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An optical functional analysis of amorphous semiconductors 2012

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An optical functional analysis of amorphous semiconductors by Sanjida Begum Minar B.Sc., Bangladesh University of Engineering and Technology, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) December 2012 c© Sanjida Begum Minar 2012 Abstract Models for the spectral dependence of the real and imaginary components of the dielectric function, appropriate for the case of amorphous semiconduc- tors, are considered. In the first phase of this analysis, from an empirical expression for the imaginary part of the dielectric function, this expression corresponding to that of Jellison and Modine [G. E. Jellison, Jr. and F. A. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Applied Physics Letters, vol. 69, pp. 371-373, 1996], a closed-form expression for the real part of the dielectric function is deter- mined using a Kramers-Kronig transformation. The resultant expression for the real component of the dielectric function corresponds with that of the model of Jellison and Modine. The subsequent comparison with experiment is found to be satisfactory. Then, in the latter stage of this analysis, through the application of a Kramers-Kronig transformation on an empirical model for the imaginary part of the dielectric function, this model stemming from ii Abstract a model for the distributions of electronic states, the spectral dependence of the real part of the dielectric function is determined. Fits with the results of experiment, taken over the near-infrared, visible, and near-ultraviolet, are also found to be satisfactory. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Theoretical basis for analysis . . . . . . . . . . . . . . . . . . 15 iv Table of Contents 2.1 The optical response of materials . . . . . . . . . . . . . . . . 15 2.2 The spectral dependence of the imaginary part of the dielectric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Three critical assumptions . . . . . . . . . . . . . . . . . . . . 19 2.4 A momentum matrix element formulation . . . . . . . . . . . 24 2.5 A dipole matrix element formulation . . . . . . . . . . . . . . 28 2.6 Free electron model of the DOS functions . . . . . . . . . . . 31 2.7 The spectral dependence of the JDOS function . . . . . . . . 34 2.8 Kramers-Kronig relations . . . . . . . . . . . . . . . . . . . . 39 2.9 Review of models . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.10 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 49 3 The model of Jellison and Modine . . . . . . . . . . . . . . . 53 3.1 Empirical models for the optical functions associated with an amorphous semiconductor . . . . . . . . . . . . . . . . . . . . 53 3.2 The Lorentz model . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Model for the optical functions . . . . . . . . . . . . . . . . . 57 3.4 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 60 v Table of Contents 3.4.2 Root calculation through expansion of the denomina- tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.3 Calculation of the coefficients . . . . . . . . . . . . . . 63 3.4.4 Grouping the terms in the partial fraction expression . 71 3.4.5 Summary of the partial fraction expansion . . . . . . . 75 3.4.6 Evaluation of the constants c3, c4, c5, and c6 . . . . . 76 3.4.7 Main integration . . . . . . . . . . . . . . . . . . . . . 85 3.4.8 Summation by parts . . . . . . . . . . . . . . . . . . . 90 3.4.9 Calculation of term 3: . . . . . . . . . . . . . . . . . . 95 3.4.10 Summation of term 1, term 2 and term 3 . . . . . . . . 97 3.4.11 Final Tauc-Lorentz expression for the model of Jellison and Modine . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 Comparing the results of the Jellison and Modine model with experimental data . . . . . . . . . . . . . . . . . . . . . . . . 99 4 A physically based Kramers-Kronig consistent model for the optical functions associated with amorphous semiconductors 105 4.1 Evaluating the JDOS function from the distributions of elec- tronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Empirical model for the DOS functions . . . . . . . . . . . . 107 vi Table of Contents 4.3 JDOS evaluation and analysis . . . . . . . . . . . . . . . . . . 115 4.4 Modeling the optical functions associated with a-Si . . . . . . 119 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.6 Implications of a non-unity 1∞ . . . . . . . . . . . . . . . . . 137 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 vii List of Tables 3.1 The “best-fit” a-Si modeling parameter selections employed for the Tauc-Lorentz model of Jellison and Modine [15] for fits to experimental data corresponding to EGE a-Si, PECVD a-Si I, and PECVD a-Si II. . . . . . . . . . . . . . . . . . . . . . . 101 4.1 The “best-fit” a-Si DOS modeling parameter selections em- ployed for the purposes of this analysis. These modeling pa- rameters relate to Eqs. (2.17), (2.45), and (4.8) . . . . . . . . 133 viii List of Figures 1.1 Semiconductor and flat panel display shipments plotted as a function of date. This data was obtained from the 2005 Infor- mation Society Technologies proposal for advancement [10]. . . 4 1.2 A two-dimensional schematic representation of the atomic dis- tributions associated with hypothetical crystalline, microcrys- talline, and amorphous semiconductors. This figure is from Nguyen [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 A schematic representation of the light intensity as a function of the penetration depth, z, with light normally incident upon the material propagating from the left. This figure is from Thevaril [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Variations in the relative intensity, I(z)/Io, with the penetra- tion depth, z, for different selections of the optical absorption coefficient, α. This figure is after Orapunt [14]. . . . . . . . . . 9 ix List of Figures 1.5 A representative optical transition, from an occupied elec- tronic state in the valence band to an unoccupied electronic state in the conduction band. . . . . . . . . . . . . . . . . . . 11 2.1 The possible optical transitions within c-Si and a-Si. This figure is after Jackson et al. [16]. . . . . . . . . . . . . . . . . 22 2.2 The occupancy function, f(E), as a function of energy, E, for various selections of temperature. The Fermi energy level, EF , is the reference energy level, i.e., EF = 0 eV. . . . . . . . . . . 23 2.3 A schematic representation of a three-dimensional potential energy well. This figure is from Thevaril [13]. . . . . . . . . . . 32 2.4 The valence band DOS function, Nv(E), and the conduction band DOS function, Nc(E), plotted as a function of energy, E, determined through Eqs. (2.23) and (2.24), respectively, assuming the nominal DOS modeling parameter selections, Nvo = Nco = 2 × 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These values are representative of the case of a-Si. . . . . 36 x List of Figures 2.5 The JDOS function, J(E), plotted as a function of the pho- ton energy, E, determined through Eq. (2.25), assuming the nominal DOS modeling parameter selections, Nvo = Nco = 2× 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These val- ues are representative of the case of a-Si. This plot is depicted on a linear scale. . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 The JDOS function, J(E), plotted as a function of the pho- ton energy, E, determined through Eq. (2.25), assuming the nominal DOS modeling parameter selections, Nvo = Nco = 2× 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These val- ues are representative of the case of a-Si. This plot is depicted on a logarithmic scale. . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Contours of integration for Eq. (2.30). . . . . . . . . . . . . . 42 2.8 The three experimental data sets, i.e., EGE a-Si [30], PECVD a-Si I [31], and PECVD a-Si II [32], for the real part of the dielectric function, 1(E), as a function of the photon energy, E. 51 xi List of Figures 2.9 The three experimental data sets, i.e., EGE a-Si [30], PECVD a-Si I [31], and PECVD a-Si II [32], for the imaginary part of the dielectric function, 2(E), as a function of the photon energy, E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the EGE a-Si expermental data set [30]. These experimental results are from Piller [30]. The fits to experiment are shown with the solid lines. The experimental data, EGE a-Si [30], is depicted with the solid points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the PECVD a-Si I experimen- tal data set [31]. These experimental results are from Synow- icki [31]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si I [31], is depicted with the solid points. . . . . . . . . . . . . . . . . . . . . . . . . . . 103 xii List of Figures 3.3 A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the PECVD a-Si II experimen- tal data set [32]. These experimental results are from Ferluato et al. [32]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si II [32], is depicted with the solid points. . . . . . . . . . . . . . . . . . . . . . . . 104 4.1 A linear plot of the valence band DOS function as a function of energy, E, for various selections of γv. The DOS modeling parameters, Nvo and Ev, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. . . . . . . . . . . . . . . . . . 111 4.2 A logarithmic plot of the valence band DOS function as a func- tion of energy, E, for various selections of γv. The DOS model- ing parameters, Nvo and Ev, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. . . . . . . . . . . . . . . . . . 112 xiii List of Figures 4.3 A linear plot of the conduction band DOS function as a func- tion of energy, E, for various selections of γc. The DOS model- ing parameters, Nco and Ec, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. . . . . . . . . . . . . . . . . . 113 4.4 A logarithmic plot of the conduction band DOS function as a function of energy, E, for various selections of γc. The DOS modeling parameters, Nco and Ec, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. . . . . . . . . . . 114 4.5 A linear plot of the JDOS function as a function of the photon energy, E, for various selections of γv and γc. The DOS mod- eling parameters are nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for the purposes of this plot. This figure is after O’Leary [17]. . . . . . . . . . . 116 xiv List of Figures 4.6 A logarithmic plot of the JDOS function as a function of the photon energy, E, for various selections of γv and γc. The DOS modeling parameters are nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for the purposes of this plot. This figure is after O’Leary [17]. . . 117 4.7 A linear plot of the functional dependence of the square-root of the JDOS function on the photon energy, E, for various se- lections of γv and γc. The DOS modeling parameters are nom- inally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for this plot. This figure is after O’Leary [17].118 4.8 A plot of the dependence of γv on γc. Experimental results from Sherman et al. [23], Rerbal et al. [34], Teidje et al. [35], and Winer and Ley [36] are depicted. A modeling result, ob- tained through a fit with experimental data, from O’Leary [17], is also depicted on this plot. This figure is after Thevaril and O’Leary [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 xv List of Figures 4.9 The spectral dependence of the JDOS function. The a-Si JDOS results of Jackson et al. [16] are depicted with the solid points. The JDOS fit result, obtained by setting Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV, is depicted with the solid line. . . . . . . . . . . 122 4.10 R2(E) as a function of the photon energy, E, determined using Eq. (4.7), depicted with the solid lines. For the purposes of this fit, R2o is set to 10 Å2 and Ed is set to 3.4 eV. The experimental results of Jackson et al. [16] are depicted with the solid points. . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.11 The spectral dependence for the imaginary part of the dielec- tric function, 2(E), determined using Eq. (2.17), whereR2(E) is determined using Eq. (4.7), on a logarithmic scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points . . . . . . . . . . 126 xvi List of Figures 4.12 The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.7), on a linear scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points . . . . . . . . . . 127 4.13 R2(E) as a function of the photon energy, E, determined using Eqs. (4.7) and (4.8), are depicted with the solid lines. For the purposes of this fit, R2o is set to 10 Å2 and Ed is set to 3.4 eV. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of the Eq. (4.8) plot. The experimental results of Jackson et al. [16] are depicted with the solid points. 129 xvii List of Figures 4.14 The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.8), on a logarithmic scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points . . . . . . . . . . . . . . . . . . . . . . . 130 4.15 The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.8), on a linear scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points . . . . . . . . . . . . . . . . . . . . . . . 131 xviii List of Figures 4.16 A “best-fit” of the model results with the results of experiment corresponding to EGE a-Si [30]. These experimental results are from Piller [30]. The fits to experiment are shown with the solid lines. The experimental data, EGE a-Si [30], is depicted with the solid points. . . . . . . . . . . . . . . . . . . . . . . . 134 4.17 A “best-fit” of the model results with the results of experi- ment corresponding to PECVD a-Si I [31]. These experimen- tal results are from Synowicki [31]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si I [31], is depicted with the solid points. . . . . . . . . . . 135 4.18 A “best-fit” of the model results with the results of experiment corresponding to PECVD a-Si II [32]. These experimental results are from Ferluato et al. [32]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si II [32], is depicted with the solid points. . . . . . . . . . . 136 xix List of Symbols 1 real part of the dielectric function 2 imaginary part of the dielectric function α optical absorption coefficient I(z) intensity of light at position z Io intensity of light at position z = 0 n(E) spectral dependence of the refractive index k(E) spectral dependence of the extinction coefficient q electron charge m electron mass V illuminated volume ~η polarization vector of the incident light E photon energy xx List of Symbols Ev valence band band edge EvT critical energy at which the exponential and square-root distribu- tions interface in the valence band Ec conduction band band edge EcT critical energy at which the exponential and square-root distribu- tions interface in the conduction band Eg energy gap γv breadth of the valence band tail γc breadth of the conduction band tail |v〉 single-spin electronic state associated with the valence band |c〉 single-spin electronic state associated with the conduction band ~P momentum operator P 2(E) aggregate momentum matrix element squared average R2(E) aggregate dipole matrix element squared average ~R the dipole operator ρA density of silicon atoms xxi List of Symbols EF Fermi energy level Nv(E) valence band density of states function Nc(E) conduction band density of states function Nvo valence band density of states prefactor Nco conduction band density of states prefactor J(E) joint density of states function E(t) time-dependent electric field J(t) time-dependent current density Σ(t) time-dependent conductivity Eo resonance energy of a bound electron at which the transition of the electron occurs between two atomic states C breadth of the region of anomalous dispersion A Lorentz oscillator strength Ed characteristic energy xxii List of Acronyms a-Si amorphous silicon c-Si crystalline silicon µc-Si microcrystalline silicon DOS density of states JDOS joint density of states EGE electron-gun evaporated PECVD plasma enhanced chemical vapor deposition xxiii Acknowledgements I am very thankful to my supervisor, Dr. Stephen Karrer O’Leary, for his guidance, full support, and constant encouragement throughout my thesis. I am grateful to him for keeping faith in me. His valuable advice always helped me to progress. I owe particular thanks Dr. Mario Beaudoin, Research Associate, Man- ager of the AMPELs Advanced Nanofabrication Facility, UBC, for guiding me so wonderfully during the lab training period and helping me for enlarging my vision of science. xxiv To my family xxv Chapter 1 Introduction For the past half-century, electronics has played an important role in shaping the course of human development. In advanced industrial economies, such as Canada’s, electronics has become a critical adjunct to modern life. Electronic devices are ubiquitous. They are found in houses, offices, recre- ational settings, vehicles, and appliances. They are used in communication systems, control systems, sensor systems, and in a plethora of other systems. With each passing year, new applications for electronic devices are conceived. Thus, the role that electronics plays in modern society continues to broaden. The transistor is the fundamental building block upon which the electron- ics revolution has been built. While the first microprocessors, developed in the early 1970s, were comprised of about a thousand transistors, at the time of writing, 2012, the standard personal computer, available in most offices and homes, has in excess of a billion transistors. In the near term future, this 1 Chapter 1. Introduction is only expected to exponentially increase, as it has for the past four decades; Moore [1] prophesized this exponential increase in the number of transistors on a processor in 1965, and, thus far, Moore’s prophesy, or Moore’s law, as this projection is now commonly referred to as, has held true. The electronics revolution found its genesis with the development of the first transistor in 1947 [2, 3]. Transistors are comprised of conductors, insula- tors, and semiconductors. The fabrication of the first transistor was achieved as a result of a detailed and quantitative understanding of the material prop- erties of these different materials. While the properties of conductors and insulators were relatively well understood by 1947, the understanding of the material properties of semiconductors was more rudimentary; at the time, semiconductors were considered more of a laboratory curiosity. Since that time, however, fundamental progress in electronics has been achieved through continued improvements in the understanding of the material properties of the materials used within electronic devices. The next generation of elec- tronic device will undoubtedly rely upon an even greater understanding of these material properties. Accordingly, interest in the material properties of conductors, insulators, and particularly semiconductors, remains intense. In conventional electronics, technological improvements have been achieved 2 Chapter 1. Introduction as a consequence of making the constituent electronic devices, i.e., the tran- sistors, smaller, faster, cheaper, and more reliable. There is, however, another class of electronic device that requires size in order to properly function. Flat panel displays [4, 5] and scanners [6], acting at the interface between the hu- man and electronic worlds, are two such examples. Digital x-ray imagers [7] and photovoltaic solar cells [8, 9] are two other examples. These electronic devices are commonly referred to as examples of large area electronic devices. Starting out as a niche field in the late 1960s, large area electronics now con- stitutes a sizeable and growing fraction of the overall electronics enterprise; see Figure 1.1 [10]. In conventional electronics, the focus is on producing electronic devices with sub-micron feature sizes. In contrast, for large area electronics, the focus is on depositing materials as thin-films over large substrates, i.e., of the order of a square-meter [11]. For large area electronics, the uniformity of the resultant thin-films and the expense of the deposition process are of paramount concern. Crystalline silicon (c-Si), the workhorse of conventional electronics, can not be deposited as a thin-film. Accordingly, alternative electronic materials must be employed instead. Microcrystalline silicon (µc- Si) and amorphous silicon (a-Si) are two alternate silicon-based materials 3 Chapter 1. Introduction 1985 1990 1995 2000 2005 2010 0 50 100 150 200 250 300 Year M a rk et ($ U S B il li o n s) Semiconductors Flat Panel Displays Figure 1.1: Semiconductor and flat panel display shipments plotted as a function of date. This data was obtained from the 2005 Information Society Technologies proposal for advancement [10]. 4 Chapter 1. Introduction commonly used in the large area electronics field. The distinction between c-Si, µc-Si, and a-Si relates to the underlying dis- tribution of atoms. In c-Si, the atoms are arranged in a periodic and ordered manner. In µc-Si, however, small crystallites, randomly oriented and sepa- rated by grain boundaries, are distributed throughout the volume. In a-Si, all residual crystalline order is absent, there being variations in the short- range bonding lengths and bonding angles. In Figure 1.2 [12], a schematic two-dimensional representation of the distribution of atoms within hypotheti- cal crystalline, microcrystalline, and amorphous semiconductors is presented. The disorder that is present, in both µc-Si and a-Si, plays an important role in shaping the resultant properties of these materials. While disorder is present within the atomic distributions of µc-Si and a-Si, there is still a considerable amount of order that is present in both of these materials. For both µc-Si and a-Si, most silicon atoms are bonded to four other silicon atoms, as in the case of c-Si. The bonding lengths and bonding angles, while exhibiting variations, are, on average, similar to those exhibited by c-Si. Accordingly, as the properties of a given material are primarily determined by its short-range order, many of the properties of these materials are similar to those of c-Si. The disorder that is present, 5 Chapter 1. Introduction Crystalline semiconductor structure Polycrystalline semiconductor structure Amorphous semiconductor structure Figure 1.2: A two-dimensional schematic representation of the atomic distri- butions associated with hypothetical crystalline, microcrystalline, and amor- phous semiconductors. This figure is from Nguyen [12]. 6 Chapter 1. Introduction however, will lead to some important differences. These will be the focus of the subsequent analysis. When a material is exposed to light, it will respond. Consider, for exam- ple, a beam of light incident upon a semi-infinite slab of material, as seen in Figure 1.3 [13]. Ignoring reflections from the slab, if the material comprising the slab is absorbing, assuming that the material within the slab is homo- geneous, it is found that the intensity of the light exponentially attenuates as it propagates through the absorbing slab. In particular, it can be shown that the intensity of the light within the slab I(z) = Io exp(−αz), (1.1) where I(z) denotes the intensity of the light at position z, Io represents the incident light intensity, and α denotes the optical absorption coefficient; it is assumed that z = 0 corresponds to the interface between the vacuum and the absorbing slab. Representative attenuations of the light intensity, for a number of selections of the optical absorption coefficient, α, are depicted in Figure 1.4 [14]. The absorption that occurs within a semiconductor arises as a conse- quence of optical transitions from the occupied electronic states in the va- lence band to the unoccupied electronic states in the conduction band. In 7 Chapter 1. Introduction Figure 1.3: A schematic representation of the light intensity as a function of the penetration depth, z, with light normally incident upon the material propagating from the left. This figure is from Thevaril [13]. 8 Chapter 1. Introduction 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Penetration Depth (µm) R el a ti v e In te n si ty α=104 cm−1 α=105 cm−1 α=106 cm−1 Figure 1.4: Variations in the relative intensity, I(z)/Io, with the penetration depth, z, for different selections of the optical absorption coefficient, α. This figure is after Orapunt [14]. 9 Chapter 1. Introduction Figure 1.5, a representative optical transition is depicted. The optical re- sponse of a material depends critically upon the availability of electronic states and on the probability of the optical transitions between such elec- tronic states. The distributions of valence band and conduction band elec- tronic states determines the range of possible optical transitions. The optical transition matrix elements, that couple the electronic states between which optical transitions occur, determine the probability of such optical transitions occurring. The optical response of a material may be characterized in terms of the spectral dependence of the optical absorption coefficient, α(E), where E denotes the photon energy of the photons within the beam of incident light. In a defect-free, disorderless semiconductor, this optical absorption coefficient terminates abruptly at the energy gap. Accordingly, measurements of the optical absorption spectrum allow one to determine the corresponding energy gap. In addition, the spectral dependence of the optical absorption coefficient beyond the energy gap provides information as to how the electronic states are distributed and on what the probability of the optical transitions between such states are. When disorder is introduced, however, the situation is more complicated, as there are now electronic states encroaching into the otherwise 10 Chapter 1. Introduction incoming photon h̄ω conduction band valence band energy gap Figure 1.5: A representative optical transition, from an occupied electronic state in the valence band to an unoccupied electronic state in the conduction band. 11 Chapter 1. Introduction empty energy gap region. Accordingly, the optical absorption spectrum no longer terminates abruptly at the energy gap. In fact, the definition of the energy gap itself becomes the subject of some controversy. While α(E) is often experimentally measured, it is the real and imaginary parts of the dielectric function, 1(E) and 2(E), respectively, that are the more usual focus of theoretical attention. The complex dielectric function may be expressed as the sum over its real and imaginary components, i.e., ̃(E) = 1(E) + i2(E), (1.2) where 1(E) and 2(E) denote the real and imaginary components of this dielectric function, respectively, and i = √−1. The optical absorption coeffi- cient, α(E), may be related to the imaginary part of the dielectric function, 2(E), through the relationship α(E) = E ~cn(E) 2(E), (1.3) where ~ and c represent the reduced Planck’s constant and the speed of light in a vacuum, respectively, and where n(E) denotes the spectral dependence of the refractive index, where the complex refractive index, ñ(E) = n(E) + ik(E), (1.4) 12 Chapter 1. Introduction k(E) being the corresponding extinction coefficient. It is noted that the complex refractive index is related to the complex dielectric function through the relationship ̃(E) = ñ2(E). (1.5) From Eqs. (1.2), (1.4), and (1.5), one can thus conclude that 1(E) = n 2(E)− k2(E), (1.6) and 2(E) = 2 n(E) k(E). (1.7) These functions, α(E), 1(E), 2(E), n(E), and k(E), are collectively referred to as the optical functions, as they quantify the nature of the optical response of a given material. In this thesis, the optical response of amorphous semiconductors will be the focus of analysis. Following a presentation of the theoretical background required of this work, the analysis begins with a theoretical justification for the empirical model of Jellison and Modine [15], in which Kramers-Kronig consistent empirical expressions for 1(E) and 2(E) are formulated. Then, a model for the optical functions, i.e., 1(E) and 2(E), stemming from a model for the distributions of valence band and conduction 13 Chapter 1. Introduction band electronic states, is developed. Finally, these models will be employed in order to fit some relevant experimental data. Conclusions, regarding the suitability of these models, will then be drawn from these fits. The main new results presented in this thesis pertain to the theoretical justification of the Kramers-Kronig consistent empirical expressions for 1(E) and 2(E) corresponding to the empirical model of Jellison and Modine [15] and the new model for these optical functions stemming from a model for the distributions of valence band and conduction band electronic states. This thesis is organized in the following manner. In Chapter 2, the the- oretical foundations for this analysis are provided. Then, in Chapter 3, a theoretical justification for the Kramers-Kronig consistent model of Jellison and Modine [15] is formulated. An alternate Kramers-Kronig model for the spectral dependence of 1(E) and 2(E), stemming from a model for the un- derlying distributions of valence band and conduction band electronic states, is then developed in Chapter 4. Finally, the conclusions of this analysis are presented in Chapter 5. 14 Chapter 2 Theoretical basis for analysis 2.1 The optical response of materials The optical response of a material is primarily shaped by the distri- butions of valence band and conduction band electronic states and by the magnitude of the optical transition matrix elements which couple the elec- tronic states between which optical transitions occur. The distributions of valence band and conduction band electronic states determines what optical transitions can occur in a given material. The magnitude of the optical tran- sition matrix elements, however, determine the probability of such optical transitions occurring. The primary goal of this analysis is the development of Kramers-Kronig consistent models for the spectral dependence of the real and imaginary components of the dielectric function, 1(E) and 2(E), re- spectively, for the case of amorphous semiconductors. In order to achieve this goal, the role that each of these factors plays in shaping the optical 15 2.1. The optical response of materials response of an amorphous semiconductor must be well understood. In this chapter, the theoretical foundations for the work presented in this thesis are provided. Initially, an expression for the spectral dependence of the imaginary part of the dielectric function, 2(E), in terms of the valence band and conduction band energy levels and the magnitude of the corresponding optical transition matrix elements, is presented. Then, through the introduc- tion of a number of suitable assumptions, a relationship between the valence band and conduction band distributions of electronic states and the spectral dependence of 2(E) is developed. Two particular formulations for this re- lationship are developed: (1) a momentum matrix element formulation, and (2) a dipole matrix element formulation, these formulations being related to each other. For each of these formulations, an aggregate matrix element is employed. The joint density of states (JDOS) function, J(E), is intro- duced as a corollary to this analysis. The Kramers-Kronig relations are then presented, these relating the spectral dependence of 1(E) with 2(E) and the spectral dependence of 2(E) with 1(E). A number of models, for both 1(E) and 2(E), are then presented. Finally, three sets of experimental data, corresponding to different samples of a-Si, are presented, these experimental data sets being used in the subsequent analysis. 16 2.1. The optical response of materials This chapter is organized in the following manner. In Section 2.2, an expression for the spectral dependence of the imaginary part of the dielectric function, 2(E), in terms of the valence band and conduction band energy levels and the magnitude of the corresponding optical transition matrix el- ements, is presented. Then, in Section 2.3, three critical assumptions are introduced, these allowing one to relate the spectral dependence of 2(E) with that of J(E). In Section 2.4, a momentum matrix element formulation for the relationship between the imaginary part of the dielectric function, 2(E), and the distributions of the valence band and conduction band elec- tronic states is developed, the JDOS function and the aggregate momentum matrix element being introduced as a byproduct of this particular formu- lation. A dipole matrix element formulation for the relationship between the imaginary part of the dielectric function, 2(E), and the distributions of the valence band and conduction band electronic states is then devel- oped in Section 2.5, the aggregate dipole matrix element being proposed as a byproduct of this analysis. The free electron model, for the distributions of electronic states, is then presented in Section 2.6. The spectral depen- dence of this JDOS function, assuming free electron distributions of valence band and conduction band electronic states, is then presented in Section 2.7. 17 2.2. The spectral dependence of the imaginary part of the dielectric function The Kramers-Kronig relationships between the real and imaginary parts of the dielectric function, 1(E) and 2(E), respectively, are discussed in Sec- tion 2.8, the time-symmetry requirements demanded of these relations being featured. A number of models for the spectral dependence of the optical functions, 1(E) and 2(E), are then presented in Section 2.9. Finally, ex- perimental a-Si data sets, corresponding to 1(E) and 2(E), are presented in Section 2.10, these data sets being used in the subsequent analysis. 2.2 The spectral dependence of the imaginary part of the dielectric function Optical transitions occur from occupied electronic states in the valence band (initial states) to unoccupied electronic states in the conduction band (final states). Within the framework of a one-electron picture, assuming linear optical response, Jackson et al. [16] assert that the imaginary part of the dielectric function, 2(E) = ( 2piq~ mE )2 2 V ∑ v,c |~η · ~P v,c|2δ(Ec − Ev − E), (2.1) where q denotes the electron charge, m represents the electron mass, V is the illuminated volume, ~η is the polarization vector of the incident light, 18 2.3. Three critical assumptions and E is the photon energy; the δ(·) terms refer to the Dirac delta function. Ev and Ec denote the energy levels of representative valence band and con- duction band electronic states, |v〉 and |c〉, respectively. ~P v,c represents the momentum matrix element that couples these particular electronic states in an optical transition process; in quantum mechanics, this matrix element, ~P v,c = 〈c|~P |v〉, where ~P denotes the momentum operator. The |~η · ~P v,c|2 operation refers to taking the complex modulus squared of the complex num- ber ~η · ~P v,c. The summation in Eq. (2.1) is performed over all of the occupied valence band and unoccupied conduction band single-spin electronic states. 2.3 Three critical assumptions Eq. (2.1) expresses the spectral dependence of the imaginary part of the dielectric function, 2(E), in terms of the valence band and conduction band energy levels and the magnitude of the corresponding momentum matrix elements. This expression applies to both the crystalline and amorphous cases. In order to relate 2(E) with the underlying distributions of valence band and conduction band electronic states for the specific case of amorphous semiconductors, three critical assumptions are introduced into the analysis: 19 2.3. Three critical assumptions (1) it is assumed that the incident light is unpolarized, (2) it is assumed that the disorder that is present in an amorphous semiconductor completely relaxes the momentum conservation rules, and (3) it is assumed that zero- temperature statistics apply. The consequences of each approximation is outlined subsequently. The absence of polarization simplifies Eq. (2.1). With random polar- ization, assuming that all of the significant momentum matrix elements are of the same order of magnitude, the average value of |~η · ~P v,c|2 reduces to 1 3 |Pv,c|2, where |Pv,c| denotes the amplitude of the momentum matrix element. As a result, Eq. (2.1) reduces to 2(E) = (2piq) 2 ~2 m2E2 2 3V ∑ v,c |Pv,c|2δ(Ec − Ev − E). (2.2) In a crystalline semiconductor, momentum must be conserved in an op- tical transition. In an amorphous semiconductor, however, the momentum conservation rules are relaxed. That is, optical transitions can occur from every occupied state (of a particular spin) in the valence band to every un- occupied state (of the same spin) in the conduction band; spin-flips are not permitted during an optical transition. As a consequence, in a crystalline semiconductor, the |Pv,c| values vary greatly, depending upon whether mo- mentum is conserved or not, while in an amorphous semiconductor, |Pv,c| is 20 2.3. Three critical assumptions essentially constant for all cases. This greatly simplifies the analysis. The contrast between c-Si and a-Si is instructive. Consider the case of N silicon atoms. For the case of c-Si, from a given single-spin state in the valence band, four optical transitions are possible; there are N 2 possible values of crystal momentum for the case of c-Si. For the case of a-Si, however, 2N optical transitions are possible. These possible optical transitions are depicted in Figure 2.1. If the overall “optical strength” of c-Si and a-Si is similar, it seems likely that the |Pv,c| matrix elements associated with the small number of possible c-Si optical transitions will be of a much greater magnitude than that of their more numerous a-Si counterparts. Later on in this analysis, a normalized matrix element is introduced in order to allow for a direct comparison between the c-Si and a-Si matrix elements, where the weighting factor is directly proportional to the ratio between the number of allowed optical transitions for the c-Si and a-Si cases. The final assumption that is introduced into this analysis is that of zero- temperature statistics. In Figure 2.2, the occupancy function, f(E) = 1 1 + exp(E−EF KBT ) , (2.3) is depicted as a function of an energy, E, for a variety of different tempera- ture selections, EF denoting the Fermi energy level. It is noted that as the 21 2.3. Three critical assumptions k En er gy EF c−Sia−Si 2ρAV 4 Figure 2.1: The possible optical transitions within c-Si and a-Si. This figure is after Jackson et al. [16]. 22 2.3. Three critical assumptions −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0 0.25 0.5 0.75 1 0 K Electron Energy (eV) P ro b a b il it y o f O cc u p a ti o n 300 K 800 K EF Figure 2.2: The occupancy function, f(E), as a function of energy, E, for various selections of temperature. The Fermi energy level, EF , is the reference energy level, i.e., EF = 0 eV. 23 2.4. A momentum matrix element formulation temperature goes to zero that this function, sometimes referred to as the Fermi-Dirac function, reduces to a step function, where the step occurs at the Fermi energy level, EF . For the case of semiconductors, considering that the Fermi energy level, EF , is typically in the middle of the energy gap, this implies that at zero temperature, all of the valence band electronic states are fully occupied and all of the conduction band electronic states are completely unoccupied. This assumption simplifies the subsequent analysis. 2.4 A momentum matrix element formulation It will be assumed that the overall “optical strength” of c-Si is identical to that of a-Si. As was discussed earlier, if this is the case it would seem rea- sonable to expect that the small number of non-zero c-Si momentum matrix elements will be larger in magnitude than their much more numerous a-Si counterparts. In order to allow for a direct contrast between these matrix elements, one could conceive of normalizing the a-Si momentum matrix el- ements by a factor that is proportional to the ratio between the number of allowed optical transitions, i.e., 2N 4 . Letting ρA denote the density of silicon 24 2.4. A momentum matrix element formulation atoms within a-Si, this normalization factor becomes 2ρAV 4 . In order to allow for a direct relationship between 2(E) and the dis- tributions of valence band and conduction band electronic states, one can introduce an aggregate momentum matrix element P 2(E) ≡ ρAV 2 ∑ v,c |Pv,c|2δ(Ec − Ev − E)∑ v,c δ(Ec − Ev − E) , (2.4) where the ρAV 2 prefactor allows for a direct comparison between the momen- tum matrix elements associated with c-Si and a-Si. From Eqs. (2.2) and (2.4), it can now be shown that 2(E) = (2piq) 2 ~2 m2E2 2 3V ( 2 ρAV ) P 2(E) ∑ v,c δ(Ec − Ev − E). (2.5) The term ∑ v,c δ(Ec − Ev − E) provides for the number of possible optical transitions between the occupied valence band and unoccupied conduction band electronic states separated by the energy E. Defining the JDOS func- tion, J(E) ≡ 4 V 2 ∑ v,c δ(Ec − Ev − E), (2.6) from Eq. (2.5), it can now be seen that 2(E) = (2piq) 2 ~2 m2E2 2 3V V 2 4 ( 2 ρAV ) P 2(E) J(E). (2.7) 25 2.4. A momentum matrix element formulation Thus, the spectral dependence of 2(E) reduces to a product of a prefactor, the reciprocal of E2, the aggregate momentum matrix element, and the JDOS function, J(E). The valence band density of states (DOS) function, Nv(E), whereNv(E)∆E represents the number of valence band electronic states between [E,E+∆E], per unit volume, may be expressed as the sum over all of the valence band single-spin electronic states divided by the volume, i.e., Nv(E) = 2 V ∑ v δ(E − Ev), (2.8) where the factor of 2 arises as a consequence of the fact that each elec- tronic state is a single-spin state, i.e., there are two possible spins. Similarly, the conduction band DOS function, Nc(E), where Nc(E)∆E represents the number of conduction band electronic states between [E,E + ∆E], per unit volume, may be expressed as the sum over all of the conduction band single- spin electronic states divided by the volume, i.e., Nc(E) = 2 V ∑ c δ(E − Ec), (2.9) where the factor of 2 arises as a consequence of the fact that each electronic state is a single-spin state. Assuming that all valence band states are fully occupied and that all 26 2.4. A momentum matrix element formulation conduction band states are completely unoccupied, i.e., the zero-temperature assumption, it may be shown that the JDOS function may be expressed as an integral over the product of the valence band and conduction band DOS functions, i.e., J(E) = ∫ ∞ −∞ Nv(ξ)Nc(E + ξ)dξ. (2.10) From Eqs. (2.5) and (2.6), one can thus conclude that 2(E) = (2piq)2 3ρA ~2 m2E2 P 2(E) J(E). (2.11) For a wide variety of amorphous semiconductor experimental analyzes, it has been assumed that the aggregate momentum matrix element, P 2(E), exhibits a constant spectral dependence with respect to energy, E. This is referred to as the constant momentum matrix element assumption. This assumption, which found its origins in the analysis at Tauc et al. [18], was widely accepted by researchers in the field until the early 1980s. Despite this acceptance, it lacks a theoretical justification. Moreover, Cody et al. [19] demonstrated that experimental results interpreted within the framework of this assumption are fundamentally inconsistent. Accordingly, other models for the spectral dependence of 2(E) are sought. 27 2.5. A dipole matrix element formulation 2.5 A dipole matrix element formulation Cody et al. [19] provided experimental data which challenged the valid- ity of the constant momentum matrix element assumption. For the purposes of their study, Cody et al. [19] prepared a large number of a-Si samples under identical conditions, but with different film thicknesses. They then measured the spectral dependence of the optical functions using transmission and re- flectance measurements on these samples. Cody et al. [19] noted that while fits to this data, determined assuming a constant momentum matrix ele- ment, are satisfactory, they become greatly improved if one instead assumes a constant dipole matrix element. Moreover, such an assumption removes a film thickness dependence artifact that was noted by Cody et al. [19] within the framework of the constant momentum matrix element assumption. Ac- cordingly, it is instructive to recast our formulation in terms of the spectral dependence of the dipole matrix element. Taking into account the experimental evidence provided by Cody et al. [19], Jackson et al. [16] recast Eq. (2.1) into a form in which the matrix elements are instead expressed in terms of the dipole matrix elements as opposed to the momentum matrix elements, i.e., ~Rv,c = 〈c| ~R|v〉, where ~R denotes the 28 2.5. A dipole matrix element formulation dipole operator. From the commutator relations, Jackson et al. [16] find that ∑ v,c |η · P v,c|2δ(Ec − Ev − E) = ( mE ~ )2∑ v,c |η ·Rv,c|2δ(Ec − Ev − E). (2.12) As before, if the incident light is unpolarized, the average value of |~η · ~Rv,c|2 reduces to 1 3 |Rv,c|2, where |Rv,c| denotes the amplitude of the dipole matrix element. Accordingly, Jackson et al. [16] conclude that 2(E) = (2piq) 2 2 3V ∑ v,c |Rv,c|2δ(Ec − Ev − E). (2.13) Introducing an aggregate dipole matrix element, defined and normalized in a manner similar to that introduced earlier for the case of the momentum matrix formalism, Jackson et al. [16] proposed that, R2(E) ≡ ρAV 2 ∑ v,c |Rv,c|2δ(Ec − Ev − E)∑ v,c δ(Ec − Ev − E) , (2.14) where, as before, ρA denotes the atomic density. For this definition, it is noted that Eq. (2.13) may be represented as 2(E) = (2piq) 2 2 3V ( 2 ρAV ) R2(E) ∑ v,c δ(Ec − Ev − E). (2.15) From Eqs. (2.6) and (2.15), one can thus conclude that 2(E) = (2piq)2 3ρA R2(E) J(E). (2.16) 29 2.5. A dipole matrix element formulation That is, the spectral dependence of the imaginary part of the dielectric func- tion, 2(E), is proportional to the product of a prefactor, R2(E), and J(E). Given that the density of the silicon atoms within a-Si is typically around 4.4 ×1022 cm−3 [17], it can thus be seen that, for the specific case of a-Si, Eq. (2.16) may be re-expressed as 2(E) = 4.3× 10−45 R2(E) J(E), (2.17) whereR2(E) is in units of Å2 and J(E) is in the units of cm−6eV−1 [16, 17]. In light of the experimental results of Cody et al. [19], the spectral dependence of R2(E) is often assumed to be constant. This is referred to as the constant dipole matrix element assumption. It is noted, from Eqs. (2.11) and (2.16), that the relationship between the aggregate momentum matrix element squared average and the aggregate dipole matrix element squared average is found to be P 2(E) = m2E2 ~2 R2(E). (2.18) Eq. (2.18) allows one to relate the one formulation for the spectral dependence of the imaginary part of the dielectric function, 2(E), with the other. 30 2.6. Free electron model of the DOS functions 2.6 Free electron model of the DOS functions The spectral dependence of the JDOS function depends upon a con- volution over the valence band and conduction band DOS functions; recall Eq. (2.10). Accordingly, the determination of these functions is of paramount concern. While the exact form of these DOS functions depends critically upon the exact potential distribution, the free electron DOS function provides a useful limiting result that is often employed for theoretical benchmarking purposes. Consider the electronic states within a cubic box, of dimensions L×L×L, surrounded by an infinitely high potential barrier, as seen in Figure 2.3. It will be assumed that the potential within the box itself is nil. From quantum mechanics, for steady-state conditions, the wavefunctions associated with the corresponding bound electron states may be determined from Schrödinger’s equation, i.e., ~2 2m 52 Ψ(~r) + V (~r)Ψ(~r) = EΨ(~r), (2.19) where ~, m, V (~r), and E represent the reduced Planck’s constant, the mass of the electron, the potential energy, and the electron energy, respectively. 31 2.6. Free electron model of the DOS functions Figure 2.3: A schematic representation of a three-dimensional potential en- ergy well. This figure is from Thevaril [13]. 32 2.6. Free electron model of the DOS functions 52 denotes the mathematical operator ∂ ∂x2 + ∂ ∂y2 + ∂ ∂z2 in three-dimensions. It may be shown that Ψ(~r) = Ψnx,ny ,nz(x, y, z) = ( 2 L )3/2 sin (pinx L x ) sin (piny L y ) sin (pinz L z ) , (2.20) where nx, ny, and nz are positive integers, associated with the electron motion in the different directions; these are the quantum numbers. The energy of the corresponding electronic states may be found by substituting the obtained wavefunction, i.e., Eq. (2.20), back into the Schrödinger’s equation, i.e., Enx,ny ,nz = ~2 2m [(pinx L )2 + (piny L )2 + (pinz L )2] , (2.21) where Enx,ny ,nz represents the energy associated with the nx, ny, and nz wavefunction, Ψnx,ny ,nz(x, y, z), the labeling integers, nx, ny, and nz, exactly corresponding to the aforementioned quantum numbers. In the continuum limit, i.e., for E sufficiently large, the corresponding DOS function may be shown to be N(E) =  √ 2 m3/2 pi2~3 √ E, E ≥ 0 0, E < 0 . (2.22) 33 2.7. The spectral dependence of the JDOS function Eq. (2.22) is often referred to the free electron DOS model; within the well, the potential is nil, i.e., the electrons are “free”. This form for the DOS function is often assumed in analyzes of amorphous semiconductors. 2.7 The spectral dependence of the JDOS function It is often useful to employ the free-electron model for the valence band and conduction band DOS functions in the evaluation of the JDOS func- tion. This model suggests that the distribution of electronic states exhibits a square-root functional dependence on the energy above the conduction band minimum and below the valence band maximum. Following O’Leary [17], one can set the valence band DOS function Nv(E) =  0, E > Ev Nvo √ Ev − E, E ≤ Ev , (2.23) 34 2.7. The spectral dependence of the JDOS function and the conduction band DOS function Nc(E) =  Nco √ E − Ec, E ≥ Ec 0, E < Ec , (2.24) where Nvo and Nco represent the valence band and conduction band DOS prefactors, respectively, and Ev and Ec represent the valence band and con- duction band band edges; these DOS functions are also known as the Tauc DOS functions. The resultant distributions of electronic states, for this spe- cific case, are depicted in Figure 2.4, nominal values of Nvo, Nco, Ev, and Ec being employed for the purposes of this analysis; in particular, Nvo and Nco are both set to 2 ×1022 cm−3 eV−3/2, Ev is set to 0 eV, and Ec is set to 2 eV, these values being representative of the case of a-Si. Through the use of Eq. (2.10), it may be shown that J(E) = NvoNco  pi 8 (E − Eg)2, E ≥ Eg 0, E < Eg , (2.25) where Eg ≡ Ec−Ev represents the energy gap of the material. The spectral dependence of the corresponding JDOS function is depicted in Figures 2.5 and 2.6. This is known as the Tauc JDOS function after the pioneering work 35 2.7. The spectral dependence of the JDOS function −0.5 0 0.5 1 1.5 2 2.5 0 5 10 15 Energy (eV) D O S (× 10 21  cm − 3 e V− 1 ) N v (E) N c (E) Figure 2.4: The valence band DOS function, Nv(E), and the conduction band DOS function, Nc(E), plotted as a function of energy, E, determined through Eqs. (2.23) and (2.24), respectively, assuming the nominal DOS modeling parameter selections, Nvo = Nco = 2 × 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These values are representative of the case of a-Si. 36 2.7. The spectral dependence of the JDOS function 1.2 1.6 2 2.4 2.8 0 0.4 0.8 1.2 1.6 N vo  = N co  = 2 × 1022 cm−3eV−3/2 Eg = 2 eV Photon Energy (eV) JD O S ×  10 44  (cm − 6 e V− 1 ) Figure 2.5: The JDOS function, J(E), plotted as a function of the photon energy, E, determined through Eq. (2.25), assuming the nominal DOS mod- eling parameter selections, Nvo = Nco = 2 × 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These values are representative of the case of a-Si. This plot is depicted on a linear scale. 37 2.7. The spectral dependence of the JDOS function 0.5 1 1.5 2 2.5 3 1034 1036 1038 1040 1042 1044 N vo  = N co  = 2 × 1022 cm−3eV−3/2 Eg = 2 eV Photon Energy (eV) JD O S (cm − 6 e V− 1 ) Figure 2.6: The JDOS function, J(E), plotted as a function of the photon energy, E, determined through Eq. (2.25), assuming the nominal DOS mod- eling parameter selections, Nvo = Nco = 2 × 1022 cm−3 eV−3/2, Ev = 0 eV, and Ec = 2 eV. These values are representative of the case of a-Si. This plot is depicted on a logarithmic scale. 38 2.8. Kramers-Kronig relations of Tauc et al. [18]. 2.8 Kramers-Kronig relations As a consequence of causality, there is a relationship between the spectral dependence of the real and imaginary parts of the dielectric function, 1(E) and 2(E), respectively. Consider a time-dependent electric field, E(t), ap- plied to a solid that produces a current density, J(t), in response. The variations with respect to position of these quantities are neglected. If only weak fields are considered, then a general linear relationship between E and J exists, where J(t) = ∫ ∞ −∞ Σ(t− t′) E(t′)dt′, (2.26) Σ(t) denoting the time-dependent conductivity. Through a Fourier transform on Eq. (2.26), one obtains J (ω) = σ(ω)ξ(ω), (2.27) 39 2.8. Kramers-Kronig relations where J (ω), ξ(ω), and σ(ω) are the Fourier transforms of J , E, and Σ, respectively, i.e., J (ω) = ∫ ∞ −∞ J(t)eiωtdt, (2.28a) ξ(ω) = ∫ ∞ −∞ E(t)eiωtdt, (2.28b) and σ(ω) = ∫ ∞ −∞ Σ(t)eiωtdt. (2.28c) It is also noted that the inverse Fourier transforms become J(t) = 1 2pi ∫ ∞ −∞ J (ω)e−iωtdω, (2.29a) E(t) = 1 2pi ∫ ∞ −∞ ξ(ω)e−iωtdω, (2.29b) and Σ(t) = 1 2pi ∫ ∞ −∞ σ(ω)e−iωtdω. (2.29c) In order to further the analysis, two critical assumptions have to be made: (1) Σ(t) is zero for negative values of its argument, and (2) J(t) = 0 for t < t0, where t0 is the earliest value of t ′ for which E(t′) 6= 0. The first assumption leads to the fact that σ(ω) is an analytic function in the upper half of the complex ω plane. According to Cauchy’s theorem, the integral in Eq. (2.29c) should vanish for σ(ω) being analytic in the upper half plane. It may thus 40 2.8. Kramers-Kronig relations be shown that σ(ω0) = 1 2pii ∮ C1 σ(ω) ω − ω0dω. (2.30) At first, the contour of integration of Eq. (2.30) may be chosen to be a small circle, C1, about ω0, as shown in Figure 2.7. It is known that the distortion of C1 into a larger contour of integration, C2, does not change the result; see Figure 2.7, i.e., σ(ω0) = 1 2pii ∮ C1 σ(ω) ω − ω0dω = 1 2pii ∮ C2 σ(ω) ω − ω0dω. (2.31) Assuming that σ(ω) → 0 for |ω| → ∞, then only contributions from the real axis play a role in determining the integral in Eq. (2.30). Considering that ωr is the frequency along the real axis, then it may be shown that σ(ω0) = 1 2pii ∫ ∞ −∞ σ(ωr) ωr − ω0 dωr. (2.32) It is noted that ω0 may be expressed as ω0 = ω + iω ′. (2.33) Since lim ω′→0 1 ωr − ω − iω′ = P 1 ωr − ω + ipiδ(ωr − ω), (2.34) 41 2.8. Kramers-Kronig relations C1 C2 ωi ωr ωo Imaginary Real Figure 2.7: Contours of integration for Eq. (2.30). 42 2.8. Kramers-Kronig relations where P denotes the principal value, from Eqs. (2.32) and (2.34), it is seen that σ(ω) = 1 pii P ∫ ∞ −∞ σ(ωr) ωr − ω dωr. (2.35) From Eq. (2.35), it is seen that σ(ω) is a complex quantity, i.e., σ(ω) = σ1(ω) + iσ2(ω). (2.36) Thus, it can be seen that σ1(ω) = 1 pi P ∫ ∞ −∞ σ2(ωr) ωr − ω dωr, (2.37a) and σ2(ω) = − 1 pi P ∫ ∞ −∞ σ1(ωr) ωr − ω dωr. (2.37b) That is, σ1(ω) and σ2(ω) are Hilbert transforms of each other. As a real E(t) produces a real J(t), then the time-dependent conductivity, σ(t), must also be real. This requires that [σ1(ωr) + iσ2(ωr)] ∗ = σ(−ωr). (2.38) As a result, σ1(ωr) = σ1(−ωr), (2.39a) and σ2(ωr) = −σ2(−ωr). (2.39b) 43 2.8. Kramers-Kronig relations So, from Eq. (2.37a), it is seen that σ1(ω) = 1 pi P [∫ 0 −∞ σ2(ωr) ωr − ω dωr + ∫ ∞ 0 σ2(ωr) ωr − ω dωr ] , = 1 pi P ∫ ∞ 0 σ2(ωr) [ 1 ωr − ω + 1 ωr + ω ] dωr, = 2 pi P ∫ ∞ 0 ωrσ2(ωr) ω2r − ω2 dωr. (2.40) Similarly, σ2(ω) = −2ω pi P ∫ ∞ 0 σ1(ωr) ω2r − ω2 dωr. (2.41) Eqs. (2.40) and (2.41) constitute the Kramers-Kronig relations, applied to conductivity. It is also possible to develop Kramers-Kronig relations for the real and imaginary parts of the dielectric function. Since (ω) = 0 + i σ(ω) ω , (2.42) it may be shown that 1(ω) = 1− [ σ2(ω) 0ω ] , (2.43a) and 2(ω) = σ1(ω) 0ω . (2.43b) 44 2.8. Kramers-Kronig relations Thus, 1(ω) = 1 + 2 pi P ∫ ∞ 0 ωr2(ωr) ω2r − ω2 dωr, (2.44a) and 2(ω) = 2 piω P ∫ ∞ 0 ω2r [1− 1(ωr)] ω2r − ω2 dωr. (2.44b) So, the real part of the dielectric function, 1(E), is related to the imaginary part of the dielectric function, 2(E), through the Kramers-Kronig transfor- mation, 1(E) = 1∞ + 2 pi P ∫ ∞ Eg u2 (u) u2 − E2 du, (2.45) where 1∞ should ideally be set to unity. Similarly, the imaginary part of the dielectric function, 2(E), is related to the real part of the dielectric function, 1(E), through the Kramers-Kronig transformation 2(E) = 2 piE P ∫ ∞ 0 u2[1− 1(u)] u2 − E2 du. (2.46) The conditions developed in Eqs. (2.39a) and (2.39b) mandate that 1(E) = 1(−E), (2.47a) and 2(E) = −2(−E). (2.47b) 45 2.9. Review of models These are referred to as the time-symmetry requirements mandated of the Kramers-Kronig relations. 2.9 Review of models Within the framework of the momentum matrix element and the dipole matrix element formulations, the imaginary part of the dielectric function, 2(E), may be expressed as the product of the corresponding aggregate op- tical transition matrix element, the JDOS function, and some other function of the photon energy. The optical transition matrix elements play an im- portant role in shaping the spectral dependence of the imaginary part of the dielectric function, 2(E). Models for the spectral dependence of the imag- inary part of the dielectric function, 2(E), have been developed assuming a constant momentum matrix element, i.e., a momentum matrix element that is independent of the photon energy, E. Examples include the model of Forouhi and Bloomer [20], the model of McGahan et al. [21], and the model of Jellison and Modine [15]. This assumption of a constant momentum matrix element was first sug- gested by Tauc et al. [18]. In fact, Tauc et al. [18] suggested that the zero- 46 2.9. Review of models intercept of the plot of √ αE as a function of E, where α represents the corresponding optical absorption coefficient, allows one to define the “opti- cal bandgap” of these materials if the aggregate momentum matrix element is constant and the valence band and conduction band DOS functions are square-root in form, i.e., of the form presented in Eqs. (2.23) and (2.24). Based upon a linear response formalism for 2(E), Tauc et al. [18] derived an expression for 2(E) that characterizes the absorption edge of amorphous semiconductors. They concluded that 2(E) ∝ (E − Eg) 2 E2 , (2.48) for E > Eg, where Eg denotes the corresponding “optical bandgap”. Eq. (2.48) may be used to describe the interband optical transition mechanism for the optical response of such a material, and thus, 2(E) = 0 for E < Eg. Recalling Eq. (1.7), i.e., 2(E) = 2n(E)k(E), where n(E) represents the spectral dependence of the refractive index and k(E) denotes the spectral dependence of the extinction coefficient, Forouhi and Bloomer [20] suggest an expression for the extinction coefficient kFB(E) = A(E − Eg)2 E2 −BE + C , (2.49) where A, B, and C are treated as empirical fitting parameters. The term 47 2.9. Review of models (E −Eg)2 in Eq. (2.49) represents the assumed form for the JDOS function of the material itself, and arises from the assumed square-root valence band and conduction band DOS functions; recall Eq. (2.25). This formulation was then used in order to fit experimental optical functions corresponding to amorphous diamond-like carbon films [15, 21, 22, 24]. Unfortunately, as was pointed out by McGahan and Woollam [21, 22], the Forouhi and Bloomer [20] model is unable to describe the spectral dependence of these optical functions correctly. In particular, the Forouhi and Bloomer [20] formulation provides for negative band gaps, a clearly unphysical result [15, 22]. McGahan et al. [21] made some modifications to the Forouhi and Bloomer [20] model, and this was shown to successfully fit several n and k experimental data sets. Unfortunately, the Forouhi and Bloomer [20] and McGahan et al. [21] formulations have several fundamental problems, these being nicely summa- rized by Jellison and Modine [15]. They are as listed below: • While experiment indicates that 2(E)→ 0 as E →∞, both the models of Forouhi and Bloomer [20] and McGahan et al. [21] instead approach non-zero values for 2(E) as E →∞. • Both the models of Forouhi and Bloomer [20] and McGahan et al. [21] fail to satisfy the Kramers-Kronig time-symmetry requirements, 48 2.10. Experimental results i.e., Eqs. (2.47a) and (2.47b), which mandate that k(−E) = −k(E). (2.50) Jellison and Modine [15] developed a new parameterized model for the optical functions associated with an amorphous semiconductor that does not have the problems listed above. In particular, they developed a Tauc-Lorentz expression for the imaginary part of the dielectric function, 2(E), based on the Tauc JDOS function [18, 25], i.e., Eq. (2.25), and the Lorentz oscilla- tor [26]. They then determined the corresponding spectral dependence of the real part of the dielectric function, 1(E), using a Kramers-Kronig trans- formation, i.e., Eq. (2.45). As with the models of Forouhi and Bloomer [20] and McGahan et al. [21], the model of Jellison and Modine [15] is cast within the framework of the constant momentum matrix element assumption. 2.10 Experimental results In the subsequent analysis, models for the spectral dependence of the real and imaginary components of the dielectric function, 1(E) and 2(E), respec- tively, are fit to a-Si optical function experimental data, and from these fits, the “best” model parameters are determined, i.e., the modeling parameters 49 2.10. Experimental results that produce the “best” fits with the corresponding experimental data. The experimental a-Si optical function data considered is from Piller [30], Synow- icki [31], and Ferluato et al. [32], these experimental results corresponding to samples of electron-gun evaporated (EGE) a-Si, and plasma enhanced chem- ical vapor deposition (PECVD) prepared a-Si; the experimental data from Piller [30] was obtained from a sample of EGE a-Si, the experimental data from Synowicki [31] was obtained from a sample of PECVD a-Si, and the experimental data from Ferluato et al. [32] was obtained from a sample of PECVD a-Si, these samples being referred to as EGE a-Si, PECVD a-Si I, and PECVD a-Si II, respectively. These three sets of experimental data are shown in Figures 2.8 and 2.9, the spectral dependence of the real part of the dielectric function being depicted in Figure 2.8 and the spectral dependence of the imaginary part of the dielectric function being depicted in Figure 2.9. The EGE a-Si [30] experimental data set was collected from an electron-gun evaporated a-Si sample, with silicon as the substrate [30]. For the case of the PECVD a-Si I [31] experimental data sets, fused silica was used as the substrate [31]. For the case of the PECVD a-Si II [32] experimental data sets, glass was used as the substrate [32]. 50 2.10. Experimental results 1 2 3 4 5 6 7−10 0 10 20 30 40 50 Photon Energy (eV) ε 1 EGE a−Si [30] PECVD a−Si I [31] PECVD a−Si II [32] Figure 2.8: The three experimental data sets, i.e., EGE a-Si [30], PECVD a- Si I [31], and PECVD a-Si II [32], for the real part of the dielectric function, 1(E), as a function of the photon energy, E. 51 2.10. Experimental results 1 2 3 4 5 6 7−10 0 10 20 30 40 50 Photon Energy (eV) ε 2 EGE a−Si [30] PECVD a−Si I [31] PECVD a−Si II [32] Figure 2.9: The three experimental data sets, i.e., EGE a-Si [30], PECVD a-Si I [31], and PECVD a-Si II [32], for the imaginary part of the dielectric function, 2(E), as a function of the photon energy, E. 52 Chapter 3 The model of Jellison and Modine 3.1 Empirical models for the optical functions associated with an amorphous semiconductor For the purposes of materials characterization and device design, empir- ical models for the optical functions of an amorphous semiconductor have been sought. Forouhi and Bloomer [20] and McGahan et al. [21] have pro- posed empirical models for such functions. Unfortunately, these models have a number of shortcomings. While experiment indicates that as E → ∞, 2(E) → 0, both the models of Forouhi and Bloomer [20] and McGahan et 53 3.1. Empirical models for the optical functions associated with an amorphous semiconductor al. [21] yield non-zero values for 2(E) as E → ∞. The models of Forouhi and Bloomer [20] and McGahan et al. [21] have the additional problem of being obtained using unphysical expressions for 2(E). In particular, the pro- posed models for 2(E) do not terminate exact at the energy gap. Finally, both the models of Forouhi and Bloomer [20] and McGahan et al. [21] fail to satisfy the time-symmetry requirements mandated of the Kramers-Kronig relations; recall Eqs. (2.47a) and (2.47b). In 1996, Jellison and Modine [15] developed a new parameterized model for the optical functions associated with an amorphous semiconductor that remedies some of the problems found with these previous models. In this approach, a physically-based model for the imaginary part of the dielectric function, 2(E), is proposed. This model includes a Tauc JDOS function. i.e., Eq. (2.25), and a Lorentzian momentum matrix element. Then, through the use of a Kramers-Kronig transformation on 2(E), i.e., Eq. (2.45), the spectral dependence of the real part of the dielectric function, 1(E), is de- termined. This chapter is organized in the following manner. In Section 3.2, the Lorentz model is described. In Section 3.3, Jellison and Modine’s model for the spectral dependence of the real and imaginary components of the 54 3.2. The Lorentz model dielectric function, 1(E) and 2(E), respectively, is presented [15]. Then, in Section 3.4, an explicit determination of the spectral dependence of 1(E) from the use of a Kramers-Kronig transformation on 2(E), i.e., Eq. (2.45), is provided. Finally, a fit with the results of experiment is presented in Section 3.5. 3.2 The Lorentz model According to the Lorentz model, an atom with electrons bounded by the nucleus may be considered in the same way as a small mass bounded to a large mass by a spring. The motion of an electron bound to the nucleus may be described by m d2r dt2 +mC dr dt +mω20r = −qEloc, (3.1) where m denotes the electron mass, q represents the electron charge, and Eloc is the driving local electric field acting on the electron. It may be shown that r̃ = −eEloc m 1 (ω20 − ω2)− iCω , (3.2) 55 3.2. The Lorentz model and the induced dipole moment is p̃ = e2Eloc m 1 (ω20 − ω2)− iCω . (3.3) There is a linear relationship between p̃ and Eloc if the displacement r is assumed to be sufficiently small. That is, p̃ = α̃(ω)Eloc, (3.4) where α̃(ω) is the frequency-dependent atomic polarizability. From Eqs. (3.3) and (3.4), it may be shown that α̃(ω) = e2 m 1 (ω20 − ω2)− iCω . (3.5) The complex dielectric function, ̃, may be expressed in terms of the complex polarizability, α̃, by ̃ = 1 + 4piNα̃, (3.6) where N denotes the number of atoms per unit volume. From Eqs. (3.5) and (3.6), it may be shown that ̃ = 1 + 4piNe2 m 1 (ω20 − ω2)− iCω . (3.7) Since the complex dielectric function, ̃, may be expressed as the sum over its real and imaginary components, i.e., ̃ = 1 + i2, (3.8) 56 3.3. Model for the optical functions where 1 and 2 denote the real and imaginary components of the dielectric function, respectively, it can be seen that 1 = 1 + 4piNe2 m ω20 − ω2 (ω20 − ω2)2 + C2ω2 , (3.9a) and 2 = 4piNe2 m Cω (ω20 − ω2)2 + C2ω2 , (3.9b) Jellison and Modine [15] used this simple Lorentz model in their proposed parameterized model for the spectral dependence of the imaginary part of the dielectric function, 2, shown in the subsequent analysis. 3.3 Model for the optical functions Within the framework of the momentum matrix element formulation, i.e., Eq. (2.1), Jellison and Modine [15] propose a Tauc-Lorentz expression for the imaginary part of the dielectric function, 2(E), i.e., 2(E) is essentially proportional to the product of a Tauc JDOS function and a Lorentz oscillator. Specifically, Jellison and Modine [15] suggest that 2TL(E) =  0, E ≤ Eg AEoCE (E2 − E2o)2 + C2E2 (E − Eg)2 E2 , E > Eg , (3.10) 57 3.3. Model for the optical functions where the four modeling parameters, Eo, C, Eg, and A, are in units of energy. Here, Eo denotes the resonance energy of a bound electron at which the transition of the electron occurs between two atomic states, A represents the oscillator strength, C is the breadth of the region of anomalous dispersion, and Eg is the energy gap. Jellison and Modine [15] determine the real part of the dielectric function, 1(E), using a Kramers-Kronig transformation on 2(E). That is, 1(E) = 1∞ + 2 pi P ∫ ∞ Eg u2 (u) u2 − E2 du, (3.11) where P represents the Cauchy principal part of the integral. An additional fitting parameter, 1∞, is included in order to obtain the desired fits; in the actual Kramers-Kronig relations, 1∞ is unity. This integral may be solved, in closed-form, to yield 1(E) = 1∞ + 1 2 A pi C ζ4 aln αEo ln [ E2g + E 2 o + αEg E2g + E 2 o − αEg ] − A piζ4 aa tan Eo [ pi − tan−1 ( 2Eg + α C ) + tan−1 (−2Eg + α C )] +2 AEo piαζ4 Eg [ E2 − γ2] [pi + 2 tan−1(2(γ2 − E2g ) αC )] −AEoC piζ4 E2 + E2g E ln |Eg − E| (Eg + E) + 2AEoC piζ4 Eg ln  [(Eg + E)|Eg − E|]√ (E2o − E2g )2 + E2gC2)  , (3.12) 58 3.3. Model for the optical functions where aln = (E 2 g − E2o)E2 + E2gC2 − E2o(E2o + 3E2g ), (3.13a) aa tan = (E 2 − E2o)(E2o + E2g ) + E2gC2, (3.13b) ζ4 = [(E2 − E2o)2 + C2E2], (3.13c) χ = √ 4E2o − C2, (3.13d) and γ = √ E2o − C2 2 , (3.13e) Eo, C, Eg, and A being as defined earlier. The two closed-form expressions for the optical functions, 1(E) and 2(E), respectively, i.e., Eqs. (3.10) and (3.12), provide a Kramers-Kronig consistent model for the optical functions of an amorphous semiconductor. This model avoids many of the problems associated with the models of For- oughi and Bloomer [20] and McGahan et al. [21]. That is, 2(E) approaches zero as E → ∞, the model for 2(E) is physically based, and finally, these optical function expressions obey the time-symmetry requirements mandated of the Kramers-Kronig relations, i.e., Eqs. (2.47a) and (2.47b). Jellison and Modine [15] then employed this model in order to fit a variety of experimen- tal optical function data for the spectral dependence of 1(E) and 2(E), and 59 3.4. Derivation drew a number of conclusions into the character of the materials considered. 3.4 Derivation 3.4.1 Definitions From Eqs. (3.10) and (3.11), it is seen that 1(E) = 1∞ + 2AEoC pi P ∫ ∞ Eg (u− Eg)2 (u2 − E2)[(u2 − E2o)2 + C2u2] du.(3.14) Letting f(u) = (u− Eg)2 (u2 − E2)[(u2 − E2o)2 + C2u2] , (3.15) the analysis starts with a partial fraction expansion of this polynomial ex- pression in terms of the roots of its denominator. As the denominator is a 6th-order polynomial, it has six roots, two of which may be seen imme- diately, from inspection, to be E and −E, respectively. Accordingly, the corresponding partial fraction expansion may be expressed as f(u) = A1 u+ E + A2 u− E + A3 u− r1 + A4 u− r2 + A5 u− r3 + A6 u− r4 ,(3.16) where A1, A2, A3, A4, A5, and A6 are the co-efficients of the partial fraction expansions, r1, r2, r3, and r4 being the other four, as yet unknown, roots of the denominator of f(u). 60 3.4. Derivation 3.4.2 Root calculation through expansion of the denominator The roots of f(u) may be determined by setting (u2 − E2o)2 + C2u2 = 0. (3.17) Expanding Eq. (3.17) fully, one can see that u4 − u2(2E2o − C2) + E4o = 0. Factoring the u2 term, this equation reduces to u2 = 2E2o − C2 ± √ (2E2o − C2)2 − 4E4o 2 , which leads to u2 = E2o − C2 2 ± iC 2 √ 4E2o − C2. Noting that u2 = (√ 4E2o − C2 2 ± iC 2 )2 , it is seen that u = ± (√ 4E2o − C2 2 ± iC 2 ) . 61 3.4. Derivation Letting a = √ 4E2o − C2 2 , and b = C 2 , it is seen that the four roots of f(u) are r1 = a+ ib, (3.18a) r2 = a− ib, (3.18b) r3 = −a+ ib, (3.18c) and r4 = −a− ib. (3.18d) Through inspection, the following relationships are seen amongst these roots, i.e., r2 = r ∗ 1, r3 = −r∗1, r4 = −r1, r1 = −r∗3, r2 = −r3, 62 3.4. Derivation and r4 = r ∗ 3. 3.4.3 Calculation of the coefficients The coefficients, A1, A2, A3, A4, A5, and A6, may be determined using the usual approach for a partial fraction expansion. That is, for the partial fraction expansion, f(z) = ∑ i Ai z − zi , (3.19) if all the roots, zi, are distinct, then it may be seen that Ai = (z − zi)f(z)|z=zi . (3.20) Noting that all of the roots associated with the denominator of f(u) are distinct, employing the approach suggested in Eq. (3.20), the coefficients A1, A2, A3, A4, A5, and A6 will be determined. For the case of these coefficients, a polar notation is employed, i.e., the imaginary number p+ iq = √ p2 + q2 exp ( i tan−1 [ q p ]) , (3.21) may instead be expressed as p+ iq = √ p2 + q2∠ tan−1 [ q p ] , (3.22) 63 3.4. Derivation where this last expression corresponds to polar notation. That is, every complex number is expressed in this particular polar form. An alternate form for this polar notation abs (p+ iq) = √ p2 + q2, (3.23) and arg (p+ iq) = tan−1 [ q p ] , (3.24) may be employed. Determination of A1 A1 = f(u)(u+ E)|u=−E, = (−E − Eg)2 (−E − E)[(E2 − E2o)2 + C2E2] , = − (E + Eg) 2 2E[(E2 − E2o)2 + C2E2] . (3.25) Determination of A2 A2 = f(u)(u− E)|u=E, = (E − Eg)2 (E + E)[(E2 − E2o)2 + C2E2] , = (E − Eg)2 2E[(E2 − E2o)2 + C2E2] . (3.26) 64 3.4. Derivation Determination of A3 A3 = f(u)(u− r1)|u=r1 , = (r1 − Eg)2 (r21 − E2)(r1 − r2)(r1 − r3)(r1 − r4) , = (r1 − Eg)2 (r21 − E2)(r1 − r∗1)(r1 + r∗1)(r1 + r1) , = (r1 − Eg)2 (r21 − E2)(r1 − r∗1)(r1 + r∗1)2r1 . (3.27) Now, by substituting r1 = a + ib, the factors in the numerator and the denominator of Eq. (3.27) may be determined and expressed in polar form. For the numerator, this yields (r1 − Eg)2 = (a+ ib− Eg)2, = (a− Eg)2 − b2 + i2b(a− Eg), = √ ((a− Eg)2 − b2)2 + 4b2(a− Eg)2∠ tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] , = [ (a− Eg)2 + b2 ] ∠ tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] . The polar form for the other factors in the denominator of Eq. (3.27) may now be expressed as r21 − E2 = √ (a2 − b2 − E2)2 + 4a2b2∠ tan−1 [ 2ab a2 − b2 − E2 ] , 2(r1 − r∗1)(r1 + r∗1) = 8ab∠ [pi 2 ] , and 65 3.4. Derivation r1 = √ a2 + b2∠tan−1 [ b a ] . So, the coefficient, A3, may be expressed in polar form as A3 = [(a− Eg)2 + b2]∠ [ tan−1 2b(a−Eg) (a−Eg)2−b2 ] √ (a2 − b2 − E2)2 + 4a2b2∠ [tan−1 2ab a2−b2−E2 ] × 1√ a2 + b2∠ [ tan−1 b a ] [ 8ab∠pi 2 ] , = [(a− Eg)2 + b2] 8ab[ √ (a2 − b2 − E2)2 + 4a2b2][√a2 + b2] ×∠ [ tan−1 2b(a− Eg) (a− Eg)2 − b2 − tan −1 2ab a2 − b2 − E2 − tan −1 b a − pi 2 ] . Thus, it is seen that abs(A3) = [(a− Eg)2 + b2] 8ab[ √ (a2 − b2 − E2)2 + 4a2b2][√a2 + b2] , (3.28a) and that arg(A3) = −pi 2 + tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] − tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b a ] . (3.28b) 66 3.4. Derivation Determination of A4 A4 = f(u)(u− r2)|u=r2 , = (r2 − Eg)2 (r22 − E2)(r2 − r1)(r2 − r3)(r2 − r4) , = (r∗1 − Eg)2 (r∗21 − E2)(r∗1 − r1)(r1 + r∗1)(r∗1 + r∗1) , = (r∗1 − Eg)2 2r∗1(r ∗2 1 − E2)(r∗1 − r1)(r1 + r∗1) . (3.29) Now, by substituting r∗1 = a − ib, the factors in the numerator and the denominator of Eq. (3.29) may be determined and expressed in polar form. For the numerator, this yields (r∗1 − Eg)2 = (a− ib− Eg)2, = √ ((a− Eg)2 − b2)2 + 4b2(a− Eg)2∠− tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] , = [(a− Eg)2 + b2]∠− tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] . The polar form for the other factors in the denominator of Eq. (3.29) may be expressed as r∗21 − E2 = √ (a2 − b2 − E2)2 + 4a2b2∠− tan−1 [ 2ab a2 − b2 − E2 ] , 2(r∗1 − r1)(r1 + r∗1) = 8ab∠ [−pi 2 ] , and r∗1 = √ a2 + b2∠− tan−1 [ b a ] . 67 3.4. Derivation Thus, abs(A4) = [(a− Eg)2 + b2] 8ab[ √ (a2 − b2 − E2)2 + 4a2b2][√a2 + b2] , (3.30a) and arg(A4) = pi 2 − tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] + tan−1 [ 2ab a2 − b2 − E2 ] + tan−1 [ b a ] . (3.30b) Through inspection, it is seen that abs(A4) = abs(A3) and that arg(A4) = − arg(A3), i.e., A3 = A ∗ 4. Determination of A5 A5 = f(u)(u− r3)|u=r3 , = (r3 − Eg)2 (r23 − E2)(r3 − r1)(r3 − r2)(r3 − r4) , = (r3 − Eg)2 (r23 − E2)(r3 − r∗3)(r3 + r∗3)(r3 + r3) , = (r3 − Eg)2 (r23 − E2)(r3 − r∗3)(r3 + r∗3)2r3 . (3.31) For this equation, for the numerator, (r3 − Eg)2 = (−a+ ib− Eg)2, = √ ((a+ Eg)2 − b2)2 + 4b2(a+ Eg)2∠− tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] , = [(a+ Eg) 2 + b2]∠− tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] . 68 3.4. Derivation Similarly, for the denominator, r23 − E2 = √ (a2 − b2 − E2)2 + 4a2b2∠− tan−1 [ 2ab a2 − b2 − E2 ] , 2(r3 − r∗3)(r3 + r∗3) = 8ab∠ [−pi 2 ] , and r3 = √ a2 + b2∠ tan−1 [ b −a ] . Thus, abs(A5) = [(a+ Eg) 2 + b2] 8ab[ √ (a2 − b2 − E2)2 + 4a2b2][√a2 + b2] , (3.32a) and arg(A5) = pi 2 − tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] + tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b −a ] . (3.32b) Determination of A6 A6 = f(u)(u− r4)|u=r4 , = (r4 − Eg)2 (r24 − E2)(r4 − r1)(r4 − r2)(r4 − r3) , = (r∗3 − Eg)2 (r∗23 − E2)(r∗3 − r3)(r∗3 + r∗3)(r∗3 + r3) , = (r∗3 − Eg)2 (r∗23 − E2)(r∗3 − r3)(r3 + r∗3)2r∗3 . (3.33) 69 3.4. Derivation Here, the numerator of Eq. (3.33) yields (r∗3 − Eg)2 = (−a− ib− Eg)2, = [(a+ Eg) 2 + b2]∠ tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] . Similarly, the terms in the denominator of Eq. (3.33) yield r∗23 − E2 = √ (a2 − b2 − E2)2 + 4a2b2∠ tan−1 [ 2ab a2 − b2 − E2 ] , 2(r∗3 − r3)(r3 + r∗3) = 8ab∠ [pi 2 ] , and r∗3 = √ a2 + b2∠− tan−1 [ b −a ] . Thus, abs(A6) = [(a+ Eg) 2 + b2] 8ab[ √ (a2 − b2 − E2)2 + 4a2b2][√a2 + b2] , (3.34a) and arg(A6) = −pi 2 + tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] − tan−1 [ 2ab a2 − b2 − E2 ] + tan−1 [ b −a ] . (3.34b) Through inspection, it is also seen that abs(A5) = abs(A6) and that arg(A5) = − arg(A6), i.e., A5 = A ∗ 6. 70 3.4. Derivation 3.4.4 Grouping the terms in the partial fraction expression In Section 3.4.3, it was seen that A3 = A ∗ 4 and that A5 = A ∗ 6. These rela- tions may be used in order to group the terms associated with the coefficients in the partial fraction expansion. Grouping the terms associated with A3 and A4 The terms associated with A3 and A4 in the partial fraction expansion may be combined as follows: A3 u− r1 + A4 u− r2 = abs(A3)∠(arg(A3)) u− a− ib + abs(A4)∠arg(A4)) u− a+ ib , = abs(A3)∠(arg(A3))√ (u− a)2 + b2∠ [− tan−1 ( b u−a )] + abs(A3)∠− [arg(A3)]√ (u− a)2 + b2∠ [tan−1 ( b u−a )] , = abs(A3)√ (u− a)2 + b2∠ [ arg(A3) + tan −1 ( b u− a )] + abs(A3)√ (u− a)2 + b2∠− [ arg(A3) + tan −1 ( b u− a )] . 71 3.4. Derivation Letting, x = arg(A3) + tan −1 ( b u− a ) , and k = abs(A3)√ (u− a)2 + b2 , it may be seen that A3 u− r1 + A4 u− r2 = k∠x+ k∠− x, = k(eix + e−ix), = 2k cosx, = 2 abs(A3)√ (u− a)2 + b2 cos [ arg(A3) + tan −1 ( b u− a )] , = 2 abs(A3)√ (u− a)2 + b2 cos [arg(A3)] u− a√ (u− a)2 + b2 −2 abs(A3)√ (u− a)2 + b2 sin [arg(A3)] b√ (u− a)2 + b2 , = 2abs(A3) cos [arg(A3)] u− a (u− a)2 + b2 −2abs(A3) sin [arg(A3)] b (u− a)2 + b2 . 72 3.4. Derivation Letting c3 = 2 abs(A3) cos[arg(A3)], and c4 = −2 abs(A3) sin[arg(A3)], it is also seen that, A3 u− r1 + A4 u− r2 = c3 u− a (u− a)2 + b2 + c4 b (u− a)2 + b2 . (3.35) 73 3.4. Derivation Grouping the terms associated with A5 and A6 The terms associated with A5 and A6 in the partial fraction expansion may be combined as follows: A5 u− r3 + A6 u− r4 = abs(A5)∠ [arg(A5)] u+ a− ib + abs(A5)∠− [arg(A5)] u+ a+ ib , = abs(A5)∠ [arg(A5)]√ (u+ a)2 + b2∠ [− tan−1 ( b u+a )] + abs(A5)∠− [arg(A5)]√ (u− a)2 + b2∠ [tan−1 ( b u+a )] , = abs(A5)√ (u+ a)2 + b2 ∠ [ arg(A5) + tan −1 ( b u+ a )] + abs(A5)√ (u+ a)2 + b2 ∠− [ arg(A5) + tan −1 ( b u+ a )] , = 2 abs(A5)√ (u+ a)2 + b2 cos [ arg(A5) + tan −1 ( b u+ a )] , = 2 abs(A5)√ (u+ a)2 + b2 cos [arg(A5)] u+ a√ (u+ a)2 + b2 −2 abs(A5)√ (u+ a)2 + b2 sin [arg(A5)] b√ (u+ a)2 + b2 , = 2abs(A5) cos [arg(A5)] u+ a (u+ a)2 + b2 −2abs(A5) sin[arg(A5)] b (u+ a)2 + b2 . Letting c5 = 2 abs(A5) cos[arg(A5)], and 74 3.4. Derivation c6 = −2 abs(A5) sin[arg(A5)]. it is seen that, A5 u− r3 + A6 u− r4 = c5 u+ a (u+ a)2 + b2 + c6 b (u+ a)2 + b2 . (3.36) 3.4.5 Summary of the partial fraction expansion From Eqs. (3.16), (3.35), and (3.36), it is thus seen that f(u) = A1 u+ E + A2 u− E + A3 u− r1 + A4 u− r2 + A5 u− r3 + A6 u− r4 , = A1 u+ E + A2 u− E + c3 u− a (u− a)2 + b2 + c4 b (u− a)2 + b2 +c5 u+ a (u+ a)2 + b2 + c6 b (u+ a)2 + b2 , (3.37) where A1 = −(E + Eg)2 2E[(E2 − E2o)2 + C2E2] , (3.38a) A2 = (E − Eg)2 2E[(E2 − E2o)2 + C2E2] , (3.38b) c3 = 2 abs(A3) cos[arg(A3)], (3.38c) c4 = −2 abs(A3) sin[arg(A3)], (3.38d) c5 = 2 abs(A5) cos[arg(A5)], (3.38e) 75 3.4. Derivation and c6 = −2 abs(A5) sin[arg(A5)]. (3.38f) 3.4.6 Evaluation of the constants c3, c4, c5, and c6 Calculation of c3 From Eq. (3.38c), it is seen that c3 = 2abs(A3) cos[arg(A3)], (3.39) where, from Eq. (3.28b), arg(A3) = −pi 2 + tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] − tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b a ] . Letting x = tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] , y = tan−1 [ −2ab a2 − b2 − E2 ] , and z = tan−1 [−b a ] , 76 3.4. Derivation it is seen that arg(A3) = −pi 2 + x+ y + z. Noting that cos[arg(A3)] = cos(−pi 2 + x+ y + z) = sin(x+ y + z), it may be seen that cos[arg(A3)] = cosx cos y sin z − sinx sin y sin z + sinx cos y cos z + cosx sin y cos z. Letting x = tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] , one observes that cosx = (a− Eg)2 − b2 (a− Eg)2 + b2 , and sinx = 2b(a− Eg) (a− Eg)2 + b2 . Similarly, letting y = tan−1 [ −2ab a2 − b2 − E2 ] , 77 3.4. Derivation one observes that cos y = a2 − b2 − E2√ (a2 − b2 − E2)2 + (2ab)2 , and sin y = −2ab√ (a2 − b2 − E2)2 + (2ab)2 . Letting z = tan−1 [−b a ] , it is seen that cos z = a√ (a2 + b2 , and sin z = −b√ (a2 + b2 . Substituting the results for abs(A3), as well as the cosine and sine of x, y, and z, respectively, into Eq. (3.39) yields c3 = 2 abs(A3) cos[arg(A3)], 78 3.4. Derivation c3 = 2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × −b× [((a− Eg)2 − b2)(a2 − b2 − E2)− 2b(a− Eg)(−2ab)] + a× [2b(a− Eg)(a2 − b2 − E2) + ((a− Eg)2 − b2)(−2ab)]  , = 2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2b(a− Eg)[a(a2 − b2 − E2)− 2ab2] − ((a− Eg)2 − b2)[b(a2 − b2 − E2) + 2a2b]  . Calculation of c4 From Eq. (3.38d), it is seen that c4 = −2 abs(A3) sin[arg(A3)], (3.40) where, from Eq. (3.28b), arg(A3) = −pi 2 + tan−1 [ 2b(a− Eg) (a− Eg)2 − b2 ] − tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b a ] . 79 3.4. Derivation Similarly, it is seen that sin[arg(A3)] = sin(−pi 2 + x+ y + z) = − cos(x+ y + z) = −[cosx cos y cos z − sinx sin y cos z − sinx cos y sin z − cosx sin y sin z]. Thus, c4 = −2 abs(A3) sin[arg(A3)], = 2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × a× [((a− Eg)2 − b2)(a2 − b2 − E2)− 2b(a− Eg)(−2ab)] − b× [−2b(a− Eg)(a2 − b2 − E2)− ((a− Eg)2 − b2)(−2ab)]  , = 1 4ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2b(a− Eg)[b(a2 − b2 − E2) + 2a2b] + +((a− Eg)2 − b2)[a(a2 − b2 − E2)− 2ab2]  . 80 3.4. Derivation Calculation of c5 From Eq. (3.38e), c5 = 2 abs(A5) cos[arg(A5)], (3.41) where, from Eq. (3.32b), arg(A5) = pi 2 − tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] + tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b −a ] . Letting x = tan−1 [ −2b(a+ Eg) (a+ Eg)2 − b2 ] , y = tan−1 [ 2ab a2 − b2 − E2 ] , and z = tan−1 [−b −a ] , it is seen that arg(A5) = pi 2 + x+ y + z. Noting that cos [arg(A5)] = cos( pi 2 + x+ y + z) = − sin(x+ y + z), 81 3.4. Derivation it may also be seen that cos [arg(A5)] = −[cosx cos y sin z − sinx sin y sin z + sinx cos y cos z + cosx sin y cos z]. Letting x = tan−1 [ −2b(a+ Eg) (a+ Eg)2 − b2 ] , one observes that cosx = (a+ Eg) 2 − b2 (a+ Eg)2 + b2 , and sinx = −2b(a+ Eg) (a+ Eg)2 + b2 . Similarly, letting y = tan−1 [ 2ab a2 − b2 − E2 ] , one observes that cos y = a2 − b2 − E2√ (a2 − b2 − E2)2 + (2ab)2 , and sin y = 2ab√ (a2 − b2 − E2)2 + (2ab)2 . 82 3.4. Derivation Letting z = tan−1 [−b −a ] , it is seen that cos z = −a√ (a2 + b2 , and sin z = −b√ (a2 + b2 . Substituting the results of abs(A5), as well as the cosine and sine of x, y, and z into Eq. (3.41), yields c5 = 2 abs(A5) cos[arg(A5)], = −2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × −b× [((a+ Eg)2 − b2)(a2 − b2 − E2)− (−2b(a+ Eg)) 2ab] − a× [−2b(a+ Eg)(a2 − b2 − E2) + ((a+ Eg)2 − b2) 2ab]  , = 2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × −2b(a+ Eg)[a(a2 − b2 − E2)− 2ab2] + ((a+ Eg) 2 − b2)[b(a2 − b2 − E2) + 2a2b]  . 83 3.4. Derivation Calculation of c6 From Eq. (3.38f), c6 = −2 abs(A5) sin[arg(A5)], (3.42) where, from Eq. (3.32b), arg(A5) = pi 2 − tan−1 [ 2b(a+ Eg) (a+ Eg)2 − b2 ] + tan−1 [ 2ab a2 − b2 − E2 ] − tan−1 [ b −a ] . Similarly, sin[arg(A5)] = sin( pi 2 + x+ y + z) = cos(x+ y + z), = cosx cos y cos z − sinx sin y cos z − sinx cos y sin z − cosx sin y sin z. Thus, it may be seen that c6 = −2 abs(A5) sin[arg(A5)], = −2 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × −a× [((a+ Eg)2 − b2)(a2 − b2 − E2)− (−2b(a+ Eg)) 2ab] − b× [−2b(a+ Eg)(a2 − b2 − E2)− ((a+ Eg)2 − b2) 2ab]  , 84 3.4. Derivation = 1 4ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2b(a+ Eg)[b(a 2 − b2 − E2) + 2a2b] + ((a+ Eg) 2 − b2)[a(a2 − b2 − E2)− 2ab2]  . 3.4.7 Main integration So, from Eq.(3.14), it is seen that 1(E) = 1∞ + 2AEoC pi P ∫ ∞ Eg (u− Eg)2 (u2 − E2)[(u2 − E2o)2 + C2u2] du, = 1∞ + 2AEoC pi P ∫ ∞ Eg f(u) du. (3.43) From the partial fraction expansion of Eq. (3.37), this yields ∫ ∞ Eg f(u) du = ∫ ∞ Eg A1 u+ E du+ ∫ ∞ Eg A2 u− E du + ∫ ∞ Eg c3 u− a (u− a)2 + b2 du + ∫ ∞ Eg c4 b (u− a)2 + b2 du + ∫ ∞ Eg c5 u+ a (u+ a)2 + b2 du + ∫ ∞ Eg c6 b (u+ a)2 + b2 du, 85 3.4. Derivation In order to avoid improper integrals, the integrals will all be taken to t, and then t will go to infinity, i.e., t→∞. Thus, it can be seen that =⇒ ∫ t Eg f(u) du = ∫ t Eg A1 u+ E du+ ∫ t Eg A2 u− E du + ∫ t Eg c3 u− a (u− a)2 + b2 du + ∫ t Eg c4 b (u− a)2 + b2 du + ∫ t Eg c5 u+ a (u+ a)2 + b2 du + ∫ t Eg c6 b (u+ a)2 + b2 du. (3.44) Sequentially integrating the terms in Eq. (3.44), it is thus seen that∫ t Eg A1 u+ E du = A1 ∫ t Eg 1 u+ E du, = A1 [ln(u+ E)] t Eg , = A1 [ln(t+ E)]− A1 ln(Eg + E), (3.45) and ∫ t Eg A2 u− E du = A2 [ lim →0 ∫ E− Eg 1 u− E du+ lim→0 ∫ t E+ 1 u− E du ] , = A2 [ lim →0 ln |u− E|E−Eg + lim→0 ln |u− E| t E+ ] , = A2 [ lim →0 ln | − | ] − A2 [ln |Eg − E|+ ln |t− E|] −A2 [ lim →0 ln || ] , = A2 [ln |t− E|]− A2 ln |Eg − E|. (3.46) 86 3.4. Derivation Now, for ∫ t Eg c3 u− a (u− a)2 + b2 du, letting w = (u− a)2 + b2, =⇒ dw = 2(u− a) du, it may be seen that ∫ t Eg c3 u− a (u− a)2 + b2 du = c3 2 ∫ (t−a)2+b2 (Eg−a)2+b2 1 w dw, (3.47) = c3 2 [ ln[(t− a)2 + b2]] −c3 2 ln[(Eg − a)2 + b2]. (3.48) Again, for ∫ t Eg c4 b (u− a)2 + b2 du, letting (u− a) = b tan θ, =⇒ du = b sec2 θ dθ, 87 3.4. Derivation it is seen that c4 ∫ t Eg b (u− a)2 + b2 du = c4 ∫ t Eg b sec2 θ b2 tan2 θ + b2 dθ, = c4 [θ] t Eg , = c4 [ tan−1 ( u− a b )]t Eg , = c4 [ pi 2 − tan−1 ( Eg − a b )] . [as t→∞] (3.49) Similarly, ∫ t Eg c5 u+ a (u+ a)2 + b2 du = c5 2 [ ln[(t+ a)2 + b2] ] −c5 2 ln[(Eg + a) 2 + b2], (3.50)∫ t Eg c6 b (u+ a)2 + b2 du = c6 [ pi 2 − tan−1 ( Eg + a b )] . (3.51) From Eq. (3.44), it is seen that ∫ t Eg f(u)du = A1 [ln(t+ E)]− A1 ln(Eg + E) +A2 [ln |t− E|]− A2 ln |Eg − E| 88 3.4. Derivation + c3 2 [ ln[(t− a)2 + b2]]− c3 2 ln[(Eg − a)2 + b2] +c4 [ pi 2 − tan−1 ( Eg − a b )] + c5 2 [ ln[(t+ a)2 + b2] ]− c5 2 ln[(Eg + a) 2 + b2] +c6 [ pi 2 − tan−1 ( Eg + a b )] . (3.52) Letting φ1 = ln(t+ E)]− ln(Eg + E), φ2 = ln |t− E| − ln |Eg − E|, φ3 = ln[(t− a)2 + b2]− ln[(Eg − a)2 + b2], φ4 = pi 2 − tan−1 ( Eg − a b ) , φ5 = ln[(t+ a) 2 + b2]− ln[(Eg + a)2 + b2], and φ6 = pi 2 − tan−1 ( Eg + a b ) , Eq. (3.52) leads to ∫ ∞ Eg f(u)du = A1φ1 + A2φ2 + c3 2 φ3 + c5 2 φ5 + c4φ4 + c6φ6. (3.53) 89 3.4. Derivation 3.4.8 Summation by parts For simplicity, the summation, depicted in Eq. (3.53), is performed in parts such that term 1 = A1φ1 + A2φ2, term 2 = c3 2 φ3 + c5 2 φ5, term 3 = c4φ4 + c6φ6. From Eq. (3.53), it is thus seen that ∫ ∞ Eg f(u)du = A1φ1 + A2φ2︸ ︷︷ ︸ term 1 + c3 2 φ3 + c5 2 φ5︸ ︷︷ ︸ term 2 + c4φ4 + c6φ6︸ ︷︷ ︸ term 3 . Calculation of term 1 term 1 = A1φ1 + A2φ2, = −(E + Eg)2 2E[(E2 − E2o)2 + C2E2] φ1 + (E − Eg)2 2E[(E2 − E2o)2 + C2E2] φ2, = E2 + E2g 2E[(E2 − E2o)2 + C2E2] [φ2 − φ1]− 2EEg 2E[(E2 − E2o)2 + C2E2] [φ2 + φ1], = E2 + E2g 2E[(E2 − E2o)2 + C2E2] [φ2 − φ1]− Eg [(E2 − E2o)2 + C2E2] [φ2 + φ1], = E2 + E2g 2Eζ4 [φ2 − φ1]− Eg ζ4 [φ2 + φ1], 90 3.4. Derivation where φ2 − φ1 = ln |t− E| − ln |Eg − E| − ln(t+ E)] + ln(Eg + E), = ln |t− E| t+ E + ln (Eg + E) |Eg − E| , = ln (Eg + E) |Eg − E| , [as t→∞] φ2 + φ1 = ln |t− E| − ln |Eg − E|+ ln(t+ E)]− ln(Eg + E), = ln[(t+ E)|t− E|]− ln[(Eg + E)|Eg − E|]. 91 3.4. Derivation Calculation of term 2: term 2 = c3 2 φ3 + c5 2 φ5, = 1 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2b(a− Eg)[a(a2 − b2 − E2)− 2ab2]φ3 − ((a− Eg)2 − b2)[b(a2 − b2 − E2) + 2a2b]φ3 + −2b(a+ Eg)[a(a2 − b2 − E2)− 2ab2]φ5 + ((a+ Eg) 2 − b2)[b(a2 − b2 − E2) + 2a2b]φ5  , = 1 8ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2ba[a(a2 − b2 − E2)− 2ab2][φ3 − φ5] − (a2 + E2g − b2)[b(a2 − b2 − E2) + 2a2b][φ3 − φ5] + −2bEg[a(a2 − b2 − E2)− 2ab2][φ3 + φ5] + 2aEg[b(a 2 − b2 − E2) + 2a2b][φ3 + φ5]  , 92 3.4. Derivation = 1 8a(E2o − C24 + C 2 4 )[(E2o − C24 − C 2 4 − E2)2 + 4(E2o − C24 )C 2 4 ] × 2a2(a2 − 3b2 − E2)[φ3 − φ5] − (a2 − b2 + E2g )(3a2 − b2 − E2)[φ3 − φ5] + [−2abEg(a2 − b2 − E2) + 2abEg 2b2][φ3 + φ5] + [2abEg(a 2 − b2 − E2) + 2abEg 2a2][φ3 + φ5]  , = 1 8a E2o [(E 2 − E2o)2 + C2E2] × [2(E2o − C2 4 )(E2o − C2 − E2)][φ3 − φ5] − [(E2o − C2 2 + E2g )(3E 2 o − C2 − E2)][φ3 − φ5]  + Eg 2[(E2 − E2o)2 + C2E2] [φ3 + φ5], = −E4o + E2gC2 + E2gE2 − E2oE2 − 3E2oE2g 8a E2o [(E 2 − E2o)2 + C2E2] [φ3 − φ5] + Eg 2[(E2 − E2o)2 + C2E2] [φ3 + φ5], = (E2g − E2o)E2 + E2gC2 − E2o(E2o + 3E2g ) 8a E2o [(E 2 − E2o)2 + C2E2] [φ3 − φ5] + Eg 2[(E2 − E2o)2 + C2E2] [φ3 + φ5], = aln 4χ E2o ζ 4 [φ3 − φ5] + Eg 2 ζ4 [φ3 + φ5], 93 3.4. Derivation where φ3 − φ5 = ln[(t− a)2 + b2]− ln[(Eg − a)2 + b2] − ln[(t+ a)2 + b2] + ln[(Eg + a)2 + b2], = ln (t− a)2 + b2 (t+ a)2 + b2 + ln [(Eg + a) 2 + b2] [(Eg − a)2 + b2] , = ln [E2g + a 2 + b2 + 2aEg] [E2g + a 2 + b2 − 2aEg] , [as t→∞] = ln [E2g + E 2 o + 2aEg] [E2g + E 2 o − 2aEg] , = ln [E2g + Eo 2 + χEg] [E2g + Eo 2 − χEg] , [recall that, χ = √ 4E2o − C2 = 2a] φ3 + φ5 = ln[(t− a)2 + b2]− ln[(Eg − a)2 + b2] + ln[(t+ a)2 + b2]− ln[(Eg + a)2 + b2], = ln[[(t− a)2 + b2][(t+ a)2 + b2]] − ln[[(Eg − a)2 + b2][(Eg + a)2 + b2]], = ln[[(t− a)2 + b2][(t+ a)2 + b2]] − ln[(E2g + E2o + χEg)(E2g + E2o − χEg)], = ln[[(t− a)2 + b2][(t+ a)2 + b2]]− ln[(E2o − E2g )2 + E2gC2)]. 94 3.4. Derivation 3.4.9 Calculation of term 3: term 3 = c4φ4 + c6φ6, = 1 4ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] × 2b(a− Eg)[b(a2 − b2 − E2) + 2a2b]φ4 + ((a− Eg)2 − b2)[a(a2 − b2 − E2)− 2ab2]φ4 + 2b(a+ Eg)[b(a 2 − b2 − E2) + 2a2b]φ6 + ((a+ Eg) 2 − b2)[a(a2 − b2 − E2)− 2ab2]φ6  , = 2b2a(3a2 − b2 − E2) + a(a2 + E2g − b2)(a2 − 3b2 − E2) 4ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] [φ4 + φ6] + 2b2Eg(3a 2 − b2 − E2) + 2a2Eg(a2 − 3b2 − E2) 4ab(a2 + b2)[(a2 − b2 − E2)2 + 4a2b2] [φ6 − φ4], = 2b2(3a2 − b2 − E2) + (a2 + E2g − b2)(a2 − 3b2 − E2) 2C E2o [(E 2 − E2o)2 + C2E2] [φ4 + φ6] + 2Eg[b 2(3a2 − b2 − E2) + a2(a2 − 3b2 − E2) χ C E2o [(E 2 − E2o)2 + C2E2] [φ6 − φ4], = C2 2 (3E2o − C2 − E2) + (E2o − C 2 2 + E2g )(E 2 o − C2 − E2) 2C E2o [(E 2 − E2o)2 + C2E2] [φ4 + φ6] + 2Eg[ C2 4 (3E2o − C2 − E2) + (E2o − C 2 4 )(E2o − C2 − E2)] χ C E2o [(E 2 − E2o)2 + C2E2] [φ6 − φ4], 95 3.4. Derivation = E4o − E2E2o + E2oE2g − E2gE2 − E2gC2 2C E2o [(E 2 − E2o)2 + C2E2] [φ4 + φ6] + 2Eg[−E2oC22 + E4o − E2oE2] χ C E2o [(E 2 − E2o)2 + C2E2] [φ6 − φ4], = −[(E2 − E2o)(E2o + E2g ) + E2gC2] 2C E2o [(E 2 − E2o)2 + C2E2] [φ4 + φ6] + 2Eg[E 2 − E2o + C 2 2 ] χ C [(E2 − E2o)2 + C2E2] [φ4 − φ6], = −aa tan 2C E2o ζ 4 [φ4 + φ6] + 2Eg[E 2 − γ2] χCζ4 [φ4 − φ6]. where φ4 + φ6 = pi 2 − tan−1 ( Eg − a b ) + pi 2 − tan−1 ( Eg + a b ) , = pi − tan−1 ( Eg + a b ) + tan−1 (−Eg + a b ) , = pi − tan−1 ( 2Eg + 2a C ) + tan−1 (−2Eg + 2a C ) , = pi − tan−1 ( 2Eg + χ C ) + tan−1 (−2Eg + χ C ) , φ4 − φ6, = pi 2 − tan−1 ( Eg − a b ) − pi 2 + tan−1 ( Eg + a b ) , = tan−1 ( 2ab b2 − a2 + E2g ) , = tan−1 ( χC −2(a2 − b2 − E2g ) ) , = pi 2 + tan−1 ( 2(a2 − b2 − E2g ) χC ) , [ recall that, tan−1 (y x ) = pi 2 + tan−1 (−x y )] 96 3.4. Derivation = pi 2 + tan−1 ( 2(E2o − C 2 2 − E2g ) χC ) , = pi 2 + tan−1 ( 2(γ2 − E2g ) χC ) . 3.4.10 Summation of term 1, term 2 and term 3 term 1+term 2+term 3 = E2 + E2g 2Eζ4 [φ2 − φ1]− Eg ζ4 [φ2 + φ1] + aln 4χE2oζ 4 [φ3 − φ5] + Eg 2.ζ4 [φ3 + φ5] + −aa tan 2CE2oζ 4 [φ4 + φ6] + 2Eg[E 2 − γ2] χCζ4 [φ4 − φ6], = E2 + E2g 2Eζ4 ln [ (Eg + E) |Eg − E| ] − Eg ζ4 ln [(t+ E)|t− E|] + Eg ζ4 ln[(Eg + E)|Eg − E|] + aln 8aE2oζ 4 ln [ [E2g + E 2 o + χEg] [E2g + E 2 o − χEg] ] + Eg 2ζ4 ln[[(t− a)2 + b2][(t+ a)2 + b2]] − Eg 2ζ4 ln[(E2o − E2g )2 + E2gC2)] − aa tan 2CE2oζ 4 [ pi − tan−1 ( 2Eg + χ C ) + tan−1 (−2Eg + χ C )] + 2Eg[E 2 − γ2] χCζ4 [ pi 2 + tan−1 ( 2(γ2 − E2g ) χC )] , [here t→∞] 97 3.4. Derivation = aln 4χE2oζ 4 ln [ E2g + E 2 o + χEg E2g + E 2 o − χEg ] − aa tan 2CE2oζ 4 [ pi − tan−1 ( 2Eg + χ C ) + tan−1 (−2Eg + χ C )] + Eg[E 2 − γ2] χCζ4 [ pi + 2 tan−1 ( 2(γ2 − E2g ) χC )] −E 2 + E2g 2Eζ4 ln [ |Eg − E| (Eg + E) ] + Eg ζ4 ln  [(Eg + E)|Eg − E|]√ (E2o − E2g )2 + E2gC2)  . 3.4.11 Final Tauc-Lorentz expression for the model of Jellison and Modine Now, from Eq. (3.43), 1(E) = 1∞ + 2AEoC pi P ∫ ∞ Eg f(u)du, = 1∞ + 2AEoC pi [term 1 + term 2 + term 3], = 1∞ + 2AEoC pi aln 4χE2oζ 4 ln [ E2g + E 2 o + χEg E2g + E 2 o − χEg ] −2AEoC pi aa tan 2CE2oζ 4 [ pi − tan−1 ( 2Eg + χ C ) + tan−1 (−2Eg + χ C )] + 2AEoC pi Eg[E 2 − γ2] χCζ4 [ pi + 2 tan−1 ( 2(γ2 − E2g ) χC )] −2AEoC pi E2 + E2g 2Eζ4 ln [ |Eg − E| (Eg + E) ] + 2AEoC pi Eg ζ4 ln  [(Eg + E)|Eg − E|]√ (E2o − E2g )2 + E2gC2)  . 98 3.5. Comparing the results of the Jellison and Modine model with experimental data =⇒ 1(E) = 1∞ + 1 2 A pi C ζ4 aln χEo ln [ E2g + E 2 o + χEg E2g + E 2 o − χEg ] − A piζ4 aa tan Eo [ pi − tan−1 ( 2Eg + χ C ) + tan−1 (−2Eg + χ C )] +2 AEo piχζ4 Eg[E 2 − γ2] [ pi + 2 tan−1 ( 2(γ2 − E2g ) χC )] −AEoC piζ4 E2 + E2g E ln [ |Eg − E| (Eg + E) ] + 2AEoC piζ4 Eg ln  [(Eg + E)|Eg − E|]√ (E2o − E2g )2 + E2gC2)  , this being the same result as that in Eq. (3.12), all terms being defined in the same manner. 3.5 Comparing the results of the Jellison and Modine model with experimental data The parameterized model for the optical functions, 1(E) and 2(E), of Jellison and Modine [15] is now fit with the results of experiment. The exper- imental results that are considered include that of Piller [30], Synowicki [31], and Ferluato et al. [32], these experimental results corresponding to the EGE a-Si, PECVD a-Si I, and PECVD a-Si II experimental data sets mentioned 99 3.5. Comparing the results of the Jellison and Modine model with experimental data in Section 2.10, respectively; these experimental data sets are further de- scribed in Section 2.10. The approach that is adopted for each experimental fit involves the selection of the modeling parameters that “best-fits” the cor- responding experimental data set. This is achieved through a selection of modeling parameters, a determination of the corresponding spectral depen- dencies of the optical functions, 1(E) and 2(E), and a comparison with the results of experiments. The “best-fit” is obtained through visual inspec- tion. Through a sweep over a range of parameter values, the parameter selections that “best-fit” the corresponding experimental optical functions is determined. It is noted that, in all cases, the results of the model of Jellison and Modine [15] agree with that of experiment over most of the spectrum considered. The “best-fit” parameter values, corresponding to the different experimental fits, are tabulated in Table 3.1. The resultant fits, and the corresponding experimental data sets, are depicted in Figures 3.1, 3.2, and 3.3, these corresponding to the experimental results of EGE a-Si, PECVD a-Si I, and PECVD a-Si II, respectively. 100 3.5. Comparing the results of the Jellison and Modine model with experimental data Table 3.1: The “best-fit” a-Si modeling parameter selections employed for the Tauc-Lorentz model of Jellison and Modine [15] for fits to experimental data corresponding to EGE a-Si, PECVD a-Si I, and PECVD a-Si II. parameter (units) EGE a-Si [30] PECVD a-Si I [31] PECVD a-Si II [32] A (eV) 125 196 203 Eo (eV) 3.43 3.64 3.72 C (eV) 2.54 2.3 2.1 Eg (eV) 1.21 1.51 1.68 1∞ 1.0 1.0 0.65 101 3.5. Comparing the results of the Jellison and Modine model with experimental data 1 2 3 4 5−10 0 10 20 30 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 EGE a−Si [30] Figure 3.1: A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the EGE a-Si expermental data set [30]. These experimental results are from Piller [30]. The fits to experiment are shown with the solid lines. The experimental data, EGE a-Si [30], is depicted with the solid points. 102 3.5. Comparing the results of the Jellison and Modine model with experimental data 1 2 3 4 5−10 0 10 20 30 40 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 PECVD a−Si I [31] Figure 3.2: A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the PECVD a-Si I experimental data set [31]. These experimental results are from Synowicki [31]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si I [31], is depicted with the solid points. 103 3.5. Comparing the results of the Jellison and Modine model with experimental data 1 2 3 4 5−10 0 10 20 30 40 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 PECVD a−Si II [32] Figure 3.3: A “best-fit” of the Jellison and Modine model with the results of experiment corresponding to the PECVD a-Si II experimental data set [32]. These experimental results are from Ferluato et al. [32]. The fits to exper- iment are shown with the solid lines. The experimental data, PECVD a-Si II [32], is depicted with the solid points. 104 Chapter 4 A physically based Kramers-Kronig consistent model for the optical functions associated with amorphous semiconductors 4.1 Evaluating the JDOS function from the distributions of electronic states One weakness associated with the models of Forouhi and Bloomer [20], McGahan et al. [21], and Jellison and Modine [15], is that they do not pro- 105 4.1. Evaluating the JDOS function from the distributions of electronic states vide for a direct connection with the underlying distributions of electronic states. Even the model of Jellison and Modine [15], while being considerably improved over that of Forouhi and Bloomer [20] and McGahan et al. [21], fails to provide for a direct relationship between the underlying valence band and conduction band DOS functions, Nv(E) and Nc(E), respectively, with the spectral dependence of the optical functions, 1(E) and 2(E); Jellison and Modine [15] take a physically based model for the spectral dependence of the imaginary part of the dielectric function, 2(E), and employ a Kramers- Kronig transformation in order to obtain the spectral dependence of the real part of the dielectric function, 1(E). A direct relationship between the func- tional dependencies of these optical functions, 1(E) and 2(E), with the form of the distributions of electronic states, would allow for the gleaning of insights into the underlying electronic properties from experimental mea- surements of the spectral dependence of these optical functions. In this chapter, an empirical model for the valence band and conduction band DOS functions, Nv(E) andNc(E), is proposed, this model capturing the fundamental essence of an amorphous semiconductor, i.e., exponential distri- butions of tail states and square-root distributions of band states. Then, the corresponding joint density of states (JDOS) function is evaluated through 106 4.2. Empirical model for the DOS functions the use of Eq. (2.10). Through the application of a model for the aggregate optical transition matrix element, the corresponding spectral dependence of the imaginary part of the dielectric function, 2(E), is determined. A Kramers-Kronig transformation is then employed in order to determine the corresponding spectral dependence of the real part of the dielectric function, 1(E). A comparison with the results of experiment is then shown. This chapter is organized in the following manner. In Section 4.2, an empirical model for the DOS functions is presented. Then, in Section 4.3, the corresponding JDOS function, J(E), is determined. In Section 4.4, a model for the spectral dependence of the dipole optical matrix element is presented. Finally, in Section 4.5, a fit with the results of experiment, for the spectral dependence of real and imaginary parts of the dielectric function, 1(E) and 2(E), respectively, is shown. Finally, the implications of a non- unity 1∞ are described in Section 4.6. 4.2 Empirical model for the DOS functions There is general consensus that the DOS functions associated with an amorphous semiconductor, i.e., Nv(E) and Nc(E), exhibit square-root func- 107 4.2. Empirical model for the DOS functions tional dependencies in the band regions and exponential functional depen- dencies in the tail regions. Following O’Leary [29], one can thus set Nv (E) = Nvo  √ Ev − EvT exp ( EvT − Ev γv ) exp ( Ev − E γv ) , E > EvT √ Ev − E, E ≤ EvT , (4.1) and Nc (E) = Nco  √ E − Ec, E ≥ EcT √ EcT − Ec exp ( Ec − EcT γc ) exp ( E − Ec γc ) , E < EcT , (4.2) whereNvo andNco denote the valence band and conduction band DOS prefac- tors, respectively, Ev and Ec represent the valence band and conduction band band edges, γv and γc are the breadths of the valence band and conduction band tails, EvT and EcT being the critical energies at which the exponential and square-root distributions interface. It is clear, from Eqs. (4.1) and (4.2), that both Nv (E) and Nc (E) are continuous functions of energy. In order to further simplify these DOS functions, it will be further as- sumed that the derivatives of these DOS functions, Nv (E) and Nc (E), are also continuous functions of energy, i.e., these functions are smooth func- 108 4.2. Empirical model for the DOS functions tions. For this to be the case, it may be shown that EvT = Ev − γv/2 and that EcT = Ec + γc/2 [28]. With these simplifications, Eq. (4.1) reduces to Nv (E) = Nvo  √ γv 2 exp ( −1 2 ) exp ( Ev − E γv ) , E > EvT √ Ev − E, E ≤ EvT , (4.3) and Eq. (4.2) reduces to Nc (E) = Nco  √ E − Ec, E ≥ EcT √ γc 2 exp ( −1 2 ) exp ( E − Ec γc ) , E < EcT , (4.4) It is noted that in the disorderless limit, i.e., when γv → 0 and γc → 0, these DOS functions reduce to Nv (E)→ Nvo  0, E > Ev √ Ev − E, E ≤ Ev , (4.5) and Nc (E)→ Nco  √ E − Ec, E ≥ Ec 0, E < Ec , (4.6) respectively, these being the Tauc DOS functions, i.e., Eqs. (2.23) and (2.24), respectively. 109 4.2. Empirical model for the DOS functions The valence band DOS function, Nv(E), is depicted on a linear scale in Figure 4.1, Nvo and Ev being nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this analysis, i.e., the valence band band edge forms the reference energy for this plot. A variety of selections of γv are made, these spanning over the range of values representative of amorphous semiconductors. It is clearly seen that for energies below the valence band band edge, Ev, with the tail breadth set to zero, the valence DOS function terminates abruptly, i.e., there are no electronic states within the gap. For finite γv, however, a distribution of tail states encroaches into the gap region, the amount of encroachment increasing with γv. In Figure 4.2, this plot is depicted on a logarithmic scale, the same parameter selections as that employed in Figure 4.1 also being used. The conduction band DOS function, Nc(E), is depicted on linear and logarithmic scales in Figures 4.3 and 4.4, respectively, Nco and Ec being nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this analysis. As with the valence band case, greater γc corresponds to greater encroachment into the energy gap. 110 4.2. Empirical model for the DOS functions −0.2 −0.1 0 0.1 0.2 0 2 4 6 8 γ v  = 0 meV γ v  = 25 meV γ v  = 50 meV γ v  = 100 meV N vo  = 2×1022 cm−3eV−3/2 Energy (eV) D O S (× 10 21  cm − 3 e V− 1 ) Figure 4.1: A linear plot of the valence band DOS function as a function of energy, E, for various selections of γv. The DOS modeling parameters, Nvo and Ev, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. 111 4.2. Empirical model for the DOS functions −0.2 −0.1 0 0.1 0.2 1017 1018 1019 1020 1021 1022 Energy (eV) D O S (cm − 3 e V− 1 ) γv= 100 meV γ v = 50 meV γ v = 25 meV γ v = 0 meV N vo  = 2×1022 cm−3eV−3/2 Figure 4.2: A logarithmic plot of the valence band DOS function as a function of energy, E, for various selections of γv. The DOS modeling parameters, Nvo and Ev, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. 112 4.2. Empirical model for the DOS functions −0.5 −0.3 −0.1 0 0.1 0.3 0.5 0 2 4 6 8 10 γ c  = 0 meV γ c  = 25 meV γ c  = 50 meV γ c  = 100 meV N co  = 2×1022 cm−3eV−3/2 Energy (eV) D O S (× 10 21  cm − 3 e V− 1 ) Figure 4.3: A linear plot of the conduction band DOS function as a function of energy, E, for various selections of γc. The DOS modeling parameters, Nco and Ec, are nominally set to 2 ×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. 113 4.2. Empirical model for the DOS functions −0.5 −0.3 −0.1 0 0.1 0.3 0.5 105 1010 1015 1020 γ c = 0 meV γ c = 25 meV γ c = 50 meV γ c = 100 meV N co  = 2×1022 cm−3eV−3/2 Energy (eV) D O S (cm − 3 e V− 1 ) Figure 4.4: A logarithmic plot of the conduction band DOS function as a function of energy, E, for various selections of γc. The DOS modeling parameters, Nco and Ec, are nominally set to 2×1022 cm−3 eV−3/2 and 0.0 eV, respectively, for the purposes of this plot. This figure is after O’Leary [17]. 114 4.3. JDOS evaluation and analysis 4.3 JDOS evaluation and analysis The JDOS function, J(E), corresponding to the empirical model for the DOS functions, i.e., with Nv(E) and Nc(E) as set in Eqs. (4.3) and (4.4), respectively, is now evaluated using Eq. (2.10). This JDOS function may be used in order to determine the number of possible optical transitions between the valence band and the conduction band electronic states. In Figure 4.5, the JDOS function is depicted on a linear scale for a variety of selections of γv and γc, the DOS modeling parameters being nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV; for all cases, γv is set equal to γc. This plot is shown on a logarithmic scale in Figure 4.6. Eq. (2.25) suggests that the functional dependence of √ J(E) on the photon energy, E, for energies well above the band gap, allows one to de- termine the corresponding energy gap, Eg. In Figure 4.7, a linear plot of the functional dependence of √ J(E) on the photon energy, E, is depicted. It is clearly seen that for γv = γc = 0, that the √ J(E) function termi- nates abruptly at the band gap edge. For finite γv and γc, however, a tail in the JDOS function encroaches into the gap region, this tail representing the amount of disorder that is present. 115 4.3. JDOS evaluation and analysis 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8 1 1.2 N vo  = N co  = 2 × 1022 cm−3eV−3/2 Eg = 1.7 eV γ v  = γ c  = 100 meV γ v  = γ c  = 50 meV γ v  = γ c  = 0 meV Photon Energy (eV) JD O S ×  10 44  (cm − 6 e V− 1 ) Figure 4.5: A linear plot of the JDOS function as a function of the photon energy, E, for various selections of γv and γc. The DOS modeling parameters are nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for the purposes of this plot. This figure is after O’Leary [17]. 116 4.3. JDOS evaluation and analysis 0.5 1 1.5 2 2.5 1034 1036 1038 1040 1042 1044 N vo  = N co  = 2 × 1022 cm−3eV−3/2 Eg = 1.7 eV γ v  = γ c  = 100 meV γ v  = γ c  = 50 meV γ v  = γ c  = 0 meV Photon Energy (eV) JD O S (cm − 6 e V− 1 ) Figure 4.6: A logarithmic plot of the JDOS function as a function of the photon energy, E, for various selections of γv and γc. The DOS model- ing parameters are nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for the purposes of this plot. This figure is after O’Leary [17]. 117 4.3. JDOS evaluation and analysis 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 N vo  = N co  = 2 × 1022 cm−3eV−3/2 Eg = 1.7 eV γ v  = γ c  = 100 meV γ v  = γ c  = 50 meV γ v  = γ c  = 0 meV Photon Energy (eV) √ J D O S × 1 0 2 2 (c m − 3 eV − 1 / 2 ) Figure 4.7: A linear plot of the functional dependence of the square- root of the JDOS function on the photon energy, E, for various selec- tions of γv and γc. The DOS modeling parameters are nominally set to Nvo = Nco = 2 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, and Ec = 1.7 eV for this plot. This figure is after O’Leary [17]. 118 4.4. Modeling the optical functions associated with a-Si 4.4 Modeling the optical functions associated with a-Si In order to determine the spectral dependence of the a-Si optical func- tions, 1(E) and 2(E), the spectral dependence of the JDOS function as- sociated with this material must be known. With a model for the JDOS function, in conjunction with a model for the spectral dependence of the ag- gregate dipole matrix element, R2(E), the spectral dependence of the imag- inary part of the dielectric function, 2(E), may be determined through the use of Eq. (2.17). Then, through the use of a Kramers-Kronig transforma- tion, i.e., Eq. (2.45), the spectral dependence of the real part of the dielectric function, 1(E), may be determined. For the purposes of this analysis, the spectral dependence of J(E) and R2(E) associated with a-Si will be independently modeled. Each of these models will then be contrasted with the corresponding results of experiment, these a-Si experimental results being from Jackson et al. [16]. From the spec- tral dependence of J(E) and R2(E), the spectral dependence of 2(E) will be determined, and compared with that of experiment. Following this analysis, fits with the results of experiment corresponding to other samples of a-Si will 119 4.4. Modeling the optical functions associated with a-Si be performed, it being expected that some variations in the DOS modeling parameters will occur from sample-to-sample, these variations representing the changes in the physics of each sample. Thevaril and O’Leary [33] plotted the experimentally determined a-Si valence band tail breadth parameter, γv, as a function of the conduction band tail breadth parameter, γc, for the experimental data of Sherman et al. [23], Rerbal et al. [34], Teidje et al. [35], and Winer and Ley [36]. Fit values, obtained by O’Leary [17], were also considered. This experimental data, and the fit data, is depicted in Figure 4.8. It is noted that, in all cases, the valence band tail breadth parameter, γv, exceeds that associated with the conduction band, γc. Thevaril and O’Leary [28] demonstrated that when γv exceeds γc, the optical absorption coefficient is primarily determined by γv. Thus, many of the optical properties may be determined neglecting the conduction band tail states, i.e., assuming that γc is equal to zero. For the purposes of this analysis, γc will be set to zero for all cases. In Figure 4.9, a plot of the experimental a-Si JDOS results of Jackson et al. [16] is depicted. This result is directly contrasted with that deter- mined using the numerical evaluation procedure for the JDOS function for the DOS modeling parameters set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, 120 4.4. Modeling the optical functions associated with a-Si 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 γ v =γ c γ c  (meV) γ v  (m eV ) O’Leary [17] Sherman et al. [23] Rerbal et al. [34] Teidje et al. [35] Winer and Ley [36] Figure 4.8: A plot of the dependence of γv on γc. Experimental results from Sherman et al. [23], Rerbal et al. [34], Teidje et al. [35], and Winer and Ley [36] are depicted. A modeling result, obtained through a fit with experimental data, from O’Leary [17], is also depicted on this plot. This figure is after Thevaril and O’Leary [33]. 121 4.4. Modeling the optical functions associated with a-Si 0 1 2 3 4 5 1038 1040 1042 1044 Photon Energy (eV) JD O S (cm − 6 e V− 1 ) Jackson et al. [16] Figure 4.9: The spectral dependence of the JDOS function. The a-Si JDOS results of Jackson et al. [16] are depicted with the solid points. The JDOS fit result, obtained by setting Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV, is depicted with the solid line. 122 4.4. Modeling the optical functions associated with a-Si Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV. It is noted that the fit is reasonably satisfactory, except for low JDOS values, where defect absorption influences the results and high photon energies, where the non-parabolicity of the bands plays a role influencing the form of the JDOS function. This a-Si JDOS function will be used in the subsequent analysis. For the case of a-Si, Jackson et al. [16] experimentally determined the spectral dependence of the aggregate dipole matrix element squared aver- age, R2(E). They found that for photon energies below 3.4 eV, this matrix element remains essentially constant, while for photon energies in excess of 3.4 eV, it falls off with the algebraic dependence E−5. Based on these obser- vations, Thevaril and O’Leary [28] modeled R2(E) as R2(E) = R2o  ( Ed E )5 , E ≥ Ed 1, E < Ed , (4.7) where Ed is the characteristic energy to whichR2(E) is constant, i.e.,R2(E) = R2o, where, for the case of the a-Si experimental data of Jackson et al. [16], R2o = 10Å2. This spectral dependence for R2(E), proposed by Thevaril and O’Leary [28], is contrasted with the experimental R2(E) results of Jackson et al. [16] in Figure 4.10. The corresponding spectral dependence of 2(E), 123 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 6 7 10−1 100 101 102 103 Photon Energy (eV) R 2 (Å 2 ) Jackson et al. [16] 10 Ed Figure 4.10: R2(E) as a function of the photon energy, E, determined using Eq. (4.7), depicted with the solid lines. For the purposes of this fit, R2o is set to 10 Å2 and Ed is set to 3.4 eV. The experimental results of Jackson et al. [16] are depicted with the solid points. 124 4.4. Modeling the optical functions associated with a-Si determined using Eq. (2.17), is depicted on a logarithmic scale in Figure 4.11 and on a linear scale in Figure 4.12. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. It is also compared with the experimental 2(E) results of Jackson et al. [16]. While the comparison with the results of experiment is reasonably sat- isfactory for the case of the logarithmic plot, i.e., Figures 4.11, for the lin- ear plot, i.e., Figure 4.12, it is seen that the spectral dependence of 2(E) does not follow the a-Si experimental results for photon energies in excess of 3.4 eV. From Figure 4.12, it is also seen that the spectral dependence of 2(E) exhibits a sharp peak at Ed, which does not accord with experimental observation. The formalism for the dipole matrix element squared average, R2(E), expressed in Eq. (4.7), is responsible for this unphysical sharp peak in the spectral dependence of the imaginary part of the dielectric function, 2(E). In order to avoid this problem created with an abrupt transition model for the spectral dependence of R2(E), R2(E) is instead remodeled as R2(E) = R 2 o[ 1 + ( E Ed )2x] 12 , (4.8) 125 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 6 10−4 10−2 100 102 Photon Energy (eV) ε 2 Ed Jackson et al. [16] Figure 4.11: The spectral dependence for the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.7), on a logarithmic scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points 126 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 60 5 10 15 20 25 30 35 Photon Energy (eV) ε 2 Ed Jackson et al. [16] Figure 4.12: The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.7), on a linear scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points 127 4.4. Modeling the optical functions associated with a-Si where x represents the asymptotic polynomial rate of attenuation, i.e., for the case of the experimental observations of Jackson et al. [16], it is found that x = 5. It is noted that for photon energies E << Ed, R2(E) = R2o. For photon energies greater than Ed, however, R2(E) asymptotically approaches R2o(EdE )x. The spectral dependence of this remodeled R2(E) is contrasted with the experimental R2(E) results of Jackson et al. [16] and the spectral dependence of R2(E), proposed by Thevaril and O’Leary [28], i.e., Eq. (4.7), in Figure 4.13. The spectral dependence of 2 associated with the a-Si, evaluated using Eq. (2.17), where R2(E) is determined using Eq. (4.8), is depicted on a logarithmic scale in Figure 4.14 and on a linear scale in Figure 4.15. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this analysis. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of this plot. The experimental 2(E) results of Jackson et al. [16] are depicted with the solid points. It is noted that the fit is reasonably satisfactory for both cases. The evaluation of the real part of dielectric function, 1(E), may be directly determined from a Kramers-Kronig transformation, i.e., Eq. (2.45). Numerical integration is employed for the 128 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 6 7 10−1 100 101 102 103 Photon Energy (eV) R 2 (Å 2 ) 10 Ed Eq. (4.8) (x = 6.6) Eq. (4.7) (x = 5) Jackson et al. [16] Figure 4.13: R2(E) as a function of the photon energy, E, determined using Eqs. (4.7) and (4.8), are depicted with the solid lines. For the purposes of this fit, R2o is set to 10 Å2 and Ed is set to 3.4 eV. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of the Eq. (4.8) plot. The experimental results of Jackson et al. [16] are depicted with the solid points. 129 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 6 10−4 10−2 100 102 Photon Energy (eV) ε 2 Jackson et al. [16] Figure 4.14: The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.8), on a logarithmic scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points 130 4.4. Modeling the optical functions associated with a-Si 1 2 3 4 5 60 5 10 15 20 25 30 35 Photon Energy (eV) ε 2 Ed Jackson et al. [16] Figure 4.15: The spectral dependence of the imaginary part of the dielectric function, 2(E), determined using Eq. (2.17), where R2(E) is determined using Eq. (4.8), on a linear scale. The DOS modeling parameters are set to Nvo = Nco = 2.48 ×1022 cm−3 eV−3/2, Ev = 0.0 eV, Ec = 1.68 eV, and γv = 40 meV for the purposes of this plot. The asymptotic polynomial rate of attenuation, x, is set to 6.6 for the purposes of this plot. The experimental results of Jackson et al. [16] are depicted with the solid points 131 4.5. Results purposes of this analysis. 4.5 Results Three sets of experimental results are used in order to validate the afore- mentioned model for the spectral dependence of dielectric functions; these are the same experimental data sets as that considered earlier. In Figures 4.16, 4.17, and 4.18, the model results are compared with the experimental EGE a-Si data set [30], the PECVD a-Si I data set [31], and the PECVD a-Si II data set [32], respectively. The “best-fit” is obtained through visual inspec- tion. For every case, this model perfectly fits with these three different sets of experimental data. The fitting parameters values are tabulated in Table 4.1. As this model is directly related to the DOS modeling parameters, it is possible to extract the underlying fundamental properties related to the underlying distributions of electronic states from knowledge of the spectral dependence of these optical functions. This information could help to predict the behavior of the optical response over a broad range of energies and can guide the fabrication of optical devices. 132 4.5. Results Table 4.1: The “best-fit” a-Si DOS modeling parameter selections employed for the purposes of this analysis. These modeling parameters relate to Eqs. (2.17), (2.45), and (4.8) parameter (units) EGE a-Si [30] PECVD a-Si I [31] PECVD a-Si II [32] Nvo (cm −3eV−3/2) 1.88× 1022 2.33× 1022 2.42× 1022 Nco (cm −3eV−3/2) 1.88× 1022 2.33× 1022 2.42× 1022 Ev (eV) 0.0 0.0 0.0 Ec (eV) 1.21 1.51 1.68 Eg (eV) 1.21 1.51 1.68 Ed (eV) 3.42 3.64 3.72 γv (meV) 83 40 50 1∞ 1.0 1.0 0.65 Ro2(Å2) 10 10 10 x 5.87 6.6 6.6 133 4.5. Results 1 2 3 4 5 6 7 −5 0 5 10 15 20 25 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 EGE a−Si [30] Figure 4.16: A “best-fit” of the model results with the results of experi- ment corresponding to EGE a-Si [30]. These experimental results are from Piller [30]. The fits to experiment are shown with the solid lines. The exper- imental data, EGE a-Si [30], is depicted with the solid points. 134 4.5. Results 1 2 3 4 5 6−10 0 10 20 30 40 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 PECVD a−Si I [31] Figure 4.17: A “best-fit” of the model results with the results of experiment corresponding to PECVD a-Si I [31]. These experimental results are from Synowicki [31]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si I [31], is depicted with the solid points. 135 4.5. Results 1 2 3 4 5−10 0 10 20 30 40 Photon Energy (eV) ε 1 ,  ε 2 ε1 ε2 PECVD a−Si II [32] Figure 4.18: A “best-fit” of the model results with the results of experiment corresponding to PECVD a-Si II [32]. These experimental results are from Ferluato et al. [32]. The fits to experiment are shown with the solid lines. The experimental data, PECVD a-Si II [32], is depicted with the solid points. 136 4.6. Implications of a non-unity 1∞ 4.6 Implications of a non-unity 1∞ Ideally, 1∞ should be equal to unity when all the electronic transitions are considered in the model for the spectral dependence of the imaginary part of the dielectric function, 2(E) [32]. Ferlauto et al. [32] experienced that 1∞ departs from unity depending upon the band gap differences amongst the a- Si samples. They also commented that 1∞ is less than unity for pure a-Si samples. From Tables 3.1 and 4.1, it is seen that for the EGE a-Si [30] and the PECVD a-Si I [31] samples, 1∞ is unity, whereas for the PECVD a-Si II [32] sample, it is less than unity, which is itself a pure a-Si material. 137 Chapter 5 Conclusions Models for the spectral dependence of the real and imaginary components of the dielectric function, 1(E) and 2(E), respectively, appropriate for the case of amorphous semiconductors, are considered. In the first phase of this analysis, from an empirical expression for imaginary part of the dielectric function, this expression corresponding with that of the model of Jellison and Modine [15], a closed-form expression for the real part of the dielectric function is determined using a Kramer-Kronig transformation. The resultant expression for the real and imaginary components of the dielectric function corresponds with the model of Jellison and Modine [15]. The comparison with experiment is found to be satisfactory. Then, in the latter stage of the analysis, through the application of a Kramers-Kronig transformation on an empirical model for the imaginary part of the dielectric function, this model stemming from a model for the distributions of electronic states, the spectral 138 Chapter 5. Conclusions dependence of the real part of the dielectric function. Fits with the results of experiment are also found to be satisfactory. 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