T H E E F F E C T O F B A S E G R A D I N G O N T H E G A I N A N D H I G H F R E Q U E N C Y P E R F O R M A N C E O F A l G a A s / G a A s H E T E R O J U N C T I O N B I P O L A R T R A N S I S T O R S Simon Chak M a n Ho B . A . Sc. (Hons.) University of Br i t i sh Columbia A THESIS S U B M I T T E D IN PART IAL F U L F I L L M E N T OF 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 M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming . to the required standard T H E UNIVERS ITY OF BRITISH C O L U M B I A August 1989 © Simon Chak M a n Ho, 1989 In presenting this thesis in partial fulfilment 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 Engineering The University of British Columbia 2356 Main Mall Vancouver, Canada Date: Abstract A comprehensive, one-dimensional, analytical model of the graded-base A l G a A s / G a A s heterojunction bipolar transistor is presented, and used to examine the influence of base grading on the current gain and the high frequency performance of a device with a conventional pyramidal structure. Grading is achieved by varying the A l mole fraction x linearly across the base to a value of zero at the base-collector boundary. Recombination in the space-charge and neutral regions of the device is modeled by considering Shockley-Read-Hal l , Auger and radiative processes. Owing to the different dependencies on base grading of the currents associated with these recombination mechanisms, the base current is minimized, and hence the gain reaches a maximum value, at a moderate level of base grading (x = 0.1 at the base-emitter boundary). The maximum improvement in gain, with respect to the ungraded base case, is about four-fold. It is shown that the reduction in base transit time due to increased base grading leads to a 60 % improvement in fx, in the most pronounced case of base grading studied (x = 0.3 at the base-emitter boundary). The implications this has for improving / I n a x v ia increases in base width and base doping density are also examined. Final ly , comparisons between predictions of the model and experimental data from fabricated devices reported in the literature are made. Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement x 1 Introduction 1 1.1 Background 1 1.2 Advantages of Base Grading 3 1.3 Main Features of the Model 4 1.4 Overview of the Thesis 5 2 M o d e l Development 6 2.1 Ideal Abrupt Heterojunction Energy-Band Model 6 2.2 Thermionic-Diffusion Model 10 2.2.1 Electron and Hole Thermionic-Emission Currents at the E-B Junction 10 2.2.2 Emitter and Collector Hole Currents 13 2.2.3 Electron Diffusion Current in the Base 17 2.3 Emitter-Base Junction Grading 24 2.4 Emitter-Base Tunneling Current 27 2.4.1 Barrier Transparency 31 2.5 Recombination and Generation Currents 35 in 2.5.1 Shockley-Read-Hall Recombination Process 36 2.5.2 Radiat ive Recombination Process 40 2.5.3 Auger Recombination Process 41 2.5.4 Generation Process 43 2.6 Parameters for GaAs and A l ^ G a ^ i A s 46 2.6.1 Effective Density of States 46 2.6.2 Bandgap and Electron Affinity 48 2.6.3 Effective Mass 52 2.6.4 Dielectric Constant 53 2.6.5 Mobi l i t y and Diffusion Coefficient 55 2.6.6 Minor i ty Carrier Lifetimes 57 2.7 High Frequency Performance of H B T s 64 2.7.1 Cutoff Frequency 65 2.7.2 M a x i m u m Frequency of Oscillation 75 2.7.3 Modified Collector Structures 77 3 Results and Discussion 81 3.1 Emit ter and Collector Currents 83 3.2 Base Current Components 86 3.3 D C Current Ga in 90 3.4 Transit T ime Components 94 3.5 Effects of Base W i d t h and Base Doping on fT and / m ; i x 98 4 Comparison with Experimental Data 101 4.1 Case I: Current Gain and Cutoff Frequency 101 4.2 Case II: High Frequency Characteristics 107 iv 5 Summary 112 5.1 Conclusions 112 5.2 Considerations for Future Work 113 References 115 A C o d i n g Scheme for the Tunnel ing Factor 130 B New Effective Densities of States 132 C Fermi-Dirac Integral Rat io F1/2{n) / F_l/2{r]) 134 D Derivation of Transit T i m e Delays from the Hybrid-7r Equivalent C i r -cuit 135 v List of Tables 2.1 Relative dielectric constants for GaAs and AlAs 54 2.2 Parameters of low-field mobilities for GaAs as used in Eq. (2.167). . . . 57 2.3 Parameters for the pyramidal heterojunction bipolar transistor 66 4.1 Epitaxial layer structure parameters for fabricated HBTs (Case I). . . . 102 4.2 Structural parameters for the base layer (Case I) . 102 4.3 Epitaxial layer structure parameters for fabricated HBTs (Case II). . . . 108 4.4 Fabricated device dimensions (Case II) 108 4.5 Measured and calculated fr and / m a x (Case II) 109 vi List of Figures 1.1 Base grading profile 4 2.1 Energy-band diagram of an ideal abrupt n-p heterojunction at thermal equilibrium 7 2.2 Energy-band diagram of the n-p emitter-base heterojunction under for-ward bias 11 2.3 Energy-band diagram of the p-n base-collector junction under reverse bias 15 2.4 Schematic of charge flows in a heterojunction bipolar transistor 21 2.5 Energy-band diagram of the graded emitter-base junction 25 2.6 Direction of thermionic-emission and tunneling current components in the conduction band of the emitter-base junction 27 2.7 Tunneling through the conduction-band spike of the abrupt emitter-base junction 31 2.8 Tunneling through the conduction-band spike of the graded emitter-base junction 33 2.9 Energy-band and potential diagram of the emitter-base junction at for-ward bias, with a linearly varying intrinsic Fermi level in the emitter-base depletion region. . 37 2.10 Band structure of GaAs. .' . . 48 2.11 Compositional dependence of the T, X, and L interband energy gaps. . 49 vii 2.12 Collection of experimental minority carrier lifetime data of GaAs for electrons (open symbols) and holes (solid symbols) at low doping densi-ties 60 2.13 Collection of experimental radiative lifetime data of GaAs for various doping concentrations and a corresponding least squares fit 61 2.14 The pyramidal heterojunction transistor structure 66 2.15 A simplified hybrid fl" circuit model for a transistor, with the emitter and collector terminals short-circuited 67 2.16 Equivalent circuit resistances for the emitter layers and emitter-base junction 69 2.17 Equivalent circuit resistances for the intrinsic and buffer regions of the collector. 74 2.18 Equivalent circuit resistances and capacitances for the base and base-collector junction 76 2.19 Heterojunction transistor structure with an implant-damaged external collector 78 3.1 Dependence of collector current density on base-emitter voltage for dif-ferent amounts of base grading, with VBC — —3 Y (broken lines) and JG = 0 (solid lines) 84 3.2 Dependence of emitter current density on base-emitter voltage for dif-* ferent amounts of base grading 85 3.3 Dependence of base current components on base-emitter voltage for the case of Xbe = 0.1 87 3.4 Dependence of base current components on A) mole fraction for J t - = 103 A / c m 2 88 viii 3.5 Dependence of base current components on Al mole fraction for Jc — IO" 4 A/cm 2 89 3.6 Dependence of DC current gain on collector current density for different amounts of base grading 91 3.7 Dependence of DC current gain on emitter junction grading width for different amounts of base grading, with Jc = 103 A/cm 2 93 3.8 Dependence of fx and / m a x on collector current density for different amounts of base grading 95 3.9 Dependence of transit time components on collector current density for xbe = 0.3 96 3.10 Dependence of transit time components on Al mole fraction at base-emitter junction for Jc = 2 x 104 A/cm 2 97 3.11 Dependence of fx and / m a x on base width for Jc = 2 x 104 A/cm 2 . . . . 99 3.12 Dependence of fx and / m a x on base doping concentration for Jc = 2 x 104 A/cm 2 100 4.1 Dependence of experimental and calculated current gain on base built-in field for / c = 6x 10~2 A and a base thickness of 1000 A 103 4.2 Dependence of experimental and calculated current gain on base thick-ness for Ic — G x 10 - 2 A and a base built-in field of 8 kV/cm 104 4.3 Dependence of cutoff frequency on collector current for VCE = 2 V. . . ' . 106 4.4 Schematic structure of an HBT with a proton-implanted external collec-tor layer and a single collector electrode . 107 4.5 Dependence of fx on collector-emitter voltage for Jc = 4 x 104 A/cm 2 . . Ill D. l Hybrid-7T circuit model of the transistor with short-circuited emitter and collector terminals 136 ix Acknowledgement I would like to sincerely thank my supervisor Dr. David L. Pulfrey for his generous support and guidance during this project. I am also deeply indebted to him for his efforts in writing and publishing an excerpt of this thesis. Special thanks are due to Dr. R. K. Surridge of Bell-Northern Research, Ottawa for his encouragement of this work and for specifying the parameters and structure of the HBT device studied in this thesis. I would like to express my sincere appreciation to my colleagues Haosheng Zhou, Oonsim Ang, and Allan Laser with whom I have had invaluable discussions. Finally, this thesis would not be possible without the patience and encouragement of my family. x Chapter 1 Introduction 1.1 Background A Heterojunction Bipolar Transistor (HBT) is a bipolar transistor in which the emitter and the base are semiconductors having different bandgaps. It has long been rec-ognized that HBTs have a number of potential advantages over conventional bipolar transistors [1,2,3]. The wider-bandgap emitter creates a potential barrier that greatly suppresses the reverse injection of charge from base to emitter, resulting in near unity injection efficiencies and thus very high gains. Experimental AlGaAs/GaAs HBTs with common-emitter current gains in excess of 1000 have been recorded [4,5,6]. Since the high injection efficiency is achieved independently of the base and emitter dopings, one can make the doping density of the base very high and that of the emitter low to de-crease the base spreading resistance and the emitter-base capacitance, thus improving the high frequency properties of the transistor. Early HBTs were developed mostly for the Ge-GaAs system [2,7] and, because of the immaturity of the technology, had very little practical use. With the emergence of epitaxial growth technologies for III-V com-pounds, development of HBTs has advanced rapidly. In the last ten years, the majority of HBTs have been based on the AlGaAs/GaAs system and fabricated using molecular beam epitaxy (MBE) or metal-organic chemical vapor deposition (MOCVD). The high electron mobility and the large bandgap of GaAs and AlGaAs make HBTs fabricated from these materials specially attractive for device applications at high frequency and 1 Chapter 1. Introduction 2 high temperature. The recent improvement in HBT performance has been very rapid. In 1985, a typical HBT had a short-circuit, unity current gain, cutoff frequency fx — 11 GHz and a unit power gain cutoff frequency / m a x = 7 GHz [8]. Within a few years, fx and / m a x values of over 100 GHz were reported [9,10]. Much of the improvement in the high frequency operation can be attributed to the use of proton or oxygen implantation in the external collector regions of the transistor to reduce the collector capacitance, and to the use of base grading, that is, varying the Al composition in the AL-Ga^As base [9],[ll]-[14]. Recent investigations into novel collector structures [10] and nonalloyed ohmic contacts [15,16] promise even further improvements in the high frequency performance of HBTs. These rapid advances in experimental HBT performance increase the need for HBT models, particularly ones that are useful for circuit and device design. Many of the recent HBT models require extensive computations. These include one-dimensional [17,18,19] and two-dimensional [20] numerical models, and one-dimensional Monte Carlo simulation [21], all of which are based on carrier transport by drift and diffusion. There are also less complex analytical models based on the charge-control relations [22,23,24]. The models, however, which are best suited for circuit design or for relating I-V characteristics to device and material parameters, are Ebers-Moll models. Some very simple Ebers-Moll model formulations have been used to relate the theoretical and measured offset voltages and potential energy spikes in AlGaAs/GaAs HBTs [25,26,27]. The over simplification of these models, however, limits their usefulness. Only two more comprehensive Ebers-Moll model formulations have been reported, one by Grinberg et al. [28] and another one by Lundstrom [29]. Except for Lundstrom's Ebers-Moll model, none of the above models deal with base grading, which is employed more and more frequently in today's HBTs. Even Lundstrom's treatment of base grading was simple Chapter 1. Introduction 3 since he assumed no base recombination. Furthermore, no detailed analysis of the ef-fect of base grading on HBT performances have been reported. In this thesis, we will present a more comprehensive Ebers-Moll model that incorporates base grading and various recombination processes, and use this to give a detailed analysis of the effect of base grading on the DC current gain and on the high frequency response of n-p-n AlGaAs/GaAs HBTs. 1.2 Advantages of Base G r a d i n g As first pointed out by Kroemer [30], the base region of a HBT can be graded to introduce a strong quasi-electric field into the base to aid the minority carrier transport, and thus improve the base transit time rB. Generally, only a small degree of base grading is needed to reduce the base transit time to a small fraction of the total signal propagation delay. Hence, a larger drift field in the base obtainable by a greater degree of base grading will not further improve the cutoff frequency fr significantly. However, a small base transit time can be traded off for a much thicker base region, which would have a much lower base resistance. A reduction in the base resistance in turn increases the unity power cutoff frequency / m t t X . Another advantage of base grading is that the quasi-electric field created by grading of the base sweeps the minority carriers rapidly across the base, reducing the amount of recombination in the external base region and around the periphery of the emitter. Since carrier recombination in these regions is known to cause the emitter size effect (degradation of current gain as emitter size is scaled down) [31,32], base grading sup-presses this effect [33]. Chapter 1. Introduction 4 A l M O L E F R A C T I O N 0.3 0-2 0.1 0.0 E M I T T E R ND=5*101 7cm3! NA=3x101 9cm~3 C O L L E C T O R | N D = 5x10 i y3* 1 ° 1 8 c rrf 3 D I S T A N C E Figure 1.1: The profile of the Al mole fraction (i.e., x in Al,;Gai_,;As). The four cases of base grading are referred to as Xbt = 0.3, 0.2, 0.1, 0, where x^ is the Al mole fraction at the base-emitter metallurgical boundary. 1.3 M a i n Features of the M o d e l The main feature of our model is, of course, the inclusion of base grading in an Ebers-Moll representation of the H B T . The base composition profiles which are considered are illustrated in Figure 1.1. Note that the variation of Al mole fraction in the base is assumed to be linear, hence the amount of base grading is determined by the Al mole fraction in the base at the emitter-base interface. The possibility of grading the emitter conduction-band spike is also allowed for. The derivations of the emitter and collector currents are based on the thermionic and tunneling current representation of Grinberg et al. [28], but extended to incorporate base grading. In computing the other currents in the device, particular attention is paid to recombination in both the quasi-neutral base region and the emitter-base space charge region, and to generation .Chapter 1. Introduction 5 in the collector-base space charge region. Inclusion of these current components in the model allows a useful extension of Lundstrom's [29] Ebers-Moll formulation for HBTs to be realized. Three processes of recombination-generation in the space charge and quasi-neutral regions of the device are considered, namely: Shockley-Read-Hall, Auger, and radiative. Many of the material parameters for A l I G a i _ I A s are taken from the device analysis program S E D A N III [34], and Fermi-Dirac statistics are used. 1.4 Overview of the Thesis In Chapter 1, we have briefly described the progress in performance of Heterojunction Bipolar Transistors in recent years. The advantages of HBTs, specially those gained from base grading, are outlined. The main features of our new model for the H B T are also summarized. In Chapter 2, the model is presented in detail. In Chapter 3, current gain, I-V characteristics, and the high frequency figures-of-merit computed from our model are analyzed and the results are discussed. In Chapter 4, experimental data taken from the literature are compared with the theoretical values calculated from our model. Finally, conclusions and recommendations are presented in Chapter 5. Chapter 2 M o d e l Development 2.1 Ideal A b r u p t Heterojunction E n e r g y - B a n d M o d e l Two very important parameters used in our analytical model are the built-in potential and the depletion-layer width of the emitter-base heterojunction. For an ideal abrupt heterojunction, these two parameters are designated respectively by the total energy-band bending, qV^ = qV^n + qV^, and total depletion-layer width, WT, as shown in the energy-band diagram in Figure 2.1. The subscripts 1 and 2 represent, respectively, the wide-gap and narrow-gap semiconductors, or, in a n-p-n HBT, the emitter and the base. Anderson [7] has shown that for an abrupt heterojunction the depletion-layer width and capacitance can be obtained by solving Poisson's equation for the step junction on either side of the interface, with the full depletion assumption (n = p ~ 0): d2V qND dx2 ti d2V qNA for -dn < x < 0 (2.1) for 0 < x < dp (2.2) dx2 6 2 For a graded-base or a graded-junction HBT, however, the above treatment is com-plicated by the Al composition dependency of the dielectric constant. The base side depletion-layer width of a typical HBT is very small because of the usually high base doping level, thus the change in dielectric constant at the base side due to base grading is negligible. In any case, the dielectric constant of Al^Ga^a-As changes, according to our calculation, by less than 10% as the Al mole fraction varies from 0.1 to 0.3. For 6 Chapter 2. Model Development 7 1 1 1 • X - '-e-i the difference of electron affinities, and Vn and Vp are the separations of the electron and hole quasi-Fermi levels from, respectively, the conduction band in the emitter and the valence band in the base. and Vp are related to the electron and hole equilibrium concentrations and effective densities of states1 by [36, p. 27] qVn = fcrin(^) (2.12) qVp = kT ln (^y^j (2.13) where n n o and ppo are the equilibrium majority carrier concentrations in the emitter and base respectively. At equilibrium the intrinsic carrier concentration of semiconductor 2 is given by nf2 = Nc2 A V 2 e*p(—Eg 2/kT), hence qVu , , r ^ ^ ) + ( X j . , l ) + t r i n ( S _ ) _ ^ 1 ^ T ( + «• - ( £ ) 'In order to incorporate Fermi-Dirac statistics into these formulations, a "new1' effective density of states for conduction band and valence band should be used instead; see Subsection 2.6.1 for more details. Chapter 2. Model Development 10 Note that for a homojunction, Nc2 = Nci a n d X2 = Xi> and Eq. (2.14) reduces to the usual expression for a homojunction. For the graded-base HBT, X2, ^ 2) and NC2 are calculated from the parameters of the base at the emitter-base interface. 2.2 Thermionic-Diffusion M o d e l In this section, we will derive the basic static Ebers-Moll current-voltage relationship for the HBT. Our model is based on the "Thermionic-Field-Diffusion" model of Grinberg et al. [28] but extended to include surface recombination velocities of the contacts, base grading, and a more accurate formulation of space charge region recombination-generation currents. The effects due to series resistance, high-level injection and hot-electrons are neglected. Following the traditional practice in Ebers-Moll formulation, we derive separately the hole currents at the two junctions, the electron diffusion current in the base, and the space charge region recombination-generation currents. Emitter-base junction grading and field-emission tunneling are described in later sections. Without loss of generality, only n-p-n HBTs are considered, so that N&, for example, refers implicitly to the doping concentration in the base. 2.2.1 Electron and Hole Thermionic-Emiss ion Currents at the E - B Junc-tion « The carrier transport across the abrupt emitter-base junction can be treated with a simple thermionic-emission model. Consider the n-p emitter-base heterojunction shown in Figure 2.2. Note that the conduction band in the base has a pronounced maximum near the emitter-base junction due to grading of the base. It is assumed thai the amount of conduction band lowering at the depletion edge x — 0 is negligible so that Chapter 2. Model Development 11 Figure 2.2: Energy-band diagram of the n-p emitter-base heterojunction under forward bias. the potential drop across the base depletion region, Vp*, remains the same as if there were no base grading. The net electron thermionic-emission current density injected from the emitter to the base can be described as the difference of two oppositing electron fluxes over the conduction-band spike [29]: Jm = -q vTnB [n(xE) e-«v^kT - n(0) e~^kT} (2.15) where qVTi and AEn are the electron potential energy barriers shown in Figure 2.2, n(xE) and n(0) are the electron carrier concentrations at the depletion edges, and v T l l E , the average i-direction electron thermal velocity in the emitter, is given by kT (2-16) 2nmnE with m*nE being the electron effective mass in the emitter. Chapter 2. Model Development 12 Assuming low-level injection, the electron carrier concentration at x = xE is ap-proximated by n{xE) c± n{xE) = nB0 exp[{qVhi - AEc)/kT] (2.17) The bar indicates the equi l ibr ium condition and nB0 is the equi l ibr ium electron con-centrat ion in the base at i = 0. Referring to Figure 2.2, one can write qVTi = q{Vbi-VBE-VT2) = q{Vbi-VBE)- AEC + AEn (2.18) where VBE is the applied potential across the emitter-base junct ion . A l so by defining n(0) = rJB0 + n(0) (2.19) where denotes excess carrier concentration, E q . (2.15) becomes JT„ = - q v T n E e - * E " ? k T [ n B o e ^ - A E ° » k T e ^ ^ = - 9 v T n J s e - A J S - / t r [ W B 0 ( e ' v " / t r - l ) - M 0 ) ] (2-20) Not al l of the electron carriers transported across the heterojunction is due to thermionic-emission; a port ion of it is due to field-emission tunneling through the conduction-band spike. This effect can be accounted for by replacing vTnE w i th vTnB In in E q . (2.15), 7„ being the tunneling factor [28]. The derivation of 7„ , which is always greater than or equal to 1, is described in another section. Hence, E q . (2.20) becomes Jm = -q vTnB 7» e-*EnlkT [nBQ (e?kT \pE ( c < W * r _ 1 } _ p { x E ) ] ( 2 2 7 ) 2.2.2 Emit ter and Collector Hole Currents Hole carrier transport across the emitter is governed by the diffusion process and the proper boundary conditions at the emitter contact and depletion edge. For simplicity, in deriving the hole diffusion current in the emitter we neglect the effect of grading of the emitter junction and contact. Under the static, time-independent, field-free condition, the continuity equation 1 dJ„(x) p(x) 2.28 q dx TPE and the equation for the hole diffusion current in the emitter Jp(x) = - q D p E ^ - (2.29) Chapter 2. Model Development 14 combine to give the differential equation d2p{x) p(x) dx2 LpE = o (2.30) where p(x) is the excess hole concentration at position x in the emitter, and LpE — \jDPE TpE is the hole minor i ty carrier diffusion length, DPE and rpE being the minori ty carrier diffusion coefficient and lifetime, respectively, in the emitter. The general solution of E q . (2.30) is I x \ I x p(x) = ki exp --— ) + k2 exp j , . - r . j , (2-31) \ ^pE) \ LipE J which is substituted into E q . (2.29) to give the following equations for the hole diffu-sion current density evaluated at the emitter boundaries x — xE and i = WEE (see Figure 2.2): MXE) = JP(WEE) = qDpE LpE s'mh(WE/LpE) p for electrons and holes by Jn = -qpnnVn (2.41) JP = -<7MPpV0 p (2.42) where pn and pp are the electron and hole drift mobilities in the base, and n and p are the base electron and hole carrier concentrations. Based on Kroemer's approximation that V^p ~ 0, Eq. (2.41) can be rewritten as Jn = qpnnV{p-(j>n) (2.43) For the nondegenerate case, the pn product is given by (2.44) pn = n] exp where n, is the intrinsic carrier concentration in the base and is position-dependent due to base grading. Taking the gradient of Eq. (2.44) and substituting it into Eq. (2.43), we obtain the following expression for the base electron current density [37]: NA dx\ n ? ( i ) / Here, we have assumed that p ~ NA in the P-type heavily doped base, and used the nondegenerate Einstein relation D„B — kT pn/q. The electron diffusion coefficient in the base, DNB, is assumed constant at a value appropriate to material of the Al mole Chapter 2. Model Development 18 fraction as exists in the center of the base. The base grading is taken to be linear, so the bandgap is given by AE Eg{x) = Eg0-—±x (2.46) where Eg0 is the bandgap at x = 0, that is, at the emitter-base depletion edge on the base side (see Figure 2.3), XB is the total base width, and AEg is the difference in Eg between the values at the two metallurgical junctions which define the base. The intrinsic carrier concentration can be expressed as n]{x) = Nc{x)Nv{x)e-Eo^l kT = ae fx (2.47) where n*0Nc{x)Nv{x) a = Nco Nvo f " A qAEg kT XB In the above equation, Nc and Ny denote the effective densities of states in the con-duction and valence band respectively, and the subscript zero refers to conditions at x — 0. It is safe to assume that "a" changes much more slowly than e-^ 1, or more precisely, |± ^ | 5 5) where nBW is the electron equilibrium concentration in the base at x = W. For n(0) one must match J„(0) with the electron thermionic-emission current density of Eq. (2.21) which is rewritten here as JTn = -zn \nB0 (e* v™/ kT - 1) - n(0)| (2.56) where zn - qvTnE in e "i When evaluated at x — 0, Eq. (2.54) reduces to Jn{0)=yn[2th(W)-anh{0)} (2.57) where qDnB "n = (rx-f)er* W -(r2-f)e n W Equating the current densities of Eqs. (2.56) and (2.57) and solving for h(0) gives m =*y.W)+ -1) (2M) zn + anyn Substituting Eqs. (2.55) and (2.58) into Eq. (2.57) yields the following expression for the emitter electron current density: UO) = - ( ""y" / ) nm (e^kT - 1) - —— TxBw (e*V^kT - 1) \1 + anyn/znJ ar, (2.59) Chapter 2. Model Development 21 Figure 2.4: Schematic of charge flows in a heterojunction bipolar transistor. Similarly the collector electron current density can be found by evaluating Eq. (2.54) at x = W: Jn{W) = yn [bn h(W) - 2t eu w n(0)] (2.60) where k = ( r ! - / ) e ' ' * - ( » • , - / ) f w Finally, replacing h(W) and n(0) in Eq. (2.60) by their known equivalents in Eqs. (2.55) and (2.58) leads to Jn{W) = - 1 + a n yn/zni \nB0(e"v^ kT-l) 6„ + ( a n bn-4t 2 e 2> w)yn/zn 2t e 2fW nBW{z'v*clkT -1)J (2.61) Chapter 2. Model Development 22 The main electron and hole current components formulated so far are shown schem-atically in Figure 2.4. What we call the emitter and collector electron current densities, Jn(0) and JN{W), are actually electron current densities entering the quasi-neutral base and the base-collector depletion region respectively. JP{XE) is the hole current density back-injected into the quasi-neutral emitter and —Jp{xc) is the hole current density entering the base-collector depletion region from the collector. The latter current den-sity is shown with a negative sign because in our derivation the positive sense is from the P material to the N material. The term |J„(0) — Jn(W)| represents the part of the base current density due to recombination in the quasi-neutral base. JR is the re-combination current density in the base-emitter depletion region for the forward-biased base-emitter junction and JQ is the space charge region generation current density for the reverse-biased collector-base junction. The total DC emitter and collector current densities, Jg and Jc, are drawn with arrows indicating the direction of charge flow under normal operating conditions. From Figure 2.4 we may write Substituting Eqs. (2.35), (2.40), (2.59) and (2.61) into Eqs. (2.62) and (2.63) leads to the following Ebers-Moll expressions: JE = -Jn(0) + Jp{xE) + JR (2.62) Jc = -Jn{W) - Jp{xc) + JG (2.63) Jc = A 2 1 ( e ^k T -1) + An (e« v^/* T -1) + A22 (e" v^ kT -1) + JR 1) + JG (2.64) (2.65) where Chapter 2. Model Development 23 122 bn + (anbn-4t2e 2° w)yn/zn Vn nBW — JpC cosh •>pC, 1 + O n Vn/Zn All the symbols used have been defined earlier. The equilibrium carrier concentrations nB0 and nBw can be calculated from the intrinsic carrier concentration and the base doping density: nB0 — U2Q/NA, ™ B W = n2W/NA. The base current density is simply the difference of the emitter and collector current densities, i.e., JE — Jc-It is instructional to show that the Ebers-Moll equations (2.64) and (2.65) do reduce to those predicted by the conventional diffusion model for the case of a simple homo-junction transistor. This would require that the electron energy barrier AEn < 0 and l A ^ I » kT (see Figure 2.2). For the hole energy barrier, the inequality AEP » kT should still apply for homojunction transistors. Assuming also that the contacts are perfectly ohmic, i.e., SPE — Spc —• oo, we would have the simplied expressions 9 DpE VE RE = l J, zn —> OO Jpc pE LpE smh{WE/LpE) 9 Dpc Pc LpC s inh (Wc /LpC) Of course, no compositional grading is possible with a homostructure transistor so many of the earlier expressions are also simplified: / = 0, nB0 = nBw = nB, s = 0, t = l/LnB, 2 i . ( w \ , nB \ ^nB / nBJ * 2 smh{W/LnB) Inserting these equations into the Ebers-Moll coefficients of Eqs. (2.64) and (2.65) and making the reasonable assumption that WE » LpE and Wc ^> Lpc, the emitter and collector current densities reduce to , qDpEpE JE = q DnB nB / W - coth LnB VL„£) 9 DnB n B qVBE/kT LnB smh.{W[LnB) LpE 1) (2.66) Chapter 2. Model Development 24 Jc = q DnB nB ,qVBB/kT 1) LnB s'mh(W/LnB) qDnBnB — coth q DpC pc Lpc 1) + JG (2.67) which are the normal diffusion-model Ebers-Moll current density equations [38, p. 260). 2.3 Emitter-Base Junction Grading The presence of the conduction-band spike in the emitter-base junction of an HBT is usually regarded as an undesirable feature. For example, it is known that such an electron-blocking barrier can cause a substantial drop of emitter injection efficiency [22]. Another disadvantage is that the potential notch accompanying the barrier at the base side tends to confine injected electrons and therefore enhances recombination losses [3]. The conduction-band spike also creates a high emitter-base turn-on voltage in HBTs [3]. In light of these drawbacks, it makes sense to utilize some form of grading of Al composition in the emitter-base junction to reduce the conduction-band spike. In fact, a number of modeling schemes for emitter grading have been published already. Many of these are simple models in which the expressions for the electron and hole currents are slightly modified according to the amount of emitter grading [28,39,40]. Cheung et al. [41] developed a simplified version of the generalized model of Oldham and Milnes [42] and used it to calculate the conduction-band profiles of a p-n heterojunction for different grading widths. In still other models, extensive numerical simulations were employed [43,44] to investigate the effects of emitter grading on HBT characteristics. It has been found that, in the case of graded emitters, not only were the current gain increased and the turn-on voltage reduced but the cutoff frequency, fr, was also higher. Hayes et al. [35], using the model of Cheung et al. [41], also found that emitter-graded Chapter 2. Model Development 25 Figure 2.5: Energy-band diagram of the graded emitter-base junction. H B T s exhibited lower offset voltages (possibly due to lower emitter turn-on voltages). On the other hand, the conduction-band barrier may act as a "ballistic launching ramp", injecting into the base electrons with a high kinetic energy [3,39,45]. Provided that most of the electrons remain in the lower conduction-band valley, the average electron will speed across the base with a very high velocity. However, the recent trend of experimental HBTs [46,47,48] appears to lean towards the use of emitter grading. In our model, we follow the simple linear grading scheme of Grinberg et al. [28]. The basic idea is to modify the electron energy barrier parameter AEn which appears in Eq. (2.21). Consider the band diagram of a graded emitter-base junction shown in Figure 2.5. The grading is linear and it applies only to the emitter conduction band. Since the valence band is assumed to be unaffected by grading, the hole energy barrier Chapter 2. Model Development 26 AEP can be simply expressed as AEP = qVT1 + AEV (2.68) where VJI is the potential drop across the emitter depletion layer of a non-graded junction and AEy is the valence-band energy difference of the emitter and base. If Wg is the emitter grading width, then the amount of conduction band lowering can be found by evaluating Eq. (2.3) at x = —Wg: qvT1 = — — [-f-yygdn\ q 2ND [{dn - W,)> - dl] (2.69) 26! Since gV^ = qVT1 - qV^ (see Figure 2.5) and gVT1 = q 2NDd 2n/2e1 (from Eq. (2.5)), we have qv^ = K - w*)* (2-7°) The electron energy barrier parameter can be written as AEn = AEc-qVT2-qV^ = AEC - {qVT - gVT1) - qV% = AEc-qVr + qV^ (2.71) where qVT is equal to q {Vbi — VBE). Note that Eq. (2.70) does not apply when the grading width is greater than the depletion width dn. If Wg > dn, the barrier qVjr * should become zero. In general then we can write AEn = AEc + q (VBE - Vbi + VLX) (2.72) where £J±{dn-Wgy for W, dn Chapter 2. Model Development 27 E = Ex i E E = E' I qVn2 AEn f ' E*, the integration limits appeared in Eq. (2.84) are ETI = qV^i and E* = qVn — AEc\ (ii) for large junction grading, i.e., ETI < E*, no tunneling is possible, hence in = 1 (or set ETI — E*). It may also be possible that the potential notch (see Figure 2.6) falls below the energy level of the conduction band in the quasi-neutral emitter, i.e., E* < 0. In this case, we set E* = 0 because no tunneling can occur below the reference energy level E = 0. In order to solve for the integral in Eq. (2.84) one needs to know the equation for the barrier transparency D{E). Although Grinberg et al. [28] had already published their result for D(E) (derivation not given), our own derivation shows a slightly different Chapter 2. Model Development 31 0 4 dr B A S E E M I T T E R x 0 Figure 2.7: Tunneling through the conduction-band spike of the abrupt emitter-base junction. expression for D{E) in the case of a graded emitter junction. In the following subsection we will show in detail the derivation of the barrier transparency based on the work of Stratton and Padovani [50,51,52]. 2.4.1 Barr i er Transparency A. Abrupt Junction Consider first the abrupt heterojunction shown in Figure 2.7. In general, the expres-sion for barrier transparency for an arbitrary potential barrier shape is given by [50] where a — 2 y/2m{/h, ml being the effective electron mass in the emitter and h being the reduced Planck constant. The barrier potential energy (x) is given by (2.85) 4>{x) (dn - X ) 2 (2.86) 2ci Chapter 2. Model Development 32 From Figure 2.7 we see that the tunneling electrons leave the barrier at x = 0 but enter it at some arbitrary position x 2 in the emitter depending on the energy of the electrons. Thus the lower integration limit in' Eq. (2.85) is zero. Before solving for the integral in Eq. (2.85), it is useful to know the solution of the following integral: jja{dn-x)2-b\^ 2dx = ^ = {-y/i~{dn-x)^/a(dn-x) 2-b + b ln y/a (4 - x) + ^/a{dn-x) 2-b] ) ' (2.87) J J o Comparing the integrals in Eqs. (2.85) and (2.87), we see that a = q 2 Nu/2e-l and b = E. Also we know that an arbitrary tunneling energy level is related to x 2 by E = a(dn — x 2 ) 2 and that the energy barrier is given by ETI = adn. Therefore Eq. (2.85) becomes - \nD{E) = | [ E \n^E + y/Er~i\/ET1 -E-E ln \[E~T~\ + \J En - E j (2.88) Defining an energy 2q ND hq / ND a f 2ti 2 V ci m i and a dimensionless variable X = E/ETI, the barrier transparency given by Eq. (2.88) can be rewritten as D(E) = D{X) = exp ( En Eoo X - X In . Vx t ]) (2.89) B. Graded Junction For a linearly graded junction such as the one shown in Figure 2.8, the barrier may be divided into two parts: a normal potential barrier between x = dn and x = Wg, and Chapter 2. Model Development 33 0 E A ETI = QVTI B A S E E M I T T E R x -* 0 Figure 2.8: Tunneling through the conduction-band spike of the graded emitter-base junction. a graded barrier between x = Wg and x = 0. To solve for the barrier transparency, the integral in Eq. (2.85) must be integrated from x = z 3 to x — x2. The integration is done separately for the two divided barriers. For the first potential barrier (i.e., barrier I in Figure 2.8) we have the integral Again using Eq. (2.87) and noting the relationships a = g 2 Np/2ei , b = E, E = a (dn — x 2 ) 2 , and ETI = a(dn — Wg)7, the above integral is readily solved. The solution turns out to be exactly the same as Eq. (2.89), i.e., For the graded barrier (i.e., barrier II), the barrier potential energy is given by (2.90) Chapter 2. Model Development 34 — (dn-x) + — X - ± E C (2.91) The integral in Eq. (2.85), integrating from x = £ 3 to Wg, is J X q ND , , AJ^C * 3 I 1/2 dx J X X3 » , / . * ^ l - 6 2d where 2]j q2ND Wg dn 1/2 dx (2.92) Eq. (2.92) can be changed to a form similar to that of Eq. (2.87) by rewriting the integrand as \(dn - y)2 - 6]1 / 2. This leads to -dy = yjq2 ND/2e1 dx, and Eq. (2.92) becomes } (dn -y)+ yj{dn - y ) 2 - b x=x3 x=W„ (2.93) The above integral can be solved with the help of Eq. (2.87). Noting also that ' 1 a / 2 C l 2 V q2 ND oo V K - y ) 2 - 6 | x=x3 y/E-E - 0 V(d„ -y)2-b = y/Er^E Eq. (2.93) reduces to - i - (ft ln Vb + \ j E T l - E + b yjETl - E - b ln L/£ri - E + b + J ETi - E In W . ' + 1 + W7 . _ " > (2.94 E-E VT^XyJl -X + b/ETl E- b/E b/E Tl j (2.94 Chapter 2. Model Development 35 where X — E / ETI- The term is a function of X : E + €tAEl En ETi 2NDq2WgETl ETl 2 / . \ 2 AEc (dn - 1 dn 1 /AEC 2 V jE'n ( i r ) ' - 1 - 1 (2.95) In deriving E q . (2.95) we have used the relationship ETI — q 2 ND {dn — W / 5 ) 2 / 2 e 1 . To obtain the barrier transparency for a graded junct ion, we simply sum the solu-tions of the two integrals in E q . (2.90) and (2.92). In summary, the barrier transparency for a graded junct ion is given by ETI D(E) = D(X) = exp { - - £ [I(X) + I I (X) ]} (2.96) where 1(X) = V r a - X l n [ * ± 3 F ] X > 0 X = 0 I I ( X ) y/l-X + f[X) + ^ A=Y f(X) + o f(X) = 0 VT=Xjl-X+f{X)-f(X) ln 1 - X For an abrupt junct ion , i.e., Wg = 0, I I ( X ) = 0. In Append ix A the tunneling fac-tor given by E q . (2.84), which contains the expression for the barrier transparency (Eq . (2.96)), is reorganized in a form appropriate for coding. 2.5 Recombinat ion and Generation Currents The emitter-base space charge region recombination current Jp in Eq. (2.64) and the collector-base space charge region generation current Jg in Eq. (2.65) are computed from JR -G = q I J SCR Udx (2.97) Chapter 2. Model Development 36 where U is the net recombination rate which, in general, consists of several recombina-tion rates from different recombination processes. Three major types of recombination process are considered here: Shockley-Read-Hall (SRH), radiative, and Auger. While SRH recombination is usually the dominant process in Si bipolar transistors, it may not be so for GaAs/AlGaAs HBTs. Radiative recombination is known to be important in most III-V compounds with direct energy gaps (i.e., GaAs), and Auger recombination is important in materials with high doping concentrations (such as the base region of an HBT). We will model all three recombination processes. 2.5.1 Shockley-Read-Hall Recombination Process The basic SRH recombination process normally assumes one trapping energy level in the bandgap [53,54,55]. The single-level recombination rate is given by Us** = — T — t,._J^~n\ , / „ . _ „ . M (2-98) n + m exp (^p 1 )] + rno [p + n, exp where rpo and rno are, respectively, the minority carrier lifetime in highly extrinsic N-type and P-type material, due to single-level recombination, Et is the energy level of the recombination-generation centers, and = qtjj is the intrinsic Fermi level. Note that the intrinsic carrier concentration n, in Eq. (2.98) is position-dependent since the materials in the emitter and in the base are different. From now on it is implicit that rii — rii(x). Equation (2.98) may be written in terms of various electrostatic potentials with the use of the following expressions: pn = n, exp \ -j^r j (2-99) n — n, exp p = n, exp (2.100) (2.101) Chapter 2. Model Development 37 Figure 2.9: Energy-band and potential diagram of the emitter-base junction at forward bias, with a linearly varying intrinsic Fermi level in the emitter-base depletion region. Here, xp is the intrinsic Fermi potential, n and p are, respectively, the electron and hole quasi-Fermi potentials, and VBE = p — 4>n is the applied emitter-base voltage. Thus USRH = ^exp - *i±±)] + ^ e x p [& ( * ± * t - * ) ] + 2exp ( - ^ ) cosh + In (2.102) Following an approach similar to that of Choo [56], ip(x) is assumed to vary linearly across the forward biased emitter-base heterojunction depletion region. In Figure 2.9, the intrinsic Fermi potential and the electron and hole quasi-Fermi potentials are drawn. The intrinsic Fermi potentials in the quasi-neutral emitter (i < — dn) and in the quasi-neutral base (x > dp) are denoted by ipn and ipp respectively. Without loss of generality, Chapter 2. Model Development 38 we set ipn — 0 as a potential reference. Therefore — [x + dn) WBE K T n> (2.103) where WBE refers to the total emitte-base depletion width. In the quasi-neutral regions, we may approximate ND ~n = n,i exp I ^ (ipn - n) NA~p = ni2 exp ^ [p - V>P) The subscripts 1 and 2 refer to the emitter and base respectively. We define (2.104) (2.105) 0 = ^ r b k - i M / tin i t 2_ \ g^BJS (2.106) which is derived from Eqs. (2.104) and (2.105) and the relationship VBE = p — r Substituting Eq. (2.106) into Eq. (2.103), the latter becomes Since tpn = 0, Eq. (2.104) leads to Q A. l U i l (2.107) (2.108) As a result, J L f _ ^ L ± A fcrA 2 ^ / , , x , n « l QVBE — (x + dn)-ln— - — e ™BE (2.109) where a — dn + ln AT D BE BE f i i i 2kT Chapter 2. Model Development 39 Note also that dn/WBE = NA/(NA + ND). After Eq. (2.109) is substituted into Eq. (2.102), the latter is used in Eq. (2.97) to solve for the SRH recombination current density. The integration must be performed separately for the emitter and the base because the minority carrier lifetimes and the intrinsic carrier concentrations are different in the two regions: 2qWBE sinh ["„,., fx=o dz ni2 r x=d" dz J SRH R where nn r«=o dz r*=* Ti Jx=-dn Z2 + 2z6j -f 1 T2 Jx=0 z 2 + 2z&2 + l (2.110) Tl = y/Tpoi Tnol ?2 = y/Tpo2 Tno2 , / qVBE\ u(Et-Et bx = exp ( — n t ^ ) cosh | — — h ln 62 - exp In Eq. (2.110), the substitution z = 2kT <1VBE 2kT kT c o s h ( ^ r ^ + l n ' 'pol 'nol. !Tpo2 1~no2, 'pol exp e v w - \WBE was made for the first integral, and the substitution x + a exp V Tno2 ' \WBE was made for the second integral. The solution of'Eq. (2.110) is 2qWBE sinh J SRH R — — f{bi) + —g(b2) T\ T2 where f(h) = ^ tan" ( z 0 1 - z i ) v / | l - ^ | ] + 2 1 Z o i + t l ( Z i + Z o , ) Zl+1 201 + 1 b\ # 1 b\ = l (2.111) Chapter 2. Model Development 40 g(b2) = 1-^1 tan - l («3-«na) y/ll-tjl l + « 2 * 0 2 + * 2 ( 2 2 + Z 0 2 ) 1 2(12 + 1 Z 2 + 1 *1 = 1 anc 2 l = ^01 z 2 = z 02 — 9 exp ^~ exp(a) Tpol Tnol dn + a exp(a) lTpo2 T~no2 Note that dp/WBE = 1 - = ND/{NA + ND). Due to a lack of experimental data we let Et — Ei (the effect of this is to produce a maximum recombination rate), TVO\ — rpo2, and rno\ = r n 0 2 . Under these conditions oi = o2 = - ( ^ / — + ) exp ' po 2kT 2.5.2 Radiative Recombination Process Under low injection conditions, the rate by which radiative recombination exceeds thermal generation is given by [57] Urad = B{np- n]) (2.112) where B is the radiative constant. The non-equilibrium pn product for the emitter-base junction is related to the emitter-base voltage by Eq. (2.99) which is substituted into Eq. (2.112). The latter is then used in Eq. (2.97) to solve for the recombination current density due to the radiative process. The result is JR" = q (e" v^ kT - 1) (dn B, n]x + dp B2 n 2i2) (2 .113) Chapter 2. Model Development 41 The subscripts 1 and 2 again denote the emitter and the base, respectively, while dn i and dp are the depletion-layer widths as shown in Figure 2.9. 2.5.3 Auger Recombination Process Two Auger recombination processes for Al IGa 1_ IAs are usually recognized [58]. The first process, known as the CHSH process, occurs when a conduction-band (CB) elec-tron recombines with a heavy-hole-band (HB) hole. The subsequent release of energy causes a light hole in the spin-splitoff band (SB) to transfer to the heavy-hole band. Since this process involves two holes and one electron, its recombination rate under low injection conditions is given by Rp = Cpnp2 (2.114) where Cv is the Auger coefficient for the CHSH process. The second process, known as the CHCC process, is similar to the CHSH process except that the energy released from the electron-hole recombination is given to an electron in the conduction band. In this case, the recombination rate is Rn = Cnn2p (2.115) where Cn is the Auger coefficient for the CHCC process. In thermal equilibrium, Auger recombination is balanced exactly by the converse process of generation of electron-hole pairs by electrons and light holes. For the CHCC process, the generation rate is proportional to the electron concentration, n, since it depends only on the number of electrons present [59, p. 271], thus Gn = C„npn (2.116) Chapter 2. Model Development 42 Similarly the generation rate for the CHSH process is proportional to the hole concen-tration, p, since it depends only on the number of light holes present: Gp = Cpnpp (2.117) In non-equilibrium, the net Auger recombination rate is given by UAug = (Rn + Rp)-(Gn + Gp) - (Cnn + Cpp){np-np) (2.118) Substituting the expression np = n\ and those in Eqs. (2.99) to (2.101) into Eq. (2.118), the latter becomes UAug = n? (e«v"/*r - 1) {C n exp p + n \ cD exp exp q ( p + n kT \ 2 rp + (2.119) The relationship VBE = n was used in the above equation. As in Subsection 2.5.1, we assume that t/>(i) varies linearly across the forward biased emitter-base junction. Letting z = exp q , P + n kT V 2 exp 0 WBE {x + a) and c — \JCn Cp, where 6 and a are defined in Eqs. (2.106) and (2.109), and substituting Eq. (2.119) into Eq. (2.97), yields TAug q £ nf e V W » r ( e , v „ / * r _ 1 ) e ^ + l j W B B d z 2qWBE e" v^f kT sinh ( ^ ) 0z 2qWBE z*vB*l kT s i n h ( ^ ) [*" 3 ci (1 + — | dz + / ni2 (nfj ci ( Z 0 J - 2 X ) - ( — - ) c2 1 + —A dz + nf2 c 2 (z2 - 202) - ( — — Zl ZQ2 (2.120) Chapter 2. Model Development 43 where Ci = \JCn\ Cpi, c2 = yJCn2 Cp2, 21 = ^expVvVB-Edn + aj zoi = exp(a) 9 1 = V ? e x p f e d p + a ! 2 0 2 = y~ exp (a) The total emitter-base space charge region recombination current density is the sum of Eqs. (2.111), (2.113) and (2.120). 2.5.4 Generat ion Process In the reverse-biased collector-base junction, the most dominant generation process is of the SRH type. To illustrate this, we assume that under large reverse bias the applied junction voltage, VBc, is negative and its magnitude is at least several kTjq inside the space charge layer. For the SRH process, this means that in the denominator of Eq. (2.102) the first two terms are small compared with the last term. Assuming Et — Ei, Eq. (2.102) can be approximated by USRH * — ^ — ( e « v » ° / f c r - 1) (2.121) rno -r rpo The SRH generation current density over the entire collector-base depletion region is JgRH — —q I USRH dx JC-B SCR ^nc ^pc q ni WBC ~ — — — (2.122) '"no Tpo Chapter 2. Model Development 44 where WBC is the collector-base depletion-layer width. In deriving Eq. (2.122) we have assumed that the collector-base junction is essentially a homojunction, i.e., n, ~ constant, and that under the normal range of reverse bias qVBC/kT > x *2 \Cn n, Cp NA exp 2kT J and 2 sinh(gVBc/2itr) ~ - exp(-qVBC/2kT), Eq. (2.124) reduces to JG — q e l V B O / 2 k T W B c ln(^p) + <}\VBC\ kT qn 2WBC(CnND + CpNA) I n ( ^ ) ~ kT (2.125) To see how the three generation current density components compare with one another, we need some typical values for some of the parameters. For GaAs, rno ~ 10 - 9 s, rp0 10~8s, n, ~ 10 6 cm - 3 , Cn Cp < 10 - 3 0 cm 6 /s and B 10~ 9cm 3/s are Chapter 2. Model Development 45 used. Typical values for the collector and base doping concentrations are 1016 and 1019cm~3 respectively. Comparing the Auger generation current density with the SRH generation current density, we note that JG U9 = ni{CnND + CpNA){Tno + Tpo) < m cp NA TPO < (106) (1(T30) (1019) (10~8) < 1(T13 A similar comparison between the radiative and the SRH generation current densities shows Trad JG rSRH JG = ri, B (r n o + Tpo) < TliB Tpo < (io 6) (10- 9)(io-8) < i o - 1 1 Both the Auger and the radiative generation current densities are many orders of magnitude lower than the SRH generation current density. Therefore the total generation current density in the collector-base depletion region is approximately JTOT „ JSRH = { n w ^ + n.c d c ) [ l _ tqVBClkT) ( 2 1 2 6 ) t Tn0 T T p 0 where n,£ and n,c are, respectively, the base and collector intrinsic carrier concentra-tions, which are different because of base grading, and dg and dc are, respectively, the base and collector depletion-layer widths obtained using the usual depletion ap-proximation. By restricting JC to suitably low values (see Chapter 3), base widening due to donor neutralization by electrons entering the collector (Kirk Effect) need not Chapter 2. Model Development 46 be considered. The collector-base voltage, VBC, must be less than or equal to zero. Note that under injection conditions, Eq. (2.126) is no longer valid, but the fact that it overestimates the generation current is of no consequence as Jc ) ( 2 , 1 3 3 ) where m* and m* are the effective masses for electrons and holes respectively, and h is the Planck's constant. The new effective densities of states should replace the effective * densities of states in all the formulations unless stated otherwise. The intrinsic carrier concentration can be computed from the pn product at thermal equilibrium under which the Fermi level is constant, i.e., Epn = Epp, hence m = y/np = y/NcNy exp{-Eg/2k,T) (2.134) where the bandgap Eg = Ec — Ey • Chapter 2. Model Development 48 4 INDIRECT 1 X 0 o c u w -1 - 2 - 3 tr/o o K 2TT/0 [111] T [lOo] Figure 2.10: Band structure of GaAs with the energy E plotted as a function of mo-mentum wave vector k along the [100] and [111] directions [34]. 2.6.2 B a n d g a p and Electron Affinity The energy band structure of GaAs with the electron energy plotted as a function of momentum wave vector k is shown in Figure 2.10. The figure shows three local conduction-band minima ( r 6 , X$) located at k = 0 and the zone boundaries along the two crystal momentum directions, and a global valence-band maximum (r8 lo-cated at k = 0. Three interband energy gaps, defined as the difference between the local conduction-band minimum and the global valence-band maximum (i.e., Tg), are identified and designated appropriately as T, X, and L. The bandgap, on the other hand, is defined as the difference between the lowest conduction-band minimum and the highest valence-band maximum. In GaAs, a direct bandgap material, both the global conduction-band minimum and the global valence-band maximum occur at the same value of k (k = 0) . In other words, the bandgap of GaAs is the same as its T Chapter 2. Model Development 49 3.2 I I I I T — r ~ \ i i 3 0 T - M 7 K l j « 3.018. 2.8 / -2.6 m ml < e >• o oe w z ml 2.4 2.2 2.0 1.8 1.6 -£$•1,900 x (T- Pp.* — j * ! ' 2 * 8 . . 1.4 t 9 -££•1.424 l l l l • i i i i 0 O.t 0.2 O J 0.4 OS 0.6 0.7 0.8 0.9 10 60AS MOLE FRACTION A1AS.X A i AS Figure 2.11: Compositional dependence of the T, X, and L interband energy gaps [34]. interband energy gap. However, in A l x G a 1 _ I A s , the conduction band varies so that when the Al mole fraction x > 0.45 the X conduction-band minimum at the [100] zone boundary drops below the T conduction-band minimum at the zone center. In this case, AlsGa^xAs becomes an indirect bandgap material and the bandgap is given by the X interband energy gap. Figure 2.11 shows the variation of the three interband energy gaps with composition. The equations that describe these energy gaps, in (eV), as a function of composition, are as follows [60] f 1.424 + 1.247 x 0 < i < 0.45 ETg(x) = ~ ~ (2.135) { 1.424 + 1.247i + 1.147 (i - 0.45)2 0.45 < x < 1.0 E$(x) = 1.708 + 0.642 x (2.136) E?(x) = 1.900 + 0.1251 + 0.143 x2 (2.137) Chapter 2. Model Development 50 Obviously from Figure 2.11, the bandgap of A ^ G a ^ A s , Eg(x), is equal to either Eg(x) or Ef(x), whichever one is smaller. The above equations for energy gap are formulated for high-purity GaAs and AlsGax-sAs at 297 K. Bandgap, however, is known to shrink with doping concen-tration. Casey and Stern [61] have measured the doping dependence of bandgap in p-GaAs and arrived at the following empirical expression: Eg (eV) = 1.424 - 1.6 x 1 0 - V / 3 (2.138) where p is the P-type doping concentration in cm - 3 . For lack of better data, we assume this formula is applicable also to A ^ G a ^ A s of both P-type and N-type doping. The general formula for Eg in A^Ga^^As is therefore Eg{x,N) = m i n ( £ [ ( x ) , £ f (x)) - 1.6 x 1 0 - 8 N1?3 (2.139) where TV is the net doping concentration in c m - 3 , min( ) means "the minimum of", and Eg is in eV. According to Anderson's model [7], the difference in electron affinities between GaAs and Alj;Gai_xAs equals the difference of conduction bands in the respective materials, i.e., A x = AEc- The electron affinity for GaAs is about 4.07 [36]. Acknowledging that electron affinity decreases as Al mole fraction increases, we can express the dependence of the electron affinity on composition as X{x) = 4.07 - AEc{x) (2.140) The conduction-band offset is generally assumed to be linearly proportional to the difference of energy gaps: AEC = Qt AEg (2.141) The constant Qe was determined by Dingle for x = 0.2 to be about 0.85 [62, p. 21.]. This value was accepted and confirmed by a few early papers [63,64]. However, in Chapter 2. Model Development 51 1984, Miller et al. [65] found that the energy gap discontinuity AEg divides more equally between the conduction and valence band offsets. Subsequently a large number of workers remeasured AEc and AEv using a variety of techniques and found that Qe was indeed smaller than previously expected. The majority of recent works indicates that Qe is between 0.60 and 0.67 for x < 0.45 [66]-(76|. When x > 0.45, AljGa^As becomes an indirect material and Eq. (2.141) no longer holds. In this case it is more meaningful to measure AEy and express it as a function of the gamma energy gap offset [77,78]: AEV = Qv AETg (2.142) where Qv is a constant. However there is also evidence that the valence-band offset is linearly proportional to the Al mole fraction [74,79]. Assuming that Eq. (2.142) is valid and Qv = 0.36 (equivalently Qe = 0.64 for x < 0.45, a value used in SEDAN III [34]), we can derive the conduction-band offset AEc as a function of composition using the method proposed by Hill and Ladbrooke [78]. The energy gap discontinuity must divide between the conduction and valence band offsets, hence AEC = AEg - AEV = AEg-QvAE] = [Et(x) - 1.424] -QV\E]- 1.424] (2.143) In obtaining Eq. (2.143), we have used 1.424 eV for the energy gap of GaAs. For x < 0.45, Eg{x) = £[(x) , thus Eq. (2.143) becomes AEC = ( 1 ( x ) - 1.424] = Q e (1-247 x) = 0.7981 x for 0 < x < 0.45 (2.144) Chapter 2. Model Development 52 The formula for E v9(x) is given by Eq. (2.135). For x > 0.45, Eg(x) = Ef(x)\ therefore substituting Eqs. (2.135) and (2.137) into Eq. (2.143) yields AEC = 0.392 + 0.048 x - 0.27 x 2 for 0.45 < x < 1.0 (2.145) 2.6.3 Effective Mass For GaAs and A ^ G a ^ A s there is an electron effective mass associated with each of the three conduction band valleys. These electron effective masses are also-known as the density of states effective masses and are given by [60] m Tn = (0.067 + 0.083 x) mo (2.146) m\\ = (0.55+ 0.12 x) m 0 (2.147) m * = (0.85 -0.07 x) mo (2.148) where mo is the electron rest mass and x is the Al mole fraction. The above equations are formulated based on a linear extrapolation of the density of states effective masses of GaAs and AlAs. The overall electron effective mass is derived from the assumption that the total electron concentration is equal to the sum of the electron concentrations in the three conduction-band valleys [80], that is n = n r + nL + nx (2.149) 4 For x < 0.45, the T valley is the lowest in energy among the three conduction-band valleys, T, L, X, and Eq. (2.149) can be written as " c e x p ( ^ £ ) = AG exp ( ^ ^ ) + N$ exp + N$ exp EF — Ec\ ,rr IEp — Ec , , - I + 1\X exn kT EF - [Ec + AE?-rY kT EF - {Ec + AEj;-*)1* (2.150) Chapter 2. Model Development 53 where AEjf~r = E\ - Evg and AEf~r = Ex -ETg. Since the effective density of states for the conduction band is in general given by Nc = 2 2nmnkT\ h2 (2.151) Eq. (2.150) reduces to K ) 3 / 2 + « ) 3 / * exp ( - ^ - ) + {mXn)Z'2 exp ( - A E ? ~ ^ ] kT kT 2/3 x < 0.45 (2.152) where mn is the overall electron effective mass. For x > 0.45, the X valley is the lowest in energy. Thus, the overall electron effective mass is K ) > / ! exp M ^ ) + exp L^L) + 1 2/3 x > 0.45 (2.153) where AETg~x = ETg - Ef and AELg~x = ELg - Ef. The hole effective mass for A l i G a i ^ A s is taken as [60] m* - (0.48 + 0.31 x) m 0 (2.154) 2.6.4 Dielectric Constant The dielectric constant of AlxGaj.a-As can be calculated by assuming that the dipoles in the alloy are divided into a fraction x of type-1 dipoles and a fraction 1 — x of type-2 dipoles where x is the A l mole fraction and 1, 2 refer to AlAs and GaAs respectively [81]. The dipoles are characterized by polarizability a. In order to take into account any change in a due to changing equilibrium interatomic spacing, it is assumed that a is proportional to the volume of the unit cell. Therefore, representing a, ai and Q 2 as the polarizabilities of the alloy, AlAs and GaAs, respectively, we can write oc = x ai — + (1 - i ) a 2 — (2.155) Chapter 2. Model Development 54 Table 2.1: Dielectric constants for GaAs and AlAs. material ei GaAs 10.9 13.1 AlAs 8.12 10.06 where V,Vi, and V2 are the unit cell volumes of the alloy, AlAs, and GaAs, respectively. The dielectric constant e is related to polarizability by the Clausius-Mossotti equation [82, p. 382] f — 1 4-7T (2.156) c - 4T „ = — > Nidi c + 2 3 ^ where is the number per unit volume of type-i dipoles having polarizability a,. In a pure type-1 compound, its dielectric constant ej is related to Qj by Eq. (2.156) but with the summation sign dropped. A similar relationship exists between e2 and a2. These relationships are substituted into Eq. (2.155) to give e - 1 NV / c i - l \ , NV where e and N are the dielectric constant and the number of dipoles per unit volume for the alloy. Clearly all the N V products cancel out. Hence, solving for e, , '+2[*teO + (i-*>(fm)l ,2158, which applies to both high and low frequency. The values for £x (AlAs) and e2 (GaAs) for both high and low frequencies are listed in Table 2.1 [34]. Chapter 2. Model Development 55 2.6 . 5 M o b i l i t y and Diffusion Coefficient The mobility p for electrons or holes is, in general, related to the effective mass m* and the relaxation time r by [38, p. 25] q r M = " ~ (2.159) In III-V semiconductors, r is primarily dominated by polar optical phonon scattering which has the form [80] " ~ ^ ( i - i ) where and e/ are the high and low frequency dielectric constants for A l x G a i _ x A s ob-tainable from Eq. (2.158). Substituting Eq. (2.160) into Eq. (2.159) yields M oc (mT 3 / 2 f - - - ) " 1 (2.161) For AljjGax-jcAs, the mobility of holes is simply [80] T oc / . -v- (2.160) Vp{NT,x) * ? ( * ) - < r l { x ) < 2- 162) where Mp,GaAa(-Wr) is the doping-dependent hole mobility in GaAs with NT being the total doping concentration. The expression for the electron mobility of AlxGax-jAs is more complicated because it depends on the number of carriers in the direct and indirect conduction-band valleys. According to [80], the effective electron mobility can be calculated by weighting the direct and indirect mobilities by their respective electron populations: pn{NT,x) = rdpd + (1 - rd)m (2.163) where d and i refer to, respectively, direct and indirect, and rd is the fraction of elec-tron concentration in the direct valley (i.e., T valley) as opposed to the total electron Chapter 2. Model Development 56 concentration and is given by rid + rt{ Mr + TIL + rix 1 l + (nL + nx)/nr exp kT ^ ' e x p f - ^ kT (2.164) In obtaining Eq. (2.164), the expressions for the various electron concentrations de-rived in Subsection 2.6.3 were used. The direct (GaAs) and indirect (AlAs) elec-tron mobilities, pd and Pi, can be derived from Eq. (2.161). For the direct valley, the electron effective mass is simply the electron effective mass of the T valley, i.e., m*ni = mjj. The electron effective mass for the indirect valleys may be approximated by (m;,) 3/ 2 ~ ( m £ ) 3 / 2 + (m*) 3 / 2 . The formulations are Pi = < ( g = o) 3/2 Pn 1014 c m - 3 . For simplicity, we let Tsrh = rno for P-type GaAs and TSRH — rpo for N-type GaAs. The actual values of r n o and TPO are very process dependent. From a collection of published data on majority carrier diffusion length of GaAs [84]-[88], we calculated the corresponding electron and hole carrier lifetimes at various doping densities, as shown in Figure 2.12. Note that only data at relatively low doping concentrations are used because at higher doping concentrations radiative and Auger recombinations cause the effective minority carrier lifetime to go down. From Figure 2.12, we estimated for GaAs that TPO ~ 2 x 10 - 8 s and rno ~ 5.5 x 10"9 s. These two values are assumed applicable to A ^ G a i . j A s since there is not enough reliable data to show how the SRH lifetime actually varies with A l composition. Under low-level injection conditions, the radiative lifetime TR is given by [57] = WW) (2175) where B is the radiative constant as introduced in Subsection 2.5.2. If the minority carrier density is neglected, then T-aTH (2176) where TV = NA or TV = TVp depending on whether the material is P or N type, respec-tively. In Figure 2.13, a collection of experimental data for TR for GaAs is plotted as a function of doping'density. These data are obtained from experimentally measured radiative lifetimes and radiative constants [60],[89]-[94]. A least squares fit of the TR data produces a dependence of r R on TV slightly different from that predicted by Eq. (2.176). However, Eq. (2.176) can still hold if we assume that the radiative constant (cm 3/s) has a small dependence on the doping density given as follows: B{N) = 1.204706 x KT 7 TV"0 1 6 7 7 5 6 1 7 (2.177) Chapter 2. Model Development 60 • [84] Casey et al. O [85] Wright et al. A [86] Aukerman et al. • [87] Hwang • [88] Ashley & Biard ~ 20 ns O "° O r-o ~ 5.5 ns • , , i i i i i i i i I i i i i i i i 11 i i i I I I I 10" 10'6 10" 1018 N-type or P-type Doping Density (cm" 5) Figure 2.12: Collection of experimental minority carrier lifetime data of GaAs for electrons (open symbols) and holes (solid symbols) at low doping densities. Chapter 2. Model Development 61 Net Doping Density (cm - 3 ) Figure 2.13: Collection of experimental radiative lifetime data of GaAs for various doping concentrations and a corresponding least squares fit. Chapter 2. Model Development 62 It is also assumed that B is independent of the type of dopant. Moss et al. had also shown that the radiative lifetime for direct-bandgap materials can be expressed as a function of bandgap, dielectric constant, and effective masses of electrons and holes, all of which vary with Al composition [95, p. 205]. Since the radiative constant is related to the radiative lifetime by Eq. (2.175), the following expression is obtained: N P Using mn ~ 0.067 m 0 , m* ~ 0.48 m 0 , Eg s 1.424 eV, and er ~ 13.1, for GaAs, the following expression for B as a function of composition is produced. € (xYl* E (xY B(x, N) = 3.0367 x H T 3 '-±L B(N) (2.179) "io L mo J where B(N) is the doping-dependent radiative constant for GaAs given by Eq. (2.177). For AUGai-xAs , the radiative constant decreases with Al composition because the increase in effective masses is greater than the increase in bandgap. The computation of the Auger lifetime depends on which of the two Auger recombi-nation processes, C H S H or C H C C (see Subsection 2.5.3), is in effect. Since the CHSH process is dominant in P-type material and the C H C C process in N-type material, the Auger lifetime can be expressed, at low injection levels, as [96, p. 557] Ta ~ n m N-type material (2.180) Cn ND r — ——j in P-type material (2.181) CPNA Analytic expressions for the Auger coefficients, in cm6/s, as a function of Al mole fraction are given by Takeshima as follows [58] Cn = Cn0 exp(a„ T + bn T 2) (2.182) Cp = C p 0 exp(apr + 6 p T 2 ) (2.183) Chapter 2. Model Development 63 where C n 0 = (1.960 - 11.361 + 31.37 i J ) x 10_ a n = (0.8714 + 0.88 x - 6.36 x 2) x 10 bn = (-0.03655 - 0.0638 x + 0.562 x 2) x 10 C p 0 = (9.786 - 36.35 x + 111.6 x 2) x 10" a p = (1.045 - 0.408 x - 1.64 x2) x 10 bp = (-0.0774 + 0.0371 x + 0.127 x2) x 10 -32 -32 - 2 - 2 -4 (cm 6/s) (K- 2 ) (cm6/s) IK" 1) ( K - 2 ) and the temperature T is set to 300 K. Takeshima's equations were made to fit theoret-ically calculated Cn and Cp for A ^ G a ^ A s in the Al mole fraction range 0 < x < 0.2. Uncertain as to whether Eqs. (2.182) and (2.183) are applicable to x > 0.2, we assume that under this condition Cn and Cp take on the values computed at x = 0.2. The effect of the presence of interface traps due to lattice mismatch at the emitter-base interface is to reduce the bulk lifetime in the active layer. For a single-heterojunction bipolar transistor, this may be represented by [97] where T0 is the bulk lifetime and is the same as that defined in Eq. (2.171), d is the thickness of the active layer, and fx = S,NTL/D, SINT being the interface recombina-tion velocity and D, L being the diffusion coefficient and diffusion length respectively. Letting L = \Ab D, Eq. (2.184) can be expressed as 1 (2.184) T — 1 + 6/tanh(d/L) T T0 y/r0 D tanh(d/y/T0 D) (2.185) Comparing this to Eq. (2.171), we see that TINT — (2.186) Chapter 2. Model Development 64 where den — \/T0 D tanh(d/y/T0 D) is the effective active layer width. The interface recombination velocity is given by [98] Nsao (2.187) where vTH is the average thermal velocity, Nsg is the interface trap density, and o is the interface traps' capture cross section. The density of interface traps can be calculated by assuming that each atom terminating an edge dislocation constitutes a recombination center and is given by [99] N„ = 4 * 2 2.188 a\a\ where a\ and a2 are the lattice constants of Alj;Gai_j;As and GaAs respectively (ax can be interpolated from agaAs = 5.6533 A and CLMAS = 5.6605 A, the lattice constants of GaAs and AlAs respectively). Nelson found that for Alo.5Gao.5As, Nss ~ 1.6 x 1012 cm - 2 and S,NT ~ 500 cm/s [98]. Because of this, we can rewrite Eq. (2.187) in terms of the lattice constants in A as 2 2 SINT = 1.26 x 107 C l ~ ° 2 (cm/s) (2.189) The resulting S,NT is less than 1000 cm/s for the AlGaAs/GaAs heterojunction inter-face. 2.7 H i g h Frequency Performance of H B T s Because of the high electron mobility of III-V materials and the typical low base re-sistance, the Heterojunction Bipolar Transistor, especially of N-P-N design, can po-tentially operate at very high frequencies, making it very attractive for high speed microwave applications. In this section, we shall describe in detail the calculations of Chapter 2. Model Development 65 two widely used ngures-of-merit that characterize the high frequency performance of an HBT: (i) the cutoff frequency, fa, and (ii) the maximum frequency of oscillation, / m a x - The computations of fa and / m a x depend on the device structure and physical parameters. In the derivations that follow, we shall concentrate specifically on the pyramidal HBT structure shown in Figure 2.14 whose geometrical and doping density parameters are given in Table 2.3. These values pertain to the prototype HBT device being developed at Bell-Northern Research, Ottawa. The parameters Wg and xbt shown in Table 2.3 are the emitter junction grading width and the Al mole fraction at the base-emitter junction respectively. 2.7.1 Cutoff Frequency The cutoff frequency /r , also known as the gain-bandwidth product, is defined as the frequency at which the common-emitter short-circuit current gain is unity. The cutoff frequency is usually evaluated from the total emitter-to-collector transit time, rec, using the expression Sr = - i - (2.190) 2TTTEC For microwave transistors, this total transit time consits of four delay times [100]: Tec =TB + TB + TSCR + Tc (2.191) where rE = emitter charging time rB = base transit time TSCR — collector space charge region transit time rc — collector charging time Chapter 2. Model Development 66 5£ = 2^ m Figure 2.14: The pyramidal heterojunction transistor structure. Table 2.3: Parameters for the pyramidal heterojunction bipolar transistor. Layer # Material Thickness (A) Doping (cm"3) Al or In composition X emitter cap 1 n + - I n I G a i _ I A s 300 1 x 1 0 1 9 0.6 • 2 n + -In a ; Ga 1 _j ; As 300 1 x 1 0 i y 0.6-0 linear 3 n + -GaAs 1000 3 x 1 0 1 8 0 emitter grading 4 n-AlxGax-j-As 500 5 x 1 0 1 7 0-0.3 linear emitter 5 n - A l I G a x _ I A s 1500 - Wg 5 x 1 0 1 7 0.3 emitter grading 6 n - A l z G a ^ A s wg 5 x 1 0 1 7 0 . 3 - X ( , E linear base 7 p+'AlxGai-jAs 1000 3 x 1 0 1 9 £( , , . -0 linear collector 8 n-GaAs 4000 5 x 1 0 1 6 0 collector buffer 9 n + -GaAs 4000 3 x 1 0 1 8 0 Chapter 2. Model Development 67 b ». f t 6 ' b' ( w V — t c V.'. c O • -0 e e Figure 2.15: A simplified hybrid n circuit model for a transistor, with the emitter and collector terminals short-circuited. A. Emitter Charging Time The emitter charging time, rB, is really a time constant representing the delay in the input of a common-emitter circuit. Consider the simplified hybrid ir model with the collector and emitter short-circuited shown in Figure 2.15. Here, Miller's Theorem has been applied so that the effective capacitance C is the sum of the emitter and the collector junction capacitances, i.e., C = CEJ + CQJ. The other important elements shown are the base spreading resistance rw», the transconductance gm, the emitter * differential resistance r e, and the dc common-base short-circuit gain a 0. The common-emitter short-circuit current gain, /?, is simply the ratio of the currents ic and ib. Since gm — a0/re and the dc current gain f30 = a 0/(l - a0) > 1, the current gain, derived from Figure 2.15, can be written as follows: 0 = ~ = ic gmrt/(l-a0) H 1 + j wCr e / ( l - a0) Chapter 2. Model Development 68 Po l+jtore (A> + 1)C l + i w r e / ? 0 C Putting u> = wE and |/?| = 1 gives (2.192) ' E — OJr. ~ reC = re{CEj + Ccj) (2.193) To be more accurate, the emitter series resistance should also be included, thus the emitter charging time becomes (refer to Appendix D) rE = re {CEj + CCj) + {REC + REX + REI) CCj (2.194) The emitter series resistance has three components: the contact resistance and the bulk resistance of the extrinsic (cap) and intrinsic layers. With reference to Figure 2.16, the emitter contact resistance is given by REC = ^ f - (2-195) J E LiE where pcE is the specific contact resistivity of the emitter. For non-alloyed N-type contacts using graded InGaAs, pcE ~ 5 x 10~8 $7 cm2 (n+ = 1.5 x 1019 cm - 3) [15]. The extrinsic emitter resistance, REx, refers to the resistance of the emitter cap which is composed of three layers: a top n + Ino.6Gao.4As layer, a n + GaAs layer, and a graded layer in between. Using the subscript notations for the layers in Figure 2.16 and denoting N for doping density and W for layer thickness, we can write REX — (Pcapl ^capl + P c a p l 2 ^ c a p H + Pcxxp2 Wcap2)/SE LE (2.196) where Chapter 2. Model Development 69 Figure 2.16: Equivalent circuit resistances for the emitter layers and emitter-base junc-tion. Chapter 2. Model Development 70 1 P c a p l 2 — M n c a p 2 are the electron mobilities of In0.eGa0.4As, Ino.3Gao.7As, and G a A s , respectively. Note that a constant value is used for the electron mobil i ty of the graded layer. The first two electron mobilities are calculated by linearly interpo-lating between the intrinsic electron mobil i ty of InAs and G a A s (33000 c m 2 / V s and 8500 c m 2 / V s respectively) and assuming that the mobi l i ty of I n I G a i _ I A s has the same doping dependence as G a A s . The intrinsic emitter resistance, REI, is equal to the sum of the resistances of the Alo.3Gao.7As intrinsic emitter (layer #5) and the graded layer immediately above it (layer #4), i.e., REI = [PEI WEI + PE2 {WE - XE)}/SE LE (2.197) where 1 PEI = —rT 1 PE2 = — 7 7 q NE2 P*nE2 and p,nEi, P-nE2 are the electron mobilities of Alo.15Gao.85As and Alo.3Gao.7As respec-tively, and XE is the depletion-layer w id th in the emitter (obtainable using E q . (2.6)). The emitter junct ion differential resistance, r e , is defined as 1 dVBE dVBE re = vBB SELe dJE (2.198) V B B dIE The differentiation of VBE w i th respective to JE can be performed from E q . (2.64) using the method of finite difference calculus. The emitter and collector junct ion capacitances, derived from E q . (2.9), are CE, = F S ' L ' ' " ' (2.199) (•E 1 €B A B E Chapter 2. Model Development 71 Cc, = (2.200) Ac + Aflc where XBE and XBC are the base depletion-layer width next to the base-emitter and base-collector interface respectively, and Xc is the depletion-layer width in the collector. The dimensions LB and SCD a r e defined in Figure 2.14. In deriving the emitter charging time in Eq. (2.193), we have omitted the emitter diffusion capacitance, CD, which normally should be part of the effective capacitance C shown in Figure 2.15. It can be shown, however, that the time delay due to r e and Co is essentially equivalent to the base transit time rB [38, p. 188]. By definition, the emitter diffusion capacitance is C° = §TE <2-201> The excess minority charge in the base, QB, is equal to rBIN(W), where / n (W) is the electron diffusion current at the edge of the quasi-neutral base near the collector. For thin-base transistors, IN{W) ^ IE, the emitter current. Thus, Eq. (2.201) becomes C ^ r B - ^ = ^ (2.202) dVBE re or r e CD ~ rB. B. Base Transit Time The time for electrons to cross the quasi-neutral base region can be calculated from the known distribution of base excess electron density and the collector electron current density. By definition, (2.203) Chapter 2. Model Development 72 where W is the quasi-neutral base width, and h(x) and «/„(W) are obtainable from Eqs. (2.53) and (2.54). The result of the integration is TB ~ DnB \{ri-f)^"-\rt-f)Ce*») (2-2°4) where C = -C2jCx ~ e2tW and Cu C2, ru r 2 , t are denned in Eq. (2.53). When the amount of base grading is large (f2 ~> 4/LnB), Eq. (2.204) reduces to Kroemer's expression [37], i.e., (W2/2DnB) x (2kT/AE3). When there is no base grading (/ = 0) and WjLnB < 1, Eq. (2.204) reduces to the usual expression W2/2DnB. The above expression for rB is only an estimate; it becomes inaccurate under extreme conditions. First, under high-current conditions, the effective base width increases due to the Kirk effect [102], causing Eq. (2.204) to underestimate the base transit time. Second, when the built-in field due to base grading is high enough, the velocity of the carriers saturates, which may place a lower limit on rB if ballistic effects do not occur. Third, ballistic transport of carriers can occur in a very thin base. Accurate modeling of the latter two effects would require more sophisticated modeling schemes, an exam-ple being Monte-Carlo simulation [103]. C. Collector Space Charge Region Transit Time The time delay for transport through the base-collector space charge layer is given by ihe usual expression [104, p. 35] TSCR = (2.205) where WBc is the width of the base-collector space charge region and v„ is the saturation velocity for electrons in GaAs. The factor 1/2 in Eq. (2.205) is due to the inclusion of an additional delay between ac collector current and ac emitter current [105, pp. 321-336]. In Eq. (2.205) it is assumed that the electrons traverse the collector depletion Chapter 2. Model Development 73 region at a constant saturated velocity. In reality, electron velocity is not constant throughout the collector depletion region and a transiently high electron velocity can exist near the base side of the collector depletion region [21,106]. When electrons with unsaturated velocities enter the collector depletion region, the high electric field they experience causes their velocities to overshoot. As the electrons pick up more energy from the electric field, they transfer from the T valley to the low-velocity L valley and eventually their velocities saturate. Normally the velocity overshoot region occupies only a fraction of the total collector depletion region and therefore Eq. (2.205) should provide a reasonable estimate of r s c n , although new collector structures have been proposed to increase the velocity overshoot region, thereby reducing the effective rSOR [10,107]. The electron drift velocity in GaAs is a slowly decreasing function of electric field when the electric field exceeds ~ 100 kV/cm. Using the doping density parameters for the base and collector listed in Table 2.3 and assuming a reverse-biased voltage of — 3 V, we estimated that the average electric field in the collector depletion region is about 120 kV/cm which corresponds to a t/, ^ 7.5 x 106 cm/s [108]. D. Collector Charging Time This is the time delay caused by charging the collector junction capacitance through the collector series resistance and is given by rc = {Rec + RCB + Rei) CCj (2.206) where Rec RCB, RCI are the resistances of the collector contacts, n + buffer layer, and intrinsic n layer, respectively, and CCj is given by Eq. (2.200). With reference to Chapter 2. Model Development 74 Figure 2.17: Equivalent circuit resistances for the intrinsic and buffer regions of the collector. Figure 2.17, the collector contact resistance is [109] is the sheet resistance of the n + collector buffer layer, pnbuf being the electron mobility of the buffer layer, and pcc is the collector specific contact resistivity. In general, the specific contact resistivity of both N-type and P-type GaAs is a function of doping concentration and the type of contact metal used. For simplicity, we leave the specific contact resistivity (base or collector) as a device input parameter. Measured contact resistivities of good Au/Ni /Au-Ge alloyed ohmic contacts to heavily doped N-type GaAs can be as low as 1 x 10 - 6 f l cm 2 [110]. The lateral resistance of the buffer layer is given by [ i l l , p. 217] (2.207) where 1 q NbuS pnbxl{wbu{ RCB — R'cB + CB Chapter 2. Model Development 75 Rsbuf ScD Rsbuf &BC , — 1 yz.zuo) 12 Lc 2 Lc The first term in Eq. (2.208) corresponds to the part of the buffer layer immediately underneath the intrinsic collector. The second term corresponds to the two small buffer strips between the collector electrode and the edge of the intrinsic collector layer, which are connected in parallel as far as the collector current is concerned. The resistance of the vertical intrinsic collector is given by Re, = "C {™C' 7 X C ) (2.209) where 1 Pc q NCI PnC [inC and Xc are the electron mobility and depletion width in the intrinsic collector layer, and Nci and WCi are the doping density and thickness of the intrinsic collector layer. 2.7.2 M a x i m u m Frequency of Oscillation The maximum oscillation frequency, / m a x , is the frequency at which the unilateral gain becomes unity. The following simple approximation for / m a x is often used [36, p. 164]: =i^kc, <2-2io> where fx is the cutoff frequency, Rb is the base resistance, and Cc is the collector capacitance. Due to the distributed nature of the actual base resistance and collec-tor capacitance, a more accurate result can be obtained by considering an effective Ri, Cc product [2,101,109]. Assuming the base resistance and collector capacitance are distributed in the way shown in Figure 2.18, the effective Rb Cc is (Rt Cc)eff = Cci (RBI + RBX + RBC) + Ccx ^Bc^j + CCc RBC (2.211) Chapter 2. Model Development 76 BASE-COLLECTOR n + n -2RBC> 2RBx 2RBi 2RBX <2RBC •>cc Ccx j CCi | Ccx | Cgc 7 1 Figure 2.18: Equivalent circuit resistances and capacitances for the base and base-collector junction. where RBI, RBX, RBC are, respectively, the intrinsic, extrinsic, and contact resistance of the base, and Cci, Ccx, Ccc are the collector-base junction capacitances underneath the intrinsic, extrinsic, and contact regions of the base. The lateral base resistances of the intrinsic and extrinsic regions are given by [ i l l , p. 217] R'sB $E RBI — RBX = 12 LB RsB SEB 2 LB (2.212) (2.213) where R'SB RSB q NB UpB [WB - XBc - XBE) 1 qNBpPB {wB - xBC) Chapter 2. Model Development 77 are the sheet resistance of the intrinsic and extrinsic bases, NB is the base doping density, WB is the thickness of the base layer, and p,PB is the base hole mobility taken as a constant at a value appropriate to material of the Al mole fraction as exists in the center of the base. The expression for the base contact resistance is similar to that of Eq. (2.207), namely where a value of 3 x 10 - 6 fl cm2 is used for the base specific contact reistivity, pC£ [112]. This value of pCB for a 3 x 1019 cm - 3 doped base layer is also consistent with the theoretical value calculated using the Schottky-barrier tunneling model [13]. The three distributed collector capacitances are calculated as follows, assuming a base-collector homojunction: Cc, = (2.215) 2 SEB Lb ec tooi^ C c x ~ ~ y T1T~ (2.216) Ccc = (2.217) Note that the sum of the above three capacitances equals the total collector junction capacitance as given by Eq. (2.200). 2.7.3 Modif ied Collector Structures All the formulations described in this section so far are based on a simple transistor structure shown in Figure 2.14. Newer and more practical HBTs often have implant-damage external collector regions for reducing the base-collector capacitance. Others may have only a single collector contact. It turns out that only minor modifications to some of the earlier equations are needed to take in account these two cases. Chapter 2. Model Development 78 Figure 2.19: Heterojunction transistor structure with an implant-damaged external collector. Chapter 2. Model Development 79 The external regions of the intrinsic collector layer, that is, the hatched regions shown in Figure 2.19, can be made highly resistive or intrinsic wi th either proton or oxygen implantat ion [11,113]. More importantly, the base-collector capacitance in these regions is reduced and independent of applied voltage. It has been shown that an oxygen fluence of 8 x 1 0 1 3 c m - 2 [113] or a proton fluence of 5 x 10 1 2 c m - 2 [11] would produce an external base-collector capacitance of 0.2 fF/^um 2 . W i t h reference to Figure 2.19, SCD is n o w defined as the intrinsic n collector layer wid th excluding the implanted isolation regions; in this case, SCD — &E- The total n collector layer w id th is denoted by a new variable: SET = SE + 2 (SEB + SB) (2.218) The expression for the resistance of the buffer collector layer, i.e., E q . (2.208), must now be changed to R c b = Rs^S^ + i W + SBT - SCD^ ( 2 2 i 9 ) The distr ibuted capacitances Ccx and Ccc-, as a result of proton or oxygen implan-tat ion, become C c c = 2 SB LB (2 x 10~ 8 F / c m 2 ) (2.220) C c x = 2 SEB LB (2 x 1 0 - 8 F / c m 2 ) (2.221) The total collector junct ion capacitance originally given by E q . (2.200) must now be calculated from the distr ibuted collector capacitances, i.e., Ccj — Ccc + Ccx + Cci (2.222) For a single collector electrode structure, only the expressions for the collector con-tact resistance Rcc a n d the collector buffer resistance RcB are needed to be modified. Chapter 2. Model Development 80 For Rcc, Eq. (2.207) is changed to = VPcC flsbuf c Q t h / ^ / W | ( 2 2 2 3 ) £c V V Pec / and for RCB, Eq. (2.219) is changed to Res = + i W ( s*c + S"T-S<»>) ( 2. 2 24) Chapter 3 Results and Discussion In this thesis, we have favored simplicity in employing a one-dimensional model for the derivation of the H B T current equations and a quasi-two-dimensional model for the formulation of the high frequency figures-of-merit. One reason for using a simple model is that we want to obtain some reasonable estimates of the device performance without resorting to the use of extensive computations like those required by two-dimensional and Monte Carlo models. More importantly, we want to investigate the qualitative effects of base grading and various intrinsic recombination mechanisms on the device performance of HBTs. Before we present the results in this chapter, two limitations about the present model should be clarified. First, the present model does not include any of the effects that occur at high injection levels, namely, base push-out due to the Kirk effect [102] and emitter and base resistance voltage drops. To avoid complications due to the Kirk effect, Jc is restricted to values below the transition current density Ji, which, from numerical analysis, is identified as being the threshold for a sudden increase in r e c , and corresponding drop in fx [114]. This critical current density is given by where NCi is the doping density of the intrinsic collector (5 x \0 lb cm 3) and vm is the drift velocity in the collector space charge region under the field Em, which is given by Ji - qNciv. (3-1) VW + IVBCI WCI (3.2) 81 Chapter 3. Results and Discussion 82 where Vbi is the base-collector junction built-in voltage (~ 1.4 V) and WCi is the width of the intrinsic collector layer (4000 A). The results presented in this chapter are computed for the H B T structure parameters shown in Table 2.3 and for VBC = —3 V , which yields Em = 1 x 105 V /cm. At such fields the carrier velocity will saturate, giving vm — vs ~ 107 cm/s, although velocity overshoot effects may take vm as high as 1.5 x 107 cm/s [12]. Thus, a reasonable value for J\ would appear to be 105 A / c m 2 . As the effects of base push-out occur within a very narrow collector current range about Ji [114], the results presented here should be valid for Jc ~ 5 x 104 A / c m 2 . It is significant that a number of experimental results for AlGaAs/GaAs HBTs with Nci = 5x 1016 c m - 3 give no indication of base push-out occuring at collector current densities as high as 4-5 x 104 A / c m 2 [9,12,115] and 1.5 x 105 A / c m 2 [14]. The second limitation of the present model is that only intrinsic recombination mechanisms are modeled, that is, recombinations at the surface of the emitter periphery and in the external base are neglected. The surface recombination around the emitter periphery can make a significant contribution to the base current [116] and cause the emitter size effect (degradation of current gain as emitter size is scaled down) [31] in non-graded-base HBTs. Unfortunately, this surface recombination, because of its two-dimensional nature, is very difficult to incorporate into our model. However, in newer HBTs with better passivated surfaces [117] and graded bases [33], surface recombination is often suppressed. Unless specifed- otherwise, all the calculations presented in this chapter are based on the pyramidal H B T structure shown in Figure 2.14 whose geometrical and doping parameters are given in Table 2.3. The surface recombination velocities at all contacts are taken to be infinite and the actual base-emitter junction is taken to be ungraded. The base-collector reverse biased voltage is set to —3 V . Chapter 3. Results and Discussion 83 3.1 Emi t t er and Collector Currents Recalling from Eqs. (2.64) and (2.65), the emitter and collector current densities for the H B T are JE = ^ n ( e ? W * r - l ) + ^ i 2 K B c / f c r - l ) + JR (3.3) Jc = Atl (e^ kT - 1) + A22 (e< v°°' kT - 1) + JG (3.4) It is worth noting that the coefficients An and A2X are not exactly equal. This is beacuse in the derivation of the electron diffusion current in the base in Subsection 2.2.3, an approximation was made in assuming that the variable "a" in n,(x) = a e ' z is constant. However, the difference in An and A2i is very small so that in essence the reciprocity rule still holds.3 The dependence of the collector current density on the base-emitter voltage for different amounts of Al mole fraction is illustrated in Figure 3.1. The solid lines are drawn for JG = 0; these curves converge to A22 at VBE = 0, which has a magnitude of about 3 x 10~18 A / c m 2 and is nearly indepenedent of the A l mole fraction at the base-emitter metallurgical boundary, i j e . The flat broken line at 5.4 x 10 - 1 0 A / c m 2 is the value of Jc; the solid lines would converge to for VBC = — 3 V . The magnitude of the collector current density decreases as the amount of base grading increases. This is directly attributable to a smaller saturation current density, i.e., A2X, for a larger xi,e. The saturation current density can be obtained by extrapolating the solid lines in Figure 3.1 to VBE = 0. For examples, A2l ~ 3 x 10 - 2 0 A / c m 2 for i(,e = 0 and A2\ ~ 2 x 10 - 2 4 A / c m 2 for i(,e = 0.3. The variation of A2X with xbe depends, in a complicated way, on the changes in n,o, / , and A.£7n; however, the major contribution to the decrease of A2X with increasing is the exponential decline of the intrinsic 3If the assumption of constancy of "a" is applied in the actual calculation of ni(x), then it follows that new — " B o e 2 , V V a n ^ thus, as reciprocity demands, AX2 — —A2i-Chapter 3. Results and Discussion 8 4 6 — 2 0 - f 1 i i I I I I 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1.2 1.4 1.6 Base-Emitter Voltage (V) Figure 3.1: Dependence of collector current density on base-emitter voltage for different amounts of base grading, with VBC = -3 V (broken lines) and JG = 0 (solid lines). carrier concentration, n, 0 , which is about 1000 times smaller for Alo.3Gao.7As than that for GaAs. It can be seen from Figure 3.1 that the slopes of the JC-VBE curves are different for different amounts of base grading. A measure of these slopes shows that the ideality factors are 1.08, 1.05, 1.02, and 1.01 for xhe = 0, 0.1, 0.2, and 0.3, respectively. The deviation from unity ideality factors is a direct result of the heterostructural nature of the emitter-base junction. For a heterojunction, the electron energy barrier AEn is Chapter 3. Results and Discussion 85 6-1 Base-Emitter Voltage (V) Figure 3.2: Dependence of emitter current density on base-emitter voltage for different amounts of base grading. positive and it increases as the applied junction voltage rises. The effect of increasing AEn is to reduce -the rate at which Jc varies with VBE-, and thus the ideality factor. For a more heterostructural junction (i.e., xie = 0), the magnitude of AEn is larger and is therefore a stronger function of VBE\ thus the resulting ideality factor is also larger. When x&e = 0.3, AEn is negative and it will not affect Jc- In this case, the emitter-base junction behaves like a homojunction and the ideality factor is almost one. Chapter 3. Results and Discussion 86 The dependence of the emitter current density on the base-emitter voltage for dif-ferent amounts of A l mole fraction is shown in Figure 3.2. At the high current end, JE is essentially equal to Jc- At lower current densities, JE becomes basically indepen-dent of Al mole fraction and changes more slowly with VBE- In this current density range, the ideality factor is about 2, which suggests the emitter current is dominated by SRH recombination in the emitter-base depletion region. The fact that the current is independent of base grading at low current levels indicates that much of the SRH recombination occurs in the depletion layer of the emitter where the A l composition is constant. 3.2 Base Current Components The base current density can be obtained by simply subtracting Jc from JE using the expressions given by Eqs. (3.3) and (3.4), i.e., JB = JE — Jc- The base current density is composed primarily of five components: the neutral base recombination current density represented by |./n(0) ~ "^n(W)|i the hole current density back-injected into the emitter, Jp(xE); the Shockley-Read-Hall, radiative, and Auger recombination current densities in the emitter-base depletion region denoted, respectively, by JRRH, JR*, and JftUa. The dependence of the various base current density components on VBE for the case of xbe = 0.1 is shown in Figure 3.3. As indicated in the previous section, SRH recombination current is the strongest base current component at low biases. Above VBE 1.2 V , the neutral base recombination current and the back-injected hole current surpass the SRH recombination current. At still higher base-emitter biases, back injection of holes into the emitter dominates. Both the radiative and Auger recombination currents also exceed the SRH recombination current at the high end of the bias range. Incidently, the neutral base recombination is due primarily to the Chapter 3. Results and Discussion 87 2 0 ^ -2-E TOTAL Q C Q> Q O 1 1.3 Base-Emitter Voltage (V) 1.6 Figure 3.3: Dependence of base current components on base-emitter voltage for the case of xbe = 0.1. radiative recombination process since the radiative lifetime is smaller than the Auger and SRH lifetimes.for the given base doping level of 3 x 10 1 9 c m - 3 . The variation of the base current density components with A l mole fraction is shown in Figure 3.4 for Jc = 103 A / c m 2 and in Figure 3.5 for Jc — 10 4 A / c m 2 . If the space-charge currents were independent of base grading, one would expect JRRH , JTRad and J^U9 to increase with xbe because, in order to maintain Jc constant, VBE must increase with xbe as well. This is not seen except for the SRH recombination current in Chapter 3. Results and Discussion 88 100CH o.oi-i 1 1 0.0 0.1 0.2 0.3 Al Mole Fraction Figure 3.4: Dependence of base current components on Al mole fraction for Jc = 103 A / c m 2 . the case of small collector current density (Figure 3.5). In fact, for both high and low collector current densities, the space-charge currents in general decrease with increasing Al mole fraction except for xbt > 0.2. In the latter case, the exponential dependence on voltage of the space-charge currents causes them to eventually increase with xbe. The large reduction of the radiative and Auger recombination currents with increasing Al mole fraction for xbe < 0.2 indicates that these two recombination processes occur mostly in the base side of the emitter-base depletion region where the bandgap increases Chapter 3. Results and Discussion 89 Al Mole Fraction Figure 3.5: Dependence of base current components on A l mole fraction for Jc = IO"4 A / c m 2 . with base grading. On the other hand, at low collector currents, SRH recombination occurs mostly in the emitter side of the emitter-base depletion region; at high collector currents, however, the contribution from the base side of the emitter-base depletion region becomes significant. From Figure 3.4 and Figure 3.5, the Auger recombination current is in general always less than the radiative recombination current even for a base doping density as high as 3 x 10 1 9 c m - 3 . At low collector currents, SRH recombination is the dominant Chapter 3. Results and Discussion 90 space-charge recombination process for all degrees of base grading. At high collector currents, radiative recombination is the strongest space-charge recombination process for xbe < 0.1. For higher xbe, the large base bandgap at the emitter-base interface causes both JrRad and to drop below J | H H . Base grading is also seen to be effective in reducing the neutral base recombination current. This is, however, overshadowed by the large increase of back-injected flow of holes to the emitter due to the increasingly homojunction-like nature of the emitter-base junction. In fact, at high collector currents, the total base current is due mostly to this hole current for xbe > 0.1. 3.3 D C Current G a i n The computation of the D C current gain is done using the expression The D C current gain as a function of collector current density and base grading is shown in Figure 3.6. All four curves show a relatively flat j3 for a wide range of high collector current densities. At low collector current densities, the contribution of the recombination current in the base-emitter depletion region is usually larger than the useful diffusion current of minority carriers across the base. The current gain may be written as [36, p. 143] Jc exp(qVBE/kT) P — ~T 0 C JB exp(qVBE/mkT) = exp qVjBE kT V m oc 41 ™) (3.6) where m is a constant that determines the rate of exponential increase of the recombi-1 /2 nation current with bias. For SRH recombination, m ~ 2 and (3 oc Jj by Eq. (3.6). Chapter 3. Results and Discussion 91 10001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000 Col lector Current Density ( A/cm 2 ) Figure 3.6: Dependence of D C current gain on collector current density for different amounts of base grading. But, for radiative recombination, m ~ 1 and /? is independent of Jc- In Figure 3.6, the 1/2 current gain for xbe = 0.1 and 0.2 exhibits the Jc' dependence for Jc ~ 0.01 A / c m 2 . For xbt = 0.3, a similar trend for the current gain can be seen for Jc ~ 0.001 A / c m 2 . SRH recombination is clearly an important process at low collector currents for the graded-base devices. Since /? remains relatively constant down to a collector current density as low as 10 - 4 A / c m 2 for the non-graded base case, the base current must be composed mostly of space-charge radiative recombination current for Jc > 10~4 A / c m 2 . Chapter 3. Results and Discussion 92 We know that this is true for Jc = 103 A / c m 2 from Figure 3.4. For Jc ~ l O - 4 A / c m 2 , base current comes mainly from SRH recombination in the base-emitter depletion re-gion as is evident in Figure 3.5. As bias increases, the barrier that blocks the flow of holes into the emitter decreases. The resulting increase in the back-injected hole current raises the overall base current and thus reduces the current gain. When xbe = 0.3, the back-injected hole current is so much larger than the other base current components that the current gain is essentially given by Jp{XE) But both Jc and Jp(xE) vary as exp(qVBE/kT), so /? is essentially independent of Jc as indicated in Figure 3.6. For xbe = 0, the fall in /? with increasing Jc cannot be due to the increase in the back-injected hole current as it contributes only a fraction of the total base current even at a strong bias (see Figure 3.4). It is due, rather, to a slight increase of the energy barrier AEn as VBE rises since current gain is proportional to exp(-AEJkT) [3]. It can be seen in Figure 3.6 that the current gain reaches a maximum value of around 180 when the A l mole fraction is 0.1. This is directly related to the total base current curve shown in Figure 3.4 where a minimum base current is seen when Xbe = 0.1. As the amount of base grading is increased, various recombination currents go down as a result of widening of the base bandgap, causing the current gain to rise. Further increase in the amount of base grading renders the emitter-base junction more homojunction-like and allows the back-injected flow of holes to increase. The result is a diminution of the emitter injection efficiency and, correspondingly, the current gain. The current gain values shown in Figure 3.6 are not very high because we have not employed any emitter-base junction grading. If junction grading were applied, the * Chapter 3. Results and Discussion 93 1000 1_ o 03 , , 1 , 1 0 20 40 60 80 100 Emitter Junction Grading Width (angstrom) Figure 3.7: Dependence of D C current gain on emitter junction grading width for different amounts of base grading, with Jc = 103 A / c m 2 . electron flow to the base should increase substantially, thereby increasing the emitter injection efficiency %and thus the current gain. In Figure 3.7, we have plotted the current gain as a function of emitter-base junction grading width WG and Al mole fraction. The collector current density is held at 103 A / c m 2 which produces a depletion-layer width in the emitter of 200-250 A depending on the degree of base grading. As expected, current gain increases with WG but levels off at WG ~ 100 A, about half of the total depletion-layer width in the emitter. The current gain is independent of junction grading for Chapter 3. Results and Discussion 94 Xf,e = 0.3 because, in this case, there is no conduction-band spike to begin with. The increase of current gain is about 4-5 fold when Wg > 100 A. 3.4 Transit T i m e Components In experimental H B T research, great efforts are being made to reduce various parasitic resistances and capacitances of an H B T in order to increase its high frequency per-formance. Two examples are the use of non-alloyed ohmic contacts to reduce contact resistances [15,16] and the use of oxygen and proton implantations to reduce the ex-ternal base-emitter capacitance [11,113]. Most of the parasitic resistances, calculated using the equations in Section 2.7 and the physical parameters listed in Table 2.3, are bias-independent, i.e., i ? £ C = 0.50 n REX = 0.13O RBC = 37.49 fl RBx = 7.00 O RBi = 15.62 fl R C C = 5.38 fl R C B = 6.89 fl R C I = 0.21 fl These resistances are computed for the case of maximum base grading. The three base resistances are not completely independent of bias, but only change by a few tens of milliohms as VBE changes from 0.1 to 1.7 V . The collector capacitances, computed for VBE = — 3 V in the case of maximum base grading, are C c c = 9.51fF C C x = 0.48fF Cc,i = 3.17fF totaling 13.16 fF. The remaining components, REI, re and CEJ, depend on the base-emitter bias voltage. The intrinsic emitter resistance changes from 1.07 fl to 1.55 fl over the VBE range of 0.1 to 1.7 V . The other two components vary more substantially with bias: re changes from 2.6 kfl to 1.3 fl and C'Ej changes from 42 fF to 107 fF in the VBE range of 1.55 to 1.75 V , assuming again maximum base grading. At the most Chapter 3. Results and Discussion 95 100 1000 10000 100000 Collector Current Density (A/cm 2) Figure 3.8: Dependence of fr and / m a x on collector current density for different amounts of base grading. ideal bias, the total emitter resistance is quite.small (no more than 4 fl) due largely to a very small emitter ohmic resistance that results from the use of a non-alloyed emitter ohmic contact. Although CEJ increases significantly with V B E , the overall emitter charging time should decrease with bias unless VBE approaches very close to the built-in emitter-base potential. The large base contact resistance compared to the other two base resistive components implies that the quality of the base contact can strongly affect the highest achievable / m a x of an H B T . Chapter 3. Results and Discussion 96 0.1 " i i i i i i 1 11 — — i i i i i i 11 i i i i i i i i 100 1000 10000 100000 Collector Current Density ( A / c m 2 ) Figure 3.9: Dependence of transit time components on collector current density for xht = 0.3. We have plotted /V, / m a x and the transit time components as functions of the collector current density in Figure 3.8 and Figure 3.9. Although the transit time plot of Figure 3.9 is for the case of maximum base grading only (i.e.,xie = 0.3), the variations of the transit time components shown are quite representative of all degrees of base grading. In Figure 3.8 the increasing trend of fa and / m a x with Jc up to a collector current density of 104 A / c m 2 is due principally to the reduction in the base-emitter differential resistance with increasing VBE. This results in a rapid decrease in the Chapter 3. Results and Discussion 97 I l 0.00 0.05 0.10 0.15 0.20 Al Mole Fraction 0.25 0.30 Figure 3.10: Dependence of transit time components on Al mole fraction at base-emitter junction for Jc = 2 x 104 A / c m 2 . emitter charging time re as indicated in Figure 3.9. As the other time delay components are essentially independent of VBE, IT and thus / m a x reach their maximum values at the point of the minimum value of rE. The emitter charging time will eventually increase at higher VBE because of the increase in the base-emitter junction capacitance, CEj. The cutoff frequency, fx, appears in the expression for / m a x (Eq. (2.210)) as a square root term. This fact, coupled with the independence of the effective base-collector Ri, Cc time constant on VBE, causes / m a x to be less dependent than fx on Jc-Chapter 3. Results and Discussion 98 Figure 3.8 also shows that fx and / m a x improve with base grading. A plot of the transit time components as a function of Al mole fraction at the base-emitter junction for Jc = 2 x 104 A / c m 2 (see Figure 3.10) shows that this is due almostly entirely to the reduction of base transit time with base grading. This is in turn due, of course, to the aiding field in the base and, to a very small extent because of the thin base, the associated decrease in neutral base recombination shown in Figure 3.4. Note that the reduction in TB is about 6 times, corresponding roughly to a decrease from W 2/2DB in the non-graded base case to (W 2/2 DB) x (2kT/ AEG) in the highly graded base case (see Subsecton 2.7.1). The actual increase in fx is only about 60 % because the collector-base depletion region transit time TSNR, remains high. Judging from Figure 3.10, even a small degree of base grading (i.e., xbe = 0.1) is sufficient to significantly reduce the total transit time and leave TSCR the principal intrinsic limitation to attainment of high fx. 3.5 Effects of Base W i d t h and Base Doping on fT and / m a x Base grading not only creates a quasi-neutral electric field that accelerates minority carriers through the base but also in the process reduces the chances of neutral base recombination. This effect can be seen in Figure 3.11 where fx and fmax are plotted against base width for Jc = 2 x 104 A / c m 2 in the case of xbe — 0 and 0.3. The rapid decline in fx as the base widens is due to the increase in the base transit time rB. Since neutral base recombination further increases the base transit time, the decline in fx is more pronounced when there is no base grading. The effect on / m a x is lessened by the reductions in intrinsic and external base resistance. In fact, at base widths less than about 1500 A, where the influence of rB on TCC is slight, the reduction in base resistance actually leads to an increase in / m a x with WB (see Eq. (2.210)). * Chapter 3. Results and Discussion 99 10 I 1 1 1 1 i • 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 J 0 1000 2000 3000 4000 5000 Base Width (angstrom) Figure 3.11: Dependence of fT and / m a x on base width for Jc — 2 x 104 A / c m 2 . The effect of base doping density NB on fx and / m a x is illustrated in Figure 3.12. Here, the decline in fT with NB is mostly due to a decrease in diffusion coefficient of the base, DB, and'correspondingly an increase in rB. However, when the base doping density is very high ( 1020 c m - 3 ) , the increase in neutral base recombination, notably in the non-graded base case, further raises rB. As a result, a steeper decline in fx is seen at very high doping densities when there is no base grading. The increase in neutral base recombination with NB arises from the strong doping density dependence of radiative and Auger recombinations. The peculiar structure shown in the fx curve Chapter 3. Results and Discussion 100 Base Doping Density (cm - 3) Figure 3.12: Dependence of fT and / m a x on base doping concentration for Jc = 2 x 104 A/cm 2 . for Xbe — 0.3 is due to the fact that, at Jc = 2 x 104 A/cm 2 , rE is still significant at the lower base doping densities, and its decline with NB counteracts the increase of rB with NB- In contrast to the case of / m a x versus the base width in Figure 3.11, the diminution in fx with NB is not sufficient to bring about an associated decrease in /max- The reduction in base resistance is the dominant effect here, resulting in the increase of / m a x with NB as depicted in Figure 3.12. Chapter 4 Comparison with Experimental Data In this chapter, our graded-base H B T model is put to the test of predicting the per-formance of real devices. We have selected from the recent literature two papers (both from the N T T group) where experimental results for fabricated HBTs were presented. The first paper [118], published in 1985, was mainly a study of the effects of base grad-ing and base width on current gain. The variation of current gain cutoff frequency, fr, with collector current was also presented. In the more recent second paper [13], HBTs of different sizes were fabricated using a proton-implanted external collector layer and high frequency measurements were made. 4.1 Case I: Current Gain and Cutoff Frequency The epitaxial layer structure parameters for the fabricated HBTs are shown in Table 4.1 [118]. For the current gain measurements, the HBTs used have a relatively large emitter area (48 x 48/zm2). Although no other horizontal dimensions for the fabricated HBTs were given, they are, in this case, not required since the model we use to calculate the current gain is one-dimensional. Note that in Table 4.1 the emitter-base junction was graded parabolically from x = x\,e to x — 0.3 over a distance of 300 A to provide a smooth conduction band edge at the interface. The contact side of the emitter layer was similarly graded. In our model, however, grading on both sides of the emitter is linear. Two sets of HBTs were used to study the effects of base grading and base width 101 Chapter 4. Comparison with Experimental Data 102 Table 4.1: Epitaxial layer structure parameters for fabricated HBTs (Case I). Layer Material Thickness (A) Doping (cm"3) Al composition X emitter cap n + -GaAs 1500 5 x 1018 0 emitter grading n - A l x G a 1 _ I A s 300 5 x 1017 0-0.3 parabolic emitter n - A l I G a 1 _ I A s 900 5 x 1017 0.3 emitter grading n - A l I G a 1 _ I A s 300 5 x 1017 0.3-ij,e parabolic base p + - A l I G a 1 _ I A s wB 1 x 1019 Xbe-0 linear collector n-GaAs 3000 5 x 1016 0 collector buffer n + -GaAs 5000 3 x 1018 0 Table 4.2: Structural parameters for the base layer (Case I). H B T Set #1 H B T Set #2 WB (A) 1000 1000, 1500, 2200, 3000 Xbe 0, 0.025, 0.067, 0.1 0.067, 0.1, 0.147, 0.2 EBF (kV/cm) 0, 3, 8, 12 8 on current gain. The first set of HBTs has a fixed base width of 1000 A but different amounts of base grading so that each device has a specific built-in field, EM, in the base. The second set of HBTs has both different base widths and degrees of base grading but a constant built-in field of 8 kV/cm. The structure parameters for the base layer for both sets of devices are summarized in Table 4.2. The results of current gain vs. built-in field and inverse base thickness for Ic = 6 x 10 - 2 A are shown, respectively, in Figure 4.1 and Figure 4.2. The open circles indicate esperimental values. The solid lines are generated from a computer program based on our model. The theoretical current gain values are also calculated for a collector current Ic = Jc x (emitter area) = 6 x 10 - 2 A . The emitter area is used because of the one-dimensinal nature of our model. However, the exact current level is not important in this case as we have found that the theoretical Chapter 4. Comparison with Experimental Data 103 250-1 2 0 0 -O 1 5 0 O o 100-5 0 -5 10 Base Built-in Field (kV/cm) 15 Figure 4.1: Dependence of experimental and calculated current gain on base built-in field for / c = 6x 10"2 A and a base thickness of 1000 A. current gain values remain almost the same for at least two orders of magnitude about lc = 6 x 10 - 2 A. In the original paper, it was estimated that the minority (electron) lifetime in the base was about 105 ps and was nearly independent of Al composition. This value of electron lifetime in the base was used in our program. The experimental and calculated values of current gain are quite similar, indicating that our model does predict at least the correct order of magnitude of current gain. More importantly, our model also predicted the increasing trend of current gain in Chapter 4. Comparison with Experimental Data 104 oH 1—— 1 1 1 0 1000 2000 3000 4000 Base Thickness (angstrom) Figure 4.2: Dependence of experimental and calculated current gain on base thickness for Ic = 6 x 10_J A and a base built-in field of 8 kV/cm. response to a increasing built-in field and a decreasing base thickness. The effects on current gain of base grading and base thickness were explained, in the original paper, by the base transport factor dependence on the built-in field and base thickness. This explanation is consistent with the analyses done in Chapter 3. As the built-in field increases or the base width decreases, the amount of quasi-neutral base recombination is reduced. In Figure 3.4 of Chapter 3, we see that indeed the quasi-neutral base recombination current component, |^ n(0) — «/n(W)|, constitutes a significant portion of Chapter 4. Comparison with Experimental Data 105 the total base current for Al mole fractions of 0-0.1. A decrease in quasi-neutral base recombination is equivalent to an increase in the base transport factor, both leading to a rise in current gain. However, our analyses earlier had shown that other base current components were also responsible for changes in the current gain. For example, as shown in Figure 3.4, the radiative recombination current in the emitter-base depletion region is actually the dominant base current component for very small x\,t. Therefore, in Figure 4.1, the increase in current gain for built-in field from 0 k V / c m to 3 kV/cm (corresponding to xbe = 0 to xie = 0.025) is due mostly likely to a reduction of emitter-base junction radiative recombination. In Table 4.2, we also see that the HBTs with the two largest base thickness have a relatively large xbe (•> 0.1) and thus a large back-injected flow of holes from the base to emitter. This suggests that the small current gains at large base widths shown in Figure 4.2 are the result of this large back-injected hole current. A much smaller H B T was fabricated for the cutoff frequency measurement. The transistor used in this experiment had two emitter fingers, each 4.5 pm wide and 10 pm long. A base width of 3000 A and a base-grading parameter xbe = 0.2 were used to give a built-in field of 8 kV/cm. Since not enough device structure information was given in the paper for the calculation of fx, we have made some educated guesses: SE — 9 ^ni, LE = 10 pm (i.e., an effective emitter area of 90 pm2), SB = Sc = 5 /zm, LB — LC = 10 /urn, SEB — 0.5 pm, and SBC = 1 M m - The meaning of these dimensions are described in Figure 2.17 and Figure 2.18. The emitter and collector ohmic metals used for the N-type layers were both A u G e / N i / T i / A u , i.e., an alloyed ohmic metal. Since the doping densities of the emitter and the collector ohmic contact layers are about the same, we estimated that pCE ~ pcc — 1 x 10~6 fl cm 2 (110). For the base ohmic contact, we arbitrarily chose pcB = 5 x 10 - 5 f lcm 2 , but the calculation of fx would not be affected by it. Chapter 4. Comparison with Experimental Data 106 Figure 4.3: Dependence of cutoff frequency on collector current for VCE = 2 V. The collector current dependence of cutoff frequency for VCE = 2 V is shown in Figure 4.3. The open circles are the experimental data and the solid line corresponds to the calculated values. The experimental and theoretical values match surprisingly well given that rough estimates were made for many of the device structure parameters. Both the experimental and theoretical fr values vary at about the same rate in the low IC region and converge to around 20 GHz in the high IC region. The observed cutoff frequency of 20 GHz corresponds to a emitter-to-collector transit time, rec, of 8 ps. From our program, we also found that the base transit time r B , is about 3.4 ps, Chapter 4. Comparison with Experimental Data 107 S.I. Figure 4.4: Schematic structure of an HBT with a proton-implanted external collector layer and a single collector electrode. which constitutes still a significant portion of the total transit time. Part of the reason for this is that the base layer of this particular transistor is quite thick (WB = 3000 A) and, even with a base-grading parameter x\,t as high as 0.2, the resulting built-in field is only 8 kV/cm. In Table 4.2, an E^ of 8 kV/cm corresponds to, for WB = 1000 A, a base-grading parameter Xj, e of only 0.067. Another reason for the relatively large rB is that the collector space charge region transit time, which normally constitutes a very large portion of the total transit time, is reduced because of the use of a constant V C £ . 4.2 Case II: H i g h Frequency Characteristics The HBTs used for experimental comparison in this section were fabricated using a proton-implanted external collector layer and a single collector electrode [13]. A cross section of the fabricated HBT is illustrated schematically in Figure 4.4. The Si0 2 Chapter 4. Comparison with Experimental Data 108 Table 4.3: Epitaxial layer structure parameters for fabricated HBTs (Case II). Layer Material Thickness (A) Doping (cm"3) Al composition x emitter cap n + -GaAs 2000 5 x 10 1 8 0 emitter grading n - A l I G a 1 _ I A s 300 5 x 101 7 0-0.3 parabolic emitter n-AljGai-rAs 900 5 x 10 1 7 0.3 emitter grading n-AljGax-jjAs 300 5 x 10 1 7 0.3-0.1 parabolic base p + - A l I G a i _ a ; A s 1000 4 x 10 1 9 0.1-0 linear collector n-GaAs 6000 5 x 10 1 6 0 collector buffer n + -GaAs 5000 3 x 10 1 8 0 Table 4.4: Fabricated device dimensions (Case II). ABB (Mm2) ABc (Mm2) A-BCJA-EB PEB/AEB (Mm"1) Tr 1 1 x 10 3 x 12 3.60 2.2 Tr 2 2 x 5 4 x 7 2.80 1.4 Tr 3 2 x 10 4 x 12 2.40 1.2 Tr 4 5 x 10 7 x 12 1.68 0.6 sidewall that separates the base electrode and the emitter is less than 0.2 pm; in our program, a value of 0.2 pm for SEB w a s used. As a result of proton implantation, the base-collector capacitance for the implanted regions is about 0.2 fF//zm 2 (see Subsec-tion 2.7.3). The epitaxial film structure for the devices is shown in Table 4.3. Note that both junction and base grading were employed. Four HBTs of different emitter-base and base-collector areas (AEB and ABC, respectively) were made and their dimensions are shown in Table 4.4. The four transistors are numbered in order of decreasing collector to emitter junction area ratio [ABC/AEB) and emitter perimeter to area ratio [PEB/AEB]-All four devices have a fixed base electrode width of about 0.8 pm. Dimensions of the Chapter 4. Comparison with Experimental Data 109 Table 4.5: Measured and calculated fx and / m a x (Case II). Experimental Calculated (Implanted) (Implanted) (Unimplanted) fT (GHz) / m « (GHz) h (GHz) / m « (GHz) IT ( G H Z T /max (GHz) Tr 1 35 70 41.2 89.0 39.2 72.0 Tr 2 45 70 41.0 67.2 39.6 58.9 Tr 3 50 70 41.2 67.5 39.9 59.1 Tr 4 40 42 39.3 39.9 38.4 37.7 collector electrode and the spacing between the collector electrode and the intrinsic collector were not given; we guessed they were about 1 pm and 5 pm respectively. The emitter and collector ohmic contacts were alloyed types; their contact resistivities were both assumed, as in the previous section, to have a value of 1 x 10 - 6 f lcm 2 . The base contact resistivity was given in the original paper as 2.5 x 10~6 f lcm 2 . Measured and calculated fx and / m a x values at VCE = 2 V and Jc = 4 x 104 A / c m 2 for the four HBTs are shown in Table 4.5. Also included in Table 4.5 are the calculated fx and / m a x values for the unimplanted HBTs. The measured fx and / m a x values for the proton-implanted HBTs are quite consistent with the calculated values, with the exception of the measured / m a x value for transistor Tr 1. In contrast to the somewhat scattered measured fx data, the calculated fx values for the proton-implanted HBTs are more or less independent of device size. An analysis of the calculated transit times shows that only the collector charging time, r c , changes substantially among the HBTs, ranging from r c = 0.17 ps for transistor Tr 1 to rc = 0.43 ps for transistor Tr 4. The increase in r c with device size is due mainly to an increase in the collector junction capacitance. However, since TQ amounts to less than 15 % of the total transit time, Chapter 4. Comparison with Experimental Data 110 the resulting fx values change very l i t t le . The calculated / n i a x values for the proton-implanted H B T s match very closely to the measured ones for transistors T r 2, T r 3 and Tr 4. The low / m a x value for transistor T r 4 is due understandably to a large collector junct ion capacitance resulting from a large base-collector junct ion area and a large base spreading resistance resulting from a large base current path. Since the cross section for base charge flow is twice as large for transistor T r 3 as for transistor Tr 2, the base spreading resistance of transistor T r 3 is half that of transistor Tr 2. O n the other hand, the collector junct ion capacitance of transistor Tr 3 is about twice that of transistor T r 2 because transistor T r 3 has about twice the base-collector junct ion area of that of transistor T r 2. A s a result, both transistors T r 2 and Tr 3 have about the same effective R/, Cc product and thus / m a x - Using the same arguments as above, transistor T r 1 should produce a significantly larger / m a x than the other transistors, as indicated by the calculated / m a x values in Table 4.5. It is unclear why this is not seen in the measured / m a x of transistor T r 1. The improvement gained in / m a x as a result of proton implantat ion is obvious when comparing the calculated / m a x values for the implanted and umimplanted cases in Table 4.5. The improvement seen in / m a x increases as the collector to emitter junct ion areas ratio increases. Th i s is reflective of the fact that a larger collector to emitter junct ion area ratio means a larger por t ion of the base-collector junct ion area may be subjected to proton implantat ion. The dependence of fx on the collector-emitter voltage VCE at Jc — 4 x 10 4 A / c m 2 for a proton-implanted 2 pm x 5 emitter H B T (transistor T r 2) is shown in F i g -ure 4.5. The solid line indicates the calculated fx and the open circles correspond to the experimental values. As in the case of transistor T r 2 in Table 4.5, the calculated fx values shown in Figure 4.5 are consistently lower than the measured values by ^ 4 G H z . B o t h the measured and the calculated fx decrease w i th increasing VCE, implying that Chapter 4. Comparison with Experimental Data 111 50 N X CD >^ o c 3 40 cr 3 O 30 H calculated O experimental Collector-Emitter Voltage (V) Figure 4.5: Dependence of fT on collector-emitter voltage for Jc = 4 x 10 4 A / c m 2 . the collector space charge region transit time presents a strong influence on / j . Chapter 5 Summary 5.1 Conclusions In this thesis the current gain characteristics of the graded-base AlGaAs/GaAs n-p-n Heterojunction Bipolar Transistor have been examined in detail using a comprehensive one-dimensional analytical model. The HBT's high frequency characteristics have also been studied through a quasi-two-dimensional model for pyramid-structured devices. The following conclusions can be drawn for this work: 1. Base grading, by increasing the bandgap in the base, reduces the Shockley-Read-Hall, radiative and Auger recombinations in the neutral base region as well as in the emitter-base depletion region. But, as the Al mole fraction on the base side of the base-emitter junction increases, the hole-blocking property of the junction is lost and the back-injection current increases. This leads to an optimum value of base grading at which current gain may be maximized. In the device considered here, maximum gain occurred with an A l mole fraction of 0.1 at the base-emitter junction. The improvement in gain with respect to the ungraded case was about four-fold. 2. Base grading improves the values of fT and / m a x by reducing the base transit time. In the device considered here, the use of maximum base grading, i.e., the Al mole fraction at the base-emitter junction being 0.3, increases fT by about 60 % and / m a x by about 20 %. If the transistor is operated at a sufficiently high 112 Chapter 5. Summary 113 current level (Jc ^ 2 x 104 A/cm 2 ) , an increase in fr of about 30 % and in / m a x of about 15 % can be obtained utilizing only an Al mole fraction of 0.1 at the base-emitter junction. 3. Base grading also mitigates the reduction in fr that occurs when either the base width or the base doping density is increased. The former is particularly significant because it allows the use of a thicker base layer to reduce the base spreading resistance without too much degradation in jr. 4. Once rD is reduced by base grading, the major contributor to the overall delay time, and hence / r , is the transit time through the base-collector space charge region. 5.2 Considerations for Future Work There are two obvious areas in which the present model needs to be improved. First, to make the model valid for very high current densities, the Kirk effect and emitter and base resistance voltage drops should be considered. 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(2.84) is made: (A.l) The barrier transparency, of Eq. (2.96) can be written in a shorter form, such as D(X)=exp{-^g(X)) (A.2) where g(X) = I(X) + II(X). Letting a = ET1/kT and r = E*/ET1, Eq. (A.l) becomes kT l n = 1 + exp oo ag{X) -aX dX (A.3) A further simplification can be made by letting & = — = —— (A 4) E00 100 hq V ND { ' ' Since ti and ND are usually expressed in cm units, the factor 1/100 in Eq. (A.4) is required to make b dimensionless. The final form for the tunneling factor is l n = 1 + a J\xp{-a[bg{X) + X- 1}} dX (A.5) Since ETI = qVTl = q 'qND 130 Appendix A. Coding Scheme for the Tunneling Factor 131 and E* =qVT1- AEC = q y the lower integration limit r may be written as qNpdj 2ci - AEC 2 n (A.6) dn- 9 which must be between zero and one. If r < 0 (i.e., E* < 0), then r is set to zero (i.e., set E* = 0). On the other hand, 7„ is set to one, that is, no tunneling current, if junction grading is large enough to make ETI = 0 or WG > dn, or if r > 1. When AEc < 0, tunneling is not possible because no conduction-band spike exists, and junction grading should not affect the barrier energy q V r i . At the end of Section 2.3, it was stated that if AEc < 0, VT1 should be replaced by Vj-i- This is no longer necessary as r will be greater than one according to Eq. (A.6) and consequently the tunneling factor will be equal to one. Appendix B New Effective Densities of States It was defined in Eqs. (2.127)-(2.128) and Eqs. (2.130)-(2.13l) that n = NcFl/2{r,n) = Nc exp(r?n) (B.l) P = Nv F1/2(Vp) = Nv exp(Vp) (B.2) The new effective densities of states can be rearranged so that they are functions of carrier concentrations only: N ° = IxpJ^) = explF-^n/Nc)) ( R 3 ) K = e ^ f o j = exp[Fj2(p/Nv)} { B A ) For a N-type semiconductor, N£ is computed from Eq. (B.3) using n c± Np, and Nv ~ Ny. The latter approximation is made because for a N-type semiconductor r/p 5 (B.5) 132 Appendix B. New Effective Densities of States 133 At rj = 5, Eq. (B.5) gives Fi/2 ^ 8.82 and a relative error of about 0.25 % which decreases as r\ increases. Since F i / 2 ( r7 ) is an increasing function, the following approx-imation for Fy}2(z) can be made: \M~4 ) " T f o r ^ > 9 (B.6) For z = FI/2(T7) < 9, the following Joyce-Dixon series approximation [120] is used: n = ln(z) + az + b z2 + czs + dz4 (B.7) where a ~ 0.35355391 6 ~ -4 .9500897 x 10~ 3 c ~ 1.483857 x I O - 4 d ~ - 4 . 4 2 5 6 8 X I O - 6 Our calculations indicate that Eq. (B.7) produces a relative error £ 0.1 % for z < 9. The independent variable z may be expressed as a function of doping density using Eqs. (B.l) and (B.2): z = n/Nc ~ ND/NC and z = p/Nv ~ NA/NV for N-type and P-type semiconductors, respectively. Appendix C Fermi-Dirac Integral Rat io Fi/2(y)/F- 1 / 2 ( 7 ? ) One of the properties of Fermi-Dirac integrals is that Fi-M = ^jFi(ri) (CI) or more specifically, F_i/ 2(»7) = dFi/2(rj)/dri. From Eq. (B.7) of Appendix B, one can infer that F_ 1 / 2(r?) = + a + 2bz + 3cz 2 + 4dz3^J (C.2) where z = Fi/2{n). Thus, the Fermi-Dirac integral ratio is given by 5 1 / 2^l = l + az + 2bz 2 + 3 c z 3 + 4dz 4 (C.3) and z ^ Np/Nc, N^/Ny for N-type and P-type semiconductors, respectively. Eq. (C.3) is valid for rj < 4 where the relative error is less than 0.1 %. For 17 < —10, both Fi/2(n) and F_i/ 2(r/) approach exp(r/) and therefore Fi/2(r])/F_x/2(r)) ~ 1. For n > 4, an asymptotic expression for F_i/2(»7) exists [34]: F _ 1 / t ( „ = + ( c . 4 ) which has a relative error less than 0.7 %. In this range of rf, 134 Appendix D Derivat ion of Transit T i m e Delays from the Hybrid-7r Equivalent Circuit The hybrid-7T model of the transistor is shown in Figure D . l . Many of the circuit elements have already appeared in Figure 2.15 which is essentially a simplified ver-sion of the circuit shown here. While we have included the emitter-to-base diffusion capacitance, Cp, in the hybrid-7r circuit of Figure D . l , the collector-to-base diffusion capacitance is neglected since it is much smaller than the collector junction capacitance under the normal bias conditions. The resistive elements Rc and RE are, respectively, the collector and the emitter series resistance which include bulk and contact resis-tances. From Figure D . l , *i = l 2 + * 3 = jU){CD + CEj)Vb,e,+juCCj[VVe'-Ve>e>) (D.l) and Vc,e, = -{ieRE + icRc) • - -ic [RE + RC) = -{gm V W - t s ) {RE + Rc) -9m V W {RE + Rc) (D.2) Eq. (D.2) is obtained by applying the approximations: ie ~ ic and t 3 (CD + CEi) + % CCj + (RE + Rc) Ccj] The frequency where |/?| = 1 is the cutoff frequency, fT, and can by obtained from Eq. (D.7) by noting that a 0 ^ 1 and that the second term in the denominator of Eq. (D.7) is much greater than one. Thus, 1 _ 1 U)j> 2 7T ff = re(CEj + CCj) + RECCj + reCD + Rc CCj (D.8) = Te + rB + rc (D.9) On the right hand side of Eq. (D.8), the sum of the first two terms corresponds to the emitter charging time r B , the third term corresponds to the base transit time rD, and the last term corresponds to the collector charging time (compare to Eqs. (2.194), (2.202) arid (2.206), respectively). The collector space charge region transit time, which is not included in the derivation here, should also be part of the total transit time.