that such an event occurs [28]. In thecase of rolling a six sided dice, for example, event A can refer to obtainingan even number from the roll, the corresponding P[A] being 1/2.252.9. Set theory2.9 Set theorySet theory outlines the behaviour of the elements in a set and providesmeans of characterizing them individually and collectively. Various types ofset operations may be used to describe the interaction of sets, which in turnplay an important role in defining the rules and use of probability theory[29]. Some of the basic set operations have the following definitions:i. Null set : An empty set with no possible events is referred to as thenull set, denoted by φ.ii. Complement: The complement of a set S represents all elements inthe complete set of elements that are not included in the set S, whichbelongs to the universal set, denoted by Sc.iii. Union: The union of two events, A and B, is a set of all the outcomesthat are in either events A or B, denoted by A∪B.iv. Intersection: The intersection of two events, A and B, is a set ofoutcomes that occur in both events A and B, denoted by A∩B.The set operations are shown in the Venn diagrams depicted in Figure2.5.2.10 The axioms of probabilityFor a probabilistic model to be valid, the sample-space, S, has to bemodeled in a manner such that the events corresponding to a given exper-iment comply with the axioms of probability. The probability of each of262.10. The axioms of probabilitySA B(a) A∩B represents a null set.SAAc(b) Ac represents complement of A.SA B(c) A∪B represents union.SA B(d) A∩B represents intersection.Figure 2.5: Venn diagrams of set operations represented by the shaded re-gion. The online version of this figure is depicted in color.272.11. Random variablesthe events, P[A] for event A, must conform with the following axioms ofprobability theory [31]:i. The probability of a given event must be greater than or equal to zero,i.e., non-negative, i.e., P[A]≥0.ii. The probability of the sample-space itself, S, must equal to unity, i.e.,P[S] = 1.iii. The addition of mutually exclusive events of a given sample-space isequal to their sum. That is, P[A∪B] = P[A]+P[B], where A and B aredisjoint events and ∪ denotes the union operation.Various probabilistic rules are derived from the axioms of probability,these providing the basis for probabilistic models [28]. These probabilisticmodels make it possible to use probabilistic reasoning for the calculation ofthe probabilities associated with various events.2.11 Random variablesIf each element of the sample-space of a given probabilistic experiment isassigned a scalar number, the resultant numbers are referred to as randomvariables. In the experiment of flipping a coin, for example, with the sample-space elements being either a head or a tail, random variables can be allottedas one for the head and zero for the tail. A random variable is consideredto be discrete if its values are in a finite or in a countable range, such as inan experiment of flipping a coin. This can include a set of values that havean infinite range with countable values, such as the set of natural numbers282.12. The probability density functionwhich has an infinite range of numbers which are discrete. In contrast, forthe case of experiments with ‘continuous’ sample values, the correspondingrandom variables may assume values over a continuous range. Continuousrandom variables have an uncountable number of possible values, such as inan experiment of the time it takes people to cross a road. The continuousrandom variable takes into account all of the points associated within a givenrange. Random variables can also be considered mixed if it has both discreteand continuous components [29]. Figure 2.6 demonstrates an example of howrandom variables may be mapped from the underlying sample-space.2.12 The probability density functionThe probability associated with each of the values of the discrete randomoutcomes of an experiment may be expressed as a function, referred to asthe probability mass function (PMF) [29]. Considering the discrete randomvariable, X, the PMF states the probability of it taking on a certain value ofx. Shown in Figure 2.7 is an example of the PMF, pX , for the experiment offlipping a coin, where the sample-space consists of the outcomes ‘Head’ and‘Tail’, and the random variable x1 is assigned based on the outcome ‘Head’and x2 based on the outcome ‘Tail’.A counterpart to the PMF that is valid for continuous random variablesis the probability density function (PDF). However, for continuous randomvariables, the probability for the random variables to take a definite valueis zero due to its uncountable nature. Therefore, considering the continuousrandom variable, X, the PDF states the probability of it falling within a292.12. The probability density functionABCDRandom Variable, X𝑥1 𝑥2 𝑥3 𝑥4𝑥X(x) =x1 for Event A,x2 for Event B,x3 for Event C,x4 for Event D.Figure 2.6: The mapping of events A, B, C and D in a sample-space to anumber line representing the random variables, X(x). The online version ofthis figure is depicted in color.302.12. The probability density function1(head)2(tail)0.51PMFxxxFigure 2.7: PMF of an experiment of flipping a coin. Note that the PMF isnot defined at any other points other than x1 and x2. The online version ofthis figure is depicted in color.312.12. The probability density functionrange of values of x. The area under the curve of the PDF over a range isthe probability of the random variable to be within that range. The PDFis a non-negative function and the sum over all the possible probabilitiesin the complete range, that is the total area under the curve of the PDFfunction, must be unity [29]. The equation below shows an example of theprobability of the random variable X to take values in the range from x1and x2, where the PDF of X is fx(X), i.e.,P[x1 ≤ X ≤ x2] =∫ x2x1fx(u)du. (2.9)For both discrete and continuous random variables, it is possible to definethe cumulative distribution function (CDF). The CDF of a given randomvariable, X, is the cumulative probability for the random variable to take avalue up until x [28]. The CDF of a given random variable, X, is denotedby FX(x). For discrete random variables, the CDF may be expressed as,FX(x) = P [X ≤ x] =∑k≤xpx(k), (2.10)while for continuous random variables, the CDF may be expressed as,Fx(x) =x∫−∞fX(u)du. (2.11)The PDF, for both discrete and continuous random variables, can hence beexpressed as,fx(x) =dFX(x)dx. (2.12)322.13. Distributions of continuous random variablesThe CDF is a monotonically increasing function that starts at zero andhas a final value of unity as the probability of the entire sample-space mustbe unity. The definition for the CDF is the same for both discrete andcontinuous random variables, and this allows for calculations to be done forall types of random variables. For discrete random variables, the CDF takesthe form of a monotonically increasing function, that increases in steps,while for continuous random variable, it is a non-decreasing function [28].The following section looks into the characteristics of the PDF and CDF forvarious common distributions of continuous random variables.2.13 Distributions of continuous randomvariablesVarious continuous random variables that follow similar distributions canbe classified in terms of their PDF. Some examples of popular distributionsare listed below [32].Uniform distributionUniformly distributed random variables have the following PDF,fx(x) =1b−a , if a ≤ x ≤ b,0, elsewhere,(2.13)332.13. Distributions of continuous random variablesand CDF,FX(x) =0 for x < a,x−ab−a for a ≤ x ≤ b,1, if x > b.(2.14)This represents a finite number of random variables that have an equallylikely probability of occurring in the range between a and b. The uniformdistribution is often used as a reference distribution due to the equal prob-ability it represents for each range of random variables. Subsequently, it isutilized in random number generators and in the transformation of otherdistributions, which will be further explored in Chapter 3. The graphs forthe PDF and CDF are shown in Figure 2.8.Exponential distributionThe exponential distribution has the following PDF;fx(x) =λe−λx, x ≥ 00, x < 0.(2.15)where λ represents the rate and is always greater than 0. The exponentialdistribution has the following CDF;FX(x) =1− e−λx, x ≥ 00, x < 0.(2.16)The exponential distribution represents a distribution which follows thePoisson process, i.e., a random process which increments with respect to342.13. Distributions of continuous random variables01b−aa b xf (x)(a) Uniform distribution PDF. Uniform Distribution PDF SVG c©IkamusumeFanCC BY-SA 3.00 a b x1F (x)(b) Uniform distribution CDF. Uniform cdf c©IkamusumeFan CC BY-SA 3.0Figure 2.8: PDF and CDF of an uniform distribution. The online versionof this figure is depicted in color.352.13. Distributions of continuous random variablesa rate. One of its unique properties is its memory-less characteristic, whichmeans that the probability is not affected by prior results and the proba-bility of each interval of the increment is newly started. The exponentialdistribution is often used to represent natural processes with a rate, suchas the radioactive decay of materials. The PDF and CDF curves of theexponential distribution are shown in Figure 2.9.Gaussian distributionThe Gaussian distribution, also known as the normal distribution, hasthe following PDFfx(x) =1√2piσ2e−(x−µ)22σ2 , (2.17)and CDFFX(x) =12[1 + erf(x− µσ√2)], (2.18)where µ is the mean about which the PDF is symmetric, and σ is thestandard deviation of the PDF. The PDF of the Gaussian distribution is abell shaped distribution that is symmetric about its mean, µ. The standarddeviation, σ, attributes the spread of the PDF; the smaller the standarddeviation, the more abrupt the transition from zero to unity for the CDF.This is shown in Figure 2.10. The Gaussian distribution is an importantdistribution, and it can be found to occur in many natural processes. Usingthe basis of the Central Limit Theorem, which states that it is possible todescribe the probability of something if there are a sufficiently large numberof events available, regardless of not being able to define the probabilityof the individual event, it is possible to describe events with the Gaussiandistribution. The Central Limit Theorem is discussed further in the next362.13. Distributions of continuous random variables(a) Exponential distribution PDF. Exponential pdf c©Skbkekas, CC BY 3.0.(b) Exponential distribution CDF. Exponential cdf c©Skbkekas, CC BY 3.0.Figure 2.9: PDF and CDF of exponential distribution with different valuesof λ. The online version of this figure is depicted in color.372.14. Central limit theoremsection of this chapter.Cauchy distributionThe Cauchy distribution has a similar PDF to the Gaussian distribution,and it also follows the same bell shaped curve. However, the Cauchy distri-bution does not have an expected value, i.e., it does not exist. Its varianceis undefined and it has no mean. The PDF is,fx(x) =1piγ[1 +(x−x0γ)2] (2.19)The corresponding CDF is;FX(x) =1piarctan(x− x0γ)+12(2.20)It is an interesting distribution that is useful for modeling the ratio of twonormal random variables, and exponential mapping. Figure 2.11 shows ex-amples of the PDF and CDF of the Cauchy distribution.2.14 Central limit theoremThe central limit theorem effectively describes why the Gaussian distri-bution can be attributed to various circumstances, especially those found innatural processes. The theorem identifies that when considering the sum ofa series of independently and identically distributed random variables witha finite mean and variance, as the number of random variables increases, theCDF approaches that of a Gaussian distribution [33]. Considering Sn to be382.14. Central limit theorem(a) Gaussian distribution PDF. Normal Distribution PDF Inductiveload in thepublic domain.(b) Gaussian distribution CDF. Normal Distribution CDF Inductiveload in thepublic domain.Figure 2.10: PDF and CDF of Gaussian distribution with varying mean, µ,and standard deviation, σ The online version of this figure is depicted incolor.392.14. Central limit theorem(a) Cauchy distribution PDF. Cauchy pdf c©Skbkekas CCBY 3.0(b) Cauchy distribution CDF. Cauchy cdf c©Skbkekas CCBY 3.0Figure 2.11: PDF and CDF of Cauchy distribution. The online version ofthis figure is depicted in color.402.14. Central limit theoremthe sum of n random variables,Sn = X1 +X2 + ...+Xn, (2.21)the mean of the sequence is then the sum of that associated with the in-dividual random variables, i.e., nµ, where µ is the identical mean of eachof the random variables. Similarly, the expected value for the sequences isnσ2. Thus, normalizing the sequence, Sn, the normalized random variablecan be defined asZn =Sn − nµσ√n. (2.22)The CDF of this random variable is then P[Zn ≤ z], which, as n approachesinfinity, becomeslimn→∞P[Zn ≤ z] =1√2pi∫ z−∞e−x22 dx. (2.23)This, in turn, is comparable to the Gaussian CDF, seen in Eq. (2.18).41Chapter 3The Monte Carlo method3.1 OverviewMonte Carlo simulations have a long and rich tradition of utility withinthe physical sciences and engineering. In a Monte Carlo simulation, oneattempts to simulate the seemingly random permutations that occur eitherin a physical process or a system. Typically, one uses a random number gen-erator in order to attempt to sample over the entire sample-space of a givenexperiment, the amount of time spent at any point in the aforementionedsample-space being roughly proportional to the probability of that particu-lar realization of the experiment. Typically, the Monte Carlo method allowsfor the sampling of a simulated random process, obtaining results from mul-tiple realizations of the experiment in question [34]. The results of such asimulation are useful in estimating the outcomes of seemingly random cir-cumstances and can be used in order to obtain solutions to difficult problemswithout requiring the use of analytical approaches [35, 36].For the purposes of this analysis, Monte Carlo simulations are used inorder to examine how disorder shapes the distribution of potential fluctu-ations within a disordered semiconductor. In this case, the realizations ofthe “experiment” in question correspond to specific distributions of defect423.2. The random number generatorpotentials, the overall potential corresponding to the sum over these defectpotentials. The overall potential is then a random variable, its value beinga function of both the distribution of defect sources and the potential pro-file associated with each such source. Clearly, an infinite number of suchdistributions exists. In order for my results to be statistically meaningful, Iwill perform the “experiment” a large number of times and determine thestatistics based on the outcomes of these multiple realizations. With thedistribution of potential fluctuations determined, it will then be possible toevaluate the corresponding DOS functions, which is the ultimate aim of thisthesis.This chapter is organized in the following manner. Section 3.2 looksinto how random number generators are used as a basis to form a modelof a random experiment. The following section, i.e., Section 3.3, discusseshow results acquired from the sampling over the sample-space can be usedin order to evaluate the corresponding probability distribution functions.Then, Section 3.4 examines how a random number generator may be usedin order to simulate any arbitrary probability distribution function. Finally,in Section 3.5, I provide examples of how the Monte Carlo method may beused and the results obtained.3.2 The random number generatorMonte Carlo simulations require the use of a random number generator.Such a generator will produce randomly distributed values over a givenrange. Thus, the random number generator can be considered to be a digital433.2. The random number generatorversion of a tool that will create randomness [37]. Real-life examples ofrandomness include the shuffling of a deck of cards or the tossing of a pairof dice. A random number generator provides a series of numbers which donot follow any recognizable pattern, i.e., the next number in such a sequencecan not be predicted from the previous sequence of numbers. Thus, the setof numbers produced from such a random number generator appears tofollow a random nature [38]. This section will explore how random numbergenerators are implemented and how they can be utilized as a modeling tool.Unfortunately, in reality, it is not possible to generate a sequence oftruly random numbers algorithmically. Computational algorithms that aimto generate random numbers must follow a deterministic set of instructions,and hence, they are actually pseudo-random number generators. Typically,a seed value is utilized in an algorithm that applies this value to a math-ematical formula that is aimed at generating a sequence of numbers thatappear to be random [37]. How random a pseudo-random number generatoractually is has been the subject of much analysis through the years [38, 39].Most random number generators produce a distribution of random num-bers that are uniformly distributed over some set range. The correspondingPDF has been seen in Chapter 2, and is known as the uniform distributionfunction. As any range of data, i.e., between a and b, can be scaled fromthat between 0 and 1, most random number generators focus on generatingrandom numbers between 0 and 1 and then map those numbers into therequired range of numbers, i.e., through a uniform mapping, one can takethe uniform distribution between 0 and 1 and map it into a uniform distri-bution between a and b, where a and b are arbitrary numbers; it is implicitly443.2. The random number generatorassumed that b is greater than a. The algorithm used typically requires theuser to input a seed value or automatically chooses a seed where, as long asthe seed value and the formulae utilized to generate the sequence of num-bers is unknown to the user, it can serve as a sufficient random numbergeneration tool [38].In order to chose this initial seed value, various methods can be applied,such as utilizing the value of the time during which the process is initiated.The value obtained is then mathematically manipulated, and from an itera-tive process of such manipulations, a sequence of seemingly random numbersmay be obtained. Strategies that have been employed include that suggestedby Herrero-Collantes et al. [39], in which a generator may utilize the timereceived from the seed value to process the next number in the seeminglyrandom sequence. Various forms of algorithms exist, these ensuring thatthe numbers generated are unpredictable. Typically, each realization of therandom sequence requires the use of a different seed value, which in turn,produces different random sequences [37]. Henceforth, the pseudo-randomnumber generators will be referred to simply as random number generators.The different algorithms that are in use and have been implementedare all subject to certain criteria that make the random number generatora reliable one for use. A commonly used structure of a random numbergenerator uses the linear congruential method [39]. The random numbergenerator initiates with the following sequence,Xn+1 = (aXn + c) modm, n ≥ 0, (3.1)453.2. The random number generatorwhere Xi is the i -th output, X0 is the initial seed value, a is considered asthe multiplier, and m is referred to as the modulus. The sequence producedwill eventually repeat, the length of such a random sequence being known asthe period length of the generator. However, the seed and multiplier valuesused should be able to produce random number sequences sufficient for theexperiment at hand [38].It is instructive to evaluate the performance of such a random numbergenerator. This may be seen through an evaluation of the distribution ofvalues that such a random number generator produces between 0 and 1.For example, in Figure 3.1, I depict the distribution of numbers that agiven random number generator produces, between 0 and 1, that fall into 25uniformly distributed intervals, each interval ranging over 4% of the overallrange of numbers considered in this analysis. The counts depicted on they-axis display the number of realizations within a given interval. I considera variable number of realizations of this “experiment” for the different cases.For Figure 3.1a, I only consider 100 realizations. It is noted that the countdistribution is not very uniform. For Figure 3.1b, however, I consider 1,000realizations. It is noted that much more uniformity occurs, i.e., there is lessvariations from interval-to-interval. Finally, in Figure 3.1c, I consider 10,000realizations. Clearly, this distribution is much more uniform. Ultimately,with an infinite number of realizations, a completely uniform distributionwill emerge, this being known as the continuum limit [40]. Such a randomnumber generator is employed for the purposes of this thesis.463.2. The random number generator0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Number generated012345678Counts(a) 100 realizations.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Number generated0102030405060Counts(b) 1,000 realizations.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Number generated050100150200250300350400450Counts(c) 10,000 realizations.Figure 3.1: Histogram of 100, 1,000, and 10,000 numbers from a randomnumber generator using Matlab distributed over 25 bins of equal size between0 and 1. The online version of this figure is depicted in color.473.3. Empirical distribution function3.3 Empirical distribution functionFor the purposes of executing a Monte Carlo simulation, once the ran-dom number generator has been utilized in order to produce random real-izations, it is required for the simulation to be sampled. The sampling ofeach realization produces an empirical distribution function, which in thecontinuum limit, should approach that of the actual distribution function[40]. The simulation is realized multiple times in the Monte Carlo process. Ifthe outcomes of the “experiment” correspond to a finite number of possiblerealizations, i.e., the sample-space consists of a finite number of points, bydetermining the number of realizations of a given point in the sample-space,in the continuum limit, the probability of each possible outcome may beevaluated. For the case of a discrete random variable, i.e., a number as-signed to each point in the sample-space, the PMF may be determined. Forthe general treatment of random variables of either a discrete or continuousnature, the CDF is the most useful function, as it avoids the singularitiesthat plague its PDF counterpart [28].Within the context of a Monte Carlo simulation, for the case of a scalaroutput corresponding to such a simulation, the i -th realization may be ex-pressed as xi [41]. The empirical CDF may thus be written asFxemp(z) =1MM∑i=1u(z − xi), (3.2)where M denotes the number of realizations and u( · ) represents the step-483.4. Random number generators of other distributionsfunction, i.e.,u(z) =1, z > 0,0, z < 0.(3.3)In the continuum limit, i.e., M → ∞, this empirical distribution functionshould approach the actual distribution function.At this juncture, it is useful to consider the uniform number generator. InFigure 3.2, I plot the empirical CDF associated with the output of a randomnumber generator for the case of the random numbers uniformly distributedbetween 0 and 1. Note that as the number of realizations approaches infinity,that the staircase behaviour found for the cases of M = 100, M = 1, 000,etc., approaches the ideal result corresponding to the uniform distributionbetween 0 and 1. This is to be expected in the continuum limit, i.e., aslimM→∞Femp(x)→ F (x). (3.4)3.4 Random number generators of otherdistributionsThus far, I have considered random number generators that ideally pro-duce a uniform distribution of numbers over a given range. It is often thecase, however, that one must instead provide for a sequence of randomnumbers that are instead distributed in a non-uniform manner. In Chap-ter 2, Section 2.13, for example, in addition to providing for the uniformdistribution, a number of non-uniform distributions were considered. For493.4. Random number generators of other distributions0 0.2 0.4 0.6 0.8 1x00.20.40.60.81F(x)Empirical CDFM = 100M = 1,000M = 10,000Figure 3.2: The empirical CDF obtained from uniformly distributed randomnumbers from a random number generator with M = 100, 1,000 and 10,000iterations. The online version of this figure is depicted in color.503.4. Random number generators of other distributionsexample, the exponential distribution is a non-uniform distribution. Othernon-uniform distributions introduced in this section include the Gaussiandistribution and the Cauchy distribution. There are an infinite number ofsuch distributions, each with its own PDF and CDF. Typically, the genera-tion of a non-uniform distribution is performed through the transformationof a uniform distribution into a non-uniform distribution, through a pro-cess called the Inverse Transform method [42]. Intuitively, this may be donethrough either dilating or contracting the scale of the corresponding uniformdistribution.The transformation of a uniformly distributed random number generatorinto an arbitrarily distributed random number generator may be facilitatedby examining the corresponding CDFs in the Inverse Transform method [40].Consider, for example, the uniform and exponentially distributed CDFs, asseen in Figures 2.8 and 2.9. A properly designed uniform random numbergenerator will ideally generate a uniform distribution of numbers, which,in the continuum limit, will approach the uniform distribution depicted inFigure 3.1; in this case, I restrict the numbers to be between 0 and 1,this being the usual case for a random number generator. In order for anexponential distribution to be created, each number generated through theuniform random number generator must be mapped into a correspondingnumber through a scaling transformation. This must be done such that thetransformed CDF conforms with the exponential CDF seen in Chapter 2;recall page 37.For the general case of transforming a random number generator result,with an output distributed uniformly between 0 and 1, into a random num-513.4. Random number generators of other distributionsber generator with an output distribution, Y , characterized with the CDFFY (z), the transformation may be expressed asxi = FY (yi), (3.5)where xi is the i -th initially produced uniformly distributed random numberand yi is the corresponding i -th transformed random number [43]. Thereason for this equivalence is that the accumulated probabilities of such aselection are equivalent to each other, i.e., the uniform distribution functionFx(xi) = xi; (3.6)see Figure 3.3. Thus, the i -th selection of the random number yi4 may beexpressed asyi = F−1y (xi). (3.7)Finally, we present an example of such a random number generator usingthe specific example of the exponential distribution. For the λ parameterset equal to unity, the CDF of the exponential distribution isFY (z) =1− e−z, z ≥ 00, z < 0. (3.8)4Some technical challenges arise when the inverse presents a challenge, i.e., can not besolved analytically [40].523.5. Monte Carlo simulationsThus, the transformation represented in Eq. (3.7) becomes,xi = 1− e−yior⇒ yi = ln(11− xi).(3.9)The evaluation of the corresponding CDF distribution corresponding toY in the continuum limit, i.e. for M set to 10,000, is depicted in Figure 3.4The corresponding analytical expression is also depicted. The differencesare found to be negligible, as would be expected.3.5 Monte Carlo simulationsThe Monte Carlo method is a simulation approach that may be used forthe modeling of random systems. It is a computational tool that is usedin order to sample over the entire set of possible realizations of a givenexperiment. If the experiment is performed enough times, some sense as tothe probability of a given event will emerge in the continuum limit. TheMonte Carlo method has been proven to be a useful tool for the evaluationof solutions where the cost of an analytical solution is prohibitive. It is awidely used method in many areas, including risk analysis in business andin materials physics [34].In this section, the basic structure of the algorithm used in a Monte Carlosimulation is discussed. Then, several examples of how it may be utilizedare provided. How this method may be applied for the determination ofproperties of disordered semiconductors, with respect to achieving the aimof the thesis, are discussed in the following chapter.533.5. Monte Carlo simulations𝑥𝑦𝑖𝐹𝑌01𝑥𝑥𝑖𝐹𝑥01Figure 3.3: Transformation of the CDF. The online version of this figure isdepicted in color.0 1 2 3 4 5 6 7 8 9y00.10.20.30.40.50.60.70.80.91F(y)Empirical CDFEmpirical CDFTheoretical CDFFigure 3.4: CDF of the distribution obtained from a transformation of theuniform distributed random numbers to exponentially distributed randomnumbers. The online version of this figure is depicted in color.543.5. Monte Carlo simulationsThe Monte Carlo simulation approach requires a sampling of the sample-space outcomes of a given experiment in proportion to the probability of eachsuch outcome. To do this, the “experiment” is realized a large number oftimes, the continuum limit being approached as the number of times overwhich the “experiment” is performed approaches infinity. Through countingthe number of realizations of a given event in sample-space, the probabilityof such an event can be evaluated [40]. The random number generatorunderlies such an approach.Monte Carlo simulations may be applied to a variety of problems. Agreat example of this is the determination of the value of pi from the randomdistributions of points over a plane [44]. To do this, consider a circle witha radius r placed in a square with sides 2r, as depicted in Figure 3.5. Thearea of the circle is, of course, pir2, and the area of the square within whichthe circle is placed is 4r2. If I distribute points uniformly over the plane, theprobability that such a point will be in the circle is pir24r2or pi4 , i.e., roughly78%. Thus, by evaluating the fraction of points in the circle as a function ofthe number of points, in the continuum limit, i.e., as the number of pointsgoes to infinity, the fraction of points inside the circle should approach pi4 .Such an experiment is depicted in Figure 3.6. Alternatively, plotting thisfraction multiplied by four should yield pi as the number of realizations,M , approaches the continuum limit. Such a plot is depicted in Figure 3.7Note that a transient occurs as the continuum limit is approached. Thecorresponding distribution of points, for the cases of 100, 1,000, and 10,000realizations, is depicted in Figure 3.6. It is noted that in the continuumlimit this distribution of points is essentially uniform.553.5. Monte Carlo simulationsAnother example of the use of the Monte Carlo approach is the findingof the solution of a problem without the need for an analytical solution canbe seen with regards to the integration of functions. Consider, for example,the following definite integral,I =∫ baf(x).Using M randomly selected points over the interval [a, b], this integral maybe expressed as,I =(b− a)MM∑i=1f(Xi), (3.10)where Xi are the uniformly randomly distributed random points between aand b, the index i denoting the i -th point [45]. Using this method, Figure 3.8depicts the result of this approximation for the integration of f(x) = x overthe limits 0 to 1 with increasing values of M . Similarly, the approximatevalue of the integration comes closer to the corresponding analytical value asthe number of iterations is increased. Once again, the result becomes exactin the continuum limit, i.e., as the number of points approaches infinity.563.5. Monte Carlo simulations𝑟2𝑟Figure 3.5: A circle of radius r within a square of sides 2r. The onlineversion of this figure is depicted in color.573.5. Monte Carlo simulations(a) 100 points, pi = 3.2 (b) 1,000 points, pi = 3.144(c) 10,000 points, pi = 3.112 (d) 100,000 points, pi = 3.1468Figure 3.6: Monte Carlo simulation for finding the value of pi using 100,1,000, 10,000 and 100,000 realizations where the blue dots represent thepoints counted within the circle. The online version of this figure is depictedin color.583.5. Monte Carlo simulations0 1 2 3 4 5 6 7 8 9 1010400.511.522.533.544.55Mπ Figure 3.7: A Monte Carlo evaluation of pi as the number of realizationsapproaches infinity. The online version of this figure is depicted in color.593.5. Monte Carlo simulations0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000M0.150.20.250.30.350.40.450.50.550.6IEmpirical AverageTheoritical MeanFigure 3.8: Monte Carlo integration approximation for∫ 10 f(x)dx, wheref(x) = x, as the number of realizations approaches infinity. The onlineversion of this figure is depicted in color.60Chapter 4Analysis4.1 OverviewDisordered semiconductors exhibit a range of properties that are quitedistinct from those of their crystalline counterparts. Many of these differ-ences are accountable when one considers the different forms of disorderthat are present within a disordered semiconductor. Unlike in a defect-freecrystalline semiconductor where the DOS functions associated with the con-duction band and valence band terminate abruptly at the band edges, in adisordered semiconductor, these DOS functions encroach into the otherwiseempty gap region [10]. This encroachment into the gap region may be under-stood from the distribution of potential fluctuations. Accordingly, interestin evaluating the distribution of potential fluctuations remains considerable.I am going to be treating this problem within the framework of a MonteCarlo approach. The advantage offered through the use of a Monte Carlosimulation approach is its ability to solve problems without necessitatinga closed-form solution, closed-form solutions sometimes proving infeasible.As will be seen later, for this particular case, a Monte Carlo simulation willadequately serve my needs. A comparison with analytical results, acquiredfor special cases, will be made later on in order to benchmark the accuracy614.1. Overviewof my simulation results.To model the distribution of potential fluctuations, I treat a disorderedsemiconductor within the framework of a one-electron effective mass ap-proach. The material is treated as being comprised of an ensemble of uni-formly and independently distributed point defects, each such point defectcontributing to the overall potential, i.e., the overall potential may be ex-pressed as a linear sum over the potentials associated with these individualpoint defects. Variations in the positions of the point defects and variationsin the nature of the interactions with these point defects, i.e., the contribu-tion of each point defect to the overall potential, contribute to fluctuationsin the evaluation of the overall potential. The characterization of these fluc-tuations is the goal of this analysis. I then will use these distributions ofpotential fluctuations in order to assess their impact on the correspondingDOS functions. Representative random potential profiles, corresponding toboth the conduction band and valence band, are depicted in Figure 4.1. Thecorresponding DOS functions will be determined within the framework ofthe semi-classical approach of O’Leary et. al. [13].624.1. OverviewFigure 4.1: Representative random potentials corresponding to that foundwithin a disordered semiconductor. The online version of this figure is de-picted in color.634.2. The DOS functions for a single electron in a random potential profileThis chapter is organized in the following manner. In Section 4.2, Idiscuss how the DOS function may be evaluated for a single electron ina random potential profile. Then, in Section 4.3, I present my approachto modeling the random potential. The Monte Carlo approach, that isemployed for the purposes of this analysis, is then discussed in Section 4.4.The Monte Carlo results are then presented and discussed in Section 4.5.Analogous closed-form analytical results are then featured in Section 4.6.Finally, the impact of distributions of potential fluctuations on the DOSfunctions are explored in Section 4.7.4.2 The DOS functions for a single electron in arandom potential profileAn exact solution for the problem of a single electron in a random po-tential profile has proven difficult to acquire. A number of approximatesolutions have been proffered [24, 25, 46]. Most of these solutions drawupon the statistics associated with the underlying random potential. In theinterests of simplicity, and in order to acquire analytical solutions, the as-sumption of Gaussian statistics is usually employed. Indeed, many analyzeshave been performed within the framework of a Gaussian approximation,wherein the distribution of potential fluctuations is simply set to that of aGaussian distribution. Usually, the use of a Gaussian distribution is justi-fied by invoking the central limit theorem, wherein there are a large numberof overlapping sources of disorder, and these add up in order to produce aGaussian distribution of potential fluctuations.644.3. Modeling the random potentialI believe that this assumption may be overly broad and not applicableto all cases. Accordingly, I believe that there is some merit in developing amodel for the evaluation of the distribution of potential fluctuations from afirst principles perspective rather than simply invoking the Gaussian approx-imation. In this analysis, I will therefore provide a framework within whichI can explicitly determine the distribution of potential fluctuations, with-out the need to simply draw upon a specific form. My evaluation of thesestatistics assumes that the different sources of disorder within a disorderedsemiconductor are randomly and uniformly distributed over the volume ofthe semiconductor.Within such a theoretical framework, the distribution of potential fluc-tuations is, of course, dependent upon the density of the sources of disorderand the exact nature of the contribution of that particular source of disorderto the overall potential at any given point within the disordered semicon-ductor. I have found that this surprisingly simple problem is in actual facta challenge to address, particularly if one expects a closed-form solution,owing to a number of complications, these being described later.4.3 Modeling the random potentialI follow the course of analysis pursued by O’Leary et al. [13] That is,within the framework of a one-electron effective mass approach, the disorderthat exists within a disordered semiconductor is treated as being that corre-sponding to a number of point sources of disorder, these point sources beinguniformly and independently distributed within the volume. The compo-654.3. Modeling the random potentialnents of the material without sources of disorder may be viewed of as beingwith perfect order. Associating each point source of disorder within thedisordered semiconductor with a potential fluctuation, the conduction bandpotential may thus be expressed as the sum over these potential fluctuations,i.e.,Vc(~R) = V′c (~R) + V 0c , (4.1)where the sum over the potential fluctuations is V′c (~R) and V 0c is the conduc-tion band potential in the absence of disorder. The sum over the potentialfluctuations,V′c (~R) =∑mvmc (~R− ~rm), (4.2)where ~rm represents the location of the m-th defect and vmc ( · ) depicts thecorresponding potential; in order to simplify the analysis, I will assume thatthe potential profiles associated with each point defect are spherically sym-metric, i.e., that vmc ( ·) is purely a function of the distance between the pointdefect and the point at which the overall potential is determined. As thesources of disorder are uniformly and independently distributed, given thateach individual point source of disorder may have its own unique potentialprofile, it may be seen that V′c (~R) is a random variable that depends onlocation. The determination of the statistics of V′c (~R) corresponds to thegoal of this analysis.Thus far, I have presented a formulation for the conduction band poten-tial. A corresponding analysis may be pursued for the valence band potentialusing an analogous formalism. In order to simplify matters, I will focus onthe conduction band case, noting that the valence band analysis may be664.3. Modeling the random potentialperformed in the same manner. It should be noted, however, that the im-pact of disorder on the different bands may be quite distinct, a fluctuationin the conduction band potential not necessarily being seen in the valenceband. The impact of defects on the different bands requires an analysis thatgoes far beyond the scope of the work presented here.I treat the point defects as being uniformly and independently dis-tributed over the volume of the semiconductor. A two-dimensional dis-tribution of uniformly and independently distributed points is depicted inFigure 4.2a. Another distribution of such points is depicted in Figure 4.2b.The potential value at any point in the plane is a function of the exact dis-tribution of point defects, and their corresponding potential profiles. Thevalue of the overall potential at any given point changes according to theparticular realization of this uniform and independent distribution of pointsover the plane. For the case that I am interested in, i.e., a uniform andindependent distribution of points over a volume, a similar impact occurs.Through simulating a large number of such realizations, a distribution ofpotential fluctuations may be determined. In the limit that the number ofrealizations is infinite, i.e., in the continuum limit, an exact distribution ofpotential fluctuations may be acquired.I have developed a general algorithm for the determination of the distri-bution of potential fluctuations that may be used for our evaluations of thedistribution of potential fluctuations. In particular, the CDF is evaluatedusing the empirical CDF definition, where I note that in the continuumlimit, i.e., as the number of realizations is increased towards infinity, theempirical CDF approaches the actual CDF. As was mentioned earlier, in674.3. Modeling the random potential-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5(a) 100 uniformly distributed points.-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5(b) 100 uniformly distributed points.Figure 4.2: Uniformly distributed points over a two-dimensional plane. Notethat these distributions are distinct. The online version of this figure isdepicted in color.684.3. Modeling the random potentialorder to ease the analysis, the potential profiles associated with each pointdefect are assumed to be spherically symmetric in nature, i.e., it takes on aform of v(r), where r denotes the radial distance from the point-of-origin,the origin being the point of which the overall potential is evaluated.The sensitivity of the results to variations in the potential form is ex-plored. For the purposes of this particular analysis, I consider three typesof defect potential profile. The first is “a power law” of the form v(r) = 1rγ ,where γ denotes the exponent; the electrostatic potential between two pointcharges in a vacuum, for example, exhibits this kind of potential for γ setto unity. The only problem with this potential profile is that it can exhibitsingularities associated with a defect potential being very close to the pointat which the overall potential is evaluated. In order to assess the robust-ness of my approach to the form of the potential, the second type of defectpotential profile that I deal with is of the form v(r) = exp (−kr), where kcorresponds to an exponential attenuation factor. The third potential profileis the abrupt potential barrier, i.e.,v(r) =1, r ≤ a0, r > a,where a denotes the radius of this particular potential profile. Pre-factors infront of these potentials, and variations in the density of the point defects,merely scale the results without changing their fundamental character, andtherefore, will not be considered.It should be noted that although I only deal with these three potential694.4. Monte Carlo approachprofile types, the algorithm that I employ may be used for any type of defectpotential profile. The selection of the three particular profiles that I haveconsidered was motivated by the need to examine how the results vary overwhat might be reasonably expected for a disordered semiconductor.4.4 Monte Carlo approachIf the point defects are uniformly and independently distributed overa volume, where the volume extends to infinity, and if the potential profilecorresponding to each point defect does not vary, then the distribution of po-tential fluctuations should be invariant to location. This suggests that I canevaluate the distribution of potential fluctuations at the origin. Consider, forexample, Figure 4.3a, which corresponds to the same uniform distributionof points depicted earlier in Figure 4.2a. Associating a potential profile witheach point defect, an exact value of the overall potential may be evaluated.In Figure 4.3b, I do the same thing for the uniform distribution of pointsshown in Figure 4.2b. Once again, associating a potential profile with eachpoint defect, an exact value of the overall potential may be acquired, thisvalue being distinct from that determined corresponding to Figure 4.3a.Each evaluation of the overall potential corresponds to a particular re-alization of this problem. Through performing this ‘experiment’ multipletimes, i.e., M times, one will acquire a distribution of values for the overallpotential. This will allow for an evaluation of the empirical CDF, Femp( · ).In the limit that the number of realizations approaches infinity, the contin-uum limit will be achieved, and the empirical CDF will approach the actual704.4. Monte Carlo approach-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5(a) 100 uniformly distributed points with the origin clearlymarked. The arrow shows the point at which the overallpotential is evaluated.-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5(b) 100 uniformly distributed points with the origin clearlymarked. The arrow shows the point at which the overallpotential is evaluated.Figure 4.3: Uniformly distributed points over a two-dimensional plane. Notethat these distributions are distinct. The arrows point to where the overallpotential is evaluated. The online version of this figure is depicted in color.714.4. Monte Carlo approachCDF. Understanding how many realizations are required in order to achievethe continuum limit will be one of the goals of this analysis.One technical challenge that presents itself is the fact that the evaluationof the overall potential requires the consideration of all point defects overan infinite volume. Of course, from a computational perspective, this cannot be achieved. Thus, I am going to suggest performing the evaluation ofthe distribution of potential fluctuations corresponding to a finite volumeand then examine what happens as this volume is increased, i.e., examininghow the evaluation of the overall distribution of potential fluctuations variesas the boundary conditions of the problem are changed. In particular, Inote that the overall potential evaluated at the center of a cube that hasdimensions L × L × L should approach the exact solution as L approacheslarge values. So, I am going to be examining how the results that areacquired are dependent upon the dimensions of the cube. This is referred toas a boundary effect, and I seek dimensions that are large enough in order toprovide results that are effectively independent of L but small enough so thatthe required computations are manageable. If the results are independent ofL, that will demonstrate that the details of the boundary are not changingthe results, as would be expected.As was discussed earlier, the initial step that is required in the pursuitof a Monte Carlo simulation involves the setting-up of the random numbergenerator to mimic the random nature of the problem at hand. In thecase of this analysis, I am modeling the random distribution of potentialfluctuations found in disordered semiconductors. Once the model for theevaluation of the overall potential has been set up, the simulation is run724.5. Monte Carlo resultsmultiple times in order to obtain results that approach the continuum limit.4.5 Monte Carlo resultsIn this section, I present results obtained from my evaluation of thedistribution of potential fluctuations from the Monte Carlo simulations thatare performed. There are a number of variables that are considered for eachsimulation. Variables that are changed include:1. the boundary of the volume over which the simulations are considered,2. the potential profile corresponding to each point defect,In light of the fact that the exact distribution function itself should beindependent of the boundaries, I first examine the sensitivity of the resultsto the boundary conditions, noting that I want the smallest volume overwhich I can perform my distribution evaluations for the results to appear tobe essentially independent of the boundaries. In order to simplify matters,I consider the distribution of potential fluctuations at the origin of a cube,of dimensions L × L × L, for the purposes of this analysis. Such a cube isdepicted in Figure 4.4.As has been mentioned before, three representative potential profiles areconsidered for the purposes of this analysis. In particular, I consider point734.5. Monte Carlo resultsLLLEvaluation pointFigure 4.4: The point where the overall potential is evaluated, at the originof a cube, of dimensions L × L × L. The online version of this figure isdepicted in color.744.5. Monte Carlo resultsdefect potential profiles of the formv(r) =1rγ,v(r) = exp (−kx),and v(r) =1, r < a0, r > a.Results corresponding to each of these potential profiles, with variations inthe parameters related to each potential, are presented next.4.5.1 The power-law potentialI first consider the potential of the randomly distributed point defects ina three-dimensional space to bev(r) =1rγ(4.3)where r is the radius from the point defect. Other parameters consideredin the simulations include the number of points per unit volume, λ, theboundary length, L, and the exponent, γ. Figure 4.5 depicts the form ofthis potential, for various values of γ. It is noted that large exponents leadto rapid attenuation in v(r) for r greater than unity.In order to get started, I first enumerate the variables that I considerfor each simulation. These are M , the number of realizations of the MonteCarlo simulations, γ, the exponent, L, the dimensions of the cube over whichthis overall potential is distributed, and λ, the number of points per unit754.5. Monte Carlo results0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.511.522.533.544.55=1=4=6=8v(r)rFigure 4.5: The dependence of the potential profile in Eq. (4.3) as a functionof r, for various exponent selections. The online version of this figure isdepicted in color.764.5. Monte Carlo resultsvolume. These are tabulated in Table 4.1. As it merely contributes toscaling, for the purposes of this analysis, I will set the number of points perunit volume, λ, to 100 for all cases. The initial thing to examine is whereTable 4.1: The simulation variables for power-law potential profile.Name DescriptionM Number of realizations for Monte Carlo simulationγ exponentL boundary lengthλ number of points per unit volumethe potential profile stabilizes with respect to the boundary of the cube, i.e.,L. This means that I must first find the boundary length at which the affectof the point defects beyond the dimensions of the cube does not have anyfurther affect on the overall potential. As observed in the potential profilesdepicted in Figure 4.5, as the value of the exponent, γ, increases, the affectof potentials beyond the range of the cube should be less intense. Clearly,in light of the dependence of the form of v(r) on r for the various values ofthe exponent, γ, the L required will be dependent on γ.I start by considering the case of γ set to unity, i.e., the case of elec-trostatics in a vacuum. In order to allow for rapid analysis, the number ofrealizations, M , is set to 100, and the cube boundary dimensions, L, areset to 1, 2, 5, 10, 15, and 20. The evaluated empirical CDF is depicted inFigure 4.6. One thing to note is that the CDF acquired corresponding tothe lower values of L are always upper bound by those acquired for highervalues, as excluding the consideration of point defects beyond the bound-aries of a smaller cube will only lead to the overall potential being reduced,774.5. Monte Carlo results103 104 10500.10.20.30.40.50.60.70.80.91L=1L=2L=5L=10L=15L=20V(r)=1/r , =1, M=100, = 100F(x)xFigure 4.6: Monte Carlo simulations with M = 100 realizations for the caseof γ set to 1. λ is set to equal to 100 points per unit volume, for all cases. Lis set to 1, 2, 5, 10, 15, and 20. The online version of this figure is depictedin color.784.5. Monte Carlo resultsand hence, the CDF rising up to unity earlier. Certainly, this is observedhere, as it is in all other cases.It is noted that as the cube volume dimension, L, is increased, that theCDF associated with this particular distribution of potential fluctuationsdoes not really stabilize. Instead, it just shifts to the right. This lack of sta-bility in the distribution of potential fluctuations arises as a consequence ofthe increasing number of contributors to the overall potential. Consideringa spherical region, of radial dimensions, r, each shell between r and r + ∆rwill have, on average, λ4pir2∆r point defects, overpowering the reduced po-tential contribution corresponding to each such point defect, i.e., v(r) = 1rγ .For the three-dimensional case, a plausible calculus reveals that exponentsin excess of three, i.e., γ > 3, lead to stable distributions. This may be seenby noting that the overall potential sum, i.e.,v =N∑i=1v(ri), (4.4)where N corresponds to the number of point defects contributing to theoverall potential, i.e., on average, N = λ4pi3 R3, where R denotes the radiusof the sphere within which the sum over Eq. (4.4) is to be performed. In thecontinuum limit, the sum in Eq. (4.4) may be converted into an integral.v =∫ Lεv(r)λ4pir2 drv =∫ Lελrγ4pir2 dr=∫ Lε4piλrγ−2dr(4.5)794.5. Monte Carlo resultswhich may be shown to yield a finite result when γ − 2 > 1, i.e., γ > 3.The lower-limit of this integral, ε, remains ambiguous owing to a break-down in this continuum approach to evaluating the sum in Eq. (4.4) as thedimensions approach the order of the inter-point-defect spacing.As it has been plausibly argued that γ > 3 will produce stable results, Icontinue the analysis by considering the cases of γ = 4, γ = 6, and γ = 8.Figure 4.7 shows the impact of L selections of 0.25, 0.5, 1, 2, 5, and 10 onthe case of the power-law potential with the exponent, γ, set to 4, observedover M = 1,000 realizations. It can be seen that for L values beyond 5,the distribution becomes stable. Therefore, it can be assumed that thepotential for γ = 4 stabilizes at boundary lengths of L = 5. Looking back atthe potential for γ = 4 from Figure 4.5, it can be seen that this observationis consistent with the observation found in Figure 4.5, that the contributionof points beyond r = 5 play little role in contributing to the overall potential.The same analysis may be performed for the case of γ = 6. Figure 4.8shows the affect of the boundary conditions for this case. Figure 4.8 showsthe impact of the L selections of 0.25, 0.5, 1, 2, and 5 on the potential withthe exponent, γ = 6, observed over M = 1, 000 realizations. It can be seenthat for L values beyond 1, the distribution remains stable. Therefore, itcan be assumed that the potential for γ = 6 stabilizes at boundary lengthsof L = 1. It is noted that this is less than that found for the case of γ = 4,undoubtedly due to the decrease in the contributions of the far-off pointdefects found for this potential; recall Figure 4.5.Finally, I look into the affect of the boundary conditions for potentialswith γ = 8. Figure 4.9 shows the impact of the L selections of 0.25, 0.5, 1,804.5. Monte Carlo results104 105 10600.10.20.30.40.50.60.70.80.91L=0.25L=0.5L=1L=2L=5L=10V(r)=1/r , =4, M=1,000, = 100F(x)xFigure 4.7: Monte Carlo simulations with M = 1, 000 realizations for thecase of γ set to 4. λ is set equal to 100 points per unit volume, for all cases.L is set to 0.25, 0.5, 1, 2, 5, and 10. The online version of this figure isdepicted in color.814.5. Monte Carlo results106 107 10800.10.20.30.40.50.60.70.80.91L=0.25L=0.5L=1L=2L=5V(r)=1/r , =6, M=1,000, = 100F(x)xFigure 4.8: Monte Carlo simulations with M = 1, 000 realizations for thecase of γ set to 6. λ is set equal to 100 points per unit volume, for all cases.L is set to 0.25, 0.5, 1, 2, and 5. The online version of this figure is depictedin color.824.5. Monte Carlo results107 108 10900.10.20.30.40.50.60.70.80.91L=0.25L=0.5L=1L=5V(r)=1/r , =8, M=1,000, = 100F(x)xFigure 4.9: Monte Carlo simulations with M = 1, 000 realizations for thecase of γ set to 8. λ is set equal to 100 points per unit volume, for all cases.L is set to 0.25, 0.5, 1, and 5. The online version of this figure is depictedin color.834.5. Monte Carlo resultsand 5 on the potential with the exponent, γ = 8, observed over M = 1, 000realizations. It can be seen that for L values beyond 0.5, the distributionremains stable. Therefore, it can be assumed that the potential for γ = 8stabilizes at boundary lengths of L = 0.5. It is noted that this is lessthan that found for the case of γ = 6, undoubtedly due to the decrease inthe contributions of the far-off point defects found for this potential; recallFigure 4.5.The fact that the cube boundary extent required in order to stabilizethe distribution diminishes as the exponent is increased can be accountedfor by noting that the impact of the far-flung point sources of disorder di-minished as the exponent is increased; recall Figure 4.5. The estimates forthe minimum boundary scales required in order to achieve stability, for thevarious exponents considered in this analysis, are depicted in Figure 4.10.Now that the boundary lengths required in order to achieve stability havebeen established, it is instructive to examine the approach to the continuumlimit. I did not consider the approach to the continuum limit when establish-ing the boundaries of the cube, as one can get away with smaller numbers ofrealizations if one is merely interested in determining the boundary lengthsrequired in order to achieve stability in the distribution function. Figure4.11 shows the distribution for M = 10, 50, 100, 1,000, 10,000 and 100,000realizations for γ = 4 for L set to 5; L = 5 is selected as this corresponds tothe onset of stability for the case of γ set to 4. A similar plot for γ = 6 isshown in 4.12 for L = 1; once again, L = 1 is selected as this correspondsto the onset of stability for the case of γ set to 6. Finally, for the case ofγ = 8, a similar analysis is performed, for L set to 0.5. L = 0.5 is selected844.5. Monte Carlo resultsas this corresponds to the onset of stability for the case of γ set to 8. Fromthese diagrams, it is clearly observed that for M = 100,000 iterations, theresults are the most accurate, i.e., they are closest to the continuum limit.Unfortunately, in terms of computational load, they are the most expensiveresults.4.5.2 Exponential potentialNext, I consider the potential profile of the randomly distributed pointdefects in a three-dimensional space to be of the formv(r) = exp (−kr), (4.6)where r is the radius from the point defect and k is some extinction coeffi-cient. Other parameters considered in the simulation include the number ofpoints per unit volume, λ, the boundary length, L, and the extinction coef-ficient, k. Figure 4.14 depicts the form of the exponential potential profileas a function of r for various values of k. It is noted that as the k valueincreases, the potential experiences a quicker drop.The variables that I consider for each simulation are shown in Table 4.2.The variables include M , the number of realizations of the Monte Carlosimulation, k, the extinction coefficient, L, the dimensions of the cube overwhich this overall potential is distributed, and λ, the number of points perunit volume. Similar to the power-law potential profile case, I set the numberof points per unit volume, λ, to be 100 for all simulations.Again, I first examine where the acquired empirical CDF stabilizes with854.5. Monte Carlo results01234564 6 8Stabilizing length, LFigure 4.10: An estimate for the minimum boundary lengths required inorder to achieve stability for the power-law potential profile, i.e., Eq. (4.3),as a function of the exponent, γ.Table 4.2: The simulation variables for exponential potential profile.Name DescriptionM Number of realizations for Monte Carlo simulationk extinction coefficientL boundary lengthλ number of points per unit volume864.5. Monte Carlo results104 105 106x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 10M = 50M = 100M = 1,000M = 10,000M = 100,000V(r)=1/r , = , L=5, = 1004Figure 4.11: Monte Carlo simulations with 10, 50, 100, 1,000, 10,000, and100,000 realizations for the case of γ set to 4. λ is set equal to 100 pointsper unit volume for all cases. L is set to 5. The online version of this figureis depicted in color.874.5. Monte Carlo results105 106 107x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 10M = 50M = 100M = 1,000M = 10,000M = 100,000V(r)=1/r, =6, L=1, = 100Figure 4.12: Monte Carlo simulations with 10, 50, 100, 1,000, 10,000, and100,000 realizations for the case of γ set to 6. λ is set equal to 100 pointsper unit volume for all cases. L is set to 1. The online version of this figureis depicted in color.884.5. Monte Carlo results106 107 108 109x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 10M = 50M = 100M = 1,000M = 10,000M = 100,000V(r)=1/r, =8, L=0.5, = 100Figure 4.13: Monte Carlo simulations with 10, 50, 100, 1,000, 10,000, and100,000 realizations for the case of γ set to 8. λ is set equal to 100 pointsper unit volume for all cases. L is set to 0.5. The online version of thisfigure is depicted in color.894.5. Monte Carlo results0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91k=1k=5k=10v(r)rFigure 4.14: The dependence of the potential profile in Eq. (4.6) as afunction of r, for various k selections. The online version of this figure isdepicted in color.904.5. Monte Carlo resultsrespect to the boundary of the cube, i.e., L. As was noted for the power-lawcase, the boundary length required in order to achieve stability will dependon the exact nature of the potential profile, i.e., on the extinction coefficient,k, greater k implying that one can get away with reduced cube boundaries. Iobserve how the dimensions of the cube affect the empirical CDF evaluationfor k values of 1, 5 and 10. Each of the simulations are run for M = 1, 000realizations for varying L values. Figures 4.15, 4.16 and 4.17 depict theresults for the k selections of 1, 5, and 10, respectively.It can be seen that for k set to 1, for L values beyond 15, the distributionstabilizes. For k set to 5, however, a shift is observed, and the L value atwhich the potential stabilizes is seen to be L = 5. Finally, for the k value of10, the potential distribution stabilizes for L beyond 2. These observationscan be compared with the potential profile seen in Figure 4.14, where highervalues of k mean that the far-flung point defects will contribute less to theoverall potential. In Figure 4.18, the minimum boundary lengths, L, areplotted as a function of the extinction coefficient, k.914.5. Monte Carlo results0 500 1000 1500 2000 2500 3000 3500 400000.10.20.30.40.50.60.70.80.91L=1L=5L=10L=15L=20v(r)=exp(-kr), k = 1, M = 1000, = 100xF(x)Figure 4.15: Monte Carlo simulations with M = 1, 000 realizations for thecase of k set to 1. λ is set equal to 100 points per unit volume, for all cases.L is set to 1, 5, 10, 15 and 20. The online version of this figure is depictedin color.924.5. Monte Carlo results0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91L=1L=2L=5L=10L=15v(r)=exp(-kr), k = 5, M = 1,000, = 100F(x)xFigure 4.16: Monte Carlo simulations with M = 1, 000 realizations for thecase of k set to 5. λ is set equal to 100 points per unit volume, for all cases.L is set to 1, 2, 5, 10, and 15. The online version of this figure is depictedin color.934.5. Monte Carlo results0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91L=0.5L=1L=2L=5L=10v(r)=exp(-kr), k = 10, M = 1,000, = 100F(x)xFigure 4.17: Monte Carlo simulations with M = 1, 000 realizations for thecase of k set to 10. λ is set equal to 100 points per unit volume, for all cases.L is set to 0.5, 1, 2, 5, and 10. The online version of this figure is depictedin color.944.5. Monte Carlo results0246810121416k = 1 k = 5 k = 10Stabilizing length, LFigure 4.18: An estimate for the minimum boundary lengths required inorder to achieve stability for the exponential potential profile, i.e., Eq. (4.6),as a function of the extinction coefficient, k.954.5. Monte Carlo resultsNow that the boundary conditions for the potential have been deter-mined for the different k values, I observe how the number of realizationsaffect the distribution at the point of stabilization. To do this, I simulateeach of the potentials for M = 50, 100, 1,000, and 10,000 realizations. Fork = 1, with L = 15, the result is shown in Figure 4.19. Due to the largedimensions required to reach stabilization point for k = 1, I use M = 10, 000realizations as the highest number of realizations used in order to ease theprocess of the simulation. From the distributions, I can conclude that 10,000realizations is sufficient to provide an accurate result, i.e., it is reasonablyclose to the continuum limit.Next, I look into the affect of the number of realizations for k = 5 atL = 5. The distribution is plotted for M = 10, 50, 100, 1,000, 10,000 and100,000 realizations, shown in Figure 4.20. It can be seen that for 10,000realizations and beyond, the distribution produced coincides and producesaccurate results. Similarly, Figure 4.21 depicts the distributions for k = 10and L = 2. A similar conclusion can be drawn that 10,000 iterations andbeyond produces sufficiently accurate results. The simulations provides themost accurate results for M = 100, 000 realizations, i.e., the higher thenumber of realizations, the more accurate results produced. Unfortunately,from a computational perspective, these are the most expensive results.964.5. Monte Carlo results0 500 1000 1500 2000 2500 3000 3500 4000x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 50M = 100M = 1,000M = 10,000V(r)=exp(-kr), k=1, L=15, = 100Figure 4.19: Monte Carlo simulations with k value of 1, at the stabilizingdimension, L = 15, for various number of realizations, M , i.e., 50, 100,1,000, and 10,000. The online version of this figure is depicted in color.974.5. Monte Carlo results0 5 10 15 20 25 30 35x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 10M = 50M = 100M = 1,000M = 10,000M = 100,000V(r)=exp(-kr), k=5, L=5, = 100Figure 4.20: Monte Carlo simulations with the k value of 5, at the stabilizingdimension, L = 5, for various number of realizations, M , at 10, 50, 100,1,000, 10,000 and 100,000. The online version of this figure is depicted incolor.984.5. Monte Carlo results0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x00.10.20.30.40.50.60.70.80.91F(x)Empirical CDFM = 10M = 50M = 100M = 1,000M = 10,000M = 100,000V(r)=exp(-kr), k=10, L=2, = 100Figure 4.21: Monte Carlo simulations with the k value of 10, at the stabi-lizing dimension, L = 2, for various number of realizations, M , at 10, 50,100, 1,000, 10,000 and 100,000. The online version of this figure is depictedin color.994.5. Monte Carlo results4.5.3 Step function potentialThe third potential profile that I consider for the randomly distributedpoint defects is of the formv(r) =1, r < a0, r > a., (4.7)where a is the radius of the of the potential. Other parameters considered inthese simulations include the number of points per unit volume, λ, which Iwill set at 100 points per unit volume for all simulations, and the boundarylength, L. Figure 4.22 depicts the abrupt potential for various values of a. Itcan be observed that the boundary set for the potential contributes to howfar the affect of the potential can have, beyond which it abruptly ceases.Table 4.3 shows the variables that I consider for each of the simulations.Table 4.3: The simulation variables for step function potential profile.Name DescriptionM Number of realizations for Monte Carlo simulationa potential boundaryL boundary lengthλ number of points per unit volumeAs will be seen later, the abrupt potential profile exhibits a form forwhich beyond the radius a, no point defects will contribute to the overallpotential profile. Accordingly, there is really no need to perform the bound-ary analysis that I have performed for the other cases. Moreover, as will beseen later, a closed-form expression for the PDF and CDF functions may1004.5. Monte Carlo results0 0.5 1 1.5 2 2.5 300.511.522.53a=2a=1a=0.5V(r)rFigure 4.22: The step function potential profile for various a selections. Theonline version of this figure is depicted in color.1014.6. Closed-form analytical resultsbe easily acquired for this particular case. The approach to the continuumlimit, however, is instructive.I observe how the number of realizations affect the distribution. To dothis, I simulate each of the potentials for M = 10, 50, 100, 1,000, 10,000,and 100,000 realizations for a = 0.5 at L = 2, shown in Figure 4.23. Next,I look into the affect of the number of realizations for a = 1 at L = 3. Thedistribution is plotted for M = 10, 50, 100, 1,000, 10,000 and 100,000 real-izations, shown in Figure 4.24. It can be seen that for 1,000 realizations andbeyond, the distribution produced coincides and produces accurate results.Similarly, Figure 4.25 depicts the distributions for a = 2 and L = 4. Asimilar conclusion can be drawn that 1,000 iterations and beyond producessufficiently accurate results. The simulations provides the most accurateresults for M = 100, 000 realizations, i.e., the higher the number of real-izations, the more accurate results. Unfortunately, from a computationalperspective, they are also the most expensive.4.6 Closed-form analytical resultsI now seek closed-form analytical results for the distribution of poten-tial fluctuations. The purpose of this analysis is two-folded. First, I aimto explore as to whether or not I can acquire a general expression for thedistribution of potential fluctuations independent of the exact form of thepotential profile, v(r). Second, I aim to develop closed-form analytical re-sults, corresponding to special cases, against which I can benchmark theaccuracy of my Monte Carlo results. As closed-form analytical results are1024.6. Closed-form analytical results0 20 40 60 80 10000.20.40.60.81M=10M=50M=100M=1,000M=10,000M=100,000v(r)=1, a = 0.5, L = 2xF(x)Figure 4.23: Monte Carlo simulations with a value of 0.5, at stabilizingdimension, L = 2 for various number of realizations, M , at 10, 50, 100,1,000, 10,000 and 100,000. The online version of this figure is depicted incolor.1034.6. Closed-form analytical results0 50 100 150 200 250 300 350 400 450 500 55000.10.20.30.40.50.60.70.80.91M=10M=50M=100M=1,000M=10,000M=100,000v(r)=1, a = 1, L = 3xF(x)Figure 4.24: Monte Carlo simulations with a value of 1, at stabilizing di-mension, L = 3 for various number of realizations, M , at 10, 50, 100, 1,000,10,000 and 100,000. The online version of this figure is depicted in color.1044.6. Closed-form analytical results0 500 1000 1500 2000 2500 3000 3500 4000 4500 500000.10.20.30.40.50.60.70.80.91M=10M=50M=100M=1,000M=10,000M=100,000v(r)=1, a = 2, L = 4xF(x)Figure 4.25: Monte Carlo simulations with a value of 2, at stabilizing di-mension, L = 4 for various number of realizations, M , at 10, 50, 100, 1,000,10,000 and 100,000. The online version of this figure is depicted in color.1054.6. Closed-form analytical resultsmuch more difficult to obtain than those acquired through Monte Carlosimulations, they allow for less flexibility in the choice of the underlying po-tential profile. For the purposes of this analysis, I will only consider the caseof v(r) = 1rγ and the abrupt spherical potential, as these are the only poten-tials for which closed-form solutions may be easily acquired. The followinganalysis is based on work presented earlier by O’Leary et al. [13].4.6.1 On the distribution of pointsFundamentally, the value of the overall potential corresponding to anyparticular distribution of randomly distributed points, i.e., sources of disor-der, is intimately related to the number of such points within the locality atwhich this potential is evaluated and to the form of the defect potential pro-file. Prior to the evaluation of the distribution of potential fluctuations, itis therefore instructive to consider the statistics associated with the numberof points that are present within a certain volume. I perform this analysiswithin the framework of probabilistic considerations.In order to ease the analysis, let us first consider the one-dimensionalcase. Consider, for example, N points distributed uniformly and indepen-dently along a line that ranges between [0, L]. In the statistical character-ization of the distribution of points along such a line, it is instructive toevaluate the statistics associated with the number of points within a givensub-section of the line, say of length l, where l < L. The probability thatn points are in the interval of length l, can be shown to occur according to1064.6. Closed-form analytical resultsthe binomial distribution, i.e.,P [n points in l] =(Nn)pn(1− p)N−n (4.8)where(Nn)=N !(N − n)!n! =N(N − 1)(N − 2)...(N − (n− 1))n!(4.9)and where p denotes the probability that a given randomly selected point isin the aforementioned subsection, i.e., p = lL ; note that p is less than unityfor this case. Therefore, the probability that there are n points within a lineof length, l, may be expressed asP [n points in l] =(Nn)(lL)n(1− lL)N−n, (4.10)where this expression is exactly the same as Eq. (4.8). This expressionfor the probability arises as a consequence of combinatorics and its validityrequires statistical independence and uniformity in the distribution of points.A representative uniform and independent distribution of points along a line,between 0 and L, is depicted in Figure 4.26. Note that if the points are trulydistributed uniformly, that the location of l should not matter to the validityof Eq. (4.10).4.6.2 The Poisson point processI examine the case for which the distribution of points extends over aninfinite range, i.e., from −∞ to +∞. Of course, for this case, the total1074.6. Closed-form analytical results(a) N points distributed uniformly and independently between 0 and L. For thisspecific case, N = 10.(b) A region of interest, between 0 and l, along a line between 0 and L.Figure 4.26: Uniformly distributed points on a line. The online version ofthis figure is depicted in color.1084.6. Closed-form analytical resultsnumber of points, N , tends to infinity. In order to avoid unpleasant issuesassociated with infinities, I instead define a uniform point process, where λdenotes the ‘average’ density of points per unit length. Expressed in termsof our previously developed binomial formalism, the density of points, λ,may be expressed as the limit of N divided by L as L approaches infinity,i.e.,λ = limL→∞NL(4.11)I can now employ the binomial expansion formalism, i.e., Eq. (4.10).My aim is to determine the PMF associated with such a process in order toderive an expression valid for the case of infinite range, i.e., L→∞, with afinite λ. It is noted that in the limit of L→∞, thatP [n points in l] =limL→∞N(N − 1)(N − 2)...(N − (n− 1))n!(lL)n(1− lL)N−n= limL→∞(NL )(NL − 1L)(NL − 2L)...(NL − (n−1)L )n!ln(1− lL)N−n,(4.12)It is noted that each of the terms of N−mL , where m is a finite integer lessthan N , reduces to λ, i.e.,limL→∞NL= λ,limL→∞NL− 1L= λ,limL→∞NL− (n− 1)L= λ.(4.13)1094.6. Closed-form analytical resultsFor the final term, I simply sety =(1− lL)N−n. (4.14)Taking the logarithm on both sides, it is seen thatln y = ln(1− lL)N−n,= (N − n) ln(1− lL),(4.15)Noting thatln (1− x) = −x− x22− x33− ..., (4.16)it can be seen thatln y = (N − n)(− lL− l22L2− ...). (4.17)Taking the limit that L→∞, it can be seen thatlimL→∞ln y = −λl, (4.18)and thus, it is seen that in the limit as L→∞,y = exp (−λl). (4.19)Accordingly, it is thus shown thatP [n points in l] =(λl)ne−λln!. (4.20)1104.6. Closed-form analytical resultsThis result, i.e., Eq. (4.20), is the classical Poisson point process statistics.It is noted that this is the PMF associated with the Poisson point process,i.e., Eq. (4.20). This may be demonstrated by noting that,exp (x) = 1 +x1!+x22!+ ..., (4.21)and it may therefore be shown thatexp (λl) =∞∑n=0(λl)nn!, (4.22)and that therefore,∞∑n=0P [n points in l] = 1. (4.23)This verifies that Eq. (4.20) is indeed a valid PMF.The Poisson point process generates a discrete random variable, i.e., thenumber of points is a discrete quantity. That is, the number of points in agiven interval over a line can only be described using non-negative integervalues. Thus, the PDF associated with the number of points, Nl, foundwithin the interval of length l, may be expressed asfNl(z) =∞∑n=0(λl)nn!e−λlδ(z − n), (4.24)where δ( · ) corresponds to the delta function; as was mentioned in Section2.2, owing to the discrete nature of the random variables involved, deltafunctions are to be expected in the PDF. Nevertheless, as delta functionscan be integrated, i.e., the integration over such a function will produce a1114.6. Closed-form analytical resultsfinite result, the corresponding CDF results are well behaved and exhibitproperties that are consistent with those laid down in Section 2.2. In par-ticular, noting that∫ ∞−∞δ(z−z0)dz = u(z − z0), (4.25)where u( · ) denotes the step function, i.e.,u(z) = 1, z ≥ 00, z < 0 , (4.26)it may be seen thatFNl(z) =∞∑n=0(λl)nn!e−λlu(z − n). (4.27)It is instructive to depict the PMF and the CDF associated with thePoisson point process. In Figure 4.27, I plot the PMF associated with thePoisson point process, for a number of λl values. It is noted that for lowvalues of λl, the PMF is heavily skewed towards low n values. This makessense, as low λl corresponds to fewer points on average being distributed overthe segment l. In contrast, as λl is increased, a higher number of points inthe segment l become more probable. As a result, a shift in the PMF to theright occurs.The corresponding CDF functions are depicted in Figure 4.28. Notethat, as expected, these CDF functions exhibit a stair-case type behaviour,owing to the discrete nature of the random variable. Once again, I see a1124.6. Closed-form analytical results0 1 2 3 4 5 6 7 8 9 1000.050.10.150.20.250.30.350.4l=1l=2l=5f(x)xFigure 4.27: The PMF of a Poisson point process for different values of λl.The online version of this figure is depicted in color.1134.6. Closed-form analytical results-2 0 2 4 6 8 1000.10.20.30.40.50.60.70.80.91l=1l=2l=5F(x)xFigure 4.28: The CDF of a Poisson point process for different values of λl.The online version of this figure is depicted in color.1144.6. Closed-form analytical resultsshift towards higher n values as λl is increased.It is instructive to evaluate the mean and variance associated with therandom process, Nl. Noting that the mean is the expected value of Nl,Nl = E[Nl], (4.28)it may be expressed as the weighted sum over the Poisson statistics, i.e.,Nl =∞∑n=0n(λl)nn!e−λl = λl. (4.29)The variance, instead, may be determined through an evaluation of thesecond order moment, i.e.,E[N2l ] =∞∑n=0n2(λl)nn!e−λl= (λl)2 + λl.(4.30)Noting that the varianceσ2 = E[N2l ]− (E[Nl])2, (4.31)it is thus seen thatσ2 = λl. (4.32)The Stirling approximationIn many applications of probability theory, it is useful to have an ana-lytical closed form approximation for n! For large numbers, such as those1154.6. Closed-form analytical resultsencountered in statistical mechanics, the Stirling approximation, which for-mally may be expressed aslnn! = n lnn− n+O(lnn),has found utility, where O denotes the big-O notation. It is noted thatwithin the framework of this approximation,n! '√2pin(ne)n,lnn! = ln (√2pin) + ln[(ne)n],lnn! =12ln (2pin) + n ln(ne),=12ln (2pin) + n lnn− n.Therefore, given thatP [n in l] =(λl)nn!e−λl, (4.33)it may be shown thatP [n in l] =(λl)nelnn!e−λl,= (λl)ne− ln (n!)e−λl,= (λl)ne−[n lnn−n+12ln (2pin)]e−λl,= eln (λl)n−n ln (n)+n− 12ln (2pin)−λl.(4.34)It is noted that the exponent of Eq. (4.34) may be expressed asf(n) = n ln (λl)− n ln (n) + n− 12ln (2pin)− λl. (4.35)1164.6. Closed-form analytical resultsTaking the derivative of Eq. (4.35) with respect to n, it is seen that thisexponent achieves a maximum when n = λl. Taking a Taylor series aboutthis maximum, including terms up to 2nd-order, it is seen that a “Gaussian”distribution form is achieved.4.6.3 Analytical result for the PDFThus far, my analysis has only focused on a distribution of points alonga line. The Poisson point process may be generalized to deal with the moregeneral cases of two and three-dimensions. For the two-dimensional case,for a uniformly and independently distributed points over an area A, thecorresponding Poisson point process may be shown to beP[n points in A] =(λA)ne−λAn!, (4.36)where λ denotes the density of points over the area. Alternatively, for thethree-dimensional case, for uniformly and independently distributed pointsover a volume V , the corresponding Poisson point process may be shown tobeP[n points in V ] =(λV )ne−λVn!, (4.37)where λ denotes the density of points over the volume. Given that theabrupt potential profile, i.e., Eq. (4.7), leads to a random variable that isequal to the number of points within the radius a, it may be seen that the1174.6. Closed-form analytical resultscorresponding PDFfv(z) =∞∑n=0(λ4pi3 a3)nn!e−λ4pi3a3δ(z − n). (4.38)The resultant closed-form CDF isFv(z) =∞∑n=0(λ4pi3 a3)nn!e−λ4pi3a3u(z − n), (4.39)where u( · ) is the step-function.The approach that I adopt is that employed earlier by Chandrasekhar[47] and, more recently, by Sousa and Silvester [48]. In particular, I evaluatethe distribution of potential fluctuations corresponding to point centers overa finite volume of radius R and then let the radius go to infinity. That is, Ievaluate the distribution of potential fluctuations corresponding toVR =∑ri≤Rv(ri), (4.40)where v(·) denotes the point defect potential fluctuation and ri represents theseparation between the origin and the i -th point defect. The characteristicfunction approach is employed for the purposes of this analysis. For therandom variable z, the characteristic function isΦz(ω) = E[eiωz], (4.41)1184.6. Closed-form analytical resultswhere, by definition,Φz(ω) =∫ ∞−∞eiωufz(u)du. (4.42)The corresponding PDF may be found by taking the inverse Fourier trans-form, i.e.,fz(y) =12pi∫ ∞−∞Φz(ω)e−iωzdw. (4.43)The characteristic function approach provides a powerful tool for the evalua-tion of the PDF. It is found in many applications of science and engineering.The characteristic function for the potential, VR, may be determined,i.e.,ΦVR(ω) = E[eiωVR ]. (4.44)Conditioning this probability on the number of point defects in the volumeof radius R, i.e.,ΦVR(ω) = E[E[eiωVR |k in VR]], (4.45)it may be seen that,ΦVR(ω) =∞∑k=0e−λ43piR3(λ43piR3)kk!E[eiωVR |k in VR] (4.46)Noting the evaluation of E[eiωVR |k in VR] assumes that the location ofeach defect potential is independently distributed, and noting, that eachpoint defect is uniformly distributed over the volume, i.e., in essence, thelocation of point defects are independent and identically distributed randomvariables. The function E[eiw∑ki=1 v(ri)] may be expressed as (E[eiwv(r)])k,1194.6. Closed-form analytical resultswhere E[eiwv(r)]) corresponds to the characteristic function associated withan individual point defect within the radius R. If each point defect is uni-formly distributed (in the statistical sense) over the volume, this may beevaluated usingE[eiwv(r)] =∫ R03r2R3eiwv(r)dr. (4.47)Thus, for Eq. (4.43) and Eq. (4.44)ΦVR(ω) =∞∑k=0e−λ43piR3(λ43piR3)kk!(∫ R03r2R3eiωv(r)dr)k. (4.48)This leads toΦVR(ω) =∞∑k=0e−λ43piR3k!(λ43piR3∫ R03r2R3eiωv(r)dr)k, (4.49)which may be more succinctly expressed asΦVR(ω) = exp(λ43piR3∫ R03r2R3eiωv(r)dr − λ43piR3). (4.50)Expressing Eq. (4.50) in a more compact form, it is seen thatΦVR(ω) = exp(λ43piR3(∫ R03r2R3eiωv(r)dr − 1))(4.51)which can be even more succinctly expressed asΦVR(ω) = exp(λ43piR3∫ R03r2R3(eiωv(r) − 1)dr). (4.52)1204.6. Closed-form analytical resultsNoting that as R→∞, ΦVR(ω)→ ΦV (ω), it can be seen thatΦV (ω) = exp(λ4pi∫ ∞0r2(eiωv(r) − 1)dr). (4.53)Recalling Eq. (4.43), it is thus seen thatfz(u) =12pi∫ ∞−∞Φz(ω)e−iwzdω=12pi∫ ∞−∞exp(λ4pi∫ ∞0r2(eiwv(r) − 1)dr − iωz)dω,(4.54)which can be more compactly expressed asfz(z) =12pi∫ ∞−∞exp(λ∫ (eiwv(~r) − 1)d~r − iωz)dω. (4.55)This expression, which corresponds to Eq. (A4) of O’Leary et al [27], for thecase of one-type of point-defect profile, applies independent of the form ofthe potential profile. Thus, the first stated goal of this analysis, i.e., the ac-quisition of a general expression for the distribution of potential fluctuations,has been achieved. There are some peculiar cases that require convergenceconsiderations, however.In order to further simplify the expression, I now must make some as-sumptions related to the specific form for the point defect potential profiles.In particular, I assume:1. that v(r) is monotonically decreasing, where limr→0 v(r) = ∞ andlimr→∞ v(r) = 02. that limr→∞ r3v(r) = 0.1214.6. Closed-form analytical resultsSubject to these conditions, it can be shown, through integration by parts,that the principal term in Eq. (4.52)∫ R0r2(eiwv(r) − 1)dr=[r33(eiwv(r) − 1)]R0−∫ R0r33iωv′(r)eiwv(r)dr(4.56)It is noted thatr33(eiwv(r) − 1)∣∣∣R0=R33(eiwv(R) − 1), (4.57)and that as R→∞, thatR33(eiwv(R) − 1)→ 0. (4.58)Thus, for R→∞∫ R0r2(eiwv(r) − 1)dr → −∫ ∞0r33iωv′(r)eiwv(r)dr. (4.59)Thus, it is seen thatΦV (ω) = exp(−λ4pi∫ ∞0r33iωv′(r)eiωv(r)dr). (4.60)Defining t = v(r), it may be seen thatΦV (ω) = exp(+λ43piiω∫ ∞0[v−1(t)]3eiωtdt), (4.61)which is generally applicable as long as the potential profile has an inverse.1224.6. Closed-form analytical resultsFor the specific case of the power-law potential, i.e,v(r) =1rγ,noting thatv−1(r) = r−1γ ,it may be seen that Eq. (4.61) reduces toΦV (ω) = exp(i43piωλ∫ ∞0t−αeiwtdt)(4.62)where α = 3γ . Following Feller, this integral may be evaluated asΦV (ω) = exp(−43piλΓ(1− α)e− ipiα2 |w|α), (4.63)which is known is known to correspond to a stable distribution functionfor 0 < α < 1. This implies that stable distributions may be achieved for0 < 3γ < 1 or γ > 3, confirming the voracity of my plausibility argumentprovided in Section 4.5.1, recall page 75.It can be demonstrated, for the specific case of α = 1/2, i.e., γ = 6, thatthe corresponding CDF may be expressed asFv(z) = erfc(pi32 4λ2√z). (4.64)where erfc(x) = e−x2x√pi∑∞n=0(−1)n (2n−1)!!(2x2)n . This closed-form expression allowsfor a direct benchmarking with the results acquired from over the Monte1234.7. DOS evaluationCarlo simulations. In particular, for Monte Carlo simulations using M =100,000 iterations, the number of points per unit volume, λ, being set to 100,the boundary length, L, of the cube set to 1, Figure 4.29 demonstrates howthe two CDFs compare. From the plot, I can conclude that the analyticalCDF and the Monte Carlo acquired CDF coincide completely, or at least tothe extent that any differences are visible. This can be used to show thatthe results obtained from the simulations are satisfactory.One other special case is that of the abrupt potential profile, i.e., Eq.(4.7). Going back to Eq. (4.55), for this special case, it is seen thatΦV (ω) = exp(λ(eiw − 1)4pia33). (4.65)It is noted that this characteristic function is a periodic function, with aperiod of 2pi. Thus, a Fourier series can be set up. Thus, the correspondingPDF isfV (z) =∞∑n=0(λ4pia33)nn!e−λ4pi3a3δ(z − n), (4.66)which is exactly the same result as that shown in Eq. (4.38).4.7 DOS evaluationWith the distribution of potential fluctuations evaluated, I can now ex-amine how such a distribution impacts upon the corresponding form of theDOS function. For the purposes of this analysis, I employ the semi-classicalapproach of O’Leary et al. [13]. This method, which follows the semi-classical approach of Kane [49], takes the local conduction band DOS to be1244.7. DOS evaluation0 1 2 3 4 5 6 7 8 9 1010700.10.20.30.40.50.60.70.80.91Monte Carlo CDFAnalytical CDFV(r)=1/r , = 6F(x)xFigure 4.29: Plot of CDF obtained analytically with that obtained froma Monte Carlo simulation. The online version of this figure is depicted incolor.1254.7. DOS evaluationequal to that obtained using the free-electron approximation, i.e., in a localregion, the conduction band DOS function may be seen to beNc(E) =√2m∗3/2cpi2}3√E − Vc(~R), E ≥ Vc(~R)0, E < Vc(~R). (4.67)where Vc(~R) is the potential at ~R, and m∗c is the effective mass of electronsin the band; this is in the same form as the free-electron DOS shown in Eq.(2.8).In order to evaluate the overall conduction band DOS function, I averagethis local form for the conduction-band DOS function over the distributionof potential fluctuations. Accordingly the averaged conduction band DOSfunction may be expressed asNc(E) =√2m∗3/2cpi2}3∫ E−∞√E − zfVc(z)dz, (4.68)where fVc(z)dz is the PDF for Vc, i.e., it is the distribution of potentialfluctuations. Thus, the conduction band DOS function is dependent on theCDF of the potential fluctuations, which can be obtained through MonteCarlo simulations.I start the analysis by noting that there are no experimental resultsthat are presently available wherein one can explicitly resolve for the exactform of the distribution of potential fluctuations. Accordingly, I will pursuethis component of the analysis using two distinct approaches. First, I willexamine the impact of disorder on the form of the DOS function. This will be1264.7. DOS evaluationpursued within the framework of the Gaussian distribution approximation,i.e., it will be assumed thatfV (z) =1√2piσ2exp(−(z − n)22σ2), (4.69)where I will set n = 0 for all cases and examine what happens to the form ofthe DOS as σ is increased. Then, I will demonstrate, using a representativedistribution of potential fluctuations that I have obtained, from a MonteCarlo simulations, that an evaluation of the corresponding DOS functionmay be performed.I start the analysis by examining how the DOS function varies as a func-tion of σ within the framework of the Gaussian approximation. In Figure4.30, I plot the DOS function as a function of the energy, with the mean setto zero, and σ at a variety of levels, i.e., σ is set to 0meV, 35meV, 50meV,and 100meV. The electron effective mass is set to the free electron mass forall cases. It is noted that as σ is increased that the DOS function furtherencroaches into the otherwise empty gap region. This becomes particularlyapparent when these results are plotted on a logarithmic scale, as shown inFigure 4.31. Another interesting feature is that as σ reduces to zero, thefree-electron model for the DOS arises, i.e.,Nc(E)→√2m∗3/2c√E − Ec, E ≥ Ec0, E < Ec,(4.70)where Ec denotes the conduction band edge. Similar results have been1274.7. DOS evaluation-200 -150 -100 -50 0 50 100 150 200Energy (meV)012345678910DOS Function (cm-3 eV-1 )1022 = 0meV = 35meV = 50meV = 100meVFigure 4.30: The DOS corresponding to a Gaussian distribution of poten-tial fluctuations for various σ selections. The free-electron result is seen tocorrespond to the σ = 0meV result. This plot is depicted on a linear scale.The online version of this figure is depicted in colour.1284.7. DOS evaluation-200 -150 -100 -50 0 50 100 150 200Energy (meV)1012101410161018102010221024DOS Function (cm-3 eV-1 ) = 0meV = 35meV = 50meV = 100meVFigure 4.31: The DOS corresponding to a Gaussian distribution of poten-tial fluctuations for various σ selections. The free-electron result is seen tocorrespond to the σ = 0meV result. This plot is depicted on a semi-logarithmetic scale. The online version of this figure is depicted in colour.1294.7. DOS evaluationobtained by O’Leary et al. [27].Finally, I depict a result computed for a distribution of potential fluc-tuations for Monte Carlo acquired results. The particular distribution ofpotential fluctuations that I focus upon is the case of the power-law profile,i.e., v(r) = 1rγ , for the exponent set to 6. The corresponding DOS functionsare depicted in Figures 4.32 and 4.33. An analogous result, determined us-ing the exactly analytical CDF Eq. (4.64), is also depicted in Figure 4.33. Adirect comparison with the free electron model was not pursued as the meanvalue is shifted for the case of γ set to 6 and λ set to 100 points per unitvolume. In principle, the form of the DOS function, corresponding to anyother distribution of potential fluctuations, could also be evaluated usingthis approach.1304.7. DOS evaluation-200 -150 -100 -50 0 50 100 150 200Energy (meV)02468101214DOS Function (cm-3 eV-1 )1022Monte CarloAnalyticalv(r)=1/r, = 6Figure 4.32: The DOS function evaluated from the Monte Carlo acquireddistribution of potential fluctuations corresponding to the power-law profile,v(r) = 1rγ , with γ set to 6 and λ set to 100 points per unit volume. Acorresponding analytically determined result is also presented. This plot isdepicted on a linear scale.1314.7. DOS evaluation-200 -150 -100 -50 0 50 100 150 200Energy (meV)1020102110221023DOS Function (cm-3 eV-1 )Monte CarloAnalyticalv(r)=1/r, = 6Figure 4.33: The DOS function evaluated from the Monte Carlo acquireddistribution of potential fluctuations corresponding to the power-law profile,v(r) = 1rγ , with γ set to 6 and λ set to 100 points per unit volume. Acorresponding analytically determined result is also presented. This plot isdepicted on a semi-logarithmic scale.132Chapter 5ConclusionsThe unique properties exhibited by disordered semiconductors impactthe field of large area electronics in many ways. These properties are, inlarge measure, related to the disorder that is present within the distributionof atoms. Unlike their crystalline counterparts, disordered semiconductorsdo not posses long-range order and their DOS functions encroach into theotherwise empty band gap region. In order to be able to draw upon theunique properties of a disordered semiconductor, one must be first able tocharacterize its fundamental properties. The DOS function plays an essen-tial role in understanding the electronic and optical properties of a givenmaterial. Unfortunately, however, due to the random nature of such a ma-terial, the determination of the DOS functions associated with a disorderedsemiconductor has proven a challenge.In this thesis, a model has been proposed wherein the distribution ofpotential fluctuations of potential fluctuations associated with a disorderedsemiconductor may be evaluated. In this model, the overall potential isevaluated by taking a sum over the potential contributions correspondingto a large number of point defects, these point defects being distributeduniformly and independently over the volume. Monte Carlo simulations were133Chapter 5. Conclusionsemployed in order to acquire solutions. A general analytical solution, for thedistribution of potential fluctuations, was acquired, explicit solutions beingdetermined corresponding to specific cases. The accuracy of the Monte Carloresults was benchmarked using these special cases, and these comparisonswere shown to be quite satisfactory.With the distribution of potential fluctuations evaluated, I then exam-ined the impact upon the DOS functions. Focusing on the specific caseof the conduction band, the impact of disorder was first probed by exam-ining how the form of DOS function varies with the amount of disorder.For the purposes of this particular component of the analysis, the Gaussiandistribution was employed. Then, it was demonstrated that DOS results,corresponding to the Monte Carlo acquired distribution of potential fluc-tuations, could be determined, the corresponding closed-form distributionresult being exactly the same; this was done for a special case for whichboth analytical closed-form and Monte Carlo results were acquired.It is clear that much more work needs to be performed. I have developedan approach for the determination of the distribution of potential fluctua-tions and the corresponding form of the DOS functions. Unfortunately,these two components of the analysis were presented essentially separately.What I need in order to really link these two components of the analysis to-gether is a means in which one can select a specific form for the distributionof potential fluctuations. Thus far, there is a paucity of experimental datathat is available that will allow for this to be done. Eventually, however, ifsuch experimental data is made available, then the obtained DOS functioncan be computed using an experimentally confirmed distribution of poten-134Chapter 5. Conclusionstial fluctuations. This will allow for a much clearer picture to emerge as tothe nature of disordered semiconductors.135References[1] W. F. Brinkman, D. E. Haggan, and W. W. 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Thesis/Dissertation
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Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Monte Carlo analysis for the evaluation of the distribution of potential fluctuations within a disordered semiconductor
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