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Large-signal spice models for heterojunction bipolar transistors and lasers 1994
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Title | Large-signal spice models for heterojunction bipolar transistors and lasers |
Creator |
Feng, James Jun Xiong |
Date Created | 2009-02-25 |
Date Issued | 2009-02-25 |
Date | 1994 |
Description | Large-signal SPICE models for heterojunction bipolar transistors (HBTs) and semiconductor lasers are developed. For a general graded-base double heterojunction bipolar transistor (DHBT), a Full Ebers-Moll model and its simplified versions for specific HBTs have been derived from DAPHNE, a and implemented in the circuit simulator HSPICE by using its piece-wise-linear features to represent the coefficients with voltage-dependent normalized junction velocity terms, which are used to describe tunneling factors and junction barrier heights for back-injected electrons. For uniform and moderately-graded base single heterojunction bipolar transistors (SHBTs), this model can be further simplified and BJT-compatible versions of the HBT SPICE model can also be derived by using an exponential fit to the normalized junction velocity. The experimental data, forward collector current and the variation of the oscillation frequency fosc with bias voltage Vcci f°r a graded-base SHBT and two five-stage ring oscillators, respectively, can be well-fitted by simulation results from DAPHNE and the BJT SPICE model. A popular large-signal equivalent circuit model, developed by Tucker [4, 5], based on the rate equation for a single-mode semiconductor laser, has been modified, simulated and compared with experimental data. Finally, the performance of HBT-laser transmitters is also simulated to show that the models developed in this thesis have the capability of being very useful design tools for HBT-laser optoelectronic integrated circuits. [Footnote] 1 DAPHNE: An acronym for Device Analysis Program for heterojunction Numerical Evaluation, has been developed at UBC based on the work of Ho [1], Ang [2], and Laser [3]. |
Extent | 3636810 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-02-25 |
DOI | 10.14288/1.0065146 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/5043 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0065146/source |
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LARGE-SIGNAL SPICE MODELS FOR HETEROJUNCTION BIPOLAR TRANSISTORS A N D LASERS By James Jun Xiong Feng B. A. Sc, University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1994 (c) James Jun Xiong Feng, 1994 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 cf i^ctrUct-j S?iQSi4.<trit n<t"i*t>triH4 The University of British Columbia Vancouver, Canada Date tfWcX 3/> ><?H DE-6 (2/88) Abstract Large-signal SPICE models for heterojunction bipolar transistors (HBTs) and semicon- ductor lasers are developed. For a general graded-base double heterojunction bipolar transistor (DHBT), a Full Ebers-Moll model and its simplified versions for specific HBTs have been derived from DAPHNE, a and implemented in the circuit simulator HSPICE by using its piece-wise-linear features to represent the coefficients with voltage-dependent normalized junction velocity terms, which are used to describe tunneling factors and junction barrier heights for back-injected electrons. For uniform and moderately-graded base single heterojunction bipolar transistors (SHBTs), this model can be further sim- plified and BJT-compatible versions of the HBT SPICE model can also be derived by using an exponential fit to the normalized junction velocity. The experimental data, forward collector current and the variation of the oscillation frequency fosc with bias voltage Vcci f° r a graded-base SHBT and two five-stage ring oscillators, respectively, can be well-fitted by simulation results from DAPHNE and the BJT SPICE model. A popular large-signal equivalent circuit model, developed by Tucker [4, 5], based on the rate equation for a single-mode semiconductor laser, has been modified, simulated and compared with experimental data. Finally, the performance of HBT-laser transmitters is also simulated to show that the models developed in this thesis have the capability of being very useful design tools for HBT-laser optoelectronic integrated circuits. 1 DAPHNE: An acronym for Device Analysis .Program for heterojunction Numerical .Evaluation, has been developed at UBC based on the work of Ho [1], Ang [2], and Laser [3]. 11 Table of Contents Abstract ii List of Tables vii List of Figures ix Acknowledgement xiii 1 Introduct ion 1 1.1 Review of Existing Equivalent Circuit Models and Objectives of the Project 2 1.2 Overview of Thesis 3 2 S P I C E M o d e l Deve lopment for the General D H B T 5 2.1 Current Transport 6 2.1.1 Current Transport at an Abrupt Emitter-Base Heterojunction . . 6 2.2 The Full Intrinsic DC Model 9 2.2.1 Electron Diffusion Current in the Base 10 2.2.2 Emitter and Collector Hole Currents 12 2.2.3 Ebers-Moll Intrinsic DC Circuit Model 13 2.2.4 Voltage-Dependent Ebers-Moll Coefficients 14 2.3 The Full Extrinsic DC Model " . . . . 16 2.3.1 Recombination/Generation Currents 16 2.3.2 Resistances 18 2.4 Large-Signal HBT Model 18 iii 2.4.1 Junction Capacitances 18 2.4.2 Diffusion Capacitances • 19 3 Simplified Mode l s for Specific H B T s 22 3.1 The Full Model 23 3.1.1 Abrupt-Junction, Uniform-Base SHBT 24 3.1.2 Abrupt-Junction, Graded-Base SHBT 25 3.1.3 Abrupt-Junction, Uniform-Base DHBT 26 3.2 The BJT Model 27 3.2.1 BJT Intrinsic Model for the Abrupt-Junction, Uniform-Base SHBT 34 3.2.2 BJT Intrinsic Model for the Abrupt-Junction, Graded-Base SHBT 35 3.2.3 BJT Extrinsic Model 36 3.3 Comparisons of the DC Models 38 3.3.1 Gummel Plots 39 3.3.2 Collector Output Characteristics 45 4 Comparison wi th H B T Experimental D a t a 51 4.1 DC Characteristics 53 4.2 Large-Signal Characteristics for two 5-Stage Ring Oscillators 57 5 Laser Dev ice and Circuit Model ing 66 5.1 The Rate Equations for Semiconductor Lasers 66 5.2 Large-Signal Model for Lasers 70 5.2.1 Nonlinear Gain Parameter 72 5.2.2 Spontaneous Emission Coupling Coefficient 72 5.3 Steady-State Model 73 5.4 Simulation of Laser Steady-State and Large-Signal Characteristics . . . . 74 IV 5.4.1 Steady-State Characteristics 75 5.4.2 Large-Signal Characteristics " 75 5.5 Comparison of Simulated Laser Characteristics with Experimental Data . 78 5.6 Simulation of Laser Driver Transmitter Circuits 83 6 Summary 87 6.1 Conclusions 87 6.2 Recommendations for Future Work 88 Bibliography 90 Appendices 95 A Closed-Form Analyt ical Expressions for the Abrupt-Junct ion Tunneling Factor 95 A.l Error Function Approximation 95 A.2 Exponential Fitting 97 B The Interchangeable Relat ionships for Variables U s e d in Various Rate Equations 99 C Fermi Dirac Correction for the Bandgap Narrowing 100 D S P I C E Input Files 103 D.l Files Used in Chapter 3 103 D. l . l Ful lJV • . . . . 103 D.l.2 SHBTJib 105 D.l .3 B J T J V 108 D.2 Files Used in Chapter 4 109 v D.2.1 G3.B36 109 D.2.2 Ring_B36 • . . . . - 110 D.2.3 Switch_B36 112 D.3 Files Used in Chapter 5 114 D.3.1 SSXD 114 D.3.2 Tran_LD 116 D.3.3 Expe-LD 118 D.3.4 Mono 120 D.3.5 Hybrid 122 D.3.6 LDJib 124 vi List of Tables 3.1 Layer structure for the baseline HBT 24 3.2 Geometrical parameters for the baseline HBT (emitter area = 36 um2). . 24 3.3 Fitting parameters and corresponding ideality factors for eqn. 3.17. The graded-base case has the Al mole fraction changing linearly from 0.1 to 0 across the base. In all cases the bias range used for the fitting was about 1-1.55V 30 3.4 SPICE Full model parameters for the "baseline" SHBT 49 3.5 SPICE BJT model parameters for the "baseline" SHBT 49 3.6 Diode parameters for space charge region recombination currents of the "baseline" SHBT 50 4.1 Geometrical and compositional parameters for the F36 device 51 4.2 Layout parameters for the F36 devices (emitter area = 3 x 3um2) 53 4.3 Layout parameters for the F36 devices (emitter area = 2 x 2um2) 53 4.4 SPICE BJT model parameters for the F36 3 x 3um2 devices (BGN = IhmtV and base-emitter junction grading = 20 A) 54 4.5 SPICE BJT model parameters for the F36 devices (emitter area = 2 x 2um2). 59 5.1 Typical parameter values for a l.3um buried-heterostructure laser 74 5.2 SPICE parameters for a 1.3um buried-heterostructure laser 75 5.3 1.55 um laser parameters [52] 80 5.4 Geometrical and compositional parameters for the AT&T SHBT (emitter area = 5 x 9um2) [53]. Data marked with an asterisk are assumed values. 84 vii 5.5 Layout parameters for the AT&T SHBT (emitter area = 5 x 9um2) [53]. Data marked with an asterisk are assumed values ' 84 C.l Calculated values of the Fermi energy and Fermi-Dirac correction for uni- form and graded-base (xf,e = 0.1) cases with Pg = 1 x 1019cm-3 and PB = 5 x 1019cm-3 102 viii List of Figures 2.1 General energy band diagram for an abrupt-junction, graded-base DHBT in the active mode 5 2.2 Energy band diagram of, and current transport at, the abrupt emitter-base hetero junction 6 2.3 Schematic of charge flows in an HBT 10 2.4 Full intrinsic dc equivalent circuit 14 2.5 Full extrinsic equivalent circuit of HBT 19 3.1 Baseline HBT device structure cross-section 23 3.2 Dependence of anyn/zn over most of the forward-bias range of V(,e for the HBT specified in Table 3.1. Xbe is the aluminum mole fraction in the base close to the emitter-base boundary 29 3.3 Normalized junction velocity for iV£ = l x 101 7cm~3 ,PB = 1 x 10 1 9cm - 3 for a uniform base. The exponential fit uses A*e~VBE^sv', with A* = 0.005 and S=16.816 30 3.4 Normalized junction velocity for NE = 5 x \017cm~3,PB = 1 x 1019cm~3 for a uniform base. The exponential fit uses A*e~VB£/SVt, with A* = 1.618 and 5 = 6.937 31 3.5 Normalized junction velocity for NE = 5 x 101 7cm~3 ,Pe = 5 x 10 i9cm~3 for a uniform base. The exponential fit uses A*e~VBE^sv', with A* = 0.189 and S = 9.537 32 IX 3.6 Normalized junction velocity for NE = 5 X 1017cm 3 , PB — 1 x 1019cm 3 in the graded-base case (xbe — 0.1). The exponential fit uses A*e~VBE^SVt, with A* = 22.184 and S = 7.63 33 3.7 "BJT" extrinsic equivalent circuit of the HBT 36 3.8 SCR recombination current components (SRH = Shockley— Read—Hall, Aug = Auger, Rad — Radiative) in both the base-side(_B) and emitter- side(_E) of the depletion region at the BE junction for the "baseline" uniform-base SHBT 37 3.9 Comparison of Gummel plots for the baseline uniform-base SHBT 40 3.10 Comparison of Gummel plots for the baseline graded-base SHBT 41 3.11 Comparison of Gummel plots for the baseline uniform-base DHBT. . . . 42 3.12 log(7c) v s- VBE plot for the "baseline" uniform-base DHBT with VBC = 0V (.swO), - I V (.swl), -IV (.sw2), and -3V (.sw3) 43 3.13 log(7B) vs. VBE plot for the "baseline" uniform-base DHBT with VBC = 0V (.swO), -IV (.swl), -IV (.sw2), and -3V (.sw3) (top panel) and its expanded-voltage-scale version (bottom panel) 44 3.14 Comparison of collector output characteristics from the two SPICE models for the baseline uniform-base SHBT 46 3.15 Comparison of collector output characteristics from the SPICE Full and BJT models for the "baseline" graded-base SHBT with h — 2QuA (.swO), AOuA (.swl), QOuA (.sw2), 80uA (.sw3) and 100u.4 (.sw4) 47 3.16 Ic — VCE characteristic from the SPICE Full model for the "baseline" uniform-base DHBT with Ih = lOOuA (.swO), 200uA (.swl), 300uA (.sw2), 400u.4 (.sw3) and 500uA (.sw4) 48 4.1 Cross-section of BNR F36 HBT structure [34] 52 4.2 Layout of the BNR F36 HBT 52 4.3 Collect current characteristic comparison of experimental data with sim- ulation results from SPICE and DAPHNE for the various conditions. . . 55 4.4 Comparison of Gummel plots from BJT-SPICE-model and experimental data for BNR F36 device (assuming BGN = lOOmeV) 56 4.5 BNR F36 5-stage ring oscillator 57 4.6 BNR 5-stage ring oscillator measurement set-up 58 4.7 Comparison of BJT-SPICE-model and experimental data for the BNR 5-stage ring oscillator with RL = 20017 60 4.8 Comparison of BJT-SPICE-model and experimental data for the BNR 5-stage ring oscillator with RL = 400f) 61 4.9 Circuit for simulation of switching of single HBT. Note the transistor shown here is the full extrinsic device represented by the equivalent circuit in Figure 3.7 62 4.10 Simulation of switching of a single HBT with Vcc = 1.55V (.trO), 2.4V ( . tr l) , 3.19V (.tr2) and 4.76V (.tr3). Input pulse Vw and output pulse VOUT a r e represented by node voltages V(6) and V(4), respectively. . . . 63 4.11 Simulation of the ring oscillator with Ri = 400fl for the supply voltages Vcc = 1-55V (.trO), 1.6V (. tr l) , 1.70V (.tr2), and 1.80V (.tr3). The voltage is taken at the output of any of the stages in Figure 4.5. Note the increasing minimum of voltage swing as the supply voltage Vcc is reduced to low values 64 4.12 Propagation delay times vs Vcc comparisons of 21-, 11-, and 5-stage BNR ring oscillator with RL — 4000 65 5.1 Equivalent circuit model for rate equations 69 xi 5.2 Equivalent circuit model for laser diodes 71 5.3 Equivalent circuit model for laser diodes at steady state ' 73 5.4 V-I and L-I characteristics 76 5.5 L-I characteristic for (3ejj = 1 x 10 - 4 (.swl), and 1 x 1 0 - 3 (.sw2) respectively. 77 5.6 Transient analysis for the 1.3ttm buried-heterostructure laser: injection current (top panel) and optical output waveform (bottom panel). The following four cases are simulated by varying the nonlinear gain parameter (ep0). the dc bias current (hias), a n d the rise time (tr): 1) tp0 = 2.97W~1, hias = 1-1 Ith, and tT — 150ps (.trO); 2) ePo = 5 W _ 1 , Ibias = 1.1 Ith, and tr = 150ps ( . tr l) ; 3) tPo = 5 W - 1 , Ibias = 1.2Ith, and tr = 150ps (.tr2); 4) tPo = 5H / _ 1 , Ibia, = 1.2/ifc, and tr = 80ps (.tr3) 79 5.7 Simulated optical power waveform (top panel) and injection current (bot- tom panel) at 1.7Gb/s 81 5.8 Experimental [52] optical power waveforms at 1.7Gb/s 82 5.9 (a) Monolithic ECL HBT-laser transmitter [53]. (b) Hybrid ECL HBT- laser transmitter. The inductor in (b) represents the connection impedance: a value of 3 nH was used 85 5.10 Monolithic vs. hybrid ECL HBT-laser transmitters: optical output wave- forms (top panel) and injection currents (bottom panel) 86 A.l Comparison of the g(X) and the second-order polynomial vs. X 96 A.2 The emitter-base junction tunneling factor calculated by DAPHNE, the error function expression (zj = 1.3616, x2 = 0.468, and x3 = —0.8158), and the exponential fitting (ej = 244.55, e2 = 18.6, and e$ = —6.88). . . . 97 xn Acknowledgement I would like to sincerely thank my supervisor Dr. David L. Pulfrey for his excellent guidance, understanding and support throughout the course of this work. I am also indebted to him for his efforts in the preparation of a paper based on the work done in this thesis. I would like to express my thanks to Simon Ho, Oon Sim Ang and Allan Paul Laser for developing the HBT device simulation program, DAPHNE, which has been used extensively in my analyses, also to Dr. Hao-Sheng Zhou for modeling the space-charge- region transit-time, and to Tony St. Denis, Shawn Searles, Stephen Grant, and Bahram Ghodsian for their contributions to the device modeling, equivalent circuit modeling and invaluable discussions. Special thanks are due to Dr. John Sitch, Dr. R.K. Surridge, Dr. Juntao Hu, and Dr. T. Lester of Bell-Northern Research, Ottawa for specifying the geometrical and compositional parameters, and measurement results for the HBT devices and circuits studied in this thesis. I would also like to thank Dr. Tom Tiedje of the Physics Department, UBC for offering a course on semiconductor lasers, Gregory Buriey of the Astronomy Department for allowing access to the experimental data in his thesis. I would like to express my sincere appreciation to Dr. Qing Zhong Liu of TR Labs, Edmonton for many helpful discussions. Useful information on equivalent circuit modeling of semiconductor lasers, provided by Alvin Loke and Raymond Yip, is also acknowledged. I would also like to thank Dr. Mike Jackson for his introduction to Greyory Buriey, to Dave Gagne and Rob Ross for the generous computer system help, and also to Benny xiii Tsou for proofreading. Financial support provided by Bell-Northern Research, the Solid-State"Optoelectron- ics Consortium, and the Natural Sciences and Engineering Research Council is greatfully acknowledged. Finally, this thesis would have been impossible without the support and encourage- ment of my family, my uncle Man Hong Fung's family, my great aunt Pui Ling Lem's family, close friends, and friends in the Solid State group. xiv Chapter 1 Introduct ion The high-speed, high-current-drive and low-noise capabilities make heterojunction bipo- lar transistors (HBTs) very promising candidates for applications in optoelectronic com- munication systems. HBTs have already been monolithically integrated with photode- tectors to produce optical receiver front-ends operating at 1.5 um with data rates as high as lOGb/s [6]. In this application the high-frequency and low-noise [7] properties of HBTs were exploited. Further development of receiver circuitry using small-signal models for HBTs [3, 8] can be expected. For other components in future lightwave systems, e.g. multiplexers, demultiplexers, decision circuits and laser drivers, it is the large-signal prop- erties of HBTs which are important. To develop these circuits it follows that large-signal simulations tools for HBTs are required. SPICE would be a very convenient tool as it is widely used and easily obtainable. The difficulty in incorporating a useful, physics-based HBT model in SPICE is that the Ebers-moll coefficients are voltage-dependent [2, 9], and therefore, the HBT would not appear to be able to be properly described by the BJT macromodule in SPICE. However, the incentive to use the existing B J T macromodule is so great that it provided the motivation for this thesis, namely: to examine the physics of operation of the HBT and find a way in which the equations describing its performance could be cast in a form suitable for their implementation in existing versions of SPICE, i.e., without having to write new source code. With a large-signal model for the HBT available it would be possible to design, via circuit simulation, circuits for the large-signal components of transmission systems [10]. 1 Chapter 1. Introduction 2 Here at UBC, there is a particular interest in HBT/laser driver/transmitter circuits as these are under investigation via a contract with the Solid-State Optoelectronic Consor- tium (SSOC) at NRC, Ottawa. Therefore, another aspect of the work pursued in this thesis was the development of a large-signal model for semiconductor lasers. Having such a model for lasers and a similar one for HBTs would greatly facilitate the design of a successful driver/transmitter circuit. 1.1 R e v i e w of Exist ing Equivalent Circuit Models and Object ives of the Projec t To be of universal appeal, the physics-based equivalent circuit model parameters should relate to the geometrical, compositional and known-electrical parameters directly. Em- pirical models do not meet this criterion as they rely on parameter extraction from exper- imental data for specific HBTs [11, 12, 13]. Previously-reported physics-based HBT mod- els are not suitable for innovative design because of: 1) making over-simplified assump- tions, for example, characterizing the carrier transport across the heterojunction by drift- diffusion [14]; 2) neglecting the neutral base recombination [15, 16, 17]; 3) not showing how to account for the voltage dependencies of the tunneling factor and the barrier height for back injection of electrons [16, 18, 19]; 4) or failing to present a complete physical pic- ture of carrier-transport mechanisms under a wride range of structures. None of them dealt with the technologically-important case of graded-base HBTs. For Ebers-Moll like mod- els, Grinberg et al. first derived the Thermionic-Field-Diffusion model for thermionic- emission and tunneling currents across the conduction band "spike" for uniform-base single heterojunction bipolar transistors (SHBTs) [20]. Lundstrom introduced an expo- nential bias-dependent junction velocity to characterize the thermionic-emission current across the "spike", and extended the analysis to uniform-base DHBTs and graded-base Chapter 1. Introduction 3 SHBTs [21]. And Teeter included the transit-time effects [22]. DAPHNE, a numerical model developed at UBC for a general graded-junction and graded-baseTJHBT, incor- porates the tunneling current into this bias-dependent junction velocity term [1, 2], re- combination current to account for the neutral base recombination, Shockley-Read-Hall, Auger and radiative processes in the space-charge-region (SCR) [1, 23], and calculation of the maximum frequency of oscillation [3]. The first objective of this project is to develop complete and simple large-signal equivalent circuit models for the devices that can be modeled by DAPHNE. The only large-signal SPICE model for semiconductor lasers to date was developed by Tucker to analyze the performance of direct modulation, and of the electrical para- sitics and chirping [4, 5]. However, this model used the voltage of the output node to represent the photon density, which cannot be measured, and also needed to introduce a normalization constant to avoid numerical overflow [4]. The second objective of this project is to modify this model by replacing the photon density by the output power to achieve a more useful representation of the laser with better accuracy and convergence. The third objective of this project is to demonstrate that these models will be useful in HBT-laser optoelectronic integrated-circuit design. 1.2 Overview of Thesis In Chapter 1, wre have described the needs for physics-based large-signal equivalent circuit models for HBTs and laser diodes, and then briefly reviewed the existing SPICE models and objectives of this project. In Chapter 2, Full intrinsic dc, extrinsic dc and large-signal equivalent circuit models are developed to account for the voltage dependencies of the Ebers-Moll coefficients, the SCR recombination/generation currents at the junction and the parasitics. Chapter 1. Introduction 4 In Chapter 3, the Full model is simplified for specific HBTs. By using an exponen- tial fit to the normalized junction velocity, the simplest BJT versions for uniform and moderately-graded-base HBTs are also derived. Forward Gummel plots and collector- output characteristics are simulated to highlight the accuracy of these simplified models. In Chapter 4, both dc and large-signal experimental data for HBTs designed and fabricated at Bell Northern Research (BNR) are compared with the simulated results from our model. In Chapter 5, a large-signal equivalent circuit model for semiconductor lasers is de- rived, simulated and verified by comparison with the response of an actual laser; then the performance of HBT-laser monolithic and hybrid transmitters are also simulated. In Chapter 6, conclusions and recommendations for future work are presented. Finally, in Appendix A, closed-form analytical expressions for an abrupt junction tun- neling factor are derived using either an error function approximation or an. exponential fitting; in Appendix B, the interchangeable relationships for variables used in various rate equations expressed in terms of photon population P, photon density S and out- put power P0 are given; in Appendix C, the Fermi Dirac correction calculation for the bandgap narrowing is derived; and in Appendix D, SPICE input files are listed. Chapter 2 SPICE Model Development for the General DHBT Efi nn qVBE / f AEnE Efnp i AEnC \l \ Efp Emitter *JE o Base Efi nn I qVBC Collector W XJC * c Figure 2.1: General energy band diagram for an abrupt-junction, graded-base DHBT in the active mode. In this chapter, a large-signal equivalent circuit model is developed for an abrupt-junction, graded-base double heterojunction bipolar transistor (DHBT) with the general energy band diagram shown in Figure 2.1. Current transport across the heterojunctions based 5 Chapter 2. SPICE Model Development for the General DEBT 6 on the Thermionic-Field-Diffusion model [20] is briefly reviewed. Full intrinsic dc, ex- trinsic dc and large-signal equivalent circuit models are then developed from DAPHNE, which employs a one-dimensional model for the derivation of the current equations and a quasi-two-dimensional model for the formulation of the parasitic components needed to calculate the high frequency figures-of-merit [1, 2, 3]. 2.1 Current Transport 2.1.1 Current Transport at an Abrupt Emit ter-Base Heterojunct ion ^ jTnE A. JT1 Figure 2.2: Energy band diagram of, and current transport at, the abrupt emitter-base heterojunction. Chapter 2. SPICE Model Development for the General DHBT 7 The model for current transport across the abrupt emitter-base junction is based on the Thermionic-Field-Diffusion model of Grinberg et al. [20]. The various current compo- nents are shown in Figure 2.2. JT\ and Jx2 are the thermionic emission current densities while JF\ and JE2 are the tunneling current densities. Assuming low-level injection, the net electron current density injected from the emitter to the base is given by [9] JTnE = -qvTnElnEe-AE"E/kT[nBO(ev^v'-l)-h(0)} = -qSE[nBo(eVBE/Vt-l)-n(0)} (2.1) where VTnE is the average electron thermal velocity in the emitter in a direction normal to the junction, -ynE is the tunneling factor to account for the transport of electrons through the narrow emitter-base junction "spike" due to tunneling, AEnE is the electron potential energy barrier as shown in Figure 2!2, n^o is the equilibrium electron concentration at the depletion edge (x = 0, see Figure 2.1) of the base, Vt is the thermal voltage, VBE is the applied potential across the emitter-base junction, n(0) is the excess concentration at the depletion edge of the base and SE = VTnE7nEe~AE"E^kT is defined as the emitter-base junction velocity and follows the original formulation of Lundstrom [21], but with the incorporation of the tunneling factor 7„. The various components of JrnE are given by / kT VTnE = 27rm *nE AEnE = 6EVBE + AEnEo • NQBENEPB* nlBENCE tENE AEnE0 = ( X ' B E - X J S ) ( 1 -6E)-SEkT]n(—5—^—-) SE = CBEPB + tENE — n ' B0 (i <^ PB where eE is the permittivity of the emitter, CBE is the permittivity of the base at the emitter side, \E is the electron affinity of the emitter, XBE is the electron affinity of Chapter 2. SPICE Model Development for the General DEBT 8 the base at the emitter side, n,Bo is the intrinsic carrier concentration in the base at the depletion edge (x = 0), riiBE is the intrinsic carrier concentration jn the base at the metallurgical emitter-base junction, NCBE is the effective density of states in the conduction band of the base at the emitter side, NCE is the effective density of states in the conduction band of the emitter, m*£ is the electron effective mass in the emitter, NE is the doping concentration of the emitter and PB is the doping concentration of the base. The tunneling factor 7 „ E is given by [1] InE = 1 + y JEm D(E)exp(- — )dE (2.3) where ET\ = qVri = q{VbiE - VBE) - AECE + AEnE AECE = XBE - XE- (2.4) The lower limit of the integral in (2.3) is given by either E* = En — AECE when the energy level of the conduction band in the quasi-neutral emitter is lower than the potential notch, or by 0 when when the potential notch falls below the energy level of the conduction band in the quasi-neutral emitter as is the case in Figure 2.2. In the above equations, V^E is the built-in potential for the base-emitter junction, En is the electron potential energy barrier shown in Figure 2.2, AECE is the conduction band discontinuity for the base-emitter junction, and D(E) is the barrier transparency. D(E) is given by [1] D(E) = exp{-^[Vl^X -Xln(l + "%*)]} (2.5) where X = f, Ex=!fJJ^. (2.6) En * V eEmnE Chapter 2. SPICE Model Development for the General DHBT 9 2.2 T h e Full Intrinsic D C Model In this section, the electron diffusion current in the base and the hole currents in the emit- ter and collector are discussed. An analytical Ebers-Moll SPICE model for the graded- junction and graded-base DHBT is then derived from the equations used in DAPHNE. The voltage-dependent Ebers-Moll coefficients are then simplified into a form which can be implemented into circuit analysis programs with piece-wise-linear features. In the analytical equivalent circuit model, the following assumptions are made: 1. The base is highly doped, which means that basewidth modulation can be ignored and the quasi-neutral base width is constant and effectively equal to the thickness of the base W. 2. The emitter and collector are considered to be wide, so that in these regions, the quasi-neutral thicknesses equal the metallurgical thicknesses WE and Wc, respec- tively. 3. The emitter and collector surface contacts are considered to be ohmic. 4. The effective densities of states in the conduction and valence bands of the base are assumed to remain the same, regardless of the base grading. 5. The intrinsic carrier concentrations at the depletion edges of the base, U,BO and TCtBvr, are assumed constant and equal to the values at the metallurgical junctions, i.e., riiBo = n,BE and n,Bvv = riiBC, for the emitter and collector edges of the base, respectively. Here, the emitter current shown in Figure 2.3 is redefined to be positive in the opposite direction to that of previous work [1, 2] so that the Ebers-Moll coefficients are always positive. Chapter 2. SPICE Model Development for the General DEBT 10 W E W B Wc JE JR.BE Jp(xE) Jn(0) Jn(W) JNB JR/G.BC Jp(xC) JC c XE JB I O B W xc Figure 2.3: Schematic of charge flows in an HBT. 2.2.1 Electron Diffusion Current in the Base The expressions for the emitter and collector electron current densities at the depletion edges x = 0 and x = W shown in Figure 2.3 are: Jn(0) = - o „ ( C v « > v ' - l ) + a 1 2 ( e v * * ' v ' - l ) Jn(W) = _ Q 2 1 ( e v ^ / v ' - l ) + Q 2 2 ( e v ' ^ / v ' - l ) where the Ebers-Moll coefficients ctij for the electron currents are [2] (2.7) AoVn BoVn A-wVn BwVn Oi l = , "12 = , a 2 1 = , "22 = (2.8) Chapter 2. SPICE Model Development for the General DEBT 11 and z = (zc + bnyn){zZ + anyn)-(2tynesW)2 A0 = z*nB0[an{zZ + Kyn) - 4*2yne2sVV] B0 = 2tZnZnUBW Aw = zt2n zn nBoe Bw = z^nBw[bn(z^ + anyn)-4t2yne2sW}. (2.9) The components of the coefficients above are AEg EgBE ~ EgBC f kTW kTW . JPLlB+4 f t = , a = - , ri=s + t, r2 = s - t 6LnB I 2 2 7T n»B0 - _ niBW - 2sW "BO = -77—, riBW — —7;— = riBoe rB rB an = (ri-fy*w-(r2-fy>w bn = (ri-f)er>w-(r2-f)eT>w qDnB Vn = oT\W pTiW ^n = °vTnE~fnEe kT , 2^ = 9 u r n C 7 n C e *r (2.10) where E9BE and £?flBc are the energy bandgaps in the base at the emitter and collector sides respectively, LnB is the electron minority carrier diffusion length in the base, Dns is the electron diffusion coefficient in the base, W is the thickness of the quasi-neutral region of the base, Pg is the doping concentration in the base, ngw is the equilibrium electron concentration at the depletion edge (1 = W, see Figure 2.1) of the base, riisw is the intrinsic carrier concentration at the depletion edge (x = W) of the base, and vjnc, 7 n c , and AEnc are the thermal velocity, tunneling factor, and electron potential energy, respectively, for the collector-base junction. Chapter 2. SPICE Model Development for the General DEBT 12 2.2.2 Emi t ter and Collector Hole Currents The emitter hole current, JP(XE), shown in Figure 2.3 can be found by equating the hole diffusion current in the emitter and the hole thermionic emission current across the emitter-base junction. With an infinite emitter surface recombination velocity, JP(XE) is given by [l] JP(xE) = JpE(eVBE/Vt - 1) (2.11) where ^ c o t M g ) JpE LpE V£pE ' ZE f ^ c o t h ( p ^ ) ZE = 1 + ^ -±^ (2.12) VTpE where pE is the equilibrium hole concentration in the emitter, LPE is the hole diffusion length in the emitter, DPE is the hole diffusivity in the emitter, VTPE is the average hole thermal velocity in the emitter in a direction normal to the junction, and WE is the thickness of the emitter. Assuming infinite thermal velocity for holes at the base-emitter junction, JPE becomes: J „ E = i ^ * c o t h ( ^ ) . (2.13) •LpE LpE This back-injected hole current is small because the intrinsic carrier concentration in the emitter is small. Similarly, the collector hole current, Jp(xc), shown in Figure 2.3 is given by JP(xc) = JPc(eVBc/V'-l) (2.14) where qDPcPc ^(WCs / 0 1 C N JpC = —y co th ( -—) (2.15) LVC Lpc where pc is the equilibrium hole concentration in the collector, Lpc is the hole diffusion length in the collector, Dpc is the hole diffusivity in the collector, and Wc is the thickness Chapter 2. SPICE Model Development for the General DEBT 13 of the collector. From (2.11) and (2.14), it can be seen that the hole currents can be represented by two diodes with equations of the form Ipi = JpiAE(eVB'/N'v' - 1) i = E,C (2.16) where the ideality factors NE and Nc are equal to unity and AE is the emitter area. 2.2.3 Ebers-Moll Intrinsic D C Circuit M ode l Since the neutral base recombination current I^B is INB = AE\Jn{W) - Jn(0)] = INBE + INBC (2.17) where INBE = AE(au-a2l){eVBE>Vt-l) INBC = AE(a22 - a12)(ev»c'v' - 1) (2.18) the three terminal currents in the HBT (see Figure 2.3) can be expressed as IE = AE[Jn{0) - Jp(xE)} =-INBE-IP(XE) +IEC - Ice Ic = AE[-Jn(W) - Jp(xc)] = -INBC -IP(XC) - IEC + Ice IB = -IE-IC (2-19) where the electron transport currents are: Ice = AEa21(ev*E'Vt - 1) IEC = AEau(ev»c?v' - 1). (2.20) Therefore, representing the hole currents at the junctions by the diode elements Dl and D2 and the electron diffusion currents by current sources, the Ebers-Moll intrinsic dc model can be represented by the equivalent circuit of Figure 2.4. The circuit for the Chapter 2. SPICE Model Development for the General DEBT 14 traditional Ebers-Moll representation of a homojunction transistors can be depicted in the same form as that of Figure 2.4, but note that in the HBT case, the a^ coefficients implicit in the current sources are not constant but are voltage-dependent. B IB D1 Ic D2 f t ) INBC • ) INBE Ice ( k K lEC IE Figure 2.4: Full intrinsic dc equivalent circuit. 2.2.4 Vol tage-Dependent Ebers-Moll Coefficients We also define where „ Z C\ Co C3 ZCZE SNE SNC SNESNC (2.21) On2/n bnyn c\ = , c2 = — , c3 - r 2 D2nB qVTnE qVTnC LnBVTnEVTnC Chapter 2. SPICE Model Development for the General DHBT 15 SNE = lnEe-AE^kT, SNC = lnce-AE"c'kT ( 2 .22) where SHE and Sjvc are defined as normalized junction velocities at the emitter-base junction and collector-base junction, respectively. Their voltage-dependence poses the principal problem as regards producing SPICE parameters for heterojunction devices. Since the product of the effective densities of states in the conduction and valence band, NcNy, is assumed to be constant in the base, the ctij coefficients can be rewritten as 2tnBoe2sWyn Q 1 2 = Q 2 1 = nB0anyn + UBQ-^— " i i a n« n + ,2 q nB Q— _ nyn L.' vTnCSNC = nB0 — nBwbnyn + nBw^§— Q 2 2 — n L2BvTnESNE -_ 0 . = nBW y • (2.23) The four current sources of Figure 2.4 for the neutral base recombination and electron transport can be transformed into the form INBE = i ^ ( l + ^ ) ( e v « / * - l ) Z dNC j h,NBC n , d2 u VBC/V, nN INBC = — y — ( 1 + 7;—)(e BCI - 1) Ice = t-fie^'-l) he = ^£(eVBC'Vt-l) (2.24) where IS,F = 2tnB0yne2sWAE IS,NBE = nB0yn(an - 2te s )AE Chapter 2. SPICE Model Development for the General DEBT 16 IS,NBC = nBwyn(bn - 2t)A£ dx = iliafli vTncyn{an - 2te2°w) d2 = HKL»BJ n^. (2.25) VTnEVnibn ~ 2t) Therefore the circuit of Figure 2.4 can be implemented in those versions of SPICE which allow the voltage-dependent ctij coefficients to be described algebraically with piece-wise- linear descriptions of the voltage-dependence of the normalized junction velocities. This model has been implemented in HSPICE, a commercial version of SPICE marketed by Meta-Software, Inc, and is called the Full, intrinsic dc model. 2.3 T h e Full Extrinsic D C Mode l To turn the Full intrinsic dc model into a Full extrinsic dc model, it is necessary to add circuit elements to account for the space-charge-region (SCR) recombination-generation currents at the junctions and the parasitic resistances. The detailed recombination cur- rent model used in DAPHNE is based on that of Ho [1] and Searles [23], and includes Shockley-Read-Hall, Auger and radiative processes. Parasitic resistances can be calcu- lated from DAPHNE for single-sided [2, 3], pyramidal [9] and rectangular [24] structures. 2.3.1 Recombinat ion /Genera t ion Currents For the case of recombination in the emitter-base depletion region, it has been pointed- out that the splitting of the electron quasi-Fermi level AEjn at the junction must be considered if the base-side recombination current is not to be grossly overestimated [25]. However, the likely components of current (SRH, Auger and radiative) in both sides of the depletion region (see JR,BE in Figure 2.3) can still be represented by diode-like Chapter 2. SPICE Model Development for the General DHBT 17 expressions, i.e., [23] T r, NEniBE AE<-6BEVt,iF, 6BEvBE-*Bfn JSRH,B = Cs e v< e vt TnOK-iE JSHH,E = Cs^eW* IT VBE-AEfn JAU9,B = CsniBEApPBe vt JAU9,E = Csn2lEAnNEeVBE/v< T CSVhlE 2 D , vBE-*Bfn JRad,B = — T J — n i B E B p 6 E e v< T CsVbiE 2 c c ^ ,o oe \ jRad,E = — 7 } — r i i E B n d B E e v' (2.26) Vt where n T/ , ZqtBE ^s = vt &BE = PBSEVKE £BEPB CBEPB + tENE Tl' J? AEi = Vt ln(——) + XBE - XE riiBE AEjn = Ejnn - Ejnp = Vt ln( — — ) VTnEVVnB,E^NE w WnB,E = LnBt&nh(-—) •LnB T = y/TnOTpo (2.27) where Ejnn and Efnp are quasi-Fermi levels shown in Figures 2.1-2.2, rn 0 and rp0 are the electron and hole minority carrier lifetimes in the p-type material and n-type material, respectively, An and Ap are the electron and hole Auger coefficients, respectively, and Bn and Bv are the electron and hole radiative recombination coefficients, respectively. For the base-collector junction of a single heterojunction bipolar transistor (SHBT), the standard expression for the depletion region recombination/generation current (see JR/G,BC in Figure 2.3) in a homojunction applies, [26] i.e., 7 qniBC (WB}SCR , Wc,SCR\f_v„nfvt -x , 9 0 S x JBC,SRH = —7i—( 1" )( e ~ 1) [Z.Zb) Z Tp0 r n 0 Chapter 2. SPICE Model Development for the General DHBT 18 where the depletion-region widths WB,SCR and WC,SCR are evaluated at a fixed bias. Recombination/generation currents JR^BE and JR/G,BC are represented by two diodes D3 and D4 respectively in Figure 2.5. 2.3.2 R e s i s t a n c e s The parasitic resistances can be simply found from DAPHNE by adding-up the intrinsic, extrinsic, and contact portions of the emitter, base and collector resistances under zero- bias conditions, i.e., RE = REI + REX 4- REC RB = RBI + RBX + RBC Rc = Rci + Rcx + Rcc (2.29) Expressions for these resistances for single-sided, pyramidal, and rectangular device struc- tures can be found in [2, 3, 9, 24]. 2.4 La rge -S igna l H B T M o d e l In this section, a Full extrinsic model for large-signal simulation (see Figure 2.5) is ob- tained by adding junction capacitances and diffusion capacitances into the Full extrinsic dc model. 2.4.1 J u n c t i o n C a p a c i t a n c e s In the Full version of the SPICE model, the junction capacitances can be included in the diodes Dl and D2 by supplying the zero-bias values and the junction built-in potentials. This data can be obtained from DAPHNE, i.e., CJE = C E J ( O ) , CJC = C C J ( O ) , VJE = Vbie, and VJC = Vbic. The SPICE parameters for the forward-bias Chapter 2. SPICE Model Development for the General DEBT 19 IC RC B IB RB UDC j CJC A D4 ^ D2 (. t XDE , CJE V D3 INBC i ) Ice( D1 ( I ) INBE RE IE IEC Figure 2.5: Full extrinsic equivalent circuit of HBT. depletion capacitor coefficient, FC, the base-emitter junction grading coefficient, MJE, and the base-collector junction grading coefficient, MJC, can also be included to model the non-ideal junction capacitances. 2.4.2 Diffusion Capacitances In the normal, active mode of operation the principal stored minority carrier charge resides in the base and the base-collector space charge region [27]. This charge arises from a change in VBE ar"d is represented in SPICE by a capacitor CDE-, which is characterized Chapter 2. SPICE Model Development for the General DEBT 20 by a forward transit time TF, i.e., TF = TB + TBC,scR (2.30) where rg is the base transit time and TBC,SCR is the base-collector depletion-region signal delay time. To properly model operation in the saturation regime, a reverse transit t ime TR must also be defined, and is computed from (2.30) but using TB and TBC,SCR as evaluated for the device operating in the inverse active mode. In the Full version of the SPICE model, the diffusion capacitances are represented by current-controlled capacitors connected between the appropriate terminals, i.e., CDE = TFIcc/NFVt CDC = rRIEC/NRVt (2.31) where NF and NR are the ideality factors for the base-emitter and base-collector junc- tions, respectively. Base Transit T i m e The base transit time is given by 1 fw TB = T iw\ / ~<in(.x)dx (2-32) Jn[W ) Jo where ft(x) = C\trxX•\-C2^"2X is the excess electron concentration, and Jn(W) is calculated from [9] Jn(x) = qDnB[(ri - . O d e ' 1 * + (r2 - f)C2eT>x] (2.33) with _ n(W) - n ( 0 ) e ^ n(0)e'*w - h(W) , 1 2c'w 'sinh(<H0 ' 2esM/sinh(iVK) ' ^ " j Chapter 2. SPICE Model Development for the General DEBT 21 The result of the integration is r -- 1 h 2*n-(WQ + qHn(0) B rir2DnB[ 2th(0)e*°w - bnh(Wy { } where . ( w ) = 2 t y n ^ e 2 ^ n B 0 ( e V f l £ / V i _ i ) (2.36) After some algebra, (2.35) reduces to Z£ * c o s h ( W ) + £ sinh(<W0 sinh(^W) •^nS ^ e 2 K K te? be = r B l + ^ (2.37) AE„f7 with the bias-dependent base-collector junction velocity 5"c = ^Tnce vt ^nc = VTUCSNC- For SHBTs, 5c is assumed to be infinite and T# becomes constant. Note also, that if the SHBT has a uniform base and W <C LnB, (2.37) gives the familiar value (777—) for TB- Base-Col lector Deple t ion Region Transit T ime The signal delay time through the base-collector depletion region for a uniform carrier velocity is [28] TBCSCR = 0.5WBC'SCR = 0.5rt (2.38) Vsat where WBC,SCR is the full depletion-region width at the base-collector junction (calculated at some fixed bias), vsat is the saturation velocity for electrons in the space charge region and Tt is the transit-time. However, in AlGaAs/GaAs HBTs the velocity profile is likely to be highly nonuniform due to velocity overshoot effects, and for conventional collector structures, TBCSCR/TI is about 0.4 [29]. Chapter 3 Simplified Models for Specific HBTs The objectives of this chapter are to simplify the Full intrinsic dc model and base transit time expression to describe specific abrupt-junction HBTs such as: uniform-base SHBT, graded-base SHBT, and uniform-base DHBT, so that the SPICE parameters can be expressed in terms of geometrical, compositional and known-electrical parameters. As device designers would begin with these parameters, the SPICE models developed here should be very useful in rapidly assessing the impact of device design changes on circuit performance. To further increase the utility of the HBT SPICE models, a model which can be implemented in the public-domain versions of SPICE is also described in this chapter. This model is called the BJT model and works for abrupt-junction SHBTs with either a uniform-base or a moderately-graded-base. The results for the various SPICE models are compared with the dc results simulated by DAPHNE, using the data in Tables 3.1 and 3.2 and assuming a pyramidal structure (see Figure 3.1) [1]. The data in Table 3.1 represents a typical "baseline" device, such as has been used previously in analyses of reciprocity [30] and space charge region recombination [23]. The electrode spacings in Table 3.2 are typical of state-of-art technology [31]. 22 Chapter 3. Simplified Models for Specific HBTs 23 Se Sub-collector Figure 3.1: Baseline HBT device structure cross-section. 3.1 T h e Full Mode l For SHBTs, the thermal velocity for electrons at the base-collector junction is assumed infinite. Therefore, the current source INBE in (2.24) reduces to lNBE=^(eV^V'-l). (3.1) The Z term in (2.21) is also redefined as Z = l+Wn = 1 + SNE (3-2) Chapter 3. Simplified Models for Specific HBTs Layer Emitter Base Collector Sub-collector Material n — AlxGa\-xAs p+ — GaAs n~ — AlxGai~xAs n + — GaAs Thickness A 4000 1000 4000 4000 Doping (cm~3) 5 x 1017 1 x 1019 5 x 1016 1 x 1018 Al composition 0.3 0 or 0.1-0 0 or 0.3 0 Table 3.1: Layer structure for the baseline HBT. Se sb Sc 6.0um A.bum 3.5urn ^biso Seb Sbc 0.75um 0.25um O.burn Le U Lc 6.0um 4.5um 3.0um Table 3.2: Geometrical parameters for the baseline HBT (emitter area = 36 um2). For uniform-base HBTs, the components an, bn and yn in (2.10) are 2 W a„ = bn - -—cosh(-—) J-inB LinB Vn qDn B 2 s i n h ( T ^ ) ' 3.1.1 Abrupt -Junct ion , Uniform-Base S H B T The a%1 coefficients in (2.23) are (3 « i i = O-nVnnB 2tynnB Q 1 2 = a2\ = &22 = -^[OnJ/n + 7^ £ J Z L nBVTnE^NE and the SPICE parameters in (2.22) and (2.25) are Cl = DnBcoth(^) LnB^TnE (3 Chapter 3. Simplified Models for Specific HBTs 25 _ qnBDnBAE S ' F ~ T • -L/ W \ LnBsmhij^) qnBDnB[cosh(-^)-l]AE h,NBE = h,NBC = ~ . ,,nw \ LnBsmhij^) DnBsmh(^) LnBVTnE[cOsh(^-) - 1] d2 = r ~ n °r2Lw[ , • (3-5) Also the current gain BF for the base transport is constant and equal to: BF=-^-= I (3.6) JS,NBE COsh( j^) - 1 The base forward transit time in (2.37) for the abrupt-junction, uniform-base SHBT is rB = | ^ [ c o s h ( - ^ ) - 1] L>nB LnB w2 2DnB 3.1.2 Abrupt-Junction, Graded-Base SHBT The components of the a,j coefficients in (2.10) are <zn = e^ w[2tcosh(tW) + fsmh(tW)} bn = e*w[2t cosh(tW) - f smh{tW)} if W < LnB. (3.7) y n = * D «B . ( 3 . 8 ) 2e£wsinh(*W) The o.ij coefficients in (2.23) are OnS/n^BO on = Z 2tynnBW Q12 = a 2 i = - " B H ' f t qD2nB 1 ,o QN Z L„BVTnE3NE and the SPICE parameters in (2.22) and (2.25) are DnB[tcoth(tW) + -L) VTnE Chapter 3. Simplified Models for Specific HBTs 26 _ gnBWDnBtAE 5 ' F ~ e$wsmh{tW) f teiw IS,NBE = qnBoDnBAE[tcoth(tW) + -2 sinh(iW') IS,NBC = qnBWDnBAE[tcoth(tW)- j ~ -] ^ e^ w s'mh(tW) Dn B L2nBvTnE[t coth(tW) - i - , « ]' e* sinh(tW') + fet(l + ^ - e ' W ) 3.1.3 Abrupt-Junction, Uniform-Base DHBT The Q,-j coefficients in (2.23) are " B r , 9-Pnfi i Qll = -ITianVn + J^ J J Z L nBvTnCONC 2tynnB 0-12 = c*2i- Z " B r „ .. , 9 ^ •^ LnBvTnESNE and the SPICE parameters in (2.22) and (2.25) are c\ = DnBcoth(^) LnBVTnE A C2 = DnBcoth(^) LnB^TnC (3.10) Also the current gain BF for the base transport is constant and equal to B F = - ^ - = i U j - . (3.11) /S,TVB£ i cosh(W) + | smh(tW) - teiw The base transit time in (2.37) for the abrupt-junction, graded-base SHBT is L2nB t cosh(W) + { sinh(W) ,. TB = ~D7B{ ^ 1] 2DnB P n B - ^ ^ 1 f()r W < L n B . ( 3 > 1 2 ) e2 a22 = - ^ [ a n y n + r 2 v n* ] (3.13) Chapter 3. Simplified Models for Specific HBTs 27 C3 IS,F D 2 n B LlBVTnEVTnC qnBDnBAE LnB sinh(-j^-) nB ' qnBDnBAE[cos\\{~-) - 1] = Is ^ * - 'S,NBE = *S,NBC = • i , W \ L ^ s m h ^ ) d, = DnBS-mh{^) LnBVTnc[cOsh(j—) - 1] A ^ s i n h ^ ) di = J , uw B\ i r (3-14) -'nB ' Also the current gain BF for the base transport is p P _ JS,F _ 1 BFSHBT m - x ^ - IS..BE " [ C o s h ( i ) - 1](1 + j j L ) - (1 + ^ Y (6Ab> The base transit time in (2.37) for the abrupt-junction, uniform-base DHBT is L2„n , , , TT,X •. sinh(il'F) TB = jf^[cOsh(tW) - 1] + —-^ 1 DnB tJNC W2 W X T „ r ~ T F ^ + ^ lfW<LnB = rBi + ^ - (3.16) where rB\ — -^— and TB2 = ~ ~ • Lundstrom [21] also derived a similar expression and pointed out that the increase in TB is caused by the conduction band spike at the collector-base junction, which makes Sc <C vjuC, and therefore, confines the carriers in the base. 3.2 The B J T Mode l The Full SPICE model described in Section 3.1 is likely to give an accurate representation of the HBT, but at the expense of considerable execution time. The presence of the voltage-controlled current sources in the model (see Figure 2.4), and the need for piece- wise representation of the data to describe the tunneling and the junction barrier height, Chapter 3. Simplified Models for Specific HBTs 28 does not make for a compact circuit model that one could envisage using in the design of complex HBT circuits. It also demands that commercial versions of SPICE, such as HSPICE, must be used as only these support the piece-wise and customer-specified equation features. In this section, the Full model is reduced to a much simpler model, one that can be represented by the BJT macromodule already present in all versions of SPICE. Such a model should be of great use to designers and should also help avoid problems such as "the convergence problem" and "the time step too small problem" which can result from simulation of circuits using models as complicated as that in Figure 2.4. This simplification can be brought about by approximating the Z term in (3.2) and representing the voltage dependencies of the tunneling factor and the junction barrier height by a simple diode ideality factor. First, Figure 3.2 for the SHBTs described in Table 3.1 shows that ^^^ ^> 1 is valid over most of the forward-bias regime, provided that the aluminum mole fraction at the base-emitter boundary does not exceed about 0.1. This condition allows the Z term to be written as ^—. Secondly, the normalized junction velocity is represented by a simple exponential expression, i.e., SNE = A"e-VBE/SVt (3.17) where A* and S are fitting parameters. Bandgap narrowing in the base was also taken into account by utilizing recent GaAs data [32] (see Appendix C) and making an appropriate correction for the use of Fermi-Dirac statistics[33] (see Table C.l in Appendix C). The best-fit values, by minimizing the average relative errors, for A* and S (and corresponding forward ideality factor NF) for a range of doping densities, for both uniform and graded bases, are listed in Table 3.3. This relationship for SNE can be made to be reasonably accurate over the bias range of general interest for the uniform-base case and low base- grading cases, as shown in Figures 3.3-3.6. Chapter 3. Simplified Models for Specific HBTs 29 1 0 0 0 0 0 ' • ' • ' • ' ' 0.0 0.5 1.0 1.5 VBE (V) Figure 3.2: Dependence of anyn/zn over most of the forward-bias range of He f° r the HBT specified in Table 3.1. Xbe is the aluminum mole fraction in the base close to the emitter-base boundary. Chapter 3. Simplified Models for Specific HBTs 30 NE {cm-3) 1 x 1017 5 x 1017 5 x 1017 PB (cm-3) 1 x 1019 1 x 1019 5 x 1019 uniform base A' 0.005 1.618 0.189 S 16.816 6.937 9.537 NF 1.063 1.168 1.117 graded base A' 0.101 22.184 2.045 5 18.202 7.63 11.237 NF 1.058 1.151 1.098 Table 3.3: Fitting parameters and corresponding ideality factors for eqn. 3.17. The graded-base case has the Al mole fraction changing linearly from 0.1 to 0 across the base. In all cases the bias range used for the fitting was about 1-1.55V. 5.5 x10 4.5 •D I 3.5 to E o ^ 3 CD c CO 2.5- 1.5 0.9 NE=1e+17PB=1e+19 DAPHNE Exponential fitting 1.1 1.2 1.3 1.4 1.5 1.6 Vbe (V) Figure 3.3: Normalized junction velocity for NE = 1 x 1017cm"3,P.B = 1 x 1019cm 3 for a uniform base. The exponential fit uses A*e~VBElSVt, with A" = 0.005 and S=16.816. Chapter 3. Simplified Models for Specific HBTs 31 x 1 0 ~ T5 N 4 16 E O ID 3 c CO 2- NE=5e+17PB=1e+19 DAPHNE Exponential fitting 1.1 1.2 1.3 Vbe (V) 1.4 1.5 1.6 Figure 3.4: Normalized junction velocity for NE = 5 x 1017cm 3 ,Pg = 1 x 1019cm 3 for a uniform base. The exponential fit uses A*e~VBE^sv', with A* = 1.618 and S = 6.937. Chapter 3. Simplified Models for Specific HBTs 32 3.5 x10"' 2.5 "O ~ 2 to E o c ^ 1 . 5 c 0.5- NE=5e+17PB=5e+19 - DAPHNE -- Exponential fitting 1.1 1.2 1.3 Vbe (V) 1.4 1.5 1.6 Figure 3.5: Normalized junction velocity for NE = 5 X 1017cm 3, Pg = 5 x 1019cm 3 for a uniform base. The exponential fit uses A"e~VBElsVl, with A* = 0.189 and S — 9.537. Chapter 3. Simplified Models for Specific HBTs 33 0.14 0.12 0.1 S" <D N |0 .08 o c^ Q) -0.06 0.04 0.02 • v i i i . | ... | , \ \ NE=5e+17PB=1e+19 \ \ \ \ \ - DAPHNE \ \ \ \ \ \ V x -- Exponential fitting NN >& \ \ y y N. \ >. \ \ \ \ . S X. N. \ N \ V \ . V x . X. >v *v ^ \ ^ v ^ ^ V ^ ^ ^ . 1 1 1 I I 1 1.1 1.2 1.3 Vbe (V) 1.4 1.5 1.6 Figure 3.6: Normalized junction velocity for NE = 5 X 1017cm 3 , Pg = 1 x 1019cm 3 in the graded-base case (xbe = 0.1). The exponential fit uses A*e~VBE'SVt, with A* = 22.184 and S = 7.63. Chapter 3. Simplified Models for Specific HBTs 34 3.2.1 B J T Intrinsic Mode l for the Abrupt-Junct ion , Uniform-Base S H B T The four current sources for the neutral base recombination and electron transport in the Full intrinsic dc model in (2.24) become: INBE = He-^/^^/v, - 1) w = H(eWV(-i) Ice = ISe-y™?sv'(ev^v<-l) IEC = ISe-VB^SVt(eVBc/Vt-l) (3.18) where IS _ qnBoVTnE2te2sW' AE ^ BF 2te2sW a„ - 2te2sW BR = T ^ V ^ I B A - ( 3 > 1 9 ) DnB By restricting the modes of operation to normal active and normal saturation, i.e., VBE > VBC to ignore e~VBE^SVt in the IEC expression, the Ebers-Moll terms above reduce to INBE = ^(eV°°/NFV' - 1) INBC = ^(eVBc/NRV'-l) Icc = IS(ev*s<NFV<-l) IEC = IS(eVsc/Vt - 1). (3.20) The new Ebers-Moll equations can be implemented by the regular SPICE macromodel for a BJT. The model parameters are qnBVTnEAEA* IS cosh ( ^ - ' Chapter 3. Simplified Models for Specific HBTs 35 NF = ( i - ^ r 1 NR = 1 1 BF = c o s h f ^ ) - 1 BR = L l B V ^ (3.21) J n f l where NF is the new ideality factor which takes account of the voltage-dependent tun- neling factor and the junction barrier height of the heterojunction. 3.2.2 BJT Intrinsic Model for the Abrupt-Junction, Graded-Base SHBT Following the same formulation as for the abrupt-junction, uniform-base SHBT, the BJT model parameters for an abrupt-junction, graded-base SHBT with the base Al mole fraction at the base-emitter junction not exceeding about 0.1 are T„ qnBQVTnEtiwtAEA* *cosh(W) + £sinh(<WO NF = ( l - ^ ) - 1 NR = 1 BF = 4. 1-W It* BR = ,, l ,r \w- (3-22) t cosh(tW) + { sm\i{tW) - teiw L2nBVTnEA*t smh{tW)DnBei] For base gradings with an Al mole fraction at the base-emitter junction of greater than about 0.1 (see Figure 3.2), (3.23) cannot be used because the condition ^^ >> 1 is no longer valid. In such cases the Full model of Section 3.1 must be used. Chapter 3. Simplified Models for Specific HBTs 36 3.2.3 B J T Extrinsic Mode l To turn the BJT intrinsic dc modelinto a large-signal model (see Figure 3.7), two diodes to account for the recombination-generation currents in the space-charge-regions and resistances to account for emitter, base, and collector resistances are added. Capacitances can be specified in the SPICE BJT model on the model line for the transistor Q. RB B D4 D3 RC ^ Q RE Figure 3.7: "BJT" extrinsic equivalent circuit of the HBT. With the assumption SNE = A*e VBB/SV,^ t ] i e e i e c t r o n quasi-Fermi level splitting becomes AE '» = Vt ln( wB A J + ^ T (3"23) Chapter 3. Simplified Models for Specific HBTs 37 0.2 0.6 0.8 1 VBE (V) Figure 3.8: SCR recombination current components (SRH = Shockley — Read — Hall, Aug = Auger, Rad = Radiative) in both the base-side(-B) and emitter-side(-E) of the depletion region at the BE junction for the "baseline" uniform-base SHBT. and the components of the SCR recombination current at the base side in (2.26) can be rewritten as diode equations J. S,SRH,B = ct NErHBEVTnEWnB,EA* *Ej-*BF,vhi TnOV-iEDnB •e v> nSRH,B = Js,Aug,B — nAug,B ~ &BE - Nfn CSn*BEApPBVTnEWnBtEA* Dn B 1 Chapter 3. Simplified Models for Specific HBTs 38 T CsVbiE 2 Dc vTnEWnB,EA Js,Rad,B = —77—n i B E±fpdE Vt L>nB nRad,B = ^ r ^ (3.24) where Nfn = l/S. (3.25) The dominant component of the depletion-region recombination-generation current, for the baseline abrupt-junction, uniform-base SHBT (see Table 3.1), is due to Shockley- Read-Hall recombination in the emitter side of the junction (see Figure 3.8), which can be represented by one diode. This diode is D3 in Figure 3.7. The diode D4 represents SRH recombination in the base-collector SCR. Now that the likely components of current (SRH, Auger and radiative) in both sides of the depletion region can all be represented by diode-like expressions [23] (see (2.26) and (3.25)), their inclusion in the equivalent circuit is simply a matter of adding more diode components in parallel with D3 and D4. SCR recombination currents were not included in the Gummel plot and collector-output characteristic simulations for the baseline HBTs considered in this thesis. 3.3 Comparisons of the D C Models The Gummel plots simulated by the Full and BJT dc versions of the HBT SPICE models are compared with the results from DAPHNE'for the baseline uniform-base SHBT and the graded-base SHBT. The two SPICE models are further compared via the collector-output characteristic. For the DHBT, there is no BJT model since the Ebers-Moll coefficients are dependent on both VBE a n d VBC- Therefore, only the Full model result is compared with that obtained from DAPHNE. The forward Gummel plot with various VBC biases and the collector-output characteristic are also simulated. The accord between the results Chapter 3. Simplified Models for Specific HBTs 39 for the various models and analyses presented in this section highlights the accuracy of both the Full and BJT versions of the HBT SPICE model. The values used for the material parameters are[23]: rn0 = 50ps; Tp0 = 200ps; An = 7.99 x 1(T3 2cm6s-1; Ap = 1.12 x l O ^ c m 6 ^ 1 ; Bn = 1.29 x K T ^ c m V 1 ; Bp = 7.82 x 10 - 1 1 cm3s~1. Specific contact resistivity values taken are: pc£ = 5 x 10_8Qcm2 for the emitter, pcs = 3 x 10~6f2cm2 for the base, and pcc = 10_6ficm2 for the collector. 3.3.1 G u m m e l Plots The Full and BJT dc versions of the HBT SPICE models are compared with the results from DAPHNE in Gummel plots for the baseline uniform-base SHBT (see Figure 3.9) and graded-base SHBT (see Figure 3.10). Both the SPICE models show very good agreement with DAPHNE. For the uniform-base DHBT (see Figure 3.11), the Full SPICE model also shows very good agreement with DAPHNE. The deviations of the SPICE results from that of the DAPHNE results at very low currents are due to the setting of a current minimum of 10 fA for the SPICE simulations. In Figures 3.12 and 3.13, the forward base currents and collector currents have been plotted for the same DHBT with various base-collector reverse bias voltages (VBC = 0, — 1, — 2, and — 3V) using the Full extrinsic dc model. The collector current increases with the increase of the reverse bias due to the lowering of the barrier height, AEnc, in Figure 2.1, i.e. as the base-collector hetero junction becomes less blocking to the flow of electrons injected into the base from the emitter. Another effect of the reduced amount of blocking is the reduction in excess electron concentration in the base. This leads to less neutral-base recombination and, therefore, a reduction in IB (see Figure 3.13). Chapter 3. Simplified Models for Specific HBTs 40 10 | 1 1 1 1 1 1 r Vbe (V) Figure 3.9: Comparison of Gummel plots for the baseline uniform-base SHBT. Chapter 3. Simplified Models for Specific HBTs 41 "i r - DAPHNE o Full SPICE model -. BJT SPICE model -i 1 1 r Figure 3.10: Comparison of Gummel plots for the baseline graded-base SHBT. Chapter 3. Simplified Models for Specific HBTs 42 10u -i r "i 1 1 1 1 r - DAPHNE o Full SPICE model 0.6 0.7 0.8 0.9 Figure 3.11: Comparison of Gummel plots for the baseline uniform-base DHBT. Chapter 3. Simplified Models for Specific HBTs 43 G U M M E L P L O T S FOR THE B A S E L I N E D H B T CWITH VBC • 0. - 1. - 2. - 3V ) 1 G - D E C 9 3 2 2 : 5 1 : 2 G •3.0 H . 0 •5.0 -B . 0 7.0 8.0 9 . 0 10.0 -11.0 12.0 -13.0 DHBT_SP.SWO PARCLOG10CIC ~ A •••'^£ DHBT_SP.SW1 '^_Q PAR C LOG 1 0 C I C DHBT_SP.SW2 PARCLOG10CIC Q DHBT_SP .SW3 PARCLOG10CIC BOO . 0M 1.20 VOLTS CLIN] Figure 3.12: log( / c ) vs. VBE plot for the "baseline" uniform-base DHBT with VBC = OV (.swO), - I V (.swl), -IV (.sw2), and - 3 V (.sw3). Chapter 3. Simplified Models for Specific HBTs 44 GUMMEL PLOTS FOR THE BASELINE DHBT CWITH VBC - 0. - 1. - 2. - 3V ] 1G-DEC93 22:51 : 2G DHBT_SP .SWO — PARCLOG10CAB A DHBT_SP .SW1 — PARCL0G1OCAB E DHBT_SP.SW2 PARCL0G1OCAB DHBT_SP.SW3 PARCLOG1OCAB -3.30 3 .350 -3.40 3.450 S .-' 3.50 .J. 1.480 . . . I . . . i . . . i . . . . . 1.490 1.50 VOLTS CLIN] ...I....I...J....I. 1.510 DHBT_SP.SWO PARCLOG10C AB - DHBT_SP.SW1 PARCL0G10CAB DHBT_SP.SW2 PARCLOG10CAB O DHBT_SP .SW3 PARCL0G1 0CAB <• ,J 1 .520 Figure 3.13: log(7e) vs. VBE plot for the "baseline" uniform-base DHBT with VBC = OV (.swO), —IV (.swl), — 2V (.sw2), and —3V (.sw3) (top panel) and its ex- panded-voltage-scale version (bottom panel). Chapter 3. Simplified Models for Specific HBTs 45 3.3.2 Collector Output Characteristics The two SPICE models are further compared in the collector-output characteristics shown in Figures 3.14 and 3.15. The current gains for the base transport have been shown to be the same in (3.6) and (3.22) for both the uniform-base SHBT SPICE models. They are also the same for both graded-base SHBT SPICE models (see eqns (3.11) and (3.23)). For the uniform-base SHBT, the neutral base recombination current is the dominant component of the base current and the current gain can be approximated by the current gain for the base transport. Both models show excellent agreement, because the error introduced by the exponential fitting of the normalized junction velocity is then cancelled in the current gain calculation. However, for the graded-base SHBT, the neutral base recombination current decreases dramatically with the base grading and the hole current at the emitter-base junction becomes dominant [9]. As both models represent the hole current by similar diode equations, the base current in both models is the same. However, the Full model gives a more accurate expression for Ic, and this is the reason why the sets of curves in Figure 3.15 are slightly different. The collector output characteristic of the baseline uniform-base DHBT is shown in Figure 3.16. There is no constant collector current regime because the increasing reverse bias at the collector-base junction (as VQE increases) keeps lowering the barrier height AEnc (see Figure 2.1), i.e., Ic keeps increasing. The SPICE model parameters used in the preceding simulations, using the input files given in Appendix D.l (see Files Ful lJV, SHBTJib and B J T J V ) , are tabulated in Tables 3.4-3.6, for the abrupt-junction, uniform-base SHBT case. Chapter 3. Simplified Models for Specific HBTs 46 BJT IB=500 uA -- Full IB=400 uA IB=300 uA IB=200 uA IB=100uA 0.5 1 1.5 VCE (V) 2.5 Figure 3.14: Comparison of collector output characteristics from the two SPICE models for the baseline uniform-base SHBT. Chapter 3. Simplified Models for Specific HBTs 47 I C - V C E C H A R A C T E R I S T I C S F O R G B S H B T C X B E - 0 . 1 ] 31 - D E C 9 3 1 5 ! 1 2 : 19 18 . OM IB . OM 14 . OM 12 . OM 1 0 . OM 8 . OM G . OM 4 . OM 2 . OM 0 . B J T _ I V . S W O I C V J C A F U L L _ I V . S W O I C V J C D B J T _ I V . S W 1 '=* . I C V J C Q F U L L - I V . S W 1 I C V J C — «> B J T _ I V . S W 2 •V I C V J C -. v- jjjC F U L L - I V . S W 2 I C V J C _ M _ B J T _ I V . S W 3 I C V J C F U L L _ I V . S W 3 I C V J C I B J T _ I V . S W 4 I C V J C D F U L L - I V . S W 4 I C V J C 0 . i i i i I i i i i I i i i i I i i i i I i i i i J 1.0 2 . 0 3.0 4.0 V O L T S C L I N ) 5.0 Figure 3.15: Comparison of collector output characteristics from the SPICE Full and BJT models for the "baseline" graded-base SHBT with Ib = 20uA (.swO), AOuA (.swl), 60uA (.sw2), 80uA (.sw3) and lOOuA (.sw4). Chapter 3. Simplified Models for Specific HBTs 48 I-V C H A R A C T E R I S T I C S FOR THE B A S E L I N E DHBT 1 S - D E C 9 3 2 2 : 2:55 I DHBT_SP.SWO /- I C VI C A / ~ D H B T _ S P .SW2 i - I C V J C • ' . . _ Q D H B T _ S P .SW3 i I i i i I i J 4.0 G . 0 B.0 VOL T S C L I N ) 9.0 Figure 3.16: Ic — VCE characteristic from the SPICE Full modelior the "baseline" uni- form-base DHBT with Ib = lOOuA (.swO), 200uA (.swl), 300uA (.sw2), AOOuA (.sw3) and 500uA (.sw4). Chapter 3. Simplified Models for Specific HBTs 49 C\ c2 c3 di d2 IsF IsNBE IsNBC IsE Isc NE Nc IsCRE 0.389 0 0 0 0.765 1.065E-24 A 3.71E-26 A 3.71E-26 A 1.913E-31 A 1.726E-24 A 1 1 2.22E-19 A ISCRC N3 N4 RE RB Re TF TR CjEO CJCO vbtE vbtC 1.76E-13 A 2 2 1.180 164.140 45.320 2.6 ps 8.2 ps 5.76E-14 F 2.14E-14 F 1.59 V 1.31 V Table 3.4: SPICE Full model parameters for the "baseline" SHBT. Is NF NR BF BR IsE Isc NE Nc IsCRE ISCRC 4.43E-24 A 1.168 1 28.71 60.8 1.913E-31 A 1.726E-24 A 1 1 2.22E-19 A 1.76E-13 A JV3 JV4 RE RB Re TF TR CjEO CJCO VbxE vbxC 2 2 1.18 0 164.14 0 45.32 0 2.6ps 8.2ps 5.76e-14 F 2.14e-14 F 1.59 V 1.31 V Table 3.5: SPICE BJT model parameters for the "baseline" SHBT. Chapter 3. Simplified Models for Specific HBTs 50 A* Cs IsRH,B IsRH,E lAug,B *Aug,E lRad,B lRad,E 1.618 1.12E-26 Ccm 7.94E-24 A 2.22E-19 A 1.18E-29 A 1.97E-39 A 6.26E-28 A 3.42E-34 A 5 NsRH,B NsRH,E NAug,B * * Aug,E NRad,B NRad,E 6.937 1.367 2 1.168 1 1.168 1 Table 3.6: Diode parameters for space charge region recombination currents of the "base- line" SHBT. Chapter 4 Comparison with HBT Experimental Data The most significant conclusion drawn from Section 3.2.3 is that the dc behavior of a SHBT can be well-simulated by the macromodel of the conventional homojunction transistor, found in all versions of SPICE, by introducing a new diode-ideality factor, NF, to account for the voltage dependencies of the tunneling factor and the barrier height at an abrupt heterojunction. In this chapter, the experimental dc and large-signal data, obtained from Bell-Northern Research (BNR) in Ottawa for an abrupt-junction, graded- base SHBT and ring oscillator circuits, are compared with the simulation results obtained from DAPHNE and the BJT model. The calculations presented in this chapter are for devices representing BNR F36 2 x 2um2 and 3 x 3um2 rectangular HBT structures shown in Figures 4.1 and 4.2 [34], whose geometrical, compositional and layout parameters are given in Tables 4.1, 4.2, and 4.3 [34, 35, 36]. Layer Emitter cap Emitter grading Emitter Base Collector Sub collector Buffer Material n+ — GaAs n — AlxGai-xAs n — AlxGai-xAs p+ — AlxGa\^xAs n~ — GaAs n + — GaAs GaAs Thickness (A) 2000 300 300 800 4000 3000 500 Doping (cm-3) 3.5 x 1018 5 x 1017 5 x 1017 1 x 1019 5 x 1016 3 x 1018 0 Al composition 0 0.05-0.3 0.3 0.1-0 0 0 0 Table 4.1: Geometrical and compositional parameters for the F36 device. 51 Chapter 4. Comparison with HBT Experimental Data 52 V///////////A V/A V////// Emitter cap Emitter Wcap r-*" Isolation v i Y/^Z//////^/////A Base ! 1 Collector i i i ! Sub-collector i i 1 i \ r W e , w Wc ' Wbuf SI substrate Figure 4.1: Cross-section of BNR F36 HBT structure [34]. Sbc Sb Lc J _ L i Collector 1 1 : • i ! ; Le i i i_ Emitter i ! Sc Lb Base JL Sbiso K s e - H r Seb 1u Figure 4.2: Layout of the BNR F36 HBT. Chapter 4. Comparison with HBT Experimental Data 53 Se sb Sc S.Oum 6.0um 3.0um ^biso Seb Sbc l.Oum 0.5um l.Oum Le U Lc 3.0um 6.0um b.Oum Table 4.2: Layout parameters for the F36 devices (emitter area = 3 x 3um2). Se Sb sc 2.0um 5.0um 3.0um ^biso Seb Sbc l.Oum 0.5um l.Oum. Le Lb U 2.0um 5.0um 4.0um Table 4.3: Layout parameters for the F36 devices (emitter area = 2 x 2um2) . 4.1 D C Characterist ics The forward Gummel plots using the B JT extrinsic model are compared with results from both the detailed device-analysis program DAPHNE and experimental data in Figure 4.3. Bandgap narrowing (BGN) needs to be considered for the highly-doped base and is taken to be 75meV here (see Appendix C). Carrier lifetimes of 50ps [37] and 200p.s are taken for electrons and holes, respectively. The SPICE and DAPHNE results for the abrupt-heterojunction simulation reproduce very closely the form of the measured data. Considering the uncertainties in the values of some of the parameters used in the simulations, and in the correctness of specifying the heterojunction as being absolutely abrupt, the agreement between measurement and simulation is very good. A near-perfect match of the data can be achieved if a small amount (20A) of grading is allowed in the base-emitter junction itself (see Figure 4.3), or the amount of bandgap narrowing is increased to lOOmeV (see Figure 4.4). The former possibility is probably more realistic, although the value we have used for the apparent bandgap narrowing may be in error Chapter 4. Comparison with HBT Experimental Data 54 as we have taken data for GaAs when, in fact, the base material close to the emitter is actually Alo.iGa0$As. The effect of the introduction of junction grading is to alter A* and S from 22.184 and 7.630 to 205.761 and 6.369, respectively. These changes modify IS and NF in (3.23) to the extent necessary to produce the excellent fit shown in Figure 4.3. BR is also changed, but its effect on Ic in the forward, active mode is negligible. The input file used in the simulation of the circuit for the graded base-emitter junction case is given in Appendix D.2 (see File G3JB36). The SPICE BJT model parameters for the 3 x 3um2 devices are listed in the Tables 4.4. The collector ideality factors, extracted from the measured (1.18) and simulated (1.186) data, suggest that the collector current is limited by thermionic/tunneling across the emitter-base junction rather than by diffusion in the base layer. Otherwise, the collector ideality factor should be close to 1. IS NF NR BF BR IsE Isc NE Nc IsCRE IsCRC 1.49E-24 A 1.186 1 122.7 381 2.81E-32 A 3.17E-25 A 1 1 4.34E-20 A 4.25E-14 A N3 N4 RE RB Re 2 2 11.38 ft 109.10 ft 31.06 ft Table 4.4: SPICE BJT model parameters for the F36 3 x 3um2 devices (BGN = IhmeV and base-emitter junction grading = 20 A). However, there is a discrepancy between the measured and simulated base current at the low forward VBE bias. Three possible reasons for the difference are: 1) The fitted A* and S values are not exact at low bias voltage VBE', 2) the surface recombination current at the exposed extrinsic base surface, which increases exponentially with the base-emitter Chapter 4. Comparison with HBT Experimental Data 55 1 0 CD O M o O 1 0 10" 0 . 5 — Experimental data O S P I C E , graded BE junction - D A P H N E , abrupt BE junction^ O S P I C E , abrupt BE junction """ 1.0 1.5 VB E (V) 2 . 0 Figure 4.3: Collect current characteristic comparison of experimental data with simula- tion results from SPICE and DAPHNE for the various conditions. bias voltage with an ideality factor of 2, is not considered in our model; 3) the measured 7g — VBE data is not unique, as considerable variation was observed in measurements on devices on the same chip. (No such variation was recorded for 7c). Chapter 4. Comparison with HBT Experimental Data 56 10" 10' 10 < O 10" 10"' 10 10 CD D Extrinsic model • Experimental data ,-, 'c ^ * , O . ^ i * * 3 * 1 ! * n n • • • * n n ft n ft <=> n • i • • mfca • 1.0 1.2 1.4 VBE (V) 1.6 1.8 Figure 4.4: Comparison of Gummel plots from BJT-SPICE-model and experimental data for BNR F36 device (assuming BGN = lOOmeV). Chapter 4. Comparison with HBT Experimental Data 57 4.2 Large-Signal Characteristics for two 5-Stage Ring Oscillators vcc R L j ! R L u RL RL RL CI M-, C2+C15 C3+C16 C7+C17 C9+C18 C0+C13 C14 Figure 4.5: BNR F36 5-stage ring oscillator. In order to examine the usefulness and accuracy of the BJT model for performing large- signal transient analysis, a five-stage ring oscillator was simulated and the results com- pared with experimental data. The HBTs were BNR's F36 devices, as described previously, with an emitter area of 2 x 2um2. Each inverter stage in the oscillator was a RTL gate, and circuits with load resistors of 200H and 400fi were investigated. The ring oscillator circuit is shown in Figure 4.5, along with the parasitic capacitances which were extracted from the layout by John Sitch of BNR, using that company's extraction software [38]. John Sitch also provided the experimental data on the variation of the oscillation frequency with bias voltage Vcc- This data was obtained using the arrangement shown in Figure 4.6. A spectrum analyzer was coupled to the ring via the LC-tee shown, and allowed the fundamental oscillation frequency to be observed directly. The input file used in the simulation of the circuit is given in Appendix D (file Ring_B36). The SPICE BJT model parameters for the 2 x2uro2 devices are listed in Table 4.5 and were as used in the Gummel plot investigations (see previous section), augmented Chapter 4. Comparison with HBT Experimental Data - O HP 856 B 2.5-22 GHz SPECTRUM ANALYZER Figure 4.6: BNR 5-stage ring oscillator measurement set-up. by the forward and reverse transit times TF and TR. These were computed, according to (2.30), from DAPHNE, with the base transit time component being evaluated at VBE = 0.5V. At this voltage rg has become essentially constant [9]. To obtain TR, the HBT was simulated in the inverse mode of operation [27], with the actual collector being taken as the emitter and vice versa. Note that TR is about 10 times larger than TF . This is because the built-in field of the graded-base now opposes the electron flow from "emitter" to "collector", and also, because the spike at the new base-collector junction now presents a barrier to electron flow from the base. The simulated and experimental data are compared in Figures 4.7 and 4.8. Consid- ering all the factors that affect the oscillation frequency fosc, the agreement between theory and experiment is remarkable good. Certainly, all the essential features of the experimental data are reproduced by the simulations. The curves can be broken down into three regions: a low-bias regime where Jose falls slightly with \rcc\ a mid-bias regime where fosc increases with bias; and a high-bias Chapter 4. Comparison with HBT Experimental Data 59 IS NF NR BF BR IsE lsc NE Nc IsCRE ISCRC 6.62E-25 A 1.186 1 122.7 381 1.25E-32 A 1.41E-25 A 1 1 1.93E-20 A 1.89E-14 A N3 N4 RE RB Re TF TR CjEO CJCO vbiE Vbic 2 2 25.6 n 137.4 n 35.9 n 1.57 ps 13.78 ps 6.32E-15 F 1.49E-14 F 1.633 V 1.309 V Table 4.5: SPICE BJT model parameters for the F36 devices (emitter area = 2 x 2um2). regime where fosc decreases as Vcc is further increased. To explain these features, the switching characteristics of a single HBT were examined using the input file of Appendix D (file Switch_B36), which describes the circuit shown in Figure 4.9. In the simulations the load capacitor represents the parasitic capacitance, and the upper and lower limits of the input pulse were adjusted for each Vcc, in accordance with the results from the simulation of the full circuit of Figure 4.5. The results are shown in Figure 4.10 for Vcc = 1-55, 2.4, 3.19, and 4.76V. As VJN goes low, the base-emitter capacitor discharges through the base resistance RB- This lowers the voltage VBE at the junction, so turning off the transistor, allowing VOUT to rise as the output capacitance is charged from Vcc via RL. AS long as the device is not in saturation this process is rapid. Evidently, saturation is not reached until Vcc ~ 4.76V (see the bottom trace of Figure 4.10), at which point there is a discernible delay before the output rises, and the low-high component of the propagation delay T ^ increases. However, as far as the high-low component of the propagation delay is concerned, the device turns-on faster and faster as Vcc increases (see Figure 4.10). This is because the collector current is larger, which in turn means Chapter 4. Comparison with HBT Experimental Data 60 6 5.8 5.6 5.4 N X f • (A O LL 4.8 4.6 4.4 4.2 - - - - 0 °0 - X 1 o Measured * Simulated o o * * 1 K 0 1 o - . » - - - 6 - 2.5 Vcc(V) Figure 4.7: Comparison of BJT-SPICE-model and experimental data for the BNR 5-stage ring oscillator with RL = 200fL that the base current is larger so the transistor can be turned-on quicker via the charging of the input capacitance through i?jg. Therefore, in summary, as Vcc rises, Tdih remains more or less constant until the device enters saturation, and Tdhi decreases. Thus the overall propagation delay time, rpd = {Tdih + Tdhi)!^ [39], decreases with Vcc until V/jv (via Vcc) is s o large that the device enters saturation when in the ON-state. After this point rvd increases with Vcc- This explains the mid- and high-bias portions of Figures 4.7 and 4.8. At low bias, the turn-on time is so long (see top trace of Figure 4.10), due to the low VBE, that the output may not have chance to settle before the device is turned-off by the transition caused by the returning pulse from the ring. Under such circumstances, as Vcc decreases, this arresting of the fall of VOUT occurs at a higher and higher voltage (see Figure 4.11 for Vcc =1-55, 1.60, 1.70 and 1.80V), giving the impression of an increase in oscillation frequency, such Chapter 4. Comparison with HBT Experimental Data 61 5.5 4.5 N X u 10 3.5 2. fl - - - * * QiO i i o Measured * Simulated X K 0 * O o 1 1 K 0 1 K 0 ... 1 — - • * o 1 T X 0 1 T - • • K o 1 X - o 2.5 3.5 Vcc (V) 4.5 5.5 Figure 4.8: Comparison of B JT-SPICE-model and experimental data for the BNR 5-stage ring oscillator with RL = 400H. as is observed (see Figures 4.7 and 4.8). It should be possible to remove this artefact of the measurement circuit by making a larger ring, so allowing the voltage levels at any one stage to settle before the propagating signal returns and switches the device. To investigate this, results for 21-, 11-, and 5-stage oscillators are presented in Figure 4.12 while the low-bias anomaly is not completely removed, it does become progressively less evident as the length of the ring is increased. Chapter 4. Comparison with HBT Experimental Data 62 Vcc FU- N/OUT VlNH VlNLO / \ ' Cparasitic Figure 4.9: Circuit for simulation of switching of single HBT. Note the transistor shown here is the full extrinsic device represented by the equivalent circuit in Figure 3.7. Chapter 4. Comparison with HBT Experimental Data 63 5-STflGE F3G 2*2 RING OSCILLATOR CR 1 •400 ) USING BJT MODEL 1 2-M0R94 13:45:32 V L 0 I L N T V L 0 I L N T V L 0 I L N T V L 0 I L N T I B36_4_D.TR0 _I VC4 710 . OM 0 . 100.OP 200.OP 300.OP 400.OP TIME CLIN) 400.OP Figure 4.10: Simulation of switching of a single HBT with Vcc = 1.55V (.trO), 2.4V (.trl), 3.19V (.tr2) and 4.76V (.tr3). Input pulse Vm and output pulse VOUT are represented by node voltages V(6) and V(4), respectively. Chapter 4. Comparison with HBT Experimental Data 64 5-STAGE F3G 2»2 RING OSCILLATOR CRl-tOOD USING BJT MODEL 3 - MAR94 15: 3:35 3 0 0 . OM, Q N 1 . ON I B3E_t.TR0 ; V C 4 — A I B36.H.TR1 •- Q B 3 G _ 4 . TR2 , - V C <4 \ ± O \— B3G.1.TR3 \: V CI V » - 1 . 2 ON 1 .1 ON 1 .G ON TIME (LIN) I i 1 . 8 ON 2 . ON Figure 4.11: Simulation of the ring oscillator with RL = 400fi for the supply voltages Vcc = 1.551' (.trO), 1.6V (. tr l) , 1.70V (.tr2), and 1.80V7 (.tr3). The voltage is taken at the output of any of the stages in Figure 4.5. Note the increasing minimum of voltage swing as the supply voltage Vcc is reduced to low values. Chapter 4. Comparison with HBT Experimental Data 65 50.0 40.0 - S 30.0 20.0 10.0 ' I o® D 0 0 o © 0 8 I 9 0 I 8 o o I D 0 I I o 5-stage ° 11-stage o 21-stage - o o o o I 1.0 2.0 3.0 4.0 Vcc (V) 5.0 6.0 Figure 4.12: Propagation delay times vs Vcc comparisons of 21-, 11-, and 5-stage BNR ring oscillator with RL = 400fl. Chapter 5 Laser Dev ice and Circuit Model ing In this chapter, a large-signal equivalent circuit model is developed based on the rate equations for a single-mode semiconductor laser. Current-voltage and light-current char- acteristics, and large-signal response are simulated. The large-signal transient analysis results are also compared with experimental data. Finally, the simulated large-signal responses of monolithic and hybrid integrated HBT-laser transmitters are compared. 5.1 T h e Rate Equations for Semiconductor Lasers The rate equations for a single-mode semiconductor laser with a uniform carrier distri- bution can be written as [40, 41] —77 = GP h i?sp,o at TP dn I n GP dt qVa Te Va d ^ a ° r r i \ *i rxi\ — = y [ r v i ( n - T i o ) - — J (5.1) where P is the photon population (for the longitudinal mode oscillating at the frequency u) inside the laser cavity, G is the net rate of stimulated emission (mode gain), rp is the photon lifetime, Rsp,o is the spontaneous emission rate into the lasing mode, n is the electron density in the active region, / is the current in the active region, q is the electronic charge, Va is the active region volume, Te is the spontaneous carrier lifetime, 66 Chapter 5. Laser Device and Circuit Modeling 67 <f> is the phase of the optical field, ao is the linewidth enhancement factor at the mode frequency o>, T is the confinement factor, v3 is the group velocity, a is the gain coefficient, and n0 is the carrier density at transparency. The spontaneous emission rate into the lasing mode Rsp,o is [40] _ PVspnVq where /? is the spontaneous emission coupling coefficient and r)sp is the internal sponta- neous quantum efficiency defined as VsP = (5.3) The carrier-recombination rate 7e is 7 e = - = (AnT + Bn + Cn2) = — + — + (5.4) where AnT is the nonradiative recombination rate, B is the radiative recombination coeffi- cient, C is the Auger recombination coefficient, and the r's are the corresponding carrier lifetimes. In order to derive an equivalent circuit model to represent the rate equations, (5.1) can be rewritten as T, dn qVan _, _, at Te qGP = q^- + 3-P-qRsp,o at TP — = —Tvgan - —{Yvgan0 + — ) . (5.5) If we define the spontaneous recombination current Isp = 2-^2-, the stimulated emission current Ist,m — qGP [5] and qRsPyo = j3ejflsp with /?e// = (3risP, the rate equations above can be written in a convenient electrical-analog form, i.e., ' dt sp 4- T A- T • T J i p T ^sttr Chapter 5. Laser Device and Circuit Modeling 68 dP P •Istim — C/p/i + — Pefjlsp at Kph d<f> — = hihp - hQ (5.6) where Cph represents the photon storage and its numerical value equals that of the electronic charge q, Rph represents the photon loss and its numerical value equals ^ , hi = ao2vf,aTe and h0 = 2^-(Tvgan0 + —). The time derivative of Isp models charge storage in the active region [5] and is equivalent to the diffusion capacitance of a diode. Mode gain nonlinearity needs to be included to model the dynamic response of the semiconductor laser [41]. A simple phenomenological functional form G = Tvga(n — n 0 ) ( l — epP) [42] is used and is valid at low power levels such that tpP <C 1. ep is the nonlinear gain parameter to be used when the rate equations are expressed in terms of P (rather than in terms of the photon density S). Therefore, the stimulated emission current can be modeled as 7 s t im = qGP = b(Isp — 7spo)(l — epP)P with b = v^aTe, and J _ qVaTOQ JspO — T e • The output power P0, rather than the photon population P , which is used in the above rate equations, and the photon density S, which was chosen by Tucker, has been chosen to be represented by the node voltage for two reasons: 1) the output power can be measured; 2) its numerical value is in the range of milliwatts (mW), so the range of numerical errors can be kept approximately to the same order for all circuit variables [43]; 3) there is no need to introduce a normalization constant to avoid divergences during numerical evaluation. The output power emitted from each facet related to the photon population is [40] P0 = l-hvvgamP = pP (5.7) where h is Plank's constant, v is the lasing frequency, a m is the facet loss, and p is the average output power (W) for each photon. Thus, the rate equations for the circuit Chapter 5. Laser Device and Circuit Modeling model (see Figure 5.1) are modified to I = dISr : j , ~r -«sp T J- stim + Isv + h dP0 P0 I stim = ^ph~T V "5 Pejjlsp at Kph d * - h T h —— — riiisp — no at IsUm = b( Isp - Isp0 ) (1 - tPo P0 ) P0 with Cph = §, Rph = f , ePo = ¥ , and b = ^ = v - f e - I d I sp -^ Xe — ! i d\ T stim ' • ! bp f i i (3 'sp v. / ' eff 'ph -rdt 4> C=1 F Figure 5.1: Equivalent circuit model for rate equations. The electric field at the laser output is [44] E(t) = JP0exP{j<t>{t)) Chapter 5. Laser Device and Circuit Modeling 70 Since complex calculation cannot be done in SPICE, the real and imaginary parts of the electric field are calculated separately. The laser chirp is A,(l) = lft (5.U» SPICE input parameters needed to simulate the circuit of Figure 5.1 are Cph Rph ePo Self b JspO fel ^0 = = = = = = = = £ P TPP 9 £P P PVsp 2Tare Vah~fa>m qVan0 ctoTvgaTe 2qVa —(Tv gan 0 + —) I Tp (5.11) 5.2 Large-Signal Mode l for Lasers When the active layer thickness d is much smaller than the electron diffusion length Ln, the electron density becomes constant along the entire active layer [45]. For high-level injection, the electron density n can be related to the junction voltage V} by [46] n = ne[exp(qVJ/nLkT) — 1] (5.12) Chapter 5. Laser Device and Circuit Modeling 71 where ne is the equilibrium electron density and the ideality factor ni ss 2 for AlGaAsjGaAs and InP/InGaAsP laser diodes [4, 5]. Therefore, the static I-V characteristic of the het- erojunction can be modeled by a simple Shockley diode Isp = IsL[exp(qVj/nLkT) - 1] (5.13) where ISL — ?K"e / T e is the heterojunction saturation current [47, 5]. The equivalent circuit now becomes as shown in Figure 5.2. The series resistor RSL represents the voltage drop outside the depletion region. The depletion capacitance CJL can also be included in the specification of the diode [43, 46]. I R S L dlsp — Te-~ ( i Istim sp ( O P * ls> R ph ^ ^ G p h T d\ C=1 F Figure 5.2: Equivalent circuit model for laser diodes. The additional SPICE input parameters needed to simulate the circuit of Figure 5.2 are: ISL, " L , RSL, CJL{0), Vt,x, where V^L is the built-in potential of the heterojunction. Chapter 5. Laser Device and Circuit Modeling 72 5.2.1 Nonl inear Gain Parameter Spatial hole burning and spectral hole burning are two of the most important gain satu- " ration phenomena [48]. The relative importance of lateral spatial hole burning increases as almost the square of the cavity width [49]. The spatial hole burning effective gain saturation coefficient es,spa is given by [42] CS ' Spa = 2(l + [^-]2) ( 5 , 1 4 ) where w is the active-region width. The subscript S refers to the calculation for the rate equations in terms of photon density S , rather than the photon population P , and tptSpa = y-£s,spa- Agrawal [41, 50] has derived the gain saturation coefficient due to spectral hole-burning. For low power levels, it becomes 1 huT €p ^ ~WS- 2e0nngVaIs ( 5 ' 1 5 J where to is the permittivity of free space, ng is the group refractive index, n is.the effective mode index, and Is is the saturation intensity. Is is related to the intraband relaxation times by I. = h2/[u2Tin{Tc + Tv)] (5.16) where u is the dipole moment, TC, T„, and T{n are the intraband relaxation times for electrons, holes and polarization respectively [50]. tp is taken to be the sum of epiSpe and tp<spa in our model. 5.2.2 Spontaneous Emission Coupling Coefficient There is some disagreement over /?'s numerical value. In practice, /3 is often treated as a fitting parameter [40] and has a value in the range of 1 0 - 3 — 10 - 5 . In our model, /? is calculated using a simple formula for the conventional semiconductor laser [51] " = s£r <5-17> r>Vanth Chapter 5. Laser Device and Circuit Modeling 73 This expression gives a value of /? = 1.25 x 10 for the laser considered in this work. 5.3 S teady-Sta te Mode l In steady-state, the equivalent circuit in Figure 5.2 becomes as shown in Figure 5.3. The voltage-current relationship is V = V3 + RSLI (5.18) where V is the measured voltage across the laser diode. The light-current relationship can be derived from (5.8) as P0 = Rph[I-Isp(l-(3e}f)}. (5.19) Therefore, Rp^ in our model is the slope of the light-current curve (and called the slope efficiency). I RSL Istim V] J - ^ l sp t ) Peff U p R ph Figure 5.3: Equivalent circuit model for laser diodes at steady state. Chapter 5. Laser Device and Circuit Modeling 74 5.4 Simulation of Laser Steady-State and Large-Signal Characteristics The data in Table 5.1 are used to calculate the SPICE parameters for the rate equa- tions in this section. These data represent typical values for a InP/InGaAsP buried heterostructure laser designed to operate at 1.3 um. The resulting SPICE parameters for the laser diode as used in the simulation presented here, are given in Table 5.2. Parameter Cavity length Active-region width Active-layer thickness Confinement factor Effective mode index Group refractive index Line-width enhancement factor Facet loss Internal loss Gain constant Carrier density at transparency Nonradiative recombination rate Radiative recombination coefficient Auger recombination coefficient Threshold carrier population Threshold current Carrier lifetime at threshold Photon lifetime Dipole moment Intraband relaxation time for electrons Intraband relaxation time for holes Intraband relaxation time for polarization Symbol L w d r n ng a0 OLm Ct,nt a n0 •ri-nr B C Nth Ith Te TV U Tc Ty Tin Value 250 um 2 um 0.2 um 0.3 3.4 4 5 45cm-1 40 cm"1 2.5 x 10-16 cm2 1 x 1018 cm~3 1 x 108 s-1 1 x 10"10 cm3/s 3 x 10-29 cm6/s 2.14 x 108 15.8 mA 2.2 ns 1.6 ps 9 x lO-27cmC 0.3 ps 0.07 ps 0.1 ps Reference [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [40] [41] [41] [41] [41] Table 5.1: Typical parameter values for a l.3um buried-heterostructure laser. Chapter 5. Laser Device and Circuit Modeling 75 Te Cph Rph £Po Peff b •IspO h h0 ISL rit RsL CJL(0) VblL 2.2 ns 6.2 pF 0.253 n 2.97 W~l 1.25 x lO"5 483 W~l 7.22xlO-M 1.95 x 1014{rad/A) 3.0 x 10u(rad) 3.0 x 10~UA 2 9 Ct 10 pF 1.9 V Table 5.2: SPICE parameters for a 1.3um buried-heterostructure laser. 5.4.1 S teady-Sta te Characteristics The simulated V-I and L-I characteristics are shown in Figure 5.4. Since the spontaneous coupling coefficient (iejj is usually used as an empirical fitting parameter, values of 1 x 10 - 4 and 1 x 10~3 are also chosen to simulate the L-I characteristic, and the results are shown in Figure 5.5. It shows that the larger the /?e / / value, the softer the L-I curve. The SPICE input file is given in Appendix D.3 (see File SS_LD). 5.4.2 Large-Signal Characteristics The large-signal response for the laser diode operating at 2Gb/s writh the modulation current lm0& = 16m A (a square wave pulse which gives an output power of about 5mW) is simulated for the following cases: (see Figure 5.6). 1. ePo = 2.97W7-1, Iua, = 1-Uth, and tT = 150ps (.trO); 2. tPo = 5W~\ Ibtas = l.Uth, and tr = 150ps ( . trl); Chapter 5. Laser Device and Circuit Modeling 76 V-I AND L-I CHARACTERISTICS 2-JAN94 23:33:29 1.750; 1.50- 1.250 1 .0 750 . 0M 500 . 0M 250 . 0M 0 . 1 0 . 0M 8 . OM G . OM 4 . OM 2 . OM l _ J . . 1 . .1. . J . . I . .L . .1. . J . . I . I . . J . . 1 . . 1 . .1. . I . . 1 . .1. . J. . I . 1 . . I . . J .. I . . I . . J. SS_LD.SW2 VC2 .I..J.....I.. 10 . OM 20.OM 30.OM AMPS C L I N D . . 1 . . 1 . . J . . i . - i . . J . . j . - i . 40 . OM 50.OM GO . OM Figure 5.4: V-I (top) and L-I (bottom) characteristics. Chapter 5. Laser Device and Circuit Modeling 77 L-I CHARACTERISTICS 2-JAN91 23i33:29 V u L T 1 I N 1 . OM _ 3 .750M - 3 .50M _ 3 .250M - 3 . OM - "" 2 .750M - 2.50M - ' " 2 .250M - 2 . OM - ' 1 .750M _ 1 .5 OM - 1 .250M . 1 . OM - 750 . OU - 500 . OU - 250 . OU - 0. 0 . 1 5 . OM . 1 1 0 . OM AMPS 15. OM CLIN) 20 . OM 25.OM 30 SS_LD.SW1 VC2 i SS_LD.SW2 VC2 Q Figure 5.5: L-I characteristic for /?e// = 1 x 10 (.swl), and 1 x 10 (.sw2) respectively, Chapter 5. Laser Device and Circuit Modeling 78 3. tPo = bW'\ Ibias = 1.2Ith, and tr = 150ps (.tr2); 4. ep0 = 5 W _ 1 , /bias = 1.21th, and *r = 80ps (.tr3). where 7tlas is the dc bias current. The fall time tj equals the rise time tT in our simulation. A RC circuit (Rsm0oth = 1^ and Csmooth = lOpF) is used to smooth the corner of the input signal /,-„ (Iaas -+• I mod)- The simulated results show that the larger the nonlinear gain parameter (ep0 in cases 1 and 2), the higher the dc bias (hias in cases 2 and 3), and the larger the rise time (tr in cases 3 and 4), the smaller the optical relaxation spikes. It can be seen that by varying these values one can qualitatively predict the responses exhibited by actual laser diodes. The SPICE input file is given in Appendix D.3 (see File Tran.LD). 5.5 Comparison of Simulated Laser Characteristics wi th Exper imenta l D a t a In this section, a large-signal optical waveform is simulated at 1.7Gb/s (see Figure 5.7) and compared with the experimental data (see Figure 5.8) from Gregory Burley of UBC [52]. The parameters used in this simulation are listed in Table 5.3. The SPICE input file is given in Appendix D.3 (see File Expe_LD). In the simulation, h = l-llth, Imod = 20mA, Pejf = 10 - 4 , RSL = lfy and tr = tj = 230ps are assumed and the ISL, NL, C 'JL(O), Hii, values are the same as those listed in Table 5.2. The simulated optical output shows good agreement with the experimental result. Chapter 5. Laser Device and Circuit Modeling 79 T R A N S I E N T A N A L Y S I S FOR LASER O P E R A T I N G AT 2 G B I T / S 2-JAN94 2 2 : 2 9 : 25 TRAN_LD .TRO I CV1 - T R A N . L D . T R 1 -3 ICV1 Q -~ TRAN_LD.TR2 I CV1 Q T R A N - L D . T R 3 I CV1 •5 . ON 5 . ON 5.50N G . ON TIME CLIN) G .50N 7 . ON 7 . ON Figure 5.6: Transient analysis for the 1.3um buried-heterostructure laser: injection cur- rent (top panel) and optical output waveform (bottom panel). The following four cases are simulated by varying the nonlinear gain parameter (ep0), the dc bias current (hias), and the rise time (tT): 1) ePo = 2 .97W - 1 , /6ms = l.Uth, and tr = 150ps (.trO); 2) tPo ~ 5VF -1 , hias = 1-1/tft, and tT = 150p.s (.trl); 3) tPo = 5W~l, h,a> = l-27f^, and tr = 150ps (.tr2); 4) tPo = 5 W _ 1 , Ibias = 1.2/t/l, and tr = 80ps (.tr3). Chapter 5. Laser Device and Circuit Modeling 80 Parameter Cavity length Active-region width Active-layer thickness Confinement factor Line-width enhancement factor Facet loss Internal loss Gain constant Carrier density at transparency Nonradiative recombination rate Radiative recombination coefficient Auger recombination coefficient Group velocity Gain compression parameter Symbol L w d r CtQ am <*int a n0 •A-nr B C V9 ts Value 250 um 2 um 0.2 urn 0.4 5 50cm"1 50 cm~x 2.5 x 10~16 cm2 1 x 1018 cm'3 1.5 x 108 s"1 0.9 x 10"1 0 cm3/s 9 x 10~29 cm6/s 8.5 x 109 cm Is 5 x l 0 - 1 7 c m 3 Table 5.3: 1.55 um laser parameters [52]. Chapter 5. Laser Device and Circuit Modeling 81 Figure 5.7: Simulated optical power waveform (top panel) and injection current (bottom panel) at 1.7Gb/s . Chapter 5. Laser Device and Circuit Modeling 82 Figure 5.8: Experimental [52] optical power waveforms at 1.7Gb/s. Chapter 5. Laser Device and Circuit Modeling 83 5.6 Simulation of Laser Driver Transmitter Circuits Now that large-signai equivalent circuit models for both HBTs and laser diodes have been developed, it is interesting to simulate the performance of HBT-laser transmitters. Of particular interest is the monolithic integrated version of this circuit as this has the potential for improved performance over the usual hybrid circuits. The most successful monolithically-integrated HBT-laser circuit to date is the one from AT&T Bell Labs which demonstrated operation at 5Gb/s [53]. Details of the transistors used in this circuit were given in sufficient detail for them to be simulated by our programs. However, the required laser parameters were not given. Therefore, for the purpose of demonstrating the capabilities of our model, we use HBT parameters from Reference [53], as given in Tables 5.4 and 5.5, and laser parameters as given earlier in Table 5.3. To complete the specification of the HBT, it was necessary to infer values for RE-, the emitter resistance, and r , the minority carrier lifetime. The base-emitter junction of the HBT was quoted as having a series resistance of about 10ft [53], therefore, RE was taken as 10Q. The current gain was specified to be 40 at 35 mA [53]. In order to fit this gain, the carrier lifetime for electrons was taken to be 41 ps. This is a reasonable value as it is close to the value of 28 ps which has been used for similar transistors [7]. Evidence that we are simulating the HBT correctly came from the fact that our predicted fj at a bias of VBE — 0.8V was 30 GHz, in agreement with the measured value [53]. Also satisfying is the fact that the reported and simulated turn-on voltages for the base-emitter junction were the same (0.7 V). . The actual circuits simulated are shown in Figures 5.9 (a) and (b). Figure 5.9 (b) is a hybrid IC version of Figure 5.9 (a) and is of interest as this will be the circuit which the author will develop at NRC as part of a SSOC contract. The simulations were performed with a modulation current of 20 mA, which results in an output power of 5 mW, a dc Chapter 5. Laser Device and Circuit Modeling 84 Layer Emitter cap Emitter Base Collector Sub collector Material n + — InGaAs* n- InP p+ — InGaAs n~ — InGaAs n + — InGaAs Thickness A 1500* 1500* 1000 5000 4000* Doping (cm~3) 1 x 1019 4 x 1017 8 x 1018 1 x 1016 1 x 1019 In composition 0.53* 0 0.53* 0.53* 0.53* Table 5.4: Geometrical and compositional parameters for the AT&T SHBT (emitter area = 5 x 9K???2) [53]. Data marked with an asterisk are assumed values. se sb Sc 5.0um 2.0um 13.0u???* Jbiso Seb Sbc 0 um* 2.0um* 2.0um* u U Lc 9.0um 23.0um 23.0um* Table 5.5: Layout parameters for the AT&T SHBT (emitter area = 5 x 9um2) [53]. Data marked with an asterisk are assumed values. bias current of l.2Ith (about 23.3m.4), and the rise t ime tr = IbOps at 1.7Gb/s. The input injection current pulses and output optical waveforms shown in Figure 5.10 (see Files Mono , Hybrid and LDJib in Appendix D.3) indicate that both circuits do function as electro-optic converters at 1.7Gb/s. As expected, the inductance (L = 3nH was used in the simulation) of the coupling in the hybrid case does introduce some oscillations of the switching current pulse. The results give confidence that the models developed in this thesis will prove useful in HBT-laser IC design. Chapter 5. Laser Device and Circuit Modeling 85 Vref Vs (- - } - - j i Q1 * (a) — f c _\ i Q2 j > Laser • — ' "j I (b) -p>- Vsig Q3 Ibias Vref Vs • < = • • /">r.>"A Laser " Q Ibis Q1 Q2 Vsig Q3 VEE(-) VEE(-) Figure 5.9: (a) Monolithic ECL HBT-laser transmitter [53]. (b) Hybrid ECL HBT-laser transmitter. The inductor in (b) represents the connection impedance: a value of 3 nH was used. Chapter 5. Laser Device and Circuit Modeling 86 SIMULATION FOR ECL T R A N S M I T T E R S AT 1.7GB/S 20-MAR94 2 2 : 5 9 : 2 8 V 0 L T L I N 7 S 5 1 3 2 1 OM H OM -" ~ OM ~ ^ OM - OM r OM ir OM IT 25 . OM 20 . OM 15 . OM Z 10.OM - 5 . OM _ MONO .TRO VCPO —: h 2 H Y B R I D . T R O VCPO Q iR^A^i • 5 . 0 MONO .TRO I CV20 h H Y B R I D .TRO -. I C V 2 0 . _ - B ~B r i i i i i i i i i i i i i i i *\ . 5 0 N 5 . ON 5 . 5 0 N G . ON 4 , 5 0 N T I M E C L I N ) G . ON Figure 5.10: Monolithic vs. hybrid ECL HBT-laser transmitters: optical output wave- forms (top panel) and injection currents (bottom panel). Chapter 6 Summary 6.1 Conclusions In this thesis, various intrinsic dc, extrinsic dc, and large-signal equivalent circuit models for graded-base HBTs have been developed from DAPHNE; and a large-signal equivalent circuit model for laser diodes has been derived based on the single-mode rate equations. The following conclusions can be drawn from this work: 1. In contrast to the traditional Ebers-Moll representation of a homojunction BJT, the intrinsic Ebers-Moll coefficients for a HBT consist of voltage-dependent junction velocity terms to describe tunneling factors and junction barrier heights. These co- efficients can be simplified into a form which allows their implementation in circuit simulators with piece-wise-linear features. Full intrinsic dc, extrinsic dc, and large- signal models can then be developed by adding circuit elements to account for the SCR recombination/generation currents at the junction, the parasitic resistances, and the parasitic capacitances. 2. The Full model can be further simplified for specific, abrupt-junction HBTs such as the uniform-base SHBT, graded-base SHBT, and uniform-base DHBT so that the SPICE parameters can be expressed in terms of geometrical, compositional, and known-electrical parameters. The most significant conclusion is that BJT versions of the HBT SPICE model for SHBTs can also be derived by using an exponential fit 87 Chapter 6. Summary 88 to the normalized junction velocity. Simulated forward Gummel plots and collector- output characteristics highlight the accuracy of both SPICE models. 3. The experimental dc data and large-signal data, obtained from BNR for an abrupt- junction, graded-base SHBT and two five-stage ring oscillators, respectively, can be well-fitted by simulation results obtained from DAPHNE and the BJT SPICE model. The fit between the measured data and the simulated data for the collec- tor current is particularly good; the collector ideality factors, extracted from the measured (1.18) and simulated (1.186) data, suggest that the collector current is limited by thermionic/tunneling across the emitter-base junction rather than by diffusion in the base. For ring oscillators, the variation of the oscillation frequency fosc with bias voltage Vcc is well-accounted for by the model; and all the essential features of the experimental data can be reproduced and explained. 4. Steady-state and large-signal simulations show that our large-signal SPICE model for lasers can qualitatively predict the response exhibited by actual laser diodes. 5. By using the HBT SPICE model and the laser SPICE model, the performance of HBT-laser transmitters can be simulated. The preliminary results obtained here suggest that the models developed in this thesis will be useful in HBT-laser optoelectronic integrated-circuit design. 6.2 R e c o m m e n d a t i o n s for Future Work Now that large-signal equivalent circuit models have been developed for HBTs and semi- conductor lasers, future projects using these models might be: 1. Design, optimize and verify the performance of HBT-laser transmitters. Chapter 6. Summary 89 2. Incorporate effects such as thermal effects, transit-time effects, high-level injec- tion and base-collector breakdown into the SPICE modeling for HBTs; develop physically-based models for bandgap narrowing and conduction band lowering. 3. Develop accurate models for specific laser structures; and incorporate thermal ef- fects into the model. 4. The interaction of the electrical and optical circuits needs to be accounted for in optoelectronic integrated-circuit design [54]. 5. Analyze an entire optoelectronic communication channel by including SPICE mod- els for dispersive fibres and receivers [55]. Bibliography [1] S. C. M. Ho, "The Effect of Base Grading on the Gain and High Frequency Perfor- mance of AlGaAs/GaAs Heterojunction Bipolar Transistors," M.A.Sc thesis, Uni- versity of British Columbia, Aug. 1989. [2] O-S. Ang, "Modeling of Double Heterojunction Bipolar Transistors," M.A.Sc thesis, University of British Columbia, July 1990. [3] A. P. Laser, "Calculation of the Maximum Frequency of Oscillation for Microwave Heterojunction Bipolar Transistors," M.A.Sc thesis, University of British Columbia, June 1990. [4] R. S. Tucker, "Large-signal circuit model for simulation of injection-laser modulation dynamics", IEE Proc, vol. 128, Pt. I, pp. 180-184, Oct, 1981. [5] R. S. Tucker and I. P. Kaminow, "High-Frequency Characteristics of Directly Modu- lated InGaAsP Ridge Waveguide and Buried Heterostructure Lasers," J. Lightwave Technology, vol. 2, pp. 385-393, Aug. 1984. [6] S. Chandrasekhar, L. M. Lunardi, A. H. Gnauck, G. J. Qua, D. Ritter, R. A. Hamm and M. B. Panish, "A lOGbit/s OEIC Photoreceiver Using InP/InGaAs Heterojunc- tion Bipolar Transistors," Electronics Lett., vol. 28, pp. 466-468, 1992. [7] Q. Z. Liu, D. L. Pulfrey and M. K. Jackson, "Analysis of the Transistor-Related Noise in Integrated Pin-HBT Optical Receiver Front-Ends," IEEE Trans. Electron Devices, vol. 40, pp. 2204-2210, Dec. 1993. [8] A. P. Laser and D. L. Pulfrey, "Reconciliation of Methods for Estimating fmax for Microwave Heterojunction Transistors," IEEE Trans. Electron Devices, vol. 38, pp. 1685-1692, Aug. 1991. [9] S. C. M. Ho and D. L. Pulfrey, "The Effect of Base Grading on the Gain and High Frequency Performance of AlGaAs/GaAs Heterojunction Bipolar Transistors," IEEE Trans. Electron Devices, vol. 36, pp. 2173-2182, Oct. 1989. [10] K. Runge, D. Daniel, R. D. Standley, J. L. Gimlett, R. B. Nubling, R. L. Pierson, S. M. Beccue, K-C. Wang, N-H. Sheng, M-C. F. Chang, D. M. Chen, and P. M. Asbeck, "AlGaAs/GaAs HBT IC's for High-Speed Lightwave Transmission Systems," IEEE Jour. Solid-State Circuits, vol. 27, pp. 1332-1339, Oct. 1992. 90 Bibliography 91 [11] C. T. Matsuno, A. K. Sharma, and A. K. Oki, "A Large-Signal HSPICE Model for the Heterojunction Bipolar Transistor," IEEE Trans. Microwave Theory and Techniques, vol. 37, pp. 1472-1475, Sept. 1989. [12] M. E. Kim, "GaAs Heterojunction Bipolar Transistor (HBT) Device and IC Tech- nology for High Performance Analog/Microwave, Digital, and A/D Conversion Ap- plications," SPIE High-Speed Electronics and Device Scaling, vol. 1288, pp. 9-20, 1990. [13] M. E. Hafizi, C. R. Crowell and M. E. Grupen, "The DC Characteristics of GaAs/AlGaAs Heterojunction Bipolar Transistors with Application to Device Mod- eling," IEEE Trans. Electron Devices, vol. 37, pp. 2121-2129, Oct. 1990. [14] J. P. Bailbe, A. Marty, G. Rey, J. Tasselli, and A. Bouyahyaoui, "Electrical Behavior of Double Heterojunction NpN GaAlAs/GaAs/GaAlAs Bipolar Transistors", Solid- State Electronics vol. 28, pp. 627-638, 1985. [15] B. R. Ryum and I. M. Abdel-Motaleb, "A Gummel-Poon Model for Abrupt and Graded Heterojunction Bipolar Transistors (HBTs)," Solid-State Electronics, vol. 33, pp. 869-880, 1990. [16] C. N. Huang and I. M. Abdel-Motaleb, "Gummel-Poon Model for Single and Double Heterojunction Bipolar Transistors," IEE Proceedings-G, vol. 138, pp. 165-169, Apr. 1991. [17] C. D. Parikh and F. A. Lindholm, "A New Charge-Control Model for Single- and Double-Heterojunction Bipolar Transistors," IEEE Trans. Electron Devices, vol. 39, pp. 1303-1311, June 1992. [18] J. J. Liou and J. S. Yuan, "Physics-Based Large-Signal Heterojunction Bipolar Tran- sistor Model for Circuit Simulation," IEE Proceedings-G, vol. 138, pp. 97-103, Feb. 1991. [19] P. C. Grossman and J. Choma, "Large Signal Modeling of HBT's Including Self- Heating and Transit Time Effects," IEEE Trans. Microwave Theory and Techniques, vol. 40, pp. 449-463, Mar. 1992. [20] A. A. Grinberg, M. S. Shur, R. J. Fischer, and H.Morkog, "An Investigation of the Effect of Graded Layers and Tunneling on the Performance of AlGaAs/GaAs Heterojunction Bipolar Transistors," IEEE Trans. Electron Devices, vol. 31, pp. 1758-1765, Dec. 1984. [21] M. S. Lundstrom, "An Ebers-Moll Model for the Heterostructure Bipolar Transis- tor," Solid-State Electronics, vol. 29, pp.1173-1179, 1986. Bibliography 92 D. A. Teeter, J. R. East, R. K. Mains and G. I. Haddad, "Large-Signal Numerical and Analytical HBT Models," IEEE Trans. Electron Devices, vol. 40, pp. 837-845, May 1993. S. Searles and D. L. Pulfrey, "An Analysis of Space-Charge-Region Recombination in HBTs," IEEE Trans. Electron Devices, to appear, Apr. 1994. A. St. Denis, "HBT Modeling for Device Design: Quarterly Report No. 1," CRC, Ottawa, Dec. 1992. D. L. Pulfrey and S. Searles, "Electron Quasi-Fermi Level Splitting at the Base- Emitter Junction of AlGaAs/GaAs HBTs," IEEE Trans. Electron Devices, vol. 40, pp. 1183-1185, June 1993. D. L. Pulfrey and N. G. Tarr, Introduction to Microelectronic Devices, Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1989, pp. 360-361. I. E. Getreu, Modeling the Bipolar Transistor, Beaverton, Oregon: Tektronix Inc., 1976, pp. 34-39. D.J. Roulston, Bipolar Semiconductor Devices, Toronto: McGraw-Hill Publishing Company, 1990, pp. 239-240. H. Zhou and D. L. Pulfrey, "Computation of Transit and Signal Delay Times for the Collector Depletion Region of GaAs-Based HBTs," Solid-State Electronics, vol. 35, pp. 113-115, 1992. A. St. Denis, D. L. Pulfrey and A. Marty, "Reciprocity in Heteroj unction Bipolar Transistors," Solid-State Electronics vol. 35, pp. 1633-1637, 1992. R. K. Surridge, Bell-Northern Research, Ottawa, Private communication, Oct. 1992. M. S. Lundstrom, M. E. Klausmeier-Brown, M. R. Melloch, R. K. Ahrenkiel and B. M. Keyes, "Device-Related Material Properties of Heavily Doped Gallium Arsenide," Solid-State Electronics, vol. 33, pp. 693-704, 1990. S. C. Jain and D. J. Roulston, "A Simple Expression for Band Gap Narrowing (BGN) in Heavily Doped Si, Ge, GaAs and GexSii_x Strained Layers," Solid-State Electronics, vol. 34, pp. 453-465, 1991. A. St. Denis, Private communication, Nov. 1993. J. Hu, Bell-Northern Research, Ottawa, Private communication, Sept. 1993. T. Lester, Bell-Northern Research, Ottawa, Private communication, Oct. 1993. Bibliography 93 [37] A. P. Heberle, U. Strauss, W. W. Riihle, K. H. Bachem, T. Lauterbach and N. Haegle, "Minority-Carrier Lifetime in Heavily Doped GaAs:C," Jap. J. Appl. Phys., Part 1, no. lB, pp. 495-497, 1993. [38] J. Sitch, Bell-Northern Research, Ottawa, Private communication, Sept. 1993. [39] D. A. Hodges and H. G. Jackson, Analysis and Design of Digital Integrated Circuits (2nd Edn.), New York: McGraw-Hill, 1988, p. 222. [40] G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers, New York: Van Nostrand Reinhold Company Inc., 1991, chapter 6. [41] G. P. Agrawal, "Effect of Gain and Index Nonlinearities on Single-Mode Dynamics in Semiconductor Lasers," IEEE Trans. Quantum Electronics, vol. 26, pp. 1901-1909, Nov. 1990. [42] R. S. Tucker and D. J. Pope, "Circuit Modeling of the Effect of Diffusion on Damping in a Narrow-Stripe Semiconductor Laser," IEEE Trans. Quantum Electronics, vol. 19, pp. 1179-1183, July 1983. [43] D. S. Gao, S. M. Kang, R. P. Bryan, and J. J. Coleman, "Modeling of Quantum-Well Lasers for Computer-Aided Analysis of Optoelectronic Integrated Circuits," IEEE Trans. Quantum Electronics, vol. 26, pp. 1206-1216, July 1990. [44] A. Elrefaie, "Computer-Aided Modeling, and Simulation of Lightwave Communica- tion Systems," LEOS Newsletter, vol. 7, pp. 19-20, Feb. 1993. [45] Y. Yamamoto, S. Machida, and 0 . Nilsson, "Squeezed-State Generation by Semi- conductor Lasers," in "Coherence, Amplification, and Quantum Effects in Semicon- ductor Lasers," Y. Yamamoto Ed., pp. 502-505, New York: John Wiley & Sons, Inc. 1991. [46] R. S. Tucker and D. J. Pope, "Microwave Circuit Models of Semiconductor Injection Lasers." IEEE Trans. Microwave Theory and Techniques, vol. 31, pp. 289-294, Mar. 1983. [47] R. S. Tucker, "High-Speed Modulation of Semiconductor Lasers," J. Lightwave Tech- nology, vol. LT-3, pp. 1180-1192, Dec. 1985. [48] J. E. Bowers, B. R. Hemenway, A. H. Gnauck and D. P. Wilt, "High-Speed InGaAsP Constricted-Mesa Lasers," IEEE Trans. Quantum Electronics, vol. 22, pp. 833-843, June 1986. Bibliography 94 [49] P. A. Morton, R. F. Ormondroyd, J. E. Bowers and M. S. Demokan, "Large-Signal Harmonic and Intermodulation Distortions in Wide-bandwidth GalnAsP Semicon- ductor Lasers," IEEE Trans. Quantum Electronics, vol. 25, pp. 1559-1567, June 1989. [50] G. P. Agrawal, "Spectral Hole-Burning and Gain Saturation in Semiconductor Lasers: Strong-Signal Theory," J. Appl. Phys., vol. 63, pp. 1232-1235, Feb. 1988. [51] G. Bjork, A. Karlsson and Y. Yamamoto, "On the Linewidth of Microcavity Lasers," Appl. Phys. Lett. vol. 60, pp. 304-306, Jan. 1992. [52] G. S. Burley, "Effect of Laser Chirping on Lightwave System Performance," M.A.Sc thesis, Queen's Universitj', Sept. 1987. [53] K. Y. Liou, S. Chandrasekhar, A. G. Dentai, E. C. Burrows, G. J. Qua, C. H. Joyner, and C. A. Burrus, "A 5 Gb/s Monolithically Integrated Lightwave Transmitter with 1.5 um Multiple Quantum Well Laser and HBT Driver Circuit," IEEE Photonics Technology Letters, vol. 3, pp. 928-930, Oct. 1991. [54] S. lezekiel, "Equivalent Circuit Modeling," in "Compound Semiconductor Device Modelling," C. M. Snowden and R. E. Miles Eds, pp. 149-169, London: Springer- Verlag, 1993. [55] M. K. Jackson, Private communication, Oct. 1993. Appendix A Closed-Form Analytical Expressions for the Abrupt-Junction Tunneling Factor In Appendix A of Ho [1], the tunneling factor for an abrupt junction is simplified by introducing a dimensionless variable X = E/ETI- This simplified form for an abrupt junction (Wg = 0) is with 7„ = 1 + af exp{-a[bg{X) + X- !}}dX (A.i; 9\A) - ti x=o aNrrd2 EJIE = qVTlE = q ( 1 ^ ) = q(VblE-VBE)(l~6E) ZeE a = ETlE/kT b _ kT_ _ xkT eEm*nE E00~25qh\j NE F* r = £-. (A.2) A.I Error Function Approximation g(X) can be fitted by a second-order polynominal M0 + M\X + M2X2 for the range 0 < A' < 1 (see Figure A.I) and 7„ can be approximated by error functions as 7n = 1 + ^ e a X 2 { e r / [ v W l + x3)] - erf[y/^Xl{r + x3)]} (A.3) 95 Appendix A. Closed-Form Analytical Expressions for the Abrupt-Junction Tunneling Facior96 where y/bM~2 , M1 ..M\ . , . 1 2M2 l4M2 4bM2 2-3 •k^ + b (A.4) M0 = 0.9384, Mi = -1.7413 and M2 = 0.8021 are used in our calculation. 1 0.9 0.8 0.7 0.6 £-0.5 0.4 0.3 0.2 0.1 0 \ - - - - - - 1 1 1 1 \ V \ - Numerical calculation \ s \ s -- 2nd order polynomial fitting \ x \ \ \ \ NN. N X N X X { N X X X 1 1 1 1 ' ' 0 0.2 0.4 0.6 X 0.8 Figure A.l : Comparison of the g(X) and the second-order polynomial, vs. X. Appendix A. Closed-Form Analytical Expressions for the Abrupt-Junction Tunneling Factor97 A.2 Exponent ia l Fit t ing Tunneling factors for forward bias can also be fitted by exponential functions such as VBE exe <* v< + .e3. (A.5) The comparison of the tunneling factor calculated by DAPHNE, the error function ap- proximation and the exponential fitting for the emitter-base junction of the baseline uniform-base SHBT is shown in Figure A.2. t o 40 35 30 £25 E E « O20 15 10 5 T^ \ \ \ \ \ \ \ \ \ o. 1 \ N X 1 1 \ \ s i i i i 1 NE=5e+17PB=1e+19 - Numerical calculation ~ Error function expression -. Exponential function expression N o v I I I I - - - - 0.9 1.1 1.2 Vbe (V) 1.3 1.4 1.5 1.6 Figure A.2: The emitter-base junction tunneling factor calculated by DAPHNE, the error function expression (xi = 1.3616, x<i = 0.468, and £3 = —0.8158), and the exponential fitting ( d = 244.55, e2 = 18.6, and e3 = -6 .88) . Appendix A. Closed-Form Analytical Expressions for the Abrupt-Junction Tunneling Factor98 Finally, the tunneling expression can be fitted by a simpler exponential form, namely This one is essentially the same as the exponential fitting used for the normalized junction velocities in Chapter 3, Section 3.2. Appendix B The Interchangeable Relationships for Variables Used in Various Rate Equations The variables used in rate equations expressed in terms of photon population P , photon density S and output power P0 correspond to each other by the following relationships Po = PP p £Po ep 9P = = *— zp P r r VJS gs = vga (B.l) where gp and gs are differential gain constants. 99 A p p e n d i x C Fermi Dirac Correction for the Bandgap Narrowing Bandgap narrowing (BGN) is also taken into account by utilizing the apparent (or ef- fective) bandgap narrowing data from measurements on GaAs [32] (66meV for PB = 1 x 10 1 9 cm - 3 and 70meV for Pg = 5 x 10 1 9cm - 3) . The pn product written in terms of the effective bandgap narrowing A£"° is A Pa pn = n ? 0 e x p ( - ^ f ) = n2ie (C.l) where nte is the effective intrinsic carrier concentration, n,o is the intrinsic carrier con- centration for high purity GaAs and AlxGa!_xAs. Appropriate correction [33] needs be made for the fact that Fermi-Dirac statistics are used in DAPHNE. In Section 2.6.1 and Appendix B of Ho [1], the "new effective densities of states" for the conduction band and valence band are defined as »*<*> = * s ? <a2) for highly-doped P-type GaAs and AlxGai_xAs material, as used in the base. Nc and AV are the effective densities of states for the conduction band and valence band, respectively, Fi^iv) is the Fermi-Dirac integral of order one half, r]n = EFnkjE<~ and T\V = v~kT Fp. The carrier concentrations are also defined as n = NcF1/2{r)n) ~ Ncexp(r]n) P = NvFi/2(r)p)~PB. (C.3) 100 Appendix C. Fermi Dirac Correction for the Bandgap Narrowing 101 The effective carrier concentration can also be written as n]e = pn = Nc exp(r)n)NvF1/2(rjp) = NcNvexV[~{Es0-TAEG)}F1/2(r,p)eM-VP) = ",oexP(-^-)exP( kT ) (C.4) with AEag = AEg + AEFD (C.5) where EgQ is the bandgap for high purity GaAs and AlxGa!_xAs, AEQ is the bandgap narrowing, and AEFD is the Fermi-Dirac correction. Therefore, the Fermi-Dirac correction is AEFD = kTln[F1/2(riP)exp(-r,p)] (C.6) with h where 777* is the effective mass for the holes and F^~,\ is the inverse Fermi-Dirac integral function (see Appendix B of Ho [1]). Calculated values of the Fermi energy and Fermi- Dirac correction are listed in in Table C.l. Appendix C. Fermi Dirac Correction for the Bandgap Narrowing 102 PB{cm~3) m* (meV) Ev — Efp (meV) AEFD (meV) uniform base 1 x 1019 5 x 1019 0.48 15.5 -10.8 97.3 -50.9 graded base 1 x 1019 5 x 1019 0.511 12.1 -9.8 90.6 -46.6 Table C.l: Calculated values of the Fermi energy and Fermi-Dirac correction for uniform and graded-base (xbe = 0.1) cases with Pg = 1 x 1019cm~3 and Pg = 5 x 1019cm-3. Appendix D SPICE Input Files D . l Files Used in Chapter 3 D . l . l FullJV IC-VCE c h a r a c t e r i s t i c s using f u l l e x t r i n s i c dc model . l i b ' . . / d c / s h b t l i b ' shbtinput .param cur=100u .param cur ren t= ' cu r ' VCE pc 0 IB 0 bp DC='current' Rb 3 2 'Rbase' Re pe 1 'Remit' Re 5 4 'Rcol ' VJB bp 3 DC=0.0 VJE 0 pe DC=0.0 VJC pc 5 DC=0.0 . l i b ' . . / d c / s h b t l i b ' Sne eZinv Zinv 0 vol='l/(l+c/v(Sne))' vZinv Zinv Zinva dc=0.0 rZinv Zinva 0 1 ed da 0 vol= ' l+d/v(Sne) ' eAIl All 0 poly(2) Zinv 0 da 0 0 0 0 0 1 vAIl All Alia dc=0.0 rAIl Alia 0 1 fl 2 1 poly(2) vZinv vI2 0 0 0 0 1 f2 2 4 poly(2) vAIl vll 0 0 0 0 1 F3 4 1 poly(2) vZinv VI3.1 0 0 0 0 1 F4 1 4 poly(2) vZinv VI3_2 0 0 0 0 1 D2 2 1 DEH Dl 2 4 DCH 103 Appendix D. SPICE Input Files 104 .lib '../dc/shbtlib' intrinsic_dc models .DC VCE 0 3 0.01 .width out=132 .print dc I(VJC) .option tnom=27 itll=1000 ingold=2 epsmin=le-32 .nodeset v(2)=1.38 v(4)=0.1 .option post probe .option nomod nopage notop noelck .alter .param current='2*cur' .alter .param current='3*cur' .alter .param current='4*cur' .alter .param current='5*cur' .end Appendix D. SPICE Input Files D.1.2 SHBTJib •Library file for baseline SHBT case * .lib shbtinput .param c=0.389 d=0.765 .param ISF=1.065e-24 ISNBE=3.7lE-26 ISNBC=3.71E-26 .param ISE=1.91e-31 ISC=1.73e-24 .param remit=1.18 rbase=164.14 rcol=45.32 .param vt=0.0259 .endl shbtinput .lib Sne eSne Sne 0 pwl +-3.00000e+00 +-2.95000e+00 +-2.90000e+00 +-2.8'5000e+00 +-2.80000e+00 +-2.75000e+00 +-2.70000e+00 +-2.65000e+00 +-2.60000e+00 +-2.55000e+00 +-2.50000e+00 +-2.45000e+00 +-2.40000e+00 +-2.35000e+00 +-2.30000e+00 +-2.25000e+00 +-2.20000e+00 +-2.15000e+00 +-2.10000e+00 +-2.05000e+00 +-2.00000e+00 +-1.95000e+00 +-1.90000e+00 +-1.85000e+00 +-1.80000e+00 +-1.75000e+00 +-1.70000e+00 +-1.65000e+00 +-1.60000e+00 +-1.55000e+00 +-1.50000e+00 +-1.45000e+00 +-1.40000e+00 (1) 2 1 1.62641e+02 47348e+02 33463e+02 20860e+02 09421e+02 90414e+0'i 8.96244e+01 8.10824e+01 7.33356e+01 6.63113e+01 99434e+01 41718e+01 89417e+01 42034e+01 99116e+01 60250e+01 25064e+01 93215e+01 2.64396e+01 2.38323e+01 2.14743e+01 1.93422e+01 1.74149e+01 56732e+01 40999e+01 26789e+01 13960e+01 02382e+01 19350e+00 8.25131e+00 7.40184e+00 6.63625e+00 5.94653e+00 1. 1. 1. 1. 9. 5. 5. 4. 4. 3. 3. 3. 2. Appendix D. SPICE Input Files +-1.35000e+0( +-1.30000e+0( +-1.25000e+0< +-1.20000e+0( +-1.15000e+0( +-1.10000e+0( +-1.05000e+0( +-1.00000e+0( +-9.50000e-0 +-9.00000e-0 +-8.50000e-0 +-8.00000e-0 +-7.50000e-0: +-7.00000e-0" +-6.50000e-0: +-6.00000e-0: +-5.50000e-0: +-5.00000e-0: +-4.50000e-0: +-4.00000e-0: +-3.50000e-0: +-3.00000e-0: +-2.50000e-0: +-2.00000e-0: +-1.50000e-0: +-i.oooooe-o: +-5.00000e-0: +-2.28983e-lE +5.00000e-02 +1.00000e-01 +1.50000e-01 +2.00000e-01 +2.50000e-01 +3.00000e-01 +3.50000e-01 +4.00000e-01 +4.50000e-01 +5.00000e-01 +5.50000e-01 +6.00000e-01 +6.50000e-01 +7.00000e-01 +7.50000e-01 +8.00000e-01 +8.50000e-01 +9.00000e-01 +9.50000e-01 +1.00000e+00 +1.05000e+00 +1.10000e+00 +1.15000e+00 3, 5.32540e+00 3, 4.76628e+00 3, 4.26320e+00 3, 3.81073e+00 ), 3.40398e+00 3, 3.03850e+00 3, 2.71028e+00 3, 2.41565e+00 L, 2.15134e+00 L, 1.91434e+00 L, 1.70197e+00 L, 1.51179e+00 L, 1.34157e+00 L, 1.18934e+00 L, 1.05328e+00 L, 9.31760e-01 L, 8.23316e-01 L, 7.26616e-01 L, 6.40460e-01 L, 5.63768e-01 L, 4.95563e-01 L, 4.34966e-01 L, 3.81183e-01 L, 3.33501e-01 L, 2.91276e-01 L, 2.53930e-01 I, 2.20942e-01 5, 1.91843e-01 1.66213e-01 1.43674e-01 1.23886e-01 1.06544e-01 9.13749e-02 7.81340e-02 6.66015e-02 5.65810e-02 4.78967e-02 4.03913e-02 3.39244e-02 2.83704e-02 2.36174e-02 1.95657e-02 1.61262G-02 1.32198e-02 1.07759e-02 8.73205e-03 7.03252e-03 5.62797e-03 4.47465e-03 3.53388e-03 2.77152e-03 Appendix D. SPICE Input Files +1.20000e+00, +1.25000e+00, +1.30000e+00, +i.35000e+00, +1.40000e+00, +1.45000e+00, +1.50000e+00, +1.55000e+00, RSne Sne 0 1 .endl Sne 2, 1. 1, 9, 7. 5. 3. 2. . 15759e- ,66588e- ,27373e- ,61795e- . 13799e- , 15979e- ,55938e- . 18039e- 03 -03 -03 -04 04 04 -04 04 .lib intrinsic_dc_models eI2 11 0 2 1 1 vI2 11 12 dc=0.0 dI2 12 0 dI2h eI3_2 13 0 2 4 1 vI3_2 13 14 dc=0.0 dI3_2 14 0 dI3_2h eI3_l 17 0 2 1 1 vI3_l 17 18 dc=0.0 dI3_l 18 0 dI3_lh ell 15 0 2 4 1 vll 15 16 dc=0.0 dll 16 0 dllh .model dI2h d(is«'ISNBE') .model dI3 2h d(is=,ISF') .model dI3 lh d(is='ISF') .model dllh d(is=' ISNBC) .model den d(is='ISE') .model dch d(is=,ISCJ) .endl intrinsic_dc_models Appendix D. SPICE Input Files D.1.3 B J T J V IC-VCE characteristics for SHBT using bjt model .param cur=100u .param current='cur' VCE pc 0 IB 0 bp DC='current' VJB bp 3 DC=0.0 VJE 0 pe DC=0.0 VJC pc 5 DC=0.0 ql 5 3 pe bjt .model bjt npn is=4.43e-24 NF=1.168 bf*28.71 br=60.8 +ISC=1.73e-24 ISE«1.91e-31 NC=1 NE=1 +re=1.18 rc=45.32 rb=164.14 .DC VCE 0 3 0.01 .width out=132 .print dc I(VJC) .option tnom=27 itll=1000 ingold=2 epsmin=le-32 .nodeset v(2)=1.32 v(4)=0.1 .option post probe .alter .param current='2*cur' .alter .param current='3*cur' .alter .param current='4*cur' .alter .param current='5*cur' .end Appendix D. SPICE Input Files D . 2 Files U s e d in Chapter 4 D .2 .1 G3_B36 Forward gummel p l o t f o r F36 3x3 d e v i c e s u s i n g b j t model VBE 7 0 VBC 7 8 DC=0 VJB 5 2 DC=0.0 VJE 4 1 DC=0.0 VJC 6 3 DC=0.0 RE 0 4 11.37 RB 7 5 109.1 RC 8 6 31 .06 q l 3 2 1 b j t D3 2 1 DESCR D4 2 3 DCSCR .model b j t npn I s=1 .49e -24 NF»1.186 NR=1 BF=122.7 BR=381 +ISE=2.81e-32 ISC=3.16e-25 NE=1 NC=1 .param I s c r e = 4 . 3 4 e - 2 0 I s c r c = 4 . 2 5 e - 1 4 N3=2 N4=2 .model DESCR D(IS='ISCRE' n= 'N3 ' ) .model DCSCR D(IS='ISCRC' n= 'N4 ' ) .DC VBE 1.0 1.8 0.026 .width out=132 .print I(VJC) I(VJB) .option dcon=l gmindc=le-14 gmin=le-14 .option post probe .option tnom=27 ingold=2 .end Appendix D. SPICE Input Files D.2.2 Ring_B36 5-stage F36 2*2 ring oscillator (rl=400) using BJT model .param vcc_r=1.55 .subckt cell n_Vcc 2 1 0 rl n_Vcc 2 400 .subckt shbt 8 7 0 VJB 5 2 DC=0.0 VJE 4 1 DC=0.0 VJC 6 3 DC=0.0 RE 0 4 25.6 RB 7 5 137.4 RC 8 6 35.9 ql 3 2 1 bjt D3 2 1 DESCR D4 2 3 DCSCR .model bjt npn Is=6.62e-25 NF=1.186 NR=1 BF=122.7 BR=381 +ISE=1.25e-32 ISC=1.41e-25 NE=1 NC=1 +tf=1.57ps vje=1.633 vjc=1.309 cje=6.32e-15 cjc=1.49e-14 +tr=13.78ps .param Iscre=l.93e-20 Iscrc=l.89e-14 N3=2 N4=2 .model DESCR D(IS='ISCRE' n='N3') .model DCSCR D(IS='ISCRC n='N4') .ends xshbtl 2 1 0 shbt .ends vcc n_Vcc 0 dc 'vcc_r' xl n_Vcc 4 6 0 cell x2 n_Vcc 5 4 0 cell x3 n_Vcc 8 5 0 cell x4 n_Vcc 10 8 0 cell x5 n_Vcc 6 10 0 cell CO 6 0 6.7213e-15 cl n.Vcc 6 3.4831e-15 c2 4 0 1.9974e-15 c3 5 0 1.9974e-15 Appendix D. SPICE Input Files 111 c7 8 0 1.9974e-15 c9 10 0 1.9974e-15 cl3 0 6 7.3172e-15 cl4 0 n.Vcc 3.4485e-14 cl5 0 4 1.0269e-15 cl6 0 5 1.0269e-15 cl7 0 8 1.0269e-15 cl8 0 10 1.0269e-15 .width out=80 .ic v(4)=0.6 .tran lps 2ns uic .option tnom=27 ingold=2 epsmin=le-32 .option nomod nopage notop noelck .print tran v(4) v(5) I(xl.xshbtl.VJC) .option probe post .alter .param vcc_r=1.60 .alter .param vcc_r=1.70 .alter .param vcc_r=1.80 .alter .param vcc_r=2.0 .alter .param vcc_r=2.4 .alter .param vcc_r=2.8 .alter .param vcc_r=3.19 .alter .param vcc_r=3.74 .alter .param vcc_r=4.21 .alter .param vcc_r=4.76 .alter .param vcc_r=5.23 .end Appendix D. SPICE Input Files 112 D.2.3 Switch_B36 5-stage F36 2*2 r ing o s c i l l a t o r (rl=400) using BJT model .param vcc_r=1.55 high=1.54 low=l . l l .subckt cell n_Vcc 2 1 0 rl n_Vcc 2 400 .subckt shbt 8 7 0 VJB 5 2 DC=0.0 VJE 4 1 DC=0.0 VJC 6 3 DC=0.0 RE 0 4 25.6 RB 7 5 137.4 RC 8 6 35.9 ql 3 2 1 bjt D3 2 1 DESCR D4 2 3 DCSCR .model bjt npn Is=6.62e-25 NF=1.1862 NR=1 BF=122.7 BR=381 +ISE=1.25e-32 ISC=1.41e-25 NE=1 NC=1 +tf=1.57ps vje=1.633 vjc=1.309 cje=6.32e-15 cjc=1.49e-14 +tr=13.78ps .param Iscre=l.93e-20 Iscrc=l.89e-14 N3=2 N4=2 .model DESCR D(IS='ISCRE' n='N3') .model DCSCR D(IS='ISCRC' n='N4') .ends xshbtl 2 1 0 shbt .ends vcc n_Vcc 0 dc 'vcc.r' xl n.Vcc 4 6 0 cell c2 4 0 1.9974e-15 cl5 0 4 1.0269e-15 Vin 6 0 pulse(low high 0 2ps 2ps lOOps 204ps) .width out=80 .tran 0.lps 408ps .option tnom=27 ingold=2 epsmin=le-32 Appendix D. SPICE Input Files 113 . p r in t t r an v(4) v(6) .option probe post . a l t e r .param vcc_r=2.4 high=1.95 low=0.539 . a l t e r .param vcc_r=3.19 high=2.25 low=0.586 . a l t e r .param vcc_r=4.76 high=2.79 low=0.789 .end Appendix D. SPICE Input Files D . 3 Files U s e d in Chapter 5 D.3 .1 SS_LD V-I and L-I c h a r a c t e r i s t i c s .param t e = 2 . 2 2 e - 0 9 Cph=6.20e-12 Rph=2.53e-01 .param p_over=2.58e-08 NL_Po=2.97e+00 b e t a _ e f f = l . 2 5 e - 0 5 .param b=4.83e+02 I_sp0=7.22e-03 .param hl=1.95e+14 h0=3.00e+12 .param I_SL=3e-14 NL=2 R_SL=9 Cjo_L=10pf Vbi_L=1.9 . subck t LD 1 2 3 •Electrical section Drad 1 4 dlaser VSTAR 4 3 DC=0.0 Fdiff 1 3 VIC 1 *Subcircuit #1 for generating the derivative of current Hdiff 10 3 VSTAR 1 *Cdiff 11 3 'te' VIC 10 11 DC=0.0 *model of diode .model dlaser d is='I_SL' n='NL' cj='Cjo_L' pb='Vbi_L' * OPTICAL SECTION GG 1 2 POLY(2) 2 3 5 3 0.0 0.0 0.0 0.0 1 0.0 0.0 '-NL_Po' FSP 3 2 VSTAR 'beta.eff RP 2 3 'Rph' *CP 2 3 'Cph' * SUBCIRCUIT # 2 HDUM2 5 3 VSTAR '-b*I_sp0' 'b' RDUMMY1 5 3 1 .Ends * Main circuit XI 3 2 0 LD R 1 3 R_SL * Input Current Appendix D. SPICE Input Files I i n 0 1 .DC I i n 0 0.06 5e-5 * Optical power .Print DC V(l) .PRINT dc par('V(2,0))') .option nomod nopage notop noelck .OPTION LIMPTS=5000 PIVT0L=lE-30 LVLTIM=2 ING0LD=2 .OPTION probe post .alter .param b e t a _ e f f = l e - 0 4 . a l t e r .param b e t a _ e f f = l e - 0 3 .END Appendix D. SPICE Input Files D.3 .2 TranJLD Transient analysis for laser operating at 2 gbit/s input .param t e = 2 . 2 2 e - 0 9 Cph=6.20e-12 Rph=2.53e-01 .param NL_Po=2.97e+00 be t a_e f f=1 .25e -05 .param b=4.83e+02 I_sp0=7.22e-03 .param hl=1.95e+14 h0=3.00e+12 .param I_SL=3e-14 NL=2 R_SL=9 Cjo_L=10pf Vbi_L«1.9 .param R_smooth=l C_smooth=10pF .param Ith=15.8m I_b='1.l*Ith' I_m=0.016 .param period=1000ps rise=150ps fall='rise' dur='period/2- .subckt LD 1 2 3 •Electrical section Drad 1 4 dlaser VIsp 4 3 DC=0.0 Fdiff 1 3 VIC 1 •Subcircuit #1 for generating the derivative of current Hdiff 10 3 VIsp 1 Cdiff 11 3 'te; VIC 10 11 DC=0.0 *Model of diode .model dlaser d is='I_SL' n='NL' cj='Cjo_L' pb='Vbi_L' * Optical section GG 1 2 P0LY(2) 2 3 5 3 0.0 0.0 0.0 0.0 1 0.0 0.0 *-NL_Po' FSP 3 2 VIsp 'beta.eff RP 2 3 'Rph' CP 2 3 'Cph' * Subcircuit # 2 HDUM2 5 3 VIsp '-b*I_sp0' 'b' RDUMMY1 5 3 1 .Ends * Main circuit XI 3 2 0 LD R 1 3 R_SL Appendix D. SPICE Input Files 117 •Transient analysis U N 0 inl PULSE('I_b' 'I_b+I_m' 0.0ns 'rise' 'fall' 'dur' 'period') R_corner inl in 'R_smooth' C.corner in 0 'C_smooth' VI in 1 DC=0.0 * Optical power .TRAN 5ps 10.00NS ONS UIC .IC V(2)=0 .option nomod nopage notop noelck tnom=27 .OPTION LIMPTS=5000 PIVT0L=lE-30 LVLTIM=2 ING0LD=2 .PRINT TRAN V(2) LVl(IIN) I(VI) .print par('(hl*I(xl.VIsp)-hO)/6.28') .OPTION probe post .alter .param NL_Po=5e+00 .alter .param NL_Po=5e+00 I_b='1.2*Ith' .alter .param NL_Po=5e+00 I_b='l\2*Ith' .param period=1000ps rise=80ps fall='rise' dur='period/2-rise' .END Appendix D. SPICE Input Files D.3.3 Expe_LD Simulated optical output waveform and injection current .param te=1.65e-09 Cph=5.87e-12 Rph=2.00e-01 .param NL_Po=7.34e+00 beta_eff=1.0e-04 .param b=5.14e+02 I_sp0=9.71e-03 .param hl=2.19e+14 h0=4.25e+12 .param I_SL=3e-14 NL=2 R_SL=1 Cjo_L=10pf Vbi_L=1.9 .param R_smooth=2 C_smooth=10pF .param Ith=19.4m I_b«'Ith*l.l' I_m=0.020 .param period=1176ps rise=230ps fall='rise' dur='period/2 .subckt LD 1 2 3 * •Electrical section * Drad 1 4 dlaser VIsp 4 3 DC=0.0 Fdiff 1 3 VIC 1 •Subcircuit #1 for generating the derivative of current Hdiff 10 3 VIsp 1 Cdiff 11 3 'te; VIC 10 11 DC=0.0 •model of diode .model dlaser d is='I_SL' n='NL' cj='Cjo_L' pb='Vbi_L' • OPTICAL SECTION GG 1 2 P0LY(2) 2 3 5 3 0.0 0.0 0.0 0.0 1 0.0 0.0 '-NL_Po FSP 3 2 VIsp 'beta.eff RP 2 3 'Rph' CP 2 3 'Cph' • SUBCIRCUIT # 2 HDUM2 5 3 VIsp '-b*I_sp0' 'b' RDUMMY1 5 3 1 .Ends • Main circuit Appendix D. SPICE Input Files 119 XI 3 2 0 LD R 1 3 R.SL •Transient analysis U N 0 inl PULSE('I_b' 'I_b+I_m' 0.0ns 'rise' 'fall' 'dur' 'period') R_corner inl in 'R_smooth' C.corner in 0 'C_smooth' VI in 1 DC=0.0 •Optical power .TRAN lps 6.00NS ONS UIC .IC V(1)=0 V(2)=0 .PRINT TRAN V(2,0) LVl(IIN) I(V1) .print par(' (hl^Kxl .VIsp)-h0)/6.28') .OPTION LIMPTS=5000 PIVT0L=lE-30 LVLTIM=2 ING0LD=1 .OPTION probe post .alter .param R_SL=2 .alter .param R_SL=3 .alter .param R_SL=4 .alter .param R_SL=5 .alter .param R_SL=1 .param period=1176ps rise=260ps fall='rise' dur='period/2-rise' .END Appendix D. SPICE Input Files D.S.4 Mono Monolithic integration of HBT-laser: 1.7 Gb/s .param rf=-2.0 ee=-6 .param low=-2.4 high=-1.6 biasv=1.05 .param Ith=19.4m I_b='Ith*l.2' .param period=1176ps rise=150ps fall='rise' dur='period/2-rise' .param delay='period/2+2.5e-9' .lib 'ld_lib' laserinput .lib 'ld_lib' laser_active Rs 1 R_SL_1 'R_SL' xl R_SL_1 2 Lb_l Id Lb Lb_l 3 O.ln EPo Po 0 Vol='v(2,Lb_l)' vlO 0 20 dc 0 v20 0 1 dc 0 v30 18 19 dc 0 vee 11 0 dc ' ee' vrf 13 0 dc »rf» .subckt shbtl col base emit RE emit 1 10 RB base 2 90.25 RC col 3 3.89 ql 3 2 1 bjt D3 2 1 DESCR D4 2 3 DCSCR .model bjt npn Is=1.126e-13 NF=1.218 NR=1 BF=39.26 BR=2.51 +ISE=4.373e-24 ISC=2.134e-12 NE=1 NC=1 +TF=2.6ps TR=43.5ps +CJE=8.52e-14 CJC=8.59e-14 vje=0.9394 vjc=0.5721 .param Iscre=2.6e-15 Iscrc=l.57e-07 N3=2 N4=2 .model DESCR D(IS='ISCRE' n='N3') .model DCSCR D(IS='ISCRC n='N4') .ends xql 3 13 18 shbtl xq2 20 17 18 shbtl xq3 19 12 11 shbtl Appendix D. SPICE Input Files 121 vin 17 0 pulse('low' 'high' 'delay' 'rise' 'fall' 'dur' 'period') vbia 12 11 dc 'biasv' ilaser Lb_l 1 dc 'I_b' .width out=80 .tran 0.2ps 6ns Ons uic .print tran i(V20) v(Po) .option limpts=5000 .option gmin=le-12 gmindc=le-12 tnom=27 fs=0.01 ft=0.1 .option nomod nopage notop noelck .option post probe .end Appendix D. SPICE Input Files D.3.5 Hybrid Simulation for 1.7 Gb/s for hybrid transmitter .param r i= -2 .0 ee=-6 .param low=-2.4 high=-1.6 biasv=1.05 .param Ith=19.4m I_b= ' I th* l . 2 ' .param period=1176ps rise=150ps fall='rise' dur='period/2-riseJ .param delay='period/2+2.5e-9' .lib 'ld_lib' laserinput .lib 'ld.lib' laser_active Rs 1 R_SL_1 'R_SL' xl R_SL_1 2 Lb_l Id Lb Lb_l 3 3n EPo Po 0 Vol='v(2,Lb_l)' vlO 0 20 dc 0 v20 0 1 dc 0 v30 18 19 dc 0 vee 11 0 dc 'ee' vrf 13 0 dc 'rf .subckt shbtl col base emit RE emit 1 10 RB base 2 90.25 RC col 3 3.89 ql 3 2 1 bjt D3 2 1 DESCR D4 2 3 DCSCR .model bjt npn Is=1.126e-13 NF=1.2177 NR=1 BF=39.26 BR=2.51 +ISE=4.373e-24 ISC=2.134e-12 NE=1 NC=1 +TF=2.6ps TR=43.5ps +CJE=8.52e-14 CJC=8.59e-14 vje=0.9394 vjc=0.5721 .param Iscre=2.6e-15 Iscrc=l.57e-07 N3=2 N4=2 .model DESCR D(IS='ISCRE' n='N3') .model DCSCR D(IS='ISCRC n='N4') .ends xql 3 13 18 shbtl xq2 20 17 18 shbtl xq3 19 12 11 shbtl Appendix D. SPICE Input Files 123 vin 17 0 pulseClow' 'high' 'delay' 'rise' 'fall' 'dur' 'period') vbia 12 11 dc 'biasv' ilaser Lb_l 1 dc 'I_b' .width out=80 .tran 0.2ps 6ns Ons uic .print tran i(V20) v(Po) .option limpts=5000 .option gmin=le-12 gmindc=le-12 tnom=27 fs=0.01 ft=0.1 .option nomod nopage notop noelck .option post probe .end Appendix D. SPICE Input Files D.8.6 LDJib Laser parameter from Mr. Greg Burley .lib laserinput .param te=1.65e-09 Cph=5.87e-12 Rph=2.00e-01 .param p_over=2.72e-08 NL_Po=7.34e+00 beta_eff=1.40e-05 .param b=5.14e+02 I_sp0=9.71e-03 .param hl=2.19e+14 h0=4.25e+12 .param I_SL=3e-14 NL=2 R_SL=9 Cjo_L=10pf Vbi_L=1.9 .endl laserinput .lib laser.active .subckt LD 1 2 3 •Electrical section Drad 1 4 dlaser VIsp 4 3 DC=0.0 Fdiff 1 3 VIC 1 •Subcircuit #1 for generating the derivative of current Hdiff 10 3 VIsp 1 Cdiff 11 3 'te' VIC 10 11 DC=0.0 •model of diode .model dlaser d is='I_SL' n='NL' cj=,Cjo_L' pb='Vbi_L' • OPTICAL SECTION GG 1 2 P0LY(2) 2 3 5 3 0.0 0.0 0.0 0.0 1 0.0 0.0 '-NL_Po' FSP 3 2 VIsp 'beta.eff' RP 2 3 'Rph' CP 2 3 'Cph' • SUBCIRCUIT # 2 HDUM2 5 3 VIsp '-b*I_sp0' 'b' RDUMMY1 5 3 1 .Ends .endl laser_active
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