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Compact models for the high-frequency characteristics of modern bipolar transistors Vaidyanathan, Mani 1998

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C O M P A C T M O D E L S F O R T H E H I G H - F R E Q U E N C Y C H A R A C T E R I S T I C S O F M O D E R N B I P O L A R T R A N S I S T O R S By Mani Vaidyanathan B . A . S c , The University of Waterloo, 1990 M.A.Sc. , The University of Waterloo, 1992 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F E L E C T R I C A L A N D C O M P U T E R E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A • December 1998 © Mani Vaidyanathan, 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical and Computer Engineering The University of British Columbia 2356 Main Mall Vancouver, British Columbia Canada V6T 1Z4 Date: A b s t r a c t Modern bipolar transistors are characterized by shrinking dimensions (now on the order of a mean-free path length for carrier scattering), reduced parasitics (particularly in het-erojunction devices), and increasing cutoff frequencies (now over 100 GHz). As a result, the classical models used for transistor analysis and design, many of which were originally formulated over 40 years ago, are based upon assumptions that are no longer valid. This thesis deals with the reexamination and improvement of such models, particularly those used to describe the high-frequency characteristics. A new method of describing high-frequency carrier transport through the base of a bipolar transistor, known as the "one-flux method," is critically analyzed. It is shown that the basic one-flux equations are essentially equivalent to the classical drift-diffusion equations, and that the use of the one-flux approach to describe high-frequency transport in modern thin-base devices is essentially equivalent to employing the usual drift-diffusion equations with appropriately chosen boundary conditions. It is pointed out that while the flux approach does provide both compact, analytical expressions and useful aids for visualization, there is an inherent difficulty that exists in deriving values for the required backscattering coefficients on a rigorous, physically correct basis. A solution of the Boltzmann transport equation (BTE) in the base, and for high-frequency input signals, is carried out in order to obtain a fundamental, physical insight into the effects of carrier transport on the high-frequency operation of modern thin-base (or "quasi-ballistic") transistors, and to test the merit of recently suggested one-flux expressions for the intrinsic high-frequency characteristics of such devices. It is shown that both the common-base current gain and the dynamic distribution function are affected by a "ballistic" degradation mechanism, in addition to a "diffusive" degradation mechanism, and that, as a result, expressions from the one-flux approach alone cannot adequately model the device characteristics. Expressions which involve a combination of the one-flux expressions with the well-known expressions of Thomas and Moll are suggested for the forward characteristics, and these are then shown to agree with the B T E solutions. Expressions for the reverse parameters are derived by applying the "moving boundary approach" of Early and Pritchard to the basic one-flux equations of Shockley. ii Expressions for the extrapolated maximum oscillation frequency (commonly denoted / m a x ) of modern heterojunction bipolar transistors (HBTs) are systematically developed from a general-form, high-frequency equivalent circuit. The circuit employs an arbitrary network to model the distributed nature of the base resistance and collector-base junc-tion capacitance, and includes the parasitic resistances of the emitter and collector. The values of / m a x as found by extrapolation of both Mason's unilateral gain and the max-imum available gain to unity, at —20 dB/decade, are considered. It is shown that the / m a x of modern H B T s can be written in the form / m a x = yj/r/87r(/?C)eff, where is the common-emitter, unity-current-gain frequency, and where (RC)es is a general t ime constant that includes not only the effects of base resistance and collector-base junction capacitance, but also the effects of the parasitic emitter and collector resistances, and the device's dynamic resistance (given by the reciprocal of the transconductance). Simple expressions are derived for (/?C)eff, and these are applied to two state-of-the-art devices recently reported in the literature. It is demonstrated that, in modern H B T s , (7?C)efr can differ significantly from the effective base-resistance-collector-capacitance product conventionally assumed to determine / m a x -i i i Contents Abstract ii List of Tables vii List of Figures viii Publications ix Acknowledgments x 1 Introduction 1 1.1 The Transistor 1 1.2 This Thesis 5 1.2.1 Intrinsic Characteristics 5 1.2.2 Extrinsic Characteristics 7 2 Appraisal of the One-Flux Method 9 2.1 Introduction 9 2.2 Comparison of the Flux and D D E Approaches 10 2.3 Equivalence of Recent Short-Base Analyses 16 2.4 Use of the Flux Method to Infer Carrier Mobility in a Forward-Biased Barrier 22 2.4.1 Critique of Lundstrom and Tanaka 23 2.4.2 Recent Work 27 2.4.3 General Comments 28 2.5 Conclusions 29 iv 3 Expressions for the High-Frequency Characteristics of Quasi-Ballistic Bipolar Transistors 31 3.1 Introduction 31 3.2 B T E and Flux Approaches 32 3.2.1 B T E Approach 32 3.2.2 Flux Approach 34 3.3 Forward Parameters 36 3.3.1 Effects of Ballistic Versus Diffusive Decay 36 3.3.2 Expressions for the Forward Parameters 45 3.3.3 Additional Considerations 48 3.4 Reverse Parameters 52 3.5 Conclusions 56 4 Expressions for the Extrapolated / m a x of Heterojunction Bipolar Transistors 57 4.1 Introduction 57 4.2 Equivalent Circuit for Analysis 59 4.3 Two-Port Parameters 64 4.3.1 Expressions for the Two-Port Parameters 64 4.3.2 Verification Using SPICE 68 4.4 Extrapolated / m a x 73 4.4.1 New Expressions for / m a x • 73 4.4.2 Algebraic Simplification 75 4.4.3 Discussion 76 4.4.4 Application 77 4.5 Conclusions 81 v 5 Conclusions 82 5.1 Appraisal of the One-Flux Method 82 5.2 Expressions for the High-Frequency Characteristics of Quasi-Ballistic Bipolar Transistors 83 5.3 Expressions for the Extrapolated / m a x of Heterojunction Bipolar Transistors 86 5.4 Future Work 87 A Current-Voltage Relation of a Mott Barrier in the Thermionic-Emission Limit 89 B Comparison of the One-Flux and Balance Equations 91 B . l Balance Equations 91 B.2 One-Flux Equations 93 B.3 Discrepancies 94 B.4 Source of the Discrepancies 94 B.5 Modified One-Flux Equations 96 C Base-Collector Network for a Conventional HBT Structure 97 D Derivation of Two-Port Relations 101 Bibliography 103 v i List of Tables 4.1 Parameter values for the devices described in [59] and [60] 69 4.2 Values of (RC)^ and (RC)°£ for the device described in [59] 78 4.3 Values of (RC)^ and (RC)°g for the device described in [60] 78 vii List of Figures 1.1 Concepts of the junction transistor 3 1.2 First record of diffusion theory 4 2.1 Carrier fluxes at a thin slab 11 2.2 Distribution function in the flux method 13 2.3 Flux and D D E methods for a short-base transistor 17 2.4 Magnitude plots of /3 for a short-base device 21 2.5 Conduction-band profile of a Mott barrier 24 3.1 Common-base current gain from the B T E and flux approaches 37 3.2 Phase angle of the dynamic distribution function 40 3.3 Plots of microscopic quantities versus position 42 3.4 Common-emitter current gain from the different approaches 44 3.5 Common-base current gain from the B T E and Thomas-Moll form . . . . 47 3.6 Forward, common-base, admittance parameters 49 3.7 Reverse, common-base, admittance parameters 55 4.1 Hybrid-7T and T equivalent circuits 60 4.2 Simplified form of the hybrid-vT circuit 62 4.3 Power gains versus frequency for the device described in [59] 72 4.4 {RC)%B and [RC)f£ versus bias for the device described in [59] 80 C . l Base-collector network for a conventional H B T structure 98 V l l l Publications • I hereby declare that I am the sole author of this thesis. In accordance with the thesis guidelines of the University of British Columbia, I hereby also declare that this thesis contains material previously published (or in press) in the following articles written by me. M . Vaidyanathan and D. L. Pulfrey, "An appraisal of the one-flux method for treat-ing carrier transport in modern semiconductor devices," Solid-State Electronics, vol. 39, pp. 827-832, June 1996. M . Vaidyanathan and D. L. Pulfrey, "Effects of quasi-ballistic base transport on the high-frequency characteristics of bipolar transistors," IEEE Transactions on Electron Devices, vol. 44, pp. 618-626, Apri l 1997. M . Vaidyanathan and D. L. Pulfrey, "Extrapolated / m a x of heterojunction bipolar tran-sistors," IEEE Transactions on Electron Devices, ms. 3407, accepted for publication October 5, 1998. IX Acknowledgments I owe profound gratitude to my Ph.D. supervisor, Professor D. L. Pulfrey. Throughout the course of this work, Professor Pulfrey was always available for discussions, regardless of the outlook, and he remained patient and flexible, allowing me to work at my own pace, even when progress was slow. Professor Pulfrey's devotion to his students, and his conscientious effort to play an active role in their research, are hallmarks of his character. He is a man of great integrity, a true gentleman, scholar, and teacher, and it is an honor to have him as my guru. I also owe special thanks to Professor D . J . Roulston of the University of Waterloo, my former M.A.Sc. supervisor. Even though my background had been in computer engineering, Professor Roulston accepted me as his student in the field of electronics. I deeply appreciate the many hours he spent with me, teaching me the fundamentals of the area, and helping me develop research skills. Without him, my doctoral studies would never have been possible. I would like to thank the members of my Ph.D. supervisory committee, Professors M. K. Jackson and T. Tiedje, both of whom kept me honest at my oral exams with challenging questions. I additionally thank Professor Jackson for allowing me to audit his photonics courses, and for the interest he has shown in my future success. I would also like to thank Professor S. Luryi of the State University of New York, Stony Brook, for his involvement as external examiner, and Professors N. A. F. Jaeger, R. R. Johnson, and J. R. Marti for their roles on my examining committees. I am especially grateful to my friends and colleagues. I thank my coworker, Tony St. Denis, for his friendship and assistance. Tony and I shared many interests, and the numerous technical discussions in which we engaged were very helpful. I also thank my friends, Prasad Gudem, Steve Pye, and Andrew Sarangan. Over the years, I have benefited immensely from their advice, and I have learned a great deal about solving problems from each of them. This work would not have been possible without my family. I thank my parents for their support and encouragement, and my older brother, Ganesh, for his unending wisdom. I thank Prabha for her advice and for the revitalizing meals she prepared whenever I visited, and I thank Saumya and Akul for the joy and laughter they have brought to all our lives. I gratefully acknowledge scholarship support from NSERC, the UBC Killam program, BC TEL, and Hughes Aircraft of Canada. I also acknowledge MICRONET for funding the research described in this thesis. M A N I V A I D Y A N A T H A N xi Chapter 1 Introduction 1.1 The Transistor About 50 years ago, on June 30, 1948, a press conference was held at the headquarters of Bell Laboratories in order to announce an important new invention. The statement came from the lab's director of research [1, p. 8]: We have called it the Transistor, because it is a resistor or semiconductor device which can amplify electrical signals as they are transferred through it. The transistor (transfer resistor) itself had actually been born months earlier, on December 23, 1947, when John Bardeen and Walter Brattain, in a group led by William Shockley, demonstrated the first "point-contact" device to key lab managers [2]. By early 1951, the "junction transistor" had emerged, conceived and developed by Shockley as an improvement over the delicate point-contact approach [3]. Today, electronic systems employ a variety of solid-state devices, many of which bear little resemblance to those created by Bardeen, Brattain, and Shockley. However, state-of-the-art bipolar junction transistors (BJTs) and heterojunction bipolar transis-tors (HBTs) are direct descendants of Shockley's original device [3], and they continue to be used in a number of applications, including digital and mixed-signal circuits, wireless communications, and high-bit-rate, fiber-optic systems [4]. Generally speaking, modern BJTs and HBTs work on the same basic principles as the original junction transistors, a fact which is best illustrated by considering three concepts stated by Shockley [3, pp. 598-599] as being essential to the invention of the 1 2 junction transistor: (1) minority-carrier injection into the base layer which increases exponen-tially with forward emitter bias; (2) application of reverse voltage at the collector junction; (3) favorable geometry and doping levels so as to obtain good emitter-to-collector efficiency. These three concepts, which were first recorded together in Shockley's laboratory note-book in 1948, as shown in Figure 1.1, continue to describe the basic operation of present-day BJTs and HBTs, with the only exception being that the third item is augmented, in modern devices, by the ability to use advanced structures and processing techniques (which make available features such as the polysilicon emitter, heterojunctions, and compositionally variable material) to trade off more-than-adequate emitter efficiency for speed-enhancing characteristics. The fundamental conceptual similarity between modern bipolar transistors and the original junction transistors of the '50s and '60s has contributed to a somewhat surprising situation: many of the theoretical models formulated in those decades have continued to apply to new devices, despite the rapid technological advancements that have occurred over the years. For example, consider Shockley's diffusion theory for minority carriers. The first record of this theory was made in 1947, and is shown in Figure 1.2. The complete theory was published in 1949 [5], but it has continued to provide the basis for treating minority-carrier base transport for almost 5 decades. With the passing of the 50th anniversary of the transistor, devices are continuing to emerge with shrinking dimensions (now comparable to a mean-free path for scattering), reduced parasitics (particularly in HBTs), and increasing cutoff frequencies (in excess of 100 GHz). Despite a continued conceptual similarity with earlier transistors, these new devices have physical characteristics that are beginning to undermine the assumptions on which classical transistor models are based. Therefore, these classical descriptions, such as the diffusion theory originally presented in [5]—a theory which assumes "large" device dimensions, that is, dimensions much greater than a mean-free path length—must be carefully reconsidered and updated. Figure 1.1: The notebook entries of William Shockley, dated January 23, 1948, which record the combination of the three concepts (mentioned in the text) for the junction transistor. (Taken directly from [3, Fig. 20].) 4 Figure 1.2: The notebook entry of William Shockley, dated April 24, 1947, consisting of the first record of diffusion theory for minority electrons in a p-type layer. This record assumes a large reverse bias for the junction at x = 0 and a p-region width much greater than \ZTTT. The complete theory, including a description of base transport that continues to be used today, was published by Shockley in 1949 [5]. (Taken directly from [3, Fig. 10].) 5 1.2 This Thesis This thesis focuses on compact models for the high-frequency characteristics of modern bipolar transistors. Classical descriptions of the high-frequency behavior are reexamined and amended, leading to a set of well-founded, compact expressions that more soundly describe the high-frequency operation of modern devices. Chapters 2 and 3 deal with the intrinsic high-frequency characteristics, and Chapter 4 deals with the extrinsic char-acteristics. The conclusions of the thesis are summarized in Chapter 5. 1.2.1 Intrinsic Characteristics Background The intrinsic characteristics of bipolar transistors are traditionally found by applying classical drift-diffusion theory (or, in the absence of electric fields, classical diffusion theory) to describe base transport. Solutions of the drift-diffusion equation (DDE) not only form the theoretical basis for transistor modeling at high frequencies, but are also used to obtain values for the so-called "non-quasi-static" parameters employed in circuit simulators. These parameters, such as the "excess-phase" parameter in SPICE or the "delay-time" parameters in H I C U M [6], are needed to improve the simulated response whenever the input signals (large or small) change rapidly, and are known to have an impact on the performance of various circuits, including mixers, ECL-type circuits, and various amplifiers [7, pp. 1-6, 2-18]. With transistor dimensions continuing to shrink, Si and SiGe devices with very narrow base widths (250 to 500 angstroms) and high cutoff frequencies (from 50 to over 100 GHz) are now being reported [8]—[11]. For such devices, where the base width becomes comparable to the collision-free path length of the minority carriers, use of the drift-diffusion equation becomes questionable, and, strictly speaking, one needs to solve the Boltzmann transport equation (BTE). A simple question then arises: What types of error are incurred by continuing to employ the D D E in such devices, where the base transport is said to be "quasi-ballistic"? The question cited above was first addressed by Alam, Tanaka, and Lundstrom in [26], and then subsequently by Alam, Schroter, and Lundstrom in [27]; it was also cited as 6 an "issue of increasing importance" by Schroter in his short course [7, p. 7-2] on bipolar transistor modeling for high-speed circuit design and process development, presented at the IEEE Bipolar Circuits and Technology Meeting of 1996. Research Undertaken and Main Results In [26] and [27], the authors employed a so-called "one-flux approach" in place of the D D E in order to obtain the high-frequency characteristics of thin-base bipolar transistors, and found significant differences from the traditional D D E results. The first step in the research undertaken was to come to grips with the one-flux approach as a means of treating carrier transport in modern devices. A general appraisal of the technique, including a comparison with the traditional D D E method, was carried out. The details of the appraisal, published in [12], are presented in Chapter 2; the main conclusion is that the one-flux approach is essentially equivalent to employing the D D E , only with different mathematical boundary conditions. The basic equivalence between the one-flux and D D E approaches prompted a closer look at the effects of quasi-ballistic base transport on the high-frequency behavior, and this was accomplished by using the method of Grinberg and Luryi [31], [47] to solve the Boltzmann transport equation. The details of the B T E investigation, published in [13], are presented in Chapter 3. The B T E solutions not only show that the one-flux approach misses some of the key physics behind high-frequency, short-base transport, but also indicate that it erroneously predicts the device characteristics, including the so-called "excess phase" of the common-emitter current gain. It is suggested that in order to match the B T E results, one must employ an appropriate combination of the one-flux expressions with the well-known expressions of Thomas and Moll [48], and this approach is then demonstrated to be successful. The result of the work is a new set of expressions for the intrinsic device characteristics, which are valid from very low to very high frequencies, and which replace the traditional expressions derived over 40 years ago [45], [46] and used since that time. 7 1.2.2 Extrinsic Characteristics Whereas the intrinsic device behavior at high frequencies depends on carrier-transport effects within the device, the extrinsic characteristics and circuit performance depend additionally on the device parasitics. It is well known that device parasitics have been shrinking in value, particularly in HBTs. The particular question addressed in this thesis is the following: Given the vastly reduced values of parasitics in state-of-the-art heterojunction bipolar transistors, can one properly calculate the device " / m a x " — a figure of merit almost universally used to characterize the potential high-speed circuit performance of bipolar transistors—by employing the usual, well-known expression that was originally derived over 40 years ago [53]-[56] for the purposes of characterizing the homojunction devices of the time? This is, in fact, a long-standing and widespread question that has hitherto not been properly treated. The findings of the research, which have been accepted for publication [14], are pre-sented in Chapter 4. It is shown that it is possible to write the / m a x of modern HBTs in the familiar form / m a x = y / r / 8 i , ( f i C , ) e f r , where fr is the common-emitter, unity-current-gain frequency, provided that (RC)en is taken to be a general time constant that includes not only an effective product of base resistance and collector-base junction capacitance, but also terms involving the parasitic emitter resistance, the parasitic collector resistance, and the device's dynamic resistance (given by the reciprocal of the transconductance). The values of / m a x as found by extrapolation (at —20 dB/decade) of both Mason's uni-lateral gain [66] and the maximum available gain [69] are considered, and expressions are derived for (RC)e([ corresponding to each of these power gains. The expressions are fully validated using SPICE, and by considering two state-of-the-art HBTs recently reported in the literature [59], [60], it is illustrated that the value of (i?C)eff can differ considerably from that of the effective base-resistance-collector-capacitance product con-ventionally assumed to determine / m a x . The derived expressions for (i?C)eff are valid for an arbitrary breakup of the base resistance and collector-base junction capacitance within the equivalent circuit of the H B T , and all of the assumptions (which will typically be true) needed in the derivation are clearly- stated. Reasonably compact expressions for 8 the extrinsic two-port characteristics of the device at high frequencies are also found in the course of the work, and these may be useful for parameter-extraction purposes. The results of this research should be extremely useful in identifying speed-limiting parasitics in modern HBTs, and for designing these devices for optimum / m a x and hence optimum high-speed circuit performance. Chapter 2 Appraisal of the One-Flux Method 2.1 Introduction In the numerical analysis of semiconductor devices, it is now well recognized that the classical drift-diffusion equation (DDE) is beginning to lose validity as device dimensions continue to shrink [15], [16]. Generally, to obtain a proper picture of carrier transport within the small-dimension or high-field regions of modern devices, one has to solve the Boltzmann transport equation (BTE) , often by resorting to Monte Carlo simulations [15]. Recently, McKelvey's one-flux method [17]—[24] has resurfaced as a simplified alternative to this involved task, and it has been applied to various transport problems that are relevant to the modeling of modern bipolar transistors [25]-[28]. Specifically, the flux method has been used to describe quasi-ballistic base transport under both static (dc or bias) [25] and dynamic (ac or small-signal) [26], [27] conditions—yielding new expressions for the intrinsic characteristics of modern thin-base devices—and to address the long-standing issue of carrier mobility in a forward-biased barrier [28]. It is important that the results obtained with the flux method be viewed in the proper context, and this requires a general evaluation of the technique. This chapter presents an overall appraisal of the one-flux method. The basic one-flux equations are compared with the usual drift-diffusion equations, the short-base results found in [25]-[27] are compared with those obtainable from the D D E [29], and the ap-proach taken in [28] is critically examined. This work serves as a necessary prelude to Chapter 3, where the small-signal, one-flux results derived in [26] and [27], which 9 10 are of primary interest to this thesis, are more thoroughly examined using a solution to the B T E . In Section 2.2, it is shown that for a bulk region in which a small electric field is present, where the backscattering coefficients required by the flux method are known [20], use of the flux method is essentially equivalent to solving the usual continuity and drift-diffusion equations under low-level injection. In Section 2.3, this equivalence is then explicitly demonstrated by showing that the short-base flux analyses presented in [25]-[27] are equivalent to Hansen's [29] recent approach of employing the D D E with appropriate boundary conditions. In Section 2.4, use of the flux method within the region of high, built-in electric field of a forward-biased barrier is considered. Such an application of the flux method requires knowledge of the backscattering coefficients. In [28], the authors suggest values for these coefficients, and then use them to find a field-dependent carrier mobility within the barrier. The physical bases for the choices made, and hence the resulting conclusions regarding mobility, are questioned, and the discussion highlights a basic problem inherent to the flux approach. Section 2.5 summarizes the conclusions of this chapter. 2.2 C o m p a r i s o n of the F l u x a n d D D E A p p r o a c h e s The basic idea of the one-flux method can be understood by considering Figure 2.1, where the position- and time-dependent carrier fluxes are shown at a slab of thickness Az . The slab itself is assumed to be characterized by an absorption coefficient S*(z), an isotropic generation rate g(z,t), and backscattering coefficients ('(z) and ((z) for the right- and left-directed fluxes a(z,t) and b(z,t), respectively. Assuming that the carrier distribu-tion function is formed by abutting two hemi-Maxwellian velocity distributions at the lattice temperature T , as shown in Figure 2.2, and that the slab thickness Az is small, and 11 a(z,t) b(z,t) C'Az CAz a(z +Az,t ) b(z +Az,t ) z + Az Figure 2.1: Carrier fluxes at a slab of thickness A z . Indicated within the slab are the fractions ( ' A z and CAz of the incident fluxes a(z, t) and b(z + A z , t), respectively, that are backscattered. The fraction S*Az of each incident flux that recombines within the slab, and the contribution (g/2)Az of flux generated within the slab to each of the right-and left-directed flows, are not shown. 12 considering a small time interval At, one can write a simple conservation equation: a(z,t + At) — a(z,t) Az = VT + a(z,t)[l - ('(z)Az - S*(z)Az]At - a(z + Az,t)At + b(z + Az,t)((z)AzAt + 9-^-AzAt, (2.1) where VT = ylksTfirm* is the thermal velocity, ks is Boltzmann's constant, and m* is the effective mass. The left side of (2.1) represents the total change in the number per unit area of right-going carriers within the slab during At, and this change is taken into account by the terms on the right side of (2.1) as follows: the first term represents the number of incoming, right-going carriers that are not reversed and that do not recombine; the second term accounts for the right-going carriers that exit the slab on the right; the third term represents the number of incoming, left-going carriers that are reversed; and the fourth term accounts for the number of right-going carriers that are generated. Writing a similar equation for the change in the number of left-going carriers within the slab during At, and taking the limit Az, At —)• 0, one obtains the following system of equations1 governing the fluxes a(z,t) and b(z,t): (l/vT)(da/dt) + da/dz = -('{z)a + ((z)b - 8*(z)a + g(z, t)/2, (2.2) -(l/vT)(db/dt) + db/dz = -('{z)a + ((z)b + 5*(z)b - g(z, t)/2. (2.3) Use of the one-flux method to solve a transport problem amounts to finding a solution to (2.2) and (2.3) for the left- and right-directed fluxes; once these have been found, information on the carrier concentration n(z, t) — [a(z, t) -\-b(z,t)]/vT and the net carrier flux F(z,t) = a(z,t) — b(z,t) can easily be obtained. 'The generation terms appearing in (2.2) and (2.3) have been omitted in [26, eqs. (3), (4)], and the backscattering coefficients for the right- and left-directed fluxes are interchanged in [26, eq. (4)]. Equations (2.2) and (2.3) of this work are correct. 13 b = n vy n + v r velocity Figure 2.2: The assumed form of the distribution function / in the flux method, formed by abutting two counter-directed hemi-Maxwellians at the lattice temperature T. The left- and right-directed fluxes consistent with this distribution function are a = n+v? and b — TI~VT, where n + and n~ are the carrier concentrations corresponding to the positive and negative parts of / , respectively [30, pp. 272-273]. 14 Subtracting (2.3) from (2.2), one gets a result similar to the continuity equation: dn 1 dJn / \ r* , / *\ a7 5~ = 9th(z) -0 n v T + g (z, t) at q oz = gop(z,t) - rn, (2.4) where Jn = —qF is the current density and q is the magnitude of the electronic charge, the generation rate g(z,t) = gth(z) + g0p{z, t) is broken up into thermal and optical parts, and rn = 8*HVT — fi'th(z) is the net recombination rate. Identifying the thermal generation rate as gth(z) — no(z)/TREC, where no(z) is the equilibrium electron concentration and r r e c is the usual carrier lifetime, and noting that the absorption coefficient for systems obeying Boltzmann statistics is [17] r = — , (2.5) VTTREC equation (2.4) reduces to the usual continuity equation under low-level injection: dn \ d J n A n -wr 7 — = gop , (2.6) at q dz T r e c where A n = n — no is the excess electron concentration. Adding (2.2) and (2.3), one gets a result similar to the drift-diffusion equation: j _ C - C VT F VT dn 1 dJn ( e ) 7 s n ~ V + C + **) £ q ( C + C + S*) dz vT(C + c + s*)dt' 1 0 where S is the electric field. For a bulk region in which a small electric field is present,2 and for systems obeying Boltzmann statistics, the backscattering coefficients have been derived by McKelvey and Balogh [20] as and 2 A n electric field £ is defined to be small if u*€ < 2vT, where u* is the "total carrier mobility" determined by both scattering and recombination effects [20]. 15 where fj,n0 a n d Dn0 refer to the usua l , low-f ie ld , b u l k values of m o b i l i t y a n d d i f fu s iv i ty . 3 I f the q u a n t i t y Dno/vT is i d e n t i f i e d 4 as be ing ( a p p r o x i m a t e l y ) the mean-free t i m e between col l i s ions r c o i , and i t is assumed tha t Tool «C Tree (2.10) and ^ « i - . (2.11) t hen us ing (2.5), (2.8), a n d (2.9), equa t ion (2.7) reduces to the usua l dr i f t -dif fusion equa t ion : dn Jn = qfinonS + qDn0 — . (2.12) oz T h e es tab l i shed a lgebra ic equiva lence of the f lux equat ions (2.2) a n d (2.3) to the c o n t i n u i t y a n d dr i f t -dif fusion equat ions (2.6) a n d (2.12) c l ea r ly ind ica tes the contex t i n w h i c h one mus t v i e w results o b t a i n e d w i t h the one-flux m e t h o d : unless the car r ie r l i f e t ime is c o m p a r a b l e to the mean-free t i m e for sca t t e r ing or the current changes a p p r e c i a b l y i n a l eng th of t i m e r c o i , the one-flux m e t h o d , when a p p l i e d to a b u l k region i n w h i c h a s m a l l e lec t r ic field is present, w i l l y i e l d results tha t c o u l d be o b t a i n e d us ing the usua l c o n t i n u i t y and dr i f t -di f fus ion equat ions; moreover , th i s equivalence appl ies even to s m a l l - d i m e n s i o n regions, tha t is , to regions w h i c h have a spa t i a l extent on the order of a mean-free p a t h l eng th . S h o c k l e y [18] first p o i n t e d out th i s equiva lence for the t ime- independen t case, and also f o r m u l a t e d b o u n d a r y cond i t ions for the D D E t r ea tmen t i n t e rms of those app l i cab l e to the f lux m e t h o d . 3The original formulation by McKelvey and Balogh [20] assumed transport of holes, for which the signs of the second terms in (2.8) and (2.9) are - and +, respectively. 4Using Dn0 = 2vTlsc/3 and r c o i = lsc/2vT [30, pp. 470, 485], where / s c refers to the mean-free path length for scattering and 2vT is the mean speed, one obtains TCO1 = (3 /4) /J n 0 /^ ~ Duo/v?. 16 2.3 Equivalence of Recent Short-Base Analyses S i m p l i f i e d t r ea tments of t r anspor t w i t h i n a short-base t rans i s tor i n the absence of e lec t r i c fields, u s ing b o t h the f lux m e t h o d [25]-[27] a n d the D D E w i t h app rop r i a t e b o u n d a r y cond i t ions [29], have been shown to agree c losely w i t h a so lu t ion to the B o l t z m a n n t r anspor t equa t ion [31]. It is easy to demons t ra t e tha t the results of the f lux a n d D D E approaches are essent ia l ly equiva lent . C o n s i d e r , for e x a m p l e , the s ta t i c co l lec tor cur rent as a func t i on of base w i d t h . T h e v i s u a l i z a t i o n of the p r o b l e m i n t e rms of the flux m e t h o d is shown i n F i g u r e 2.3(a). T h e base is cons idered to be a b u l k region of w i d t h WB u p o n w h i c h k n o w n car r ie r fluxes of a(0) = (n*E/2)vT and 6(14 7B) — 0 are i nc iden t , where n*E/2 is the e lec t ron concen t r a t i on i n the r igh t -go ing , h e m i - M a x w e l l i a n d i s t r i b u t i o n in jec ted in to the base f r o m the e m i t t e r . T h e base is t h e n t aken to be charac te r i zed by compos i t e ref lect ion a n d t r a n s m i s s i o n coefficients o b t a i n e d f r o m previous so lu t ions to (2.2) a n d (2.3) for b u l k regions of a r b i t r a r y w i d t h . Fo r a field-free base of w i d t h WB, and neg lec t ing r e c o m b i n a t i o n , the ref lect ion and t r ansmi s s ion coefficients are the same for left- and r i gh t -d i r ec t ed fluxes and g iven , respect ive ly , by [17], [25, eqs. (33)] 1 + V R Dn0/WB (2.13) Dn0/WB and t% = l-r°Bl (2.14) where VR = vT/2 is the so-cal led R i c h a r d s o n ve loc i ty . F r o m F i g u r e 2.3(a), one t h e n gets the fo l l owing for the u n k n o w n fluxes 6(0) a n d a(Ws) i n t e rms of rB, t°B, a n d the k n o w n fluxes a(0) = (n*E/2)vT a n d b(WB) = 0: 6(0) = r ° a(0) + t%b(WB) = r°B(nE/2)vT, (2.15) a{WB) = t°Ba(0) + r°Bb{WB) = tB(n*E/2)vT. (2.16) 17 a(0) = (n* E /2)v T b(0) a ( W B ) b ( W B ) = 0 0 ^ z W F (a) ( n£/2 ) v T n L v T n R v T W F (b) F igure 2.3: F l u x (a) and D D E (b) v isual izat ions of the short-base t ransport p rob lem. In (a), t ransport w i th in the base is governed by (2.2) and (2.3) for a(z) and b(z). In (b), t ransport is de te rmined by (2.6) and (2.12) for n(z) and Jn(z). 18 T h e current can then easily be found using either of the fol lowing equat ions: Jn = -q[a(0) - 6(0)] = -q[l - rB](nE/2)vT, (2.17) Jn = -q[a(WB) - b(WB)] = -q[t0B]{n*E/2)vT. (2.18) T h e D D E picture of the s i tuat ion is shown in F igure 2.3(b), f rom which one has the fol lowing boundary condit ions for the electron concentrat ion and current: n(0) = n*E/2 + nL, (2.19) n(WB) = nR, (2.20) ^n(O) = -q(n*E/2 - nL)vT, (2.21) Jn(WB) = -q(nR)vr, (2.22) where UL and nR refer to the (unspecified) left- and right-going electron concentrat ions at z.= 0 and z = WB, respectively. C o m b i n i n g (2.19) and (2.20) wi th (2.21) and (2.22) yields the bounda ry cond i t ions 5 suggested by Hansen [29, eqs. (5), (6)]: J n ( 0 ) = -q[nE - n(0)}vT, (2.23) Jn{WB) = -q[n(WB)]vT. (2.24) These boundary condit ions can be used wi th the usual solut ion Jn = M 0 ) = MWB) = - ^ - ° [ " ( 0 ) - » ( ^ ) ] WB to (2.6) and (2.12) in order to obta in an expression for the current. It is easy to verify that this procedure yields a result for Jn which is ident ical to that found f rom (2.17) or (2.18): WB + Dn0/vR ' 5 T h e boundary conditions in (2.23) and (2.24) should be contrasted with the well-known conditions n(0) = n*E and TI(WB) = 0 that are conventionally employed. 19 The results for the common-emitter current gain from the flux and DDE approaches can also be shown to be equivalent. Using a flux analysis very similar to the dc case, the ac current gain (3 for a field-free base can be found as a function of the radian frequency ui [26, eq. (28)]: 13 = I N 1 _ 1 cosh(A/eWB) - 1 + ( - sinh(A/cWB) K J 2sinh* ™?S. where K and A are defined by + ( - ) s i n h ( X K W B ) - 1 (2.27) K X = 1 + 1 + 1 /TTec + jw' (2.28) (2.29) However, invoking the inequality (2.10), and assuming u <C l / r c o i in accord with (2.11), equations (2.28) and (2.29) imply that K = Dn0Tr, D nO and 1 + jUTri XK = — , VT TT L: nO (2.30) (2.31) where L* = <*J(Dn0TTec)/(1 + ju>TIec) is the complex diffusion length for electrons. With these, the current gain in (2.27) reduces to 0 2 sinh2 WB 2L: + DnO vTL*N sinh 'WB I - i (2.32) which is the result obtained by Hansen [29, eq. (14)]. For the device considered in [26], [29], and [31], one has ''"rec ~ 1000TCOI and 1/2,TTTCO\ ~ 1 THz, so that con-dition (2.10) is well met and condition (2.11) only starts to become unsatisfied at a frequency of about 100 GHz; thus, it is perhaps not surprising that the plots in [26, 20 Fig. 2] and [29, Fig. 4], which compare \0\ from (2.27) and (2.32) to the B T E solu-tion of Grinberg and Luryi [31], are identical, even when the base width is chosen to equal one mean-free path length. Figure 2.4 of this work shows \(3\ versus cyclic fre-quency / = U>/2TV from (2.27) and (2.32) using the same parameter values as in [26], [29], and [31]; as expected, the predicted values are the same up to (and, in this case, beyond) / = 100 GHz. Shown also in Figure 2.4, for comparison, are values of \/3\ determined by the usual expression [32, pp. 224-244] 2si„h> ^ \2L* (2.33) which is the result obtained by solving the continuity and drift-diffusion equations with conventional boundary conditions (rather than those suggested by Hansen). It can be concluded from the above equivalences that the results of recent applications of the flux method to study short-base transport [25]-[27] reduce to those obtainable from a solution (employing the appropriate boundary conditions) of the usual continuity and drift-diffusion equations [29], provided conditions (2.10) and (2.11) are satisfied. However, a few important points regarding this result should be borne in mind. First, as mentioned above, note the need to employ appropriate boundary conditions when applying the D D E . For the short-base transport problem, these are the conditions suggested by Hansen, as given by (2.23) and (2.24). More generally, one must apply boundary conditions that are consistent with those used in the flux approach. As indicated earlier, Shockley [18] has formulated such boundary conditions for various situations, and these typically differ from those conventionally employed with the DDE. Hereafter, in order to avoid ambiguity, whenever boundary conditions are employed that are consistent with those used in the flux approach, such as the boundary conditions suggested by Shockley or Hansen, the D D E method shall be referred to as being "Hansen's D D E approach," or a "modified D D E approach," or the "DDE approach with appropriate boundary conditions"; on the other hand, when conventional boundary conditions are employed, the terms "conventional D D E approach" or "classical D D E approach" will be used. Second, it should be noted that in terms of visualization and bookkeeping, there are advantages that can be gained 21 FREQUENCY (Hz) Figure 2.4: Magnitude plots of the common-emitter current gain /3 from (2.27) and (2.32) versus cyclic frequency / = U>/2?T. Shown also, for comparison, are values found from the conventional expression (2.33). The results from (2.27) and (2.32) are identical and shown by the solid lines, and the results from (2.33) are shown by the dashed lines. The parameter values used for the plot are / s c = 500 A, = 2 x 107 cm/s, r r e c = 0.125 ns, and Dno = 67 cm 2/s. 22 by using the flux method: specifically, since the flux method deals separately with the right- and left-going carrier flows, as opposed to the net current as in the D D E treatment, it affords a better mental image of transport problems; additionally, the analysis of transport across a series of device regions can be conveniently handled in the flux method by establishing a so-called "scattering matrix" for each region, the elements of which are derived from the composite reflection and transmission coefficients of the region, and then by systematically cascading the individual matrices [33]. Finally, with respect to conditions (2.10) and (2.11), which are required for the flux and D D E approaches to be formally identical, it should be noted that while one can expect (2.10) to be true in almost all practical situations, the same cannot be said about (2.11). For devices with very narrow base widths [WB ~ 4c), frequency responses will extend beyond the values for which (2.11) is strictly true, namely, beyond / = 0.1/27TT c o[. While not apparent from the particular plots in Figure 2.4, at such high frequencies, there generally may be some differences in the values of ac parameters derived from the flux approach and Hansen's D D E approach. Further comments regarding this last point will be reserved until Section 3.3.3. 2.4 Use of the Flux Method to Infer Carrier Mobility in a Forward-Biased Barrier The issue of what form the carrier mobility takes in a forward-biased barrier is long standing. In the past [34]-[38], several workers have addressed the problem by solving the Boltzmann equation, or a set of derived moment equations, within the junction-field region; however, each made a different set of simplifying assumptions, and various conclusions about the form of the mobility and diffusivity were reached. In an attempt to resolve the issue, Lundstrom and Tanaka [28] have recently applied the flux method to a Mott barrier. In order to complete the appraisal of the flux approach, it is worth examining their analysis. 23 2.4.1 Critique of Lundstrom and Tanaka [28] Neglecting recombination, one can use (2.7) to write the mobility \xn and diffusivity Dn of the drift-diffusion equation in terms of the backscattering coefficients of the flux method: C ' - C C' + C (2-34) D n = -f^- (2-35) It follows that if one could derive expressions for £' and £, then fin and Dn would be spec-ified. In the context of the Mott barrier shown in Figure 2.5, Lundstrom and Tanaka [28] suggest that the backscattering coefficients for a small slab of width Az are < = it <2-36> and l - ( l - C / A z ) e x p ( - 9 A I / / f c B r ) ^ Az ' 1 > where AV is the potential drop across the slab.6 These choices can be disputed: • The expression for £' is that for a system obeying Boltzmann statistics in the absence of an electric field [17]; it is not obvious why the same expression should be expected to be valid for electrons accelerated by the field in the barrier, which is, in fact, a high-field region. The choice for (' is justified in [28] by the claim that as long as the potential drop AV is much less than ksT/q, then it can have little effect on backscattering. This overlooks the fact that if AV is very small, then so too will be the slab width Az, and thus the field £ = —AV/Az is likely to be high and have a strong effect on • The expression for ( is based on one suggested by Price [39]. However, Price considered the entire junction-field region, not a thin slab within the barrier. In the context of the Mott barrier of Figure 2.5, where the boundaries A and B follow 5The expression given for C is believed to be the intended form of equation (6b) in [28]. 24 E e(A) = Epm + <t>Bn 1 h A d F i g u r e 2.5: T h e c o n d u c t i o n - b a n d profi le of a M o t t bar r ie r ( i n w h i c h the e lec t r i c f ie ld is cons tan t ) . T h e assumed negat ive d i r ec t i on for the f ie ld , a n d the expressions (2.36) a n d (2.37) for the backsca t t e r ing coefficients of r ight - a n d lef t -going carr iers , are consis-tent w i t h the d iscuss ion in the text of [28]. Efm is the F e r m i leve l i n the m e t a l , Ec(z) is the c o n d u c t i o n - b a n d edge, and Fn(z) is the q u a s i - F e r m i level for e lectrons i n the semi -conduc to r . T h e c o n d u c t i o n - b a n d edge at z = A is g iven by EC(A) = Epm + $y3 n , where $ B „ is the bar r ie r height seen by electrons i n the m e t a l . 1 ^ z B 25 Price's notation, Price suggested that [39, eq. (130)] T = —exp(qVA/kBT), j nB (2.38) where the symbols are as follows: (j and (j refer to the composite backscattering probabilities for fluxes in the positive and negative z-directions, respectively, for the entire junction-field region between A and B; nA and nB refer to the electron concentrations at A and B, respectively; and VA is the voltage applied across the junction, taken as being positive under forward bias. Price further suggested that the ratio nA/nB be replaced by its equilibrium value of exp(—qVli/kBT), where Vu is the built-in barrier potential. Using this so-called "detailed-balance argument," equation (2.38) then yields Berz [40]. While it is reasonable to assume that (2.39) might continue to hold for small values of (j over the entire junction-field region, there is no justification for replacing (j, Cj> and the total junction potential Vj = Vii ~ VA in this equation by their respective slab values of (Az, ('Az, and A V to obtain (2.37) for (. Both Henisch [41, p. 11] and McKelvey et al. [17] have noted the powerful nature of the flux method, but have also pointed out the considerable difficulty in deriving expressions for the nonmeasurable backscattering coefficients on a rigorous, physically correct basis. McKelvey and Balogh [20] were able to derive the low-field values (2.8) and (2.9), but only by equating the predictions of the flux method to those following from the known form of the distribution function under bulk, low-field conditions. It is likely that establishing (' and ( within the high, built-in field of a forward-biased barrier would similarly require an a priori knowledge of the detailed transport physics, which, if known, could then be used to obtain pLn and Dn directly. While Lundstrom and Tanaka have suggested (2.36) and (2.37) for (' and £, the physical bases for these expressions are not clear. (j = 1 - (1 - (j)exp[-q(Vbi - VA)/kBT}. (2.39) When (j = 0, this is consistent with the thermionic-emission result derived by 26 The field-dependent mobility and diffusivity suggested in [28] are obtained by us-ing (2.36) for assuming AV <C kBT/q and expanding (2.37) to write ( = ^ ( 1 - ^ ) , (2.40) 2Dn0 \ vT J and then substituting into (2.34) and (2.35), which yields I+ \£(J,no/VT\ and 1 + l ^ n o / u r l These expressions are validated in [28] by the fact that they allow the DDE to be in-tegrated over the entire field region, from z = B to z = A, with the physically cor-rect boundary condition = +Jn/qvT [40], to yield a correct expression for Jn in the thermionic-emission limit. However, this justifies neither the assumptions made in ob-taining (2.41) and (2.42), nor the use of the DDE over the entire junction-field region. The latter point is particularly germane because, in the thermionic-emission limit, the distri-bution function changes drastically within a mean-free path of the metal-semiconductor interface (over the so-called  11 kT layer"), thereby invalidating the near-equilibrium as-sumption upon which the DDE is based [40]. Furthermore, as suggested by Rhoderick and Williams [42, p. 104], and as shown in Appendix A, it is possible to obtain the cor-rect expression for Jn in the thermionic-emission limit by employing the DDE with the low-field values fino and Dno of mobility and diffusivity, from z = B to the edge of the kT layer at z = d, together with the physical boundary condition n(d) — (+2Jn/qvT)e2 derived by Berz [40, eq. (23)], where the voltage drop across the kT layer is taken to be 2kBT/q [40] and e is the base of natural logarithms; this would seem to indicate that the correct mobility and diffusivity within the barrier, except in the kT layer, are the field-independent values /x„o and Dno, contrary to the assertions of [28]. It thus appears that the use of the field-dependent expressions suggested by Lundstrom and Tanaka, which is tantamount to dividing the usual drift and diffusion terms by 1 + \£/J.no/vT\, 27 can only be viewed as an empirical, phenomenological modification of the DDE, sim-ilar to such a modification suggested in the past7 [43]; correspondingly, the suggested expressions for the backscattering coefficients, equations (2.36) and (2.40), can only be viewed as phenomenological values needed to make the flux method consistent with this empirical modification of the DDE. While this ad hoc correction to the DDE does allow it to be solved with a physically correct boundary condition at the metal-semiconductor interface, to yield the correct expression for the current in the thermionic-emission limit, it does not follow that the use of this modified DDE, or, equivalently, the use of the flux method with the corresponding values of backscattering coefficients, should provide a valid, general description of carrier transport within forward-biased barriers. 2.4.2 Recent Work [44] As part of an invited journal paper on the continued use of the DDE in modern devices, Assad, Banoo, and Lundstrom have very recently performed numerical simulations of transport in a forward-biased, metal-semiconductor barrier [44]. Assad et al. employed a generalized "m-flux" method to solve the Boltzmann transport equation, and then used the solution to extract the rigorous carrier mobility. While their simulations do appear to show that the carrier mobility is reduced within the barrier region, this numerical result alone does not validate the arguments originally presented by Lundstrom and Tanaka [28], and a physical justification for the reduction in mobility is still lacking. 7If one assumes that \dF„/dz\ < \dEc/dz\ within the junction-field region, then employing the relations n(z) = Ncexp{[F„(z) - EC(Z)]/IIBT} and q£ = dEc/dz, one has 1 + \Sfino/vr\ = 1 + \Dno{dn/dz)/nvr\, and the use of (2.41) and (2.42) can be seen to be equivalent to using a so-called "concentration-dependent diffusivity," D„ = D„o/[l + \Dno(dn/dz)/nvT\]- The idea of dividing the usual drift and diffusion terms in the DDE by a factor such as 1 + \Dn0(dn/dz)/nvT\ was first empirically sug-gested by Persky [43, eq. (7)], with the hope that the modified DDE would yield more physical solutions in the presence of strong concentration gradients. 28 2.4.3 General Comments The above critique of Lundstrom and Tanaka [28] highlights the substantial difficulty involved in establishing values for the one-flux backscattering coefficients on a rigorous, physically correct basis, a problem that is generally encountered in the flux approach. Even the bulk, field-free and low-field values are based on a visualization of the dis-tribution function (shown in Figure 2.2) which is strictly incompatible with the forms that actually occur in practice. The field-free value (2.36) was originally obtained by McKelvey et al. [17] by performing a suitable averaging procedure, over a perfect hemi-Maxwellian, to find the mean distance in the z-direction that a carrier travels prior to experiencing a collision, and the subsequent value was then assumed to apply through-out a field-free region; however, this approach rejects the fact that a hemi-Maxwellian impinging upon a slab in the positive z-direction will not emerge on the other side as a perfect hemi-Maxwellian, 8 and that this emerging distribution will therefore see an effec-tive backscattering coefficient which is strictly different from the original. In the low-field case, McKelvey and Balogh [20] obtained the values (2.8) and (2.9) by comparing results from the flux method with those found from the well-known form of a displaced, full Maxwellian for the distribution function in a bulk, low-field region; however, such a com-parison rejects the fact that a displaced Maxwellian is strictly incompatible with the assumed one-flux form of abutted hemi-Maxwellians. Shockley [18] was the first to point out that these types of inconsistencies were necessarily inherent to the flux approach; interestingly, while making note of the problems, he came up with the same results for the backscattering coefficients as in [17] and [20], but did so mathematically, rather than physically, by simply demanding (in the time-independent case) that the one-flux equa-tions reduce to the usual continuity and drift-diffusion equations under the condition of negligible recombination, a technique which emphasizes the essential equivalence of the two sets of transport equations. The problems discussed here are obviously attributable to the fact that the one-flux 8This occurs because carriers in the incident distribution that are traveling tangentially to the surface are more likely to be backscattered, whereas those traveling perpendicular to the surface will penetrate more deeply, causing the emerging distribution to be "focused" in the positive ^-direction. 29 approach is an approximate analysis method. Nevertheless, the cited inconsistencies can-not be completely overlooked, and they provide an indication that any results obtained with the flux method, while perhaps yielding powerful aids to visualization and compact, analytical expressions, must generally be checked and refined using more rigorous, micro-scopic, and physically self-consistent descriptions of transport, especially in those cases where one can expect the usual (continuity and drift-diffusion) transport equations to fail. It is precisely this point that provides the main motivation for the work described in Chapter 3, where the one-flux expressions derived in [26] and [27], discussed earlier in Section 2.3, are more carefully examined and improved by means of an exact solution to the Boltzmann transport equation. 2.5 Conclusions The following conclusions can be drawn from this assessment of the one-flux method as a means of treating carrier transport in modern devices: 1. For a bulk region in which a small electric field is present, provided that the mean-free time for scattering TCO\ is much less than the carrier lifetime r r e c , and provided that the current does not change appreciably in a length of time r c o i , the one-flux equations are algebraically equivalent to the usual continuity and drift-diffusion equations, and this equivalence applies even if the region has small dimensions. 2. Recent applications of the flux method to short-base transistors [25]-[27], which yield results that agree closely with solutions to the Boltzmann equation [31], are essentially equivalent to Hansen's [29] approach of using the drift-diffusion equation with appropriate boundary conditions. 3. Use of the flux method in the region of high, built-in electric field within a forward-biased junction is curtailed by a lack of knowledge of the backscattering coefficients for the right- and left-directed fluxes. While recent efforts [28] have attempted to establish relationships for these coefficients, the choices of a field-free value for car-riers moving down the potential-energy barrier, and a modified thermionic-emission 30 result for carriers moving up the barrier, have no obvious physical bases. The corre-sponding result of a reduced carrier mobility within a forward-biased junction also has no obvious physical basis. The problem of establishing values for the nonmea-surable backscattering coefficients on a rigorous, physically correct basis is inherent to the flux approach, and serves as an indication that results obtained with this method should generally be checked and refined using more detailed descriptions of carrier transport. These results not only yield a basic understanding of the one-flux method, but also provide an impetus for the investigation undertaken in Chapter 3. Chapter 3 Expressions for the High-Frequency Characteristics of Quasi-Ballistic Bipolar Transistors 3.1 Introduction The modeling of the intrinsic, small-signal, high-frequency characteristics of bipolar tran-sistors is traditionally based upon a solution to the minority-carrier diffusion equation in the base [32, Ch. 5], [45], [46]. For base widths in the quasi-ballistic regime, that is, for base widths comparable to the collision-free path length of the minority carriers, the diffusion equation breaks down [31], and one cannot use the conventional expressions. Grinberg and Luryi [31], [47] recently investigated the problem by deriving an analytical solution to the Boltzmann transport equation (BTE) , and emphasized the distinctive na-ture of high-frequency transport in purely diffusive versus purely ballistic devices. Alam, Tanaka, and Lundstrom [26] later found simple expressions, using the one-flux approach, that reproduce certain features of the B T E solution with great accuracy. Subsequently, Alam, Schroter, and Lundstrom [27] showed that these same expressions predict signifi-cantly different values, when compared to the classical results [32, Ch. 5], for the forward characteristics of modern thin-base transistors; given the conclusions of Chapter 2, this result suggests that quasi-ballistic transport effects may be present in these devices to an extent that requires more careful treatment. In this chapter, the nature of high-frequency transport in short-base devices is more closely examined by employing the B T E approach of Grinberg and Luryi [31], [47]. Com-31 32 pact expressions are developed that correctly yield both the magnitude and phase of all the forward characteristics, as predicted by the B T E , up to the intrinsic transit frequency. Expressions for the reverse small-signal parameters, starting from the basic one-flux equa-tions of Shockley [18], are also derived. The result is a complete set of expressions for the intrinsic small-signal parameters of quasi-ballistic transistors, valid from very low to very high frequencies. In Section 3.2, the B T E and flux approaches employed to determine the forward char-acteristics are briefly reviewed; details on the B T E approach can be found in [31] and [47], and additional details on the flux approach are available in [26], [27], and Chapter 2 of this thesis. In Section 3.3, it is shown that ballistic effects on the carrier transport play a significant role in the degradation of the current gain at high frequencies, even for quasi-ballistic base widths, and expressions for the forward characteristics that account for this degradation are found by combining the one-flux expressions with those of Thomas and Moll [48]. In Section 3.4, the moving-boundary approach of Early [45] and Pritchard [32, pp. 237-244], [46] is applied to the basic one-flux equations to find expressions for the reverse parameters. Section 3.5 summarizes the conclusions of this chapter. 3.2 B T E and Flux Approaches 3.2.1 B T E Approach Following the approach of Grinberg and Luryi [31], [47], it is assumed that the base is field free and homogeneous in the scattering parameters, and that the distribution function is inhomogeneous only in the z-direction, and a solution to the B T E is sought in the form /(r, k, t) = / ( z , k, u) + f(z, k, u) exp(jwi), (3.1) where r is the coordinate vector, the base lies between z = 0 and z = WB, t is time, / represents the static (dc or bias) part of the distribution function, / represents the complex amplitude of the dynamic (ac or small-signal) part of the distribution function,1 !The convention of explicitly distinguishing between static and dynamic quantities is only used in this chapter; elsewhere in this thesis, the nature of each symbol is obvious from the context. 33 u) is the radian frequency of the applied small signal, and u is the cosine of the angle 6 between the wave vector k and the z-axis. Parameterizing the collision integrals with a scattering length lsc(k), the B T E then splits into two equations, one for the static part / ( z , k, u), and, of interest in this work, one for the complex ac amplitude / ( z , k, u): df(z,k,u) f(z,k,u) _ f0(z,k) U dz l*(k,u) lsc(k) ' [ ' where fo(z, k) = (1/2) / ( z , k, u) du is the "angle-averaged" part of / , and l*(k,u) and ltot(k) are defined by 1 jcom* 1 + 7-TTT (3-3) and i*(k,u>) hk itot(k) i l l , N + 7 7T\' (3-4) Uot{k) lsc(k) lcp{k) where m* is the effective mass and lcp(k) is a characteristic length associated with capture processes. For homojunction devices, f(z,k,u) is subject to the boundary conditions of an injected hemi-Maxwellian at the base-emitter junction (z = 0), /(<),*,u > 0) = / f (0,fc ,u) = ^ ^ e x p kBT (3.5) and of a perfect sink at the base-collector junction (z = WB), f(WB,k,u<0) =f-(WBlk,u) =0, (3.6) where the symbols are as follows: q is the magnitude of the electronic charge; kB is Boltzmann's constant; T is the temperature; n*E = nBoGX-p(qVBE/kBT) is twice the electron concentration in the static hemi-Maxwellian distribution injected into the base from the emitter, where nBo is the equilibrium electron concentration in the base and VBE is the applied bias voltage across the base-emitter junction; A^c is the effective density of states; = h2k2/2m* is the electron energy; and VBE is the applied signal voltage across the base-emitter junction. With these boundary conditions on the inbound parts of f(z,k,u), equation (3.2) can be converted into an integral equation for /o(z,fc); once 34 found, the solution for fo(z, k) can then be used to obtain the various quantities of interest, as outlined in [31] and [47], such as the distribution function f(z,k,u), the signal-charge concentration n(z) = (1/47T 3) / f(z, k, u) dk, and the signal current Jn(z) = (-q/47v:i)f(hkz/m*)f(z,k,u)dk. It should be noted that the actual inbound distributions may differ slightly from the forms assumed in (3.5) and (3.6). The validity of (3.5) depends on the Bethe condition [31, eq. (6)] being well satisfied, and (3.6) neglects the small amount of flux backscattered from the collector-base, space-charge region. Nevertheless, equations (3.5) and (3.6) are reasonable approximations, and they are exactly consistent with the boundary conditions assumed in previous treatments of high-frequency, short-base transport [26], [27], [31]. It should also be noted that, strictly speaking, the use of a scattering length lsc(k) to simplify the collision integrals is only rigorously valid for elastic and isotropic scattering, such as by acoustic phonons; exact inclusion of inelastic and anisotropic processes, such as scattering by optical phonons and charged impurity centers, leads to much more involved equations, and precludes an analytical approach [31]. However, the latter processes are less important when there are no hot electrons [47], which is the case for homojunction devices and heterojunction devices with graded emitter-base junctions; moreover, results from this approach for the static problem, even with the use of a constant scattering length /SC(A;) = / s c , are in agreement with more rigorous simulations performed for silicon that do explicitly account for these processes [49]. Hence, in this work, equation (3.2) will be used with constant lengths lsc(k) = lsc and lcp{k) = 'cP, together with the ideal boundary conditions (3.5) and (3.6), to examine the dynamic problem, an approach that should be reasonably valid at least for silicon homojunction devices, and the focus shall be on improving the one-flux expressions in [26] and [27] to account for high-frequency, quasi-ballistic transport effects predicted by this simplified B T E . 3.2.2 Flux Approach Details on the small-signal, one-flux approach of examining the forward characteristics are available in [26] and [27], and a general appraisal of the flux method was presented in Chapter 2 of this thesis. Here, it is noted only that by assuming a distribution function 35 consisting of abutted hemi-Maxwellians, and by solving the basic one-flux equations, high-frequency transport can be characterized by transmission and reflection coefficients for a slab of width z [26, eqs. (10)-(14)]: f(z) = cosh(A/cz)+ K2 + l 2K sinh(A«z) - l R(z) = f(z) where AC and A are denned by and K 2 - l 2K sin \I(\KZ) XD + 1 nO 1 ju + — , TTecVT vT (3.7) (3.8) (3.9) (3.10) and Dn0 = (8kBT/9nm*)^2lsc is the low-field diffusivity, r r e c = {irm*/8kBT)^2lcp is the electron lifetime, and VT = y2kBT/nm* is the thermal velocity. Straightforward analysis then allows the forward- and backward-going fluxes a(z) and b(z) at a point z to be written in terms of R, T, and the inbound flux 5(0) = n*EVTqVBE/2kBT: . f(z)a(0) alz) = =——— , , 1 - R(WB - z) R{z) b{z) = R(WB-z)a(z). (3.11) (3.12) The fluxes a(z) and b(z) can then be used to find the signal-charge concentration h(z) = [a(z) + b(z)]/vT and the signal current Jn(z) = -q[a(z) - b(z)]. The common-base and common-emitter current gains can be written in terms of f(WB) and R(WB): a — 0 f(WB) 1 - R(WB) a 1 \ - i - i cosh(A K iy B ) + ( - sinh(AKW B) AC / 1 - a 2sinh 2 ( ^ £ j + Q ) s i n h ( A A c i y B ) (3.13) (3.14) 36 3.3 Forward Parameters 3.3.1 Effects of Ballistic Versus Diffusive Decay Shown in Figure 3.1(a) are plots of |a | versus cyclic frequency / = o>/27r from the B T E and flux approaches. Parameter values used for this and all subsequent computations in this chapter are VT — 2 x 107 cm/s, / s c = 500 A, and lcp = 1000/sc. Evidently, for base widths in the quasi-ballistic regime (WB ~ 4 c ) , the flux expression (3.13) signif-icantly overestimates |or| at high frequencies; for example, for WB = l 4 o the cv-cutoff frequency (defined to be that at which |a | has fallen to l / \ / 2 times its low-frequency value) is overestimated by about 40 percent. The difference between the results from the two methods is emphasized in Figure 3.1(b), which shows plots of the phase trajectory (Re a, Im a) for the WB = HSc case. These results are somewhat surprising, given that the one-flux expressions match other aspects of the B T E solution very closely, in both the static [25] and dynamic [26] cases. However, the discrepancies can be understood by considering the change in the physical nature of the decay of \a\ as the base width shrinks and the transport changes from being diffusive to ballistic, and by noting the ef-fects of the decay mechanism on the phase angle of the forward-going part of the dynamic distribution function. In the case of diffusive transport, the mechanism by which |c*| degrades is well known. At high frequencies, carriers injected at the emitter side of the base during a half-period TT/U have insufficient time to diffuse to the collector during the subsequent half-period. Thus, as the frequency increases, an increasing fraction of these injected carriers return to the emitter rather than reach the collector. In terms of the steady-state signal current, this primarily has the effect of increasing |J n(0)|, the magnitude of the current at the emitter side of the base. As a result, |or| = |J n (M /B) | / | - /n(0) | degrades. With this decay mechanism, the two halves of the dynamic distribution function / retain an essentially hemi-Maxwellian shape, the form assumed by the flux approach. The forward-going half at a point z may thus be expressed as f(z,k,u>0) = f+(z,k,n) = ^l exp Nc r l k B T \ (3.15) Figure 3.1: Behavior of the common-base current gain a versus frequency from the B T E (solid lines) and flux (dashed lines) approaches. Part (a) shows the magnitude response for different values of the base width. Part (b) shows the computed phase trajectory (Re cv, Im a) up to / = 1 THz for the WB = USC case. Parameter values used for this and all subsequent computations in this chapter are the same as those in recent studies [26], [29], [31] of short-base transport: VT = 2 x 107 cm/s, / s c = 500 A, and lCD = 1000/ SC * 38 where h+(z) is the forward-going, signal-charge concentration at z. It follows from (3.15) that the phase angle of f+(z,k,u) is constant with respect to k and u and given by lf+(z,k,u) = Zn+(z), (3.16) and that the signal velocity of the forward-going ensemble is V  + (z) EE f(hkz/m*)f+(z,k,u)dk ff+(z,k,u)dk - VT- (3.17) In the case of ballistic transport, as pointed out in [47], |a| degrades because of the scatter in the energies (speeds) and angles of the inbound electrons. Members of the injected ensemble, that have different normal components of velocity, arrive at the collector at different times. In the frequency domain, this means that the carriers at the collector are out of phase with each other; that is, at high frequencies, f+(WB, k, u) does not have a constant phase angle with respect to k and u. This can be seen by considering (3.2) in the ballistic limit, which reads - a hk m* dz' (3.18) and then noting that the solution to this for u > 0 may be written as f+(z, k, u) = / + (0 , k, u) exp —jzurni u hk (3.19) Since f+(0,k,u) given by (3.5) has zero phase by definition, this means that the phase of f+(z,k,u) is lf+(z,k,u) = —zum uhk ' (3.20) and, unlike the value specified by (3.16), this varies appreciably with k and u for large enough w, especially for z = WB. Using (3.19), the magnitude of the signal charge at the 3 9 co l lec to r t h e n fol lows: \n+(WB)\ = ^ J f+(WB,k,u)dk ——z / / k2f+(0,k,u)exp(-j\VBuJm*/uhk)dudk Z7T J0 Jo (3.21) A t h i g h frequencies, exp(— jWBu>m*fuhk) becomes a r a p i d l y o s c i l l a t i n g e x p o n e n t i a l , de-creas ing the c o n t r i b u t i o n to the in teg ra l . A s a resul t , | f i + ( j y B ) | "washes ou t " as the f requency increases, decreas ing | J n ( W B ) | ( the m a g n i t u d e of the s igna l cur rent at the co l lec tor side of the base) a n d degrad ing | a | = | « / n ( W s ) | / | - / n ( 0 ) | . E a c h of the above decay m e c h a n i s m s , w h e n present, can be expec t ed to become signif icant o n l y w h e n / ~ 1/2TTTB, where TB is the base t rans i t t i m e . O n e m i g h t also conc lude , b y e x a m i n i n g the behav io r of rB as a f unc t i on of base w i d t h [26], [49], t h a t the b a l l i s t i c decay effect w o u l d not be i m p o r t a n t unless WB <C / s c . H o w e v e r , i t tu rns ou t tha t th is m e c h a n i s m is p l a y i n g a s ignif icant role even for WB ~ / s c , a n d th is is the m a i n reason for the discrepancies shown i n F i g u r e 3.1. E v i d e n c e of th is is presented i n F igu re s 3.2 and 3.3. F i g u r e 3.2 shows the d i s to r t ions i n Z / + r e su l t ing f r o m the spread i n the inc iden t angles of the i n b o u n d ensemble (further d i s t o r t i o n w i l l also be present due to the spread i n the inc iden t energies) . P l o t t e d is lf+(z, yf2m*kBT/h, u) versus z a n d u , tha t is , the phase angle of / + , for a thermal -average car r ie r h a v i n g energy — kBT, as a f unc t i on of u = cos 9, for each p o s i t i o n z a long the base. P l o t s are shown for WB = 3 4 c , WB = 1 4 c , a n d WB — / s c / 2 , w i t h the phase of / + eva lua ted at the t r ans i t f requency ft = l/2nTB, where [25], [29], [49] Wl WR 2Vn0 vT For WB = 3 / s c , the phase of / + is r e l a t i v e l y cons tant , w h i c h is consis tent w i t h ( 3 . 1 6 ) . O n the o ther h a n d , for WB = l / s c a n d WB = 4 c / 2 , the phase of / + c l ea r ly varies w i t h respect t o u, w h i c h is consis tent w i t h ( 3 . 2 0 ) , a n d / + is c l ea r ly not a h e m i - M a x w e l l i a n , e spec ia l ly near the co l lec tor . Hence , i n these cases, the b a l l i s t i c decay m e c h a n i s m is l i k e l y to be qu i t e s igni f icant . < -100-Figure 3.2: Phase angle /.f+(z,y/2m*kBT/h,u) of the dynamic distribution function versus normalized position z (in units of the base thickness) and cosine of angle u = cos 9 for (a) WB = 3/ s c, (b) WB = 1/ s c , and (c) WB — / s c / 2 . Each plot shows the phase of at the transit frequency ft = 1/2TTTB, where rB is given by (3.22). 42 1.5 1.4 in > Q 1.2 UJ N _ i < 2 1.1 o 1.0 0.9 0.0 0.2 0.4 0.6 0.8 1.0 POSITION IN UNITS OF BASE THICKNESS (c) Figure 3.3: Normalized magnitude of (a) the forward-going, signal-charge concentration |n + (z) /n + (0) | , (b) the net signal current | Jn(z)/qa(0)\, and (c) the forward-going signal velocity \V+(Z)/VT\ versus position z (in units of the base thickness), at different frequen-cies about / = ft, from the B T E (solid lines) and flux (dashed lines) approaches. A l l plots are for the WB — l l s c case, for which ft = 1/27TTB = 364 GHz. The approximate frequency corresponding to each curve is labeled on the plots. The results at 1 THz, which is well above ft, are included only to help illustrate the effects of the different decay mechanisms discussed in the text. 43 Figure 3.3 shows magnitude plots of the forward-going, signal-charge concentration n+(z), the net signal current Jn{z), and the forward-going signal velocity v+(z) versus po-sition z, at different frequencies about / = ft. The plots are shown for the WB = HSC case, which was chosen for illustration, although similar results were obtained for WB = 4 c / 2 . Figure 3.3(a) shows |re +(H /a)| from the B T E approach decreasing at high frequencies, consistent with the ballistic decay mechanism, and Figure 3.3(b) shows that, as a result, while there is an increase in |J„(0) | corresponding to the diffusive decay mechanism, there is also a significant decrease in | « / N ( W B ) | - The flux solutions in Figure 3.3(a) do not predict the reduction in |n +(l4 ye)|, and hence | < / N ( W B ) | in Figure 3.3(b) is overesti-mated. Note, however, that the overestimation of | « / T I ( W B ) | = 9 | " + ( W / B ) | | £ + ( V I / B ) | is not as large as might be expected from a consideration of | n + (Ws) | alone. This is because, as shown in Figure 3.3(c), the flux method underestimates the value of | V + ( W B ) | . The B T E solution shows an increase in the magnitude of the signal velocity with frequency, which can be attributed to the fact that the signal charge "washes away" for carriers with low u (traveling at large angles with respect to the z-axis) and low k (low speeds) faster than for those with high u (traveling at small angles) and high k (high speeds).2 Nevertheless, the decrease in | n + (WB) | does dominate, and the resulting lower value for | < / T I ( W B ) | from the B T E is the primary reason for the differences in Figure 3.1. One important consequence of overestimating the decay of \a\ is shown in Figure 3.4, where the magnitude and phase of the common-emitter current gain have been plotted for the WB = l / s c case. While \(3\ is correctly predicted by the flux expression (3.14), a fact already demonstrated in [26], note that the so-called "excess phase" of /?, namely, the phase of (3 above and beyond 90 degrees at / ~ 1/2TTTB, an important high-frequency parameter, is significantly overestimated. For WB = llsc, the discrepancy at f = ft is 12 degrees, and for WB = 4 c / 2 , it increases to 16 degrees; for WB — 3/ s c , the difference drops to 5 degrees, and it is insignificant for WB > 3/ s c. Generally speaking, the magnitude and phase of the intrinsic device characteristics at high frequencies (/ ~ 1 / 2 7 t t s ) can only be found by solving the charge-transport 2 T h e magnitude of the signal velocity can be thought of as the average speed associated with those velocity channels that carry the signal current. Since the current at high frequencies is effectively carried by channels with higher u and k, the magnitude increases. 44 Figure 3.4: Magnitude and phase of the common-emitter current gain /3 versus frequency for the WB = 11 computed from the B T E (solid lines) and flux (dashed lines) approaches. The magnitude responses are virtually indistinguishable. The open circles show the values from the Thomas-Moll form (3.24). 45 equations. Given that this is the case, it would seem worthwhile to find a simple way to more closely match the B T E results for a and (3 in Figures 3.1 and 3.4. 3.3.2 Expressions for the Forward Parameters In [26], the authors recognized that the one-flux expressions would break down when transit-time delays associated with the velocity distribution of the injected carriers be-came important, which is the CcLSG^  3jS shown above, even for quasi-ballistic base widths. They suggested that their results might be generalized by integration over the differ-ent particle streams of the inbound ensemble. Unfortunately, while this method works for purely ballistic transport (WB/ISC —>• 0), for which the particle streams can be considered to be noninteracting [26, eq. (33)], it cannot be used in the quasi-ballistic case (WB/lsc ~ 1), and another approach must be taken. Expressions for the current gains a and j3 can be found by employing the approach suggested by Thomas and Moll [48]. Using very general arguments based only on network theory, and without making any assumptions about the underlying transport physics, Thomas and Moll derived the following representations for the current gains: where K is a parameter, ao is the low-frequency magnitude of a, and fa is the a-cutoff frequency. These equations are generally very accurate for f < fa, provided only that K > 0.8, a0 > 0.9, and Z/3(/ = fa) does not greatly exceed 90 degrees [48], all of which will be true in practical cases. A value for Qf0 to use in (3.23) and (3.24) can be found by setting u = 0 in (3.13): where (2.30) and (2.31) have been used to write K\U=0 = vTLn/Dn0 and A / c | w = 0 = 1/Ln a (3.24) (3.23) (3.25) with Ln = y/DN0TTEC. On the other hand, no analytical expressions are available for K 46 and fa- However, it was empirically noted from the B T E simulations of this work that for the range of base widths one can expect to find in modern short-base devices, namely, 3 4 c > WB > 4 c / 2 , and for different values of the material parameters VT and 4 c ? the frequency fa may be found using / a « 1 . 1 2 / « = 1.12/27TTB, (3.26) where TB is specified by (3.22). A value for K then follows from the relation4 [48, eq. (38)] a0Ja 1 .1 Za0 Figure 3 . 5 shows plots of the magnitude and phase of a computed from (3.23) employing these values, superimposed on the B T E results. In all cases, the agreement is excellent at frequencies / < fa. Equation (3.24) for j3 was also found to work well, as shown, for example, for the WB = 1 4 c case, in Figure 3 .4. While preserving the magnitude response, equation (3.24) also correctly predicts the phase, with the maximum phase error for f < ft being less than 1 degree; for WB = 3 / s c and WB — 4 c / 2 , the same was found to be true, with maximum phase errors of less than 0 . 5 and 2 degrees, respectively. It should be pointed out that, for a given device, any effects arising from the nonidealities discussed in Section 3 .2 . 1 (with respect to the assumed boundary conditions and the description of carrier scattering) could be taken into account by modifying the factor relating ft and fa in (3.26); nevertheless, the actual value should be close to that following from the simplified B T E approach,5 which can hence serve as a useful guideline, and the forms (3.23) and (3.24) for the current gains should still be valid. Expressions for the forward, common-base, admittance parameters can be found by 3For example, it was found that (3.26) remained true, and that (3.23) and (3.24) worked equally well, when the values vT = 1 x 107 cm/s and / s c = 150 A, which are typical for the base of a modern silicon device [49], were chosen instead of the values specified in the caption of Figure 3.1. 4By comparison, in the case of purely diffusive transport (WB/ISC oo), it can be shown analytically that a0 = a0D = sech(WB/LN), fa = 1.22/2TTTBD, and K = l/1.22a0D, where TBD = WB/2Dn0 is the base transit time in the diffusive limit [48], [50]. 5The BTE result for fa should be contrasted with that following from the flux approach. Whereas the BTE yields fa « 1.12/t for 3JSC > WB > W 2 , the flux expression (3.13) yields /„ = 1.32/t for WB = 3/sc, fa = 1.61ft for WB = l / s c , and fa = 2.34/t for WB = lK/2. 47 FREQUENCY (Hz) (a) FREQUENCY (Hz) (b) Figure 3.5: Magnitude (a) and phase (b) of the common-base current gain a versus frequency for different values of the base width. The solid lines show results from the B T E and the stippled lines show values computed from the Thomas-Moll form (3.23). For each value of base width, the Thomas-Moll form agrees with the B T E values at least up to fa, the cv-cutoff frequency. 48 combining the one-flux and Thomas-Moll expressions. The input current J n(0) is unaf-fected by the ballistic decay mechanism discussed previously, and the input admittance ynu = —Jn(Q)*A-/V~BE may therefore be written, using the flux approach, as _ q2nEvTA y n b i ~ 2kBT cosh(AAcWB) + (l/K)smh(\KWB) cosh(\KWB) + [(K2 + l)/2/c] smh(\KWB) (3.28) where A is the cross-sectional device area. A very accurate form for the transadmittance then follows by noting that _ +Jn(WB)A V2\u = Yr = ~aynbi, (3.29) VBE and then using a as given by the Thomas-Moll form (3.23). These expressions were found to predict both the magnitude and phase of the B T E results very well up to at least fa, as shown, for example, for the WB = l l s c case, in Figure 3.6. Shown also in Figure 3.6 is the value of the transadmittance following from the flux analysis alone: _ -q2nEvTA m b l ~ 2kBT cosh(A/cWB) + ( — j sm\\(\KWB) (3.30) The noticeable differences between the values predicted by (3.30) and the B T E results at high frequencies, particularly in the magnitude response, are those responsible for the discrepancies in Figures 3.1 and 3.4. Equation (3.29), with the employment of the correct form (3.23) for the current gain, is hence preferable. 3.3.3 Additional Considerations While the effects of the ballistic decay mechanism cause the distribution function to be-come distorted at high frequencies, and while this causes the flux approach to become invalid, there is another factor which should be considered. As shown in Appendix B, even if one assumes that the distribution function consists purely of hemi-Maxwellians, the usual one-flux equations are not consistent with the true balance (moment or con-servation) equations derived from the B T E ; it turns out that use of the usual one-flux approach amounts to employing a momentum balance equation with an incorrect mo-49 180 160 140 co 120 CD CD CD LU C/3 < X Q. 100 80 60 40 20 0 10 Phase of y21bi Phase of y 1 1 b i .10 10 10" 10 FREQUENCY (Hz) 2.0 1.5 1.0 120.5 LU Q Z) Z CD < Q LU N O Figure 3.6: Magnitude and phase of the forward, common-base, admittance parameters ynbi and y2m versus frequency for the WB = 1 4 c case, for which fa = 408 GHz. The magnitude values are for the normalized admittances \ynbi/9m\ and |y2i6t/<7m|, where gm is the low-frequency transconductance found by evaluating (3.30) at u> = 0. The solid lines show results from the B T E , and the stippled lines show values from (3.28) and (3.29). The dashed lines show values for y2m predicted by (3.30). 50 mentum relaxation term. The condition under which this discrepancy is important is given by (B.16); for the small-signal problem, this condition is satisfied at frequencies / > 0.1/27rrcoi ~ 100 GHz, precisely where the usual one-flux approach disagrees with the B T E . Given that this is the case, it is worth briefly considering two alternative approaches for finding compact expressions for the forward parameters. The first approach6 is to consider the solution of (B.8) and (B.9), the proper balance equations for a distribution function made up of hemi-Maxwellians. For the small-signal problem of interest in this work, this amounts to considering the solutions to the following equations: —n . . 1 dJn ^ - - ^ 7 ' (3-31) Tree q dz ' J - n d h (2 jn — quNQ— — dz \7T D nO L VT juJn, (3.32) where the appropriate boundary conditions to be used, corresponding to (3.5) and (3.6), are as follows [29, eqs. (5), (6)]: J n(0) = -q[hE - h(0)]vT, (3.33) Jn(WB) = -q[h(WB)]vT. (3.34) Unfortunately, it turns out that while this approach does yield expressions that provide improvement over those found from the usual one-flux method, significant discrepancies with the B T E results still occur (for example, for l / s c > WB > / s c / 2 , the resulting expressions continue to overestimate both the a-cutoff frequency by at least 20 percent and the excess phase of /? by up to 10 degrees). Despite the use of a momentum relaxation term that is consistent with the hemi-Maxwellian visualization, the distortion of the distribution function at high frequencies (due to the ballistic decay effect) dominates, and this method therefore remains inadequate. The second approach is to consider the neglect of the momentum relaxation term 6This approach is equivalent to employing the modified (corrected) one-flux equations (B.22) and (B.23) together with the boundary conditions 5(0) = n*EviqVBE I^BT and 1(WB) = 0. 51 altogether, that is, to consider the solution of (3.31) and the usual diffusion equation: Jn = qDn0^, (3.35) dz where (3.33) and (3.34) are still the appropriate boundary conditions to be used.7 It turns out that this approach does not yield expressions that match all of the forward parameters found from the B T E ; for example, the resulting expression for ymi yields a phase that disagrees with the B T E at / = fa by up to 10 degrees for l / s c > WB > 4c/2. However, the expression for the common-emitter current gain from this approach, P 2 s i n h 2 f ^ U ^ f s i n h f ^ 1 - 1 \2L*J vTL*n \L*t (3.36) does match both the magnitude and phase of the corresponding B T E result, and hence it may be used instead of the Thomas-Moll form (3.24); the common-base current gain then also follows from the identity a = P/(P + 1). This is a very surprising result, since the distortion of the distribution function (from the hemi-Maxwellian shape) at high frequencies means that the true balance equations corresponding to the B T E are the involved ones following from (B. l ) and (B.7): —n • ~ 1 dJn = 3 ™ ~ ( 3 - 3 7 ) ( f c p ) q dz ^ + 2' ' dz " dz ~ ~ dh ^_ „du ,; ,, , ~ Jn = qDn— + 2 / / n n— - {(Tm))juJn, (3.38) where p,n = q{{fm))/m*, u = W/h, Dn = (2u/q)p,n, and where l / ( f c p ) , l / ( ( f m ) ) , and W are defined by the right sides of (B.3), (B.4), and (B.6) with f(z,k,u,t) replaced by f(z,k,u,u>). Obviously, the neglect of the last two terms in (3.38) is compensated by the simultaneous assumption that Dn(z,u>) = Dno (and, to a much less extent, the assumption that (f c p ) = r r e c ) , such that (3.31) and (3.35) yield results for the current gains that are in agreement with (3.37) and (3.38), that is, in agreement with the B T E . 7This is the approach originally suggested by Hansen [29], although the only forward parameter he considered was |/?|. A discussion of Hansen's approach, including the choice of appropriate boundary conditions, can be found in Section 2.3. 52 3.4 Reverse Parameters In order to complete the intrinsic admittance matrix, expressions are needed for the reverse parameters. Under the assumption that the collector-base junction acts as a perfectly absorbing boundary for charge exiting the base, and noting that negligible charge is injected into the base from the collector for operation of the device in the forward-active mode, the reverse parameters are determined solely from the effect of base-width modulation due to depletion-region widening at the collector-base junction. In order to obtain expressions for the reverse parameters, one thus has to apply a "moving boundary condition" to the transport equations at the collector edge of the base [32, pp. 237-244], [45], [46]. For short-base devices, expressions for the reverse parameters can most conveniently be obtained by applying this boundary condition to the one-flux equations. Following the results for the forward parameters, it should be kept in mind that the expression for the reverse transadmittance found from this approach will not be entirely accurate at transit-time frequencies and quasi-ballistic base widths. On the other hand, the expression for the output admittance should not suffer from this limitation. 8 Neglecting generation, and combining the static and dynamic solutions to the time-dependent, one-flux equations (2.2) and (2.3), the total right- and left-moving fluxes at a point z may be written 9 as [18, eqs. (3), (4)] a(z) = A^oo exp(ez) + B exp(-ez) + [CRoo exp(ez) + D exp(-ez)] exp(jut), (3.39) b(z) = Aexp(ez) + BR^exp^—ez) + [Cexp(ez) + DR,*, exp(-ez)] exp(jujt), (3.40) where A, B, C, and D are arbitrary constants to be determined by the moving boundary 8The cited inaccuracies of the flux solutions will occur regardless of whether one employs the usual forms (2.2) and (2.3) or the modified forms (B.l) and (B.2) of the one-flux equations; therefore, for simplicity, the usual forms are chosen to derive the reverse parameters. 9The use of the symbol "R^" in (3.39) and (3.40) follows the notation of Shockley [18]. R^ and are the static and dynamic reflection coefficients, respectively, for a slab of infinite length. 53 conditions, and where e = XK, RQO = ux/[2D no(A-fe) + i>7'], £ - e|w=oj a n d ^oo = Rco\u=o-For the purpose of obtaining the reverse parameters, the signal voltage at the emitter-base junction is set to zero, and the signal voltage VBc at the collector-base junction is assumed to cause a moving depletion-region boundary and hence a time-varying base width: dWR ~ WB = WB + Wb exp(jut) = WB + -ITT^VBC exp(jut). (3.41) dVBC At the location of the emitter-base junction z = 0, equation (3.39) is then subject to the boundary condition of an inbound flux that has only a static component (h*E/2)vT, which gives — * = AR^ + B + [CRoo + D] exp{jut). (3.42) At the location of the moving boundary z = WB, equation (3.40) is subject to the boundary condition of a total inbound flux that is zero, which gives 0 = Aexp(eWB) + BR^ exp(-eW B ) + [Cexp(cW B ) + DRooexpi-eWB^expiJut). (3.43) Using (3.41) for WB, assuming that WB is sufficiently small for linear, small-signal op-eration, that is, assuming JeVVel <C 1 and |eWs| <C 1, and expanding the exponentials in (3.43) and neglecting the higher-order terms, one then obtains 0 = [A exp(eWB) + BR^ exp(-eWB)} + [AeWB exp(eWB) - BR^eWs exp( -e iy B ) + Cexp(eWB) + DRoo exp(- tW B ) ] exp(jut). (3.44) By equating the time-independent and time-dependent parts of the left and right sides of (3.42) and (3.44), one then obtains four equations that can be solved for the constants A, B, C, and D; these can then be substituted back into (3.39) and (3.40) to find the total fluxes at any point. The reverse transadmittance can be found by evaluating (3.39) and (3.40) at z = 0 54 and extracting the time-dependent parts to find the net signal current at the fixed emitter boundary: _= -Jn{0)A _ —qA[C + DRQO] VBC VBC Employing the solutions for C and D, this can be worked out to find that (3.45) J/126» -qenEvTWBA VI w Roo{i - Rio) {ei-RlQe2){el-R?00ei)\ (3.46) where Vw = WB/(dWB/dVBc), £\ = exp(eWB), e2 = exp(-eWB), &\ = exp(eWB), and e2 = exp(—eWB). The output admittance can be found by evaluating (3.39) and (3.40) at z = WB = WB + WB&xp(ju>t), expanding the exponentials and neglecting the higher order terms, and extracting the time-dependent parts to get the net signal current at the moving collector boundary: 2/226i = _ +Jn{WB)A -qA VBC VBC {eWB[ARoo exp(EWB) - B exp(-eWB)] + CR^ exp{eWB) + D exp( -cW B )} . (3.47) Employing the solutions for A, B, C, and D, this can be worked out to find that V22bi --qenEvTWBA 2V w 2R00R00{e~1 - e2) - (R^ + l ) ( g 1 - ^ g 2 ) (ex - Rle,)^ - Rl~e2) (3.48) Figure 3.7 shows plots of the magnitude and phase of (3.46) and (3.48) for WB = 1 4 c , along with the following conventional results [32, p. 241], [45], [46]: Vl2bi 2/226; = -gDn0nEWBA LnL*nVw +qDn0hEWBA LnL*V\ w csch csch 'W, B csch coth 'WB (3.49) (3.50) The differences in magnitude at low frequencies, between the flux and conventional ex-pressions, are a reflection of the fact that short-base devices have an increased Early volt-age [51], [52] resulting from a static charge distribution which is trapezoidal rather than triangular. The behavior of the phases is the same as that for the forward parameters. 55 180 160 140 120 100 80 60 40 20 0 10" Phase of y1 2 bj 10 Phase of y22bi 10a 10 1 V22bi I 1 y i2b i i 10 FREQUENCY (Hz) 10 11 10 120.1 Figure 3.7: Magnitude and phase of the reverse, common-base, admittance parameters J/i26t and j/22i>t versus frequency for the WB = llSc case, for which fa = 408 GHz. The magnitude values are for the normalized admittances \y\2biVw/n*EA\ and \y22tiVwIn*EA\. The long-dashed lines show values from the one-flux expressions (3.46) and (3.48), and the short-dashed lines show values from the conventional expressions (3.49) and (3.50). 56 3.5 Conclusions The following conclusions can be drawn from this study of high-frequency transport in bipolar transistors with quasi-ballistic base widths: 1. The ballistic mechanism of decay in the common-base current gain becomes im-portant even for base widths in the quasi-ballistic regime, that is, for base widths WB comparable to a scattering length / s c . At high frequencies, the dynamic distri-bution function develops a varying phase in k-space, and this tends to degrade the collector current and hence the current gain in addition to any degradation result-ing from the diffusive decay mechanism. This effect must be taken into account to accurately predict both the magnitude and phase of all the forward small-signal parameters up to the transit frequency ft. 2. Using a combination of the one-flux and Thomas-Moll expressions, one can take the above effect into account, and accurately model the magnitude and phase of all the forward parameters: the common-base current gain; the common-emitter current gain; the common-base input admittance; and the common-base transadmittance. Equations (3.23) through (3.29) for these parameters are accurate at least up to ft, provided 3/ s c > WB > 4 c / 2 . For wider bases (WB > 3 4 c ) , the ballistic decay mechanism becomes unimportant, and the one-flux expressions alone are sufficient to predict these quantities. 3. By applying the moving-boundary approach of Early [45] and Pritchard [46], [32, pp. 237-244] to the basic one-flux equations of Shockley [18], one can obtain expres-sions for the reverse small-signal parameters. These are (3.46) and (3.48). Following the behavior of the forward parameters, some error can be expected in (3.46) for the reverse transadmittance due to transit-time effects when WB ~ 4 c and f ~ ft-Equation (3.48) for the output admittance should not suffer from this limitation. The simple expressions of this work should be useful for the compact modeling of modern bipolar transistors at high frequencies, and serve as replacements for the traditional expressions [32, Ch. 5], [45], [46] for devices with quasi-ballistic base widths. Chapter 4 Expressions for the Extrapolated /max of Heterojunction Bipolar Transistors 4.1 Introduction The extrapolated maximum oscillation frequency, namely, the frequency / m a x found by extrapolating the device power gains to unity, at —20 dB/decade, from lower frequen-cies, is often quoted and used as a high-speed figure of merit for heterojunction bipolar transistors (HBTs). It is common practice to express / m a x in the form where is the extrapolated, common-emitter, unity-current-gain frequency, and where rj, refers to the base resistance and Cc refers to the collector-base junction capaci-tance. Equation (4.1) is based on similar relations that were originally derived and used in the 1950s and '60s [53]—[56]. However, in those works, which considered the per-formance of the homojunction devices of the time, assumptions were made that are no longer applicable to modern III-V HBTs, such as the devices described in [57]-[64]: (a) the base resistance and collector-base junction capacitance were represented as single, lumped elements; (b) the parasitic resistances of the emitter and collector, ree and rcc, were excluded; and (c) the base resistance was taken to be large compared to the dynamic resistance l/gm = ksT/qlc, where gm is the transconductance, kB is Boltzmann's con-(4.1) 57 58 stant, T is the temperature, q is the electronic charge, and Ic is the dc collector current. This raises the question of whether (4.1) can be used to accurately predict the / m a x of modern transistors. Laser and Pulfrey [65] found that (4.1) was accurate for the particular HBTs they con-sidered, provided that Mason's unilateral power gain [66] was extrapolated to unity from frequencies below those at which transit-time effects became pronounced; however, they did not directly address point (a), (b), or (c). In a study that focused on homojunction devices, Vaidyanathan and Roulston [67] resolved some of the ambiguity in (4.1) with respect to point (a) by showing that the r jC c product should be replaced by an effective product (r;,Cc)eff that depends on the choice of power gain used to find / m a x , and they suggested values for (r^Cc)eff in terms of the individual components of base resistance and collector-base junction capacitance; however, they did not address points (b) and (c). Re-cently, Kurishima [68] examined the validity of (4.1) for modern HBTs, with the particu-lar aim of addressing point (a); however, as will be shown in Section 4.3 and Appendix C of this work, the basis of the analysis there is questionable, and, moreover, the main conclusions, regarding the roles of the intrinsic and extrinsic parts of the collector-base junction capacitance in determining / m a x , n a c l already been reached in [67]. In this chapter, starting from a general-form equivalent circuit that employs an arbi-trary network to represent the distributed nature of the base resistance and collector-base junction capacitance, and which includes the parasitic emitter and collector resistances, expressions are derived for the extrapolated / m a x of modern HBTs; the values of / m a x as defined by extrapolation of both Mason's unilateral power gain [66] and the maximum available gain [69] to unity are considered. It is shown that the extrapolated / m a x can be written in the form (4.1), provided that the product r jC c is replaced by a general time constant (/?C)efr, where (i?C)efr not only involves an effective product (rbCc)ef[ of base resistance and collector-base junction capacitance, but also includes the effects of the parasitic resistances ree and rcc of the emitter and collector, and the dynamic resis-tance l/gm. The derived expressions are validated using SPICE simulations, and it is demonstrated that, for modern HBTs, the value of (i?C)efr can differ considerably from that of (ri,Cc)eff. 59 In Section 4.2, the general-form equivalent circuit that forms the basis for the analysis is developed, and the limitations of this circuit are pointed out. In Section 4.3, the circuit is used to obtain expressions for the two-port parameters of an H B T at high frequencies. In Section 4.4, the two-port relations are used to find new expressions for the extrapolated /max, and these are then applied to two devices recently reported in the literature [59], [60]. Section 4.5 summarizes the conclusions of this work. 4.2 Equivalent Circuit for Analysis Figure 4.1 shows forms of the hybrid-TT and T circuits conventionally used to model mi-crowave HBTs at high frequencies. The Early effect is neglected, since it is only important for determining the power gain at low frequencies [65, Fig. 5]. The base resistance and collector-base junction capacitance are each represented by n parts: r i , r 2 , . . . , r „ and C i , C2, • • •, Cn, where = J2k=i rk is the total base resistance and CJC = Z)fc=i Cfc is the total collector-base junction capacitance. The parasitic emitter and collector re-sistances are modeled as the lumped elements ree and rcc, and Cje is the emitter-base junction capacitance. Since the focus of this work is on III-V HBTs, no account is made for collector-substrate capacitance, and it should therefore be noted that the results of this work may not be directly applicable to SiGe HBTs, where the collector-substrate capacitance may be appreciable, and where it may have a significant influence on the high-frequency characteristics of the device. The other parameter values in Figure 4.1, as discussed1 in [65], are as follows: a' is a transport factor, given by exp{-ju;[(l - m)TB + r c ]} , (4.2) where u is the radian frequency, a0 is the value of the common-base current gain at low frequencies, rB is the base transit time, TC is the collector-base, space-charge-region, signal-delay time, and m is an empirical fitting factor; z'v = r'e/(l — a') is the dynamic base-emitter impedance, where r'e = l/g'e = a0/gm is the dynamic emitter resistance; and Cde = mg'eTB is the portion of the forward storage capacitance associated with the emitter ]It is cautioned that the nomenclature used in this chapter is not identical to that in [65] or [67]. a = a0 sin(tJTc) U)TC 60 (b) Figure 4.1: Forms of the (a) hybrid-7r and (b) T equivalent circuits conventionally used to model microwave HBTs at high frequencies. The definitions of the elements are given in the text. 61 lead. With this choice of element values, the hybrid-7r and T circuits are indistinguishable from the terminals; in this work, the hybrid-7r will be used. Circuits such as those in Figure 4.1 attempt to offer an improved model of the intrinsic transistor (shown boxed) by employing a precise form of the transport factor a', as given by (4.2). However, considerable simplification can be achieved by using the single-zero approximation for a': a , « a o { l - j w [ ( l - m ) r B + r c ]} , (4.3) which will agree with the more involved form in (4.2) when " < S K l - m U + T B ] - ( 4 ' 4 ) With this simplification, the internal portion of the circuit in Figure 4.1(a) simplifies to that in Figure 4.2(a), where Cdc = <7m[(l — m)TB + Tc] is the portion of the forward storage capacitance associated with the collector lead, rv = = l/g'e(l — ao) is the dynamic base-emitter resistance, and ym = gm — juiCdc is the complex transadmittance. Reinserting the other elements from Figure 4.1(a) then yields the circuit of Figure 4.2(b), where CV = + Cje is the total capacitance between base and emitter, with C& = Cde + Cdc being the total amount of forward storage capacitance. The use of the circuit in Figure 4.2(a) to model the internal transistor is equivalent to employing Pritchard's general-form approximations for the forward characteristics; this can be seen by comparing the expressions for the forward, common-base, admittance pa-rameters from the circuit with those in [32, eqs. (6.20), (6.82)]. As noted by Pritchard [32, pp. 247-250], this method is not theoretically exact,2 but does offer the practical advan-tage of being applicable to a wide variety of transistors, and, with a proper choice of the parameter m, should yield a sufficiently adequate representation of the intrinsic de-vice such that the extrinsic (terminal) characteristics of the circuit in Figure 4.2(b) are 2 A theoretically exact representation of the intrinsic device over all frequencies requires the use of "exact" expressions resulting from a solution of the charge-transport equations. For example, for a uniform-base device having a quasi-ballistic base width, the appropriate expressions are those derived in Chapter 3, namely, equations (3.28) and (3.29). 62 Ci + # * t o C 7^r C<jc Cj e (a) YmVTr 6 e rn r2 ri B o — V - 1 • • • A r ^ V v n v 2 b V i Ci + C -o-'cc YmV^ e <> 6 E (b) Figure 4.2: (a) Simplified model for the intrinsic transistor, (b) Equivalent circuit to be used in the analysis. The definitions of the elements are given in the text. 63 reasonably valid over a wide range of frequencies, satisfying the condition u < —, (4.5) TB or that specified by (4.4), whichever is more restrictive. Traditionally, and especially for common-emitter applications, the best choice of m (to minimize the overall error in the extrinsic characteristics) is taken to be that which yields the proper phase of the common-emitter current gain at high frequencies (u ~ 1/TB). For a uniform-base device with purely diffusive base transport, this best choice is m = 5/6, as originally suggested by Pritchard [32, p. 274], not m = 2/3, which is sometimes used [68], [70]. For other devices, the best value of m depends on the nature of vertical transport through the base at high frequencies.3 The best choice of values for the elements r i , r 2 , . . . , r n and C i , C 2 , . . . , C n w i U de-pend on the exact physical structure of the H B T in question; appropriate values for a conventional structure are suggested in Appendix C. To keep the results general, these elements will be left unspecified in the analysis, with the exception that, in this work, the phenomenon of lateral, high-frequency, ac current crowding [71], [72] will be neglected. This places another upper limit on UJ, beyond which the representation in Figure 4.2(b) will not be strictly valid: w < u; a c , (4.6) where u; a c is given by (C.l) in Appendix C. For frequencies above u a c , the intrinsic contribution to is actually a frequency-dependent complex number. While this can be taken into account by making n a frequency-dependent impedance, it will be assumed that (and r 2 , r 3 , . . . , rn) are constant and purely resistive. Some of the effects of ac crowding on high-frequency power gain are examined in [54] and [67]. The right sides of conditions (4.4)-(4.6) represent upper frequency limits, beyond which vertical and lateral carrier-transport effects will begin to render the circuit of Fig-3For devices where the base transport is quasi-ballistic, the work in Section 3.3.2 can be used to deduce the best value of m; it turns out m = 1/1.12 = 0.89. For devices having an electric field in the base, the value of m will depend on the nature (strength and positional dependence) of the field. 64 ure 4.2(b) invalid. However, despite these limitations, this circuit should still provide at least a first-order representation of the H B T at high frequencies, and, with the restric-tions indicated by (4.4)-(4.6) borne in mind, will be used as the basis for the analysis in this work. 4.3 Two-Port Parameters 4.3.1 Expressions for the Two-Port Parameters Even with the simplified form of the H B T circuit in Figure 4.2(b), it is not possible to come up with compact analytical expressions for the two-port parameters, and hence the device power gains, that are valid over all frequencies and for all possible circuit-element values. Therefore, in this work, following a method similar to that used in [32, pp. 250-251], [53], [54], [56], and [67], a set of assumptions, regarding the relative sizes of the circuit-element values and the range of frequencies (within the high-frequency, rolloff region) to be considered, will be used to keep the analysis manageable. Aside from these required assumptions, the derivation of the two-port parameters is essentially algebraic, and only the final results will be given. In order to obtain the two-port parameters of the circuit of Figure 4.2(b) between the terminals B, c, and e (that is, excluding the resistances ree and rcc), the following assumptions will be employed: • The element values are assumed to satisfy the relations (4.7) A> > i , (4.8) (4.9) where fa = a0/(l - a0) is the low-frequency value of the common-emitter current gain. Each of these should easily be satisfied by modern HBTs at typical bias points in the forward-active mode. 65 • The frequency is assumed to be high enough to satisfy the condition u>2 » wii (4.10) where up = ^/(CV + Cjc) is the —3 dB frequency for the common-emitter current gain in the absence of r e e and rcc. While this places a lower limit on u>, it should be remembered that the power gain is not likely to be in the rolloff region until condition (4.10) is satisfied. • The frequency is assumed to be low enough to satisfy the restrictions where ut = gm/iC^ + Cjc) is the extrapolated value of the common-emitter, unity-current-gain frequency4 in the absence of ree and rcc, and where A = ri(C2 + C 3 + • • • + Cn) + r2{C3 + C4 + • • • + Cn) + • • • + rn^{Cn) (4.14) is a time constant related to the distributed base-collector network. These re-strictions are required in order to neglect higher-order terms (in u) that arise when the base-collector network is represented with n > 1. As will be illustrated (see Figure 4.3), despite the use of (4.11)—(4.13), and of other similar restrictions on the upper value of u> to be introduced later (as needed), the resulting two-port relations will remain valid over a wide enough frequency range for the main purpose of this work, that is, for finding the extrapolated / m a x from those frequencies where the power gain exhibits a single-pole rolloff. Using each of these assumptions as necessary to drop the less important terms (as outlined 4 It should be noted that in Chapter 3, where the device characteristics due only to base transport were considered, the definition Up = 1/TB was used, which is different from the one appropriate for this chapter. u2 < (l/rbbCjc)2, u2 < (l/ArbbCv), OJ2 « (wt/A), (4.11) (4.12) (4.13) 66 in Appendix D ) , the final results for the common-emitter hybrid parameters are as follows: hn » rbb + - (4.15) , _ gmgv gmA + Czc jgm iA->a\ h " K ^ c i , c7, Zc~t'  ( 4 ' 1 6 ) l * ' " % + { j & + £ ) i a C < " ( 4 ' 1 7 ) L 9m.Cjc ( gm.g-K \ • n tAia\ h 2 2 * ~ c ~ r + v + ^ c j j 3 u C ^  ( 4 - 1 8 ) where the symbols Cnt = CV + Cjc, Czc = Cdc + Cjc, and 7 = (Cde + Cje - gmrxCjc)/C„t have been introduced for convenience, with rx = - L [n (C 2 + C3 + • • • + Cnf + r2(C3 + C4 + • • • + Cn)2 + • • • + r n _! (C n ) 2 ] (4.19) ^ jc defined as a resistance related to the distributed base-collector network. The hybrid parameters of the circuit of Figure 4.2(b) between the external terminals B, E, and C (that is, including the elements ree and rcc) can now be written in terms of the above results using standard network theory: H22 = h229h, (4.20) //12 = hi20h + reeH22, (4.21) H2l = h2l0h-reeH22, (4.22) #n = h n + {h12/h22)(H2l - h21) + r e e ( l + H21), (4.23) where 6h = [1 + (ree + rcc)h22\~l. To simplify (4.20)-(4.22), the following additional assumptions are needed: • The values of r e e and rcc are restricted to be not too large: ( r e e + r c c ) « (4.24) ( r„ + r J«(^)( r„ +i.). (4.25) 67 • By analogy with (4.11)—(4.13), the frequency of extrapolation is restricted to be low enough to satisfy the conditions 1 (ree + VCC)2C2 u2 « 1 + 9 m V e LO2 < A ( r e e + rcc)C^ (4.26) (4.27) (4.28) Substituting (4.16)-(4.18) into (4.20)-(4.22), and employing each of the stated assump-tions as needed to drop the less important terms, after some manipulation, one can obtain the following results: Re(ff 2 2) #12 UTCjc \ree + — ) \ 9m) + j u C J c l U B U J T { r e \ + 1 / 9 m ) [ LO2 + U>TC-Tit A luT(reeCn - rccCjc) Cjc 9 m #21 - uTreetjc - £ -(4.29) LO (4.30) (4.31) where the new symbol £ is given by £ = LOT (A + + + r c c ) C ^ 9n (4.32) and where LOB — UT/PO, with LOT being the extrapolated, common-emitter, unity-current-gain frequency in the presence of r e e and r c c , given (as usual) by — - — + (r e e + r c c ) (A c = J- + (r e e + rcc )Cjc. LOT ut gm (4.33) The use of (4.15)-(4.18) and (4.31) in (4.23) leads to a very untidy expression. Substantial 68 simplification results if one additionally assumes to2 « to2, (4.34) even though this then makes (4.13) and (4.28) somewhat redundant. With this expedi-ency, equation (4.23) can be reduced to get (LOHLOT \ I 1 \ Re(Hn) ~ ( ^ - ^ UTTeeCjc - £J \ ree + — ) + rbb + Tee + —7; LOT(ree + r c c )A . (4.35) 4.3.2 Verification Using SPICE In order to confirm the validity of the above expressions for the two-port parameters, they were compared with a SPICE simulation of the circuit in Figure 4.2(b), employing equivalent-circuit parameter values for the H B T recently described in [59, Fig. 1]; these parameter values can be found in Table 4.1. In performing the comparison, the following device gains were considered: the maximum stable gain (MSG), for frequencies at which the device is potentially unstable [73, eq. (2)]; the maximum available gain (MAG) , for frequencies at which the device is unconditionally stable [69]; a power gain figure of merit 5 Go [69]; Mason's unilateral power gain U [66]; and the current gain \H2\\2. The power gains are given by the following expressions: rr _ l # 2 i + #121 2 A[Re(Hn)Re(H22) + lm(H21)lm(Hl2)Y 1 ' °j M S G = \H2l/H12\, (4.37) f _ l # 2 l | 2 . . ° 4Re( t f n )Re( i / 2 2 ) - 2Re(ff 2 1 /7 1 2 ) ' { ' M A G = F(C)G0, (4.39) 5 G 0 can easily be shown to be the power gain of the transistor driving a load admittance that is conjugately matched to the open-circuit output admittance H22-69 PARAMETER UNITS REF. [59] REF. [60] n — 2 2 ft 4.18 12.2 0 1.8 5.5 fF 3.4 4.7 c 2 fF 10.4 3.2 0 6.2 14.9 '"cc ft 16 11.1 C e^ fF 112 208 (*Q — 0.9697 0.982 m — 0.833 0.833 TB ps 1.84 0.14 TC ps 0.31 1.52 gm mO 231 446 Table 4.1: Parameter values for the devices described in [59] and [60]. 70 where C = 2G0 #12 (4.40) #21 is the well-known Linvill stability factor, and where F{C) 2(1 - y/1 - C2) C2 (4.41) The results of the comparison are presented in Figure 4.3. In viewing these results, it should be remembered that, for the chosen device, the H B T circuit itself is not rigorously valid above 85 GHz; this limit, marked in the figure, arises from carrier-transport effects, as previously discussed in Section 4.2. Figure 4.3(a) shows all of the gains in the high-frequency, rolloff region; the solid lines are results from SPICE, and these agree with the experimental data in [59, Fig. 2], and the open symbols are gain values found by employing (4.15)-(4.18) directly in (4.20)-(4.23) for the /^-parameters. The agreement is excellent, up to at least 140 GHz, beyond which (4.12) and (4.13) are no longer satisfied for the chosen device. Figure 4.3(b) shows only M S G / M A G (for clarity); the solid line is from SPICE, the open symbols are values found by employing the reduced relations (4.29)-(4.31) and (4.35) of this work, and the stippled line shows values found by using the two-port relations6 recently suggested in [68, eqs. (17)]. As shown, the relations in [68] lead to incorrect values for M S G / M A G over much of the rolloff region. It turns out that even the agreement at frequencies / ~ 15 GHz is fortuitous, and the two-port parameters themselves, when found from the expressions in [68], are erroneous over the entire rolloff range; for example, the values of Re(#n) and Im(#i 2), both of which strongly influence the power gains, are in error by at least 40% for the chosen device. These errors arise because higher-order terms (in u) were arbitrarily neglected [68, eqs. (14)] when the two-port relations were derived in [68], yielding results that are not generally correct; for instance, when n = 1 and ree,rcc = 0, it is obvious that aQ —>• 1 implies R e ( # n ) —>• at all frequencies, whereas the relations in [68] imply (in the notation of the present 6In employing these two-port relations, the base-collector network was first reduced using the pre-scription given in [68, Fig. 3]. 71 72 Figure 4.3: Plots of power gain versus cyclic frequency for the device described in [59]. In part (a), the solid lines are values resulting from a SPICE simulation of the circuit in Figure 4.2(b), and the open symbols are values found by employing (4.15)—(4.18) directly in (4.20)-(4.23) for the two-port parameters. Part (b) shows only values of M S G / M A G ; the solid line represents results from SPICE, the open symbols are results found by employing (4.29)-(4.31) and (4.35) for the //-parameters, and the stippled line shows M S G / M A G computed using the relations for the two-port parameters in [68, eqs. (17)]. In both (a) and (b), the dashed vertical line indicates the frequency beyond which conditions (4.4) and (4.5) are no longer valid; there was insufficient information in [59] to check condition (4.6). 73 work) that Re(Hn) ->• rbb - (Cde + Cje)/gmGKf As a result, the conditions of validity for these two-port relations, and for the corresponding expression for / m a x [68, eq. (25)], are open to question. By contrast, the two-port relations of this work depend only on the very reasonable assumptions stated in the previous subsection, and Figure 4.3(b) shows that these relations are in agreement with SPICE; the predicted values of the H-parameters, and hence the computed values of M S G / M A G (and of U and Go, although these are not shown), start to become significantly incorrect for the chosen device only when / > 55 GHz, once both (4.28) and (4.34) are violated. The two-port relations of this work can hence justifiably be used to find an expression for the extrapolated /max-4.4 Extrapolated / m a x 4.4.1 New Expressions for / m a x In the literature, the maximum oscillation frequency is found by extrapolation of both U and M A G to 0 dB. These extrapolated values shall be denoted by / ^ a x and f ^ i ^ i respectively. While (4.36) for U is reasonably simple, equation (4.39) for M A G is some-what involved. Therefore, in this work, an expression will be found for / ^ a x , the value of /max determined by extrapolating the quantity Go in (4.38) to 0 dB, and then it will be assumed7 that ~ /mL- While this is a good, assumption on theoretical grounds [32, pp. 495-499], it should be cautioned that M A G may not converge to the single-pole rolloff of GQ at all bias points and for all devices; in such cases, one cannot meaningfully write /maxG ~ /max' a n < ^ o n e should not use the expressions below, based on Go, as indicators of the corresponding quantities based on M A G . Under the assumptions that have been made, equations (4.30) and (4.31) imply | # 2 i + # i 2 | 2 ~ | # 2 i | 2 ~ ( ^ ) 2 - (4.42) Using (4.42) for the numerators of (4.36) and (4.38), and using (4.29)-(4.31) and (4.35) for 7Similar assumptions have been made in the past [32, pp. 499-500], [67, eq. (10)], [70, eq. (16)]. Alternatively, in their original derivations, Pritchard [54] and Lindmayer [56] employed the simplifying assumption that the transistor would be driven by a source having a purely resistive internal impedance, and they used a corresponding expression for MAG that differs somewhat from (4.39). 74 the denominators, and doing the algebra without invoking any assumptions, one finds that the terms in LOB drop out, and that the remaining terms imply a high-frequency power gain of the form u>x/4u>2(.ftC)eff, so that fn h 87r(/?C)eff ' with (/?C)eff given for the power gains U and Go by the following: (J2C)S = (nCc) G0 eff uTreeC jc (A - rxC]C) ur(ree + r c c )C. jc A + + 1 + uTrccC jc 2 2 Urn TeeCjc n. ^JCI (4.43) = (nCc)ueS - [uTreeCjc)(A - rxClc) + [ioTrccCjc] (ree + —) Cjc, (4.44) (4.45) where (rbCc)^H and (rbCc)^ are effective base-collector time constants (consistent with those previously suggested [67] for finding / m a x ) specified by (nCc)^ = rbbCjc - A, A 2 ' (rhCc)^ = ruCj (4.46) (4.47) and where, in finally writing down each of (4.44) and (4.45), condition (4.7) was used to retain only the most important terms in addition to (rbCc)e{{. When inspecting (4.44) and (4.45), it may be helpful to recall the following: LOT is the extrinsic unity-current-gain frequency, which can be found by measurement or calculated using (4.33); A is a time constant and r\ is a resistance related to the distributed base-collector network, defined by (4.14) and (4.19), respectively; C%c = gm[(l — m)ra +TC] + Cjc is the total capacitance associated with the collector lead, and was defined below (4.18); and the other symbols, which have their usual meanings, were denned in conjunction with Figure 4.2. 75 4.4.2 Algebraic Simplification Equations (4.44) and (4.45) have been derived under assumptions that will be valid for virtually all III-V HBTs. However, these expressions are admittedly somewhat more tedious than desirable. The complexity occurs mainly because of the distributed nature of the base-collector network in Figure 4.2(b); when the network is not distributed, one has n = 1, (rtC c)eff = rbbCjc, A = 0, and r\ = 0, and a number of the terms vanish. Alternatively, a similar simplification can be achieved if the unity-current-gain frequency is not greatly affected by r e e and rcc. Given that rbbCjc, r\Cjc, and A are all time constants of similar magnitude, then provided the condition LOT < (4.48) is true, the second term in (4.44) can be dropped in relation to the first, yielding (rtC)Sr ~ (rbCc)"s + [uTrccCJC] (ree + — ) Cjc. \ 9m J (4.49) If in addition to (4.48), the condition LOT < is also met, then it is reasonable to write (4.50) + LOT Y "f- 2rccCjc -j- reeCjc Cjc, (4.51) where the second and third terms of (4.45) have been dropped in relation to the first, and where the fourth term has additionally been simplified. Unlike the previous assumptions made in Section 4.3, conditions (4.48) and (4.50) may not always be strictly satisfied. However, unless these conditions are seriously violated, equations (4.49) and (4.51) will have merit, and can at least be used for a first approximation, with the more involved forms (4.44) and (4.45) always available if more accuracy is needed. In this regard, it may be useful to note that the use of (4.49) and (4.51) amounts to simply neglecting 76 the distributed nature of the base-collector network everywhere except in the (rbCc)ef[ terms; that is, the use of (4.49) and (4.51) is tantamount to assuming that r e e and rcc "see" a base-collector network with n = 1, and that the effects of re > 1 can be taken into account solely by employing (rbCc)eK in place of the total time constant rbbCjc. 4.4.3 Discussion It is possible to make some useful observations from the derived expressions for (RC)efi. It is sufficient for this discussion to refer to the simplified relations (4.49) and (4.51); similar statements can be made even if one uses the more detailed forms (4.44) and (4.45). The presence of ree and rcc in the results for (RC)eg should not be surprising, since power is dissipated in these parasitics just as it is in rbb. While (RC)^S is particularly sensitive to the presence of r c c , as shown by the last term of (4.49), ree contributes directly to (RC)^, as shown by the second term of (4.51). The presence of gm (both directly and through LOT) in the expressions means that both (RC)^ and (RC)f^ will exhibit a bias dependence, independent of any bias-dependent changes in the device parasitics. Each of the terms in (RC)^S and (RC)fg is essentially proportional to Cjc, a con-sequence of the fact that the output admittance is proportional to Cjc, and hence that the power gain is inversely proportional to Cjc. While conventional design focuses on reducing both rbb and Cjc for improved performance, this demonstrates that a reduction in Cjc will generally be more effective. In the absence of ree and rcc, one finds (RC)^ = (rbCc)efr', on the other hand, 1 r _ 0 = (nCc)% + \ (^) (f) C3C, (4.52) r e e,r Cc-U Z \Unt/ \9mJ where C„t w a s previously defined below (4.18) as the total device capacitance. While Csc/Cvt < 1, the contribution specified by the second term on the right side of (4.52) can still be appreciable, especially at low bias; it was originally neglected [54, eqs. (6), (7)], [56, eq. (11)] on the assumption that rbb >^ l/gm. While each of these observations is important, the one most readily apparent from the derived expressions is worth emphasizing: when rbb ree, rcc,l/gm, as can be the case in 77 modern III-V HBTs, the latter elements may significantly contribute to the overall time constant determining / m a x . As will now be shown, the analytical expressions of this work allow one not only to determine whether this is indeed the case, but also to isolate the individual speed-limiting components. 4.4.4 Application In order to validate the expressions derived for (RC)eB, and to illustrate some of the points just discussed, they were applied to two state-of-the-art HBTs recently reported in the literature [59], [60, Table II], the parameters for which are given in Table 4.1. SPICE was run on the circuit of Figure 4.2(b), and (i?G)eff = / r / 8 7 r / m a x was extracted using the SPICE value of extrapolated fr, which agreed with that predicted by (4.33), and the SPICE value of extrapolated / r a a x ; the extrapolations were performed from a frequency near the start of the power-gain rolloff region, but after —20 dB/decade slopes were well established. (RC)es was also calculated using the simplified expressions (4.49) and (4.51); for the chosen devices, the more involved forms (4.44) and (4.45) yield negligibly improved values. Table 4.2 presents the results for the device described in [59], and Table 4.3 presents the results for the device described in [60]. Results are presented for various values of r e e and rcc. With regard to Go and M A G , for each of the cases considered, it was found that M A G did converge to the single-pole rolloff of Go, and hence that /maAxG ~ /max a n d (RC)™tfAG « {RC)% however, for definiteness, the SPICE values in Tables 4.2 and 4.3 were obtained by extrapolating Go. Equation (4.46) yields values of {rbCc)^B equal to 39 and 101 fs for the devices de-scribed in [59] and [60], respectively. Using these as reference values, the results for {RC)^B in Tables 4.2 and 4.3 illustrate the following points: • For a wide range of values of r e e and r c c , the analytical expression for (RC)^B works very well in matching the SPICE results. • When r e e , r c c = 0, the value of (RC)^B reduces to that of {rbCc)^B, as expected. • In the presence of appreciable r e e and rcc, particularly when rcc is substantial, and as shown by the values in Table 4.2, (RC)^B can differ significantly from (rbCc)^B. 78 CASE SOURCE (RC) e u f f (fs) (RC)S (^ ) ree = 0 rCc = 0 SPICE 39 68 Expression 39 68 r e e = 6.2 f2 rcc = 0 SPICE 39 110 Expression 39 112 ree = 0 rcc = 16 Cl SPICE 44 71 Expression 44 72 ree = 6.2 0 r c c = 16 SPICE 51 117 Expression 50 115 Table 4.2: Values of (RC)^ and (RC)% for the device described in [59] from SPICE and the simplified expressions (4.49) and (4.51). CASE SOURCE (RC) e u f f (fs) (RC)Sf (^) ree - 0 rcc = 0 SPICE 100 126 Expression 101 127 r e e = 14.9 Ct rcc = 0 SPICE 99 182 Expression 101 186 ree = 0 rcc = 11.1 ft SPICE 101 125 Expression 101 127 ree = 14.9 0 rcc = 11.1 n SPICE 105 184 Expression 106 186 Table 4.3: Values of (RC)^S and (RC)% for the device described in [60] from SPICE and the simplified expressions (4.49) and (4.51). J 79 Equation (4.47) yields values of (r( ,C c )^ equal to 61 and 120 fs for the devices de-scribed in [59] and [60], respectively. Using these as reference values, the results for (RC)tf[ in Tables 4.2 and 4.3 illustrate the following points: • For a wide range of values of ree and r c c , the analytical expression for (RC)^g works very well in matching the SPICE results. • When r e e , rcc = 0, the value of (RC)f£ does not reduce to that of (n,C c)2f, which is consistent with (4.52); for example, in Table 4.2, one has (RC)f^ = 68 fs whereas (r6Cc)S = 61 fs. • In the presence of appreciable r e e and rcc, particularly when ree is substantial, (RC)eH can differ significantly from {r\,Cc)^. To show how (Z?C)eff varies with bias, the values of (4.49) and (4.51) have been plot-ted in Figure 4.4, for the device described in [59], as a function of gm = qlc/ksT, while holding all other parameter values in Table 4.1 constant; for comparison, SPICE values are also shown. For a wide range of values of gm, the analytical expressions are in agree-ment with SPICE, and, as shown, both (RC)^ and (flC)g? increase as gm is reduced. The rapid increase in (RC)^ at low bias is due to the last term in (4.52); however, while (RC)^ did increase rapidly and was correspondingly degraded, it should be noted that for the low values of gm where this occurred, it was found (unlike for large gm) that the device did not become stable until relatively very high frequencies (where GQ and M A G were close to 0 dB), so that one could not meaningfully write « / ^ ° x as described earlier. A l l of the above results illustrate the utility of the expressions of this work. For exam-ple, with knowledge only of the unity-current-gain frequency and the device parasitics, the analytical expression for (RC)^ shows that the collector resistance contributes sig-nificantly to the degradation of the / ^ a x of the device in [59] (leading to an additional delay, depending on bias, of between 11 and 26 fs over and above the 39 fs which results from the effective base-resistance-collector-capacitance product), whereas the / ^ a x of the device in [60] is only marginally affected by r c c ; similarly, the expression for (RC)^ 80 500 400 HI 300 O 200 LU Value for U \ \ \ \ \ \ \ \ Value for G 0 100 10 -3 i<r 101 TRANSCONDUCTANCE (mhos) 10u Figure 4.4: Values of effective time constant (Z?C)efr versus transconductance gm = qIc/kBT for the device described in [59]. The dashed lines are values from the simplified expressions (4.49) and (4.51), and the solid symbols are values from SPICE. 81 shows that ree plays a major role in degrading the f^lx of both the devices considered. By providing the capability to readily make such observations, the expressions of this work should be very helpful in optimizing the design of modern III-V HBTs. 4.5 Conclusions The main conclusions from this study of the extrapolated / m a x of heterojunction bipolar transistors (HBTs) are as follows: 1. By making very reasonable assumptions that will be satisfied by virtually all III-V HBTs, it is possible to show that the extrapolated /max can be written in the form /max = 'fr/87v(RC)ea, where is the common-emitter, unity-current-gain frequency, and where (RC)eR is a general time constant that takes into account all the device parasitics. Equations (4.44) and (4.45) are the rigorous expressions for (RC)eff for the power gains U and Co (or M A G ) , respectively. However, the simplified forms of (jRC)eff in (4.49) and (4.51) should be adequate for at least a first approximation. 2. When rbb ^> ree,rcc,l/gm, where r^, r e e , and rcc refer to the parasitic resistances of the base, emitter, and collector, and where gm is the transconductance, the value of (7?C)eff c a n differ significantly from the effective base-resistance-collector-capacitance product conventionally assumed to determine / m a x - Moreover, (i?C)eff exhibits a bias dependence, independent of any bias-dependent changes in the de-vice parasitics. 3. Application of the derived expressions for (RC)ef[ to two state-of-the-art devices recently reported in the literature [59], [60] demonstrates both their accuracy and utility in identifying speed-limiting components. Chapter 5 Conclusions This thesis has focused on compact models for the high-frequency characteristics of mod-ern bipolar transistors. Several new conclusions were reached regarding the modeling of both the intrinsic and extrinsic characteristics of state-of-the-art devices. These conclu-sions are gathered together and summarized below. 5.1 Appraisal of the One-Flux Method A technique, known as the "one-flux method," which has recently been used [26], [27] to examine the intrinsic high-frequency characteristics of modern thin-base devices, was critically appraised in Chapter 2. The idea of the flux approach is to assume that the distribution function is formed with abutted hemi-Maxwellians, and then to seek solutions for the right- and left-going fluxes a(z,t) and b(z,t) as a function of position z and time t, where the fluxes are governed by the basic one-flux equations: (l/vT)(da/dt) + da/dz = -C(z)a + ((z)b - S*(z)a + g(z,t)/2, -(l/vT)(db/dt) + db/dz = -C(z)a + C(z)b + 6*(z)b - g(z, t)/2, where VT is the thermal velocity (equal to twice the Richardson velocity), 5*(z) is the absorption coefficient, g(z,t) is the generation rate, and ('(z) and ((z) are the backscat-tering coefficients for the right- and left-going fluxes, respectively. For a bulk region with a small electric field £, provided that the mean-free time for collisions TCO\ is much 82 83 less than the recombination lifetime r r e c , and provided that the current does not change appreciably in a time TCO\, the above one-flux equations are algebraically equivalent to the usual drift-diffusion equations: dn 1 dJn n — n0 3t q dz S°P T r e c dn Jn = qfJ-nOnt + qOn0 — , oz where q is the electronic charge, gop is the optical generation rate, n = [a(z, t)-\-b(z, t)}/vr is the electron concentration, no is the electron concentration at equilibrium, Jn = —q[a(z,t) — b(z, t)] is the electron current, and nn0 and Dn0 are the usual, low-field values of mobility and diffusivity, respectively. This equivalence applies even to regions with small dimensions (comparable to a mean-free path length). Use of the flux approach to determine the intrinsic characteristics of modern thin-base devices, in both the static [25] and dynamic (high-frequency) [26], [27] cases, is essentially equivalent to employing the usual drift-diffusion equations with appropriately chosen boundary conditions [29]. The flux approach suffers from an inherent difficulty that exists in deriving values for the nonmeasurable backscattering coefficients on a rigorous, physically correct basis. In this regard, inconsistencies within the flux approach can be cited, and these provide an indication that any results obtained from the technique, while perhaps yielding useful aids to visualization and compact, analytical expressions, must generally be checked and refined using more detailed descriptions of carrier transport, especially in those cases where one can expect the usual (continuity and drift-diffusion) transport equations to fail. 5.2 Expressions for the High-Frequency Character-istics of Quasi-Ballistic Bipolar Transistors With the results of the appraisal of the flux method borne in mind, a closer look at the high-frequency characteristics of thin-base transistors was undertaken. This was accomplished by using the method of Grinberg and Luryi [31], [47] to solve the Boltzmann 84 transport equation (BTE) for devices where the base width WB is comparable to the minority-carrier scattering length / s c , that is, for devices where the base transport is "quasi-ballistic." The results were presented in Chapter 3. When the base transport is quasi-ballistic, the common-base current gain is degraded at high frequencies by not only a "diffusive" decay mechanism, but also a "ballistic" decay mechanism. The ballistic decay effect causes the dynamic distribution function to develop a varying phase in wave-vector space, and this tends to "wash out" the signal charge and signal current at the collector. Since the one-flux method cannot account for this decay effect, the expressions developed in [26] and [27] cannot completely model the intrinsic high-frequency characteristics of quasi-ballistic devices. The intrinsic forward parameters can be modeled by combining results from the flux method with the well-known expressions of Thomas and Moll [48]. The common-base current gain and the common-emitter current gain can be written as a function of radian frequency u by using the Thomas-Moll forms: a •exp{j[(K-l)/K](uTB/u)}, 0 = 1 +J(UTB/1S) a0exp{j[(K - 1)/VK](LOTB/IS)} (l-<x0){l+j[uTB/K(l-ao)is}y where the base transit time TB and the low-frequency, common-base current gain a0 can be found using the one-flux expressions Wl WB 2D nO VT and cv0 = vrLn with Ln = y/DN0TREC, and where K = l/va0 with the value of v ^ 1.12 empirically obtained from B T E solutions. The common-base input admittance is unaffected by the 85 ballistic decay effect and can be found from the one-flux method alone, and is given by cosh(A/cWB) + (l//Qsinh(A/dy B) cosh(\KWB) + [{K2 + 1)/2K] smh(\KWB)\ ' where the new symbols are as follows: A is the cross-sectional device area; kB is Boltz-mann's constant; T is the temperature; n*E = nBoexr)(qVBE/kBT), where nBo is the equilibrium electron concentration in the base and VBE is the base-emitter bias voltage; and K and A are given by the relations K = \J\ -f vj/\Dno and A = l / r r e c U T + j^/v?. An accurate form for the transadmittance, including the effects of ballistic decay, then follows by writing _ q2nEvTA 2/216; = -CM/116; , and using a from the Thomas-Moll form. All of these expressions are accurate for radian frequencies at least up to l / r B , provided 3/ s c > WB > lsc/2. For wider bases (WB > 3/ sc), the ballistic decay mechanism becomes unimportant, and the one-flux expressions alone are sufficient to predict the forward parameters. Expressions for the intrinsic reverse parameters can be found by applying the "moving-boundary" approach of Early [45] and Pritchard [46] to the one-flux equations of Shock-ley [18]. The reverse transadmittance and output admittance are given, respectively, by 2/126; -qen*EvTWBA w Roo(l - Rio) L ( c 1 - J ^ 0 e 2 ) ( g i - / ? o g 2 ) J and 2/226: -qenEvTWBA 2Vi w 2f l o c . f l ( X , (g 1 - e 2) - (Rl + l ) (g x - ^ g 2 ) (e"! - Rle^er - R2^) -where the symbols not previously defined are as follows: WB is the static base width; Vw = WB/(dWB/dVBc) is a base-width modulation factor with VBQ being the base-collector voltage; e = XK and R^ = vT/[2Dno(\ + i) + vT] are defined dynamic quantities; £ = £| w =o and Roo = i?oo|u/=o are defined static quantities; and e~\ = exp(eWg), e2 = 86 exp(—EWB), ei = exp(eWB), and e2 = exp(—£WB) are defined exponential quantities. Following the behavior of the forward parameters, some error can be expected in the result for y\iu due to transit-time effects when WB ~ 4c and 10 ~ l / r ^ . The result for y 226i should not suffer from this limitation. While the intrinsic characteristics depend on the nature of transport through the base, the extrinsic high-frequency characteristics depend additionally on the device parasitics. The extrapolated maximum oscillation frequency (/max) is a figure of merit commonly used to characterize the extrinsic high-frequency performance of heterojunction bipolar transistors (HBTs). By properly considering the effects of all the device parasitics, new expressions for the / m a x of state-of-the-art HBTs were derived and validated in Chapter 4. Starting from forms of the hybrid-7r and T circuits conventionally used to model III-V HBTs at high frequencies, a general-form equivalent circuit amenable to analysis can be developed. In the circuit, the total base resistance rjj, and the total collector-base junction capacitance C J C , are each represented by n parts, r j , r 2 , . . . , rn and C\, C 2 , . . . , C„ (where 7*1 and C\ denote the intrinsic components), and the parasitic resistances of the emitter and collector are included as the lumped elements ree and rcc. An empirical parameter m accounts for transit-time effects, and is used as part of a general-form representation of the intrinsic device. By making suitable assumptions, all of which will typically be true, the circuit can be utilized to develop simplified expressions for the extrinsic hybrid parameters of an H B T at high frequencies. These expressions can then be employed to find the high-frequency power gain and / m a x . The / m a x of modern HBTs can be written in the form where fT is the common-emitter, unity-current-gain frequency, and where (RC)e{{ is a 5.3 Expressions for the Extrapolated / m a x of Heterojunction Bipolar Transistors 87 general time constant that accounts for all the device parasitics. For the values of / m a x defined by extrapolation of Mason's unilateral gain ([/) [66] and the maximum available gain ( M A G or G0) [69] to unity, at —20 dB/decade, the values of (RC)es are given by the following approximate expressions: where the new symbols are as follows: wj = 27r/r is the radian, common-emitter, unity-current-gain frequency; gm is the transconductance; C^c = Cjc + gm[(l —  M ) T B + T~C]  LS the total capacitance associated with the collector lead, where Tc is the signal-delay time for the collector; and (rbCc)^s = rbbCJC — A and ( n C c ) ^ = rbbCjc — A / 2 are effective base-collector time constants, with A = Yle=i{re 2~2k=e+i Ck}-In modern HBTs [59], [60] having relatively low values of base resistance rbb, that is, values that are comparable to those of the parasitic resistances r e e and rcc of the emitter and collector, and to the dynamic resistance l/gm, the value of (i?C)eff can differ significantly from that of (rbCc)ett-5.4 Future Work Projects for future consideration include the following: 1. The work in Chapter 3 assumes a field-free base region. Modern devices are often designed with a built-in field in the base in order to aid electron transport and improve device speed. By solving the Boltzmann transport equation for dynamic operation in the presence of an electric field, the effects of quasi-ballistic trans-port on the high-frequency characteristics of such "graded-base" or "built-in-field" devices could be investigated. Dignam and Grinberg [74] have already outlined a method that could be used to perform this task, and the results could then be compared with the one-flux expressions (in the presence of a field) given in [26]. (RC\ 88 2. Heterojunction devices may have a conduction-band spike present at the emitter-base junction. For such devices, the boundary conditions on the electron concen-tration and current at the edges of the base differ from those in homojunction and graded-gap heterojunction transistors (where the band spike is absent). It would be interesting to examine how such boundary conditions affect the high-frequency behavior. Some work has already been done on this topic in both the diffusive [75] and ballistic [47] limits. An examination in the quasi-ballistic case would require not only the incorporation of the different boundary conditions into the solution, but also a technique to solve the dynamic Boltzmann equation in the presence of more realistic scattering mechanisms (applicable to the materials typically used in the construction of abrupt-junction HBTs) than those assumed in Chapter 3. 3. The work in Chapter 4 could be extended to include the effects of collector-substrate capacitance, so that the results would be applicable to Si-based HBTs. The key is to find a simple way to incorporate the substrate capacitance without greatly complicating the analysis or results. Appendix A Current-Voltage Relation of a Mott Barrier in the Thermionic-Emission Limit As mentioned in Section 2.4.1, it is possible to derive the proper current-voltage relation of a Mott Barrier, in the thermionic-emission limit, by employing the D D E with the usual values of mobility and diffusivity. In Figure 2.5, let A correspond to the point z = 0 and let B correspond to z = w. Let the condition for thermionic-emission be satisfied, and recall that the drift-diffusion equation can be written as follows [42, pp. 92-93]: Jn = q^nonS + qDn0(dn/dz) = q/j,n0n(dFn/dz), (A. l ) where n(z) = Nc exp{[Fn(z) - Ec(z)]/kBT} (A.2) and Nc is the effective density of states for the conduction band. Then, substituting (A.2) into ( A . l ) , and integrating from z = w to z = d, one gets r kBTpn0Nc{exp[gVA/kBT} - exp[Fn(<J)/fcBr]} f?exp[Ec(z)/kBT]dz ' [ A - 6 ) where Fn(w) = qVA and Epm = 0. Using the boundary condition derived by Berz [40, 89 90 eq. (23)], one can write Jn = [qvTexp(-qAVkT/kBT)/2} X n(d) = [qvTexp{-qAVkT/kBT)/2] x Ncexp{[Fn(d) - Ec{d)]/kBT} = qvRNcexp{[Fn(d) - Ec{d) - qAVkT]/kBT} = qvRNcexp{[Fn(d) - $Bn]/kBT}, (AA) where AVkr = V(d) — V(0) = 2kBT/q [40] is the potential drop across the kT layer. Eliminating Fn(d) between (A.3) and (A.4) yields j _ qNcvRexp(-$BnlkBT) exp(qVA/kBT) n ~ l + [qvR/^n0kBT]exp(-^Bn/kBT) ffexp[Ec(z)/kBT]dz' [ ' ' For the Mott barrier of Figure 2.5, this can be rewritten in terms of the constant electric field £, by noting that Ec(z) = q£z + $ B „ , Ed = — AVkr, and £w = VA — Vii, to get , _ qNcvRexp(-$Bn/kBT) exp(qVA/kBT) " ~ 1 + [vR/fin0£]{exp[-q(Vbi - VA)/kBT] - exp[-qAVkT/kBT}Y [ ' ' However, since the condition for thermionic-emission is satisfied, it is true that [42, p. 102] Vno£ vR, and hence (A.6) reduces to Jn = qNcvRexp{-$Bn/kBT)exp{qVA/kBT). (A.7) Equation (A.7), which has been derived by employing the D D E with the low-field mobility and diffusivity, from the end of the junction-field region to the edge of the kT layer, is the correct expression for the current in the thermionic-emission limit [42, p. 96]. Appendix B Comparison of the One-Flux and Balance Equations As indicated in Section 3.3.3, a discrepancy exists between the one-flux method and the balance equations derived from the B T E . 1 This can be demonstrated by carefully comparing the two approaches. B . l Balance Equations Neglecting generation for simplicity, the prescription2 given in [76, Sec. 5.3.2] or [77, Ch. 5] may be used to write down the time-dependent balance equations. One finds that the zeroth-order balance equation is just the continuity equation, -n dn 1 dJn ( B . l ) (r c p ) dt q dz' and that the first-order balance equation, known as the "momentum balance equa-tion (MBE) ," can be written in the form n\((rm)) ( (r c p )>; m* + m* dz dt ' ^ JThe "balance equations" are exact statements of conservation derived from the B T E . Finding a solution to the balance equations is tantamount to solving the BTE, although this approach is not possible in practice unless some simplifying assumptions are first made regarding the expected form of the distribution function. 2The prescription given in [76] additionally neglects recombination. 91 92 where £ is the electric field and the other symbols not defined prior to Section 3.3.3 are as follows: (B.3) (B.4) (B.5) (B.6) 1 f[l/Tcp(k))f(z,k,U,t)dk ( T c p ) f f(z,k,u,t)dk 1 I[kz/rm(k)]f(z,k,u,t)dk f kzf(z,k,u,t)dk 1 _ f[kz/rcp(k)]f(z,k,u,t)dk « T c p » fkj(z,k,u,t)dk w 1 f h2k2z r l , = 4n*J 2 m . / ( * , M , * ) < f c , where, for the assumed models of carrier capture (recombination) and scattering used in Chapter 3, r c p(k) = rcp{k) = lcp/(hk/m*) and r m (k) = rsc(k) = lsc/(hk/m*). In these equations, (r c p) is the average capture time, ((rm)) is a specially defined, average time between collisions, known as the average "momentum relaxation time," ((T c p)) is a spe-cially defined, average time associated with capture processes, and W is the (z-directed) kinetic energy density. Assuming / s c >C / c p , the term 1/((T c p)) in (B.2) can be neglected in comparison with l /((r m )) . Using this simplification, and rearranging the remaining terms, the M B E may be written as follows: Jn = qpnn{£ + £') + qDn^ - « r m » ^ , (B.7) where fxn = q((Tm))/m* is the mobility, £' = (2/q)(du/dz) is the "energy-gradient" field with u = W/n being the 2-directed kinetic energy per carrier, and Dn = (2u/q)fin is the diffusivity. If one now assumes that the distribution function consists of abutted hemi-Maxwellians, as illustrated in Figure 2.2, then it is easy to show3 that (B.l) and (B.7) 3This can be done by writing / ± = (2n±{z,t)/Nc) exp[-Ek/kBT], evaluating the right sides of (B.3), (B.4), and (B.6), and employing the results in (B.l) and (B.7). 93 reduce to the following equations: —n dn 1 dJn T r e c 8t q dz' ^ dn / 2 \ Jn = qi^nOnt + qUnO-7^ ~ \ Z) D nO L VT J (B.8) ^ • < B - 9 > where Dno and r r e c are defined (in terms of / s c and / c p ) below (3.10), and where fj,no — qDn0/kBT. B.2 One-Flux Equations In Section 2.2, the basic one-flux equations were compared with the continuity and drift-diffusion equations. It was found that use of the flux approach is equivalent to solving generalized continuity and drift-diffusion equations: dn 1 dJn T _ C'-C vT P VT dn 1 dJn n ~ V + C + 8*) £nt + V + C + **) dz vT((' + ( + 5*) dt> ( B - U j where generation has again been neglected, 8* is the absorption coefficient, and (' and ( are the backscattering coefficients for the right- and left-going fluxes, respectively. Employing the usual value for the absorption coefficient, (B.12) VTTCI one finds that (B.10) is obviously identical to (B.8). On the other hand, using the usual (bulk, low-field) values for the backscattering coefficients, C' = ^ ( l + — ) (B.13) 2Dn0 \ vT J and 94 and neglecting 8* (for consistency with the omission of l / ( ( r c p ) ) from the M B E ) , equa-tion ( B . l l ) becomes Jn = q^nont + qDn0-az D. nO 9Jn Hi' <R15> B.3 Discrepancies Comparing (B.9) with (B.15), it is evident that there is a discrepancy4 in the term involv-ing dJn/dt: in the M B E (B.9), this term, known as the "momentum relaxation term," is multiplied by the rigorous momentum relaxation time corresponding to a distribution function made up of hemi-Maxwellians, namely, r m 0 = (2/ir)[Dno/vT]; on the other hand, in (B.15), derived from the one-flux equations, this term is multiplied by an approximate collision time TC OI = Dno/vT ^ r m 0 . As a result, even if the distribution function retains a purely hemi-Maxwellian form, the one-flux approach will not coincide with the balance equations (and hence the BTE) when applied to time-dependent problems in which the momentum relaxation term is significant, that is, when applied to problems in which the following condition is true: ^ > H . ( B . i 6 ) B.4 Source of the Discrepancies As derived in Section 2.2, the basic one-flux equations appear to be simple statements of particle conservation corresponding to a distribution function constructed from abutted hemi-Maxwellians. It is therefore somewhat surprising that these equations are incon-4It may be worth pointing out that another discrepancy, with respect to the effects of carrier cap-ture, exists between the balance equations and the one-flux approach. If the term 1/((Tcp)) is not dropped from (B.2), and if S* is not omitted from (B.ll), then (assuming time-independent conditions for simplicity) both (B.9) and (B.15) should be rewritten in the form dn Jn = qu*0n£ + qD*0 — , where D*0 = (kBT/q)jj.*0 and where u*0 is a "total" mobility that includes the effects of both capture and scattering processes. One finds that the balance-equation approach leading to (B.9) employs a total mobility specified by l//i*0 = (Z/2)(kBT/qvT)[l/lSc + l / ' cp ] , while the one-flux approach leading to (B.15) uses the (strictly) different value given by l / / / * 0 = (3/2)(kBT/qvT)[l/lsc + (4/3)(l//cp)]. However, since it is typically true that / s c <g Z c p , it should be added that this discrepancy has little practical significance. 95 sistent with the balance equations, the true conservation equations derived from the B T E . However, it should be remembered that while the velocity vj does character-ize the z-directed flux arising from a hemi-Maxwellian, there is a spread in both the energy (velocity or wave vector) and the angle of the carriers in such a distribution; nowhere is this dispersion taken into account in writing the one-flux equations, and this must be the source of the discrepancies. In support of this idea, it is easy to show that the one-flux equations do coincide (exactly) with the balance equations when all of the right-going carriers and all of the left-going carriers are assumed to be traveling in a collimated, monoenergetic beam along the z-direction, that is, when there is no spread in the distribution function: / + (z , fc ,u , t ) oc 5{u-l)S{k-kT), (B.17) f~(z,k,u,t) oc 5(u+l)S(k-kT), (B.18) where 6 refers to the Dirac-Delta function and kr = m*VT/h~- This can be seen by writing / in the form / = 47r 3n+ 6(kx) S(ky) S(kz - kT) + A7T3n- S(kx) S(ky) 5(kz + fcT), (B.19) and then employing (B.3)-(B.6) in (B.l) and (B.2), which yields the following results for the balance equations: —n dn 1 dJn (B.20) Tree 5 Ot q dz ' dJn dn Jn = q^nS + qD*nS — dz v\ j dt' (B.21) where r r e c <5 = lCp/vr is the appropriate recombination time, and /x*^ = qltot/m*VT and DnS = VThot a re the proper values of "total" mobility and diffusivity for the Delta-function distribution. If one now employs (B.12)-(B.14), and replaces T r e c , Dn0, and fj,n0 with T r e C ( $ , Dns = vrhc, and fins — qlsc/m*VT, respectively, then a little algebra will show that (B.10) and ( B . l l ) coincide precisely with (B.20) and (B.21). 96 B.5 Modified One-Flux Equations For the case where / is given by hemi-Maxwellians, it is possible to write a set of modified one-flux equations that effectively correct the discrepancy with respect to the momentum relaxation term. 5 These may be expressed as follows: where a generation rate g(z,t) has been reintroduced for generality and where x  iS a constant; choosing X = 1 returns the usual one-flux equations. It is easy to verify that subtracting (B.23) from (B.22), neglecting g(z,t) as before, and using the usual value of 5* given by (B.12), yields (B.8). Similarly, adding (B.22) and (B.23), neglecting 8* as before, choosing x = 2/7T, and using the usual values of (' and ( given by (B.13) and (B.14), one obtains (B.9), as required. 5Equations (B.22) and (B.23) do not correct the discrepancy discussed in footnote 4. Appendix C Base-Collector Network for a Conventional H B T Structure Figure C.l(a) shows a schematic cross-section of a conventional (mesa-type) H B T struc-ture, with the components of base resistance and collector-base junction capacitance labeled. The components of base resistance are as follows: rbc is the base contact resis-tance, given by [78] rbc = (RSBLT/H) co th(5 B /Lx) , where RSB is the sheet resistance of the base layer, SB is the length of the base contact in the direction of base current flow, H is the length of the base contact perpendicular to the base current flow, and LT — \J PCB I RSB is the so-called "transfer length," with pcB being the contact resistiv-ity for the base contact; rbx = RSBSEBIH is the extrinsic base resistance, where SEB is the distance from the edge of the base contact to the start of the intrinsic base; and fbi = RSBSE/^H is the intrinsic base resistance value in the absence of current-crowding effects, where SE is the length of the base under the emitter. As discussed in Section 4.2, strictly speaking, the use of a frequency-independent value for rbi requires u < w a c , where [67, eq. (19)], [71], [72] 3utkBTH qtlsBdElc is the frequency at which ac current crowding begins to become significant. The com-ponents of the collector-base junction capacitance illustrated in Figure C.l(a) are as follows: Cjcc = CASBH, CJCX = CASEBH, and Cjd = CASEH, where CA is the depletion capacitance per unit area for the base-collector junction. Figure C.l(b) shows the electrical base-collector network for the structure described 97 98 S B — S E B — » - | - « — S E — Emitter Contact Base Contact Collector (a) B o-g dx g dx g dx r dx r dx r dx g, rbx/2 rbx/2 g// rbi A/ o — V - r A / — 0 — V c o- -o c c dx c dx x = 0 c dx -JCX -JCI X = S F (b) J 1 1 rcv l"bc — T c v g/ Tbx/2 l"bx/2 g// Tbi A - A _ A - A A V v u V c Qcx C » c jci x = 0 x = S B F i g u r e C . l : (a) S c h e m a t i c cross-sect ion of a conven t iona l (mesa- type) H B T s t ruc ture , w i t h the componen t s of base resis tance a n d col lector-base j u n c t i o n capac i t ance l abe led . T h e d r a w i n g is not to scale. Fo r s i m p l i c i t y , the deta i l s of the co l lec tor a n d subco l lec to r are not shown . T h e l eng th of the s t ruc tu re i n to the page is a ssumed to be H. (b) E l e c t r i c a l base-col lec tor ne twork for the s t ruc ture , (c) R e d u c e d base-col lector ne twork . T h e ele-m e n t values to the left of B' i n par ts (b) a n d (c) are d iscussed i n the t ex t . 99 above. The topology to the right of B' follows from [67, Fig. 8]. The region to the left of B' (between x = 0 and x = SB) is represented by distributing Cjcc over the Berger transmission-line model [78] that gives rise to rj,c, where g = H/pcB, R = RSB/H, and c = Cjcc/SB • The effects of capacitance at the base contact itself, which would otherwise require capacitive components in parallel with the g elements in Figure C.l(b), will be neglected. This is a good approximation for typical contacts. The case where these components are important, and where r&c should be replaced by a frequency-dependent value Zbc, has been addressed in [79]. At the frequencies of interest, it is easy to show that the network in Figure C.l(b) reduces to that in Figure C.l(c), where the circuit to the left of B' in Figure C.l(b) is replaced by the simple T circuit involving the components rcv, rj,c — rcv, and Cjcci with rcv = PCB/HSB being the purely vertical part of the base contact resistance. The circuit to the left of B' can be considered as a three-terminal network, with terminals B, B', and c. By the properties of a three-terminal network [80, pp. 104-111], it is then sufficient to show that the circuits to the left of B' in Figures C.l(b) and C.l(c) have the same two-port characteristics, with port 1 defined as that between B and c, and port 2 as that between B' and c. Solving the line equations for the region between x = 0 and x — SB in Figure C.l(b), with the boundary condition i(0) — 0, one can get the following open-circuit impedance parameters for the two-port: z " = (C3) ]CC z» = ZcZ> (c-4) JCC _(RSB/H)coth(SBr) rcv j • 1 1 + ju>rcvCjcc toCjcc where T = \Jr(g + jcvc) = (1 /LT)yjl + jwrcvCjcc. Note that the results for z u and z2\ could even be written down simply by inspection of the network in Figure C.l(b), and that z\2 = z2\ by the property of reciprocity [80, pp. 95-96]. The result for z22 can be 100 further simplified by employing condition (4.11), which allows the imaginary parts of the first two terms on the right side of (C.5) to be neglected in comparison with the magnitude of the third, and the real parts to be approximated by rbc and —rcv, respectively, so that z22 ~ rbc - rcv . (C.6) Inspection of the reduced two-port in Figure C.l(c) reveals that it has the same values of z u , z2\i and 212 as in (C.2)-(C.4), and the same z 2 2 as in (C.6), as required. It should be noted that a reduction of the network to the left of B' is also suggested in [68]. However, the reduced 7r-network given there [68, Fig. 2] cannot be correct, as can readily be seen, for example, by noting that it is not consistent with the simple (and exact) results for z2ii and Z12 given above. The overall network of Figure C.l(c) fits the general form used in the H B T circuit of Figure 4.2(b), and specifies the appropriate values of r\, r 2 , . . . , rn and C i , C 2 , . . . , C n for a conventional H B T structure, where, in this particular case, n = 3. Appendix D Derivation of Two-Port Relations To obtain (4.15) and (4.16), consider the circuit of Figure 4.2(b) with c shorted to e (vce = 0) and a signal source connected between B and e. Writing out the expressions for each of the node voltages v\,v2,..., u n + i , and using conditions (4.7), (4.9), and (4.12) to simplify the results, it is possible to obtain, for £ = 1,2,..., n + 1, the following expression: vt ~ vn e-i 1 + {9n + j w C w t ) ]TVfc (D.l) k=i By inspection of the circuit, one can also write the following relations for the terminal quantities: VBe = u n + i , (D.2) n *c = VmVn - ^2(juCe)ve, (D.3) »B = vw(gv + juiC*) + ^2(juCt)ve. (D.4) Using (D.l) for vt, and taking the ratio of (D.2) to (D.4), yields an expression for hu = (vBe/«B)|v c e =o, and using conditions (4.7), (4.9), (4.10), and (4.11) to simplify the result, after some manipulation, one can obtain (4.15). Using (D.l) for vt, and taking the ratio of (D.3) to (D.4), yields an expression for h2i = (* c/*B)k«=o, and using conditions (4.8)-(4.11) and (4.13) to simplify the result, after some manipulation, one can get (4.16). To obtain (4.17) and (4.18), consider the circuit of Figure 4.2(b) with B open (iB = 0) and a signal source connected between c and e. For this configuration, let z\ rep-resent the equivalent impedance between nodes c and b due to the ladder network 101 102 formed by C l 5 C2, • • •, Cn and r i , r 2 , . . . , r n . The value of z\ is then defined by the following equations: ZX = (1/jud) || Zx, * i = r i + [(l/juC2) || z2], ^2 = r2 + [(1/J'WC 3 ) || * 3 ] , i (D .5) Zn-2 = rn-2 + [(1/juCn-i) || Z n - l ] , *n- l = rn.i + l/juCn. Starting with k — n — 1 and writing these out, and repeatedly using condition (4.11) to simplify the results, one finds _ rk{Ck+i + Ck+2 + ••• + Cn)2 + rk+l(Ck+2 + Ck+3 + • • • + Cnf + • • • + rn_x{Cn)2 (Cfc+l + Cfc+2 + • • • + Cn) (D.6) w ( C f c + i + (7 f c + 2 + • • • + C„) for fc = n — 1, re — 2 , . . . , 1, and *A « r A - (D .7) where r\ is specified by (4.19). 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