Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00082438/00001
## Material Information- Title:
- Effects of grain boundaries in polysilicon-on-insulator (SOI) MOSFETS
- Creator:
- Ortiz-Conde, Adelmo, 1956-
Fossum, Jerry G. (*Thesis advisor*) Lindholm, Frederik (*Reviewer*) Li, Sheng S. (*Reviewer*) Burk, Dorothea E. (*Reviewer*) Varma, Arun K. (*Reviewer*) - Publication Date:
- 1985
- Language:
- English
- Physical Description:
- vii, 155 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Charge density ( jstor )
Drains ( jstor ) Electric current ( jstor ) Electric potential ( jstor ) Electrons ( jstor ) Grain boundaries ( jstor ) Modeling ( jstor ) Narrative devices ( jstor ) Silicon ( jstor ) Threshold voltage ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Grain boundaries ( lcsh ) Metal oxide semiconductor field-effect transistors ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- This dissertation presents physical models that describe the effects of grian boundaries on the steady-state current-voltage characteristics of large- and small-grain polysilicon (SOI: Si-on-SiO2) MOSFETS. These models, which are supported experimentally, reveal that the grain boundaries can control the drain current and hence the electrical properties of the MOSFET. Interpretations of measurements based on single-crystal MOSFET theory can therefore result in erroneous parameter evaluations and misconceptions regarding the underlying physics. The models developed herein enable proper interpretations of measurements and facilitate optimal design of the devices. The models for the large-grain polysilicon SOI MOSFET predict: (a) an effective turn-on characteristic in the linear-region, controlled by the grain boundaries, that occurs beyond the strong-inversion threshold voltage, and henceforth defines the "carrier mobility threshold voltage" and the effective field effect carrier mobility; (b) nearly exponential dependence on the (front) gate voltage, defined by the properties of the grain boundaries, for moderate-inversion conductance, and (c) that a grain boundary near the drain can control the conduction properties for all (week-to-strong) inversion conditions in all (linear-to-saturation) regions of operation. The models for the small-grain polysilicon SOI MOSFET predict: (a) the anomalous leakage current (OFF state), which is attributed to field emission via grain-boundary traps in the (front) surface depletion region at the drain; (b) that the gate-voltage swing for the subthreshold drain current (ON state) depends strongly on the grain-boundary properties and weakly on the charge coupling between the front and back gates; (c) that the effective threshold voltage (ON state) depends strongly on grain-boundary properties and on the charge coupling between the front and back gates; and (d) the device design modifications to control and reduce the leakage current, the gate-voltage swing, and the effective threshold voltage.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1985.
- Bibliography:
- Bibliography: leaves 148-154.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Adelmo Ortiz-Conde.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Adelmo Ortiz-Conde. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000876113 ( ALEPH )
14706270 ( OCLC ) AEH3691 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

EFFECTS OF GRAIN ROUNDARIES IN POLYSILICON-ON-INSULATOR (SO) MOSFETS By ADELMO ORTIZ-CONDE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REOIJIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 TENGO EL INMENSO PLACER DE DEDICAR ESTA TESIS A MIS PADRES ALICIA CONDE-BRANDT DE ORTIZ ADELMO ORTIZ PINERO ACKNOWLEDGMENTS I would like to thank all those who helped me in one way or another to make this work possible. I wish to express my sincere gratitude to my guru, Dr. Jerry G. Fossum, for his invaluable guidance, encouragement and assistance in all phases of this dissertation. It has been my privilege and my pleasure to have been his student. I am thankful to Professors Fredrik A. Lindholm, Dorothea E. Burk, Sheng S. Li, and Arun K. Varma for their participation on my supervisory committee. I also thank Professor Arnost Neugroschel for his help during the experimental part of this work and Professor Eugene R. Chenette for guiding my steps during the beginning of my graduate work. I want to express my appreciation to Drs. Hon Wai Lam, Ravishankar Sundaresan, Hisashi Shichijo, and Sanjay Ranerjee of Texas Instruments, Inc., for technological support. I would especially like to thank my friends Drs. Hyung-Kiu Lim and Ravishankar Sundaresan, former graduate students, for the many insightful discussions. My interaction with them has been a very gratifying learning experience. I would like to also thank my other colleagues and friends, Dr. Franklin Gonzalez, Bruce Rushing, Dr. HsingLiang Lu, Victor de la Torre, Tae-Wong Jung, Robert McDonald, Suy-Young Yung, Surya Veeraraghavan, Dr. Jean Andrian, Dr. Ganesh Kousik, Arthur Van Rheenen, Dr. Saeid Tehrani-Nikoo, and Juin-Jei Liou for helpful comments and encouragements. I am grateful to Ms. Carole Boone for her excellent work in editing and typing this dissertation. I cannot in words express my thanks to my former professors at the Universidad Simn ?ollvar, Drs. Pierre Schmidt, Gustavo Roig, Paul Esqueda, and Francisco Garcia, for all they have done in support of my graduate work. I am infinitely indebted to ny parents and family for their incredible support and encouragement throughout my graduate school career. The financial support of The Consejo Nacional de Investigaciones Cientificas y Tecnologicas (CONICIT), Naval Research Laboratory (NRL), and the University of Florida Center-of-Excellence Program is gratefully acknowledged. TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS . . . .iii ABSTRACT . . .****** ********* *****.V vi1 CHAPTER ONE INTRODUCTION . .01 TWO LINEAR-REGION CONDUCTANCE OF LARGE-GRAIN POLYSILICON MOSFETS.11 2.1 Introduction . . o. o. . 1 2.2 Linear-Region Conductance in Strong Inversion. 14 2.2.1 Intragrain Electron Distribution in Channel . 14 2.2.2 Grain-Boundary Potential Barrier in Channel.16 2.2.3 Channel Conductance. . .24 2.3 Linear-Region Conductance in Moderate Inversion.31 2.4 The Significance of Grain Boundary Orientation.35 2.5 Experimental Support and Discussion . 38 2.5.1 Support for the Strong Inversion Analysis . 39 2.5.2 Support for the Moderate Inversion Analysis.45 2.6 Summary . 46 THREE CURRENT-VOLTAGE CHARACTERISTICS OF LARGE-GRAIN POLYSILICON MOSFETS. . . . . . . 49 3.1 Introduction . . 49 3.2 Analysis oo .o . o. -._ .5 3.2.1 Formalism. . . . . .53 3.2.2 Numerical Solution . . 59 3.3 Experimental Support and discussion .o. . 65 3.4 Summa ry . . . . . 67 FOUR ANOMALOUS LEAKAGE CURRENT OF SMALL-GRAIN POLYSILICON MOSFETS. . . . 69 4.1 Introduction . . 69 4.2 Leakage Current Model . . 73 4.3 Summary . . 94 FIVE SUBTHRESHOLD BEHAVIOR OF THIN-FILM SMALL-GRAIN POLYSILICON MOSFETS . *. .97 5.1 Introduction. . . 97 5.2 Analysis. , . . . . . . . . . . .,. 100 5.2.1 Formalism. . . . . .102 5.2.2 Numerical Solution. . -. . . . 108 5.3 Experimental Results and Discussion. . . 121 5.4 Summary. . o _ . . . . . .o. 125 SIX SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . 128 6.1 Summary and Conclusions. 128 6.2 Recommendations for Further Research .131 APPENDICES A THE CHARGE TRAPPED AT THE GRAIN BOUNDARY IN TERMS OF THE OUASI-FERMI LEVEL . . . . . . . 134 B THE FOUNDATION OF A CHARGE-SHEET MODEL FOR THE THIN-FILM MOSFET. . . o . 136 C FORTRAN COMPUTER PROGRAM TO CALCULATE THE CHARGE DENSITY IN A THIN-FILM SMALL-GRAIN POLYSILICON MOSFET. REFERENCES . . 148 BIOGRAPHICAL SKETCH. . . . 155 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFECTS OF GRAIN BOUNDARIES IN POLYSILICON-ON-INSULATOR (SOI) MOSFETS Ry ADELO ORTIZ-CONDE August 1985 Chairman: Jerry G. Fossum Major Department: Electrical Engineering This dissertation presents physical models that describe the effects of grain boundaries on the steady-state current-voltage characteristics of large- and small-grain polysilicon (SOI: Si-on-SiO2) MOSFETs. These models, which are supported experimentally, reveal that the grain boundaries can control the drain current and hence the electrical properties of the MOSFET. Interpretations of measurements based on single-crystal MOSFET theory can therefore result in erroneous parameter evaluations and misconceptions regarding the underlying physics. The models developed herein enable proper interpretations of measurements and facilitate optimal design of the devices. The models for the large-grain polysilicon SOI MOSFET predict: (a) an effective turn-on characteristic in the linear-region, controlled by the grain boundaries, that occurs beyond the strong-inversion threshold voltage, and henceforth defines the "carrier mobility theshold voltage" and the effective field effect carrier mobility; (b) a nearly exponential dependence on the (front) gate voltage, defined by the properties of the grain boundaries, for moderate-inversion conductance, and (c) that a grain boundary near. the drain -can control the conduction properties for all (weak-to-strong) inversion conditions in all (linearto-saturation) regions of operation. The models for the small-grain polysilicon SOl MOSFET predict: (a) the anomalous leakage current (OFF state), which is attributed to field emission via grain-boundary traps in the (front) surface depletion region at the drain; (b) that the gate-voltage swing for the subthreshold drain current (ON state) depends strongly on the grainboundary properties and weakly on the charge coupling between the front and back gates; (c) that the effective threshold voltage (ON state) depends-strongly on grain-boundary properties and on the charge coupling between the front and back gates; and (d) the device design modifications to control and reduce the leakage current, the gatevoltage .swing, and the effective threshold voltage. viii CHAPTER ONE INTRODUCTION Because of the advantages of dielectric isolation and threedimensional (3-D) integration [GI80; LA82; MA85], there is much interest in SOI (silicon-on-insulator) MOSFETs. The advantages of these devices compared with the single-crystal counterpart are [LA821: (a) increased circuit speed due to reduced parasitic capacitance; (b) superior hardness to transient radiation; and (c) elimination of'latch-up, which is of fundamental importance when the feature sizes in CMOS (complementary metal-oxide-semiconductor) technology are scaled to smaller dimensions. Today, CMOS is the dominant technology for VLSI (very large scale integration) because of low power consumption, superior noise margins, better compatibility with analog circuits, and reduced vulnerability to soft errors. The most promising SOl technologies for VLSI are: SOl formed by high-dose ion implantation [HE841, SOI using porous silicon [BA841, silicon-on-sapphire (SOS) [SA84], beam recrystallization of polysiliconon-silicon dioxide FLASO], and as-deposited LPCVD (low-pressure chemical vapor deposition) polysilicon-on-silicon dioxide [MA85i. The first two of these technologies yield dielectrically isolated single-crystal silicon, but they have disadvantages. The SOl formed by high-dose ion implantation technology requires excessive capital costs for equipment, and the SOl formed by the porous silicon technology is not compatible with the subsequent process [LA82]. The SOS technology also produces dielectrically isolated single-crystal silicon, but it has not been widely accepted because of fundamental material limitations [LA82] thatimpede the realization of high-quality silicon-on-sapphire. The beamrecrystallization SOT technology, which yields large-grain polysilicon (> 1 um), is of practical interest because of the relatively good performance of the devices fabricated with it compared with that of the single-crystal counterpart [LASO; TS81; C083,84]. The as-deposited LPCVD SOl technology, which produces small-grain polysilicon (< 0.1 um), is also of practical interest because of the circuit applications [MA84,851 that do not require stringent performance of all the devices, e.g., CMOS memories, and because of the simplicity of the fabrication. Because of the practicality of the last two technologies, it is of primary importance to model the large- and small-grai.n polysilicon SOT MnSFETs. Most of the previous research and development of SOl has emphasized either the technology, i.e., the recrystallization process [LARO; LE81; NG81; TSA81; N1831 or the grain-boundary passivation [KA80; SH84; MA851, or the recrystall ized silicon [GE82; MA82; SC831, i.e., its characteristic defects and grain boundaries. Little work [KA72; DE80,82; LEV82; COV82,83,84] has been done on the characterization and modeling of devices in large- and small-grain polysilicon SOl films, which is essential if SOl integrated circuits are to be optimally developed. Such characterization of the large- and small-grain polysilicon SOT MOSFET must include a description of the charge coupling between the front and back gates [LI83b,84a,84b], and must account for the influence of grain boundaries on the electrical characteristics, which is the subject of this dissertation. The inversion-mode large-grain polysilicon SOl MOSFET, illustrated in Fig. 1.1, presents a relatively good performance compared with that of the single-crystal counterpart [LA80; TS81; C083, 841, but it has the disadvantage of requiring the additional recrystallization step. The accumulation-mode small-grain (as deposited) polysilicon SOl MOSFET, shown in Fig. 1.2, does not require the recrystallization step, but it is inferior to the single-crystal counterpart, especially because of anomalous high leakage current and exceptionally high gate-voltage swing FON82; SH841. To improve the performance of the small-grain SOT MOSFET, for applications that do not require single-crystal silicon device characteristics (e.g. load elements for a dense static RAM), grainboundary passivation (e.g., via hydrogenation [KA80; SH841) has been successfully used rMA84; MA85]. Unlike the large-grain polysilicon device, the small-grain polysilicon device can be designed to be operated in either the accumulation- or inversion-mode because the film body (grains) is completely depleted of free carriers, facilitated by grain-boundary trapping. The purpose of this dissertation is to develop physical models for the effects of grain boundaries in large- and small-grain polysilicon SOT MOSFETs, which are useful for the prediction and optimization of VG f . Polysilicon or Metal SiO0 I U I Si Substrate VG b Fig. 1.1 Cross-section of the four terminal n-channel inversion-mode heam-recrystallized (large-grain) polysilicon SOI MOSFET. The -terminal voltages are referenced to the source-voltage (vs 0 0). Vs =0 rrnrrimi~~rrrrrni~ Polysilicon Film . . ! . . . . . . . . . S i.0 * :::::::: . 2: . . . . . . . . . . . . . . . . . V)' VGb Pig. 1.2 Basic structure of the four-terminal p-channel accumulationmode LPCVD (small-grain) polysilicon SOl MOSFET. device performance in SOl integrated circuits. Chapters Two and Three concern the large-grain polysilicon device, and Chapters Four and Five concern the small-grain polysilicon device. The major contributions made in this dissertation are: (1) the modeling of the effects of grain boundaries for all regions of operation in large-grain polysilicon SOT MOSFETs; (2) the physical characterization of the anomalous leakage current (OFF state) in small-grain polysilicon SOl MOSFETs; (3) the numerical modeling of the subthreshold drain current and the threshold voltage (ON state) of thin-film small-grain - polysilicon SOT MOSFETs; (4) the development of the foundation of a charge-sheet model [BR78,811 for the thin-film single-crystal SOT MOSFET; (5) the experimental support for the developed models from measurements of representative SOI MOSFETs. We derive in Chapter Two a theoretical description of the linearregion drain current of the large-grain polysilicon SOl MOSFET, which is valid for all inversion levels and accounts for arbitrary orientation of the grain boundaries. The corresponding channel conductance shows an effective turn-on characteristic controlled by the grain boundaries that occurs beyond the strong-inversion threshold. Henceforth the carrier mobility threshold voltage, which exceeds the actual one, and the effective carrier mobility, which is typically higher than the actual (intragrain) one, are defined. For sufficiently high gate voltage, the grain-boundary potential barrier is low enough that the channel conductance is not significantly influenced by the boundaries. For gate voltages lower than the carrier mobility threshold voltage, the conductance varies nearly exponentially with the gate voltage and depends strongly on the grain-boundary properties. Grain boundaries perpendicular to the carrier flow in the channel produce the strongest effects on the conductance. To support this analysis and to stress its practicality, we compare model, predictions with measured currentvoltage-temperature characteristics of laser-recrystallized SOI MOSFETs fabricated at Texas Instruments FLA83]. The theoretical-experimental agreement is good, and in addition to indicating properties of the grain boundaries in these devices, it exemplifies how the mobility threshold voltage and the effective carrier mobility can be easily misinterpreted as the actual threshold voltage and mobility when conventional MOSFET theory is used as the basis for interpreting electrical measurements of SOI MOSFETs. Such misinterpretations can obscure essential criteria for achieving optimal designs of SOI devices and integrated circutis. For example, our physical analysis reveals that in particular cases grain boundaries can actually benefit the SOI MOSFET performance by producing an unusually high transconductance. This suggests, in contrast to the general belief, that optimal designs may not require elimination of all grain boundaries. We describe in Chapter Three a physical model for the currentvoltage characteristics of the large-grain polysilicon SOl MOSFET in all regions of operation. The essence of this model is an accounting for sizable, position-dependent voltage drops across the grain boundaries that-can occur when the device isdriven out of the linear-region. The carrier transport through the grain boundaries (viz., over potential barriers-created by carrier trapping) is then nonlinear, and the channel conduction depends ;on how the grain boundaries are distributed between the source and the drain. Although our model accounts for any number of grain boundaries in the channel, we apply it herein to the most likely case (in beam-recrystallized VLSI) of an SOI MOSFET with only one grain boundary. We emphasize the importance of the position of the grain boundary, as well as its electrical properties, in defining the currentvoltage characteristics. Model calculations, supported by limited experimental results, show that grain boundaries tend to decrease the drain current of large-grain polysilicon SOl MOSFETs, but can increase the transconductance. We develop in Chapter Four a physical model for the anomalous leakage current (OFF state) in small-grain polysilicon SOl MOSFETs based on field emission via grain-boundary traps. To support this model, we compare its predictions with measured data from p-channel accumulationmode and n-channel inversion-mode devices FSU84; SH85]. Good correlation is shown, and field emission at the back surface is suggested as the mechanism underlying the minimization of the leakage current at relatively low values of front gate voltage. Insight regarding the physics underlying the anomalously strong drain and gate voltage dependences is readily provided by the model, and implies design criteria to control the leakage current in small-grain polysilicon MOSFETs. We derive in Chapter. Five a theoretical description., of the. suhthreshold drain current and the threshold voltage (ON state) in the thin-film small- grain polysi.licon S(l. M(ISFET, --which reveals-,the physical.influence of grain boundaries in the channel, and the charge coupling between the front and back gates. The main results of this model, supported by experimental results, are: the gate-voltage swing depends strongly on grain-boundary properties and weakly on the charge-coupling effects; the threshold voltage depends strongly on grain-boundary properties and charge-coupling effects; the charge-coupling effects decrease as the trap density, the thickness of the film, or the doping concentration increases. We summarize in Chapter Six the main conclusions and accomplishments of this dissertation. We also suggest in this chapter further related research. We show in Appendix A that the electron charge trapped at a grain boundary (in an n channel) can be expressed in terms Of the electron quasi-Fermi level for any grain-boundary voltage drop. This result, which was used in Chapter Three, indicates that a previous assumption rRA78aI, which establishes that the charge trapped at the grain boundary is independent of the grain-boundary voltage drop, is generally invalid. In Chapter Three, we have also avoided the use of another classical, but generally invalid assumption [MU61] that a (constant) fraction of the thermionically emitted electrons are captured by the grain-boundary traps. As a first step towards the development of a practical model for integrated circuit design with SOI MOSFETs, we present in Appendix R the foundation of' a -charge-sheet model -[RR78,81] for the thin-film singlecrystal silicon MOSFET. In Appendix C we include the computer program used in Chapter Five to calculate the subthreshold drain current and the threshold voltage (ON state) in the thin-film small-grain polysilicon MOSFET. The program is based on a "two-dimensional" bisection method [BU81i that we developed. Although this method is not as fast computationally as Newton-Raphson FBU81], we use it because it avoids the problems of convergence that typically occur when Newton-Raphson [BU81] is applied to complex problems. CHAPTER TWO LINEAR-REGION CONDUCTANCE OF LARGE-GRAIN POLYSILICON MOSFETs 2.1 Introduction We derive in this chapter a theoretical description of the linearregion drain current of the large-grain polysilicon SOI MOSFET, which reveals the physical influence of grain boundaries in the channel. The corresponding channel conductance is described in terms of the (front) gate voltage, the device parameters, and the grain and grain-boundary properties. We restrict our analysis to cases in which the polysilicon film is not completely depleted between the front and back surfaces. We initially assume in Section 2.2 strong inversion, and that the grain boundaries in the channel are perpendicular to the carrier flow; but we generalize the analysis in Sections 2.3 and 2.4 by removing these two as5umptions respectively. The model comprises the following physics: (a) the quantummechanical description [HS79] of the carrier distribution in the inversion layer, which implies an average carrier density and its dependence on the gate voltage that can be modeled based on the classical solution [C070; SZ11; (b) the two-dimensional potential variation near a grain boundary in the channel, which when approximated by coupled one-dimensional solutions of Poisson's equation defines the grain-houndary barrier height resulting from carrier trapping [BA78a,b]; and (c) the description of the carrier transport through the grain boundary, assumed to be predominantly thermionic emission over the potential barrier rP1791. To obtain closed-form expressions for the channel conductance, which give physical insight and facilitate the development of SOl MOSFET models suitable for computer-aided circuit analysis, simplifying assumptions are made and justified. The resulting strong-inversion channel-conductance model of Section 2.2 shows an effective turn-on characteristic controlled by the grain boundaries that occurs beyond the strong-inversion threshold. Henceforth the carrier mobility threshold voltage, which exceeds the actual one, and the effective carrier mobility, which is typically higher than the actual (intragrain) one, are defined. For sufficiently high gate voltage, the grain-boundary potential barrier is low enough that the channel conductance is not significantly influenced by the boundaries. Thus the intragrain mobility, which can be affected by surface scattering [SU80, controls the conductance at high gate voltages. In Section 2.3 we extend the analysis to account for moderate- and weak-inversion levels. For gate voltages lower than the carrier mobility threshold voltage, we find that the conductance varies nearly exponentially with the gate voltage, and that the gate-voltage swing needed to reduce the conductance by one order of magnitude is strongly dependent on the properties of the grain boundaries. We account, in Section 2.4 for -arbitrary orientation -of the grain boundaries. This analysis is of interest because of the possibility of controlling [MA8Z; :TSA82; N183; SC83] the predominant -grain-boundary orientation in devices fabricated in recrystallized polysilicon. We find that grain boundaries perpendicular to the carrier flow in the channel maximize the grain-boundary effects on the conductance. In contrast, grain boundaries parallel to carrier flow in the channel do not affect the conductance. To support the analysis and to stress its practicality, we compare in Section 2.5 model predictions with measured current-voltagetemperature characteristics of laser-recrystallized SOl _.MOSFETs fabricated at Texas Instruments FLA83i. The theoretical-experimental agreement is good, and in addition to indicating properties of the grain boundaries in these devices, it exemplifies how the mobility threshold voltage and the effecti-ve carrier mobility can be easily misinterpreted as the actual threshold voltage and mobility when conventional MOSFET theory is used as the basis for interpreting electrical measurements of SOl MOSFETs. Such misinterpretations can obscure essential criteria for achieving optimal designs of SOI devices and integrated circuits. For example, our physical analysis reveals that in particular cases grain boundaries can actually benefit the SOT MOSFET performance by producing an unusually high transconductance. This suggests, in contrast to the general belief, that optimal designs may not require elimination of all grain boundaries. 2.2 Linear-Region Conductance in Strong Inversion We assume, based on studies [RA78a,b; P179] of majority-carrier transport through silicon grain boundaries at room temperature, that thermionic emission of carriers over the grain-boundary potential barrier 'Ro underlies the predominant influence of the boundary on the channel conductance of SOI MOSFETs, and that TBo results from carrier trapping at localized grain-boundary states. The trapping and TBo are characterized by a two-dimensional solution of Poisson's equation in the channel. Before we discuss this solution and the corresponding thermionic-emission current, we must consider the intragrain carrier distribution in the channel and its dependence on the gate bias, which define 'Bo* We refer to the four-terminal n-channel inversion-mode large-grain polysilicon SOl MOSFET illustrated in Fig. 1.1, and we assume that the grain boundaries in the channel are perpendicular to the electron flow. 2.2.1 Intragrain Electron Distribution in Channel Because the inversion layer thickness xi is very narrow (on the order of the electron de Broglie wavelength), the true electron distribution n(x) in the channel (away from grain boundaries) must be described quantum-mechanically FHS791. This description follows from a self-consistent solution of the Schrodinger equation and Poisson's equation. The result differs markedly from the classical solution [C070, SZ81] based on Poisson's equation and Maxwell-Boltzmann statistics: xi is narrower and n(x) is more uniform FHS79]. However the inversion-layer areal charge density, x. -0 = q f n(x)dx (2.1) n 0 where x = 0 represents the Si-SiO2 interface, is predicted well -by the classical solution. The analyses suggest a simplification in the description of n(x) and its dependence on the (front) gate voltage VGf. We define an average electron density R over the effective portion of the inversion layer, O x4 Xi(eff), as revealed by the quantum-mechanical solution, but we use the classical solution, ncl(x), to convey the VCf dependence. We find that Xi(eff) is described well by Xi(eff) cl ocl qf n (x)dx: -0.9 0 (2.2) 0 n where Ocl is given by (2.1) with n(x) replaced by nCl(x); that is, about 90% of the inversion-layer charge is contained within a region in which n = and in which virtually all the channel current flows. Then we define R by cl qnxi(eff) = -0.9 0n (2.3) Numerical evaluations of Xi(eff) reveal that it is not strongly dependent on VGf, that it decreases with increasing film doping density NA, and that typically it is quite narrow. For example, when NA = In16 cm-3, Xi (eff) = 120 A. Corresponding calculations of R defined by (2.3) are plotted -versus (VGf -- V-rf) in Fig.2.1 for-different values of NA and for an oxide thickness tof of 600 A; VTf is the threshold voltage : - - .-- .---3 that earrespotds, torthe_ -nset,,of strong- tnverstorIn-[l(o NAT.- As -implied -by (2.3), Fig. -21 shows -that ff increases with increasing VGf and with increasing NA. We find that Xi(eff) and 5 as defined by (2.2) and (2.3) in terms of the classical solution [C070; SZ81] are generally consistent with the actual electron distribution given by the quantum-mechanical solution rHS79J. 2.2.2 Grain-Roundary Potential Barrier in Channel For strong-inversion _conditions_ within the grains, .electron trapping at localized grain-boundary states produces potential barriers that - affect the electron transport along the channel. The barrier formation is similar to that. at grain boundaries in bulk polysilicon [R A78b, - F08 21 except '-in the "channel- To is .influenced,. by_ VGf as described by the two-dimensional form of Poisson's equation. We consider the potential variation near a grain houndary in the channel as shown in Fig. 2.2. We assume that away from the grain boundary (y > yd) the electric field is vertical (in the x-direction), and, n(x) .is well approximated by n .over the effective inversion layer, o( x 4 Xi(eff), as discussed in the preceding subsection. In this region (I), Poisson's equation simplifies to x(2.4) 4 x 1017 3S- x 1017 3 x 1017 2.5 x 1017 1.5 x . . 0.5S x. VGf - VTf (V) Fig. 2.1 Calculated average electron density in channel versus (front) gate voltage for several film doping densities. Vs =0 i: s G f V !D re I-a0 N eff) L _V g + n+ P Grain Boundaries Fig. 2.2 Cross-section of effective inversion layer showing typical grain and grain boundaries, which are assumed to he perpendicular to the electron flow. where T is the electrostatic potential. In the vicinity of the grain boundary (y < Yd), a horizontal (y-direction) component of the electric field is prdduced-by the electrons trapped at the grain boundary. We assuine that the trapping nearly depletes this region (II) of free electrons; hence + 3 T q NA "(2.5) ax ayIWe note that this depletion approximation [BA78b; P1791 is valid provided TBo is sufficiently high: high enough in fact, we assume, that the grain boundaries significantly affect the channel conductance. We discuss the validity of this assumption in Subsection 2.2.3. Assuming that, analogous to the gradual-channel approximation FSZ81], the trapped electrons at the grain boundary typically create only a small perturbation on the x-component of electric field, we can write << (2.6) a x I ax II ay II where the partial derivatives are evaluated anywhere in the regions indicated. We justify this assumption by noting that the subsequent solution we obtain is consistent with it when (VGf - VTf) > TFo' which is usually true for strong-inversion conditions. Hence (2.6) implies an approximate solution to the two-dimensional problem defined by (2.4) and (2.5), which is obtained by coupling two one-dimensional solutions. The corresponding approximation for TB derives combination of (2.4)-(2.6), which yields SII s with the boundary conditions T(x'y = yd) :pl(x) from the (2.7) (2.8) and (2.9) In (2.8) T(x) is the intragrain (region I) potential variation in the channel, which is given by the one-dimensional solution -of Poisson's equation and the Schrodinger equation [HS791. We now identify Yd as the grain-boundary depletion-region width, and we note that our analysis applies only when the grains are not completely depleted. The solution to (2.7)-(2.9) is T (x,y) II - (y-) + (x) and hence -2 qn5yd Ro = UI(x) - '(xO) II= Tn o S (2.10) (2.11) 3T9 Y=Yd To complete the description of Bo,' we must express yd in terms of known parameters. This expression is implied by the conservation of charge in the vicinity of the grain boundary: O :- 2q yd , (2.12) which equates the areal density of charge trapped at the grain boundary, OGR, to the electron charge density removed to form the (two) adjacent depletion regions. In writing (2.12) we have implicitly assumed that the electrons are trapped within Xi(eff), which is commensurate with our previous assumptions. The trapped charge density depends on the distribution in the energy gap of localized grain-boundary states (acceptor-type since QGB < 0) [P1791. It is reasonable to approximate this. distribution by a delta function FBA78b; P179; F082; LUlI], yielding NST states (traps) per unit area at an energy level ET. Then 0 = I ST_ (2.13) GR 1 xlE T - E FY=.O 1 +2exp kT where EF is the Fermi level and the factor of 1/2 reflects the (spin) degeneracy of the localized states. The position of EF relative to ET is defined by YRo and the electron density in region I, i.e., : [ET E E +-T In( ) (2.14) FE - Fy=O lT " il + q'Bo - q n- where Ei is the intrinsic Fermi level (virtually at midgap) and ni is the intrinsic carrier density in silicon. Thus (2.11)-(2.14) implicitly describe ThBo in terms of the grainboundary parameters NST and (ET - Ei), and of R, which depends on VGf and the MOSFET properties as described in Suhsection 2.2.1. Numerical calculations of TBo are plotted versus (VGf - VTf) in Fig. 2.3 for NA : 1016 cm"3, tof = 600 A , two representative values of NST, 'I011 and 1012 cm-2, and three positions of ET in the energy gap. In all cases, for VGf sufficiently high, TRo decreases with increasing Vrf. This can be explained by noting that under these conditions virtually all the grain-boundary states- (within Xi(eff)) -are filled, ,and hence OGR- :-qNST is independent of VGf. Therefore since R increases with VGf (see Fig. 2.1), Yd concomitantly decreases as described by (2.12), which implies through (2.11) that 'Ro also decreases: qN T RO S T (2.1 5) Ss5 However when VGf is low,'Bo is nearly insensitive to VGf. This is because the grain-boundary states are not completely filled, and hence EF is near ET, which virtually fixes T Bo as described by (2.14). We note in Fig. 2.3 that for NST = 1011 cm'2, T8o is less than 10 mV when (VGf - VTf) exceeds about 0.1 V. Thus although our depletion approximation is invalid for these conditions, we surmise that 'Ro is low enough that the grain boundaries do not significantly affect the channel conductance. However for NST = 1012 cm"3, 'Po is high enough, VCf - v.f (V) Calculated grain boundary potential barrier versus (front) gate voltage for two representative grain-boundary trap densities and three energy levels. WO note that the low values of 'Ro calculated for N1ST : I0 cm"2 are probably inaccurate because of the invalidity of the depletion approximation (2.5). Nevertheless the curve is useful because it indicates when T3o is low enough that the grainboundary effect on channel conduction is insignificant. Fig. 2.3 even when (VGf - VTf) is relatively large, to validate the depletion approximation and to strongly influence the channel conductance as we describe in the next section. 2.2.3 Channel Conductance The physical basis for the influence of grain boundaries on the channel conductance is the interaction between electrons flowing from source to drain and the potential barriers at the boundaries. Although quantum-mechanical tunneling of electrons through the barrier may be significant at low temperatures FLU81] and diffusion of electrons is important when the barrier is low [C082, 83, 84], we assume (at room temperature) that thermionic emission of electrons over the barrier T co is the predominant grain boundary transport mechanism [P179). Then if the drain voltage VD is low enough (linear region) that the voltage drop across a grain boundary Vgb is much smaller than 2kT/q, and if TRo > kT/q, the emitted current density is [BA78b] qA*T exp( kT n V (2.16) gb kNC e gb where A* is the effective Richardson constant [SZ81] for electrons (- 250 A/cm2/K2) and NC is the effective density of states in the conduction band (- 2.9 x 1019 cm-3 at 300' K). Since the current in the channel is continuous from source to drain, the drain current ID can be expressed by the integral of (2.16) over the (effective) cross-sectional area of the channel: I D :Z xi(eff) Jgb (2.17) where Z is the channel width. The combination of (2.16) and (2.17) gives ID as a function of Vgb. To obtain ID as a function of V we simply equate the sum of the voltage drops along the channel to Vn. If we assume that the channel comprises N grains of equal length yg separated by (Ng - 1) identical grain boundaries (see Fig. 2.2), then VD =(Ng -l)Vgb + N V (2.18) gb gg where V is the voltage drop across a grain, which assuming that the carrier transport in the grain is by drift [SZ81] is Vyq - yd (2.19) Yg nY Vg 31, ng 1Qn I ID(219 in the linear region. In (2.19) ung is the intragrain electron mobility, the dependence of which on VGf and on device parameters can he given empirically [SU80]. Combining (2.3) and (2.16)-(2.19), we obtain IF(VGf,VD) for the SOT MOSFET in the linear region (V < (Ng - 1) 2kT/q). If we assume that Yg " Yd, which is valid in typical recrystallized SOl MOSFETs, then our result simplifies to Z ~ -J-qIlOnlVn ID ng V (2.20) 1 + (Ng-) kN'n4nq exp Ro(V f) (2.20 0. 9LA*T kT 1 where L = Ngyg is the channel length. In (2.20) On is given by the strong-inversion condition -Qn Cof(VGf - VTf) , (2.21) where Cof is the (front) gate oxide capacitance. The influence of the grain boundaries on ID is reflected by the second term in the denominator of (2.20), which depends of VGf through Bo[f(VGf - VTf)] as described in Subsections 2.2.1 (Fig. 2.1) and 2.2.2 (Fig. 2.3). If the number of grains N constituting the channel is one, vis-a-vis, if there are no grain boundaries in the channel, then (2.20) reduces to the corresponding result of conventional MOSFET theory FSZ81I. Furthermore if T'Bo is sufficiently low, because of low NST and/or high VGf (see Fig. 2.3), then the same result obtains. We note that (2.20), which because of the model assumptions is strictly valid only when TRo > kT/q, will correctly give the conventional current at high VGf only if the pre-exponential coefficient is much less than unity. With this insight then, (2.20) facilitates a self-consistency check for our model assumptions (2.5) and (2.16). We find that when the grain boundaries are influential, TBo is generally high enough that the assumptions are valid. In deriving (2.20) we have neglected thermionic field emission (tunneling) through Ro, and we have ignored the possible existence of a significant grain-boundary scattering potential barrier [LU81 through which the electrons must tunnel to traverse the boundary. The tunneling can be predominant at low temperatures, but at room temperature and above it is generally insignificant [P179; LU81]. We also neglected diffusion of electrons through 'Bo, which is important only when 'Ro is low rC083]. When TBo is high enough that the grain boundaries significantly affect ID, the diffusion can be ignored. To illustrate the grain-boundary effects described by (2.20), we plot in Figs. 2.4 and. 2.5 calculations of the linear-region channel conductance (gA ID/V)) versus (VGf - VTf) for several values of Ng and NST. In Fig. 2.4 we let Ng vary from one to 200 grains, and we use typical-. values for the remaining parameters: NST = 101.2 cm2 .at ET = Ei; NA = 1016 cm-3, tof = 600A , Z = L = 40 um; we also specify a (front) fixed oxide charge density Off = q(1011 cm"2), which defines i ng and its dependence on VGf ISU80]. We see that as Ng increases, g decreases and the plots become inflected, in general accord with recent measurements of laser-annealed SOT MOSFETs [LE81; C0841. The plots show apparent threshold voltages that are higher than VTf and transconductances (gm m ID/'VGf = VP9g/ VGff) that imply effective electron mobilities (via the conventional MOSFET theory [SZ81]) which can differ from i ng. The apparent threshold voltage is actually a '.'carrier mobility threshold voltage" (V ) at which n becomes high enough that'TRo begins to diminish with increasing VGf (see Fig. 2.3) as described by (2.15). For VGf >> V, Ro is too low to significantly affect If); that is, theT 'Ro term in (2.20) is negligible, and gm is defined hyu ng. Note however in Fig. 2.4 that the plots for .0 1.5 VGI -VTf (V) Fig. 2.4 Calculated linear-region channel conductance versus (front) gate voltage for several numbers of grains constituting the channel. The broken portions of the curves for Ng=20, 100, and 200 are inaccurate because of the invalidity of (2.2n) as discussed in Subsection 2.2.3. The N 4 I curve is inaccurate for VGf near VTf because of the invalidity of the strong-inversion relationship (2.21). 9 x 10-5 8 x 10-5 7 x 10-9 6 x 10-5 ,5 x 10-5 4 x 10-5 3 x10-5 2xiO 10- 5 0 VGf - VTf (V) Fig. 2.5 Calculated linear-region channel conductance versus (front) gate voltage for several grain-boundary trap densities. N very large become erroneous when VGf >> V because, as we discussed previously, the pre-exponential coefficient in (2.20) is not much less than unity. When V f > V , g is typically higher than that 11 corresponding tO u ng. We stress that the high effective electron mobility implied by gm is defined predominantly by the properties of the grain boundaries. Measured ID(VGf,VD) characteristics of SOl MOSFETs can thus be misleading because of the nonlinear effects of grain boundaries as we discuss in the next section. Additional calculations reveal that V depends on NA and tof; it 11 decreases with increasing NA and it increases with increasing tof. These dependences reflect, for a given (VGf - VTf), the dependences of Fn, which controls Ro, on NA shown in Fig. 2.1 and on tof implied by -0n Cof. The plots of g versus (VGf - VTf) in Fig. 2.5 for NST ranging from loll to 2 x 1012 cm"2 were calculated from (2.20) for the same device parameter values used to derive the plots in Fig. 2.4. We let Ng = 2 (one grain boundary) to simplify the physical interpretation of the results. The same type of inflection seen in Fig. 2.4 is noted in Fig. 2.5 for NST > l0l cm-2. For the device considered, if NST is much lower than 1012 cm- 2, the grain boundary is virtually ineffective; whereas if NST is higher than 1012 cm"2, the grain boundary severely affects (lowers) the channel conductance. Similar calculations have been made for different values of ET. In this case of the n-channel MOSFET, we find that as ET approaches the conduction-hand edge, T RO diminishes and the grain boundary becomes ineffective. As ET moves towardmidgap and below, the grain-boundary effect materializes as indicated in Fig. 2.5. This dependence on ET reflects the electron occupancy of the grain-boundary states, which has been described elsewhere FF082]. These results for a monoenergetic trap density could be used to infer corresponding results for different trap distributions in the energy gap. 2.3 Linear-Region Conductance in Moderate Inversion To extend the analysis described in Section 2.2 to the moderate inversion region of operation, we must remove the strong-inversion approximation (2.21). If the polysilicon film is not completely depleted between the front and back surfaces, we can neglect the chargecoupling effects [BA8O; LI83b]. Then for all inversion conditions [TSI82], 0n =0s -0b (2.22) where qP n 2 1/2 r T -A F-) (2.23) is the (areal) charge density in the silicon and - _qP f 1/2 0 : -0 r-- i (2.24) r KT is the depletion-region charge density. In (2.23) and (2.24), *sf is the front hand bending and Or = [2kTesNA]I/2 has been defined to make a compact notation. The relationship between 'sf and VGf is defined by f - 05 qNsf Gf VFR : - C C sf o~f of where V f is the front-gate flatband voltage [N182], which includes a FR contribution from fast surface states at the Si-SiO2 interface, the density Nsf (cm2eV-1) of which is assumed to be uniform in the energy gap. The On(VGf) dependence in (2.20) is now defined by (2.22)-(2.25). .To illustrate the grain-boundary effects in moderate inversion described by (2.20) and (2.22)-(2.25), we plot in Fig. 2.6 and 2.7 calculations of the linear-region channel conductance versus VGf for several values of ET and Nsf. To facilitate a later comparison between experimental and theoretical results (see Section 2.5.2), we set VTf 0, which defines Vf through (2.25). In Fig. 2.6 we let (ET-Ei) vary from 0 to 0.22 eV, and we use typical values for the remaining parameters: NA = 2xl016cm-3, which implies Xi(eff) = 80A, .1ng 380 cm2/v-sec, tof = 600 A , Z = L = 40 u m, NST = I012cm'2, and Ng = 50. We see that the conductance presents a nearly exponential dependence on Vf for the lower-Vr~f( 1-4 0 .2 .4 .6 .8 1.0 1.2 1.4 Fig. 2.6 Measured (points) and calculated (curves) linear-region (VO 50 mV) conductance versus front-gate voltage of an n-channel SOI MOSFET in laser-recrystallized polysilicon at room temperature. The measurements were made with the hack gate biased at -40 V. The calculations were done for different grain-boundary trap energy levels as indicated and with the fast surface-state density at the front Si-SiO2 interface equal to zero. Note that VTf = 0 V. VG f (V) Fig. 2.7 Measured (points) and calculated (curves) linear-region (VD : 50 mV) conductance versus front-gate voltage of an n-channel SOl MOSFET in laser-recrystallized polysilicon at room temperature. The measurements were made with the hack gate biased at -40 V. The calculations were done for different fast surface-state densities at the front Si-SiO2 interface as indicated and with the grain-boundary trap energy level at 0.2 eV above midgap. The plots of g versus VGf in Fig. 2.7 for Nsf ranging from 0 to 5 x 10ll cm-2eV-1 were calculated for (ET-Ei) = 0.2 eV and the same remaining parameters values used to derive the plots" in-Fig. 2.6. We see that S is nearly independent of Nsf although g decreases as Nsf increases. We conclude this subsection by stressing that the drain current in the lower-VGf, or "submobility-threshold" (VGf < V) regions of operation, presents a nearly exponential dependence with respect to VGf, and that the gate-voltage swing needed to reduce ID by one order-ofmagnitude is strongly dependent on the properties of the grain boundaries. 2.4 The Significance of Grain Boundary Orientation The studies in Section 2.3 and 2.4 have been based on the assumption that the grain boundaries are perpendicular to the carrier flow in the channel. In this Section, we generalize the analysis to account for arbitrary orientation of the grain boundaries. This generalization is of interest because of the possibility of controlling IMA82; TSA82; N183; SC831 the predominant grain-boundary orientation in devices fabricated in recrystallized polysilicon. We consider a (straight) grain boundary arbritrarily oriented in the channel as shown in Fig. 2.8. The drain current can be expressed, to first order, as in = Ilf + I D( (2.26) H L grain boundary Fig. 2.8 Illustration of arbitrary grain-boundary orientation in channel. where IDf is the component that flows in the grain-boundary-free portion (Z-Zb) of the channel and IDh is the component that flows in the portion (Zb) containing the grain boundary. Note in Fig. 2.8 that Zb is defined by Z and L, and a , the angle between the grain boundary and the z-direction. In the linear region (strong inversion), (Z-Zb) l ff L ngCof(VGf - Tf)VD "(2.27) The characterization of I Ob depends on a complicated twodimensional electron transport problem. To derive a crude approximation, we assixne that the electron current density Jgb through the grain boundary (via thermionic emission) is perpendicular to it. Then, analogous to (2.16), qA*T e p Bn gh exp(- kT (2.28) where n = cos(0 )y - sin(e )z is the unit vector normal to the grain boundary. We further assume that away from the grain boundary the electrons flow predominantly in the y-direction. Then to ensure current continuity from source to drain, we must have Zb (.9 flb =cos 6 Xi(eff) n * Jgh " (2.29) Using (2.28) and (2.29) and following the derivation in Section 2.2, we obtai n Zb _ U n g C of (V Gf-VTf )V 0 Db + cxp . (2.30) 1 ep.9LA*T xc The combination of (2.26), (2.27) and (2.30) then describes approximately, for strong-inversion conditions in the linear region, the significance of the grain-boundary orientation illustrated in Fig. 2.8. The cos(e) in (2.30), as well as the Zb(e) dependence, convey this significance. If e > 00, then Zb < Z and the grain-houndary effect is ameliorated. If 6:= .90' (grain boundary parallel to electron flow), then Zh = 0 and the grain boundary does not affect the channel conductance (although it may enhance source-drain leakage current via other mechanisms). 2.5 Experimental Support and Discussion To support the analysis in this chapter and to identify critical aspects of it with regard to SOl device and integrated circuit design, we measured linear-region In(VGf,VD,T) characteristics of four-terminal SO1 MOSFETs (n-channel) fabricated at Texas Instruments [LAR3]. The polysilicon film is 0.5 um thick and was laser-recrystallized after being deposited via LPCVD on a 1-um-thick layer of silicon-dioxide, which had been thermally grown on a silicon substrate. The film was doped by ion implantation of boron that yielded N A = 2 x 1016 cm3 near the front surface and NA~ 1015 cm-3 at the back surface [LAR3]. The front gate is n+ polysilicon and Cof = 5.8 x 10-8 F/cm2 Of (tof = 600 A). Large devices (Z = L = 40 um) were selected to preclude small-geometry effects [AK82]. To avoid complications due to the charge coupling between the front and hack gates [LI83b], a high negative voltage (- 40 V) was applied to the back gate to ensure accumulation at the back Si-SiO2 interface and to fix VTf. The Io(VGf) dependence was measured with VD := 50 mV at three temperatures (240 C, 70' C, and 1000 C). 2.5.1 Support for the Strong Inversion Analysis The corresponding channel conductance characteristics g(V~f,T) of a particular device, which typify the characteristics of identically processed devices, are plotted in Fig. 2.9. The basic shape of these plots is the same as that of the theoretical curves in Figs. 2.4 and 2.5, which implies qualitative support for our analysis. (The experimental curves and Figs. 2.4 - 2.5 should not be compared quantitatively because the parameter values used in the calculations are not necessarily the actual values.) The support for (2.20) is demonstrated by examination of the measured g(VGf,T) characteristics within particular ranges of Vrf. For high VGf (> V), g is defined by the numerator of (2.20); the grainboundary effect is negligible. Thus as in the case of conventional MOSFETs [SZSII, the carrier mobility (Ing) follows from the slope of g(VG'f), i.e., from gm, and the threshold voltage (VTf) is given by the 4 x 10-0 7 6 3 x 10-5 2 x 10- 5 10- 0 0 0.5 1 1.5 2 2.5 VG (VI Fig. 2.9 Measured linear-region channel conductance. versus (front) gate voltage of n-channel SOI MOSFET in laser-recrystallized polysilicon [LA83] at three temperatures. The threshold voltage is fixed by the back-gate voltage [LI83b], which was set at -40V to ensure accumulation at the back Si-SiO2 interface. linear extrapolation of the characteristic to the VGf axis. From Fig. 2.9 we thereby get VTf: 0.10 V and Ung 380 cm2/V-sec at 240 C. This vaiue ofPng is low, and hence implies excessive scattering at the polysilicon surface, due possibly to high Off [SUSO]. The low Ung does not reflect decreased transconductance due to a high surface electric field FSUSO, which we observed only at values of VGf higher than those in Fig. 2.9. These interpretations are supported hy the temperature dependence of g in the high-VGf region. We see in Fig. 2.9 a weak dependence of VTf on T and a negative temperature coefficient for png, which are consistent with the g(T) characteristics of conventional silicon MOSFETs [LE81; GA75]. 7 We see from Fig. 2.9 that VTf is considerably less than the electron mobility threshold voltage V . Thus there is a significant range of VGf (VTf< VGf < ) in which the grain boundaries suppress In. In this case the TBo term in (2.20) is much greater than unity, i.e., Vgb >> Vg, and hence g- exp(-OyBo/kT). As long as VGf < VU, TBo is high and does not vary significantly with VGf (see Fig. 2.3). The positive temperature coefficient for g thus predicted is consistent with the measured conductance plotted in Fig. 2.9 in this region. When VGf > , 'Ro decreases with increasing VGf (see Fig. 2.3), and hence g increases. To analytically describe this increase and to estimate V we use the approximate T8o(n) dependence in (2.15) and the strong-inversion relationship (2.21). The cnmbination of (2.15), (2.20), and (2.21) yields a g(VGf) characteristic that exhibits an inflection point where gm is maximum. The theoretical and experimental plots in Figs. 2.4, 2.5 and 2.9 imply that this maximum is broad. Therefore we approximate the actual characteristic by the linear function Z C( (2.31) g n(eff)Cof(VGf - V ) which is tangent to the actual g(VGf) curve af the inflection point. This function then analytically defines A/ and the effective fieldeffect electron mobility1ln(eff) due to the grain boundaries. The value of VGf at the inflection point is defined by equating to zero the ,second derivative of (2.20) with respect to VGf, using (2.3), (2.15), and (2.21). We find that at this value, the denominator of (2.20) is two. Thus (2.31) describes the tangent to g(VGf) at the point where the TFRo term in the denominator of (2.20) is unity. This tangent yields (for Ng > 1.) 9 q3 x N2 q xi(eff) ST 8kTe sCo V = V f+ 8 es Cof (2.32) n 0.9A*TL kNe ng (Ng-1) and ng { 2+1n 0.9A*TL ) (2.33) Un(eff) 4 { +I kN &ng (Ng-Il We note that the weak dependence of xi(eff) on VGf has been ignored in the derivation of (2.32) and (2.33). Thus Vu in (2.32) is evaluated by assuring a representative value for xi(eff), which depends on NA as discussed in Subsection 2.2.1. We stress that (2.32) and (2.33), which are based on analytic simplifications of our more general analysis described in Section 2.2, are merely estimates of V and pn(eff)" However they are useful in describing the functional dependences of g and gm on device parameters and temperature, and hence will facilitate SOI MOSFET design and computer-aided SOI circuit analysis. We see from (2.33) that the effective electron mobility is typically higher than u ng depending on L, Ng. and T. The measured g(VGf) characteristics plotted in Fig. 2.9 when interpreted using (2.31) yield p n(eff) 530 cm2/V-sec at 240 C, which is considerably higher than ung. The negative temperature. coefficient for in(eff) implied by the data in Fig. 2.9 is consistent with (2.33), which shows that the temperature dependence is defined primarily by that of ung" Using the measured value of u n(eff) mentioned above and (2.33), we find that Ng 50 grains. Since L = 40 um, this implies a crude estimate of about 1 um for the average grain size (yg), which is not unreasonable for the laser-recrystallized polysilicon film rLA83]. We note finally that the dependence of u n(eff) on L suggested by (2.33) is consistent with measurements FNG81] of (effective) electron mobility in laserrecrystallized MOSFETs having different channel lengths. For a given Yg (=L/N with Ng > 1), un(eff) increases as L is reduced from many times yg toward yg. The electron mobility threshold voltage as described in (2.32) is strongly dependent on NST and T, as well as on NA through VTf FLI83h] and Xi(eff). The inverse dependence of Xi(eff) on NA described in Subsection 2.2.1 implies that the difference between V,, and VTf decreases as NA increases. The predicted direct dependence on NST is consistent with observed decreases in the (apparent) threshold voltage of polysilicon MOSFETs resulting from hydrogenation [KA80], which is known to reduce NST. The inverse dependence of V on T suggested by (2.32) is corrohorated by the measured g(VGf,T) data plotted in Fig. 2.9. At 240 C, the measurements when interpreted using (2.31) imply V = 0.55 V, whereas VTf 0.10 V. The difference between Vu and VTf, based on (2.32), indicates that NST: 1 x 1012 cm-2 (where the traps are near midgap). We conclude -this -subsection bY -stressing two significant conclusions drawn from it. First, because (2.31), which is of the same form as the linear-region conductance expression for the conventional MOSFET FSZ81I, empirically describes well an appreciable region of the g(Vrf) characteristic for the SOI MOSFET, V. and un(eff) can be easily misinterpreted as VTf and Ung. Such misinterpretations, which evidently have been made in some previous work., can lead to misconceptions regarding SOI and can impede the development of optimal SOI devices and integrated circuits. Second, even though grain boundaries are effective in defining the channel conductance of SOI MOSFETs, the transconductance can be higher than that of the conventional counterpart; the grain boundaries are actually beneficial in this regard. Thus perhaps optimal designs of SOl MOSFETs may not require complete elimination of grain boundaries. 14 t5$o'port for the Moderate Inversion Analysis -In-.subsection 2.5.1 we estimated, for a typical device, that the hr d voTtage- defi ne-d by the linear extrapolation-6f the measured' 14(V i 0 - liHn the grain boundaries are insignificant (VGf >> V ) is Tf 0. 1 V, and that the electron mobility defined by the slope of the extrapolation is p ng 380 cm /V-sec. From g(VGf) that is affected by Ith-e ï¿½r~in boundaries (VGf > V ), we measured, based on our model, V = 0.5 V, NST= 1012 cm-2 (for ET assumed to be at midgap), and N 9-="50. -Note that typically Vf = VTf + nkT/q with n = 3-5 depending on NA and Cof [TS82b]. Thus our strong-inversion measurements imply Vf- which is consistent with calculations based on (2.22),- (2.25). *We stress -that the difference between VT and Vjf can be ignored for the strrng-in-Vrsion analysis because (VGf - VTf) >> kT/q. " T4ep1bt in Fig. 2.6 the g(VGf) characteristic of a typical device mIea.sured--a.a room temperature. Note especially the lower-VGf (< V ) -'-data,iWich show a nearly exponential dependence on VGf. For comparison we also show in Fig. 2.6 theoretical g(VGf) curves that were numerically derived from (2.20) and (2.22) - (2.25) using the parameter values given above -and NA = 2 x 1016 cm-3. We varied ET and let Nsf = 0, which = -f-rom7f =) and (2.25) implies Vf =-1.9 V. The calculated g(VGf) .c.iiaa-teristics also are nearly exponential for low VGf, even though the inversion level is not weak. (In weak inversion, the conductance of single-crystal MOSFETs is exponentially dependent on the gate voltage because On is [TS82h; S14721.) This dependence is due primarily to the exp(qTBo/kT) term in (2.20) as implied by the strong dependence of S ('i.e.-, the inverse slope) on ET. As ET moves from midgap (= Ei) toward the conduction band, S increases; when (ET-Ei) = 0.2 eV, the measured S is'-mo'deled Well. -Thus 'the energy level of the grain-boundary traps significantly affects the channel conductance below the electron mobility threshold (VGf < V ) We illustrate in Fig. 2.7 the effect of Nsf on the g(VGf) characteristic. The theoretical curves plotted were derived using the same parameter values for Fig. 2.6 and (ET-Ei) = 0.2 eV. For each value of Nsf, VfB was calculated from (2.25) using VTf . Increasing Nf tends to suppress the conductance for intermediate values (_ V ) of VGf, but does not significantly affect S. By comparing the calculated curves with the measured data, we crudely estimate that Nsf 10 cm-2eV1. Measurements at different temperatures (T = 240C, 70'C, and 100'C) indicate that, for intermediate VGf, both g(VGf) and S increase with increasing T. As T increases from 240C to 100'C, S increases from 0.25 V/decade to 0.34 V/decade and, at VGf = V = 0.5 V, g increases 1V from 1.3 x 10-6 u to 4.5 x 10-6u . These changes.are consistent with (2.20) in which, for relatively low VGf, the exp(qiBO/kT) term defines the predominant dependence on temperature. 2.6 Summary A physical model that describes the effects of grain boundaries on channel conductance in SOl MOSFETs has been developed and supported experimentally. These effects originate when electrons (n-channel MOSFET) are trapped at localized grain-boundary states, thereby creating potential barriers that influence the flow of electrons from source to drain. The electron trapping depends on the degree of inversion in the channel and hence on the gate voltage. For sufficiently high Vf Bo is low enough that the grain boundaries are inconsequential with regard to g and gm" However for lower VGf, the grain boundaries can predominantly control g and gm and can define: (a) an effective turn-on (linear-region) characteristic that occurs well beyond the stronginversion threshold as illustrated in Figs. 2.4 and 2.5; and (b) a nearly exponential dependence with gate voltage, as shown in Figs. 2.6 and 2.7, for moderate inversion conditions. 7 .The effective turn-on characteristic, described generally by (2.20) and approximated by (2.31), is actually a reflection of the "carrier mobility turn-on", which is controlled by the grain boundaries. it defines the electron mobility threshold voltage V, which exceeds VTf, and the effective electron mobility l1n(eff), which is typical-y higher than the actual (intragrain) mobility ung. Evidently measurements of V. and un(eff) have been previously misinterpreted as determinations of VTf. and ung. Subsequent erroneous conclusions regarding SOT can inhibit the development of optimal SOl devices and integrated circuits, which, based on our analysis, possibly need not nor should not be completely void of grain boundaries. For moderate-inversion conditions, the drain current, which is controlled by the grain boundaries, varies nearly exponentially with gate voltage and the gate-voltage swing needed to reduce the drain current by one order-of-magnitude depends strongly on the properties of the grain boundaries, especially the grain-boundary trap level, and on the properties of the Si-SiO2 interface, i.e., the fast surface-state density. Grain boundaries perpendicular to the carrier flow in the channel maximizes the grain-boundary effects on the conductance as described by (2.30). In contrast, grain boundaries parallel to carrier flow in the channel does not affect the conductance.(although it may enhance sourcedrain leakage current via other mechanisms). CHAPTER THREE CURRENT-VOLTAGE CHARACTERISTICS OF LARGE-GRAIN POLYSILICON MOSFETs 3.1 Introduction In this. chapter, we describe extensions of our previous work that yield a physical model for the steady-state current-voltage characteristics of the large-grain polysilicon SOl MOSFETs in all regions of operation. The essence of the extensions is an accounting for sizable, position-dependent voltage drops across the grain boundaries that can occur when the device is driven out of the linear region. The carrier transport through the grain boundaries (viz., over potential barriers created by carrier trapping) is then nonlinear, and the channel conduction depends on how the grain boundaries are distributed between the source and the drain. Although our model accounts for any number of grain boundaries in the channel, we apply it herein to the most likely case (in beam-recrystallized VLSI) of an SOI MOSFET with only one grain boundary. We emphasize the importance of the position of the grain boundary, as well as its electrical properties, in defining the current-voltage characteristics. As in the previous chapter, we assume that thermionic-emission theory adequately describes the carrier transport over the grainboundary potential harriers. Unlike the previous chapter, the use of the thermionic-emission theory is not well established because the applied voltage to the grain boundary is much greater than 2kT/q. The previous analyses [MU61; BA78b] of this problem for polysilicon resistors are based on assumptions which are generally invalid, e.g., that a (constant) fraction of the thermionically emitted electrons are captured by the grain boundary traps FMU61], or that the charge trapped at the grain boundary is independent of the grain-boundary voltage drop [RA78a]. The former assumption is invalid because the rate of the bandto-trap recombination process is proportional [SZ81] to the concentration of unoccupied traps and not to the current. The latter assumption is -invalid because the charge trapped at the grain boundary can be expressed (see Appendix A) in terms of the electron quasi-Fermi level, and therefore, it depends on the grain-boundary voltage drop. We avoid the use of these invalid assumptions by using the physically reasonable approximation that the electron quasi-Fermi level is nearly flat on the emitting side of the grain boundary. These potential barriers, which result from trapped inversion-layer charge, decrease with increasing inversion level, and hence are modulated by the gate voltage and vary along the channel when the drain voltage is high. Consequently grain boundaries near the drain, where the inversion level is weakest, are most influential. To properly account for the inversion-level dependence, we necessarily base our analysis on a MOSFET model [RR78, 81] that is applicable for all inversion levels. Model calculations, supported by limited experimental results, show that grain boundaries generally tend to decrease the conductance (drain current) of SOl MOSFETs, but can increase the transconductance. Grain boundaries having a trap density comparable to that (~1012 cm-2) estimated for typical high-angle boundaries in beam-recrystallized SOl can, when located near the drain, significantly affect the currentvoltage characteristics of the SOl MOSFET in all regions of operation. The grain-boundary effect is enhanced as the channel length is shortened. . . .3.2 Analysis We refer to the n-channel, enhancement-mode large-grain polysilicon SOl MOSFET illustrated in Fig. 1.1. To emphasize the grain-boundary effects, we assume that the polysilicon film' is not completely depleted between the front and back surfaces so that charge-coupling effects FLI83b] can be ignored (vis-a-vis, the back gate is inconsequential). We initially assume that the (front) channel comprises Ng grains separated by (Ng - 1) identical grain boundaries (surfaces) perpendicular to the carrier (electron) flow. Later we analyze the likely case of a single grain boundary in the channel (Ng = 2), emphasizing the importance of its position. The energy-band diagram at the jth [1< j < (N - 1)1 grain boundary, counted from source to drain, is illustrated in Fig. 3.1 for the cases of zero drain voltage (VD) and of VD > 0. When VD = 0, electrons trapped at localized grain-boundary states produce the potential harrier 'Bo (at each grain boundary), which q B Ev EC EFn . electron energy J~L4J30 EFn , jth grain boundary (a) VD=O t electron energy (b) VD:-O Fig. 3.1 Energy-hand diagram at jth grain boundary for drain voltage equal to (a) and greater than (b) zero. 53 is determined by the inversion level, vis-a-vis, the I. (front) gate voltage VGf, as we described in the previous chapter. When VD > 0, a voltage Vgbj is -dropped- -across--the jth grain boundary, skewing the energy-hand diagram as illustrated. If Vghj is large enough, it produces significant changes in the (areal) density of charge OCRj trapped at the grain boundary and in the inversion levels in the adjacent grains. 3.2.1 Formalism From Fig. 3.1, for VD > 0, 11 r Vgbj =@ -j +Bj -Bj(.1 where Tj rnd i are the potential barriers on the left and right sides of the jth grain boundary, and l and j are the electron quasi-Fermi potentials in the left and right adjacent grains. The average electron densities in the adjacent inversion layers are n niexp(qp /kT) (3.2) Fr : niexp(qp r/kT) (3.3) where ni is the intrinsic carrier density in silicon. The densities in (3.2) and (3.3) are related to the inversion layer (areal) charge densities On on the left and right by (2.3). The electron transport is controlled by the gate and drain voltages through the dependence of On on VGf and VD. To characterize this dependence, as well as the intragrain current, we use the charge-sheet model [rBR78, BR81], which is applicable for all levels of inversion. At an arbitrary (intragrain) point y in the channel, 0n(Y) = (Y) - b(Y) (3.4) where _rc sf (Y) n i2 ryS( VY]1/ 0s(Y)- kT - I + (-rA)2expF.Fsf(y) V(y)]]}112 (3.5) is the charge density in the silicon and c- sf(y) 1/2 Obh(Y) -Qr- kT 1] (3.6) is the depletion-region charge density. In (3.5) and (3.6), Tisf is the band bending (normal to the front surface), V is the difference between the electron and hole quasi-Fermi potentials [V(O) = 0, V(L) = V0 where L is the channel length], and Or = [2kThsNA]1/2. The band bending is related to VGf by 0s(Y) = -Cof[VGf - vf -f f(y) (3.7) To complete the description of the energy-band diagram in Fig. 3.1, we ensure that charge is conserved in the vicinity of the grain boundary S 2: IBj /2s Bj (3.8) which equates the charge trapped at the jth grain boundary to the electron charge removed to form the adjacent depletion regions. (We assume the regions are virtually depleted of free electrons.) Because the inversion layer is void of holes, the electron capture and emission rates for the grain-boundary traps must be equal in the steady state, and hence OGj can be expressed in terms of the electron quasi-Fermi level EFnj at the jth grain boundary (see Appendix A): 0 ~-qN ST(39 OGBj 1 +1exp(ET-n . (3.9) I +- 2 - ) In (3.9), (ET-EFni) depends on Vgbj as suggested by Fig. 3.1. This dependence is, in general, complicated and can be defined only when the electron transport mechanism(s) is specified. Although many theories regarding carrier transport through grain boundaries have been purported (e.g., thermionic emission, diffusion, thermionic field emission), none can be verified unequivocally because of the complex, variable nature of the grain boundaries. Thus to avoid undue model complexity, we assume, as in the previous chapter, that the predominant transport mechanism is thermionic emission over the potential barrier. This simplifying assumption is physically reasonable at and above room temperature where thermionic field emission is not probable, and for substantial (nontrivial) barrier heights, which render diffusion less significant. The thermionic-emission model, which in fact has functional dependences similar to the diffusion model, is further consistent with the depletion approximation, and hence with it yields insightful results commensurate with the uncertain nature of the grain boundaries. Referring to Fig. 3.1, we note that if there is a net left-to-right transport of electrons predominantly by thermionic emission, then EFn can be assumed to be nearly flat on the left side of the grain boundary; Vgbj is dropped predominantly on the right side where the (net) emitted electrons drift away from the grain boundary. Thus in (3.9),. . E E ) (E E1 +q1 (.0 T Fnj T Ei) - + Bj(3.10) where (ET-Ei) gives generally the position of the traps in the energy gap. We stress that dEFn/dy at the grain boundary is not related to the current because of the assumptions that the carrier transport is described by thermionic emission theory and not by diffusion theory. We have now described, in (3.1) - (3.10), how the energy-band diagram at a grain boundary changes to reflect the voltage drop Vgbj. Ry using physically reasonable approximations, we have avoided the use of a classical, but generally invalid assumption [MU61] that a (constant) fraction of the thermionically emitted electrons are captured by the grain-boundary traps. This commonly used assumption in fact overly defines the grain-boundary transport problem because the rate of the-band--to-trap recombination process is proportional [SZ81] to the concentration of unoccupied traps and not to the current. We have furthermore- not used another common assumption [BA78a] that QGBj is independent of Vgbj, which is also generally invalid as indicated by Our model for the steady-state current-voltage characteristics of the large-grain polysilicon MOSFET is completed by: (a) equating the drain current I to the net thermionic-emission current defined by the perturbed energy-band diagram at each grain boundary; (b) equating In to the current defined by the charge-sheet model [BR78] applied to each 9rjatd-(c) summing all the grain-boundary and grain voltage drops to VD. The net thermionic-emission current density over the potential harrier at the jth grain boundary (Fig. 3.1) is [BA78b] 1*T 1 . -.g b . . A*T2 I e - - njexp(- __ )]q (3.11) where A* is the effective Richardson constant for electrons in silicon . 25 A/cm2-K2) and NC is the effective density of states in the .---onduction-band (- 2.9 x 1019 cm-3 at 300ï¿½K). Thus for all j, I D = ZXi(eff) gbj (3.12) where Z is the channel width. We assume xi(eff) (-i00 A) is constant, independent of position and bias; n reflects changes in the local channel conductivity. With (2.3) and (3.1) - (3.10), (3.11) and (3.12) relate ID to Vgbj for the (Ng - 1) grain boundaries. Using the charge-sheet model [BR78], we now express ID as a function of the band bending at the left and right sides of each grain. For the kth (1< k< N ) grain with length Ygk (L = Ygl + + YgN ), ID Z kTp n r ) ( I q Cf { kT(V f r 1 q r 2 D 2q ofE3 of " sfk (3 .3kT)sfk 1(I 2 2 r /2, Q r ),r 3/2 1 ) 3/2] k) -7 TT - f (Tfk3/ - (fk)J.3 + sfk Cof s where ung is the electron mobility in the intragrain channel. The combination of (3.5), (3.7), and (3.13) gives ID as a function of the voltage drop V Vr V 1 (the variation in V) along the channel in the gk k k kth grain. Since V1 V .r + V _ for 2 1 k Ng, and V = 0 and k k-i gh(j-1) fo9 g n V1 0an r . VD, we have related ID to Vgk for all the Ng grains. Vg The final relationship needed to define ID(VD,VGf) is (Ng9-I) N D j=1 gbj k=l gk (3.14) 3.2.2 Numerical Solution The current-voltage characteristic is evaluated numerically by solving simultaneously the nonlinear system of equations described- by (2.3) and (3.1) - (3.14), for all j and k. Instead of solving directly this nonlinear system of equations, we obtain the solutions by first assigning values for ID and VGf, and then calculating the corresponding value of VD. The advantage of this method is that we avoid the typical convergence problems of the iterative methods [BU81] for. solving nonlinear system of equations because we only solve many nonlinear independent equations with one variable. To illustrate the - predictions of our model, we apply it to a typical (but thick) SOI MOSFET for which N = 1016 cm"3, C - 5.8 x ACof - . 10-8 F/cm2 (the gate oxide thickness is 600A), Z = L 40um, and ung 700 cm2/V-sec. To emphasize the most likely case (in beamr.ecrystallized SOI VLSI), we let Ng = 2 (one grain boundary). We plot in Fig. 3.2, for NST = 1012 cm-2, ET = Ei (traps at midgap), and Ygl = Yg2 = L/2 (grain boundary at middle of channel), the calculated IF}(Vj) characteristics for several values of (VGf - VTf); the threshold voltage VTf is the value of VGf yielded by (3.5) and (3.7) when Tsf(V = 0) is Sff twice the Fermi potential of the silicon film body. We note that VFR does not need to be specified because it is related to VTf through (3.5) and (3.7) evaluated at y=O. For comparison we also plot (dashed curves) corresponding characteristics for Ng = I (no grain boundary). The grain boundary reduces I; as in the linear region (see Chapter Two), its effect is most significant at low VGf. In the saturation region, 60 25 N=2 L VGf--VTf 1.5 V y91 = y92= NST 1012 C-2 20- Z=L=40 i.m 10 1.0 V ///'/.75 V .5 V 00 .5 1.0 1.5 VD (V) Fig. 3.2 Calculated current-voltage characteristics (solid curves) for typical large-grain polysilicon SOT MOSFET with one grain boundary at middle of channel. Without the grain boundary, the dashed curves derive. ID(sat) can be substantially limited, although VD(sat) is virtually unaffected since V(y=L) always equals the drain voltage. The grain-boundary effect is strongly dependent on NST. To illustrate this dependence, we plot in Fig. 3.3 the square-root of ID(sat) versus (VGf - VTf) for NST ranging from 0 (no grain boundary) to 2 x 1012 cm-2. As NST increases, the grain-boundary potential barrier increases, and hence a larger part of VD(sat) must be dropped across the boundary to enable ID(sat) to flow through it. For NST high, ID(sat) is reduced considerably even for VGf high. Because the grain-boundary potential barrier increases as the adjacent intragrain inversion level decreases, the effect of the grain boundary will, for VD > 0, be stronger if the boundary is closer to the drain. To emphasize this important position dependence, we show in Fig. 3.4 how the calculated ID(sat)(VGf) characteristic is altered as the grain boundary, with NST = 1.2 x 1012 cm2, is shifted toward the drain. Since Qn 0 near the saturated drain, a grain boundary there is influential regardless of how high VGf is. A grain boundary near the source however is significant only for VGf low. We also see in Fig. 3.4 that the position of the grain boundary is irrelevant for (VGf - VTf) < .6 V because the inversion level remains nearly constant along the channel for this low VGf. The grain-boundary effect illustrated in Figs. 3.2 - 3.4 is enhanced as the channel length is shortened. This enhancement is demonstrated in Fig. 3.5 where we plot the calculated IF(sat)(VGf) characteristic for different L with Z/L = 1 and Yg2 = L/4. The 62 6 6Ng= 2 Y91 = Y92=4 Z=L=40 im 4 7 1.2x 1012 cr02. 2- 1012 crii2 8x 1611 ClT 2, /2 x1O12 cm-2 0 .5 1.0 15 VGf, VTf (V) Fig. 3.3 Calculated dependence of drain saturation current, versus gate voltage, on grain-boundary (at middle of channel) trap density. 63 6 N9=2 NST=1.2 x 1012 cm2 "/ / Z=L= 40 iJ.m // // / / 4 / S/ /\//-/ no grain / 20 1i m 0--Q boundary / - 2 1OJm 7 g2= 1.m 7 7 7 0I 0 .5 1.0 1.5 VGfVTf (V) Fig. 3.4 Calculated dependence of drain saturation current, versus gate voltage, on grain-boundary position along channel. 64 6 / Ng=2 Yg2 ï¿½ / / NST=I.2 x 10 cm/ , 4 z . //// n grain --///rn 2- boundary/. // _Om / / 10 ir 00 .5 1.0 1.5 VG-VTf (V) Fi g. 3.5 Calculated dependence of drain saturation current, versus gate voltage, on channel length with grain boundary L/4 from d rai n. reduction in current with decreasing channel length results because the constraint on ID defined by (3.11) and (3.12) is independent of L, and hence Vgb must increase to support higher current densities in the channel. To further stress the significance of grain boundaries in largegrain polysilicon SOl MOSFETs, we plot in Fig. 3.6 the calculated transconductance in the saturation region, gm(sat) a ID(sat)/aVGf, for one grain boundary in the middle of the channel having different values of NST. Depending on VGf, gm(sat) can be lower or higher than that for the grain-boundary-free (NsT : 0) counterpart. At low VGf, below the "mobility threshold" (V{) the grain boundary virtually inhibits current; thus gm(sat) - 0. As VGf increases, the grain-boundary effect is diminished as the intragrain channel conductance is enhanced, thereby producing unusually high transconductance (like in the linear region analysis of Chapter Two). At high VGf, the grain-boundary effect tends to subside, and gm(sat) approaches that corresponding to NST = 0. 3.3 Experimental Support and Discussion To provide experimental support for the analysis, we measured current-voltage characteristics of large-grain polysilicon SOl MOSFETs described in Section 2.5. We find that the measured ID(sat) is smaller than that of the theoretical calculations of the corresponding single-crystal counterpart (NG :1) , and that this relative difference increases as VGf decreases. This result implies qualitative support for our analysis. E 20- 1.2x102 cm 2 1012 -2c _8 x 1 11 c m 2 1 .5 X l 12 c m -: 0 .5 1.0 1.5 VGf-VTf (V) Fig. 3.6 Calculated saturation-region transconductance versus gate voltage and grain-boundary (at middle of channel) trap density. Unfortunately, quantitative support is not obtained because Ng is not known exactly. Additional support for our analyses has been presented by Colinge et al. [C083], who developed a technique to control the location of the grain boundaries in SOI MOSFETs. They fabricated two transistors with the same geometry, one beside the other, one of them with a perpendicular grain boundary at the middle of the channel, and the other without grain boundary. They found that ID(sat) for the transistor with a grain boundary is smaller than that of the transistor without a grain boundary. We conclude this section by stressing three significant'conclusions drawn from this analysis. First, because the ID(VD) characteristics of SOI MOSFETs resemble that of the single crystal counterpart, the device parameters can be easily misinterpreted by using direct MOSFETs theory. Such misinterpretations, which evidently have been made in some previous work, can lead to misconceptions regarding SOl and can impede the development of optimal SOT devices and integrated circuits. Second, because of the variation in the degree of inversion along the channel produced by VD, a grain boundary close to drain affects IDsat even at high VGf. Third, because part of VD(sat) is dropped across the grain boundaries, ID(sat) is reduced. 3.4 Summary Using simplifying, but physically reasonable assumptions, we have modeled the effects of grain boundaries on the steady-state current- voltage characteristics of large-grain polysilicon SOl MOSFETs. We have assumed that the predominant transport mechanism is thermionic emission over the potential barrier,and we have avoided the use of previous generally invalid assumptions [MIJ61; BA78a]. The complexity of the model is commensurate with the uncertain and variable nature of the grain boundaries, but its predictions are in general accord with experimental results. Basically the model shows that grain boundaries tend to reduce the MOSFET conductance (ID), but can increase or decrease the transconductance. Although the grain-boundary effects are most apparent at low gate voltages, they can be quite significant at higher gate voltages ;when the drain voltage-is high, e.g., in the saturation region. Grain boundaries close to the drain are most effective. The effects are enhanced as the channel length is shortened. We have measured current-voltage characteristics of both [LA831 laser- (Ng >> 1) and graphite-strip-heater- (Ng 2) recrystallized SOl MOSFETs, and have found general agreement with the model predictions. Because M g is not known exactly, it is difficult to make more quantitative comparisons. A main conclusion of our work is that a single grain boundary in the channel can significantly affect the electrical properties of an SII MOSFET. Thus although the grain size of beam-recrystallized SOl is large, the grain boundaries, with randomly varying properties, can pose problems regarding yield, reproducibility, and reliability of SOl VLSI that cannot be ignored. CHAPTER FOUR ANOMALOUS LEAKAGE CURRENT OF SMALL-GRAIN POLYSILICON MOSFETs 4.1 Introduction Recent laboratory achievements [MA84; MA85] imply that the first commercial adaptation of three-dimensional integration may be stacked CMOS VLSI memory chips in which one of the complementary transistors (usually p-channel) is fabricated in a- layer of LPCVD polysilicon on silicon dioxide. Grain-boundary passivation (e.g., via hydrogenation FSH841) is required to render the polysilicon MOSFET performance acceptable for the circuit application, although single-crystal silicon device characteristics are not needed. The polysilicon transistor is inferior to the single-crystal counterpart, especially because of anomalous high leakage current and exceptionally high gate-voltage swing [0N82; SH84]. In this chapter we model the OFF-state leakage current of the small-grain polysilicon SOl MOSFET, which we theorize is controlled by grain-boundary traps. By qualitative deduction, we identify a plausible physical mechanism underlying the leakage current, and then show that it is consistent with the anomalously strong dependences on the gate and drain voltages that have been observed [0N82; SH84; MA85]. Such physical insight can aid the design of polysilicon MOSFETs to control and minimize the leakage. A typical set of measured current-voltage characteristics of an unpassivated LPCVD polysilicon MOSFET is shown in Fig. 4.1. The particular device is p-channel and operates in the accumulation mode. The back gate and the source are grounded. In the OFF state (front-gate voltage VGf> 0), the film body (grains) is completely depleted of free carriers, facilitated by grain-boundary trapping of holes. The front surface is inverted for VGf sufficiently high (> -0), facilitated by positive charge at the interface. The leakage current (IL = ID in the OFF region) increases exponentially with VGf and as a power (> 1) of the drain voltage VD. The device characterized in Fig. 4.1 is long-channel (32 1m), which means- that the anomalous IL(VIf,VD) is not a' shortchannel effect [AK82]. Other measurements [MA85] reveal that IL is virtually independent of the (long) channel length, and that the same anomalies obtain for other polysilicon MOSFET structures, e.g., the n-channel inversion-mode device. Passivation of the grain boundaries in hydrogen plasma [SH84] reduces IL by two or three orders of magnitude, but does not remove the strong dependences on VGf and VD. To physically model IL, we first deduce the most plausible mechanism producing the leakage by qualitatively eliminating the possibilities of other significant mechanisms. Although this deduction is not rigorous, we support the model by demonstrating correlation between its IL(VGf,VD) predictions and measured data. A rigorous corroboration, which would require comprehensive analyses of all the possible mechanisms and extensive measurements of special test structures, is not feasible. 10701 1 1 1 1 1 1 1 1 1 10 0 10- -0.05 V 12 - - ' -5.0 5.0 VGf (V) Fig. 4.1 Measured current-voltage characteristics of an unpassivated LPCVD polysilicon MOSFET (p-channel, accumulfpion-m5ode; Z = 128 pro, L = 32 pin, t = 500 A, NA = 10ï¿½ cim- ). The polysilicon film is O.1% pm-thick and was deposited via LPCVD on a 0.5 pm-thick layer of silicon-dioxide. The back gate and the source are grounded. 10- Possibly significant leakage mechanisms in the polysilicon MOSFET are: (a) space-charge-limited flow [R173; SC82] of holes from source to drain through the (depleted) film body; (b) thermal emission of carriers [SZ811, via grain-boundary traps, in the depletion region near the drain; (c) field-enhanced (Poole-Frenkel [GR82]) thermal emission in the drain depletion region; (d) impact ionization (avalanching) [DU78] in the drain depletion region; (e) band-band field emission (tunneling) [R173] in the drain depletion region; and (f) field emission via grainboundary traps [GR84], or possibly metal precipitates [LEF82], in the drain depletion region. The measured independence of IL on (long) channel length [MA85] rules out space-charge-limit-ed flow.- The strong observed dependences of IL on VGf and VD imply that thermal-emission current, which depends only weakly on VD, is not significant. The observed saturation of IL with increasing VGf in Fig. 4.1 is inconsistent with predominant Poole-Frenkel emission or avalanching. Furthermore the electric field in the drain depletion region, for the values of VGf and VD used in the measurements (Fig. 4.1), is not high enough to produce significant avalanching or band-band tunneling. We are thus left with field emission through grain-boundary traps, or metal precipitates, as the most plausible mechanism underlying the observed IL(VGf,VD). The strong dependence on VGf further implies that the predominant field emission occurs near the front surface, in fact between the p+ drain and the n inversion layer where the electric field is highest. If the hack surface is inverted, significant field emission can occur there also. Our analysis of the field emission emphasizes grain-boundary traps, not metal precipitates, for two main reasons. First, neutron activation analyses [SH85] of the LPCVD polysilicon reveal concentrations (< 1013 cm-3) of metallic impurities comparable to those in bulk silicon and much lower than the grain-boundary trap density. Second, the grain boundaries getter metallic impurities, which tends to prevent the formation of metal precipitates. In the next section, we develop an analytic model for the trapassisted field-emission current in the LPCVD polysilicon MOSFET. To support the model, we compare its predictions of IL(VGf,VD) with measured data !from p-channel accumulation-mode! and n-channel inversionmode devices. Good correlation is shown, and field emission at the back surface is suggested as the mechanism underlying the minimization of the leakage current at relatively low values of VGf like that illustrated in Fig. 4.1. Insight regarding the physics underlying the anomalously strong IL(VGf,VD) dependences is readily provided by the model, and implies design criteria to control IL in polysilicon MOSFETs. 4.2 Leakage Current Model To develop a physical model for the leakage current in the LPCVD polysilicon MOSFET, we consider the p-channel accumulation-mode device, the basic structure of which is illustrated in Fig. 1.2. The leakage current in other polysilicon devices, e.g., the inversion-mode MOSFET [0N82], can be described basically by the same model as we discuss later. The accumulation-node device is of interest because it can be designed to have reasonably low threshold voltages [SH84; MA85]. Such design depends on the complete depletion of the small (- 1O00A) grains in the body via carrier trapping at localized grain-boundary states [BA78a; SH84]. The OFF state of the device then obtains in the absence of an accumulation layer under the gate; the surface is either depleted or inverted. Our model is based on the latter, more prevalent surface condition, and hence describes the anomalous strong dependences of the leakage current on the gate and drain voltages [0N82; SH84]. As discussed in Section 4.1, the leakage current is assumed to originate via field emission through grain-boundary traps (bound states) at the drain junction. .Because of the complete depletion of the body, this emission occurs predominantly at the surface under the gate in the depletion region between the p+ drain and n inversion layer. This conclusion is consistent with the observed strong dependence of the leakage current on the gate voltage. Although band-band tunneling in this surface region would occur only at very high electric fields [R173], substantial field emission can occur at low fields because of the high density of grain boundary traps in the LPCVD polysilicon. To enable an analytic description of this emission, we assume that the traps are monoenergetic (at ET in the energy gap) and uniformly distributed in space FKA72; DE80; DE84] (with trap density NT = 2NsT/dG where NST is the grain-boundary areal trap density and dG is the average columnar grain size). The critical region of the device described above is illustrated in Fig. 4.2. It is the depletion region near the surface between the p+ 75 Polyslllcon Gate n InvearsI on Layer y a- :ae. arm-, Depletion P+ Region. Drain Fig. 4.2 Critical depletion region between the drain and the inversion layer with important electric field components in (4.11)(4.13) indicated. drain and the n inversion layer. Because of the lateral diffusion of the drain under the gate, the inversion extends into the drain region with complicated geometry. Thus the effective cross-sectional area of the n-p+ junction is difficult to define. The electric field in this region, which governs the field emission, is two-dimensional [FR69; TE84] (for wide channels) depending on both the gate and drain voltages, VGf and VD. Following a previous analysis EFR69] of the bulk MOSFET in the saturation region, we will treat this complex twodimensional problem empirically, which enables us to model the field emission processes as occurring predominantly parallel to the surface (in the y-direction). This simplification is commensurate with device ambiguities, yet yields a model that is consistent with experimental results and hence is insightful. The energy-band diagram in the'region, including the grain-boundary trap level, is sketched (versus y) in Fig. 4.3. Maintaining the degree of complexity implied above, we assume that the electric field in the y-direction, Fy = (dEi/dy)/q, is constant in the depletion (barrier) region, and that the electrons (or holes) tunnel through the barrier via the traps at constant energy [GA70; R0771. The leakage current derives then from the combined net field emission of holes, ITV, from the traps to the valence band in the drain region, and of electrons, ITC, from the traps to the conduction band in the inversion region. For traps within an incremental width (dy) of the depletion region (see Fig. 4.3) [GA70; R077; GR84], Electron Energy y ECp ET EVp EFp V ,, I " i I y(Ecn) Y(Evp) Fig. 4.3 Electron energy-band diagram along the surface between the (n) inversion layer and the (p+) drain. dITV q(l - fT)NTZXedy (4.1) TV T TV and qf TNT ZXe dY d(-ITc) TTC (4.2) TC the rates of the respective inverse processes are negligibly small for the nonequilibrium conditions of interest (e.g., IVD1 >> kT/q). In (4.1) and (4.2), Z is the channel width of the MOSFET, xe is an effective depth of the junction region (see Fig. .4.2), fT is the electron occupancy factor of the traps, and TTV and TTC are the time constants for hole and electron tunneling respectively, which depend on Fy and ET as we discuss later. In the steady state, the number of trapped electrons is constant, and hence in the absence of significant thermal emission, dITV = d(-ITc). Equating (4.1) and (4.2) then, we have f T TC (4.3) T TTC +TTV The incremetal field-emission current is given by the combination of (4.1) or (4.2) with (4.3). The total (leakage) current is then expressed by integrating the result over that portion of the depletion region across which valence band-trap-conduction band carrier transitions at constant energy are possible (see Fig. 4.3): Y(Evp) _ dy (4.4) L ZXeNT y(E TC+ TV Since we assume that Fy is constant in the depletion region, F D E Vp E ECn Fy qy(Evp) - Y(Ecn)] (4.5) where D is the potential barrier height and. W is the width. The tunneling time constants are thus independent of y also, and (4.4) can be written as 1 ) V N 1 (4.6) IL ZXNT(T + T TV F( TC TV y Because the leakage current is low, little voltage is dropped across the inversion layer; VD is dropped across the drain depletion region as indicated in Fig. 4.3: EFp - EFn qIVr! 1 (4.7) For JVDI greater than a few tenths of a volt then, we note that the quasi-Fermi level separation in (4.7) equals approximately (Evp - ECOn) in (4.6). Thus I-- qZx) ï¿½ (4 .8) L TC + TTV y The time constants TTC and TTV reflect the probability per unit time that a trapped carrier will tunnel through a triangular barrier, defined by ET as shown in Fig. 4.3, to its respective band. Based on the WK8 approximation rR077; SZ81; GR84], 4(2mp)1/2(F - EV3/2 T TV T OVexp[ 3qhFy T (4.9) and 4(2mnI/2(E C- ET)3/2 T TC T OCexp[ 3qhFy T (4.10) where mp and mn are the appropriate [LU72; GR841 effective masses for the tunneling holes and electrons (mp - mn = 0.2 m0 [GR84] where m0 is the free electron mass), and where Tov. and TOC are effective carrier transit times in the valence and conduction bands, which we assume to be constants [LU721. For a parabolic barrier [R077; SZ81], the numerical constant in the exponential argument is different (lower). The effect of the barrier shape on IL can thus be studied by varying the effective masses in (4.9) and (4.10), which in fact cannot be unequivocally defined [LU72; GR84]. We stress that the field-emission current IL in (4.8) is predominant because of the high density NT of traps that increase substantially the tunneling probability, conveyed by (4.9) and (4.10), over that for hand-band tunneling [SZ81]. To complete the model for IL defined by (4.8)-(4.10), we must describe Fy in terms of VGf and VD. As discussed with reference to Fig. 4.2, the electric field in the depletion region at the surface, and indeed the two-dimensional region itself are virtually impossible to describe analytically. However an empirical description, commensurate with our one-dimensional fieldemission model, can be given based on a previous analysis [FR69] of the bulk MOSFET in the saturation region. With reference to Fig. 4.2 three "components" of Fy can be identified [FR69]: Fy"=F+F2+F3 . (4.1) In the empirical representation (4.11), F, is the electric field that exists in the absence of the gate; i.e., F1 is due to the depletion charge in the reverse-biased drain junction. The presence of the gate produces fringing electric fields F2 and F3: F2 is due to the gate-drain potential drop and F3 is due to the potential difference between- the drain end of the inversion layer and the gate. Following [FR69], which has been supported experimentally. [BR811, we assume that F2 and F3 are proportional to the respective normal electric fields at the surface (see Fig. 4.2): F2 = aFD (4.12) 2 S F3 13 F (4.13) whereas and a are constant "field-fringing factors". At the p+ drain side of the depletion region, the surface potential is nearly zero, and D) C of + f F -4V - -V -p S VGf D -PMS +C (4.14) s of where Cof is the .(front) gate oxide capacitance, Off is the fixed charge + density at the (front) Si-SiO2 interface, and P is the gate-p+ silicon work function difference. For an n+ polysilicon 'gate, + P -E g/q where Eg is the silicon energy gap. At the inversion-layer side of the depletion region, F Cof - + off E-q) (4.15) S j-s (VGf cfMS o f -q since the potential drop between the end of the (strong) inversion layer and the source region (VS = 0) is about Eg/q. Using the one-dimensional analysis of the reverse-biased p-n junction [SZ81], we express F1 : D ]1/2 (4.16) s as an average value of the y-dependent electric field in the depletion region between the p+ drain and the n inversion layer. In (4.16), n is defined by (2.3) and (2.21), Cof f (VGf " VTf (4.17) qxi (eff) where Xi(eff) is the inversion layer thickness (- 100 A) defined in Section 2.2.1 and VTf is the strong-inversion threshold voltage of the MOS structure, the characterization of which depends not only on the structural properties but also on the polysilicon film properties, e.g., NST, ET, and dG [KA72J. The potential barrier height D (see Fig. 4.3) is defined by VD and the surface potential in the inversion layer. For strong inversion, E SD g + IVDI ï¿½ (4.18) The combination of (4.8)-(4.18) gives an analytic description of IL and its dependences on VGf and VD for the p-channel accumulation-mode polysilicon MOSFET. The control of IL by the grain boundaries is conveyed by NT (= 2NsT/dG) and ET (relative to the band edges) in the i + model. The parameters Cof, VTf, MS' Qff' and xi(eff) are defined hy the MOSFET structure, or can be estimated. The field-fringing factors a and 0 , and xe must be estimated or evaluated through comparisons of model predictions and measured data; they control the relative significance of VGf and VD in determining IL. The transit times tOV and TOC associated with the field emission are in effect normalization factors for the tunneling time constants; they have been characterized from first principles [LU721, although not accurately. The model therefore actually predicts the normalized IL(VGf,VD) dependences; absolute measured values however could be matched by assigning proper values to-r0V and-TOC. The model does indeed predict the general IL(VGf,VD) dependences exemplified in Fig. 4.1. To demonstrate this correlation and to indicate the physical insight afforded, we plot in Fig. 4.4 the calculated leakage current versus VGf and VD for a device similar to the one measured. Quantitative comparisons of the theoretical and experimental characteristics should not be made because actual values of many of the model parameters are unknown. In the calculations, we used Z = 128 um (L is irrelevant) and Cof = 6.9 x 10-8 F/cm2 (= ï¿½of/tof with tox = 500 A) corresponding to the device measured. We chose xe = 500 A, greater than xi(eff) but less than the extent of the lateral drain diffusion, and a = 0.2 and 8 = 0.6, crude values based on the bulk MOSFET model [FR69]. We let TrOV = TOC-- To : 10-12 sec, a crude estimate derived experimentally [GR84], and NT : 2 x 1017 cm-3, which corresponds to NST : 1012 cm"2 and dG = 1000 A. We put ET at midgap. We characterized VTf = VTO - Qff/Cof with Qff/q 1012 cm2 and VTO i i V, which yield VTf -2 V. [We stress that VTf is the strong-inversion threshold voltage, in contrast to the turn-ON threshold voltage (for strong accumulation), which is about -4 V as indicated in Fig. 4.1.] From (4.6) we note that IL is directly proportional to the factor ZxeNT/tO, and hence changes in these parameters simply alter the magnitude of IL and not shape of the semi-logarithmic IL(VGf,VD) characteristics in Fig. 4.4. Additional calculations show also that the 101u --I 10-14 1 10 8 6 4 2 0 VGt (V) Fig. 4.4 Calculated leakage current versus (front) gate and drain voltages for a p-channel accumulation-mode LPCVD polysilicon MOSFET. shape of IL(VGf) is not strongly dependent on a and a. Variations in the oxide capacitance Cof, however, produce significant changes in IL(VGf), primarily because of its influence on F1 described by (4.16) i and (4.17). Similar changes are produced by variations in VTf, or VTO as shown by the calculations plotted in Fig. 4.5. The leakage is most sensitive to VGf just above threshold for strong inversion; well above threshold (high VGf), Fy in (4.11) is high and the tunneling time constants TTV and TTC in (4.9) and (4.10) tend toward minimum values (TO), thereby causing IL to approach a value independent of VGf. The calculated dependence of IL on the trap level ET is illustrated in Fig. 4.6. As can be inferred from (4.9) and (4.10), traps near midgap are most effective in the field-emission processes; shallow traps do not facilitate carrier tunneling to the opposite band. We note that ET could possibly be inferred from measurements of the slope of the IL(VGf) characteristic. Although the model predictions can he brought into close agreement with the measured leakage current by altering parameter values, the benefit of doing so is questionable because of the uncertainty in the actual physical structure and parameters of the device. For example, variations in the shape of the potential barrier in the drain depletion region, and/or in the effective masses in (4.9) and (4.10) can result in considerable changes in the predicted IL(VGf,VD) characteristics. Such changes are illustrated in Fig. 4.7 where we plot the calculated IL(VGf,VD) with mp = mn varying from 0.1 mo to 0.5 Mo. We note however from the calculations plotted in Figs. 4.4-4.7 that the measured I " ,-14 1U 10 0 VG f (V) Fig. 4.5 Calculated variation of IL(VGf) for different (stronginversion) threshold voltages V1f = VTO- Off/Cof (with Off/Cof = 2.3 V). 10 6 - . VTO=-3V I I I . . I0"10 0 88 10-F6 - (Ec-ET) E9 E/2 VD=-5v 10 10 8 6 4 2 0 VGf (V) Fig. 4.6 Calculated dependence of IL(VGf) on the grain-boundary trap energy level ET. 1 VGf (V) Fig. 4.7 Calculated dependence of effective masses mp = mn- IL(VGf) on tunneling carrier 1076 10710 -J IL(VGf,VD) characteristics in Fig. 4.1 could be simulated well by the model with physically reasonable parameter values. The plots in Figs. 4.4-4.7, as well as the data in Fig. 4.1 which indeed reflect the general IL(VGfVD) dependences seen in a variety of LPCVD polysilicon MOSFETs, show that IL varies predominantly exponentially with VGf (> VTf) for a given value of VD. This dependence results from the exponential dependence of TTV and TTC on Fy expressed in (4.9) and (4.10). The dependence. of IL on VD is also strong, as emphasized by the calculations of Fig. 4.4 replotted in Fig. 4.8. For a given value of VGf (> VT), IL varies as IVDIm where m - 1-10. This dependence follows from (4.8) and the implicit dependences on VD of Fy and T TC and TTV. Note that m decreases with increasing VGf and jVf)j. The predicted m vs. VGf variation can be seen in the measured data in Fig. 4.1, although the m vs. IVDI variation cannot. This discrepancy appears to be due to (trap-assisted) avalanching in the drain depletion region that causes the measured leakage current to increase abruptly as IVDI approaches -10 V. We note further that the measured m (1-5) for all the devices is lower than that calculated, which could indicate that the effective masses in (4.9) and (4.10) are lower than 0.2 m0 as we assumed, or that the potential barrier is more parabolic than linear. Additional calculations reveal that the effective masses must be reduced by about an order of magnitude to bring m down to the measured value. Then to retain theoretical-experimental agreement for the absolute value of IL, the effective carrier transit times in (4.9) and (4.10) must be reduced by three to four orders of magnitude. Higher values of a and lower values of a also weaken the dependence of IL on VI). 10-11 1013 -VD (V) Fig. 4.8 Calculated I (VGf,Vn) characteristics dependence on rain voltage V. emphasizing -J 10 the Additional support for the field-emission model for IL is obtained from consideration of the influence of the back-gate bias Vrb. This influence is related to the minimization of the leakage current at low VGf depicted in Fig. 4.1. As VGf is reduced to switch the device from OFF to ON (accumulation), the measured leakage current reaches a minimum value, whereas the calculated current continues to be reduced monotonically. This simply indicates that the field emission in our model is insignificant at this point, and the actual minimum leakage current is produced by another mechanism, possibly field emission *via grain-boundary traps near the back surface where positive interfacial charge could indeed cause inversion with the back gate grounded (VGb : 0). The minimum measured currents in Fig. 4.1 show a strong dependence on VD like our model predicts, and hence we surmise that they result from field emission near the hack surface. We note further that. the value of VGf at which the minimum IL obtains decreases as VD increases. This implies that the dependence on VD of the field-emission current at the back surface is weaker than that of the front-surface current, which, based on our model, is a result of the thicker back-gate (underlying) oxide. Measured ID(VGf,VGb) characteristics for a hydrogenated n-channel inversion-mode LPCVD thin-film polysilicon MOSFET, with VD 5V, are plotted in Fig. 4.9. For VGf << 0 (OFF region), ID (= L) is independent of VGb and increases exponentially with IVGfI in accordance with our model. For low VGf (near the minimum 10), with VCb < 0 which implies that the back surface is accumulated, 10 increases with IVGbI |

Full Text |

PAGE 1 ())(&76 2) *5$,1 %281'$5,(6 ,1 32/<6,/,&2121,168/$725 62,f 026)(76 %\ $'(/02 257,=&21'( $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$ PAGE 2 7(1*2 (/ ,10(162 3/$&(5 '( '(',&$5 (67$ 7(6,6 $ 0,6 3$'5(6 $/,&,$ &21'(%5$1'7 '( 257,= $'(/02 257,= 3,21(52 PAGE 3 $&.12:/('*0(176 a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Â¯YDU 3UV 3LHUUH 6FKPLGW *XVWDYR 5RLJ 3DXO (VTXHGD DQG )UDQFLVFR *DUFLD IRU DOO WKH\ KDYH GRQH LQ VXSSRUW RI P\ JUDGXDWH ZRUN fÂµ , DP LQILQLWHO\ LQGHEWHG WR P\ SDUHQWV DQG IDPLO\ IRU WKHLU LQFUHGLEOH VXSSRUW DQG HQFRXUDJHPHQW WKURXJKRXW P\ JUDGXDWH VFKRRO FDUHHU 7KH ILQDQFLDO VXSSRUW RI 7KH &RQVHMR 1DFLRQDO GH ,QYHVWLJDFLRQHV &LHQWÂ¯ILFDV \ 7HFQROÂµJLFDV &21,&,7f 1DYDO 5HVHDUFK /DERUDWRU\ 15/f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Â‘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fÂ¬3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ ())(&76 2) *5$,1 %281'$5,(6 ,1 32/<6,/,&2121,168/$725 62,f 026)(76 5\ $'(/02 257,=&21'( $XJXVW &KDLUPDQ -HUU\ * )RVVXP 0DMRU 'HSDUWPHQW (OHFWULFDO (QJLQHHULQJ 7KLV GLVVHUWDWLRQ SUHVHQWV SK\VLFDO PRGHOV WKDW GHVFULEH WKH HIIHFWV RI JUDLQ ERXQGDULHV RQ WKH VWHDG\VWDWH FXUUHQWYROWDJH FKDUDFWHULVWLFV RI ODUJH DQG VPDOOJUDLQ SRO\VLOLFRQ 62, 6L RQ6L M!f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f DQ HIIHFWLYH WXUQRQ FKDUDFWHULVWLF LQ WKH OLQHDUUHJLRQ FRQWUROOHG E\ WKH JUDLQ ERXQGDULHV WKDW RFFXUV EH\RQG WKH VWURQJLQYHUVLRQ WKUHVKROG YROWDJH DQG KHQFHIRUWK GHILQHV WKH FDUULHU PRELOLW\ WKHVKROG YL L PAGE 8 Â« YROWDJH DQG WKH HIIHFWLYH ILHOG HIIHFW FDUULHU PRELOLW\ Ef D QHDUO\ H[SRQHQWLDO GHSHQGHQFH RQ WKH IURQWf JDWH YROWDJH GHILQHG E\ WKH SURSHUWLHV RI WKH JUDLQ ERXQGDULHV IRU PRGHUDWHLQYHUVLRQ FRQGXFWDQFH DQG Ff WKDW D JUDLQ ERXQGDU\ QHDU WKH GUDLQ FDQ FRQWURO WKH FRQGXFWLRQ SURSHUWLHV IRU DOO ZHDNWRVWURQJf LQYHUVLRQ FRQGLWLRQV LQ DOO OLQHDU WRVDWXUDWLRQf UHJLRQV RI RSHUDWLRQ 7KH PRGHOV IRU WKH VPDOOJUDLQ SRO\VLOLFRQ 62, 026)(7 SUHGLFW Df WKH DQRPDORXV OHDNDJH FXUUHQW 2)) VWDWHf ZKLFK LV DWWULEXWHG WR ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV LQ WKH IURQWf VXUIDFH GHSOHWLRQ UHJLRQ DW WKH GUDLQ Ef WKDW WKH JDWHYROWDJH VZLQJ IRU WKH n VXEWKUHVKROG GUDLQ FXUUHQW 21 VWDWHf GHSHQGV VWURQJO\ RQ WKH JUDLQn ERXQGDU\ SURSHUWLHV DQG ZHDNO\ RQ WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV Ff WKDW WKH HIIHFWLYH WKUHVKROG YROWDJH 21 VWDWHf GHSHQGV VWURQJO\ RQ JUDLQERXQGDU\ SURSHUWLHV DQG RQ WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV DQG Gf WKH GHYLFH GHVLJQ PRGLILFDWLRQV WR FRQWURO DQG UHGXFH WKH OHDNDJH FXUUHQW WKH JDWH YROWDJH VZL QJ DQG WKH HIIHFWLYH WKUHVKROG YROWDJH Y L L L PAGE 9 &+$37(5 21( ,1752'8&7,21 %HFDXVH RI WKH DGYDQWDJHV RI GLHOHFWULF LVRODWLRQ DQG WKUHH GLPHQVLRQDO 'f LQWHJUDWLRQ >*, /$ 0$@ WKHUH LV PXFK LQWHUHVW LQ 62, VLOLFRQRQLQVXODWRUf 026)(7V 7KH DGYDQWDJHV RI WKHVH GHYLFHV FRPSDUHG ZLWK WKH VLQJOHFU\VWDO FRXQWHUSDUW DUH >/$@ Df LQFUHDVHG FLUFXLW VSHHG GXH WR UHGXFHG SDUDVLWLF FDSDFLWDQFH Ef VXSHULRU KDUGQHVV WR WUDQVLHQW UDGLDWLRQ DQG Ff HOLPLQDWLRQ RInODWFKXS ZKLFK LV RI IXQGDPHQWDO LPSRUWDQFH ZKHQ WKH IHDWXUH VL]HV LQ &026 FRPSOHPHQWDU\ PHWDOR[LGHVHPLFRQGXFWRUf WHFKQRORJ\ DUH VFDOHG WR VPDOOHU GLPHQVLRQV 7RGD\ &026 LV WKH GRPLQDQW WHFKQRORJ\ IRU 9/6, YHU\ ODUJH VFDOH LQWHJUDWLRQf EHFDXVH RI ORZ SRZHU FRQVXPSWLRQ VXSHULRU QRLVH PDUJLQV EHWWHU FRPSDWLELOLW\ ZLWK DQDORJ FLUFXLWV DQG UHGXFHG YXOQHUDELOLW\ WR VRIW HUURUV 7KH PRVW SURPLVLQJ 62, WHFKQRORJLHV IRU 9/6, DUH 62, IRUPHG E\ KLJKGRVH LRQ LPSODQWDWLRQ >+(@ 62, XVLQJ SRURXV VLOLFRQ >%$@ VLOLFRQRQVDSSKLUH 626f >6$@ EHDP UHFU\VWDOOL]DWLRQ RI SRO\VL LFRQ RQVLOLFRQ GLR[LGH >/$@ DQG DVGHSRVLWHG /3&9' ORZSUHVVXUH FKHPLFDO YDSRU GHSRVLWLRQf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f LV RI SUDFWLFDO LQWHUHVW EHFDXVH RI WKH UHODWLYHO\ JRRG SHUIRUPDQFH RI WKH GHYLFHV IDEULFDWHG ZLWK LW FRPSDUHG ZLWK WKDW RI WKH VLQJOHFU\VWDO FRXQWHUSDUW >/$ 76 @ 7KH DVGHSRVLWHG /3&9' 62, WHFKQRORJ\ ZKLFK SURGXFHV VPDOOJUDLQ SRO\VLOLFRQ XUQf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f SRO\VLOLFRQ 62, 026)(7 VKRZQ LQ )LJ GRHV QRW UHTXLUH WKH UHFU\VWDOOL]DWLRQ VWHS EXW LW LV LQIHULRU WR WKH VLQJOHFU\VWDO FRXQWHUSDUW HVSHFLDOO\ EHFDXVH RI DQRPDORXV KLJK OHDNDJH FXUUHQW DQG H[FHSWLRQDOO\ KLJK JDWHYROWDJH VZLQJ UR1 6+@ 7R LPSURYH WKH SHUIRUPDQFH RI WKH VPDOOJUDLQ 62, 026)(7 IRU DSSOLFDWLRQV WKDW GR QRW UHTXLUH VLQJOHFU\VWDO VLOLFRQ GHYLFH FKDUDFWHULVWLFV HJ ORDG HOHPHQWV IRU D GHQVH VWDWLF 5$0f JUDLQn ERXQGDU\ SDVVLYDWLRQ HJ YLD K\GURJHQDWLRQ >.$ 6+@f KDV EHHQ VXFFHVVIXOO\ XVHG >0$ 0$6@ 8QOLNH WKH ODUJHJUDLQ SRO\VLOLFRQ GHYLFH WKH VPDOOJUDLQ SRO\VLOLFRQ GHYLFH FDQ EH GHVLJQHG WR EH RSHUDWHG LQ HLWKHU WKH DFFXPXODWLRQ RU LQYHUVLRQPRGH EHFDXVH WKH ILOP ERG\ JUDLQVf LV FRPSOHWHO\ GHSOHWHG RI IUHH FDUULHUV IDFLOLWDWHG E\ JUDLQERXQGDU\ WUDSSLQJ 7KH SXUSRVH RI WKLV GLVVHUWDWLRQ LV WR GHYHORS SK\VLFDO PRGHOV IRU WKH HIIHFWV RI JUDLQ ERXQGDULHV LQ ODUJH DQG VPDOOJUDLQ SRO\VLOLFRQ 62, 026)(7V ZKLFK DUH XVHIXO IRU WKH SUHGLFWLRQ DQG RSWLPL]DWLRQ RI PAGE 12 Â« )LJ &URVVVHFWLRQ RI WKH IRXU WHUPLQDO QFKDQQHO LQYHUVLRQPRGH EHDPUHFU\VWDO L]HG ODUJHJUDLQf SRO\VLOLFRQ 62, 026)(7 7KH WHUPLQDO YROWDJHV DUH UHIHUHQFHG WR WKH VRXUFH YROWDTH 9V f PAGE 13 3LJ %DVLF VWUXFWXUH RI WKH IRXUWHUPLQDO SFKDQQHO DFFXPXODWLRQn PRGH /3&9' VPDOJUDLQf SRO\VLOLFRQ 62, 026)(7 PAGE 14 GHYLFH SHUIRUPDQFH LQ 62, LQWHJUDWHG FLUFXLWV &KDSWHUV 7ZR DQG 7KUHH FRQFHUQ WKH ODUJHJUDLQ SRO\VLOLFRQ GHYLFH DQG &KDSWHUV )RXU DQG )LYH FRQFHUQ WKH VPDOOJUDLQ SRO\VLOLFRQ GHYLFH 7KH PDMRU FRQWULEXWLRQV PDGH LQ WKLV GLVVHUWDWLRQ DUH f WKH PRGHOLQJ RI WKH HIIHFWV RI JUDLQ ERXQGDULHV IRU DOO UHJLRQV RI RSHUDWLRQ LQ ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7V f WKH SK\VLFDO FKDUDFWHUL]DWLRQ RI WKH DQRPDORXV OHDNDJH FXUUHQW 2)) VWDWHf LQ VPDOOJUDLQ SRO\VLOLFRQ 62, 026)(7V f WKH QXPHULFDO PRGHOLQJ RI WKH VXEWKUHVKROG GUDLQ FXUUHQW DQG WKH WKUHVKROG YROWDJH 21 VWDWHf RI WKLQILOP VPDOOJUDLQ SRO\VLOLFRQ 62, 026)(7V f WKH GHYHORSPHQW RI WKH IRXQGDWLRQ RI D FKDUJHVKHHW PRGHO >55@ IRU WKH WKLQILOP VLQJOHFU\VWDO 62, 026)(7 f WKH H[SHULPHQWDO VXSSRUW IRU WKH GHYHORSHG PRGHOV IURP PHDVXUHPHQWV RI UHSUHVHQWDWLYH 62, 026)(7V :H GHULYH LQ &KDSWHU 7ZR D WKHRUHWLFDO GHVFULSWLRQ RI WKH OLQHDU UHJLRQ GUDLQ FXUUHQW RI WKH ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7 ZKLFK LV YDOLG IRU DOO LQYHUVLRQ OHYHOV DQG DFFRXQWV IRU DUELWUDU\ RULHQWDWLRQ RI WKH JUDLQ ERXQGDULHV 7KH FRUUHVSRQGLQJ FKDQQHO FRQGXFWDQFH VKRZV DQ HIIHFWLYH WXUQRQ FKDUDFWHULVWLF FRQWUROOHG E\ WKH JUDLQ ERXQGDULHV WKDW RFFXUV EH\RQG WKH VWURQJLQYHUVLRQ WKUHVKROG +HQFHIRUWK WKH FDUULHU PRKLOLW\ WKUHVKROG YROWDJH ZKLFK H[FHHGV WKH DFWXDO RQH DQG WKH HIIHFWLYH FDUULHU PRELOLW\ ZKLFK LV W\SLFDOO\ KLJKHU WKDQ WKH DFWXDO LQWUDJUDLQf RQH DUH GHILQHG )RU VXIILFLHQWO\ KLJK JDWH YROWDJH WKH JUDLQKRXQGDU\ SRWHQWLDO EDUULHU LV ORZ HQRXJK WKDW WKH FKDQQHO PAGE 15 FRQGXFWDQFH LVQRW VLJQLILFDQWO\ LQIOXHQFHG E\ WKH ERXQGDULHV )RU JDWH YROWDJHV ORZHU WKDQ WKH FDUULHU PRELOLW\ WKUHVKROG YROWDJH WKH FRQGXFWDQFH YDULHV QHDUO\ H[SRQHQWLDOO\ ZLWK WKH JDWH YROWDJH DQG GHSHQGV VWURQJO\ RQ WKH JUDLQERXQGDU\ SURSHUWLHV *UDLQ ERXQGDULHV SHUSHQGLFXODU WR WKH FDUULHU IORZ LQ WKH FKDQQHO SURGXFH WKH VWURQJHVW HIIHFWV RQ WKH FRQGXFWDQFH 7R VXSSRUW WKLV DQDO\VLV DQG WR VWUHVV LWV SUDFWLFDOLW\ ZH FRPSDUH PRGHOV SUHGLFWLRQV ZLWK PHDVXUHG FXUUHQW YROWDJHWHPSHUDWXUH FKDUDFWHULVWLFV RI DVHUUHFU\VWDOOL]HG 62, 026)(7V IDEULFDWHG DW 7H[DV ,QVWUXPHQWV >/$@ 7KH WKHRUHWLFDOH[SHULPHQWDO DJUHHPHQW LV JRRG DQG LQ DGGLWLRQ WR LQGLFDWLQJ SURSHUWLHV RI WKH JUDLQ ERXQGDULHV LQ WKHVH GHYLFHV LW H[HPSOLILHV KRZ WKH PRELOLW\ WKUHVKROG YROWDJH DQG WKH HIIHFWLYH FDUULHU PRELOLW\ FDQ EH HDVLO\ PLVLQWHUSUHWHG DV WKH DFWXDO WKUHVKROG YROWDJH DQG PRELOLW\ ZKHQ FRQYHQWLRQDO 026)(7 WKHRU\ LV XVHG DV WKH EDVLV IRU LQWHUSUHWLQJ HOHFWULFDO PHDVXUHPHQWV RI 62, 026)(7V 6XFK PLVLQWHUSUHWDWLRQV FDQ REVFXUH HVVHQWLDO FULWHULD IRU DFKLHYLQJ RSWLPDO GHVLJQV RI 62, GHYLFHV DQG LQWHJUDWHG FLUFXWLV )RU H[DPSOH RXU SK\VLFDO DQDO\VLV UHYHDOV WKDW LQ SDUWLFXODU FDVHV JUDLQ ERXQGDULHV FDQ DFWXDOO\ EHQHILW WKH 62, 026)(7 SHUIRUPDQFH E\ SURGXFLQJ DQ XQXVXDOO\ KLJK WUDQVFRQGXFWDQFH 7KLV VXJJHVWV LQ FRQWUDVW WR WKH JHQHUDO EHOLHI WKDW RSWLPDO GHVLJQV PD\ QRW UHTXLUH HOLPLQDWLRQ RI DOO JUDLQ ERXQGDULHV :H GHVFULEH LQ &KDSWHU 7KUHH D SK\VLFDO PRGHO IRU WKH FXUUHQW YROWDJH FKDUDFWHULVWLFV RI WKH ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7 LQ DOO UHJLRQV RI RSHUDWLRQ 7KH HVVHQFH RI WKLV PRGHO LV DQ DFFRXQWLQJ IRU VL]DEOH SRVLWLRQGHSHQGHQW YROWDJH GURSV DFURVV WKH JUDLQ ERXQGDULHV PAGE 16 WKDW FDQ RFFXU ZKHQ WKH GHYLFH LV GULYHQ RXW RI WKH OLQHDU UHJLRQ 7KH FDUULHU WUDQVSRUW WKURXJK WKH JUDLQ ERXQGDULHV YL] RYHU SRWHQWLDO EDUULHUVFUHDWHG E\ FDUULHU WUDSSLQJf LV WKHQ QRQOLQHDU DQG WKH FKDQQHO FRQGXFWLRQ GHSHQGV RQ KRZ WKH JUDLQ ERXQGDULHV DUH GLVWULEXWHG EHWZHHQ WKH VRXUFH DQG WKH GUDLQ $OWKRXJK RXU PRGHO DFFRXQWV IRU DQ\ QXPEHU RI JUDLQ ERXQGDULHV LQ WKH FKDQQHO ZH DSSO\ LW KHUHLQ WR WKH PRVW OLNHO\ FDVH LQ EHDPUHFU\VWDOOL]HG 9/6,f RI DQ 62, 026)(7 ZLWK RQO\ RQH JUDLQ ERXQGDU\ :H HPSKDVL]H WKH LPSRUWDQFH RI WKH SRVLWLRQ RI WKH JUDLQ ERXQGDU\ DV ZHOO DV LWV HOHFWULFDO SURSHUWLHV LQ GHILQLQJ WKH FXUUHQW YROWDJH FKDUDFWHULVWLFV 0RGHO FDOFXODWLRQV VXSSRUWHG E\ OLPLWHG H[SHULPHQWDO UHVXOWV VKRZ WKDW JUDLQ ERXQGDULHV WHQG WR GHFUHDVH WKH GUDLQ FXUUHQW RI ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7V EXW FDQ LQFUHDVH WKH WUDQVFRQGXFWDQFH :H GHYHORS LQ &KDSWHU )RXU D SK\VLFDO PRGHO IRU WKH DQRPDORXV OHDNDJH FXUUHQW 2)) VWDWHf LQ VPDOOJUDLQ SRO\VLOLFRQ 62, 026)(7V EDVHG RQ ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV 7R VXSSRUW WKLV PRGHO ZH FRPSDUH LWV SUHGLFWLRQV ZLWK PHDVXUHG GDWD IURP SFKDQQHO DFFXPXODWLRQn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f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f FDQ EH H[SUHVVHG LQ WHUPV RI WKH HOHFWURQ TXDVL)HUPL OHYHO IRU DQ\ JUDLQKRXQGDU\ YROWDJH GURS 7KLV UHVXOW ZKLFK ZDV XVHG LQ &KDSWHU 7KUHH LQGLFDWHV WKDW D SUHYLRXV DVVXPSWLRQ >5$D@ ZKLFK HVWDEOLVKHV WKDW WKH FKDUJH WUDSSHG DW WKH JUDLQ ERXQGDU\ LV LQGHSHQGHQW RI WKH JUDLQERXQGDU\ YROWDJH GURS LV JHQHUDOO\ LQYDOLG ,Q &KDSWHU 7KUHH ZH KDYH DOVR DYRLGHG WKH XVH RI DQRWKHU FODVVLFDO EXW JHQHUDOO\ LQYDOLG DVVXPSWLRQ >08@ WKDW D FRQVWDQWf IUDFWLRQ RI WKH WKHUPLRQLFDOO\ HPLWWHG HOHFWURQV DUH FDSWXUHG E\ WKH JUDLQERXQGDU\ WUDSV PAGE 18 $V D ILUVW VWHS WRZDUGV WKH GHYHORSPHQW RI D SUDFWLFDO PRGHO IRU LQWHJUDWHG FLUFXLW GHVLJQ ZLWK 62,026)(7V ZH SUHVHQW LQ $SSHQGL[ % WKH IRXQGDWLRQ RI D FKDUJHVKHHW PRGHO >55@ IRU WKH WKLQILOP VLQJOHn FU\VWDO VLOLFRQ 026)(7 ,Q $SSHQGL[ & ZH LQFOXGH WKH FRPSXWHU SURJUDP XVHG LQ &KDSWHU )LYH WR FDOFXODWH WKH VXEWKUHVKROG GUDLQ FXUUHQW DQG WKH WKUHVKROG YROWDJH 21 VWDWHf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f JDWH YROWDJH WKH GHYLFH SDUDPHWHUV DQG WKH JUDLQ DQG JUDLQERXQGDU\ SURSHUWLHV :H UHVWULFW RXU DQDO\VLV WR FDVHV LQ ZKLFK WKH SRO\VLOLFRQ ILOP LV QRW FRPSOHWHO\ GHSOHWHG EHWZHHQ WKH IURQW DQG EDFN VXUIDFHV :H LQLWLDOO\ DVVXPH LQ 6HFWLRQ VWURQJ LQYHUVLRQ DQG WKDW WKH JUDLQ ERXQGDULHV LQ WKH FKDQQHO DUH SHUSHQGLFXODU WR WKH FDUULHU IORZ EXW ZH JHQHUDOL]H WKH DQDO\VLV LQ 6HFWLRQV DQG E\ UHPRYLQJ WKHVH WZR DVVXPSWLRQV UHVSHFWLYHO\ 7KH PRGHO FRPSULVHV WKH IROORZLQJ SK\VLFV Df WKH TXDQWXP PHFKDQLFDO GHVFULSWLRQ >+6@ RI WKH FDUULHU GLVWULEXWLRQ LQ WKH LQYHUVLRQ OD\HU ZKLFK LPSOLHV DQ DYHUDJH FDUULHU GHQVLW\ DQG LWV GHSHQGHQFH RQ WKH JDWH YROWDJH WKDW FDQ EH PRGHOHG EDVHG RQ WKH FODVVLFDO VROXWLRQ >& 6% Ef WKH WZRGLPHQVLRQDO SRWHQWLDO YDULDWLRQ QHDU D JUDLQ ERXQGDU\ LQ WKH FKDQQHO ZKLFK ZKHQ DSSUR[LPDWHG E\ FRXSOHG RQHGLPHQVLRQDO VROXWLRQV RI 3RLVVRQnV HTXDWLRQ GHILQHV WKH PAGE 20 fÂµ Â« JUDLQKRXQGDU\ EDUULHU KHLJKW UHVXOWLQJ IURP FDUULHU WUDSSLQJ >%$DE@ DQG Ff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f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n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g5R XQGHUOLHV WKH SUHGRPLQDQW LQIOXHQFH RI WKH ERXQGDU\ RQ WKH FKDQQHO FRQGXFWDQFH RI 62, 026)(7V DQG WKDW g5R UHVXOWV IURP FDUULHU WUDSSLQJ DW ORFDOL]HG JUDLQERXQGDU\ VWDWHV 7KH WUDSSLQJ DQG gJ DUH FKDUDFWHUL]HG E\ D WZRGLPHQVLRQDO VROXWLRQ RI 3RLVVRQnV HTXDWLRQ LQ WKH FKDQQHO %HIRUH ZH GLVFXVV WKLV VROXWLRQ DQG WKH FRUUHVSRQGLQJ WKHUPLRQLFHPLVVLRQ FXUUHQW ZH QXVW FRQVLGHU WKH LQWUDJUDLQ FDUULHU GLVWULEXWLRQ LQ WKH FKDQQHO DQG LWV GHSHQGHQFH RQ WKH JDWH ELDV ZKLFK GHILQH n)J4 :H UHIHU WR WKH IRXUWHUPLQDO QFKDQQHO LQYHUVLRQPRGH ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7 LOOXVWUDWHG LQ )LJ DQG ZH DVVXPH WKDW WKH JUDLQ ERXQGDULHV LQ WKH FKDQQHO DUH SHUSHQGLFXODU WR WKH HOHFWURQ IORZ ,QWUDJUDLQ (OHFWURQ 'LVWULEXWLRQ LQ &KDQQHO %HFDXVH WKH LQYHUVLRQ OD\HU WKLFNQHVV LV YHU\ QDUURZ RQ WKH RUGHU RI WKH HOHFWURQ GH 5URJOLH ZDYHOHQJWKf WKH WUXH HOHFWURQ GLVWULEXWLRQ Q[f LQ WKH FKDQQHO DZD\ IURP JUDLQ ERXQGDULHVf PXVW EH GHVFULEHG TXDQWXPPHFKDQLFDO O\ U+6O 7KLV GHVFULSWLRQ IROORZV IURP D VHOIFRQVLVWHQW VROXWLRQ RI WKH 6FKURGLQJHU HTXDWLRQ DQG 3RLVVRQnV HTXDWLRQ 7KH UHVXOW GLIIHUV PDUNHGO\ IURP WKH FODVVLFDO VROXWLRQ >& 6=EDVHG RQ 3RLVVRQnV HTXDWLRQ DQG 0D[ZHO%ROW]PDQQ VWDWLVWLFV [A LV QDUURZHU DQG Q[f LV PRUH XQLIRUP >+6@ +RZHYHU WKH LQYHUVLRQOD\HU DUHDO FKDUJH GHQVLW\ PAGE 23 L 2 T Q[fG[ f Â‘ R ZKHUH [ 2 UHSUHVHQWV WKH LQWHUIDFH LV SUHGLFWHG ZHOOE\ WKH FODVVLFDO VROXWLRQ 7KH DQDO\VHV VXJJHVW D VLPSOLILFDWLRQ LQ WKH GHVFULSWLRQ RI Q[f DQG LWV GHSHQGHQFH RQ WKH IURQWf JDWH YROWDJH :H GHILQH DQ DYHUDJH HOHFWURQ GHQVLW\ Â³ RYHU WKH HIIHFWLYH SRUWLRQ RI WKH LQYHUVLRQ OD\HU [ [AA DV UHYHDOHG E\ WKH TXDQWXPPHFKDQLFDO VROXWLRQ EXW ZH XVH WKH FODVVLFDO VROXWLRQ QFA[f WR FRQYH\ WKH 9ILI GHSHQGHQFH :H ILQG WKDW [cHIIf LV GHVFULEHG ZHOO E\ LHIIfF T QF [fG[ FO f ZKHUH2A 4Qf LV JLYHQ E\ f ZLWK Q[f UHSODFHG E\ QFA[f WKDW LV DERXW b RI WKH LQYHUVLRQOD\HU FKDUJH LV FRQWDLQHG ZLWKLQ D UHJLRQ LQ ZKLFK Q Â³ DQG LQ ZKLFK YLUWXDOO\ DOO WKH FKDQQHO FXUUHQW IORZV 7KHQ ZH GHILQH Â³ E\ A[LHIIf &Q@ f 1XPHULFDO HYDOXDWLRQV RI [AHIIA UHYHDO WKDW LW LV QRW VWURQJO\ GHSHQGHQW RQ 9A WKDW LW GHFUHDVHV ZLWK LQFUHDVLQJ ILOP GRSLQJ GHQVLW\ 1D DQG WKDW W\SLFDOO\ LW LV TXLWH QDUURZ )RU H[DPSOH ZKHQ FUU7A $ &RUUHVSRQGLQJ FDOFXODWLRQV RI Â³ GHILQHG E\ PAGE 24 \ \ n n Â‘ f DUH SORWWHG YHUVXV 9A 9\If LQ )LJ IRU GLIIHUHQW YDOXHV RI DQG IRU DQ R[LGH WKLFNQHVV W4I RI $ 9MI LV WKH WKUHVKROG YROWDJH n >/,EO KDW FRUUHVSRUWWLV ARfÂ¬ WKHaRQVHWRI VWURULJn fÂ¬WQYHUVWRU7rfÂ¬>KFO` nm 1A@ $V LPSOLHG E\ f )LJ VKRZV WKDW Q LQFUHDVHV ZLWK LQFUHDVLQJ 9*I DQG ZLWK LQFUHDVLQJ 1A :H ILQG WKDW DQG Â³ DV GHILQHG E\ f DQG f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nV HTXDWLRQ :H FRQVLGHU WKH SRWHQWLDO YDULDWLRQ QHDU D JUDLQ ERXQGDU\ LQ WKH FKDQQHO DV VKRZQ LQ )LJ :H DVVXPH WKDW DZD\ IURP WKH JUDLQ ERXQGDU\ \ ! \UIf WKH HOHFWULF ILHOG LV YHUWLFDO LQ WKH [GLUHFWLRQf DQG Q[f LV ZHO DSSUR[LPDWHG E\ Â³ RYHU WKH HIIHFWLYH LQYHUVLRQ OD\HU Q [ [MHIIf DV GLVFXVVHG LQ WKH SUHFHGLQJ VXEVHFWLRQ ,Q WKLV UHJLRQ ,f 3RLVVRQnV HTXDWLRQ VLPSOLILHV WR 9 fÂ§ [ fÂ¯ U 9 V f PAGE 25 Â« nO & r VM nWL Â¾ r )LJ "O &DOFXODWHG DYHUDJH HOHFWURQ GHQVLW\ LQ FKDQQHO YHUVXV IURQWf JDWH YROWDJH IRU VHYHUDO ILOP GRSLQJ GHQVLWLHV PAGE 26 r r )LJ &URVVVHFWLRQ RI HIIHFWLYH LQYHUVLRQ OD\HU VKRZLQJ W\SLFDO JUDLQ DQG JUDLQ ERXQGDULHV ZKLFK DUH DVVXPHG WR EH SHUSHQGLFXODU WR WKH HOHFWURQ IORZ PAGE 27 Â« ZKHUH LV WKH HOHFWURVWDWLF SRWHQWLDO ,Q WKH YLFLQLW\ RI WKH JUDLQ ERXQGDU\ \ \Gf D KRUL]RQWDO \GLUHFWLRQf FRPSRQHQW RI WKH HOHFWULF ILHOG LV SURGXFHG E\ WKH HOHFWURQV WUDSSHG DW WKH JUDLQ ERXQGDU\ :H DVVXPH WKDW WKH WUDSSLQJ QHDUO\ GHSOHWHV WKLV UHJLRQ ,,f RI IUHH HOHFWURQV KHQFH S V \ Â ,, t \ LL eB 1 HV $ f :H QRWH WKDW WKLV GHSOHWLRQ DSSUR[LPDWLRQ >%$E 3,@ LV YDOLG SURYLGHG g%R LV VXIILFLHQWO\ KLJK KLJK HQRXJK LQ IDFW ZH DVVXPH WKDW WKH JUDLQ ERXQGDULHV VLJQLILFDQWO\ DIIHFW WKH FKDQQHO &RQGXFWDQFH :H GLVFXVV WKH YDOLGLW\ RI WKLV DVVXPSWLRQ LQ 6XEVHFWLRQ $VVXPLQJ WKDW DQDORJRXV WR WKH JUDGXDOFKDQQHO DSSUR[LPDWLRQ >6=@ WKH WUDSSHG HOHFWURQV DW WKH JUDLQ ERXQGDU\ W\SLFDOO\ FUHDWH RQO\ D VPDOO SHUWXUEDWLRQ RQ WKH [FRPSRQHQW RI HOHFWULF ILHOG ZH FDQ ZULWH A A , A ! _ UR_ [ L G[ _LL \ LL f ZKHUH WKH SDUWLDO GHULYDWLYHV DUH HYDOXDWHG DQ\ZKHUH LQ WKH UHJLRQV LQGLFDWHG :H MXVWLI\ WKLV DVVXPSWLRQ E\ QRWLQJ WKDW WKH VXEVHTXHQW VROXWLRQ ZH REWDLQ LV FRQVLVWHQW ZLWK LW ZKHQ 9ILI 9\If ! ZKLFK LV XVXDOO\ WUXH IRU VWURQJLQYHUVLRQ FRQGLWLRQV +HQFH f LPSOLHV DQ DSSUR[LPDWH VROXWLRQ WR WKH WZRGLPHQVLRQDO SUREOHP GHILQHG E\ f DQG f ZKLFK LV REWDLQHG E\ FRXSOLQJ WZR RQHGLPHQVLRQDO VROXWLRQV PAGE 28 7KH FRUUHVSRQGLQJ DSSUR[LPDWLRQ IRUnLnJ4 GHULYHV IURP WKH FRPELQDWLRQ RI ff ZKLFK \LHOGV D" fÂ§" \ LL DV H V ZLWK WKH ERXQGDU\ FRQGLWLRQV f "[\ \Gf "ML[f f DQG \ f \ \U ,Q f I M[f LV WKH LQWUDJUDLQ UHJLRQ ,f SRWHQWLDO YDULDWLRQ LQ WKH FKDQQHO ZKLFK LV JLYHQ E\ WKH RQHGLPHQVLRQDO VROXWLRQ RI 3RLVVRQnV HTXDWLRQ DQG WKH 6FKURGLQJHU HTXDWLRQ >+6@ :H QRZ LGHQWLI\ \G DV WKH JUDLQERXQGDU\ GHSOHWLRQUHJLRQ ZLGWK DQG ZH QRWH WKDW RXU DQDO\VLV DSSOLHV RQO\ ZKHQ WKH JUDLQV DUH QRW FRPSOHWHO\ GHSOHWHG 7KH VROXWLRQ WR ff LV r[\f ,,D \\GA r LA } LQf DQG KHQFH 75R A;f 7[f X TÂ³\G f PAGE 29 7R FRPSOHWH WKH GHVFULSWLRQ RI 7J ZH PXVW H[SUHVV \A LQ WHUPV RI NQRZQ SDUDPHWHUV 7KLV H[SUHVVLRQ LV LPSOLHG E\ WKH FRQVHUYDWLRQ RI FKDUJH LQ WKH YLFLQLW\ RI WKH JUDLQ ERXQGDU\ TQ\G f ZKLFK HTXDWHV WKH DUHDO GHQVLW\ RI FKDUJH WUDSSHG DW WKH JUDLQ ERXQGDU\ WR WKH HOHFWURQ FKDUJH GHQVLW\ UHPRYHG WR IRUP WKH WZRf DGMDFHQW GHSOHWLRQ UHJLRQV ,Q ZULWLQJ f ZH KDYH LPSOLFLWO\ DVVXPHG WKDW WKH HOHFWURQV DUH WUDSSHG ZLWKLQ ZKLFK LV FRPPHQVXUDWH ZLWK RXU SUHYLRXV DVVXPSWLRQV 7KH WUDSSHG FKDUJH GHQVLW\ GHSHQGV RQ WKH GLVWULEXWLRQ LQ WKH HQHUJ\ JDS RI ORFDOL]HG JUDLQERXQGDU\ VWDWHV DFFHSWRUW\SH VLQFH f >3,@ ,W LV UHDVRQDEOH WR DSSUR[LPDWH WKLV GLVWULEXWLRQ E\ D GHOWD IXQFWLRQ >%$E 3 ) /8@n \LHOGLQJ 1A\ VWDWHV WUDSVf SHU XQLW DUHD DW DQ HQHUJ\ OHYHO (\ 7KHQ f *% ZKHUH (S LV WKH )HUPL OHYHO DQG WKH IDFWRU RI UHIOHFWV WKH VSLQf GHJHQHUDF\ RI WKH ORFDOL]HG VWDWHV 7KH SRVLWLRQ RI (S UHODWLYH WR (\ LV GHILQHG E\ r DQG WKH HOHFWURQ GHQVLW\ LQ UHJLRQ , LH Â³ f PAGE 30 ZKHUH (A LV WKH LQWULQVLF )HUPL OHYHO YLUWXDOO\ DW PLGJDSf DQG Qc LV WKH LQWULQVLF FDUULHU GHQVLW\ LQ VLOLFRQ 7KXV ff LPSOLFLWO\ GHVFULEH A%R LQ WHUPV RI WKH JUDLQ ERXQGDU\ SDUDPHWHUV 1A\ DQG (\ (Af DQG RI Â³ ZKLFK GHSHQGV RQ 9J\ DQG WKH 026)(7 SURSHUWLHV DV GHVFULEHG LQ 6XEVHFWLRQ 1XPHULFDO FDOFXODWLRQV RI 7 5R DUH SORWWHG YHUVXV 9J\ 9\\f LQ )LJ IRU r FPfÂ¯A W4I $ WZR UHSUHVHQWDWLYH YDOXHV RI 1\ nA DQG A FQIA DQG WKUHH SRVLWLRQV RI (\ LQ WKH HQHUJ\ JDS ,Q DOO FDVHV IRU 9*\ VXIILFLHQWO\ KLJK AU GHFUHDVHV ZLWK LQFUHDVLQJ 9J\ 7KLV FDQ EH H[SODLQHG E\ QRWLQJ WKDW XQGHU WKHVH FRQGLWLRQV YLUWXDOO\ DOO WKH JUDLQERXQGDU\ VWDWHV ZLWKLQ [AH\\Af DUH ILOOHG DQG KHQFH 55 LV LQGHSHQGHQW RI 9J\ 7KHUHIRUH VLQFH Â³ LQFUHDVHV ZLWK 9J\ VHH )LJ f \UI FRQFRPLWDQWO\ GHFUHDVHV DV GHVFULEHG E\ f ZKLFK LPSOLHV WKURXJK f WKDWg5R DOVR GHFUHDVHV t Q V f +RZHYHU ZKHQ 9J\ LV ORZ<5R LV QHDUO\ LQVHQVLWLYH WR 9J\ 7KLV LV EHFDXVH WKH JUDLQERXQGDU\ VWDWHV DUH QRW FRPSOHWHO\ ILOOHG DQG KHQFH (S LV QHDU (\ ZKLFK YLUWXDOO\ IL[HV 9 DV GHVFULEHG E\ f :H QRWH LQ )LJ WKDW IRU 16\ FPA 75R LV OHVV WKDQ P9 ZKHQ 9J\ 9\\f H[FHHGV DERXW 9 7KXV DOWKRXJK RXU GHSOHWLRQ DSSUR[LPDWLRQ LV LQYDOLG IRU WKHVH FRQGLWLRQV ZH VXUPLVH WKDW 75R LV ORZ HQRXJK WKDW WKH JUDLQ ERXQGDULHV GR QRW VLJQLILFDQWO\ DIIHFW WKH FKDQQHO FRQGXFWDQFH +RZHYHU IRU 1J\ r FPfÂ¯A 75R LV KLJK HQRXJK PAGE 31 Â« )LJ &DOFXODWHG JUDLQ ERXQGDU\ SRWHQWLDO EDUULHU YHUVXV IURQWf JDWH YROWDJH IRU WZR UHSUHVHQWDWLYH JUDLQERXQGDU\ WUDS GHQVLWLHV DQG WKUHH HQHUJ\ OHYHOV :R QRWH WKDW WKH ORZ YDOXHV RI < 5R FDOFXODWHG IRU 1c\ FP DUH SUREDEO\ LQDFFXUDWH EHFDXVH RI WKH LQYDOLGLW\ RI WKH GHSOHWLRQ DSSUR[LPDWLRQ f 1HYHUWKHOHVV WKH FXUYH LV XVHIXO EHFDXVH LW LQGLFDWHV ZKHQ PAGE 32 HYHQ ZKHQ 9JA 9MIf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f WKDW WKHUPLRQLF HPLVVLRQ RI HOHFWURQV RYHU WKH EDUULHU 75R LV WKH SUHGRPLQDQW JUDLQ ERXQGDU\ WUDQVSRUW PHFKDQLVP >3,@ 7KHQ LI WKH GUDLQ YROWDJH 9J LV ORZ HQRXJK OLQHDU UHJLRQf WKDW WKH YROWDJH GURS DFURVV D JUDLQ ERXQGDU\ LV PXFK VPDOOHU WKDQ N7T DQG LI A5R ! nA7T WKH HPLWWHG FXUUHQW GHQVLW\ LV >%$E@ -JE fÂ¯ A H[3 TS 5R ! X cUUfÂ¬ Y f ZKHUH $r LV WKH HIIHFWLYH 5LFKDUGVRQ FRQVWDQW >6=@ IRU HOHFWURQV $FPA.Af DQG 1J LV WKH HIIHFWLYH GHQVLW\ RI VWDWHV LQ WKH FRQGXFWLRQ KDQG [ A FPaA DW r .f 6LQFH WKH FXUUHQW LQ WKH FKDQQHO LV FRQWLQXRXV IURP VRXUFH WR GUDLQ WKH GUDLQ FXUUHQW ,J FDQ EH H[SUHVVHG E\ WKH LQWHJUDO RI f RYHU WKH HIIHFWLYHf FURVVVHFWLRQDO DUHD RI WKH FKDQQHO PAGE 33 9 [LHIIf -JE f ZKHUH = LV WKH FKDQQHO ZLGWK 7KH FRPELQDWLRQ RI f DQG f JLYHV ,' DV D IXQFWLRQ RI YJEr 7R REWDLQ ,T DV D IXQFWLRQ RI 9S ZH VLPSO\ HTXDWH WKH VXP RI WKH YROWDJH GURSV DORQJ WKH FKDQQHO WR 95 ,I ZH DVVXPH WKDW WKH FKDQQHO FRPSULVHV 1J JUDLQV RI HTXDO OHQJWK \J VHSDUDWHG E\ 1J f LGHQWLFDO JUDLQ ERXQGDULHV VHH )LJ f WKHQ 9Q 1 f9 1 9 Q J JE J J f ZKHUH 9J LV WKH YROWDJH GURS DFURVV D JUDLQ ZKLFK DVVXPLQJ WKDW WKH FDUULHU WUDQVSRUW LQ WKH JUDLQ LV E\ GULIW >6=@ LV \T n \G K _ QJ Q f LQ WKH OLQHDU UHJLRQ ,Q f YQJ LV WKH LQWUDJUDLQ HOHFWURQ PRELOLW\ WKH GHSHQGHQFH RI ZKLFK RQ 9JI DQG RQ GHYLFH SDUDPHWHUV FDQ EH JLYHQ HPSLULFDOO\ >68@ &RPELQLQJ f DQG ffÂ§f ZH REWDLQ ,Q9JA9Sf IRU WKH 62, 026)(7 LQ WKH OLQHDU UHJLRQ 9Q 1J f N7Tf ,I ZH DVVXPH WKDW \J !! \G ZKLFK LV YDOLG LQ W\SLFDO UHFU\VWDOOL]HG 62, 026)(7V WKHQ RXU UHVXOW VLPSO L ILHV WR ,Q 8ZQ 9U 7Â“ Tf NgQT /$r7 H[SI A 5RA*IA N7 @ f PAGE 34 Â« ZKHUH / LV WKH FKDQQHO OHQJWK ,Q f 4Q LV JLYHQ E\ WKH VWURQJLQYHUVLRQ FRQGLWLRQ n4Q r &RI9*I Y7If f ZKHUH &4I LV WKH IURQWf JDWH R[LGH FDSDFLWDQFH 7KH LQIOXHQFH RI WKH JUDLQ ERXQGDULHV RQ ,S LV UHIOHFWHG E\ WKH VHFRQG WHUP LQ WKH GHQRPLQDWRU RI f ZKLFK GHSHQGV RI WKURXJK fÂ¬)%RQÂ³9*I 9Af@ DV GHVFULEHG LQ 6XEVHFWLRQV )LJ f DQG )LJ f ,I WKH QXPEHU RI JUDLQV 1J FRQVWLWXWLQJ WKH FKDQQHO LV RQH YLVDYLV LI WKHUH DUH QR JUDLQ ERXQGDULHV LQ WKH FKDQQHO WKHQ f UHGXFHV WR WKH FRUUHVSRQGLQJ UHVXOW RI FRQYHQWLRQDO 026)(7 WKHRU\ U6=% + )XUWKHUPRUH LI < J LV VXIILFLHQWO\ ORZ EHFDXVH RI ORZ 1J\ DQGRU KLJK 9J\ VHH )LJ f WKHQ WKH VDPH UHVXOW REWDLQV :H QRWH WKDW f ZKLFK EHFDXVH RI WKH PRGHO DVVXPSWLRQV LV VWULFWO\ YDOLG RQO\ ZKHQ I 5T ! N7T ZLOO FRUUHFWO\ JLYH WKH FRQYHQWLRQDO FXUUHQW DW KLJK 9JI RQO\ LI WKH SUHH[SRQHQWLDO FRHIILFLHQW LV PXFK OHVV WKDQ XQLW\ :LWK WKLV LQVLJKW WKHQ f IDFLOLWDWHV D VHOIFRQVLVWHQF\ FKHFN IRU RXU PRGHO DVVXPSWLRQV f DQG f :H ILQG WKDW ZKHQ WKH JUDLQ ERXQGDULHV DUH LQIOXHQWLDO YLV JHQHUDOO\ KLJK HQRXJK WKDW WKH DVVXPSWLRQV DUH YDOLG ,Q GHULYLQJ f ZH KDYH QHJOHFWHG WKHUPLRQLF ILHOG HPLVVLRQ WXQQHOLQJf WKURXJK 75 DQG ZH KDYH LJQRUHG WKH SRVVLEOH H[LVWHQFH RI D VLJQLILFDQW JUDLQERXQGDU\ VFDWWHULQJ SRWHQWLDO EDUULHU >/8Â³O@ WKURXJK PAGE 35 ZKLFK WKH HOHFWURQV PXVW WXQQHO WR WUDYHUVH WKH ERXQGDU\ 7KH WXQQHOLQJ FDQ EH SUHGRPLQDQW DW ORZ WHPSHUDWXUHV EXW DW URRP WHPSHUDWXUH DQG DERYH LW LV JHQHUDOO\ LQVLJQLILFDQW >3, /8@ :H DOVR QHJOHFWHG GLIIXVLRQ RI HOHFWURQV WKURXJK g5R ZKLFK LV LPSRUWDQW RQO\ ZKHQ LV ORZ >&@ :KHQ g %R LV KLJK HQRXJK WKDW WKH JUDLQ ERXQGDULHV VLJQLILFDQWO\ DIIHFW ,S WKH GLIIXVLRQ FDQ EH LJQRUHG ! 7R LOOXVWUDWH WKH JUDLQERXQGDU\ HIIHFWV GHVFULEHG E\ f ZH SORW LQ )LJV DQG FDOFXODWLRQV RI WKH OLQHDUUHJLRQ FKDQQHO FRQGXFWDQFH J$ ,'95f YHUVXV 9*\ 9\\f IRU VHYHUDO YDOXHV RI 1J DQG 1mM\ ,Q )LJ ZH OHW 1J YDU\ IURP RQH WR JUDLQV DQG ZH XVH W\SLFDO YDOXHV IRU WKH UHPDLQLQJ SDUDPHWHUV 1\ r" FP" DW )\ (M rp FPfÂ¯ W4\ $ = / XP ZH DOVR VSHFLI\ D IURQWf IL[HG R[LGH FKDUJH GHQVLW\ 2\\ TA FPAf ZKLFK GHILQHV X QJ DQG LWV GHSHQGHQFH RQ 9J\ >68@ :H VHH WKDW DV 1J LQFUHDVHV J GHFUHDVHV DQG WKH SORWV EHFRPH LQIOHFWHG LQ JHQHUDO DFFRUG ZLWK UHFHQW PHDVXUHPHQWV RI ODVHUDQQHDOHG 62, 026)(7V >/( &@ 7KH SORWV VKRZ DSSDUHQW WKUHVKROG YROWDJHV WKDW DUH KLJKHU WKDQ 9\\ DQG WUDQVFRQGXFWDQFHV JP $ DLAD9J\ 9SJ9J\f WKDW LPSO\ HIIHFWLYH HOHFWURQ PRELOLWLHV YLD WKH FRQYHQWLRQDO 06)(7 WKHRU\ >6=@f ZKLFK FDQ GLIIHU IURPXQJ 7KH DSSDUHQW WKUHVKROG YROWDJH LV DFWXDOO\ D FDUULHU PRELOLW\ WKUHVKROG YROWDJH 9A f DW ZKLFK Â³ EHFRPHV KLJK HQRXJK WKDWIA EHJLQV WR GLPLQLVK ZLWK LQFUHDVLQJ 9J\ VHH )LJ f DV GHVFULEHG E\ f )RU !! g5R LV WRR ORZ WR VLJQLILFDQWO\ DIIHFW ,Q WKDW LV WKHI5R WHUP LQ f LV QHJOLJLEOH DQG JP LV GHILQHG E\ \ 1RWH KRZHYHU LQ )LJ WKDW WKH SORWV IRU PAGE 36 )LJ &DOFXODWHG OLQHDUUHJLRQ FKDQQHO FRQGXFWDQFH YHUVXV IURQWf JDWH YROWDJH IRU VHYHUDO QXPEHUV RI JUDLQV FRQVWLWXWLQJ WKH FKDQQHO 7KH EURNHQ SRUWLRQV RI WKH FXUYHV IRU 1 DQG DUH LQDFFXUDWH EHFDXVH RI WKH LQYDOLGLW\ RI\f DV GLVFXVVHG LQ 6XEVHFWLRQ 7KH 1 FXUYH LV LQDFFXUDWH IRU 9SI QHDU 97I EHFDXVH RI WKH LQYDOLGLW\ RI WKH VWURQJLQYHUVLRQ UHODWLRQVKLS f PAGE 37 LOOf )LJ &DOFXODWHG OLQHDUUHJLRQ FKDQQHO FRQGXFWDQFH YHUVXV IURQWf JDWH YROWDJH IRU VHYHUDO JUDLQKRXQGDU\ WUDS GHQVLWLHV PAGE 38 P 1J YHU\ ODUJH EHFRPH HUURQHRXV ZKHQ 9J\ !! 9\ EHFDXVH DV ZH GLVFXVVHG SUHYLRXVO\ WKH SUHH[SRQHQWLDO FRHIILFLHQW LQ f LV QRW PXFK OHVV WKDQ n XQLW\ :KHQ ! ?7 JP LV W\SLFDOO\ KLJKHU WKDQ WKDW FRUUHVSRQGLQJ WRXQJ :H VWUHVV WKDW WKH KLJK HIIHFWLYH HOHFWURQ PRELOLW\ LPSOLHG E\ JP LV GHILQHG SUHGRPLQDQWO\ E\ WKH SURSHUWLHV RI WKH JUDLQ ERXQGDULHV 0HDVXUHG 7QY*I 9Jf FKDUDFWHULVWLFV RI 62, 026)(7V FDQ WKXV EH PLVOHDGLQJ EHFDXVH RI WKH QRQOLQHDU HIIHFWV RI JUDLQ ERXQGDULHV DV ZH GLVFXVV LQ WKH QH[W VHFWLRQ $GGLWLRQDO FDOFXODWLRQV UHYHDO WKDW ,7 GHSHQGV RQ DQG W4I LW GHFUHDVHV ZLWK LQFUHDVLQJ DQG LW LQFUHDVHV ZLWK LQFUHDVLQJ W\ 7KHVH GHSHQGHQFHV UHIOHFW IRU D JLYHQ 9J\ 9\\f WKH GHSHQGHQFHV RI Â³ ZKLFK FRQWUROV g 5T RQ VKRZQ LQ )LJ DQG RQ W4\ LPSOLHG E\rQfÂ¯&RI 7KH SORWV RI J YHUVXV 9J\ 9\\f LQ )LJ IRU 1\ UDQJLQJ IURP WR [ A FPA ZHUH FDOFXODWHG IURP f IRU WKH VDPH GHYLFH SDUDPHWHU YDOXHV XVHG WR GHULYH WKH SORWV LQ )LJ :H OHW 1J RQH JUDLQ ERXQGDU\f WR VLPSOLI\ WKH SK\VLFDO LQWHUSUHWDWLRQ RI WKH UHVXOWV 7KH VDPH W\SH RI LQIOHFWLRQ VHHQ LQ )LJ LV QRWHG LQ )LJ IRU 1J\ ! A FPA )RU WKH GHYLFH FRQVLGHUHG LI 1J\ LV PXFK ORZHU WKDQ A FPfÂ°A WKH JUDLQ ERXQGDU\ LV YLUWXDOO\ LQHIIHFWLYH ZKHUHDV LI 1V\ L6 K LJKHU WKDQ A FPfÂ¯A WKH JUDLQ ERXQGDU\ VHYHUHO\ DIIHFWV ORZHUVf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n f AI WKH SRO\VLOLFRQ ILOP LV QRW FRPSOHWHO\ GHSOHWHG EHWZHHQ WKH IURQW DQG EDFN VXUIDFHV ZH FDQ QHJOHFW WKH FKDUJH FRXSOLQJ HIIHFWV >%$ /E@ 7KHQ IRU DOO LQYHUVLRQ FRQGLWLRQV >76,@ f ZKHUH f $ LV WKH DUHDOf FKDUJH GHQVLW\ LQ WKH VLOLFRQ DQG f PAGE 40 LV WKH GHSOHWLRQUHJLRQ FKDUJH GHQVLW\ ,Q f DQG f LSVAU LV WKH IURQW EDQG EHQGLQJ DQG S >N7H61$@A KDV EHHQ GHILQHG WR PDNH D FRPSDFW QRWDWLRQ 7KH UHODWLRQVKLS EHWZHHQ IV\ DQG LV GHILQHG E\ 9*I 9)% nrVI r6 r T1VI & & A VI XRI /RI 67 f ZKHUH 9S5 LV WKH IURQWJDWH IODWEDQG YROWDJH >1,@ ZKLFK LQFOXGHV D FRQWULEXWLRQ IURP IDVW VXUIDFH VWDWHV DW WKH 6L6L&A LQWHUIDFH WKH GHQVLW\ 1\ FQIAHY0 RI ZKLFK LV DVVXPHG WR EH XQL IRUP LQ WKH HQHUJ\ JDS 7KH Q9IL\f GHSHQGHQFH LQ f LV QRZ GHILQHG E\ ff fÂµ7R LOOXVWUDWH WKH JUDLQERXQGDU\ HIIHFWV LQ PRGHUDWH LQYHUVLRQ GHVFULEHG E\ f DQG ff ZH SORW LQ )LJ DQG FDOFXODWLRQV RI WKH OLQHDUUHJLRQ FKDQQHO FRQGXFWDQFH YHUVXV IRU VHYHUDO YDOXHV RI (\ DQG 1\ 7R IDFLOLWDWH D ODWHU FRPSDULVRQ EHWZHHQ H[SHULPHQWDO DQG WKHRUHWLFDO UHVXOWV VHH 6HFWLRQ f ZH VHW 9\\ ZKLFK GHILQHV 9SJ WKURXJK f ,Q )LJ ZH OHW (M(\f YDU\ IURP WR H9 DQG ZH XVH W\SLFDO YDOXHV IRU WKH UHPDLQLQJ SDUDPHWHUV [OAFP ZKLFK LPSOLHV $ XQJ FPYVHF W4\ $ = / XUQ 16\ O2AFPA DQG 1J :H VHH WKDW WKH FRQGXFWDQFH SUHVHQWV D QHDUO\ H[SRQHQWLDO GHSHQGHQFH RQ IRU WKH RZHU95\^ ?A f UHJLRQ DQG WKDW WKH FRUUHVSRQGLQJ JDWH YROWDJH VZLQJ 6 QHHGHG WR UHGXFH ,S E\ RQH RUGHURIPDJQLWXGH LQFUHDVHV A5R DV (\ LQFUHDVHV 7KLV GHSHQGHQFH LV GXH SULPDULO\ WR WKH H[S A\ f WHUP LQ f ZKLFK LV GRPLQDQW !! f PAGE 41 )LJ "IL 0HDVXUHG SRLQWVf DQG FDOFXODWHG FXUYHVf OLQHDUUHJLRQ 9A P9f FRQGXFWDQFH YHUVXV IURQWJDWH YROWDJH RI DQ QFKDQQHO 62, 026)(7 LQ DVHUUHFU\VWDOOL]HG SRO\VLOLFRQ DW URRP WHPSHUDWXUH 7KH PHDVXUHPHQWV ZHUH PDGH ZLWK WKH KDFN JDWH ELDVHG DW 9 7KH FDOFXODWLRQV ZHUH GRQH IRU GLIIHUHQW JUDLQERXQGDU\ WUDS HQHUJ\ OHYHOV DV LQGLFDWHG DQG ZLWK WKH IDVW VXUIDFHVWDWH GHQVLW\ DW WKH IURQW 6L6L2 LQWHUIDFH HTXDO WR ]HUR 1RWH WKDW 9\\ 9 PAGE 42 / )LJ 0HDVXUHG SRLQWVf DQG FDOFXODWHG FXUYHVf OLQHDUUHJLRQ 9' P9f FRQGXFWDQFH YHUVXV IURQWJDWH YROWDJH RI DQ QFKDQQHO 62, 026)(7 LQ DVHUUHFU\VWDOL]HG SRO\VLOLFRQ DW URRP WHPSHUDWXUH 7KH PHDVXUHPHQWV ZHUH PDGH ZLWK WKH EDFN JDWH ELDVHG DW 9 7KH FDOFXODWLRQV ZHUH GRQH IRU GLIIHUHQW IDVW VXUIDFHVWDWH GHQVLWLHV DW WKH IURQW 6L6L2S LQWHUIDFH DV LQGLFDWHG DQG ZLWK WKH JUDLQERXQGDU\ WUDS HQHUJ\ OHYHO DW H9 DERYH PLGJDS PAGE 43 7KH SORWV RI J YHUVXV LQ )LJ IRU 1V\ UDQJLQJ IURP WR [ FPfÂ¯AH9O ZHUH FDO&XODWHG IRU (\(Mf H9 DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV YDOXHV XVHG WR GHULYH WKH SORWV LQ )LJ :H VHH WKDW 6 LV QHDUO\ LQGHSHQGHQW RI 1\ DOWKRXJK J GHFUHDVHV DV 1V\ LQFUHDVHV :H FRQFOXGH WKLV VXEVHFWLRQ E\ VWUHVVLQJ WKDW WKH GUDLQ FXUUHQW LQ WKH ORZHU9JI RU VXEPRELOLW\WKUHVKROG 9J\ 9Af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f JUDLQ ERXQGDU\ DUELWUDULO\ RULHQWHG LQ WKH FKDQQHO DV VKRZQ LQ )LJ 7KH GUDLQ FXUUHQW FDQ EH H[SUHVVHG WR ILUVW RUGHU DV QI E f PAGE 44 / Â« )LJ ,OOXVWUDWLRQ RI DUELWUDU\ JUDLQERXQGDU\ RULHQWDWLRQ LQ FKDQQHO PAGE 45 ZKHUH ,SA LV WKH FRPSRQHQW WKDW IORZV LQ WKH JUDLQERXQGDU\IUHH SRUWLRQ ==Ef RI WKH FKDQQHO DQG ,JE LV WKH FRPSRQHQW WKDW IORZV LQ WKH SRUWLRQ =Ef FRQWDLQLQJ WKH JUDLQ ERXQGDU\ 1RWH LQ )LJ WKDW =E LV GHILQHG E\ = DQG / DQG WKH DQJOH EHWZHHQ WKH JUDLQ ERXQGDU\ DQG WKH ]GLUHFWLRQ ,Q WKH OLQHDU UHJLRQ VWURQJ LQYHUVLRQf AIrn /bJ&RI9*I a 97IA9' fÂ« f 7KH FKDUDFWHUL]DWLRQ RI ,SE GHSHQGV RQ D FRPSOLFDWHG Â‘ WZR GLPHQVLRQDO HOHFWURQ WUDQVSRUW SUREOHP 7R GHULYH D FUXGH DSSUR[LPDWLRQ ZH DVVXPH WKDW WKH HOHFWURQ FXUUHQW GHQVLW\ -JE WKURXJK WKH JUDLQ ERXQGDU\ YLD WKHUPLRQLF HPLVVLRQf LV SHUSHQGLFXODU WR LW 7KHQ DQDORJRXV WR f JE T$r7 N1F H[S A %2} $ fÂ§Q9JE f $$$ ZKHUH Q FRV f\ VLQ f] LV WKH XQLW YHFWRU QRUPDO WR WKH JUDLQ ERXQGDU\ :H IXUWKHU DVVXPH WKDW DZD\ IURP WKH JUDLQ ERXQGDU\ WKH HOHFWURQV IORZ SUHGRPLQDQWO\ LQ WKH \GLUHFWLRQ 7KHQ WR HQVXUH FXUUHQW FRQWLQXLW\ IURP VRXUFH WR GUDLQ ZH PXVW KDYH = $ fÂ§ r+E fÂ° FRV f [LHIIf Q r AJE f PAGE 46 8VLQJ f DQG f DQG IROORZLQJ WKH GHULYDWLRQ LQ 6HFWLRQ ZH RKWDLQ Â‘ Â‘ , 2E UbJ&RI9*I97IA N1AL n/$r7n H[S^7U@ &6f 7KH FRPELQDWLRQ RI f f DQG f WKHQ GHVFULEHV DSSUR[LPDWHO\ IRU VWURQJLQYHUVLRQ FRQGLWLRQV LQ WKH OLQHDU UHJLRQ WKH VLJQLILFDQFH RI WKH JUDLQERXQGDU\ RULHQWDWLRQ LOOXVWUDWHG LQ )LJ 7KH FRV f LQ f DV ZHOO DV WKH =Ef GHSHQGHQFH FRQYH\ WKLV VLJQLILFDQFH ,I ! r WKHQ =E = DQG WKH JUDLQERXQGDU\ HIIHFW LV DPHOLRUDWHG ,Ik r JUDLQ ERXQGDU\ SDUDOOHO WR HOHFWURQ IORZf WKHQ =E DQG WKH JUDLQ ERXQGDU\ GRHV QRW DIIHFW WKH FKDQQHO FRQGXFWDQFH DOWKRXJK LW PD\ HQKDQFH VRXUFHGUDLQ OHDNDJH FXUUHQW YLD RWKHU PHFKDQLVPVf TS %R f ([SHULPHQWDO 6XSSRUW DQG 'LVFXVVLRQ 7R VXSSRUW WKH DQDO\VLV LQ WKLV FKDSWHU DQG WR LGHQWLI\ FULWLFDO DVSHFWV RI LW ZLWK UHJDUG WR 62, GHYLFH DQG LQWHJUDWHG FLUFXLW GHVLJQ ZH PHDVXUHG OLQHDUUHJLRQ ,S9ILI9Q7f FKDUDFWHULVWLFV RI IRXUWHUPLQDO 62, 026)(7V QFKDQQHOf IDEULFDWHG DW 7H[DV ,QVWUXPHQWV >/$@ 7KH SRO\VLOLFRQ ILOP LV SP WKLFN DQG ZDV ODVHUUHFU\VWDOOL]HG DIWHU EHLQJ GHSRVLWHG YLD /3&9' RQ D SPWKLFN OD\HU RI VLOLFRQGLR[LGH ZKLFK KDG EHHQ WKHUPDOO\ JURZQ RQ D VLOLFRQ VXEVWUDWH 7KH ILOP ZDV GRSHG E\ LRQ LPSODQWDWLRQ RI ERURQ WKDW \LHOGHG [ FPfÂ¯QHDU WKH IURQW VXUIDFH DQG 1Aa A FPA DW WKH EDFN VXUIDFH >/$@ PAGE 47 7KH IURQW JDWH LV Q SRO\VL LFRQ DQG &4I [ )FP W $f /DUJH GHYLFHV = / SPf ZHUH VHOHFWHG WR SUHFOXGH VPDOOJHRPHWU\ HIIHFWV >$.@ 7R DYRLG FRPSOLFDWLRQV GXH WR WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV >/,E@ D KLJK QHJDWLYH YROWDJH 9f ZDV DSSOLHG WR WKH EDFN JDWH WR HQVXUH DFFXPXODWLRQ DW WKH EDFN 6L6Le LQWHUIDFH DQG WR IL[ 9MI 7KH ,S9ILIf GHSHQGHQFH ZDV PHDVXUHG ZLWK 9S P9 DW WKUHH WHPSHUDWXUHV r & r & DQG r &f 6XSSRUW IRU WKH 6WURQJ ,QYHUVLRQ $QDO\VLV 7KH FRUUHVSRQGLQJ FKDQQHOFRQGXFWDQFH FKDUDFWHULVWLFVJ9JI7f RI D SDUWLFXODU GHYLFH ZKLFK W\SLI\ WKH FKDUDFWHULVWLFV RI LGHQWLFDOO\ SURFHVVHG GHYLFHV DUH SORWWHG LQ )LJ 7KH EDVLF VKDSH RI WKHVH SORWV LV WKH VDPH DV WKDW RI WKH WKHRUHWLFDO FXUYHV LQ )LJV DQG ZKLFK LPSOLHV TXDLWDWLYH VXSSRUW IRU RXU DQDO\VLV 7KH H[SHULPHQWDO FXUYHV DQG )LJV VKRXOG QRW EH FRPSDUHG TXDQWLWDWLYHO\ EHFDXVH WKH SDUDPHWHU YDOXHV XVHG LQ WKH FDOFXODWLRQV DUH QRW QHFHVVDULO\ WKH DFWXDO YDOXHVf 7KH VXSSRUW IRU f LV GHPRQVWUDWHG E\ H[DPLQDWLRQ RI WKH PHDVXUHG J9*I7f FKDUDFWHULVWLFV ZLWKLQ SDUWLFXODU UDQJHV RI YILIr )RU KLJK 9JI !! 9 f J LV GHILQHG E\ WKH QXPHUDWRU RI f WKH JUDLQn KRXQGDU\ HIIHFW LV QHJOLJLEOH 7KXV DV LQ WKH FDVH RI FRQYHQWLRQDO 026)(7V >6= WKH FDUULHU PRELOLW\ XQJf IROORZV IURP WKH VORSH RI J9ILIf LH IURP JP DQG WKH WKUHVKROG YROWDJH 9MIf LV JLYHQ E\ WKH PAGE 48 8f )LJ 0HDVXUHG OLQHDUUHJLRQ FKDQQHO FRQGXFWDQFH YHUVXV IURQWf JDWH YROWDJH RI QFKDQQHO 62, 026)(7 LQ ODVHUUHFU\VWDO L]HG SRO\VLOLFRQ >/$@ DW WKUHH WHPSHUDWXUHV 7KH WKUHVKROG YROWDJH LV IL[HG E\ WKH EDFNJDWH YROWDJH >O,E@ ZKLFK ZDV VHW DW 9 WR HQVXUH DFFXPXODWLRQ DW WKH EDFN 6L6L2" LQWHUIDFH PAGE 49 OLQHDU H[WUDSRODWLRQ RI WKH FKDUDFWHULVWLF WR WKH 9JI D[LV )URP )LJ ZH WKHUHE\ JHWB9A r 9 DQG \QJ FPA9VHF DW r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f FKDUDFWHULVWLFV RI FRQYHQWLRQDO 6LOLFRQ 026)(7V >/( *$@ :H VHH IURP )LJ WKDW 9MI LV FRQVLGHUDEO\ OHVV WKDQ WKH HOHFWURQ PRELOLW\ WKUHVKROG YROWDJH 9A 7KXV WKHUH LV D VLJQLILFDQW UDQJH RI 9JI 9MI 9JI A f LQ ZKLFK WKH JUDLQ ERXQGDULHV VXSSUHVV ,Q ,Q WKLV FDVH WKH 9 J4 WHUP LQ f LV PXFK JUHDWHU WKDQ XQLW\ LH 9JE !! 9J DQG KHQFH Jm H[SJ3%RN7f $V ORQJ DV 9JI 9\ n"%T LV KLJK DQG GRHV QRW YDU\ VLJQLILFDQWO\ ZLWK 9JI VHH )LJ f 7KH SRVLWLYH WHPSHUDWXUH FRHIILFLHQW IRU J WKXV SUHGLFWHG LV FRQVLVWHQW ZLWK WKH PHDVXUHG FRQGXFWDQFH SORWWHG LQ )LJ LQ WKLV UHJLRQ :KHQ 9JI ! ?A 9%T GHFUHDVHV ZLWK LQFUHDVLQJ 9JI VHH )LJ f DQG KHQFH J LQFUHDVHV 7R DQDO\WLFDOO\ GHVFULEH WKLV LQFUHDVH DQG WR HVWLPDWH 9A ZH XVH WKH DSSUR[LPDWH :f GHSHQGHQFH LQ f DQG WKH VWURQJLQYHUVLRQ UHODWLRQVKLS f 7KH FRPELQDWLRQ RI f f DQG f \LHOGV D J9JIf FKDUDFWHULVWLF WKDW H[KLELWV DQ LQIOHFWLRQ SRLQW ZKHUH JP LV PD[LPXP 7KH WKHRUHWLFDO DQG H[SHULPHQWDO PAGE 50 SORWV LQ )LJV DQG LPSO\ WKDW WKLV PD[LPXP LV EURDG 7KHUHIRUH ZH DSSUR[LPDWH WKH DFWXDO FKDUDFWHULVWLF E\ WKH OLQHDU IXQFWLRQ f ZKLFK LV WDQJHQW WR WKH DFWXDO J9JIf FXUYH DWn WKH LQIOHFWLRQ SRLQW 7KLV IXQFWLRQ WKHQ DQDO\WLFDOO\ GHILQHV Y DQG WKH HIIHFWLYH ILHOG HIIHFW HOHFWURQ PRELOLW\SQHIIf GXH WR WKH JUDLQ ERXQGDULHV 7KH YDOXH RI 95I DW WKH LQIOHFWLRQ SRLQW LV GHILQHG E\ HTXDWLQJ WR ]HUR WKH VHFRQG GHULYDWLYH RI f ZLWK UHVSHFW WR 9AI XVLQJ f f DQG f A :H ILQG WKDW DW WKLV YDOXH WKH GHQRPLQDWRU RI f LV WZR 7KXV f GHVFULEHV WKH WDQJHQW WR J95If DW WKH SRLQW ZKHUH WKHn)SJ WHUP LQ WKH GHQRPLQDWRU RI f LV XQLW\ 7KLV WDQJHQW \LHOGV IRU 1J !f fÂ§ N7H & V RI f $r7/ N1G:! DQG f :H QRWH WKDW WKH ZHDN GHSHQGHQFH RI [AHIAM RQ 95I KDV EHHQ LJQRUHG LQ WKH GHULYDWLRQ RI f DQG f 7KXV 9\ LQ f LV HYDOXDWHG E\ DVVWPLQJ D UHSUHVHQWDWLYH YDOXH IRU [AA ZKLFK GHSHQGV RQ DV PAGE 51 GLVFXVVHG LQ 6XEVHFWLRQ :H VWUHVV WKDW f DQG f ZKLFK DUH EDVHG RQ DQDO\WLF VLPSOLILFDWLRQV RI RXU PRUH JHQHUDO DQDO\VLV GHVFULEHG LQ 6HFWLRQ DUH PHUHO\ HVWLPDWHV RI 9\ DQG Y}QHIIfr +RZHYHU WKH\ DUH XVHIXO LQ GHVFULELQJ WKH IXQFWLRQDO GHSHQGHQFHV RI J DQG JP RQ GHYLFH SDUDPHWHUV DQG WHPSHUDWXUH DQG KHQFH ZLOO IDFLOLWDWH 62, 026)(7 GHVLJQ DQG FRPSXWHUDLGHG 62, FLUFXLW DQDO\VLV :H VHH IURP f WKDW WKH HIIHFWLYH HOHFWURQ PRELOLW\ LV W\SLFDOO\ KLJKHU WKDQ XQJ GHSHQGLQJ RQ 1J DQG 7 7KH PHDVXUHG JY*If FKDUDFWHULVWLFV SORWWHG LQ )LJ ZKHQ LQWHUSUHWHG XVLQJ f \LHOG m FP 9VHF DW r & ZKLFK LV FRQVLGHUDEO\ KLJKHU WKDQ XQJ 7KH QHJDWLYH WHPSHUDWXUH FRHIILFLHQW IRU XQHIIf LPSOLHG E\ WKH GDWD LQ )LJ LV FRQVLVWHQW ZLWK f ZKLFK VKRZV WKDW WKH WHPSHUDWXUH GHSHQGHQFH LV GHILQHG SULPDULO\ E\ WKDW RI \QJ 8VLQJ WKH PHDVXUHG YDOXH RI \AIIf PHQWLRQHG DERYH DQG f ZH ILQG WKDW 1J JUDLQV 6LQFH / SP WKLV LPSOLHV D FUXGH HVWLPDWH RI DERXW \P IRU WKH DYHUDJH JUDLQ VL]H \Jf ZKLFK LV QRW XQUHDVRQDEOH IRU WKH ODVHUUHFU\VWDOOL]HG SRO\VLOLFRQ ILOP >/$@ :H QRWH ILQDOO\ WKDW WKH GHSHQGHQFH RI ZQHIIf RQ / VXJJHVWHG E\ f LV FRQVLVWHQW ZLWK PHDVXUHPHQWV >1*@ RI HIIHFWLYHf HOHFWURQ PRELOLW\ LQ ODVHU UHFU\VWDOO L ]HG 026)(7V KDYLQJ GLIIHUHQW FKDQQHO OHQJWKV )RU D JLYHQ \J /1 ZLWK 1J ! f LQFUHDVHV DV / LV UHGXFHG IURP PDQ\ WLPHV \J WRZDUG \J 7KH HOHFWURQ PRELOLW\ WKUHVKROG YROWDJH DV GHVFULEHG LQ f LV VWURQJO\ GHSHQGHQW RQ 1AM DQG 7 DV ZHOO DV RQ WKURXJK 9MI >/,K@ DQG [LHIIf r 7KH LQYHUVH GHSHQGHQFH RI RQ 1$ GHVFULEHG LQ PAGE 52 6XEVHFWLRQ LPSOLHV WKDW WKH GLIIHUHQFH EHWZHHQ 9 DQG GHFUHDVHV DV LQFUHDVHV 7KH SUHGLFWHG GLUHFW GHSHQGHQFH RQ 1AM LV FRQVLVWHQW ZLWK REVHUYHG GHFUHDVHV LQ WKH DSSDUHQWf WKUHVKROG YROWDJH RI SRO\VLOLFRQ 026)(7V UHVXOWLQJ IURP K\GURJHQDWLRQ >.$@ ZKLFK LV NQRZQ WR UHGXFH 1AM 7KH LQYHUVH GHSHQGHQFH RI 9\ RQ 7 VXJJHVWHG E\ f LV FRUURERUDWHG E\ WKH PHDVXUHG J94I7f GDWD SORWWHG LQ )LJ $W r & WKH PHDVXUHPHQWV ZKHQ LQWHUSUHWHG XVLQJ f LPSO\ 9\ 9 ZKHUHDV 9 7KH GLIIHUHQFH EHWZHHQ 9\ DQG 9MI EDVHG RQ f LQGLFDWHV WKDW [ FP ZKHUH WKH WUDSV DUH QHDU PLGJDSf :H FRQFOXGH WKLV VXEVHFWLRQ E\ VWUHVVLQJ WZR VLJQLILFDQW FRQFOXVLRQV GUDZQ IURP LW )LUVW EHFDXVH f ZKLFK LV RI WKH VDPH IRUP DV WKH OLQHDUUHJLRQ FRQGXFWDQFH H[SUHVVLRQ IRU WKH FRQYHQWLRQDO 026)(7 >6=@ HPSLULFDOO\ GHVFULEHV ZHOO DQ DSSUHFLDEOH UHJLRQ RI WKH JY5If FKDUDFWHULVWLF IRU WKH 62, 026)(7 9A DQG XQHIIf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eLJFW4IRSRUW IRU WKH 0RGHUDWH ,QYHUVLRQ $QDO\VLV ,QVXEVHFWLRQ ZH HVWLPDWHG IRU D W\SLFDO GHYLFH WKDW WKH WKKÂ«Â«K6OGW YR7WDJH GHILQHG E\ WKH L QHDU H[WUDSRO DW LRQ RI WKH PHDVXUHG WI9JIfWKH JUDLQ ERXQGDULHV DUH LQVLJQLILFDQW 9J\ } 9 f LV 9MI 9 DQG WKDW WKH HOHFWURQ PRELOLW\ GHILQHG E\ WKH VORSH RI WKH H[WUDSRODWLRQ LV XSJ FP 9VHF )URP J9J\f WKDW LV DIIHFWHG E\ WKH JUDLQ ERXQGDULHV 9J\ ! ?Af ZH PHDVXUHG EDVHG RQ RXU PRGHO 9 9 16MD r FP IRU (\ DVVXPHG WR EH DW PLGJDSf DQG 1J n1RWH WKDW W\SLFDOO\ 9M\ 9\\ QN7T ZLWK Q r GHSHQGLQJ RQ DQG FRI >76E@ 7KXV RXU VWURQJLQYHUVLRQ PHDVXUHPHQWV LPSO\ 9\\ ZKLFK LV FRQVLVWHQW ZLWK FDOFXODWLRQV EDVHG RQ f f Â‘:HAVWUHVVLWKDW WKH GLIIHUHQFH EHWZHHQ 9\ DQG 9M\ FDQ EH LJQRUHG IRU WKH WURQJ LQYHUVLRQ DQDO\VLV EHFDXVH 9J\ 9\\f !! N7T 7HaS7RW LQ )LJ WKH J9J\f FKDUDFWHULVWLF RI D W\SLFDO GHYLFH PHÂ£VXUHGDWURRP WHPSHUDWXUH 1RWH HVSHFLDOO\ WKH ORZHU9J\ 9\f naGDWDaZKLFK VKRZ D QHDUO\ H[SRQHQWLDO GHSHQGHQFH RQ Y*I fÂµ )RU FRPSDULVRQ ZH DOVR VKRZ LQ )LJ WKHRUHWLFDO J9J\f FXUYHV WKDW ZHUH QXPHULFDOO\ Dr Â‘ mUU m%UD GHULYHG IURP f DQG f f XVLQJ WKH SDUDPHWHU YDOXHV JLYHQ DERYH DQG [ r FPfÂ¯A :H YDULHG (\ DQG OHW 1VI ZKLFK DQG f LPSOLHV 9SS 9 7KH FDOFXODWHG J9J\f nFKDUDFWHULVWLFV DOVR DUH QHDUO\ H[SRQHQWLDO IRU ORZ 9J\ HYHQ WKRXJK WKH LQYHUVLRQ OHYHO LV QRW ZHDN ,Q ZHDN LQYHUVLRQ WKH FRQGXFWDQFH RI VLQJOHFU\VWDO 026)(7V LV H[SRQHQWLDOO\ GHSHQGHQW RQ WKH JDWH YROWDJH EHFDXVH S LV >76E 6:@f 7KLV GHSHQGHQFH LV GXH SULPDULO\ WR WKH H[SFI\ JN7f WHUP LQ f DV LPSOLHG E\ WKH VWURQJ GHSHQGHQFH RI 6 PAGE 54 LH WKH LQYHUVH VORSHf RQ (\ $V (\ PRYHV IURP PLGJDS (f WRZDUG WKH FRQGXFWLRQ EDQG 6 LQFUHDVHV ZKHQ (\(f m H9 WKH PHDVXUHG 6 LV PRGHOHG ZHOO 7KXV WKH HQHUJ\ OHYHO RI WKH JUDLQERXQGDU\ WUDSV VLJQLILFDQWO\ DIIHFWV WKH FKDQQHO FRQGXFWDQFH EHORZ WKH HOHFWURQ PRELOLW\ WKUHVKROG 9TM Af :H LOOXVWUDWH LQ )LJ WKH HIIHFW RI 1V\ RQ WKH J9J\f FKDUDFWHULVWLF 7KH WKHRUHWLFDO FXUYHV SORWWHG ZHUH GHULYHG XVLQJ WKH VDPH SDUDPHWHU YDOXHV IRU )LJ DQG (\(Af H9 )RU HDFK YDOXH RI 1V\ 9S% ZDV FDOFXODWHG IURP f XVLQJ 9A ,QFUHDVLQJ 1V\ WHQGV WR VXSSUHVV WKH FRQGXFWDQFH IRU LQWHUPHGLDWH YDOXHV a 9\f RI Y*I fÂ¬ EXW GRHV QRW VLJQLILFDQWO\ DIIHFW 6 %\ FRPSDULQJ WKH FDOFXODWHG FXUYHV ZLWK WKH PHDVXUHG GDWD ZH FUXGHO\ HVWLPDWH WKDW a FP H9 0HDVXUHPHQWV DW GLIIHUHQW WHPSHUDWXUHV 7 r& r& DQG r&f LQGLFDWH WKDW IRU LQWHUPHGLDWH 9J\ ERWK J9J\f DQG 6 LQFUHDVH ZLWK LQFUHDVLQJ $V 7 LQFUHDVHV IURP r& WR r& 6 LQFUHDVHV IURP 9GHFDGH WR 9GHFDGH DQG DW 9 9 J LQFUHDVHV IURP [ fÂ¯p X WR [ p X 7KHVH FKDQJHV DUH FRQVLVWHQW ZLWK f LQ ZKLFK IRU UHODWLYHO\ ORZ 9J\ WKH H[SLFI)SN7f WHUP GHILQHV WKH SUHGRPLQDQW GHSHQGHQFH RQ WHPSHUDWXUH 6XPPDU\ $ SK\VLFDO PRGHO WKDW GHVFULEHV WKH HIIHFWV RI JUDLQ ERXQGDULHV RQ FKDQQHO FRQGXFWDQFH LQ 62, 026)(7V KDV EHHQ GHYHORSHG DQG VXSSRUWHG H[SHULPHQWDOO\ 7KHVH HIIHFWV RULJLQDWH ZKHQ HOHFWURQV QFKDQQHO 026)(7f DUH WUDSSHG DW ORFDOL]HG JUDLQERXQGDU\ VWDWHV WKHUHE\ FUHDWLQJ PAGE 55 SRWHQWLDO EDUULHUV WKDW LQIOXHQFH WKH IORZ RI HOHFWURQV IURP VRXUFH WR GUDLQ 7KH HOHFWURQ WUDSSLQJ GHSHQGV RQ WKH GHJUHH RI LQYHUVLRQ LQ WKH FKDQQHO DQG KHQFH RQ WKH JDWH YROWDJH )RU VXIILFLHQWO\ KLJK 9*I} r%R LV ORZ HQRXJK WKDW WKH JUDLQ ERXQGDULHV DUH LQFRQVHTXHQWLDO ZLWK UHJDUG WR J DQG JA +RZHYHU IRU ORZHU 9*I WKH JUDLQ ERXQGDULHV FDQ SUHGRPLQDQWO\ FRQWURO J DQG JP DQG FDQ GHILQH Df DQ HIIHFWLYH WXUQRQ OLQHDUUHJLRQf FKDUDFWHULVWLF WKDW RFFXUV ZHOO EH\RQG WKH VWURQJ LQYHUVLRQ WKUHVKROG DV LOOXVWUDWHG LQ )LJV DQG DQG Ef D QHDUO\ H[SRQHQWLDO GHSHQGHQFH ZLWK JDWH YROWDJH DV VKRZQ LQ )LJV DQG IRU PRGHUDWH LQYHUVLRQ FRQGLWLRQV 7KH HIIHFWLYH WXUQRQ FKDUDFWHULVWLF GHVFULEHG JHQHUDOO\ E\ f DQG DSSUR[LPDWHG E\ f LV DFWXDOO\ D UHIOHFWLRQ RI WKH FDUULHU PRELOLW\ WXUQRQ ZKLFK LV FRQWUROOHG E\ WKH JUDLQ ERXQGDULHV ,W GHILQHV WKH HOHFWURQ PRELOLW\ WKUHVKROG YROWDJH 9\ ZKLFK H[FHHGV 9MI DQG WKH HIIHFWLYH HOHFWURQ PRELOLW\ X ZKLFK LVfÂ¯W\SLFDOO\ KLJKHU WKDQ WKH DFWXDO LQWUDJUDLQf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Â« SURSHUWLHV RI WKH 6L6Â2 LQWHUIDFH LH WKH IDVW VXUIDFHVWDWH GHQVLW\ *UDLQ ERXQGDULHV SHUSHQGLFXODU WR WKH FDUULHU IORZ LQ WKH FKDQQHO PD[LPL]HV WKH JUDLQERXQGDU\ HIIHFWV RQ WKH FRQGXFWDQFH DV GHVFULEHG E\ f ,Q FRQWUDVW JUDLQ ERXQGDULHV SDUDOOHO WR FDUULHU IORZ LQ WKH FKDQQHO GRHV QRW DIIHFW WKH FRQGXFWDQFH Â‘ DOWKRXJK LW PD\ HQKDQFH VRXUFH GUDLQ OHDNDJH FXUUHQW YLD RWKHU PHFKDQLVPVf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f LV WKHQ QRQOLQHDU DQG WKH FKDQQHO FRQGXFWLRQ GHSHQGV RQ KRZ WKH JUDLQ ERXQGDULHV DUH GLVWULEXWHG EHWZHHQ WKH VRXUFH DQG WKH GUDLQ $OWKRXJK RXU PRGHO DFFRXQWV IRU DQ\ QXPEHU RI JUDLQ ERXQGDULHV LQ WKH FKDQQHO ZH DSSO\ LW KHUHLQ WR WKH PRVW OLNHO\ FDVH LQ EHDPUHFU\VWDOOL]HG 9/6,f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f IUDFWLRQ RI WKH WKHUPLRQLFDOO\ HPLWWHG HOHFWURQV DUH FDSWXUHG E\ WKH JUDLQ ERXQGDU\ WUDSV >08@ RU WKDW WKH FKDUJH WUDSSHG DW WKH JUDLQ ERXQGDU\ LV LQGHSHQGHQW RI WKH JUDLQERXQGDU\ YROWDJH GURS >5$D@ 7KH IRUPHU DVVXPSWLRQ LV LQYDOLG EHFDXVH WKH UDWH RI WKH EDQG WRWUDS UHFRPELQDWLRQ SURFHVV LV SURSRUWLRQDO >6=@ WR WKH FRQFHQWUDWLRQ RI XQRFFXSLHG WUDSV DQG QRW WR WKH FXUUHQW 7KH ODWWHU DVVXPSWLRQ LV LQYDOLG EHFDXVH WKH FKDUJH WUDSSHG DW WKH JUDLQ ERXQGDU\ FDQ EH H[SUHVVHG VHH $SSHQGL[ $f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fÃ¼ 0RGHO FDOFXODWLRQV VXSSRUWHG E\ OLPLWHG H[SHULPHQWDO UHVXOWV VKRZ WKDW JUDLQ ERXQGDULHV JHQHUDOO\ WHQG WR GHFUHDVH WKH FRQGXFWDQFH GUDLQ FXUUHQWf RI 62, 026)(7V EXW FDQ LQFUHDVH WKH WUDQVFRQGXFWDQFH *UDLQ ERXQGDULHV KDYLQJ D WUDS GHQVLW\ FRPSDUDEOH WR WKDW aA FPAf HVWLPDWHG IRU W\SLFDO KLJKDQJOH ERXQGDULHV LQ EHDPUHFU\VWDOOL]HG 62, FDQ ZKHQ ORFDWHG QHDU WKH GUDLQ VLJQLILFDQWO\ DIIHFW WKH FXUUHQW YROWDJH FKDUDFWHULVWLFV RI WKH 62, 026)(7 LQ DOO UHJLRQV RI RSHUDWLRQr 7KH JUDLQERXQGDU\ HIIHFW LV HQKDQFHG DV WKH FKDQQHO OHQJWK LV VKRUWHQHG Q IWLDO\6Â6 U U Â‘ :H UHIHU WR WKH QFKDQQHO HQKDQFHPHQWPRGH ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7 LOOXVWUDWHG LQ )LJ 7R HPSKDVL]H WKH JUDLQERXQGDU\ HIIHFWV ZH DVVXQH WKDW WKH SRO\VLOLFRQ ILOPnLV QRW FRPSOHWHO\ GHSOHWHG EHWZHHQ WKH IURQW DQG EDFN VXUIDFHV VR WKDW FKDUJHFRXSOLQJ HIIHFWV >/,E@ FDQ EH LJQRUHG YLVDYLV WKH EDFN JDWH LV LQFRQVHTXHQWLDOf :H LQLWLDOO\ DVVXPH WKDW WKH IURQWf FKDQQHO FRPSULVHV 1J JUDLQV VHSDUDWHG E\ 1J f LGHQWLFDO JUDLQ ERXQGDULHV VXUIDFHVf SHUSHQGLFXODU WR WKH FDUULHU HOHFWURQf IORZ /DWHU ZH DQDO\]H WKH OLNHO\ FDVH RI D VLQJOH JUDLQ ERXQGDU\ LQ WKH FKDQQHO 1J f HPSKDVL]LQJ WKH LPSRUWDQFH RI LWV SRVLWLRQ 7KH HQHUJ\EDQG GLDJUDP DW WKH MWK > M 1J f@ JUDLQ ERXQGDU\ FRXQWHG IURP VRXUFH WR GUDLQ LV LOOXVWUDWHG LQ )LJ IRU WKH FDVHV RI ]HUR GUDLQ YROWDJH 9Tf DQG RI 9T ! :KHQ 9T HOHFWURQV WUDSSHG DW ORFDOL]HG JUDLQERXQGDU\ VWDWHV SURGXFH WKH SRWHQWLDO EDUULHU7J DW HDFK JUDLQ ERXQGDU\f ZKLFK PAGE 60 fÃ¼ )LJ )QHUJ\EDQG GLDJUDP DW MWK JUDLQ ERXQGDU\ IRU GUDLQ YROWDJH HTXDO WR Df DQG JUHDWHU WKDQ Ef ]HUR PAGE 61 LV GHWHUPLQHG E\ WKH LQYHUVLRQ OHYHO YLVDYLVWKH IURQWf JDWH YROWDJH 9JI DV ZH GHVFULEHG LQ WKH SUHYLRXV FKDSWHU :KHQ 95 ! D YROWDJH 9JAM LV GURSSHGfÂ§DFURVVfÂ§WKH MWK JUDLQ ERXQGDU\ VNHZLQJ WKH HQHUJ\EDQG GLDJUDP DV LOOXVWUDWHG ,I 9JKM LV ODUJH HQRXJK LW SURGXFHV VLJQLILFDQW FKDQJHV LQ WKH DUHDOf GHQVLW\ RI FKDUJH IL5M WUDSSHG DW WKH JUDLQ ERXQGDU\ DQG LQ WKH LQYHUVLRQ OHYHOV LQ WKH DGMDFHQW JUDLQV )RUPDOLVP )URP )LJ IRU 95 ! 9JEM rfÂ¬EM EM f U ZKHUH 7SM DQGn3SM DUH WKH SRWHQWLDO EDUULHUV RQ WKH OHIW DQG ULJKW VLGHV 3 RI WKH MWK JUDLQ ERXQGDU\ DQG I! DQG ! DUH WKH HOHFWURQ TXDVL)HUPL SRWHQWLDOV LQ WKH OHIW DQG ULJKW DGMDFHQW JUDLQV 7KH DYHUDJH HOHFWURQ GHQVLWLHV LQ WKH DGMDFHQW LQYHUVLRQ OD\HUV DUH Â³O QH[STM! N7f f Â³M QH[STff -N7f f ZKHUH QA LV WKH LQWULQVLF FDUULHU GHQVLW\ LQ VLOLFRQ 7KH GHQVLWLHV LQ f DQG f DUH UHODWHG WR WKH LQYHUVLRQ OD\HU DUHDOf FKDUJH GHQVLWLHV 4Q RQ WKH OHIW DQG ULJKW E\ f PAGE 62 7KH HOHFWURQ WUDQVSRUW LV FRQWUROOHG E\ WKH JDWH DULG GUDLQ YROWDJHV WKURXJK WKH GHSHQGHQFH RI 4Q RQ 9*I DQG 9' 7R FKDUDFWHUL]H WKLV GHSHQGHQFH DV ZHOO DV WKH LQWUDJUDLQ FXUUHQW ZH XVH WKH FKDUJHVKHHW QRGH >%5 %5ZKLFK LV DSSOLFDEOH IRU DOO OHYHOV RI LQYHUVLRQ $W DQ DUELWUDU\ LQWUDJUDLQf SRLQW \ LQ WKH FKDQQHO Q\f 4V\f E\f f ZKHUH F_)F[\f Q L R V\f A 67 AL:AVIÂ¯\f 9\f@@`Â f $ LV WKH FKDUJH GHQVLW\ LQ WKH VLOLFRQ DQG A VI E\ffÂ¯rUIfÂ§77a @ f LV WKH GHSOHWLRQUHJLRQ FKDUJH GHQVLW\ ,Q f DQG f LV WKH EDQG EHQGLQJ QRUPDO WR WKH IURQW VXUIDFHf 9 LV WKH GLIIHUHQFH EHWZHHQ WKH HOHFWURQ DQG KROH TXDVL)HUPL SRWHQWLDOV >9f 9/f ZKHUH / LV WKH FKDQQHO OHQJWK@ DQG S >N7H 7KH EDQG EHQGLQJ LV UHODWHG WR 9*IU E\ RV\f FRI>YILI 95 nVI\fO f PAGE 63 7R FRPSOHWH WKH GHVFULSWLRQ RI WKH HQHUJ\EDQG GLDJUDP LQ )LJ ZH HQVXUH WKDW FKDUJH LV FRQVHUYHG LQ WKH YLFLQLW\ RI WKH JUDLQ ERXQGDU\ UFS Â³9 OA 8*%M 8&)VM %Mf ZKLFK HTXDWHV WKH FKDUJH WUDSSHG DW WKH MWK JUDLQ ERXQGDU\ WR WKH HOHFWURQ FKDUJH UHPRYHG WR IRUP WKH DGMDFHQW GHSOHWLRQ UHJLRQV :H DVVLPH WKH UHJLRQV DUH YLUWXDOO\ GHSOHWHG RI IUHH HOHFWURQVf %HFDXVH WKH LQYHUVLRQ OD\HU LV YRLG RI KROHV WKH HOHFWURQ FDSWXUH DQG HPLVVLRQ UDWHV IRU WKH JUDLQERXQGDU\ WUDSV PXVW EH HTXDO LQ WKH VWHDG\ VWDWH DQG KHQFH FDQ EH H[SUHVVHG LQ WHUPV RI WKH HOHFWURQ TXDVL)HUPL OHYHO (SQM DW WKH MWK JUDLQ ERXQGDU\ VHH $SSHQGL[ $f *%M T1 67 M H[S (7a()QM Â‘ N7 f ,Q f (7(SQMf GHSHQGV RQ 9JEM DV VXJJHVWHG E\ )LJ 7KLV GHSHQGHQFH LV LQ JHQHUDO FRPSOLFDWHG DQG FDQ EH GHILQHG RQO\ ZKHQ WKH HOHFWURQ WUDQVSRUW PHFKDQLVPVf LV VSHFLILHG $OWKRXJK PDQ\ WKHRULHV UHJDUGLQJ FDUULHU WUDQVSRUW WKURXJK JUDLQ ERXQGDULHV KDYH EHHQ SXUSRUWHG HJ WKHUPLRQLF HPLVVLRQr GLIIXVLRQ WKHUPLRQLF ILHOG HPLVVLRQf QRQH FDQ EH YHULILHG XQHTXLYRFDOO\ EHFDXVH RI WKH FRPSOH[ YDULDEOH QDWXUH RI WKH JUDLQ ERXQGDULHV 7KXV WR DYRLG XQGXH PRGHO FRPSOH[LW\ ZH DVVXPH DV LQ WKH SUHYLRXV FKDSWHU WKDW WKH SUHGRPLQDQW WUDQVSRUW PHFKDQLVP LV WKHUPLRQLF HPLVVLRQ RYHU WKH SRWHQWLDO EDUULHU 7KLV VLPSOLI\LQJ DVVXPSWLRQ LV SK\VLFDOO\ UHDVRQDEOH DW DQG DERYH URRP WHPSHUDWXUH ZKHUH PAGE 64 WKHUPLRQLFn ILHOG HPLVVLRQ LV QRW SUREDEOH DQG IRU VXEVWDQWLDO QRQWULYLDOf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f HPLWWHG HOHFWURQV GULIW DZD\ IURP WKH JUDLQ ERXQGDU\ 7KXV LQ f HU (7 :fÂ° (7 (L! ! ZKHUH (\(cf JLYHV JHQHUDOO\ WKH SRVLWLRQ RI WKH WUDSV LQ WKH HQHUJ\ JDS :H VWUHVV WKDW G(SQG\ DW WKH JUDLQ ERXQGDU\ LV QRW UHODWHG WR WKH FXUUHQW EHFDXVH RI WKH DVVXPSWLRQV WKDW WKH FDUULHU WUDQVSRUW LV GHVFULEHG E\ WKHUPLRQLF HPLVVLRQ WKHRU\ DQG QRW E\ GLIIXVLRQ WKHRU\ :H KDYH QRZ GHVFULEHG LQ f f KRZ WKH HQHUJ\EDQG GLDJUDP DW D JUDLQ ERXQGDU\ FKDQJHV WR UHIOHFW WKH YROWDJH GURS 9JAM %\ XVLQJ SK\VLFDOO\ UHDVRQDEOH DSSUR[LPDWLRQV ZH KDYH DYRLGHG WKH XVH RI D FODVVLFDO EXW JHQHUDOO\ LQYDOLG DVVXPSWLRQ >08@ WKDW D FRQVWDQWf IUDFWLRQ RI WKH WKHUPLRQLFDOO\ HPLWWHG HOHFWURQV DUH FDSWXUHG E\ WKH JUDLQERXQGDU\ WUDSV 7KLV FRPPRQO\ XVHG DVVXPSWLRQ LQ IDFW RYHUO\ GHILQHV WKH JUDLQERXQGDU\ WUDQVSRUW SUREOHP EHFDXVH WKH UDWH RI PAGE 65 WKHED"WGWRWUDS UHFRPELQDWLRQ SURFHVV LV SURSRUWLRQDO >6=@ WR WKH FRQFHQWUDWLRQ RI XQRFFXSLHG WUDSV DQG QRW WR WKH FXUUHQW :H KDYH IXUWKHUPRUH QRW XVHG DQRWKHU FRPPRQ DVVXPSWLRQ >%$D@ WKDW 2JJM LV LQGHSHQGHQW RI YJEM! ZKLFK LV DOVR JHQHUDOO\ LQYDOLG DV LQGLFDWHG E\ f 2XU PRGHO IRU WKH VWHDG\VWDWH FXUUHQWYROWDJH FKDUDFWHULVWLFV RI WKH ODUJHJUDLQ SRO\VLOLFRQ 026)(7 LV FRPSOHWHG E\ Df HTXDWLQJ WKH GUDLQ FXUUHQW ,4 WR WKH QHW WKHUPLRQLFHPLVVLRQ FXUUHQW GHILQHG E\ WKH SHUWXUEHG HQHUJ\EDQG GLDJUDP DW HDFK JUDLQ ERXQGDU\ Ef HTXDWLQJ ,T WR WKH FXUUHQW GHILQHG E\ WKH FKDUJHVKHHW PRGHO >%5@ DSSOLHG WR HDFK iUVÂIFGGFf VXPPLQJ DOO WKH JUDLQERXQGDU\ DQG JUDLQ YROWDJH GURSV WR 9 7KH QHW WKHUPLRQLFHPLVVLRQ FXUUHQW GHQVLW\ RYHU WKH SRWHQWLDO EDUULHU DW WKH MWK JUDLQ ERXQGDU\ )LJ f LV >%$E@ fÂ§ $r7 U U ,}@ -JEM ccÂ> QM H[S 7U ! n QMH[S WK Qf ZKHUH $r LV WKH HIIHFWLYH 5LFKDUGVRQ FRQVWDQW IRU HOHFWURQV LQ VLOLFRQ a $FP . f DQG 1T LV WKH HIIHFWLYH GHQVLW\ RI VWDWHV LQ WKH FRQGXFWLRQEDQG [ r FPfÂ¯A DW r.f 7KXV IRU DOO M Â‘ O?f =[LHIIf-JEM f ZKHUH = LV WKH FKDQQHO ZLGWK :H DVVXPH [AIA a $f LV FRQVWDQW LQGHSHQGHQW RI SRVLWLRQ DQG ELDV Â³ UHIOHFWV FKDQJHV LQ WKH ORFDO PAGE 66 FKDQQHO FRQGXFWLYLW\ :LWK f DQG f f f DQG f UHODWH ,Q WR 9JAM IRU WKH 1J f JUDLQ ERXQGDULHV 8VLQJ WKH FKDUJHVKHHW PRGHO >%5@ ZH QRZ H[SUHVV ,T DV D IXQFWLRQ RI WKH EDQG EHQGLQJ DW WKH OHIW DQG ULJKW VLGHV RI HDFK JUDLQ )RU WKH NWK N 1Af JUDLQ ZLWK OHQJWK \JA / \JM fÂ§A\J1Jf N7Y 7' V JN -RI ^& 7)9*I ZALIN AVINA Â‘ L0AAVINA WRO ?L T? &IWRU ? fÂ± ?L IVIN` 7)n 7 PAGE 67 1XPHULFDO 6ROXWLRQ 7KH FXUUHQWYROWDJH FKDUDFWHULVWLF LV HYDOXDWHG QXPHULFDOO\ E\ VROYLQJ VLPXOWDQHRXVO\ WKH QRQOLQHDU V\VWHP RI HTXDWLRQV GHVFULEHG E\ f DQG f f IRU DOO M DQG N ,QVWHDG RI VROYLQJ GLUHFWO\ WKLV QRQOLQHDU V\VWHP RI HTXDWLRQV ZH REWDLQ WKH VROXWLRQV E\ ILUVW DVVLJQLQJ YDOXHV IRU ,J DQG Y*I} DQG WKHQ FDOFXODWLQJ WKH FRUUHVSRQGLQJ YDOXH RI 9J 7KH DGYDQWDJH RI WKLV PHWKRG LV WKDW ZH DYRLG WKH W\SLFDO FRQYHUJHQFH SUREOHPV RI WKH LWHUDWLYH PHWKRGV >%8@ IRU VROYLQJ QRQOLQHDU V\VWHP RI HTXDWLRQV EHFDXVH ZH RQO\ VROYH PDQ\ QRQOLQHDU LQGHSHQGHQW HTXDWLRQV ZLWK RQH YDULDEOH 7R LOOXVWUDWH WKH SUHGLFWLRQV RI RXU PRGHO ZH DSSO\ LW WR D W\SLFDO EXW WKLFNf 62, 026)(7 IRU ZKLFK ORrp FQ7A &4\ [ fÂ¯A )FPA WKH JDWH R[LGH WKLFNQHVV LV $f = / SP DQG SQJ FPA9VHF 7R HPSKDVL]H WKH PRVW OLNHO\ FDVH LQ EHDP UHFU\VWDOOL]HG 62, 9/6,f ZH OHW 1J RQH JUDLQ ERXQGDU\f :H SORW LQ )LJ IRU 16\ A FPA (\ (c WUDSV DW PLGJDSf DQG \JM \J / JUDLQ ERXQGDU\ DW PLGGOH RI FKDQQHOf WKH FDOFXODWHG ,J9Jf FKDUDFWHULVWLFV IRU VHYHUDO YDOXHV RI 9J\ 9\\f WKH WKUHVKROG YROWDJH 9\\ LV WKH YDOXH RI 9*\ \LHOGHG E\ f DQG f ZKHQ 7V\9 f LV WZLFH WKH )HUPL SRWHQWLDO RI WKH VLOLFRQ ILOP ERG\ :H QRWH WKDW 9S5 GRHV QRW QHHG WR EH VSHFLILHG EHFDXVH LW LV UHODWHG WR 9\\ WKURXJK f DQG f HYDOXDWHG DW \ )RU FRPSDULVRQ ZH DOVR SORW GDVKHG FXUYHVf FRUUHVSRQGLQJ FKDUDFWHULVWLHV IRU 1J QR JUDLQ ERXQGDU\f 7KH JUDLQ ERXQGDU\ UHGXFHV ,J DV LQ WKH OLQHDU UHJLRQ VHH &KDSWHU 7ZRf LWV HIIHFW LV PRVW VLJQLILFDQW DW ORZ 9JI ,Q WKH VDWXUDWLRQ UHJLRQ PAGE 68 Â« )LJ &DOFXODWHG FXUUHQWYROWDJH FKDUDFWHULVWLFV VROLG FXUYHVf IRU W\SLFDO ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7 ZLWK RQH JUDLQ ERXQGDU\ DW PLGGOH RI FKDQQHO :LWKRXW WKH JUDLQ ERXQGDU\ WKH GDVKHG FXUYHV GHULYH PAGE 69 ,SVDWf FDQ VXEVWDQWLDOO\ OLPLWHG DOWKRXJK 9SAVDWA LV YLUWXDOO\ XQDIIHFWHG VLQFH 9\ /f DOZD\V HTXDOV WKH GUDLQ YROWDJH 7KH JUDLQERXQGDU\ HIIHFW LV VWURQJO\ GHSHQGHQW RQ 1A\ 7R LOOXVWUDWH WKLV GHSHQGHQFH ZH SORW LQ )LJ WKH VTXDUHURRW RI ,S VDWf YHUVXV Y*I YMIf ARUA167 UDQLQ IURP QR JUDLQ ERXQGDU\f WR [ r FPA ?V Q M LQFUHDVHV WKH JUDLQERXQGDU\ SRWHQWLDO EDUULHU LQFUHDVHV DQG KHQFH D ODUJHU SDUW RI 9'VDWA PXVW EH GURSSHG DFURVV WKH ERXQGDU\ WR HQDEOH ,RVDWf WR IORZ WKURXJK LW )RU 1A\ KLJK ,QVDWf UHGXFHG FRQVLGHUDEO\ HYHQ IRU 9*I KLJK %HFDXVH WKH JUDLQERXQGDU\ SRWHQWLDO EDUULHU LQFUHDVHV DV WKH DGMDFHQW LQWUDJUDLQ LQYHUVLRQ OHYHO GHFUHDVHV WKH HIIHFW RI WKH JUDLQ ERXQGDU\ ZLOO IRU 9S ! EH VWURQJHU LI WKH ERXQGDU\ LV FORVHU WR WKH GUDLQ 7R HPSKDVL]H WKLV LPSRUWDQW SRVLWLRQ GHSHQGHQFH ZH VKRZ LQ )LJ KRZ WKH FDOFXODWHG ,TVDWf9AIf FKDUDFWHULVWLF LV DOWHUHG DV WKH JUDLQ ERXQGDU\ ZLWK [ r AA LV VKLIWHG WRZDUG WKH GUDLQ 6LQFH 4Q a QHDU WKH VDWXUDWHG GUDLQ D JUDLQ ERXQGDU\ WKHUH LV LQIOXHQWLDO UHJDUGOHVV RI KRZ KLJK LV $ JUDLQ ERXQGDU\ QHDU WKH VRXUFH KRZHYHU LV VLJQLILFDQW RQO\ IRU 9J\ ORZ :H DOVR VHH LQ )LJ WKDW WKH SRVLWLRQ RI WKH JUDLQ ERXQGDU\ LV LUUHOHYDQW IRU 9IL\ 9\\f 9 EHFDXVH WKH LQYHUVLRQ HYHU UHPDLQV QHDUO\ FRQVWDQW DORQJ WKH FKDQQHO IRU WKLV ORZ 9*\ 7KH JUDLQERXQGDU\ HIIHFW LOOXVWUDWHG LQ )LJV LV HQKDQFHG DV WKH FKDQQHO OHQJWK LV VKRUWHQHG 7KLV HQKDQFHPHQW LV GHPRQVWUDWHG LQ )LJ ZKHUH ZH SORW WKH FDOFXODWHG ,QVDWf A*IA FKDUDFWHULVWLF IRU GLIIHUHQW / ZLWK =/ DQG \A a / 7KH PAGE 70 )LJ &DOFXODWHG GHSHQGHQFH RI GUDLQ VDWXUDWLRQ FXUUHQW YHUVXV JDWH YROWDJH RQ JUDLQERXQGDU\ DW PLGGOH RI FKDQQHOf WUDS GHQVLW\ PAGE 71 3LJ &DOFXODWHG GHSHQGHQFH RI GUDLQ VDWXUDWLRQ FXUUHQW YHUVXV JDWH YROWDJH RQ JUDLQERXQGDU\ SRVLWLRQ DORQJ FKDQQHO PAGE 72 )LJ &DOFXODWHG GHSHQGHQFH RI GUDLQ VDWXUDWLRQ FXUUHQW YHUVXV JDWH YROWDJH RQ FKDQQHO OHQJWK ZLWK JUDLQ ERXQGDU\n / IURP GUDLQ PAGE 73 fÃ¼ UHGXFWLRQ LQ FXUUHQW ZLWK GHFUHDVLQJ FKDQQHO OHQJWK UHVXOWV EHFDXVH WKH FRQVWUDLQW RQ ,S GHILQHG E\ f DQG f LV LQGHSHQGHQW RI / DQG KHQFH 9JA PXVW LQFUHDVH WR VXSSRUW KLJKHU FXUUHQW GHQVLWLHV LQ WKH FKDQQHO 7R IXUWKHU VWUHVV WKH VLJQLILFDQFH RI JUDLQ ERXQGDULHV LQ ODUJH JUDLQ SRO\VLOLFRQ 62, 026)(7V ZH SORW LQ )LJ WKH FDOFXODWHG WUDQVFRQGXFWDQFH LQ WKH VDWXUDWLRQ UHJLRQ PVDWf A VDWfA*Ir ARU RQH JUDLQ ERXQGDU\ LQ WKH PLGGOH RI WKH FKDQQHO KDYLQJ GLIIHUHQW YDOXHV RI 1J\ 'HSHQGLQJ RQ 9J\ JPVD\f FDQ EH ORZHU RU KLJKHU WKDQ WKDW IRU WKH JUDLQERXQGDU\IUHH 16\ f FRXQWHUSDUW $W ORZ 9J\ EHORZ WKH PRELOLW\ WKUHVKROG A f WKH JUDLQ ERXQGDU\ YLUWXDOO\ LQKLELWV FXUUHQW WKXV JPVD\f a $V 9J\ LQFUHDVHV WKH JUDLQERXQGDU\ HIIHFW LV GLPLQLVKHG DV WKH LQWUDJUDLQ FKDQQHO FRQGXFWDQFH LV HQKDQFHG WKHUHE\ SURGXFLQJ XQXVXDOO\ KLJK WUDQVFRQGXFWDQFH OLNH LQ WKH OLQHDU UHJLRQ DQDO\VLV RI &KDSWHU 7ZRf $W KLJK 9J\ WKH JUDLQERXQGDU\ HIIHFW WHQGV WR VXEVLGH DQG JPVD\f DSSURDFKHV WKDW FRUUHVSRQGLQJ WR 16\ ([SHULPHQWDO 6XSSRUW DQG 'LVFXVVLRQ 7R SURYLGH H[SHULPHQWDO VXSSRUW IRU WKH DQDO\VLV ZH PHDVXUHG FXUUHQWYROWDJH FKDUDFWHULVWLFV RI ODUJHJUDLQ SRO\VLOLFRQ 62, 026)(7V GHVFULEHG LQ 6HFWLRQ :H ILQG WKDW WKH PHDVXUHG ,TVDWf LV VPDOOHU WKDQ WKDW RI WKH WKHRUHWLFDO FDOFXODWLRQV RI WKH FRUUHVSRQGLQJ VLQJOHFU\VWDO FRXQWHUSDUW 1T f DQG WKDW WKLV UHODWLYH GLIIHUHQFH LQFUHDVHV DV 9J\ GHFUHDVHV 7KLV UHVXOW LPSOLHV TXDOLWDWLYH VXSSRUW IRU RXU DQDO\VLV PAGE 74 )LJ &DOFXODWHG VDWXUDWLRQUHJLRQ WUDQVFRQGXFWDQFH YHUVXV JDWH YROWDJH DQG JUDLQERXQGDU\ DW PLGGOH RI FKDQQHOf WUDS PAGE 75 8QIRUWXQDWHO\ TXDQWLWDWLYH VXSSRUW LV QRW REWDLQHG EHFDXVH 1J LV QRW NQRZQ H[DFWO\ $GGLWLRQDO VXSSRUW IRU RXU DQDO\VHV KDV EHHQ SUHVHQWHG E\ &ROLQJH HW DO >&@ ZKR GHYHORSHG D WHFKQLTXH WR FRQWURO WKH ORFDWLRQ RI WKH JUDLQ ERXQGDULHV LQ 62, 026)(7V 7KH\ IDEULFDWHG WZR WUDQVLVWRUV ZLWK WKH VDPH JHRPHWU\ RQH EHVLGH WKH RWKHU RQH RI WKHP ZLWK D SHUSHQGLFXODU JUDLQ ERXQGDU\ DW WKH PLGGOH RI WKH FKDQQHO DQG WKH RWKHU ZLWKRXW JUDLQ ERXQGDU\ 7KH\ IRXQG WKDW ,Q IRU WKH WUDQVLVWRU ZLWK XVDWf D JUDLQ ERXQGDU\ LV VPDOOHU WKDQ WKDW RI WKH WUDQVLVWRU ZLWKRXW D JUDLQ ERXQGDU\ Â‘ :H FRQFOXGH WKLV VHFWLRQ E\ VWUHVVLQJ WKUHH VLJQLILFDQWnFRQFOXVLRQV GUDZQ IURP WKLV DQDO\VLV )LUVW EHFDXVH WKH ,S9Sf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f ERXQGDULHV ,Q LV UHGXFHG VDWf 6XPPD U\ 8VLQJ VLPSOLI\LQJ EXW SK\VLFDOO\ UHDVRQDEOH DVVXPSWLRQV ZH KDYH PRGHOHG WKH HIIHFWV RI JUDLQ ERXQGDULHV RQ WKH VWHDG\VWDWH FXUUHQW PAGE 76 fÃ¼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f EXW FDQ LQFUHDVH RU GHFUHDVH WKH WUDQVFRQGXFWDQFH $OWKRXJK WKH JUDLQERXQGDU\ HIIHFWV DUH PRVW DSSDUHQW DW ORZ JDWH YROWDJHV WKH\ FDQ EH TXLWH VLJQLILFDQW DW KLJKHU JDWH YROWDJHV ZKHQ WKH GUDLQ YROWDJHLV KLJK HJr LQ WKH VDWXUDWLRQ UHJLRQ *UDLQ ERXQGDULHV FORVH WR WKH GUDLQ DUH PRVW HIIHFWLYH 7KH HIIHFWV DUH HQKDQFHG DV WKH FKDQQHO OHQJWK LV VKRUWHQHG 9LH KDYH PHDVXUHG FXUUHQWYROWDJH FKDUDFWHULVWLFV RI ERWK >/$@ ODVHU 1J !! f DQG JUDSKLWHVWULSKHDWHU 1J f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n 5HFHQW ODERUDWRU\ DFKLHYHPHQWV >0$ 0$@ LPSO\ WKDW WKH ILUVW FRPPHUFLDO DGDSWDWLRQ RI WKUHHGLPHQVLRQDO LQWHJUDWLRQ PD\ EH VWDFNHG &026 9/6, PHPRU\ FKLSV LQ ZKLFK RQH RI WKH FRPSOHPHQWDU\ WUDQVLVWRUV XVXDOO\ SAKDQQHOf LV IDEULFDWHG LQ D OD\HU RI /3&92 SRO\VLOLFRQ RQ VLOLFRQ GLR[LGH *UDLQERXQGDU\ SDVVLYDWLRQ HJ YLD K\GURJHQDWLRQ >6+@f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f WKH ILOP ERG\ JUDLQVf LV FRPSOHWHO\ GHSOHWHG RI IUHH FDUULHUV IDFLOLWDWHG E\ JUDLQERXQGDU\ WUDSSLQJ RI KROHV 7KH IURQW ? VXUIDFH LV LQYHUWHG IRU VXIILFLHQWO\ KLJK ! af IDFLOLWDWHG E\ SRVLWLYH FKDUJH DW WKH LQWHUIDFH 7KH OHDNDJH FXUUHQW ,_B ,T LQ WKH 2)) UHJLRQf LQFUHDVHV H[SRQHQWLDOO\ ZLWK 9JI DQG DV D SRZHU ! f RI WKH GUDLQ YROWDJH 95 OKH GHYLFH FKDUDFWHUL]HG LQ )LJ LV ORQJFKDQQHO XPf ZKLFK PHDQV WKDW WKH DQRPDORXV ,>B9JI9Tnf LVn QRW D VKRUW FKDQQHO HIIHFW >$.@ 2WKHU PHDVXUHPHQWV >0$@ UHYHDO WKDW LV YLUWXDOO\ LQGHSHQGHQW RI WKH ORQJf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ff SUHGLFWLRQV DQG PHDVXUHG GDWD $ ULJRURXV FRUURERUDWLRQ ZKLFK ZRXOG UHTXLUH FRPSUHKHQVLYH DQDO\VHV RI DOO WKH SRVVLEOH PHFKDQLVPV DQG H[WHQVLYH PHDVXUHPHQWV RI VSHFLDO WHVW VWUXFWXUHV LV QRW IHDVLEOH PAGE 79 9* 9f )LJ 0HDVXUHG FXUUHQWYROWDJH FKDUDFWHULVWLFV RI DQ XQSDVVLYDWHG /3&9' SRO\VLOLFRQ 026)(7 SFKDQQHO DFFXPXODWLRQPRGH = XUQ / SP W I $ 1$ FPf 7KH SRO\VLOLFRQ ILOP LV 2OESPWKLFN DQG ZDV GHSRVLWHG YLD /3&92 RQ D SPWKLFN OD\HU RI VLOLFRQGLR[LGH 7KH EDFN JDWH DQG WKH VRXUFH DUH JURXQGHG PAGE 80 3RVVLEO\ VLJQLILFDQW OHDNDJH PHFKDQLVPV LQ WKH SRO\VLOLFRQ 026)(7 DUH Df VSDFHFKDUJHLPLWHG IORZ >5 6&@ RI KROHV IURP VRXUFH WR GUDLQ WKURXJK WKH GHSOHWHGf ILOP ERG\ Ef WKHUPDO HPLVVLRQ RI FDUULHUV >6=@ YLD JUDLQERXQGDU\ WUDSV LQ WKH GHSOHWLRQ UHJLRQ QHDU WKH GUDLQ Ff ILHOGHQKDQFHG 3RROH)UHQNHO >*5@f WKHUPDO HPLVVLRQ LQ WKH GUDLQ GHSOHWLRQ UHJLRQ Gf LPSDFW LRQL]DWLRQ DYDODQFKLQJf >'8@ LQ WKH GUDLQ GHSOHWLRQ UHJLRQ Hf EDQGEDQG ILHOG HPLVVLRQ WXQQHOLQJf >5,@ LQ WKH GUDLQ GHSOHWLRQ UHJLRQ DQG If ILHOG HPLVVLRQ YLD JUDLQn ERXQGDU\ WUDSV >*5@ RU SRVVLEO\ PHWDO SUHFLSLWDWHV >/()@ LQ WKH GUDLQ GHSOHWLRQ UHJLRQ 7KH PHDVXUHG LQGHSHQGHQFH RI A RQ ORQJf FKDQQHO OHQJWK >0$@ UXOHV RXW 6SDFHFKDUJHOLPLWHG IORZÂ‘fÂ¯nÂ‘Â‘7KH VWURQJ REVHUYHG GHSHQGHQFHV RI A RQ 9JI DQG 9T LPSO\ WKDW WKHUPDOHPLVVLRQ FXUUHQW ZKLFK GHSHQGV RQO\ ZHDNO\ RQ 9T LV QRW VLJQLILFDQW 7KH REVHUYHG VDWXUDWLRQ RI A ZLWK LQFUHDVLQJ 9TA LQ )LJ LV LQFRQVLVWHQW ZLWK SUHGRPLQDQW 3RROH)UHQNHO HPLVVLRQ RU DYDODQFKLQJ )XUWKHUPRUH WKH HOHFWULF ILHOG LQ WKH GUDLQ GHSOHWLRQ UHJLRQ IRU WKH YDOXHV RI 9JI DQG 9' XVHG LQ WKH PHDVXUHPHQWV )LJ f LV QRW KLJK HQRXJK WR SURGXFH VLJQLILFDQW DYDODQFKLQJ RU EDQGEDQG WXQQHOLQJ :H DUH WKXV OHIW ZLWK ILHOG HPLVVLRQ WKURXJK JUDLQERXQGDU\ WUDSV RU PHWDO SUHFLSLWDWHV DV WKH PRVW SODXVLEOH PHFKDQLVP XQGHUO\LQJ WKH REVHUYHG ,>B 94I 9G f 7KH VWURQJ GHSHQGHQFH RQ 9JI IXUWKHU LPSOLHV WKDW WKH SUHGRPLQDQW ILHOG HPLVVLRQ RFFXUV QHDU WKH IURQW VXUIDFH LQ IDFW EHWZHHQ WKH S GUDLQ DQG WKH Q LQYHUVLRQ OD\HU ZKHUH WKH HOHFWULF ILHOG LV KLJKHVW ,I WKH KDFN VXUIDFH LV LQYHUWHG VLJQLILFDQW ILHOG HPLVVLRQ FDQ RFFXU WKHUH DOVR PAGE 81 2XU DQDO\VLV RI WKH ILHOG HPLVVLRQ HPSKDVL]HV JUDLQERXQGDU\ WUDSV QRW PHWDO SUHFLSLWDWHV IRU WZR PDLQ UHDVRQV )LUVW QHXWURQ DFWLYDWLRQ DQDO\VHV >6+@ RI WKH /3&9' SRO\VLOLFRQ UHYHDO FRQFHQWUDWLRQV A FPf RI PHWDOOLF LPSXULWLHV FRPSDUDEOH WR WKRVH LQ EXON VLOLFRQ DQG PXFK ORZHU WKDQ WKH JUDLQERXQGDU\ WUDS GHQVLW\ 6HFRQG WKH JUDLQ ERXQGDULHV JHWWHU PHWDOOLF LPSXULWLHV ZKLFK WHQGV WR SUHYHQW WKH IRUPDWLRQ RI PHWDO SUHFLSLWDWHV ,Q WKH QH[W VHFWLRQ ZH GHYHORS DQ DQDO\WLF PRGHO IRU WKH WUDS DVVLVWHG ILHOGHPLVVLRQ FXUUHQW LQ WKH /3&9' SRO\VLOLFRQ 026)(7 7R VXSSRUW WKH PRGHO ZH FRPSDUH LWV SUHGLFWLRQV RI A9'f ZLWK PHDVXUHG GDWD UIURP SFKDQQHO DFFXPXODWLRQPRGHnDQG QFKDQQHO LQYHUVLRQ PRGH GHYLFHV *RRG FRUUHODWLRQ LV VKRZQ DQG ILHOG HPLVVLRQ DW WKH EDFN VXUIDFH LV VXJJHVWHG DV WKH PHFKDQLVP XQGHUO\LQJ WKH PLQLPL]DWLRQ RI WKH OHDNDJH FXUUHQW DW UHODWLYHO\ ORZ YDOXHV RI Y*I LNH WKDW LOOXVWUDWHG LQ )LJ ,QVLJKW UHJDUGLQJ WKH SK\VLFV XQGHUO\LQJ WKH DQRPDORXVO\ VWURQJ A9TA9Tf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r GHVLJQHG WR KDYH UHDVRQDEO\ ORZ WKUHVKROG YROWDJHV >6+ 0$@ 6XFK GHVLJQ GHSHQGV RQ WKH FRPSOHWH GHSOHWLRQ RI WKH VPDOO a $f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f DW WKH GUDLQ MXQFWLRQ %HFDXVH RI c WKH FRPSOHWH GHSOHWLRQ RI WKH ERG\ WKLV HPLVVLRQ RFFXUV SUHGRPLQDQWO\ DW WKH VXUIDFH XQGHU WKH JDWH LQ WKH GHSOHWLRQ UHJLRQ EHWZHHQ WKH S GUDLQ DQG Q LQYHUVLRQ OD\HU 7KLV FRQFOXVLRQ LV FRQVLVWHQW ZLWK WKH REVHUYHG VWURQJ GHSHQGHQFH RI WKH OHDNDJH FXUUHQW RQ WKH JDWH YROWDJH $OWKRXJK EDQGEDQG WXQQHOLQJ LQ WKLV VXUIDFH UHJLRQ ZRXOG RFFXU RQO\ DW YHU\ KLJK HOHFWULF ILHOGV >5,@ VXEVWDQWLDO ILHOG HPLVVLRQ FDQ RFFXU DW ORZ ILHOGV EHFDXVH RI WKH KLJK GHQVLW\ RI JUDLQ ERXQGDU\ WUDSV LQ WKH /3&9' SRO\VLOLFRQ 7R HQDEOH DQ DQDO\WLF GHVFULSWLRQ RI WKLV HPLVVLRQ ZH DVVXPH WKDW WKH WUDSV DUH PRQRHQHUJHWLF DW (\ LQ WKH HQHUJ\ JDSf DQG XQLIRUPO\ GLVWULEXWHG LQ VSDFH >.$ '( '(@ ZLWK WUDS GHQVLW\ 1\ 1A\G4 ZKHUH 16\ LV WKH JUDLQERXQGDU\ DUHDO WUDS GHQVLW\ DQG G* LV WKH DYHUDJH FROXPQDU JUDLQ VL]Hf 7KH FULWLFDO UHJLRQ RI WKH GHYLFH GHVFULEHG DERYH LV LOOXVWUDWHG LQ )LJ ,W LV WKH GHSOHWLRQ UHJLRQ QHDU WKH VXUIDFH EHWZHHQ WKH S PAGE 83 $ )LJ &ULWLFDO GHSOHWLRQ UHJLRQ EHWZHHQ WKH GUDLQ DQG WKH LQYHUVLRQ OD\HU ZLWK LPSRUWDQW HOHFWULF ILHOG FRPSRQHQWV LQ f f LQGLFDWHG PAGE 84 GUDLQ DQG WKH Q LQYHUVLRQ OD\HU %HFDXVH RI WKH ODWHUDO GLIIXVLRQ RI WKH GUDLQ XQGHU WKH JDWH WKH LQYHUVLRQ H[WHQGV LQWR WKH GUDLQ UHJLRQ ZLWK FRPSOLFDWHG JHRPHWU\ 7KXV WKH HIIHFWLYH FURVVVHFWLRQDO DUHD RI WKH QS MXQFWLRQ LV GLIILFXOW WR GHILQH 7KH HOHFWULF ILHOG LQ WKLV UHJLRQ ZKLFK JRYHUQV WKH ILHOG HPLVVLRQ LV WZRGLPHQVLRQDO >)5 7(@ IRU ZLGH FKDQQHOVf GHSHQGLQJ RQ ERWK WKH JDWH DQG GUDLQ YROWDJHV DQG 94 )ROORZLQJ D SUHYLRXV DQDO\VLV >)5@ RI WKH EXON 026)(7 LQ WKH VDWXUDWLRQ UHJLRQ ZH ZLOO WUHDW WKLV FRPSOH[ WZR GLPHQVLRQDO SUREOHP HPSLULFDOO\ ZKLFK HQDEOHV XV WR PRGHO WKH ILHOG HPLVVLRQ SURFHVVHV DV RFFXUULQJ SUHGRPLQDQWO\ SDUDOOHO WR WKH VXUIDFH L Q WKH \GLUHFWLRQf 7KLV VLPSOLILFDWLRQ LV FRPPHQVXUDWH ZLWK GHYLFH DPELJXLWLHV \HW \LHOGV D PRGHO WKDW LV FRQVLVWHQW ZLWK H[SHULPHQWDO UHVXOWV DQG KHQFH LV LQVLJKWIXO 7KH HQHUJ\EDQG GLDJUDP LQ WKH UHJLRQ LQFOXGLQJ WKH JUDLQERXQGDU\ WUDS OHYHO LV VNHWFKHG YHUVXV \f LQ )LJ 0DLQWDLQLQJ WKH GHJUHH RI FRPSOH[LW\ LPSOLHG DERYH ZH DVVXPH WKDW WKH HOHFWULF ILHOG LQ WKH \GLUHFWLRQ )\ G(AG\OT LV FRQVWDQW LQ WKH GHSOHWLRQ EDUULHUf UHJLRQ DQG WKDW WKH HOHFWURQV RU KROHVf WXQQHO WKURXJK WKH EDUULHU YLD WKH WUDSV DW FRQVWDQW HQHUJ\ >*$ 5 7KH OHDNDJH FXUUHQW GHULYHV WKHQ IURP WKH FRPELQHG QHW ILHOG HPLVVLRQ RI KROHV ,\9 IURP WKH WUDSV WR WKH YDOHQFH EDQG LQ WKH GUDLQ UHJLRQ DQG RI HOHFWURQV ,\* IURP WKH WUDSV WR WKH FRQGXFWLRQ EDQG LQ WKH LQYHUVLRQ UHJLRQ )RU WUDSV ZLWKLQ DQ LQFUHPHQWDO ZLGWK G\f RI WKH GHSOHWLRQ UHJLRQ VHH )LJ f >*$ 5 *5@ PAGE 85 )LJ (OHFWURQ HQHUJ\EDQG GLDJUDP DORQJ WKH VXUIDFH EHWZHHQ WKH Qf LQYHUVLRQ OD\HU DQG WKH S f GUDLQ PAGE 86 GL 79 TO I7f17=[HG\ f DQG GfÂ¬,7&ffÂ¯ TI717=[HG\ W7& f WKH UDWHV RI WKH UHVSHFWLYH LQYHUVH SURFHVVHV DUH QHJOLJLEO\ VPDOO IRU WKH QRQHTXLOLEULXP FRQGLWLRQV RI LQWHUHVW HJ _9'_ !! N7Tf ,Q f DQG f = LV WKH FKDQQHO ZLGWK RI WKH 026)(7 [H LV DQ HIIHFWLYH GHSWK RI WKH MXQFWLRQ UHJLRQ VHH )LJ f I\ LV WKH HOHFWURQ RFFXSDQF\ IDFWRU RI WKH WUDSV DQG W\9 DQG WMT DUH WKH WLPH FRQVWDQWV IRU KROH DQG HOHFWURQ WXQQHOLQJ UHVSHFWLYHO\ ZKLFK GHSHQG RQ )\ DQG (\ DV ZH GLVFXVV ODWHU n ,Q WKH VWHDG\ VWDWH WKH QXPEHU RI WUDSSHG HOHFWURQV LV FRQVWDQW DQG KHQFH LQ WKH DEVHQFH RI VLJQLILFDQW WKHUPDO HPLVVLRQ GO\\ G,\Ff (TXDWLQJ f DQG f WKHQ ZH KDYH I 7 7& W7& 7 79 f 7KH LQFUHPHWDO ILHOGHPLVVLRQ FXUUHQW LV JLYHQ E\ WKH FRPELQDWLRQ RI f RU f ZLWK f 7KH WRWDO OHDNDJHf FXUUHQW LV WKHQ H[SUHVVHG E\ LQWHJUDWLQJ WKH UHVXOW RYHU WKDW SRUWLRQ RI WKH GHSOHWLRQ UHJLRQ DFURVV ZKLFK YDOHQFH EDQGWUDSFRQGXFWLRQ EDQG FDUULHU WUDQVLWLRQV DW FRQVWDQW HQHUJ\ DUH SRVVLEOH VHH )LJ f PAGE 87 Â« ,/ O=[H17 r9 G\ \(&Qf W. ;79 f 6LQFH ZH DVVXPH WKDW ) LV FRQVWDQW LQ WKH GHSOHWLRQ UHJLRQ I a Â/ (9S a (&Q \ Z T/\(YSf \(&Qff ZKHUH !' LV WKH SRWHQWLDO EDUULHU KHLJKW DQG : LV WKH ZLGWK 7KH WXQQHOLQJ WLPH FRQVWDQWV DUH WKXV LQGHSHQGHQW RI \ DOVR DQG f FDQ EH ZULWWHQ DV fÂµO =[H17 7& ; 79 ffÂµ Â9t ( &Q f %HFDXVH WKH OHDNDJH FXUUHQW LV ORZ OLWWOH YROWDJH LV GURSSHG DFURVV WKH LQYHUVLRQ OD\HU 9T LV GURSSHG DFURVV WKH GUDLQ GHSOHWLRQ UHJLRQ DV LQGLFDWHG LQ )LJ )S ( )Q T_9U f )RU _ 9T , JUHDWHU WKDQ D IHZ WHQWKV RI D YROW WKHQ ZH QRWH WKDW WKH TXDVL)HUPL OHYHO VHSDUDWLRQ LQ f HTXDOV DSSUR[LPDWHO\ (9S (&Qf LQ f 7KXV 7& 7 79 f W ,/ T=[A f PAGE 88 7KH WLPH FRQVWDQWV W \J DQG [\\ UHIOHFW WKH SUREDELOLW\ SHU XQLW WLPH WKDW D WUDSSHG FDUULHU ZLOO WXQQHO WKURXJK D WULDQJXODU EDUULHU GHILQHG E\ (\ DV VKRZQ LQ )LJ M WR LWV UHVSHFWLYH EDQG %DVHG RQ WKH :.% DSSUR[LPDWLRQ >5 6= *5@ Prf(\ ( f W79V 79H[3> TK) A rf DQG Prf(F (\f W7&a 7&H;S TK) A f ZKHUH DQG PQ DUH WKH DSSURSULDWH >/8 *5@ HIIHFWLYH PDVVHV IRU WKH WXQQHOLQJ KROHV DQG HOHFWURQV PS PQ PJ >*5@ ZKHUH P4 LV WKH IUHH HOHFWURQ PDVVf DQG ZKHUH [J\DQG [JJ DUH HIIHFWLYH FDUULHU WUDQVLW WLPHV LQ WKH YDOHQFH DQG FRQGXFWLRQ EDQGV ZKLFK ZH DVVXPH WR EH FRQVWDQWV >/8@ )RU D SDUDEROLF EDUULHU >5 6=@ WKH QXPHULFDO FRQVWDQW LQ WKHnH[SRQHQWLDO DUJXPHQW LV GLIIHUHQW ORZHUf 7KH HIIHFW RI WKH EDUULHU VKDSH RQ ,_B FDQ WKXV EH VWXGLHG E\ YDU\LQJ WKH HIIHFWLYH PDVVHV LQ f DQG f ZKLFK LQ IDFW FDQQRW EH XQHTXLYRFDOO\ GHILQHG >/8 *5@ :H VWUHVV WKDW WKH ILHOGHPLVVLRQ FXUUHQW ,/ LQ f LV SUHGRPLQDQW EHFDXVH RI WKH KLJK GHQVLW\ 1\ RI WUDSV WKDW LQFUHDVH VXEVWDQWLDOO\ WKH WXQQHOLQJ SUREDELOLW\ FRQYH\HG E\ f DQG f RYHU WKDW IRU EDQGEDQG WXQQHOLQJ >6=@ 7R FRPSOHWH WKH PRGHO IRU ,/ PAGE 89 Â« GHILQHG E\ ffZH PXVW GHVFULEH )\ LQ WHUPV RI 9*I DQG 9' $V GLVFXVVHG ZLWK UHIHUHQFH WR )LJ WKH HOHFWULF ILHOG LQ WKH GHSOHWLRQ UHJLRQ DW WKH VXUIDFH DQG LQGHHG WKH WZRGLPHQVLRQDO UHJLRQ LWVHOI DUH YLUWXDOO\ LPSRVVLEOH WR GHVFULEH DQDO\WLFDOO\ +RZHYHU DQ HPSLULFDO GHVFULSWLRQ FRPPHQVXUDWH ZLWK RXU RQHGLPHQVLRQDO ILHOG HPLVVLRQ PRGHO FDQ EH JLYHQ EDVHG RQ D SUHYLRXV DQDO\VLV >)5@ RI WKH EXON 026)(7 LQ WKH VDWXUDWLRQ UHJLRQ :LWK UHIHUHQFH WR )LJ WKUHH FRPSRQHQWV RI )\ FDQ EH LGHQWLILHG >)5@ ÂÂf ,Q WKH HPSLULFDO UHSUHVHQWDWLRQ f )M LV WKH HOHFWULF ILHOG WKDW H[LVWV LQ WKH DEVHQFH RI WKH JDWH LH )A LV GXH WR WKH GHSOHWLRQ FKDUJH LQ WKH UHYHUVHELDVHG GUDLQ MXQFWLRQ 7KH SUHVHQFH RI WKH JDWH SURGXFHV IULQJLQJ HOHFWULF ILHOGV )Â DQG ) ) LV GXH WR WKH JDWHGUDLQ SRWHQWLDO GURS DQG ) LV GXH WR WKH SRWHQWLDO GLIIHUHQFH EHWZHHQWKH GUDLQ HQG RI WKH LQYHUVLRQ OD\HU DQG WKH JDWH )ROORZLQJ >)5@ ZKLFK KDV EHHQ VXSSRUWHG H[SHULPHQWDOO\ >%5@ ZH DVVXPH WKDW ) DQG ) DUH SURSRUWLRQDO WR WKH UHVSHFWLYH QRUPDO HOHFWULF ILHOGV DW WKH VXUIDFH VHH )LJ f f W PAGE 90 ) fÂ¯ r)6 f ZKHUHD DQGH DUH FRQVWDQW ILHOGIULQJLQJ IDFWRUV $W WKH S GUDLQ VLGH RI WKH GHSOHWLRQ UHJLRQ WKH VXUIDFH SRWHQWLDO L V QHDUO\ ]HUR DQG IGa B a * 2I fÂ«9*I fÂ¯ 9' A06 &4IA f ZKHUH &4I LV WKH IURQWf JDWH R[LGH FDSDFLWDQFH 2II LV WKH IL[HG FKDUJH GHQVLW\ DW WKH IURQWf 6L6L&A LQWHUIDFH DQG LV WKH JDWHS VLOLFRQ ZRUN IXQFWLRQ GLIIHUHQFH )RU DQ Q SRO\VLOLFRQ JDWH A06 a fÂ¯(J$O ZKHUH (J 6 WOLH VLOLFRQ HQHUJ\ JDS $W WKH LQYHUVLRQOD\HU VLGH RI WKH GHSOHWLRQ UHJLRQ )r 6 H 2I 9 *I r06 & 4II (Q II Bef RI T f VLQFH WKH SRWHQWLDO GURS EHWZHHQ WKH HQG RI WKH VWURQJf LQYHUVLRQ OD\HU DQG WKH VRXUFH UHJLRQ 9mM f LV DERXW (JT 8VLQJ WKH RQHGLPHQVLRQDO DQDO\VLV RI WKH UHYHUVHELDVHG SQ MXQFWLRQ >6=@ ZH H[SUHVV ) f V DV DQ DYHUDJH YDOXH RI WKH \GHSHQGHQW HOHFWULF ILHOG LQ WKH GHSOHWLRQ UHJLRQ EHWZHHQ WKH S GUDLQ DQG WKH Q LQYHUVLRQ OD\HU ,Q f Â³ LV GHILQHG E\ f DQG f PAGE 91 nRI T[ LHIIf 9 *I 97If f ZKHUH [AJIIf LV WKH LQYHUVLRQ OD\HU WKLFNQHVV a $f GHILQHG LQ L 6HFWLRQ DQG 9\\ LV WKH VWURQJLQYHUVLRQ WKUHVKROG YROWDJH RI WKH 026 VWUXFWXUH WKH FKDUDFWHUL]DWLRQ RI ZKLFK GHSHQGV QRW RQO\ RQ WKH VWUXFWXUDO SURSHUWLHV EXW DOVR RQ WKH SRO\VLOLFRQ ILOP SURSHUWLHV HJ 1VW (\ DQG G* >.$@ 7KH SRWHQWLDO EDUULHU KHLJKW S VHH )LJ f LV GHILQHG E\ 9S DQG WKH VXUIDFH SRWHQWLDO LQ WKH LQYHUVLRQ OD\HU )RU VWURQJ LQYHUVLRQ f 7KH FRPELQDWLRQ RI ff JLYHV DQ DQDO\WLF GHVFULSWLRQ RI ,_B DQG LWV GHSHQGHQFHV RQ 9J\ DQG 9S IRU WKH SFKDQQHO DFFXPXODWLRQPRGH SRO\VLOLFRQ 0*6)(7 7KH FRQWURO RI E\ WKH JUDLQ ERXQGDULHV LV FRQYH\HG E\ 1\ Â1A\GSf DQG (\ UHODWLYH WR WKH EDQG HGJHVf LQ WKH L ' PRGHO 7KH SDUDPHWHUV &4\ 9\\ c!ccc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}9'A GHSHQGHQFHV DEVROXWH PHDVXUHG YDOXHV KRZHYHU FRXOG EH PDWFKHG E\ DVVLJQLQJ SURSHU YDOXHV W274\ DQG744 7KH PRGHO GRHV LQGHHG SUHGLFW WKH JHQHUDO A9HIrAf GHSHQGHQFHV H[HPSOLILHG LQ )LJ 7R GHPRQVWUDWH WKLV FRUUHODWLRQ DQG WR LQGLFDWH WKH SK\VLFDO LQVLJKW DIIRUGHG ZH SORW LQ )LJ WKH FDOFXODWHG OHDNDJH FXUUHQW YHUVXV 9J\ DQG 9J IRU D GHYLFH VLPLODU WR WKH RQH PHDVXUHG 4XDQWLWDWLYH FRPSDULVRQV RI WKH WKHRUHWLFDO DQG H[SHULPHQWDO FKDUDFWHULVWLFV VKRXOG QRW EH PDGH EHFDXVH DFWXDO YDOXHV RI PDQ\ RI WKH PRGHO SDUDPHWHUV DUH XQNQRZQ ,Q WKH FDOFXODWLRQV ZH XVHG = XP / LV LUUHOHYDQWf DQG &4\ [ A )FPA H\WI ZLWK W4; $f FRUUHVSRQGLQJ WR WKH GHYLFH PHDVXUHG :H FKRVH [H $ JUHDWHU WKDQ [cHIIf EXW OHVV WKDQ WKH H[WHQW RI WKH ODWHUDO GUDLQ GLIIXVLRQ DQG D DQG FUXGH YDOXHV EDVHG RQ WKH EXONn 06)(7 PRGHO >)5@ :H OHW WJ\ WJJ WT A VHF D FUXGH HVWLPDWH GHULYHG H[SHULPHQWDOO\ >*5@ DQG 1\ [ A FQ7A ZKLFK FRUUHVSRQGV WR 1\ A FUUIA DQG GJ $ :H SXW (\ DW PLGJDS :H FKDUDFWHUL]HG YM\ 9\J 4\\&\ ZLWK 4\\T A FPfÂ¯A DQG 9\J L B L 9 ZKLFK \LHOG 9\\ 9 >:H VWUHVV WKDW 9\\ LV WKH VWURQJLQYHUVLRQ WKUHVKROG YROWDJH LQ FRQWUDVW WR WKH WXUQ21 WKUHVKROG YROWDJH IRU VWURQJ DFFXPXODWLRQf ZKLFK LV DERXW 9 DV LQGLFDWHG LQ )LJ @ )URP f ZH QRWH WKDW ,_B LV GLUHFWO\ SURSRUWLRQDO WR WKH IDFWRU =[H1\U J DQG KHQFH FKDQJHV LQ WKHVH SDUDPHWHUV VLPSO\ DOWHU WKH PDJQLWXGH RI DQG QRW VKDSH RI WKH VHPLORJDULWKPLF FKDUDFWHULVWLFV LQ )LJ 3GGLWLRQDO FDOFXODWLRQV VKRZ DOVR WKDW WKH PAGE 93 )LJ FLOLDWHG OHDNDJH FXUUHQW WDJHV IRU D SFKDQQHO DFF 6)(7 YHUVXV IURQWf XPXO DWLRQPRULH JDWH OSFYG DQG GUDLQ SRO\VLOLFRQ PAGE 94 VKDSH RI ,_B94If QRW VWURQJO\ GHSHQGHQW RQ D DQGH 9DULDWLRQV LQ WKH R[LGH FDSDFLWDQFH &4I KRZHYHU SURGXFH VLJQLILFDQW FKDQJHV LQ ,O9*If SULPDULO\ EHFDXVH RI LWV LQIOXHQFH RQ )M GHVFULEHG E\ f DQG f 6LPLODU FKDQJHV DUH SURGXFHG E\ YDULDWLRQV LQ 9MI RU 9MT DV VKRZQ E\ WKH FDOFXODWLRQV SORWWHG LQ )LJ 7KH OHDNDJH LV PRVW VHQVLWLYH WR 9JI MXVW DERYH WKUHVKROG IRU VWURQJ LQYHUVLRQ ZHOO DERYH WKUHVKROG KLJK 9TAf )\ LQ f LV KLJK DQG WKH WXQQHOLQJ WLPH FRQVWDQWV WM9 DQG WMT LQ f DQG f WHQG WRZDUG PLQLPXP YDOXHV WTf WKHUHE\ FDXVLQJ WR DSSURDFK D YDOXH LQGHSHQGHQW RI 9*I 7KH FDOFXODWHG GHSHQGHQFH RI RQ WKH WUDS OHYHO (M LV LOOXVWUDWHG LQ )LJ $VFDQ EH LQIHUUHG IURP f DQG f WUDSV QHDU PLGJDS DUH PRVW HIIHFWLYH LQ WKH ILHOGHPLVVLRQ SURFHVVHV VKDOORZ WUDSV GR QRW IDFLOLWDWH FDUULHU WXQQHOLQJ WR WKH RSSRVLWH EDQG :H QRWH WKDW (M FRXOG SRVVLEO\ EH LQIHUUHG IURP PHDVXUHPHQWV RI WKH VORSH RI WKH ,/94If FKDUDFWHULVWLF $OWKRXJK WKH PRGHO SUHGLFWLRQV FDQ EH EURXJKW LQWR FORVH DJUHHPHQW ZLWK WKH PHDVXUHG OHDNDJH FXUUHQW E\ DOWHULQJ SDUDPHWHU YDOXHV WKH EHQHILW RI GRLQJ VR LV TXHVWLRQDEOH EHFDXVH RI WKH XQFHUWDLQW\ LQ WKH DFWXDO SK\VLFDO VWUXFWXUH DQG SDUDPHWHUV RI WKH GHYLFH )RU H[DPSOH YDULDWLRQV LQ WKH VKDSH RI WKH SRWHQWLDO EDUULHU LQ WKH GUDLQ GHSOHWLRQ UHJLRQ DQGRU LQ WKH HIIHFWLYH PDVVHV LQ f DQG f FDQ UHVXOW LQ FRQVLGHUDEOH FKDQJHV LQ WKH SUHGLFWHG ,L-Y*IfÂ¬9'f FKDUDFWHULVWLFV 6XFK FKDQJHV DUH LOOXVWUDWHG LQ )LJ ZKHUH ZH SORW WKH FDOFXODWHG ,,9*I9Tf ZLWK PS PQ YDU\LQJ IURP P4 WR P4 :H QRWH KRZHYHU IURP WKH FDOFXODWLRQV SORWWHG LQ )LJV WKDW WKH PHDVXUHG PAGE 95 }IFM nLZ LQS fÂµ : )LJ &DOFXODWHG YDULDWLRQ RI A*IA IrU GLIIHUHQW VWURQJ LQYHUVLRQf WKUHVKROG YROWDJHV 0MU 9WT 2II&QI ZLWK 2IIARI 9f PAGE 96 )LJ IL &DOFXODWHG GHSHQGHQFH RI ,/9*If RQ WKH JUDLQERXQGDU\ WUDS HQHUJ\ OHYHO (\ PAGE 97 9* 9f )LJ &DOFXODWHG GHSHQGHQFHA RI ,/9ILIf Q WXQQHOLQJ FDUULHU HIIHFWLYH PDVVHV PS PQr / PAGE 98 Â« c/Y*I}9>Mf FKDUDFWHULVWLFV LQ )LJ FRXOG EH VLPXODWHG ZHOO E\ WKH PRGHO ZLWK SK\VLFDOO\ UHDVRQDEOH SDUDPHWHU YDOXHV 7KH SORWV LQ )LJV DV ZHOO DV WKH GDWD LQ )LJ ZKLFK LQGHHG UHIOHFW WKH JHQHUDO ,A*I }9'A GHSHQGHQFHV VHHQ LQ D YDULHW\ RI /3&9' SRO\VLOLFRQ 026)(7V VKRZ WKDW ,_B YDULHV SUHGRPLQDQWO\ H[SRQHQWLDOO\ ZLWK 9J\ ! 9\\f IRU D JLYHQ YDOXH RI 9S 7KLV GHSHQGHQFH UHVXOWV IURP WKH H[SRQHQWLDO GHSHQGHQFH RI I\\ DQG [\J RQ )\ H[SUHVVHG LQ f DQG f 7KH GHSHQGHQFH RI ,>B RQ 9S LV DOVR VWURQJ DV HPSKDVL]HG E\ WKH FDOFXODWLRQV RI )LJ UHSORWWHG LQ )LJ )RU D JLYHQ YDOXH RI 9J\ ! 9\f A YDULHV DV _9J_P ZKHUH P a 7KLV GHSHQGHQFH IROORZV IURP f DQG WKH LPSOLFLW GHSHQGHQFHV RQ 9S RI )\ DQG W\J DQG 7\9 1RWH WKDW P GHFUHDVHV ZLWK LQFUHDVLQJ 9J\ DQG _9SM_ 7KH SUHGLFWHG P YV 9J\ YDULDWLRQ FDQ EH VHHQ LQ WKH PHDVXUHG GDWD LQ )LJ DOWKRXJK WKH P YV _9T_ YDULDWLRQ FDQQRW 7KLV GLVFUHSDQF\ DSSHDUV WR EH GXH WR WUDSDVVLVWHGf DYDODQFKLQJ LQ WKH GUDLQ GHSOHWLRQ UHJLRQ WKDW FDXVHV WKH PHDVXUHG OHDNDJH FXUUHQW WR LQFUHDVH DEUXSWO\ DV ,94_ DSSURDFKHV a 9 :H QRWH IXUWKHU WKDW WKH PHDVXUHG P f IRU DOO WKH GHYLFHV LV ORZHU WKDQ WKDW FDOFXODWHG ZKLFK FRXOG LQGLFDWH WKDW WKH HIIHFWLYH PDVVHV LQ f DQG f DUH ORZHU WKDQ P4 DV ZH DVVXPHG RU WKDW WKH SRWHQWLDO EDUULHU LV PRUH SDUDEROLF WKDQ OLQHDU $GGLWLRQDO FDOFXODWLRQV UHYHDO WKDW WKH HIIHFWLYH PDVVHV PXVW EH UHGXFHG E\ DERXW DQ RUGHU RI PDJQLWXGH WR EULQJ P GRZQ WR WKH PHDVXUHG YDOXH 7KHQ WR UHWDLQ WKHRUHWLFDOH[SHULPHQWDO DJUHHPHQW IRU WKH DEVROXWH YDOXH RI ,_B WKH HIIHFWLYH FDUULHU WUDQVLW WLPHV LQ f DQG f PXVW EH UHGXFHG E\ WKUHH WR IRXU RUGHUV RI PDJQLWXGH +LJKHU YDOXHV RI D DQG ORZHU YDOXHV RI DOVR ZHDNHQ WKH GHSHQGHQFH RI A RQ 9Q PAGE 99 )LJ &DOFXODWHG 0Y*I}YQf FKDUDFWHULVWLFV HPSKDVL]LQJ WKH GHSHQGHQFH RQ GUDLQ YROWDJH 9S PAGE 100 $GGLWLRQDO VXSSRUW IRU WKH ILHOGHPLVVLRQ PRGHO IRU ,_B LV REWDLQHG IURP FRQVLGHUDWLRQ RI WKH LQIOXHQFH RI WKH EDFNJDWH ELDV 9JA 7KLV LQIOXHQFH LV UHODWHG WR WKH PLQLPL]DWLRQ RI WKH OHDNDJH FXUUHQW DW ORZ 9JI GHSLFWHG LQ )LJ $V 9JI LV UHGXFHG WR VZLWFK WKH GHYLFH IURP 2)) WR 21 DFFXPXODWLRQf WKH PHDVXUHG OHDNDJH FXUUHQW UHDFKHV D PLQLPXP YDOXH ZKHUHDV WKH FDOFXODWHG FXUUHQW FRQWLQXHV WR EH UHGXFHG PRQRWRQLFDOO\ 7KLV VLPSO\ LQGLFDWHV WKDW WKH ILHOG HPLVVLRQ LQ RXU PRGHO LV LQVLJQLILFDQW DW WKLV SRLQW DQG WKH DFWXDO PLQLPXP OHDNDJH FXUUHQW LV SURGXFHG E\ DQRWKHU PHFKDQLVP SRVVLEO\ ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV QHDU WKH EDFN VXUIDFH ZKHUH SRVLWLYH LQWHUIDFLDO FKDUJH FRXOG LQGHHG FDXVH LQYHUVLRQ ZLWK WKH EDFN JDWH JURXQGHG 9*E f 7KH PLQLPXP PHDVXUHG FXUUHQWV LQ )LJ VKRZ D VWURQJ GHSHQGHQFH RQ 9J OLNH RXU PRGHO SUHGLFWV DQG KHQFH ZH VXUPLVH WKDW WKH\ UHVXOW IURP ILHOG HPLVVLRQ QHDU WKH EDFN VXUIDFH :H QRWH IXUWKHU WKDW WKH YDOXH RI 9JI DW ZKLFK WKH PLQLPXP ,_B REWDLQV GHFUHDVHV DV 9J LQFUHDVHV 7KLV LPSOLHV WKDW WKH GHSHQGHQFH RQ 9J RI WKH ILHOGHPLVVLRQ FXUUHQW DW WKH EDFN VXUIDFH LV ZHDNHU WKDQ WKDW RI WKH IURQWVXUIDFH FXUUHQW ZKLFK EDVHG RQ RXU PRGHO LV D UHVXOW RI WKH WKLFNHU EDFNJDWH XQGHUO\LQJf R[LGH 0HDVXUHG ,J9JIY*Ef FKDUDFWHULVWLFV IRU D K\GURJHQDWHG QFKDQQHO LQYHUVLRQPRGH /3&9' WKLQILOP SRO\VLOLFRQ 026)(7 ZLWK 9J 9 DUH SORWWHG LQ )LJ )RU 9JI 2)) UHJLRQf ,J Af LV LQGHSHQGHQW RI 9JA DQG LQFUHDVHV H[SRQHQWLDOO\ ZLWK _9JI_ LQ DFFRUGDQFH ZLWK RXU PRGHO )RU ORZ 9JA QHDU WKH PLQLPXP ,Jf ZLWK 9JA ZKLFK LPSOLHV WKDW WKH EDFN VXUIDFH LV DFFXPXODWHG ,J LQFUHDVHV ZLWK _9JA_ PAGE 101 Â« )LJ 0HDVXUHG FXUUHQWYROWDJH FKDUDFWHULVWLFV RI D K\GURJHQDWHG ,3&9' SRO\VLOLFRQ 026)(7 QFKDQQHO LQYHUVLRQQRGH = LRQ XP / X Q WRI $ [ FPf 7KH SRO\VLOLFRQ ILOP LV YPWKL FN DQG ZDV GHSRVLWHG YLD /3&9' RQ D XPWKLFN OD\HU RI VLOLFRQGLR[LGH PAGE 102 DQG FRXOG EH VLPXODWHG ZHOO E\ RXU ILHOGHPLVVLRQ PRGHO DSSOLHG WR WKH EDFN VXUIDFH A9TA9Tf 7KH GHSHQGHQFH RI WKH EDFNVXUIDFH OHDNDJH FXUUHQW RQ f LV QRW DV VWURQJ DV ,>B9*If LQ )LJV DQG EHFDXVH RI WKH WKLFNHU EDFNJDWH R[LGH ! $f DQG WKH KLJK ! f WKUHVKROG YROWDJH IRU DFFXPXODWLRQ 1RWH WKDW ,Q DOVR LQFUHDVHV ZLWK 94E ! f EHFDXVH RI EDFNVXUIDFH LQYHUVLRQ 7KXV WKH GDWD LQ )LJV DQG DUH FRQVLVWHQW ZLWK RXU ILHOG HPLVVLRQ PRGHO IRU OHDNDJH FXUUHQW 7KH PLQLPL]DWLRQV RI WKH ,J94If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f )LHOG HPLVVLRQ PAGE 103 QHDU WKH EDFN VXUIDFH FDQ EH SUHYDOHQW LQ WKH WKLQILOP GHYLFH ZKHQ WKH IURQW VXUIDFH LV GHSOHWHG $OWKRXJK WKH PRGHO ZDV GHYHORSHG EDVHG RQ WKH SFKDQQHO DFFXPXODWLRQPRGH GHYLFH VKRZQ LQ )LJ LW LV LQ HVVHQFH DSSOLFDEOH WR RWKHU SRO\VLOLFRQ 026)(7 VWUXFWXUHV )RU H[DPSOH LQ WKH QFKDQQHO LQYHUVLRQPRGH GHYLFH WKH SUHGRPLQDQW OHDNDJH FXUUHQW IORZV ZKHQ WKH IURQWf VXUIDFH LV DFFXPXODWHG LW UHVXOWV IURP ILHOG HPLVVLRQ WKURXJK JUDLQERXQGDU\ WUDSV EHWZHHQ WKH S DFFXPXODWLRQ OD\HU DQG WKH Q GUDLQ ,Q GHVFULELQJ WKH OHDNDJH LQ WKLV GHYLFH VWUXFWXUDO GLIIHUHQFHV EHWZHHQ LW DQG WKH GHYLFH LQ )LJ PXVW RI FRXUVH EH DFFRXQWHG IRU )RU H[DPSOH ZLWK WKH EDFN JDWH JURXQGHG ZLWK WKH VRXUFHf WKH WHQGHQF\ WRZDUG LQYHUVLRQ DW WKH EDFN VXUIDFH RI WKH QFKDQQHO WUDQVLVWRU VXJJHVWV D GLIIHUHQW FRPSRQHQW RI OHDNDJH FXUUHQW GULIWGLIIXVLRQ RI HOHFWURQV IURP VRXUFH WR GUDLQ LQ WKH EDFN FKDQQHOf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f LV UHGXFHG %DVHG RQ WKH PRGHO )A FDQ EH UHGXFHG E\ Df GHFUHDVLQJ WKH GRSLQJ GHQVLW\ LQ WKH GUDLQ Ef LQFUHDVLQJ WKH JDWH R[LGH WKLFNQHVV DQG Ff XVLQJ D S JDWH IRU WKH SFKDQQHO 026)(7f 7KH PRGLILFDWLRQ Df FDQ EH PAGE 104 DFFRPSOLVKHG E\ XVLQJ DQ /'' /LJKWO\ 'RSHG 'UDLQVRXUFHf VWUXFWXUH ZKLFK LV ZLGHO\ XVHG WR UHGXFH WKH KRWHOHFWURQ HIIHFWV LQn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Â‘ U BFU ,Q WKH SUHYLRXV FKDSWHU ZH SUHVHQWHG D SK\VLFDO PRGHO IRU WKH 2)) VWDWH OHDNDJH FXUUHQW EDVHG RQ ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV LQ WKH GHSOHWLRQ UHJLRQ DW WKH GUDLQ ,Q WKLV FKDSWHU ZH DQDO\]H WKH 21VWDWH FRQGXFWLRQ SURSHUWLHV RI WKH WKLQILOP /3&9' VPDOOJUDLQf SRO\VLOLFRQ 026)(7 :H GHVFULEH IRU DOO ELDV FRQGLWLRQV WKH FKDQQHO FKDUJH GHQVLW\ DW WKH VRXUFH LQ WHUPV RI WKH JUDLQKRXQGDU\ SURSHUWLHV DQG WKH GHYLFH SDUDPHWHUV DFFRXQWLQJ IRU WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV >/,E %$@ %DVHG RQ WKLV GHVFULSWLRQ ZH FKDUDFWHUL]H WKH GUDLQ FXUUHQW DVVXPHG WR EH SUHGRPLQDQWO\ GLIIXVLRQ LQ WKH VXEWKUHVKROG UHJLRQ DQG WKH WKUHVKROG YROWDJH 7KH PRGHO SUHGLFWLRQV DUH VXSSRUWHG E\ FXUUHQWYROWDJH PHDVXUHPHQWV DQG SK\VLFDO LQVLJKW UHJDUGLQJ WKH HIIHFWV RI K\GURJHQDWLRQ RQ WKH JUDLQ ERXQGDULHV LV UHYHDOHG PAGE 106 / 3UHYLRXV DQDO\VHV >.$ '( /(D & /,ED %$@ RI WKLQILOP 62, 026)(7V KDYH DFFRXQWHG IRU WKH HIIHFWV RI JUDLQ ERXQGDULHV RU WKH HIIHFWV RI WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV EXW WKH\ KDYH QRW DFFRXQWHG IRU ERWK 0RVW RI WKH UHVHDUFK >/(D @ FRQFHUQLQJ WKH HIIHFWV RI JUDLQ ERXQGDULHV LQ 026)(7V KDV IRFXVHG RQ ODUJHJUDLQ UHFU\VWDOOL]HGf SRO\VLOLFRQ 7KH HDUO\ ZRUN RI .DPL QV >.$@ IRU VPDOOJUDLQ SRO\VLOLFRQ 026)(7V FRQFHQWUDWHG PDLQO\ RQ WKH WKUHVKROG YROWDJH ZLWKRXW DQDO\]LQJ WKH GUDLQ FXUUHQW 7KH UHFHQW ZRUN RI 'HSS HW DO >'(@ IRU VPDOOJUDLQ SRO\VLOLFRQ 026)(7V VWXGLHG WKH GUDLQ FXUUHQW DQG WKH WKUHVKROG YROWDJH KXW OLNH .DPL QVn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f D QXPHULFDO VROXWLRQ RI WKH RQHGLPHQVLRQDO 3RLVVRQ HTXDWLRQ ZLWK PL[HG ERXQGDU\ FRQGLWLRQV DW WKH IURQW DQG EDFN LQWHUIDFHV ZKLFK DFFRXQWV IRU FKDUJH WUDSSHG DW JUDLQ ERXQGDULHV Ef WKH DVVXPSWLRQ EDVHG RQ SUHYLRXV ZRUN >.$ PAGE 107 '(i@ 7KDW WKH JUDLQ ERXQGDULHV LQ VPDOOJUDLQ SRO\VLOLFRQ FDQ EH DQDO\]HG DV WUDSV XQLIRUPO\ GLVWULEXWHG LQ VLOLFRQ Ff WKH GHVFULSWLRQ KUnWKÂ‹7nVWLE WKUHVKROG GUDLQ FXUUHQW ZKLFK LV DVVXPHG WR EH SUHGRPLQDQWO\ GLIIXVLEIIIRU GUDLQ YROWDJHV VXEVWDQWLDOO\ KLJKHU WKDQ N7T DQG Gf WKH FKDUDFWHUL]DWLRQ RI WKH WKUHVKROG YROWDJH REWDLQHG IURP WKH GHULYHG GHSHQGHQFH RI WKH FKDQQHO FKDUJH GHQVLW\ RQ WKH JDWH YROWDJH 6L QHHWKH VSDFHGHSHQGHQFH RI WKH HOHFWULF ILHOG DQG WKH SRWHQWLDO LQ WKH ILOP LV QRW QHHGHG >&@ WR GHILQH WKH DUHDO FKDUJH GHQVLWLHV LQ WKH 026)(7 ZH GR QRW VROYH 3RLVVRQnV HTXDWLRQ FRPSOHWHO\ ,QVWHDG ZH XVH LW DQG WKH ERXQGDU\ FRQGLWLRQV WR IRUPXODWH D QRQOLQHDU V\VWHP RI HnTÂ¾DWLÂ¯RP ZKLFK ZH VROYH QXPHULFDOO\ WR GHVFULEH WKH HOHFWULF ILHOG DQG SRWHQWLDO DW WKH IURQW DQG EDFN VXUIDFHV 7R REWDLQ WKH QXPHULFDO VROXWLRQ ZH GHYHORS D WZRGLPHQVLRQDO ELVHFWLRQ PHWKRG >8@ n$OWKRXJK WKLV PHWKRG LV QRW DV IDVW FRPSXWDWLRQDOO\ DV 1HZWRQ5DSKVRQ In%QfÂ¬a: XVH LW EHFDXVH LW DYRLGV WKH SUREOHPV RI FRQYHUJHQFH WKDW W\SLFDOO\ RFFXU ZKHQ 1HZWRQ5DSKVRQ LV DSSOLHG WR FRPSOH[ SUREOHPV 0DLQ FRQFOXVLRQV GUDZQ IURP WKH DQDO\VLV DUH Df WKH VXEWKUHVKROG JDWHYROWDJH VZLQJ GHSHQGV VWURQJO\ RQ JUDLQERXQGDU\ SURSHUWLHV DQG ZHDNO\ RQ WKH IURQWJDWHEDFNJDWH FKDUJH FRXSOLQJ HIIHFWV Ef WKH WKUHVKROG aYEOWDJH GHSHQGV VWURQJO\ RQ JUDLQERXQGDU\ SURSHUWLHV DQG RQ FKDUJHFRXSOLQJ HIIHFWV Ff WKH FKDUJHFRXSOLQJ HIIHFWV GHFUHDVH DV WKH JUDL QERXQGDU\ WUDS GHQVLW\ WKH WKLFNQHVV RI WKH ILOP RU WKH ILOP GRSLQJ FRQFHQWUDWLRQ LQFUHDVHV 7R VXSSRUW WKH DQDO\VLV DQG WR VWUHVV LWV LPSRUWDQFH ZH FRPSDUH LQ 6Â«FWLRQ PRGHO SUHGLFWLRQV ZLWK PHDVXUHG FXUUHQWYROWDJH PAGE 108 Â« FKDUDFWHULVWLFV RI WKLQILOP /3&9' SRO\VLOLFRQ 026)(7n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nV WKDW WKH JUDLQERXQGDU\ WUDSV FDQ EH DFFRXQWHG IRU DV LI WKH\ ZHUH XQLIRUPO\ GLVWULEXWHG LQ VLOLFRQ :H UHIHU WR WKH IRXU WHUPLQDO SFKDQQHO DFFXPXODWLRQPRGH 62, 026)(7 LOOXVWUDWHG LQ )LJ DQG ZH DFFRXQW LQ FRQWUDVW ZLWK SUHYLRXV ZRUNV >.$ '(@ IRU WKH FKDUJHFRXSOLQJ HIIHFWV >/,ED 5$@ EHWZHHQ WKH IURQW DQG EDFN VXUIDFHV :H ILUVW DQDO\]H WKH VXEWKUHVKROG FXUUHQWYROWDJH FKDUDFWHULVWLFV )RU ZHDNDFFXPXODWLRQ FRQGLWLRQV DQG IRU GUDLQ I YROWDJHV 9T UHODWLYH WR WKH VRXUFHf VXEVWDQWLDOO\ JUHDWHU WKDQ N7T WKH KROHV LQ WKH FKDQQHO IORZ SUHGRPLQDQWO\ E\ GLIIXVLRQ >/,D@ 7KHUHIRUH WKH GUDLQ FXUUHQW JRYHUQHG E\ RQO\ WKH IURQW VXUIDFH PAGE 109 Â« FRQGXFWLRQ WKH EDFN VXUIDFH LV DVVXPHG WR EH GHSOHWHG RU LQYHUWHGf LV >/,D 6:@ = N7 QV / XSI T rSI f ZKHUH = LV WKH FKDQQHO ZLGWK / LV WKH FKDQQHO OHQJWK 3SI LV KROH PRELOLW\ WKH FKDUDFWHUL]DWLRQ RI ZKLFK ZH GLVFXVV ODWHUf DQG M!I LV WKH KROH DFFXPXODWLRQOD\HU DUHDO FKDUJH GHQVLW\ DW WKH VRXUFH ,Q ZULWLQJ f ZH KDYH DVVXPHG WKDW WKH DFFXPXODWLRQOD\HU FKDUJH GHQVLW\ DW WKH GUDLQ LV PXFK VPDOOHU WKDQ 4SI ZKLFK REWDLQV LQ ZHDN DFFXPXODWLRQ RSHUDWLRQ %DVHG RQ FODVVLFDO 026)(7 WKHRU\ >&@ ZH H[SUHVV 4SI IRU DOO DFFXPXODWLRQ FRQGLWLRQV DV V SI VI TSS f , fÂ§ R U f ZKHUH LfVI LV WKH IURQW VXUIDFH SRWHQWLDO DW WKH VRXUFH SSf LV WKH H[FHVV KROH FRQFHQWUDWLRQ DQG ) LV WKH HOHFWULF ILHOG QRUPDO WR G[ WKH VXUIDFH 1RWH WKDW WKH DSSOLFDELOLW\ RI f UHTXLUHV WKDW S S4 2IL$ f DW VRPH SRLQW LQ WKH ILOP ZKLFK LV FRQVLVWHQW ZLWK RXU DVVXPSWLRQ WKDW WKH EDFN VXUIDFH LV QRW DFFXPXODWHG )RU JLYHQ IURQW DQG EDFN JDWH YROWDJHV DQG 9ILE LV FKDUDFWHUL]HG E\ D RQHGLPHQVLRQDO VROXWLRQ RI 3RLVVRQnV HTXDWLRQ ZLWK PL[HG ERXQGDU\ FRQGLWLRQV DW WKH IURQW DQG KDFN LQWHUIDFHV ,QVWHDG RI VROYLQJ WKLV GLIIHUHQWLDO HTXDWLRQ FRPSOHWHO\ IRU WKH VSDFHGHSHQGHQFH PAGE 110 RI ) DQG?_! >%$@ ZH XVH f DQG HYDOXDWH WKH ERXQGDU\ FRQGLWLRQV DQG WKH EDFN VXUIDFH SRWHQWLDO LAf E\ VROYLQJ D V\VWHP RI QRQOLQHDU HTXDWLRQV DV GHVFULEHG LQ WKH IROORZLQJ VXEVHFWLRQV )RUPDOLVP :H UHIHU WR WKH RQHGLPHQVLRQDO UHSUHVHQWDWLRQ RI WKH 026)(7 VKRZQ LQ )LJ DSSOLHG DW WKH VRXUFH 8VLQJ WKLQILOP 026)(7 WKHRU\ >/,ED@ ZH ZULWH WKH IROORZLQJ PL[HG ERXQGDU\ FRQGLWLRQV 9*I 75 / HV)VI AVI fÂ¯& 67 9 f DQG 9*E fÂµ 9)5 rVE H V)VE &RE f ZKHUH )JI DQG )V> DUH WKH IURQW DQG EDFNVXUIDFH HOHFWULF ILHOGV &4I I I DQG &4E DUH WKH IURQW DQG EDFNJDWH R[LGH FDSDFLWDQFHV DQG 9SJ AII E K 2IE fÂ§ DQG 9SJ fÂ§ DUH WKH IURQW DQG WKH EDFNJDWH RI RE FRQYHQWLRQDOf IODWEDQG YROWDJHV DQG DUH WKH IURQW DQG WKH EDFN JDWHERG\ ZRUNIXQFWLRQ GLIIHUHQFHV DQG 2II DQG IA DUH WKH IL[HG FKDUJH GHQVLWLHV DW WKH IURQW DQG EDFN 6L6L2 LQWHUIDFHV UHVSHFWLYHO\ :LWK f DQG f 3RLVVRQnV HTXDWLRQ G[ V f PAGE 111 fÂ§Â« [ )LJ 2QH GLPHQVLRQDO UHSUHVHQWDWLRQ RI WKH WKLQILOP 026)(7 DSSOLHG DW WKH VRXUFH $OWKRXJK WKH IURQW JDWH LV W\SLFDOO\ SRO\VLOLFRQ LW LV ODEHOHG 0 WR VLPSOLI\ WKH QRWDWLRQ PAGE 112 ZKHUHS LV WKH FKDUJH GHQVLW\ S TS Q 1D Q_T 1Df f FRXOG EH VROYHG QXPHULFDOO\f WR \LHOG )[f DQGA[f DQG WKXV S[f DQG 4rI LQ f ,Q f 1$ LV WKH DFFHSWRU GRSLQJ FRQFHQWUDWLRQ 3 QA+3O7Uf + 3H[S NI! f LV WKH KROH FRQFHQWUDWLRQ Q QH[Saaf Q4H[SAf LV WKH HOHFWURQ FRQFHQWUDWLRQ 1 7$ 7$ fÂµ (7$n()Q OH[SfÂ§fÂ§f f f LV WKH FRQFHQWUDWLRQ RI RFFXSLHG QHJDWLYHf JUDLQERXQGDU\ DFFHSWRU WUDSV 1\A DW D PRQRHQHUJHWLF OHYHO (MA DQG 1 7' 72 ()a(7' H[S b,+f f LV WKH FRQFHQWUDWLRQ RI HPSW\ SRVLWLYHf JUDLQERXQGDU\ GRQRU WUDSV 1MT DW D PRQRHQHUJHWLF OHYHO (MA (S LV WKH )HUPL OHYHO DW WKH VRXUFHf 7KH WUDS GHQVLWLHV 1\$ DQG 1\' DUH UHODWHG WR WKH JUDLQERXQGDU\ VXUIDFH PAGE 113 6WDWH DUHDOf GHQVLW\ 1J\ E\ 1\ 1J\GJ ZKHUH GJ LV WKH DYHUDJH JUDLQ VL]H DV GHVFULEHG LQ WKH SUHYLRXV FKDSWHU :LWK UHIHUHQFH WR f WKHQ ZH VWUHVV WKDW S4 DQG Q QISf DUH WKH FDUULHU GHQVLWLHV WKDW ZRXOG REWDLQ LQ WKH ILOP LI S ,QVWHDG RI VROYLQJ f GLUHFWO\ ZH XVH LW WR GHULYH WZR HTXDWLRQV WKDW ZKHQ FRPELQHG ZLWK f DQG f \LHOG VXIILFLHQW LQIRUPDWLRQ WR FKDUDFWHUL]H 4SI9J\ 9JEf 7R GR WKLV ZH PXVW ILUVW H[SUHVV S LQ WHUPV RI :H UHODWH (S WR E\ (S ( Tf! )R TS f ZKHUH f S LV WKH )HUPL SRWHQWLDO LQ WKH ILOP ZKHQ S LH ZKHQ S S DQG Q Q4 `fS N7TfO QSQMf 1RZ VXEVWLWXWLQJ f LQWR f DQG f ZH REWDLQ DQG f f ZKHUH ([ ( fÂµ TE 7$ L .MD$B H[S N7 f PAGE 114 DQG fÂ± aA7'aA )R ? .M'$ H[S N7 f f )LQDOO\ FRPELQLQJ ff f DQG f ZH JHW SIRf T3RH[S Af Q4H[SAUf 1IW 1 7' A7$ O.7'H[SAUf O.7$H[S AUf A A 1RZ IROORZLQJ FRQYHQWLRQDO 026)(7 WKHRU\ >&@ ZH UHZULWH 3RLVVRQnV HTXDWLRQ GIB G_! b W f H f DQG WKHQ LQWHJUDWH IURPWVI WRW WR REWDLQ ) )I *Wf *WIf f ZKHUH Jrf 6 f PAGE 115 Â« 6XEVWLWXWLQJ f LQWR f DQG SHUIRUPLQJ WKH LQWHJUDWLRQ ZH JHW *rnf IA^3>H[S Âf @ QR>H[SAf @ 1$ N7 17'AN7 fÂ¬ RJA .7'H[S^NIAn A7$AN7fÂ¬ A7$H[Aa N7AA n f ,QWHJUDWLRQ RI f XVLQJ D VHSDUDWLRQ RI YDULDEOHV WHFKQLTXH WKHQ \LHOGV [ I $ f AVI )I*_!VIf.LL! f ZKLFK GHILQHV WKH VSDWLDO GHSHQGHQFH RI W[f LQ WHUPV RI DQG (YDOXDWLRQ RI f DQG f DW WKH EDFN LQWHUIDFH [A W AVK! DQG ) )VWf GHILQHV WZR UHODWLRQVKLSV WKDW ZLWK f DQG f FRQVWLWXWH D V\VWHP RI HTXDWLRQV IRUL^!VI L_!VE )VI )E )VI *rVI! Â‘ VnrVE! ! DQG r n3 VE GS VEfÂ¬9AIRA*A f f PAGE 116 ZKHUH WE LV WKH ILOP WKLFNQHVV :H VWUHVV WKDW fÂ«WKH VLPXOWDQHRXV VROXWLRQ RI f f f DQG f \LHOGV L_!VI LI!VE )VI DQG )VE ZLWKRXW UHTXLULQJ WKH HYDOXDWLRQ RI WKH [GHSHQGHQFH RI ) DQG } ZKLFK LQ IDFW FRXOG EH REWDLQHG E\ XVLQJ f DQG f 7KH VROXWLRQ ZH REWDLQ LQ SDUWLFXODU DQG )I IRU JLYHQ DQG 9JE FDQ WKHQ EH XVHG LQ FRPELQDWLRQ ZLWK f DQG f WR JLYH SI9JI 9ILKf ZKLFK LV YDOLG IRU DOO OHYHOV RI DFFXPXODWLRQ 1XPHULFDO 6ROXWLRQ 7KH V\VWHP RI HTXDWLRQV f f f DQG f FDQ EH UHGXFHG WR WZR HTXDWLRQV LQ ZKLFK WKH XQNQRZQ YDULDEOHV DUH rVI DQG LL}VE E\ VXEVWLWXWLQJ f DQG f LQWR f DQG f UHVSHFWLYHO\ & ,AY 6 *In9)E rVIQ *rVIf UBNY 9 / 2Y*E Y 7% fÂ«AVEA AVE!r AVIAVEr f DQG rVE ^ Â‘ rVI fÂµrVE! r ! V PAGE 117 ZKHUH JAW_! VEf DQG A VI A VEf KDYH EHHQ GHILQHG WR PDNH D FRPSDFW QRWDWLRQ:H VROYH f DQG f XVLQJ D WZR GLPHQVLRQDO ELVHFWLRQ PHWKRG $QDORJRXV WR WKH FODVVLFDO ELVHFWLRQ PHWKRG >%8@ WKH DOJRULWKP RI WKH WZRGLPHQVLRQDO ELVHFWLRQ PHWKRG LV Df VHOHFW D UDQJH AVE &rVE AVE VEf LQ ZKLFK WKH VROXWLRQ W_!VI}LSVEf H[LVWV Ef FDOFXODWH LS WKH FRUUHVSRQGLQJ UDQJH RI L_!VI ?_!rI?S_If E\ VROYLQJ LWVI}nOnVEf DQG JAn_}VAAJMff Ff VXEGLYLGH WKH UDQJH RI AVE LQWR WZR SDUWV UHSHDWLQJ Ef WR GHILQH WKH FRUUHVSRQGLQJ VXEGLYLVLRQ RI WKH LM!VI UDQJH Gf GHWHUPLQHLQ ZKLFK SDUW RI WKH VXEGLYLGHG UDQJH WKH VROXWLRQ H[LVWV E\ FKHFNLQJ IRU JIRrI MSrEf [ JIR _I Â_!JEf Hf UHSHDW Ff DQG Gf XQWLO WKH VROXWLRQ LV GHULYHG DV DFFXUDWHO\ DV GHVLUHG 2QFHAI DQGAVE DW WKH VRXUFHf KDYH EHHQ FDOFXODWHG )I DQG )VE DUH HYDOXDWHG XVLQJ f DQG f UHVSHFWLYHO\ 7KHQ 4SI9AI9*Ef LV REWDLQHG E\ QXPHULFDOO\ HYDOXDWLQJ WKH LQWHJUDO LQ f DIWHU VXEVWLWXWLQJ f DQG f LQWR LW 7KH )RUWUDQ FRPSXWHU SURJUDP KDV EHHQ OLVWHG LQ $SSHQGL[ & 7R LOOXVWUDWH WKH SUHGLFWLRQV RI RXU PRGHO ZH DSSO\ LW WR D W\SLFDO WKLQILOP DFFXPXODWLRQPRGH SFKDQQHO /3&9' SRO\VLOLFRQ 026)(7 IRU ZKLFK &RI [ r )FP W4I $f &RE [ )FP W4E $f = SP / SP DQG3SA FP9VHF ZKLFK LV UHSUHVHQWDWLYH RI DQ XQSDVVLYDWHG GHYLFH >6+@ :H SORW LQ )LJ FDOFXODWLRQV RI AA9JAf IRU 9AS9AJf 9 FKRVHQ WR HQVXUH QR DFFXPXODWLRQ DW WKH EDFN VXUIDFH DQG IRU VHYHUDO YDOXHV RI 1MT :H VHW PAGE 118 rSI LQ &/ D r _ R Q_ )LJ &DOFXODWHG QRUPDOL]HG KROH DFFXPXODWLRQOD\HU DUHDO FKDUJH GHQVLW\ DW WKH VRXUFH YHUVXV IURQW JDWH YROWDJH IRU VHYHUDO GRQRU JUDLQERXQGDU\ WUDS GHQVLWLHV 7KH FDOFXODWLRQV DUH QRUPDOL]HG WR IDFLOLWDWH D ODWHU GLVFXVVLRQ RI WKH ZHDN DFFXPXODWLRQ FXUUHQW LQ f PAGE 119 ,OO (M' (Q DQG OHW 1MA :H XVHG W\SLFDO YDOXHV IRU O2AnFQIA DQG WK XP 7R IDFLOLWDWH D ODWHU GLVFXVVLRQ RI WKH ZHDNDFFXPXODWLRQ M 9I r I mL , rfÂµ W r ? FXUSAQW LS f ZH KDYH SORWWHG LQ )LJ D QRUPDOL]HG YHUVLRQ RI A ZKLFK LQ IDFW LV ,S LQ f :H VWUHVV WKDW WKH 2SI9JIf FD7RXWDWLRQV DUH YDOLG IRU DOO DFFXPXODWLRQ OHYHOV $V LQ WKH EXON 026)(7 >& %5@ 46I9JIf LQ )LJ VKRZV WZR GLVWLQFW GHSHQGHQFHV H[SRQHQWLDO IRU ORZ 9JIZHDN DFFXPXODWLRQf DQG OLQHDU IRU KLJK 9JI VWRQJ DFFXPXODWLRQf :H QRWH WKDW WKH VXEWKUHVKROG JDWHYROWDJH VZLQJ 6 $ 9JI QHHGHG WR FKDQJH 2SI RU ,T RQH RUGHU RI PDJQLWXGHf LQFUHDVHV VWURQJO\ ZLWK 1\S )RU H[DPSOH 6 P9 IRU 1MTAA,2A FPa ZKLFK FRUUHVSRQGV WR 1J\ [ A FQ7A LI GJ b $f ZKHUHDV IRU 1\S 6 P9 PXFK OHVV WKDQ WKDW IRU DQ LQYHQVLRQPRGH GHYLFH >6: %5@f 7KH FDOFXODWHG 61\f GHSHQGHQFH UHIOHFWV WKH ODUJHU YDULDWLRQ LQ WUDSSHG FKDUJH GHQVLW\ FRUUHVSRQGLQJ WR fÂµ ? r r L ? D $fÂ¬On VI FDXVHG E\ KLJKHU 1\S :H L ,OÂ¼VWUDWH LQ )LJ WKH FDOFXODWHG GHSHQGHQFH RI 4AI RQ (\S AIDFA\4A= [ FP DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV XVHG LQ WKH SUHYLRXV ILJXUH :H VHH WKDW 6 GHFUHDVHV DV (MT LQFUHDVHV 7KLV GHSHQGHQFH LV H[SODLQHG E\ QRWLQJ WKDW KLJKHU (MT UHVXOWV LQ VPDOOHU FKDQJH LQ WUDSSHG FKDUJH IRU D JLYHQ $L_}V\ $GGLWLRQDO FDOFXODWLRQV UHYHDO WKDW 6 LV RQO\ ZHDNO\ GHSHQGHQW RQ 1\$ 7KLV UHVXOW LV H[SODLQHG E\ WKH IDFW WKDW RQO\ KROH WUDSV FDQ VLJQLILFDQWO\ DIIHFW WKH KROH DFFXPXODWLRQ SURFHVV LQ WKH 026)(7 7R LOOXVWUDWH WKH GHSHQGHQFH RI 2SI RQ WKH FKDUJHFRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV ZH SORW LQ )LJ FDOFXODWLRQV RI 2SI YHUVXV & PAGE 120 SI $f fÂµ r )LJ &DOFXODWHG QRUPDOL]HG KROH DFFXPXODWLRQOD\HU DUHDO FKDUJH GHQVLW\ DW WKH VRXUFH YHUVXV IURQW JDWH YROWDJH IRU VHYHUDO GRQRU JUDLQERXQGDU\ WUDS HQHUJ\ OHYHOV PAGE 121 Â« JDWH YROWDJHV PAGE 122 9TA IRU WZR YDOXHV RI FKRVHQ WR SUHFOXGH DFFXPXODWLRQ DW WKH EDFN VXUIDFHf 1\ [ A FP DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV XVHG LQ )LJ :H VHH RQO\ D ZHDN GHSHQGHQFH RI 6 RQ 9TA EHFDXVH WKH FKDUJH FRXSOLQJ DIIHFWV WKH WRWDO WUDSSHG FKDUJH EXW QRW LWV YDULDWLRQ ZLWK L_}V\ &RQVLVWHQW ZLWK WKHVH UHVXOWV DGGLWLRQDO FDOFXODWLRQV VKRZ WKDW 6 LV DOVR RQO\ ZHDNO\ GHSHQGHQW RQ WA 7KH SORWV LQ )LJV LPSO\ WKDW WKH VWURQJDFFXPXODWLRQ D WKUHVKROG YROWDJH 9\\ LV DIIHFWHG E\ WKH FKDUJH FRXSOLQJ EHWZHHQ WKH IURQW DQG EDFN JDWHV DV ZHOO DV E\ WKH JUDLQERXQGDU\ WUDSV :H D B HYDOXDWH 9\\ IURP WKH OLQHDU H[WUDSRODWLRQ RI WKH FDOFXODWHG 4A\I9A\f nIRU VWURQJ DFFXPXODWLRQ WR WKH 9TA D[LV DV LV GRQH IRU WKH FRQYHQWLRQDO 026)(7 >&@ 7KXV IRU VWURQJ DFFXPXODWLRQ 4SI &RI9*I 97If fÂ¬ f D 7R LOOXVWUDWH WKH SUHGLFWHG GHSHQGHQFH RI 9\\ RQ WKH WUDS D I GHQVLWLHV ZH SORW LQ )LJ 9\\ 9S%f YHUVXV 1\S IRU GLIIHUHQW 1\A DQG WKH VDPH UHPDLQLQJ GHYLFH SDUDPHWHU YDOXHV XVHG WR GHULYH WKH SORWV LQ )LJ :H VHH WKDW _9\\_ LQFUHDVHV DV 1\A LQFUHDVHV DQG WKDW IRU ODUJH 1\' ! [ O2A FPf _9\\_ LV QHDUO\ SURSRUWLRQDO WR 1A >.$ '(@ :H QRWH WKDW _9\\_ GHFUHDVHV DV 1\A LQFUHDVHV EHFDXVH RI WKH FRPSHQVDWLQJ HIIHFW RI WKH DFFHSWRU WUDSV RQ WKH GRQRU WUDSV D 7KH FDOFXODWHG GHSHQGHQFH RI 9\\ RQ WKH WUDS OHYHOV LV LOOXVWUDWHG LQ )LJ LQ ZKLFK ZH OHW 1\S 1\A [ A FP DQG XVHG WKH VDPH UHPDLQLQJ SDUDPHWHUV DV XVHG LQ WKH SUHYLRXV ILJXUHV :H VHH WKDW _9\\_ PAGE 123 fÂµ V 1WG FPf )LJ &DOFXODWHG IURQW VWURQJDFFXPXODWLRQ WKUHVKROG YROWDJH YHUVXV JUDLQERXQGDU\ GRQRU WUDS GHQVLW\ IRU WZR JUDLQKRXQGDU\ DFFHSWRU WUDS GHQVLWLHV PAGE 124 )LJ &DOFXODWHG IURQW VWURQJDFFXPXODWLRQ WKUHVKROG YROWDJH YHUVXV JUDLQKRXQGDU\ GRQRU WUDS HQHUJ\ OHYHO IRU GLIIHUHQW JUDLQn ERXQGDU\ DFFHSWRU WUDS HQHUJ\ OHYHOV PAGE 125 Â« YB LQFUHDVHV DV (\$ DQG (\J LQFUHDVH 7KH 9\\(\Jf GHSHQGHQFH LV H[SODLQHG E\ QRWLQJ WKDW WKH GRQRU WUDSV WHQG WR EH PRUH LRQL]HG YL] PRUH HIIHFWLYH ILQ nWUDSSLQJ KROHV DV (\J LQFUHDVHV 1RWH WKDW DV (\J Â¯ O D n Â‘ fÂµÂ‘Â‘Â‘ n DSSURDFKHV WKH FRQGXFWLRQ EDQG 9\\ VDWXUDWHV DQG WKH GHYLFH EHKDYHV Â‘ n fÂ¬ n n > Â‘ D VLPLODUO\ WR DQ LQYHUVLRQPRGH 026)(7 $QDORJRXVO\ WKH bbf GHSHQGHQFH LV LQWHUSUHWHG E\ REVHUYLQJ WKDW WKH DFFHSWRU WUDSV WHQG WR O L L EH OHVV LRULL]HG!DQG OHVV HIIHFWLYH LQ FRPSHQVDWLQJ 1\J DV (\$ LQFUHDVHV 7KH FKDUJHFRXSOLQJ HIIHFWV EHWZHHQ WKH IURQW DQG EDFN JDWHV GHFUHDVH HLWKHU DV WKH WUDS GHQVLW\ IRU WKH ILOP WKLFNQHVV LQFUHDVHV D 7R HPSKDVL]H WKH IRUPHU GHSHQGHQFH ZH SORW LQ )LJ 9\\ YHUVXV 95K IRU GLIIHUHQW 1\J ? fÂ«1\$ DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV XVHGnLQ D )LJ :H VHH WKDW 9\\ LV LQGHSHQGHQW RI RQO\ IRU ODUJH WUDS GHQVLWLHV 1\$ 1\J !r FPrf 7KLV LV H[SODLQHG E\ QRWLQJ WKDW WKH VSDFHFKDUJH UHJLRQ DVVRFLDWHG ZLWK HDFK JDWH GHFUHDVHV LQ ZLGWK DULG WKH FKDUJHFRXSOLQJ HIIHFWV GLPLQLVK DV WKH WUDS GHQVLWLHV LQFUHDVH 7R fÂ° D HPSKDVL]H WKH ODWWHU GHSHQGHQFH QRWHG DERYH ZH SORW LQ )LJ 9\\ YHUVXV IRUAWMM UDQJLQJ IURP \P WR \P 1\$ 1\J [ L' FP DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV XVHG LQ )LJ $V LPSOLHG E\ SUHYLRXV ZRUN >/,E@ ZH VHH WKDW 97I LV LQGHSHQGHQW RI 9*E RQO\ IRU WKLFNILOP GHYLFHV WA ! \Pf 7KH WKUHVKROG YROWDJH GHSHQGV RQ 1$ LI LW LV VXIILFLHQWO\ KLJK ,I 1D LV PXFK KLJKHU WKDQ WKH WUDS GHQVLW\ WKH GHYLFH WHQGV WR EHKDYH DV WKH VLQJOHFU\VWDO FRXQWHUSDUW :H SORW LQ )LJ FDOFXODWHG 9"I YHUVXV IRU 1$ UDQJLQJ IURP r FPr WR r FPr DQG WKH VDPH UHPDLQLQJ SDUDPHWHUV :H QRWH WKDW IRU 1$ ! r FPr !! A PAGE 126 Â« , &DOFXODWHG IURQW VWURQJDFFXPXODWLRQ WKUHVKROG YROWDJH YHUVXV EDFN JDWH YROWDJH IRU GLIIHUHQW JUDLQERXQGDU\ WUDS GHQVLWLHV 7KH SORWV ZHUH GHULYHG IRU WKH UDQJH RI IRU ZKLFK f LV DSSOLFDEOH )LJ PAGE 127 r & )LJ &DOFXODWHG IURQW VWURQJDFFXPXO DWLRQ WKUHVKROG YROWDJH YHUVXV EDFN JDWH YROWDJH IRU VHYHUDO ILOP WKLFNQHVVHV PAGE 128 Â« Y*E YeJ! 9f I )LJ &DOFXODWHG IURQW VWURQJDFFXPXODWLRQ WKUHVKROG YROWDJH YHUVXV KDFN JDWH YROWDJH IRU VHYHUDO DFFHSWRU GRSLQJ FRQFHQWUDWLRQV PAGE 129 1MDf 9MI LV ZHOO GHVFULEHG E\ SUHYLRXV VLQJOHFU\VWDO DQDO\VLV >/,E@ DQG IRU 1A O2A FQIA 1A 1\S 1\Af} 9MI LV LQGHSHQGHQW RI 1A ([SHULPHQWDO 5HVXOWV DQG 'LVFXVVLRQ 7R VXSSRUW WKH DQDO\VLV LQ 6HFWLRQ DQG WR LGHQWLI\ FULWLFDO DVSHFWV RI LW ZLWK UHJDUG WR 62 ,' GHYLFH DQG LQWHJUDWHG FLUFXLW GHVLJQ ZH GLVFXVV LQ WKLV VHFWLRQ PHDVXUHG ,JL9J\ 9G 9TAf FKDUDFWHULVWLFV RI SFKDQQH7 DFFXPXODWLRQPRGH DQG QFKDQQHO LQYHUVLRQn PRGH VPDOOJUDLQ SRO\VLOLFRQ 026)(7nV GHVFULEHG LQ WKH SUHYLRXV FKDSWHU 7KH ASFKDQQHO DFFXPXODWLRQPRGH SRO\VLOLFRQ 026)(7nV DUH XQSDVVLYDWHG ,QL WKHVH GHYLFHV WKH SDUDPHWHUV RI LQWHUHVW DUH WA \ P W4I $ WA $ 1A rp FPfÂ°A = \P DQG / \P 0HDVXUHG FKDUDFWHULVWLFV >6+@ RI D SDUWLFXODU GHYLFH ZKLFK W\SLI\ WKH FKDUDFWHULVWLFV RI LGHQWLFDOO\ SURFHVVHG ` GHYLFHV ZHUH SORWWHG LQ )LJ 7KH DQRPDORXV LQFUHDVH ZLWK 9J\ RI WKH OHDNDJH FXUUHQW LQ WKH 2)) VWDWH YJI ! f ZDV DWWULEXWHG LQ &KDSWHU )RXU WR ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV LQ WKH IURQWf VXUIDFH GHSOHWLRQ UHJLRQ DW WKH GUDLQ ,Q WKH 21 VWDWH 9ILI f WKH ZHDNDFFXPXO DWLRQf FXUUHQW LQFUHDVHV QHDUO\ H[SRQHQWLDOO\ ZLWK 9J\ DV FKDUDFWHUL]HG E\ RXU PRGHO 7KH KLJK JDWHYROWDJH VZLQJ 9GHFDGHf DQG WKH KLJK DFFXPXODWLRQ WKUHVKROG YROWDJH 9f DUH VLPLODU WR WKRVH FKDUDFWHULVWLFV SUHGLFWHG E\ RXU WKHRUHWLFDO FDOFXODWLRQV IRU 1\Q A FPfÂ¯ 7KLV ODUJH YDOXH RI 1\S LV QRW XQUHDVRQDEOH EHFDXVH WKLV GHYLFH LV XQSDVVLYDWHG >6+ PAGE 130 *UDLQERXQGDU\ SDVVLYDWLRQ YLD K\GURJHQDWLRQ GUDPDWLFDOO\ LPSURYHV WKH ,A9TA9Tf FKDUDFWHULVWLFV RI /3&9' SRO\VLOLFRQ 026)(7V >6+@ :LWK UHIHUHQFH WR WKH FKDUDFWHULVWLFV RI D W\SLFDO XQSDVVLYDWHG DFFXPXODWLRQPRGH SFKDQQHO WUDQVLVWRU LQ )LJ SODPD K\GURJHQDWLRQ D W\SLFDOO\ UHGXFHV 6 E\ D IDFWRU RI WKUHH RU IRXU DQG 0MI E\ D IDFWRU RI DERXW WZR >6+@ 5HIHUULQJ WR )LJ DQG WKH DQDO\VLV LQ 6HFWLRQ ZH WKXV LQIHU WKDW WKH K\GURJHQDWLRQ UHGXFHV WKH JUDLQ ERXQGDU\ WUDS GHQVLW\ E\ DERXW DQ RUGHU RI PDJQLWXGH WRaA FPp $OWKRXJK WKH DQDO\VLV LQ 6HFWLRQ ZDV GHYHORSHG IRU DQ DFFXPXODWLRQPRGH GHYLFH LW FDQ EH DSSOLHG ZLWK PLQRU PRGLILFDWLRQ WR DQ LQYHUVLRQPRGH GHYLFH %HFDXVH W\SLFDOO\ WKH JUDLQERXQGDU\ WUDS GHQVLW\ H[FHHGV WKH GRSLQJ GHQVLW\ WKH EHKDYLRU RI DQ LQYHUVLRQPRGH GHYLFH LV VLPLODU WR WKDW RI WKH DFFXPXODWLRQPRGH GHYLFH DV PRGHOHG LQ 6HFWLRQ 7KH QFKDQQHO LQYHUVLRQPRGH SRO\VLOLFRQ 026)(7nV ZH PHDVXUHG ZHUH JUDLQERXQGDU\SDVVLYDWHG >6+@ E\ D SODVPD RI K\GURJHQ 7KHVH GHYLFHV GLIIHU IURP WKH SFKDQQHO DFFXPXODWLRQPRGH GHYLFHV LQ WKH IROORZLQJ SDUDPHWHUV WI XP &4I [ p )FPA W4I $f [ A FPp = SP DQG / XP :H SORW LQ )LJ PHDVXUHG ,IfY*I}Y*Ef FKDUDFWHULVWLFV IRU D UHSUHVHQWDWLYH GHYLFH ZLWK 9T 9 7KH ZLGH UDQJH RI XVHG 9 WR 9f HQVXUHG WKDW WKH EDFN VXUIDFH FKDUJH FRQGLWLRQ ZDV YDULHG IURP LQYHUVLRQ WR ZHDNDFFXPXODWLRQ ,Q DFFRUG ZLWK WKH FDOFXODWHG UHVXOWV SORWWHG LQ )LJ WKH GDWD D LQ )LJ VKRZ WKDW 6 LV QHDUO\ LQGHSHQGHQW RI 9JA DQG WKDW 9A LV ORZHUHG VOLJKWO\ DV 9TA LQFUHDVHV 7KH YDOXH RI 6 9GHFDGHf LV PAGE 131 & )LJ 0HDVXUHG GUDLQ FXUUHQW YHUVXV IURQW JDWH YROWDJH RI QFKDQQHO LQYHUVLRQARGH /3&9' SRO\VLOLFRQ 62, 026)(7 IRU GLIIHUHQW EDFN JDWH YROWDJHV n PAGE 132 VPDOOHU WKDQ WKDW PHDVXUHG IRU WKH XQSDVVLYDWHG GHYLFH LQ )LJ DQG LV FRQVLVWHQW ZLWK FDOFXODWLRQV LQ 6HFWLRQ f FRUUHVSRQGLQJ WR 1MA a $ FPfÂ° 2XU LQIHUUHG UHGXFWLRQ LQ WUDS GHQVLW\ DQ RUGHU RI PDJQLWXGHf SURGXFHG E\ WKH K\GURJHQDWLRQ LV FRQVLVWHQW ZLWK SUHYLRXV VWXGLHV RI JUDLQERXQGDU\ SDVVLYDWLRQ >6+@ 8VLQJ WKH PHDVXUHG OLQHDUUHJLRQ VWURQJLQYHUVLRQ ,T9TAf FKDUDFWHULVWLF RI WKH SDVVLYDWHG QFKDQQHO GHYLFH ZH FDOFXODWH 9\\ 9 IURP WKH OLQHDU H[WUDSRODWLRQ WR WKH 9J\ D[LV DQG XQI FPA9VHF IURP WKH VORSH 7KLV VPDOOHU YDOXH RI 9\\ FRPSDUHG ZLWK WKDW RI WKH XQSDVVLYDWHG GHYLFHV LV FRQVLVWHQW ZLWK SUHYLRXV H[SHULPHQWDO UHVXOWV >6+ 0$@ DQG DFFRUGLQJ WR WKH WKHRUHWLFDO DQDO\VLV RI 6HFWLRQ LQGLFDWHV 1\Aa A FPA FRQVLVWHQW ZLWK RXU LQIHUHQFH IURP PHDVXUHPHQWV RI 6 2XU PHDVXUHPHQWV DV ZHOO DV SUHYLRXV H[SHULPHQWDO UHVXOWV >6+ 0$@ LQGLFDWH WKDW WKH FDUULHU PRELOLW\ LQ VPDOOJUDLQ SRO\VLOLFRQ 026)(7V LQFUHDVHV E\ DOPRVW DQ RUGHU RI PDJQLWXGH GXH WR WKH JUDLQ ERXQGDU\ SDVVLYDWLRQ K\GURJHQDWLRQf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cLUUHVSHFWLYH RI WKH SRWHQWLDO EDUULHUV ZKLFK DUH QHJOLJLEOH IRU KLJK JDWH YROWDJH 7R SK\VLFDOO\ PRGHO WKLV PRELOLW\ WKHQ D EDVLF VWXG\ RI FDUULHU WUDQVSRUW WKURXJK WKH JUDLQ ERXQGDU\ ZRXOG EH QHFHVVDU\ Â‘ YF Â‘ Â‘ 6XPPDU\ fÂµU Â‘ c $ SK\VLFDO PRGHO WKDW GHVFULEHV WKH HIIHFWV RI JUDLQ ERXQGDULHV DQG RI IURQW JDWHEDFN JDWH FKDUJH FRXSLQJ RQ WKH fÂµ21VWDWH FRQGXFWLRQ RI WKH WKLQILOP /3&92 VPDOOJUDLQf SRO\VLOLFRQ 026)(7 KDV EHHQ GHYHORSHG 7KH PRGHO LV EDVHG RQ D ULJRURXV QXPHULFDO GHVFULSWLRQ RI WKH FKDQQHO FKDUJH GHQVLW\ DW WKH VRXUFH IRU DOO ELDV FRQGLWLRQV RI RSHUDWLRQ LQ WHUPV RI WKH JUDLQERXQGDU\ SURSHUWLHV DQG WKH GHYLFH SDUDPHWHUV &RUURERUDWLRQ RI WKH PRGHO ZDV REWDLQHG E\ VKRZLQJ FRQVLVWHQF\ EHWZHHQ PHDVXUHG DQG SUHGLFWHG GHSHQGHQFHV RI WKH GUDLQ FXUUHQW RQ WKH IURQW DQG EDFN JDWH YROWDJHV LQ ERWK SDVVLYDWHG K\GURJHQDWHGf DQG XQSDVVLYDWHG GHYLFHV $OWKRXJK Â‘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Â« WKRXJK LV WKH LPSOLFDWLRQ WKDW WKH JDWHYROWDJH VZLQJ ZLQ EH UHGXFHG DV 1? LQFUHDVHV 1A !! 1\S DQG 1\Af EHFDXVH WKH GHYLFH ZLOO WHQG WR EHKDYH DV WKH VLQJOHFU\VWDO FRXQWHUSDUW Â‘ H PAGE 136 &+$37(5 6,; 6800$5< &21&/86,21 $1' 5(&200(1'$7,216 6XPPDU\ DQG &RQFOXVLRQV :H KDYH GHYHORSHG DQG VXSSRUWHG H[SHULPHQWDOO\ SK\VLFDO PRGHOV WKDW GHVFULEH WKH HIIHFWV RI JUDLQ ERXQGDULHV RQ WKH VWHDG\VWDWH FRQGXFWLRQ SURSHUWLHV RI ODUJH DQG VPDOOJUDLQ SRO\VLOLFRQ 62,f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f SRO\VLOLFRQ 62, 026)(7 7KH PRGHO VKRZV WKDW WKH JUDLQ ERXQGDULHV GHILQH D QHDUO\ H[SRQHQWLDO GHSHQGHQFH RI WKH FRQGXFWDQFH RQ JDWH YROWDJH IRU PRGHUDWH PAGE 137 LQYHUVLRQ FRQGLWLRQV DQG DQ HIIHFWLYH WXUQRQ LQHDUUHJLRQf FKDUDFWHULVWLF WKDW RFFXUV ZHOO EH\RQG WKH VWURQJLQYHUVLRQ WKUHVKROG 7KLV WXUQRQ FKDUDFWHULVWLF LPSOLHV WKH FDUULHU PRELOLW\ WKUHVKROG YROWDJH ZKLFK H[FHHGV WKH VWURQJLQYHUVLRQ WKUHVKROG DQG WKH HIIHFWLYH ILHOGHIIHFW FDUULHU PRELOLW\ ZKLFK LV W\SLFDOO\ KLJKHU WKDQ WKH DFWXDO LQWUDJUDLQf PRELOLW\ 7KH PRGHO DOVR DFFRXQWV IRU DUELWUDU\ RULHQWDWLRQ RI WKH JUDLQ ERXQGDULHV DQG SUHGLFWV WKDW JUDLQ ERXQGDULHV SHUSHQGLFXODU WR WKH FDUULHU IORZ LQ WKH FKDQQHO PD[LPL]HV WKH JUDLQn ERXQGDU\ HIIHFWV RQ WKH FRQGXFWDQFH &RUURERUDWLRQ RI WKH PRGHO ZDV REWDLQHG E\ VKRZLQJ JRRG DJUHHPHQW EHWZHHQ H[SHULPHQWDO DQG WKHRUHWLFDO UHVXOWV ,Q &KDSWHU 7KUHH ZH GHVFULEHG D SK\VLFDO QXPHULFDO PRGHO EDVHG RQ UHDVRQDEOH DVVXPSWLRQV IRU WKH FXUUHQWYROWDJH FKDUDFWHULVWLFV RI WKH ODUJHJUDLQ EHDP UHFU\VWDOOL]HGf SRO\VLOLFRQ 62, 026)(7 YDOLG IRU DOO ELDV FRQGLWLRQV 7KH PRGHO LQGLFDWHV WKDW JUDLQ ERXQGDULHV WHQG WR UHGXFH WKH GUDLQ FXUUHQW EXW FDQ LQFUHDVH RU GHFUHDVH WKH WUDQVFRQGXFWDQFH WKDW WKH JUDLQERXQGDU\ HIIHFWV FDQ EH VLJQLILFDQW DW KLJK JDWH YROWDJH ZKHQ WKH GUDLQ YROWDJH LV KLJK HJ LQ WKH VDWXUDWLRQ UHJLRQ WKDW JUDLQ ERXQGDULHV FORVH WR WKH GUDLQ DUH PRVW HIIHFWLYH DQG WKDW WKH JUDLQERXQGDU\ HIIHFWV DUH HQKDQFHG DV WKH FKDQQHO OHQJWK LV VKRUWHQHG 7KH PRGHO KDV EHHQ VXSSRUWHG E\ FXUUHQW YROWDJH PHDVXUHPHQWV RI ODVHU DQG JUDSKLWHVWULSKHDWHUUHFU\VWDOOL]HG 62, 026)(7V ,Q &KDSWHU )RXU ZH GHYHORSHG DQ DQDO\WLF PRGHO IRU ILHOG HPLVVLRQ YLD JUDLQERXQGDU\ WUDSV LQ WKH VXUIDFH GHSOHWLRQ UHJLRQ DW WKH GUDLQ WR PAGE 138 H[SODLQ WKH 2))VWDWH FRQGXFWLRQ DQRPDORXV OHDNDJH FXUUHQWf REVHUYHG LQ VPDOOJUDLQ /3&9'f SRO\VLOLFRQ 62, 026)(7V &RUURERUDWLRQ RI WKH PRGHO ZDV JLYHQ E\ VKRZLQJ FRQVLVWHQF\ EHWZHHQ PHDVXUHG GDWD DQG SUHGLFWHG GHSHQGHQFHV RIWKH ILHOGHPLVVLRQ FXUUHQW RQ WKH JDWH DQG GUDLQ YROWDJHV 7KH PRGHO LPSOLHV WKDW WKH OHDNDJH FXUUHQW FDQ EH UHGXFHG E\ GHFUHDVLQJ WKH JUDLQERXQGDU\ WUDS GHQVLW\ YLD K\GURJHQDWLRQ IRU H[DPSOHf RU WKH HOHFWULF ILHOG 7KH UHGXFWLRQ LQ WKH HOHFWULF ILHOG FDQ EH DFFRPSOLVKHG E\ Df GHFUHDVLQJ WKH GRSLQJ GHQVLW\ LQ WKH GUDLQ Ef LQFUHDVLQJ WKH JDWH R[LGH WKLFNQHVV DQG Ff XVLQJ D S JDWH IRU WKH SFKDQQHO 026)(7f ,Q &KDSWHU )LYH ZH GHVFULEHG D SK\VLFDO PRGHO IRU WKH 21VWDWH FRQGXFWLRQ RI WKH WKLQILOP VPDOOJUDLQ /3&9'f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f DQG XQSDVVLYDWHG GHYLFHV $FFRUGLQJ WR RXU PRGHO WKH JDWHYROWDJH VZLQJ DQG WKH WKUHVKROG YROWDJH FDQ EH UHGXFHG E\ GHFUHDVLQJ WKH JUDLQn ERXQGDU\ WUDS GHQVLWLHV ZKLFK FDQ EH DFFRPSOLVKHG E\ JUDLQERXQGDU\ SDVVLYDWLRQ RU E\ LQFUHDVLQJ WKH GRSLQJ GHQVLW\ PAGE 139 n Â‘ ,Q $SSHQGL[ $ ZH VKRZHG WKDW WKH FKDUJH WUDSSHG DW WKH JUDLQ ERXQGDU\ FDQ EH H[SUHVVHG LQ WHUPV RI WKH TXDVL)HUPL OHYHO IRU DQ\ JUDLQERXQGDU\ YROWDJH GURS n7KLV UHVXOW ZDV XVHGfÂ¬ LQ &KDSWHU 7KUHH WR DYRLG DfÂ°SUHYLRXV JHQHUDOO\ LQYDOLG DVVXPSWLRQ >%$D@ ZKLFK HVWDEOLVKHV WKDW WKH FKDUJHfÂ¬ WUDSSHG DW WKHfÂ¬ JUDLQ ERXQGDU\ LV LQGHSHQGHQW RI WKH JUDLQERXQGDU\ YROWDJH GURS a Â‘ ,Q $SSHQGL[ ZH SUHVHQWHG WKH IRXQGDWLRQ RI D FKDUJHVKHHW PRGHO >%5@ IRU WKH WKLQILOP VLQJOHFU\VWDO VLOLFRQ 026)(7 n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f LQ WKH WKLQILOP VPDOO JUDLQ SRO\VLOLFRQ 026)(7 fÂ¯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f SRO\VLOLFRQ 62, 026)(7V 7KLV ORZ PRELOLW\ FPA9VHFf GHSHQGV VWURQJO\ RQ WKH JUDLQERXQGDU\ WUDS GHQVLWLHV LW LQFUHDVHV E\ DOPRVW RQH RUGHU RI PDJQLWXGH DV WKH WUDS GHQVLWLHV GHFUHDVH DERXW RQH RUGHU RI PDJQLWXGH PAGE 141 Â« :H FDQQRW H[SODLQ WKLV PRELOLW\ E\ VXUIDFHURXJKQHVV VFDWWHULQJ >68` RU &RXORPE VFDWWHULQJ >68@ ZLWK IL[HG R[LGH RU LQWHUIDFHVWDWH FKDUJH EHFDXVH VPDOOJUDLQ /3&92f SRO\VLOLFRQ UHVLVWRUV >1 .5@ DOVRn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fÂ§ 9 Y Â¯ $33(1',; $ n Y A 7+( &+$5*( 75$33(' $7 7+( *5$,1 %281'$5< ,1 7(506 2) 7+( 48$6,)(50, /(9(/ $FFRUGLQJ WR WKH 6KRFNH\5HDG+DOO 65+f WKHRU\ >6=@ WKH UDWH RI FDSWXUH RI HOHFWURQV IURP WKH FRQGXFWLRQ EDQG WR WKH JUDLQERXQGDU\ VWDWHV LV UF FQ167O If $Of ZKHUH I LV WKH SUREDELOLW\ RI RFFXSDWLRQ RI WKH JUDLQERXQGDU\ VWDWHV F LV D SURSRUWLRQDOLW\ FRQVWDQW DQG Q LV WKH HOHFWURQ FRQFHQWUDWLRQ DW WKH JUDLQ ERXQGDU\ (I ( Q QLH[S $f ,Q $f (SQM DQG (A DUH WKH TXDVL)HUPL OHYH DQG LQWULQVLF OHYH DW WKH MWK JUDLQ ERXQGDU\ 7KH UDWH RI HPLVVLRQ RI HOHFWURQV IURP WKH JUDLQERXQGDU\ VWDWHV WR WKH FRQGXFWLRQ EDQG LV >6=@ UH H167I $f PAGE 143 ZKHUH H LV WKH FRUUHVSRQGLQJ SURSRUWLRQDOLW\ FRQVWDQW ZKLFK LV UHODWHG WR F E\ WKH XVH RI WKH FRQYHQWLRQDO 65+ WKHRU\ HWH H FQLH[S r\If fÂµ $f 6LQFH WKH LQYHUVLRQ OD\HU LV YRLG RI KROHV WKH UDWH RI FDSWXUH DQG HPLVVLRQ RI HOHFWURQV PXVW EH HTXDO U fÂµr UJf LQ VWHDG\ VWDWH 7KH FRPELQDWLRQ RI WKLV FRQGLWLRQ DQG $f$f JLYHV I r $f H[SfÂ§cAQMf )LQDO O\ WKLV UHVXOW DQG n r*%M O167I \LHOG WR f Â‘ PAGE 144 $33(1',; % 7+( )281'$7,21 2) $ &+$5*(6+((7 02'(/ )25 7+( 7+,1),/0 026)(7 ,Q WKLV DSSHQGL[ ZH SUHVHQW WKH IRXQGDWLRQ RI D FKDUJHVKHHW PRGHO IRU WKH QFKDQQHO LQYHUVLRQPRGH WKLQILOP VLQJOHFU\VWDO 026)(7 7KH XWLOLW\ RI WKLV PRGHO IROORZV IURP WKH HOLPLQDWLRQ RI WKH QHHG WR QXPHULFDOO\ HYDOXDWH WKH LQWHJUDO IRU WKH IURQWf FKDQQHO FKDUJH GHQVLW\ QI VI Âž+ ) G_} %Of 6LPLODU WR WKH FKDUJHVKHHW PRGHO >%5@ RI WKH EXON 026)(7 ZH ZULWH WKH FRUUHVSRQGLQJ DSSUR[LPDWLRQ RI QI DV r H ) M H ) QID V VIGD V VI %f ZKHUH )6I LV WKH HOHFWULF ILHOG DW WKH IURQW VXUIDFH GHVFULEHG LQ 6HFWLRQ DQG )6IGD LV WKH HOHFWULF ILHOG DW WKH IURQW VXUIDFH REWDLQHG IURP XVLQJ WKH UHILQHG GHSOHWLRQ DSSUR[LPDWLRQ >.,@ %\ UHILQHG GHSOHWLRQ DSSUR[LPDWLRQ ZH PHDQ WKDW LQ WKH FDOFXODWLRQ RI )VIGD WKH FKDUJH GHQVLW\ LQ WKH ILOP LV DSSUR[LPDWHG E\ QHJOHFWLQJ RQO\ WKH PLQRULW\HOHFWURQ FKDUJH WKH KROH FKDUJH LV DFFRXQWHG IRU PAGE 145 SD T3 QDf f :H ILQG WKDW WKH FRQWULEXWLRQ RI S LQ %Of LV QHFHVVDU\ WR DYRLG ODUJH HUURUV LQ WKH FKDUJHVKHHW PRGHO Q 7KH SURFHGXUH WR FDOFXODWH )VIGD LV GHVFULEHG DV IROORZV )LUVW ZH FDOFXODWH WVI DQG L_!6E IURP WKH ULJRURXV DQDO\VLV LQ 6HFWLRQ 6HFRQG ZH VXEVWLWXWH %fLQWR f WR JHW WKH FRUUHVSRQGLQJ DSSUR[LPDWLRQ RI *AAf fÂµ Â‘ U fÂµ I/1IO&H[S$ L ;@ %f V 7KLUG ZH FRPELQH f DQG %f WR REWDLQ DQ LQWHJUDO HTXDWLRQ ZKLFK GHILQHV )IGD 7 fÂµ fÂ¬ VE D WE , f rVI A) *L_f If*W f VIGD DnUVI Dn VEn 7R LOOXVWUDWH WKH DFFXUDF\ RI WKLV PRGHO ZH FRPSDUH LWV SUHGLFWLRQ 4QID ZLWK WKH H[DFW YDOXH 4QI GHVFULEHG LQ %Of :H SORW LQ )LJ %O WKH UHODWLYH HUURU (5525 AQID9 QI [ b %f TS VI YHUVXV fÂ§eMfÂ§ IRU YDULRXV GRSLQJ OHYHOV DQG WKH VDPH XVHG WR GHULYH WKH SORWV LQ )LJ :H ILQG WKDW OHVV WKDQ b UHPDLQLQJ SDUDPHWHUV WKH PD[LPXP HUURU LV PAGE 146 (5525 bf )LJ %O 7KH UHODWLYH HUURU LQ WKH IURQWf LQYHUVLRQ FKDQQHO FKDUJH GHQVLW\ YHUVXV IURQWf EDQG EHQGLQJ IRU VHYHUDO GRSLQJ GHQVLWLHV 7KH FRUUHVSRQGLQJ RQVHW IRU VWURQJLQYHUVLRQ >n-nTI _fU$ N7T ORJ1IOQf@ IRU HDFK GHQVLW\ LV LQGLFDWHG PAGE 147 L $OWKRXJK WKLV FKDUJHVKHHW PRGHO IRU WKH WKLQILOP 026)(7 DFFXUDWHO\ GHVFULEHV WKH LQYHUVLRQOD\HU DUHDO FKDUJH GHQVLW\ LW PXVW EH IXUWKHU VLPSOLILHG WR KDYH LPPHGLDWH SUDFWLFDO DSSOLFDWLRQ 1HYHUWKHOHVV ZH FRQVLGHU WKDW WKH LGHDV SUHVHQWHG LQ WKLV DSSHQGL[ ZLOO FRQWULEXWH WR WKH GHYHORSPHQW RI D FRPSOHWH DQG SUDFWLFDO PRGHO IRU WKH WKLQILOP 026)(7 n n n n LO PAGE 148 RRRRRRRRU} $33(1',; & )2575$1 &20387(5 352*5$0 72 &$/&8/$7( 7+( &+$5*( '(16,7< ,1 $ 7+,1),/0 60$//*5$,1 32/<6,/,&21 026)(7 rrrr} mLWrrrrnrrnrnr fÂµfÂµÂ‘rrfÂµr fÂµIWrrrrrrrrrrnr rrÂ‘r rfÂµrrrnmfÂµrfÂµrrfÂ¬rrrrrnrrrnrrrrIWrr mLrnrrrnrrnrnr}rr 352*5$0 86(' 72 &$/&8/$7( 7+( $&&838/$7,21 25 7+( ,19(56,21&+$5*( 2616,7< $7 7+( 6285&( ,1 $ 7+,1),/0 60$//*5$,1 32/<6,/,&21 026)(7 )25 *,9(1 *$7( %,$6 )25 ,19(56,21fÂ§022e 86( 7+( )5217 ,17(5)$&(r )25 $&&808/$7,2102&( 86( 7+( $&; ,17(5)$&( $6 7+( 0$,1 ,17(5)$&( (/(&75,& ),6/!f 7+( 5$1*(6 2) 86) $12 86%r ,1 68%5287,1( 7(67r0$< 1((' 72 %( $2-8676' )25 ($&+ 3$57,&8/$5 &21',7,21 0.6 6<67(0 ,6 86(' rrrrrrrrrrrrrrrrrrrrrrr rrÃ¦rÃ¦rÃ¦ ArAÂ‘rrrA rrrrrrrr rrfÂµrrrnÃ¦ r &20021 =&f 4 ( (6 r 51Lr( &%p( &)( 51$r( =/r 502%}( & (7r(726Lf $12 67$}(7$(Lf LQ r9 7r( 9*)rfÂµ (7$D2 (7& 517$2( 5($rf 5172 :5,7(rf 517&r (72r517$r(7$ =fr51$ =fr517' =fV51Lr(;3(72rDf =`517$ =f}51Lre;3r(7$f =fr51Lr51L =fD6;3%r67$f =!r(;3Dr67&f & &$/&8/$7,21 2) 534 +2/( &21&(175$7,21 ,1 0.6 ! &$// &532532rrf =fD53 :5,76rf7r51$&)r&% :5,7(r`=/r502% PAGE 149 Â« =&fD51L53frr =&fDr4r53e6rf = f D =&!D!r/*=f! =fD&(6 Â‘Â‘ =&fD7% c =fD4r51$r766 B =fD&3r&)r51$rf(6rr4r51Lr51Lf =&fD66rr51$f&r4r51Lr51Lf =f}&r&r%f(6rr4r51$f =fD(6rf&r4r51$f =&fD&)(6 =fD= =fDmr4r51$(6 )r=f n :5U7m\nf 5322) rn rfÂµfÂµfÂµ Â¯ Â‘ X =&nf}9*) nÂ‘ Â‘Â‘ fÂµÂ‘Â‘Â‘ YAA I& ),1' 7+( &219(1,(17n5$1*(6 2) 86)n Â12 86% )25 7+( fÂ¯7:2 ',0(16,21$/ & %,6(&7,21 067+&2 &$// 7(6786)/\86)0A86%/A86%0f 9*%r\ =fD9*% & B &$/&8/$7,21 2)86) $12 86% 6< 62/9,1* 7:2 1Â•1/,1($5 (48$7,216r & n 21( 2) 7+(0 ,6 $1 ,17(*5$/ (48$7,21 &$// +,6586)r86%r86)/r86)0r86/r86%0f 86)/D86) (6)r&2)(6fr9*)86)f =&D863 = fD(6) 66% &2%66!r&9*D86%f :5,7(Uf 86)86V(6)e66 43%D4r532r7,17%86DrDr&f 5,'D=/r50r43% :5,7(!f9*)9*r43r5,' ,)5,4/((*2 72 &217,18( *2 72 )50$7&(f )250$7r 7IOfDnefÂ« 1$Pfn(r &23)Pf r6r r &3rfn(&f )250$7r =/ r6r r 0 096(&f Dn(f )250$76 86)n)rr 86%Dr)rfÂ« (6)}r(n (6%n6f )250$7r 9*3n)rr 9*Dn)r r 4)%n6rn ,'Dr(f )250$7 r 532 &rf ( 6 n 2)2f r ( rf )250$7& 172PfDn(rn &(726Lf4n)r 17$Pfn6 9 (7$eLf4r)f (12 68%5287,1( 766786)/r86)0r86/r86%,2 & ),12 7+( &219(1,(17 5$1*(6 2) 86) Â12 86% )520 66))86)f)8f! &$// &(0$;8)0$;rrf :5,7(rf 8)0$; 86)/D8)0$; 86)0r8)0$;( 86/DfÂ° 860D PAGE 150 Â« )250$7 n 86)LWD[ r n(f 5(7851 (12 Â‘ 68%5287,1( +,645 ;<$<0,1<0$;f & &$/&8/$7,21 23 *; PAGE 151 \ \PD[ 5(7851 )250$7 r +,'5 7+(5( ,6 12 62/87,21 )25 ;< ,1 7+,6 ,17(59$/rf 612 F 68%5287,1( n +,65;$<,; f Â UfÂ¬ fÂ« & &$/&8/$7,21 2) ; )25 $ *,9(1 < )520 *5;<,;f 86,1* %,6(&7,21 07+ & ,; 2(),1(2 7+( (4 72(n62/9(' ',0(16,21 f ,7( $r*5$<,;f ,) $2642f *& 72 r*5<,;f ,) (42f *2 72 $}$$6 $2f r$%6 f ;r$r ,) ;/7&f *2 72n e 6($5&+ )25 $ &219(1,(17 ,17(59$/ ; $5$18fÂµffrfÂ¯$f ; *5;<,;f ,) ;2(4f *2 72 ;r[$6;! Â‘ ; $r; ,) ;/7}f *2 72 &217,18( :5,7( f 6723 }; 4 $fD $ $fDV PAGE 152 &20021 =4f ,) &,(4 *2 72 *5I7,177;<&22f=6f 5(7851 & )25 *;r PAGE 153 5(7851 (1' )81&7,21 4198! &20021 = B Â‘ rNf Â‘ fÂµ rfÂµ fÂµ ? XVIr]f fÂ§ (6)=f &19r(;3r8ff6457)8f)863f(6)r(6)ff 5(7851 (12 )81&7,21 7,17%86)86%1f & &$/&8/$7,21 2) 7+( +2/( $&&808/$7,&1/$<(5 &+$5*( '(16,7< $7 7+( 6$&. & ,17()$&6 :( 86( 7+( 75$3(=2,'$/ 58/( 72 ,17(*5$7( 7,17%r fÂµ 51r1 Â‘ nÂ‘fÂµfÂµfÂµÂ‘r Â‘ nÂ‘r r86)86%f51 1IV1 fÂµ Â‘ A Â‘ 5r86 fÂµn fÂµ n -1 7,17r7,1743%5f 5D52 &217,18( 7,17r&7,174386)f*3 86%f fÂ«fr Â‘ nLn < Y 7"?I62,' 5HWXUQ n (12 )81&7,21 43%8f &20021 =f r f 86)r=f (6)rf 43r;3r8ff6457)8f)86)f6)r66)ff 5(7851 fÂµ (12 68%5287,1( &53;$f fÂµ & &$/&8/$7,21 2) 7+( +2/( &21&(175$7,21 532f 7+$7 :28/' 7$,1 ,) 7+( & ),/0 ,6 1(875$/ :J 86( 7+( %,6(&7,21 0(7+22 2,0(16,21 f ,7(r $r&14$f A A ,) $2f *2 72 r&14f ,3 f *2 72 & $r$2$6$2f r$6f ;/r$ r ,) ;/74f *2 72 & 6($5&+ )25 $ &219(1,(17 ,17(59$/ r ;r$5$184ffr$f ;&r&1;f ,) ;(42f *2 72 ;r;$%6;f ;r$r; ,) ;/7f *2 72 &217,18( H PAGE 154 :5,7( f 6723 Â‘ r; $'$ &$f} PAGE 155 Â« ,) ;/7f *2 72 &217,18( :5,7( rf 6723 fÂµ %}; U $fm$ $f} 7mff <6r(0 PAGE 156 F R 5()(5(1&(6 >$.@ $NHUV / $ DQG 6DQFKH] 7KUHVKROG 9ROWDJH 0RGHOV RI 6KRUW 1DUURZ DQG 6PDOO *HRPHWU\ 026)(7nV $ 5HYLHZ 6ROLG 6WDWH (OHFWURQ YRO SS -XO\ 75$DO %DFFDUDQL * 0 ,PSURQWD 5 5LFFR DQG 3 )HULD ,9 &KDUDFWHULVWLFV RI 3RO\FU\VWDOOLQH 6LOLFRQ 5HVLVWRUV 5HY 3K\V $SSO YRO SS 'HF U%$E@ %DFFDUDQL * 5 5LFFR DQG * 6SDGLQL 7UDQVSRUW SURSHUWLHV RI SRO\FU\VWDOOLQH VLOLFRQ ILOPV $SSO 3K\V YRO SS 1RY >%$ %DUWK 3 : 'LHOHFWULF ,VRODWLRQ 7HFKQRORJ\ IRU %LSRODU DQG 026 ,QWHJUDWHG &LUFXLWV 7HFKQLFDO 5HSRUW 1R 6(/-%$ 6WDQIRUG (OHFWURQLFV /DERUDWRULHV 6WDQIRUG 8QLYHUVLW\ 0DUFK >5$@ %DUWK 3 : 3 5 $SWH DQG % $QJHOÂ¯ 7KH 0,6,0)(7 LQ 7KLQ 6HPLFRQGXFWRUV /D\HUV 'HSOHWLRQ$SSUR[LPDWLRQ 0RGHO RI ,9 &KDUDFWHULVWLFV ,((( 7UDQV (OHFWURQ 'HYLFHV YRO (' SS 'HF >5$@ %DXGUDQG + ( +DPDGWR DQG -/ $PDOULF $Q ([SHULPHQWDO DQG 7KHRUHWLFDO 6WXG\ RI 3RO\FU\VWDOOLQH 7KLQ )LOP 7UDQVLVWRUV 6ROLG6WDWH (OHFWURQ YRO SS 'HF >%$@ 5DXPJDUW + 5 & )U\H ) 3KL LSS DQG + /HDP\ 'LHOHFWULF ,VRODWLRQ 8VLQJ 3RURXV 6LOLFRQ 056 6\PSRVLD 3URF YRO SS >%5@ %UHZV 5 $ &KDUJH6KHHW 0RGHO RI WKH 026)(7 6ROLG 6WDWH (OHFWURQ YRO SS )HE 7555O@ 5UHZV 5 3K\VLFV RI WKH 026 WUDQVLVWRU LQ ' .DKQJ (G $SSOLHG 6ROLG 6WDWH 6FLHQFH 6XSSO $ 1HZ PAGE 157 >&@ &REEROG 5 6 & 7KHRU\ DQG $SSOLFDWLRQV RI )LHOG(IIHFW 7UDQVLVWRUV 1HZ PAGE 158 Â«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f YRO SS 1RY .DPLQV 7 , )LHOGHIIHFWV LQ 3RO\FU\VWDOOLQH6LOLFRQ )LOPV 6ROLG6WDWH (OHFWURQ YRO SS -XO\ .DPLQV 7 , DQG 3 0DUFRX[ +\GURJHQDWLRQ RI 7UDQVLVWRUV )DEULFDWHG LQ 3RO\FU\VWDOOLQH6LOLFRQ )LOPV ,((( (OHFWURQ 'HYLFH /HWW YRO ('/ SS $XJXVW Âž@ .LP ' 0 $ 1 .KRQGNHU 6 6 $KPHG DQG 5 5 6KDK 7KHRU\ RI &RQGXFWLRQ LQ 3RO\VLOLFRQ 'ULIW'LIIXVLRQ $SSURDFK LQ &U\VWDOOLQH$QRUSKRXV &U\VWDOOLQH 6HPLFRQGXFWRU 6\VWHP 3DUW , 6PDOO 6LJQDO 7KHRU\ ,((( 7UDQV (OHFWURQ 'HYLFHV YRO (' SS $SULO n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fÂ¯ & PAGE 159 >/$@ >/(5,@ >/()@ >/(9@ >/,D@ 7/ D@ >/,E@ >/ ,E@ 7/8@ &/8@ >0$@ /DP + : $ ) 7DVFK -U DQG 5 ) 3LQL]]RWWR 6LOLFRQ RQLQVXODWRU IRU 9/6, DQG 9+6,& n LQ 9/6, (OHFWURQLFV 0LFURVWUXFWXUH 6FLHQFH 9RO 1HZ PAGE 160 >0$6 0DOKL 6 ' 6 + 6KLFKLMR 6 . %DQHUMHH 5 6XQGDUHVDQ 0 (ODK\ * 3 3ROODFN : ) 5LFKDUGVRQ $ + 6KDK / 5 +LWH 5 + :RPDFN 3 . &KDWWHUMHH DQG + : /DP &KDUDFWHULVWLFV DQG 7KUHH'LPHQVLRQDO ,QWHJUDWLRQ RI 026)(7nV LQ 6PDOO*UDLQ /3&9' 3RO\FU\VWDOOLQH 6LOLFRQ ,((( 7UDQV (OHFWURQ 'HYLFHV YRO (' SS )HE >08@ 0XHOOHU 5 . &XUUHQW )ORZ $FURVV *UDLQ %RXQGDULHV LQ Q7\SH *HUPDQLXP , $SSO 3K\V YRO SS $SULO >1*@ 1J . . * . &HOOHU ( , 3RYLORQLV 5 & )U\H + /HDP\ DQG 6 0 6]H (IIHFWV RI *UDLQ %RXQGDULHV RQ /DVHU &U\VWDOOL]HG 3RO\6L 026)(7nV ,((( (OHFWURQ 'HYLFHV /HWW YRO ('/ SS 'HF ; >1 @ 1LFROOLDQ ( + DQG 5 %UHZV 026 0HWDO 2[LGH 6HPLFRQGXFWRUf 3K\VLFV DQG 7HFKQRORJ\ 1HZ PAGE 161 Â«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a %,2*5$3+,&$/ 6.(7&+ Â‘Â‘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Â« , FHUWLI\ WKDW ,n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n n fÂ« 3URIHVVRU RI 0DWKHPDWLFV 7QLVGLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQGZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW 'HDQ &ROOHJH RI (QJLQHHULQJ 'HDQ *UDGXDWH 6FKRRO EFFECTS OF GRAIN ROIINOARIES IN POLYSILICON-ON-INSULATOR (SOI) MOSFETS By ADELMO ORTIZ-CONDE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 TENGO EL INMENSO PLACER DE DEDICAR ESTA TESIS A MIS PADRES ALICIA CONDE-BRANDT DE ORTIZ ADELMO ORTIZ PIONERO - ACKNOWLEDGMENTS ~ I would like to thank all those who helped me in one way or another to make this work possible. I wish to express my sincere gratitude to my guru, Dr. Jerry G. Fossum, for his invaluable guidance, encouragement and assistance in all phases of this dissertation. It has been my privilege and my pleasure to have been his student. I am thankful to Professors Fredrik A. Lindholm, Dorothea E. Burk, Sheng S. Li, and Arun K. Varma for their participation on ny supervisory committee. I also thank Professor Arnost Neugroschel for his help during the experimental part of this work and Professor Eugene R. Chenette for guiding my steps during the beginning of my graduate work. I want to express my appreciation to Drs. Hon Wai Lam, Ravishankar Sundaresan, Hisashi Shi chi jo, and Sanjay Ranerjee of Texas Instruments, Inc., for technological support. I would especially like to thank my friends Drs. Hyung-Kiu Lim and Ravishankar Sundaresan, former graduate students, for the many insightful discussions. My interaction with them has been a very gratifying learning experience. I would like to also thank my other colleagues and friends, Dr. Franklin Gonzalez, Bruce Rushing, Dr. Hsing- Liang Lu, Victor de la Torre, Tae-Wong Jung, Robert McDonald, Suy-Young Yung, Surya Veeraraghavan, Dr. Jean Andrian, Dr. Ganesh Kousik, Arthur i i i Van Rheenen, Dr. Saeid Tehrani-Ni koo, and Juin-Jei Liou for helpful comments and encouragements. I am grateful to Ms. Carole Roone for her excellent work in editing and typing this dissertation. I cannot in words express my thanks to my former professors at the Universidad Sirm/n Rol iva r, Drs. Pierre Schmidt, Gustavo Roig, Paul Esqueda, and Francisco Garcia, for all they have done in support of my graduate work. - I am infinitely indebted to my parents and family for their incredible support and encouragement throughout my graduate school career. The financial support of The Consejo Nacional de Investigaciones CientÃficas y TecnolÃ³gicas (CONICIT), Naval Research Laboratory (NRL), and the University of Florida Center-of-Excellence Program is gratefully acknowledged. TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS iii ABSTRACT..... vii CHAPTER ONE INTRODUCTION 1 TWO LINEAR-REGION CONDUCTANCE OF LARGE-GRAIN POLYSILICON MOSFETS..11 2.1 Introduction 11 2.2 Linear-Region Conductance in Strong Inversion 14 2.2.1 Intragrain Electron Distribution in Channel 14 .. . 2.2.2 Grain-Boundary Potential Barrier in Channel 16 2.2.3Channel Conductance... 24 2.3 Linear-Region Conductance in Moderate Inversion 31 2.4 The Significance of Grain Boundary Orientation 35 2.5 Experimental Support and Discussion .38 2.5.1 Support for the Strong Inversion Analysis 39 2.5.2 Support for the Moderate Inversion Analysis 45 2.6 Summary 46 THREE CURRENT-VOLTAGE CHARACTERISTICS OF LARGE-GRAIN POLYSILICON MOSFETS i 49 3.1 Introduction 49 3.2 Analysis 51 3.2.1 Formalism 53 3.2.2 Numerical Solution 59 3.3 Experimental Support and Discussion 65 3.4 Summary 67 FOUR ANOMALOUS LEAKAGE CURRENT OF SMALL-GRAIN POLYSILICON MOSFETS 69 4.1 Introduction 69 4.2 Leakage Current Model 73 4.3 Summary 94 v FIVE SUBTHRESHOLO BEHAVIOR OF THIN-FILM SMALL-GRAIN POLYSILICON MOSFETS . ..97 5.1 Introduction 97 5.2 Analysis.... 100 5.2.1 Formalism 102 5.2.2 Numerical Solution... 108 5.3 Experimental Results and Discussion 121 5.4 Summary.... 125 SIX SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 128 6.1 Summary and Conclusions 128 6.2 Recommendations for Further Research 131 APPENDICES A THE CHARGE TRAPPED AT THE GRAIN BOUNDARY IN TERMS OF THE OUASI-FERMI LEVEL 134 B THE FOUNDATION OF A CHARGE-SHEET MODEL FOR THE THIN-FILM MOSFET. 136 C FORTRAN COMPUTER PROGRAM TO CALCULATE THE CHARGE DENSITY IN A THIN-FILM SMALL-GRAIN POLYSILICON MOSFET. 140 REFERENCES 148 BIOGRAPHICAL SKETCH 155 VI Abstract of Dissertation Presented to the Graduate School of the University of Florida inâ€™Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFECTS OF GRAIN BOUNDARIES IN POLYSILICON-ON-INSULATOR (SOI) MOSFETS Ry ADELMO ORTIZ-CONDE August 1985 Chairman: Jerry G. Fossum Major Department: Electrical Engineering This dissertation presents physical models that describe the effects of grain boundaries on the steady-state current-voltage characteristics of large- and small-grain polysilicon (SOI: Si -on-Si 0-?) MOSFETs. These models, which are supported experimentally, reveal that the grain boundaries can control the drain current and hence the electrical properties of the MOSFET. Interpretations of measurements based on single-crystal MOSFET theory can therefore result in erroneous parameter evaluations and misconceptions regarding the underlying physics. The models developed herein enable proper interpretations of measurements and facilitate optimal design of the devices. The models for the large-grain polysilicon SOI MOSFET predict: (a) an effective turn-on characteristic in the linear-region, controlled by the grain boundaries, that occurs beyond the strong-inversion threshold voltage, and henceforth defines the "carrier mobility theshold vi 1 voltage" and the effective field effect carrier mobility; (b) a nearly exponential dependence on the (front) gate voltage, defined by the properties of the grain boundaries, for moderate-inversion conductance, and (c) that a grain boundary near the drain can control the conduction properties for all (weak-to-strong) inversion conditions in all (linear- to-saturation) regions of operation. The models for the small-grain polysilicon SOI MOSFET predict: (a) the anomalous leakage current (OFF state), which is attributed to field emission via grain-boundary traps in the (front) surface depletion region at the drain; (b) that the gate-voltage swing for the subthreshold drain current (ON state) depends strongly on the grainÂ¬ boundary properties and weakly on the charge coupling between the front and back gates; (c) that the effective threshold voltage (ON state) depends strongly on grain-boundary properties and on the charge coupling between the front and back gates; and (d) the device design modifications to control and reduce the leakage current, the gate- voltage swi ng, and the effective threshold voltage. v i i i CHAPTER ONE INTRODUCTION Because of. the advantages of dielectric isolation and three- dimensional (3-D) integration [6180; LA82; MA851, there is much interest in SOI (silicon-on-insulator) MOSFETs. The advantages of these devices compared with the single-crystal counterpart are [LA82]: (a) increased circuit speed due to reduced parasitic capacitance; (b) superior hardness to transient radiation; and (c) elimination of'latch-up, which is of fundamental importance when the feature sizes in CMOS (complementary metal-oxide-semiconductor) technology are scaled to smaller dimensions. Today, CMOS is the dominant technology for VLSI (very large scale integration) because of low power consumption, superior noise margins, better compatibility with analog circuits, and reduced vulnerability to soft errors. The most promising SOI technologies for VLSI are: SOI formed by high-dose ion implantation [HE84], SOI using porous silicon [BA84], silicon-on-sapphire (SOS) [SA84], beam recrystallization of polysilicon- on-silicon dioxide [LA80], and as-deposited LPCVD (low-pressure chemical vapor deposition) polysilicon-on-silicon dioxide [MA85]. The first two of these technologies yield dielectrically isolated single-crystal silicon, but they have disadvantages. The SOI formed by high-dose ion implantation technology requires excessive capital costs for equipment, and the SOI formed by the porous silicon technology is not compatible 1 2 with the subsequent process [LA82]. The SOS technology also produces dielectrically isolated single-crystal silicon, but it has not been widely accepted because of fundamental material limitations [LA82] thatâ€™ impede the realization of high-quality silicon-on-sapphire. The beam- recry stall izati on SOI technology, which yields large-grain polysilicon (> 1 urn), is of practical interest because of the relatively good performance of the devices fabricated with it compared with that of the single-crystal counterpart [LA80; TS81; 0083,84], The as-deposited LPCVD SOI technology, which produces small-grain po.lysilicon (< 0.1 urn), is also of practical interest because of the circuit applications [MA84,85] that do not require stringent performance of all the devices, e.g., CMOS memories, and because of the simplicity of the fabrication. Recause of the practicality of the last two technologies, it is of primary importance to model the large- and small-grain polysilicon SOI MOSFETs. Most of the previous research and development of SOI has emphasized either the technology, i.e., the recrystallization process [LA80; LE81; Nfiftl; TSA81; NI83] or the grain-boundary passivation [KA80; SH84; MA85], or the recrystallized silicon [GE82; MA82; SC83], i.e., its characteristic defects and grain boundaries. Little work [KA72; 0E80,82; LEV82; C0V82,83,84] has been done on the characterization and modeling of devices in large- and small-grain polysilicon SOI films, which is essential if SOI integrated circuits are to be optimally developed. 3 Such characterization of the large- and small-grain polysilicon SOI MOSFET must include a description of the charge coupling between the front and back gates [L183b,84a,84b], and must account for the influence of grain boundaries on the electrical characteristics, which is the subject of this dissertation. The inversion-mode large-grain polysilicon SOI MOSFET, illustrated in Fig. 1.1, presents a relatively good performance compared with that of the single-crystal counterpart [LA80; TS81; C083, 84], but it has the disadvantage of requiring the additional recrystal 1ization step. The accumulation-mode small-grain (as deposited) polysilicon SOI MOSFET, shown in Fig. 1.2, does not require the recrystallization step, but it is inferior to the single-crystal counterpart, especially because of anomalous high leakage current and exceptionally high gate-voltage swing [0N82; SH84]. To improve the performance of the small-grain SOI MOSFET, for applications that do not require single-crystal silicon device character!'sties (e.g. load elements for a dense static RAM), grainÂ¬ boundary passivation (e.g., via hydrogenation [KA80; SH84]) has been successfully used [MA84; MA85]. Unlike the large-grain polysilicon device, the small-grain polysilicon device can be designed to be operated in either the accumulation- or inversion-mode because the film body (grains) is completely depleted of free carriers, facilitated by grain-boundary trapping. The purpose of this dissertation is to develop physical models for the effects of grain boundaries in large- and small-grain polysilicon SOI MOSFETs, which are useful for the prediction and optimization of 1.1 Cross-section of the four terminal n-channel inversion-mode beam-recrystallized (large-grain) polysilicon SOI MOSFET. The terminal voltages are .referenced to the source voltaqe (Vs = 0). 5 Pig. 1.2 Basic structure of the four-terminal p-channel accumulationÂ¬ mode LPCVD (smal1-grain) polysilicon SOI MOSFET. 6 device performance in SOI integrated circuits. Chapters Two and Three concern the large-grain polysilicon device, and Chapters Four and Five concern the small-grain polysilicon device. The major contributions made in this dissertation are: (1) the modeling of the effects of grain boundaries for all regions of operation in large-grain polysilicon SOI MOSFETs; (2) the physical characterization of the anomalous leakage current (OFF state) in small-grain polysilicon SOI MOSFETs; (3) the numerical modeling of the subthreshold drain current and the threshold voltage (ON state) of thin-film small-grain polysilicon SOI MOSFETs; (4) the development of the foundation of a charge-sheet model [BR78,81] for the thin-film single-crystal SOI MOSFET; (5) the experimental support for the developed models from measurements of representative SOI MOSFETs. We derive in Chapter Two a theoretical description of the linear- region drain current of the large-grain polysilicon SOI MOSFET, which is valid for all inversion levels and accounts for arbitrary orientation of the grain boundaries. The corresponding channel conductance shows an effective turn-on characteristic controlled by the grain boundaries that occurs beyond the strong-inversion threshold. Henceforth the carrier mohility threshold voltage, which exceeds the actual one, and the effective carrier mobility, which is typically higher than the actual (intragrain) one, are defined. For sufficiently high gate voltage, the grain-boundary potential barrier is low enough that the channel 7 conductance is not significantly influenced by the boundaries. For gate voltages lower than the carrier mobility threshold voltage, the conductance varies nearly exponentially with the gate, voltage and depends strongly on the grain-boundary properties. Grain boundaries perpendicular to the carrier flow in the channel produce the strongest effects on the conductance. To support this analysis and to stress its practicality, we compare model- predictions with measured current- voltage-temperature characteristics of 1aser-recrystal 1 ized SOI MOSFETs fabricated at Texas Instruments [LA83]. The theoretical-experimental agreement is good, and in addition to indicating properties of the grain boundaries in these devices, it exemplifies how the mobility threshold voltage and the effective carrier mobility can be easily misinterpreted as the actual threshold voltage and mobility when conventional MOSFET theory is used as the basis for interpreting electrical measurements of SOI MOSFETs. Such misinterpretations can obscure essential criteria for achieving optimal designs of SOI devices and integrated circutis. For example, our physical analysis reveals that in particular cases grain boundaries can actually benefit the SOI MOSFET performance by producing an unusually high transconductance. This suggests, in contrast to the general belief, that optimal designs may not require elimination of all grain boundaries. We describe in Chapter Three a physical model for the current- voltage characteristics of the large-grain polysilicon SOI MOSFET in all regions of operation. The essence of this model is an accounting for sizable, position-dependent voltage drops across the grain boundaries 8 that can occur when the device is driven out of the linear region. The carrier transport through the grain boundaries (viz., over potential barriers created by carrier trapping) is then nonlinear, and the channet conduction depends on how the grain boundaries are distributed between the source and the drain. Although our model accounts for any number of grain boundaries in the channel, we apply it herein to the most likely case (in beam-recrystallized VLSI) of an SOI MOSFET with only one grain boundary. We emphasize the importance of the position of the grain boundary, as well as its electrical properties, in defining the current- voltage characteristics. Model calculations, supported by limited experimental results, show that grain boundaries tend to decrease the drain current of large-grain polysilicon SOI MOSFETs, but can increase the transconductance. We develop in Chapter Four a physical model for the anomalous leakage current (OFF state) in small-grain polysilicon SOI MOSFETs based on field emission via grain-boundary traps. To support this model, we compare its predictions with measured data from p-channel accumulationÂ¬ mode and n-channel inversion-mode devices [SU84; SH85], Good correlation is shown, and field emission at the back surface is suggested as the mechanism underlying the minimization of the leakage current at relatively low values of front gate voltage. Insight regarding the physics underlying the anomalously strong drain and gate voltage dependences is readily provided by the model, and implies design criteria to control the leakage current in small-grain polysilicon MOSFETs. 9- We derive in Chapter Five a theoretical description of the subthreshold drain current and the threshold voltage (ON state) in the thin-film small-grain polysilicon SOI MOSFET, which reveals the physical' influence of grain boundaries in the channel, and the charge coupling between the front and back gates. The main results of this model, supported by experimental results, are: the gate-voltage swing depends strongly on grain-boundary properties and weakly on the charge-coupling effects; the threshold voltage depends strongly on grain-boundary properties and charge-coupling effects; the charge-coupling effects decrease as the trap density, the thickness of the film, or the doping concentration increases. . .. 4 ^ ^ ^ ^^ ^ * We summarize in Chapter Six the main conclusions and accomplishments of this dissertation. We also suggest in this chapter further related research. We show in Appendix A that the electron charge trapped at a grain boundary (in an n channel) can be expressed in terms of the electron quasi-Fermi level for any grain-boundary voltage drop. This result, which was used in Chapter Three, indicates that a previous assumption [RA78a], which establishes that the charge trapped at the grain boundary is independent of the grain-boundary voltage drop, is generally invalid. In Chapter Three, we have also avoided the use of another classical, but generally invalid assumption [MU61] that a (constant) fraction of the thermionically emitted electrons are captured by the grain-boundary traps. 10 As a first step towards the development of a practical model for integrated circuit design with SOI MOSFETs, we present in Appendix B the foundation of a-charge-sheet model [RR78,81] for the thin-film singleÂ¬ crystal silicon MOSFET. In Appendix C we include the computer program used in Chapter Five to calculate the subthreshold drain current and the threshold voltage (ON state) in the thin-film small-grain polysilicon MOSFET. The program is based on a "two-dimensional" bisection method [RU81] that we developed. Although this method is not as fast computationally as Newton-Raphson [RU81], we use it because it avoids the problems of convergence that typically occur when Newton-Raphson [BU81] is applied to complex problems. â– CHAPTER TWO LINEAR-REGION CONDUCTANCE OF LARGE-GRAIN POLYSILICON MOSFETs 2.1 Introduction We derive in this chapter a theoretical description of the linear- region drain current of the large-grain polysilicon SOI MOSFET, which reveals the physical influence of grain boundaries in the channel. The corresponding channel conductance is described in terms of the (front) gate voltage, the device parameters, and the grain and grain-boundary properties. We restrict our analysis to cases in which the polysilicon film is not completely depleted between the front and back surfaces. We initially assume in Section 2.2 strong inversion, and that the grain boundaries in the channel are perpendicular to the carrier flow; but we generalize the analysis in Sections 2.3 and 2.4 by removing these two assumptions respectively. The model comprises the following physics: (a) the quantum- mechanical description [HS79] of the carrier distribution in the inversion layer, which implies an average carrier density and its dependence on the gate voltage that can be modeled based on the classical solution [C070; S7.R11; (b) the two-dimensional potential variation near a grain boundary in the channel, which when approximated by coupled one-dimensional solutions of Poisson's equation defines the 11 grain-houndary barrier height resulting from carrier trapping [BA78a,b]; and (c) the description of the carrier transport through the grain boundary, assumed to be predominantly thermionic emission over the potential barrier [PI79], To obtain closed-form expressions for the channel conductance, which give physical insight and facilitate the development of SOI MOSFET models suitable for computer-aided circuit analysis, simplifying assunptions are made and justified. The resulting strong-inversion channel-conductance model of Section 2.2 shows an effective turn-on characteristic controlled by the grain boundaries that occurs beyond the strong-inversion threshold. Henceforth the carrier mobility threshold voltage, which exceeds the actual one, and the effective carrier mobility, which is typically higher than the actual (intragrain) one, are defined. For sufficiently high gate voltage, the grain-boundary potential barrier is low enough that the channel conductance is not significantly influenced by the boundaries. Thus the intragrain mobility, which can be affected by surface scattering [SU801, controls the conductance at high gate voltages. In Section 2.3 we extend the analysis to account for moderate- and weak-inversion levels. For gate voltages lower than the carrier mobility threshold voltage, we find that the conductance varies nearly exponentially with the gate voltage, and that the gate-voltage swing needed to reduce the conductance by one order of magnitude is strongly dependent on the properties of the grain boundaries. 13 We account in Section 2.4 for arbitrary orientation of the grain boundaries. This analysis is of interest because of the possibility of controlling [MA82; TSA82; N183; SCH3] the predominant grain-boundary orientation in devices fabricated in recrystallized polysilicon. We find that grain boundaries perpendicular to the carrier flow in the channel maximize the grain-boundary effects on the conductance. In contrast, grain boundaries parallel to carrier flow in the channel do not affect the conductance. To support the analysis and to stress its practicality, we compare in Section 2.5 model predictions with measured current-voltage- temperature characteristics of laser-recrystallized SOI . MOSFETs fabricated at Texas Instruments [LA83], The theoretical-experimental agreement is good, and in addition to indicating properties of the grain boundaries in these devices, it exemplifies how the mobility threshold voltage and the effective carrier mobility can be easily misinterpreted as the actual threshold voltage and mobility when conventional MOSFET theory is used as the basis for interpreting electrical measurements of SOI MOSFETs. Such misinterpretations can obscure essential criteria for achieving optimal designs of SOI devices and integrated circuits. For example, our physical analysis reveals that in particular cases grain boundaries can actually benefit the SOI MOSFET performance by producing an unusually high transconductance. This suggests, in contrast to the general belief, that optimal designs may not require elimination of all grain boundaries. 14 2.2 Linear-Region Conductance in Strong Inversion We assume, based on studies [RA78a,b; PI79] of majority-carrier transport through silicon grain boundaries at room temperature, that thermionic emission of carriers over the grain-boundary potential barrier YpQ underlies the predominant influence of the boundary on the channel conductance of SOI MOSFETs, and that 'FRo results from carrier trapping at localized grain-boundary states. The trapping and are characterized by a two-dimensional solution of Poisson's equation in the channel. Before we discuss this solution and the corresponding thermionic-emission current, we mist consider the intragrain carrier distribution in the channel and its dependence on the gate bias, which define Vqq. We refer to the four-terminal n-channel inversion-mode large-grain polysilicon SOI MOSFET illustrated in Fig. 1.1, and we assume that the grain boundaries in the channel are perpendicular to the electron flow. 2.2.1 Intragrain Electron Distribution in Channel Because the inversion layer thickness xn- is very narrow (on the order of the electron de Rroglie wavelength), the true electron distribution n(x) in the channel (away from grain boundaries) must be described quantum-mechanically [HS791. This description follows from a self-consistent solution of the Schrodinger equation and Poisson's equation. The result differs markedly from the classical solution [C070, SZ81J based on Poisson's equation and Maxwel1-Boltzmann statistics: x^ is narrower and n(x) is more uniform [HS791. However the inversion-layer areal charge density, 15 -O = q / n(x)dx n n (2.1) where x = O represents the Si-SiC^ interface, is predicted well by the classical solution. The analyses suggest a simplification in the description of n(x) and its dependence on the (front) gate voltage Vfif. We define an average electron density Ã± over the effective portion of the inversion layer, 0< x< xi(eff)Â» as revea^ed ttle quantum-mechanical solution, but we use the classical solution, ncl(x), to convey the dependence. . We find that, x^(eff) is described well by /i(eff)cl q/ nc (x)dx 0 -0.9 0 cl (2.2) whereO^ (=.0n) is given by (2.1) with n(x) replaced by nc^(x); that is, about 90% of the inversion-layer charge is contained within a region in which n = Ã± and in which virtually all the channel current flows. Then we define Ã± by ^xi (eff) = â€œ0*9 Op â€¢ (2.3) Numerical evaluations of x^eff^ reveal that it is not strongly dependent on VGf, that it decreases with increasing film doping density NA, and that typically it is quite narrow. For example, when = 1016 cmâ€œ^, = 120 A . Corresponding calculations of Ã± defined by (2.3) are plotted versus (V^ - Vyf) in Fig. 2.1 for different values of NA and for an oxide thickness tQp of 600 A ; is the threshold voltage f.LI83b] that corresponds to the 'onset of strong- Inversion" [nc^(0) - ' Na3. As implied by (2.3), Fig. 2.1 shows that Ã± increases with increasing VGp and with increasing NA. We find that and Ã± as defined by (2.2) and (2.3) in terms of the classical solution [C070; SZ81] are generally consistent with the actual electron distribution given by the quantum-mechanical solution [HS79]. 2.2.2 Grain-Roundary Potential Barrier in Channel For strong-inversion .conditions. within the grains, -electron trapping at localized grain-boundary states produces potential barriers that affect the electron transport along the channel. The barrier formation is similar to that, at grain boundaries in bulk polysilicon [RA78b; F082} except^in the channel is influenced by as described by the two-dimensional form of Poisson's equation. We consider the potential variation near a grain boundary in the channel as shown in Fig. 2.2. We assume that away from the grain boundary (y > yd) the electric field is vertical (in the x-direction), and n(x) is wel1_approximated _by Ã± .over the effective inversion layer, 0 < x < x-Â¡(eff), as discussed in the preceding subsection. In this region (I), Poisson's equation simplifies to = r- (2.4) ro 17 Ã© Fig. 2.1 Calculated average electron density in channel versus (front) gate voltage for several film doping densities. * * Fig. 2.2 Cross-section of effective inversion layer showing typical grain and grain boundaries, which are assumed to be perpendicular to the electron flow. 19 where Y is the electrostatic potential. In the vicinity of the grain boundary (y < yd), a horizontal (y-direction) component of the electric field is produced by the electrons trapped at the grain boundary. We assime that the trapping nearly depletes this region (II) of free electrons; hence 9a We note that this depletion approximation [BA78b; PI793 is valid provided fis sufficiently high: high enough in fact, we assume, that the grain boundaries significantly affect the channel conductance. We discuss the validity of this assumption in Subsection 2,2.3. Assuming that, analogous to the gradual-channel approximation [SZB1], the trapped electrons at the grain boundary typically create only a small perturbation on the x-component of electric field, we can write II d2? 3y2 ii = R- Â£ S (2.5) 9 ^ 9 | 9 ^ , 2 . 2 << rr~ 9 x i 8x In 3 y ii (2.6) where the partial derivatives are evaluated anywhere in the regions indicated. We justify this assumption by noting that the subsequent solution we obtain is consistent with it when (VGf - Vjf) > , which is usually true for strong-inversion conditions. Hence (2.6) implies an approximate solution to the two-dimensional problem defined by (2.4) and (2.5), which is obtained by coupling two one-dimensional solutions. 20 The corresponding approximation forTg denves ^Tom the combination of (2.4)-(2.6), which yields a2*? Â¡7 II -7" s (2.7) with the boundary conditions * (x,y = yd) = T j.(x) (2.8) and ay_ 3y y=y. (2.9) In (2.8) Y j(x) is the intragrain (region I) potential variation in the channel, which is given by the one-dimensional solution of Poisson's equation and the Schrodinger equation [HS79]. We now identify yd as the grain-boundary depletion-region width, and we note that our analysis applies only when the grains are not completely depleted. The solution to (2.7)-(2.9) is *(x,y) 11 ~ lf~ ^y"yd^ + Y I^x) (2.in) and hence TRo -Tj(x) -f(x,0) qny; 11 (2.11) 21 To complete the description of Tg0, we must express yd in terms of known parameters. This expression is implied by the conservation of charge in the vicinity of the grain boundary: Â°RR = " 2qf5yd 5 (2.12) which equates the areal density of charge trapped at the grain boundary, 0qR, to the electron charge density removed to form the (two) adjacent depletion regions. In writing (2.12) we have implicitly assumed that the electrons are trapped within x.Â¡(eyy), which is commensurate with our previous assumptions. The trapped charge density depends on the distribution in the energy gap of localized grain-boundary states (acceptor-type since < 0) [PI79]. It is reasonable to approximate this distribution by a delta function [8A78b; P179; F082; LU81]', yielding nst states (traps) per unit area at an energy level Ey. Then -qN ST GR 1 + j exp *-ET â€œ EFjy=0 kT (2.13) where Ep is the Fermi level and the factor of 1/2 reflects the (spin) degeneracy of the localized states. The position of Ep relative to Ey is defined by ^ Bo and the electron density in region I, i.e., Ã±: [ET ~ EF^y=0 ~ [ET " Ei] + ^Bo ' kq ln(nâ€œ^ (2.14) 22 where is the intrinsic Fermi level (virtually at midgap) and n.Â¡ is the intrinsic carrier density in silicon. Thus (2.11)-(2.14) implicitly describe ^Bo i" terms of the grain- boundary parameters N$y and (Ey - E^), and of Ã±, which depends on VGy and the MOSFET properties as described in Subsection 2.2.1. Numerical calculations of are plotted versus (Vgy - Vyy) in Fig. 2.3 for = 10â„¢ cm"^, tQf = 600 A , two representative values of NSy, '10^ and 10^ cm"^, and three positions of Ey in the energy gap. In all cases, for Vgy sufficiently high, ^ r0 decreases with increasing Vgy. This can be explained by noting that under these conditions virtually all the grain-boundary states (within are filled, and hence 0fiR = ~qNgy is independent of Vgy. Therefore since Ã± increases with Vgy (see Fig. 2.1), y^ concomitantly decreases as described by (2.12), which implies through (2.11) that TRo also decreases: 8Â£ s n (2.15) However when Vgy is low,'FRo is nearly insensitive to Vgy. This is because the grain-boundary states are not completely filled, and hence Ep is near Ey, which virtually fixes Tgo as described by (2.14). We note in Fig. 2.3 that for NSy = 1011 cnT^, TRq is less than 10 mV when (Vgy - Vyy) exceeds about 0.1 V. Thus although our depletion approximation is invalid for these conditions, we surmise that TRo is low enough that the grain boundaries do not significantly affect the channel conductance. However for NRy = 10^ cm-^, TRo is high enough, 23 Ã© Fig. 2.3 Calculated grain boundary potential barrier versus (front) gate voltage for two representative grain-boundary trap densities and three energy levels. Wo note that the low values of f calculated for MqT = 10^ cmâ€œ^ are probably ta i uu i atcu i ui inaccurate because of the invalidity of the depletion approximation (2.5). Neverthel ess the curve is useful because it indicates when Vg0 is low enough that the grain- boundary effect on channel conduction is insignificant. 24 even when (Vg^ - Vy^) is relatively large, to validate the depletion approximation and to strongly influence the channel conductance as we describe in the next section. 2.2.3 Channel Conductance The physical basis for the influence of grain boundaries on the channel conductance is the interaction between electrons flowing from source to drain and the potential barriers at the boundaries. Although quantum-mechanical tunneling of electrons through the barrier may be significant at low temperatures [LU81] and diffusion of electrons is important when the barrier is low [C082, 83, 84], we assume (at room temperature) that thermionic emission of electrons over the barrier is the predominant grain boundary transport mechanism [PI791. Then if the drain voltage Vq is low enough (linear region) that the voltage drop across a grain boundary Vgh is much smaller than 2kT/q, and if TRo > kT/q, the emitted current density is [8A78b] qA*T kNc exp(- -<* Ro kT â€¢) Ã± V gb (2.16) where A* is the effective Richardson constant [SZ81] for electrons {- 250 A/cm2/K^) and Ng is the effective density of states in the conduction band (- 2.9 x 10^g cm-^ at 300Â° K). Since the current in the channel is continuous from source to drain, the drain current ID can be expressed by the integral of (2.16) over the (effective) cross-sectional area of the channel: 25 V ' 1 xi(eff) Jgb (2.17) where Z is the channel width. The combination of (2.16) and (2.17) gives ID as a function of Vgb* To obtain Iq as a function of V^, we simply equate the sum of the voltage drops along the channel to Vn. If we assume that the channel comprises Ng grains of equal length yg separated by (Ng - 1) identical grain boundaries (see Fig. 2.2), then + N V g g (2.18) where Vg is the voltage drop across a grain, which assuming that the carrier transport in the grain is by drift [SZ81] is yq ' 2yd Zi 910 | ng 1 n 1 (2.19) in the linear region. In (2.19) vng is the intragrain electron mobility, the dependence of which on Vq^ and on device parameters can be given empirically [SU80]. Combining (2.3) and (2.16)-(2.19), we obtain In(Vgf,VD) for the SOI MOSFET in the linear region (Vn < (Ng - 1) 2kT/q). If we assume that yg >> yd, which is valid in typical recrystallized SOI MOSFETs, then our result simplifies to In = rÂ«ng!Â°n!Vn 1 + 0.9LA*T expf- kT (2.20) 26 where L = Ngyg is the channel length. In (2.20) Qn is given by the strong-inversion condition Gf (2.21) where is the (front) gate oxide capacitance. The influence of the grain boundaries on Ip is reflected by the second term in the denominator of (2.20), which depends of through rÃ±(Vgf - V-j.^)] as described in Subsections 2.2.1 (Fig. 2.1) and 2.2.2 (Fig. 2.3). If the number of grains Ng constituting the channel is one, vis-a-vis, if there are no grain boundaries in the channel, then (2.20) reduces to the corresponding result of conventional MOSFET theory rsZBll. Furthermore i f gQ is sufficiently low, because of low N^j and/or high Vgf (see Fig. 2.3), then the same result obtains. We note that (2.20), which because of the model assumptions is strictly valid only when V Ro > kT/q, will correctly give the conventional current at high only if the pre-exponential coefficient is much less than unity. With this insight then, (2.20) facilitates a self-consistency check for our model assumptions (2.5) and (2.16). We find that when the grain boundaries are influential, Tis generally high enough that the assumptions are valid. In deriving (2.20) we have neglected thermionic field emission (tunneling) through Â¥ R , and we have ignored the possible existence of a significant grain-boundary scattering potential barrier [LU81] through 27 which the electrons must tunnel to traverse the boundary. The tunneling can be predominant at low temperatures, but at room temperature and above it is generally insignificant [PI79; LU811. We also neglected diffusion of electrons through YRo, which is important only when YRq is low [C083]. When YRo is high enough that the grain boundaries significantly affect IR, the diffusion can be ignored. â€¢ X To illustrate the grain-boundary effects described by (2.20), we plot in Figs. 2.4 and. 2.5 calculations of the linear-region channel conductance (gA ID/V^) versus (VRy - Vyy) for several values of Ng and N$y. In Fig. 2.4 we let Ng vary from one to 200 grains, and we use typical' values for the remaining -parameters: N$y- 10-? cm'^.at Fy = E.j; = 10*6 cm-3, tQy = 600 A , Z = L = 40 urn; we also specify a (front) fixed oxide charge density Oyy = q(10^ cm-^), which defines u ng and its dependence on Vgy [SU80], We see that as Ng increases, g decreases and the plots become inflected, in general accord with recent measurements of laser-annealed SOI MOSFETs [LE81; C084]. The plots show apparent threshold voltages that are higher than Vyy and transconductances (gm A 3 V^ag/av^y) that imply effective electron mobilities (via the conventional M0SFET theory [SZ81]) which can differ fromy^. The apparent threshold voltage is actually a "carrier mobility threshold voltage" (Vy ) at which Ã± becomes high enough thatYbegins to diminish with increasing VRf (see Fig. 2.3) as described by (2.15). For >> , YRq is too low to significantly affect IR; that is, theYRQ term in (2.20) is negligible, and gm is defined by y . Note however in Fig. 2.4 that the plots for 28 Fig. 2.4 Calculated linear-region channel conductance versus (front) gate voltage for several numbers of grains constituting the channel. The broken portions of the curves for N =20, 100, and 200 are inaccurate because of the invalidity ofy(2.20) as discussed in Subsection 2.2.3. The N = 1 curve is inaccurate for near Vjf because of the yinvalidity of the strong-inversion relationship (2.21). g (u) 29 Fig. 2.5 Calculated linear-region channel conductance versus (front) gate voltage for several grain-houndary trap densities. 30 Ng very large become erroneous when Vgy >> Vp because, as we discussed previously, the pre-exponential coefficient in (2.20) is not much less than unity. When is typically higher than that corresponding toung. We stress that the high effective electron mobility implied by gm is defined predominantly by the properties of the grain boundaries. Measured In(VGfÂ»VD^ characteristics of SOI MOSFETs can thus be misleading because of the nonlinear effects, of grain boundaries as we discuss in the next section. Additional calculations reveal that depends on and tof; it decreases with increasing NA and it increases with increasing t0y. These dependences reflect, for a given (V^ - Vyf), the dependences of Ã±, which controls T R , on shown in Fig. 2.1 and on tQy implied -Â°n â€œ Cof The plots of g versus - Vyy) in Fig. 2.5 for N^y ranging from 10-*--*- to 2 x 10^ cm"^ were calculated from (2.20) for the same device parameter values used to derive the plots in Fig. 2.4. We let Ng = 2 (one grain boundary) to simplify the physical interpretation of the results. The same type of inflection seen in Fig. 2.4 is noted in Fig. 2.5 for N^j > 1011 cm"-. For the device considered, if N^y is much Ip p lower than lO111 cm" , the grain boundary is virtually ineffective; whereas if Nst is hi gher than 10^ cmâ€œ^, the grain boundary severely affects (lowers) the channel conductance. Similar calculations have been made for different values of Ey. In this case of the n-channel MOSFET, we find that as Ey approaches the conduction-band edge, TR diminishes and the grain boundary becomes ineffective. As Ey 31 0 moves toward midgap and below, the grain-boundary effect materializes as indicated in Fig. 2.5. This dependence on Ey reflects the electron occupancy of the grain-boundary states, which has been described elsewhere [F082]. These results for a monoenergetic trap density could be used to infer corresponding results for different trap distributions in the energy gap. 2.3 Linear-Region Conductance in Moderate Inversion To extend the analysis described in Section 2.2 to the moderate inversion region of operation, we must remove the strong-inversion approximation.' (2.21). If the polysilicon .film is not completely depleted between the front, and back surfaces, we can neglect the charge- coupling effects [BA80; L183b]. Then for all inversion conditions [TSIR21, 0 (2,22) where (2.23) is the (areal) charge density in the silicon and (2.24) 32 is the depletion-region charge density. In (2.23) and (2.24)^sf is the front band bending and 0p = [2kTÂ£ has been defined to make a compact notation. The relationship between Â¥sy and is defined by VGf ' VFB **sf â– >, . qNsf . C + C ^ sf uof Lof ST (2.25) where vj.R is the front-gate flatband voltage [NI82], which includes a contribution from fast surface states at the Si-SiC^ interface, the density Nsf (cnf^evM of which is assumed to be uniform in the energy gap. The Op(Vfif) dependence in (2.20) is now defined by (2.22)-{2.25). ! To illustrate the grain-boundary effects in moderate inversion described by (2.20) and (2.22)-(2.25), we plot in Fig. 2.6 and 2.7 calculations of the linear-region channel conductance versus VRy for several values of Ey and N$y. To facilitate a later comparison between experimental and theoretical results (see Section 2.5.2), we set Vyf = 0, which defines Vpg through (2.25). In Fig. 2.6 we let (Ej-Ey) vary from 0 to 0.22 eV, and we use typical values for the remaining parameters: = 2xl0^6cm-3, which implies = 80 A, ung = 380 cm3/v-sec, tQy = 600 A , Z = L = 40 um, NSy = lO^cm"^, and Ng = 50. We see that the conductance presents a nearly exponential dependence on for the 1 ower-VRy( <\^ ) region, and that the corresponding gate- voltage swing S needed to reduce Ig by one order-of-magnitude increases q^Ro as Ey increases. This dependence is due primarily to the exp(-^ ) term in (2.20) which is dominant (>> 1). # Fig. ?.fi Measured (points) and calculated (curves) linear-region (V^ = 50 mV) conductance versus front-gate voltage of an n-channel SOI MOSFET in 1aser-recrystall i zed polysilicon at room temperature. The measurements were made with the hack gate biased at -40 V. The calculations were done for different grain-boundary trap energy levels as indicated and with the fast surface-state density at the front Si-SiO2 interface equal to zero. Note that Vyy = 0 V. 34 Fig. 2.7 Measured (points) and calculated (curves) linear-region (Vq = 50 mV) conductance versus front-gate voltage of an n-channel SOI MOSFET in 1aser-recrystallized polysilicon at room temperature. The measurements were made with the back gate biased at -40 V. The calculations were done for different fast surface-state densities at the front Si -SiOj? interface as indicated and with the grain-boundary trap energy level at 0.2 eV above midgap. 35 The plot's of g versus in Fig. 2.7 for Nsf ranging from 0 to 5 x 1011 cnT^eV"! were calculated for (Ey-E^) = 0.2 eV and the same remaining parameters values used to derive the plots in Fig. 2.6. We see that S is nearly independent of Nsy although g decreases as Nsy increases. We conclude this subsection by stressing that the drain current in the lower-Vgf, or "submobility-threshold" (VGy < Vy) regions of operation, presents a nearly exponential dependence with respect to V^, and that the gate-voltage swing needed to reduce Iq by one order-of- magnitude is strongly dependent on the properties of the grain boundaries,. 2.4 The Significance of Grain Boundary Orientation The studies in Section 2.3 and 2.4 have been based on the assumption that- the grain boundaries are perpendicular to the carrier flow in the channel. In this Section, we generalize the analysis to account for arbitrary orientation of the grain boundaries. This generalization is of interest because of the possibility of controlling [MA82; TSA82; NI83; SC831 the predominant grain-boundary orientation in devices fabricated in recrystallized polysilicon. We consider a (straight) grain boundary arbritrarily oriented in the channel as shown in Fig. 2.8. The drain current can be expressed, to first order, as !nf + *Db (8.26) 36 H L Fig. 2.8 Illustration of arbitrary grai n-bounrlary orientation in channel. 37 where 1^ is the component that flows in the grain-boundary-free portion (Z-Zb) of the channel and 1^ is the component that flows in the portion (Zb) containing the grain boundary. Note in Fig. 2.8 that Z^ is defined by Z and L, and 0 , the angle between the grain boundary and the z-direction. In the linear region (strong inversion), (2.27) The characterization of In(:) depends on a complicated twoÂ¬ dimensional electron transport problem. To derive a crude approximation, we assume that the electron current density Jg^ through the grain boundary (via thermionic emission) is perpendicular to it. Then, analogous to (2.16), (2.28) where n = cos(0 )y - sin(0)z is the unit vector normal to the grain boundary. We further assume that away from the grain boundary the electrons flow predominantly in the y-direction. Then to ensure current continuity from source to drain, we must have Db cos(0 ) xi(eff) n â€¢ J (2.29) 38 Using (2.28) and (2.29) and following the derivation in Section 2.2,we ohtain I Db L y ngCof ^VGf'VTf ^VD kN(f 1 + 0T9ÃœÃ*f exp^_kTâ€œ^ co?(0) The combination of (2.26), (2.27) and (2.30) then describes approximately, for strong-inversion conditions in the linear region, the significance of the grain-boundary orientation illustrated in Fig. 2.8. The cos(9 ) in (2.30), as well as the Zb(0) dependence, convey this significance. If 0 > 0Â°, then Zb < Z and the grain-boundary effect is ameliorated. If 0 = 90Â° (grain boundary parallel to electron flow), then Zh = 0 and the grain boundary does not affect the channel conductance (although it may enhance source-drain leakage current via other mechanisms). qp Bo. (2.30) 2.5 Experimental Support and Discussion To support the analysis in this chapter and to identify critical aspects of it with regard to SOI device and integrated circuit design, we measured linear-region Ip(Vfif,Vn,T) characteristics of four-terminal SOI MOSFETs (n-channel) fabricated at Texas Instruments [LA83]. The polysilicon film is 0.5 ym thick and was 1 aser-recrystallized after being deposited via LPCVD on a 1-urn-thick layer of silicon-dioxide, which had been thermally grown on a silicon substrate. The film was doped by ion implantation of boron that yielded = 2 x 10 cmâ€œJ near the front surface and ~ 10^ cm-^ at the back surface [LA83], 39 . -8 2 The front gate '' is n polysilicon and CQf - 5.8 x 10â€œ F/cm (t.^ = 600 A ). Large devices (Z = L = 40 ym) were selected to preclude small-geometry effects [AK82]. To avoid complications due to the charge coupling between the front and back gates [LI83b], a high negative voltage (- 40 V) was applied to the back gate to ensure accumulation at the back Si-Siinterface and to fix Vyf. The Ip(Vfif) dependence was measured with V^ = 50 mV at three temperatures (24Â° C, 70Â° C, and 100Â° C). 2.5.1 Support for the Strong Inversion Analysis The corresponding channel- conductance characteristics,g(Vfiy,T) of a particular device, which typify the characteristics of identically processed devices, are plotted in Fig. 2.9. The basic shape of these plots is the same as that of the theoretical curves in Figs. 2.4 and 2.5, which implies qualitative support for our analysis. (The experimental curves and Figs. 2.4 - 2.5 should not be compared quantitatively because the parameter values used in the calculations are not necessarily the actual values.) The support for (2.20) is demonstrated by examination of the measured g(Vfif,T) characteristics within particular ranges of vGf For high VGf (>> V ), g is defined by the numerator of (2.20); the grain- boundary effect is negligible. Thus as in the case of conventional MOSFFTs [SZ811, the carrier mobility (vng) follows from the slope of g(VRf), i.e., from gm, and the threshold voltage (Vyf) is given by the 9 (U) C x 10" 5 x 10" 4 x 10- 3 x 10â€œ 2 x IQ- 10- Fig. 2.9 Measured linear-region channel conductance versus (front) gate voltage of n-channel SOI MOSFET in 1 aser-recrystal1ized polysilicon [LA83] at three temperatures. The threshold voltage is fixed by the back-gate voltage [LI83b.l, which was set at -40V to ensure accumulation at the back Si -Si 02 interface. 41 linear extrapolation of the characteristic to the Vgf axis. From Fig. 2.9 we thereby get = 0.10 V and ung - 380 cm^/V-sec at 24Â° C. This value of ung is low, and hence implies excessive scattering at the polysilicon surface, due possibly to high Off [SU80]. The lowyng does not reflect decreased transconductance due to a high surface electric field [SU801, which we observed only at values of Vgf higher than those in Fig. 2.9. These interpretations are supported by the temperature dependence of g in the high-Vgf region. We see in Fig. 2.9 a weak dependence of Vyf on T and a negative temperature coefficient for yng, which are consistent with the g(T) characteristics of conventional silicon MOSFETs [LE81; - . â€¢ We see from Fig. 2.9 that Vyf is considerably less than the electron mobility threshold voltage V^ . Thus there is a significant range of Vgf (Vyf < Vgf < ^ ) in which the grain boundaries suppress In. In this case the Y Ro term in (2.20) is much greater than unity, i.e., Vgb >> Vg, and hence gÂ« exp( -gF Bo/kT). As long as Vgf < , Y Ro is high and does not vary significantly with Vgf (see Fig. 2.3). The positive temperature coefficient for g thus predicted is consistent with the measured conductance plotted in Fig. 2.9 in this region. When Vgf > \^ , Y Ro decreases with increasing Vgf (see Fig. 2.3), and hence g increases. To analytically describe this increase and to estimate V^ , we use the approximate TRo(n) dependence in (2.15) and the strong-inversion relationship (2.21). The combination of (2.15), (2.20), and (2.21) yields a g(Vgf) characteristic that exhibits an inflection point where gm is maximum. The theoretical and experimental 42 plots in Figs. 2.4, 2.5 and 2.9 imply that this maximum is broad. Therefore we approximate the actual characteristic by the linear function (2.31) which is tangent to the actual g(Vgf) curve at' the inflection point. This function then analytically defines V and the effective field- effect electron mobilityyn(eff) due to the grain boundaries. The value of Vfif at the inflection point is defined by equating to zero the .second derivative of (2.20) with respect to V^f, using (2.3), (2.15), and (2.21). We find that at this value, the denominator of (2.20) is two. Thus (2.31) describes the tangent to g(VRf) at the point where theY^ term in the denominator of (2.20) is unity. This tangent yields (for Ng > 1) â€” - - 8k Te C , s of (2.32) 0.9A*TL and (2.33) We note that the weak dependence of x^ef1rj on has been ignored in the derivation of (2.32) and (2.33). Thus Vy in (2.32) is evaluated by assuming a representative value for which depends on as 43 discussed in Subsection 2.2.1. We stress that (2.32) and (2.33), which are based on analytic simplifications of our more general analysis described in Section 2.2, are merely estimates of Vy and un(efy). However they are useful in describing the functional dependences of g and gm on device parameters and temperature, and hence will facilitate SOI MOSFET design and computer-aided SOI circuit analysis. We see from (2.33) that the effective electron mobility is typically higher than vng depending on L, Ng, and T. The measured 9(vfif) characteristics plotted in Fig. 2.9 when interpreted using (2.31) 2 yiel d \i n(eff) - 530 cm /V-sec at 24Â° C, which is considerably higher than ung. The negative temperature coefficient for un(eff) implied. by the data in Fig. 2.9 is consistent with (2.33), which shows that the temperature dependence is defined primarily by that of ung. Using the measured value of un(eff) mentioned above and (2.33), we find that Ng = 50 grains. Since L = 40 urn, this implies a crude estimate of about 1 urn for the average grain size (yg), which is not unreasonable for the laser-recrystall ized polysilicon film [LA83], We note finally that the dependence of un(eff) on L suggested by (2.33) is consistent with measurements [N681] of (effective) electron mobility in laser- recrystall ized MOSFETs having different channel lengths. For a given yg (=L/N with Ng > 1), increases as L is reduced from many times yg toward yg. The electron mobility threshold voltage as described in (2.32) is strongly dependent on N$y and T, as well as on through Vyy [LI83h] and x.j(eff). The inverse dependence of x^eyy^ on NA described in 44 Subsection 2.2.1 implies that the difference between -V --and decreases as increases. The predicted direct dependence on N^j is consistent with observed decreases in the (apparent) threshold voltage of polysilicon MOSFETs resulting from hydrogenation [KA80], which is known to reduce N^j. The inverse dependence of Vy on T suggested by (2.32) is corroborated by the measured g(VQf,T) data plotted in Fig. 2.9. At 24Â° C, the measurements when interpreted using (2.31) imply Vy = 0.55 V, whereas = 0.10 V. The difference between Vy and Vjf, based on (2.32), indicates that = 1 x 10 cm (where the traps are near midgap). ..... We conclude this subsection by .stressing two , significant conclusions drawn from it. First, because (2.31), which is of the same form as the linear-region conductance expression for the conventional MOSFET [SZ81], empirically describes well an appreciable region of the g(vRf) characteristic, for the. SOI MOSFET, V^.. and un(eff) can be easily misinterpreted as V-^ andung. Such misinterpretations, which evidently have been made in some previous work^ can lead to misconceptions regarding SOI and can impede the development of optimal SOI devices and integrated circuits. Second, even though grain boundaries are effective in defining.the channel conductance of SOI MOSFETs, the transconductance can be higher than that of the conventional counterpart; the grain boundaries are actually beneficial in this regard. Thus perhaps optimal designs of SOI MOSFETs may not require complete elimination of grain boundaries. 45 -ftiSsgctSQflport for the Moderate Inversion Analysis In-.subsection 2.5.1 we estimated, for a typical device, that the ' ihreshSldt voTtage defined by the linear extrapolation oâ€˜f the measured â– gl'/gf)â€^wfien the grain boundaries are insignificant (Vgy >> V ) is = 0.1 .V, and that the electron mobility defined by the slope of the extrapolation is u = 380 cm /V-sec. From g(Vgy) that is affected by the. grain boundaries (Vq^ ,> ^), we measured, based on our. model, 12 -2 = 0.5 V, a 10 cm (for Ey assumed to be at midgap), and Nâ€'="-50. 'Note that typically Vjp = + nkT/q with n = 3-5 depending on NA-and CQf [TS82b], Thus our strong-inversion measurements imply Vy^- -â– Â£!, which is consistent with calculations based on (2.22). - (2.25). We^stressi-that the difference between Vy and Vjy can be ignored for the strong^inversion analysis because (Vgf - Vyy) >> kT/q. ,â€˜''f'~_Tle "plot in Fig. 2.6 the g(Vgy) characteristic of a typical device measured,^.room temperature. Note especially the lower-Vgf (< Vp) data,"which show a nearly exponential dependence on vGf â€¢ For comparison we also show in Fig. 2.6 theoretical g(Vgf) curves that were numerically â– derived from (2.20) and (2.22) - (2.25) using the parameter values given above and = 2 x 10^Â® cm-^. We varied Ey and let NSf = 0, which -...â€”from Vyy = -0 and (2.25) implies VpR Â« -1.9 V. The calculated g(Vgy) characteristics also are nearly exponential for low Vgf, even though the inversion level is not weak. (In weak inversion, the conductance of single-crystal MOSFETs is exponentially dependent on the gate voltage because 0n is [TS82b; SW72].) This dependence is due primarily to the exp(cfFg0/kT) term in (2.20) as implied by the strong dependence of S 46 (i .e. , the inverse slope) on Ey. As Ey moves from midgap (= E. ) toward the conduction band, S increases; when (Ey-E..) Â« 0.2 eV, the measured S is modeled well. .Thus the energy level of the grain-boundary traps significantly affects the channel conductance below the electron mobility threshold (V^y < ^ ). We illustrate in Fig. 2.7 the effect of Nsy on the g(Vgy) characteristic. The theoretical curves plotted were derived using the same parameter values for Fig. 2.6 and (Ey-E.j) - 0.2 eV. For each value of Nsy, VpB was calculated from (2.25) using Vyy = 0. Increasing Nsy tends to suppress the conductance for intermediate values (~ V ) of V^y, but does not significantly affect S. By comparing the calculated curves 11 -2 -1 with the measured data, we crudely estimate that Nsf ~ 10 cm eV Measurements at different temperatures (T = 24Â°C, 70Â°C, and 100Â°C) indicate that, for intermediate V^y, both g(Vgy) and S increase with increasing T. As T increases from 24Â°C to 100Â°C, S increases from 0.25 V/decade to 0.34 V/decade and, at Vgy = V = 0.5 V, g increases from 1.3 x 10â€œÂ® u to 4.5 x 10"Â® u . These changes, are consistent with (2.20) in which, for relatively low V^y, the expicf? p0/kT) term defines the predominant dependence on temperature. 2.6 Summary A physical model that describes the effects of grain boundaries on channel conductance in SOI MOSFETs has been developed and supported experimentally. These effects originate when electrons (n-channel MOSFET) are trapped at localized grain-boundary states, thereby creating 47 potential barriers that influence the flow of electrons from source to drain. The electron trapping depends on the degree of inversion in the channel and hence on the gate voltage. For sufficiently high VGfÂ» ^Bo is low enough that the grain boundaries are inconsequential with regard to g and g^. However for lower VGy, the grain boundaries can predominantly control g and gm and can define: (a) an effective turn-on (linear-region) characteri stic that occurs well beyond the strong- inversion threshold as illustrated in Figs. 2.4 and 2.5; and (b) a nearly exponential dependence with gate voltage, as shown in Figs. 2.6 and 2.7, for moderate inversion conditions. The effective turn-on characteristic, described generally by (2.20) and approximated by (2.31), is actually a reflection of the â€œcarrier mobility turn-on", which is controlled by the grain boundaries. It defines the electron mobility threshold voltage Vy, which exceeds Vyy, and the effective electron mobility un(eff), which is typically higher than the actual (intragrain) mobility ung. Evidently measurements of Vp andun(eff) have been previously misinterpreted as determinations of Vyy andung. Subsequent erroneous conclusions regarding SOI can inhibit the development of optimal SOI devices and integrated circuits, which, based on our analysis, possibly need not nor should not be completely void of grain boundaries. For moderate-inversion conditions, the drain current, which is controlled by the grain boundaries, varies nearly exponentially with gate voltage and the gate-voltage swing needed to reduce the drain current by one order-of-magnitude depends strongly on the properties of 48 the grain boundaries, especially the grain-boundary trap level, and on thÃ© properties of the Si-Sii^ interface, i.e., the fast surface-state density. ' ' Grain boundaries perpendicular to the carrier flow in the channel maximizes the grain-boundary effects on the conductance as described by (2.30). In contrast, grain boundaries parallel to carrier flow in the channel does not affect the conductanceâ– (although it may enhance source- drain leakage current via other mechanisms). CHAPTER THREE CURRENT-VOLTAGE CHARACTERISTICS OF LARGE-GRAIN POLYSILICON MOSFETs 3.1 Introduction In this, chapter, we describe extensions of our previous work that yield a physical model for the steady-state current-voltage characteristics of the large-grain polysilicon SOI MOSFETs in all regions of operation. Hie essence of the extensions is an accounting for sizable, position-dependent voltage drops across the grain boundaries that can occur when the device is driven out of the linear region. The carrier transport through the grain boundaries (viz., over potential barriers created by carrier trapping) is then nonlinear, and the channel conduction depends on how the grain boundaries are distributed between the source and the drain. Although our model accounts for any number of grain boundaries in the channel, we apply it herein to the most likely case (in beam-recrystall ized VLSI) of an SOI MOSFET with only one grain boundary. We emphasize the importance of the position of the grain boundary, as well as its electrical properties, in defining the current-voltage characteristics. As in the previous chapter, we assume that thermionic-emission theory adequately describes the carrier transport over the grain- boundary potential barriers. Unlike the previous chapter, the use of 49 50 the thermionic-emission theory is not well established because the applied voltage to the grain boundary is much greater than 2kT/q. The previous analyses [MU61; BA78b] of this problem for polysilicon resistors are based on assumptions which are generally invalid, e.g., that a (constant) fraction of the thermionically emitted electrons are captured by the grain boundary traps [MII61], or that the charge trapped at the grain boundary is independent of the grain-boundary voltage drop [RA78a], The former assumption is invalid because the rate of the band- to-trap recombination process is proportional [SZ81] to the concentration of unoccupied traps and not to the current. The latter assumption is -invalid because the charge trapped at the grain boundary can be expressed (see Appendix A) in terms of the electron quasi-Fermi level, and therefore, it depends on the grain-boundary voltage drop. We avoid the use of these invalid assumptions by using the physically reasonable approximation that the electron quasi-Fermi level is nearly flat on the emitting side of the grain boundary. These potential barriers, which result from trapped inversion-layer charge, decrease with increasing inversion level, and hence are modulated by the gate voltage and vary along the channel when the drain voltage is high. Consequently grain boundaries near the drain, where the inversion level is weakest, are most influential. To properly account for the inversion-level dependence, we necessarily base our analysis on a MOSFET model [RR78, 81] that is applicable for all inversion levels. 50 51 â‚¬ Model calculations, supported by limited experimental results, show that grain boundaries generally tend to decrease the conductance (drain current) of SOI MOSFETs, but can increase the transconductance. Grain boundaries having a trap density comparable to that (~10^ cm"^) estimated for typical high-angle boundaries in beam-recrystallized SOI can, when located near the drain, significantly affect the current- voltage characteristics of the SOI MOSFET in all regions of operation. The grain-boundary effect is enhanced as the channel length is shortened. 3.2 Analysis - ...râ€¢ We refer to the n-channel, enhancement-mode large-grain polysilicon SOI MOSFET illustrated in Fig. 1.1. To emphasize the grain-boundary effects, we assume that the polysilicon film' is not completely depleted between the front and back surfaces so that charge-coupling effects [LI83b] can be ignored (vis-a-vis, the back gate is inconsequential). We initially assume that the (front) channel comprises Ng grains separated by (Ng - 1) identical grain boundaries (surfaces) perpendicular to the carrier (electron) flow. Later we analyze the likely case of a single grain boundary in the channel (Ng = 2), emphasizing the importance of its position. The energy-band diagram at the jth [1 < j < (Ng - 1)] grain boundary, counted from source to drain, is illustrated in Fig. 3.1 for the cases of zero drain voltage (VQ) and of Vq > 0. When Vq = 0, electrons trapped at localized grain-boundary states produce the potential barrier TgQ (at each grain boundary), which 52 Fig. 3.1 Fnerqy-band diagram at jth grain boundary for drain voltage equal to (a) and greater than (b) zero. 53 is determined by the inversion level, vis-a-vis, the (front) gate voltage Vgf, as we described in the previous chapter. When Vq > 0, a voltage Vgbj i s -droppedâ€”across-the jth grain boundary, skewing the energy-band diagram as illustrated. If Vghj is large enough, it produces significant changes in the (areal) density of charge Â°GBj trapped at the grain boundary and in the inversion levels in the adjacent grains. 3.2.1 Formalism From Fig. 3.1, for > 0, gbj -<Â¡>r + <â€¢ J Bj - Y Bj (3.1) 1 r where T R ^ andfp^ are the potential barriers on the left and right sides 1 P of the jth grain boundary, and 3 J potentials in the left and right adjacent grains. The average electron densities in the adjacent inversion layers are -1 n. niexp( qi> j /kT) (3.2) Ã±j = n.exp(qj> J/kT) (3.3) where n^ is the intrinsic carrier density in silicon. The densities in (3.2) and (3.3) are related to the inversion layer (areal) charge densities Qn on the left and right by (2.3). 54 The electron transport is controlled by the gate arid drain voltages through the dependence of Qn on VGf and VD. To characterize this dependence, as well as the intragrain current, we use the charge-sheet node! [BR78, BR81J, which is applicable for all levels of inversion. At an arbitrary (intragrain) point y in the channel, 0n(y) = Qs(y) - 0b(y) (3.4) where c|Fcx-(y) n. 0 -i #o 0s(y) - -0^ S(T - 1 + (^iW^sfÃy) - V(y)]]}1/Â¿ (3.5) A is the charge density in the silicon and ^ sf 1/2 0b(y)â€œ--Â°rfâ€”TT~ "1] - (3.6) is the depletion-region charge density. In (3.5) and (3.6), is the band bending (normal to the front surface), V is the difference between the electron and hole quasi-Fermi potentials [V(0) = 0, V(L) = where L is the channel length], and 0p = [2kTe The band bending is related to VGfr by os(y) - -cof[vfif - VR -4'sf(y)l (3.7) 55 To complete the description of the energy-band diagram in Fig. 3.1, we ensure that charge is conserved in the vicinity of the grain boundary 5ssr -r-2:VrV/:! - -2 electron charge removed to form the adjacent depletion regions. (We assime the regions are virtually depleted of free electrons.) Because the inversion layer is void of holes, the electron capture and emission rates for the grain-boundary traps must be equal in the steady state, and hence 0GRj* can be expressed in terms of the electron quasi-Fermi level Epnj at the jth grain boundary (see Appendix A): -qNsT 1 + j exp(â€” (3.9) In (3.9), (ET-Epnj) depends on Vgbj as suggested by Fig. 3.1. This dependence is, in general, complicated and can be defined only when the electron transport mechanism(s) is specified. Although many theories regarding carrier transport through grain boundaries have been purported (e.g., thermionic emission, diffusion, thermionic field emission), none can be verified unequivocally because of the complex, variable nature of the grain boundaries. Thus to avoid undue model complexity, we assume, as in the previous chapter, that the predominant transport mechanism is thermionic emission over the potential barrier. This simplifying assumption is physically reasonable at and above room temperature where 56 thermionic field emission is not probable, and for substantial (nontrivial) barrier heights, which render diffusion less significant. The thermionic-emission model, which in fact has functional dependences similar to the diffusion model, is further consistent with the depletion approximation, and hence with it yields insightful results commensurate with the uncertain nature of the grain boundaries. Referring to Fig. 3.1, we note that if there is a net left-to-right transport of electrons predominantly by thermionic emission, then Epn can be assimed to be nearly flat on the left side of the grain boundary; is dropped predominantly on the right side where the (net) emitted electrons drift away from the grain boundary. Thus in (3.9), where (Ej-E^) gives generally the position of the traps in the energy gap. We stress that dEpn/dy at the grain boundary is not related to the current because of the assumptions that the carrier transport is described by thermionic emission theory and not by diffusion theory. We have now described, in (3.1) - (3.10), how the energy-band diagram at a grain boundary changes to reflect the voltage drop Vg^-. Ry using physically reasonable approximations, we have avoided the use of a classical, but generally invalid assumption [MU61] that a (constant) fraction of the thermionically emitted electrons are captured by the grain-boundary traps. This commonly used assumption in fact overly defines the grain-boundary transport problem because the rate of 57 theâ€”band-to-trap recombination process is proportional [SZ81] to the concentration of unoccupied traps and not to the current. We have furthermore not used another common assumption [BA78a] that Oggj is independent of Vghj> which is also generally invalid as indicated by {3-.-9) Our model for the steady-state current-voltage characteristics of the large-grain polysilicon MOSFET is completed by: (a) equating the drain, current IQ to the net thermionic-emission current defined by the perturbed energy-band diagram at each grain boundary; (b) equating In to the current defined by the charge-sheet model [BR78] applied to each 9ratofcasddÂ¡(c) summing all the grain-boundary and grain voltage drops to V The net thermionic-emission current density over the potential barrier at the jth grain boundary (Fig. 3.1) is [BA7Rb] 9bj ' "d eXP(' â€œkT ' ' "jeXp(' â€œkT >1 (3-U) where A* is the effective Richardson constant for electrons in silicon 2 2 (= 250 A/cm -K ) and is the effective density of states in the conduction-band (- 2.9 x 10*9 cm-9 at 300Â°K). Thus for all j, â– !0 = Zxi(eff)Jgbj (3.12) where Z is the channel width. We assume x^^ (-100 A) is constant, independent of position and bias; Ã± reflects changes in the local 58 channel conductivity. With (2.3) and (3.1) - (3.10), (3.11) and (3.12) relate In to Vg^j for the (Ng - 1) grain boundaries. Using the charge-sheet model [BR78], we now express Iq as a function of the band bending at the left and right sides of each grain. For the kth (1< k< N ) grain with length ygk (L = yg^ + â€”- + ygNg)- i klu , i i o *D â€œ q 9 Cof I ^ + l$*VGf "VFB'-'^sfk "â€™sfk* " ' - - â– T^l'Ifk)372 - ^s<"<'3,2- (3.13) + (^)1/2Â¿[(^fk)1/2 - (^fk)1/2J where u ng is the electron mobility in the intragrain channel. The combination of (-3.5), (3.7), and (3.13) gives Iq as a function of the p 1 voltage drop (the variation in V) along the channel in the Ip 1 kth grain. Since Vk = Vj^ + vgb(j=k-l) for 2 < k < Ng> and = 0 and p VN = VD> we have related In to Vqk for all the Nq grains. 9 The final relationship needed to define I[)(V[),VQf) is V D (M -D = I j=l (3.14) 59 3.2.2 Numerical Solution The current-voltage characteristic is evaluated numerically by solving simultaneously the nonlinear system of equations described by (2.3) and (3.1) - (3.14), for all j and k. Instead of solving directly this nonlinear system of equations, we obtain the solutions by first assigning values for Iq and Vgy, and then calculating the corresponding value of Vq. The advantage of this method is that we avoid the typical convergence problems of the iterative methods [BU81] for solving nonlinear system of equations because we only solve many nonlinear independent equations with one variable. To illustrate the predictions of our model, we apply it to a typical (but thick) SOI MOSFET for which = 10^ cnf^, C0y = 5.8 x 10"^ F/cm2 (the gate oxide thickness is 600 A), Z = L = 40 pm, and ung = 700 cm2/V-sec. To emphasize the most likely case (in beam- r.ecrystallized SOI VLSI), we let Ng = 2 (one grain boundary). We plot in Fig. 3.2, for N$y = 1012 cm-2, Ey = E.Â¡ (traps at midgap), and ygl = yg2 = L/2 (grain boundary at middle of channel), the calculated In(Vn) characteristics for several values of (Vgy - Vyf); the threshold voltage Vyf is the value of Vgy yielded by (3.5) and (3.7) when twice the Fermi potential of the silicon film body. We note that does not need to be specified because it is related to Vyy through (3.5) and (3.7) evaluated at y=0. For comparison we also plot (dashed curves) correspondÃng characteristies for Ng = 1 (no grain boundary). The grain boundary reduces IG; as in the linear region (see Chapter Two), its effect is most significant at low Vgf. In the saturation region, 60 Ã© Fig. 3.2 Calculated current-voltage characteristics (solid curves) for typical large-grain polysilicon SOI MOSFET with one grain boundary at middle of channel. Without the grain boundary, the dashed curves derive. 61 If)(sat) can be substantially limited, although VD(sat) is virtually unaffected since V(y=L) always equals the drain voltage. The grain-boundary effect is strongly dependent on N<Â¡y. To illustrate this dependence, we plot in Fig. 3.3 the square-root of I0(sat) versus (V^ - Vyy) for^NSy ranging from 0 (no grain boundary) to 2 x 1012 cm"2. As Nst increases, the grain-boundary potential barrier increases, and hence a larger part of vg(say) must be dropped across the boundary to enable Iq(sat) to ^ow trough it. For N^y high, In(sat) reduced considerably even for VGf high. Because the grain-boundary potential barrier increases as the adjacent intragrain inversion level decreases, the effect of the grain boundary will, for Vq > 0, be stronger if the boundary is closer to the drain. To emphasize this important position dependence, we show in Fig. 3.4 how the calculated Iq(sat)(vGf) characteristic is altered as the grain boundary, with NSy = 1.2 x 10*2 cm"2, is shifted toward the drain. Since Qn ~ 0 near the saturated drain, a grain boundary there is influential regardless of how high is. A grain boundary near the source however is significant only for VGy low. We also see in Fig. 3.4 that the position of the grain boundary is irrelevant for (Vfiy - Vyy) < .6 V because the inversion level remains nearly constant along the channel for this low VGy. The grain-boundary effect illustrated in Figs. 3.2 - 3.4 is enhanced as the channel length is shortened. This enhancement is demonstrated in Fig. 3.5 where we plot the calculated In(sat)(^Gf) characteristic for different L with Z/L = 1 and y^ - L/4. The 62 0 Fig. 3.3 Calculated dependence of drain saturation current, versus gate voltage, on grain-boundary (at middle of channel) trap density. 63 0 Pig. 3.4 Calculated dependence of drain saturation current, versus gate voltage, on grain-boundary position along channel. 64 # Fig. 3.5 Calculated dependence of drain saturation current, versus gate voltage, on channel length with grain boundary' L/4 from drain. 65 reduction in current with decreasing channel length results because the constraint on ID defined by (3.11) and (3.12) is independent of L, and hence Vg^ must increase to support higher current densities in the channel. To further stress the significance of grain boundaries in large- grain polysilicon SOI MOSFETs, we plot in Fig. 3.6 the calculated transconductance in the saturation region, gm(sat) ^ 3 *o(sat/3^Gfâ€™ ^or one grain boundary in the middle of the channel having different values of Ngy. Depending on gm(say) can be lower or higher than that for the grain-boundary-free (NSy = 0) counterpart. At low Vq^, below the "mobility threshold" (^ ) the grain boundary virtually inhibits current; thus gm(sat) ~ 0. As Vq^ increases, the grain-boundary effect is diminished as the intragrain channel conductance is enhanced, thereby producing unusually high transconductance (like in the linear region analysis of Chapter Two). At high Vgy, the grain-boundary effect tends to subside, and gm(say) approaches that corresponding to N^y = 0. 3.3 Experimental Support and Discussion To provide experimental support for the analysis, we measured current-voltage characteristics of large-grain polysilicon SOI MOSFETs described in Section 2.5. We find that the measured Ip(say) is smaller than that of the theoretical calculations of the corresponding single-crystal counterpart (Nq = 1), and that this relative difference increases as VGf decreases. This result implies qualitative support for our analysis. 66 * Fig. 3.6 Calculated saturation-region transconductance versus gate voltage and grain-boundary (at middle of channel) trap density. 67 Unfortunately, quantitative support is not obtained because Ng is not known exactly. Additional support for our analyses has been presented by Colinge et al. [C083], who developed a technique to control the location of the grain boundaries in SOI MOSFETs. They fabricated two transistors with the same geometry, one beside the other, one of them with a perpendicular grain boundary at the middle of the channel, and the other without grain boundary. They found that IG ^ for the transistor with a grain boundary is smaller than that of the transistor without a grain boundary. ; We conclude this section by stressing three significant'conclusions drawn from this analysis. First, because the I^Vp) characteristics of SOI MOSFETs resemble that of the single crystal counterpart, the device parameters can be easily misinterpreted by using direct MOSFETs theory. Such misinterpretations, which evidently have been made in some previous work, can lead to misconceptions regarding SOI and can impede the development of optimal SOI devices and integrated circuits. Second, because of the variation in the degree of inversion along the channel produced by Vn, a grain boundary close to drain affects In even at sat high VGf. Third, because part of Vn ^ is dropped across the grain (sat) boundaries, In is reduced, (sat) 3.4 Summa ry Using simplifying, but physically reasonable assumptions, we have modeled the effects of grain boundaries on the steady-state current- 68 voltage characteristics of large-grain polysilicon SOI MOSFETs. We have assumed that the predominant transport mechanism is thermionic emission over the potential barrier,and we have avoided the use of previous generally invalid assumptions [MIJ61; BA78a]. The complexity of the model is commensurate with the uncertain and variable nature of the grain boundaries, but its predictions are in general accord with experimental results. Basically the model shows that grain boundaries tend to reduce the MOSFET conductance (Ip), but can increase or decrease the transconductance. Although the grain-boundary effects are most apparent at low gate voltages, they can be quite significant at higher gate voltages when the drain voltage-is high, e.g.'f in the saturation region. Grain boundaries close to the drain are most effective. The effects are enhanced as the channel length is shortened. We have measured current-voltage characteristics of both [LA83] laser- (Ng >> 1) and graphite-strip-heater- (Ng 2) recrystallized SOI MOSFETs, and have found general agreement with the model predictions. Because Mg is not known exactly, it is difficult to make more quantitative comparisons. A main conclusion of our work is that a single grain boundary in the channel can significantly affect the electrical properties of an SOI MOSFET. Thus although the grain size of beam-recrystallized SOI is large, the grain boundaries, with randomly varying properties, can pose problems regarding yield, reproducibility, and reliability of SOI VLSI that cannot be ignored. 0 CHAPTER FOUR ANOMALOUS LEAKAGE CURRENT OF SMALL-GRAIN POLYSILICON MOSFETs 4.1 Introduction Recent laboratory achievements [MA84; MA85] imply that the first commercial adaptation of three-dimensional integration may be stacked CMOS VLSI memory chips in which one of the complementary transistors (usually p-channeV) is fabricated in a layer of LPCVO polysilicon on silicon dioxide. Grain-boundary passivation (e.g., via hydrogenation [SH84]) is required to render the polysilicon MOSFET performance acceptable for the circuit application, although single-crystal silicon device characteristics are not needed. The polysilicon transistor is inferior to the single-crystal counterpart, especially because of anomalous high leakage current and exceptionally high gate-voltage swing [0N82; SH84]. In this chapter we model the OFF-state leakage current of the small-grain polysilicon SGI MOSFET, which we theorize is controlled by grain-boundary traps. By qualitative deduction, we identify a plausible physical mechanism underlying the leakage current, and then show that it is consistent with the anomalously strong dependences on the gate and drain voltages that have been observed [0N82; SH84; MA85], Such physical insight can aid the design of polysilicon MOSFETs to control and minimize the leakage. 69 70 A typical set of measured current-voltage characteristics of an unpassivated LPCVD polysilicon MOSFET is shown in Fig. 4.1. The particular device is p-channel and operates in the accumulation mode. The back gate and the source are grounded. In the OFF state (front-gate voltage VGf > 0), the film body (grains) is completely depleted of free carriers, facilitated by grain-boundary trapping of holes. The front surface is inverted for VGf sufficiently high (> ~0), facilitated by positive charge at the interface. The leakage current (I|_ = Iq in the OFF region) increases exponentially with Vgf and as a power (> 1) of the drain voltage VR. lhe device characterized in Fig. 4.1 is long-channel (32 un), which means that the anomalous Ij_(vGfÂ»Vq) is not â€™a short- channel effect [AK82]. Other measurements [MA85] reveal that Ij_ is virtually independent of the (long) channel length, and that the same anomalies obtain for other polysilicon MOSFET structures, e.g., the n-channel inversion-mode device. Passivation of the grain boundaries in hydrogen plasma [SH84] reduces 1^ by two or three orders of magnitude, but does not remove the strong dependences on VGf and Vq. To physically model 1^, we first deduce the most plausible mechanism producing the leakage by qualitatively eliminating the possibilities of other significant mechanisms. Although this deduction is not rigorous, we support the model by demonstrating correlation between its (vGf,vd) predictions and measured data. A rigorous corroboration, which would require comprehensive analyses of all the possible mechanisms and extensive measurements of special test structures, is not feasible. Fig. 4.1 Measured current-voltage characteristies of an unpassivated LPCVD polysilicon MOSFET (p-channel, accumulation-mode; Z = 128 un, L = 32 um, t f = 500 A, NA = 1016 cm-3). The polysilicon film is O.lbpm-thick and was deposited via LPCVfl on a 0.5 y m-thick layer of silicon-dioxide. The back gate and the source are grounded. 72 Possibly significant leakage mechanisms in the polysilicon MOSFET are: -(a) space-charge-1 invited flow [R173; SC82] of holes from source to drain through the (depleted) film body; (b) thermal emission of carriers [SZ81], via grain-boundary traps, in the depletion region near the drain; (c) field-enhanced (Poole-Frenkel [GR82]) thermal emission in the drain depletion region; (d) impact ionization (avalanching) [DU78] in the drain depletion region; (e) band-band field emission (tunneling) CR1733 in the drain depletion region; and (f) field emission via grainÂ¬ boundary traps [GR84], or possibly metal precipitates [LEF82], in the drain depletion region. The measured independence of 1^ on (long) channel length [MA85] rules out space-charge-limited flow.' The -strong observed dependences of 1^ on Vgf and VD imply that thermal-emission current, which depends only weakly on V^, is not significant. The observed saturation of 1^ with increasing Vg^ in Fig. 4.1 is inconsistent with predominant Poole-Frenkel emission or avalanching. Furthermore the electric field in the drain depletion region, for the values of Vg^ and VQ used in the measurements (Fig. 4.1), is not high enough to produce significant avalanching or band-band tunneling. We are thus left with field emission through grain-boundary traps, or metal precipitates, as the most plausible mechanism underlying the observed Il(VGf,VD^ * The strong dependence on VGf further implies that the predominant field emission occurs near the front surface, in fact between the p+ drain and the n inversion layer where the electric field is highest. If the hack surface is inverted, significant field emission can occur there also. 73 Our analysis of the field emission emphasizes grain-boundary traps, not metal precipitates, for two main reasons. First, neutron activation analyses [SH85] of the LPCVD polysilicon reveal concentrations (< 10l3 cm"3) of metallic impurities comparable to those in bulk silicon and much lower than the grain-boundary trap density. Second, the grain boundaries getter metallic inpurities, which tends to prevent the formation of metal precipitates. In the next section, we develop an analytic model for the trap- assisted field-emission current in the LPCVD polysilicon MOSFET. To support the model, we compare its predictions of 1^(vRfâ€™VD) W1t:h measured data from p-channel accumulation-mode'and n-channel inversion-- mode devices. Good correlation is shown, and field emission at the back surface is suggested as the mechanism underlying the minimization of the leakage current at relatively low values of VGf like that illustrated in Fig. 4.1. Insight regarding the physics underlying the anomalously strong Ij_(VGf,VD) dependences is readily provided by the model, and implies design criteria to control I|_ in polysilicon MOSFETs. 4.2 Leakage Current Model To develop a physical model for the leakage current in the LPCVD polysilicon MOSFET, we consider the p-channel accumulation-mode device, the basic structure of which is illustrated in Fig. 1.2. The leakage current in other polysilicon devices, e.g., the inversion-mode MOSFET [0N82], can be described basically by the same model as we discuss later. The accumulation-mode device is of interest because it can be 74 designed to have reasonably low threshold voltages [SH84; MA85]. Such design depends on the complete depletion of the small (~ 1000 A) grains in the body via carrier trapping at localized grain-boundary states [8A78a; SH84]. The OFF state of the device then obtains in the absence of an accumulation layer under the gate; the surface is either depleted or inverted. Our model is based on the latter, more prevalent surface condition, and hence describes the anomalous strong dependences of the leakage current on the gate and drain voltages [0N82; SH84]. As discussed in Section 4.1, the leakage current is assumed to originate via field emission through grain-boundary traps (bound states) at the drain junction. Because of the complete depletion of the body, this emission occurs predominantly at the surface under the gate in the depletion region between the p+ drain and n inversion layer. This conclusion is consistent with the observed strong dependence of the leakage current on the gate voltage. Although band-band tunneling in this surface region would occur only at very high electric fields [RI73], substantial field emission can occur at low fields because of the high density of grain boundary traps in the LPCVD polysilicon. To enable an analytic description of this emission, we assume that the traps are monoenergetic (at Ey in the energy gap) and uniformly distributed in space [KA72; DE80; DE84] (with trap density Ny = 2N^y/dQ where N$y is the grain-boundary areal trap density and dG is the average columnar grain size). The critical region of the device described above is illustrated in Fig. 4.2. It is the depletion region near the surface between the p+ 75 Fig. 4.2 Critical depletion region between the drain and the inversion layer with important electric field components in (4.11)- (4.13) indicated. 76 drain and the n inversion layer. Because of the lateral diffusion of the drain under the gate, the inversion extends into the drain region with complicated geometry. Thus the effective cross-sectional area of the n-p+ junction is difficult to define. The electric field in this region, which governs the field emission, is two-dimensional [FR69; TE84] (for wide channels) depending on both the gate and drain voltages, VGf and VQ. Following a previous analysis [FR69] of the bulk MOSFET in the saturation region, we will treat this complex two- dimensional problem empirically, which enables us to model the field emission processes as occurring predominantly parallel to the surface (in the y-direction). This simplification is commensurate with device ambiguities, yet yields a model that is consistent with experimental results and hence is insightful. The energy-band diagram in the region, including the grain-boundary trap level, is sketched (versus y) in Fig. 4.3. Maintaining the degree of complexity implied above, we assume that the electric field in the y-direction, F = (dE^/dyl/q, is constant in the depletion (barrier) region, and that the electrons (or holes) tunnel through the barrier via the traps at constant energy [GA70; R077]. The leakage current derives then from the combined net field emission of holes, IyV, from the traps to the valence band in the drain region, and of electrons, IyC, from the traps to the conduction band in the inversion region. For traps within an incremental width (dy) of the depletion region (see Fig. 4.3) [GA70; R077; GR84], 77 Fig. 4.3 Electron energy-band diagram along the surface between the (n) inversion layer and the (p ) drain. 78 and di TV q(l - fT)NTZxedy **TV (4.1) qfTNTZxedy d('ITC)= t 1 ^ T TC (4.2) the rates of the respective inverse processes are negligibly small for the nonequilibrium conditions of interest (e.g., |VQ| >> kT/q). In (4.1) and (4.2), Z is the channel width of the MOSFET, x0 is an effective depth of the junction region (see Fig. 4.2), fy is the electron occupancy factor of the traps, and tyV and tjq are the time constants for hole and electron tunneling respectively, which depend on Fy and Ey as we discuss later. In the steady state, the number of trapped electrons is constant, and hence in the absence of significant thermal emission, dlyy = d(*"Iyq) . Equating (4.1) and (4.2) then, we have TC tTC + T TV (4.3) The incremetal field-emission current is given by the combination of (4.1) or (4.2) with (4.3). The total (leakage) current is then expressed by integrating the result over that portion of the depletion region across which valence band-trap-conduction band carrier transitions at constant energy are possible (see Fig. 4.3): 79 IL= qZXgNj / dy y(ECn) Ttc +ttv (4.4) Since we assume that Fy is constant in the depletion region, f ~ EVp ~ ECn y " H- gLy (E Vp) - y(ECn)J (4.5) where tunneling time constants are thus independent of y also, and (4.4) can be written as ZxeNT(f TC + T TV -x^ (4.6) Because the leakage current is low, little voltage is dropped across the inversion layer; Vg is dropped across the drain depletion region as indicated in Fig. 4.3: EFp " EFn â€œ qlVnl ' (4*7^ For |Vg| greater than a few tenths of a volt then, we note that the quasi-Fermi level separation in (4.7) equals approximately (EVp - ECn) in (4.6). Thus TC + T TV -)(- IL - qZxeNT(f (4.8) 80 The time constants t yC and x yy reflect the probability per unit time that a trapped carrier will tunnel through a triangular barrier, defined by Ey as shown in Fig. 4.3, to its respective band. Rased on the WKB approximation [R077; SZ81; GR84], 4(2mV/2(ET - E )3/2 tTV s T 0VexP[ 3qhF ^ (4*9) and ec - tTC2 T0CexP[ 3^hF ^ (4.10) 'k 'k where mp and mn are the appropriate [LU72; GR84] effective masses for the tunneling holes and electrons (m = mn = 0.2 mQ [GR84] where miq is the free electron mass), and where XQy.and xqq are effective carrier transit times in the valence and conduction bands, which we assume to be constants [LU72], For a parabolic barrier [R077; SZ81], the numerical constant in the-exponential argument is different (lower). The effect of the barrier shape on Iy can thus be studied by varying the effective masses in (4.9) and (4.10), which in fact cannot be unequivocally defined [LU72; GR84], We stress that the field-emission current IL in (4.8) is predominant because of the high density Ny of traps that increase substantially the tunneling probability, conveyed by (4.9) and (4.10), over that for band-band tunneling [SZ81], To complete the model for IL 81 Ã© defined by (4.8)-(4.10), we must describe Fy in terms of Vgf and Vq. As discussed with reference to Fig. 4.2, the electric field in the depletion region at the surface, and indeed the two-dimensional region itself are virtually impossible to describe analytically. However an empirical description, commensurate with our one-dimensional field- emission model, can be given based on a previous analysis [FR69] of the bulk MOSFET in the saturation region. With reference to Fig. 4.2 three "components" of Fy can be identified [FR69]: (4.11) In the empirical representation (4.11), Fj is the electric field that exists in the absence of the gate; i.e., F^ is due to the depletion charge in the reverse-biased drain junction. The presence of the gate produces fringing electric fields ?2 ancl ^3: 1S due t0 9ate-drain potential drop and F3 is due to the potential difference between- the drain end of the inversion layer and the gate. Following [FR69], which has been supported experimental ly. [BR81], we assume that F2 and F3 are proportional to the respective normal electric fields at the surface (see Fig. 4.2): (A.12) 82 F3 Â» *F* (4.13) where a and g are constant "field-fringing factors". At the p+ drain side of the depletion region, the surface potential is nearly zero, and (4.14) where CQf is the (front) gate oxide capacitance, Off is the fixed charge 4* density at the (front) Si -Si O2 interface, and is the gate-p+ silicon work function difference. For an n+ polysilicon gate, 4* ^MS ~ â€œ^g/q where Eg is the silicon energy gap. At the inversion-layer side of the depletion region, F1 = -2! (V hS s lvGf s *MS + C Qff En ff _ J3L) of < (4.15) since the potential drop between the end of the (strong) inversion layer and the source region (V$ = 0) is about Eg/q. Using the one-dimensional analysis of the reverse-biased p-n junction [SZ81], we express F1 = ^D]1/2 (4.16) s as an average value of the y-dependent electric field in the depletion region between the p+ drain and the n inversion layer. In (4.16), Ã± is defined by (2.3) and (2.21), 83 'of qx i(eff) V1 ) VTf (4.17) where x^yf) is the inversion layer thickness (~ 100 A) defined in i Section 2.2.1 and Vyy is the strong-inversion threshold voltage of the MOS structure, the characterization of which depends not only on the structural properties but also on the polysilicon film properties, e.g., Nst, Ey, and dG [KA72]. The potential barrier height is defined by and the surface potential in the inversion layer. For strong inversion, E *0 â€œ q + >v0l â€¢ M.1Â») The combination of (4.8)-(4.18) gives an analytic description of I|_ and its dependences on VGy and Vp for the p-channel accumulation-mode polysilicon MOSFET. The control of by the grain boundaries is conveyed by Ny (= 2Ngy/dQ) and Ey (relative to the band edges) in the i D+ model. The parameters CQy, Vyy, and 0, and xe must be estimated or evaluated through comparisons of model predictions and measured data; they control the relative significance of Vfiy and in determining 1^. The transit times Tgy and tqC associated with the field emission are in effect normalization factors for the tunneling time constants; they have been characterized from first principles [LU72], although not accurately. The model 84 therefore actually predicts the normalized Â»VD ^ dependences; absolute measured values however could be matched by assigning proper values to t qV andxgg. The model does indeed predict the general 1^(,Vq) dependences exemplified in Fig. 4.1. To demonstrate this correlation and to indicate the physical insight afforded, we plot in Fig. 4.4 the calculated leakage current versus Vgy and Vp for a device similar to the one measured. Quantitative comparisons of the theoretical and experimental characteristics should not be made because actual values of many of the model parameters are unknown. In the calculations, we used Z = 128 um (L is irrelevant) and CQf = 6.9 x 10â€œÂ® F/cm^ (= e0f/t0f with tQX = 500 A ) corresponding to the device measured. We chose xe = 500 A, greater than x.Â¡(eff) but less than the extent of the lateral drain diffusion, and a = 0.2 and 3 = 0.6, crude values based on the bulk' M0SFET model [FR69], We let x gy = x gg = xg = 10"^ sec, a crude estimate derived experimentally [GR84], and Nj = 2 x 10^ cm"^, which corresponds to Ngy = 10^ cnT^ and dG = 1000 A. We put Ey at midgap. We characterized vjy = Vyg - Qff/C0y with Qff/q = 10^ crrf^ and Vyg = 0 i _ i V, which yield Vyy = -2 V. [We stress that Vyy is the strong-inversion threshold voltage, in contrast to the turn-ON threshold voltage (for strong accumulation), which is about -4 V as indicated in Fig. 4.1.] From (4.6) we note that Iy is directly proportional to the factor ZxeNyA g, and hence changes in these parameters simply alter the magnitude of Iy and not shape of the semi-1 ogarithmi c I{_ (vGf * VD ^ characteristics in Fig. 4.4. Additional calculations show also that the 85 Fig. 4.4 P , kage c.rrÂ«* versus *Â«s$ SSlftr U1^â€™ SCCUâ€œ mosfet. 86 shape of I|_(VGf) is not strongly dependent on a and e . Variations in the oxide capacitance CQf, however, produce significant changes in II(vGf)* Primarily because of its influence on Fj described by (4.16) and (4.17). Similar changes are produced by variations in or Vyg as shown by the calculations plotted in Fig. 4.5. The leakage is most sensitive to VGf just above threshold for strong inversion; well above threshold (high V^), Fy in (4.11) is high and the tunneling time constants tjV and tjq in (4.9) and (4.10) tend toward minimum values (tq), thereby causing II to approach a value independent of VGf. The calculated dependence of IL on the trap level Ej is illustrated in Fig. 4.6.- As - can be inferred from (4.9) and (4.10), traps near midgap are most effective in the field-emission processes; shallow traps do not facilitate carrier tunneling to the opposite band. We note that Ej could possibly be inferred from measurements of the slope of the I|_(VGf) characteristic. Although the model predictions can be brought into close agreement with the measured leakage current by altering parameter values, the benefit of doing so is questionable because of the uncertainty in the actual physical structure and parameters of the device. For example, variations in the shape of the potential barrier in the drain depletion region, and/or in the effective masses in (4.9) and (4.10) can result in considerable changes in the predicted ^(Vq^.Vq) characteristics. Such changes are illustrated in Fig. 4.7 where we plot the calculated Il(VGf,Vo) with nip = mn varying from 0.1 mQ to 0.5 mQ. We note however from the calculations plotted in Figs. 4.4-4.7 that the measured 87 # â€¢ _ -Â»*-Â» â– f^f> T ; / W â€¢ _ â€¢ . Fig. 4.5 Calculated variation of (vGf^ fÂ°r different (strong- inversion) threshold voltages V= Vtq - Qff/C0f (with Off/CQf = 2.3 V). 88 Fig. 4.fi Calculated dependence of Ij_(vGf^ on the grain-boundary trap energy level Ey. VG( (V) Fig. 4.7 Calculated dependence^ of IL(Vfif) 0n tunneling carrier effective masses mp = mn* 90 IL(VGf'Â»Vd) characteristics in Fig. 4.1 could be simulated well by the model with physically reasonable parameter values. The plots in Figs. 4.4-4.7, as well as the data in Fig. 4.1 which indeed reflect the general lL(VGf,VD) dependences seen in a variety of LPCVD polysilicon MOSFETs, show that Ip varies predominantly exponentially with (> Vyy) for a given value of Vp. This dependence results from the exponential dependence of Tyy and tjq on Fy expressed in (4.9) and (4.10). The dependence- of Ip on Vg is also strong, as emphasized by the calculations of Fig. 4.4 replotted in Fig. 4.8. For a given value of Vgy (> Vy), Ip varies as |Vyj|m where m ~ 1-10. This dependence follows from (4.8) and the implicit dependences on Vg of Fy and tyg and tyy. Note that m decreases with increasing Vgy and )Vq|. The predicted m vs. Vgy variation can be seen in the measured data in Fig. 4.1, although the m vs. |Vg| variation cannot. This discrepancy appears to be due to (trap-assisted) avalanching in the drain depletion region that causes the measured leakage current to increase abruptly as IVQ | approaches ~ 10 V. We note further that the measured m (1-5) for all the devices is lower than that calculated, which could indicate that the effective masses in (4.9) and (4.10) are lower than 0.2 mQ as we assumed, or that the potential barrier is more parabolic than linear. Additional calculations reveal that the effective masses must be reduced by about an order of magnitude to bring m down to the measured value. Then to retain theoretical-experimental agreement for the absolute value of Ip, the effective carrier transit times in (4.9) and (4.10) must be reduced by three to four orders of magnitude. Higher values of a and lower values of B also weaken the dependence of Ip on Vn. ^ (V) S ic*Lcul*t * (V Â°n L/Gf, l/, r'aK>vn) n V0]f chd *d9e V *ct*r1 'St 7CS *Â»pft *Sif 7 Hg 92 Additional support for the field-emission model for I|_ is obtained from consideration of the influence of the back-gate bias Vg^. This influence is related to the minimization of the leakage current at low Vqjt depicted in Fig. 4.1. As Vgf is reduced to switch the device from OFF to ON (accumulation), the measured leakage current reaches a minimum value, whereas the calculated current continues to be reduced monotonically. This simply indicates that the field emission in our model is insignificant at this point, and the actual minimum leakage current is produced by another mechanism, possibly field emission via grain-boundary traps near the back surface where positive interfacial charge- could indeed cause inversion with the back gate grounded (VGb = 0). The minimum measured currents in Fig. 4.1 show a strong dependence on Vp like our model predicts, and hence we surmise that they result from field emission near the back surface. We note further that- the value of Vgf at which the minimum IÂ¡_ obtains decreases as Vp increases. This implies that the dependence on Vp of the field-emission current at the back surface is weaker than that of the front-surface current, which, based on our model, is a result of the thicker back-gate (underlying) oxide. Measured ^(Vgf.Vgb) characteristics for a hydrogenated n-channel inversion-mode LPCVD thin-film polysilicon MOSFET, with Vp = 5V, are plotted in Fig. 4.9. For Vgf << 0 (OFF region), Ip (= 1^) is independent of Vg^ and increases exponentially with |Vgf| in accordance with our model. For low Vgf (near the minimum Ip), with Vg^ < 0 which implies that the back surface is accumulated, Ip increases with |Vg^| 93 Ã© Fig. 4.9 Measured current-voltage characteristics of a hydrogenated IPD/n polysilicon MOSFET (n-channel, inversion-node; Z = ion um, L = 10 u m, tof = 400 A , NA = 3 x 1016 cm"-3). The polysilicon film is 0.17 um-thick and was deposited via LPOVD on a 0.Sym-thick layer of silicon-dioxide. 94 and could be simulated well by our field-emission model applied to the back surface, ^(Vq^.Vq). The dependence of the back-surface leakage current on (< 0) is not as strong as IL(VGf) in Figs. 4.1 and 4.9 because of the thicker back-gate oxide (> 5000 A) and the high (> 0) threshold voltage for accumulation. Note that In also increases with VGb (> 0) because of back-surface inversion. Thus the data in Figs. 4.1 and 4.9 are consistent with our field- emission model for leakage current. The minimizations of the Ig(Vgf) characteristics reflect significant back-surface conduction, which for the OFF condition at the back can be accounted for by field emission via grain-boundary traps. 4.3. Summary An analytic model for field emission via grain-boundary traps in the surface depletion region at the drain has been developed to explain the anomalous leakage current observed in LPCVD polysilicon MOSFETs. To derive an insightful model, simplifying approximations, commensurate with the uncertain device morphology, were made. Corroboration of the model was given by showing consistency hetween measured data and predicted dependences of the field-emission current on the gate and drain voltages. Such consistency was demonstrated by using physically reasonable model parameter values. The analyses imply that field emission via grain-boundary traps in the drain depletion region near the surface could be the predominant mechanism controlling the leakage current in the LPCVD polysilicon MOSFET (for Vgf > Vjf). Field emission 95 near the back surface can be prevalent in the thin-film device when the front surface is depleted. Although the model was developed based on the p-channel accumulation-mode device shown in Fig. 1.2, it is in essence applicable to other polysilicon MOSFET structures. For example, in the n-channel inversion-mode device, the predominant leakage current flows when the (front) surface is accumulated; it results from field emission through grain-boundary traps between the p accumulation layer and the n+ drain. In describing the leakage in this device, structural differences between it and the device in Fig. 1.2 must of course be accounted for. For example, with the back gate grounded (with the source) the tendency toward inversion at the back surface of the n-channel transistor suggests a different component of leakage current (drift/diffusion of electrons from source to drain in the back channel) than that discussed for the p-channel transistor, but one that saturates as the drain voltage is increased and the channel is pinched off [BR81; SZ81]. This suggestion is supported by measured data in [0N82]. Our model for the trap-assisted field emission implies how the leakage current in polysilicon MOSFETs can be controlled and reduced. Obviously reducing the grain-boundary trap density, via hydrogenation for example, will decrease IL. Less obvious though is the implication that I|_ will decrease as the electric field in (4.11) is reduced. Rased on the model, F^ can be reduced by (a) decreasing the doping density in the drain, (b) increasing the gate oxide thickness, and (c) using a p+ gate (for the p-channel MOSFET). The modification (a) can be 96 accomplished by using an LDD (Lightly Doped Drain-source) structure, which is widely used to reduce the hot-electron effects in' bulk MOSFETs [0G80], with perhaps an optimal doping density [TE84]. We have experimentally observed that the leakage current of polysilicon MOSFETs is indeed reduced when the doping level in the source/drain region is decreased. The oxide spacer used to realize the LDD structure has the additional benefit of reducing the lateral diffusion of source/drain dopants, which is important in realizing short-channel-length devices. It is interesting to note that it has also been observed leakage current-voltage characteristics [SH85] for bulk MOSFETs with buried n+ source/drains similar to those shown in this chapter for polysilicon MOSFETs. Transistors having a buried n+ source/drain have a high defect density near the drain junction because of the proximity of the buried- oxide periphery. These traps could be responsible for field emission in these bulk MOSFETs like the grain-boundary traps in a polysilicon transistor, thereby leading to strong dependences of the leakage current on gate and drain voltages. CHAPTER FIVE SURTHRESHOLD BEHAVIOR OF THIN-FILM SMALL-GRAIN POLYSILICON MOSFETs 5.1 Introduction The main disadvantages of the small-grain polysilicon MOSFET compared with the single-crystal counterpart are the anomalous high leakage current and the exceptionally high gate-voltage swing [0N82; SH84], both of which can be controlled sufficiently to permit viable circuit applications. - i, â€¢- . i. _cr; In the previous chapter we presented a physical model for the OFF- state leakage current based on field emission via grain-boundary traps in the depletion region at the drain. In this chapter we analyze the ON-state conduction properties of the thin-film LPCVD (small-grain) polysilicon MOSFET. We describe, for all bias conditions, the channel charge density at the source in terms of the grain-boundary properties and the device parameters, accounting for the charge coupling between the front and back gates [LI83b; BA83], Rased on this description, we characterize the drain current, assumed to be predominantly diffusion, in the subthreshold region and the threshold voltage. The model predictions are supported by current-voltage measurements, and physical insight regarding the effects of hydrogenation on the grain boundaries is revealed. 97 98 Previous analyses [KA72; DE80,82; LE82a; C082,83,84; LI83b,84a; BA83] of thin-film SOI MOSFETs have accounted for the effects of grain boundaries or the effects of the charge coupling between the front and back gates, but they have not accounted for both. Most of the research [LE82a; 0082,83,84] concerning the effects of grain boundaries in MOSFETs has focused on large-grain (recrystallized) polysilicon. The early work of Kamins [KA72] for small-grain polysilicon MOSFETs concentrated mainly on the threshold voltage without analyzing the drain current. The recent work of Depp, et al. [DE80,82] for small-grain polysilicon MOSFETs studied the drain current and the threshold voltage, but like Kami ns's work, it did not analyze the subthreshold drain current and the consequences of fabricating the devices in thin films. We derive in Section 5.2 theoretical descriptions of the subthreshold drain current and the threshold voltage of the thin-film small-grain polysilicon MOSFET, which reveal the physical influence of grain boundaries in the channel and of the charge coupling between the front and back gates. The results are described in terms of the front- and back-gate voltages, the device parameters, and the grain-boundary properties. The analysis is used as a basis for discussing the superthreshold current-voltage characteristics, with emphasis on the grain-boundary and charge-coupling effects. The analysis comprises the following: a) a numerical solution of the one-dimensional Poisson equation with mixed boundary conditions at the front and back interfaces, which accounts for charge trapped at grain boundaries; b) the assumption, based on previous work [KA72; 99 DE80,S2] 'that the grain boundaries in small-grain polysilicon can be analyzed as traps uniformly distributed in silicon; c) tbe description characterization of the threshold voltage obtained from the derived dependence of the channel charge density on the gate voltage. :_-Since;the space-dependence of the electric field and the potential in the film is not needed [C070] to define the areal charge densities in the MOSFET, we do not solve Poisson's equation completely. Instead we use it and the boundary conditions to formulate a nonlinear system of e'^uat'ikrrrÃ©, : solution, we develop a "two-dimensional" bisection method [BU81]. Although this method is not as fast computationally as Newton-Raphson fB081â€˜ijn,r"Wer use it because it avoids the problems of convergence that typically occur when Newton-Raphson is applied to complex problems. Main conclusions drawn from the analysis are: a) the subthreshold gate-voltage swing depends strongly on grain-boundary properties, and weakly on the front-gate-back-gate charge coupling effects; b) the thTesHOT d~vo 11age depends strongly on grain-boundary properties and on charge-coupling effects; c) the charge-coupling effects decrease as the grai n-boundary trap density, the thickness of the film, or the film doping concentration increases. To support the analysis and to stress its importance, we compare in Section 5.3 model predictions with measured current-voltage 100 characteristics of thin-film LPCVD polysilicon MOSFET's fabricated at Texas Instruments, Inc. [SU84; SH85], The theoretical-experimental agreement is good, and in addition to indicating properties of the grain boundaries in these devices, it exemplifies how the charge-coupling effects must be accounted for to avoid errors in interpreting electrical measurements. Such errors can obscure essential criteria for achieving optimal designs of SOI devices and integrated circuits. The theoretical-experimental studies reveal very low carrier mobility in the channel of the LPCVD polysilicon MOSFET, which is increased by hydrogenation. Based on the results, physical mechanisms possibly responsible for the low mobility are discussed. ; Â¡ 5.2 Analysis We assume, based on studies [KA72; DE80,82] of small-grain polysilicon MOSFET's, that, the grain-boundary traps can be accounted for as if they were uniformly distributed in silicon. We refer to the four- terminal p-channel accumulation-mode SOI MOSFET illustrated in Fig. 1.2; and we account, in contrast with previous works [KA72; DE80,82] for the charge-coupling effects [LI83b,84a; RA83] between the front and back surfaces. We first analyze the subthreshold current-voltage characteristics. For weak-accumulation conditions and for drain voltages Vq (relative to the source) substantially greater than kT/q, the holes in the channel flow predominantly by diffusion [LI33a]. Therefore, the drain current, governed by only the front surface 101 conduction (the back surface is assumed to be depleted or inverted), is [LI83a; SW72] Jio pf q s pf (5.1) where Z is the channel width, L is the channel length, Ppf is hole mobility (the characterization of which we discuss later), and 0^ is the hole accumulation-layer areal charge density at the source. In writing (5.1), we have assumed that the accumulation-layer charge density at the drain is much smaller than Qpf, which obtains in weak- accumulation operation.- Based on classical MOSFET theory [C070], we express Qpf, for all accumulation conditions, as Â°pf â– i ^sf q(p-p ) d|> (5.2) where tsf is the front surface potential at the source, (p-p0) is the excess hole concentration, and F = --jp-is the electric -field normal to dx the surface. Note that the applicability of (5.2) requires that p=pQ (4>A. 0) at some point in the film, which is consistent with our assumption that the back surface is not accumulated. For given front and back gate voltages and Vfib, O^f is characterized by a one-dimensional solution of Poisson's equation with mixed boundary conditions at the front and hack interfaces. Instead of solving this differential equation completely for the space-dependence 102 of F and-\|> [BA80.83] we use (5.2) and evaluate the boundary conditions and the back surface potential ip^) by solving a system of nonlinear equations as described in the following subsections. 5.2.1 Formalism We refer to the one-dimensional representation of the MOSFET shown in Fig. 5.1, applied at the source. Using thin-film MOSFET theory [LI83b,84a] we write the following mixed boundary conditions: VGf =*sf (5.3) of and VGb â€¢ VFR 8*sb e sFsb Cob (5.4) where Fgf and Fs[5 are the front- and back-surface electric fields, CQf f f and CQ^ are the front- and back-gate oxide capacitances, and Vpg = Â«j^ ^ff b h Ofb - 0â€” and Vpg =
of ob
RETURN
ENO
FUNCTION TINTT(USF,US3,N)
C CALCULATION OF THE INTEGRAL OF FTB(R) FROM USF TO USB USING TRAPEZOIDAL.
C THIS HAS TO BE EQUAL TO T3 (SEE SC. C 5.253 ).
TINTT*a.
RN*N
0=Â» |