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An analysis of the optical response of amorphous semiconductors with distributions of defect states Chowdhury, Shamsul Azam 2014

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An analysis of the optical response ofamorphous semiconductors withdistributions of defect statesbyShamsul Azam ChowdhuryB.Sc., Bangladesh University of Engineering and Technology (BUET), 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)February 2014c? Shamsul Azam Chowdhury, 2014AbstractDefects play an important role in shaping the optical response of a semi-conductor material. In this thesis, models for the spectral dependence ofthe imaginary part of the dielectric function, for the specific case of de-fective amorphous semiconductors, are considered. Within the frameworkof a joint density of states functional analysis, closed-form expressions arederived for the imaginary part of the dielectric function, with defect statestaken into account. Both valence band and conduction band defect modelsare considered in this analysis. The role that the different types of opti-cal transitions play in shaping the corresponding optical response are alsoclosely examined. Using the derived models, the spectral dependence of theoptical absorption coefficient is compared with that of experiment for defectabsorption influenced samples of amorphous silicon. The fits of these modelswith the results of experiment are found to be satisfactory.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: Background material . . . . . . . . . . . . . . . . . 112.1 The optical response . . . . . . . . . . . . . . . . . . . . . . . 11iiiTABLE OF CONTENTS2.2 The imaginary part of the dielectric function . . . . . . . . . 132.3 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . 142.4 The relationship between the optical functions and the DOSfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 A free electron model for the DOS function . . . . . . . . . . 242.6 A review of models for the density of state functions . . . . . 26Chapter 3: DOS and JDOS analysis: Single defect band model 373.1 DOS and JDOS analysis . . . . . . . . . . . . . . . . . . . . . 373.2 Conduction band defect model . . . . . . . . . . . . . . . . . 383.2.1 case 1 subcase 1 . . . . . . . . . . . . . . . . . . . . . 493.2.1.1 JVBB-CBB (~?) . . . . . . . . . . . . . . . . . . 493.2.1.2 JVBB-CBT (~?) . . . . . . . . . . . . . . . . . . 513.2.1.3 JVBB-CBD (~?) . . . . . . . . . . . . . . . . . . 533.2.1.4 JVBT-CBB (~?) . . . . . . . . . . . . . . . . . . 543.2.2 Case 1 subcase 2 . . . . . . . . . . . . . . . . . . . . . 553.2.2.1 JVBB-CBT (~?) . . . . . . . . . . . . . . . . . . 553.2.2.2 JVBB-CBD (~?) . . . . . . . . . . . . . . . . . . 563.2.2.3 JVBT-CBB (~?) . . . . . . . . . . . . . . . . . . 573.2.2.4 JVBT-CBT (~?) . . . . . . . . . . . . . . . . . . 583.2.3 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60ivTABLE OF CONTENTS3.2.3.1 JVBB-CBD (~?) . . . . . . . . . . . . . . . . . . 603.2.3.2 JVBT-CBB (~?) . . . . . . . . . . . . . . . . . . 613.2.3.3 JVBT-CBT (~?) . . . . . . . . . . . . . . . . . . 623.2.3.4 JVBT-CBD (~?) . . . . . . . . . . . . . . . . . . 633.3 Valence band defect model . . . . . . . . . . . . . . . . . . . . 65Chapter 4: Application of the conduction band defect andvalence band defect models to the analysis of theoptical response of a-Si . . . . . . . . . . . . . . . . 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Comparison between models . . . . . . . . . . . . . . . . . . . 784.3 Comparison with experiment: the conduction band defectmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 Comparison with experiment: the valence band defect model 924.5 On the uniqueness of the fits . . . . . . . . . . . . . . . . . . 97Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 98References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100vList of TablesTable 2.1 The nominal DOS modeling parameters correspondingto a-Si. These parameter selections, employed for theempirical DOS models described in this section, arefrom Thevaril [4]. . . . . . . . . . . . . . . . . . . . . . 31Table 3.1 The nominal a-Si DOS modeling parameter selectionsemployed for the purposes of this analysis. . . . . . . . 43Table 3.2 The nominal a-Si DOS modeling parameter selectionsemployed for the purposes of this analysis. . . . . . . . 68Table 4.1 The DOS modeling parameter selections for the con-duction band defect model associated with a-Si, em-ployed for the purposes of the fit to the experimen-tal data of Stolk et al. [22], Remes? [23], and Jack-son et al. [6]; the corresponding fits are shown in Fig-ures 4.5, 4.6, and 4.7, respectively. . . . . . . . . . . . 87viLIST OF TABLESTable 4.2 The DOS modeling parameter selections for the va-lence band defect model associated with a-Si, employedfor the purposes of the fit to the experimental data ofStolk et al. [22], Remes? [23], and Jackson et al. [6]; thecorresponding fits are shown in Figures 4.8, 4.9, and4.10, respectively. . . . . . . . . . . . . . . . . . . . . . 95viiList of FiguresFigure 1.1 The number of transistors on a microprocessor. Thisfigure is after Hadi [1]. . . . . . . . . . . . . . . . . . 3Figure 1.2 Semiconductor and flat panel display shipments, plot-ted as a function of date. This data was obtainedfrom the 2005 Information Society Technologies pro-posal for advancement [2]. . . . . . . . . . . . . . . . 6Figure 1.3 A schematic depiction of the light intensity as a func-tion of the penetration depth, z. It is assumed thatthe interface between the vacuum and the material isat z = 0, and that the incident light propagates fromthe left. Reflections from the surface are neglected.This figure is after Thevaril [4]. . . . . . . . . . . . . 8Figure 2.1 The number of possible optical transitions within c-Siand a-Si. This figure is after Jackson et al. [6]. . . . . 17viiiLIST OF FIGURESFigure 2.2 The occupancy function, f(E), as a function of theenergy, E, for three different temperature selections.The reference energy level, i.e., the Fermi energy level,EF , is set to 0 eV for the purposes of all of the plots. 18Figure 2.3 An electron confined within a cubic volume, of di-mensions L?L?L, surrounded by infinite potentialsbarriers. . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.4 The valence band and conduction band DOS func-tions, plotted as a function of energy, E, determinedusing the DOS model of Tauc et al. [11]. . . . . . . . 29Figure 2.5 The valence band and conduction band DOS func-tions, plotted as a function of energy, E, determinedusing the DOS model of Chen et al. [12]. . . . . . . . 30Figure 2.6 The valence band and conduction band DOS func-tions, plotted as a function of energy, E, determinedusing the DOS model of Redfield [13]. . . . . . . . . . 33Figure 2.7 The valence band and conduction band DOS func-tions, plotted as a function of energy, E, determinedusing the DOS model of Cody [14]. . . . . . . . . . . 34ixLIST OF FIGURESFigure 2.8 The valence band and conduction band DOS func-tions, plotted as a function of energy, E, determinedusing the DOS model of O?Leary et al. [15]. . . . . . . 36Figure 3.1 The valence band and conduction band DOS func-tions associated with a-Si. The critical energies in theDOS functions, i.e., EvT , EcT , and EcD , are clearlymarked with the dashed lines and the arrows. Allof the possible optical transition are shown with thearrows. The DOS modeling parameters are as set inTable 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.2 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for ~? > EcT ? EvT .The DOS modeling parameters are as set in Table 3.1. 44Figure 3.3 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for EcD?EvT < ~? <EcT ?EvT . The DOS modeling parameters are as setin Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.4 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for ~? < EcD ? EvT .The DOS modeling parameters are as set in Table 3.1. 48xLIST OF FIGURESFigure 3.5 The valence band and conduction band DOS func-tions associated with a-Si. The critical energies in theDOS functions, i.e., EvT , EcT , and EcD , are clearlymarked with the dashed lines and the arrows. Allthe possible optical transitions are shown with thearrows. The DOS modeling parameters are as set inTable 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 3.6 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for ~? > EcT ? EvT .The DOS modeling parameters are as set in Table 3.2. 69Figure 3.7 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for EcT?EvD < ~? <EcT ?EvT . The DOS modeling parameters are as setin Table 3.2. . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.8 The factors in the JDOS integrand, Nv(E) andNc(E+~?), as a function of energy, E, for ~? < EcT ? EvD .The DOS modeling parameters are as set in Table 3.2. 73xiLIST OF FIGURESFigure 4.1 The fitting of the spectral dependence of JDOS func-tion. The experimental a-Si JDOS result of Jacksonet al. [6] is depicted with solid points. The JDOSfit result, obtained by setting Nvo = Nco = 2.48 ?1022cm?3eV?3/2, Ev = 0.0 eV, Ec = 1.68 eV, and?v = 40 meV, is depicted with the solid line. Thisfigure is after Minar [21]. . . . . . . . . . . . . . . . . 77Figure 4.2 A comparison between two JDOS functions, J(E), as-sociated with a-Si, determined through an evaluationof Eq (2.13), are shown in this figure. For the defect-free model, the DOS functions, Nv(E) and Nc(E),are as specified in Eqs. (3.1) and (3.2), respectively.For a model with defect states included, Nv(E) andNc(E) are as specified in Eqs. (3.45) and (3.46), re-spectively. EcT?EvD and EcT?EvT , critical energiesin the JDOS analysis, are clearly marked with thedashed lines and the arrows. The DOS modeling pa-rameters are as set in Table 3.1. . . . . . . . . . . . . 79xiiLIST OF FIGURESFigure 4.3 The fractional contributions to the overall JDOS func-tion associated with the various types of a-Si opti-cal transitions for the conduction band defect model.EcD ? EvT and EcT ? EvT , critical energies in theJDOS analysis, are clearly marked with the dashedlines and the arrows. The DOS modeling parametersare as set in Table 3.1. . . . . . . . . . . . . . . . . . 80Figure 4.4 The fractional contributions to the overall JDOS func-tion associated with the various types of a-Si op-tical transitions for the valence band defect model.EcT ? EvD and EcT ? EvT , critical energies in theJDOS analysis, are clearly marked with the dashedlines and the arrows. The DOS modeling parametersare as set in Table 3.2. . . . . . . . . . . . . . . . . . 82xiiiLIST OF FIGURESFigure 4.5 The optical absorption spectrum, ?(~?), associatedwith a-Si. The experimental a-Si data set of Stolk etal. [22] is depicted with solid points; these experimen-tal data points corresponds to the ?relaxed a-Si? datapoints, depicted in Figure 13(a) of Stolk et al. [22].The solid line corresponds the conduction band defectmodel fitting to this experimental data, by setting theDOS modeling parameters to the selections specifiedin Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . 88Figure 4.6 The optical absorption spectrum, ?(~?), associatedwith a-Si. The experimental a-Si data set of Remes? [23]is depicted with solid points; these experimental datapoints correspond to the ?standard GD-a? data points,depicted in Figure 5.2 of Remes? [23]. The solid linecorresponds the conduction band defect model fittingto this experimental data, by setting the DOS mod-eling parameters to the selections specified in Table 4.1. 89xivLIST OF FIGURESFigure 4.7 The imaginary part of dielectric function spectrum,2, associated with a-Si. The experimental a-Si dataset of Jackson et al. [6] is depicted with solid points;these experimental data points correspond to the ?log-arithmic plot of 2? data points, depicted in Figure3(b) of Jackson et al. [6]. The solid line correspondsto the conduction band defect model fitting to thisexperimental data, by setting the DOS modeling pa-rameters to the selections specified in Table 4.1. . . . 91Figure 4.8 The optical absorption spectrum, ?(~?), associatedwith a-Si. The experimental a-Si data set of Stolk etal. [22] is depicted with solid points; these experimen-tal data points correspond to the ?relaxed a-Si? datapoints, depicted in Figure 13(a) of Stolk et al. [22].The solid line corresponds the valence band defectmodel fitting to this experimental data, by settingthe DOS modeling parameters to the selections spec-ified in Table 4.2. . . . . . . . . . . . . . . . . . . . . 93xvLIST OF FIGURESFigure 4.9 The optical absorption spectrum, ?(~?), associatedwith a-Si. The experimental a-Si data set of Remes? [23]is depicted with solid points; these experimental datapoints correspond to the ?standard GD-a? data points,depicted in Figure 5.2 of Remes? [23]. The solid linecorresponds the valence band defect model fitting tothis experimental data, by setting the DOS modelingparameters to the selections specified in Table 4.2. . . 94Figure 4.10 The imaginary part of dielectric function spectrum,2, associated with a-Si. The experimental a-Si dataset of Jackson et al. [6] is depicted with solid points;these experimental data points correspond to the ?log-arithmic plot of 2? data points, depicted in Figure3(b) of Jackson et al. [6]. The solid line correspondsthe valence band defect model fitting to this experi-mental data, by setting the DOS modeling parame-ters to the selections specified in Table 4.2. . . . . . . 96xviList of SymbolsR2(E) aggregate dipole matrix element squared averageP2(E) aggregate momentum matrix element squared average?v breadth of the valence band tail?c breadth of the conduction band tail?vD breadth of the valence band defect distribution?cD breadth of the conduction band defect distributionEc conduction band band edgeNco conduction band density of states prefactorN?co conduction band tail prefactorNc(E) conduction band density of states functionEvT critical energy at which the exponential and square-rootdistributions interface in the valence bandEvD critical energy at which the exponential and defectdistributions interface in the valence bandxviiList of SymbolsEcT critical energy at which the exponential and square-rootdistributions interface in the conduction bandEcD critical energy at which the exponential and defectdistributions interface in the conduction band?A density of silicon atomsq electron chargeme electron massEg energy gapEF Fermi energy levelV illuminated volume2 imaginary part of the dielectric functionI(z) intensity of light at position zI0 intensity of light at position z=0? optical absorption coeffieientE Photon energyn(E) Refractive index|v? single spin electronic state associated with the valence band|c? single spin electronic state associated with the conduction bandEv valance band band edgeNvo valence band density of states prefactorxviiiList of SymbolsNv(E) valence band density of states functionN?vo valence band tail prefactorxixList of Acronymsa-Si amorphous siliconc-Si crystalline siliconDOS density of StatesJDOS joint density of statesVBB valence band bandVBT valence band tailVBD valence band defectCBB conduction band bandCBT conduction band tailCBD conduction band defectxxAcknowledgementsThis thesis would not have been possible without the help and supportof my colleagues, friends, and family.I am heartily thankful to my supervisor, Dr. Stephen O?Leary, for bothinspiring and teaching me. His constant guidance, support and encourage-ment helped me throughout my thesis.xxiTo my familyxxiiChapter 1IntroductionThe semiconductor industry, as we know it today, can trace its originsback to a series of innovations that occurred in the mid-20th Century at BellTelephone Laboratories in Murray Hill, New Jersey. By that time, vacuumtube technology had been fully developed, and such tubes were deployedin all aspects of the telephone network; vacuum tubes were then used inorder to electronically process electrical signals. Electro-mechanical relays,allowing for automatic telephone dialing and switching over such networks,were also available and also widely deployed. Thus, at the time, to thetechnical lay-person at least, it may have appeared that the technology wasavailable in order to address all communication needs for the foreseeablefuture.As far as the management of Bell Telephone Laboratories was concerned,however, difficulties still persisted. Vacuum tubes consume large amountsof electrical power and have a relatively short mean-time-to-failure. Electro-1Chapter 1. Introductionmechanical relays, while offering many enticing capabilities, operated at lim-ited speeds. In addition, they were also inherently prone to failure. Therewas a realization that the limitations inherent to these technologies wouldeventually limit future developments in telephony. Thus, a group was com-missioned in 1945 in order to find a solid-state alternative to both vacuumtubes and electro-mechanical relays that will offer the same functionality butat a reduced cost and with greater reliability. This group, led by the bril-liant but mercurially tempered William B. Shockley, focused upon a classof hitherto unexplored materials, semiconductors, owing to their peculiarand promising properties. This effort paid off handsomely in short order,the demonstration of the first semiconductor-based transistor occurring onChristmas Eve of 1947.The developments in electronics that have occurred since that time haveprimarily arisen as a consequence of a detailed and quantitative understand-ing of the material properties of the materials used within the constituentelectron devices found within electronic systems. As a consequence, thematerial properties of conductors, insulators, and dielectrics, the materialsfrom which these constituent electron devices are fabricated, has been thefocus of tremendous interest for many years. This understanding has al-lowed for a shrinking in the dimensions of these electron devices as well as a2Chapter 1. Introductionconcomitant increase in the number of electron devices packed onto a givenchip. In fact, in 1965, Gordon E. Moore, the founding president of Intel,suggested that the number of transistors on a given chip would double every18 months for the next decade; see Figure 1.1 [1]. Moore?s law, as this 1965projection is now referred to as, has held for almost 5 decades now, andthe present expectation is that it will continue to hold for at least anotherdecade. As the dimensions of the devices employed approach fundamentallimits, however, it is clear that a deeper understanding of the material prop-erties of these materials will be required if Moore?s law is to continue to holdinto the future. As a consequence, research into the material properties ofsemiconductors continues with great intensity.Progress in conventional electronics has typically been achieved by mak-ing the constituent electron devices smaller, faster, more reliable, and lessexpensive. There is, however, another class of electron device that requiressize in order to be useful. Scanners and displays, which act at the interfacebetween the human world and its digital counterpart, must be of a certainsize in order to perform properly. Solar cells and x-ray imagers also must beof a certain size in order to perform properly. These devices are all referredto as examples of large area electronic devices.3Chapter 1. Introduction1970 1975 1980 1985 1990 1995 2000 2005 2010103104105106107108109Number of transistors on a microprocessorYearFigure 1.1: The number of transistors on a microprocessor. This figure isafter Hadi [1].4Chapter 1. IntroductionStarting off as a niche field of inquiry in the late 1960s, large area elec-tronics is now an important field in its own right, with commercial sales nowconstituting a substantial fraction of the overall commercial semiconductormarket; see Figure 1.2 [2]. While the focus in conventional electronics is onfabricating devices with sub-micron device features, in large area electron-ics the focus instead is on depositing thin-films uniformly and inexpensivelyover substrates of the order of a square-meter in dimensions [3]. Crystallinesilicon (c-Si), the dominant material used in conventional electronics, cannot be used as a material for thin-film deposition, and thus, other materialsmust be used instead. Amorphous silicon (a-Si) and microcrystalline silicon(?c-Si) are often used for this purpose. As they are disordered forms ofsilicon, the electronic properties of these materials are rather distinct fromthose of their crystalline counterparts. It is the quantitative understandingof these differences that has been the focus of materials work for a greatmany years.For many of the device applications considered within the large areaelectronics field, the response of the underlying electronic materials to lightplays a decisive role in determining the corresponding device performance.This is particularly true for solar cell devices and digital flat panel x-rayimage detectors, the response to light being a fundamental performance5Chapter 1. Introduction1985 1990 1995 2000 2005 2010050100150200250300350SemiconductorsFlat Panel DisplaysYearMarket ($ US Billions)Figure 1.2: Semiconductor and flat panel display shipments, plotted as afunction of date. This data was obtained from the 2005 Information SocietyTechnologies proposal for advancement [2].6Chapter 1. Introductionmetric associated with both applications. Accordingly, the optical responseof the materials used within large area electron devices continues to be animportant focus of research.When a material is exposed to light, i.e., when light passes through agiven material, the intensity of the light decreases with the depth into thematerial. Consider, for example, a beam of light perpendicularly incidentupon a semi-infinite slab of a material, as depicted in Figure 1.3 [4]. If thematerial properties are uniform, as the light passes through the material, itsintensity will diminish exponentially. In the absence of reflections from thesurface of the material, the intensity of the beam of light within the materialcan be expressed as a function of the penetration depth into the material,z, i.e.,I(z) = Io exp(??z), (1.1)where Io denotes the intensity of the light at the surface of the material,z = 0, and ? represents the optical absorption coefficient [5]. The opticalabsorption coefficient, ?, determines how far light can penetrate into a ma-terial before it is fully absorbed. This coefficient exhibits a strong spectraldependence on the incident photon energy, E, and from this dependence,the nature of the optical transitions that occur within the semiconductormay be determined.7Chapter 1. IntroductionFigure 1.3: A schematic depiction of the light intensity as a function of thepenetration depth, z. It is assumed that the interface between the vacuumand the material is at z = 0, and that the incident light propagates fromthe left. Reflections from the surface are neglected. This figure is afterThevaril [4].8Chapter 1. IntroductionThe experimental determination of the spectral dependence of the opti-cal absorption coefficient, i.e., the determination of ?(E), will allow one todetermine: (1) the energy gap of the material, (2) the nature of this energygap, i.e., whether it be direct or indirect, and (3) provide some insights intothe amount of disorder that is present within the material. With referenceto the third point mentioned, it should be noted that in a disorderless crys-talline semiconductor, at zero-temperature, this optical absorption spectrumterminates abruptly at the energy gap, while in an amorphous or microcrys-talline semiconductor, the optical absorption spectrum will encroach into theotherwise empty gap region, the amount of encroachment corresponding tohow much disorder is present. From a detailed study into the nature of thisencroachment, insights into the character of the underlying distributions ofelectronic states may be gleaned.In this thesis, an empirical model for the distribution of electronic statesassociated with an amorphous semiconductor, which includes defect states,will be developed, and the corresponding spectral dependence of the opticalabsorption spectrum will be determined. The distribution of defect stateswithin an amorphous semiconductor will be modeled by adding broad ex-ponential tails of defect states onto a traditional model for the distributionof electronic states in the absence of defect states. The corresponding op-9Chapter 1. Introductiontical absorption spectrum is then determined through an evaluation of thecorresponding joint density of states (JDOS) function, this being obtainedthrough a convolution over the valence band and conduction band distribu-tions of electronic states and the use of a model for the relationship betweenthe spectral dependence of the imaginary part of the dielectric function andthat of the corresponding JDOS function. The results of these theoreticalinvestigations are then compared with those of experiment, and from thiscomparison, the modeling parameters are determined. For the purposes ofthis analysis, the primary focus is on the case of hydrogenated amorphoussilicon (a-Si:H), the most widely deployed amorphous semiconductor today.This thesis is organized in the following manner. In Chapter 2, thetheoretical background for this analysis is presented. Then, in Chapter 3, amodel for the distribution of electronic states, one which includes treatmentof the distributions of defect states, is introduced. The resultant spectraldependence of the optical absorption coefficient, ?(E), is then evaluated inChapter 4, a comparison with the results of experiment, for the specific caseof a-Si:H, being provided. Finally, the conclusions of this analysis, and somerecommendations for further research, are presented in Chapter 5.10Chapter 2Background material2.1 The optical responseThe response of a given material to light, i.e., its optical response, isoften considered in materials characterization studies. Optical transitions,from occupied valence band electronic states to unoccupied conduction bandelectronic states, determine the nature of this response. The distribution ofelectronic states, the occupancy of these states, and the probability that agiven optical transition between such states can occur at all, play roles inshaping the exact form of this response. The central goal of this analysisis to develop a model for the spectral dependence of the optical absorptioncoefficient, ?(E), that may be used in order to interpret the physical meaningof the optical properties associated with amorphous semiconductors. Thiswill be achieved through the development of an empirical model for thedistributions of valence band and conduction band electronic states, the112.1. The optical responsedistributions of defect states being explicitly included in this model, andthe determination of the corresponding JDOS function from this model.Given that the JDOS function is directly related to the imaginary part ofthe dielectric function, 2(E), and the spectral dependence of the opticalabsorption coefficient, ?(E), this will allow one to relate the underlyingdistributions of electronic states with the corresponding optical response.Through a fit with the results of experiment, the underlying form of thedistribution of electronic states may thus be determined.In this chapter, the background required for this work is provided. Theanalysis begins with an expression for the spectral dependence of the imag-inary part of the dielectric function in terms of the energy levels of the in-dividual states associated with the valence and conduction bands. Then, anumber of simplifying assumptions are introduced, within the framework ofa dipole matrix element based formalism, a relationship between the imagi-nary part of the dielectric function and the density of states (DOS) functionsbeing developed. A free electron DOS model is then presented. Finally, abrief review of the development of a number of empirical DOS models, ap-propriate for the treatment of amorphous semiconductors, is then provided.This chapter is organized in following manner. In Section 2.2, an ex-pression for the spectral dependence of the imaginary part of the dielectric122.2. The imaginary part of the dielectric functionfunction, 2(E), is presented. This expression is given in terms of the valenceand conduction band electronic state energy levels and the correspondingoptical transition matrix elements. Then, three simplifying assumptions areintroduced in Section 2.3, these assumptions allowing one to simplify theexpression for the imaginary part of the dielectric function for the specificcase of a-Si. In Section 2.4, a dipole matrix element based formalism isintroduced in order to develop a relationship between the imaginary part ofthe dielectric function, 2(E), and the valence and conduction band DOSfunctions. An aggregate dipole matrix element arises as a byproduct of thisparticular formalism. Then, in Section 2.5, the free electron model, for theDOS functions, is introduced. Finally, in Section 2.6, a brief review on thedevelopment of models for the DOS functions associated with an amorphoussemiconductor, is provided.2.2 The imaginary part of the dielectric functionOptical transitions occur between the occupied valence band electronicstates and the unoccupied conduction band electronic states. For a linearresponse, within the framework of a one-electron perspective, Jackson etal. [6] assert that the imaginary part of the dielectric function,132.3. Simplifying assumptions2(E) =(2piq~meE)2 2V?v,c|~?. ~Pv,c|2?(Ec ? Ev ? E), (2.1)where q denotes the electron charge, me is the free electron mass, V is theilluminated volume, ~? is the polarization vector of the incident light, and Eis the incident photon energy, Ev and Ec corresponding to representative va-lence band (initial state) and conduction band (final state) electronic states,respectively. The matrix element, ~Pv,c, which is defined as ~Pv,c = ?c |~P | v?,where ~P denotes the momentum operator, |v? is a representative valenceband electronic state, and |c? is a representative conduction band electronicstate, couples the electronic states between which an optical transition mayoccur, i.e., it determines the probability of an optical transition betweensuch states. The summation in Eq. (2.1), performed within the frameworkof the aforementioned one-electron perspective, is performed over all of theoccupied valence band and unoccupied conduction band electronic states.2.3 Simplifying assumptionsThe imaginary part of the dielectric function, 2(E), is expressed inEq. (2.1) in terms of the valence band and conduction band energy levels,Ev and Ec, respectively, and the magnitude of the corresponding momentummatrix elements, ~Pv,c. This expression is generally applicable, for both the142.3. Simplifying assumptionsamorphous and crystalline semiconductor cases. For the specific case ofan amorphous semiconductor, however, the imaginary part of the dielectricfunction, 2(E), may be related to the distribution of valence band andconduction band electronic states by invoking three critical assumptions:(1) the incident light is unpolarized, (2) the momentum conservation rulesare relaxed, and (3) zero-temperature statistics apply. The consequence ofeach of these assumptions is discussed subsequently.The use of unpolarized light simplifies Eq. (2.1). Assuming random po-larization, the directional average of |~?. ~Pv,c|2 reduces to 13 |Pv,c|2, where|Pv,c| denotes the amplitude of the momentum matrix element. As a conse-quence, Eq. (2.1) reduces to2(E) =(2piq~meE)2 23V?v,c|Pv,c|2?(Ec ? Ev ? E). (2.2)Within a crystalline semiconductor, momentum is conserved during anoptical transition. In contrast, for the case of an amorphous semiconductor,the momentum conservation rules are relaxed. That is, from every occu-pied valence band electronic state, optical transitions are possible to everyunoccupied conduction band electronic state, assuming that the spin statesare coincident, Jackson et al. [6] asserting that spin-flips are not allowedduring such an optical transition. As a consequence, the momentum matrixelement, |Pv,c|, is constant for all possible optical transitions in the amor-152.3. Simplifying assumptionsphous case, while it may vary greatly for the crystalline semiconductor case.This relaxation in the momentum conservation rules allows for a furthersimplification in the subsequent analysis.In order to understand how a relaxation in the conservation rules impactsupon the allowed optical transitions, consider the case of N silicon atoms.For the case of c-Si, from every occupied valence band electronic states, thereare four possible optical transitions, while for the case of a-Si, 2N opticaltransitions are possible, as the momentum conservation rules are relaxed.The number of possible optical transitions, for both cases, are stylisticallydepicted in Figure 2.1.The third simplifying assumption is that of zero-temperature occupationstatistics. The occupancy function, i.e., the probability of a given electronicstate being occupied, also referred to as the Fermi-Dirac distribution func-tion [10], may be expressed asf(E) = 11 + exp(E?EFkbT ). (2.3)The dependence of this occupancy function on temperature is depicted inFigure 2.2, three different temperature levels being considered. Here, EFdenotes the Fermi energy level, kb is the Boltzmann constant, and T isthe temperature. It is noted that at absolute zero-temperature, i.e., atT = 0 K, this function behaves as a step-function, where the step is at the162.3. Simplifying assumptionsa?Si c?SiEF2?AVkEnergyFigure 2.1: The number of possible optical transitions within c-Si and a-Si.This figure is after Jackson et al. [6].172.3. Simplifying assumptions?0.3 ?0.2 ?0.1 0 0.1 0.2 0.300.250.50.751EF300 K800 K0 KElectron Energy (eV)Probability of occupationFigure 2.2: The occupancy function, f(E), as a function of the energy, E,for three different temperature selections. The reference energy level, i.e.,the Fermi energy level, EF , is set to 0 eV for the purposes of all of the plots.182.4. The relationship between the optical functions and the DOS functionsFermi energy level itself, i.e., at E = EF . For the case of semiconductors,at T = 0 K, the occupancy probability is unity for the valence band statesand zero for the conduction band states, i.e., all the valence band electronicstates are fully occupied and all the conduction band electronic states arefully unoccupied. This assumption simplifies the subsequent analysis.2.4 The relationship between the opticalfunctions and the DOS functionsIn order to smooth the variations in |Pv,c| that occur from optical tran-sition to optical transition, researchers have found it useful to introduce anaggregate matrix element, P2(E), which corresponds to the average value of|P2v,c| over all of the optical transitions that occur at energy E. It was oftensimply assumed that P2(E) is independent of the photon energy, E. Codyet al. [7] attempted to validate this constant momentum matrix elementassumption with some experimental data. For this purpose, they prepareda large number of a-Si:H samples of different thicknesses. They observedthat although the spectral dependence of the optical functions can be sat-isfactorily explained assuming a constant P2(E), the fit with experimentconsiderably improves when one instead assumes a constant dipole matrix192.4. The relationship between the optical functions and the DOS functionselement. Moreover, a thickness dependence artifact, observed in the exper-imental results of Cody et al. [7], disappears when one adopts the constantdipole matrix element assumption. Based on these experimental observa-tions of Cody et al. [7], Jackson et al. [6] casts Eq. (2.1) in terms of thedipole-matrix elements rather than in terms of the momentum-matrix ele-ments. In particular, from the commutator relations, Jackson et al. [6] findthat?v,c|~?.~Pv,c|2?(Ec?Ev?E) =(meE~2)2?v,c|~?. ~Rv,c|2?(Ec?Ev?E), (2.4)where ~Rv,c = ?c| ~R|v? is the dipole-matrix element and ~R is the dipoleoperator. As was mentioned previously, for the case of a-Si, if the incidentlight is unpolarized, the directional average of |~?. ~Rv,c|2 reduces to 13 |Rv,c|2,where |Rv,c| denotes the amplitude of the dipole matrix element. As aconsequence2(E) = (2piq)223V?v,c|Rv,c|2?(Ec ? Ev ? E),= (2piq)2 23V [R?(E)]2?v,c?(Ec ? Ev ? E). (2.5)So far, the derived expression, i.e., Eq. (2.5), is applicable for both c-Si and a-Si. In order to facilitate a direct comparison between the matrixelements associated with these two different materials, the dipole matrixelements should be normalized by a factor that is proportional to the ra-202.4. The relationship between the optical functions and the DOS functionstio between the numbers of allowed optical transitions, corresponding toeach material, i.e., 2N4 , so that the overall ?optical strength? remains con-stant. Letting ?A denote the atomic density of the Si atoms within a-Si, thisnormalization factor becomes 2?AV4 . Following this assumption, Jackson etal. [6] introduced a normalized average dipole matrix elementR2(E) = ?AV2 [R?(E)]2, (2.6)where [R?(E)]2 is the dipole matrix element squared averaged over all opticaltransitions separated by the energy E. This matrix element may be definedas[R?(E)]2 ??v,c?(Ec ? Ev ? E)|Rv,c|2?v,c?(Ec ? Ev ? E). (2.7)Using the relationship defined in Eq. (2.6), it is noted that Eq. (2.5) can berepresented as2(E) = (2piq)223V( 2?AV)R2(E)?v,c?(Ec ? Ev ? E). (2.8)The last term,?v,c?(Ec ?Ev ?E), in Eq. (2.8) provides information on thenumber of possible optical transitions between the occupied valence bandand the unoccupied conduction band states, separated by energy E. Defin-ing the JDOS function,J(E) ? 4V?v,c?(Ec ? Ev ? E), (2.9)212.4. The relationship between the optical functions and the DOS functionsfrom Eqs. (2.8) and (2.9), one can thus conclude that2(E) =(2piq)23?AR2(E)J(E). (2.10)The valence band DOS function, Nv(E), where Nv(E)?E represents thenumber of one electron valence band states, between energies [E,E + ?E],per unit volume, may be expressed asNv(E) =2V?v?(E ? Ev), (2.11)the sum being taken over all of the valence band single-spin electronic statesdivided by the volume, the factor of 2 representing the fact that each elec-tronic state considered is single-spin, i.e., there are two possible spins. Sim-ilarly, one can define the conduction band DOS function, Nc(E), whereNc(E)?E represents the number of one electron conduction band states,between energies [E,E + ?E], per unit volume, it can be expressed asNc(E) =2V?c?(E ? Ec). (2.12)As before, in Eq. (2.12), the factor 2 represents the fact that each electronicstate is single-spin, i.e., there are two possible spins.For zero-temperature statistics, i.e., assuming that all of the valenceband electronic states are fully occupied and that all of the conductionband electronic states are completely unoccupied, the JDOS function maybe expressed as an integral over the valence band and the conduction band222.4. The relationship between the optical functions and the DOS functionsDOS functions. That is,J(E) =? ???Nv(?)Nc(E + ?)d?. (2.13)For the specific case of a-Si, given that the density of Si atoms within a-Si isaround 4.4? 1022 cm?3 [8], it can be seen that Eq. (2.10) may be expressedas2(E) = 4.3? 10?45R2(E)J(E), (2.14)where R2(E) is in unit of A?2 and J(E) is in the unit of cm?6eV?1 [6].The spectral dependence of the optical absorption coefficient can be de-termined [4] by noting that,?(E) = En(E)c~2(E), (2.15)where n(E) denotes the spectral dependence of the index of refraction, crepresents the speed of the light in vacuum, and ~ is the reduced Plank?sconstant. The spectral dependence of the refractive index, n(E), is deter-mined by fitting a tenth-order polynomial to the experimental results ofKlazes et al. [9]; the experimental data considered for determining n(E)corresponds to that presented in Figure 4 of Klazes et al. [9]; this techniquewas used previously by Thevaril [4].232.5. A free electron model for the DOS function2.5 A free electron model for the DOS functionConsider a three-dimensional cubic box, of dimensions L?L?L, withinwhich an electron is confined, as depicted in Figure 2.3. According to quan-tum mechanics, for steady-state conditions, the wavefunctions associatedwith this confined electron can be determined through the solution of thesteady-state, three-dimensional Schro?dinger?s equation [10], i.e.,?2??x2 +?2??y2 +?2??z2 +2me~2(E ? V )?(x, y, z) = 0, (2.16)where me, E, and V denote the free electron mass, the electron energy, andthe potential energy, respectively. If it is assumed that the electron is totallyfree within the cubic box itself, i.e., V=0 for 0 ? x ? L, 0 ? y ? L, and0 ? z ? L, and if the potential outside the box is infinite, then it can beshown that,?nxnynz(x, y, z) =( 2L)2/3sin(nxpixL)sin(nypiyL)sin(nzpizL), (2.17)where nx, ny, and nz are positive integers associated with the electron?smotion in the x, y, and z directions, respectively, denoting the correspondingquantum numbers. The energy of the corresponding electronic states maybe found by substituting this solution for the wavefunction, i.e., Eq. (2.17),back into Schro?dinger?s equation. It may thus be shown that242.5. A free electron model for the DOS functionXyz V(x,y,z)=0-inside boxV(x,y,z)=? -outside boxLLLFigure 2.3: An electron confined within a cubic volume, of dimensions L?L? L, surrounded by infinite potentials barriers.252.6. A review of models for the density of state functionsEnxnynz =~22me[(nxpiL)2+(nypiL)2+(nzpiL)2], (2.18)where Enxnynz represents the energy level corresponding to the wavefunc-tion, ?nxnynz(x, y, z). It is clear that these energy levels depend on threequantum numbers, i.e., nx, ny, and nz. In the continuum limit, i.e., for largevalues of energy, E, the associated DOS function may be expressed asN (E) =????????2 m3/2epi2~3?E, E ? 00, E < 0. (2.19)It is seen that this DOS function has no dependence on the cubic di-mensions, i.e., L, in of itself. That is, when L??, i.e., when the confinedelectron becomes totally free, the DOS function still remains the same. Thissquare-root DOS function, often referred to as the free-electron DOS model,provides the basis for the subsequent analysis.2.6 A review of models for the density of statefunctionsThe nature of the disorder that is present within a-Si has been the focusof investigation for many years. It is widely assumed that disorder plays animportant role in shaping the optical response of an amorphous semicon-ductor, i.e., it produces a distribution of electronic states that encroaches262.6. A review of models for the density of state functionsinto the otherwise empty gap region. The distribution of electronic statesthat encroach into the gap region are generally referred to as tail states. Inorder to provide a theoretical basis for understanding the nature of the opti-cal absorption spectrum associated with an amorphous semiconductor, theDOS functions associated with these materials must be known. For manyyears, the exact role that disorder plays in influencing the form of the DOSfunctions has been under investigation. A brief review of the DOS models,that have been proposed in the past, is provided next.Based on the free-electron DOS model, in 1966, Tauc et al. [11] proposedan empirical DOS model for both the valence band and conduction bandDOS functions. In particular, Tauc et al. [11] assumes thatNv (E) =???????Nvo?Ev ? E, E ? Ev0, E > Ev, (2.20)and thatNc (E) =???????Nco?E ? Ec, E ? Ec0, E < Ec, (2.21)where Nvo and Nco denote the valence band and conduction band DOS pref-actors, respectively, Ev and Ec represent the valence band and conductionband band edges, and E is the energy. The resultant DOS functions, for thenominal selections of DOS modeling parameters tabulated in Table 2.1, are272.6. A review of models for the density of state functionsdepicted in Figure 2.4. This model forms the basis for the determination ofthe optical gap associated with an amorphous semiconductor.In 1981, an improved model of Tauc et al. [11] was devised by Chen etal. [12], in which an exponential distribution of tail states is splined ontoa square-root distribution of valence band states; Chen et al. [12] only as-sumed exponential tail states for the valence band DOS function, Nv(E),the conduction band DOS function, Nc(E), being exactly same as that ofTauc et al. [11]. Chen et al. [12] thus assume thatNv (E) =???????Nvo?Ev ? E, E ? EvN?vo exp(Ev?E?v), E > Ev, (2.22)and thatNc (E) =???????0, E < EcNco?E ? Ec, E ? Ec, (2.23)where N?vo represents the valence band tail prefactor and ?v denotes thevalence band tail breadth. All other parameters are the same as definedearlier. The resultant DOS functions, for the nominal selections of the DOSmodeling parameters tabulated in Table 2.1, are depicted in Figure 2.5.There is general consensus that the optical properties of a semiconductorare greatly influenced by both the valence band and the conduction bandtails. Following Chen et al. [12], Redfield [13] proposed a new model where282.6. A review of models for the density of state functions?0.5 0 0.5 1 1.5 212345678910 x 1021N c ( E )N v ( E )Energy (eV) DOS (cm?3 eV ?1 )Figure 2.4: The valence band and conduction band DOS functions, plottedas a function of energy, E, determined using the DOS model of Tauc etal. [11].292.6. A review of models for the density of state functions?0.5 0 0.5 1 1.5 212345678910 x 1021N c ( E )N v ( E )Energy (eV)DOS (cm?3 eV ?1 )Figure 2.5: The valence band and conduction band DOS functions, plottedas a function of energy, E, determined using the DOS model of Chen etal. [12].302.6. A review of models for the density of state functionsTable 2.1: The nominal DOS modeling parameters corresponding to a-Si.These parameter selections, employed for the empirical DOS models de-scribed in this section, are from Thevaril [4].Parameter (unit) Tauc et al. Chen et Redfield Cody O?Leary[11] al. [12] [13] [14] et al. [15]Nvo (cm?3eV?3/2) 2? 1022 2? 1022 - 2? 1022 2? 1022Nco (cm?3eV?3/2) 2? 1022 2? 1022 - 2? 1022 2? 1022N?vo (cm?3eV?3/2) - 5? 1021 5? 1021 - -N?co (cm?3eV?3/2) - - 5? 1021 - -Ev (eV) 0 0 0 0 0Ec (eV) 1.7 1.7 1.7 1.7 1.7Ev ? EvT (meV) - - - - 25EcT ? Ec (meV) - - - - 13.5?v (meV) - 50 50 50 50?c (meV) - - 27 - 27the conduction band tail states are also included. However, Redfield [13]assumes constant distributions of valence band and conduction band bandstates. That is, Redfield [13] assumes thatNv (E) =???????N?vo, E ? EvN?vo exp(Ev?E?v), E > Ev, (2.24)and thatNc (E) =???????N?co, E ? EcN?co exp(E?Ec?c), E < Ec, (2.25)where N?co represents the conduction band tail prefactor and ?c denotes theconduction band tail breadth. All other parameters are the same as definedearlier. The resultant DOS functions, for the nominal selections of the DOS312.6. A review of models for the density of state functionsmodeling parameters tabulated in Table 2.1, are depicted in Figure 2.6.Later on, a further improved model was developed, based on the modelsof Chen et al. [12] and Redfield [13]. In 1984, Cody [14] proposed a newmodel, where it is assumed that the exponential tail states occur belowthe square-root band band edge, Ev, differing from Chen et al. [12] in thesense that Chen et al. [12] assume an exponential band tail splined onto aterminated square-root band state. That is, Cody [14] asserts thatNv (E) =???????Nvo?Ev ? E, E ? Ev ? 32?vNvo?32?v exp(?32)exp(Ev?E?v), E > Ev ? 32?v, (2.26)and thatNc (E) =???????Nco?E ? Ec, E ? Ec0, E < Ec, (2.27)where all the DOS modeling parameters are the same as defined earlier. Theresultant DOS functions, for the nominal selections of the DOS modelingparameters tabulated in Table 2.1, are depicted in Figure 2.7.The empirical DOS model of Cody [14] was further improved by O?Learyet al. [15]. In 1997, they proposed a new model, where it was assumed:(1) square-root distributions of band states and exponential distributions oftail states, for the valence band and conduction band, and (2) the valenceband and conduction band DOS functions, Nv(E) and Nc(E), and their322.6. A review of models for the density of state functions?0.5 0 0.5 1 1.5 21234567 x 1021N c ( E )N v ( E )Energy (eV) DOS (cm?3 eV ?1 )Figure 2.6: The valence band and conduction band DOS functions, plottedas a function of energy, E, determined using the DOS model of Redfield [13].332.6. A review of models for the density of state functions?0.5 0 0.5 1 1.5 212345678910 x 1021N c ( E )N v ( E )Energy (eV) DOS (cm?3 eV ?1 )Figure 2.7: The valence band and conduction band DOS functions, plottedas a function of energy, E, determined using the DOS model of Cody [14].342.6. A review of models for the density of state functionsderivatives, are continuous at the critical energies at which the square-rootdistributions and exponential distributions interface. That is, O?Leary etal. [15] assert thatNv (E) = Nvo????????Ev ? E, E ? Ev ? ?v2??v2 exp(?12)exp(Ev?E?v), E > Ev ? ?v2, (2.28)and thatNc (E) = Nco????????E ? Ec, E ? Ec + ?c2??c2 exp(?12)exp(E?Ec?c), E < Ec + ?c2, (2.29)where all of the DOS modeling parameters are the same as defined ear-lier. The resultant DOS functions, for the nominal selections of the DOSmodeling parameters tabulated in Table 2.1, are depicted in Figure 2.8.352.6. A review of models for the density of state functions?0.5 0 0.5 1 1.5 212345678910 x 1021N c ( E )N v ( E )Energy (eV) DOS (cm?3 eV ?1 )Figure 2.8: The valence band and conduction band DOS functions, plottedas a function of energy, E, determined using the DOS model of O?Leary etal. [15].36Chapter 3DOS and JDOS analysis:Single defect band model3.1 DOS and JDOS analysisIt has already been established that defects play an important role inshaping the optical response of a semiconductor. Defects also influencethe corresponding electronic properties. Understanding how defects are dis-tributed within the energy gap of a semiconductor is of critical importance tounderstanding how a material responds to light and in evaluating how such aresponse impacts upon the performance of a given device. Defect states willproduce a distinctive broadening of the JDOS function associated with anamorphous semiconductor in the sub-gap region and this will shape the formof the JDOS function, J(E), the imaginary part of the dielectric function,2(E), and the optical absorption spectrum, ?(E).373.2. Conduction band defect modelIn this chapter, two empirical models for the valence band and conduc-tion band DOS functions, Nv(E) and Nc(E), are presented, defect statesbeing included in these models. Assuming square-root distributions of bandstates and exponential distributions of tail states, the distribution of defectvalence band and conduction band states will be treated through the intro-duction of an additional exponential tail that is broader than that associatedwith the intrinsic tail. The contributions to the JDOS function, attributableto the various possible types of optical transitions, are then evaluated.This chapter is organized in the following manner. In Section 3.2, adetailed conduction band defect model is presented. Then, in Section 3.3, adetailed valence band defect model is presented.3.2 Conduction band defect modelThe analysis is performed within the framework of a general empiricalmodel for the DOS functions, Nv(E) and Nc(E), that captures the basic ex-pected features. For the case of a-Si:H, in the absence of defect states, thereis a general consensus that Nv(E) and Nc(E) exhibit square-root functionaldependencies in the band regions and exponential functional dependenciesin the tail regions. Following O?Leary [16], one can thus set383.2. Conduction band defect modelNv (E) = Nvo??????????Ev ? EvT exp(EvT?Ev?v)exp(Ev?E?v)E > EvT?Ev ? E, E ? EvT,(3.1)andNc (E) = Nco??????????E ? Ec, E ? EcT?EcT ? Ec exp(Ec?EcT?c)exp(E?Ec?c), E < EcT,(3.2)where Nvo and Nco denote the valence band and conduction band DOSprefactors, respectively, Ev and Ec represent the valence band and conduc-tion band band edges, ?v and ?c are the breadths of the valence band andconduction band tail distributions, EvT and EcT being the critical energiesat which the exponential and square-root distributions interface; it shouldbe noted that this model implicitly requires that Ev ? EvT ? 0 and thatEcT ?Ec ? 0. From Eqs. (3.1) and (3.2), it is noted that Nv(E) and Nc(E)are continuous functions of energy.To include the defect states, a new course of analysis is performed, sim-ilar to that employed by Shur and Hack [17]. In the model of Shur andHack [17], the conduction band DOS below the band edge is comprised oftwo exponentially varying distributions, i.e., a narrow exponential distribu-tion, corresponding to the intrinsic (without defects) tail states, and a broad393.2. Conduction band defect modelexponential distribution, corresponding to the defect states. A slightly dif-ferent model, which captures the same spirit as that of Shur and Hack [17],but allows for an easier analysis, is employed here. This empirical model forthe DOS functions captures the basic expected features of defect states. Inparticular, a distribution of conduction band defect (CBD) states is addedonto the distribution of valence band band (VBB), valence band tail (VBT),conduction band band (CBB), and conduction band tail (CBT) states thatare considered by Malik and O?Leary [18]. For this added distribution of de-fect states, optical transitions from the valence band band to the conductionband defect (VBB-CBD) states and from the valence band tail to the con-duction band defect states (VBT-CBD), are to be considered, in addition tothe VBB-CBB, VBB-CBT, VBT-VBB, and VBT-CBT optical transitions.The conduction band defect model is given byNv (E) = Nvo??????????Ev ? EvT exp(EvT?Ev?v)exp(Ev?E?v)E > EvT?Ev ? E, E ? EvT,(3.3)403.2. Conduction band defect modelandNc (E) = Nco??????????????????????????????E ? Ec, E ? EcT?EcT ? Ec exp(Ec?EcT?c)exp(E?Ec?c), EcD ? E < EcT?EcT ? Ec exp(Ec?EcT?c)exp(EcD?Ec?c)exp(E?EcD?cD), E < EcD,(3.4)where ?cD represents the breadth of the conduction band defect distributionand EcD is the critical energy at which the conduction band exponential anddefect distributions interface. All other parameters are the same as definedearlier; as with the defect-free model, i.e., Eqs. (3.1) and (3.2). It is furtherassumed that ?cD ? ?c and that EcT > EcD for the purposes of this analysis.The same nominal a-Si DOS modeling parameters, tabulated in Table 1 ofThevaril and O?Leary [19], are employed throughout the analysis. TheseDOS modeling parameters are tabulated in Table 3.1.From Figure 3.1, it is clear that there are six different types of op-tical transitions that occur. These are: (1) VBB-CBB optical transitions,(2) VBB-CBT optical transitions, (3) VBB-CBD optical transitions, (4) VBT-CBB optical transitions, (5) VBT-CBT optical transitions, and (6) VBT-CBD optical transitions.413.2. Conduction band defect model?0.5 0 0.5 1 1.5 2 2.51016101810201022VBB VBT CBD CBT CBBE v T E c D E c TVBB?CBBVBB?CBTVBT?CBBVBT?CBTVBB?CBDVBT?CBDEnergy (eV)Density of States (cm?3 eV?1 )Figure 3.1: The valence band and conduction band DOS functions associatedwith a-Si. The critical energies in the DOS functions, i.e., EvT , EcT , andEcD , are clearly marked with the dashed lines and the arrows. All of thepossible optical transition are shown with the arrows. The DOS modelingparameters are as set in Table 3.1.423.2. Conduction band defect modelTable 3.1: The nominal a-Si DOS modeling parameter selections employedfor the purposes of this analysis.ParameterParameters(units) ValueNvo (cm?3eV?3/2) 2? 1022Nco (cm?3eV?3/2) 2? 1022Ev (eV) 0Ec (eV) 1.7?v (meV) 50?c (meV) 27Ev ? EvT (meV) 35EcT ? Ec (meV) 35?cD (meV) 80Ec ? EcD (meV) 200In order to understand how these different types of optical transitionscontribute to overall optical response of this material, it is necessary tofind the JDOS contributions corresponding to these six types of opticaltransitions. For this analysis, three different cases are considered, i.e.,1. case 1 (subcase 1) EvT > EcD ? ~? and EvT > EcT ? ~?,2. case 1 (subcase 2) EvT > EcD ? ~? and EvT < EcT ? ~?, and3. case 2 EvT < EcD ? ~?.The first condition, EvT > EcD ? ~?, implies that ~? > EcD ? EvT ,and EvT > EcT ? ~? implies that ~? > EcT ? EvT . That is, it is assumedthat ~? > EcT ? EvT . The factors in the JDOS integrand are plotted as afunction of E for this case in Figure 3.2. For the case of ~? > EcT ? EvT ,it is clear that VBB-CBB, VBB-CBT, VBB-CBD, and VBT-CBB optical433.2. Conduction band defect modelh?? > E c T ? E v TN c (E + h??)N v (E)E c D ? h??E c T ? h??E v TVBBCBDVBBCBTVBBCBBVBTCBBEnergy DOSFigure 3.2: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for ~? > EcT ? EvT . The DOS modeling parametersare as set in Table 3.1.443.2. Conduction band defect modeltransitions contribute to the overall JDOS function. These contributions tothe JDOS function are denoted JVBB-CBB (~?), JVBB-CBT (~?), JVBB-CBD (~?),and JVBT-CBB (~?), respectively, and correspond to the integrations of theJDOS integrand, Nv(E)Nc(E + ~?), between energies EcT ? ~? and EvT ,EcD ? ~? and EcT ? ~?, ?? and EcD ? ~?, and EvT and ?. The JDOScontributions to the overall JDOS function, attributable to these differenttypes of optical transitions, are enumerated below. That is,JVBB-CBB (~?) =?EvTEcT?~?Nv (E)Nc (E + ~?) dE,JVBB-CBT (~?) =?EcT?~?EcD?~?Nv (E)Nc (E + ~?) dE,JVBB-CBD (~?) =? EcD?~???Nv (E)Nc (E + ~?) dE,JVBT-CBB (~?) =??EvTNv (E)Nc (E + ~?) dE,JVBT-CBT (~?) = 0,andJVBT-CBD (~?) = 0.The second condition, EvT > EcD ? ~?, implies that ~? > EcD ? EvT .This suggests that EvT < EcT?~?. That is, EcT?EvT > ~? or EcD?EvT <~? < EcT?EvT . The factors in the JDOS integrand are plotted as a functionof E for this case in Figure 3.3. For the case of EcD ? EvT < ~? < EcT ?EvT , from Figure 3.3, it is clear that the contributions to the overall JDOSfunction, attributable solely to the VBB-CBT, VBB-CBD, VBT-CBT, and453.2. Conduction band defect modelE c D ? E v T < h?? < E c T ? E v TN c (E + h??)N v (E)E c D ? h?? E c T ? h??E v TVBBCBDVBBCBTVBTCBTVBTCBBEnergy DOSFigure 3.3: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for EcD?EvT < ~? < EcT?EvT . The DOS modelingparameters are as set in Table 3.1.463.2. Conduction band defect modelVBT-CBB optical transitions, i.e., JVBB-CBT (~?), JVBB-CBD (~?), JVBT-CBT (~?),and JVBT-CBB (~?), respectively, correspond to the integrations of the JDOSintegrand, Nv(E)Nc(E+~?), between energies EcD ?~? and EvT , ?? andEcD?~?, EvT and EcT?~?, and EcT?~? and?. The JDOS contributionsto the overall JDOS function, attributable to these different types of opticaltransitions, are enumerated next. It is found thatJVBB-CBT (~?) =?EvTEcD?~?Nv (E)Nc (E + ~?) dE,JVBB-CBD (~?) =?EcD?~???Nv (E)Nc (E + ~?) dE,JVBT-CBT (~?) =?EcT?~?EvTNv (E)Nc (E + ~?) dE,JVBT-CBB (~?) =??EcT?~?Nv (E)Nc (E + ~?) dE,JVBB-CBB (~?) = 0,andJVBT-CBD (~?) = 0.The third condition, EvT < EcD?~?, implies that ~? < EcD?EvT . Thefactors in the JDOS integrand are plotted as a function of E for this case inFigure 3.4. For the case of ~? < EcD ?EvT , from Figure 3.4, it is clear thatthe contributions to the overall JDOS function attributable solely to theVBB-CBD, VBT-CBB, VBT-CBT, and VBT-CBD optical transitions, i.e.,JVBB-CBD (~?), JVBT-CBB (~?), JVBT-CBT (~?), and JVBT-CBD (~?), respectively,correspond to the integration of the JDOS integrand, Nv(E)Nc(E + ~?),473.2. Conduction band defect modelh?? < E c D ? E v TN c (E + h??)N v (E)E c D ? h?? E c T ? h??E v TVBBCBDVBTCBDVBTCBTVBTCBBEnergy DOSFigure 3.4: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for ~? < EcD ? EvT . The DOS modeling parametersare as set in Table 3.1.483.2. Conduction band defect modelbetween energies ?? and EvT , EcT?~? and?, EcD?~? and EcT?~?, andEvT and EcD ? ~?. The JDOS contributions to the overall JDOS function,attributable to these different types of optical transitions, are enumeratednext. That isJVBB-CBD (~?) =?EvT??Nv (E)Nc (E + ~?) dE,JVBT-CBB (~?) =??EcT?~?Nv (E)Nc (E + ~?) dE,JVBT-CBT (~?) =?EcT?~?EcD?~?Nv (E)Nc (E + ~?) dE,JVBT-CBD (~?) =?EcD?~?EvTNv (E)Nc (E + ~?) dE,JVBB-CBB (~?) = 0,andJVBB-CBT (~?) = 0.3.2.1 case 1 subcase 1For the first condition, there are four possible types of optical transitions.In this analysis, the contributions to the overall JDOS function, attributableto those different types of optical transitions, will be derived in closed form.3.2.1.1 JVBB-CBB (~?)From Figure 3.2, JVBB-CBB (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBB optical transitions, may be evaluated493.2. Conduction band defect modelby integrating the JDOS integrand between energies EcT?~? and EvT . Thatis,JVBB-CBB (~?) =? EvTEcT?~?Nv (E)Nc (E + ~?) dE,= NvoNco? EvTEcT?~??EV ? E?E + ~? ? Ec dE.Letting u = Ev ? E, it is seen thatJVBB-CBB (~?) = NvoNco? Ev?EvTEv?EcT+~??u?~? ? (Ec ? Ev)? u d(?u),= NvoNco? Ec?Eg?EcT+~?Ev?EvT?u?~? ? Eg ? u du.(3.5)Again letting u = (~? ? Eg)z, it is found thatJVBB-CBB (~?) = NvoNco? Ec?Eg?EcT+~?~??EgEv?EvT~??Eg?(~? ? Eg)z?(~? ? Eg)? (~? ? Eg)z(~? ? Eg)z dz= NvoNco(~? ? Eg)2? ~??Eg?(EcT?Ec)~??EgEv?EvT~??Eg?z?1? z dz,= NvoNco(~? ? Eg)2 Y(Ev ? EvT~? ? Eg, 1? EcT ? Ec~? ? Eg),where the dimensionless functionY(z1, z2) ????????? z2z1?x?1? x dx, 0 ? z1 ? z2 ? 10, otherwise. (3.6)503.2. Conduction band defect modelThis is the same function that is defined in Eq. (11) of O?Leary [20]. Thesolution for this function is given as,Y(z1, z2) =???????Z(z2)?Z(z1), 0 ? z1 ? z2 ? 10, otherwise,whereZ(z) =? z0?x?1? x dx. (3.7)Through substituting x = sin2(?), it can be shown thatZ(z) = 14 sin?1(?z)? 14?z?1? z[1? 2z]. (3.8)3.2.1.2 JVBB-CBT (~?)From Figure 3.2, JVBB-CBT (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBT optical transitions, may be evaluatedby integrating the JDOS integrand between energies EcD?~? and EcT?~?.That is,JVBB-CBT (~?) =? EcT?~?EcD?~?Nv (E)Nc (E + ~?) dE,= NvoNco?EcT ? Ec exp(Ec ? EcT?c)? EcT?~?EcD?~??Ev ? E exp(E + ~? ? Ec?c)dE? ?? ?integral. (3.9)513.2. Conduction band defect modelFor this integral, if one assumes that u = Ev ? E,=? Ev?EcT+~?Ev?EcD+~??u exp(Ev ? u+ ~? ? Ec?c)d(?u),=? Ev?EcD+~?Ev?EcT+~??u exp(Ev + ~? ? Ec?c)exp(? u?c)du.Letting u?c = z, this integral transforms into= ?3/2c exp(Ev + ~? ? Ec?c)? Ev?EcD+~??cEv?EcT+~??c?z exp (?z) dz,= ?3/2c exp(Ev + ~? ? Ec?c)[Y(Ev ? EcT + ~??c)?Y(Ev ? EcD + ~??c)],where the dimensionless function,Y(z) =? ?z?x exp (?x) dx, (3.10)Using integration by parts, this integration reduces toY(z) =?z exp (?z) +? ?z12?x exp (?x) dx. (3.11)By substituting x = t2, one can obtainY(z) =?z exp (?z) +?pi2 erfc(?z), (3.12)where the complementary error function is defined aserfc(z) ? 2?pi? ?zexp (?x2) dx. (3.13)523.2. Conduction band defect modelThus,JVBB-CBT (~?) = NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(Ev + ~? ? Ec?c)?3/2c[Y(Ev ? EcT + ~??c)?Y(Ev ? EcD + ~??c)].(3.14)3.2.1.3 JVBB-CBD (~?)From Figure 3.2, JVBB-CBD (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBD optical transitions, may be evaluatedby integrating the JDOS integrand between energies?? and EcD?~?. Thatis,JVBB-CBD (~?) =? EcD?~???Nv (E)Nc (E + ~?) dE,= NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)? EcD?~????Ev ? E exp(E + ~? ? EcD?cD)dE? ?? ?integral.(3.15)This integral has a similar form to that found in Eq (3.9). Following thesame procedure, this integral yields= ?3/2cD exp(Ev + ~? ? EcD?cD)Y(Ev + ~? ? EcD?cD), (3.16)533.2. Conduction band defect modelwhere Y(z) is the dimensionless function defined in Eq. (3.12). Thus,JVBB-CBD (~?) = NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)?3/2cD exp(Ev + ~? ? EcD?cD)Y(Ev + ~? ? EcD?cD).(3.17)3.2.1.4 JVBT-CBB (~?)From Figure 3.2, JVBT-CBB (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBB optical transitions, may be evaluatedby integrating the JDOS integrand between energies EvT and ?. That is,JVBT-CBB (~?) =? ?EvTNv (E)Nc (E + ~?) dE,= NvoNco?Ev ? EvT exp(EvT ? Ev?v)? ?EvT?E + ~? ? Ec exp(Ev ? E?v)dE? ?? ?integral.(3.18)For this integral, if one assumes that u = E + ~? ? Ec, one obtainsexp(Ev ? Ec + ~??v) ? ?EvT+~??Ec?u exp(? u?v)du.543.2. Conduction band defect modelLetting u?v = z, this yields= ?3/2v exp(Ev ? Ec + ~??v) ? ?EvT+~??Ec?v?z exp (?z) dz,= ?3/2v exp(Ev ? Ec + ~??v)Y(EvT + ~? ? Ec?v), (3.19)where Y(z) is the dimensionless function, defined in Eq. (3.12). Thus,JVBT-CBB (~?) = NvoNco?Ev ? EvT exp(EvT ? Ev?v)exp(Ev ? Ec + ~??v)?3/2v Y(EvT + ~? ? Ec?v).(3.20)3.2.2 Case 1 subcase 2For the second condition, there are four possible types of optical tran-sitions that have definite values. For this analysis, the contributions to theoverall JDOS function, attributable to these different types of optical tran-sitions, will be derived in closed form.3.2.2.1 JVBB-CBT (~?)From Figure 3.3, JVBB-CBT (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBT optical transitions, may be evaluatedby integrating the JDOS integrand between energies EcD?~? and EvT . Thatis,553.2. Conduction band defect modelJVBB-CBT (~?) =? EvTEcD?~?Nv (E)Nc (E + ~?) dE,= NvoNco?EcT ? Ec exp(Ec ? EcT?c)? EvTEcD?~??Ev ? E exp(E + ~? ? Ec?c)dE? ?? ?integral. (3.21)This integral has a similar form to that of Eq (3.9). Following the sameprocedures, this integral yields,= ?3/2c exp(Ev + ~? ? Ec?c) [Y(Ev ? EvT?c)?Y(Ev + ~? ? EcD?c)],(3.22)where Y(z) is the dimensionless function defined in Eq. (3.12). Thus,JVBB-CBT (~?) = NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(Ev + ~? ? Ec?c)(3.23)?3/2c[Y(Ev ? EvT?c)?Y(Ev + ~? ? EcD?c)]. (3.24)3.2.2.2 JVBB-CBD (~?)From Figure 3.3, JVBB-CBD (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBD optical transitions, may be evaluatedby integrating the JDOS integrand between energies?? and EcD?~?. That563.2. Conduction band defect modelis,JVBB-CBD (~?) =? EcD?~???Nv (E)Nc (E + ~?) dE,= NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)? EcD?~????Ev ? E exp(E + ~? ? EcD?cD)dE? ?? ?integral. (3.25)This integral has exactly the same form to that of Eq (3.15). Thus, theresult reduces toJVBB-CBD (~?) = NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)(3.26)?3/2cD exp(Ev + ~? ? EcD?cD)Y(Ev + ~? ? EcD?cD), (3.27)where Y(z) is the dimensionless function defined in Eq. (3.12).3.2.2.3 JVBT-CBB (~?)From Figure 3.3, JVBT-CBB (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBB optical transitions, may be evaluatedby integrating the JDOS integrand between energies EcT?~? and?. That573.2. Conduction band defect modelis,JVBT-CBB (~?) =? ?EcT?~?Nv (E)Nc (E + ~?) dE,= NvoNco?Ev ? EvT exp(EvT ? Ev?v)? ?EcT?~??E + ~? ? Ec exp(Ev ? E?v)dE? ?? ?integral. (3.28)This integrand has a similar form to that of Eq (3.18). Following the sameprocedure, this integral yields= ?3/2v exp(Ev ? Ec + ~??v)Y(EcT ? Ec?v), (3.29)where Y(z) is the dimensionless function, defined in Eq. (3.12). Thus,JVBT-CBB (~?) = NvoNco?Ev ? EvT exp(EvT ? Ev?v)exp(Ev ? Ec + ~??v)?3/2v Y(EcT ? Ec?v). (3.30)3.2.2.4 JVBT-CBT (~?)From Figure 3.3, JVBT-CBT (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBT optical transitions, may be evaluatedby integrating the JDOS integrand between energies EvT and EcT?~?. Thatis,JVBT-CBT (~?) =? EcT?~?EvTNv (E)Nc (E + ~?) dE,583.2. Conduction band defect model= NvoNco?Ev ? EvT?EcT ? Ecexp(EvT ? Ev?v)exp(Ec ? EcT?c)? EcT?~?EvTexp(Ev ? E?v)exp(E + ~? ? Ec?c)dE,= NvoNco?Ev ? EvT?EcT ? Ec exp(Ec ? EcT?c)exp(EvT ? Ev?V)exp(Ev?v+ ~? ? Ec?c)? EcT?~?EvTexp(E?c? E?v)dE? ?? ?integral.(3.31)For this integral, two types of conditions are considered.For the specific case for which, ?c = ?v, this integral reduces to? EcT?~?EvTdE= EcT ? ~? ? EvT .In contrast, for ?c 6= ?v, the integral reduces to? EcT?~?EvTexp(E?c? E?v)dE,593.2. Conduction band defect modelwhich is found to be= 1(1?c? 1?v)[exp((EcT ? ~?)( 1?c? 1?v))? exp(EvT( 1?c? 1?v))].(3.32)3.2.3 Case 2For the third condition, there are four possible types of optical transitionto consider. In this analysis, the contributions to the overall JDOS function,attributable to these different types optical transitions, will be obtained.3.2.3.1 JVBB-CBD (~?)From Figure 3.4, JVBB-CBD (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBB-CBD optical transitions, may be evaluatedby integrating the JDOS integrand between energies ?? and EvT . That is,JVBB-CBD (~?) =? EvT??Nv (E)Nc (E + ~?) dE,= NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)? EvT???Ev ? E exp(E + ~? ? EcD?cD)dE? ?? ?integral. (3.33)603.2. Conduction band defect modelThis integrand has a similar form to that of Eq (3.9). Following the sameprocedure, this integral yields= ?3/2cD exp(Ev + ~? ? EcD?cD)Y(Ev ? EvT?cD), (3.34)where Y(z) is the dimensionless function, defined in Eq. (3.12). Thus,JVBB-CBD (~?) = NvoNco?EcT ? Ec exp(Ec ? EcT?c)exp(EcD ? Ec?c)?3/2cD exp(Ev + ~? ? EcD?cD)Y(Ev ? EvT?cD). (3.35)3.2.3.2 JVBT-CBB (~?)From Figure 3.4, JVBT-CBB (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBB optical transitions, may be evaluatedby integrating the JDOS integrand between energies EcT?~? and?. Thatis,JVBT-CBB (~?) =? ?EcT?~?Nv (E)Nc (E + ~?) dE,= NvoNco?Ev ? EvT exp(EvT ? Ev?v)? ?EcT?~??E + ~? ? Ec exp(Ev ? E?v)dE? ?? ?integral. (3.36)613.2. Conduction band defect modelThis integral has exactly the same form as that associated with Eq (3.28).Thus, this contribution may be expressed asJVBT-CBB (~?) = NvoNco?Ev ? EvT exp(EvT ? Ev?v)exp(Ev ? Ec + ~??v)?3/2v Y(EcT ? Ec?v), (3.37)where Y(z) is the dimensionless function, defined in Eq. (3.12).3.2.3.3 JVBT-CBT (~?)From Figure 3.4, JVBT-CBT (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBT optical transitions, may be evaluatedby integrating the JDOS integrand between energies EcD?~? and EcT?~?.That is,JVBT-CBT (~?) =? EcT?~?EcD?~?Nv (E)Nc (E + ~?) dE,= NvoNco?Ev ? EvT?EcT ? Ec exp(Ec ? EcT?c)exp(EvT ? Ev?V)exp(Ev?v+ ~? ? Ec?c)? EcT?~?EcD?~?exp(E?c? E?v)dE? ?? ?integral. (3.38)623.2. Conduction band defect modelFor this integral, two types of conditions are considered.For the specific case for which ?c = ?v, this integral reduces to? EcT?~?EcD?~?dE = EcT ? ~? ? EcD + ~?,= EcT ? EcD . (3.39)In contrast, for ?c 6= ?v, the integral reduces to? EcT?~?EcD?~?exp(E?c? E?v)dE,which is found to be= 1(1?c? 1?v)[exp((EcT ? ~?)( 1?c? 1?v))? exp((EcD ? ~?)( 1?c? 1?v))].(3.40)3.2.3.4 JVBT-CBD (~?)From Figure 3.4, JVBT-CBD (~?), i.e., the contribution to the overall JDOSfunction attributable to the VBT-CBD optical transitions, may be evaluatedby integrating the JDOS integrand between energies EvT and EcD?~?. Thatis,JVBT-CBD (~?) =? EcD?~?EvTNv (E)Nc (E + ~?) dE,= NvoNco?Ev ? EvT?EcT ? Ec exp(Ec ? EcT?c)633.2. Conduction band defect modelexp(EvT ? Ev?V)exp(EcD ? Ec?c)exp(Ev?v+ ~? ? EcD?cD)? EcD?~?EvTexp( E?cD? E?v)dE? ?? ?integral. (3.41)For this integral, two types of conditions are considered.For the specific case for which ?cD = ?v, this integral reduces to? EcD?~?EvTdE = EcD ? ~? ? EvT . (3.42)In contrast, for ?cD 6= ?v, this integral reduces to? EcD?~?EvTexp( E?cD? E?v)dE, (3.43)which is found to be= 1(1?cD? 1?v)[exp((EcD ? ~?)( 1?cD? 1?v))? exp(EvT( 1?cD? 1?v))].(3.44)643.3. Valence band defect model3.3 Valence band defect modelFor this case, a distribution of valence band defect states is consideredinstead. As with the approach for the conduction band defect distribution,for this case, a distribution of valence band defect (VBD) states is added tothe distributions of VBB, VBT, CBB and CBT states that were consideredby Malik and O?Leary [18]. For this added defect distribution, optical tran-sitions, from the valence band defect to the conduction band band (VBD-CBB) states and from the valence band defect to the conduction band tail(VBD-CBT) states, are to be considered along with the VBB-CBB, VBB-CBT, VBT-VBB and VBT-CBT optical transitions. The empirical valenceband DOS defect model may thus be expressed asNv (E) = Nvo??????????????????????????????Ev ? EvT exp(EvT?Ev?v)exp(Ev?EvD?v)exp(EvD?E?vD), E > EvD?Ev ? EvT exp(EvT?Ev?v)exp(Ev?E?v), EvD ? E > EvT?Ev ? E, E ? EvT,(3.45)andNc (E) = Nco??????????E ? Ec, E ? EcT?EcT ? Ec exp(Ec?EcT?c)exp(E?Ec?c), E < EcT,(3.46)653.3. Valence band defect modelwhere Nvo and Nco denote the valence band and conduction band DOSprefactors, respectively, Ev and Ec represent the valence band and conduc-tion band band edges, ?v and ?vD are the breadths of the valence band tailand valence band defect distributions, ?c is the breadth of conduction bandtail, EvT and EcT being the critical energies at which the exponential andsquare-root distributions interface, EvD being the critical energy at whichthe valence band exponential and defect distributions interface; it shouldbe noted that this model implicitly requires that Ev ? EvT ? 0 and thatEcT?Ec ? 0. From Eqs. (3.45) and (3.46), it is noted that Nv(E) and Nc(E)are continuous functions of energy. It is assumed that ?vD ? ?v, and thatEvD > EvT for the purposes of this analysis. The same nominal a-Si DOSmodeling parameters, tabulated in Table 1 of Thevaril and O?Leary [19], arealso employed for this analysis. These values are tabulated in Table 3.2, thecorresponding DOS functions being as depicted in Figure 3.5. From thisfigure, it is clear that there are six different types of optical transitions thatare possible: (1) VBB-CBB optical transitions, (2) VBB-CBT optical tran-sitions, (3) VBT-CBB optical transitions, (4) VBT-CBT optical transitions,(5) VBD-CBB optical transitions, and (6) VBD-CBT optical transitions.In order to understand how these optical transitions contribute to theoverall optical response of a material, it is necessary to find the JDOS663.3. Valence band defect model?0.5 0 0.5 1 1.5 2 2.51016101810201022VBB VBT VBD CBT CBBE v T E v D E c TVBB?CBBVBB?CBTVBT?CBBVBT?CBTVBD?CBBVBD?CBTEnergy (eV)Density of States (cm?3 eV?1 )Figure 3.5: The valence band and conduction band DOS functions associatedwith a-Si. The critical energies in the DOS functions, i.e., EvT , EcT , andEcD , are clearly marked with the dashed lines and the arrows. All thepossible optical transitions are shown with the arrows. The DOS modelingparameters are as set in Table 3.2.673.3. Valence band defect modelTable 3.2: The nominal a-Si DOS modeling parameter selections employedfor the purposes of this analysis.ParameterParameters(units) ValueNvo (cm?3eV?3/2) 2? 1022Nco (cm?3eV?3/2) 2? 1022Ev (eV) 0Ec (eV) 1.7?v (meV) 50?c (meV) 27Ev ? EvT (meV) 35EcT ? Ec (meV) 35?vD (meV) 130EvD ? Ev (meV) 400contribution corresponding to these six transitions. Three different casesare considered,i.e.,1. case 1 EvT > EcT ? ~?,2. case 2 EvT < EcT ? ~? < EvD ,and3. case 3 EvD < EcT ? ~?.The first condition, EvT > EcT ? ~?, implies that ~? > EcT ? EvT .The factors in the JDOS integrand are plotted as a function of E forthis case in Figure 3.6. For the case of ~? > EcT ? EvT , it is clear thatVBB-CBB, VBB-CBT, VBT-CBB, and VBD-CBB optical transitions con-tribute to the overall JDOS function. These optical transitions are denotedJVBB-CBB (~?), JVBB-CBT (~?), JVBT-CBB (~?), and JVBD-CBB (~?), respectively,683.3. Valence band defect modelh?? > E c T ? E v TN c (E + h??)N v (E)E v DE c T ? h?? E v TVBDCBBVBBCBTVBBCBBVBTCBBEnergy DOSFigure 3.6: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for ~? > EcT ? EvT . The DOS modeling parametersare as set in Table 3.2.693.3. Valence band defect modeland these correspond to the integration of the JDOS integrand, Nv(E)Nc(E+~?), between energies EcT ? ~? and EvT , ?? and EcT ? ~?, EvT and EvD ,and EvD and ?. The JDOS contributions to the overall JDOS function,attributable to these different types of optical transitions, are enumeratedbelow. That is,JVBB-CBB (~?) =?EvTEcT?~?Nv (E)Nc (E + ~?) dE,JVBB-CBT (~?) =?EcT?~???Nv (E)Nc (E + ~?) dE,JVBT-CBB (~?) =?EvDEvTNv (E)Nc (E + ~?) dE,JVBD-CBB (~?) =? ?EvDNv (E)Nc (E + ~?) dE,JVBT-CBT (~?) = 0,andJVBD-CBT (~?) = 0.The second condition, EvT < EcT ? ~? < EvD implies that EcT ?EvD <~? < EcT ? EvT . The factors in the JDOS integrand, plotted as a functionof E for this case, are depicted in Figure 3.7. For the case of EcT ? EvD <~? < EcT ? EvT , from Figure 3.7 it is clear that the contributions tothe overall JDOS function, attributable solely to the VBB-CBT, VBT-CBB, VBT-CBT, and VBD-CBB optical transitions, i.e., JVBB-CBT (~?),JVBT-CBB (~?), JVBT-CBT (~?), and JVBD-CBB (~?), respectively, correspondto the integrations of the JDOS integrand, Nv(E)Nc(E + ~?), between703.3. Valence band defect modelE c T ? E v D < h?? < E c T ? E v TN c (E + h??)N v (E)E v DE c T ? h??E v TVBDCBBVBBCBTVBTCBTVBTCBBEnergy DOSFigure 3.7: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for EcT?EvD < ~? < EcT?EvT . The DOS modelingparameters are as set in Table 3.2.713.3. Valence band defect modelenergies ?? and EvT , EcT ? ~? and EvD , EvT and EcT ? ~?, and EvD and?. The JDOS contributions to the overall JDOS function, attributable tothese different types of optical transitions, are enumerated below. That is,JVBB-CBT (~?) =?EvT??Nv (E)Nc (E + ~?) dE,JVBT-CBT (~?) =?EcT?~?EvTNv (E)Nc (E + ~?) dE,JVBD-CBB (~?) =??EvDNv (E)Nc (E + ~?) dE,JVBB-CBB (~?) = 0,andJVBD-CBT (~?) = 0.The third condition, EvD < EcT ? ~?, implies that ~? < EcT ? EvD .The factors in the JDOS integrand, plotted as a function of E for thiscase, are depicted in Figure 3.8. For the case of ~? < EcT ? EvD , fromFigure 3.8, it is clear that the contributions to the overall JDOS function,attributable solely to the VBB-CBT, VBT-CBT, VBD-CBB, and VBD-CBT optical transitions, i.e., JVBB-CBT (~?), JVBT-CBT (~?), JVBD-CBB (~?),and JVBD-CBT (~?), respectively, correspond to the integrations of the JDOSintegrand, Nv(E)Nc(E+~?), between energies ?? and EvT , EvT and EvD ,EcT ? ~? and ?, and EvD and EcT ? ~?. The JDOS contributions tothe overall JDOS function, attributable to these different types of opticaltransitions, are enumerated below. That is,723.3. Valence band defect modelh?? < E c T ? E v DN c (E + h??)N v (E)E v D E c T ? h??E v TVBDCBBVBBCBTVBTCBTVBDCBTEnergy DOSFigure 3.8: The factors in the JDOS integrand, Nv(E) and Nc(E+~?), as afunction of energy, E, for ~? < EcT ? EvD . The DOS modeling parametersare as set in Table 3.2.733.3. Valence band defect modelJVBB-CBT (~?) =?EvT??Nv (E)Nc (E + ~?) dE,JVBT-CBT (~?) =?EvDEvTNv (E)Nc (E + ~?) dE,JVBD-CBB (~?) =??EcT?~?Nv (E)Nc (E + ~?) dE,JVBD-CBT (~?) =?EcT?~?EvDNv (E)Nc (E + ~?) dE,JVBB-CBB (~?) = 0,andJVBT-CBB (~?) = 0.Analytical results, for the valence band defect model, yield similar ex-pressions to those of the conduction band defect model. One can easilyderive these expressions for the three different cases following the same pro-cedures, as discussed previously.74Chapter 4Application of theconduction band defect andvalence band defect modelsto the analysis of the opticalresponse of a-Si4.1 IntroductionThe presence of defect states will lead to a distinctive broadening exhib-ited in the sub-gap region of the imaginary part of the dielectric function,2(E), associated with a-Si. For many years, there has been some basic workperformed on understanding the optical response of a-Si. With experimen-754.1. Introductiontal optical response results for a-Si, models have been proposed to fit theresults. In Figure 4.1, a fit is shown with the experimental results of Jacksonet al. [6], where it is seen that the spectral dependence of the JDOS func-tion, J(E), does not follow the a-Si experimental result for photon energiesbelow 1.4 eV; this evaluation of the JDOS function is performed without thepresence of defect state being taken into account. Adding an exponentialdistribution of defect states onto the intrinsic exponential distribution of tailstates can account for this discrepancy.In this chapter, results produced from the proposed models, i.e., thevalence band defect model and the conduction band defect model, are fitto those of experiment. In addition to that, the underlying distribution ofdefect states that shape the resultant optical response, and the role that theindividual types of optical transitions play in determining the behavior ofthe optical response of these materials, are assessed.This chapter is organized in the following manner. In Section 4.2, acomparison between the results obtained through the proposed defect modelwith those obtained with the non-defect model is provided. Then, in Section4.4, a fit with the result of experiments for the valence band defect model, isshown. In Section 4.3, a fit with the results of experiment, for the conductionband defect model, is discussed. Finally, the uniqueness of the fits to the764.1. Introduction0 1 2 3 4 51038104010421044Photon Energy (eV)JDOS (cm?6 eV?1 )Jackson et al.Figure 4.1: The fitting of the spectral dependence of JDOS function. Theexperimental a-Si JDOS result of Jackson et al. [6] is depicted with solidpoints. The JDOS fit result, obtained by setting Nvo = Nco = 2.48 ?1022cm?3eV?3/2, Ev = 0.0 eV, Ec = 1.68 eV, and ?v = 40 meV, is depictedwith the solid line. This figure is after Minar [21].774.2. Comparison between modelsexperimental results considered is commented upon in Section 4.5.4.2 Comparison between modelsTo clearly demonstrate how the presence of defects influences the JDOSfunction, the JDOS functions, with and without defect states taken intoaccount, are plotted in Figure 4.2. The valence band defect JDOS model iscompared with defect-free JDOS model, given in Eqs (3.1) and (3.2). It isclear that, in the absence of defect states, an algebraic functional dependenceis visible in the band region and an exponential functional dependence isapparent in the sub-gap region, i.e., for ~? < EcT ? EvT , the breadth ofthis region being dominated by the tail breadth, ?v. In the valence banddefect model, with defect states included, the corresponding JDOS functionbroadens considerably at lower photon energies. This significant broadeninghappens due to optical transitions involving VBD states, i.e., ~? < EcT ?EvD , and the breadth of this region is dominated by the defect distributionbreadth, ?vD , for the nominal selection of DOS modeling parameters.In order to quantitatively investigate the influence of each type of opticaltransition on the overall JDOS function associated with a-Si, the fractionalcontributions to the JDOS function related to the conduction band defectmodel are depicted in Figure 4.3, the fractional contributions to the JDOS784.2. Comparison between models1 1.2 1.4 1.6 1.8 2 2.2 2.41035103610371038103910401041104210431044E c T ? E v TE c T ? E v Dwith defect stateswithout defect statesPhoton Energy (eV)JDOS (cm?6 eV?1 )Figure 4.2: A comparison between two JDOS functions, J(E), associatedwith a-Si, determined through an evaluation of Eq (2.13), are shown in thisfigure. For the defect-free model, the DOS functions, Nv(E) and Nc(E), areas specified in Eqs. (3.1) and (3.2), respectively. For a model with defectstates included, Nv(E) and Nc(E) are as specified in Eqs. (3.45) and (3.46),respectively. EcT?EvD and EcT?EvT , critical energies in the JDOS analysis,are clearly marked with the dashed lines and the arrows. The DOS modelingparameters are as set in Table 3.1.794.2. Comparison between models1 1.2 1.4 1.6 1.8 2 2.2 2.400.20.40.60.81E c T ? E v TE c D ? E v TVBB?CBBVBB?CBTVBT?CBBVBT?CBTVBT?CBDVBB?CBDPhoton Energy (eV)Fractional JDOSFigure 4.3: The fractional contributions to the overall JDOS function asso-ciated with the various types of a-Si optical transitions for the conductionband defect model. EcD ?EvT and EcT ?EvT , critical energies in the JDOSanalysis, are clearly marked with the dashed lines and the arrows. The DOSmodeling parameters are as set in Table 3.1.804.2. Comparison between modelsfunction related to the valence band defect model being depicted in Fig-ure 4.4.In Figure 4.3, for the conduction band defect model, the fractional con-tributions of the VBB-CBB, VBB-CBT, VBT-CBB, VBT-CBT, VBT-CBD,and VBB-CBD optical transitions to the overall JDOS function, are plottedas a function of photon energy, E. The DOS modeling parameters are set totheir nominal a-Si values for the purposes of this analysis, i.e., as set in Ta-ble 3.1. The VBB-CBB optical transition contribution to the overall JDOSfunction is shown with the solid blue line. The VBT-CBB optical transitioncontribution to the overall JDOS function is shown with the solid green line.The VBB-CBT optical transition contribution to the overall JDOS functionis shown with the solid red line. The VBT-CBT optical transition contribu-tion to the overall JDOS function is shown with the solid yellow line. TheVBB-CBD optical transition contribution to the overall JDOS function isshown with the solid purple line. The VBT-CBD optical transition contri-bution to the overall JDOS function is shown with the light blue line. Again,one should note that for the selection of ~? well above the optical gap, i.e.,~? > EcT ?EvT , the contribution to the overall JDOS function attributableto VBB-CBB optical transitions increases monotonically with increasing~?, and it is the dominant type of optical transition in this region, this814.2. Comparison between models1 1.2 1.4 1.6 1.8 2 2.2 2.400.20.40.60.81E c T ? E v TE c T ? E v DVBB?CBBVBB?CBTVBT?CBBVBT?CBTVBD?CBTVBD?CBBPhoton Energy (eV)Fractional JDOSFigure 4.4: The fractional contributions to the overall JDOS function as-sociated with the various types of a-Si optical transitions for the valenceband defect model. EcT ?EvD and EcT ?EvT , critical energies in the JDOSanalysis, are clearly marked with the dashed lines and the arrows. The DOSmodeling parameters are as set in Table 3.2.824.2. Comparison between modelscontribution being nil below the optical gap, i.e., for ~? < EcT ? EvT . Inthe sub-gap region, i.e., ~? < EcT?EvD , VBT-CBB, and VBT-CBT opticaltransitions are the dominant contributors to the overall JDOS function. Onthe other hand, VBT-CBD and VBB-CBD optical transitions make smallcontributions to the overall JDOS function. For ~? set to 1 eV, VBB-CBD,VBT-CBB, VBT-CBT, and VBT-CBD optical transitions contribute 34.45,18.62, 14.26, and 32.68 % to the overall JDOS function, respectively.In Figure 4.4, for the valence band defect model, the fractional contribu-tions of the VBB-CBB, VBB-CBT, VBT-CBB, VBT-CBT, VBD-CBB, andVBD-CBT optical transitions to the overall JDOS function, are plotted asa function of photon energy, ~?. The DOS modeling parameters are setto their nominal a-Si values for the purposes of this analysis, i.e., as setin Table 3.2. The VBB-CBB optical transition contribution to the overallJDOS function is shown with the solid blue line. The VBT-CBB opticaltransition contribution to the overall JDOS function is shown with the solidgreen line. The VBB-CBT optical transition contribution to the overallJDOS function is shown with the solid red line. The VBT-CBT opticaltransition contribution to the overall JDOS function is shown with the solidyellow line. The VBD-CBB optical transition contribution to the overallJDOS function is shown with the solid purple line. The VBD-CBT optical834.2. Comparison between modelstransition contribution to the overall JDOS function is shown with the lightblue line. One should note that, for the selection of ~? well above the opticalgap, i.e., ~? > EcT ? EvT , the contribution to the overall JDOS functionattributable to VBB-CBB optical transitions increases monotonically withincreasing ~?, and it is the dominant optical transition in this region whileother types of optical transition also contribute. The contribution of theVBB-CBB optical transitions to the overall JDOS function is nil below theoptical gap, i.e., ~? < EcT ?EvT . While VBB-CBT and VBT-CBB opticaltransitions make small contributions in this region, VBD-CBB and VBD-CBT optical transitions have no contribution to the overall JDOS functionin this region. For ~? set to 2 eV, VBB-CBB, VBB-CBT, and VBT-CBBoptical transitions contribute 79.42, 7.05, and 13.53 % to the overall JDOSfunction, respectively. In the sub-gap region, i.e., ~? < EcT ? EvD , VBD-CBB optical transitions contribute the most to the overall JDOS function.On the other hand, VBD-CBT and VBT-CBT optical transitions make asmall contribution to the overall JDOS. For ~? set to 1 eV, VBD-CBB andVBD-CBT optical transitions contribute 88.59 and 11.41 % to the overallJDOS function, respectively.844.3. Comparison with experiment: the conduction band defect model4.3 Comparison with experiment: the conductionband defect modelIn this section, results produced through the use of the conduction banddefect model, developed earlier in this thesis, are fit to the spectral de-pendence of the optical absorption coefficient associated with several defectabsorption influenced samples of a-Si. Two experimental optical absorptiona-Si data sets, and one imaginary part of the dielectric function data set, areconsidered for the purposes of this analysis. The first set of experimentaldata points corresponds to the Figure 13(a) of Stolk et al [22]. A structureof pure unhydrogenated amorphous silicon was modified by means of ionimplantation, furnace annealing, and pulse laser annealing. The sample wasannealed for 500 oC for 1 hour, and is referred to as ?relaxed a-Si?. Thesecond set of experimental data points corresponds to the ?standard GD-a?data points, depicted in Figure 5.2 of Remes? [23]. The sample was preparedby conventional 13.56 MHz glow discharge deposition at a substrate tem-perature 250 oC and annealed for 24 hours at 500 oC to evolve hydrogen.The third set of experimental data points corresponds to the ?logarithmicplot of 2? data points, depicted in Figure 3(b) of Jackson et al. [6]. Thesample was prepared by doping a 100 nm thick undoped thick film using854.3. Comparison with experiment: the conduction band defect model2 W rf power and 100 % silane onto a substrate held at 230 oC. Recallingthat the imaginary part of dielectric function2(E) = 4.3? 10?45R2(E)J(E), (4.1)and the optical absorption coefficient?(E) = En(E)c~2(E), (4.2)where n(E) denotes the spectral dependence of the index of refraction, crepresents the speed of the light in vacuum, and ~ is the reduced Plank?sconstant, one has means of evaluating these optical functions from the JDOSfunction, J(E). The spectral dependence of the refractive index, n(E), isdetermined by fitting a tenth-order polynomial to the experimental resultsof Klazes et al. [9]; the experimental data considered for n(E) correspondsto that presented in Figure 4 of Klazes et al. [9]; this technique was usedpreviously by Thevaril [4]. It will be assumed that the deviations fromthis model for n(E) are not significant. Throughout this analysis, R2(E) issimply set to 10 A?2.A reasonably satisfactory fit with the a-Si experimental data sets of Stolket al. [22] may be achieved by setting Nvo = Nco = 2.38? 1022 cm?3eV?3/2,Ev = 0 eV, Ec = 1.483 eV, Ev ? EvT = 180 meV, EcT ? Ec = 25 meV,?v = 105 meV, ?c = 60 meV, EvD ? Ev = 227 meV, and ?vD = 520 meV.864.3. Comparison with experiment: the conduction band defect modelTable 4.1: The DOS modeling parameter selections for the conduction banddefect model associated with a-Si, employed for the purposes of the fit tothe experimental data of Stolk et al. [22], Remes? [23], and Jackson et al. [6];the corresponding fits are shown in Figures 4.5, 4.6, and 4.7, respectively.Parameter (unit) Stolk et al. [22] Remes? [23] Jackson et al. [6]Nvo (cm?3eV?3/2) 2.38? 1022 2.38? 1022 2.38? 1022Nco (cm?3eV?3/2) 2.38? 1022 2.38? 1022 2.38? 1022Ev (eV) 0 0 0Ec (eV) 1.483 1.59 1.65Ev ? EvT (meV) 180 55 55EcT ? Ec (meV) 25 35 55?v (meV) 105 76 46?c (meV) 60 43 16Ec ? EcD (meV) 227 205 77?cD (meV) 520 590 685The resultant fit is shown in Figure 4.5. These DOS modeling parametersare tabulated in Table 4.1.A second reasonably satisfactory fit with the a-Si experimental data setsof Remes? [23] is achieved by setting Nvo = Nco = 2.38 ? 1022 cm?3eV?3/2,Ev = 0 eV, Ec = 1.59 eV, Ev ? EvT = 55 meV, EcT ? Ec = 35 meV,?v = 76 meV, ?c = 43 meV, EvD ? Ev = 205 meV, and ?vD = 590 meV.The resultant fit is shown in Figure 4.6. These DOS modeling parametersare tabulated in Table 4.1.The third and last fit was with the a-Si experimental data sets of Jack-son et al. [6]. A satisfactory fit is achieved by setting Nvo = Nco = 2.38 ?1022 cm?3 eV?3/2, Ev = 0 eV, Ec = 1.65 eV, Ev ? EvT = 55 meV,874.3. Comparison with experiment: the conduction band defect model1 1.2 1.4 1.6 1.8 2 2.2102103104105106Photon Energy (eV) Absorption Coefficient (cm?1)Figure 4.5: The optical absorption spectrum, ?(~?), associated with a-Si.The experimental a-Si data set of Stolk et al. [22] is depicted with solidpoints; these experimental data points corresponds to the ?relaxed a-Si?data points, depicted in Figure 13(a) of Stolk et al. [22]. The solid linecorresponds the conduction band defect model fitting to this experimentaldata, by setting the DOS modeling parameters to the selections specified inTable 4.1.884.3. Comparison with experiment: the conduction band defect model1 1.2 1.4 1.6 1.8 2 2.2101102103104105106Photon Energy (eV) Absorption Coefficient (cm?1 )Figure 4.6: The optical absorption spectrum, ?(~?), associated with a-Si.The experimental a-Si data set of Remes? [23] is depicted with solid points;these experimental data points correspond to the ?standard GD-a? datapoints, depicted in Figure 5.2 of Remes? [23]. The solid line corresponds theconduction band defect model fitting to this experimental data, by settingthe DOS modeling parameters to the selections specified in Table 4.1.894.3. Comparison with experiment: the conduction band defect modelEcT ? Ec = 55 meV, ?v = 46 meV, ?c = 16 meV, EvD ? Ev = 77 meV, and?vD = 685 meV. The resultant fit is shown in Figure 4.7. The correspondingDOS modeling parameters are also tabulated in Table 4.1.904.3. Comparison with experiment: the conduction band defect model1 1.2 1.4 1.6 1.8 2 2.210?410?310?210?1100101Photon Energy (eV) ? 2Figure 4.7: The imaginary part of dielectric function spectrum, 2, asso-ciated with a-Si. The experimental a-Si data set of Jackson et al. [6] isdepicted with solid points; these experimental data points correspond tothe ?logarithmic plot of 2? data points, depicted in Figure 3(b) of Jacksonet al. [6]. The solid line corresponds to the conduction band defect modelfitting to this experimental data, by setting the DOS modeling parametersto the selections specified in Table 4.1.914.4. Comparison with experiment: the valence band defect model4.4 Comparison with experiment: the valenceband defect modelIn this section, the valence band defect model developed in this thesisis fit to the spectral dependence of the optical absorption coefficient asso-ciated with defect absorption influenced samples of a-Si. As before, twoexperimental optical absorption a-Si data sets, and one imaginary part ofdielectric function data set, are considered for the purposes of this analysis.A reasonably satisfactory fit with the a-Si experimental data sets of Stolket al. [22] may be achieved by setting Nvo = Nco = 2.38? 1022 cm?3eV?3/2,Ev = 0 eV, Ec = 1.5 eV, Ev ? EvT = 142 meV, EcT ? Ec = 25 meV,?v = 100 meV, ?c = 60 meV, EvD ? Ev = 255 meV, and ?vD = 300 meV.The resultant fit is shown in Figure 4.8. These DOS modeling parameterscorresponding to this fit are tabulated in Table 4.2.A second reasonably satisfactory fit with the a-Si experimental data setsof Remes? [23] is achieved by setting Nvo = Nco = 2.38 ? 1022 cm?3eV?3/2,Ev = 0 eV, Ec = 1.58 eV, Ev ? EvT = 35 meV, EcT ? Ec = 35 meV,?v = 75 meV, ?c = 43 meV, EvD ? Ev = 425 meV, and ?vD = 675 meV.The resultant fit is shown in Figure 4.9. These DOS modeling parametersare tabulated in Table 4.2.924.4. Comparison with experiment: the valence band defect model1 1.2 1.4 1.6 1.8 2 2.2102103104105106Photon Energy (eV) Absorption Coefficient (cm?1 )Figure 4.8: The optical absorption spectrum, ?(~?), associated with a-Si. The experimental a-Si data set of Stolk et al. [22] is depicted with solidpoints; these experimental data points correspond to the ?relaxed a-Si? datapoints, depicted in Figure 13(a) of Stolk et al. [22]. The solid line correspondsthe valence band defect model fitting to this experimental data, by settingthe DOS modeling parameters to the selections specified in Table 4.2.934.4. Comparison with experiment: the valence band defect model1 1.2 1.4 1.6 1.8 2 2.2101102103104105106Photon Energy (eV) Absorption Coefficient (cm?1 )Figure 4.9: The optical absorption spectrum, ?(~?), associated with a-Si.The experimental a-Si data set of Remes? [23] is depicted with solid points;these experimental data points correspond to the ?standard GD-a? datapoints, depicted in Figure 5.2 of Remes? [23]. The solid line corresponds thevalence band defect model fitting to this experimental data, by setting theDOS modeling parameters to the selections specified in Table 4.2.944.4. Comparison with experiment: the valence band defect modelTable 4.2: The DOS modeling parameter selections for the valence banddefect model associated with a-Si, employed for the purposes of the fit tothe experimental data of Stolk et al. [22], Remes? [23], and Jackson et al. [6];the corresponding fits are shown in Figures 4.8, 4.9, and 4.10, respectively.Parameter (unit) Stolk et al. [22] Remes? [23] Jackson et al. [6]Nvo (cm?3eV?3/2) 2.38? 1022 2.38? 1022 2.38? 1022Nco (cm?3eV?3/2) 2.38? 1022 2.38? 1022 2.38? 1022Ev (eV) 0 0 0Ec (eV) 1.5 1.58 1.68Ev ? EvT (meV) 142 35 38EcT ? Ec (meV) 25 35 38?v (meV) 100 75 49?c (meV) 60 43 18EvD ? Ev (meV) 255 425 340?vD (meV) 300 675 475The third and last fit was with the a-Si experimental data sets of Jacksonet al. [6]. A satisfactory fit is achieved by setting Nvo = Nco = 2.38 ?1022 cm?3eV?3/2, Ev = 0 eV, Ec = 1.68 eV, Ev ? EvT = 38 meV, EcT ?Ec = 38 meV, ?v = 49 meV, ?c = 18 meV, EvD ? Ev = 340 meV, and?vD = 475 meV. The resultant fit is shown in Figure 4.10. The correspondingDOS modeling parameters are tabulated in Table 4.2.954.4. Comparison with experiment: the valence band defect model1 1.2 1.4 1.6 1.8 2 2.210?410?310?210?1100101Photon Energy (eV) ? 2Figure 4.10: The imaginary part of dielectric function spectrum, 2, asso-ciated with a-Si. The experimental a-Si data set of Jackson et al. [6] isdepicted with solid points; these experimental data points correspond tothe ?logarithmic plot of 2? data points, depicted in Figure 3(b) of Jacksonet al. [6]. The solid line corresponds the valence band defect model fittingto this experimental data, by setting the DOS modeling parameters to theselections specified in Table 4.2.964.5. On the uniqueness of the fits4.5 On the uniqueness of the fitsThe approach that has been adopted for the fitting of the experimentalresults involved systematically sweeping through a selection of DOS model-ing parameter selections until the comparison with experiment is reasonablysatisfactory. While this may be adequate for the purposes of this particularanalysis, the uniqueness of the fits, i.e., can other selections of DOS model-ing parameters produce the similar fits, may be called into question. Furtheranalysis, beyond the scope of the present thesis, will be required in order toresolve this matter.97Chapter 5ConclusionsIn this thesis, empirical models for the DOS functions, that include dis-tributions of defect states, were introduced. Then, the corresponding JDOSfunctions were evaluated. Through a fit with the results of experiment,the DOS modeling parameters were determined. The fits were found to besatisfactory.There are a number of topics that are related to this work that couldbe further explored. The defect model could be further developed to in-clude distributions of defect states associated with both bands. The use ofthis model for the consideration of other types of amorphous semiconduc-tors would also be a worthwhile pursuit. Developing means of determiningthe uniqueness of the DOS modeling parameters could also be studied fur-ther. The development of formal means of determining the underlying DOSfunctions from the corresponding optical properties, through the use of thismodel, would be worthy of further consideration. Finally, what this models98Chapter 5. Conclusionstells one about the design of device that employ these types of materialswould also be interesting to examine. These opportunities will have to beexplored in the future.99References[1] W. A. Hadi, ?The electron transport within the wide energy gap com-pound semiconductors gallium nitride and zinc oxide,? Ph. D. Thesis,University of Windsor, Windsor, Ontario, Canada, 2014.[2] ?Displays as key driver for large area electronics in intelligent environ-ments: A vision for Europe 2007+?, A proposal for the advancementof the IST thematic priority, Information Society Technologies, May2005.[3] R. A. Street, Technology and applications of amorphous silicon, NewYork: Springer-Verlag, 2000.[4] J. J. Thevaril, ?The optical response of hydrogenated amorphoussilicon,? Ph. D. Thesis, University of Windsor, Windsor, Ontario,Canada, 2011.[5] M. Shur, Introduction to electronic devices, New York: John Wileyand Sons Inc, 1996.100References[6] W. B. Jackson, S. M. Kelso, C. C. Tsai, J. W. Allen, and S.-J. Oh, ?En-ergy dependence of the optical matrix element in hydrogenated amor-phous and crystalline silicon,? Physical Review B, vol. 31, pp. 5187-5198, 1985.[7] G. D. Cody, B. G. Brooks, and B. Abeles, ?Optical absorption abovethe optical gap of amorphous silicon hydride,? Solar Energy Materials,vol. 8, pp. 231-240, 1982.[8] S. K. O?Leary, ?An empircal density of states and joint density ofstates analysis of hydrogenated amorphous silicon: a review,? Journalof Material Science: Materials in Electronics, vol. 15, pp. 401-410,2004.[9] R. H. Klazes and M. H. L. M. van den Broek and J. Bezemer andS. Radelaar, ?Determination of the optical bandgap of amorphoussilicon,? Philosophical Magazine B, vol. 45, pp. 377-383, 1982.[10] S. O. Kasap, Principles of electronic materials and devices, 3rd ed.New York: McGraw Hill Higher Education, 2005.[11] J. Tauc, R. Grigorovici, and A. Vancu, ?Optical properties and elec-tronic structure of amorphous germanium,? Physica Status Solidi,vol. 15, pp. 627-637, 1966.101References[12] W. C. Chen, B. J. Feldman, J. Bajaj, F. M. Tong and G. K. Wong,?Thermalization gap excitation photoluminescence and optical ab-sorption in amorphous silicon-hydrogen alloys,? Solid State Commu-nication, vol. 38, pp. 357-383, 1981.[13] D. Redfield, ?Energy-band tails and the optical absorption edge; thecase of a-Si:H,? Solid State Communication, vol. 44, pp. 1347-1349,1982.[14] G. Cody, ?The optical absorption edge of a-Si:H,? in Hydro-genated Amorphous Silicon, ser. Semiconductors and Semimetals, J.I. Pankovepp, Ed. Elsevier, 1984, vol. 21B, pp.11-82.[15] S. K. O?Leary, S. R. Johnson and P. K. Lim, ?The relationship betweenthe distribution of electronic states and the optical absorption spec-trum of an amorphous semiconductor: an empirical analysis,? Journalof Applied Physics, vol. 82, pp. 3334 - 3340, 1997.[16] J. J. Thevaril and S. K. O?Leary, ?A dimensionless joint density ofstates formalism for the quantitative characterization of the opti-cal response of hydrogenated amorphous silicon,? Journal of AppliedPhysics, vol. 107, pp. 083105 1-6, 2010.102References[17] M. Shur and M. Hack, ?Physics of amorphous silicon based alloy field-effect transistors,? Journal of Applied Physics, vol. 55, pp. 3831-3842,1984.[18] S. M. Malik and S. K. O?Leary, ?Optical transitions in hydrogenatedamorphous silicon,? Applied Physics Letters, vol. 80, pp. 790-792, 2002.[19] J. J. Thevaril and S. K. O?Leary, ?Defect absorption and optical tran-sitions in hydrogenated amorphous silicon,? Solid State Communica-tions, vol. 150, pp. 1851 - 1855, 2010.[20] S. K. O?Leary, ?An analytical density of states and joint densityof states analysis of amorphous semiconductors,? Journal of AppliedPhysics, vol. 96, pp. 3680-3686, 2004.[21] S.B. Minar, ?An optical functional analysis of amorphous semiconduc-tors?, Mater?s thesis, University of British Columbia, Canada, 2012.[22] P. A. Stolk, F. W. Saris, A. J. M. Berntsen, W. F. van der Weg, L. T.Sealy, R. C. Barklie, G. Krotz and G. Muller, ?Contribution of defectsto electronic, structural, and thermodynamic properties of amorphoussilicon,? Journal of Applied Physics, vol. 75, pp. 7266-7286, 1994.103References[23] Z. Remes?, ?Study of defects and microstructure of amorphous andmicrocrystalline silicon thin films and polycrystalline diamond usingoptical methods,? Ph. D. Thesis, Charles University, Prague, 1999.104

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