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Large-signal spice models for heterojunction bipolar transistors and lasers Feng, James Jun Xiong 1994

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LARGE-SIGNAL SPICE MODELS FOR HETEROJUNCTION BIPOLAR TRANSISTORS A N D LASERS By James Jun Xiong Feng B. A. S c , University of British Columbia, 1992  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF A P P L I E D 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  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  tfWcX  3/> ><?H  S?iQSi4.<trit n<t"i*t>triH4  Abstract  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 voltage Vcci f ° r  a  with bias  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 C o n t e n t s  Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgement 1  2  xiii  Introduction  1  1.1  Review of Existing Equivalent Circuit Models and Objectives of the Project  2  1.2  Overview of Thesis  3  S P I C E M o d e l D e v e l o p m e n t for t h e General D H B T  5  2.1  6  Current Transport 2.1.1  2.2  2.3  2.4  Current Transport at an Abrupt Emitter-Base Heterojunction  . .  The Full Intrinsic DC Model  6 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  The Full Extrinsic DC Model  ". . . .  16  2.3.1  Recombination/Generation Currents  16  2.3.2  Resistances  18  Large-Signal HBT Model  18  iii  3  2.4.2  Diffusion Capacitances  18 •  19 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  3.3  5  Junction Capacitances  Simplified M o d e l s for Specific H B T s  3.2  4  2.4.1  The B J T Model  27  3.2.1  B J T Intrinsic Model for the Abrupt-Junction, Uniform-Base SHBT  34  3.2.2  B J T Intrinsic Model for the Abrupt-Junction, Graded-Base SHBT  35  3.2.3  B J T Extrinsic Model  36  Comparisons of the DC Models  38  3.3.1  Gummel Plots  39  3.3.2  Collector Output Characteristics  45  Comparison with H B T Experimental Data  51  4.1  DC Characteristics  53  4.2  Large-Signal Characteristics for two 5-Stage Ring Oscillators  57  Laser D e v i c e and Circuit M o d e l i n g  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  6  5.4.1  Steady-State Characteristics  5.4.2  Large-Signal Characteristics  75 "  75  5.5  Comparison of Simulated Laser Characteristics with Experimental Data .  78  5.6  Simulation of Laser Driver Transmitter Circuits  83  Summary  87  6.1  Conclusions  87  6.2  Recommendations for Future Work  88  Bibliography  90  Appendices  95  A Closed-Form Analytical Expressions for t h e A b r u p t - J u n c t i o n Tunneling Factor A.l  95 Error Function Approximation  95  A.2 Exponential Fitting  97  B T h e Interchangeable R e l a t i o n s h i p s for Variables U s e d in Various R a t e Equations  99  C Fermi Dirac Correction for t h e B a n d g a p N a r r o w i n g  100  D S P I C E I n p u t Files  103  D.l  Files Used in Chapter 3  103  D.l.l  FullJV  •. . . .  D.l.2  SHBTJib  105  D.l.3  BJTJV  108  D.2 Files Used in Chapter 4  103  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).  3.3  Fitting parameters and corresponding ideality factors for eqn. 3.17. The  .  24  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 2um 2 )  53  4.4  SPICE BJT IhmtV  model parameters for the F36 3 x 3um2  devices (BGN =  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. vii  84  5.5  Layout parameters for the AT&T SHBT (emitter area = 5 x 9um 2 ) [53]. Data marked with an asterisk are assumed values  C.l  '  84  Calculated values of the Fermi energy and Fermi-Dirac correction for uniform and graded-base (xf,e = 0.1) cases with Pg = 1 x 10 19 cm -3 and PB = 5 x 10 19 cm- 3  102  viii  List of Figures  2.1  General energy band diagram for an abrupt-junction, graded-base DHBT in the active mode  2.2  5  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 3.3  29  Normalized junction velocity for iV £ = l x 10 1 7 cm~ 3 ,P B = 1 x 1 0 1 9 c m - 3 for a uniform base. The exponential fit uses A*e~VBE^sv',  with A* = 0.005  and S=16.816 3.4  30  Normalized junction velocity for NE = 5 x \017cm~3,PB  = 1 x 10 19 cm~ 3  for a uniform base. The exponential fit uses A*e~VB£/SVt,  with A* = 1.618  and 5 = 6.937 3.5  31  Normalized junction velocity for NE = 5 x 10 1 7 cm~ 3 ,Pe = 5 x 10 i 9 cm~ 3 for a uniform base. The exponential fit uses A*e~VBE^sv', and S = 9.537  with A* = 0.189 32  IX  3.6  Normalized junction velocity for NE = 5 X 10 17 cm  3  , PB — 1 x 10 19 cm  in the graded-base case (xbe — 0.1). The exponential fit uses  3  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 Aug = Auger, Rad — Radiative)  = Shockley—  Read—Hall,  in both the base-side(_B) and emitter-  side(_E) of the depletion region at the BE junction for the "baseline"  3.9  uniform-base SHBT  37  Comparison of Gummel plots for the baseline uniform-base SHBT  40  3.10 Comparison of Gummel plots for the baseline graded-base SHBT 3.11 Comparison of Gummel plots for the baseline uniform-base DHBT. 3.12 log(7c)  vs  41 ...  42  - 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),  4.1  400u.4 (.sw3) and 500uA (.sw4)  48  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 simulation results from SPICE and DAPHNE for the various conditions.  4.4  . .  55  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  4.8  60  Comparison of BJT-SPICE-model and experimental data for the BNR 5-stage ring oscillator with RL = 400f)  4.9  61  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  (.trl), 3.19V (.tr2) and 4.76V (.tr3). Input pulse Vw and output pulse VOUT  are  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 (.trl), 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  5.1  ring oscillator with RL — 4000  65  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  5.4  V-I and L-I characteristics  5.5  L-I characteristic for (3ejj = 1 x 1 0 - 4 (.swl), and 1 x 1 0 - 3 (.sw2) respectively. 77  5.6  Transient analysis for the 1.3ttm buried-heterostructure laser: injection  '  73 76  current (top panel) and optical output waveform (bottom panel). The following four cases are simulated by varying the nonlinear gain parameter (ep 0 ). the dc bias current (hias),  an  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 (.trl); 3) tPo = 5 W - 1 , Ibias = 1.2Ith, and tr = 150ps (.tr2); 4) tPo = 5 H / _ 1 , Ibia, = 1.2/ifc, and tr = 80ps (.tr3) 5.7  79  Simulated optical power waveform (top panel) and injection current (bottom 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 HBTlaser 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-  A.l  forms (top panel) and injection currents (bottom panel)  86  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, e 2 = 18.6, and e$ = —6.88). . . .  xn  97  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 H B T device simulation program, DAPHNE,  which has been used  extensively in my analyses, also to Dr. Hao-Sheng Zhou for modeling the space-chargeregion 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. J u n t a o Hu, and Dr.  T. Lester of Bell-Northern Research, Ottawa for specifying the geometrical and  compositional parameters, and measurement results for the H B T 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 T R 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"Optoelectronics Consortium, and the Natural Sciences and Engineering Research Council is greatfully acknowledged. Finally, this thesis would have been impossible without the support and encouragement 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  Introduction  The high-speed, high-current-drive and low-noise capabilities make heterojunction bipolar transistors (HBTs) very promising candidates for applications in optoelectronic communication systems. HBTs have already been monolithically integrated with photodetectors 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 properties 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 B J T 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 Consortium (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 E x i s t i n g Equivalent Circuit M o d e l s and O b j e c t i v e s of t h e Project  To be of universal appeal, the physics-based equivalent circuit model parameters should relate to the geometrical, compositional and known-electrical parameters directly. Empirical models do not meet this criterion as they rely on parameter extraction from experimental data for specific HBTs [11, 12, 13]. Previously-reported physics-based HBT models are not suitable for innovative design because of: 1) making over-simplified assumptions, for example, characterizing the carrier transport across the heterojunction by driftdiffusion [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 picture 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 models, 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 exponential 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  SHBTs [21]. And Teeter included the transit-time effects [22]. DAPHNE,  3  a numerical  model developed at UBC for a general graded-junction and graded-baseTJHBT, incorporates the tunneling current into this bias-dependent junction velocity term [1, 2], recombination 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 parasitics 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  O v e r v i e w of T h e s i s  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 exponential fit to the normalized junction velocity, the simplest B J T versions for uniform and moderately-graded-base HBTs are also derived. Forward Gummel plots and collectoroutput 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 derived, 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 tunneling 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 output 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 D H B T  /  f i  AEnE  AEnC  Efinn  \l \  Efnp qVBE Efp  qVBC Efinn  Base  I  Emitter  Collector  *JE  o  W  X  JC  *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  on the Thermionic-Field-Diffusion  for the General DEBT  6  model [20] is briefly reviewed. Full intrinsic  trinsic dc and large-signal equivalent circuit models are then developed from  dc, exDAPHNE,  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 2.1.1  Current Transport Current Transport at an A b r u p t E m i t t e r - B a s e H e t e r o j u n c t i o n ^  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 t h e 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„. T h e various components of JrnE are given by / VTnE  =  kT 27rm  *nE  AEnE AEnE0 SE —  =  6EVBE +  AEnEo  = (X'BE-XJS)(1 =  •  NQBENEPB*  -6E)-SEkT]n(— 5—^—-) n lBENCE  tENE CBEPB n B0  '  + tENE (i <^  PB where eE is the permittivity of the emitter, CBE is the permittivity of t h e 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\ AECE  = qVri = q{VbiE - VBE) - AECE =  + AEnE  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, En  Ex=!fJJ^. e * V EmnE  (2.6)  Chapter 2. SPICE Model Development  2.2  for the General  DHBT  9  T h e Full Intrinsic D C M o d e l  In this section, the electron diffusion current in the base and the hole currents in the emitter and collector are discussed. An analytical Ebers-Moll SPICE model for the gradedjunction 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, respectively. 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  WE  10  Wc  WB  Jn(W)  Jn(0)  JE  JC JNB  JR.BE  JR/G.BC  c  Jp(xC) Jp(xE)  XE  W  JB  xc  I  OB Figure 2.3: Schematic of charge flows in an HBT. 2.2.1  E l e c t r o n Diffusion Current in t h e B a s e  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) Jn(W)  =  - o „ ( C v « > v ' - l ) + a12(ev**'v'-l)  =  _ Q 2 1 ( e v ^ / v ' - l ) + Q22(ev'^/v'-l)  (2.7)  where the Ebers-Moll coefficients ctij for the electron currents are [2] AoVn Oil =  BoVn ,  "12 =  ,  a  A-wVn 21 =  BwVn ,  "22 =  (2.8)  Chapter 2. SPICE Model Development  for the General  DEBT  11  and z  = (zc + bnyn){zZ  anyn)-(2tynesW)2  +  + Kyn) - 4*2yne2sVV]  A0  =  z*nB0[an{zZ  B0  =  2tZnZnUBW  Aw  =  zt2 n zn nBoe  Bw  = z^nBw[bn(z^  + anyn)-4t2yne2sW}.  (2.9)  The components of the coefficients above are AEg kTW  f  .  t  =  EgBE ~ EgBC kTW  JPLlB+4 ,  f  a = - , ri=s  6LnB n  7T  where  E9BE  2 »B0  I  -  _  2 iBW  n  -  2sW  "BO  =  -77—, riBW — —7;— = riBoe rB rB  an  =  (ri-fy*w-(r2-fy>w  bn  =  (ri-f)er>w-(r2-f)eT>w  Vn  =  ^n  =  qDnB  oT\W  °vTnE~fnEe  + t, r2 = s - t  pTiW  kT ,  2^ = 9 u r n C 7 n C e  *r  (2.10)  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 t h e electron diffusion coefficient in the base, W is t h e thickness of the quasi-neutral region of the base, Pg is t h e doping concentration in t h e base, ngw is t h e 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 7 n c , and AEnc  vjnc,  are the thermal velocity, tunneling factor, and electron potential energy,  respectively, for the collector-base junction.  Chapter 2. SPICE Model Development  2.2.2  for the General  DEBT  12  E m i t t e r and Collector H o l e 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 Vg ) LpE  JpE ZE  =  £pE '  ZE f^coth(p^) 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 ( ^ ) . •LpE  (2.13)  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), JP(xc)  shown in Figure 2.3 is given by  = JPc(eVBc/V'-l)  (2.14)  where JpC  qDPcPc ^(WCs = —y coth(-—) LVC Lpc  /  CN (2.15)  0 1  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  E b e r s - M o l l Intrinsic D C Circuit M o d e l  Since the neutral base recombination current I^B is INB  = AE\Jn{W)  - J n (0)] = INBE + INBC  (2.17)  where AE(au-a2l){eVBE>Vt-l)  INBE = = AE(a22  INBC  - 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)}  Ic  =  AE[-Jn(W)  IB  =  -IE-IC  =-INBE-IP(XE)  - Jp(xc)] = -INBC  +IEC -IP(XC)  -  Ice  - IEC + Ice (2-19)  where the electron transport currents are: Ice = AEa21(ev*E'Vt IEC = AEau(ev»c?v'  - 1) - 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.  Ic  D2  f t )  INBC  Ice  B  ( k  K  lEC  IB  D1  •  ) INBE  IE  Figure 2.4: Full intrinsic dc equivalent circuit.  2.2.4  V o l t a g e - D e p e n d e n t E b e r s - M o l l Coefficients  We also define „  Z ZCZE  C\  Co  SNE  SNC  C3 SNESNC  where c\  =  On2/n  bnyn , c2 = — , c3 -  qVTnE  qVTnC  D2nB r2  LnBVTnEVTnC  (2.21)  Chapter 2. SPICE Model Development  for the General  = lnEe-AE^kT,  SNE  DHBT  SNC = lnce-AE"c'kT  15  ( 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 Q12  =  Q21 =  nB0anyn  +  UBQ-^—  "ii  an«n + ,2 q _  =  nyn  L.'  nB0  nB  Q—  vTnCSNC  —  nBwbnyn  +  nBw^§—  Q22  n  —  =  L2BvTnESNE  nBW  -_  y  •  0  .  (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 Z h,NBC  j INBC  n  dNC , d2 u  v  «/*-l)  VBC/V,  = — y — ( 1 + 7;—)(e  BCI  nN  - 1)  Ice = he  t-fie^'-l)  = ^£(eVBC'Vt-l)  (2.24)  where IS,F IS,NBE  =  2tnB0yne2sWAE  = nB0yn(an  - 2te  s  )AE  Chapter 2. SPICE Model Development  for the General  =  nBwyn(bn  dx  =  iliafli vTncyn{an - 2te2°w)  d2  =  IS,NBC  -  DEBT  HKL BJ  »  16  2t)A£  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-wiselinear 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 M o d e 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 current 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 calculated from DAPHNE  2.3.1  for single-sided [2, 3], pyramidal [9] and rectangular [24] structures.  R e c o m b i n a t i o n / G e n e r a t i o n Currents  For the case of recombination in the emitter-base depletion region, it has been pointedout 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  JSRH,B  =  AE r, NEniBE <-6BEVt,iF, v Cs e < e TnOK-iE  JSHH,E =  6  BEvBE-*Bfn v t  Cs^eW* IT VBE-AEfn v  JAU9,B  =  CsniBEApPBe  JAU9,E  =  Csn2lEAnNEeVBE/v< v CSVhlE 2 D , BE-*Bfn v —TJ—niBEBp6Ee < CsVbiE 2 c c ^ — 7 } — r i i E B n d B E e v'  T  JRad,B  =  T  jRad,E  =  t  ,o o e \ (2.26)  Vt  where n  T/  ^s  =  &BE  =  vt  ,  ZqtBE PBSEVKE £BEPB  CBEPB  + tENE Tl' J?  AEi AEjn  =  Vt ln(——) + XBE - XE riiBE  =  Ejnn - Ejnp = Vt ln(  —  —)  VTnEVVnB,E^NE  w WnB,E  =  LnBt&nh(-—) •LnB  T where Ejnn  =  y/TnOTpo  (2.27)  and Efnp are quasi-Fermi levels shown in Figures 2.1-2.2, r n 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 JBC,SRH  qniBC (WB}SCR = —7i—( Z T p0  , Wc,SCR\f_v„nfvt 1" )( e rn0  -x ~ 1)  ,90Sx [Z.Zb)  Chapter 2. SPICE Model Development  for the General DHBT  where the depletion-region widths WB,SCR  and WC,SCR  18  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  Resistances  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 zerobias 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 structures can be found in [2, 3, 9, 24].  2.4  Large-Signal H B T Model  In this section, a Full extrinsic model for large-signal simulation (see Figure 2.5) is obtained by adding junction capacitances and diffusion capacitances into the Full extrinsic dc model.  2.4.1  Junction Capacitances  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, CCJ(O),  VJE  = Vbie, and VJC  i.e., CJE  = C E J ( O ) , CJC  =  = Vbic. The SPICE parameters for the forward-bias  Chapter 2. SPICE Model Development  for the General  DEBT  19  IC  RC  UDC  B  CJC  A  D4  ^  D2 (. t  INBC  i ) Ice(  j  IB  IEC  RB XDE  , CJE  V  D3  D1 ( I ) INBE  RE IE  Figure 2.5: Full extrinsic equivalent circuit of HBT. depletion capacitor coefficient, FC, the base-emitter junction grading coefficient, and the base-collector junction grading coefficient, MJC,  MJE,  can also be included to model  the non-ideal junction capacitances.  2.4.2  Diffusion C a p a c i t a n c e s  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 time 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 junctions, respectively.  B a s e Transit T i m e The base transit time is given by  1 TB  where ft(x) = C\trxX•\-C2^"2X  fw  = T iw\ / Jn[W ) Jo  ~<in(.x)dx  is the excess electron concentration, and Jn(W)  (2-32) is calculated  from [9] Jn(x) = qDnB[(ri  - . O d e ' 1 * + (r 2 - f)C2eT>x]  (2.33)  with 1  _ n(W) - n ( 0 ) e ^ 2c' w 'sinh(<H0 '  n(0)e'*w - h(W) 2e sM/ sinh(iVK) '  , ^"  j  Chapter 2. SPICE Model Development  for the General  DEBT  21  The result of the integration is r B  1 -rir2DnB[  h  2*n-(WQ + q H n(0) 2th(0)e*°w - bnh(Wy  {  }  where  .  ( w )  =  2tyn^e2^nB0(eVfl£/Vi_i)  (2.36)  After some algebra, (2.35) reduces to Z£ * c o s h ( W ) + £ sinh(<W0 •^nS ^e2KK =  rBl  +  sinh(^W) te? be  ^  (2.37) AE„f7  with the bias-dependent base-collector junction velocity 5"c = ^Tnce  v  t ^nc =  VTUCSNC-  For SHBTs, 5 c 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 TBB a s e - C o l l e c t o r D e p l e t i o n R e g i o n Transit T i m e The signal delay time through the base-collector depletion region for a uniform carrier velocity is [28] TBCSCR  = 0.5WBC'SCR  = 0.5r t  (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 M o d e 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  +  (3-2) SNE  Chapter 3. Simplified Models for Specific  Material  Layer  HBTs  Thickness  Doping (cm~3)  A Emitter Base Collector Sub-collector  n — AlxGa\-xAs p+ — GaAs n~ — AlxGai~xAs n+ — GaAs  4000 1000 4000 4000  5 1 5 1  x x x x  Al composition  1017 1019 1016 1018  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  6.0um 4.5um 3.0um  Le  U Lc  Table 3.2: Geometrical parameters for the baseline H B T (emitter area = 36 um2). For uniform-base HBTs, the components an, bn and yn in (2.10) are 2  W -—cosh(-—)  a„ =  bn -  Vn  qDn B 2sinh(T^)'  J-inB  3.1.1  LinB  (3  Abrupt-Junction, Uniform-Base S H B T  The a%1 coefficients in (2.23) are O-nVnnB  «ii  =  Q12  =  &22 =  a2\ =  2tynnB  -^[OnJ/n + 7^ Z  £ L  and the SPICE parameters in (2.22) and (2.25) are DnBcoth(^) Cl  =  LnB^TnE  nBVTnE^NE  J  (3  Chapter 3. Simplified Models for Specific HBTs  _ SF  '  25  qnBDnBAE  ~  T  • -L/ W \  LnBsmhij^) qnBDnB[cosh(-^)-l]AE h,NBE  =  ~ . ,,nw LnBsmhij^)  h,NBC =  \  DnBsmh(^)  d2 = Lr BVTnE[cOsh(^-) ~n°r2Lw[ n  ,•  (3-5)  - 1]  Also the current gain BF for the base transport is constant and equal to:  BF=-^-=  I  JS,NBE  COsh( j^)  (3.6) - 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  w  if W < LnB.  2  2DnB 3.1.2  (3.7)  Abrupt-Junction, Graded-Base SHBT  The components of the a,j coefficients in (2.10) are e^w[2tcosh(tW)  <zn =  +  fsmh(tW)}  bn = e*w[2t cosh(tW) - f smh{tW)} y n  *D«B  =  .  ( 3  .8)  w  2e£ sinh(*W) The o.ij coefficients in (2.23) are OnS/n^BO  on  =  Q12  =  Z a2i =  2tynnBW -  qD2nB  "BH'ft Z  L„BVTnE3NE  and the SPICE parameters in (2.22) and (2.25) are DnB[tcoth(tW) VTnE  + -L)  1  ,o QN  Chapter 3. Simplified Models for Specific HBTs  _ ~  5 F  '  26  gnBWDnBtAE e$wsmh{tW) teiw sinh(iW')  f + 2-  IS,NBE  = qnBoDnBAE[tcoth(tW)  IS,NBC  = qnBWDnBAE[tcoth(tW)-  j ~ -] w e^ s'mh(tW)  ^ Dn B 2 L nBvTnE[t coth(tW) - i -  ,  «  e*  (3.10)  ]'  sinh(tW')  Also the current gain BF for the base transport is constant and equal to B F  = - ^ - = /S,TVB£  i U  j - .  (3.11)  teiw  i cosh(W) + | smh(tW) -  The base transit time in (2.37) for the abrupt-junction, graded-base SHBT is L2nB t cosh(W) + { sinh(W) TB  =  {  ~D7B 2DnB  ,. 1]  ^  W + Pfet(l )1 nB- + ^^ ^ - e '  f()r  W  <  L n B  .  (3>12)  e2  3.1.3  Abrupt-Junction, Uniform-Base D H B T  The Q,-j coefficients in (2.23) are "Br  Qll =  ,  9-Pnfi  -ITianVn + J^ Z  nBvTnCONC  2tynnB Z  =  c*2i-  a 22  =  - ^ [r a n y n +  "B „  ..  •^  ,  v 9 n^ r2  DnBcoth(^) LnBVTnE  D AnBcoth(^) =  LnB^TnC  *  ]  LnBvTnESNE  and the SPICE parameters in (2.22) and (2.25) are  C2  J  L  0-12  c\ =  i  J  (3.13)  Chapter 3. Simplified Models for Specific  HBTs  27  D2nB LlBVTnEVTnC qnBDnBAE LnB sinh(-j^-)  C3  IS,F  nB '  'S,NBE  = =  Is *S,NBC  qnBDnBAE[cos\\{~-)  • i , W^ *\ -  =  - 1]  L^smh^) DnBS-mh{^)  d,  =  di  =  LnBVTnc[cOsh(j—)  -  A^sinh^) , uwB\  J  1]  (3-14)  i r  -'nB '  Also the current gain BF for the base transport is p  ^  P  _  J  S,F  _  1  BFSHBT  - IS..BE " [ C o s h ( i ) - 1](1 + j j L ) - (1 + ^ Y  m  -x  (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] + —-^ DnB W2  ~ =  W  TF^ + ^ rBi + ^ -  where rB\ — -^— and TB2 = ~ ~ •  1  tJNC XT„  r  lfW<LnB (3.16)  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 Model  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 piecewise 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 basegrading cases, as shown in Figures 3.3-3.6.  Chapter 3. Simplified Models for Specific HBTs  29  10000  0  '  0.0  •  '  0.5  •  '  1.0  •  '  '  1.5  V BE (V) Figure 3.2: Dependence of anyn/zn over most of the forward-bias range of H e 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  NE {cm-3)  (cm-3)  A'  1 x 1017 5 x 1017 5 x 1017  1 x 1019 1 x 1019 5 x 1019  0.005 1.618 0.189  PB  HBTs  30  uniform base NF S 16.816 6.937 9.537  1.063 1.168 1.117  A'  graded base 5 NF  0.101 22.184 2.045  18.202 7.63 11.237  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  NE=1e+17PB=1e+19 4.5  DAPHNE Exponential fitting  •D  Ito 3.5 E o ^  3  CD  c CO  2.5-  1.5  0.9  1.1  1.2  1.3  1.4  1.5  1.6  Vbe (V)  Figure 3.3: Normalized junction velocity for NE = 1 x 10 17 cm" 3 ,P.B = 1 x 10 19 cm 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  x10~  NE=5e+17PB=1e+19  DAPHNE T5  N  4  Exponential fitting  16 E O  ID 3 c CO  2-  1.1  1.2  1.3  1.4  1.5  1.6  Vbe (V)  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  3.5  32  x10"'  NE=5e+17PB=5e+19  2.5  - DAPHNE  "O  ~  2  -- Exponential fitting  to  E o c ^1.5 c  0.5-  1.1  1.2  1.3  1.4  1.5  1.6  Vbe (V)  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  •v i  \  0.14  i  i  \  33  HBTs  .  |  |  ...  ,  NE=5e+17PB=1e+19 \  0.12  \  \  \ \  0.1 S"  \  - DAPHNE \  \ \ \\ Vx  -- Exponential fitting NN  <D N  >& \  |0.08  \  o c^  yy  N.  \ >.  \\  Q)  -0.06  \\. S X. N.  \  N  \  0.04 V  \ . V x. X.  >v *v ^ \ ^ v^^V^  0.02  ^ ^ . 1  1  1  I  1.1  1.2  1.3  I  1  1.4  1.5  1.6  Vbe (V)  Figure 3.6: Normalized junction velocity for NE = 5 X 10 17 cm 3 , Pg = 1 x 10 19 cm 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  3.2.1  HBTs  34  B J T Intrinsic M o d e l for t h e A b r u p t - J u n c t i o n , U n i f o r m - B a s e 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:  = He-^/^^/v, - 1)  INBE  = H(eWV(-i)  w Ice  =  IEC  =  ISe-y™?sv'(ev^v<-l) ISe-VB^SVt(eVBc/Vt-l)  (3.18)  where qnBoVTnE2te2sW' AE  _  IS  ^  2te2sW  BF  a„ -  BR  =  2te2sW  T ^ V ^ I B A -  (3>19)  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  =  INBC  =  Icc  =  IEC  =  ^(eV°°/NFV'  - 1) ^(eVBc/NRV'-l)  IS(ev*s<NFV<-l) IS(eVsc/Vt  - 1).  (3.20)  The new Ebers-Moll equations can be implemented by the regular SPICE macromodel for a B J T . The model parameters are IS  qnBVTnEAEA*  cosh ( ^ - '  Chapter 3. Simplified Models for Specific HBTs  35  NF = ( i - ^ r 1 NR  = 1  BF  = coshf^) - 1  BR  =  1  L  l  B V J  ^  (3.21)  nfl  where NF is the new ideality factor which takes account of the voltage-dependent tunneling 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 qnBQVTnEtiwtAEA*  T„  *cosh(W) + £sinh(<WO (l-^)-1  NF  =  NR  = 1  BF  =  4. 1-W  It*  t cosh(tW) + { sm\i{tW) -  teiw  L2nBVTnEA*t  BR = smh{tW)D ,, l ,rnBei\] w-  ( 3 - 22 )  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 ^^ no longer valid. In such cases the Full model of Section 3.1 must be used.  >> 1 is  Chapter 3. Simplified Models for Specific  3.2.3  36  HBTs  B J T Extrinsic M o d e 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.  RC  D4  RB B  Q ^  D3  RE  Figure 3.7: "BJT" extrinsic equivalent circuit of the HBT.  With the assumption SNE  = A*e  V  BB/SV,^  t  ] i e e i e c t r o n quasi-Fermi level splitting  becomes AE  '» = Vt ln(  wB  AJ + ^T  (3 23)  "  Chapter 3. Simplified Models for Specific  0.2  0.6  HBTs  0.8 VBE (V)  37  1  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  nSRH,B  =  Js,Aug,B  —  nAug,B  ~  *Ej-*BF,vhi  NErHBEVTnEWnB,EA* •e  TnOV-iEDnB  &BE - Nfn CSn*BEApPBVTnEWnBtEA* Dn B 1  v  >  Chapter 3. Simplified Models for Specific  HBTs  Js,Rad,B  =  CsVbiE 2 Dc —77—n i B E ±f p d E Vt  nRad,B  =  ^ r ^  T  38  v  TnEWnB,EA L>nB (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 ShockleyRead-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  C o m p a r i s o n s of t h e D C M o d e l s  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  an  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]: r n 0 = 50ps; Tp0 = 200ps; An = 7.99 x 1(T 3 2 cm 6 s- 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 1 0 - 1 1 cm3s~1.  Specific contact resistivity values taken are: pc£ = 5 x 10 _ 8 Qcm 2 for the  emitter, pcs = 3 x 10~6f2cm2 for the base, and pcc = 10 _ 6 ficm 2 for the collector.  3.3.1  Gummel 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  10  |  1  1  1  1  HBTs  40  1  1  r  Vbe (V)  Figure 3.9: Comparison of Gummel plots for the baseline uniform-base SHBT.  Chapter 3. Simplified Models for Specific  "i  r  HBTs  -i  41  1  1  r  - DAPHNE  o Full SPICE model  -. BJT SPICE model  Figure 3.10: Comparison of Gummel plots for the baseline graded-base SHBT.  Chapter 3. Simplified Models for Specific  10u  -i  "i  r  HBTs  42  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  GUMMEL  PLOTS  HBTs  FOR THE B A S E L I N E DHBT CWITH 1G-DEC93 22:51:2G  •3.0  43  V B C • 0. - 1. - 2 . - 3V )  DHBT_SP.SWO PARCLOG10CIC ~ A •••'^£  H . 0  DHBT_SP.SW1 PAR C LOG 1 0 C I C  '^_Q  DHBT_SP.SW2 PARCLOG10CIC  •5.0 Q  D H B T _ S P .SW3 PARCLOG10CIC  -B . 0  7.0  8.0  9 . 0  10.0  -11.0  12.0  -13.0 BOO . 0M  VOLTS  1.20 CLIN]  Figure 3.12: l o g ( / 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  GUMMEL  PLOTS  FOR  THE  HBTs  44  BASELINE DHBT CWITH 1G-DEC93 22:51 : 2G  VBC - 0. - 1. - 2. - 3V ]  D H B T _ S P .SWO PARCLOG10CAB  — A  D H B T _ S P .SW1 PARCL0G1OCAB  — E  DHBT_SP.SW2 PARCL0G1OCAB DHBT_SP.SW3 PARCLOG1OCAB  DHBT_SP.SWO P A R C L O G 1 0 C AB  -3.30 -  DHBT_SP.SW1 PARCL0G10CAB  3 .350 DHBT_SP.SW2 PARCLOG10CAB O -3.40  D H B T _ S P .SW3 PARCL0G1 0CAB  <• 3.450 S .-' ...I...i...i.....  3.50  1.490 1.480  .J.  1.50 VOLTS CLIN]  ...I....I...J....I.  ,J  1.510  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 expanded-voltage-scale version (bottom panel).  Chapter 3. Simplified Models for Specific  3.3.2  HBTs  45  Collector O u t p u t 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 D H B T 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 FullJV, 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  BJT  46  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  IC-VCE  CHARACTERISTICS FOR GBSHBT 31 - D E C 9 3 1 5 ! 1 2 : 1 9  47  CXBE-0.1]  18 . OM  BJT_IV.SWO ICVJC A FULL_IV.SWO ICVJC  IB . OM D 14 . OM  12 . OM  BJT_IV.SW1 '=* . I C V J C Q FULL-IV.SW1 ICVJC — «> BJT_IV.SW2 ICVJC  •V 1 0 . OM  -. vjjjC  FULL-IV.SW2  I C VJC _ M 8 . OM  G . OM  _  BJT_IV.SW3 ICVJC FULL_IV.SW3 I CVJC I BJT_IV.SW4 I CVJC  4 . OM  D FULL-IV.SW4 ICVJC  2 . OM  0. i i i i I i i i i I i i i i I i i i i I i i i iJ 1.0 2.0 3.0 4.0 0. VOLTS CLIN) 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  I-V  48  C H A R A C T E R I S T I C S FOR THE BASELINE 1S-DEC93 2 2 : 2:55  DHBT  I /-  DHBT_SP.SWO I C VI C A  / ~ i •  '  .  .  _  D H B T _ S P .SW2 I CVJC Q  D H B T _ S P .SW3  i  4.0 VOLTS CLIN)  I  G. 0  i  i  i  I  i  J  B.0 9.0  Figure 3.16: Ic — VCE characteristic from the SPICE Full modelior the "baseline" uniform-base DHBT with Ib = lOOuA (.swO), 200uA (.swl), 300uA (.sw2), AOOuA (.sw3) and 500uA (.sw4).  Chapter 3. Simplified Models for Specific HBTs  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  49  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  A* Cs IsRH,B IsRH,E lAug,B *Aug,E lRad,B lRad,E  HBTs  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  50  5  6.937  NsRH,B  1.367 2 1.168 1 1.168 1  NsRH,E NAug,B * * Aug,E NRad,B NRad,E  Table 3.6: Diode parameters for space charge region recombination currents of the "baseline" SHBT.  Chapter 4  Comparison with H B T 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, gradedbase 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  Material  Thickness  (A) Emitter cap Emitter grading Emitter Base Collector Sub collector Buffer  +  n — GaAs n — AlxGai-xAs n — AlxGai-xAs p+ — AlxGa\^xAs n~ — GaAs n+ — GaAs GaAs  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////// Emitter cap  Wcap -  r *" Isolation  v  i  Emitter  V/A  We  Y/^Z//////^/////A  r  ,w  Base ! 1  Collector  i  Wc  i  V///////////A  i  1  ! i  Sub-collector  Wbuf \  i  SI substrate  Figure 4.1: Cross-section of BNR F36 HBT structure [34].  Sbc Sb  1 1  : • i  Lc  Collector  !  Emitter  ; Le i i  Lb  Base  i_ i  J_Li  !  JL Sc  '  i  Kse-Hr Sbiso  Seb  Figure 4.2: Layout of the BNR F36 HBT.  1u  Chapter 4. Comparison  with HBT Experimental  Se  sb  Sc  S.Oum 6.0um 3.0um  ^biso Seb Sbc  53  Data  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 3um 2 ).  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 2um 2 ).  4.1  D C Characteristics  The forward Gummel plots using the B J T 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 B J T 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 3um 2 devices (BGN = and base-emitter junction grading = 20 A).  IhmeV  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  10  — O O  Experimental data S P I C E , g r a d e d B E junction D A P H N E , abrupt B E junction^ S P I C E , abrupt B E junction """  CD O  10  M o O  10"  0.5  1.0  1.5  2.0  V B E (V) Figure 4.3: Collect current characteristic comparison of experimental data with simulation 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"  i**3*1!*  D Extrinsic model • Experimental data  10'  10  .  ^  mfca •  ^  'c  <  ,-, O  * ,  O  n  10"  n n •  • • *  •  10"'  ft 10  <=>  CD  1.0  •  n  ft 10  n  n •  i 1.2  1.4  1.6  1.8  V BE (V)  Figure 4.4: Comparison of Gummel plots from BJT-SPICE-model and experimental data for BNR F36 device (assuming BGN = lOOmeV).  Chapter 4. Comparison  4.2  with HBT Experimental  Data  57  Large-Signal Characteristics for two 5-Stage R i n g Oscillators  vcc  j  RL  ! RL  RL  u  RL  CI  RL  M-, C2+C15  C3+C16  C7+C17  C14 C9+C18  C0+C13  Figure 4.5: BNR F36 5-stage ring oscillator. In order to examine the usefulness and accuracy of the BJT model for performing largesignal transient analysis, a five-stage ring oscillator was simulated and the results compared 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 VccThis 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 B J T model parameters for the 2 x2uro 2 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 T F 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 T R is about 10 times larger than T F . 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. Considering 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  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  Data  N3 N4 RE RB  Re TF TR CjEO CJCO  vbiE Vbic  59  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.  A S 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  6  1  5.8 -  o Measured  5.6 -  * Simulated  Data  60  1  . »  K  5.4 -  -  0  N  X  f •  o  -  (A O LL  0  °0  o  -  *  -  o  4.8 -  6  4.6 4.4 4.2  X  -  * 1  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  5.5  i  i  1  1 —  Data  T  -•  61  T  -••  1  K  o Measured  *  K  * Simulated  X  X  4.5  K  N  X  -  u 3.5  0  -  -  0  0  K  10  X  o  0  -  o  *  O  o  * *o QiO  2. l  f  1  1  2.5  ...  1  3.5 Vcc (V)  1  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  FUN/OUT  Cparasitic  VlNH VlNLO  /  \  '  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 I B36_4_D.TR0 _I VC4 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 710 . OM 0 .  100.OP TIME  200.OP CLIN)  300.OP  400.OP 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  5-STAGE  F3G  with HBT Experimental  2»2  RING  Data  OSCILLATOR CRl-tOOD 3 - M A R 9 4 1 5 : 3:35  64  USING  BJT  MODEL I B3E_t.TR0 ; VC 4 — A  I  B36.H.TR1  •- Q , \  B 3 G _ 4 . TR2 - V C <4 ± O \— B3G.1.TR3 \: V CI  V »  3 0 0 . OM,  I QN  1 . ON  1 . 2 ON  1 .1 ON 1 .G ON TIME (LIN)  -  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 (.trl), 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  50.0  '  I  I  65  I  I  o 5-stage ° 11-stage o 21-stage  o®  40.0 -  D  0  0 o ©  0  S  30.0 8  o o o o  9 20.0  10.0  D 0  8  I  1.0  2.0  0  o o  3.0  I  I  4.0  5.0  6.0  Vcc (V) Figure 4.12: Propagation delay times vs Vcc comparisons of 21-, 11-, and 5-stage BNR ring oscillator with RL = 400fl.  Chapter 5  Laser D e v i c e and Circuit M o d e l i n g  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 characteristics, 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 R a t e E q u a t i o n s for S e m i c o n d u c t o r Lasers  The rate equations for a single-mode semiconductor laser with a uniform carrier distribution can be written as [40, 41]  —77 at dn dt  =  h i?sp,o TP  I qVa  d  ^ —  GP n Te  a  =  GP Va  °rr i \ *i y [ r v i ( n - T i o ) - —J  rxi\ (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), r p 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 n 0 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 spontaneous quantum efficiency defined as VsP =  (5.3)  The carrier-recombination rate 7 e is 7e  = -  = (AnT + Bn + Cn2) = — + — +  (5.4)  where AnT is the nonradiative recombination rate, B is the radiative recombination coefficient, 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 at  qGP  =  q^- + at  —  =  —Tvgan  qVan Te TP  _, _,  3-P-qRsp,o - —{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., sp  ' dt  4- T  A- T •  T JipT  ^sttr  Chapter 5. Laser Device and Circuit  •Istim  Modeling  — C/p/i  dP at  +  68  P Kph  — Pefjlsp  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 =  ao v aTe 2 f,  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 — e p P ) [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 i m = qGP = b(Isp — 7 sp o)(l — epP)P  with b =  v aTe  ^ , and  J _ qVaTOQ JspO — • Te  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 Isv T+ J-hstim : j , + ~r -«sp dP0  I stim  =  ^ph~T  P0 V "5  at  Pejjlsp  Kph  d  * h T — hno —— — riiisp at IsUm = b( Isp - Isp0 ) (1 - tPo P0 ) P0  with Cph = §, Rph = f ,  e Po =  ¥  , and b = ^  I  = v-fe-  stim  d I sp - ^ Xe — ! i d\ T  ' • ! bp  f i v.  i (3 'sp /'  eff  'ph  4>  C=1 F  -rdt  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  £  =  P T  Rph ePo  =  Self  =  b = JspO  PP  =  9 £P P  PVsp 2Tare Vah~fa>m qVan0  =  ctoTvgaTe 2qVa  fel = ^0  5.2  =  —(Tv g an 0 + —) I Tp  (5.11)  Large-Signal M o d e 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 and InP/InGaAsP  AlGaAsjGaAs  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)  (5.13)  - 1]  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  RSL  Istim  dlsp — Te-~ ( i  sp  ( O P * ls>  R ph ^ ^ G p h  C=1 F T  d\  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  5.2.1  72  N o n l i n e a r Gain P a r a m e t e r  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 )  (5,14)  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 €p  ^  1 ~WS-  huT 2e0nngVaIs  (5 15J  '  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  S p o n t a n e o u s 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 — 1 0 - 5 . In our model, /? is calculated using a simple formula for the conventional semiconductor laser [51]  " = s£r r>Vanth  <5-17>  Chapter 5. Laser Device and Circuit  Modeling  This expression gives a value of /? = 1.25 x 10  5.3  73  for the laser considered in this work.  Steady-State Model  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  Up  R ph  Figure 5.3: Equivalent circuit model for laser diodes at steady state.  Chapter 5. Laser Device and Circuit Modeling  5.4  74  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 equations 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 T  V  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  Te  Cph Rph £Po  Peff b •IspO  h h0 ISL rit  RsL CJL(0) VblL  75  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 t e a d y - S t a t e 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 1 0 - 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 = 2.97W 7-1 , Iua, = 1-Uth, and tT = 150ps (.trO); 2. tPo = 5W~\  Ibtas = l.Uth, and tr = 150ps (.trl);  Chapter 5. Laser Device and Circuit Modeling  V-I  76  AND L-I C H A R A C T E R I S T I C S 2-JAN94 2 3 : 3 3 : 2 9  1.750; 1.501.250 1 .0 750 . 0M 500 . 0M 250 . 0M l _ J . . 1 . .1. . J . . I . .L . .1. . J .. I.. .I J . . 1 . . 1 . .1. . I . . 1 . .1. . J. . . I1 . . I . .  J.. I..I.. J.  0 . SS_LD.SW2 VC2 1 0 . 0M  8 . OM  G . OM  4 . OM  2 . OM .I..J.....I..  10 . OM  20.OM AMPS  . . 1 . .1.. J. . i . -i.. J . . j . - i .  30.OM C LI N D  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 C H A R A C T E R I S T I C S 2-JAN91 23i33:29 1 . OM _  SS_LD.SW1 VC2 i SS_LD.SW2 VC2 Q  3 .750M 3 .50M _ 3 .250M 3 . OM - " " 2 .750M -  V  u L T 1 I N  2.50M - ' " 2 .250M 2 . OM - ' 1 .750M _ 1 .5 OM 1 .250M . 1 . OM 750 . OU 500 . OU 250 . OU -  1 0. 0 .  5 . OM  .1 1 0. OM AMPS  15. OM CLIN)  Figure 5.5: L-I characteristic for /? e // = 1 x 10  20 . OM  25.OM  30  (.swl), and 1 x 10  (.sw2) respectively,  Chapter 5. Laser Device and Circuit  3. tPo = bW'\  Modeling  78  Ibias = 1.2Ith, and tr = 150ps (.tr2);  4. ep 0 = 5 W _ 1 , /bias = 1.21th, and *r = 80ps (.tr3). where 7t las 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 -+• Imod)- The simulated results show that the larger the nonlinear gain parameter (ep 0 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  C o m p a r i s o n of Simulated Laser Characteristics w i t h E x p e r i m e n t a 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 = 1 0 - 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  TRANSIENT  ANALYSIS  Modeling  79  FOR LASER OPERATING 2-JAN94 22:29 : 25  AT 2 G B I T / S  T R A N _ L D .TRO I CV1 - TRAN.LD.TR1 -3 I C V 1  Q -~  TRAN_LD.TR2 I CV1 Q TRAN-LD.TR3 I CV1  •5 . ON 5 . ON  5.50N TIME  G . ON CLIN)  G .50N  7 . ON 7 . ON  Figure 5.6: Transient analysis for the 1.3um 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 (ep 0 ), the dc bias current (hias), and the rise time (tT): 1) ePo = 2 . 9 7 W - 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-27 f ^, 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  Symbol  Value  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  L w d  250 um 2 um 0.2 urn 0.4 5 50cm" 1 50 cm~x 2.5 x 10~ 16 cm2 1 x 10 18 cm'3 1.5 x 10 8 s" 1 0.9 x 10" 1 0 cm3/s 9 x 10~ 29 cm6/s 8.5 x 10 9 cm Is 5 xl0-17cm3  r CtQ  am <*int  a n0 •A-nr  B C V  9  ts  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  Figure 5.8: Experimental [52] optical power waveforms at 1.7Gb/s.  82  Chapter 5. Laser Device and Circuit  5.6  Modeling  83  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  Material  Layer  Thickness  A Emitter cap Emitter Base Collector Sub collector  +  n np+ n~ n+  — InGaAs* InP — InGaAs — InGaAs — InGaAs  1500* 1500* 1000 5000 4000*  84  Doping (cm~3) 1 4 8 1 1  x x x x x  In composition  10 19 10 17 10 18 10 16 10 19  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 time 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 L D J i b 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  (a)  (- - } - -  (b)  — f c _\  -p>-  "j  •<=••  /">r.>"A Laser  Laser i  "Q  Ibias  Ibis  j i  Vref  Q1  Q2  *  j >  I •  —  '  Vref  Q1  Q2 Vsig  Vsig  Vs  Q3  VEE(-)  Vs  Q3  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  SIMULATION 7  Modeling  86  FOR ECL T R A N S M I T T E R S 20-MAR94 22:59:28  AT  1.7GB/S  OM H  M O N O .TRO VCPO —: h 2 HYBRID.TRO VCPO Q  S OM -" V 0 L  ~ 5  OM ~  1  OM -  ^  T L I N  3 OM r  ir  2  OM  1  OM IT  25 . OM  MONO . T R O I CV20 h  20 . OM  iR^A^i  HYBRID .TRO -. I C V20 ._- B  15 . OM Z  10.OM -  5 . OM _  ~B •5.0  r i *\ . 5 0 N 4 ,50N  i  i  i  i i i i 5 . ON TIME CLIN)  i  i i 5 .50N  i  i  i  i G . ON G . ON  Figure 5.10: Monolithic vs. hybrid ECL HBT-laser transmitters: optical output waveforms (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 B J T , the intrinsic Ebers-Moll coefficients for a HBT consist of voltage-dependent junction velocity terms to describe tunneling factors and junction barrier heights. These coefficients can be simplified into a form which allows their implementation in circuit simulators with piece-wise-linear features. Full intrinsic dc, extrinsic dc, and largesignal 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 collectoroutput characteristics highlight the accuracy of both SPICE models. 3. The experimental dc data and large-signal data, obtained from BNR for an abruptjunction, 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 collector 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 semiconductor 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 injection 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 effects 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 models for dispersive fibres and receivers [55].  Bibliography  [1] S. C. M. Ho, "The Effect of Base Grading on the Gain and High Frequency Performance of AlGaAs/GaAs Heterojunction Bipolar Transistors," M.A.Sc thesis, University 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. 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Yuan, "Physics-Based Large-Signal Heterojunction Bipolar Transistor 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 SelfHeating 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 Transistor," 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 BaseEmitter 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 Cliffs, New Jersey: Prentice-Hall, Inc., 1989, pp. 360-361.  Devices, Englewood  I. E. Getreu, Modeling the Bipolar Transistor, Beaverton, Oregon: Tektronix Inc., 1976, pp. 34-39. D.J. Roulston, Bipolar Semiconductor Company, 1990, pp. 239-240.  Devices, Toronto: McGraw-Hill Publishing  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. 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Phys., Part 1, n o . l B , 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 (2nd Edn.), New York: McGraw-Hill, 1988, p. 222. [40] G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Van Nostrand Reinhold Company Inc., 1991, chapter 6.  Circuits  Lasers, New York:  [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 Communication Systems," LEOS Newsletter, vol. 7, pp. 19-20, Feb. 1993. [45] Y. Yamamoto, S. Machida, and 0 . Nilsson, "Squeezed-State Generation by Semiconductor Lasers," in "Coherence, Amplification, and Quantum Effects in Semiconductor 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 nology, vol. LT-3, pp. 1180-1192, Dec. 1985.  Tech-  [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 Semiconductor 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: SpringerVerlag, 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 7„ = 1 + af  exp{-a[bg{X)  + X-  !}}dX  (A.i;  with 9\A)  -  ti  x=o 2  EJIE  aNrrd = qVTlE = q ( 1 ^ ) = ZeE a = ETlE/kT b  _  kT_ _ xkT E00~25qh\j F*  q(VblE-VBE)(l~6E)  eEm*nE NE  (A.2)  r = £-. 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  95  + x 3 )]}  (A.3)  Appendix  A. Closed-Form Analytical Expressions for the Abrupt-Junction  Tunneling Facior96  where y/bM~2 ,  M1 2M 2  ..M\ l 4M 2  .,.  1 4bM2  •k^ + b  2-3  (A.4)  M 0 = 0.9384, Mi = -1.7413 and M2 = 0.8021 are used in our calculation.  1  1  1  1  1  0.9 \  0.8 \  0.7  -  0.6 -  V  \  - Numerical calculation  \ s  \s  -- 2nd order polynomial fitting \ x  \\ \ \ NN.  £-0.5 -  NX NX  0.4 -  X{ NX XX  0.3 0.2 0.1 0  0  1  1  1  1  0.2  0.4  0.6  0.8  '  '  X  Figure A.l: Comparison of the g(X) and the second-order polynomial, vs. X.  Appendix  A.2  A. Closed-Form Analytical Expressions for the Abrupt-Junction  Tunneling Factor97  Exponential Fitting  Tunneling factors for forward bias can also be fitted by exponential functions such as V  BE  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. to  T^  1  1  i  i  i  1  -  40 35 30  \  NE=5e+17PB=1e+19 \  \ \ \ \ \ \ \ \  - Numerical calculation  o.  £25 E E « O20  -  ~ Error function expression  \ NX  \  s  -. Exponential function expression -  15 10  -  N o v  5 1  0.9  i  I  I  I  I  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, e 2 = 18.6, and e 3 = - 6 . 8 8 ) .  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 P 0 correspond to each other by the following relationships Po =  p  =  £Po  =  PP  zp P  r  ep  9P  *—  r VJS  gs  =  vga  where gp and gs are differential gain constants.  99  (B.l)  Appendix C  Fermi Dirac Correction for t h e B a n d g a p N a r r o w i n g  Bandgap narrowing (BGN) is also taken into account by utilizing the apparent (or effective) bandgap narrowing data from measurements on GaAs [32] (66meV for PB = 1 x 1 0 1 9 c m - 3 and 70meV for Pg = 5 x 10 1 9 cm - 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 concentration for high purity GaAs and Al x Ga!_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  <a2)  »*<*> = * s ?  for highly-doped P-type GaAs and Al x Gai_ x As 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 =  EFn E< kj ~  and T\V =  v  ~kT Fp. The  carrier concentrations are also defined as n  =  P =  NcF1/2{r)n)  ~  NvFi/2(r)p)~PB. 100  Ncexp(r]n) (C.3)  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)  =  = ",o ex P(-^-) ex P(  )  (C.4)  AEag = AEg + AEFD  (C.5)  kT  with  where EgQ is the bandgap for high purity GaAs and Al x Ga!_ x As, 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 FermiDirac correction are listed in in Table C.l.  Appendix C. Fermi Dirac Correction for the Bandgap Narrowing  uniform base PB{cm~ ) 1 x 1019 5 x 1019 m* (meV) 0.48 15.5 97.3 Ev — Efp (meV) AEFD (meV) -10.8 -50.9 3  102  graded base 1 x 1019 5 x 1019 0.511 12.1 90.6 -46.6 -9.8  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 10 19 cm -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 c u r r e n t = ' c u r ' VCE pc 0 IB 0 bp DC='current' Rb 3 2 'Rbase' Re pe 1 'Remit' Re 5 4 ' R c o l ' 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 v o l = ' l + d / v ( S n e ) ' 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 f2 F3 F4  2 2 4 1  1 4 1 4  poly(2) poly(2) poly(2) poly(2)  vZinv vI2 0 0 0 0 1 vAIl vll 0 0 0 0 1 vZinv VI3.1 0 0 0 0 1 vZinv VI3_2 0 0 0 0 1  D2 2 1 DEH Dl 2 4 DCH  103  Appendix D. SPICE Input Files  .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  104  Appendix D. SPICE Input Files  D.1.2  SHBTJib  •Library file for baseline SHBT case *  .lib shbtinput .param .param .param .param .param  c=0.389 d=0.765 ISF=1.065e-24 ISNBE=3.7lE-26 ISNBC=3.71E-26 ISE=1.91e-31 ISC=1.73e-24 remit=1.18 rbase=164.14 rcol=45.32 vt=0.0259  .endl shbtinput .lib Sne eSne Sne 0 pwl (1) 2 1 +-3.00000e+00 1.62641e+02 +-2.95000e+00 1.47348e+02 +-2.90000e+00 1.33463e+02 +-2.8'5000e+00 1.20860e+02 +-2.80000e+00 1.09421e+02 +-2.75000e+00 9. 90414e+0'i +-2.70000e+00 8.96244e+01 +-2.65000e+00 8.10824e+01 +-2.60000e+00 7.33356e+01 +-2.55000e+00 6.63113e+01 +-2.50000e+00 5.99434e+01 +-2.45000e+00 5.41718e+01 +-2.40000e+00 4. 89417e+01 +-2.35000e+00 4.42034e+01 +-2.30000e+00 3.99116e+01 +-2.25000e+00 3.60250e+01 +-2.20000e+00 3.25064e+01 +-2.15000e+00 2. 93215e+01 +-2.10000e+00 2.64396e+01 +-2.05000e+00 2.38323e+01 +-2.00000e+00 2.14743e+01 +-1.95000e+00 1.93422e+01 +-1.90000e+00 1.74149e+01 56732e+01 +-1.85000e+00 40999e+01 +-1.80000e+00 26789e+01 +-1.75000e+00 13960e+01 +-1.70000e+00 02382e+01 +-1.65000e+00 19350e+00 +-1.60000e+00 +-1.55000e+00 8.25131e+00 +-1.50000e+00 7.40184e+00 +-1.45000e+00 6.63625e+00 +-1.40000e+00 5.94653e+00  Appendix  D. SPICE Input Files  +-1.35000e+0(3, 5.32540e+00 +-1.30000e+0(3, 4.76628e+00 +-1.25000e+0<3, 4.26320e+00 +-1.20000e+0(3, 3.81073e+00 +-1.15000e+0(), 3.40398e+00 +-1.10000e+0(3, 3.03850e+00 +-1.05000e+0(3, 2.71028e+00 +-1.00000e+0(3, 2.41565e+00 +-9.50000e-0 L, 2.15134e+00 +-9.00000e-0 L, 1.91434e+00 +-8.50000e-0 L, 1.70197e+00 +-8.00000e-0 L, 1.51179e+00 +-7.50000e-0:L, 1.34157e+00 +-7.00000e-0"L, 1.18934e+00 +-6.50000e-0:L, 1.05328e+00 +-6.00000e-0:L, 9.31760e-01 +-5.50000e-0:L, 8.23316e-01 +-5.00000e-0:L, 7.26616e-01 +-4.50000e-0:L, 6.40460e-01 +-4.00000e-0:L, 5.63768e-01 +-3.50000e-0:L, 4.95563e-01 +-3.00000e-0:L, 4.34966e-01 +-2.50000e-0:L, 3.81183e-01 +-2.00000e-0:L, 3.33501e-01 +-1.50000e-0:L, 2.91276e-01 +-i.oooooe-o:L, 2.53930e-01 +-5.00000e-0:I, 2.20942e-01 +-2.28983e-lE5, 1.91843e-01 +5.00000e-02 1.66213e-01 +1.00000e-01 1.43674e-01 +1.50000e-01 1.23886e-01 +2.00000e-01 1.06544e-01 +2.50000e-01 9.13749e-02 +3.00000e-01 7.81340e-02 +3.50000e-01 6.66015e-02 +4.00000e-01 5.65810e-02 +4.50000e-01 4.78967e-02 +5.00000e-01 4.03913e-02 +5.50000e-01 3.39244e-02 +6.00000e-01 2.83704e-02 +6.50000e-01 2.36174e-02 +7.00000e-01 1.95657e-02 +7.50000e-01 1.61262G-02 +8.00000e-01 1.32198e-02 +8.50000e-01 1.07759e-02 +9.00000e-01 8.73205e-03 +9.50000e-01 7.03252e-03 +1.00000e+00 5.62797e-03 +1.05000e+00 4.47465e-03 +1.10000e+00 3.53388e-03 +1.15000e+00 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  2,. 15759e-03 1.,66588e--03 1,,27373e--03 9,,61795e--04 7.. 13799e-04 5., 15979e-04 3.,55938e--04 2.. 18039e-04  .endl Sne .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 .model .model .model .model .model  dI2h d(is«'ISNBE') dI3 2h d(is=,ISF') dI3 lh d(is='ISF') dllh d(is=' ISNBC) den d(is='ISE') dch d(is=,ISCJ)  .endl intrinsic_dc_models  Appendix D. SPICE Input Files  D.1.3  BJTJV  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.2 D.2.1  D. SPICE Input  Files  Files U s e d in C h a p t e r 4 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 1 0 9 . 1 RC 8 6 3 1 . 0 6 ql 3 2 1 bjt D3 2 1 DESCR D4 2 3 DCSCR .model b j t npn I s = 1 . 4 9 e - 2 4 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 = ' N 3 ' ) .model DCSCR D(IS='ISCRC' n = ' N 4 ' ) .DC VBE 1.0 1.8  0.026  .width out=132 .print I(VJC) I(VJB) .option dcon=l gmindc=le-14 .option post probe .option tnom=27 ingold=2 .end  gmin=le-14  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 x2 x3 x4 x5  n_Vcc n_Vcc n_Vcc n_Vcc n_Vcc  4 6 0 cell 5 4 0 cell 8 5 0 cell 10 8 0 cell 6 10 0 cell  CO cl c2 c3  6 0 6.7213e-15 n.Vcc 6 3.4831e-15 4 0 1.9974e-15 5 0 1.9974e-15  Appendix D. SPICE Input Files  c7 8 0 1.9974e-15 c9 10 0 1.9974e-15 cl3 cl4 cl5 cl6 cl7 cl8  0 0 0 0 0 0  6 7.3172e-15 n.Vcc 3.4485e-14 4 1.0269e-15 5 1.0269e-15 8 1.0269e-15 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  111  Appendix D. SPICE Input Files  D.2.3  Switch_B36  5 - s t a g e F36 2*2 r i n g o s c i l l a t o r (rl=400) using BJT model .param vcc_r=1.55 high=1.54 l o w = 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  112  Appendix D. SPICE Input Files  . p r i n t t r a n v(4) v(6) .option probe post .alter .param vcc_r=2.4 high=1.95 low=0.539 .alter .param vcc_r=3.19 high=2.25 low=0.586 .alter .param vcc_r=4.76 high=2.79 low=0.789 .end  113  Appendix  D.3  D. SPICE Input  Files  Files U s e d in C h a p t e r 5  D.3.1  SS_LD  V-I and L - I .param .param .param .param .param  characteristics  t e = 2 . 2 2 e - 0 9 Cph=6.20e-12 Rph=2.53e-01 p _ o v e r = 2 . 5 8 e - 0 8 NL_Po=2.97e+00 b e t a _ e f f = l . 2 5 e - 0 5 b=4.83e+02 I _ s p 0 = 7 . 2 2 e - 0 3 h l = 1 . 9 5 e + 1 4 h0=3.00e+12 I_SL=3e-14 NL=2 R_SL=9 Cjo_L=10pf Vbi_L=1.9  . s u b c k 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  Iin 0 1 .DC I i n 0 0.06 5 e - 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 .alter .param b e t a _ e f f = l e - 0 3 .END  Appendix  D.3.2  D. SPICE Input  Files  TranJLD  Transient analysis for laser operating at 2 gbit/s input .param .param .param .param .param .param  t e = 2 . 2 2 e - 0 9 Cph=6.20e-12 Rph=2.53e-01 NL_Po=2.97e+00 b e t a _ e f f = 1 . 2 5 e - 0 5 b=4.83e+02 I _ s p 0 = 7 . 2 2 e - 0 3 h l = 1 . 9 5 e + 1 4 h0=3.00e+12 I_SL=3e-14 NL=2 R_SL=9 Cjo_L=10pf Vbi_L«1.9 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  •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 .alter .param .alter .param .param .END  NL_Po=5e+00 NL_Po=5e+00 I_b='1.2*Ith' NL_Po=5e+00 I_b='l\2*Ith' period=1000ps rise=80ps fall='rise' dur='period/2-rise'  117  Appendix D. SPICE Input Files  D.3.3  Expe_LD  Simulated optical output waveform and injection current .param .param .param .param .param .param .param .param  te=1.65e-09 Cph=5.87e-12 Rph=2.00e-01 NL_Po=7.34e+00 beta_eff=1.0e-04 b=5.14e+02 I_sp0=9.71e-03 hl=2.19e+14 h0=4.25e+12 I_SL=3e-14 NL=2 R_SL=1 Cjo_L=10pf Vbi_L=1.9 R_smooth=2 C_smooth=10pF Ith=19.4m I_b«'Ith*l.l' I_m=0.020 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  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  119  Appendix D. SPICE Input Files  D.S.4  Mono  Monolithic integration of HBT-laser: 1.7 Gb/s .param .param .param .param .param  rf=-2.0 ee=-6 low=-2.4 high=-1.6 biasv=1.05 Ith=19.4m I_b='Ith*l.2' period=1176ps rise=150ps fall='rise' dur='period/2-rise' 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 v20 v30 vee vrf  0 20 dc 0 0 1 dc 0 18 19 dc 0 11 0 dc ' ee' 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  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  121  Appendix D. SPICE Input Files  D.3.5  Hybrid  Simulation for 1.7 Gb/s for hybrid transmitter .param .param .param .param .param  r i = - 2 . 0 ee=-6 low=-2.4 high=-1.6 biasv=1.05 Ith=19.4m I _ b = ' I t h * l . 2 ' period=1176ps rise=150ps fall='rise' dur='period/2-riseJ 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 v20 v30 vee vrf  0 20 dc 0 0 1 dc 0 18 19 dc 0 11 0 dc 'ee' 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  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  123  Appendix D. SPICE Input Files  D.8.6  LDJib  Laser parameter from Mr. Greg Burley .lib laserinput .param .param .param .param .param  te=1.65e-09 Cph=5.87e-12 Rph=2.00e-01 p_over=2.72e-08 NL_Po=7.34e+00 beta_eff=1.40e-05 b=5.14e+02 I_sp0=9.71e-03 hl=2.19e+14 h0=4.25e+12 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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