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The effect of base grading on the gain and high frequency performance of AlGaAs/GaAs heterojunction bipolar… Ho, Simon Chak Man 1989

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T H E EFFECT OF BASE GRADING ON T H E GAIN AND HIGH F R E Q U E N C Y P E R F O R M A N C E OF AlGaAs/GaAs  HETEROJUNCTION BIPOLAR  TRANSISTORS  Simon Chak M a n Ho B . A . Sc. (Hons.) University of B r i t i s h C o l u m b i a  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF M A S T E R OF A P P L I E D  SCIENCE  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  ENGINEERING  We accept this thesis as conforming  .  to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A  August 1989  © Simon Chak M a n H o ,  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 w i t h a conventional pyramidal structure. G r a d i n g 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 i n the space-charge and neutral regions of the device is modeled by considering ShockleyR e a d - H a l l , Auger and radiative processes. O w i n g to the different dependencies on base grading of the currents associated w i t h these recombination mechanisms, the base current is m i n i m i z e d , and hence the gain reaches a m a x i m u m value, at a moderate level of base grading (x = 0.1 at the base-emitter boundary). T h e m a x i m u m improvement in gain, with respect to the ungraded base case, is about four-fold. It is shown that the reduction i n base transit time due to increased base grading leads to a 60 % improvement i n fx, in the most pronounced case of base grading studied (x = 0.3 at the base-emitter boundary). T h e implications this has for improving /  I n a x  v i a increases i n base w i d t h and base doping  density are also examined. F i n a l l y , comparisons between predictions of the model and experimental data from fabricated devices reported i n 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  31  2.5  Barrier Transparency  Recombination and Generation Currents in  35  2.6  2.7  3  4  2.5.1  Shockley-Read-Hall Recombination Process  36  2.5.2  R a d i a t i v e Recombination Process  40  2.5.3  Auger Recombination Process  41  2.5.4  Generation Process  43  Parameters for G a A s 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  M o b i l i t y and Diffusion Coefficient  55  2.6.6  M i n o r i t y Carrier Lifetimes  57  H i g h 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  Results and Discussion  81  3.1  E m i t t e r and Collector Currents  83  3.2  Base Current Components  86  3.3  D C Current G a i n  90  3.4  Transit T i m e Components  94  3.5  Effects of Base W i d t h and Base Doping on f  T  and /  m ; i x  98  C o m p a r i s o n with E x p e r i m e n t a l D a t a  101  4.1  Case I: Current G a i n 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 S c h e m e for the T u n n e l i n g F a c t o r  130  B  N e w Effective Densities of States  132  C  F e r m i - D i r a c Integral R a t i o F {n)  D  D e r i v a t i o n of T r a n s i t T i m e D e l a y s f r o m the H y b r i d - 7 r E q u i v a l e n t C i r -  1/2  / F_ {r]) l/2  cuit  134  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)  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 /  109  m a x  vi  (Case II)  . 102  List of Figures  1.1  Base grading profile  2.1  Energy-band diagram of an ideal abrupt n-p heterojunction at thermal  4  equilibrium 2.2  7  Energy-band diagram of the n-p emitter-base heterojunction under forward bias  2.3  11  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  2.7  Tunneling through the conduction-band spike of the abrupt emitter-base junction  2.8  31  Tunneling through the conduction-band spike of the graded emitter-base junction  2.9  27  33  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.  .  37  2.10 Band structure of GaAs. .' . .  48  2.11 Compositional dependence of the T, X, and L interband energy gaps.  vii  . 49  2.12 Collection of experimental minority carrier lifetime data of GaAs for electrons (open symbols) and holes (solid symbols) at low doping densities  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 hybridfl"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 basecollector junction  76  2.19 Heterojunction transistor structure with an implant-damaged external collector 3.1  78  Dependence of collector current density on base-emitter voltage for different amounts of base grading, with VBC — —3 Y (broken lines) and J  3.2  = 0 (solid lines)  G  84  Dependence of emitter current density on base-emitter voltage for dif*  85  ferent amounts of base grading 3.3  Dependence of base current components on base-emitter voltage for the case of Xb = 0.1  87  e  3.4  Dependence of base current components on A) mole fraction for J - = t  10 A / c m 3  88  2  viii  3.5  Dependence of base current components on Al mole fraction for J — c  IO" A / c m 4  3.6  89  2  Dependence of DC current gain on collector current density for different amounts of base grading  3.7  91  Dependence of DC current gain on emitter junction grading width for different amounts of base grading, with Jc = 10 A/cm 3  3.8  Dependence of fx and /  m a x  93  2  on collector current density for different  amounts of base grading 3.9  95  Dependence of transit time components on collector current density for x  = 0.3  be  96  3.10 Dependence of transit time components on Al mole fraction at baseemitter junction for Jc = 2 x 10 A / c m 4  3.11 Dependence of fx and / 3.12 Dependence of fx and / 10 A / c m 4  4.1  on base width for Jc = 2 x 10 A/cm . . . . 4  m a x  m a x  2  99  on base doping concentration for Jc = 2 x 100  2  Dependence of experimental and calculated current gain on base built-in field for /  4.2  97  2  = 6 x 10~ A and a base thickness of 1000 A  103  2  c  Dependence of experimental and calculated current gain on base thickness 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 collector layer and a single collector electrode  . 107 = 4 x 10 A/cm . . Ill  4.5  Dependence of fx on collector-emitter voltage for J  D.l  Hybrid-7T circuit model of the transistor with short-circuited emitter and  c  collector terminals  4  2  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 decrease 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 compounds, 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 A L - G a ^ A s 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 effect 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  A d v a n t a g e s of B a s e 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-electricfieldinto the base to aid the minority carrier transport, and thus improve the base transit time r . B  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 /  mttX  .  Another advantage of base grading is that the quasi-electricfieldcreated 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 suppresses this effect [33].  Chapter  Al  1.  4  Introduction  MOLE  FRACTION  0.3 0-2 0.1 0.0  EMITTER  COLLECTOR  N =5*10 cm ! N =3x10 cm~ 17  D  3  19  A  3  | N = 5x10 y3* ° i  D  1  18c  rrf  3  DISTANCE  Figure 1.1: The profile of the A l mole fraction (i.e., x in Al,;Gai_,;As). The four cases of base grading are referred to as Xb = 0.3, 0.2, 0.1, 0, where x^ is the A l mole fraction at the base-emitter metallurgical boundary. t  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 EbersMoll representation of the H B T . The base composition profiles which are considered are illustrated in Figure 1.1. Note that the variation of A l mole fraction in the base is assumed to be linear, hence the amount of base grading is determined by the A l 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 G a i _ A s are taken from the I  I  device analysis program S E D A N III [34], and Fermi-Dirac statistics are used.  1.4  O v e r v i e w of the Thesis  In Chapter 1, we have briefly described the progress in performance of Heterojunction Bipolar Transistors in recent years. The advantages of H B T s , 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 H e t e r o j u n c t i o n 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 energyband 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 depletionlayer 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): dV  qN  dx  ti  2  for -d  D  2  dV  qN  dx  6  2  2  A  n  < x < 0  for 0 < x < d  p  (2.1) (2.2)  2  For a graded-base or a graded-junction HBT, however, the above treatment is complicated 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 -<L  1 0 W  F i g u r e 2.1:  T  1  •  X  0%  J  Energy-band diagram of an ideal abrupt n-p heterojunction at thermal  equilibrium. simplicity, the dielectric constants of the emitter and the base are assumed constant in the derivation of the depletion-layer width and  capacitance.  Accordingly, integrating Eqs. (2.1) and (2.2) twice and applying the boundary conditions that at x = —d a n d x — d n  v  the electric field E = —dVjdx  = 0 a n d that at  x = 0 the potential V is continuous and equal to 0, we obtain  qN fx \ V = - ^ - ~ \ — + d x\ 2  D  n  for -d <x<0 n  (2.3)  Chapter  2. Model  Development  f T h e parameter  8  0 < x < dr,  for  \2  2  (2.4)  VV is defined as the total potential across the p - n heterojunction i n  n o n e q u i l i b r i u m , i.e.,  V  T  = V -V bi  =  =V  a  + V  T1  T2  V(-d )-V(d ) n  q (N d D  2  N dV  +  n  «,  V  p  .  2  A  ;  , 2  '  5 )  where V is the b u i l t - i n p o t e n t i a l , V the applied voltage across the p-n j u n c t i o n , and BI  A  Vj-i a n d VT2 are the p o r t i o n of Vj supported b y semiconductors 1 a n d 2 respectively. A n o t h e r boundary c o n d i t i o n is the continuity of electric displacement at x = 0, that is ti Ei = t-iE-i. T h i s yields the relationship N  d = N d w h i c h can be c o m b i n e d w i t h  D  n  A  p  E q . (2.5) to give d  2 V ei € N qN ( N + e N) 2  T  =  n  D  €l  A  2  D  2V e e N T  dp = g  x  2  1/2  (2.6)  A  1/2  D  (2.7)  ^ (e iV , + € ^ ) A  1  I  2  A  T h e total depletion-layer w i d t h is  2V e (iVA + Wc) qN N ( N +e N) 2  W  T  =d +d n  T e i  p  A  2  D  ei  D  2  1/2  (2.8)  A  T h e depletion-layer capacitance or j u n c t i o n capacitance (per unit area), Cj, c a n be easily derived from the following capacitance formula: d  —  Cj  -  —  C[  +  C  d  n  v  — +— ei e  — 2  . 2.9  2  C[ a n d C are the depletion-layer capacitances of semiconductors 1 a n d 2 respectively. 2  S u b s t i t u t i n g E q s . (2.6) a n d (2.7) into E q . (2.9) yields qet e N 2  2V  T  N  D  A  N  -,1/2 D  +e N) 2  A  (2.10)  Chapter 2.  Model Development  9  Emitter-base junction grading and, to a lesser extent, base grading cause the E - B junction to behave more like a GaAs homojunction.  Consequently the emitter-base  junction built-in potential should decrease as argued by Hayes et al. [35] However, it is unclear how different types of grading affect the junction built-in potential. For simplicity we derive the built-in potential for the ideal abrupt heterojunction and assume that it is valid also for the cases of junction grading and base grading. From Figure 2.1, the built-in potential is equal to qV = E ht  + A E  t2  c  - q (V„ + V )  (2.11)  p  where E z is the energy bandgap of the narrow-gap semiconductor, AEc t  = X2  —  Xi> '- -i e  the difference of electron affinities, and V and V are the separations of the electron and n  p  hole quasi-Fermi levels from, respectively, the conduction band in the emitter and the valence band in the base.  and V are related to the electron and hole equilibrium p  concentrations and effective densities of states by [36, p. 27] 1  =  qV  n  qV  =  p  where n  n o  and p  po  fcrin(^)  kT  (2.12) (2.13)  ln (^y^j  are the equilibrium majority carrier concentrations in the emitter and  base respectively. At equilibrium the intrinsic carrier concentration of semiconductor 2 is given by nf = Nc2 A V 2 2  qVu  _  ^  1  , ,  ^  e  *p(—Eg /kT), hence 2  r  T  ^ ^ )  + ( X j  (  .,  l ) +  +  t  r  «• -  i  n  (  S  _ )  ( £ )  'In order to incorporate Fermi-Dirac statistics into these formulations, a "new ' effective density of states for conduction band and valence band should be used instead; see Subsection 2.6.1 for more details. 1  Chapter 2. Model Development  Note that for a homojunction, Nc2 = Nci  10  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 N 2 are C  calculated from the parameters of the base at the emitter-base interface.  Thermionic-Diffusion M o d e l  2.2  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 recombinationgeneration currents. The effects due to series resistance, high-level injection and hotelectrons 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  E l e c t r o n a n d H o l e T h e r m i o n i c - E m i s s i o n C u r r e n t s at the E - B J u n c 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 v where qV  Ti  n(x ) E  and AE  n  [n(x ) e-« ^ v  TnB  - n(0) e~^ }  kT  kT  E  (2.15)  are the electron potential energy barriers shown in Figure 2.2,  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) 2nm  nE  with  m*  nE  being the electron effective mass in the emitter.  Chapter 2. Model Development  12  A s s u m i n g low-level injection, the electron carrier concentration a t x = x is apE  proximated by n{x )  c± n{x )  E  =n  E  exp[{qV - AE )/kT]  B0  hi  (2.17)  c  T h e bar indicates the e q u i l i b r i u m condition a n d n  is the e q u i l i b r i u m electron con-  B0  c e n t r a t i o n i n the base at i = 0. Referring to F i g u r e 2.2, one can write qV i  =  T  = where V  BE  q{V -V -V ) bi  q{V -V )bi  T2  AE + AE  BE  C  (2.18)  n  is the applied potential across the emitter-base j u n c t i o n . A l s o b y defining n(0) = rJ  where  BE  + n(0)  B0  (2.19)  denotes excess carrier concentration, E q . (2.15) becomes JT„  = =  -qv -9v  e-* "? E  T  n  E  T n J s  e-  A J S  -/  k  t r  T  [n  [W  B  B 0  o  e^-  (e' " v  A  °»  E  / t r  N o t a l l of the electron carriers transported  k  T  e^^  -l)-M0)]  (2-20)  across the heterojunction is due t o  thermionic-emission; a p o r t i o n of i t is due t o field-emission t u n n e l i n g through the c o n d u c t i o n - b a n d spike. T h i s effect can be accounted for b y replacing v  TnE  with v  TnB  In  in E q . (2.15), 7 „ being the tunneling factor [28]. T h e derivation of 7 „ , w h i c h is always greater than or equal to 1, is described i n another section. Hence, E q . (2.20) becomes Jm = -q v  TnB 7  » e-*  [n  EnlkT  (e< °°' v  BQ  - 1) - n(0)]  kT  (2.21)  T h e hole thermionic-emission current injected from the base to the emitter can be derived i n a s i m i l a r fashion except that no tunneling factor is required. T h e net hole flux across the emitter-base j u n c t i o n is given by JT  P  =qv  (p(0) e-< ™l v  TrS  kT  - p(x ) e^' \ kT  E  (2.22)  Chapter 2. Model Development  and AE  where qV  P  T2  13  are the hole potential energy barriers defined in Figure 2.2, p(0)  and P(XE) are the hole carrier concentrations at the depletion edges, and  TpE  —  V  \|27rm;  (2.23) E  is the average x-direction hole thermal velocity in the emitter, with m* being the hole pE  effective mass in the emitter. Using the following equations p(0)  ^  p{0) = p- exp[{qV -rAE )/kT}  AE  =  qV  =  q (V« - V  P  E  v  - V ) + AE  BE  (2.25)  V  T7  PE + PM  (2.26)  is the equilibrium hole concentration in the emitter, E q . (2.22) becomes JT  P  = =  2.2.2  (2.24)  v  + AE  T1  P{XE) = where p  bi  E  qv qv  T  p  B  k  W  T  E  e  {  q  W  E  v  )  e* >? \p ( < W * r _ E  TpB  e^  /  k  _  kT  E  c  e^^^  T  1 }  p { x E ) ]  ( 2 2  7  )  Emitter 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)  q  p(x) 2.28  dx  T P  E  and the equation for the hole diffusion current in the emitter  J (x) p  = - q  D  p  E  ^ -  (2.29)  Chapter 2. Model Development  14  c o m b i n e to give the differential equation d p{x)  p(x)  2  dx  2  pE  =o  (2.30)  L  where p(x) is the excess hole concentration at position x i n the emitter, a n d L  \jD E P  TpE  is the hole m i n o r i t y carrier diffusion length,  D E and  r  P  pE  —  being the m i n o r i t y  pE  carrier diffusion coefficient and lifetime, respectively, i n the emitter. T h e general solution of E q . (2.30) is I x \ I x p(x) = ki exp --— ) +. k exp j , - r .j , \ ^pE) \ LipE J  (2-31)  2  w h i c h is substituted into E q . (2.29) to give the following equations for the hole diffusion current density evaluated at the emitter boundaries x — x  E  and i = W  EE  (see  F i g u r e 2.2):  M E) = X  L  qDpE s'mh(W /L )  pE  E  p(x ) E  <lD  pE  J (W ) P  =  EE  L  The approximation W  EE  p  s\nh(W /L E)  E  E  —x  [  P  ~ W  E  cosh  pE  LpE )  P(XE) - P{W ) EE  (2.32)  P{WEE)  cosh 'W X  (2.33)  E  jpE ,  was made i n o b t a i n i n g E q s . (2.32) a n d (2.33).  E  T a k i n g the surface recombination velocity for m i n o r i t y carriers, S , pE  to be finite, the  b o u n d a r y c o n d i t i o n at the emitter contact is J {W ) P  =  EE  (2.34)  qS p{W ) pE  EE  E q u a t i n g E q . (2.34) to E q . (2.33) gives a n expression for p{W ) EE  w h i c h c a n be sub-  *  s t i t u e d into E q . (2.32) so that J {x ) p  E  c a n be expressed as a function of p(x ) only. T h e E  hole thermionic-emission current density given by E q . (2.27) should be equal to this expression at J {x ). p  E  After eliminating the hole excess carrier concentration p{x ), the E  resulting hole current density at x — x  E  becomes  (2.35)  Chapter 2.  Model Development  15  F i g u r e 2.3: E n e r g y - b a n d d i a g r a m of the p - n base-collector j u n c t i o n under reverse bias, where  JpE  —  qD Lp  =  E  l*+,  E  2 E  DE  n  iE  S /D E  L  E  E  pE  pE  [L  pE  pE  +  P  S EID E P  + 1/  P  ianh(W /L ) E  pE  tanh{W /L ) E  pE  ( — —P f\ cosh u( exp ( \ kT J \, A  PE ——  J  F  qv p TpE  a n d p~ = n /N ,  p  swh{W /L )  E  j? R  pE  E  a n d ND  E  E  ]  being, respectively, the intrinsic carrier concentration  a n d the N - t y p e doping concentration i n the emitter. T h e collector-base j u n c t i o n , shown i n F i g u r e 2.3, is essentially a G a A s homojunct i o n , so i t is reasonable t o assume that the collector hole current is governed by a simple diffusion process.  A s before, we begin w i t h the continuity equation a n d the  hole diffusion current equation in the collector (Eqs. (2.28) and (2.29)), and arrive at the following equations for the hole diffusion current density evaluated at the collector  Chapter 2. Model Development  16  depletion edge x — xc and contact boundary x = W ,  assuming W  cc  J i c)  qDpc  =  x  P  Lc  smh(W /L c)  p  J (Wcc)  Lc  JC,  c  -  c  P(W )  (2.36)  rV,c  (2.37)  CC  P  p{x )  - p{W c) cosh  c  sinh(W /L c)  p  Wr  p  qDpc  =  P  c  p(xc) cosh  — xc — W :  cc  C  p  where p(x) is the excess carrier concentration (holes) in the collector evaluated at position x, and L c = yDpC T C is the minority carrier diffusion length in the collector, p  D c and r p  P  being the minority carrier diffusion coefficient and lifetime, respectively,  pC  in the collector. The boundary conditions are  P{xc) = Fc ( « ' Jp(W )  =  CC  where p  c  —n  2 c  - 1)  W f c r  (2.38) (2.39)  <jS cp(W ) p  cc  /NDC is the equilibrium hole concentration in the collector, n,o and N^c  being, repectively, the intrinsic carrier concentration and N-type doping concentration in the collector, and S E is the hole surface recombination velocity at the collector P  contact. Eqs. (2.36) to (2.39) are combined to eliminate the excess carrier concentration variables. The resulting collector hole current density evaluated at x = XQ is Jp(xc) = J c {e  qVBclkT  P  (2.40)  ~ 1) cosh  where IpC  L  y  pC  9 Dpc Pc sinh(W /L ) c  pC  L c Spc/Dpc + p  L  pC  Spc/Dp  C  tanh(W /L ) c  pC  + l/tanh(rV /L -). c  pC  Eqs. (2.35) and (2.40) represent the hole current components of the total D C emitter and collector currents (in the case of no space charge recombination and generation). To find the electron current components, one needs to solve for the electron diffusion current in the graded b  described in the next subsection.  Chapter 2. Model Development  2.2.3  17  E l e c t r o n Diffusion C u r r e n t in the B a s e  In the base of a good homostructure transistor, the hole current is small compared to the electron current. Kroemer [37] has argued that this is an even better approximation for HBTs because of their typically higher current gains. In general, the electron and hole current densities across the base region of an n-p-n transistor are related to the two quasi-Fermi levels <p and 4> for electrons and holes by n  p  J  n  J  P  =  -qp nV<t>  =  -<7M pV0  n  (2.41)  n  P  (2.42)  p  where p and p are the electron and hole drift mobilities in the base, and n and p are n  p  the base electron and hole carrier concentrations. Based on Kroemer's approximation that V ^ p ~ 0, Eq. (2.41) can be rewritten as  J = qp nV{<j> -(j> ) n  n  p  (2.43)  n  For the nondegenerate case, the pn product is given by  pn = n] exp  (2.44)  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]:  N  A  Here, we have assumed that p ~ N  A  dx\  n?(i)  /  in the P-type heavily doped base, and used  the nondegenerate Einstein relation D„B — kT p /q. The electron diffusion coefficient n  in the base, D B, is assumed constant at a value appropriate to material of the Al mole N  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  E {x) = E -—±x g  where E  g0  (2.46)  g0  is the bandgap at x = 0, that is, at the emitter-base depletion edge on the  base side (see Figure 2.3), X  is the total base width, and AE is the difference in  B  g  Eg between the values at the two metallurgical junctions which define the base. The intrinsic carrier concentration can be expressed as n]{x)  =  =  N {x)Nv{x)e- o^l E  kT  c  ae  (2.47)  fx  where a =  n* N {x)N {x) 0  c  v  Nco N qAE " kT X  vo  g  f  A  B  In the above equation, Nc and Ny denote the effective densities of states in the conduction 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-^ , or more 1  precisely, |± ^ | <C | / | , thus when Eq. (2.47) is substituted into Eq. (2.45), the latter becomes J„(x)  qD e ^{ne- ) ax  =  =  fx  fx  nB  qD  nB  l ^ - f n ^ j  (2.48)  The electron carrier concentration can be written as the sum of the excess and equilibrium electron carrier concentrations: n(x)  =  h(x) + n {x) B  Chapter 2. Model Development  19  = W +^ FT  "  E  (- ) 2  49  W h e n E q . (2.49) is substituted into E q . (2.48), the latter becomes  J {x)  = qD  n  ( ^ - f b j  nB  (- ) 2  50  w h i c h , when substituted into the following continuity equation for m i n o r i t y carrier electrons 1 dJ (x) p-iq dx  h[x) ^ r  n  o  =  (2.51)  nB  yields  dh  , dh  2  h  ,  i*~ T*-W !  -  (2  =0  B  where the electron m i n o r i t y carrier diffusion length L  = \/D  nB  52)  T B, nB being the T  nB  n  m i n o r i t y carrier lifetime i n the base evaluated at the center of the base region. T h e solution for h(x) i n the second order differential equation (2.52) is  h{x) = C e 1  rit  + Ce  (2.53)  raX  2  where r  x  =  5 +  t  r  2  f s= 2 _ h{W) - n(0) e * r  =  s— t  , y/P L + 4 t = 2 LnB _ h{0) e - h{W) 2  nB  w  riW  In the above, x = 0 a n d x = W mark the two boundaries of the quasi-neutral base as shown i n F i g u r e 2.3. S u b s t i t u t i n g E q . (2.53) into E q . (2.48) produces  J {x) n  = qD  nB  \{r - f) d e * + ( r - / ) C e * ] r  ri  x  2  2  x  (2.54)  Chapter 2. Model Development  20  The emitter electron current density is simply J (0) and the collector electron curn  rent density is J (W).  To evaluate these two current densities exactly, it is necessary  n  to obtain expressions for n(0) and n(W). The latter is simply given by h(W) = n  where n  BW  (e'W*r _ ^  BW  (  2 > 5 5  )  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  J  = -z \n (e* ™/ v  Tn  n  kT  B0  - 1) - n(0)|  (2.56)  where z - qv  i e "i  TnE  n  n  When evaluated at x — 0, Eq. (2.54) reduces to  Jn{0)=y [2th(W)-a h{0)} n  (2.57)  n  where  qD  nB  "  =  n  (rx-f)e * r  -(r -f)en W  W  2  Equating the current densities of Eqs. (2.56) and (2.57) and solving for h(0) gives m  =  *y.W)+  -1) z +ay n  n  (2M)  n  Substituting Eqs. (2.55) and (2.58) into Eq. (2.57) yields the following expression for the emitter electron current density:  UO) = - ( "/ \1 + a y"" /z J ) n (e^ y  kT  m  n  n  n  - 1) - —— TxBw  a, r  (e* ^(2.59) - 1) V  kT  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 E q . (2.54) at x = W:  Jn{W) = y [b h(W) - 2t e n  (2.60)  n(0)]  u w  n  where k =(r!-/)e''*-(»•,-/)  f  w  Finally, replacing h(W) and n(0) in E q . (2.60) by their known equivalents in Eqs. (2.55) and (2.58) leads to  J {W)  = -  n  1+ a  \n (e" ^ -l) v  B0  kT  n  y /z n  ni  6„ + ( a b -4t n  e > )y /z  2  2  n  w  n  2fW  2t e  n  n {z' *cl v  BW  kT  -1)J  (2.61)  Chapter 2. Model Development  22  The main electron and hole current components formulated so far are shown schematically in Figure 2.4. What we call the emitter and collector electron current densities, J (0) and J {W), n  N  are actually electron current densities entering the quasi-neutral base  and the base-collector depletion region respectively.  J {XE) P  back-injected into the quasi-neutral emitter and —J {xc)  is the hole current density is the hole current density  p  entering the base-collector depletion region from the collector. The latter current density 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) — J (W)| represents the part of n  the base current density due to recombination in the quasi-neutral base. JR is the recombination 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 JE  =  -J (0)  Jc  =  -J {W)  n  n  + J {x ) p  R  - J {x ) p  (2.62)  + J  E  (2.63)  + J  c  G  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: 1) + A  n  Jc  where  =  A  2  1  (e^  k  T  - 1) + A  22  ( « ^/* v  e  (e" ^ v  kT  T  - 1)  + JR  - 1) +  J  G  (2.64) (2.65)  Chapter 2. Model Development  23  b +  (a b -4t e ° )y /z 2  n  n  122  1 +  O  n  2  w  n  n  n  Vn nBW — JpC cosh  Vn/Zn  •>pC,  All the symbols used have been defined earlier. The equilibrium carrier concentrations n  B0  and n w  can be calculated from the intrinsic carrier concentration and the base  B  doping density: n  — U /NA,  ™ B W = n /N .  2  B0  The base current density is simply the  2  Q  W  A  difference of the emitter and collector current densities, i.e., JE — JcIt 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 homojunction transistor. This would require that the electron energy barrier AE  < 0 and  n  l A ^ I » kT (see Figure 2.2). For the hole energy barrier, the inequality AE  P  » kT  should still apply for homojunction transistors. Assuming also that the contacts are perfectly ohmic, i.e., S E — S c —• oo, we would have the simplied expressions P  p  R  =  E  l  9  J,pE  L  pE  z —> OO  p  E  E  9  Jc  n  Dp VE smh{W /L )  L  Dpc Pc  s inh (W  pC  pE  /L )  c  pC  Of course, no compositional grading is possible with a homostructure transistor so many of the earlier expressions are also simplified: / = 0, n  B0  t =  = nw  = n,  B  B  s = 0,  l/L , nB  i  2  a = b = -— n  .  (  cosh ( - — )  n  w  ,  \  J  L>nB  <lD  y  and  *  \ ^nB / nB  n  nB  smh{W/L )  2  nB  Inserting these equations into the Ebers-Moll coefficients of Eqs. (2.64) and (2.65) and making the reasonable assumption that W  E  » L  and W  pE  c  ^> L c, the emitter and p  collector current densities reduce to JE  =  qD  nB  LB  n  B  - coth  V L „ £ )  n  9 DnB  L  nB  /W  n  B  smh.{W[L ) nB  ,  qD p pE  L  E  qVBE/kT  1)  pE  (2.66)  Chapter 2. Model Development  Jc  =  qD n s'mh(W/L ) qD n nB  L  nB  24  B  ,qV /kT BB  1)  nB  —  nB  B  coth  qD p Lc pC  c  1) + J  G  (2.67)  p  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). O n 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 H B T s [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 AE  n  which appears  in E q . (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  AE  P  26  can be simply expressed as  AE  = qV  P  + AE  T1  (2.68)  V  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 W is the emitter grading width, then the amount of conduction band lowering g  can be found by evaluating E q . (2.3) at x = —W : g  q  v  =  T1  —  —  [-f-yy d \ g  qN  n  2  D  [{d - W,)> - dl]  Since gV^ = qV  T1  (2.69)  n  26!  - qV^ (see Figure 2.5) and gV  = q N d /2e 2  T1  (from E q . (2.5)),  2  D  n  1  we have  K - *)*  qv^ =  ( - °)  w  2  7  The electron energy barrier parameter can be written as  AE  =  n  AE -qV -qV^ c  AE  =  AEc-qVr  where qV is equal to q {V — V E). T  bi  - {qV  =  C  T2  T  - gV ) - qV% T1  + qV^  (2.71)  Note that E q . (2.70) does not apply when the  B  grading width is greater than the depletion width d . n  If W > d , the barrier qVj g  n  r  *  should become zero. In general then we can write  AE  n  where  = AE  + q (V  c  BE  £J±{d -W y n  g  0  - V + VL ) bi  for  (2.72)  X  W,<dn  for W, > d  n  Chapter 2. Model Development  27  E = Ex i AE  n  E E = E'  f  I  '  qV  X  E  <tV  qV  n2  BE  nX  Ef  p  EMITTER  BASE  Figure 2.6: Direction of thermionic-emission and tunneling current components in the conduction band of the emitter-base junction. Notice that AE may become negative under the condition of small forward bias or n  small AEc- In this case, the electron current is governed more strongly by the diffusion process as in a conventional homojunction transistor. When AEc < 0 no conductionband spike exists. That means V"^ should not be reduced by any junction grading; in this case, V i replaces V T  2.4  T1  in Eq. (2.72) or alternately W is set to zero. g  Emitter-Base Tunneling Current  In Subsection 2.2.1 we adopted a tunneling factor 7„ in the electron thermionic-emission current equation to account for the tunneling of electrons through the conductionband spike. To see how this formulation is justified, consider the energy-band diagram for the conduction band shown in Figure 2.6. The two opposing thermionic-emission current densities are represented by Jxi and Jxi, and the two opposing field-emission  Chapter 2. Model Development  28  If TT and J  (tunneling) current densities are represented by Jf i and J . E2  F  are the  net thermionic- and field-emission current densities, respectively, then the net electron current density across the abrupt junction is  JT + JF  =  TOT  (2.73) Comparing this equation with Eqs. (2.20) and (2.21), we see that the tunneling factor is given by In = 1 + T-  (2-74)  Following Chang and Sze [49], the thermionic-emission current density injected from the base to the emitter, J , T2  is obtained by integrating, in the base, over the range of  energies above the conduction-band spike:  JTI  —  A\ T AE + qV + £ f exj k Jo A' T ( qV \ ( AE \ \-kT) \-kf-) n  n2  2  n2  (2.75)  n  eXV  eXP  k  T  where A\ is the effective Richardson constant in the base. Similarly the thermionicemission current density injected from the emitter to the base is given by  JTI  —  A\T k A\T  j  exj  qV  T1  + <7V  +  nl  (  d£ (2.76)  where A\ is the effective Richardson constant in the emitter. Incidently, since the electron densities at the depletion edges are given by n(0) =  N  C2  exp(-c/V //(:r) and n{x ) r  n2  E  = Ni C  exp(-qV /kT),  and the effective Richardson  nl  constants are related to the thermal velocities by A* T 2  2  — qN v C2  TnE  and A\T  2  =  Chapter 2. Model Development  qN iV C  29  [36, p. 261], the thermionic-emission current densities given by Eqs. (2.75)  TnB  and (2.76) reduce to JT2  =  qn(0) v  JTI  =  qn{x )  exp ^ —  TnB  v  E  TnE  (2.77)  exp I — —  (2.78)  These two expressions were used earlier to obtain the net electron thermionic-emission current density given by Eq. (2.15). Using Eqs. (2.75) and (2.76) instead, the net thermionic-emission current density is  Jr  Ji  JT2  —  T  A\T k A\T  (kT)  g(Vn  |exp  (kT) exp  +Vm)  kT  g ( V r i + Vni)  kT  — /3 exp ^— 1  —  AE  n  + qV j kT  (3 exp I — kT  n  (2.79)  where j3 = A*JA\. For the field-emission component, the forward tunneling current density through the barrier, Jpi, is proportional to the barrier transparency D(E) multiplied by the occupation probability in the emitter and the vacancy probability in the base [49], i.e.,  A\T A I T JFI = -JK  fEn r^ri J E'  D{E)h{E)\\-  f {E)\dE  (2.80)  2  where f\{E) and f {E) are the Fermi-Dirac distribution functions for the emitter and 7  base respectively, ETI = qVxi, and E* — qVxi — AEc- Similarly, the backward tunneling current density is given by  A\T r r, J 2 = - J - / " D[E) f (E) [1 k JE* E  3  F  The net field-emission current density is thus  J  F  =  J  Fl  — J  F2  h[E)]dE  (2.81)  Chapter 2. Model Development  30  =  ^jff^D(E)\f (E)-0f (E)}dE 1  A\T  2  rn E  l + exp[{E +  qV )/kT] nl  0 l + exp\{E + In general, E + qV  n2  qV )/kT}  dE  (2.82)  n2  » kT and E + qV » fcT if the emitter is not too heavily doped; nl  therefore one may apply the Boltzmann approximation to Eq. (2.82):  A* *  J  =  -TL 'i A  r /  T r^ri D  f IE-  T  {  E  )  nttr\  E r i  h ( (  E + qV \ ( E + qV ) - 0 exp (kT nl  a  kT + V m\  E  - 0 exp ^-  V  - \  n2  kT  kT .  01dE dE  (2.83)  Substituting Eqs. (2.83) and (2.79) into Eq. (2.74), the tunneling factor becomes J D(E) ETl  E  l  n  1  +  e x p ( - ^ u ) dE  kTexp\-q{V  T1  exp  fEn  +  V )/kT) nl  /  E\  For a graded junction, one must replace qVn by qV , the latter is denned in T1  Section 2.3. Furthermore, two cases are possible: (i) for small and moderate junction grading, i.e., En > 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 i  n  = 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 quasineutral 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  0  31  4  d  BASE r  EMITTER  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  Barrier Transparency  A . Abrupt Junction Consider first the abrupt heterojunction shown in Figure 2.7. In general, the expression for barrier transparency for an arbitrary potential barrier shape is given by [50] (2.85)  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 <f>(x) is given by  4>{x)  2ci  (dn -  X)  2  (2.86)  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 E q . (2.85), it is useful to know the solution of the following integral:  jja{d -x) -b\^ dx 2  =  2  n  {-y/i~{d -x)^/a(d -x) -b  ^=  +  2  n  n  b ln y/a (4 - x) + ^/a{d -x) -b]  ) '  2  n  (2.87)  J Jo Comparing the integrals in Eqs. (2.85) and (2.87), we see that a = q Nu/2e- and b = E. 2  l  Also we know that an arbitrary tunneling energy level is related to x by E = a(d — x ) n  2  and that the energy barrier is given by ETI = ad . n  - \nD{E)  = |  [  \n^E  E  + y/Er~i\/E  T1  2  2  Therefore E q . (2.85) becomes  -E-E  ln \[E~T~\ +  \J En  - E  j  (2.88) Defining an energy  2q  N  a f  2ti  and a dimensionless variable X = E/ETI,  D  hq 2  /  N  D  V i i c  m  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 = d and x = W , and n  g  Chapter 2. Model Development  33  E  A  ETI =  QV  TI  0  BASE  x -*  EMITTER  0  Figure 2.8: Tunneling through the conduction-band spike of the graded emitter-base junction. a graded barrier between x = W and x = 0. To solve for the barrier transparency, the g  integral in Eq. (2.85) must be integrated from x = z to x — x . The integration is done 3  2  separately for the two divided barriers. For the first potential barrier (i.e., barrier I in Figure 2.8) we have the integral  (2.90) Again using E q . (2.87) and noting the relationships a = g N p / 2 e i , b = E, E = 2  a (d — x ) , and ETI = a(d 2  n  2  n  — W ) , the above integral is readily solved. The solution 7  g  turns out to be exactly the same as E q . (2.89), i.e.,  For the graded barrier (i.e., barrier II), the barrier potential energy is given by  2. Model  Chapter  Development  34  —  (d -x)  + —  n  - ± E  X  (2.91)  C  The integral in Eq. (2.85), integrating from x = £ to W , is 3  q N J X* 3  D  g  1/2  AJ^C  ,,  dx  I  1/2  » , / . *W^ l - 6  2]j q N  dx  2  2d  J XX3  D  (2.92)  g  where d  n  Eq. (2.92) can be changed to a form similar to that of Eq. (2.87) by rewriting the integrand as \(d - y) - 6] . This leads to -dy = yjq N /2e 2  1/2  2  n  D  1  dx, and Eq. (2.92)  becomes  (d -y)+  yj{d  n  -y) -b 2  n  x=x  }  3  (2.93)  x=W„  The above integral can be solved with the help of Eq. (2.87). Noting also that / ' 2  a  2  Vq  VK-y) -6| 2  V(d„  1  C l  N  2  oo  D  3  -y) -b 2  -0  y/E-E  x=x  =  y/Er^E  Eq. (2.93) reduces to  -i- (ft ln Vb + \ j E E-  E  VT^XyJl  T  l  - E + b yjE  -X  Tl  +  b/E  Tl  - E - b ln  E-  L/£ - E + b + J  In W .  b/E  ri  Ei  ' + 1 + W7 . _ " b/E  Tl  T  - E  (2.94 j>(2.94  Chapter 2. Model Development  where X — E / ETI-  35  The term  is a function of X :  € AEl  E  En  AEc E  t  E  Ti  +2N q W E 2  D  2  /  g  Tl  .  (dn  - 1  Tl  \ 2  d  (ir)'  n  1 /AE  - 1  -1  C  2 V jE'n  (2.95)  In deriving E q . (2.95) we have used the relationship ETI — q ND {d — W ) / 2 e . 2  /  n  2  5  1  T o o b t a i n the barrier transparency for a graded j u n c t i o n , we s i m p l y s u m the solutions of the two integrals i n E q . (2.90) and (2.92). In summary, the barrier transparency for a graded j u n c t i o n is given by  D(E)  = D(X) = exp { - -ETI £ [I(X) + I I ( X ) ] }  (2.96)  where  1(X) =  II(X)  Vra-Xln[*±3F]  X >0 X =0  VT=Xjl-X+f{X)-f(X)  ln  y/l-X  + f[X) + ^A=Y  1- X  f(X)  +o  f(X)  =0  For a n abrupt j u n c t i o n , i.e., W = 0, I I ( X ) = 0. I n A p p e n d i x A the tunneling facg  tor given b y E q . (2.84), w h i c h contains the expression for the barrier  transparency  ( E q . (2.96)), is reorganized i n a form appropriate for c o d i n g .  2.5  Recombination and Generation Currents  T h e emitter-base space charge region recombination current Jp i n Eq. (2.64) and the collector-base space charge region generation current Jg i n Eq. (2.65) are computed from JR  -G  =q  I Udx  J SCR  (2.97)  Chapter 2. Model Development  36  where U is the net recombination rate which, in general, consists of several recombination 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,._J^~ \ n  — T —  n + m exp ( ^ p ) ] + r 1  no  where r  po  and r  no  ,  [p + n, exp  /  „._„.M  (2-98)  are, respectively, the minority carrier lifetime in highly extrinsic N-  type and P-type material, due to single-level recombination, E is the energy level of t  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  =  n, exp \ -j^r j  — n, exp  p =  n, exp  (2-99) (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, <f> and <f> are, respectively, the electron and n  hole quasi-Fermi potentials, and  VE B  p  = <f> — 4> is the applied emitter-base voltage. p  n  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 < — d ) and in the quasin  neutral base (x > d ) are denoted by ip and ip respectively. Without loss of generality, p  n  p  Chapter 2. Model Development  38  we set ip — 0 as a potential reference. Therefore n  —  [x + d ) > K  BE  where W  BE  (2.103)  n  W  T  n  refers to the total emitte-base depletion width. In the quasi-neutral regions,  we may approximate N  D  ~n  N ~p A  =  n,i exp I ^  =  n  n  n  [<f> - V> )  exp ^  i2  (2.104)  (ip - <f> )  (2.105)  P  p  The subscripts 1 and 2 refer to the emitter and base respectively. We define  0  =  ^rbk-iM / tin  i t 2_ \  g^BJS  (2.106)  which is derived from Eqs. (2.104) and (2.105) and the relationship V  BE  = <f> — </> p  Substituting Eq. (2.106) into Eq. (2.103), the latter becomes (2.107) Since tp = 0, Eq. (2.104) leads to n  Q  A.  l  U  i  l  (2.108)  As a result,  f fcrA  JL  _ ^L±A  2  —  ^  /  (x  +  ,  ,  x  ,  d )-ln— n  e where d + ln n  BE  «l  Q BE V  -  — (2.109)  ™BE  a—  n  ATD  BE  fiii  2kT  r  Chapter 2. Model Development  Note also that d /W n  39  = N /(N  BE  A  + N ).  A  D  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: SRH JR  2qW  BE  ["„,., =o nn xr«=o  sinh  f  Jx=-d  Ti  Z  n  2  dz dz + 2z6j -f 1  n  r " r*=*  dz  x=d  i2  T  Jx=0  2  z + 2z& + l (2.110) 2  2  where Tl  =  y/T i  ?2  =  y/ po2  , b  =  / qV \ (E -E exp ( — ^ ) cosh | — — 2kT kT  x  6  2  T  po  nol  T  T  no2 BE  u  t  'pol  h ln  t  n t  'nol.  <1 BE V  - exp  2kT  ! po2 T  c  o  s  ( ^ r ^  h  +  l  n  '  1~no2,  In Eq. (2.110), the substitution 'pol  z=  vw  e  exp  - \W  x+ a  BE  was made for the first integral, and the substitution  V no2 T  exp '  \WBE  was made for the second integral. The solution of'Eq. (2.110) is SRH  JR  —  2qW  BE  sinh  —  T\  f{bi) +  T  —g(b ) 2  2  where tan"  f(h)  =  ^ Zl+1  (z ]+2  1  0  1  -zi) /|l-^| v  Zoi+tl (Zi+Zo,)  201 + 1  b\ # 1 b\ = l  (2.111)  Chapter 2. Model Development  1-^1  g(b ) =  40  («-«na) y/ll-tjl  tan - l  3  l +«2  2  2(12  +1  *02 + *2 ( 2 2 +  Z2  Z02)  *1  1 +1  =1  anc  T  ^01  z  pol  Note that d /W p  BE  d +a n  exp(a)  Tnol  =  2  lTpo2 z  9  exp ^~  =  2l  02  —  exp(a)  T~no2  = N /{N  = 1-  D  A  + N ). D  Due to a lack of experimental data we let E — Ei (the effect of this is to produce t  a maximum recombination rate), oi =  o  2  =  T \ VO  —  r  po2  , and r \ = no  ( ^/— +  -  r  ) exp  2.  Under these conditions  2kT  'po  2.5.2  n 0  Radiative Recombination Process  Under low injection conditions, the rate by which radiative recombination exceeds thermal generation is given by [57]  U  rad  = 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) (d B, n] + d B n ) 2  n  x  p  2  i2  (2.113)  Chapter 2. Model Development  41  The subscripts 1 and 2 again denote the emitter and the base, respectively, while d  n  i  and d are the depletion-layer widths as shown in Figure 2.9. p  2.5.3  Auger Recombination Process  Two Auger recombination processes for Al Ga _ As are usually recognized [58]. The I  1  I  first process, known as the CHSH process, occurs when a conduction-band (CB) electron 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 = C np  2  p  (2.114)  where C is the Auger coefficient for the CHSH process. The second process, known v  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  R = Cnp 2  n  n  (2.115)  where C is the Auger coefficient for the CHCC process. n  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 G = C„npn n  (2.116)  Chapter  2. Model  42  Development  Similarly the generation rate for the CHSH process is proportional to the hole concentration, p, since it depends only on the number of light holes present: G = C npp p  (2.117)  p  In non-equilibrium, the net Auger recombination rate is given by U  =  (R + R )-(G  -  (C n + C p){np-np)  Aug  n  p  n  + G)  n  p  (2.118)  p  Substituting the expression np = n\ and those in Eqs. (2.99) to (2.101) into Eq. (2.118), the latter becomes U  Aug  n? (e« "/* - 1) { C exp v  =  r  n  <t>p + <l>n \  +  exp  c  D  q  ( <t>p + <t>n  exp kT \ The relationship V  (2.119)  rp  2  = <p — <f> was used in the above equation. As in Subsection 2.5.1,  BE  p  n  we assume that t/>(i) varies linearly across the forward biased emitter-base junction. Letting exp  z=  q  ,  <f> + <f>n P  kT V  exp  2  0 W  {x + a)  BE  and c — \JC C , where 6 and a are defined in Eqs. (2.106) and (2.109), and substituting n  p  Eq. (2.119) into Eq. (2.97), yields T  Aug  nf V W » r  q£  e  2qW  BE  2qW  BE  e" ^f v  z* B*l v  kT  kT  ( e  ,v„/*r  _  sinh ( ^ ) sinh(^)  1  )  e  ^  l j W B B  +  [*"  0z  3  (nfj ci nf  2  z  ci (1 + — | dz + /  (Z J 0  c  d  2  -  2 )X  (—  n c i2  - )  (z - 202) - ( — 2  Zl  — Z  Q2  2  1 + —A dz  + (2.120)  Chapter 2. Model Development  where  Ci =  43  \JC \ C , c = yJC 2 C , n  pi  2  p2  n  = ^ VvVB-  21  exp  dn  +  E  zoi = 91  2  exp(a)  V?  =  e x p  = y~  0 2  j  a  fe  d p+ a  !  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  G e n e r a t i o n 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, V c, is negative and its magnitude is at least several kTjq B  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  E — Ei, Eq. (2.102) can be approximated by t  U  * — ^ — (e« »°/ r -r r v  SRH  no  f c r  - 1)  (2.121)  po  The SRH generation current density over the entire collector-base depletion region is Jg  RH  —  —q I  U RH S  JC-B  SCR  ^nc  ~  ^pc  q ni W  —  '"no  —  dx  BC  — T  po  (2.122)  Chapter 2. Model Development  where W  BC  44  is the collector-base depletion-layer width.  In deriving E q . (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 qV /kT BC  <C 0.  Under large reverse-bias, np «C n , so the radiative generation rate as given by 2  Eq. (2.112) reduces to U d — —Bn}.  The radiative generation current density is  ra  therefore given by  J ^qBn}W ad  G  (2.123)  BC  For the Auger process, the generation current density, under the homojunction assumption, can be easily deduced from Eq. (2.120):  2qW  t* l VBC  BC  ^ M  ^ [  {  , - .  (  )  +  (i_J.)]} „ » ,  Since  NN\ A  In  \C  exp  n,  n  *2  BC  )  n?  =V ^  Zl  gV  D  C  v  Eq. (2.124) reduces to  BC  q  J  G  —  e  l V  B  O  / 2 k T  ln(^p)  +  qn W (C N n  B  c  <}\VBC\  kT  +  2  BC  W  D  I n ( ^ )  ~  > > x  2kT J  A  and 2 sinh(gV c/2itr) ~ - exp(-qV /2kT), B  wj  exp  N  p  kT  +  CN) p  A  (2.125)  kT  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, r  no  10 - 9 s, r  p0  10~ s, n, ~ 1 0 c m , C 8  6  -3  n  C  < 10- 3 0 c m / s and B 6  p  ~  10~ cm /s are 9  3  Chapter 2. Model Development  45  used. Typical values for the collector and base doping concentrations are 10  16  and  10 cm~ respectively. Comparing the Auger generation current density with the SRH 19  3  generation current density, we note that JG  +  ni{C N  U9  =  n  D  m cp N  <  C N ){T p  A  +  no  T ) po  T  A  PO  < (10 ) (1(T ) (10 ) (10~ ) 6  < 1(T  19  30  8  13  A similar comparison between the radiative and the SRH generation current densities shows Trad  G  J  rSRH  B (r +  =  ri,  <  TliB Tpo  <  (io ) (10- )(io- )  T )  no  po  G  J  9  6  < io-  8  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  =  {  T  t  n0  T T  p  n  w  ^  +  .  n  c  d  c  )  [  l  _  tq  V lk BC  T)  (  21  2  6  )  0  where n,£ and n,c are, respectively, the base and collector intrinsic carrier concentrations, 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 approximation. By restricting J to suitably low values (see Chapter 3), base widening C  due to donor neutralization by electrons entering the collector (Kirk Effect) need not  Chapter 2. Model Development  46  be considered. The collector-base voltage, V , BC  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 <C J (W). n  2.6  P a r a m e t e r s for G a A s a n d A l , ; G a i _ A s r  The various models used to describe the physical parameters of GaAs and Al Ga _ As a:  1  :r  are presented in this section. Some of these models, for physical parameters such as bandgap, mobility, dielectric constant, effective masses, and effective density of states, are taken from the material device analysis program SEDAN III [34]. Other physical parameters such as some of the recombination lifetimes are modeled based on a collection of data from the literature. In general, these physical parameters are both composition and doping dependent. Temperature is assumed to be, in most cases, at 300 K.  2.6.1  Effective D e n s i t y of States  Under the approximation of parabolic band structure, the carrier concentration is related to the respective Fermi energy through Fermi-Dirac statistics given by n  (2.127)  P  (2.128)  where Nc and Ny are the effective densities of states for conduction and valence bands respectively, Ec and Ey are the conduction and valence band edges, Ep and EF are n  V  the electron and hole Fermi energies, and Fi/ (r?) is the Fermi-Dirac integral of order 2  one half defined as (2.129)  Chapter 2. Model Development  47  In non-degenerate semiconductors, Boltzmann statistics apply, that is, when the Fermi energy is several kT below Ec in N-type material or above Ev in P-type material, the Fermi-Dirac integral function reduces to an exponential function. Since many of the earlier derivations assumed Boltzmann statistics it would be convenient to put Eqs. (2.127) and (2.128) in a form compatible with this assumption, that is n  =  N (r, )  p  =  ^  c  exp(r? ) n  (2.130)  (T7 ) exp(T? )  (2.131)  n  P  p  where EFH-EC  E  p  We call N  c  and N  V  ~ Fp  AT*(  E  N  v  M  F / {rin)  A r  1 2  V*M  F  AJ  \  n  kf  =  ,  A T t (  =  N  v  ^ M  the "new effective densities of states" for conduction and valence  v  bands respectively. In Appendix B, they are computed and related to doping concentration. Note that N  c  and N  are also functions of the effective masses since the  v  effective densites of states are given by N c  =  2  N  =  2  v  {—7$~) [  h>  '  (2 132)  )  ( 2 , 1 3 3 )  where m* and m* are the effective masses for electrons and holes respectively, and h is the Planck's constant. The new effective densities of states should replace the effective *  densities of states in all the formulations unless stated otherwise. The intrinsic carrier concentration can be computed from the pn product at thermal equilibrium under which the Fermi level is constant, i.e., Ep = Ep , hence n  m = y/np = y/NcNy where the bandgap E = Ec — Ey • g  exp{-E /2k,T) g  p  (2.134)  Chapter 2. Model Development  48  4 INDIRECT  1 X  o c  0  u -1 w -2  -3  tr/o [111]  o K T  2TT/0  [lOo]  Figure 2.10: Band structure of GaAs with the energy E plotted as a function of momentum wave vector k along the [100] and [111] directions [34]. 2.6.2  B a n d g a p a n d E l e c t r o n Affinity  The energy band structure of GaAs with the electron energy plotted as a function of momentum wave vector k is shown in Figure 2.10. The figure shows three local conduction-band minima (r6,  X$) located at k = 0 and the zone boundaries along  the two crystal momentum directions, and a global valence-band maximum (r lo8  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  3.2  I  49  I  I  T—r  I  ~\  i  i l j « 3.018.  30  T-M7 K  / -  2.8 2.6  m 2.4 ml <  2.2  o oe  2.0 -£$•1,900  e >•  — j * ! '  (T- Pp.*  x  2  *  8  . .  w z  ml 1.8 1.6 1.4 -££•1.424 t 9  0  60AS  l O.t  l l l 0.2 O J  • OS  0.4  i 0.6  i 0.7  i 0.8  MOLE FRACTION A1AS.X  i 0.9  10 A i AS  Figure 2.11: Compositional dependence of the T, X, and L interband energy gaps [34]. interband energy gap. However, in A l G a _ A s , x  1  I  the conduction band varies so that  when the A l mole fraction x > 0.45 the X conduction-band minimum at the [100] zone boundary drops below the T conduction-band minimum at the zone center.  In this  case, A l s G a ^ x A s becomes an indirect bandgap material and the bandgap is given by the X interband energy gap. Figure 2.11 shows the variation of the three interband energy gaps with composition. The equations that describe these energy gaps, in (eV), as a function of composition, are as follows [60]  E (x) T  g  =  f 1.424 + 1.247 x  0 < i < 0.45 ~  { 1.424 + 1.247i + 1.147 (i - 0.45)  2  E$(x)  =  1.708 + 0.642 x  E?(x)  =  1.900 + 0.1251 + 0.143 x  ~  (2.135)  0.45 < x < 1.0 (2.136)  2  (2.137)  Chapter 2. Model Development  50  Obviously from Figure 2.11, the bandgap of A ^ G a ^ A s , E (x), is equal to either Eg(x) g  or Ef(x), The  whichever one is smaller. 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 concentration. Casey and Stern [61] have measured the doping dependence of bandgap in p-GaAs and arrived at the following empirical expression: E  g  (eV) = 1.424 - 1.6 x 1 0 V -  /  (2.138)  3  where p is the P-type doping concentration in c m . For lack of better data, we assume - 3  this formula is applicable also to A ^ G a ^ A s of both P-type and N-type doping. The general formula for E in A^Ga^^As is therefore g  E {x,N) g  = m i n ( £ [ ( x ) , £ f (x)) - 1.6 x  10  - 8  N? 1  3  (2.139)  where TV is the net doping concentration in c m , min( ) means "the minimum of", - 3  and E is in eV. g  According to Anderson's model [7], the difference in electron affinities between GaAs and Alj;Gai_ As equals the difference of conduction bands in the respective materials, x  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 -  AE {x)  (2.140)  c  The conduction-band offset is generally assumed to be linearly proportional to the difference of energy gaps: AE  C  = Qt AE  (2.141)  g  The constant Q was determined by Dingle for x = 0.2 to be about 0.85 [62, p. 21.]. e  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 AE divides more g  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 Q  e  was indeed smaller than previously expected. The majority of recent works indicates that Q is between 0.60 and 0.67 for x < 0.45 [66]-(76|. When x > 0.45, AljGa^As e  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]:  AE  = Q AE  V  v  (2.142)  T g  where Q is a constant. However there is also evidence that the valence-band offset is v  linearly proportional to the Al mole fraction [74,79]. Assuming that Eq. (2.142) is valid and Q = 0.36 (equivalently Q = 0.64 for v  e  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 AE  =  C  AE - AE g  =  =  V  AE -Q AE] g  v  [E (x) - 1.424] -Q \E]t  V  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, E {x) = £ [ ( x ) , thus Eq. (2.143) becomes g  AE  C  = ( 1 ( x ) =  Q (1-247 x)  =  0.7981 x  1.424]  e  for 0 < x < 0.45  (2.144)  Chapter 2. Model Development  52  The formula for E (x) is given by E q . (2.135). For x > 0.45, E (x) = Ef(x)\ v  9  g  therefore  substituting Eqs. (2.135) and (2.137) into E q . (2.143) yields  AE  = 0.392 + 0.048 x - 0.27 x  C  2.6.3  for 0.45 < x < 1.0  2  (2.145)  Effective M a s s  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  =  (0.067 + 0.083 x) mo  (2.146)  m\\  =  (0.55+ 0.12 x) m  (2.147)  m*  =  (0.85 - 0 . 0 7 x) mo  T n  0  (2.148)  where mo is the electron rest mass and x is the A l 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 + n + n L  r  (2.149)  x  4  For x < 0.45, the T valley is the lowest in energy among the three conduction-band valleys, T, L, X, and E q . (2.149) can be written as  Ec\ " c e x p ( ^ £ ) EF  —  =  E IEp — Ec , , AG exp ( ^ ^ ) - I ++ 1\X N$ exn exp ,r  F  r  + N$ exp E  F  - [Ec +  kT  AE?- Y r  - {E + AEj;-*) * 1  c  kT (2.150)  Chapter 2. Model Development  where AEjf~  r  = E\ - E  v g  53  and AEf~  = E  r  -E .  x  Since the effective density of states  T  g  for the conduction band is in general given by  N  = 2  c  2nm kT\ n  (2.151)  h  2  Eq. (2.150) reduces to 2/3  K)  3 / 2  + « )  3  /  * exp ( - ^ - )  kT  + {m ) ' exp ( X  Z  2  n  A  E  ? ~ ^  x < 0.45  ]  kT  (2.152) where m  is the overall electron effective mass. For x > 0.45, the X valley is the lowest  n  in energy. Thus, the overall electron effective mass is  K)  > / !  where AE ~ T  g  x  M ^ )  exp  = E  T  exp  +  - Ef and AE ~ L  g  x  g  = E  L g  L^L)  -  1  +  2/3  x > 0.45 (2.153)  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  2.6.4  (2.154)  0  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, a i and Q as the 2  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 GaAs AlAs  where V,Vi,  ei 13.1  10.9 8.12  10.06  and V are the unit cell volumes of the alloy, AlAs, and GaAs, respectively. 2  The dielectric constant e is related to polarizability by the Clausius-Mossotti equation [82, p. 382] f— c - 11  where  4-7T 4TT  „  (2.156) = — > Nidi c + 2 3 ^ 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 E q . (2.156) but with the summation sign dropped. A similar relationship exists between e and a . 2  2  These  relationships are substituted into E q . (2.155) to give e-1  / c i - l \  NV  ,  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,  , '+ [*teO + (-*>(fm)l 2  , ,  i  2158  which applies to both high and low frequency. The values for £x (AlAs) and e (GaAs) 2  for both high and low frequencies are listed in Table 2.1 [34].  Chapter 2. Model  2.6.5  Development  55  M o b i l i t y a n d 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]  qr  M = "~  (2.159)  In III-V semiconductors, r is primarily dominated by polar optical phonon scattering which has the form [80]  T oc  /.  -v-  (2.160)  "~^(i-i) where  and e/ are the high and low frequency dielectric constants for A l G a i _ A s obx  x  tainable from E q . (2.158). Substituting E q . (2.160) into E q . (2.159) yields  oc(mT / 3  M  2  f - - - ) "  (2.161)  1  For AljjGax-jcAs, the mobility of holes is simply [80] Vp{ T,x) N  *?(*)-<r {x)  <- )  l  2  where M ,GaAa(-Wr) is the doping-dependent hole mobility in GaAs with N  T  p  162  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:  p {N ,x) n  T  = rp d  d  + (1 - r )m  (2.163)  d  where d and i refer to, respectively, direct and indirect, and r  d  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{ + rix 1  M r + TIL  l + (n + L  n )/n x  r  ^  kT  exp  '  e  x  p  f  -  ^  (2.164)  kT  In obtaining E q . (2.164), the expressions for the various electron concentrations derived in Subsection 2.6.3 were used. tron mobilities, p  The direct (GaAs) and indirect (AlAs) elec-  and Pi, can be derived from E q . (2.161).  d  For the direct valley,  the electron effective mass is simply the electron effective mass of the T valley, i.e.,  m* = mjj. The electron effective mass for the indirect valleys may be approximated ni  by (m;,) / 3  2  ~ (m£) / 3  Pi  =  2  + (m*)  < ( g  =  3/2  . The formulations are 3/2  o)  n<  \m (x = l ) ] / + [m*(z = l ) ] ' L  (2.165)  P GaA*{N ) 3  2  3  T  -1  2  n  + [m*(x)] / 3  ^(xj-ef^x)  2  (2.166)  VnMAa  where /i ,GaA«(^r) is the doping-dependent electron mobility in GaAs, and p ,AiAs n  n  is  the electron mobility of AlAs which takes on a contant value of 294 c m / V s [34] since 2  no reliable doping-dependent data are available. Due to an increase in the number of scattering centers, mobility decreases as doping concentration increases. Measured GaAs electron and hole low-field mobilities can be fitted into an empirical formula given by [83]  p(N ) T  where the parameters p , iV 0  ref  =  Mo  1+  (N /N )° T  (2.167)  re{  and a for electrons and holes are listed in Table 2.2 [34].  Chapter 2.  Model Development  57  Table 2.2: Parameters of low-field mobilities for GaAs as used in Eq. (2.167). parameters N Mo electrons 8100 1.69 x IO holes 408.7 2.75 x 10  a 0.436 0.395  re{  17 17  The carrier diffustion coefficient, in thermal equilibrium, is given by [59, pp. 172175] D  where  ^1/2(17)  =  .  kr pM_  (2  it  168)  is defined in Eq. (2.129) and = -7=  /  7  y/n Jo  7  d  x  1 + exp(x - 77)  is the Fermi-Dirac integral of order —1/2. For electrons, D = D , fi = fi , and 77 = 2  n  T} = (E — E )/kT; F  n  n  similarly for holes, D = D , p, — p , and n = n = (E — E )/kT.  c  p  For minority carriers, n  <C  p  p  v  F  0 and both Fi^r}) and F_i/2(n) approach asymptotically  to exp(n), thus Eq. (2.168) reduces to the Einstein equation kT  D = — p  (2.169)  9  For majority carriers and under degenerate conditions, Eq. (2.168) must be used instead. An approximation scheme for calculating the ratio  i i/ (")/i '_i/2(") r  7  2  is presented  in Appendix C. »  2.6.6  M i n o r i t y C a r r i e r Lifetimes  The minority carrier lifetime is an important parameter because it determines directly the amount of carrier recombination in the neutral and space charge regions of the transistor. In the present model, the effective minority carrier lifetime for either electrons 2  Note that in reference [59]  -fi/2( ?) r  a n <  i F-i/2(v)  3 X 6  defined differently than here.  Chapter 2. Model Development  58  or holes is composed of four separate lifetimes as follows: 1 —  1  =  eft  srtH  T  r  SRH  T, R  T  R  1  T  T  A  , „, (2.170)  1NT  1 1 - +  = where T ,  1 1 + —+ —+  , (2.171)  and r are the carrier lifetimes associated with the SRH, radiative, A  and Auger recombination process respectively, and T  is the carrier lifetime due to  INT  interface traps that exist as a result of lattice mismatch at the emitter-base interface. We have also defined a lifetime r due exclusively to the first three recombination 0  processes. In general, r ff is a function of both doping density and Al mole fraction. e  The SRH lifetime, r  SRH  , is derived by considering the capture and emission rates of  electrons and holes due to a single trap level, and is given, at low injection levels, by [59, p. 275]  r  TZ +  = r  SRH  po  P + Pi  Tlx  —-1 + r n +p  ,  2.172  n+p  no  where r and r , as defined in Subsection 2.5.1, are the minority carrier lifetimes po  no  in highly extrinsic N-type and P-type material respectively, n and p are the equilibrium electron and hole carrier concentrations, and n = n, exp[(E — x  Pi = n, exp[(£', — E )/kT]  t  E{)/kT],  are the electron and hole concentrations when the Fermi  t  level falls on the trap level E . Assuming Et = Ei (i.e., a deep level) and using the t  identity np — n, we obtain, for a P-type material with doping concentration N , 2  A  N + n,-  n?/JY + n,-  A  A  ,  r  SRH  -  Tpo  n  2  /  N  A  +  N  A  + T n o  n  2  /  N  A  +  N  (2.17J)  A  and for a N-type material with doping concentration No N N  Clearly when N » n , T 2  A  SRH  D  + Uj  + n /N 2  D  n /N 2  D  T n o  D  N  D  +  +m n /N  ~ r , and when N » n , T  2  D  2  no  D  SRH  ~r . po  It can be shown  using Eqs. (2.173) and (2.174) that, for GaAs, the SRH lifetime is relatively constant  Chapter 2. Model Development  for doping densities > 10  14  and T  SRH  — r  cm  59  - 3  = r  . For simplicity, we let T  srh  no  for P-type GaAs  for N-type GaAs.  po  The actual values of r  n o  and T are very process dependent. From a collection of PO  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 T ~ 2 x 10 PO  -8  s and r  no  ~ 5.5 x 10" s. These two values are assumed applicable 9  to A ^ G a i . j A s since there is not enough reliable data to show how the S R H lifetime actually varies with A l composition. Under low-level injection conditions, the radiative lifetime T is given by [57] R  =  WW)  (2175)  where B is the radiative constant as introduced in Subsection 2.5.2. If the minority carrier density is neglected, then  - TH  T a  where TV = N  A  (2176)  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 T for GaAs is plotted as a R  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  T data produces a dependence of r on TV slightly different from that predicted by R  R  Eq. (2.176). However, E q . (2.176) can still hold if we assume that the radiative constant (cm /s) has a small dependence on the doping density given as follows: 3  B{N) = 1.204706 x K T TV" 7  01 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  r-o ~  O  "°  5.5 ns  •  O  ,  ,  10"  i  i  i  i  i i i i I  i  10'  6  i  i  i  i i i 11  i  i  i  10"  N-type or P-type Doping Density (cm" )  I I I I  10  18  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 ( c m ) -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 A l composition [95, p. 205]. Since the radiative constant is related to the radiative lifetime by E q . (2.175), the following expression is obtained: P  N  Using m  n  ~ 0.067 m , m* ~ 0.48 m , E s 1.424 eV, and e ~ 13.1, for GaAs, the 0  g  0  r  following expression for B as a function of composition is produced. € (xYl* E (xY  B(x, N) = 3.0367 x H T  '-±L B(N) (2.179) L mo J where B(N) is the doping-dependent radiative constant for GaAs given by Eq. (2.177). 3  "io  For A U G a i - x A s , the radiative constant decreases with A l 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 recombination processes, C H S H or C H C C (see Subsection 2.5.3), is in effect. Since the C H S H 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 C N n  r  (2.180)  in P-type material  (2.181)  D  — ——j CN P  N-type material  m  A  Analytic expressions for the Auger coefficients, in cm /s, as a function of Al mole 6  fraction are given by Takeshima as follows [58]  C  n  C  p  =  C  =  C  exp(a„ T + bn T )  (2.182)  exp(a r + 6 T )  (2.183)  2  n0  2  p 0  p  p  Chapter 2. Model Development  63  where C  -32 _  = (1.960 - 11.361 + 31.37 i ) x 10 J  n 0  a  (cm /s) 6  = (0.8714 + 0.88 x - 6.36 x ) x 10- 2 2  n  b = (-0.03655 - 0.0638 x + 0.562 x ) x 10-4 2  n  C  -32  = (9.786 - 36.35 x + 111.6 x ) x 10" 2  p 0  a  = (1.045 - 0.408 x - 1.64 x ) x 10 2  p  (K- ) 2  (cm /s) 6  -2  IK" ) 1  b = (-0.0774 + 0.0371 x + 0.127 x ) x 10  (K- )  2  2  p  and the temperature T is set to 300 K . Takeshima's equations were made to fit theoretically calculated C and C for A ^ G a ^ A s in the A l mole fraction range 0 < x < 0.2. n  p  Uncertain as to whether Eqs. (2.182) and (2.183) are applicable to x > 0.2, we assume that under this condition C and C take on the values computed at x = 0.2. n  p  The effect of the presence of interface traps due to lattice mismatch at the emitterbase interface is to reduce the bulk lifetime in the active layer. For a single-heterojunction bipolar transistor, this may be represented by [97] T —  1  (2.184)  1 + 6/tanh(d/L)  where T is the bulk lifetime and is the same as that defined in E q . (2.171), d is the 0  thickness of the active layer, and fx = S, L/D, NT  S  being the interface recombina-  INT  tion velocity and D, L being the diffusion coefficient and diffusion length respectively. Letting L = \Ab D, E q . (2.184) can be expressed as  T  T  0  y/r D tanh(d/y/T D) 0  (2.185)  0  Comparing this to E q . (2.171), we see that  TINT  —  (2.186)  Chapter 2. Model Development  64  where d n — \/T D tanh(d/y/T D) is the effective active layer width. e  0  0  The interface recombination velocity is given by [98]  No  (2.187)  sa  where v  TH  is the average thermal velocity, N  is the interface trap density, and o  sg  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 * a\a\  2.188  2  where a\ and a are the lattice constants of Alj;Gai_j;As and GaAs respectively (a can 2  x  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,  ~ 1.6 x 10 c m  N  12  ss  -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  S  INT  The resulting S,  NT  = 1.26 x 10  2  7 C l  2  ~°  2  (cm/s)  (2.189)  is less than 1000 cm/s for the AlGaAs/GaAs heterojunction inter-  face. 2.7  H i g h F r e q u e n c y P e r f o r m a n c e of H B T s  Because of the high electron mobility of III-V materials and the typical low base resistance, the Heterojunction Bipolar Transistor, especially of N-P-N design, can potentially 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 W and x shown g  bt  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, r , using ec  the expression Sr = - i -  (2.190)  2TTT  EC  For microwave transistors, this total transit time consits of four delay times [100]: T  ec  =T  B  + T + T B  SCR  + T  c  where  r  = emitter charging time  r  = base transit time  E  B  T  SCR  r  c  —  collector space charge region transit time  — collector charging time  (2.191)  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  # 1  emitter cap  • 2  Material  Thickness  Doping  A l or In composition  (A)  (cm" )  X  x 10  0.6  3  n -In Gai_ As n -In Ga _j As  300  1  300 1000  x 10 3 x 10  +  I  I  +  a;  1  ;  1  1 9  i y  emitter grading  4  n -GaAs n-AlxGax-j-As  emitter  5  n-Al Ga _ As  1500 -  5  emitter grading  6  n-AlzGa^As  w  x 10  1 7  5  x 10  1 7  base  7  p+'AlxGai-jAs  1000  3  x 10  1 9  collector  8  n-GaAs  4000  5  9  n -GaAs  4000  3  3  collector buffer  +  I  +  x  I  500  5  W  g  g  x 10  x 10 x 10  0 . 6 - 0 linear 0  1 8  1 7  1 6  1 8  0 - 0 . 3 linear 0.3 0.3-X(,  £(,,.-0  linear  E  linear 0 0  Chapter 2. Model Development  b  ».  f t 6  '  67  b'  c  ( w V — t  c  V.'.  -0  •  O 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, r , is really a time constant representing the delay in B  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 r », the transconductance g , the emitter m  w  *  differential resistance r , and the dc common-base short-circuit gain a . The commone  0  emitter short-circuit current gain, /?, is simply the ratio of the currents i and i . Since c  g — a /r m  0  e  b  and the dc current gain f3 = a /(l - a ) > 1, the current gain, derived 0  0  0  from Figure 2.15, can be written as follows: 0  =  i  c  ~ =  H  g r /(l-a ) m t  0  1 + j wCr /(l - a ) e  0  Chapter 2. Model Development  68  Po l+jtor  (A> + 1)C  e  (2.192)  l+iwr /? C Putting u> = w and |/?| = 1 gives e  0  E  ' E  —  OJr.  ~  r C = r {C e  e  + C)  Ej  (2.193)  cj  To be more accurate, the emitter series resistance should also be included, thus the emitter charging time becomes (refer to Appendix D)  r = r {C E  e  Ej  + C ) + {REC + REX + REI) C Cj  (2.194)  Cj  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 J  where p  cE  Li  E  (2-195) E  is the specific contact resistivity of the emitter. For non-alloyed N-type  contacts using graded InGaAs, p  cE  ~ 5 x 10~ $7 cm (n = 1.5 x 10 8  2  +  19  cm ) [15]. -3  The extrinsic emitter resistance, R x, refers to the resistance of the emitter cap E  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 +  where  Pcapl2 ^ c a p H  +  Pcxxp2  W )/S cap2  E  L  E  (2.196)  Chapter 2. Model Development  69  Figure 2.16: Equivalent circuit resistances for the emitter layers and emitter-base junction.  Chapter 2. Model Development  70  1 Pcapl2  — <? J * c a p l 2  Pcap2  1  77  —  Mncapl2  Q -' cap2 Mncap2 v  and /Xncapi,  Mncapi2> Mncap2  G a A s , respectively.  are the electron mobilities of In0.eGa0.4As, Ino.3Gao.7As, and  Note that a constant value is used for the electron m o b i l i t y of  the graded layer. T h e first two electron mobilities are calculated b y linearly interpolating between the intrinsic electron m o b i l i t y of I n A s a n d G a A s (33000 c m / V s and 2  8500 c m / V s respectively) a n d assuming that the m o b i l i t y of I n G a i _ A s has the same 2  I  I  doping dependence as G a A s . T h e intrinsic emitter resistance, REI, is equal to the s u m of the resistances of the Alo.3Gao.7As intrinsic emitter (layer #5) a n d the graded layer immediately above it (layer #4), i.e., REI = [PEI WEI + PE2 {W  - X )}/S  E  E  L  E  E  (2.197)  where  1  PEI = —r  T  1  PE2  =  — 7 7  q N 2 P*nE2 E  and p, Ei, n  P-nE2 are the electron mobilities of Alo.15Gao.85As a n d Alo.3Gao.7As respec-  tively, a n d XE is the depletion-layer w i d t h i n the emitter (obtainable using E q . (2.6)). T h e emitter j u n c t i o n differential resistance, r , is defined as  dV r = dI  e  e  T h e differentiation of V  BE  E  dV  1  BE  v  BB  (2.198)  BE  dJ  EL  S  e  E  V  B  B  w i t h respective to JE can be performed from E q . (2.64) using  the m e t h o d of finite difference calculus. T h e emitter a n d collector j u n c t i o n capacitances, derived from E q . (2.9), are CE,  =  S F  (•E  '  L  ' ' " ' 1 € A B  (2.199) B  E  Chapter 2. Model Development  71  Cc,  =  (2.200) A c + 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 S D  a  r  defined in Figure 2.14.  e  C  In deriving the emitter charging time in E q . (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 and e  Co is essentially equivalent to the base transit time r  B  [38, p. 188]. By definition, the  emitter diffusion capacitance is  ° = §T  <- >  C  2  E  The excess minority charge in the base, QB, is equal to r I (W), B N  201  where / ( W ) is the n  electron diffusion current at the edge of the quasi-neutral base near the collector. For thin-base transistors, I {W) N  ^ I, E  the emitter current. Thus, E q . (2.201) becomes C ^ r  B  - ^ dV  BE  or r C e  D  ~  = ^ r  (2.202)  e  r. B  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  ~D  TB  nB  where C = -C jC 2  x  ~ e  \{ -f)^"-\r -f)Ce*»)  and C  2tW  C, r  u  2  the amount of base grading is large (f 2  nB  and WjL  nB  2  u  2  4)  2  nB  3  -°  r , t are denned in E q . (2.53). When  ~> 4/L ),  x (2kT/AE ).  expression [37], i.e., (W /2D )  (2  t  ri  E q . (2.204) reduces to Kroemer's  When there is no base grading (/ = 0)  < 1, E q . (2.204) reduces to the usual expression  W /2D . 2  nB  The above expression for r is only an estimate; it becomes inaccurate under extreme B  conditions. First, under high-current conditions, the effective base width increases due to the Kirk effect [102], causing E q . (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 r  B  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 example 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] T R = SC  (2.205)  where W c is the width of the base-collector space charge region and v„ is the saturation B  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. 321336].  In E q . (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 r  SOR  [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 10 cm/s [108]. 6  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 r = {Rec + RCB + Rei) C c  (2.206)  Cj  where Rec RCB, RCI are the resistances of the collector contacts, n  +  buffer layer,  and intrinsic n layer, respectively, and C j is given by Eq. (2.200). With reference to C  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] (2.207) where 1  qN  p w  buS  is the sheet resistance of the n  +  nbxl{  collector buffer layer,  bu{  p buf n  being the electron mobility  of the buffer layer, and p c is the collector specific contact resistivity. In general, the c  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 A u / N i / A u - G e alloyed ohmic contacts to heavily doped N-type GaAs can be as low as 1 x 1 0  -6  flcm  2  [110].  The lateral resistance of the buffer layer is given by [ i l l , p. 217] RB C  —  R'cB +  CB  Chapter 2. Model Development  75  —  Rsbuf ScD  Rsbuf &BC  ,  1  yz.zuo)  12 Lc  2 Lc  The first term in E q . (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, = " ™ ' 7 C  {  C  X  C  (2.209)  )  where 1  Pc  q N I PnC C  [i  and Xc are the electron mobility and depletion width in the intrinsic collector  nC  layer, and Nci and W i C  are the doping density and thickness of the intrinsic collector  layer. 2.7.2  M a x i m u m F r e q u e n c y 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  where fx is the cutoff frequency, Rb is the base resistance, and C  c  capacitance.  2io  is the collector  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, C product [2,101,109]. Assuming the base resistance and collector capacitance are c  distributed in the way shown in Figure 2.18, the effective R C is b  (Rt C ) ff = Cci (RBI + RBX + RBC) + Ccx c  e  c  ^Bc^j + C c RBC C  (2.211)  Chapter 2. Model Development  2RBC>  76  2R x  2R i  B  2R  B  <2R  BX  BC  BASE-  •>cc  Ccx  j  | Ccx  Ci C  | Cgc  7  1  COLLECTOR n  +  n -  Figure 2.18: Equivalent circuit resistances base-collector junction.  and capacitances for the base and  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]  RBI  —  RBX  =  R'sB $E  12 RsB 2  L  (2.212)  B  SEB  (2.213)  L  B  where  R'SB  q NB  UpB [ B W  1  RSB  qN p B B  P  {w  B  - Xc  -  B  -  x ) BC  X ) BE  Chapter 2. Model Development  77  are the sheet resistance of the intrinsic and extrinsic bases, NB is the base doping density, W is the thickness of the base layer, and p, B is the base hole mobility taken B  P  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 cm is used for the base specific contact reistivity, p £ 2  C  [112]. This value of p B for a 3 x 10 c m doped base layer is also consistent with the 19  -3  C  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, C  c  x  Ccc  =  . )  (2  2S  L  e  215  tooi^  ~ ~ y T1T~  (2.216)  =  (2.217)  EB  b  c  Note that the sum of the above three capacitances equals the total collector junction capacitance as given by Eq. (2.200). 2.7.3  Modified 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 implantdamage 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  T h e external regions of the intrinsic collector layer, that is, the hatched regions shown i n F i g u r e 2.19, can be made highly resistive or intrinsic w i t h either  proton  or oxygen i m p l a n t a t i o n [11,113]. M o r e 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  cm  1 3  - 2  [113] or a proton fluence of 5 x 1 0  12  cm  - 2  [11] would  produce an external base-collector capacitance of 0.2 f F / ^ u m . 2  W i t h reference to F i g u r e 2.19, S D  is  C  n  o  w  defined as the intrinsic n collector layer  w i d t h excluding the i m p l a n t e d isolation regions; i n this case, S D C  — &E- T h e total n  collector layer w i d t h is denoted by a new variable: SET = S  + 2 (S  E  EB  + S)  (2.218)  B  T h e 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  +  S  +  - S ^  BT  CD  (  2  2  i  9  )  T h e distributed capacitances Ccx and Ccc-, as a result of proton or oxygen i m p l a n t a t i o n , become C  c c  =  2S  C  c x  =  2 SE  (2 x 10~ F / c m )  L  B  8  B  B  L  B  (2 x 1 0  2  - 8  F/cm ) 2  (2.220) (2.221)  T h e total collector j u n c t i o n capacitance originally given by E q . (2.200) must now be calculated from the distributed collector capacitances, i.e., Ccj — Ccc + Ccx + Cci  (2.222)  For a single collector electrode structure, only the expressions for the collector contact resistance Rcc  a  n  d the collector buffer resistance Rc  B  are needed to be modified.  Chapter 2. Model Development  For R , cc  80  Eq. (2.207) is changed to =  VPcC flsbuf  c  Q  t  h  /^  / W |  V V  £c  ( 22  2  3  )  Pec /  and for RCB, Eq. (2.219) is changed to Res =  +  i  W (s*c + S  "T- <»>) S  ( 2  . 4) 2 2  Chapter 3  Results a n d 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 twodimensional 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 , and corresponding drop in e c  fx [114]. This critical current density is given by Ji where N i C  (3-1)  qNciv.  is the doping density of the intrinsic collector (5 x \0  lb  cm ) and v 3  m  is the  drift velocity in the collector space charge region under the field E , which is given by m  VW + IVBCI WCI  81  (3.2)  Chapter 3. Results and Discussion  where V  bi  82  is the base-collector junction built-in voltage (~ 1.4 V) and W i is the C  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 E giving v  = 1 x 10 V / c m . At such fields the carrier velocity will saturate, 5  m  — v ~ 10 cm/s, although velocity overshoot effects may take v 7  m  s  m  as high as  1.5 x 10 cm/s [12]. Thus, a reasonable value for J\ would appear to be 10 A / c m . 7  5  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 J  c  ~ 5 x 10 A / c m . 4  2  It is significant that a number of experimental results for A l G a A s / G a A s H B T s with Nci = 5x 10  16  cm  - 3  give no indication of base push-out occuring at collector current  densities as high as 4-5 x 10 A / c m 4  2  [9,12,115] and 1.5 x 10 A / c m 5  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 H B T s . Unfortunately, this surface recombination, because of its twodimensional nature, is very difficult to incorporate into our model. However, in newer H B T s 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  3.1  83  E m i t t e r 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  Jc  =  A  tl  ?  (e^  W  * - l ) + ^i2K r  - 1) + A  kT  2X  -l) + J  (3.3)  R  (e< °°' v  22  It is worth noting that the coefficients An and A  B c / f c r  - 1) + J  kT  (3.4)  G  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 ' However, the difference in An and A i 2  rule still holds. The  z  is constant.  is very small so that in essence the reciprocity  3  dependence of the collector current density on the base-emitter voltage for  different amounts of A l mole fraction is illustrated in Figure 3.1. The solid lines are drawn for JG = 0; these curves converge to A  at V  22  of about 3 x 10~  A/cm  18  2  BE  = 0, which has a magnitude  and is nearly indepenedent of the A l mole fraction at the  base-emitter metallurgical boundary, i j . The flat broken line at 5.4 x 1 0  A/cm  -10  e  the value of Jc; the solid lines would converge to for VBC =  —  2  is  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., A , 2X  for a larger  xi, . The saturation current density can be obtained by extrapolating the solid lines e  in Figure 3.1 to V  = 0. For examples, A  BE  A\  ~ 2 x 10  2  2l  A/cm  -24  2  ~ 3 x 10  -20  for i(, = 0.3. The variation of A  2X  e  A/cm  for i(, = 0 and  2  e  with x  be  depends, in a  complicated way, on the changes in n,o, / , and A.£7 ; however, the major contribution n  to the decrease of A  2X  with increasing  is the exponential decline of the intrinsic  If the assumption of constancy of "a" is applied in the actual calculation of ni(x), then it follows that new — " B o ^ thus, as reciprocity demands, A 2 — —A i3  e  2  ,  V  V  a n  X  2  Chapter 3. Results and Discussion  84  6  —20 -f  1  0  0.2  i  i  I  I  I  I  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 V = - 3 V (broken lines) and J = 0 (solid lines). BC  G  carrier concentration, n , , which is about 1000 times smaller for Alo.3Gao.7As than that 0  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 x  he  = 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 AE  n  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  AE  n  is to reduce -the rate at which Jc varies with VBE-, and thus the ideality factor.  For a more heterostructural junction (i.e., x  ie  = 0), the magnitude of AE  n  is larger and  is therefore a stronger function of VBE\ thus the resulting ideality factor is also larger. When x&e = 0.3, AE  n  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 different amounts of A l mole fraction is shown in Figure 3.2. At the high current end, J is essentially equal to Jc- At lower current densities, J  E  E  becomes basically indepen-  dent of A l mole fraction and changes more slowly with V BE  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 S R H 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 J  c  from J  using the  E  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) emitter, J (x ); p  ~  "^n(W)|i the hole current density back-injected into the  the Shockley-Read-Hall, radiative, and Auger recombination current  E  densities in the emitter-base depletion region denoted, respectively, by JR , RH  and Jft . Ua  JR*,  The dependence of the various base current density components on V E B  for the case of x  be  = 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 V  BE  1.2 V , the neutral base recombination current and the back-injected hole  current surpass the S R H 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 S R H recombination current at the high end of the bias range. Incidently, the neutral base recombination is due primarily to the  3. Results and  Chapter  Discussion  87  2  E  0  ^  -2-  TOTAL  Q  C Q> Q  O 1  1.3  1.6  Base-Emitter Voltage (V) Figure 3.3: Dependence of base current components on base-emitter voltage for the case of x = 0.1. be  radiative recombination process since the radiative lifetime is smaller than the Auger and S R H lifetimes.for the given base doping level of 3 x 10 The  19  cm . - 3  variation of the base current density components with A l mole fraction is  shown in Figure 3.4 for J  = 10 A / c m and in Figure 3.5 for J 3  c  2  c  — 10  4  A / c m . If the  space-charge currents were independent of base grading, one would expect JR and  J^  U9  to increase with x  increase with x  be  be  2  , J  RH  because, in order to maintain Jc constant, V  BE  T ad R  must  as well. This is not seen except for the SRH recombination current in  Chapter 3. Results and Discussion  88  100CH  1 0.1  o.oi-i 0.0  0.2  1 0.3  Al Mole Fraction Figure 3.4: Dependence J = 10 A / c m . 3  of base current components  on A l mole fraction for  2  c  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 A l mole fraction except for x  bt  > 0.2. In the latter case, the exponential dependence  on voltage of the space-charge currents causes them to eventually increase with x . be  The large reduction of the radiative and Auger recombination currents with increasing A l mole fraction for x  be  < 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 F r a c t i o n Figure 3.5: Dependence J = IO" A / c m . 4  of base current components on A l mole fraction for  2  c  with base grading. On the other hand, at low collector currents, S R H 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  19  c m . At low collector currents, S R H recombination is the dominant - 3  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 x  be  < 0.1.  causes both J  For higher x , the large base bandgap at the emitter-base interface be  r ad R  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 emitterbase junction. In fact, at high collector currents, the total base current is due mostly to this hole current for x  be  3.3  > 0.1.  D C Current Gain  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. A l l 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]  P  J J  exp(qV /kT) exp(qV /mkT) BE  c  —  ~T  =  exp  0  C  BE  B  qVjBE  kT  V  m  oc  4  1  ™)  (3.6)  where m is a constant that determines the rate of exponential increase of the recombi1 /2  nation current with bias. For S R H 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  Collector 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 x  be  For x  bt  = 0.1 and 0.2 exhibits the J ' c  dependence for Jc ~ 0.01 A / c m .  = 0.3, a similar trend for the current gain can be seen for J  2  ~ 0.001 A / c m . 2  c  S R H recombination is clearly an important process at low collector currents for the graded-base devices. density as low as 10  Since /? remains relatively constant down to a collector current -4  A/cm  2  for the non-graded base case, the base current must be  composed mostly of space-charge radiative recombination current for Jc > 10~ A / c m . 4  2  Chapter 3. Results and Discussion  We know that this is true for J  92  = 10 A / c m from Figure 3.4. For J 3  c  ~ lO  2  c  - 4  A/cm , 2  base current comes mainly from S R H recombination in the base-emitter depletion region 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 x  be  = 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{X ) E  But both J  c  and J (x ) p  E  vary as exp(qV /kT), BE  as indicated in Figure 3.6. For x  so /? is essentially independent of J  = 0, the fall in /? with increasing J  be  c  c  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 AE  n  exp(-AEJkT)  as VBE rises since current gain is proportional to  [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  ,  0  ,  20  1  40  ,  60  80  1  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 = 10 A / c m . 3  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 W and A l mole fraction. The G  collector current density is held at 10 A / c m which produces a depletion-layer width in 3  2  the emitter of 200-250 A depending on the degree of base grading. As expected, current gain increases with W but levels off at W ~ 100 A, about half of the total depletionG  layer width in the emitter.  G  The current gain is independent of junction grading for  Chapter 3. Results and Discussion  94  Xf, = 0.3 because, in this case, there is no conduction-band spike to begin with. The e  increase of current gain is about 4-5 fold when W > 100 A. g  3.4  Transit Time 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 performance. 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 external 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  RBC = 37.49 fl R  C  C  = 5.38  fl  REX = 0.13O Rx R  C  Ri  = 7.00 O  B  B  = 6.89  = 15.62 fl  B  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 V V  BE  BE  changes from 0.1 to 1.7 V . The collector capacitances, computed for  = — 3 V in the case of maximum base grading, are  C  c  c  = 9.51fF  C ,i = 3.17fF  C x = 0.48fF  c  C  totaling 13.16 fF. The remaining components, REI, r and CEJ, depend on the basee  emitter bias voltage. The intrinsic emitter resistance changes from 1.07 fl to 1.55 fl over the V  range of 0.1 to 1.7 V . The other two components vary more substantially  BE  with bias: r changes from 2.6 kfl to 1.3 fl and C' j changes from 42 fF to 107 fF in e  the V  BE  E  range of 1.55 to 1.75 V , assuming again maximum base grading. At the most  Chapter 3. Results and Discussion  100  95  100000  10000  1000  Collector Current Density (A/cm ) 2  Figure 3.8: Dependence of fr and / of base grading.  m  a  x  on collector current density for different amounts  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  emitter charging time should decrease with bias unless V  BE  B E  , the overall  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  0.1 " 100  i  i  i  i i i  — — 1000  111  96  i  i  i i  i  i 11 10000  i  i  i  i i  ii 100000  i  Collector Current Density ( A / c m ) 2  Figure 3.9: Dependence of transit time components on collector current density for x = 0.3. ht  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.,x  ie  = 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 / current density of 10 A / c m 4  2  m a x  with Jc up to a collector  is due principally to the reduction in the base-emitter  differential resistance with increasing V . BE  This results in a rapid decrease in the  Chapter 3. Results and Discussion  0.00  I  0.05  97  0.10  l  0.15  0.20  0.25  0.30  Al Mole Fraction Figure 3.10: Dependence of transit time components on A l mole fraction at base-emitter junction for J = 2 x 10 A / c m . 2  4  c  emitter charging time r as indicated in Figure 3.9. As the other time delay components e  are essentially independent of VBE, IT and thus /  m a x  reach their maximum values at the  point of the minimum value of r . The emitter charging time will eventually increase at E  higher V  BE  because of the increase in the base-emitter junction capacitance, C j. The E  cutoff frequency, fx, appears in the expression for / term.  m a x  (Eq. (2.210)) as a square root  This fact, coupled with the independence of the effective base-collector Ri, C  c  time constant on V , BE  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  improve with base grading. A plot of the  x  transit time components as a function of A l mole fraction at the base-emitter junction for Jc = 2 x 10 A / c m 4  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 T is about 6 times, corresponding roughly to a decrease from W /2DB 2  B  in  the non-graded base case to (W /2 D ) x (2kT/ AE ) in the highly graded base case (see 2  B  G  Subsecton 2.7.1). The actual increase in fx is only about 60 % because the collectorbase depletion region transit time T ,  remains high. Judging from Figure 3.10, even  SNR  a small degree of base grading (i.e., x  be  total transit time and leave T  SCR  = 0.1) is sufficient to significantly reduce the  the principal intrinsic limitation to attainment of high  fx.  Effects of Base W i d t h a n d B a s e D o p i n g o n f  3.5  T  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 f  max  against base width for Jc = 2 x 10 A / c m 4  in the case of x  2  be  are plotted  — 0 and 0.3. The rapid  decline in fx as the base widens is due to the increase in the base transit time r . Since B  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 r  B  actually leads to an increase in /  *  m  a  x  on T is slight, the reduction in base resistance  with W  CC  B  (see E q . (2.210)).  Chapter 3. Results and Discussion  10 I  1  1  1  0  1  i  •  1  1  1  1000  99  i  1  1  1  1  2000  i  1  1  1  1  i  1 1 1 1  3000  4000  J  5000  Base Width (angstrom) Figure 3.11: Dependence of f  T  and /  The effect of base doping density N  B  Here, the decline in f  T  the base, D , B  with N  m  a  x  on base width for J  c  on fx and /  4  2  is illustrated in Figure 3.12.  is mostly due to a decrease in diffusion coefficient of  B  and'correspondingly an increase in r .  density is very high (  m a x  — 2 x 10 A / c m .  B  However, when the base doping  10 c m ) , the increase in neutral base recombination, notably 20  - 3  in the non-graded base case, further raises r . B  As a result, a steeper decline in fx is  seen at very high doping densities when there is no base grading. neutral base recombination with N  B  The increase in  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 f J = 2 x 10 A / c m .  T  4  and /  m a x  on base doping concentration for  2  c  for Xb — 0.3 is due to the fact that, at J  c  e  = 2 x 10 A/cm , r is still significant 4  2  E  at the lower base doping densities, and its decline with N  B  of r with NB- In contrast to the case of / B  the diminution in fx with N  B  m a x  counteracts the increase  versus the base width in Figure 3.11,  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 N  B  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 performance of real devices. We have selected from the recent literature two papers (both from the N T T group) where experimental results for fabricated H B T s were presented. The first paper [118], published in 1985, was mainly a study of the effects of base grading 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], H B T s 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 H B T s are shown in  Table 4.1 [118]. For the current gain measurements, the H B T s used have a relatively large emitter area (48 x 48/zm ). 2  Although no other horizontal dimensions for the  fabricated H B T s 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 emitterbase junction was graded parabolically from x = x\, to x — 0.3 over a distance of 300 A e  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 H B T s 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 H B T s (Case I). Material  Layer emitter cap emitter grading emitter emitter grading base collector collector buffer  Thickness  (A) 1500 300 900 300 w 3000 5000  n -GaAs n-Al Ga _ As n-Al Ga _ As n-Al Ga _ As +  x  I  1  I  1  I  I  1  I  p -Al Ga _ As n-GaAs n -GaAs +  I  1  B  I  +  Doping (cm" )  A l composition  3  5x 5x 5x 5x 1x 5x 3x  10 10 10 10 10 10 10  X  18  0  17  0-0.3 parabolic  17  0.3  17  0.3-ij, parabolic e  19  Xbe-0 linear  16  0 0  18  Table 4.2: Structural parameters for the base layer (Case I).  W  B  (A)  Xbe  E  BF  (kV/cm)  H B T Set #1  H B T Set #2  1000 0, 0.025, 0.067, 0.1 0, 3, 8, 12  1000, 1500, 2200, 3000 0.067, 0.1, 0.147, 0.2 8  on current gain. The first set of H B T s 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 H B T s has both different base widths and degrees of base grading but a constant built-in field of 8 k V / c m . 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 Figure 4.2.  -2  A are shown, respectively, in Figure 4.1 and  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  200-  O  1 5 0  O  o  100-  50-  5  10  15  Base Built-in Field (kV/cm) Figure 4.1: Dependence of experimental and calculated current gain on base built-in field for / = 6x 10" A and a base thickness of 1000 A. 2  c  current gain values remain almost the same for at least two orders of magnitude about  l  c  = 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  oH  1——  0  1  104  1  1000 2000 3000 Base Thickness (angstrom)  1  4000  Figure 4.2: Dependence of experimental and calculated current gain on base thickness for I = 6 x 10 A and a base built-in field of 8 kV/cm. _J  c  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) — «/ (W)|, constitutes a significant portion of n  Chapter 4. Comparison with Experimental Data  105  the total base current for A l 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\, . Therefore, t  in Figure 4.1, the increase in current gain for built-in field from 0 k V / c m to 3 k V / c m (corresponding to x  = 0 to x  be  ie  = 0.025) is due mostly likely to a reduction of emitter-  base junction radiative recombination. In Table 4.2, we also see that the H B T s with the two largest base thickness have a relatively large x  be  (•> 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 x  = 0.2 were used to  be  give a built-in field of 8 k V / c m .  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 pm ), 2  LB — L  C  SB = Sc = 5 /zm,  = 10 /urn, S B — 0.5 pm, and SBC = 1 M - The meaning of these dimensions m  E  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 p E ~ p c C  c  — 1 x 10~ fl c m (110). For the base  ohmic contact, we arbitrarily chose pcB = 5 x 1 0 would not be affected by it.  6  -5  2  f l c m , but the calculation of fx 2  Chapter 4. Comparison with Experimental Data  106  Figure 4.3: Dependence of cutoff frequency on collector current for V  CE  = 2 V.  The collector current dependence of cutoff frequency for V E = 2 V is shown in C  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 I  C  region and converge to around 20 GHz in the high I  C  region. The observed  cutoff frequency of 20 G H z corresponds to a emitter-to-collector transit time, r , of ec  8 ps. From our program, we also found that the base transit time r , is about 3.4 ps, B  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 (W = 3000 A) B  and, even with a base-grading parameter x\, as high as 0.2, the resulting built-in field t  is only 8 kV/cm. In Table 4.2, an E^ of 8 kV/cm corresponds to, for W = 1000 A, a B  base-grading parameter Xj, of only 0.067. Another reason for the relatively large r is e  B  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  C a s e II:  H i g h F r e q u e n c y 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 H B T is illustrated schematically in Figure 4.4. The S i 0  2  Chapter 4. Comparison with Experimental Data  108  Table 4.3: Epitaxial layer structure parameters for fabricated H B T s (Case II). Layer  Material  emitter cap  Thickness  Doping  A l composition  (A)  (cm" )  x  n -GaAs  2000  +  3  5 x 10  18  0 0-0.3 parabolic 0.3  emitter grading  n-Al Ga _ As  300  5 x 10  17  emitter  n-AljGai-rAs  900  5 x 10  17  emitter grading  n-AljGax-jjAs  300  5 x 10  17  0.3-0.1 parabolic  p -Al Gai_ As  1000  4 x 10  19  0.1-0 linear  n-GaAs  6000 5000  5 x 10 3 x 10  16  0 0  base  I  1  I  +  I  collector  a ;  n -GaAs  collector buffer  +  18  Table 4.4: Fabricated device dimensions (Case II). Ac  ABB  A-BCJA-EB  B  PEB/AEB  (Mm )  (Mm )  Tr 1 Tr 2 Tr 3  1 x 10 2x5 2 x 10  3 x 12 4x7 4 x 12  3.60 2.80 2.40  2.2  Tr 4  5 x 10  7 x 12  1.68  0.6  2  (Mm" )  2  1  1.4 1.2  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 S B  w  E  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 H B T s 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 /  T  (GHz)  (Case II).  Calculated  Experimental (Implanted)  f  m a x  (Implanted)  / « (GHz) m  h  (GHz)  (GHz)  /m«  ( G H Z T  IT  50  41.0 41.2  89.0 67.2 67.5  39.6 39.9  40  42  39.3  39.9  38.4  35 45  Tr 3 Tr 4  /max  39.2  70 70 70  Tr 1 Tr 2  41.2  (Unimplanted) (GHz)  72.0 58.9 59.1 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 1 0 contact resistivity was given in the original paper as 2.5 x 10~ Measured and calculated fx and /  m  a  x  6  - 6  f l c m . The base 2  flcm .  values at VCE = 2 V and J  2  c  = 4 x 10  4  A/cm  2  for the four H B T s are shown in Table 4.5. Also included in Table 4.5 are the calculated fx and /  m  a  x  values for the unimplanted H B T s . The measured fx and /  m  a  x  values for  the proton-implanted H B T s 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 H B T s are more or less independent of device size. A n analysis of the calculated transit times shows that only the collector charging time, r , changes substantially among the H B T s , c  ranging from r = 0.17 ps for transistor Tr 1 to r c  increase in r  c  c  = 0.43 ps for transistor Tr 4. The  with device size is due mainly to an increase in the collector junction  capacitance. However, since T amounts to less than 15 % of the total transit time, Q  Chapter 4. Comparison with Experimental  Data  110  the resulting fx values change very little. T h e calculated /  n i a x  values for the proton-  i m p l a n t e d H B T s m a t c h very closely to the measured ones for transistors T r 2, T r 3 a n d T r 4.  T h e low /  m a x  value for transistor T r 4 is due understandably to a large  collector j u n c t i o n capacitance resulting from a large base-collector j u n c t i o n area and a large base spreading resistance resulting from a large base current p a t h . Since the cross section for base charge flow is twice as large for transistor T r 3 as for transistor T r 2, the base spreading resistance of transistor T r 3 is half that of transistor T r 2. O n the other h a n d , the collector j u n c t i o n capacitance of transistor T r 3 is about twice that of transistor T r 2 because transistor T r 3 has about twice the base-collector j u n c t i o n area of that of transistor T r 2. A s a result, b o t h transistors T r 2 a n d T r 3 have about the same effective R/, C product and thus / c  m a x  - U s i n g the same arguments as above,  transistor T r 1 should produce a significantly larger / as indicated by the calculated / seen in the measured /  m a x  m a x  m a x  t h a n the other transistors,  values i n Table 4.5. It is unclear w h y this is not  of transistor T r 1. T h e improvement gained i n /  result of p r o t o n i m p l a n t a t i o n is obvious when comparing the calculated /  m a x  m a x  as a  values  for the i m p l a n t e d and u m i m p l a n t e d cases i n T a b l e 4.5. T h e improvement seen i n /  m a x  increases as the collector to emitter j u n c t i o n areas ratio increases. T h i s is reflective of the fact t h a t a larger collector to emitter j u n c t i o n area ratio means a larger p o r t i o n of the base-collector j u n c t i o n area m a y be subjected to p r o t o n i m p l a n t a t i o n . T h e dependence of fx on the collector-emitter voltage VCE at Jc — 4 x 10 for a p r o t o n - i m p l a n t e d 2 pm x 5  4  A/cm  2  emitter H B T (transistor T r 2) is shown i n F i g -  ure 4.5. T h e solid line indicates the calculated fx a n d the open circles correspond to the experimental values. A s i n the case of transistor T r 2 i n Table 4.5, the calculated fx values shown i n F i g u r e 4.5 are consistently lower t h a n the measured values by ^ 4 G H z . B o t h the measured and the calculated fx decrease w i t h increasing V E, i m p l y i n g that C  Chapter 4. Comparison with Experimental Data  111  50 calculated O experimental  N  X  CD  >^  o c  3  40  cr  3 O  30  H Collector-Emitter Voltage (V)  Figure 4.5: Dependence of f  T  on collector-emitter voltage for J  c  = 4 x 10  4  A/cm .  the collector space charge region transit time presents a strong influence on / j .  2  Chapter 5  Summary  5.1  Conclusions  In this thesis the current gain characteristics of the graded-base A l G a A s / G a A s 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-ReadHall, radiative and Auger recombinations in the neutral base region as well as in the emitter-base depletion region. But, as the A l 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 f  T  and /  m  a  x  by reducing the base transit  time. In the device considered here, the use of maximum base grading, i.e., the A l mole fraction at the base-emitter junction being 0.3, increases f 60 % and /  m a x  T  by about  by about 20 %. If the transistor is operated at a sufficiently high 112  Chapter 5. Summary  current level (J  113  ^ 2 x 10 A / c m ) , an increase in fr of about 30 % and in / 4  c  2  m  a  x  of about 15 % can be obtained utilizing only an A l 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 r  D  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  C o n s i d e r a t i o n s for F u t u r e W o r k  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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[87] C . J . Hwang, "Doping dependence of hole lifetime in n-type GaAs," J. Appl.  Phys., vol. 42, pp. 4408-4413, Oct. 1971.  References  126  [88] K . L . Ashley and J . R. Biard, "Optical microprobe response of GaAs diodes,"  IEEE Trans. Electron Devices, vol. ED-14, pp. 429-432, Aug. 1967. [89] G . A . Acket, W. Nijamn, and H . 't Lam, "Electron lifetime and diffusion constant in germanium-doped gallium arsenide," J. Appl. Phys., vol. 45, pp. 3033-3040, July 1974. [90] R. J . Nelson and R. G . Sobers, "Minority-carrier lifetime and internal quantum efficiency of surface-free GaAs," J. Appl. Phys., vol. 49, pp. 6103-6108, Dec. 1978. [91] F . Stern, "Calculated spectral dependence of gain in excited GaAs," J. Appl.  Phys., vol. 47, pp. 5382-5386, Dec. 1976. [92] H . Namizaki, H . K a n , M . Ishii, and A . Ito, "Current dependence of spontaneous carrier lifetime in GaAs-Gax-zAljAs double-heterostructure lasers," Appl. Phys.  Lett., vol. 24, pp. 486-487, May 1974. [93] C . B . Su and R. Olshansky, "Carrier lifetime measurement for determination of recombination rates and doping level of III-V semiconductor light sources," Appl.  Phys. Lett, vol. 41, pp. 833-835, Nov. 1982. [94] G . B . Scott, G . Duggan, and P. Dawson, "A photoluminescence study of beryllium-doped GaAs grown by molecular beam epitaxy," J. Appl. Phys., vol. 52, pp. 6888-6894, Nov. 1981. [95] T . S. Moss, G . J . Burrel, and B. Ellis, Semiconductor Optoelectronics. London, England: Butterworth, 1973. [96] M . H . Pilkuhn, "Light emitting diodes," in T . S. Moss, E d . , Handbook on Semi-  conductors: Volume 4- Amsterdam: North-Holland Publishing Co., 1981.  References  127  [97] M . Ettenberg, C . J . Nuese, and G . H . Olsen, "Interfacial recombination velocity determination in Ino.5Gao.5P/GaAs," J. Appl. Phys., vol. 49, pp. 1288-1292, Mar. 1977. [98] R. J . Nelson, "Interfacial recombination in GaAlAs-GaAs heterostructures," J.  Vac. Sci. Technol., vol. 15, pp. 1475-1477, July/Aug. 1978. [99] H . Kressel, "The application of heterojunction structures to optical devices," J.  Electron. Mater., vol. 4, pp. 1081-1141, Oct. 1975. [100] H . F . Cooke, "Microwave transistors: theory and design," Proc. IEEE, vol. 59, pp. 1163-1181, Aug. 1971. [101] D . A . Sutherland and P. D . Dapkus, "Optimizing n-p-n and p-n-p heterojunction bipolar transistors for speed," IEEE Trans. Electron Devices, vol. ED-34, pp. 367377, Feb. 1987. [102] C . T . Kirk, "A theory of transistor cut-off frequency (/r) fall-off at high current density," IEEE Trans. Electron Devices, vol. E D - 9 , pp. 164-174, Mar. 1962. [103] C . M . Maziar and M . S. Lundstrom, "On the estimation of base transit time in A l G a A s / G a A s bipolar transistors," IEEE Electron Device Lett., vol. E D L - 8 , pp. 90-92, Mar. 1987. [104] I. E . Getreu', Modeling the Bipolar Transistor. Amsterdam: Elsevier Scientific Publishing Co., 1978. [105] R. L . Pritchard, Electrical Characteristics of Transistors. New York: McGrawHill, 1967.  References  128  [106] Y . Yamauchi and T . Ishibashi, "Electron velocity overshoot in the collector depletion layer of A l G a A s / G a A s H B T s , " IEEE Electron Device Lett, vol. E D L - 7 , pp. 655-657, Dec. 1986. [107] C . M . Maziar, M . E . Klausmeier-Brown, and M . S. Lundstrom, "A proposed structure for collector transit-time reduction in A l G a A s / G a A s bipolar transistors," IEEE Electron Device Lett., vol. E D L - 7 , pp. 483-485, Aug. 1986. [108] P. M . Smith, M . Inoue, and J . Frey, "Electron velocity in Si and GaAs at very high electric fields," Appl. Phys. Lett., vol. 37, pp. 797-798, Nov. 1980. [109] S. S. Tan and A . G . Milnes, "Consideration of the frequency performance potential of GaAs homojunction and heterojunction n-p-n transistors," IEEE Trans.  Electron Devices, vol. ED-30, pp. 1289-1294, Oct. 1983. [110] T . S. Kuan, P. E . Batson, T . N . Jackson, H . Rupprecht, and E . L . Wilkie, "Electron microscope studies of an alloyed A u / N i / A u - G e ohmic contact to GaAs," J.  Appl. Phys., vol. 54, pp. 6952-6957, Dec. 1983. [Ill] A . B . Phillips, Transistor Engineering and Introduction  to Integrated Circuits.  New York: McGraw-Hill, 1962. [112] M . F . Chang, P. M . Asbeck, D . L . Miller, and K . C . Wang, "GaAs/(GaAl)As heterojunction bipolar transistors using a self-aligned substitutional emitter process," IEEE Electron Device Lett., vol. E D L - 7 , pp. 8-10, Jan. 1986. [113] P. M . Asbeck, D . L . Miller, R. J . Anderson, and F . H . Eisen, "GaAs/(Ga,Al)As heterojunction bipolar transistors with buried oxygen-implanted isolation layers,"  IEEE Electron Device Lett., vol. E D L - 5 , pp. 310-312, Aug. 1984.  References  129  [114] H . C. Poon, H . K . Gummel, and D. L . Scharfetter, "High injection in epitaxial transistors," IEEE Trans. Electron Devices, vol. ED-16, pp. 455-457, May 1969. [115] M . Madihian, K . Honjo, H . Toyoshima, and S. Kumashiro, "The design, fabrication and characterization of a novel electrode structure self-aligned H B T with a cutoff frequency of 45 GHz," IEEE Trans. Electron Devices, vol. ED-34, pp. 14191428, July 1987. [116] C. H . Henry, R. A . Logan, and F . R. Merritt, "The effect of surface recombination on current in A L . G a i _ A s heterojunctions," J. Appl. Phys., vol. 49, pp. 3530x  3542, June 1978. [117] S. Tiwari, S. L . Wright, and A . W . Kleinsasser, "Transport and related properties of (Ga,Al)As/GaAs double heterostructure bipolar junction transistors,"  IEEE  Trans. Electron Devices, vol. ED-34, pp. 185-198, Feb. 1987. [118] H . Ito and T . Ishibashi, "Effects of the graded band-gap base structure on current gain in A l G a A s / G a A s HBTs," Proc. 12th Int. Symp. GaAs and Related  Compounds, Sept. 1985, pp. 607-612. [119] J . S. Blakemore, Semiconductor Statistics. New York: Pergamon Press, 1962, Appendix C. [120] W . B . Joyce and R. W . Dixon, "Analytic approximations for the Fermi energy of an ideal Fermi gas," Appl. Phys. Lett., vol. 31, pp. 354-356, Sept. 1977.  Appendix A  C o d i n g Scheme for the T u n n e l i n g F a c t o r  Since the barrier transparency (Eq. (2.96)) is expressed as a function of the dimensionless variable X — E/EJI,  a similar change of variable for the tunneling factor of  Eq. (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)) where g(X) = I(X) + II(X). Letting a = E /kT  and r = E*/E ,  T1  l n = 1+  kT  exp  (A.2)  T1  ag{X) -aX  E q . (A.l) becomes  dX  (A.3)  oo  A further simplification can be made by letting  & = — = —— E 100 hq 00  VN  { D  (A 4) ' '  Since ti and ND are usually expressed in cm units, the factor 1/100 in E q . (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)  Since  ETI = qV  Tl  = q  'qN  130  D  + X- 1}} dX  (A.5)  Appendix A. Coding Scheme for the Tunneling Factor  131  and  E* =qV T1  AE  C  qNpdj = q y 2ci  -  AE  C  the lower integration limit r may be written as 2  n  dn  (A.6) 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 W  G  > d , or if r > 1. When AEc < 0, n  tunneling is not possible because no conduction-band spike exists, and junction grading should not affect the barrier energy q V . At the end of Section 2.3, it was stated that r i  if AEc < 0, V  T1  should be replaced by Vj-i- This is no longer necessary as r will be  greater than one according to E q . (A.6) and consequently the tunneling factor will be equal to one.  Appendix B  N e w Effective Densities of States  It was defined in Eqs. (2.127)-(2.128) and Eqs. (2.130)-(2.13l) that  n  =  P =  N F {r, )  = N  N  = N  c  l/2  n  F ( )  v  1/2  Vp  c  v  exp(r? )  (B.l)  exp( )  (B.2)  n  Vp  The new effective densities of states can be rearranged so that they are functions of carrier concentrations only:  N  °  K  =  IxpJ^)  =  explF-^n/Nc))  (  R  3  )  =  e^foj  =  exp[Fj (p/N )}  {  B  A  )  2  v  For a N-type semiconductor, N£ is computed from E q . (B.3) using n c± Np, and N  v  ~ Ny.  The latter approximation is made because for a N-type semiconductor  r/ <c 0 and Fi/ (rj ) p  2  ~ exp(r7 ). Similarly, for a P-type semiconductor, N  p  p  from E q . (B.4) using p  N,  In order to calculate N integral function, F^\(z). Fy\(z)  and N  A  c  and Ny,  c  v  is computed  z± N C  we need to evaluate the inverse Fermi-Dirac  A n inexpensive and yet quite accurate method of computing  is the use of two approximations to cover two wide ranges of z. Blakemore [119]  gave the following asymptotic expression of 4  F {r,) 1/2  = J J =  7 T  /  2  \  F\f (rj) 2  3  ( n + —J 2  132  /  for large  77:  4  for 77 > 5  (B.5)  Appendix  B.  New Effective  Densities  At rj = 5, E q . (B.5) gives Fi/  2  decreases as r\ increases. Since  of States  133  ^ 8.82 and a relative error of about 0.25 % which  Fi/ (r7) 2  is an increasing function, the following approx-  imation for Fy} (z) can be made: 2  \M~4  For z =  FI/ (T7) < 9, 2  )  for^>9  "T  (B.6)  the following Joyce-Dixon series approximation n = ln(z) + az + b z + cz 2  + dz  s  4  [120]  is used:  (B.7)  where  a  ~  0.35355391  6  ~  -4.9500897 x 10~  c  ~  1.483857 x  d  ~  -4.42568 X I O  IO  3  - 4  - 6  Our calculations indicate that E q . (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/N  c  ~ N /N  P-type semiconductors, respectively.  D  C  and z = p/N  v  ~ N /N A  V  for N-type and  Appendix C  F e r m i - D i r a c Integral R a t i o Fi/ (y)/F-  1/2(7?)  2  One of the properties of Fermi-Dirac integrals is that  Fi-M  = ^jFi(ri)  or more specifically, F_i/ (»7) = dFi/ (rj)/dri. 2  2  (CI)  From E q . (B.7) of Appendix B , one can  infer that  + a + 2bz + 3cz  F_ (r?) = 1/2  where z = Fi/ {n). 2  + 4dz ^J  (C.2)  3  Thus, the Fermi-Dirac integral ratio is given by  5 and z ^ Np/Nc,  2  1/2  ^l = l + az + 2bz + 3 c z + 4dz 2  3  (C.3)  4  N^/Ny for N-type and P-type semiconductors, respectively. E q . (C.3)  is valid for rj < 4 where the relative error is less than 0.1 %. For 17 < —10, both Fi/ (n) 2  and F_i/ (r/) approach exp(r/) and therefore Fi/ (r])/F_x/ (r)) 2  2  2  ~ 1.  For n > 4, an asymptotic expression for F_i/2(»7) exists [34]:  F  _  1 / t (  „  +  =  which has a relative error less than 0.7 %. In this range of rf,  134  ( c  .  4 )  Appendix D  D e r i v a t i o n of T r a n s i t T i m e Delays f r o m the Hybrid-7r E q u i v a l e n t C i r c u i t  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 version 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 resistances. From Figure D . l , *i  =  l  2  +  *3  + C )V , ,+juC j[VVe'-Ve>e>)  =  jU){C  =  -{i R  -  -ic [RE + RC) = -{g  D  Ej  b e  (D.l)  C  and V,, c e  •  e  +  E  i Rc) c  m  -9m  V W {RE  V W - t ) {RE s  + Rc)  + Rc)  (D.2)  E q . (D.2) is obtained by applying the approximations: i ~ i and t <C i . Substituting e  c  3  c  Eq. (D.2) into E q . (D.l) yields ti = 3 w VW {{C  D  + C ) Ei  + [1 + g 135  m  (R  E  + Rc)} C ) Cj  (D.3)  Appendix D. Derivation of Transit Time Delays from the Hybrid-ir Equivalent  b  t'»  r  w  b'  Circuitl36  * i  rwvvv—?—*—r  Figure D . l : Hybrid-TT circuit model of the transistor with short-circuited emitter and collector terminals. Since the base current is given by 1 ib = ii + *4 = »i + Vib't'  -a  0  (D.4)  and the collector current by ie = 9m Vav - t ~ g V < , 3  the current gain can be expressed as  m  b  e  (D.5)  Appendix D. Derivation of Transit Time Delays from the Hybrid-it Equivalent Circuitl37  Here, the identities g  m  — a /r 0  and /? = c*o/(l ~ <*o) were used. Finally, substituting  e  0  Eq. (D.3) into Eq. (D.6), we have /? =  ,  •  1 + J w A>  + C) + %C  (CD  Ei  Cj  +  T  (RE  The frequency where |/?| = 1 is the cutoff frequency, f , T  Eq. (D.7) by noting that a  + Rc) Ccj]  (D.7)  and can by obtained from  ^ 1 and that the second term in the denominator of  0  Eq. (D.7) is much greater than one. Thus, 1  _  1 2 7T  U)j>  ff  =  r (C j + C ) + R C  =  T  e  E  Cj  + r + r B  e  c  E  Cj  + rC e  D  + Rc C  Cj  (D.8) (D.9)  On the right hand side of E q . (D.8), the sum of the first two terms corresponds to the emitter charging time r , the third term corresponds to the base transit time r , B  D  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.  

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