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Modeling of double heterojunction bipolar transistors Ang, Oon Sim 1990

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MODELING OF DOUBLE HETEROJUNCTION BIPOLAR TRANSISTORS Oon Sim Ang B. A . Sc. (Hons.) University of British Columbia A THESIS 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 OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A July 1990 © Oon Sim Ang, 1990 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 E L E C T R I C A L E N G I N E E R I N G The University of British Columbia Vancouver, Canada r w Q 3 AUGUST 1 9 9 0 DE-6 (2/88) Abstract A one-dimensional analytical model in the Ebers-Moll formulation of a graded base dou-ble heterojunction bipolar transistor (DHBT) is developed and used to examine the effects of base grading, the emitter-base barrier and the base-collector barrier on the d.c. current gain, offset voltage and the high frequency performance of a N — A\xGa,i-xAs/p — A l ^ G a j . ^ A s ^ — A l x G a x _ r A s DHBTs. Recombination processes considered in the space charge regions and the neutral regions are: Shockley-Read-Hall, radiative and Auger. The trade-off between base-grading, which reduces the base current, and the neutral base recombination, which is brought about by varying the aluminium the junctions, results in an optimum aluminium mole fraction profile regarding the d.c. current gain. For high frequency performance, a similar trade-off to that of the d.c. sit-uation exists. In this case, the important manifestation of the increased collector-base barrier height is an increase in the base transit time. The aluminium mole fraction profile which optimises the unity gain cut-off frequency, fx, and the unity power gain cut-off frequency, / m a x ? is established. DHBTs which are symmetrical, both in aluminium mole fraction and doping concentration profiles, are shown to have low common-emitter offset voltages, Vce<0ffset . Base-grading reduces Vcefifjset in devices in which the difference between the emitter and collector aluminium mole fraction is < 0.1; otherwise, Ke,o//set increases as base-grading increases. The model is also used to examine the performance of a N - A l I G a 1 _ x A s / p - I n y G a 1 _ y A s / N - A l x G a 1 _ ; r A s D H B T . It is shown that radiative and Auger recombination limit the d.c. current gain in this device. n Table of Contents Abstract ii List of Tables vii List of Figures xii Acknowledgement xiii 1 Introduction 1 1.1 Background 1 1.2 Overview 3 2 Modeling of the Double Heterojunction Bipolar Transistor 6 2.1 Heterojunction Basics 6 2.1.1 Depletion Layer Width 6 2.1.2 Junction Capacitance 8 2.1.3 Built-in Voltage 9 2.1.4 Junction Grading 10 • 2.1.5 Current Transport at the N-p Heterojunction 12 2.2 Double Heterojunction Bipolar Transistor 14 2.2.1 Emitter-Base Junction Current 15 2.2.2 Emitter Hole Current 16 2.2.3 Base Electron Current 16 2.2.4 Base-Collector Junction Current 18 iii 2.2.5 Ebers-Moll Equations for D H B T 18 2.3 Reduction of D H B T Ebers-Moll Equations to SHBT Equations 22 2.4 Recombination/Generation Current 24 3 D . C . Results and Discussion 20 3.1 D H B T Currents and Gain 32 3.1.1 Effect of varying the collector A l content; GaAs base 33 3.1.2 Effect of varying the collector A l content; 10% graded base . . . . 34 3.1.3 Effect of base-grading; large base-collector barrier 37 3.1.4 Effect of varying the base-collector barrier 39 3.1.5 Symmetrical D H B T ; varying the ungraded base A l content . . . . 41 3.1.6 D.C. Current Gain 43 3.2 Offset Voltage 46 4 High Frequency Results and Discussion 53 4.1 Cutoff Frequency 53 4.2 The Maximum Frequency of Oscillation 56 4.3 Results and Discussion 57 4.3.1 Effect of varying the collector A l content; GaAs base 57 4.3.2 Effect of varying the collector A l content; 10% graded base . . . . 59 4.3.3 Effect of varying the base-collector barrier 61 4.3.4 Effect of base-grading; large base-collect or barrier 62 4.3.5 Symmetrical D H B T ; varying the ungraded base A l content . . . . 64 4.4 Contour Plots of Ft and Fmax 64 5 InGaAs D H B T 71 5.1 Lattice Mismatch Structures 72 iv 5.2 Material Parameters for InGaAs 73 5.2.1 Lattice Constant 73 5.2.2 Dielectric Constant 73 5.2.3 Bandgap and Electron Affinity 74 5.2.4 Effective Mass of Electrons and Holes 75 5.2.5 Mobility 75 5.3 D .C. Results and Discussion 76 5.4 High Frequency Results and Discussion 84 6 Summary 86 6.1 Conclusions 86 6.2 Future Work S7 References 89 Appendices 95 A Material Parameters for Al x Ga 1 _ ; r As and GaAs 95 A . l Bandgap and Electron Affinity 95 A.2 Effective Electron and Hole Mass 96 A.3 Dielectric Constant 96 A.4 Mobility and Diffusion Coefficient 97 A . 5 Minority Carrier Lifetimes 98 B Comparison of 1-Dimensional and 2-Dimensional H B T Modeling Re-sults. 101 B. l Introduction 101 B.2 Results and Discussion 104 B.2.1 D.C. Results and Discussion 104 B.2.2 High Frequency Results and Discussion 109 B.3 Conclusion 112 vi List of Tables 3.1 Parameters for the single-sided D H B T 30 4.1 TB as computed from the full model and from the proposed equation (4.17) 59 5.1 Layer structure of devices used by Ramberg et al. [9] 81 vii Lis t of Figures 2.1 Energy-band diagram of an ideal abrupt N-p heterojunction in forward bias. 7 2.2 Energy-band diagram of a graded N-p heterojunction 11 2.3 Current components in a forward biased N-p heterojunction 13 2.4 Energy band diagram of a double heterojunction bipolar transistor with wide gap emitter and collector in active mode of operation 15 2.5 Schematic of charge flow in D H B T 20 3.1 Single-Sided Heterojunction Bipolar Transistor 30 3.2 Pyramid Heterojunction Bipolar Transistor 31 3.3 Variation of aluminium mole fraction in a device for xe = 0.3, Xbe = 0.2, Xbc = 0.1 and xc = 0.2 31 3.4 Base and collector current density of devices #300i, i = 0,1,2,3, with VBC = —3 volts 33 3.5 Base recombination current components of devices #300i, i = 0,1,2,3, with VBC = - 3 volts and J c = 5 x 104 A / c m 2 34 3.6 Gain of devices #300z, i = 0,1,2,3, with VBC = —3 volts 35 3.7 Base and collector current density of devices #310i, i = 0,1,2,3, with VBC = — 3 volts 35 3.8 Gain of devices #310i, i = 0,1,2,3, with VBC = —3 volts 36 3.9 Base and collector current density of devices #3z03, i = 0,1,2,3, with VBC — —3 volts 37 viii 3.10 Base recombination current components of devices #3i03, i = 0,1,2,3, with V B C = - 3 volts and J c = 5 x 104 A / c m 2 38 3.11 Gain of devices #3i03, i = 0,1,2,3, with VBc = —3 volts 39 3.12 Base and collector current density of devices #32z'3, i = 0,1,2 with VBC — - 3 volts 40 3.13 Gain of devices #32i3, i = 0,1,2, with VBc = - 3 volts 40 3.14 Base and collector current density of devices #3n3, i = 0,1,2, with VBC — - 3 volts 41 3.15 Gain of devices #3n3, % = 0,1,2, with V B C = - 3 volts 42 3.16 Base recombination current components of devices #3ii3, i = 0,1,2, with VBc = - 3 volts 42 3.17 Contour plots of current gain with xe — 0.3 and xc = 0.0 43 3.18 Contour plots of current gain with xe = 0.3 and xc = 0.1 44 3.19 Contour plots of current gain with xe = 0.3 and xc = 0.2 44 3.20 Contour plots of current gain with xe = 0.3 and xc = 0.3 45 3.21 Common-emitter current-voltage characteristics 46 3.22 V c e i 0 f f 3 e t for xe = 0.3 and xc = 0.3 with collector doping of 5 x 1 0 1 6 c m - 3 . 48 3.23 Vce,offset f ° r x e — 0-3 and xc — 0.3 with collector doping of 5 x 10 1 7cm 3 . 49 3.24 Contour plot of V c e t 0 f f s e t for xe = 0.3 and xc = 0.3 with collector doping of 5 x 10 1 7 cm" 3 49 3.25 Contour plot of V c e f i j j s e t for xe = 0.3 and xc = 0.2 with collector doping of 5 x 10 1 7 cm- 3 50 3.26 Contour plot of V c e > 0 j j a e t for xe = 0.3 and xc = 0.1 with collector doping of 5 x 10 1 7 cm- 3 51 3.27 Vce^fjaet for x e = 0.3 and xc = 0.0 with collector doping of 5 x 10 1 6 and 5 x 10 1 7 cm" 3 ; x = xbe for the SHBT and x = xbe = xbc for D H B T 51 ix 4.1 Emitter series resistance 54 4.2 Equivalent circuit resistances and capacitances of D H B T 56 4.3 fx and fmax of devices #300i; i = 0,1,2,3 58 4.4 Transit time components for devices #300?; i = 0,1,2,3 58 4.5 fx and fmax of devices #310i; i = 0,1,2,3 60 4.6 Transit time components for devices #310i; i = 0,1,2,3 60 4.7 fx and fmax of devices #3i03; i = 0,1,2,3 61 4.8 Transit time components for devices #3i03; i = 0,1,2,3 62 4.9 fx and fmax of devices #32i3; i = 0,1,2 63 4.10 Transit time components for devices #32i3; i — 0,1,2 63 4.11 fx and fmax of devices #3n3; i = 0,1,2 64 4.12 Transit time components for devices #3u3; i = 0,1,2 65 4.13 fx and fmax for devices #3i00 at a collector current density of 5 x 104 A / c m 2 66 4.14 Contour plots of fx for devices #3^1 at a collector current density of 5 x 104 A / c m 2 66 4.15 Contour plots of fmax for devices at a collector current density of 5 x 104 A / c m 2 67 4.16 Contour plots of fx for devices #3ij'2 at a collector current density of 5 x 104 A / c m 2 68 4.17 Contour plots of fmax for devices #3ij2 at a collector current density of 5 x 104 A / c m 2 68 4.18 Contour plots of fx for devices #3ij3 at a collector current density of 5 x 104 A / c m 2 69 4.19 Contour plots of fmax for devices #3^3 at a collector current density of 5 x 104 A / c m 2 69 x 5.1 Critical thickness of InGaAs in the In v Gax_ y As/GaAs system 72 5.2 Experimental data of electron mobility in In yGai_ yAs[Abrahams et al. [38]] 76 5.3 Collector and base current density of devices with xe = 0.3, ybe = 0.0, xc = 0.0 and ybc = 0, 0.1, 0.15 77 5.4 Base current components of devices #30y0, 0 < y < 0.15 at a collector current of 5 x 104 A / c m 2 78 5.5 Collector and base current densities of devices #3j/?/0, y = 0, 0.02, 0.04 . 79 5.6 Gain of devices #3y?/0, y = 0, 0.02, 0.04 at a collector current of 5 x 104 A / c m 2 79 5.7 Base current components of devices #3yy0, 0 < y < 0.08 at a collector current of 5 x 104 A / c m 2 80 5.8 Contour plot of the gain for devices with xe = 0.3 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 82 5.9 Contour plot of the gain for devices with xe = 0.2 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 82 5.10 Contour plot of the gain for devices with xe = 0.1 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 83 5.11 Contour plot of the gain for devices with xe = 0.0 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 83 5.12 Comparison of the D.C. gain from simulations and the experimental results of Ramberg et al. [9] 84 5.13 Contour plot of fx for devices with xe — 0.2, xc = 0.0 at a collector current density of 5 x 104 A / c m 2 85 B . l Simulation grid for numerical solution [41] 101 xi B.2 SHBTs used for simulations 103 B.3 Gain of HBT1 and HBT2 105 B.4 Gain of HBT3 and HBT4 105 B.5 Emitter currents of HBT1 and HBT2 106 B.6 Emitter currents of HBT3 and HBT4 107 B.7 Base currents of HBT1 and HBT2 108 B.8 Base currents of HBT3 and HBT4 108 B.9 Changes in electron charge due to a small change in VBE HO B.10 fT of HBT1 and HBT2 I l l B . l l fT of HBT3 and HBT4 I l l xii Acknowledgement I would sincerely like to thank Dr. David L . Pulfrey for his guidance and support through-out the course of this work. I would also like to thank Simon Ho, Allan Laser and Hao-Sheng Zhou for the invaluable discussions and Dave Gagne for the computer system help that he has generously given. Finally, this thesis would not be possible without the support of my family. This work was funded in part by a grant from Bell Northern Research. xiii Chapter 1 Introduction 1.1 Background "A device ... in which one of the separated zones is of a semiconductor material having a wider energy gap than that of the material in the other zone." Claim 2 of U.S. Patent 2 569 347 to W. Shockley. Filed 26 June 1948. The idea of a heterojunction bipolar transistor (HBT), as shown by the quote above, is as old as the transistor itself. The first theoretical analysis of such a device with an emitter-base hetero junction by Kroemer [1] is itself three decades old. However, it is only in the last decade with the advent of metallorganic chemical vapour deposition(MOCVD) and molecular beam epitaxy (MBE) that the concept has been physically realised. Single heterojunction bipolar transistors (SHBT) with dc current gain as high as 12,500 [2], and short circuit, unity gain cutoff frequency fa and unity power gain cutoff frequency / m a x values of over 100 GHz [3, 4] have been realised in the lattice-matched A l a ; G a 1 _ r A s / G a A s system. The impressive performance of SHBTs with respect to homojunction bipolar transis-tors is due to the use of epitaxial techniques to tailor the energy bandgap in the HBTs to control the flow of electrons and holes in the devices separately and independently of each other. This flexibility is fully exploited in a bipolar device. The wider energy bandgap of the emitter in an npn device provides a large barrier to the injection of holes 1 Chapter 1. Introduction 2 from the base to the emitter. This results in near unity emitter injection efficiency and improved high frequency performance of the transistor. The improved high frequency performance is a result of the ability to have a highly doped base and/or a reduced doping in the emitter which result in a reduced base resistance and/or reduced emitter junction capacitance, without impairing the emitter injection efficiency of the transistor. Although the possiblity of having both a wider energy bandgap emitter and collector, a double heterojunction bipolar transistor (DHBT), had been recognised by Kroemer in his 1957 paper "The Theory of the Wide Gap Emitter" [1], only the advantage of a reduced junction capacitance at the base-collector was recognised. There are however other advantages that are offered by a double heterojuction bipolar transistor (DHBT) including : • emitter-collector interchangeability • an increased freedom to tailor the energy band profile of the entire device • a reduction in the offset voltage • a reduced injection of holes from the base to the collector in the saturation mode of operation. These advantages offer the promise of very high speed digital integrated circuit perfor-mance both for Emitter Coupled Logic (ECL) [5] as well as saturated Injected Integrated Logic (I 2L) [6] circuits. In terms of the material system used for HBTs, the lattice-matched A l x G a i _ x A s / G a A s system has been dominant. Lattice matching has been considered as one of the major fac-tors in obtaining heterojunctions with good performance [7]. Recently, with the advances in epitaxial growth techniques, it has been possible to grow lattice-strained structures Chapter 1. Introduction 3 within a critical layer thickness [8], without the formation of a large number of dislo-cations. This increases the number of heterojunction pairs that can be used in HBTs. InyGai-yAs/ALxGai.xAs [9] and lately S i i _ x G e x / S i [10, 11] HBTs have been fabricated and shown to be feasible. The potential advantages of using an Inj / Ga 1 _ ! / As/Al I .Gai_ I .As system over the A l x Gai_j ;As /GaAs system include: a lower trap density as the alumini-um content is reduced in Al^Ga i -^As , a higher minority carrier velocity, lower contact resistance and a reduced lateral base resistance. With the capability of the new epitaxial technologies to grow high quality epitaxial layers with different semiconductor materials on top of each other with thicknesses as thin as 10 angstroms, research on new and more complicated structures has resulted. As the complexity increases, the need for H B T models to investigate and/or optimise the performance of the novel structures intensifies. Although modeling may not be ''exact", it permits the device designer to investigate the dominating effects which determine the operation of the device. The insight gained may be more important than the actual simulation results themselve as it guides the designer towards an optimum device struc-ture. Within H B T modeling, decisions have to be made between analytical or numerical models. The trade-off between the two types of models centers around manpower, time, accuracy and ultimately cost. 1.2 Overview In this thesis, a 1-dimensional analytical model in the Ebers-Moll formulation is devel-oped for a D H B T in Chapter 2. The advantage of an Ebers-Moll formulation is that it is well suited to circuit design simulations as well as to relating I-V characteristics to device and material parameters. The main features of our model include: base grad-ing, junction grading, a detailed recombination/generation current model for the space Chapter 1. Introduction 4 charge regions, and a thermionic-field-diffusion current model for the injected current. The detailed recombination/generation current model is based on that of Ho [15], which includes Shockley-Read-Hall, Auger and radiative processes. Although there are other 1-dimensional models of the D H B T [12, 13], base grading and a detailed recombina-tion/generation model have not been previously included. In Chapter 3, a i V - A ^ G a a . ^ A s / p - A l ^ G a i . ^ A s / A ^ - A U G a a . ^ A s D H B T is simulated using the model. The material parameters for A l x G a 1 _ x A s and GaAs , see Appendix A , are obtained mostly from S E D A N [14] and Ho [15]. A n investigation of the d.c. characteristics, including the offset voltage, of the device as the aluminium mole fraction profile is varied is done. As offset voltage is an important and undesirable characteristic of HBTs, and its cause is still controversial, a section is devoted to its dependence on the aluminium mole fraction profile and the doping concentration in the collector. Contour plots of the gain and the offset voltage are presented to assist in the understanding of their dependence on the aluminium mole fraction profile. In Chapter 4, the high frequency performance of the N—AlxGa!_rAs/p—Al^Ga^^As/ N — ALrGai-sAs D H B T is characterised by the figure-of-merits fx and / m a x - Transit time components through the device are available in our model to investigate the effect on the individual transit time, and therefore the frequency, as a function of the aluminium mole fraction profile. In Chapter 5, the model is used to simulate an I n y G a ^ A s base D H B T with a ALcGa^j-As emitter and a GaAs collector; its d.c. and high frequency peformance is investigated and compared to the N — A l x G a i _ x A s / p — Al^Gax-xAs/Af — A l x G a i _ x A s D H B T . Material parameters for In^Gai.yAs are mostly obtained by interpolating the values between GaAs and InAs from the literature [16]. The gain from simulation is compared to the experimental results of Ramberg et at. [9]. Finally in Appendix B , a comparison of the results obtained from the model developed Chapter 1. Introduction 5 in this thesis and a 2-dimensional numerical model from the University of Toronto is done for four N — A l x G a i _ x A s / p — A l x G a i _ x A s / n — A l x G a ! _ x A s SHBTs. To ensure that the parameters used in the two models are similar, some of the material parameters used in our model have been changed to those of the values used at the University of Toronto. The comparison involves d.c. as well as high frequency characteristics, and provides some insight into the relative merits of 1-dimensional and 2-dimensional models. Chapter 2 Modeling of the Double Heterojunction Bipolar Transistor 2.1 Heterojunction Basics A heterojunction refers here to a pn junction formed between two dissimilar, yet lattice-matched semiconductors. The important differences between the two semiconductors are usually the band structure and the energy bandgap. Figure 2.1 shows the energy band diagram of an ideal N-p heterojunction in forward bias. The subscripts 1 refer to the wider energy bandgap semiconductor and subscript 2 refers to the narrower energy bandgap semiconductor. Notice that an uppercase N or P and a lowercase n or p is used to represent the conductivity type of the larger and smaller energy bandgap semiconductor respectively. In the following subsections, the built-in potential, the depletion layer width, the junction capacitance, the junction grading and the tunnelling current for the heterojunction are discussed. 2.1.1 Depletion Layer Width For an abrupt heterojunction with the assumption of full depletion of carriers on both sides of the heterojunction, Anderson [17] has shown that the depletion width can be obtained by solving Poisson's equation, i.e., d2V qND for —dn < x < 0 (2.1) dx2 d2V qNA for 0 < x < dp (2.2) dx2 6 Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 7 Vacuum Level Figure 2.1: Energy-band diagram of an ideal abrupt N-p heterojunction in forward bias. Solving (2.1) and (2.2) with the boundary conditions that E = —dV/dx = 0 at x - —dn and x = dp, and V = 0 at x = 0, we get for —dn < x < 0 for 0 < x < dn (2.3) (2-4) Defining VT as the total potential across the N-p heterojunction in non-equilibrium, VT = Vbi-Va = VTl + VT2 = V(-dn)-V(dp) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 8 q (NDdl NAQ 2 V 6, e. (2.5) where Vt, is the built-in potential, Va the applied voltage across the p-N junction, and Vpi and Vy 2 a r e the portions of V j supported by semiconductors 1 and 2 respectively. Assuming no sheet charge density at x = 0, et E\ — e2E2. This gives ND dn = NA dp Substituting Eq. (2.6) into Eq. (2.5) and solving for dn and dp, we get (2.6) d„ = d„ = 2VTC, e2NA q ND (el ND + e2 NA)_ 2 VT e, t2 ND mqNA{txND + i.NA) The total depletion width WT = dn + dp is given by 1 / 2 1 / 2 WT = 2VTC, e2 (NA + NDf qNAND(e1ND + e2NA)_ 1 / 2 2.1.2 Junction Capacitance (2.7) (2.8) (2.9) The junction capacitance per unit area, Cj, of the N-p heterojunction which relates the change in charges at the edges of the depletion region to the change in the junction voltage, is given by dQ dVT d(gNDdn) dVT  d(qNAdp) dVT (2.10) (2.11) (2.12) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 9 substituting for dn and dp and differentiating, we get q t, t2 NA ND 2VT(e1ND + eaNA) (2.13) 2.1.3 Built-in Voltage From Figure 2.1, qVbi = Eg2 + AEC - q(Vn + Vp) (2.14) where Vbi = built-in potential Eg2 = bandgap of narrow-gap semiconductor AEQ = conduction band discontinuity Vn = separation of electron quasi fermi level from the conduction band in N type semiconductor Vp — separation of hole quasi fermi level from the valence band where nno and pp0 are the equilibrium majority carrier concentrations in the emitter and base respectively, Nci and Nvi are the effective density of states of the conduction and the valence band of the wide-gap and narrow-gap semiconductors respectively. in p type semiconductor Vn and Vp are also given by [18, p. 27] (2.15) (2.16) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 10 At equilibrium, the intrinsic carrier concentration of the narrow- gap semiconductor is given by n*2 = NC2NV2exp(^-) (2.17) where Nc2 = effective density of states for the conduction band Nv2 = effective density of states for the valence band. Equation (2.17) can be written as Eg2 = kT\n (Ztajpi). Substituting (2.15), (2.16) and (2.17) into (2.14), we get Vbi in terms of NC2, NV2, ni2i nno, Ppo and the electron affinities (xi>X2) of the two semiconductors, i.e., , K - = ^ „ ( ^ ) + t e - ^ + * r i „ ( ^ ) = --- + *r^H'"(tr) <2'I8> Note that this reduces to the well-known result for the homojunction when Xi = Xi and iVc72 = Na. 2.1.4 Junction Grading In the preceding discussion, it has been assumed that the N-p heterojunction is abrupt which leads to the "spike" and "notch" in the energy band diagram. In a graded junction, the junction is compositionally graded so as to reduce the "spike" in the conduction band. Disadvantages of the "spike" include • reduced electron injection efficiency from the N semiconductor to the p semicon-ductor Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 11 • the resulting "notch" enchances recombination losses • higher turn-on voltage in a heterojunction bipolar transistor On the other hand, the "spike" can act as a ballistic launching ramp for the electrons injected into the base from the emitter. This may result in improved base transit time if the ballistic electrons do not transfer to the higher conduction (L,X) valleys. In the following analysis, the linear grading scheme of Grinberg et al. [19] is used to model the effect of junction grading. Figure 2.2 shows the energy-band diagram of a graded N-p heterojunction. Figure 2.2: Energy-band diagram of a graded N-p heterojunction. In this scheme, the electron energy barrier parameter, AEn, is modified by linear compositional grading over the distance Wg on the N semiconductor side of the junction. It is assumed that only the conduction energy band is affected by the grading. For Wg Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 12 = 0, we have an abrupt junction while for Wg > dni there is no "spike". If the grading width Wg is between 0 and dn, then g q2ND 2e, (dn - Wgf - dl Also, since qVTl = qVri - qVT\ and qVj\ — q2 Nod2l/2e1, qVTl = q2ND 2d {dn - Wg)2 A £ n , the electron energy barrier parameter, can be written as (2.19) (2.20) AEn = AEC - qVT2 - qVT\ = AEC - (qVT - qVrx) - qVT' = AEc-qVT + qVTl (2.21) where VT = 4 ^ (<*» -0 for Wq<dn for Wg > dn 2.1.5 Current Transport at the N-p Heterojunction The model for current transport across the N-p heterojunction is based on [20] which, in turn, is based on the Thermionic-Field- Diffusion model of Grinberg et al [19]. Figure 2.3 shows the various current components at the N-p heterojunction. The transport of electrons across the barrier occurs through thermionic emission as well as tunnelling through the narrow "spike". Jjx a n d JT2 are the thermionic emission current density components while JpI and Jp2 are the current density components due to tunnelling. Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 13 Jin Figure 2.3: Current components in a forward biased N-p heterojunction. The net electron current density injected from the N to the p semiconductor is given by [20] JTn = -qvTnl 7 n e-AE»'kT [n20 ( e ^ * 7 - 1) - n(0)] (2.22) where 7„ is the tunnelling factor to account for the transport of electrons through the narrow "spike" due to tunnelling, vTnl is the average x-directed electron thermal velocity in the N semiconductor, AEn is the electron potential energy barrier as shown in Figure 2.3,7T20 is the equilibrium electron concentration in the p semiconductor, Va is the applied forward bias and n(0) is the excess electron concentration at the depletion edge of the p semiconductor. vTnl is given by kT 2 7r m (2.23) n l where m ' j is the effective electron mass of the electron in the wide gap semiconductor. Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 14 Similarly, for the net hole current density JTp = qvTpl eAE^kT [p 1 0 (e^kT - 1) - p{Xl)] (2.24) except that there is no tunnelling factor in (2.24) as there is no tunnelling of holes through the valence band discontinuity. vTpl is given by vTpi = kT 1 (2.25) \j27rm*! where mpl is the average x-directed velocity of holes in the wide gap semiconductor. pw is the equilibrium hole concentration and p(xi) is the excess hole concentration at the depletion edge of the wide gap semiconductor respectively. AEP is the hole potential barrier as shown in Figure 2.3. 2.2 Double Heterojunction Bipolar Transistor In this section, the Ebers-Moll current-voltage relationships for the double hetero-junction bipolar transistor will be derived. The model is an extension of Ho's [15] model for a single heterojunction bipolar transistor which, in turn, is based on the model of Grinberg et al [19], extended to include base grading, the surface recombination veloci-ties of the contacts, and a more detailed formulation of space charge region recombina-tion/generation currents. Figure 2.4 shows the energy band diagram of a D H B T with a wider gap emitter and collector, biased in the active mode. The procedure to derive the Ebers-Moll equations for the D H B T is similar to that for the single H B T except for the current components at the base-collector junction which lead to a change in the concentration in the base and, ultimately, affect all the currents in the device. Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 15 Figure 2.4: Energy band diagram of a double heterojunction bipolar transistor with wide gap emitter and collector in active mode of operation. 2.2.1 Emitter-Base Junction Current From Section 2.1.5, for the emitter J?* = -Q vTnE InE e-^'kT [nB0 (e^kT - 1) - n(0)] (2.26) and Jff = q vTpE eAE^kT \pE (e<v°*<kT - 1) - p(xe)} (2.27) where the subscripts, and/or supersripts, E and EB refer to the emitter and the emitter-base junction respectively. Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 16 2.2.2 Emitter Hole Current The emitter hole current, Jp(xe) can be found by equating the hole diffusion current in the emitter and the hole thermionic emission current across the emitter. With a finite emitter surface recombination velocity, SPE, Jp(xe) is given by U*e) = J-f {e'v°°'kT - 1) cosh ( p \ (2.28) where JpE = 1 DPE PE LPE SPE/DpE + tanh(WE / LPE) LPE smh(WE/LPE) [LPESPE/DPE + 1/ tanh(WE/LPE) _ RE = 1+ J p E _ exp q vTpE pE pE is the equilibrium hole concentration in the emitter, LE the hole diffusion length, DPE the hole diffusivity in the emitter, and AEPE is as shown in Figure 2.4. 2.2.3 Base Electron Current For a graded base, the base electron current density is given by [21] where n(x) is the electron concentration, n,(x) the intrinsic carrier concentration, NA the base doping density (assumed constant), and DB the electron diffusivity (assumed constant and, for evaluation purposes, is taken as having the value appropriate for the center of the base). Taking the base grading to be linear, we have AE E,{x) — EGQ •—-2- x (2.30) where E G 0 is the bandgap at x = 0, the depletion edge in the base on the emitter-base side, and XB is the width of the base. The intrinsic carrier concentration can be written Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 17 as n]{x) = Nc(x) Nv(x) e - E ^ k T = ae' where n20Nc(x)Nv(x) a = Nco Nvo n A / q Eg kTXB where Nc and Ny are the effective densities of states in the conduction and valence band respectively. The corresponding parameters for GaAs are Nco and Nvo-(2.29) can be written as [15] Jn(x) = qDnB [(n - /) d e'*x + (r2 - /) C 2 e***] (2.31) where ra = 5 +1 r 2 =' 5 — t 3 Ci At x = 0, eq. (2.31) becomes 2 y 7/ 2 + 4 2 £„s n ( W Q - n ( 0 ) e r 2 t y 2e°w sinh(tW) h(0)eriW -h(W) 2eaWsinh{tW) UO) = yn[2th{W) - anh(0)} (2.32) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 18 where qDnB  eriW _ er2W (rx - f)e»w - (r 2 - f)e«w At x = W, eq. (2.31) becomes MW) = yn[bnh(W) - 2te2aWh{Q)\ (2.33) where bn = (n - f)er>w - (r 2 - f)er*w (2.34) 2.2.4 Base-Collector Junction Current The expression for the base-collector electron thermionic-field- emission current is similar to that at the emitter-base, i.e., J?Z = qvTnC 7 nc e-AE"c'kT [nBW (e"v^kT - 1) - h(W)} (2.35) and J% = q vTpC e^'kT \pc {e'v°°'kT - 1) - p{xe)] (2.36) where vTnC and vTpC are the average x-directed electron and hole thermal velocities, jnc the tunnelling factor for the base-collector barrier, AEnc and AEpc are as shown in Figure 2.4. 2.2.5 Ebers-Moll Equations for D H B T To evaluate Jn(0) and Jn(W), we need expressions for n(0) and h{W)._ With J„(0) and Jn{W) known, we can form the Ebers-Moll current-voltage relationship. n(0) can Vn = an = Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 19 be found by equating J„(0) and JE^ (ignoring recombination/generation current in the space charge region for now). We find that 2 ( y„n(^) + /„%B . ( e ^ - D zE + anyn where zE = qvTnBjnE e . Similarly by equating Jn(W) and , ^ ( e ^ - l ) + 2 ^ n ( 0 ) z% + &„yn where z° = qvTnC~fnC e . Substituting (2.38) into (2.37), f r m _ [*jf(*ff + U W ( e ^ ~ 1) + [KVn&Bw]^ - 1) _ 1 > (ZE + anyn)(Zc + bnyn) - (2tyne^)2 { ^ ] Similarly, solving for h(W), pty^e^n-BoKe3^ - 1) + [zcn{zE + anyn)nBW]{e^ - 1)  1 } (zE + anyn)(zO + bnyn)-(2tyne^y [ Z A U ) To get J„(0) and J„ (W) , (2.39) and (2.40) are substituted into (2.32) and (2.33) to obtain: Jn(0) = ^ J A o ( e * - 1) + Boie3^ _ 1) | (2.41) where Z = (zZ + bnyn)(zE + anyn)-(2tyne3W)2 A0 = zEnB0[4t2yne2sW - an(zZ + ynbn)} BQ = 2tz°zEnBW and Jn(W) = I { A ^ ( e ^ - 1) + ^ ( e * - 1)} (2.42) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 20 where Aw = -2tzZz*e2aWnBO Bw = zZnBW[bn(z* + anyn)-U2yne2<w} EMITTER I BASE COLLECTOR I Jp(xe) rEB 'R n :Jn(0) Jn(W) 0 T B If fBC D -JpOfc) Figure 2.5: Schematic of charge flow in D H B T . Looking at Figure 2.5 with the current convention indicated, we see that JE = -Jn(0) + JP{xE) + JR EB (2.43) where J^B is the recombination current in the space charge region of the emitter- base junction. Substituting for J„(0) and JP(XE), we get J e = {irW^)-fM<e*-1>+{-lM(e^-1)+J"B <>VRE 1VPC = Au(e * — 1) + A12(e-S^- — 1) + JR TEB (2.44) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 21 where A JPE , ( W E \ yn M2 = — ^ T-BQ Also, Jc = —Jn{W) — Jp{xc) + JR/Q where JjyG is the recombination/generation current in the base-collector space charge region. Substituting for JN{W) and Jp(xc), we get Jc = A 2 1 ( e * - 1) + A 2 2 ( e * - 1) + Jl% (2.45) where A22 = -rr- cosh ^ ^ The expressions for Jpc and i?c are similar to JPE and i?£ except for appropriate changes is the parameters. Summarizing, JE = A u ( e * - 1) + A 1 2 ( e * - 1) + j | f l T S C ' H / G Jc = A 2 1 ( e * _ 1) + A 2 2 ( e * - 1) + J | ,CG (2.47) where A JPE , /W^ N yn ^12 = —^BQ A21 = —gAw \ Rc \ -^c / 27 Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 22 Z = (z* + anyn)(zZ + bnyn)-(2tyne°w)2 A0 = z f nBQ[it2yne2sW - an(zcn + ynbn)] Aw = -Ztznzne nB0 B0 = 2tzcnzlriBW Bw = z^nBW[it2e2aWyn-bn(z^ + anyn)} 2.3 Reduction of D H B T Ebers-Moll Equations to S H B T Equations In this section, we try to reduce the D H B T Ebers-Moll equations obtained in the pre-ceding section to the case of a single heterojunction bipolar transistor. This is to obtain a first order check on the correctness of (2.47). For simplicity, we assume the contacts to be ohmic, i.e. Sp — • oo. To reduce the base-collector junction to a homoj unction, we set L\Efc = 0 and AEBC = 0. This results in 1. AEnC = -qVT2 < 0 and \AEnC\ > kT 2. AEpC = qVTi > kT These conditions result in the following expressions: oo (2.48) Rc — • 1 (2.49) Using the superscript D H and SH to refer to the D H B T and SHBT devices respectively, and substituting (2.48) and (2.49) into A?H, we get J«F. . fWE\ zEnBQ[U2yne2aW -an{z^ + ynbn)) ADH - , ( \ _ zZnBQ\Wyne"n - an{  C ° S H \ L E ) yn(zC + bnyn)(zB + anyn) RE JPE c o g h (WE RE \ LE {2tyne'wy ) - Vn** E n B 0 WF + a»y n)(l + >&) -Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 23 setting z% — • oo, A D H _ JPE , / W E \ anynnBo _ ASH ( 2 ^ An - R e cosh {L ) + 1 + S a p - A n (2-50) Similarly for A{°2H, ADH _ 2tynz%zEnBW ^12 — (z* + anVn)(z° + 6„y„) - ( 2 r y n e ^ ) 2  2tynzEnBW  \z* + anyn)(l + ^ ) - (^ff^Y Setting z% For A°H, l21 (zE + anyn){zZ + bnyn) - (2tyne^)2 2tynzEe2*wnBQ (z* + anyn){\ + ^ ) - 1 2 a a p i setting z„ —> oo, Finally for AgH, A D H = _J£ccoshfWc\ ^Bw[^2e2aWyn-bn{zE + anyn)\ l22 — (Wc\ Rc \LCJ *n(z% + anyn)(Zc + bnyn) - {2tyne^Y JpC , (Wc\ _ nBW[U2e2sWyn - bn(zE + anyn)] RcCOS \Lc) V \ z * + anyn)(l + *#) -again setting z% — • oo, and Rc = 1 A%2H = -JpC cosh (jj^j - ynnBW bn + %(anbn-At2e2°w) (2.51) A D H _ 2iynnBW _ .SH (T> A 1 2 + 2 ^ - ^ 1 2 A Z- o Z) A D H _ 2tynzcnzEe2*wnm (2.53) ADH _ 2tyne2°wnB0 _ S H 2 1 ~ ( i i aniny ~ 2 1 ^.o4j (2.55) ! + , - A*2" (2.56) It is seen that the D H B T Ebers-Moll equations do reduce to the SHBT Ebers-Moll equations as derived by Ho et al. [20]. Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 24 2.4 Recombination/Generation Current In Section 2.2.5, we derived the Ebers-Moll equations for the D H B T . In these equations, Ji{B and JRJQ were defined as the recombination current in the emitter-base space charge region and the recombination/generation current in the base-collector space charge region respectively. Usually only the generation current is considered in the base-collector space charge region. We specifically allow for the case of recombination also, in order to be able to model the D H B T in saturation. Here, we will make use of the expressions derived by Ho et al. [20] for the different recombination/generation processes that are taken into consideration, i.e., Shockley-Read-Hall, Auger and radiative. The recombination/generation currents are calculated from R/G JSCR Udx (2.57) where U is the net recombination rate of the processes that are considered. For Shockley-Read-Hall, UQRH = pn — nf 'po p + rii exp ( - ^ r1 ) (2.58) n + m e x p ( £ ^ ) ] + where r p o and r n o are the minority carrier lifetimes in highly extrinsic N and P-type material respectively. Et is the energy level of the recombination/generation centers and Ei is the intrinsic Fermi level. Assuming that the intrinsic fermi potential varies linearly across the space charge region, and taking Et = E{, r p o l = r p o 2 , and rno\ = r n o 2 , we obtain 2qWBE sinh r n ^ JSRH -i - -m v 2 k T j ,Hi ni2 , . J R = a — f{bi) + — g{b2) 6 (2.59) where tan - l (*oi-*iMi-fe?| I l+zi zoi+fci («i+zoi) 1 Zl+1 201+1 b\ = l Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 25 g(b2) = < vV-^l tan - l fr-vn) y/\l-%\ 1+Z2 ZQ2+h (22+202) 1 202 + 1 22+1 b\ = 1 and 0 = Inf^-) \NDNAJ qVi BE a = 9 J , —— dn + In + kT ND qVBE 2kT T2 = h = zx = 0^1 = *2 = 202 = = \/TP<>l T ' nol V^po2^no2 6 2 =U\/^- + \ /^) e x p(-2 VV r"° V T P ° / ^ ( 0 qVi BE 2kT 'pol exp T„oi " V WBE <Tpol dn+Ot Tno\ ITP02 Tno2 rpo2 exp(a) 6 exp WBE dp + a Tno2 exp (a) For radiative recombination, Urad = B(np-n}) (2.60) where B is the radiative constant. The radiative recombination current is given by JrRd = <Z (e'v™<kT - 1) (dn Bi n\ + dp B2 n}2) (2.61) where the subscripts 1 and 2 refer to the wide gap and narrow gap semiconductor re-spectively. dn and dp are the depletion layer widths in the N and p-type semiconductors respectively. For Auger recombination, there are two different processes that may take place. In the C H S H process, a conduction band electron recombines with a heavy-hole-band hole Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 26 which causes a light hole in the spin-splitoff band to transfer to a heavy hole band. In the C H C C process, the energy released by the recombination of the electron and the hole is absorbed by an electron in the conduction band. The recombination rate is given by UAug = (Cnn + Cpp)(np- rip) (2.62) w here C„ = Auger coefficient for the C H C C process Cp = Auger coefficient for the CHSH process (2.63) n and p are the carrier concentrations at thermal equilibrium. Assuming that the intrinsic fermi potential varies linearly across the space charge region, JR9 = q r "? e 9 V W 2 * T {e'v"'kT - 1) c (z + -) ^ dz Jz! \ Z j VZ 2 ? W B E e ' " « " T s i n h t e ) r , , „ , / ! x , „ / i v ] » r ^ ' P ' - ^ - U - r J J + ZQl Z\ { Z 2 - ZQ2) ~ ( \ Z 2 Z 0 2 (2.64) where cx = yJCn\ Cpi, c 2 = yJCn2 Cp2, z ° l = y c7 e x p ' a ' Z2 = Mexp{wrBd'+a] *02 = ^ ^ e x p ( a ) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 27 The emitter-base recombination current is given by the sum of (2.59),(2.61) and (2.64). The calculation of the base-collector space charge region recombination/ generation cur-rent JpjQ is as for the emitter-base recombination current and yields a similar equation with appropriate changes in the subscripts. When the base-collector is in strong reverse bias, the expressions for the generation current in the base-collector space charge region can be simplified. For the Shockley-Read-Hall process, using the relationship pn = n,exp(^ | , c ) , taking Et = E{ and since n and p <C n,- (2.58) becomes USRH = ( e * - 1) Tp0 "f" Tno Performing the integration over the base-collector space charge region, we get (2.65) J, SRH = _ g ( e * _ 1) "•icdpB niBdpc (2.66) . Tnol + Tp0\ Tno2 + Tpol For the radiative process, using the relationship pn = n ,exp( i ^ £ -) , (2.60) becomes URAD = Bn2 (2.67) which gives For the Auger process, JTGad = -q(Bcn]c + BBn2B). z 2 = z02 — — —e 2*r > 1 Cpi n,c Cpl NA ^ M e - ^ f » 1 Cpl NA \Cn2 niB SZB£L „ , c7^m * 1 (2.68) Chapter 2. Modeling of the Double Heterojunction Bipolar Transistor 28 also since 2 sinh(c7VBc/2/i;T) ~ — exp(—qVscl^kT), we get the simplified expression Chapter 3 D.C. Results and Discussion In this chapter, the model of the D H B T derived in Chapter 2 is used to simulate the behaviour of the H B T shown in Figure 3.1 whose parameters are given in Table 3.1. The structure shown in Figure 3.1 is a single-sided D H B T , as opposed to the more conventional pyramid structure shown in Figure 3.2. The single-sided structure was studied in this thesis because of the advantages it has over the pyramid structure, namely: an improved maximum frequency of operation due to a reduction of the base-collector parasitic capacitance and a reduction in the 4^ ratio, where Ac = area of the collector and AE — area of the emitter. A reduction in allows the D H B T to be used in the inverted mode without a major reduction in the gain and it also reduces the offset voltage [22] of the D H B T . In a D H B T , there are four parameters that relate to the aluminium mole fraction in A\xG&\_xAs. These are xe, xbe, xbc and xc which are the aluminium mole fraction in the emitter, emitter end of the base, collector end of the base and in the collector respectively. Figure 3.3 shows a diagram of the aluminium mole fraction in a particular device with base grading with xe = 0.3, xbe = 0.2, xbc = 0.1 and xc = 0.2. For simulations in this Thesis, the maximum aluminium mole fraction used is 30% and the emitter and collector material always have a wider energy bandgap than their neighbouring base material. The profile of the "spike" in the conduction band of the emitter-base heterojunction is a function of xe and xbe although it is also dependent on VBE-, the emitter doping density, 29 Chapter 3. D.C. Results and Discussion 30 n InGaAs f -6 nm C o l l e c t o r 1 um Le = Lb = Lc = 25 um 2 um 0.3 um / I E m i t t e r H # 3 » 3 n+GaAs n AlGaAs ^ n AIGaAs 10.2 um n AKSaAs E n AIGaAs n AIGaAs i n+GaAs Semi Insulating GaAs Substrate Figure 3.1: Single-Sided Heterojunction Bipolar Transistor. Table 3.1: Parameters for the single-sided D H B T . Layer Material Thickness (A) Doping (cm" 3) A l or In composition X emitter cap a n + - In y Ga 1 _ ! / As 300 1 x 10 1 9 0.6 b n + - I n y G a i _ y A s 300 1 x 10 1 9 0 . 6 - 0 linear c n + -GaAs 1000 3 x 10 1 8 0 emitter grading d n-Al^Gat.^As 500 5 x 10 1 7 0 - xe\\ne&v emitter e n-AL r Gai_. r As 1500-W s £; 5 x 10 1 7 emitter grading f n - A l ^ G a i ^ A s w g E 5 x 10 1 7 xe - X(,elinear base 9 p + - A l I G a 1 _ : c A s 1000 1 x 10 1 9 £ & e ~ ^6clhiear collector grading h n-AIGaAs 5 x 10 1 6 xc - x^linear collector i n-AlGaAs 4000-W f l C 5 x 10 1 6 X Q collector buffer 3 n + -GaAs 4000 3 x 10 1 8 0 Chapter 3. D.C. Results and Discussion 31 Emitter i Base 1 Collector Figure 3.2: Pyramid Heterojunction Bipolar Transistor. the base doping density and junction grading (if any). Similarly, the profile of the "spike" in the conduction band at the base-collector heterojunction is dependent on xbc , xc , VBC, the collector doping density, the base doping density and the junction grading. Emitter 0.3 0.2 0.1 0.0 x. Figure 3.3: Variation of aluminium mole fraction in a device for xe = 0.3, xbe = 0.2, xbc -- 0.1 and xc = 0.2. Chapter 3. D.C. Results and Discussion 32 Since the d.c. performance of the SHBT has already been discussed by Ho et al.[20], here we focus on the performance of the transistor due to the introduction of the base-collector heterojunction. In the following sections, the dependence of the base and col-lector current densities, the d.c. gain and the offset voltage on various combinations of x e , Xbe > xbc a n d xc are investigated. 3.1 DHBT Currents and Gain In this section, the d.c. currents and gains of the DHBTs are presented and discussed according to their aluminium mole fraction profile. In the Ebers-Moll equations, we derived for the emitter and collector current densities JE = A n ( e * - 1 ) + A 1 2 ( e * - 1 ) + J | f l Jc = A ^ e ^ - l ) + A22(e^ -1) + J^G. (3.1) The base current is calculated from JB = JE — JC- JB is the sum of the recombination in the neutral base (|J n(0) — Jn(W)\), the Shockley-Read-Hall, Auger and radiative re-combinations in the emitter-base space charge region and in the bulk regions, the back injection of holes from the base to the emitter and the recombination/generation current in the base-collector space charge region. However, the recombination/generation current is usually negligible. Ho et al. [20] have shown that at low VBE, SRH recombination is the dominant component while at higher VBE the neutral base recombination and the back injected hole current both exceed the SRH recombination. At even higher VBE, the back injected hole current dominates. Auger and radiative recombination are important at high collector current. The d.c. gain is calculated from 4^ . In the following sections, we use a four digit superscript to refer to the device with a particular xe, Xbe, xbc and xc. For example, Jc213 refers to the collector current in the device with xe = 0.3, xbe = 0.2, Xbc = 0.1 and xc = 0.3. Chapter 3. D.C. Results and Discussion 33 1e+06 e/ -CNJ 0.00 0.50 1.00 Emitter-Base Voltage (V) 1.50 Figure 3.4: Base and collector current density of devices #300i, i = 0,1,2,3, with VBC = —3 volts. 3 . 1 . 1 Effect of varying the collector A l content; GaAs base In this section, the effect of the base-collector "spike" is investigated by using a purely GaAs base and varying xc. Figure 3.4 shows the collector and base current densities of devices #3003, #3002, #3001 and #3000. It is seen that at low VBE, the collector current is dominated by the generation current in the base-collector space charge region which decreases as xc increases mainly due to the decrease in the intrinsic concentration in the collector. The reduction in the intrinsic concentration is approximately one order of magnitude for each 10% increase in aluminium mole fraction. For devices #3000, #3001 and #3002, their collector current is very similar for VBE > 0.5. However, the collector current of device #3003 is significantly lower than that of the other three devices. The presence of the base-collector "spike" does not seem to affect the collection of the carriers significantly until xc > 0.2. This is reflected in the much larger base current for device Chapter 3. D.C. Results and Discussion 34 3 1e+04 ~? 3 <l 1e+03 >» 1 3 CD •E 1e+02 I 3 1e+01 1 l 1 1-/' // — - " " ^ / - - " rad. - jJn(bT-Jn(W)| -Jp(Xa) — SRH i i i i i i i 0.0 0.1 0.2 0.3 X, Figure 3.5: Base recombination current components of devices #300i, i = 0,1,2,3, with VBC = - 3 volts and JC = 5 x 104 A / c m 2 . #3003 shown in Figure 3.4. In Figure 3.5, it is seen that the increase in base current is due to the increase in neutral base recombination current. In device #3003, the neutral base recombination current is the largest component of the base current. This can be visualised by thinking of the "spike" at the base-collector heterojunction as a wall that carriers in the base, without enough energy, rebound from instead of "jumping" over. Therefore, as the wall gets higher, the number of carriers that are trapped increases resulting in a higher neutral base recombination. As can be expected from the preceding discussion, the gain of the devices decreases as xc increases (see Figure 3.6). There is basically no gain for device #3003 while the highest gain in the group is approximately 100 for device #3000. 3.1.2 Effect of varying the collector A l content; 10% graded base Chapter 3. D.C. Results and Discussion 35 c b d 100.00 90.00 80.00 70.00 60.00 50.00 h 40.00 30.00 h 20.00 10.00 0.00 / > ,3000 //300l\\ 3002 3003 1e-09 1e-06 1e-03 1e+00 1e+03 1e+06 2 Collector Current Density (A/cm ) Figure 3.6: Gain of devices #30Ch, i = 0,1,2,3, with VBC 1e+06 -3 volts. c CD a +—* c cu v_ k_ 3 O sty • 3100 I J 3 1 0 1 J c l 3102 \3103 0.00 0.50 1.00 Emitter-Base Voltage (V) 1.50 Figure 3.7: Base and collector current density of devices #310i, i = 0,1,2,3, with VBC — —3 volts. Chapter 3. D.C. Results and Discussion 36 550.00r 500.00 450.00 400.00 CO O 300.00 q 250.00 Q 200.00 100.00 50.00 0.00 I I I I I -y^3100 / \ / / 3 1 o V \ / ' N \ / ' M A A // A / \ -/ \ --/ 3102 \ -3103 i i i T 1e-09 1e-06 1e-03 1e+00 1e+03 1e+06 2 Collector Current Density (A/cm ) Figure 3.8: Gain of devices #310i, i = 0,1,2,3, with VBC = — 3 volts. It is well known that base grading improves the performance of SHBTs significantly in terms of their gain and their high frequency performance. Here, the effect of a 10% base grading in a D H B T is investigated via devices similar to those studied in Section 3.1.1 except that x;,e is held at 0.1. Figure 3.7 shows the collector and base current densities of devices #3100, #3101, #3102 and #3103. The difference between the collector current of device #3103 and devices #3100, #3101 and #3102 is much less than the corresponding difference for the ungraded case (see Figure 3.4). A five fold increase in gain over the ungraded devices in Section 3.1.1, results from the 10% base grading, as shown in Figure 3.8 where the maximum gain of device #3100 is more than 500. The dependence of the gain on xc is the same as for the ungraded case. The reason for the improvement in gain with base grading is the same as for the SHBT. Base grading results in a built-in field in the base which aids the transport of carriers, electrons in this case, across the base resulting in a reduction in the neutral base recombination. The increase in X),e also Chapter 3. D.C. Results and Discussion 37 1e+06 -CM o ie-10 -Q 1e-06 -| 1e+00-~ 1e-02 -'« 1e-04 -_ 1e+02-| 1e-08 -1e-12 1e+04 -1e-14 _ / W/ 0.00 0.50 1.00 Emitter-Base Voltage (V) 1.50 Figure 3.9: Base and collector current density of devices #3i03, i = 0,1,2,3, with VBC — —3 volts. leads to a wider bandgap which results in a reduction of the Auger, radiative and SRH recombination and, therefore, to a further reduction in the base current density. 3.1.3 Effect of base-grading; large base-collector barrier In this section, we keep the base-collector heterojunction barrier constant while vary-ing Xbe- Figure 3.9 shows the collector and base current densities of the devices #3003, #3103, #3203 and #3303. Notice that as xbe increases, the collector current density decreases. This is because, as Xbe increases, the emitter-base junction becomes more like a homojunction than a heterojunction, and the back injected hole current increas-es. Thus the emitter injection efficiency is reduced. As can be seen from Figure 3.10, the back injected hole current becomes the most dominant base current after xbe~ 0.25. Comparing this to the case of the SHBT, it can be seen from Figure 3.5 (device #3000) Chapter 3. D.C. Results and Discussion 38 e o 1e+04 1e+03 1e+02 I 1e+01 c O 1e+00 1e-01 7 |Jn(0) - Jn(W)| -SRH rad. Auger J I I L 0.0 0.1 0.2 0.3 Figure 3.10: Base recombination current components of devices #3^03, i = 0,1,2,3, with VBC = - 3 volts and J c = 5 x 104 A / c m 2 . that the introduction of a large base-collector heterojunction barrier drastically increas-es the neutral base recombination current. The recombination current components in the emitter-base space charge region decrease with increasing X j e until about xbe = 0.25 when they start increasing again. The decrease in the recombination currents in the emitter-base space charge region as xbe initially increases is basically due to the increase in bandgap which reduces the intrinsic carrier concentration. Radiative recombination is the most dominant recombination process for xbe< 0.2 while SRH becomes the most dominant above x\,e = 0.2. For xbe> 0.25, the exponential dependence of the recombina-tion currents on VBE causes them to increase. In Figure 3.11, the gain of the devices is shown. It is observed that increasing the base grading increases the gain until xbe > 0.2 when further improvements are offset by a large increase in back injected hole current which results in a reduction of the gain. The gain is a maximum of 24 for device #3203. Chapter 3. D.C. Results and Discussion 39 25.00 20.00 •j= 15.00 CO O Q 10.00 5.00 0.00 1e-09 1e-06 1e-03 1e+00 1e+03 1e+06 2 Collector Current Density (A/cm ) Figure 3.11: Gain of devices #3i03, i = 0,1,2,3, with VBC = —3 volts. 3.1.4 Effect of varying the base-collector barrier In this section, xe and xc are held at 0.3 and xbe is held at 0.2 while xbc is varied. This results in increased base grading as xbc decreases. Figure 3.12 shows that the collector current increases as the base grading increases. This is to be expected, however, it is surprising that the base current density does not decrease with increasing base grading. This indicates that with xbe = 0.2, the back injected hole current is the dominant base current component with the same order of magnitude as the neutral base recombination current. A decrease in xbc results in an increase in base grading, resulting in a reduced neutral base recombination, however, it also results in a larger barrier at the base-collector heterojunction leading to an increase in the neutral base recombination current. The increase in the base current as xbc increases indicates that the effect of the notch at the base-collector junction in increasing the neutral base recombination current is greater Chapter 3. D.C. Results and Discussion E 5 CO c CD Q c CD t_ i _ o 0.50 1.00 1.50 Emitter-Base Voltage (V) Figure 3.12: Base and collector current density of devices #32z'3, i = 0,1,2 with VBC volts. 110.00 100.00 i i i 1 1 L ^3213 / \ -/ \ 90.00 -/ \ / \ 80.00 -/ \ / \ / v Gain 70.00 60.00 / */ * 3223 * q 50.00 -/ ; i / \ \ -/ \ 1 U J 40.00 - i i / \ 1 -/ \ 1 30.00 i j 3203 20.00 - i i y i '/ 10.00 / / -0.00 i i i i i i 1e-09 1e-06 1e-03 1e+00 1e+03 1e+06 2 Collector Current Density (A/cm ) Figure 3.13: Gain of devices #32i3, i = 0,1,2, with VBC — —3 volts. Chapter 3. D.C. Results and Discussion 41 1e+06 1e+04 _ 1e+02 ~§ 1e+00 - 1e-02 •m 1 e-04 c CD . - _ Q 1e-06 | 1e-08 o 1e-10 1e-12 1e-14 0.00 0.50 1.00 1.50 Emitter-Base Voltage (V) Figure 3.14: Base and collector current density of devices #3n3, i = 0,1,2, with VBC = —3 volts. than the decrease that is offered by base grading in this case. These conflicting effects lead to an optimal base grading as regards to achieving high gain (see Figure 3.13). 3.1.5 Symmetrical D H B T ; varying the ungraded base A l content The collector and base current densities of devices #3003, #3113 and #3223 are shown in Figure 3.14. Figure 3.15 shows that the gain of the devices decreases as xbe — Xbc decreases. As *^6e — *^bc decreases both the collector current and the base current increase. The former enhancement is due to the improvement in emitter injection effi-ciency (xi,e less), whereas the latter increase is due to the increase in both the neutral base recombination current (xbc less) and the emitter-base depletion region recombina-tion current (lower base band gap). Details of the base current component dependencies on Xbe = Xbc a r e given in Figure 3.16. The gain decreases as xbe = xbc decreases, showing that the neutral base recombination effect dominates over the emitter injection efficiency Chapter 3. D.C. Results and Discussion 42 c CO O 6 Q 1e-09 1e-06 1e-03 1e+00 1e+03 1e+06 2 Collector Current Density (A/cm ) Figure 3.15: Gain of devices # 3 « 3 , i = 0,1,2, with VBC = —3 volts. 3 1e+04 3 -E o 1e+03-3 -CO 1e+02-l _ cu Q 3 -» c CD 1e+01-^_ k_ 3 3 -o 3 1e-01 SRH 0.0 ... |Jn(0) - Jn(W)| rad."--..^ . J L ' - - . . A u g e r y' J I 0.1 0.2 %be - %bc 0.3 Figure 3.16: Base recombination current components of devices #3M3, i VBC = — 3 volts. 0,1,2, with Chapter 3. D.C. Results and Discussion 43 3.1.6 D.C. Current Gain In this section, contour plots of the d.c. current gain against xbe and xt,c are presented to provide a better understanding of its dependence on xe, X ( , e , xt,c and xc. Figures 3.17 , 3.18, 3.19 and 3.20 show the maximum gain plotted against xbe and xbc for xe = 0.3 and xc varying from 0.0 to 0.3 and Xbc < Xbe- Note that Figure 3.17, which is actually the case of a SHBT with xe = 0.3 and Xbc = xc = 0.0, is presented in the normal x-y plot format since there is only one variable Xbe. The highest gain obtainable is « 560 and it occurs at a aluminium mole fraction of x^e = 0.125. As expected, the lowest gain is for the case of Xbe = 0.3 where the emitter- base junction is a homojunction. The plot shows that there is an optimal base grading that maximizes the gain of the SHBT. In Figure 3.18, xe = 0.3 and xc = 0.1. The highest gain occurs again at xbe = 0.125 Chapter 3. D.C. Results and Discussion 44 0 . 3 0 0 -0 . 2 7 5 -0 . 2 5 0 -0 . 2 2 5 -0 . 2 0 0 -0 . 1 7 5 -U •a o.iso-0 . 1 2 5 -Figure 3.18: Contour plots of current gain with xe = 0.3 and xc — 0.1. 0 . 3 0 0 - 1 m 0 . 2 7 5 -0 .250-1 xbe Figure 3.19: Contour plots of current gain with xe = 0.3 and xc -- 0.2. Chapter 3. D.C. Results and Discussion 45 X 0 . 0 0 0 F I I I I 1 I I I I I I I 1 0 . 0 0 . 0 2 5 0 . 0 5 0 0 . 0 7 5 0 . 1 0 0 0 . 1 2 5 0 . 1 5 0 0 . 1 7 5 0 . 2 0 0 0 . 2 2 5 0 . 2 5 0 0 . 2 7 5 0 . 5 0 0 xbe Figure 3.20: Contour plots of current gain with xe = 0.3 and xc = 0.3. and xc = 0.0 with a value of « 510. It is interesting to note that the highest gain does not occur at x\,c = 0.1, where the base-collector would be a homo junction. The effect of the extra base grading obtained with xc = 0.0 therefore outweighs the disadvantage of having a larger "spike" at the base-collector junction. Figure 3.19 shows an interesting feature in the contour plot. A well-defined region where the gain peaks at « 320 occurs at xbe « 0.14 and xbc « 0.05. This shows again the trade-off between the advantage of base grading and the "height" of the barrier at the base-collector. The grading at the highest gain is about 9%. With a collector aluminium mole fraction of 0.3, the gain of the D H B T is the lowest of all the cases considered so far. The highest gain is « 120 which occurs at xbe = 0.175 and Xbc = 0.125. This is a grading of approximately 5%. Chapter 3. D.C. Results and Discussion 46 Figure 3.21: Common-emitter current-voltage characteristics. 3.2 Offset Voltage The offset voltage in the common emitter configuration is the collector-emitter voltage at which the collector current is zero when the emitter-base junction is forward biased. This voltage is shown in the current-volt age characteristics in Figure 3.21. The disadvantages of a high V c e > 0 j j s e t include 1. an increase in the power consumption (= V c e i 0 f f s e t x IC) in saturating logic, which limits the packing density of integrated circuits; 2. an increase in the output low voltage VOL in digital inverter applications, leading to a reduced low noise margin and output logic swing; 3. a reduction in the active region of operation in analog circuits. Many reasons for the high V c e < 0 f f a e t in HBTs have been put forward. However, it is still controversial as to which one is the prime cause of it. Haynes et al. [23] have suggested insufficient grading at the base-collector heterojunction is the major reason for Vce,0ffset-Chand et al. [24] credit the poor quality of the base-collector junction due to growth and Chapter 3. D.C. Results and Discussion 47 fabrication procedures for the Vcei0jj3et. The difference in turn-on voltages of the emitter-base and base-collector hetero junctions has also been cited as the major contributor to Vce,offset- One of the more recent papers, by Won et al. [22], cites geometrical and electrical asymmetry between the emitter and collector junctions as the dominant cause for Vcei0ffset. Geometrical asymmetry here mainly refers to the difference in area between the emitter and collector. A n analytical expression for VCe,ojfset is [23] VCE,OFFSET = REIB + — ln(4^ ) + — m ( - % - ) (3.2) q AE q ctFJes and is obtained by setting Ic = 0 in the Ebers-Moll equations and Ic = ctAEJea exp[g(VBE - IERE - IBRB)/kT] - AcJC3exp[q{VBC - IBRB - IcRc)/kT] (3.3) where RE, RB and Rc are the emitter, base and collector series resistances, and AE and Ac are emitter and collector areas respectively. In our formulation, Jc = A 2 1 ( e * - 1) + A 2 2 ( e ^ - 1) (3.4) neglecting the generation current. By setting Jc = 0, and neglecting the terms (since both VBE and VBc are both forward biased at Vce.t0ffset) we get VcE,off,et = — ln(-4^)|j c=o (3.5) q A22 A2\ and A22 are functions of VBE and VBc, therefore, an iterative method has to be used to calculate Vce<0ffset. Alternatively, by graphing the Ic vs VCE characteristic at various 7 B , the offset voltage can be obtained as shown in Figure 3.21. This is the approach used here. In all the results that follow,' the areas of the emitter and the collector are assumed to be equal. The effect of unequal areas for the emitter and collector contributes an amount equal to ~ l n ( ^ ) which can be added to the Vce,0//se< obtained in these simulations. Figure 3.22 shows the Vcefijfset of DHBTs with xe = xc = 0.3 for different 0 Chapter 3. D.C. Results and Discussion 48 e Li 0> 0 .3 0 .2 0 .0 0.1 0 . 3 0.1 0 . 0 0 .2 o o Figure 3.22: Vce<0ffset for xe = 0.3 and z c = 0.3 with collector doping of 5 x 10 1 6cm 3 . Xbe and Xbc at a collector doping of 5 x 1 0 1 6 c m - 3 . A maximum Vce,0ffset of 93mV is obtained in the case where the emitter-base junction is a homo junction while the base-collector junction is a heterojunction with Xbc = 0.0. This corresponds to the maximum junction asymmetry for the D H B T considered here. For symmetrical aluminium mole fraction at the emitter-base and base-collector, the Vcefijj3et is less than 15 mV. In fact, for Xbe = xbc = 0.2 and 0.1, Vcefi}}set is 1-4 mV only. It is seen that for X ( , e > 0.2, a small amount of base grading ( « 10%) reduces Vcei0ffSet but increasing the amount of base grading further actually increase Vce,offset- The minimum Vce,offset occurs at Xbe = 0.2 and Xbc = 0.1. In Figure 3.23 the results are for the same device as for Figure 3.22 except that the doping in the collector has been increased to 5 x 1 0 1 7 c m - 3 to make it equal to that in the emitter. It is seen that there is a general slight improvement in Vce,offset over the previous case. In Figure 3.24, a contour plot of Vce,0ffset against Xbe and Xbc for xe = xc = 0.3 is shown. It is seen that the minimum VCe,0ffaet occurs at Xbe = 0.3 and Xbc ~ Chapter 3. D.C. Results and Discussion 49 Vce,off shown are in millivolts \ 1 Emitter o o o o • • • • j o M ro CO Ql ' ~ o o O o • • • • o M ro to Collector Emitter o o o o • • • • j o M ro CO ^ ^ ^ ^ ^ o o O o • • • • o M ro to Collector Emitter o o o o • • • • j o M ro CO 14 9. o o O o • • • • o M ro to Collector Emitter o o o o • • • • j o M ro CO o o O o • • • • o M ro to Collector Figure 3.23: V c e , offset for xe = 0.3 and xc = 0.3 with collector doping of 5 x 1017cm 3 . Figure 3.24: Contour plot of VCe,ofjset for xe 5 x 10 1 7 cm- 3 . = 0.3 and xc = 0.3 with collector doping of Chapter 3. D.C. Results and Discussion 50 0.300-0.275-OJ50-0.225-Figure 3.25: Contour plot of V c e f i ] j a e t for xe = 0.3 and xc = 0.2 with collector doping of 5 x 10 1 7 cm- 3 . 0.175. As Xi,c is decreased, V c e i 0 f j a e t increases rapidly. This supports the argument of electrical asymmetry as a factor in increasing Vce,0jjset- It is also seen that base grading does contribute to reducing Vce,0ffset since the symmetrical D H B T (xbe = %bc = 0.3) does not have the lowest Vce,oSjaet- Figures 3.25 and 3.26 show the contour plots of Vce,offset for xe = 0.3, xc = 0.2 and xe = 0.3 and xc = 0.1 respectively. For Figure 3.25, there is an island of low K e , o / / « e t centered at xbe = 0.19 and Xbc = 0.09. Again, the minimum VCe,offset for this case occurs with base grading of « 10% and Vce,offset increases rapidly as Xbe ^~* 0.3 and Xbc — • 0.0. In Figure 3.26, where xe — 0.3 and xc = 0.1, the minimum VCe,offset occurs at Xbe = 0.06 and Xbc = 0.04. Finally, in Figure 3.27, V c e i 0 f f a e t for xe = 0.3 and xc = 0.0 for doping of 5 x 10 1 6 and 5 x 1 0 1 7 c m - 3 in the collector are shown. The Ke,o//set for a symmetrically doped SHBT is lower ( « 60 mV) and this difference seems to be relatively independent of Xbe-From the results thus far, it is seen that without base-grading, the lowest Vcei0ffset Chapter 3. D.C. Results and Discussion 51 0.300 0.275 0J50 0.225 0.200 0.175 0.150 . 0 0 . 0 5 x 0.100 0.075 0.050 0.025 0.000 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 xbe Figure 3.26: Contour plot of Ke,o//aet for xe = 0.3 and xc = 0.1 with collector doping of 5 x 10 1 7 cm" 3 . co 3= O CD O > 300 250 _ l l l l l ! SHBT/ / 1 1 / 200 -I 1 • 1 1 • 5e16 XJ 150 -s 100 50 0 5e16 _ 5e17 DHBT i i i i i i 0.0 0.1 0.2 X 0.3 Figure 3.27: Vce,0//se< for xe = 0.3 and xc = 0.0 with collector doping of 5 x 10 1 6 and 5 x 10 1 7 cm- 3 ; x = xbe for the SHBT and x = xbe = xbc for DHBT. Chapter 3. D.C. Results and Discussion 52 obtained is for the case where the aluminium mole fraction profile as well as the doping concentration in the emitter and collector are symmetrical. Figure 3.27 also shows that Vce,0ffset for the symmetrical D H B T is much lower than that of the SHBT (where the two junctions are asymmetrical). These results agree with the conclusion of Won et al. [22] who cite junction asymmetry as one of the major cause of high V c e , 0 f f s e t . Chapter 4 High Frequency Results and Discussion Two commonly used figures-of-merit to measure the high frequency performance of mi-crowave transistors are the cutoff frequency, fx, and the maximum frequency of oscilla-tion, / m a x . 4.1 Cutoff Frequency The figure-of-merit, fx, is defined as the frequency at which the common emitter short circuit current gain is reduced to unity. It is related to the transistor's physical and device parameters through the emitter to collector delay time r e c by *-*k <"> T e c is the sum of the delay times encountered by the carriers as they flow from the emitter to the collector. TB + TB + TSCR + rc (4.2) ' ec W here TE = emitter charging time TB = base transit time rSCR = base-collector space charge region transit time TC = collector charging time 53 Chapter 4. High Frequency Results and Discussion 54 Emitter Base Figure 4.1: Emitter series resistance. The components of r e c are calculated as follows [20]: TE = rE{CEJ + CCJ) + RECCJ (4.3) where r e is the emitter differential resistance, CEj and Cpj are the emitter and collector junction capacitances, respectively, calculated using, for example, CE — SELEeEeEB ^ ^ eEXE + CEBXEB where SE, LE are the width and length of the emitter, XE, XEB are the depletion widths in the emitter and the emitter side of the base. Ccj is calculated similarly with appropriate changes in the parameters. RE, the emitter series resistance consists of the contact resistance(i?c' = PEC ISELE), and the bulk resistance of the extrinsic (REx) and the intrinsic (REI) layers as shown in Figure 4.1. pEc is the contact specific resistance and REx and REi are calculated from the sum of the bulk resistance (pW/SL) in each of their respective layers (see Figure 4.1). Chapter 4. High Frequency Results and Discussion 55 The base transit time is obtained from which gives 1 fw — jQ -qn(X)dX (4.5) In — (4.6) 3 DNB \{r\ — / ) e r i w — {r-2 — f)C eT2 w] where C = —C 2 / C 1 ~ e2tW and C\, C 2 , n , r 2 , t are defined in Chapter 3. The time delay through the base-collector space charge region is calculated by W B C (A I\ where WBC is the width of the base-collector space charge region and vs is the saturation velocity for electrons in GaAs. The collector charging time is given by rc = (Rcc + RCB + Rci) CCJ (4.8) where and Rcc = coth (sc (4.9) i ? S b u f = ~ T ? 777- (4.10) q iVbuf ^nbuf Wbul is the sheet resistance of the collector buffer layer. Also, D RsbufScD , RsbuiSBC „ , RCB = = r — — (4.11) Lc o Lc where the first term corresponds to the lateral resistance between the collector and emitter contact and the second term is the lateral resistance immediately underneath the intrinsic emitter as shown in Figure 4.2. Chapter 4. High Frequency Results and Discussion 56 Figure 4.2: Equivalent circuit resistances and capacitances of D H B T . 4.2 The Maximum Frequency of Oscillation / m a x is the frequency at which the unilateral power gain of the transistor becomes unity. The unilateral power gain is the maximum available power gain of the transistor when the reverse gain is zero and is independent of the terminal configuration. / m a x is usually calculated from / m a x = ^ 8*(/fcC c) e / / ( 4 J 2 ) where (RbCc)efj is the effective base resistance and collector capacitance product (RB CCU = Cci (RBI + RBX + RBC) + C c x + RBc) + C c c RBC (4.13) where RBI, RBX a n d RBC are the intrinsic, extrinsic and contact resistances of the base and Cci,Ccx and Ccc are the corresponding capacitances depicted in Figure 4.2. The Chapter 4. High Frequency Results and Discussion 57 resistances are given by [25, p. 217] RBC = v ^ £ £ c o t h [sB (4.14) RBI = % ^ (4.15) RSB SEB F. . C, RBX = r (4-16) 3 L B >i! where R'SB and RSB are the sheet resistances of the intrinsic and extrinsic regions of the base, respectively. 4.3 Results and Discussion The investigation of the effect of the various aluminium mole fractions, namely Xbe, Xbc and x c , on fx and /maX of the D H B T is reported here using the same groupings as used in Chapter 3. 4.3.1 Effect of varying the collector A l content; GaAs base Figure 4.3 shows the dependence of fx and /max on the collector current density for the devices #300i, i = 0,1,2,3. It is seen that at low collector current, / m a x for all the devices is greater than fx- For devices #3002 and #3003, /max is greater than fx for all current densities. From (4.12) the criteria for fx > /max is given by rec < 4(RbCc)ejj. Since the value of (RbCc)eff, from simulations, is approximately 2.5 x 1 0 - 1 2 s, the criterion for IT > /max is r e c < 1 x 1 0 - 1 1 s. Figure 4.3 indicates that for devices #3002 and #3003, the emitter-collector transit time does not fall below 1 x 1 0 - 1 1 s. As xc increases above 0.1, fx and /max decrease significantly from maxima of 28 GHz and 21 GHz, respectively for #3003 at xc = 0.0 and a collector current of 1 x 105 A / c m 2 . From Figure 4.4, it is seen that r s increases by more than 2 orders of magnitude from device #3000 to device Chapter 4. High Frequency Results and Discussion 58 1e+01 1e+02 1e+03 1e+04 1e+05 Collector Current Density (A/cm^ Figure 4.3: fa and fmax of devices #300i; i = 0,1,2,3. 1e+01 1e+02 1e+03 1e+04 1e+05 2 Collector Current Density (A/cm ) Figure 4.4: Transit time components for devices #300i; i = 0,1,2,3. Chapter 4. High Frequency Results and Discussion 59 #3003. The exponential rise of fT and at lower collector current density is due to the exponential decrease in TE as the collector current density increases. Device #3003 does not show the exponential increase in fx and / m a x with Jc because of the dominance of TB over rE even at low Jc- TSCR and r c are approximately 2 x 10~ 1 2 s, and are relatively independent of Jc for all the devices and therefore do not contribute much to the change in r e c . The increase in rB can be attributed mainly to the increase in AEnc causing the carriers to be trapped in the base by the base-collector barrier. To a first order, it is proposed that r B can be estimated by for these groups of devices. rB values calculated from (4.17) and from simulations are shown in Table 4.1. Clearly (4.17) needs refinement, but it may be a useful starting point for including TB in computionally-efficient device design codes. Table 4.1: TB as computed from the full model and from the proposed equation (4.17) Device AEnC 7nC rB (model) rB (4.17) 3003 0 1.0 2.0 x 10" 1 2 2.0 x l O " 1 2 3001 0.0584 • 8.8 2.6 x 1 0 - " 2.2 x l O " 1 2 3002 0.1385 20.2 1.3 x 10" 1 1 1.7 x 10" 1 1 3003 0.2185 18.0 3.0 x 10- 1 0 5.1 x l O " 1 0 4.3.2 Effect of varying the collector A l content; 10% graded base With a 10% grading in the base due to having xj,e = 0.1 and xbc = 0.0, the results for fx and /max, Figure 4.5, show that the form of the curves is similar to the ungraded case (Section 4.1.1), except that higher frequencies are attained, especially for device #3103 (2 GHz, as compared to 0.5 Ghz for device #3003). It is also seen that the fx and / m a x Chapter 4. High Frequency Results and Discussion 60 Figure 4.5: fx and fmax of devices #310i; i = 0,1,2,3. Figure 4.6: Transit time components for devices #310i; i = 0,1,2,3. Chapter 4. High Frequency Results and Discussion 61 of #3103 exhibit the exponential rise at lower collector current density which indicates that its TB is much lower than that of #3003. This is clearly seen in Figure 4.6 where T3i03 6 x iQ-ii s W n e r e a s T30°3 ~ 3 x 1 0 - 1 0 5 . The simulations shows that rSCR « 2.4 x 10~ 1 2 5 for all the devices biased at VBC = -3 volts. Therefore, at high collector current density and low base-collector barrier (xc < 0.1), space charge region delay is the largest component of r e c . For xc > 0.2, rB is the largest component of rec. 4.3.3 Effect of varying the base-collector barrier The results of Figure 4.7 show that as xbe increases, fx and / m a x increase monotonically. This is different from the results for the D.C gain where the gain begins to fall after Xbe > 0.2 due to the increase in the back injected hole current as the emitter-base junction becomes more of a homojunction. However, / j and / m a x are not dependent on the back 1e+10 3 1e+09 NT n X 3 o ie+08 co cr 3 ^ 1e+07 3 1e+06 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06 2 Collector Current Density (A/cm ) Figure 4.7: fT and / m a x of devices #3i03; i = 0,1,2,3. Chapter 4. High Frequency Results and Discussion 62 1e-08 3 1e-09 «T 3 1 1e-10 o © 3 <D 1e-11 P 3 1e-12 3 1e-13 3 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06 2 Collector Current Density (A/cm ) Figure 4.8: Transit time components for devices #3i03; i = 0,1,2,3. injected hole current, and therefore they increase with base grading on account of the lower base transit time, as shown in Figure 4.8. 4.3.4 Effect of base-grading; large base-collector barrier Figure 4.9 shows that base grading does not improve fx and in all cases. The results show fx and / m a x decreasing with increasing base grading. There is only a slight decrease from device #3223 to device #3213 while there is quite a substantial decrease for device #3203. This indicates that the improvement in rB due to base grading is overshadowed by the increase in TB due to the increased barrier at the base-collector junction as X f , c decreases. The large barrier at the base-collector junction causes an increase in the concentration of excess carriers in the base which results in the increase in rB (see Figure 4.10) from (4.6). Chapter 4. High Frequency Results and Discussion 63 Chapter 4. High Frequency Results and Discussion 64 3 1e+10 3 _ 1e+09 N £ 3 >» c 1e+08 <x> 3- 3 ^ 1e+07 3 1e+06 1e+01 1e+02 1e+03 1e+04 1e+05 2 Collector Current Density (A/cm ) Figure 4.11: fa and / m a x of devices #3n3; i = 0,1,2. 4.3.5 Symmetrical D H B T ; varying the ungraded base A l content Without base grading, the effect of the base-collector barrier in increasing the excess minority carrier concentration in the base can be clearly revealed. Figure 4.11 shows that fa and / m a x decrease as Xbe = %bc decreases. Figure 4.12 show the dependence of the various transit time components on the aluminium mole fraction in the ungraded base with Jc = 5 x 104 A / c m 2 . It is clearly seen that r e c is dominated by rB for xbe = xt,c < 0.15 and is still the largest component for xbe = Xbc > 0.15. r e c ranges from a high of approximately 3 x 10~ 1 0 s to a low of 8 x 1 0 - 1 2 5 giving fa values of 0.5 GHz and 20 GHz respectively. 4.4 Contour Plots of Ft and Fmax To obtain a better feel for the effect of the various combinations of aluminium mole fraction in the emitter, the base and the collector, contour plots of fa and / m a x at a Chapter 4. High Frequency Results and Discussion 65 0.1 0.2 %be ~ %bc Figure 4.12: Transit time components for devices #3u3; i = 0,1,2. collector current of 5 x 104 A / c m 2 are presented. For all the contour plots, X(,c < Xj, e to give the desired direction of base-grading field, and x\,c is limited to less than or equal to xc. For the case of xc = 0.0, the case of a SHBT, the result is presented in the normal x-y plot shown in Figure 4.13 since there is only one variable x\,e- xe is held at 0.3 for all the contour plots while xc — 0.1, 0.2 and 0.3 for Figures 4.14 and 4.15; 4.16 and 4.17; and 4.18 and 4.19 respectively. For the SHBT, it is seen that the maximum fx of 39.5 GHz occurs at Xbe = 0.3 where base grading is the greatest. As Xbe decreases, fx decreases almost linearly at a rate of 3GHz/0.1 mole fraction of A l until about Xbe = 0.1 when the the rate increases to 11 GHz/0.1 mole fraction of A l . The reason for this dependence is that at Xbe < 0.1, TB is a major component of r e c and the decrease in TB is reflected in rec. Above Xbe = 0.1 r f l is no longer the major component of r e c and the further reduction in TB due to increased base grading does not significantly reduce T e c. / m a x is relatively independent of xbe compared to fx due to its square root dependence on fx-Chapter 4. High Frequency Results and Discussion 66 N X o 4 0 . 0 0 3 8 . 0 0 3 6 . 0 0 3 4 . 0 0 3 2 . 0 0 3 0 . 0 0 2 8 . 0 0 2 6 . 0 0 2 4 . 0 0 2 2 . 0 0 2 0 . 0 0 4 J L ftru J I I L 0.0 0.1 0.2 0.3 xbe Figure 4.13: fT and fmax for devices #3e'00 at a collector current density of 5 x 104 A / c m 2 . o 0.300 0.275 0.250 0.225 H 0.200 0.175-0.150-0.125-0 .100-0 .075 -0.050 0.025-1 0.000-1 0.0 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0 .300 ybe Figure 4.14: Contour plots of fT for devices # 3 i ; l at a collector current density of 5 x 104 A / c m 2 . Chapter 4. High Frequency Results and Discussion 67 X 0.300 0.275-0.250 0 .225 -0.200 0.175-0.150 0.125 0.100-0.075 0.050 0.025 0 . 0 0 0 - 1 I ' I I I I I I I I 0.0 0.025 0.050 0.079 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0 .300 xbe Figure 4.15: Contour plots of fmax for devices at a collector current density of 5 x 104 A / c m 2 . Figure 4.14 shows that the maximum fx of « 38 GHz occurs at high values of Xbe and low values of X(, c. This indicates that for a low base-collector barrier, the advantage of high base grading outweighs the disadvantage of a "spike" at the base-collector junction. The maximum fmAX for xc = 0.1 is about 20 GHz and it is seen that / m a x is relatively independent of xbe and X(, c. For the case of xc = 0.2 (see Figure 4.16), the maximum fx occurs at xbe = 0.3 and Xbc — 0.09. The trade-off between a high base grading and the increase in rB due to the "spike" is apparent from this result. If the advantage of base grading were unquestionable, the maximum fx would occur at Xbe = 0.3 and Xbc = 0.0. On the other hand, if the presence of a "spike" were utterly deterimental to the performance of the D H B T , the maximum fx would occur at x f c e =0.3 and Xbc = 0.2. However, since the results show that the maximum is not at either of these extremes, there is therefore a trade-off between base grading and the occurrence of the "spike". The contour plots also reveal that for Chapter 4. High Frequency Results and Discussion 68 0 . 3 0 0 -0.275-0 . 2 5 0 -0 .225 -0.0 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0 . 3 0 0 Figure 4.16: Contour plots of fr for devices #3ij2 at a collector current density of 5 x 104 A / c m 2 . 0.300 I I 0.275-0 . 2 5 0 -0 .225 -Figure 4.17: Contour plots of fmax for devices #3ij2 at a collector current density of 5 x 104 A / c m 2 . Chapter 4. High Frequency Results and Discussion 69 Figure 4.19: Contour plots of fmax for devices #3z'.;3 at a collector current density of 5 x 104 A / c m 2 . Chapter 4. High Frequency Results and Discussion 70 a particular xbe, there are usually two values of xbc that give the same fj. The larger xbc corresponds to a reduced base grading and a smaller barrier while the smaller xbc corresponds to a higher base grading but a larger barrier. The contour plot for xc = 0.3 reveals that the maximum fa ~ 28 GHz occurs at xbe = 0.3 and xbc = 0.18. Comparing the contour plots it is seen, from the reduced spacing between the contour lines, that as xc increases, fj and fmAX become more sensitive to changes in xbe and xbc. The results from the contour plots indicate that for our device, a difference in xc and xbc of « 0.1 is optimal for the high frequency performance of the D H B T . Chapter 5 InGaAs DHBT Most of the HBTs fabricated or studied to date have been based on the A l j - G a i ^ As/GaAs heterojunction. However, deep levels in AlxGax.^As [27] are likely to limit the per-formance of these HBTs. In the interest of improving the performance of HBTs, al-ternative systems of heterojunction pairs have been tried including the lattice mis-matched In^Gai-j,As/GaAs system [28, 29]. Although the In y Gai_ y As/GaAs is lattice-mismatched, it has been shown by Ramberg et al. [9] that, despite the increasing number of interface defects with increasing indium in the base, the d.c. current gain increased exponentially with base indium content. HBTs with In y Gai_ y As bases have potential advantages over HBTs with ALjGax^As or GaAs bases. As the energy bandgap of In y Ga!_ y As decreases with increasing indium mole fraction, the bandgap difference between the emitter and base can be increased. This results in an increased emitter injection efficiency which allows a higher base doping and/or a reduced aluminium content in the emitter. Strain in the In y Gai_ y As base layer, a result of the lattice-mismatch between In y Ga!_ y As and A l x G a 1 _ I A s causes a split in the valence-band degeneracy [30]. The holes appear to preferentially populate the band which has a lighter hole mass; this reduces the lateral resistivity, and hence the base resistance resulting in a possibly higher /max- The ability to base grade, and the resulting advantage, to base transport is not lost with an In y Gax_ y As base. 71 Chapter 5. InGaAs DHBT 72 0.0 0.02 0.04 0.06 0.08 0.1 Mole fraction of Indium Figure 5.1: Critical thickness of InGaAs in the I n y G a ^ A s / G a A s system. 5.1 Lattice Mismatch Structures There are some complications associated with using lattice-mismatched semiconductor pairs. A lattice-mismatched structure may be pseudomorphic or lattice-misfitted. In a pseudomorphic structure, the lattice mismatch is accommodated by coherent elastic s-train, while in a lattice-misfitted structure the mismatch is accommodated by both elastic strain as well as misfit dislocations. Misfit dislocations frequently act as recombination centers which could reduce the gain of the device drastically. The boundary between the two situations is not clear but it is usually estimated by a critical thickness derived by Matthews and Blakeslee [8]. A plot of the critical thickness against the indium mole fraction for the Inj,Gai_ y As/GaAs pair is shown in Figure 5.1. It is controversial as to whether compositional grading increases the critical thickness derived by Matthews and Blakeslee's. Ito et al. [31] have reported that compositional grading increases the critical Chapter 5. InGaAs DHBT 73 thickness by as much as a factor of 3; Ashizawa et al. [32] however, have suggested that the critical thickness based on Matthews and Blakeslee formula is a good approximation in compositionally graded structures if the average indium composition is used. Fitzger-ald et al. [33] have shown that the rate of increase of misfit dislocations is much lower than that estimated by the theory of Matthews and Blakeslee. 5.2 Material Parameters for InGaAs In the modeling of the AL.Gax_j .As/InyGai_yAs/GaAs DHBTs, most of the physical parameters for In y Gax_ y As are either interpolated [16] between GaAs and InAs, or the value for GaAs is used, as experimental or theoretical models as a function of indium mole fraction are presently unavailable. 5.2.1 Lattice Constant The lattice constant for In y Gai_ y As is interpolated between OQaAs = 5.6533 Aand ainA S = 6.0584 A [16] where acaAs a r *d ainA S are the lattice constants for GaAs and InAs. The lattice constant of In y Gai_ y As is used in the calculation of the interface recombination velocity, Sint, which appears in a new current component arising through recombination at the interface surface states (see Appendix A) . 5.2.2 Dielectric Constant The low frequency, e/, and high frequency, e/,, dielectric constants for InAs are taken as 14.6 and 12.25 respectively [16]. The dielectric constant for In y Gai_ y As is then given by e (5.1) for both high and low frequencies [14]. Chapter 5. InGaAs DHBT 74 5.2.3 Bandgap and Electron Affinity In y Ga!_ y As is a direct bandgap semiconductor for all values of y. This is different from the case of A l x G a i _ I A s which is a direct bandgap semiconductor for x < 0.45 and is an indirect bandgap semiconductor for x > 0.45 [34]. For In y Gai_ y As we have [16] ETg(y) = 1.42(4) - 1.615y + 0.555j/2 (5.2) For the L and X valleys, we use interpolation to obtain Ef(y) = 1.708 - 1.464j/ (5.3) and £ * ( y ) = 1.900 + 0.2y (5.4) where JEj(InAs) and £*( InAs) are taken as 1.47eV and 2.10eV respectively. In addition, due to the strain induced in the In y Gai_ y As layer, the bandgap of the In y Gai_ y As layer is modified [35]. The strain produces the splitting of the valence band, and the energy bandgap is now between the conduction band and the light hole band. This increases the energy bandgap of the strained In y Gax_ y As by [36] AETg(y) = 0.53y (eV) (5.5) The conduction band discontinuity AEC is assumed to be 63% [37] of the total energy bandgap difference between the emitter and the base. Knowing that x(In y Ga 1 _ y As) increases as y increases, we have X ( I n y G a i _ y A s ) = xaaAs + 0.M[Eg(GaAs) - Eg{y)} (5.6) taking a value of 4.07 eV for XGaAs [18], we get X (In y Ga x _ y As) = 4.07 + 0.63(1.615?/ - 0.555j/2) (5.7) Chapter 5. InGaAs DHBT 75 5.2.4 Effective Mass of Electrons and Holes The effective mass of the electrons for In y Gai_ y As is interpolated between GaAs and InAs to give ml = (0.067 - 0.035y)mo m£ = (0.55 - 0.264xy)mo m * = (0.85 - 0.21y)mo where m 0 is the electron rest mass and m^(InAs) = 0.032mo, m^(InAs) = 0.286mo and (InAs) = 0.64mo [38]. Since In y Ga!_ y As is a direct bandgap material for all values of y, the effective electron mass is given by [39] mn = A £ i - r \ , / A £ * - r \ l 2 / 3 (mrnf/2 + (mLn)3/2 exp + (m*) 3 / 2 exp (5.8) kT J ' v n ' V kT where m* is the overall electron effective mass. For the holes, the hole mass of InAs is taken as 0.4mo and 0.48m0 for GaAs [34]. Using interpolation again, we get m'p = 0.48 - 0.08y (5.9) 5.2.5 Mobility For the electron mobility we use the experimental data of Abrahams et al. [40] shown in Figure 5.2. The solid line is a fourth order polynomial fit to this data, i.e., Hn(y) = (3.91 - 36.76y + 184.79j/2 - 252.93y3 + 120.99y4) x 103 cm2/v.s. (5.10) A n average doping of 5 x 10 1 6 c m - 3 is taken for the data. The dependence of the electron mobility on doping concentration for In y Gai_ y As is assumed to be the same as for GaAs, since there is no data available, that is [41] Chapter 5. InGaAs DHBT 76 « 4.00 2.00 -0.00 G a A s 0.20 0.40 0.60 0.80 1.00 InAs Figure 5.2: Experimental data of electron mobility in In^Ga^j, As [Abrahams et al. [38]]. where N r e f = 5 x 10 1 6 c m - 3 , a = 0.436 and Ny is the doping concentration. In general, in III-V semiconductors, polar optical phonon scattering dominates [39] and 1 r oc (5.12) Since n = ^ and p oc (m*) - 3 / 2 — ^) \ we have for In y Ga 1 _ J / As, pcx(m*)-3/2 • (5-13) For the hole mobility, for lack of reliable experimental data, the above expression is used. 5.3 D . C . Resul ts and Discussion In this section, the model of the D H B T derived in Chapter 2 is used to simulate the D.C. performance of the I n y G a ^ A s base D H B T . The material used in the emitter is Chapter 5. InGaAs DHBT 77 1e+05 1e+03 ~1e+01 CM | 1e-01 tJe-03 | 1e-05 Q c 1e-07 CD | 1e-09 1e-11 0.00 0.50 1.00 1.50 Base-Emitter Voltage (V) Figure 5.3: Collector and base current density of devices with xe = 0.3, ybe = 0.0, xc = 0.0 and ybc = 0, 0.1, 0.15. A l r G a i _ x A s (0 < x < 0.3) while GaAs is used in the collector so as to avoid a large base-collector junction barrier which degrades the performance of the transistor. The width of the base will be kept at 1000A with a doping of 1 x 1019 c m - 3 , as for the case of the A l x G a i _ x A s base D H B T . The average indium content in the base is kept below 8%, so as to avoid the complications of high lattice mismatch at the heterojunction which may result in poor junction characteristics. Incidentally, high gain devices have been fabricated with this indium content [9]. First, we consider the set of devices with xe =0.3, ybe = 0.0, xc = 0.0 and in which ybc is varied from 0.0 to a maximum of 0.15, resulting in a maximum average indium concentration of 8% in the base. When ybc = 0.0, the device is actually an A l x G a i _ x A s SHBT which has been discussed earlier. Figure 5.3 shows the collector and base current density for devices with ybc = 0.0, 0.1 and 0.15. It is seen that the current densities Chapter 5. InGaAs DHBT 78 ^ 1e+02 E o < 5 CO c CD o Q ^ c CD t 1e+01 o JR - -radiative ..Auger -SRH -Jp(x.) 0.00 0.05 0.10 0.15 Figure 5.4: Base current components of devices #30y0, 0 < y < 0.15 at a collector current of 5 x 104 A / c m 2 are very similar; however, the insert shows that there is a slight increase in the collector and base current densities of the devices as ybc increases. Figure 5.4 shows the various components of the base current at a collector current density of 5 x 104 A / c m 2 . The recombination current components in the base-emitter space charge region and Jp(xe) are relatively independent of yt,c since xe and yt,e are constant. Similar to the Al^Ga i . ^As base D H B T , the neutral base recombination current increases as the "spike" at the base-collector heterojunction increases with increasing ybc- This results in only a small increase in JBI as the radiative recombination current is dominant, causing a small reduction in gain. The next group of devices considered have xe = 0.3 and xc — 0.0 with ybe — ybc varying from 0.0 to 0.04. The collector current densities are similar while there is a significant increase in the base current density as ybe = ybc decreases (see Figure 5.5). The resulting Chapter 5. InGaAs DHBT 1 r 0.00 0.50 1.00 1.50 Base-Emitter Voltage (V) Figure 5.5: Collector and base current densities of devices #3?/y0, y = 0, 0.02, 0.04 i U L l 1 1 1 1 1 — 0.60 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure 5.6: Gain of devices #3yy0, y = 0, 0.02, 0.04 at a collector current of 5 x 10 A / c m 2 Chapter 5. InGaAs DHBT 80 CM TX(o) - JWW) | 03 _i 2 r O 1e+01 - SRH- -5 r. 0.00 0.02 0.04 0.06 0.08 y Figure 5.7: Base current components of devices #3yy0, 0 < y < 0.08 at a collector current of 5 x 104 A / c m 2 decrease in gain, Figure 5.6, is contrary to to the expected improvement in the emitter injection efficiency due to the increased bandgap difference between the emitter and the base. The reason for the decrease in gain can be clearly seen in Figure 5.7 where radiative and Auger recombination processes are dominant and Jp(XE) is the least significant of the base current components. Therefore, any decrease in Jp(XE) due to the increased bandgap is negligible compared to the increase in radiative and Auger recombination due to the decrease in bandgap in the base. It is also obvious that in the I n ^ G a ^ A s base D H B T the increase in neutral base recombination due to the base-collector barrier does not play a domineering role, as in the AL-Gax.^As base D H B T , due to the much higher radiative and Auger recombination currents. At an indium content of 8%, the n 3 dependence of the Auger recombination current eventually causes it to become larger than the radiative recombination current which has a nf dependence. Chapter 5. InGaAs DHBT 81 In Figures 5.8, 5.9, 5.10 and 5.11 contour plots are presented of the gain of the DHBTs with xc = 0.0 and xe = 0.3, 0.2, 0.1 and 0.0 respectively at a collector current of 5 x 104 A / c m 2 . The plots reveal that the maximum gain of « 210 occurs at xe = 0.2 with j/&e = 0.0 and ybc ss 0.03. This is much lower than the gains that are obtainable with an A l x G a i _ x A s base D H B T . This indicates that the advantage of the reduced bandgap in the base does not offer the expected increase in gain due to the corresponding significant increase in radiative and Auger recombination. As a test of our model, the simulation results are compared to the experimentally measured gain of the ungraded In^Gai.yAs base devices of Ramberg et al [9]. In the experimental devices, GaAs is the material used in the emitter and the collector. The layer structure of the devices is shown in Table 5.1. Figure 5.12 presents the gain from Table 5.1: Layer structure of devices used by Ramberg et al. [9]. Material Purpose Thickness Doping GaAs cap 2000 A n+ 2 x 10 1 8 c m " 3 GaAs emitter 2000 A n 2 x 10 1 7 c m " 3 In^Gax.yAs Be setback 200 A 7T friyGax.yAs base 800 A P+ 1 x 10 1 8 c m " 3 Inj,Gai_ yAs grading of In 500 A n~ 2 x 10 1 6 c m " 3 GaAs collector 4500 A n~ 2 x 10 1 6 cm" 3 GaAs coll. contact 5000 A n+ 2 x 10 1 8 c m " 3 our simulation and the measured gain of Ramberg et al., as a function of the indium mole fraction in the base, both at a collector current density of 125 A / c m 2 . It is seen that for ybe = ybc < 0.05, our model predicts the correct dependence of the gain against the indium mole fraction in the base. Our predicted gain is higher than that of the devices of Ramberg et al. by approximately a factor of 2, which could be accounted for by surface recombination and/or contact resistances which are not taken into consideration in our Chapter 5. InGaAs DHBT 82 • i 1 i 1 1 1 i i i i 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.1S 0.18 0.20 ybe ( Mole fraction of In ) Figure 5.8: Contour plot of the gain for devices with xe = 0.3 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 . ' 1 1 1 i i i i i i i 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 ybe ( Mole fraction of In ) Figure 5.9: Contour plot of the gain for devices with xe = 0.2 , xc = 0.0 at a collector current density of 5 x 104 A / c m 2 . Chapter 5. InGaAs DHBT 83 Figure 5.11: Contour plot of the gain for devices with xe current density of 5 x 104 A / c m 2 . = 0.0 , xc = 0.0 at a collector Chapter 5. InGaAs DHBT 84 1e+03 5 2 1e+02 c 5 ca ° 2 le+01 5 2 1e+00 0.00 0.02 0.04 0.06 0.08 ybe Figure 5.12: Comparison of the D.C. gain from simulations and the experimental results of Ramberg et al. [9]. one-dimensional model. The gain of the experimental device with an 8% ungraded base is considerably higher than that of the devices with lower indium content. Ramberg et al. attribute this to an increase in the base minority carrier velocity caused by the injection of energetic electrons across the abrupt emitter-base heterojunction. They further postulate that the high-energy effects are present in devices with ybe — ybc = 0.05 but that they are masked by the limited emitter injection efficiency. As our model does not model hot-electron effects, we do not expect to see the experimentally observed dramatic rise in gain at indium mole fractions greater than 0.05. 5.4 High Frequency Results and Discussion Figure 5.13 shows the contour plot of fx as a function of ybe and ybc with xe = 0.2 and xc = 0.0 at a collector current density of 5 x 104 A / c m 2 . The maximum fj obtained is Chapter 5. InGaAs DHBT 85 0.00 0.02 0.04 0.06 0.08 0.10 0.12 O.M 0.16 0.18 0.20 ybe ( Mole fraction of In ) Figure 5.13: Contour plot of fx for devices with xe = 0.2, xc = 0.0 at a collector current density of 5 x 104 A / c m 2 . « 34 GHz which is close to the maximum value of 37 GHz obtained in the Al^Ga^^-As base D H B T . Varying xe does not change fx significantly as the resulting change in TE has a negligible effect on r e c . Base grading and the presence of the base-collector barrier have similar effects on the InyGaj.yAs base DHBTs as they have on the A l x G a i _ x A s base DHBTs . The effect of lattice strain in splitting the valence band, resulting in an increase in light hole concentration, is not modelled here as there are no data on the fractional increase in the population of the light holes as a result of the lattice strain. Such information is needed to accurately compute the effective base resistance and, thus, make, a reasonable estimate of /max- hi the absence of this information, the / m a x contour plot would look very similar to that of the fx plot shown in Figure 5.13 Chapter 6 Summary 6.1 Conclusions In this thesis, a 1-dimensional analytical model of a D H B T has been developed and used to investigate the d.c, including the common-emitter offset voltage, and the high frequency performance of a N — Al^Ga i . ^As /p — A\xGa.i_xAs/N — AL-Gai-^As DHBT. An initial study of the d.c. current gain and fx performance of a, N — A l x G a 1 _ x A s / p — In y Ga!_ y As/Af — GaAs D H B T has also been done to compare its performance to that of the A ^ G a ^ A s base D H B T . The conclusions that can be drawn from this work are: 1. The presence of the base-collector barrier: (a) reduces the collection of electrons due to an increase in the neutral base recombination current, however, for devices with less than 0.2 difference between their base and collector aluminium mole fractions, high gains and high frequency of operation are still possible; (b) reduces fx and /max due to an increase in the base transit time. When a large base-collector barrier is present, the base transit time is the dominant component of the emitter-collector delay time. 2. There is a trade-off between base-grading and the emitter-base and base-collector barrier heights which results in an optimum aluminium mole fraction profile regard-ing the d.c. current gain. For the devices considered here, for xc < 0.2, a value of xbe « 0.125 and base grading between 12%- 9% results in maximum current gains 86 Chapter 6. Summary 87 of » 500 to » 320 for xc of 0.0 to 0.2. 3. If the base-collector barrier is held constant, increasing base grading increases fx and / m a x . However, if the emitter-base barrier is held constant, there is a trade-off between the amount of base-grading and the barrier height at the base-collector which maximises fx and /max- For the devices considered here, maximum fx and /max a r e obtained when the emitter-base is a homojunction and the difference be-tween the collector and the corresponding base aluminium mole fraction is « 0.1. 4. Symmetrical DHBTs, in terms of aluminium mole fraction profile as well as doping concentration, result in devices with low V c e < 0 j f s e t . For devices with emitter and collector aluminium mole fraction which are not too different ( < 0.1 ), there is an optimum amount of base grading that minimises Ke,o//aet • For devices with a large difference in their emitter and collector aluminium mole fraction, increasing base-grading increases V c e i 0 f f s e t . 5. In an InyGai.yAs base D H B T , the radiative and Auger recombination currents limit the d.c. current gain. The maximum gain of w 210 is obtained with an emitter and collector aluminium mole fraction of 0.2 and 0.0 respectively, and a 2% base grading with GaAs at the emitter side of the base. The fx obtained is similar to those for a GaAs base D H B T . 6.2 Future Work The next stage in the development of the D H B T model is undoubtedly to verify it by comparing results from the model with actual experimental data. It will be particu-larly interesting to see whether the predicted trends in both d.c. and high frequency performance with aluminium mole fraction profile are borne out in practice. Chapter 6. Summary 88 Adaptation of the model to material systems other than A L . G a i _ x A s / G a A s and In^Gai-yAs/GaAs is another obvious future step. Of particular interest in the grow-ing field of optoelectronic integrated circuits is the In y Gai_j,As/InP system. Some refinements to the present model may be desirable. For instance, hot electron effects could be included and their effects on the d.c. and high frequency performance of the DHBTs at high currents levels should be studied. Ramberg et al. [9] have suggested that such effects are important. Also, it may be important to investigate making some improvement to the classical depletion approximation. Neglect of the mobile charge carriers in the space charge regions at high current levels may underestimate the emitter current and junction capacitances. References [1] H . Kroemer, "Theory of a wide-gap emitter for transistors," Proc. IRE, vol. 45, pp. 1535-1537, Nov. 1957. [2] H . H . Lin and S. C. Lee, "Super-gain AlGaAs/GaAs heterojunction bipolar transis-tors using an emitter edge thinning design," Appl. Phys. Lett., vol. 47, pp. 839-841, 1985. [3] M . C. F . Chang, P. M . Asbeck, K . C. Wang, G. J . Sullivan, N . H . Sheng, J . A . Higgins, and D. L. 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Milnes, "Study of n-In^Gax.yAs/n-GaAs heterojunction epilayers," J. Vac. Sci. Technol. B, vol. 5, pp. 792-795, May/Jun. 1987. [29] A . Fischer, J . N . Miller, S. S. Laderman, S. J . Rosner, and r. Hull, "Growth and Characterization of AlGaAs/InGaAs/GaAs pseudomorphic structures," J. Vac. S-ci. B, vol. 6, pp. 620-624, Mar. /Apr . 1988. [30] M . Jaffe, Y . Sekiguchi, and J . Singh, "Theoretical formalism to understand the role of strain in the tailoring of hole masses in p-type In yGai_j,As (on GaAs sub-strates) and Irio.53+xGao.47_xAs (on InP substrates) modulation-doped field-effect transistors," Appl. Phys. Lett., vol. 51, pp. 1943-1945, Dec. 1987. V References 93 [31] H. Ito, and T. Ishibashi, "GaAs/Ino.osGao^As double heterojunction bipolar tran-sistors with a lattice-mismatched base," Jpn. J. Appl. Phys., vol. 25, pp. L421-L424, May 1986. [32] Y . Ashizawa, S. Akbar, W. J . Schaff, L . F. Eastman, E. A . Fitzgerald, and D. G. Ast, "Influence of lattice misfit on heterojunction bipolar transistors with lattice-mismatched InGaAs bases," J. Appl. Phys., vol. 64, pp. 4065-4074, Oct. 1988. [33] E . A . Fitzgerald, D. G. Ast, Y . Ashizawa, S. Akbar, and L. F. Eastman, "Dislocation structure, formation, and minority-carrier recombination in A l -GaAs/InGaAs/GaAs heterojunction bipolar transistors^" J. Appl. Phys., vol. 64, pp. 2473-2487, Sept. 1988. [34] H. C. Casey, Jr. and M . B . Panish, Heterostructure Lasers, Part A: Fundamental Principles. New York: Academic Press, 1978. [35] C. P. Kuo, S. K . Vong, R. M . Cohen, and G. B. Stringfellow, "Effect of mismatch strain on band gap in III-V semiconductors," J. Appl. Phys., vol. 57, pp. 5428-5432, Jun. 1985. [36] H . Asai, and K . Oe, "Energy band-gap shift with elastic strain in Ga^Inx.^P epi-taxial layers on (001) GaAs substrates," J. Appl. Phys., vol. 54, pp. 2052-2056, Apr. 1983. [37] W. J . Schaff, L . Eastman, K . L . Kavanagh, P. D. Kirchner, G . D. Petit, and J . M . Woodall, Technical Abstract F-6, 1986 Electronic Material Conference, Amherst, M A . [38] K . Brennan, and K . Hess, "High field transport in GaAs, InP and InAs," Solid State Electronics, vol. 27, pp. 347-357, Apr. 1984. References 94 [39] J . E . Sutherland and J . R. Hauser, " A computer analysis of heterojunction and graded composition solar cells," IEEE Trans. Electron Devices, vol. ED-24, pp. 363-372, Apr. 1977. [40] M . S. Abrahams, R. Braunstein, and F . D. Rosi, "Thermal, Electrical and Optical Properties of (In,Ga)As alloys," J. Phys. Chem. Solids, vol. 10, pp. 204-210, 1959. [41] N . D. Arora, J . R. Hauser, and D. J . Roulston, "Electron and hole mobilities in silicon as a function of concentration and temperature," IEEE Trans. Electron Devices, vol. ED-29, pp. 292-295, Feb. 1982. [42] J . X u , private communication, Feb. 1990. [43] 0 . Nakajima, K . Nagata, H . Ito, T. Ishibashi, and T. Sugeta, "Suppression of emitter size effect on current gain in AlGaAs/GaAs HBTs," Jpn. J. Appl. Phys., vol. 24, pp. 1368-1369, 1985. [44] J . J . Liou, F . A . Lindholm, and D. C. Malocha, "Forward-voltage capacitance of heterojunction space-charge regions," J. Appl. Phys., vol. 63, pp. 5015-5022, May. 1988. Appendix A Material Parameters for A l x G a i _ x A s and GaAs In this appendix, the material parameters for A l x G a i _ x A s and GaAs used in the simu-lations are presented. A . l Bandgap and Electron Affinity The T, L and X interband energy gaps (eV) are given by f 1.424 + 1.247 x 0 < x < 0 . 4 5 ( 1.424 + 1.247 x + 1.147 (x - 0.45)2 0.45 < x < 1.0 Ef;(x) = 1.708 + 0.642 x (A.l) E*(x) = 1.900 +0.125 x + 0.143x2 (A.2) The p-type doping dependence of the bandgap in GaAs is taken as Eg (eV) = 1.424 - 1.6 x 10"8 p 1 / 3 . (A.3) For n-type GaAs and AfcGai^As the same dependence is assumed, therefore Eg(x,N) = m i n ( £ [ ( x ) , £ f (x)) - 1.6 x 10" 8 iV 1 / 3 (A.4) where N is the net doping concentration in c m - 3 . The electron affinity of A l x G a i _ x A s used is X(x) = 4.07 -0.7981 x (ev) 0 < x < 0.45 (A.5) 95 Appendix A. Material Parameters for AlxGai_xAs and GaAs 96 This equation is arrived at by considering Anderson's model and taking a value of 0.64 for the ratio of the conduction band offset to the total band offset between the two heterojunction pair. A.2 Effective Electron and Hole Mass The effective mass for electrons in A l x G a i _ x A s is taken as K ) 3 / 2 + (mLnf/7 exp (-^Pl + (m*)3/2 exp A E ^ kT 2 / 3 where mTn = (0.067 + 0.083 x) m 0 m Ln = (0.55 + 0.12 x)mQ m* = (0.85 - 0.07 x)m0 x < 0.45 (A.6) (A.7) (A.8) (A.9) and AEg~r = E% - Erg and AEf~T = E* - ETg. The effective mass of the electrons is arrived at by considering that the total electron concentration is a sum of the electrons in the three conduction valleys. The hole effective mass is taken as m*p = (0.48 + 0.31 x)m0 (A.10) A.3 Dielectric Constant e = i+g[«(jja)+('-«> (ga)] (A . l l ) for both low as well as high frequencies. tx and e2 are the dielectric constant for AlAs and GaAs whose values are listed in the table below. Appendix A. Material Parameters for AlxGai_xAs and GaAs 97 material £h t\ GaAs 10.9 13.1 AlAs 8.12 10.06 A.4 Mobility and Diffusion Coefficient -1 12 For holes, rm*(x = o i l 3 / 2 * " T ' x ) m [ - £ i w \ w^-eT-w^^ (A-12) where // p ,Gai4s (^ r ) is the doping-dependent hole mobility in GaAs an A r r is the total doping concentration. For electrons, m = rofts = 0) m£(x) 3/2 e A2 — 6 -1 {2 efc ( x ) - e , (x) Un,GaAs(NT) K ( x = l ) ] 3 / 2 + [m*(x = I ) ] 3 / 2 e - 1 hi -1 (A.13) (A.14) [m£(*)]3/» + [m*(*)F 2 ^ ( x ) - eT\x) ^ , A ' where fi,i and fi; are the mobilities in the direct and indirect conduction band valleys respectively, pn,GaAs(N'r) is the doping-dependent electron mobility in GaAs, and fiUtAiAs is the electron mobility of AlAs which takes on a contant value of 294 c m 2 / V s . Doping dependence of mobility is taken as Vo 1 + (NT/Nie[)" where The carrier diffusion coefficient, in thermal equilibrium is (A.15) parameters Mo iVref a electrons 8100 1.69 x 10 1 7 0.436 holes 408.7 2.75 x 10 1 7 0.395 (A.16) Appendix A. Material Parameters for AlxGa\-.xAs and GaAs 98 w here and z = Fi/2(»?) is the Fermi-Dirac integral of 1/2. Fi/2{r]) can be approximated by 4 / TT2\3/4 ^1/2(1) = 3^= h 2 + y j for 77 > 5 (A.18) A .5 Minority Carrier Lifetimes The minority carrier lifetimes for electrons or holes is calculated by 1 1 1 1 1 / . X — = + — + — + (A.19) Tett TSRH TR TA TINT = T + T- (A-2°) T0 'INT where T S R H , TR, TA are the carrier lifetimes for SRH, radiative, Auger recombination processes respectively. T ! N T is the carrier lifetime due to the interface traps present due to interface traps at lattice-mismatched heterojunction. For SRH, TSRH = TP0 _ + TN0 _ (A.21) n + p n + p where r p o ~ 2 x 10" 8 s and r n o ~ 5.5 x 1 0 - 9 s and is assumed to be applicable to both A l j ; G a i _ x A s as well as GaAs. For the radiative process, T » " l l v <A-22> where N is the net doping. B is given by B(x,N) = 3.0367 x 10~3 B(N) . (A.23) mo where mo B(N) = 1.204706 x 10~7 N'0-16775617 (A.24) Appendix A. Material Parameters for AlxGa\_xAs and GaAs 99 For Auger recombination, T a = n AT2 m N-type material (A.25) TA = CnN2D  1 C~N\ in P-type material (A.26) where Cn = Cn0 exp(a n T + 6 n T 2 ) (A.27) C p = Cpo exp(a p T + bpT2) (A.28) and C n 0 = (1.960 - 11.36 x + 31.37 x 2 ) x 10" 3 2 (cm 6/s) a n = (0.8714 +0.88 x - 6 . 3 6 x 2 ) x 10" 2 ( K " 1 ) bn = (-0.03655 - 0.0638 x + 0.562 x 2 ) x 10~4 (K~ 2 ) Cpo = (9.786 - 36.35 x + 111.6 x 2 ) x 10" 3 2 (cm 6/s) a p = (1.045 - 0.408 x - 1.64 x 2 ) x 10" 2 ( K _ 1 ) bp = (-0.0774 + 0.0371 x + 0.127x2) x 10" 4 . (K~ 2 ) which is valid for 0 < x < 0.2. For x > 0.2, the values at x = 0.2 are used. For a double heterostructure, the ratio of the effective carrier lifetime r to the bulk lifetime r 0 in a slab of thickness d is given by r sinh(rf/Z) + k[oosh(<//£) _ i] ro (1 + 66) sinh(J/L) + (6 + 6) cosh(d/L) (A.29) where L = \/r0D is the minority carrier diffusion length in the bulk material and 6 = S)NTL/D (A.30) where S)NT (i = 1,2) is the recombination velocity at the emitter-base and collector-base interface respectively, and D is the carrier diffusivity. (A.29) can be written in the form "66 smh(d/L) + 6 cosh(d/L) + 6 ' 1 _ l_ _L_ T TO T0 smh(d/L) + 6[cosh(<//I) - 1] (A.31) Appendix A. Material Parameters for AlxGa,\-xAs and GaAs 100 Comparing (A.31) to (A.20), it is seen that 1 _ 1 TINT SINT is given by SINT = 1.26 x 107 Q* (cm/s) (A.33) where ai and a 2 are the lattice constants of A l x G a i _ s A s and GaAs respectively. 0 ^ is obtained by interpolating between the lattice constant of GaAs (5.6533 A) and AlAs (5.6605 A). £ 1 6 sinh((f/v^7J) + 6 cosh^/v/r^D) + 6 sinh(<f/v/roT0 + ^ [coshU/^/f^D) - 1] (A.32) Appendix B Comparison of 1-Dimensional and 2-DimensionaI H B T Modeling Results. B . l Introduction HBTs may be modelled either analytically in 1-dimension or numerically in one, two or even three-dimensions. In analytical models, the H B T is usually divided into space charge regions and quasi-neutral regions in which approximate analytical solutions of the one-dimensional field equations (i.e. Poisson's and the continuity equations) are obtained. These solutions are then coupled at the corresponding interface to obtain the complete model. Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.102 equation and the current continuity equations for holes and electrons, are solved numer-ically under the given device configuration and given boundary conditions. To simulate the numerical model, first, the space domain is mapped onto a grid of distinct points or nodes (see Figure B . l ) ; next, some discretisation method is applied to the set of equations to yield an algebraic problem with respect to space, and finally a numerical algorithm is used to simultaneously solve for the unknown variables. To solve the problem generally involves a large amount of software development time. Even after the development of the software is completed, the running of the program takes a lot of computational time as well as computer memory. The reward for all this effort is a more accurate model, since the "exact" fundamental equations are solved numerically, which could be used to simulate a variety of devices with different structures. In this Chapter, the results of our 1-dimensional analytical model and a 2-dimensional numerical model developed at the University of Toronto are compared to investigate the differences in the terminal characteristics, the d.c. gain and fj between the two models. In terms of computational time, using the SUN 3/60, our model takes approximately a couple of seconds to obtain a set of results with 35 bias points while the University of Toronto's numerical model takes about 30 minutes, dependent on the number of nodes in the mesh, for each bias point. For comparison purposes, both models are used to simulate four SHBTs labelled HBT1, HBT2, HBT3 and HBT4 with the structure and aluminium mole fraction profiles of the emitter and base shown in Figure B.2. HBT1 has an abrupt emitter-base junction with ungraded base; HBT2 has an abrupt emitter-base junction with a graded base; HBT3 has a graded emitter-base junction with an ungraded base; and HBT4 has a graded emitter-base junction with a graded base. For a more accurate comparison, some of the numerical values in our model have been modified to be as close as possible to University of Toronto's [42]. The modifications included: Appendix B. Comparison of 1 -Dimensional and 2-Dimensional HBT Modeling Results. 103 3.5 um 0.3 um 0.7 um C o l l e c t o r Emitter n+GaAs 500 A 7x 1Q18 cm n AIGaAs 200 A 7x 1Q18 cm"3 n AIGaAs 460 A 2.5 x ld^m" 3 n AIGaAs 325 A. 2.5 x ld'cm"3 P+AIGaAs 1500 A 1 x 1Q19 cm 3 D AIGaAs 4000 A 2 x 1Q16 cm"3 n+GaAs 5000 A 5 x lo'" cm'3 0.3 um 0.7 um 0.28 e o / HBT1 ; 500 - 200 • 460 J325 •ngttrof 0.28 c o HBT2 • 500 | 200 I 460 : 325 angstrom 0.28 HBT3 Emitter Base angstrom 0.28 HBT4 Figure B.2: SHBTs used for simulations. Appendix B. Comparison of 1 -Dimensional and 2-Dimensional HBT Modeling Jesuits. 104 1. taking AEC = 0.85AE9 2. neglecting Auger and radiative recombination processes and taking r n o = 1.69 ns and Tpo = 2.4 ns 3. taking the mobility of the holes in emitter and collector to be 166 and 410 cm 2 /Vs respectively, and the mobility of the electrons to be 2034 cm 2 /Vs . 4. taking the temperature as 303 K. In addition, the doping dependence of the energy bandgap has been neglected and, per-haps very significantly, we have removed the tunneling component of the emitter current from our model. This latter step was made because the University of Toronto's model does not allow for tunneling. B.2 Results and Discussion In this section, the simulation results from the two models are presented and discussed. The results presented in this chapter have been compiled for VCE = 0.0 volts. For compactness, the superscripts U B C and U T are used to represent the results from our model and that of the University of Toronto's model respectively. B.2.1 D .C. Results and Discussion Starting with the d.c. gain of the devices, it can be seen from Figures B.3 and B.4 that /3UBC is much lower (more than 2 orders of magnitude) than fiUT for the abrupt emitter junction devices (HBT1 and HBT2) and f3UBC is greater (w 5 times) than f)UT for the graded emitter junction devices (HBT3 and HBT4). This indicates that in the case of the graded emitter, the two models are in much better agreement than for the abrupt emitter junction devices. The removal of the tunneling component in our model Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.105 Figure B.3: Gain of HBT1 and HBT2. 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure B.4: Gain of HBT3 and HBT4. Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.106 1e+01 J 1e-01 I 1e-03 ^ 1e-05 a> ••—» •»—» IS 1e-07 1e-09 UBC I I S t ' / -UT //-# - HBT2 / / -A* HBT1 * * * * * * * * Jtr — * ^ * 4 4 • I I I 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure B.5: Emitter currents of HBT1 and HBT2. leads to an underestimation of the emitter current in the abrupt emitter devices, where tunneling can be expected to be important, and this presumably leads to the differences shown. Focusing on the emitter current, it is seen (Figures B.5 and B.6) that the emitter currents for HBT3 and HBT4 for both the models are in good agreement; however, for HBT1 and HBT2, the difference between the two models is substantial. Also for both HBT1 and HBT2, the slopes for the 2 models are generally in good agreement except that at low voltage bias, our model shows a «s 2kT dependence while the University of Toronto's shows a « lkT dependence. Apparently, relative to the University of Toronto's model, we overestimate the emitter-base depletion region recombination current and underestimate the injection current. When there is a graded emitter-base junction, our model reduces basically to a dif-fusion model which, as can be seen from the results thus far, is in good agreement with Appendix B. Comparison of 1 -Dimensional and 2-Dimensional HBT Modeling Results. 107 E t c Qi I— o I — <D 'E LU 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure B.6: Emitter currents of HBT3 and HBT4. University of Toronto's model. However, when an abrupt junction is present, the two models differ significantly, which leads us to the conclusion that the modeling of the H B T at the emitter-base heterojunction is quite different. We need to pursue this further with the University of Toronto's team. Tunneling through the abrupt junction is an important feature of our model. Inclusion of it would increase the the emitter current and improve our gains, making them closer to the values predicted by the University of Toronto for HBT1 and HBT2. However, it could be that the University of Toronto's model itself should be changed as its neglect of tunneling seems unrealistic. The base currents for the devices are shown in Figures B.7 and B.8. Examining the base currents for HBT3 and HBT4, it is seen that our model underestimates the base current consistently for these devices. This can be accounted for by surface recombina-tion which is present in Alj.Gai_j .As HBTs and which is modelled in the 2-dimensional model. It is interesting to note that for HBT4, which has a graded base, the agreement Appendix B. Comparison of 1 -Dimensional and 2-Dimensional HBT Modeling Results.lOS ie+02 r r -He+02 -He-10 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure B.7: Base currents of HBT1 and HBT2. 0.80 1.00 1.20 1.40 1.60 Base-Emitter Voltage (V) Figure B.8: Base currents of HBT3 and HBT4. Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.109 between the two models is much better. This further supports the theory that surface recombination is responsible for the difference between the two models. Nakajima et al. [43] have shown that the presence of the quasi-field due to base grading reduces the effect of surface recombination at the emitter periphery and external base by more effec-tively sweeping the minority carriers across the neutral base. This effect is also seen in comparing HBT1 and HBT2. For HBT1 and HBT2, I%BC is larger than at low VBE, while at medium VBE IBBC is smaller than IBT, which is expected from the earlier discussion on surface recom-bination. The larger base current at low VBE occurs only in HBT1 and HBT2 leading us to suspect that it is related to the emitter-base heterojunction modeling as before. The levelling-off of the base current at high VBE for the 2-dimensional model can be attributed to emitter crowding effects caused by lateral base resistance at high current conditions. Again this is an effect not accounted for in our model. B. 2.2 High Frequency Results and Discussion For the 2-dimensional model of the University of Toronto, fx is calculated from * " 5 f e ( B 1 ) where gm is the transconductance and C, n is the input capacitance in the common-emitter configuration. C t n is calculated from ft <-> with Vce constant. Qn is the total electron charge in the entire device. Since the total electron charge and the total hole charge in the device are equal, C,„ can also be calculated from the total hole charge in the device. For comparison, /x calculated from (B.l) for our model is presented together with the /x from the 2-dimensional model. To calculate C, n for our model, a finite difference method is used to calculate | ^ where dQn is taken Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.110 to be the change in the effective electron charge in the neutral base region and in the emitter and collector. Thus, Emitter I Base Collector i V B E + AV B E BE A VE BE AV B E Figure B.9: Changes in electron charge due to a small change in VBE-dQn dVb be A<2n A He (QB2 - QBI) + AQE + AQC AVBE (B.3) (B.4) where QBI and QB2 are the electron charge in the quasi-neutral base at Vbei and Vbe\ + AVbe; AQE and AQc are the increase in electronic charge due to a decrease in the depletion width in the emitter and collector respectively. Figure B.9 shows the different components of the electron charge due to a small increase in VBE- Figures B.10 and B . l l show that for all cases, the fx values calculated from (B.l) for our model, are higher than that of the University of Toronto's. The higher fx from our model could possibly be attributed to the depletion approximation used in our formulation which would result in Appendix B. Comparison of 1-Dimensional and 2-Dimensional HBT Modeling Results.lll 1e+10-1e+08 1e+06 * le+04 1e+02 1e+00 1e-02 HBT1 // .<;*' / // HBT2 I'4 / /I / / / I J L '-• UT le+11 1e+10 1e+08 1e+06 1e+04 1e+02 1e+00 1e-11 1e-08 1e-05 1e-02 1e+01 Collector Current (A/cm) Figure B.10: fT of HBT1 and HBT2. 1e+11 1e+09 1e+07 N £ ie+05 1e+03 1e+01 1e-01 I I I I i i HBT3 • 4 ~ , V ' - ' -• i i • UT i i i i i i i i 1e+12 16+11 1e+09 1e+07 1e+05 1e+03 1e+01 1e-09 1e-07 1e-05 1e-03 1e-01 1e+01 Collector Current (A/cm) Figure B . l l : / T of HBT3 and HBT4. Appendix B. Comparison of 1-Dimensional and 2-Dhnensional HBT Modeling Results.\12 a lower estimation of the electron charge in the device [44]. This underestimation results in an underestimated C,„, which ultimately would result in a higher fx-B.3 Conclusion From the results and discussions in Section 6.2.1 and 6.2.2, it appears that both the d.c. gain and fx are sensitive to the nature of the emitter-base junction. The main discrepancy between the models arises in the abrupt junction devices HBT1 and HBT2. Having to neglect tunneling in our model, to comply with the conditions of the University of Toronto's model, seriously underestimates the emitter current and hence the gain. The depletion approximation, which has been used extensively in modeling bipolar devices, in the space-charge regions at high current injection may also contribute to the smaller emitter current in our model relative to that of the University of Toronto's. As is well known, the depletion approximation assumes that the carrier concentration in the space charge region is negligible compared to the ionized donor or acceptor concentration; this is definitely true for the case where the p-n junction is reverse-bias. However, when there is a high injection of carriers, the approximation might not hold as well. From Poisson's equation (2.1) we have OX1 Cj where ND is taken to be the only charge in the emitter space charge region when the depletion approximation is assumed. In a high injection condition, — n is a more correct value to take for (2.1), resulting in a lower emitter potential barrier and thus a larger emitter current. 

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