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A theoretical toolbox for the simulation and design of HBTs constructed in the Al Ga₁₋ As and Si₁₋ Ge.. Searles, Shawn 1995-12-31

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A Theoretical Toolbox for the Simulation and Design of HBTs Constructed in the AlGai..As and Sii..Ge Material Systems by  Shawn Searles, P.Eng. B.Sc.E.E., The University of Manitoba, 1987 M.Eng., Carleton University, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 July 12, 1995 © Shawn Searles, 1995  the of fulfillment partial in thesis this presenting In British of University the at degree advanced an for requirements Columbia, I agree that the Library shall make it freely available I further agree that permission for for reference and study. extensive copying of this thesis for scholarly purposes may be her his or by or department my of head the by granted of publication or copying that is It understood es. representativ my without allowed be not shall gain financial for thesis this written permission.  (Si  Department of  &/c,ot/  The University of British Columbia Vancouver, Canada  Date  / 795  Abstract A theoretical toolbox for the simulation of Heterojunction Bipolar Transistors (HBTs), in cluding the effects of tunneling, recombination, and the optimum non-linear base proffle (for the minimisation of the base transit time), is developed. The models developed are applicable to a general material system, and are analytic. Extensions specifically required by the complex Sii..Ge material system are also developed. The optimum (to minimise base transit time) base doping is found to be non-exponential, and the optimum base bandgap grading is not linear. A general transport model for HBTs, including recombination processes, is developed that accounts for the complex nature of charge transport throughout the entire device. Unique methods for opti mising HBT metrics, which cannot be employed for Bipolar Junction Transistors (BJTs), are also presented. A description of charge transport within the emitter-base Space-Charge Region (SCR), which accounts for tunneling and is not beholden to the usual drift-diffusion analysis, is devel oped. The implications of having different electron effective masses in the two sides of the hetero junction, leading to what is termed a mass boundary, is fully explored. It is found that the tunneling of electrons within the emitter-base SCR leads to a non-Maxwellian minority-particle ensemble distribution entering the neutral base. Finally, transport within SiGe HBTs is consid ered, with all of the relevant material models presented and multi-band transport models devel oped. This treatment leads to a variety of interesting conclusions regarding the operation of present-day SiGe HBTs and possible future designs.  July 12, 1995  ii  Table of Contents Abstract  .ii  Table of Contents  iii  List of Tables  v  List of Figures  vi  Acknowledgment  x  CHAPTER 1: Introduction  1  1.1  Modelling Details  4  1.2  Thesis Organisation  7  CHAPTER 2: A Multi-Regional Model for HBTs Leading to Optimisation by Current-Limited Flow  9  2.1  Bandgap Engineering  10  2.2  Regional Decoupling and Current-Limited Flow  12  2.3  Optimisation Through Current-Limited Flow  18  CHAPTER 3: Base Layer Decoupling and Optimisation  21  3.1  Independent Optimisation of  CBE, andy  23  3.2  Reducing ‘CE by Decoupling the Base from Ic  27  3.3  Optimum Base Doping Profile to Minimise ‘CB  30  3.4  The Effect of a Non-Uniform n 1 and D on the Optimum ‘CE  38  CHAPTER 4: Transport Through the EB SCR  43  4.1  Formulation of Charge Transport at the CBS  45  4.2  Incorporation of Effective Mass Changes  49  4.3  Calculation of Fr and a Unified Model for F  67  4.4.1 4.4.2 4.4.3  Analytic CBS Transport Models Analytic Model for the Standard Flux Analytic Model for the Enhancement Flux Ffe Analytic Model for the Reflection Flux Ff  70 71 78 81  4.5  The Effect of Emitter-Base SCR Control on I  84  4.6  Deviations from Maxwellian Forms and Non-Ballistic Effects  95  4.7  Conclusion  4.4  105  July 12, 1995  iii  CHAPTER 5: Recombination Currents  .107  5.1  Electron Quasi-Fermi Energy Splitting 1Ep,  109  5.2  Modelling the Recombination Processes of HBTs SRH Recombination 5.2.1 Auger Recombination 5.2.2 5.2.3 Radiative Recombination  111 112 115 116  5.3  Current Balancing with the Neutral Region Transport Currents  117  5.4  Full Model Results  121  5.5  Simple Analytic Diode Equations  128  CHAPTER 6: The Sii..GeHBT  132 135  6.2  The Effect of Strain on SiiGe Band Offsets in SiiGe  6.3  Electron Transport in Strained SiiGe  159  6.4  The Accumulation Regime Beyond the Built-In Potential  171  6.5  Conventional and Novel SiiGe HBT Structures  179  6.1  151  CHAPTER 7: Summary and Future Work  191  References  197  Appendix A: Ramped N(x) to Minimise tB  206  Appendix B: Optimum N(x) to Minimise tB  210  July 12, 1995  iv  List of Tables Table 3.1:  tB  for the four doping cases: Optimum, Ramp, Step, and Exponential  July 12, 1995  37  v  List of Figures Fig. 1.1.  Collector current for an abrupt A1GaAs HBT. 5  Fig. 2.1.  Band diagram of an HBT including a graded-base bandgap  11  Fig. 2.2.  Band diagram of the emitter-base junction within an abrupt HBT  13  Fig. 2.3.  Hypothetical HBT structure showing the physical regions that govern charge transport  14  Fig. 2.4.  The flow T that results from a series connection of six pipes  16  Fig. 2.5.  T for  Fig. 3.1.  Band diagram of both a homojunction BJT and an HBT  25  Fig. 3.2.  Emitter cap layer design to minimise RE and CBE  27  Fig. 3.3.  Optimum doping profile N(x) obtaining by numerical minimisation  33  Fig. 3.4.  The first trial function for N(x) inspired by the form suggested by Fig. 3.3  34  Fig. 3.5.  The second and third trial functions for N(x)  35  Fig. 3.6.  Step-doping proffle for N(x)  36  Fig. 3.7.  tB  Fig. 3.8.  Optimum bandgap in the base to minimise  Fig. 3.9.  The optimum stationary function y(x) which includes doping, bandgap, and bandgap reduction due to heavy doping for the minimisation of ‘CB  41  Fig. 4.1.  Abstract model of current flux within the region containing the CBS  46  Fig. 4.2.  Blow-up of theCBSfromFig. 3.1(b)  49  Fig. 4.3.  Definitions of the cylindrical momentum space coordinates for the calculation of the Jacobian Transforms from k to U-space  50  Fig. 4.4.  Domain of integration R 1 for a uniform m*  54  Fig. 4.5.  The effect that conservation of p has upon U±,i and U±, 2 when a mass boundary is placed at x =0  58  Fig. 4.6.  Domains of integration R 1 and R 2 for the enhancement case  61  Fig. 4.7.  Domains of integration R 1 and R 2 for the reflection case  62  Fig. 4.8.  Collector current for an abrupt A1GaAs HBT with 30% Al content in the emitter  71  Flux density  75  Fig. 4.9.  a three region HBT in the absence of recombination  using N(x) from Fig. 3.3, where h 1  1  —  19  2 and h h 2 is varied  37 40  tB  normalised to 4 00 As/GaAs abrupt HBT Ga 7 3 max’ for an Al July 12, 1995  vi  Fig. 4.10. Standard Flux Fc and Reflection Flux Ffr for an HBT with the parameters given near the start of this section  86  Fig. 4.11. Relative importance of Ffr to the total flux F for an HBT with the same parameters as Fig. 4.10  88  Fig. 4.12. Standard Flux Ff and Reflection Flux Ff for an HBT with the same parameters as Fig. 4.10, but with AE reduced from 0.24eV down to 0.12eV  89  Fig. 4.13. Standard Flux Fj and the Enhancement Flux Ffe for an HBT with the parameters given near the start of this section  91  Fig. 4.14. Relative importance of Fje to the total flux F for an HBT with the same parameters as Fig. 4.13  94  Fig. 4.15. Ensemble particle distributions assuming a purely thermalised thermionic injection from the peak of the CBS in Fig. 4.2  96  Fig. 4.16. Integrated ensemble distribution versus wave vector 2 k, entering the neutral base  98  Fig. 4.17. Ensemble electron distribution entering the neutral base versus k  99  .  Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier  101  Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier  102  Fig. 4.20. Relative difference between the results obtained from the methods proposed in [511 to the model for F from this chapter  104  Fig. 5.1. Fig. 5.2.  Band diagram of the EB SCR showing the effect of the abrupt heterojunction on under an applied forward bias (reprint of Fig. 2.2) Components of the collector and the base currents emphasising that ThT must equal the total of,  Fig. 5.3.  109 111  + NB + SRN.B + Aug,B + Rad,B  Energy Band diagram for the EB SCR of an HBT under equilibrium conditions  113  Fig. 5.4.  Relative error between the approximate and exact fonns given in eqn (5.27)  120  Fig. 5.5.  Bias dependence of the SCR current from the emitter side, and the three components of the SCR current from the base side  122  Gummel plot showing the importance of including the emitter- and base-SCR current components in the computation of the total base recombination current  123  Bias dependence of the current gain f3, showing the relative importance of including SCRB in the calculation of AE  125  Bias dependence of the current gain (3 for the case of Wflb increased to 5000A and t, in the SCR reduced to 5ps  125  Fig. 5.6. Fig. 5.7. Fig. 5.8.  July 12, 1995  vii  Fig. 5.9.  Effect of changing the neutral base thickness W,,, when the CBS is responsible for current-limited-flow  126  Fig. 5.10. Comparison of the recombination currents when qi is given by the depletion approximation and when it is given by the linearisation of eqn (5.11)  127  Fig. 5.11. Z-functions as computed from eqn (5.13) when using the material parameters from Section 5.4  130  Fig. 5.12. Comparison of the full model and “diode-like” expressions for the SCR currents  130  First Briulouin zone showing (in k-space) the constant energy surfaces near the bottom of the conduction band for Si and Ge  137  Fig. 6.2.  Valence bands in unstrained Sii..Ge  138  Fig. 6.3.  Commensurate growth of the Sii.Ge alloy layer to the Sii.Ge substrate, leading to a pseudomorphic alloy film  142  SiiGe bandgap when grown commensurately to a variety of substrates oriented along (100)  147  E and E conduction band energies relative to the unstrained conduction band edge for Sii..Ge commensurately grown to a variety of substrates oriented along (100)  148  E and E’ valence band energies relative to the unstrained valence band edge for SiiGe commensurately grown to a variety of substrates oriented along (100)  149  Constant energy surface plot depicting the E and E bands in Si 017 Ge 083 commensurately strained to (001) Si  150  Conduction and valence band energies including all of the band offsets for a Si 1xaPexai to a Sii.Ge heterojunction commensurately strained to a { 100) SiiGe substrate  153  E and E conduction band minima to the left and right of an abrupt heterojunction when commensurately grown atop a { 100} Sii.Ge substrate  156  Fig. 6.1.  Fig. 6.4. Fig. 6.5.  Fig. 6.6.  Fig. 6.7. Fig. 6.8.  Fig. 6.9.  Fig. 6.10. AE when Xar = Xal + 0.20, and Xal and x are varied  158  Fig. 6.11. AE when Xar = Xaj  159  +  0.20, and Xal and x are varied  Fig. 6.12. Diagram of the A conduction band minima involved in f and g intervalley scattering  162  Fig. 6.13. Equilibrium band diagram of apn-junction, showing the relevant energies and potentials  165  Fig. 6.14. Band diagram for a np-junction with a positive step potential  172  (i.e.,  AE <0)  July 12, 1995  viii  Fig. 6.15. The exact and approximate forms for Nrat and Yat from eqns (6.46)-(6.47)  177  Fig. 6.16. Diagram of the CBS that forms under the accumulation regime  178  Fig. 6.17. Critical layer thickness for a Sii..Ge layer on a { 100) Si substrate  180  Fig. 6.18. Band diagram for an HBT with 20% Ge in the base, lattice matched to Si  181  Fig. 6.19. Band diagram and Transport currents for an HBT with 25% linear grading of Ge in the base, lattice matched to Si  183  Fig. 6.20. Novel SiGe HBT based on a 20% Ge substrate  185  Fig. 6.21. Band diagram showing the conduction and valence sub-bands for an HBT where Xal = 0, Xar = 0.45, x = 0.35, NA=1x10 , and 3 cm 17 , ND=5x10 3 cm 19 Wb700A  187  Fig. 6.22. Transport currents within the various regions of the HBT given in Fig. 6.21  188  July 12, 1995  ix  Acknowledgment I would like to thank first of all, Professor Dave L. Pulfrey, my Ph.D. supervisor. I returned from industry to obtain my Ph.D. because I was interested in performing research that probed into the complex theories of solid-state device operation. Thanks to Dr. Pulfrey and his forthcoming guidance, I was able to navigate a steady course through the often turbulent waters of academic research, and attain the research goals I had planned to explore. Dr. Pulfrey provided constant en couragement to my work, offered valuable assistance, and provided me a learning experience that I know will serve me for the rest of my life. Dr. Pulfrey, however, went even further in his contri butions during the time I worked on my Ph.D. He allowed and encouraged me to pursue other life interests, so that I can proudly say that my Ph.D. research was indeed a time that touched and en riched all aspects of my life. So to you Dr. Puifrey I can only offer in return my simple but sincer est thanks.  I would also like to thank Professor Mike Jackson who provided me with many ideas throughout my Ph.D. research. Without the presence of Dr. Jackson, my Ph.D. research would not have been as interesting nor as fulfilling as is has been. I would also like to thank Professor Tom Tiedje, who offered me an excellent course in solid-state quantum mechanics; without which I could not have performed the Ph.D. research that I have done. Dr. Tiedje, you have challenged me and as a result, provided me a fundamental base from which I will solve many questions yet to come. I would also like to thank Dr. Jackson, Dr. Tiedje, Professor Nick Jaeger, Professor Matt Yedlin, Professor Jeff Young, and Professor Fred Lindholm, whose presence on my examining committees helped to ensure that my final thesis was the best it could possibly be.  Finally, I would like to thank Barbara Ippen, my wife to be on July 29th, for being a willing partner in my Ph.D. research efforts. Your caring presence has provided me a reference point that I could always count upon, no matter how hard things became during the course of my research goals.  July 12, 1995  x  CHAPTER 1 Introduction  July 12, 1995  1  The main objective of the Ph.D. research being presented in this thesis is the creation of models that will foster a deeper understanding regarding the physics surrounding a Heterojunc tion Bipolar Transistor (HBT). To this end, physically based models for the transport of charge within an HBT will be developed. These physics-based models will allow for the simulation of present-day HBT structures and novel structures for the future. By clearly identifying the relevant mechanisms by which charge transport takes place within the HBT, an optimum design for the de vice that incorporates the various compromises between competing device metrics (such as  f3, f  and RB) can be obtained. A further goal is to reduce all of the models developed within this thesis to tractable, analytic forms. By obtaining analytic models for charge transport within the HBT, circuit level models that predict device performance can be developed in step with the emergence of HBT-based Integrated Circuit (IC) processes. Finally, the models that are developed within this thesis are in general free of any details specific to a single material system. However, given the importance of the AlGai..As and Sii..Ge material systems, these two systems will be exten sively studied and will serve as the chosen material systems for all examples presented. The concept behind the HBT has been around since the time of Shockley [1]. Further, over 30 years ago, Kroemer developed much of the fundamental physics regarding the operation of the HBT [2]. However, it has not been until the last five years that industry has had the capability to manufacture HBTs with suitable yields to be commercially viable [3-5]. Also, the material re search is still continuing and has a long way to go before HBT processes achieve the maturity of technologies such as CMOS. Furthermore, with experimental results becoming more prolific, and with rapidly diminishing device dimensions, we are finding that much of the physics laid down for modelling the HBT is inadequate for describing present-day devices [6-9]. With the increasing maturity of processes for the production of HBTs, comes an increase in the need for models that predict device operation. It is now possible to manufacture HBTs with active basewidths approaching 100 A [10-121 and with features that change over distances of less than 10  A [13-14].  As device dimensions approach the atomic lattice spacing of the crystal, the  applicability of models based upon classical continuous fields becomes questionable [15]. There is already general agreement that one must consider higher order moments beyond the drift and diffusion terms in the Boltzmann Transport Equation (BTE) in order to model deep submicron de vices [16-17]. The BTE is based upon classical physical models that in general do not incorporate  July 12, 1995  2  quantum mechanical (QM) phenomena. It has been recognised that the correct modelling of tun neling, a QM effect, is of paramount importance to the correct prediction of HBT operation [1821]. Thus, models of HBTs that incorporate QM phenomena are becoming increasingly important in order to maintain accurate simulation of the HBT. The general relationship between the terminal currents and voltages of an HBT can still be predicted today by models designed for Bipolar Junction Transistors (BJTs) [22]. However, it is not always clear why we can continue to apply BJT models to HBT operation when these BJT models were developed without consideration of the physical processes that govern transport within an HBT. Presumably, the BJT model has enough degrees of freedom so that it can be ma nipulated to cover HBT operation. For example, one of the most common discrepancies found when using BiT models for HBT simulation is that the injection indices (ideality factors) for the collector and base terminal currents do not correspond to what is theoretically predicted for BJT operation [23]. Thus, in order to accurately predict HBT operation, and to further develop HBT processes so as to advance device operation, one needs to understand such things as why the col lector and base injection indices differ between an HBT and a BJT [24,25]. The Sii..Ge material system has many unique physical considerations that other systems, such as the AlGai..As material system, do not have to contend with. The unique attributes of the Sii..Ge material system are mostly due the effects of strain. Due to the large lattice mismatch be tween Si and Ge, Sii..Ge films grown on top of Sii..Ge substrates (where x y) have a large de gree of strain present within them if non-relaxed crystals with low defect density are to be manufactured. The presence of strain breaks the cubic symmetry of the crystal and changes the bulk electrical properties [26-28] of the film. By varying the Ge alloy content and the strain im parted to the SiGe film, it is possible to tailor both the bandgap and the offsets in the conduction and valence bands. Therefore, models specific to the Sii..Ge material system must be developed in order to understand charge transport within the complex band structure that develops. Finally, the reason for focussing on the Sii..Ge and A1Gai.As material systems stems from the maturity of AlGaAs devices, and the massive installed base of Si-based IC technologies that would easily admit SiGe devices. From a manufacturing standpoint the AlGai..As material system offers no redeeming features when compared to Si, save one the lack of strain. Obvious -  ly, the key to the operation of an HBT is the formation of heterojunctions between two materials  July 12, 1995  3  characterised by different bandgaps. The AlGai.As material system has essentially a fixed lat flee constant over the entire range of Al mole fraction x. For this reason, the MGai.As material system is lattice matched and will admit an arbitrary heterojunction between AlGai..As and AlGaiAs without developing a strain within one of the films. This lack of strain within the MGai.As material system helps to ensure a defect-free heterointerface that greatly facilitates the manufacture of HBTs. For this reason, most commercially available HBTs are based in the AlGai ..As material system [29]. However, most solid-state devices are Si based [30]. With the advancement of low-temperature Chemical Vapour Deposition (CVD) processing [31], the forma tion of high-quality commensurately strained Sii..Ge films is becoming commercially viable. Therefore, given the manufacturing advantages of Si, it is expected that SiGe HBTs will shortly surpass A1GaAs HBTs as the most prolific commercially available HBT [32-37].  1.1 Modelling Details Research has been conducted into the injection of electrons from the emitter into the base of AlGai..As npn HBTs [18,24,25]. The research has centred around abrupt HBTs where the het erojunction between the wide-energy-gap emitter and the narrow-energy-gap base is abrupt. In an abrupt AlGai.As HBT one finds the formation of a Conduction-Band Spike (CBS) between the emitter and the base (see Fig. 3.1). This spike, due ostensibly to differences in the electron affinity of the materials used for the formation of the emitter and the base, results in a large impediment to the flow of electrons from the emitter into the base. In fact, if the CBS were not taken into account when modelling the HBT, the collector current would be overestimated by over three orders of magnitude at room temperature (see Fig. 1.1). However, the modelling of charge transport through the CBS cannot be based upon simple thermionic injection alone. Since the width of the CBS is typically less than  boA near the top  of the spike, the occurrence of a tunneling current  cannot be neglected. Finally, it will be shown that transport through the CBS can often be the lim iting factor for the overall transport of charge within the HBT (i.e., the determination of the col lector current Ic). This occurrence of current-limited flow outside of the neutral base region will be studied and exploited for device optimisation. Therefore, the modelling of the relevant physical phenomena surrounding charge transport through the CBS, including tunneling and conservation of transverse momentum across the heterojunction in a diagonal mass tensor, will be investigated.  July 12, 1995  4  ,—  1  W3  LU  I  ;, 10  C)  9 io-  0.8  1.0  1.2  1.4  1.6  Base-Emitter Voltage VBE (V) Fig. 1.1. Collector current for an abrupt AIGaAs HBT with 30% Al content in the emitter. The emitter doping is 5x10 , and the base doping is 1x10 3 cm 17 3 (see Section 4.5 for the com cm 19 plete device details). The top curve, where CBS limitations have been neglected, is arrived at by assuming Shockley boundary conditions and considering only neutral base transport. The possibility of regions other than the neutral base controlling I is intriguing. However, from a modelling perspective, the immediate consequence of a multi-regional system controlling Ic is the question of how to join these various regions together to form one cohesive transport mod el. Furthermore, the possibility exists that under multi-regional control of I, older models, such as those for the neutral base [38], which assume that only the specific region being studied controls Ic may not longer be valid. It will be shown in Chapter 2 that there is a very simple prescription for joining up all of the multi-regional transport models into a complete transport model for the de termination of I. It will be further demonstrated in Chapter 6 that it is possible for two spatially separate regions to control I simultaneously by having essentially identical net-charge-transport capacity through both regions; the ramification of this is the inseparability of the two regions. With the general model of Chapter 2 providing the overall method to link the various physi cal regions of the HBT together, then the problem of modelling charge transport within the entire HBT is effectively decoupled into a set of models; one model for each relevant region. To this end, Chapter 3 investigates and develops models for the various regions of the HBT, including the si  July 12, 1995  5  multaneous optimisation of the base bandgap and doping profile (provisions are also made for the inclusion of bandgap narrowing due to heavy doping effects) for the minimisation of the basetransit time ‘CD. Finally, the modelling of recombination events, which lead to the formation of the base current ‘B’ is developed in Chapter 5 with the specific attributes of a heterojunction included. These various regional models essentially form a toolbox for the study of charge transport within the HBT, with the general transport model of Chapter 2 forming the blueprint for the ultimate op eration of the device. The modelling efforts presented in this thesis regarding charge transport through the EB SCR are rigorous in that no appeal has been made to drift-diffusion analysis based upon phenom enological mobility models (i.e., mobility models with an electric field dependency). Instead, models that include the quantum mechanics of charge transport, which have no appeal to said phenomenological mobility models, are analytically solved for. However, the neutral base charge transport models are based upon drift-diffusion analysis. The reason for resorting to simpler driftdiffusion analysis for the neutral base is its been found that the neutral base often does not repre sent the bottleneck to charge transport and thus does not dictate control over I ([25] and Fig. 1.1). Nevertheless, as the neutral base thickness approaches and becomes smaller than the mean free path, then a majority of the electrons will traverse the base without thermalising [39,40]. These un-thermalised, or hot, or ballistic electrons do not follow exactly the simple models of drift-diffusion contained within the BTE [16,41]. Instead, a general solution to the BTE is neces sitated. In present-day HBTs, and even in some of the emerging high performance BJTs, the under standing of hot electrons can be essential to the accurate modelling of the device’s terminal char acteristics [9,14]. The problem with general BTE solvers, such as Monte Carlo simulation, is that some important QM effects cannot be modelled. The BTE is based upon local potentials and therefore cannot include some QM effects, such as tunneling, which are inherently non-local. As was discussed and shown in Fig. 1.1, the failure to include tunneling results in a gross error re garding the transport of charge through the HBT. Section 4.6 will address the issue of merging classical BTE solvers with the models developed in Chapter 4 for charge transport through the CBS. Specifically, Section 4.6 will show that tunneling produces a considerable distortion to the minority-particle ensemble distribution entering the neutral base (deviations that are far from  July 12, 1995  6  Maxwellian or even hemi-Maxwellian). Finally, it should be noted that the use of drift-diffusion models in the neutral base will not produce gross errors like the failure to include tunneling through the CBS. Instead, drift-diffusion models can be employed in the neutral base, but with corrections that essentially amount to a 20 to 40% change to the diffusion coefficient D [42,431. Even more importantly, if the neutral base does not control I, then in terms of D.C. calculations, no error will occur if these ballistic corrections to D are neglected; however, it terms of A.C. cal culations, such as for tB, there would be an error. The final modelling effort of this thesis pertains directly to the design and simulation of SiGe HBTs. As has been alluded to, the effect of strain on the electrical characteristics of Sii..Ge films is dramatic. Chapter 6 reviews the various material models necessary for the description and study of the electrical characteristics of strained Sii.Ge. Specifically, once a review of the litera ture regarding the Sii..Ge material models is presented, a comparison to experimental results is performed, and the most consistent set of material constants selected. The final result is a com plete set of models for the calculation of the bandgap including conduction and valence band off sets. Furthermore, strained Sii.Ge results in a two-band system both for the conduction and the valence band. Chapter 6 uses the Sii..Ge material models and derives the necessary multi-band charge-transport models that are required to simulate SiGe HBTs. In fact, it is found that there is a substantial error incurred by replacing the two-band system with a single effective band. Finally, the charge-transport models are applied to the study of present-day as well as future SiGe HBT designs with some surprising results regarding operating voltages and critical layer thicknesses.  1.2 Thesis Organisation This thesis is organised into five main chapters. Chapter 2 presents a general model for the HBT that is highly abstract in nature. The main tenet of the general model in Chapter 2 is that it can contain any number of physical regions to model the HBT, including sources and sinks within each region. Chapter 2 also introduces a method of optimisation through what is termed currentlimited flow. Chapter 3 builds upon the ideas of Chapter 2 by considering specific examples of de vice optimisation that can be performed within an HBT but not a BiT. The main development in Chapter 3 is the solution for the optimum base bandgap and doping profile. Surprisingly, the opti mum doping profile is not exponential, and the optimum base bandgap is not linear. Chapter 4  July 12, 1995  7  moves on to develop the necessary models for charge transport within the emitter-base SCR. Spe cifically, models for the tunneling of electrons through the CBS, including the effect of a spatially non-uniform effective mass, are developed. Finally, Chapter 4 goes on to show the effect of tun neling on the emerging minority-carrier ensemble distribution entering the neutral base. Chapter 5 rounds out the ideas presented in Chapter 2 by developing the necessary models for the recombi nation of minority carriers within the emitter-base SCR and the neutral base. Chapter 5 concludes by using the model of Chapter 2 to bring together the various regional models of Chapters 3 through 5 for the simulation of an A1GaAs HBT. Chapter 6 builds upon the models of Chapters 4 and 5 for the simulation of SiGe HBTs. Models that include the effects of strain on the conduction and valence bands in the Sii..Ge material system are presented. Multi-band charge transport models, which include the material models of the Sii..Ge material system, are then developed. Finally, Chapter 6 brings all of the models developed within the chapter together for the study of numerous present-day and future SiGe HBT designs.  July 12, 1995  8  CHAPTER 2 A Multi-Regional Model for HBTs Leading to Optimisation by Current-Limited Flow  July 12, 1995  9  Since the invention of the Bipolar Junction Transistor (BiT) in 1948 by Brattain, Bardeen and Shockley [44], continuous improvements have been made to its operation and reliability. Nowadays, BJTs are part of nearly every manufactured product sold within the world. This con tinuous development of the design and manufacture of the BJT shows no sign of ending nor any abating in the pace at which improvements are made. The question then, is what direction or di rections will the course of BiT development take in the future? The latest innovation in the evolution of the BIT has been termed Bandgap Engineering by Capasso [45]. By altering the actual semiconductor within the active portion of the BIT, generally by forming some sort of alloy, the shape of the bandgap can be altered to provide another force to govern the motion of electrons with the device. This idea, however, is not a new one. Shockley al luded to the use of Bandgap Engineering in his BJT patent of 1948 [1], and Kroemer first pro posed the idea of using a wide-bandgap semiconductor for the emitter and a narrow-bandgap semiconductor for the base in 1957 [2]. This junction between two semiconductors with dissimi lar bandgaps is a heterojunction, and leads to the creation of a Hetero-junction Bipolar Transistor (HBT). What makes the HBT of specific interest today, is that in 1957 it was not possible to man ufacture HBTs due to the infancy of the art of semiconductor manufacture. It has only been in the late 1980’s and the 1990’s that commercially available HBTs have become feasible. Therefore, now is the time to fully explore the possibilities afforded by Bandgap Engineering to the contin ued development of the BIT.  2.1 Bandgap Engineering The force acting upon an electron/hole within a semiconductor is the sum of the electric field due to any spatially varying charge, and the field of a spatially varying conduction/valence band (EJE)  [71. The electric field due to the spatially non-uniform charge is the standard force  responsible for drift and it changes with applied bias. However, the effect of the field due to the variation of EJE is present from the construction of the device and is therefore ostensibly inde pendent of the bias conditions (much the same as the electric field that is generated in the neutral base due to a spatially varying doping is independent of bias). It is this manufactured driving force, due to the spatial change in the bandgap and the band alignments, that gives rise to Bandgap Engineering. It is possible to effect such a rapid change in EdE that the affects of the standard  July 12, 1995  10  electric field are negligible and unimportant. One can therefore expect to create HBTs with mark edly different tenninal characteristics than those possible with BITs. Finally, and most important ly, the terminal characteristics of HBTs can have a completely different dependence upon the physical construction of the device when compared to BITs. The final objective of Bandgap Engineering can be broken down into two distinct groups: techniques that provide for a slow change in E/E such that the overall electric field is modified (such as adding a gradient to EdE that aids in the transport of charge through the base) but is not overwhelmed by the engineered field; or techniques that afford extremely rapid or abrupt changes in EIE, so much so that electron/hole transport no longer depends upon the electric field due to the space-charge but is governed completely by the engineered bandgap. The first group of Bandgap Engineering techniques was applied to the newly emerging HBT in the form of an additional adding field in the base and the collector, in order to afford a more rapid transit of the electron/hole through the device [2,46,47]. Shortly thereafter, the second group of Bandgap Engineering techniques resulted in the idea of placing an abrupt downwards change in E to provide a sudden increase in the kinetic energy to the electron as it entered the base (ballistic injection; see Fig. 2.1) [12,14,48]. The aiding field in the base produced results that were expect ed; the ballistic launcher however, did not. In the end, it was the abrupt Bandgap Engineering technique that provided the most unique results in HBTs when compared to BiTs. Thus, abrupt Bandgap Engineering may be the more promising road to follow in seeking to continue the evolu tion of BITs.  Ballistic “launcher” diding Field (N) Emitter E  (P) Base  (N) Collector  Hole blocker  Fig. 2.1. The abrupt change of E in the emitter-base junction “launches” electrons into the base with a large kinetic energy. The gradual negative slope of E in the base and the collector helps to speed the electron through these regions. Finally, the abrupt change in E at the emitter-base junc tion suppresses hole back-injection into the emitter. July 12, 1995  11  2.2 Regional Decoupling and Current-Limited Flow Within any region of a solid-state device, charge flow or transport results in a spatial variation to the quasi-Fermi energy Ef When the variation in the conduction and valence band is small (small, as defined by Berz [49], is a change of less than kT over one mean-free path ?), then one can speak of a continuous spatial change in Efi and arrive at the standard drift-diffusion transport equations. How ever, when the change in the conduction or valence bands is not small, as occurs in abrupt Bandgap Engineering, then Ef does not vary in a continuous fashion but instead changes abruptly as well [50,7]. This abrupt change in Ef is due to a departure from conditions of quasi-equilibrium, where the transported current through the region is large in comparison to the equilibrium charge flows that re sult from the drift and diffusion of carriers [50,18]. To see the effect of a departure from quasi-equilibrium upon Efi examine the effects due to an abrupt change in E, as shown in Fig. 2.1. Fig. 2.2 shows what the abrupt emitter-base hetero junction would look like, including the effect of the potential energy variation due to the SpaceCharge-Region (SCR). The transport flux F is then given by the forward directed flux Ff minus the backward directed flux Fr The forward and reverse directed fluxes are [18,20,211:  =  qvn°  and  Fr  =  qvn°  =  (2.1)  qnOekT  which produces AEf  F=FfF=qn°l_e kT)=Fj(1_e kT),  (2.2)  where n 0 is the electron concentration immediately to the left of the heterojunction,  O  is the  electron concentration immediately to the right of the heterojunction that is capable of surmount ing the barrier  ii  is the ensemble average velocity of the flux (which can include tunneling),  and AE is the abrupt change in the electron quasi-Fermi-energy E. The reason for the appear ance of the term /XEp  in  eqns (2.1) and (2.2) is due to the need for n° to surmount the barrier  AE. Therefore, the abrupt change in E generates the abrupt change in  Eqn (2.2) clearly shows that as F goes towards zero, then so does tions F << Fjand F << Fr are satisfied, then AEp  In fact, if the condi  0. This is exactly what is meant by quasi-equilib  rium; as long as the total transport current merely perturbs the equilibrium fluxes, the result will be a vanishingly small  Conversely, if the transport current is not small compared to Ff and  July 12, 1995  12  F,, then iXE will become substantial. Finally, in the limit of a large AE (more than a few kT), F,. becomes very small compared to Fj, and F  Ff. Thus, it is not possible for the demanded trans  port current to exceed the available forward directed flux.  Fig. 2.2. Band diagram of the emitter-base junction showing the effect of the abrupt heterojunc c,, under an applied forward bias. ji is the solution to the Poisson equation and is there 7 tion on E fore continuous; however, the midgap energy E 1 need not be. The condition of F  Ff is termed current-limited flow, and is a manifestation whereby  quasi-equilibrium is grossly violated. The region in which current limiting has occurred responds by generating as large a  as necessary such as to reduce the demanded F to be no more than  Fp Obviously, the transport current through the entire device will be governed by the region in which current limiting has occurred. Furthermore, the physical construction of the region limiting the transport current will dictate the dependence of F, and thus F, on the applied bias. Therefore, abrupt Bandgap Engineering techniques can in principle generate regions which will govern the total transport current irrespective of any other physical portion of the device. To examine the effects of current limiting by a region, consider the hypothetical structure shown in Fig. 2.3. Fig. 2.3 shows three different but adjoining regions with a total applied bias of V across them all. Charge is transported from Region 1 to Region 2 and finally through Region 3. Let the transport current be composed of electrons, although the same argument and solution re suits if holes are considered instead. To further generalise this picture consider a sink, or recom bination process, existing in both Regions 2 and 3. Then, by the need to conserve particle flow, the electron flow must be continuous across the two boundaries separating the three regions. This July 12, 1995  13  Electron Flow  p  Hole Flow Fig. 2.3. Hypothetical HBT structure showing three physical regions that govern current transport. The applied bias is with a drop of AE and at the region boundaries. There are recombi nation processes in Regions 2 and 3 that generate currents J, 2 and J, 3 respectively. Conservation of current forces J,, , and J,, 2 1 = J,, 2 + J, 2 = J,, 3 + J,,, . Note: Ep, is assumed to be a constant. 3 procedure has been referred to as current balancing [51,52], but is generalised here to also allow for sinks (and with a simple extension, sources as well). Thus, the sink causes the electron and hole currents emanating from the region to couple together as the total electron flux entering the region must be conserved [24]. Now, the driving force in Region 1 is the full applied bias of V However, at the boundaries, one needs to consider a drop of AEp,x (where x = 1 or 2) through the region. Thus, the driving force in Region 2 is not Vbut V— AEfi. Likewise, at the second bound ary, another drop in the electron quasi-Fermi energy of AEfr 2 occurs, resulting in a driving force of V—  —  in Region 3. Using the form given in eqn (2.2) for the transport current:  n,l  2 J  1—e f 11  O —  =  3 = Jfl,  kT  (V—E (1_e 2 ) 1  3  (V—AEffl  1  —iXEf  kT  ), (2.3)  2)’  fn, 2  2 = J  (V—AE (1_e 2 ) 1  3 J,  (V—AEffll_AEffl 3 4 . 2  =  July 12, 1995  14  It is important to realise that the hole currents  represent electrons that have recombined; hence  their direction of flow as presented in Fig. 2.3 and their connection with If the J (V— AEp) functions can be expressed as i°(V)exp(—AEp 0 2 /k7), then, equating J,, 1 with J,, 3  + J,,, 3  gives: 2 LEf  kT  2 AEf,,  )=  [3(+43(]e,  which produces, after dropping the explicit dependence upon V,  =  f2  e  jO n,2 2+  J  (2.4)  3+4 3  Then, equating J,, 1 with J,, 2+2 ,, gives: 1 J 1 LEffl  1 IEf  fn,2  Ji(V)(1-e)= ( ( 2 [J, V)1e(1_e v)+4,  kT).  Using eqn (2.4) in the above, and once again dropping the explicit dependence upon V, produces: 1 iiE  e  —  kT  ,o  n,lkn,2+  —  +4 2 (J )  —  +Jp 3 n  (3+43) +l(2+3÷43)  25  The final transport current Jexiting the device is simply equal to J, . Substituting eqn (2.4) and 3 (2.5) into J,, 3 given in eqn (2.3) produces: JT(V)  =  Jfl 3  (V)  =  1 ,  +  J ( 1 V)  +  (2.6)  1 J ( 3 V)  J2 ( 2 V)  where ‘2,2+4,2 —  —  -‘2,2  and  3+43 —  —  Eqn (2.6) provides a very simple form for the ultimate transport current T emanating from the device, and extends eqn (34) in [52]. It includes all of the recombination effects of Regions 2 and 3, while allowing for a completely general relationship between the applied bias and the for ward directed flux Ff (where the fi(V) functions are F. The only stipulation placed upon the use of eqn (2.6) is that fi(V— AEp)  =  (V)exp(—AEp/k7) (as will be seen in later chapters, where eqn 0 J  (2.6) is applied, this is exactly the functional fonn that results). Therefore, to determine the trans July 12, 1995  15  port current that results from coupling three regions together, it is sufficient to calculate the forward directed fluxes through each region in isolation, and then use these results directly in eqn (2.6). It is a very simple mathematical problem to generalise eqn (2.6) to a system of N regions. To do this, simply treat Regions 1, 2 and 3 as a single super-region, with the transport current giv en by eqn (2.6) used to define J ; Regions 4 and 5 then become Regions 2 and 3 in the analysis 1 leading up to eqn (2.6). Finally, a recursive application of the above procedure gives: N+i  = i=i  where y 4  1, and  =  JJ ‘v.,(V)J  (2.7)  j=i+i  Jj+Jj ‘  J  n,  j  Eqn (2.7) is the general formula for the calculation of transport current through any multiregional HBT (of which a BJT is a subset). The ramifications of eqn (2.7) are striking and gener ally lead to current-limited flow within a single region. An examination of eqn (2.7) begins with 4 functions, which are termed the recombination loss; y the y , therefore, represents the additional 4 current that must exist in order to satisfy the recombination events within Region j. Then, the transported current through each successive region is not J but  Now, the form of eqn  (2.7) is exactly the same as that used for the calculation of a connected series of conductors. This immediately leads to the picture of a series of pipes through which a current JTmust pass (see Fig. 2.4).  Region 1  Region 2  Region 3  Region 4  Region 5  Region 6  IT  Fig. 2.4. The flow T that results from a series connection of six pipes (the flow entering only equals the flow leaving (= J) when there is no recombination in any of the regions). Obviously, the pipe in Region 4 is the most restrictive and T will accordingly be governed mostly by this region alone.  July 12, 1995  16  Looking at eqn (2.7) and letting J  <<j k  k and k can range over all N, then Re  wherej  gionj will be responsible for the current-limited flow of T and produce: JT(V)Ji(V)  (=  where  Ii  (2.8)  k(’ k=j+1  l/) is the transport efficiency of Regionj and expresses the fraction of the transport  current that is lost to recombination within the region. Eqn (2.8) is exactly the form expected from the arguments presented in Fig. 2.4. For if Region j is responsible for the current-limited flow,  T  would equal J 3 in the absence of recombination. However, each subsequent region downstream will lose  k 0  electrons to recombination. Therefore, the current  will be diminished by c’k in  each region encountered, leaving a final current of T exiting the device. This immediately leads to eqn (2.8). Thus, in a device with say six regions, if Region 3 produces the limiting flow, then T  = Finally, looking once again at eqn (2.8), recombination events upstream of Regionj play no part in the ultimate current T This is no surprise since all of the regions upstream of Regionj can supply the demanded current within Region j. However, every region from 1 to N contributes to the recombination current, and must be included in the calculation of the total hole current  4  Adding all of the recombination events together gives: N  J(V)  N  =  Then, after bringing  r J(V)  =  fl Y =  = i=1  i=1  yj  into the multiplication and letting i  N N+1  N+1N+1  1  N+1  N  JT[Yi  j=i+1  i=1  =  i’  IT  j=i+1 —  N  N+1  [I  i=lj=i+1  1 in the second term:  1N+1  N N+1  j=1  i=2j=i  HYij = JTLH,+H-  JTLHi=lj=i i’=2j=i’  Finally, since YN÷1  F  N+1  N N+1 flN+1 i’=2j=i  1 from eqn (2.7), and i’ is a dummy variable, the above reduces to: J(  (2.9)  =JT(V)[fl(V)_11.  Eqn (2.9) provides for the total hole current generated within the device. Combining both eqns (2.7) and (2.9), the total electron and hole current entering and leaving the device is known. As will almost always be the case, one region alone will dictate the transport current and lead to current-limited flow. Then, eqn (2.7) can be replaced by its approximate form, eqn (2.8), to yield after substitution into eqn (2.9): July 12, 1995  17  J(V) =JJ(V)[fl(k(v)k=1  fl  x(V)].  (2.10)  k=j+1  The results of this section are models for the total electron and hole currents entering and leaving an HBT. These models are free of essentially any restrictions upon their functional form, and can therefore be applied to a wide variety of physical processes. Furthermore, the form of the models presented is based upon a simple, modular approach, that is easy to apply to any device. The important ramification is that one region alone will tend to determine the overall transport through the entire device; creating a situation of current-limited flow. The key to achieving a situ ation of current-limited flow is the existence of a substantial AEp in one region. Finally, abrupt Bandgap Engineering techniques provide the capacity to create a situation of current-limited flow in any region of the device. In the next section and chapters to come, the concept of current-limit ed flow will be exploited in the optimisation and modelling of HBTs. In the end, eqns (2.7) and (2.9) (or their approximate forms, eqns (2.8) and (2.10) respectively), will be used to bring togeth er all of the models for each of the relevant regions of an HBT.  2.3 Optimisation Through Current-Limited Flow The main conceptual result of the last section was that one region, or physical process, will tend to dictate the transport current through the entire device. This section examines how to inten tionally design a specific region, through Bandgap Engineering techniques, to result in currentlimited flow; thereby allowing for a decoupling of T from the physical transport processes in all other regions of the device. Finally, once J’ is decoupled from a specific region, by ensuring that transport through the region is much larger than the demanded jr then one is free to optimise that specific region without affecting J Fig. 2.5 shows the transport current that would result from a hypothetical three region de vice. For case (a), Region 3 controls J under low bias and Region 2 controls under high bias; while Region 1 plays no part at all. In case (b), the transport current in Region 1 has been lowered so that Region 1, and neither Regions 2 or 3, controls  T  under all bias conditions. This demon  strates, in principle, the feasibility of engineering a specific region to be the source of current-lim ited flow, and thereby link JTto the physical process in that region alone.  Julyl2,1995  18  In order to see how optimisation can occur by engineering a specific region to be the source of current-limited flow, one begins by identifying the need for decoupling. Imagine there are two specific metrics, say Early voltage (VA) and collector resistance (Rc), that are to be optimised. If these two metrics are connected to one parameter, in this case collector doping, and the two met rics do not both move towards their optimum value with either an increase or decrease in the one parameter, then only a compromise and not a true optimum can be reached. In the example given, VA is to be maximised and Rc minimised. However, increased collector doping decreases both VA  and Rc, forcing a compromise between the two metrics to be made. If it were possible to decouple either of these metrics from the one parameter, then it would be possible (in terms of this one pa rameter only) to optimise both metrics. Therefore, decoupling the metrics from their common competing parameter is the key to removing the compromise and achieving a true optimum.  1 3 i0  1Ocase(b) 1 0.8  1.0  i2 1 1.2  1.4  1.6  Base-Emitter Voltage VBE (V) Fig. 2.5. T for a three-region HBT in the absence of recombination. The solid lines represent the maximum regional currents .1°, while the dashed lines are J For case (a), Region 1 is never the lim iting region; while for case (b), Region 1 is the source of current-limited flow. At the heart of decoupling is the separation of the transport current from the physical pro cess that is to be optimised. For if the transport current is not affected, or at least not in a detri mental fashion, then one is free to optimise the desired metric. Current-limited flow provides the necessary tool to decouple T from all regions, and therefore all physical transport processes, save Julyl2,1995  19  one. Continuing on with the example of simultaneously optimising VA and Rc, if T were decou pled from the construction of the base and collector, say by making the emitter-base SCR the source of current-limited flow, then VA would no longer depend upon the collector doping; en abling the optimisation of Rc without affecting VA. With base-width modulation no longer an is sue, in terms of the collector current and therefore VA, it would be possible to increase the intrinsic collector doping adjacent to the base and thereby reduce Rc. A further optimisation, in terms of the base-collector capacitance CBC, could also be had by placing a low-doped collector region within the CB SCR (say at 10 3 for 2000A) in order to set CBC, followed immediately by a cm 16 highly doped extrinsic collector to reduce Rc. Optimisation of competing metrics is thus achieved by first identifying the coupling parameter; then, one other region that does not contain the cou pling parameter is constructed (generally through abrupt Bandgap Engineering techniques) to be the source of current-limited flow in order to provide for the control of J- (i.e., the collector cur rent). This chapter has provided a logical course to decouple otherwise competing metrics so they may be simultaneously optimised. The tool for decoupling the competing metrics being the cre ation of current-limited flow outside of the region or regions to be optimised. It is possible to achieve current-limited flow in any given region by resorting to abrupt Bandgap Engineering tech niques. Thus, abrupt Bandgap Engineering provides the necessary tool to further optimise BJTs. Finally, all the models for the various regions of the HBT are neatly brought together through eqns (2.7) and (2.9) (or their approximate forms eqns (2.8) and (2.10)) for the calculation of the total electron and hole currents entering and leaving the device.  July 12, 1995  20  CHAPTER 3 Base Layer Decoupling and Optimisation  July 12, 1995  21  Traditionally, the base region, or more specifically the neutral base region, has determined the overall performance of the BJT. As such, the physical construction of the base is of paramount im portance to the function of the BJT At issue with the base is the fact that there are basically two de grees of freedom within the base; namely the base doping profile N(x) and the neutral base width WB (in an HBT a third parameter, namely the bandgap in the base Eg(x), is also available). Against  these two (or three) independent parameters lie numerous device metrics that are to be optimised. Obviously, with more metrics than independently controllable parameters, it is impossible to simul taneously optimise all of the metrics. Thus, an inherent compromise is forced to exist between many of the metrics, which leads to an unnecessary limit to the peak performance of the BJT. Chapter 2 dealt with the effects of abrupt Bandgap Engineering techniques upon the trans port current within an HBT. It was found that through abrupt Baudgap Engineering, it was possi ble to construct a specific region in such a fashion that the transport current T depended on this region alone; thereby decoupling T from all other regions of the device. Once T has been decou pled from all other regions of the device, save one, the task of independently optimising each re gion becomes trivial. The possibility of decoupling J- from the physical construction of the base promises to eliminate the interdependence that the base-controlled metrics have upon each other. Once the base metrics are free of each other then one can finally consider a truly optimised BJT and thus, achieve a significant improvement to the peak performance of the BJT. Parameters such as the in trinsic base sheet-resistance RBJ, base-emitter capacitance CBE, injection index y (not to be con fused with the y in Section 2.2 which is the recombination loss), Early voltage VA, base transit time ‘CB, and the base-collector capacitance CBC could then be simultaneously optimised. The key to the optimisation of these base metrics rests simply on the decoupling of T from the base by constructing one other region of the device in such a manner that it results in current-limited flow. This chapter takes the abstract concept of optimisation through current-limited flow and ap plies it to the base region. The methods used to achieve the simultaneous optimisation of the base region metrics follow directly from the prescriptions of Chapter 2. Specifically, the base metrics RBD, CBE,  ‘  VA, and  13 ‘C  are considered for optimisation. Finally, once the optimum models for  each of these metrics within the base region have been derived, they are linked together for the calculation of the total electron and hole currents by the methods derived in Chapter 2.  July 12, 1995  22  3.1 Independent Optimisation Of RB, CBE, And In the design of any transistor, the sufficient design criteria is to provide for a gain that is greater than one. However, it is generally desirable to design a gain that is much larger than one. In the case of a BJT, this translates into maximising the current gain 3  (= collector current I di  vided by the base currentlB). In current-day BITs, the manufactured materials are so pure, that for the most part, the recombination of minority carriers being transported through the neutral base represents only a small fraction of the total ‘B [53]. Therefore, [3 will depend on the injection effi ciency y of the Emitter-Base (EB) junction. For an npn BJT the EB y is given by: “fl,  where  n,B  B  (3.1)  is the electron transport current through the base, and  is the hole current injected  into the emitter (also known as hole back-injection). Using eqn (3.1), in the absence of neutralbase recombination, the gain is: I  (3.2)  =!._•  ‘‘  “p,E  Thus, [3 is maximised as y is driven towards 1; meaning that  is driven towards zero and/or n,B  is made as large as possible. In an npn BiT, n,B is inversely proportional to the base Gummel number G#B [54-56] given by: dx,  G#B  (3.3)  = where D is the electron minority carrier diffusion coefficient, WB is the neutral base width, andp is the base majority hole concentration  (= base doping N except under high-level injection  [56]). Furthermore, for a transparent emitter (an emitter where there is little hole recombination), p,E  is inversely proportional to the emitter Gummel number G#E [54-56] given by: G#E  =  i)  dx,  where D is the hole minority carrier diffusion coefficient, is the emitter majority electron concentration jection). Thus,  (3.4)  WE  is the neutral emitter width, and n  (= emitter doping NDE except under high-level in  f3 is proportional to G#E/G#B. Now, the intrinsic base sheet-resistance RBU is also  inversely proportional to G#B [54]. However, unlike the case for [3, where it is desirable to reach a maximum, RBU is to be minimised in order to improve the high-frequency operation of the BJT. July 12, 1995  23  Since RBL] and optimising  13 are both tied to the parameter G#B, and increasing G#B optimises RBU while de  13, we realise these two metrics are competing and therefore cannot be simultaneously  optimised (at least in terms of the parameter G#B). As was discussed in Section 2.3, the key to optimising two otherwise competing metrics is to identify their common parameter (in this case G#B) and remove its dependency from one of the two metrics. Continuing on with the case of optimising RBJ and  13, it would appear possible to in crease G#B and thereby minimise RBU, while also increasing G#E and thereby maximise 13. How  ever, G#E in a BJT cannot be increased because NDE is either at or very near its maximum ). Thus, without resorting to Bandgap Engineering techniques, the only 3 physical limit ( 1021 cm available parameter is G#B, meaning that a compromise has to be made between RBU and was the motivation for the first HBT; to decouple  13. This  13 from its sole dependence upon G#B.  Looking at Fig. 3.1(a), the band-diagram for a BJT shows that it is just as easy for an elec tron to enter the base as it is for a hole to enter the emitter (the two carriers see exactly the same potential barrier of Vbj  —  VBE). Therefore, the ratio of n,B to p,E  (= 13) will be proportional to the  ratio of the available number of electrons in the emitter to the available number of holes in the base  (= NDE/N  —  G#E/G#B). Now, if it were possible to alter the bandgap of the EB junction so  that the holes had to surmount a larger barrier than the electrons, then pE would be significantly reduced and  13  increased (see Fig. 3.1(b)). Finally, if Bandgap Engineering were employed to  achieve an initial 1000-fold increase in  13 (by reducing p,E through a Bandgap Engineered  then G#B could be increased 32-fold, thereby reducing RBD 32-fold, while still leaving a net 32fold increase in  13. Thus, by creating a heterojunction at the EB metallurgical junction, it is possi  ble to reduce p,E without increasing G#E. Then, the gains provided by a reduced pE are shared between an increase in  13 and a decrease in RBIJ.  The methods just described for the simultaneous optimisation of RB and 13 demonstrate the potential gains of abrupt Bandgap Engineering. However, the techniques described above did not follow the exact prescription given in Section 2.3, and thus maintain a coupling between RBU and  13.  Instead of decoupling  13 from G#B,  another degree of freedom was added to G#E; namely the  abrupt change of t1E in the valence band at the EB junction. The dependence of 13 upon G#B still exists, but p,E and thus (3, by the addition of a heterojunction within the EB SCR, now has anoth er dependence of exp(-AEJk1) [2,46,47] through the intrinsic carrier concentration in the emitter  July 12, 1995  24  i,E•  However, since  f3 still depends upon G#B, any change in G#B due to bias (such as the Early  effect [57], Kirk effect [58] or high level injection [56,59]), will still affect and generally degrade  I. The reduction of p,E through simple abrupt Bandgap Engineering is thus seen as a good first step, but falls short of the optimum case where  (a) BJT  f3 is decoupled from G#B altogether.  (b) HBT E (eV)  E (eV)  +  Fig. 3.1. (a): Band diagram of a homojunction BJT. Clearly, the potential barrier seen by a hole trying to go from the base to the emitter is the same barrier seen by an electron trying to go from the emitter to the base. (b): Band diagram of an HBT. Through abrupt Bandgap Engineering, the barrier seen by a hole trying to enter the emitter is a least AE larger than the barrier seen by an electron trying to enter the base. Also note the formation of the Conduction-Band Spike (CBS). To fully decouple 3 from G#B one looks at the spike in E at the EB junction shown in Fig. 3.1(b). This Conduction-Band Spike (CBS) occurs in HBTs where the base is made of GaAs and the emitter is made of AlGai.As [25]. The barrier to electrons entering the base lies somewhere between q(V 1  —  1 VBE) and q(V  —  VBE)  —  tSE depending on the amount of tunneling through the  CBS. In general, it is found that the drop AE is sufficient to cause the CBS to be the region of current-limited flow (this will be fully discussed in Chapter 4). Thus, T (= n,B in the absence of significant neutral-base recombination) will be governed by the physical process of transport through the CBS, and not by the transport through the neutral base. Furthermore, transport through the CBS has little dependence upon G4B (as long as the base doping is much larger than the emitter doping). Therefore, T and thus  are decoupled from G#B through the condition of  current-limited flow at the CBS. The condition of current-limited flow in the region of the CBS follows exactly the prescrip tions of Section 2.3. n,B has now been decoupled from G#B, meaning that processes connected to July 12, 1995  25  G#B such as the Early effect, Kirk effect, and high-level injection, which degraded the collector  current of BJTs, are no longer an issue for the abrupt HBT (the term abrupt refers to the abrupt change of IXE and AK , at the EB junction). With the collector current decoupled from G#B, RBL] 1 can be minimised by increasing G#B through an increase in N, while leaving  and therefore ‘y  unaffected. Before leaving this section to discuss the further optimisation of the base and collector, it should be noted that the EB junction capacitance CBE can also be minimised due to the condition of current-limited flow at the CBS. The high-frequency performance of a BiT improves as CBE decreases. Most notablyfT (the frequency at which  f, under the conditions of an A.C. short circuit  between emitter and collector, has dropped to unity) increases as CBE is reduced. Since CBE is given by: qNN CBE  =  I,, I4kVbjVBE)  where  Nrat  NAB =  NAB + NDE  ‘  .  then NDE and Nrat need to be minimised in order to reduce CBE. In a BJT, the need to maximise  ) f  forces NDE >> N, meaning that N is reduced in order to reduce CBE. Thus, CBE is connected to RBD  as well, and leads to another condition where only a compromise and not a true optimum can  be reached. CBS-limited flow in an abrupt HBT decouples 3 from N, so that RBU can be opti mised by increasing N. Finally, CBE is reduced in an abrupt HBT through the reduction of NDE (for HBTs, N  >>  NDE  so that Nrat  1). The only limit to the reduction in NDE being the point at  which a significant intrinsic emitter resistance RE begins to occur (see Fig. 3.2). This section has presented the methods to simultaneously optimise RB(J, CBE, and ‘y Opti misation of these metrics begins by decoupling from y and CBE the dependence upon N. This decoupling is afforded by the creation of current-limited flow at the CBS. With y and CBE decou pled from N, RB is optimised by increasing N. Then, CBE is optimised by reducing NDE. Fi nally, the optimisation of y depends first of all upon  p,E  (which depends heavily on SCR  recombination [24]) and secondly upon n,B (which is governed by the flow of  T  through the  CBS); EB SCR recombination, which accounts for most of p,E’ is covered in Chapter 5, while the current within the CBS is covered in Chapter 4. The optimisation afforded by the abrupt HBT in comparison to the BJT is stunning, as none of the methods discussed in this section would have been applicable to a BJT because the gain of the transistor would have been reduced below unity.  July 12, 1995  26  Emitter-side SCR edge  Fig. 3.2. The resistance of the intrinsic emitter will become considerable if NDE,1 is reduced without bound. To minimise this parasitic resistance, the width of the intrinsic emitter is only made large enough to contain the emitter extent of the EB SCR. Then, a highly doped NDE2 ex trinsic emitter is placed as a cap layer on top of the device, where the eventual contact layer is formed.  3.2 Reducing tB by Decoupling the Base from I’ Chapter 2 discussed the merits of Bandgap Engineering, where the natural evolutionary path of the BiT produces the HBT. Two Bandgap Engineering techniques were considered: techniques that created abrupt changes in E.JE leading to the creation of current-limited flow; and tech niques that created gradual changes in EdE that produced additional aiding fields for the trans port of charge. Then, Section 3.1 focussed upon the benefits of current-limited flow produced by an abrupt change of iXE within the EB SCR. This section carries on with the benefits to be de rived from current-limited flow, but delves into the second group of Bandgap Engineering tech niques  -  namely the creation of fields in the base to aid in the transport of charge through the  region. A major component of the total transit time for a BIT or HBT is still the neutral-base transit time tB. In the absence of any spatial variation to the bandgap or N, then under low level injec tion conditions, with the neutral base width WB larger than a few mean-free paths  ,  the base tran  sit time is given by the standard equation:  w B  tB  (3.6)  2D  can be reduced from the value given in eqn (3.6), without reducing WB, by introducing an aid  ing field in the base (as is shown in Fig. 2.1). BJTs where an aiding field has been placed in the base are termed drift-base transistors [60]. This aiding field implies, for an npn BJT, a downwards July 12, 1995  27  __________  slope to E in the neutral base. Before the creation of HBTs, a negative slope in E could only be achieved by varying N from a high value near the emitter-side of the neutral base, to a low value near the collector-side of the neutral base [60]. This non-uniform N(x) would indeed reduce  tB  but at the expense of having a low base doping nearest the collector; leading to a reduced magni tude of the Early voltage. Therefore, the drift-base transistor had a rather limited range of optimi sation as the aiding field was coupled in a compromising fashion to the Early voltage. Add to this the fact that the optimum N(x) was an un-manufacturable exponential, then the optimum driftbase transistor was a good idea that was generally beyond the manufacturing capabilities of the day. Enter Bandgap Engineering once again. The issue with the drift-base transistor was the low base doping near the collector. By using a graded bandgap in the base (where the bandgap is large near the emitter-side of the neutral base and small near the collector-side of the neutral base), an aiding field can be created without the need to vary N [38]. Thus, by using Bandgap Engineer ing techniques to create a gradual down-slope to E in the base, tB can be reduced without lower ing N and compromising the Early voltage. Kroemer calculated tB for a non-uniform bandgap Eg across the neutral base, and found [38]: w w In.(x) 1  p(z)  p (x)  D, (z) n (z)  B =  dzdx,  (3.7)  where n 1 is the intrinsic carrier concentration. The derivation in [38] which leads to eqn (3.7) is based upon Shockley boundary conditions. However, it is a simple extension to show that eqn (3.7) is actually quite general, and is applicable to cases where a zS.Ep is present. Finally, if a lin ear grading of the bandgap in the base is used, such that n (x)  =  n (x= 0) exp (qFx/kT , eqn  (3.7) gives:  2 w  -  AE  2 kT  (3.8)  where F = AEgI(qWB), and AEg represents the difference between the bandgap at the emitter-side of the neutral base and the bandgap at the collector-side of the neutral base. As an example, if D =  , WB 1 s 2 30cm =  =  ioooA, and AEg = 3k7  then using eqn (3.6) tB  =  i.67ps, while using eqn (3.8)  0.76ps, a 2.2-fold reduction in tB through the addition of a graded bandgap in the base.  July 12, 1995  28  The reduction of tB through a graded-base transistor is very attractive. When coupled to the fact that the Early voltage is not compromised, Bandgap Engineering in the base appears to hold nothing but gains. The only requirement of a graded-base transistor is the need to create a graded alloy in the base in order to provide for the downwards slope in E. In the case of AlGai.As/ GaAs HBTs, the bandgap is increased with an increase in the Al mole fraction x; while in Sii..Ge HBTs, the bandgap is decreased with an increase in the Ge mole fraction x. Now, in A1GaAs HBTs the Al mole fraction must remain below a maximum of x  =  0.45, for this is the  point at which the material changes from a direct to an indirect bandgap [61]. In a similar fashion, Sii..GeJSi HBTs have an upper limit of Ax  <  0.2 due to the effects of strain (this is discussed  fully in Chapter 6). Thus, an “alloy budget” exists in the HBT, meaning that a decision must be made in the allocation of alloy mole fraction among the various regions of the HBT. Therefore, a compromise must be made in the amount of Bandgap Engineering allocated to the formation of the graded-base versus all the other bandgap-engineered regions of the device. Since the heterojunction in the EB SCR provides the most important gains in terms of opti mising the metrics of the device (namely decoupling y from G#B), part of the total alloy budget must be allocated to its formation. In the case of AIGaAs HBTs, fully 66% of the maximum total alloy budget (Ax = 0.3 of a maximum 0.45) is spent in the formation of the EB heterojunction (In reality, Ax < 0.45 is a maximum upper limit that is generally reduced to 0.30 for practical applica tions. With this reduced alloy budget, the EB heterojunction would consume the entire budget). In SiGe HBTs, virtually the entire alloy budget of Ax  <  0.2 is spent in the formation of the EB het  erojunction. Therefore, irrespective of the material system used to form the HBT, little if any of the alloy budget remains for the Engineered Bandgap in the base once the EB heterojunction has been formed. This means there is little room to reduce  ‘CB  through a manipulation of the bandgap  within the base. The reduction of ‘tB is a desirable goal, even in the face of very real practical limitations. Bandgap Engineering in the base may not play a significant role due to the restricted alloy budget; but drift-base transistors, based upon a non-uniform N(x), might become plausible by the cre ation of an abrupt EB heterojunction. The reasons for abandoning drift-base transistors were: it was not possible to manufacture the steep doping profile in the base required to generate the aid ing field; and the low base doping near the collector-side of the neutral base resulted in an intoler  July 12, 1995  29  ably low Early voltage. The first problem, namely the manufacture of the highly non-uniform N(x), is no longer an issue with advanced MBE and MOCVD growth techniques. The second  problem, a decrease to the Early voltage, is solved by decoupling the collector current I from the base, so that modulations to G#B from changes to VCB no longer matter, provided punch-through is avoided, of course. Following, once again, the prescriptions of Section 2.3, I is decoupled from G#B by creating a situation of current-limited flow at the CBS formed by the EB heterojunc tion; thereby linking I to the physical transport mechanisms associated with the CBS instead of the neutral base region. With the two old problems associated with using a non-uniform N(x) for the reduction of tB solved, the optimum Nj(x) for the reduction of tB is investigated.  3.3 Optimum Base Doping Profile to Minimise tB Bandgap Engineering in the base is not really being considered in this section; however, it can be included in the optimisation without any changes in the arguments to follow (this includes effects due to a manufactured change in the bandgap and changes to the bandgap due to heavy doping effects). Starting with eqn (3.7), then after substituting p w_ 2 1 (x) n B  =  NAB  WR NAB  (x)  (z)  =  N, tB becomes:  dzdx.  (3.9)  D, (z) n (z)  If D, is taken as some average constant, then eqn (3.9) is simplified even further to become:  j  B  =  w ?n(x)  w ?NAB(z)  =J  J  ONAB(X) x 7 D,  dzdx.  (3.10)  nl(z)  Eqn (3.10) provides the functional form of tB to be minimised. Using the calculus of variations, and searching for the weak variations in N(x)In (x), then the Euler-Lagrange characteristic equation that minimises eqn (3.10) is: y dx  =  C,  (3.11)  where w rNAB  y(x)  =  t  (z)  dz,  (3.12)  n(z)  and C is an arbitrary constant. The solution of eqn (3.11) is straightforward and yields: July 12, 1995  30  (3.13)  eA2X 1 y(x) =_A  where A 1 and A 2 are arbitrary  constants.  The beauty about eqn (3.13) is it solves for both N(x)  and n (x) simultaneously. The next section will deal with non-uniform bandgap effects, so taking for now that n(x) is constant, then differentiating eqn (3.12) and substituting in eqn (3.13) gives: N(x)  = 2 AiA e A2) =  ae  (3.14)  Eqn (3.14) gives the standard exponential solution [601 for the doping profile in the base that leads to a minimum in tB. Within the confines of weak variations, a possible minimum could occur by admitting a piece-wise solution for N(x) composed of N sections whose form within each section is given by eqn (3.14). The conditions of continuity at any break-point joining two regions being [62]: and F—y’  ()  (3.15)  be continuous,  where F is the integrand that is to be made stationary, and primes denote differentiation with respect to the dependent variable x. In the case being considered, F = yly’. Then, using the exponential so lution for y(x) in eqn (3.13), and applying the second continuity condition of eqn (3.15) produces: 2 F— —  —  which must be continuous at the break-point x 0 joining the two regions. If we let Region 1 join with Region 2, where the solution in Region 1 is A 1  Z 4 e 1  eA2,2X, A2 , 1 then the above equation requires that A 1 , 2  , 2 A  =  X 1  =  and the solution in Region 2 is . Applying the first continuity 2 A  condition of eqn (3.15) at the point x = x 0 produces: = —  ay  1  1  1 =A  =  2 Y’  1 1 A Ae ° 4  1 2 A A  =  1 A  eA2X0  2 =  . 1 A  Thus, a piece-wise connection of exponentials is not admitted as a stationary solution for N(x). However, if the last equation is rewritten as A 1 A 1 eA2X0  1 2 A A  eA2XO,  then as A 2 —* 0 no re  striction is placed on the values admitted for A , and A 1 . This admitted solution for y(x) is also 2 , 1 a piece-wise and discontinuous set of constant solutions. As such, this solution for y(x) tends to wards a strong variation and care must be exercised in the absolute applicability of the weak vari ational principles used to obtain this result. With that cautionary note in mind, if the form of A , 1 2 are carefully chosen to be 2 , 1 and A /A and 2 1 a IA respectively, then as A a 2  —*  0, N(x) also be  comes a piece-wise and discontinuous set of constant solutions. July 12, 1995  31  The weak variational principles used to find the N(x) that renders  tB  stationary are con  structed in a such a manner that only y(x) be defined at the end points of the integration. Since N(x) is given by y’, there is no simple way to specify the doping values at the end points of the integration (namely the emitter and collector edges to the neutral base). Further examination of eqn (3.14) shows that there are no bounds to the value of the constant b in the exponent of the ex ponential defining N(x). In fact, by letting b ated in the base and  tB  —>  —00,  an infinitely large aiding field can be cre  will be reduced to zero. To see this, eqn (3.14) is used in (3.10) to give: 2(eb_b_ 1) tB  =  tBO  where tBo is the tB given by eqn (3.6). Clearly, as b —> en by eqn (3.16) goes to  ‘rho.  (3.16)  2 b —00,  tB  —*0. As a check, as b —>0, tB giv  Thus, no matter what N is forced to be at the emitter-side of the  neutral base, N(x) can be made to decrease at a rate such that tB is ostensibly reduced to zero. Therefore, the variational principles used to deduce that the optimum N(x) is a pure exponential are based upon an unrestricted doping at the collector-side of the neutral base It is not reasonable to allow the doping at the collector-side of the neutral base to become ar bitrarily small, even in the presence of current-limited flow at the CBS. For even though I is de coupled from G#B, RBD still depends on G#B and would become unreasonably large as b —* Eqn (3.10) is revisited, but this time  ‘CB  —00  is made stationary subject to boundary conditions upon  N(x) at the emitter- and collector-sides of the neutral base. Since there appears to be no simple way of including these boundary conditions into the variational principles, a numerical minimisa tion was constructed [63]. The results of numerical attempts to render tB stationary, subject to the boundary conditions placed upon N(x), produced a form that suggests N(x) be exponential in the middle of the base but have two constant regions attached on the ends (see Fig. 3.3). This re sult seems plausible in light of the variational analysis performed so far, where a constant was ad mitted as a solution to N(x). Even more convincing, the form being suggested from the numerical analysis is not a piece-wise connected set of exponentials (which was rejected as a pos sible stationary solution from the variational analysis), but is a piece-wise connection involving constant regions of doping, as is admissible from the variational analysis. In any event, it is clear that the boundary conditions placed upon N(x) cause the exponential solution from simple vari ational analysis to become non-stationary.  July 12, 1995  32  loW  0.0  0.25  0.50  1.0  0.75  Normalised Position in the Base x (WB) Fig. 3.3. Optimum doping profile N(x) obtaining by numerically minimisin eqn (3.10) with the boundary conditions N(x=0) 5x10 3 and N(xtWB) = 2x10 cm 18 cm 16 Using the form for N(x) suggested from the numerical work, namely exponentials sepa rated by regions of constant doping, analytic methods were employed to find the break points be tween the exponentials and the constant regions that minimised  tB.  given in Fig. 3.4, then finding the break-point h that minimises  given by eqn (3.10) produces,  ‘CB  Using the form of N(x)  after considerable algebraic manipulation with the symbolic mathematics tool MACSYMA (see Appendix A): (U1nU+1—U)lnU (U1nU+2)lnU+2(l—U) where  ‘CBO  is still the  tB  U[U(2lnU—3)+4]—1 ) tB_tBou[(Ul 1 . 3 7 u+2)lu+2(1u )](  given by eqn (3.6), h is normalised to the neutral base width WB (and  therefore ranges from 0 at the emitter-side to 1 at the collector-side of the neutral base), and U is the doping ratio given by N(x=0)IN(x=WB). The interesting thing to note about eqn (3.17) is that it depends only on the relative doping ratio U. Further, the exact same solution results (save h —>  1  —  h) if N(x) is changed, in a symmetrical fashion to that shown in Fig. 3.4,  SO  that the con  stant region occurs first followed by the exponential region. Eqn (3.17) represents the solution of the simplest form of N(x) suggested from the numerical analysis. The process described above is repeated again, but this time with the optimum form (shown in Fig. 3.3) obtained from numerical analysis. Again, substituting this form of N(x) into eqn (3.10) and minimising  ‘CB  produces, after considerable algebraic manipulation with the symbolic  mathematics tool MACSYMA (see Appendix B): July 12, 1995  33  l—h 1 h 2 =  1 lnU+2  and  tB  =  2 tBolnU+2  where, 1 h and 2 h are nonnalised to the neutral base width  WB.  =  hl, 2 tBo  (3.18)  Eqn (3.18) shows the beauty of the  symmetric form used for N(x); namely that the length of each of the constant regions is the same, and the exponential region is perfectly centred within the base. It is very simple to prove that tB given by eqn (3.18) is always smaller then that given by eqn (3.17). Therefore, the form of N(x) given in Fig. 3.3 produces a smaller ‘CB then the form given in Fig. 3.4.  C  C  0.0  0.25  0.50  0.75  1.0  Normalised Position in the Base x (WB) Fig. 3.4. The first trial function for N(x) inspired by the form suggested by Fig. 3.3. The process is continued by constructing more complex forms based upon an extension to N(x) given in Fig. 3.3. When eqn (3.10) is minimised using the N(x) given by the form shown in Fig. 3.5(a), it is possible to find a stationary result where h 1  0 (h 1  =  0 would give N(x) as  shown in Fig. 3.3). Even though N(x) given by Fig. 3.5(a) renders tB stationary, when compared to the result obtained from eqn (3.18), it does not produce the absolute minimum value for tB. In fact, taking one final progression to using the N(x) as shown in Fig. 3.5(b), a stationary result is again obtained, but it is larger still than the case shown in Fig. 3.5(a) and therefore does not pro duce the absolute minimum value for tB. Therefore, eqn (3.18), with N(x) as shown in Fig. 3.3, produces the absolute minimum in  tB  subject to the boundary conditions for the doping at the  emitter- and collector-sides of the neutral base. The most notable thing about the optimum form of N(x), as shown in Fig. 3.3, is that it is not the pure exponential the device community has been lead to believe is the optimum. This result answers the problem posed in [64,65], where the au thors used third order perturbation theory to show that an exponential was indeed stationary but it did not produce the absolute minimum for tB. July 12, 1995  34  (a)  (b)  0.25 0.0 0.50 0.75 1.0 Normalised Position in the Base x (WB)  0.25 0.0 0.50 0.75 1.0 Normalised Position in the Base x (WB)  Fig. 3.5. (a): the second trial function for N(x), which is an extension of the form shown by Fig. 3.3; (b): the final trial function for N(x). As a final consideration, it is instructive to use the N(x) suggested by the analysis sur rounding eqn (3.15). Tn the proof that showed N(x) could not be constructed of piece-wise con tinuous exponentials, it was found that N(x) could be constructed of piece-wise discontinuous constants. In the simplest case, if N(x) is constructed as shown in Fig. 3.6, then it is straight for ward to show that tB is minimised when: h  1  and  =  tB  =  tBO  U+1 2U  (3.19)  Eqn (3.19) shows that a very simple jump discontinuity, or step, in the base doping proffle at exact ly the half-way point in the neutral base, can reduce the base transit time by a factor of two when compared to the uniform base case (tBO). In fact, for any U is achieved. Still, for all relevant U,  tB  10, the full two-fold reduction in  tB  given by the step-doping case of eqn (3.19) is larger than  that achieved by the optimum-doping case of eqn (3.18). However, the step-doping case shows that even a very simple change to the base doping profile can produce a significant reduction in the transit time through the neutral base. As for the technological objection that a perfect step-doping profile is impossible to create, any deviations from a step, say due to diffusion of dopant during the thermal-cycle of the manufacturing process, will only tend to drive N(x) towards the optimum profile and reduce tB even further: this result is obvious as a spreading of the step-discontinuity increases the spatial extent of the aiding field and thereby decreases the transit time. Therefore, the step-doping profile, although not as beneficial as the optimum doping profile, still provides for a significant reduction of tB, but with very little complexity in terms of manufacturing. July 12, 1995  35  I  x=h-”  0.25 0.0 0.50 1.0 0.75 Normalised Position in the Base x (WB)  Fig. 3.6. Step-doping profile for N(x). Comparing ‘CB given by the optimum N(x) (eqn (3.18)), to the ramped N(x) (eqn (3.17)), then to the step-doping case (eqn (3.19)), and finally to the pure exponential case (eqn (3.16), with b  =  -mU), shows some interesting results (see Fig. 3.7). In all four cases as U —* 1,  3 t(  —  tBO:  this  is required and acts as a check to the validity of the four models. As was stated before, for the en tire useful range of U  (i.e., >  1),  (3.18). However, for the range 1  ‘CB  is minimised by the optimum doping proffle leading to eqn  U  , 2 7.389=e  tB  from the step-doping proffle is smaller than  that from the pure exponential profile. Thus, not only have we found out that the pure exponential is not the optimum, we have also found that for small doping ratios the step-doping profile is better than the exponential. An examination of Table 3.1 shows that as U becomes large, the pure expo nential case and the ramped case both approach the optimum case for the minimisation of tB. This result shows that the optimum-doping case initially starts out looking much like the step-doping case, then as U increases, slowly transforms itself into the pure exponential case. Finally, for U = 300, the optimum-doing case hash 1  1  —  2 h  =  0.13 and tB is only 10% less when compared to the  pure-exponential case; however, the optimum-doping case has a 49% larger Gummel number and thus a 49% smaller RBU when compared to the pure-exponential case. Clearly, the pure exponen tial case is not the optimum doping profile to use, either in terms of minimising tB or RBU. There fore, the optimum-doping case shown in Fig. 3.3 and governed by eqn (3.18) is the best basedoping profile to use in order to minimise  tB  with the smallest impact on RB.  July 12, 1995  36  1murn’i -.  0.75 j —  0.50—  Exponential C  z  0.25— “a  100  ‘2, 101  ‘7,  102  Doping Ratio U  _;-- -,-‘  \  3 i0  4 io  0.70  0.50  0.60  0.80 1.00  Break Point h 2  Fig. 3.7. tB using NAB(x) from Fig. 3.3, where h 1 1 h 2 but h 2 is varied as a parameter instead given of being by eqn (3.18). h 2 = 0.5 corresponds to the step-doping case, while h 2 = 1 corre sponds to the pure exponential case. Finally, the line drawn on the surface is the tB that results from the optimum-doping case given by eqn (3.18). —  Table 3.1: tB for the four doping cases: Optimum, Ramp, Step, and Exponential as given by eqns (3.18), (3.17), (3.19), and (3.16) respectively. NOTE: all values are given in units of tBO. U  Optimum  Ramp  Step  Exponential  3  0.65  0.69  0.67  0.72  2 7.389=e  0.50  0.54  0.57  0.57  10  0.46  0.50  0.55  0.53  30  0.37  0.40  0.52  0.42  100  0.30  0.32  0.51  0.34  300  0.26  0.27  0.50  0.29  July 12, 1995  37  There are two major restrictions placed on the use of the optimum-doping case for the mm imisation of  tB.  These two restrictions are: that the aiding field produced by the non-uniform  N(x) be small enough to neglect high field effects; and the variation in D(x) be small enough to ignore. The first requirement is not terribly restrictive, for even with a base width of ioooA, and U =  100, the aiding field is 4 l.7xlO V Icm which is acceptable for heavy-doped Si and would be at  the edge where high field effects begin to occur in heavy-doped GaAs. However, the second re quirement that Da(x) be ostensibly constant over the entire base width is much harder to accept; for even though the base region of an HBT is very heavily doped, D(x) would still have a signif icant variation with U in the range of 7 to 30. The issue of a non-uniform D(x), as well as varia tions in n(x) due to Bandgap Engineering and heavy doping, are considered in the next section. In any event,  ‘CB  will always be reduced by using a monotonically decreasing (from emitter towards  the collector) non-uniform N(x). Therefore, if exact values and not general trends are required, then the optimum-doping case presented in this section must be applied with caution if there is considerable variation in either D(x) or n(x) across the base. This section has provided for the optimum N(x), given a set of boundary condition to the neutral base, in order to minimise  tB.  It was found that the optimum N(x) only depends on the  relative doping ratio U, and not on the absolute doping given by the boundary conditions. Further more, the optimum N(x) is not the pure exponential that the device community has thought was the case, but is an augmented exponential as shown in Fig. 3.3. They key to applying the results of this section hinge on the decoupling of I from G#B afforded by the creation of current-limited flow at the CBS. Therefore, only by creating an abrupt HBT 1 can the drift-base BJT be manufac tured without a significant reduction to the Early voltage.  3.4 The Effect of a Non-Uniform n 1 and D on the Optimum tB Section 3.3 derived the optimum base doping profile for the minimisation of  ‘CB.  It was  found that if the base doping was fixed at the emitter- and collector-sides of the neutral base, then the optimum N(x) was an augmented exponential shown in Fig. 3.3 and governed by eqn (3.18). 1. It is possible to decouple I from G#B by using a varying bandgap in the base [66]. However, as was discussed in Section 3.2, the alloy budget generally prohibits any significant Bandgap Engineering in the base if an EB hetero junction is to be formed in order to control f3. Therefore, the technique of current-limited flow is the only practical method to decouple I from G#B. July 12, 1995  38  Due the arguments presented in Section 3.2, Section 3.3 found the optimum N(x) without re gard to the optimum n (x). However, due to the heavy base doping that is characteristic of HBTs, 1 bandgap narrowing will certainly cause variations to n (x) when a non-uniform N(x) is present. 1 This section will consider the joint optimisation of N(x) and n (x) in terms of minimising tB. 1 Also, the effects of a non-uniform D(x) will be discussed. tB  is given in full by eqn (3.9). If the variation of D(x) with respect to N(x) is for the mo  ment ignored, then eqn (3.10) results. Section 3.3 finds the functions y(x) that render eqn (3.10) stationary and then finds the one y(x) that minimises tB. y(x) is given by eqn (3.12), which produc es after differentiation with respect to x: y(x)=—  NAB(x) n2 (x)  Using eqn (3.11), which is the O.D.E. that renders y(x) stationary, in the above equation yields: NAB (x) n2 (x)  =  —Cy (x).  (3.20)  At this point Section 3.3 lets n (x) be a constant, which can then be absorbed into the arbitrary 1 constant C, to yield eqn (3.14). However, one could just as easily let N(x) be a constant and solve for n (x). If this is done, then all of the results of Section 3.3 are still applicable to the opti misation of n (x); for the stationary function y(x) has no dependence on either N(x) or n 1 (x). This immediately results in the optimum n (x) being given by the reciprocal to N(x) shown in Fig. 3.3, with eqn (3.18) governing the placement of h 1 and h 2 and solving for tB. The only change is that U is now given by the ratio n (x = WB) /n (x =0) (the endpoints have been inter changed to keep U> 1). If the variation in the effective density of states for E and E is ignored, then n (x)  =  (x)/kT), where AEg(x) is now defined as the difference in 8 n (x =0) exp (—AE  Eg at x relative to Eg at the emitter-side of the neutral base. Since the optimum n (x) is given by the reciprocal to N(x) shown in Fig. 3.3, and given that Fig. 3.3 is a log plot, then LS.Eg(x) looks exactly like Fig. 3.3 but it would be linear and not log (see Fig. 3.8). Therefore, just like in the op timum doping case, the optimum bandgap-graded-base HBT is not purely linear, but is the aug mented ramp shown in Fig. 3.8. There is no reason to consider a pure optimisation of either n (x) or N(x). Eqn (3.20) solves for the simultaneous optimisation of both n (x) and N(x). Thus, part of the aiding field can be created by a non-uniform N(x), and the rest of the aiding field can be created by a Band Julyl2,1995  39  gap Engineered n (x). This realisation allows the burden of generating an aiding field to be shared between two physically different parameters. By using both n (x) and N(x), far less of the alloy budget needs to be used in order to generate n (x), and a smaller decrease in N(x) will necessarily have a smaller impact on G#B and RBD. As an example, let ‘CB  =  O.StBo. This requires  , where 2 that U = 7.389=e U =  NAB(x)  n2(x)  n(x)  NAB(X)  (3.21) X=WB  Letting both N(x) and n (x) share equally in generating the aiding field gives  UNAB  (which is  the U for eqn (3.18)) equal to U,z (which is the U for n (x) shown in Fig. 3.8) which is equal to J7.389 =e. Thus, the doping in the base as well as n (x) change by only 2.7-fold, meaning that AEg is only lkT  -kTlnU 0.0  0.25  0.75  0.50  1.0  Normalised Position in the Base x (WB) Fig. 3.8. Optimum bandgap in the base to minimise tB. The bandgap at the emitter-side of the neutral base (x= 0) is the reference point. U = n (x WB) /n (x :=0), where h , h 1 2 and tB are given by eqn (3.18). So far this section has only presented the case where N(x) and n (x) are treated indepen dently of each other. This will not be the case when N is large enough to cause bandgap narrow ing that couples n (x) to N(x). Since HBTs are characterised by their very high base doping, bandgap narrowing effects need to be considered. Fortunately, the optimisation process that ren ders y(x) stationary in eqn (3.20) does not depend upon the relationship between N(x) and n (x). Indeed, using eqns (3.21) and (3.18), the optimum y(x) has exactly the same form as the  optimum N(x) shown in Fig. 3.3 (see Fig. 3.9). Therefore, with the optimum y(x) shown in Fig. 3.9, eqn (3.20) is used to solve for N(x) where n (x)  =  n (AE 8 (x) NAB (x)). ,  July 12, 1995  40  In general, the dependence that n has with respect to N will be too complex to allow for a closed-form analytic solution. In this case a possible solution process is to use an iterative ap proach where y(x) is first solved for using eqns (3.21) and (3.18). A trial function NB (x) for the actual N(x) is constructed by using h 1 and h 2 from y(x), and forcing NB (x) to take the form of Fig. 3.3 (while obeying the original doping boundary conditions). Finally, using eqn (3.20), a new N(x) is solved for using n (iE (x) , NB (x)) and y(x). This process can be repeated until lit tle change is observed in N(x). In the event that the convergence of this iterative method is too slow, then higher-order numerical methods such as Newton-Raphson iteration could be used in stead. Thus, it is a simple matter to include banclgap narrowing into the optimum base profile for the minimisation of tB, for the stationary function y(x) that defines both N(x) and n (x) is in dependent of both these functions.  NAB  (x)  n(x)  , 1 x=W  0.0  0.25  0.50  0.75  1.0  Normalised Position in the Base x (WB) Fig. 3.9. The optimum stationary function y(x) that minimises tB. The break points h 1 and h , as 2 well as the transit time tB are given by eqn (3.18) with U defined in eqn (3.21). N(x) and n (x) are solved for using y(x) in eqn (3.20) along with C = -1. y(x), as shown here, does not depend on the functional form of either n or N, but only on the boundary condition U. The last issue to tackle is the effect of a non-constant D(x) on the optimum profile found thus far. Strictly, to accomplish this minimisation, one must apply the methods of variational cal culus to eqn (3.9) directly; which leads to an O.D.E. that is not soluble in terms of any know tran scendental functions. The effect of a non-uniform D(x) is investigated numerically in [63] for large U, and the result is a solution that has elements of the stationary functions presented in this chapter, but as a whole cannot be construed as the same. However, current day BITs (and HBTs) are such that tB is an important but not dominant part of the total transit time (in the area of 30%). July 12, 1995  41  Therefore, more than a 2-fold reduction in  tB  is really not warranted as the point of diminishing  returns would be surpassed. From the results presented earlier in this section, ‘tB can be reduced 2fold with only a 2.7-fold reduction in N(x) across the base when coupled with a AEg of lkT With N(x) changing by only 2.7-fold, it is reasonable to assert that D(x) is ostensibly constant. However, if larger changes to N(x) are pursued, then the results of this chapter will certainly re duce tB, but only a full numerical optimisation will provide the true minimum [63]. This section has found the optimum base profile for the minimisation of tB when both the doping and the bandgap have been constrained at the emitter- and collector-sides of the neutral base. The optimum base profile has the form of an augmented exponential shown in Fig. 3.9, not the long-established pure exponential [60] that has been mistakenly assumed. Further, the solution presented allows for the simultaneous optimisation of N(x) and n (x), and can also include the effects of bandgap narrowing due to heavy doping. Perhaps the most interesting and startling re sult occurs by using both N(x) and n (x) to generate the aiding field in the base, thereby reduc ing the overall variation in each parameter across the neutral base. Finally, all of the models and methods presented and discussed in this chapter have no particular material system in mind. Therefore, this chapter can be applied to an HBT build in any material system (such as A1GaAs or SiGe).  July 12, 1995  42  CHAPTER 4 Transport Through the EB SCR  July 12, 1995  43  In BJTs it is customary to apply the Shockley boundary condition at both edges to the EB SCR in order to determine the quasi-Fermi levels [67]. The Shockley boundary conditions are based upon the assumption that no matter what physical process is responsible for the movement of charge through the EB SCR, the total transport current T will be very small compared to the forward and reverse directed fluxes at any point within the region. This argument follows exactly the development of Section 2.1. Applying eqn (2.2) under the conditions described in this para graph leads to AEp  0. In fact, the Shocidey boundary conditions simply state that Ep 1 and  are constant across the EB SCR. These boundary conditions allow for an enormous simplification because the exact details of the transport through the EB SCR no longer need to be understood or included in the final model for the device. By their very nature, HBTs can generate spikes (such as the CBS in Fig. 3.1) in the conduc tion and valence bands that reduce the forward directed flux. If one of these spikes is large enough, then T could be constrained by the flux through this one feature alone. Fig. 3.1(b) shows the general band diagram for HBTs built in the AlGai..As material system, where there is an abrupt heterojunction between the emitter and the base. The very nature of the sign of AE, when coupled to the fact that the emitter doping is much smaller that the base doping, produces a fea ture in E called the CB S [25]. The CBS can easily force the electrons to take a path that requires an increase in energy of nearly 240meV. To increase the electron energy by 240meV, with respect to a homojunction, would reduce the available number of electrons, and therefore the forward di rected flux, by four orders of magnitude at room temperature. A reduction by i0 4 in the forward directed flux will most certainly result in current-limited flow in the region containing the CBS. This will invalidate the quasi-equilibrium assumption of the Shockley boundary conditions. Thus, one must consider the limits imposed by the movement of charge through the CBS upon the trans port current within the EB SCR. The thermionic injection of electrons over the top of the CBS is not the only method of transport through the region. Due to the quantum mechanical nature of the electron, and the fact that the width of the CBS is typically of the same order as the de Broglie wavelength, the electron could tunnel through the CBS instead of trying to increase its energy in order to surmount the bar rier. Since a reduction in the required energy to surmount the CBS leads to an exponential in crease in the forward directed flux, tunneling and therefore the quantum mechanical nature of the  Julyl2,1995  44  electron also needs to be considered when deriving the physical models for transport through the CBS. Failure to include this tunneling current will underestimate T by up to two orders of magni tude [25] (see also Fig. 4.8). Therefore, no matter how powerful a model is used (such as Monte Carlo modelling), if tunneling is not accounted for through the CBS, the terminal characteristics of the device will be greatly underestimated. All of the previous chapters have relied on the existence of current-limited flow in one re gion of the device that is separated from both the base and the collector. Specifically, the region providing current-limited flow occurred at the EB heterojunction where the CBS is formed. Since the transport current through the device leads to I, and because current-limited flow at the CBS controls the transport current, then I is governed completely by the transport mechanisms of the CBS. Under the condition of CBS control, I has no dependence on the physical construction of either the base or the collector. By constructing the HBT in a fashion where the CBS controls I, a detailed understanding of the physics surrounding the CBS must be undertaken if one hopes to accurately predict the terminal characteristics of the device. This chapter investigates and derives models for the transport of charge through the region containing the CBS, including effects due to tunneling and a varying effective mass.  4.1 Formulation of Charge Transport at the CBS The transport of charge through the region where the CBS is formed can be found by view ing the system as a set of forward and reverse directed fluxes (Ff and F,. respectively) entering the region from opposite sides (see Fig. 4.1). If there is no source or sink of carriers within the region considered, then just like eqn (2.2) F  =  c(-x) 7 J  —  Fr(Xp), where F is the transport flux, x, is the  thickness of the SCR extending from the heterojunction into the emitter, and x is the thickness of the SCR extending from the heterojunction into the base. If at the points -x,, and x it is acceptable to state that the system is fully thermalised, based upon a local Fermi energy Ef, then the carrier distribution with respect to total energy U is: f(U)  (4.1)  U—,.L’  =  l+e  k1  where f is the Fermi-Dirac distribution function and p. is the electrochemical potential (which is usually termed the Fermi energy Ef). Using eqn (4.1) and the quantum mechanics of crystalline July 12, 1995  45  solids, the transport flux through the region containing the CBS in the x-direction can be written in the standard form [68-70]: F  =  Ff— Fr  ( U) (1 1 j•dk f 3  —  (U) 2 f  ) WU)  =  --  2q  $d3kf 3 ( 2 U) (1— f (U) 1  (2i)  —  ) WU)-j-  (4.2)  Rr  where W £I) and W1 U) are the forward and reverse directed transmission probabilities respec tively, f 2 is the Fermi-Dirac distribution at x, Rf is the 1 is the Fermi-Dirac distribution at -x,, f valid energy range considering forward flux, Rr is the valid energy range considering reverse flux, U is total energy, U is the x-directed energy, and k is three dimensional k-space.  I I  -xn  I I  xp 0 Fig. 4.1. Abstract model of current flux within the region containing the CBS. The EB hetero junction is centred at x = 0, with x being the excursion into the emitter (Region 1), and x, being the excursion into the base (Region 2). There is a flux Ff entering the region at x = -x, and another flux F,. entering from x = xi,. The net transport flux F is equal to Fj— F,. in the absence of any sinks or sources within the region.  The interpretation of eqn (4.2) is straight forward in that: there are 2(2tY 3 electron states per unit volume in k-space (including spin degeneracy); 2 (1-f (in the case of the forward direct 1 f ) ed flux) is the probability of an electron existing in Region 1 and being able to move to an empty state in Region 2; W U) is the probability of the electron moving from -x, to  with a forward  directed energy of 1J; and (1/h)(JUThk) is the group velocity of the electron [15]. As eqn (4.2) stands, the forward and reverse directed transmission probabilities are treated separately using W L1) and W U) respectively. This allows for a non-reversible system to be studied, where electron collisions with the lattice (but not with other electrons) can be included. Strictly, if coffi sions are considered that change the total energy U of the electron, and not simply its direction in k-space, then the vacancy probability 1  —  (U) (in the case of the forward directed flux) will not 2 f  depend on U, but will depend on the exit energy in Region 2. However, if any type of collision is July 12, 1995  46  considered, then W( U) and W U) will be of an extremely complex nature and would require a numerical calculation of eqn (4.2) (this could be accomplished by a Monte-Carlo simulator using non-local mathematics; however, no such simulator exists at this time). As a result, eqn (4.2) is simplified by considering collision-less or ballistic transport throughout the entire region, leading to W U)  = W U) = W(U). With the assumption of ballistic transport throughout the region  from -x, to x, and converting from k to momentum p F  =  FfFr  (= tik), eqn (4.2) yields:  = jd3pfi(U) (1— f (U) )W(U)2 h  —  px  Rf  ?Jd3pf ( 2 U) (1— f (U) 1 1 )W(U). Rr  h  (4.3)  1)X  Examining eqn (4.3) shows that if the regions of integration Rf and R were equal, then the two integrals could be reduced to one integral with an integrand of (f 1  —  )W(aUIa). One could 2 f  then identify an Ff and F,. from this integrand (which strictly speaking is not the same as that de fined in eqns (4.2) and (4.3), but for all practical situations is identical), giving: Ff $dp f (U)W(U)L 1 h  (4.4)  Rf  and Frij’d3Pf ( 2 U)W(Ux)aP_ h Rr  (4.5)  The key to the definitions of eqns (4.4) and (4.5) is the equivalence of Rf and Rr The fact that this is indeed true is proven later on in Section 4.3 once the effects of a non-uniform effective mass have been brought into the picture. The solution of Ff and F defined in eqns (4.4) and (4.5) begins by determining the transmis sion probability W(U). Strictly, W(U) must be calculated by solving the Schrodinger equation, based upon the potential profile encountered within the EB SCR. The solution of the Schrodinger equation, even for a potential obtained from the depletion approximation, is complex enough to require a numerical solution. Failure to obtain an analytic form for W(U) would hide the rich in terplay that exists between the final transport model for the CBS and the physical attributes such as doping concentration, temperature, effective mass, electron affinity, and bias conditions. An ap proximate but analytic form is thus sought for the solution of W(U). To this end, one could ap  July 12, 1995  47  peal to the asymptotic formalisms in the complex plane used by Landau and Lifshitz [71], or to the JWKB method [72], to obtain:  W(U)  =  exp [e {Pdx}]  =  exp [e {_2  f.Jv(x)  —  (4.6)  Udx}]  where V(x) is the potential profile of the CBS, and only the real part of the exponent in eqn (4.6) is retained (i.e., U, <V(x)), such that particles with energies larger than the potential energy move without any quantum mechanical reflection. Eqn (4.6) presents a simple analytic solution for W(U), where the particle mass m is in general not equal to the electron mass me, but to the more general effective mass m* that is characteristic of semiconductors. W(U) is solved for using eqn (4.6) and a V(x) obtained from the depletion approximation. Fig. 4.2 shows the CBS, which is an enlargement of Fig. 3.1(b). Since the depletion approxima tion results in a parabolic form for V(x), then one can write: V(x)  VPk(1+)  (4.7)  for-xxO,  where Vpk is the peak energy of the CBS, and the reference energy is at the bottom of the conduc tion band where x = -x,. Eqn (4.7) is appropriate to the case where the heterojunction and the met allurgical junction are coincident. The domain 0  x  x, will be considered separately so that  W(U) may be separated into two functions; one for Region 1 (WCBS(Ux)) and another for Region 2 (WN(Ux), where N stands for Notch), leading to: W(U) = WCBS(UX)WN(UX). Using eqns (4.6)-(4.8) with WN(Ux) WCBS(UX)  =  WcBs(U Vk)  =  =  (4.8)  1 produces:  ex[  xJ2mVk P  (in  where U is normalised energy in terms of Vpk (i.e., U  ( =  Jl—U’+l —  AJi  —  u  A[  J],  (4.9)  Ux/Vk). Eqn (4.9) forms the basic ker  nel for the transmission probability, and it is written in a most general form where Vpk and x have not yet been defined in terms of the material parameters and applied bias for the EB SCR. With W(U) solved for using eqn (4.8) and (4.9) (WN(Ux) will be solved for when the re gions of integration Rj and R,. are determined), Ff and F,. can be obtained once the energy disper sion relationship U(p) has been set out. The following section will determine U(p) and include the effects of a non-uniform effective mass m* that generally occurs at an abrupt heterojunction. July 12, 1995  48  Once U(p) has been determined, the regions of integration Rf and Rr are set out so that Ff and Fr can be solved for using eqns (4.4) and (4.5) in the next section.  E(eV)t  2 xp  -xn  Fig. 4.2. Blow-up of the CBS from Fig. 3.1(b), showing the various energies and their reference.  4.2 Incorporation of Effective Mass Changes In general, the two materials that form the abrupt heterojunction shown in Fig. 4.2 are char acterised by a different effective mass m*. This change in m* can either enhance or diminish the flux F in transit through the CBS when compared to the case where m* is uniform throughout the region. Failure to account for the change in m* can result in a significant error. Worse yet, this er ror is not simply a multiplicative constant as is stated by Grinberg [51], but has a dependence on the applied bias. Therefore, in solving for Ff and F using eqns (4.4) and (4.5), the dispersion rela tionship U(p) needs to be determined in concert with the effects of a non-uniform m*. Concentrating on eqn (4.4) for Fj(the exact same results will apply to eqn (4.5) for Fr), it is realised that the integration is being performed over p-space. As the entire integrand is dependent upon total energy U and x-directed energy U,, it would be beneficial to cause a change of vari ables in the domain of integration from p to U. To this end, the dispersion relationship will be tak en as parabolic, but left as a diagonal mass tensor to yield: U (p)  =  U(p) + U (p) 1  =  + +  where mx,  £J’  (4.10)  and m are the effective masses for particles that have momenta of Px py, and July 12, 1995  re 49  spectively. As before, U is the x-directed energy, and now, U is the transverse directed energy. It is also important to realise that eqn (4.10) implicitly places the energy reference at the band ex trema. A further simplification can be achieved by a change from cartesian momentum coordi nates to cylindrical momentum coordinates. Since we are considering devices that behave essentially as one-dimensional, symmetry dictates that the azimuth direction in the cylindrical system be chosen parallel to the x axis (see Fig. 4.3). This yields: p,  =  pcosO  and  P  =  psin€),  (4.11)  where eqn (4.10) has that: 2  U  2  and  2m  2  py Pz U.L=—+---— 2m 2m  (4.12)  Eqns (4.lO)-(4.12) together allow for the solution of Ff The only approximation being made is that UQ,) can be adequately described within the parabolic approximation. However, the full mass tensor has been retained (albeit in diagonal form) so that anisotropic materials such as Si, SiGe and strained semiconductors can be modelled with the results to follow in this chapter.  Pz  py  Px  Fig. 4.3. Diagram showing the definitions of the cylindrical momentum space coordinates. At present, the non-uniformity of m* has not been included, but it has also not been preclud ed. Setting aside the issues of a spatially varying m* for the moment, the integration over p is transformed to U by the Jacobian: ap Ux ae J(Px’PY’Pz IUX,O,UL  —  —  (4.13)  au ao au ap ‘‘2 ap Ux ae July 12, 1995  50  The solution of the Jacobian in eqn (4.13) rests on the definitions in eqns (4.11) and (4.12). Look ing at eqn (4.12) for the definition of U shows a dependence upon the canonical coordinate Px alone. From this realisation it immediately follows that: =  and  0  =  0.  (4.14)  Furthermore, eqn (4.12) also produces: 3p  au  =  m —f.  (4.15)  So far, eqns (4.14) and (4.15) have quickly solved for the first row of the Jacobian in eqn (4.13). Moving on to the second row and looking once again to eqn (4.12), but this time taking the definition for U± and performing partial implicit differentiation with respect to Uj, gives: p, m U±  —  m  —  mz  —  mpap mz Py U±  y  Then using eqn (4.11), which can be condensed and rewritten as  =  e, produces after 2 p tan  implicit differentiation with respect to U±: —  pyt an  (417)  Finally, substituting eqn (4.17) into (4.16), yields: e 2 mmcos 1 )‘ Z Py 2 3) m € 2 7 mcos E ) + sin  37  aU Pressing on and using p  =  (4.18)  e, but this time performing implicit differentiation with re 2 p tan  spect to (3, gives after some algebraic manipulation:  37 ap  P cos9  =  ap [cose Py sin O 2  1 sine] P  —  =  ap 1 I [cose sine  p —  sine  (4.19)  Then, returning back to eqn (4.12) for Uj and performing implicit differentiation with respect to Oyields: 0=  ae  -  mEi9  mEi(3  =  (420)  ----  37 m pae  Finally, substituting eqn (4.20) into (4.19), and using eqn (4.11) where p,  =  pcosO/sin(3, pro  duces: —  37 m  (421)  — —  9 2 mcos ( 2 3 + msin July 12, 1995  51  _________________  The second row of the Jacobian is then finished off by realising that U, as given in eqn (4.12), has no dependence upon U, which immediately produces: =  (4.22)  0.  Eqns (4.18), (4.21) and (4.22) provide the solution for the second row of the Jacobian in eqn (4.13). Moving on to the third row of the Jacobian, and substituting eqn (4.18) into (4.17) yields: e 2 mmsin Z  i  Z  )‘  (4.23)  •  O 2 P 2 mcos + msin 9 Then, substituting eqn (4.21) into (4.20) produces: —=p  mz  (4.24)  .  ‘  e 2 € + msin 2 mcos  Finally, using exactly the same logic that lead to eqn (4.22), gives: PZ  =  0.  (4.25)  The Jacobian in eqn (4.13) is solved for by using eqns (4.14), (4.15), (4.18), (4.21)-(4.25) to yield:  m  0  0  px (P,  mmcos 9 2 /p  Py Pz  ! U, € U±)  =  0  (4 26)  9+mysin 2 mzcos 9 2 9+msin mcos G 0  mmsin e 2 /p  pm  €) 2 €) 2 €) + msin 2 mcos €) + msin 2 mcos  Given the sparse nature of the matrix in eqn (4.26), the solution of the determinant quickly yields: P U, 9, U)  (PX  =  X  mm  (4 27)  Px E €)+msin 2 mcos )  Eqn (4.27) is the Jacobian that allows the integral definitions in eqns (4.4) and (4.5) to be trans formed from p to U. As will be seen shortly, this greatly facilitates the development of the models for Fjand Fr Maintaining the focus upon eqn (4.4), as set out at the start of this section, and using eqns (4.27) and (4.10) to transform fromp to U, yields: July 12, 1995  52  F  —  2q  hJR  =  mm Z i’m 2 Jfi(Ux+Ui..)W(Ux)_:i _j•dOdUdUji_ m 1 Pmcos O+msm2 9 h R 2  =  f u L,u,e,u±) ( ) (  —J h  d9dUdU 1 R  mm 2 2 mcos O+msin 0  fl(Ux+U±)WcBs(UX)WN(UX)  (4.28)  where R 1 = Rj to reflect that F 1 originates at -x, within Region 1. Eqn (4.28) is the full model for transport through the CBS. However as was stated previously, the effect of a non-uniform m* has not been included. It is instructive to pause at this point and determine, under simpler conditions, WN(Ux) and thus the region of integration R 1 before moving on to include the effect of a spatially varying m*. With m,  and m as constants throughout the system, there is no coupling between 0,  Uj, and U, so that all canonical coordinates can be considered independently of each other (this is not the case when m* is non-constant). Re-examining Fig. 4.2 shows that in the region 0< x x, the potential profile that generates WN(U) is of a strictly monotonically increasing nature (unlike the CBS within the domain -x,  x  0, which contains AE). Since we are considering a system in  which their are no collisions that could either raise or lower the particle’s total energy, the particle must emerge from the EB SCR with sufficient energy to enter into the neutral base with an energy that is above E; else one would be admitting particle transport within the forbidden bandgap. This fact allows for a considerable simplification to the definition of WN(UX); namely: WN(U)  =  (1  if  k.0  if  I  U>V X b  (4.29)  —  UX<Vb  Although strictly speaking eqn (4.29) is not the full form for WN(Ux), it captures the ultimate re sult since any particle that enters the neutral base within the forbidden bandgap (i.e., U,  <  Vb) will  within short order be attenuated to the point where it no longer carries any current. Therefore, since we are only interested in calculating the transport current, the exact form for WN(Ux) is ir relevant, and eqn (4.29) suffices as it captures the essential feature of WN(Ux). With WN( U) defined in eqn (4.29), that last task to accomplish before Ff can be solved for by eqn (4.28) is to determineR . Re-examining Fig. 4.1, it is obvious that for a particle to enter the 1 July 12, 1995  53  EB SCR at x = -x, and contribute to Ffi it must possess a positive x-directed momentum. With the energy reference shown in Fig. 4.2, a positive x-directed momentum translates into Px thennore, examination of eqn (4.12) shows thatp  0 translates into U  0. Fur  0 (This is for the case  where mx > 0 and therefore applies to electrons. To consider holes, it is best to use a negative hole energy instead of a negative hole mass so that all of the results in this chapter may be applied di rectly.). If the requirement that U  1 U  then together with U  1 will be as shown in Fig. 4.4. Vb from eqn (4.29), then R  0, and U  +  E be imposed (where E is the bandwidth for U(p)),  U  (a)  (b)  UL  /Ux+ UE  Ux  0  Ux 0 E E Vb Fig. 4.4. Domain of integration R 1 for a uniform m*. (a): case where the applied bias is such that Vb 0; (b): case where the applied bias is such that Vj, 0. Note: Fig. 4.2 defines Vb. 0  Ff can now be solved for by using eqns (4.28), (4.29), (4.9), (4.1), and the region of integra tion R 1 as shown in Fig. 4.4. Since R 1 takes into account WN(Ux), then solving eqn (4.28) yields: 2it  Ff  2 =  h  —  J 0  E-U  E  mm 2 2 mcose+msmO Z .  f  dU WCBS (U)  max(Vb,O)  f  1f dU (U+ U 1 ). 1  (4.30)  0  Examination of eqn (4.30) reveals that the integral over E) has no dependence on the results of the second and third integrals. This allows the 0 integral to be performed independently to yield: It  2it  I 0  it/2  dG  mm mZ 2 O+msin cos G  mm  =I m 2 O+msin cos G 0  Z  =  (  Stan 4Jmmtan’ qm  July 12, 1995  54  The above equation is evaluated by letting  e approach itI2 from the left, giving:  2it  I d9 0  mm Z ‘  =  9+m sin 2 m Z cos e 2 3’  2itjmy m. z  (4.31)  Eqn (4.31) solves for the anisotropic effective mass tensor and is evaluated in such a manner than all branch points of the inverse tangent are respected. Therefore, as long as one can assert that the second and third integrals of eqn (4.30) are indeed independent of (3, then one can substitute eqn (4.31) into (4.30) to obtain: Ff=  4tqJm m  3 h  E-U  E  ZJdUW(U)fdUf(U+U)  (4.32)  .  0  max(V,0)  Eqn (4.32), with the region of integration R 1 as shown in Fig. 4.4, gives us a flavour for the transport current through the CBS. The interpretation of eqn (4.32) yields: a thermalised ensemble of electrons at x = -x, (characterised by the distribution f 1 with an electrochemical potential  t  of  ) is injected to the right, towards the CBS; each electron within the ensemble is characterised 1 E, by a forward-directed energy U and a transverse directed energy U± which is random but evenly distributed in all directions; every electron then passes through the CBS with a probability of transmission given by WCBS which is dependent upon U, alone; the transverse directed portion of the electron’s energy leads to a contribution given by the geometric mean of the two transverse ef fective masses; finally, only electrons that can enter the neutral base outside of the forbidden bandgap (i.e., U  Vb), and are within the bandwidth E of the conduction band, are allowed to  contribute to the transport current. Eqn (4.32) solves for Fjunder the condition that the effective mass tensor is a constant throughout the CBS. Returning back to eqn (4.28), the main thrust of this section is continued; namely the incor poration of a spatially varying m* into the transport current. The inclusion of a non-constant m* requires that the electron energy U ( U 1 U  =  U+U  + L1)  and  be generalised to: 2 U  (4.33)  =  where energies with a subscript of 1 refer to transit within Region 1 (i.e., -x, gies with a subscript of 2 refer to transit within Region 2 (i.e., 0 <x  x  0), while ener  xv). The reason for the gen  eralisation that leads to eqn (4.33) is that the spatial change in the effective mass tensor results in a mixing of the x-directed and transverse directed energies. Therefore, one cannot maintain a toJuly 12, 1995  55  ____  tally separate view of U, and U . Now, the energy reference continues to be located at E(x=-x), 1 so that using eqn (4.12) produces: =  and  1  1 U  =  2m  1  (4.34)  + 1  mz 2  1  while 2  u  X,  —  2 —  m 2  2  Vb  and  UJ  —  2 —  2 ‘2  2 2m  2  Pz,2 mz 2 2  435  It is important to understand the exact meaning of eqns (4.33)-(4.35). To begin with, the en ergies U, U, and U± represent total energies within their respective regions. Band diagrams such as those shown in Fig. 4.2 do not show the total energy U, but instead show only U. In the event that the system possesses transverse symmetry, then the potential energy is V(x,y,z)  V(x). When  there is transverse symmetry, it is possible cast the full three dimensional problem into two decou pled one dimensional problems whose solution only depends upon U or U 1 respectively. For this reason, U, 2 is not simply given by the kinetic energy term containing Px,2’ it must also include the offset potential energy of Vb. Thus, eqn (4.35) gives the total energy U 2 located at x  =  xi,,  while eqn (4.34) gives the total energy U 1 located at x = -x. The reason for defining the energies at -x, and  Xp  being that Ff is based upon particles injected to the right from x  based upon particles injected to the left from x V(x,y,z)  =  =  -x,j, while Fr is  x. Furthermore, because the potential energy  V(x) does not vary in the transverse direction, Uj and U., 2 do not contain an offset  potential energy term. The cumbersome nature of the energy relations given by eqns (4.34) and (4.35) arise from the quantum mechanical nature of the problem. Looking back to eqn (4.3) shows the flux being calculated by an integration over p-space. Strictly speaking, quantum mechanics does not allow one to consider momentum and position simultaneously. Eqn (4.3) must be interpreted with care, because FfiS based upon a distribution in p-space located at x = -x,, while F is based upon a dis tribution in p-space located at x = xi,. Essentially, due to the slow variation of V(x) over the atomic dimensions, it is possible to cast the problem into quasi-classical form [15] where one can speak of distinct p-space distributions at largely separated positions in real space. Finally, because we transform p into U, the same concerns for p-space apply to U-space as well.  July 12, 1995  56  _____  Due to the translational invariance of the potential V(x,y,z) along the transverse direction, the transverse momentum pj commutes with the Hamiltonian of the system; leading to the con servation of p. Therefore, at the heterojunction separating Region 1 from Region 2 (i.e., at x P±, 1  P±,2 (where eqn (4.11) has p 1 py,  =  +p ) so that: 1  =  and  p,  Py,2  Since the potential energy V(x,y,z)  0),  Pz,i  Pz,2  =  =  (4.36)  Pz  ( V(x)) does not vary in the transverse direction, then P±,1  P±,2 cannot vary with x if collisions are prohibited. Using eqn (4.36) in eqns (4.34) and (4.35)  shows that Uj 1 and U±, 2 must remain constants of the motion. Therefore, eqn (4.36) must hold equally well at any x within Regions 1 and 2, and more specifically at -x and x where eqns (4.33)-(4.35) are defined.  Using eqn (4.36) in eqns (4.34) and (4.35) leads to: 2  1 UJ =  2  2  and  my 1 + 2 2 m 1  2 UJ  2  my 2 + 2 2 mz 2  =  Applying eqn (4.11) to the above yields, after a little algebraic manipulation:  1 U±, U±, 2  =  m “ 2mz2  my lmz  ‘  R(9)  where  R(G)  =  cos 1 G +m 2 sin 1 e m Z, 2  (4.37)  sin 9 cos + my, 2 9 mz 2  1  Examination of eqn (4.37) shows the necessary condition that if m, 1  =  m,i  =  =  mZ,2, then  Uj 2 IU, = 1. Eqn (4.37) represents the change in the transverse energy that must occur to con 1 serve pj in the face of a spatially varying effective mass tensor. It is instructive at this point to re veal the full implications of eqn (4.37) upon the total energy within the system. Fig. 4.5 shows the effect of eqn (4.37) when m 1  =  m,i  =  , and m, 1 m 2  =  mZ,2  =  . When m 2 m 1 <m , then 2  As will be described in the next paragraph, total energy must be conserved throughout Re gions 1 and 2. Thus, when m 1 <m , the positive difference Uj 2 1  —  Uj is transferred into U, 2  which leads to an enhancement in the forward directed flux. Conversely, when m 1 <  Uj. Thus, when m 1  >  , the negative difference U±,i 2 m  —  >  , then Ujj 2 m  2 is removed from U, U, 2 which  leads to an diminution in the forward directed flux. Since eqn (4.3) is based upon a collision-less system within Regions 1 and 2, then the total energy must be conserved at the heterojunction separating Region 1 from Region 2 (i.e., x  =  0).  Thus, 1  2  Furthermore, since there are no collisions within the two regions, the above conservation requireJuly 12, 1995  57  ment applies equally well at all x within Regions 1 and 2, and more specifically at -x and x, where eqns (4.33)-(4.35) are defined. Using the above equation in eqn (4.33) produces: (4.38) Eqns (4.38) and (4.36) are the conservation requirements imposed at the abrupt heterojunction separating Region 1 from Region 2. It is important to remember that most of the proceeding argu ments are based upon the conservation of p. This conservation can only be asserted if the Hamil tonian of the entire system has translational symmetry along the transverse spatial dimension, if the heterojunction contains a corrugation or surface roughness, then one could not assert thatp is conserved. This would lead to a considerable increase in the complexity of the model that would necessarily require a detailed view of the device at the atomic level.  (a)  I  I  (b)  Ix  I  I  I  7 -x 0 o Fig. 4.5. The effect that conservation of p has upon U, 1 and U±, 2 when a mass boundary is placed at x = 0. Using m, 1 = m,i = m 1 and m, 2 = m,2 = m 2 in eqn (4.37), then 2 1 2 IU±, m 1 U±, . (a): when m 1 m 1 <m , energy is removed from Uj 2 1 and transferred to U, 2 when moving from the left to the right; (b): when m 1 >m , energy is removed from U, 2 2 and transferred to Uj when moving from the left to the right. -xn  With eqns (4.38), (4.37), (4.35) and (4.34), the effect of a spatially varying effective mass tensor can be completed. The abrupt change to the effective masstensor, as described in Fig. 4.5, results in a mixing of U, 1 and  with U, 2 and UJ,2 when passing through the mass barrier  (i.e., heterojunction) at x = 0. This mixing, along with the assumption that there are no collisions, results in a one-to-one mapping between energy state (U,i, U±,i) in Region 1 and energy state ) in Region 2. This mapping is solved for by substituting eqn (4,37) into (4.38), giving: 2 , U, 2 (U, July 12, 1995  58  2 U  = 1 +(e)U U ,  (4.39)  1 Ux  2 + y’(€)) U, , 2 = U  (4.40)  and and U  1 + U, 1 = U 2+ U 2 , 1 = U  (4.41)  where y(9)  = 1  —  m“ 1 m ‘  my 2 mz  R(9)  and  y’(9)  = 1  —  2 m Z “ my, 1 mg,  2  1  R(€))  =  .  (4.42)  —  Finally, using this simplified form based on the function y (the notation for y was initially set forth by Christov [70,73], but has been extended here to include anisotropic effects), eqn (4.37) becomes: ii FT  ‘  J,2  1  (4.43)  = 1‘i’ =  where the explicit dependence upon 9 has been dropped for simplification. Eqn (4.41) simply as serts the fact that a collision-less system is being considered, while eqns (4.39) and (4.40) repre sent the energy mapping that occurs when crossing the heterojunction at x  = 0 from the left or  from the right respectively. Returning back to eqn (4.28) for the calculation of Ffi the integral is being performed over U-space located at x  = -x, with a domain of integration R . Using the formalisms for passing 1  through the heterojunction that were developed in eqns (4.39)-(4.43), it is important to realise that the transmission probability W(U), as defined in eqn (4.8), must be extended to: W(U)  = WcBs(UXl)WN(UX ), 2  (4.44)  for WCBS is defined in Region 1 and thus depends upon U, , while WN is defined in Region 2 1 and thus depends upon U, . However, any function that depends upon total energy U (such as the 2  Fermi-Dirac distribution function f(U)) remains unaffected by the mass barrier due to the conser vation of total energy set out in eqn (4.41). Therefore, eqn (4.29) for WN is rewritten as: WN(UX, 2)  11  if  Ux, 2 >Vb  ‘0  if  <V,, 2 U  (445)  —  , which is used for p- or U-space integrations performed at x = 1 The domain of integration R -x,, will be modified from what is shown in Fig. 4.4 by the non-uniform effective mass tensor.  One still requires that for a particle to enter the EB SCR at x  possess Px,1  0; or in terms of energy, U,i  =  -x, and contribute to Ffi it must  0. And, the requirement that U (=  +  ) , 1 U  E  (where E is the bandwidth for U(p)) is still maintained. However, eqn (4.45) imposes the condi July 12, 1995  59  tion that U, 2  1 and U±,i when eqn (4.39) is used to Vb, which results in a coupling between U,  map from Region 2 into Region 1. Therefore, using the three boundary conditions set out in this paragraph, along with eqn (4.39), yields the following boundary for R : 1 1  0,  +U U E 1 , , UX,l+YUI,l  (4.46)  Vb.  1 (which is applicable to an integration carried out at x = -x) into It is also possible to transform R 2 (which is applicable to an integration carried out at x R  =  x) by substituting eqns (4.40) and  (4.41) into (4.46) to produce the following boundary for R : 2 2 + yU U 2 , 1  0,  2+ U U, 2 , 1  B,  (447)  Vb. 2 UX  When the effective mass tensor is uniform, then eqns (4.42) and (4.37) produce y =  =0. Un  der these uniform conditions, then indeed eqn (4.46) produces the R 1 as shown in Fig. 4.4. However, when  y’  the range  —0o  0, R 1 becomes distorted from that shown in Fig. 44.1 and y’ can take on any value in (y, y’)  1. As was discussed in the examination of eqn (4.37) that lead to Fig. 4.5,  two distinctly different domains occur for  firstly, when m 1 <m 2 where 0 <‘y  1 (and —00  <  ‘y’  <  0), and energy is transferred from U±,i into U 2 which leads to an enhancement in the forward di rected flux; secondly, when m 1 Ux,2 and transferred into  >  2 where —oo <‘y <0 (and 0< y’ m  1), and energy is removed from  which leads to a reduction in the forward directed flux. Fig. 4.6 shows  1 and R R 2 for the case where y> 0, while Fig. 4.7 shows R 2 for the case where y <0. Exami 1 and R nation of Fig. 4.6 shows a focussing of R 2 towards the direction of charge flow. This is due to the en ergy transfer into 11 x,2 when passing through the heterojunction, leading to what is termed current enhancement. Conversely, examination of Fig. 4.7 shows a reflection in R 1 against the direction of charge flow. This is due to the energy removal from 2 U,, past the heterojunction, leading to what is termed current reflection. The current reflection occurs because ultimately, no carrier may enter the base within the forbidden bandgap (i.e., U, 2 < Vb). As a result of Figs. 4.6 and 4.7, care must be ex ercised in applying the integration boundary R 1 (or R ) to the solution of Ff in eqn (4.28). 2 July 12, 1995  60  (a) 2 U, E +  1 =E U, U Vb-UX  (UL1  2  —yp  I  y  0  0 0  0  ‘Vb  yE  (b)  ,  1 U,  +  Uj  =  E \, Ux2+UJ.2=E 7  E  = 2 U±  E  2 U,;,  —y,  1 —y’  2 R \uX2  o  yE  E  Fig. 4.6. Enhancement case where m 1 <rn 2 (i.e., y> 0 and y’ <0). Domains of integration R 1 and R 2 from eqns (4.46) and (4.47) for the calculation of Ff at x = -x, and x = Xp respectively: (a) the applied bias is such that Vj, 0; (b) the applied bias is such that Vb 0. Each domain of inte gration represents the ensemble of particles that contribute to 2 how the transfer of Notice in R energy from Uji into U, , due to the increasing m* in the direction of charge flow, leads to a fo 2 cussing of the particles towards the direction of charge flow.  July 12, 1995  61  _____  (a) 2 U, E  E  1 U,  +  U  +  =  E  1 =E U, E —  1 Vb—UX  Y  2 R 2 Ux  0 Vb  E  Vb  Vb-YE  1 —Y  (b)  *  2 U, E-Vb  UJ-,1 E 2 UJ_,  ZUx,1÷U±,1=E E- Vb  U 2  —Y’ —Y  1 Vb—UX  1 —Y  =  Y  +U±, U, = 2 E  R2  1 Ux  2 Ux  0 I  Vb-YE  E  ...........................................................  Vb  E  0  1—y Fig. 4.7. Reflection case where m 1>m 2 (i.e., y < 0 and y’ > 0). Domains of integration R 1 and 2 from eqns (4.46) and (4.47) for the calculation of Ff at x = -x, and x = x,, respectively: (a) the R applied bias is such that V, 0; (b) the applied bias is such that Vb 0. Each domain of integration represents the ensemble of particles that contribute to Fp Notice in R 1 how the removal of energy from U,, 2 into U±, , due to the decreasing m* in the direction of charge flow, leads to a reflection 2 of the particles against the direction of charge flow. The reflection occurs because of the necessity for particles to enter the base outside of the forbidden bandgap (i.e., U, 2 Vb, orpX2 0). July 12, 1995  62  Before eqn (4.28) is recast to include changes to m* (by including R 1 from Figs. 4.6 and 4.7) it is instructive to calculate the Jacobian that transforms integrations performed within Region 1 into those performed within Region 2. In other words, we wish to determine:  1 au au 1 1  ,  2 9, U  Uj  (U  ) 2 u  1’  L, 1  2 U  —  —  2)  —  U  2  u1 , 1 1 22 au au±,  —  Using eqns (4.40) and (4.43) produces:  1 u  = —  ‘\x2J2J  0  2 2 m m R(9) m m1  m m 2 2 “ R(O) my lmz, 1 ‘  where R(9) is defined in eqn (4.37). Using eqn (4.37) yields, after substitution into the above: U , (Ui9 1 U 2’ U±, 2) ,  =  (Ui 1 U± N 2’ U 2)  =  my,2mz2  9+mylsin mzlcos O 2  my 1 m 1 m  cos + my, 2 O 2 sin O  (448)  Finally, by combining the above Jacobian for a change in variables from Region 1 to Region 2 with the Jacobian given by eqn (4.27) for a change in variables from p to U (which in this case is subscripted to reflect calculations within Region 1), gives: ” 1 U, —  1\ 1 ,9,U±, 2 U ) ,u±, u )  (P,i’P,i’P,i” 3 1  ,e,u± u ) 2  —  1 m  my,2mz2  (449)  c Px,lmz G s 2 9 in os +my  Examination of eqn (4.49) shows it to be almost identical to the Region 1 Jacobian in eqn (4.27), but with subscripts denoting Region 2 instead of Region 1. This is to be expected because the energy versus momentum relations in Regions 1 and 2 (eqns (4.34) and (4.35) respectively) differ only by a constant of Vb, which will not result in a deformation of the differential volume element. However, the term mi/p,i and not mx,2Ipx,2 remains in eqn (4.49). The reason for this discrepancy from perfect symmetry lies in the fact that  is an ensemble of particles originating  at x = -x,. As such, it is the particle velocity at the point of origin that will dictate the current flux. Once the ensemble population is cast in phase space, then by Liouville ‘s theorem [74], the flux is conserved at all other points in phase space and must equal the current at the point of origin. Therefore, the tenn 3UIEp in eqn (4.4) for Fjremains 1 U/ap, (Px,1’mx,1) and not July 12, 1995  63  The final transport model for Fp including the effects of a non-uniform m*, is presented. For 2 andy> 0), then using eqn (4.28) with calculations based at x the enhancement case (i.e., m 1 <m  1 defined in Fig. 4.6, produces that: = -x, and R 2it  Ff=  2  fde 0  E—  E  1 m  (U± f , dU + ) U $dUlWcBsx,l)J 1  cos 1 O 2 +m 2 sin 1 e  0  max(Vb,0)  max(Vb,O)  +  (4.50)  u,,,  f  1 E—U  dU  1 WCBS (U, 1)  0  J  fu ( dU 1 11  +  1 U  1)  1 Vb—UX  The term WN(Ux,2) is equal to 1 within the domain R 1 and has been removed for clarity. Howev er, if WN does not have this simple form, then the full WN(Ux,2)  ) must remain 11 = WN(Ux,1 + yU  in eqn (4.50), where the coupling of the canonical variables forces it to remain nested within the third integral over U, . If this is the case, it may be beneficial to calculate Ff at x = x. Using R 1 2 as defined in Fig. 4.6, along with eqns (4.48) and (4.28), produces: 2it  Ff  2’  h  —  dU J d2 cos€)+m m s 1nE) J 2  o  .  2  2 WN (Ui, 2)  (u 1 f ,2 1 J dU  2  1 U  2)  WCBS (U,  1)  0  ‘yE  ‘  (4.51)  2 E—U,  E  2 m  2 Ux,  +  f  2 WN (U dU  2)  )  0  max(Vb,O)  In this case WN  1,2 f (U 1 fd U 2+ U 1, 2) WCBS (U,  ) has been left in to show its general inclusion for the 2 (= 1 within the domain R  calculation of Fp Eqn (4.51) is useful in applications where WN does not have a simple form. However, WCBS remains nested within the third integral over U 2 and cannot be easily removed , 1 due to its dependence upon U , which by way of eqn (4.40) is equal to U, 1 2  +  . 2 , 1 y’ U  It should be noted that all of the fluxes considered within this chapter are electron fluxes. Thus, to calculate conventional current densities from these fluxes (such as F , one must multiply 1 by “-1”. Finally, for the reflection case (i.e., m 1  >  m and y < 0), then using eqn (4.28) with calcula  tions based at x = -x and R 1 defined in Fig. 4.7, produces: July 12, 1995  64  r  2it  2  Ff  _  SdOm  1 m cos2 +1 1  sin 1 9 2  0  f  (4.52)  1 E—U,  E  I  1 WCBS(UX dU  [m ax(Vb,O)  dU f (U + U ) 1 )j 1 0  Vb yE 1—y -  f  —  1 E—U  dU WCBS (Ui,  max(Vb,O)  )  f  (U 1 dU 1f 1+U ) 1  Vb—UX,l  As was done with the enhancement case, the term WN(Ux,2)  ) has been 1 (= 1 within the domain R  removed for clarity. However, as is true for the enhancement case, if WN does not have this simple form, then the full WN(Ux,2)  = WN(Ux,1 + YU±,i) must remain in eqn (4.52), where the coupling  of the canonical variables forces it to stay nested within the third integral over Uj.i. If this is the case, it may be beneficial to calculate Fj at x = x. Using R 2 as defined in Fig. 4.7 along with eqns (4.48) and (4.28), produces: 2t  Ff  =  i!Id0m o  Z  cos G 2 +rn 2 sin e  W $dUx ) f + 2 l(UX jdU N(Ux 1 )WcBs(UX 2 U± ) 0  Jnax(Vb,0)  2 E-U,  0  +  f  (4.53)  2 E—U,  E  2 m  2  2 WN (U, 2) dU  min(Vb,O)  $  ( U, 1 dU 12 f 2 + U 2) WCBS (U  )  —r Again, as with the enhancement case, eqn (4.53) simplifies the problem of calculations involving a complex WN, but at the expense of making calculations of WCBS far more complex. Basically, if WN has a simple form then use either eqn (4.50) or (4.52) for the calculation of Ff under enhance ment or reflection respectively. On the other hand, if WCBS has a simple form then use either eqn (4.51) or (4.53) for the calculation of Ff under enhancement or reflection respectively. Finally, if both WN and WCBS have a complex form then little can be done to reduce the complexity of the problem. Eqns (4.50)-(4.53) present a rigorous model, that includes the effect of quantum mechanical tunneling, for the calculation of the forward flux entering a two region system with an abrupt mass- and hetero-junction in-between. These equations solve, for the first time, the transport cur rent within a complex region while allowing for an anisotropic media. As such, these equations July 12, 1995  65  represent a significant progression from the models derived by Stratton, Padovani, Christov, Crowell and Rideout [69,70,73,75-78]. The models presented here allow for all of the features found within HBT structures which were not accounted for by the aforementioned authors in their study of Schottky diodes. Furthermore, the models presented here overcome the problem encoun tered by Perlman and Feucht [79], who solved the same system but neglected tunneling. Due to the neglect of tunneling, the models in [79] have an un-physical discontinuous change when the mass boundary is placed coincidently with the potential boundary. It is important to be able to model transport through complex regions like the CBS, for in modem abrupt HBT structures this transport current is often what defines the ultimate terminal characteristics of the device. Finally, the models presented in this section have no bias toward, or any specific requirement on, any one material system. Therefore, the results of this section can be applied equally well to any material system. In concluding this section it is important to mention some cautionary comments and shed some physical insight into eqns (4.50)-(4.53). First of all, examination of eqns (4.50) and (4.52) shows the first double integral over U, 1 and Uj,i to be identical in both equations and also equal to eqn (4.30) which is for a constant m*. For this reason, this double integral is termed the standard forward flux  as this is the standard flux that would flow in the absence of the  mass barrier. The last double integral in eqn (4.50) represents an additional flux that would normally have entered the base within the forbidden bandgap, but due to the mass boundary transferring energy from Uji into U, , it is raised up into E within the base to contribute to the 2 total Ff As such, this current is termed the enhancement forward flux 1 enhance Finally, the last double integral in eqn (4.52) represents a flux that would normally have entered the base within , it is lowered into the forbidden 2 E, but due to the mass boundary removing energy from U bandgap within the base and is lost from the total Fp As such this current is termed the reflected forward flux Ff reflectS It is also important to remember that when solving eqns (4.50)-(4.53), y and y’ have a dependence upon the  e in general. Therefore, unlike eqn (4.30) (and thus FfstjJ) where  e integration can be treated as an independent multiplier to yield eqn (4.31), the calculation of  Fjepice and 1 reflect will have y(9) and y’ () nested within the integrand, making for a potentially stiff problem to solve due to the complex nature of the 0 integral.  July 12, 1995  66  4.3 Calculation of Fr and a Unified Model for F The total transport flux F is equal to Ff— F,, as is given by eqn (4.2). The models of the previ ous section, given in eqns (4.50)-(4.53), concentrate on the calculation of Ff The reason for main taining a focus upon Fj while neglecting F,, is that the two the fluxes are essentially identical, save for a change in the electrochemical potential within the distribution functionsf 1 andf 2 used to deter mine Ff and F respectively. Furthennore, under the condition of current-limited-flow due to a given region, eqn (2.2) shows that it is Ff that defines the transport current through that region. However, as was discussed in Section 4.1, before one can assert that Ff and F share a dependency that is in dicative of eqn (2.2), it is necessary to prove that the regions of integration for  and F provide for  the form given in eqn (2.2). The calculation of F,, and the ultimate proof that eqn (4.2) (and thus the transport flux through the CBS) has the form of eqn (2.2), begins by returning back to eqn (4.3). Eqn (4.3) sets out the general models for F, Ff and F,, but does not explicitly show the effect of a mass boundary. Included within eqn (4.3) is the requirement that tunneling, or any other con duction process for that matter, that moves electrons from one state to another depend upon the probability that the final state be unoccupied  (= (1  —  f)  h). Using eqn (4.3) for Ffr eqn (4.44) for  W, eqn (4.34) for U, , and the Jacobian given by eqn (4.49) to move calculations to xi,,, yields: 1 21r  (454)  2 m 2 fdUf, -_ifdU f d9 F= 2 ff(Uf) 2 mZ 2 cos 2 G+m3’, 2 sm 0 h •  00  0  0  where the superscriptf refers to functions that have their energy reference located at the bottom of the conduction band at x = -x. To arrive at the infinite extent for the region of integration it is only necessary to extend the definitions of WCBS, WN, andf 1 to implicitly account for the fact that the flux density must be zero outside of the region R 2 defined by eqn (4.47) (i.e., WCBS(Ux,1) when U, 1  0 when U, 2  0, WN(Ux,2)  (U) 1 Vb, and f  0 when U  0  E). No loss to the general  ity of these function occurs as a result of this extension. Likewise for F,, but using only the p to U Jacobian of eqn (4.27) in order to maintain the calculations at Xp. yields: 2i  00  Fr  =  mm z,2 2 fdUi, -_Jdu $ 1 de m c srn 2 0 2 os 2 0+m 3” 2 h .  —00  0  (4.55)  W’)  0  July 12, 1995  67  where the superscript r refers to functions that have their energy reference located at the bottom of the conduction band at x  =  x. Note that in eqn (4.55) the subscripts referring to Regions 1 and 2  have been interchanged to reflect the reverse direction of flow for Fr in comparison to Ff There fore, both eqns (4.54) and (4.55) have been constructed so that the integration over U-space oc curs at the point x = x. This will facilitate direct comparison between Fr and The task that remains is to recast the r-superscripted functions of eqn (4.55) into thef-super scripted functions of eqn (4.54). The only difference that exists between the f- and r-functions is their energy reference. Since E(x=x)  —  E(x=-x)  =  Vb, and there is transverse symmetry, then  using eqns (4.34) and (4.35): —V and 2 1 = U U  UI,i  =  =.u’  2 U  =  +U, 1 U  =  UVb.  (4.56)  Finally, recasting eqns (4.39) and (4.40) into r andf form, gives: =  +7’U, 2 U  and  2 U  =  +’y’U, 1 U .  (4.57)  The reason y’ and not y is used in the definition for U , is because Regions 1 and 2 are inter 2 changed for the calculation of Fr This regional interchange maintains consistency with Section 4.2 where the flux always originates in Region 1. With the interchange of Regions 1 and 2, all of the effective masses are also interchanged. Finally, observation of eqns (4.42) and (4.37) shows that interchanging the 1 and 2 subscripts maps y into y’. Since all of the functions used in eqns (4.54) and (4.55) are thermodynamically reversible (due to the fact the system is collision-less), then a general function gT(U) is the same as g(U + Vb) (where U can be either r- orf-superscript ed). Using this functional translation, along with eqns (4.57) and (4.56) gives: ) = 2 2 WBs(U WBs(U + Vb) =  =  WBs(Ul +y’Ui, 1  + Vb) =  ) 2 2 +y’U WBs(U  iii! W(U) = wrf “CBS x,1” ‘  = wifi’rir  Vb)  — —  wrf’rif VVNkLI2),  f;(U’) = f( u’ + Vb)  =  f( (if),  f( U’)  =  f{( Ui).  =  f{( U’ + Vb)  ‘  The above equations recast the r-superscripted functions into the desiredf-superscripted functions. Using the above equations, along with the fact that the probability of hole occupancy h is equal to 1  —  f, eqn (4.55) becomes: July 12, 1995  68  _  Fr  f  =  21t  (4.58)  JdUI, s dU 1 cos2:::, 2 domz, in2e f(U)  Then, the only thing left to do before a direct comparison between eqn (4.58) for Fr and eqn (4.54) for Fe can be made, is to determine the Jacobian that transforms (U , UI 1 ) into 1 ,2 2 (U U ) . Examination of eqn (4.56) shows that the only difference between points in ,U 2 ) space is a constant Vb. Since the addition of a con 2 , UI, 1 (U ) space and points in (U 1 stant does not distort the differential volume element, the Jacobian is unity. This allows eqn 1 (4.58), along with UI,  =  , to immediately transform into: 2 U, 2it  Fr  (459)  2 m 2 fdU. _fdU f 2 de f(U) 2 () mZ 2 O+m 2 s in cos 2., h 3’,  =  0  0  Comparison of eqn (4.59) for Fr and eqn (4.54) for Ff shows almost exactly the same functions; save the fact that F,. deals with transport from Region 2 to Region 1 (i.e., f( Ui’) h(( (If) Ff deals with transport from Region 1 to Region 2 (i.e., f(( U) h( U”)  ), while  ). Therefore, the transport  flux is: 21t  F  =  Ff F r —  =  2 IdUf I dUf 3 J h ‘  2  0  J  dO  0  2 mz m 2 cos O +m 2 sin O m 2 ‘  [f{( U ’) h( U) 1  —  f( U”) h{( U”) ] WBs (U ) W(U 1 ) 2  2it  2’ —i 3 J h —e  fi ‘  .1  f  0  i  .1  0  mm z’ “ cos O +m 2 sin O m 2 [f{( U)  —  f( U”) I WBs (U ) W(U 1 ) 2  .  (4.60)  Thef superscripts have been included as a reminder that the energy reference is located at the bot  tom of the conduction band at x = -x,. Eqn (4.60) completes the proof that Fj and Fr share a dependency that is indicative of eqn (2.2). It also validates the modified definitions for Ff and Fr given by eqns (4.4) and (4.5) respec tively. Eqn (4.60) is brought into exact agreement with eqn (2.2) when the fi and f2 distribution functions of eqn (4.1) are given by the Boltzmann approximation, leading to: July 12, 1995  69  _______ ____  U  (U) 1 f  -  kT  =  kT  1+e  (4.61) U  ((1) 2 f  e  =  -  2 j.t  kT  —2  1+e  kT  Eqn (4.61), under the Boltzmann approximation, produces: U  f{( U) where AE  -  —  f( U)  I2•  =  IS.Efl,  I’i  e e’  —  e  kT  f(( U) (i  —  e  kT)  (4.62)  Since AEp is a constant with respect to the canonical variables defining the  integration in eqn (4.60), then substituting eqns (4.62) and (4.54) into (4.60) gives:  F  =  Ff— Fr  =  i F1  —  e  J.  (4.63)  Thus, the transport flux through the CBS has exactly the same fonn as eqn (2.2). This will allow the models of this chapter to be used with the results of Chapter 2. Eqn (4.63) also justifies the method ology used within this chapter where Fj alone is calculated. Finally, examination of eqn (4.63) shows that it possesses two simple but fundamental requirements: as the driving force AE increas es, so does F increase; when the system is at equilibrium (AE  0), the transport flux vanishes.  4.4 Analytic CBS Transport Models Section 4.2 presented the general models for the calculation of the transport flux Ff through a complex two region system with an abrupt mass barrier in-between. The models also allow for an anisotropic effective mass tensor m*. This section will take the models of Section 4.2 (eqns (4.50)-(4.53)) and derive analytic solutions for the calculation of F through the CBS. By obtaining analytic models, and not simply resorting to numerical calculation, the rich interplay that exists between the physical attributes such as doping concentration, temperature, effective mass, elec tron affinity, and bias conditions, will be brought out for study in the final transport model of the CBS. The key component to all of the models presented in this chapter is the inclusion of the ef fects due to tunneling. Any model or simulator (such as the highly acclaimed Monte Carlo simu July 12, 1995  70  lator) that fails to account for the vast increase in transport current through the CBS due to tunneling, will be grossly inaccurate even if every conceivable scattering process and other driv ing force outside of tunneling is accounted for (see Fig. 4.8).  106  3 i0  -  tunneling and thermionic emission  100  3 io10-6  -  no tunneling, only thermionic emission  ...‘ -  90.8 io-  I I  1.0  •  1.2  I  •  •  1.4  1.6  Base-Emitter Voltage VBE (V) Fig. 4.8. Collector current for an abrupt A1GaAs HBT with 30% Al content in the emitter. The emitter doping is 5x10 , and the base doping is 1x10 3 cm 17 . Notice the large error that re 3 cm 19 suits if the tunneling current through the CBS is not accounted for. Also, the tunneling current has a bias dependence that alters the current to voltage relationship from the form exp(qV/k7) (which characterises the thermionic emission curve quite well) to exp(qVInkT), where n> 1.  4.4.1 Analytic Model for the Standard Flux With the result of eqn (4.63), the development returns to the main goal of this section; deriv ing analytic models for Ff from eqns (4.50)-(4.53). For the problems being considered, the form of WN in eqn (4.45) suggests that eqn (4.50) be used for the enhancement case (i.e., m 1 <m 2 and >  0), and eqn (4.52) be used for the reflection case (i.e., m 1  >  2 and ‘y < 0). As was discussed m  near the very end of Section 4.2, eqns (4.50) and (4.52) share a common term called 1 Jstandard (or 8 for short), plus a unique term for the enhancement case of Fjepi (or F! e for short), and a Fj unique term for the reflection case of Fj reflect (or Fj for short). These terms, using eqns (4.50) and (4.52) are: July 12, 1995  71  2it  Ff  2 =  fd9  eIdUx,1WCBSx,1) 2 c+m 1 9 2 os“ sin  o  max(Vb,O)  2it  m m 1  dO  Pe =  h  1 E-U  E  1 m  o  2  mz  max(Vb,O)  1  9+ m,  20  0  -  1 E—U  1 E—U,  1 WCBS(UX ) .201 dU  2+  o  ( f dU± + 1 ) ,(4.65) U U±,  yE  1 m  =  0  1 Vb—UX  Vb  Ffr  (U±, f dU + 1 ) U , (4.64)  WCBS(UXl) 1 J dUX J  2it  2  f  max (Vb, 0)  V,,  J —  11 dU (U f  +  U ).(4.66)  1 U,  y The derivation of the analytic models begins with Fj. Fj is the most important term, and as it wiil turn out, the essential equation for the solution of Ff,. as well. The analytic solution of eqn (4.64) for Fj begins by noting that the integrals over U, 1 and Uj contain no term with a dependence upon 0. This allows the 9 integral to be performed inde pendently, as in eqn (4.31), to yield the same result as eqn (4.32) but with m  =  and m  =  m,i. Essentially repeating eqn (4.32), but with a change to the dummy variables in eqns (4.64), yields after performing the integration over Uj using the full Fermi-Dirac distribution:  Ff  kT 4itqjm 1 mZ 1 =  1 _Ux-,.L  E  f  kT  1+e  dU WCBS (Ui) ln  E  max(Vb,O)  1+e  —  kT  The integrand above becomes vanishingly small (at an exponential rate) for large U,, allowing for a simplification by letting E Ff  —  00  to produce:  kT 4itqjm 1 m 1 =  /  ( $dUXWcBs(UX)ln,1+e  kT  -____  max(Vb, 0)  In general, even if the emitter is degenerately doped, the energies U at which the above integrand produces significant contributions to F 15 occurs at energies where U is a few kT larger than  . 1 t  This allows what is essentially an assertion of the Boltzmann approximation that leads to eqn (4.61), so that: July 12, 1995  72  00  I  4itqqm 1 kT m 1 Ff  =  —  -—  kT  ekTJdUXWCBS(Ux)e  (467)  max(Vb, 0)  Eqn (4.67) provides for the model of the standard flux, where the integrand multiplied by the lead ing constants is the standard flux density. Eqn (4.67) is now solved for by substituting in WCBS from eqn (4.9) and making a change of variables from absolute energy U to normalised energy U (where U  U/ Vpk, and Vpk is the  =  height of the CBS as defined in Fig. 4.2). Before performing these changes to eqn (4.67), the solu tion process is further facilitated by the following change of variables: ,J1—U +1 X =  =  (  2  2x l+x  2  Ji  and  —i 2 x  , —  =  2  x+l  Letting ’= 3 x=e  J1—U =th(y),  and  =  ch (y) where ch(y) is the hyperbolic cosine of y, and th(y) is the hyperbolic tangent of y. Using the above equations, along with the normalised energies from the start of the paragraph, yields for Vj,  =  5 Ff  4itqJm  lmz, 1  kT ‘k  y  xfl./2mXlVPk  i  —  e’  I dU  VPk:  VPk  th(i))  e  <  —  kT ch (y) 2  (4.68)  ‘nax(V,0)  kT  pk  + ----e Vpk  where all energies, including V, are in terms of normalised energy (i.e., Vb  =  term inside of the square brackets is the thermionic injection term where WCBS  V ‘k). The last =  1. In the event  that V> 1 (i.e., Vb> k) then the CBS is at an energy too low to effect the transport current and: F  (kT) m 1 4tqm =  2i -  e  e  Up to this point, the parameters x (which is the n-side extent of the EB SCR) and Vpk (which is the n-side portion of the potential drop across the EB SCR) have been left as is without connection to the material parameters of the device (where the device is arbitrarily chosen as an npn HBT). However, using the depletion approximation gives [24,80]:  July 12, 1995  73  ________ ________  VPk  =  V  Vp=q(l_Nraj)(Vbj_VBE)  plc  ‘  Vb  x =  =  where Nrat =  NA 2 L 1 N 2 L N A+C D  V 2 I2e  I2elVk  ND 1 6  = eN 2A  x  =  XP=qq2;= N 2 q  V. =  and  kT INANDN I In I q I 2 —  )  +  ND  (4.69)  AE q  2 is the intrinsic carrier concentration in Region 2, Vbj is the built-in potential of the junction, n ND is the emitter doping, NA is the base doping, e is the permittivity of the respective region, and VBE is the forward bias across the EB junction. The doping ratio Nrat differs slightly from that in  eqn (3.5) due to a nonuniform e. Concentrating on the case Vb  <  Vpk then using eqn (4.69) within  eqn (4.68), along with  = U+r  where  I  U  =  ND  gives:  Nrag (Vb.VBE)  Fj  = 1 rn kTk; 4qJm h  (4.70)  5.VJeim r (ch2(Up+r)PJ  (4.71)  dU e f  max (V  Nrat (Vb  —  VBE)  v  kT +-—e Vpk  where V is the thermal voltage kTIq, and U = ch (U + r). As will be shown shortly, eqn 2 (4.71) can be solved in a tractable and analytic fashion. However, the integrand within eqn (4.71) is still the flux density, and is worthy of separate investigation. It is worthwhile to note that eqn (4.71), and the transform used to obtain it, follows that of Crowell and Rideout [781 used in the development of Schottky diodes. Furthermore, U, is the V 1 normalised version of Efj4-J from The standard forward flux density  = 4qm  in 1  where the energy U (= ch (U 2  +  1  [751.  for a given energy U is:  kT Vpk;e  Nrat (Vb.VBE)  r  (Ch2(U+ r)  -th(u÷ r))  (4.72)  r)) is defined in terms of r. The energy at which the maximum  cI occurs can be found directly from eqn (4.72). In terms of the variable  r,  and given that expo  nentials are analytic functions, 4c will be at a maximum when the exponent containing r in eqn July 12, 1995  74  (4.72) is at a maximum. To this end it is found that: r  I  drch 2 (U+r)  —  , + r) 1 th(L  (UP +r) 3 2rsh  =  )  —  r  0  =  0, —UP, ±00.  (4.73)  ch (U+r)  sh(y) is the hyperbolic sine of y. Examination of the definition for U, in terms of r, shows that r has r < oo• Furthermore, when r = -U then U  a range of-Un  =  1, which coffesponds to the top of the  CBS, and when r —>0o then U =0 (it should be noted that U  <  1 deals with the tunneling of elec  trons through the CBS while U> 1 deals with thermionic injection over the CBS). The solutions of r  =  -U,, and  —oo  occur due to the mapping used to define U, in terms of i and do not represent  the absolute maximum that is being sought. Thus, the maximum  occurs when r =0 and gives:  th(U) =  max  4itqjmy, m 1 z, lkTVk P e’e h —  U  -  ‘  1.0  at  U  -r  .  =  (4.74) ch ( 2 U). p  -  VBE= 1.4V  Thermionic Injection Regime  V 9 VBE=O.  0.8 V, when VBE = O.9V-  I  V’, when  0.6  0.4  0.2  0.0 0.0  0.2  0.4  0.6  0.8  k  I  1.0  1.2  1.4  Normalised Energy U (Vpk) Fig. 4.9. Flux density fs’ normalised tO max’ for an 07 Ga abrupt HBT at two 3 j 1 Al As/GaAs different forward biases. The material parameters, the same as in Fig. 4.8, are: emitter doping ND ; base doping NA 1x10 3 cm 17 5x10 ; emitter permittivity El ll. 3 cm 19 cj; AE is 0.24eV; n 9 2 is cm Vbì is 1.67 1 V; mx,l is 0.091m 6 2.25x10 ; 3 ; Tis 300K. Note that energies U <V would enter 0 the base within the forbidden bandgap, and although displayed here are reflected in reality. July 12, 1995  75  As was also found in [78], eqn (4.74) presents a surprising result that the energy U at which the peak flux density Im occurs is independent of the applied bias. Therefore, relative to the top of the CBS, 4 ?max occurs at the same place regardless of the applied bias (see Fig. 4.9). Further consideration of U reveals the following general traits: as U (from eqn (4.70)) increas es from 0 towards infinity, U moves from 1 towards zero, and tunneling becomes increasingly dominant over thermionic emission; as ND increases, or e decreases, the width x of the CBS de creases and U,’, becomes smaller, showing that tunneling is increasing; as m,i decreases the probability of tunneling should increase, as is confirmed by the associated reduction in U; also, as temperature decreases, U becomes smaller since it is easier for electrons to tunnel through the barrier then it is to obtain enough thermal energy to pass overtop of the CBS; finally, in the limit as 11 goes to zero, the system should evolve to a state that is purely describable by classical mechanics, and it is found that U goes to 1, which indicates that there is indeed no tunneling. Therefore, the general traits of the flux density, as presented, follow physical expectations. Returning to the solution of eqn (4.71), the integration over U is converted into an integra tion over  C  .  =  Using eqn (4.73), it is found that for: r  (U+r) 2 ch  —th(Up +r),  dr drdU  =  (_2rsh(U+r)(_ch ( 3 U+r)) (U+r) 3 ch  and then eqn (4.71) becomes (under the condition that Vb —  F  4itqJm lmzl kT VPk  1 (VbZ N,.,  e  3 h  2sh(U+r)  )  =  < VPk):  —  dC 1  VBE)  Nrat (Vbx +  d r——e 1 ,  kT _V_e  —  VBE)  1 V  -  Eqn (4.75) has had the limits of integration from eqn (4.71) temporarily removed for clarity. At this point no approximations have been introduced into the solution. At issue with the solution of eqn (4.75) is that r(C) cannot be determined in closed form. If C(r) were invertible then eqn (4.75) could potentially be solved analytically. Observation of Fig. 4.9 shows that I, the integrand of eqn (4.71), has the form of a Gaussian. Indeed, Taylor series expansion about U (i.e., r  =  ‘j  is extremely symmetric and suggests that a  0) for C(r) is a potentially good approximation. Per  forming a Taylor expansion of C(r) about r = 0 up to second order produces: sh(U) 2 C=—r  ch (Ui)  —th(U)=  sh(U) dC ——2r dr ch (Ui)  Finally, substituting the above approximate equation for  C (r) back into the integral within eqn July 12, 1995  76  ____________  (4.75) yields: Nrat  T7 ( bz  —  VBE)  fdr-e j drr  —2  =  Nrat (Vbx  sh(U)  —  VBE)  th (Un)  u  e  —  fdre J  (U) 3 ch  Nrat (Vb VBE) sh (Un) 2 r UV (U) 3 ch  The above equation is simply the integration of a Gaussian, and results in an error-function solu tion. With the limits of integration from eqn (4.71) reintroduced, the solution of the above is: sh(U) 1  Nrat (Vb  —  VBE)  (ach  th (Un)  eri  +  erf  (Jmax (V,, .0))  —  uP) (4.76)  ,  a  ch (Un) where /ch3(UpUpkT  —  —  sh(U)  VP,  Eqn (4.76) solves for the integral in eqn (4.75) and produces the analytic model for Ff that ,.  is sought after. The complexity of eqn (4.76) stems mainly from the evaluation of the boundary conditions. Fig. 4.9 shows  and the boundaries of integration. As long as the majority of  is  contained within the two boundaries, then the error functions will both approach 1, and eqn (4.76) can be approximated by: 2q  sh(U)  Nra: (Vb - VBE) th(U) 1  ae  (4.77)  .  (U) 3 ch Eqn (4.77) is the simplified model for the integral in eqn (4.75), but it still contains most of the im portant features regarding CBS transport. Thus, the final and approximate models for Ff 5 are found by substituting either eqn (4.76) or eqn (4.77) respectively, into the integral of eqn (4.75) to obtain (under the condition that Vb < Vpk) (see also eqn (4.92) for low temperature considerations): Nrat(Vb  5 Ff  u  = FjsoPPPte  (ach +erf  VBE) th(U)  [erf()+  (Jmax (vi, 0) a  —  (4.78)  U  Nra: (i  VBE)  +FfsOt Ve  r  or approximately as  July 12, 1995  77  btv  sh(Up )UPt V p  z’  fsO I 1 ‘fs  1 Nrat (Vb  —  VBE) th(U)  Nrai (Vbx  11  -  t  p  qch(U,,)  —  ) 8 V  -________  TI  j’  I  mIfsOvt  where 21  —  F  4iq ‘vimy,i m z,i kT— kT 3 h  Finally, under the condition where Vj,  Ff  =  Vk:  FfSQVe  V — AE/q VbI— 8 V,  (479)  4.4.2 Analytic Model for the Enhancement Flux Ffe With the analytic model for Ff presented in eqns (4.78)-(4.79), attention is focussed upon the solution of the enhancement term Fje. Examination of eqn (4.65) shows that the integration over Ujj has a lower limit that includes y(G). Thus, unlike the solution for Fj , the G integration 5 to calculate F! e cannot be performed independently. Further, eqns (4.42) and (4.37) show that y has a complex dependence upon € that would most likely cause the final integration over 0, for the calculation of Ffe, to become analytically intractable. To alleviate this complexity an approxi mation is made. So far, all of the models presented use a general mass tensor that is diagonal with respect to the direction of transport. This general mass tensor formulation is maintained, but the mass barrier will be confined to the study of an isotropic change in the transverse direction of the mass tensor. Thus, “,2  =  ammy,1 and  mZ,2 =  ammz,1. With this approximation, then using eqns  (4.42) and (4.37) it is found that: y(0)  =  1  —  m m cos 1 O +a m m “ 2 sin 1 0 ” 1 z 1 (a m m z 2 cos + m 2 1 0 sin ,) 1 0 amy, lmz, 1 m, 2 ‘  =  1  i —  .  (4.80)  am  Eqn (4.80) reduces y (and also y’) to a constant. With this simplification, the 0 integral in eqn (4.65) can be performed independently using eqn (4.31). Then, the development of Fje will follow exactly the one for the calculation of Fj 5 but with a slight modification to the limits of integration. Therefore, using eqn (4.67), but with the limits of integration obtained from eqn (4.65), yields: —  —  4tqim ‘v y, i m z, i kT 3 h  V max (Vb, 0) kT —  7K1  ,  U  x VYCBSk  U y— i kT Y xi  0  Examination of the above equation shows that T inside of the integral can be redefined with: July 12, 1995  78  Teff  =  T  =  Ty’  =  T(1 —am)  where  2 m  am  =  2 m =  1 m  (4.81)  mz,l  Teff is then the effective temperature of the flux density. Under the enhancement case y> 0 and thus am> 1, leading to Tejf< 0. With eqn (4.81) substituted into the equation preceding it, then: Ffe  4iq.jm m 1 z, 1 kT 3  =  U  Vb max(Vb,O)  —  e”e  --i-Tf  -  eff  dUXWCBS(UX)e  (4.82)  Eqn (4.82) is the same as eqn (4.67) except the limits of integration are slightly different. However, the effective temperature of the flux density is now negative. The effect of the negative temperature  Teff is to cause an increase to the electron distribution as one proceeds to higher ener  gies. This leads to a condition of population inversion that is similar to what is found in lasers. The solution of eqn (4.82) does indeed follow the one presented for Fj, but the fact that T?ff < 0 must be accounted for. Population inversion, when combined with the fact that WCBS also in creases with increased energy, means that the peak flux density will no longer occur at an energy of U,’ given in eqn (4.74), but will instead occur at the upper energy boundary allowed into the problem. The integral inside of eqn (4.71), although derived for the solution of Fj, will solve eqn (4.82) for Fje when the limits of integration from eqn (4.82) are employed. However, it no longer makes sense to use an expansion that is centred about U, as population inversion moves the peak flux density to an energy of max(Vb, 0). Eqn (4.82) is solved by returning to eqn (4.71) and introducing lff into all relevant equations to yield (under the condition that Vb < !‘  =  Fje  Vb  m 1kTee_ 1 4qJm  max(V,,O)  V/q  r  -th(Up+r))  Vpkf dU  (4.83)  where  IN p,eff The  —  kT t,eff  jj4e 2V m 1  q  primes on the energies still denote normalisation with respect to Vpk. Unlike the development  that took eqn (4.71) into (4.75), eqn (4.83) is expanded about V. Furthermore, the condition of population inversion causes the integrand in eqn (4.83) to become basically exponential in terms of U. Remembering from eqn (4.71) that U  =  2 ( U eff + r), and dl ch  —  U  =  th(Upeff + r),  July 12, 1995  79  ________  then a Taylor expansion about max( Vb, 0) for the exponent inside of the integral of eqn (4.83), up to and including linear terms, is: V/q  V/q  Up,eff Vt,effch 2 (U+r) _th(Up,eff+T)Up,eff t,eff )  (rU  —Jl—max(V,0)), (4.84)  where rb = ach(maxVO)_UP,eff.  The final model for Ffe is arrived at by substituting eqn (4.84) into eqn (4.83) and solving. The only concern when performing this integration is to ensure that Vb  If Vi,> k’ then the  <  integral in eqn (4.83) is broken down into two integrals: one integral from 0 up to 1 (remember, normalised energies are being used so that U  =  1 corresponds to U = k); and a second integral  from 1 up to V (over which WCBS = 1). Finally for Vb Vb  Ffe  —  F  Up,jcVt,eff  e  —  <  VPk,J1—max(VI,O)  qUV  e  (‘  VPkrbmax(Vb,O)  e 1  ff 8 qUpffV —  1  b  while for Vb 5  __i_  --  Ffe —FfsO Vt,effe  1 ykT  e  48 qV  4  As a final check on the validity of the model for Fje (i.e., eqns (4.85)-(4.86)), observation of eqn (4.65) and the region of integration in Fig. 4.6 shows that as ‘y  —  0, Fj  cause when y =0 there is no mass barrier and F = Ff. Obviously, when Vb  —  0. This occurs be  0, the upper limit of  integration in eqn (4.82) is zero and the integral itself vanishes. For the case where Vb  >  0, exami  nation of eqn (4.82) shows that the terms containing y are: U(1—y) —Vb ‘kT  e The enhancement case is being considered, where 0 < y < 1. Furthermore, since the limits of inte gration have it that 0< U < Vb, then U(1  —  y)  —  Vb <—YUX <0. Therefore, the terms that makeup  the exponent of the above equation are always negative. Then, as 7 approaches 0 from the positive side, the exponent goes to negative infinity and eqn (4.82) goes to zero. The exact same develop ment occurs for eqns (4.85) and (4.86), so that the previous argument is applicable, and eqns (4.85)-(4.86) do indeed vanish as y —> 0. July 12, 1995  80  4.4.3 Analytic Model for the Reflection Flux Ff,. With the analytic model for Fje presented in eqns (4.85)-(4.86), attention is finally focussed upon the solution of the reflection term 1 ’Jr Eqn (4.66) is the general model for Ff,., and it also contains y within the Uj as well as the U, 1 integrations. Therefore, as was the case with the so lution of Fjce, the 9 integration to calculate Ffr cannot be performed independently. To simplify this problem, as was done with Ffe, the mass barrier is assumed to consist of an isotropic change in the transverse direction of the mass tensor. This allows eqn (4.80) to be used in the solution of jr In fact, using the same basic steps from eqns (4.80) to (4.82) will also solve for Fjr The only 1 change that occurs is to the upper limit of integration over U, , which will approach infinity as E 1 —>  00  (this is because y < 0 for the reflection case). The final result is: — L  ‘  fr  V Ill — --- . kT ‘1’-’ I 4T1 WI (TI ‘ I ‘‘x “CBSk”x)” J 00  I  kT 4irq vim 1 m 1 h  ,  ‘  3  ---  ff  max (Vb, 0)  Eqn (4.87) is identical to eqn (4.82) save the limits of integration. This fact occurs because of the symmetry of the problem being considered. As was stated before, Ff is the standard flux that would flow if there was no mass barrier at all. Fje on the other hand, is the flux of carriers that would normally enter the base within the forbidden bandgap (i.e., U 2  <  Vb), but due to the  mass barrier, is raised up into the conduction band to contribute to the total flux; thus the integra tion is carried out from 0  <  1 <Vb. Finally, Fj,. is the flux of carriers that would normally enter U,  the base within the conduction band (i.e., U,, > Vb), but due to the mass barrier, is lowered down 2 into the forbidden bandgap to become reflected and take away from the total flux; thus the inte gration is carried out from Vb  <  1 U,  <00.  The form of the integral for  and Fr must be the  same since the Jacobian transforms and the boundary conditions given in eqns (4.46)-(4.47) do not depend on the sign of Even though the models for Ffe and Ff,. are ostensibly identical, their analytic solutions are not. This occurs because in Ff,, Teff is positive (the same as for Ff ). In fact, eqn (4.87) is identical 5 to eqn (4.67) for Fj , except the temperature of the flux density is no longer T but T (there is 5 also a constant multiplier of exp(-VbI’1c7) that occurs in eqn (4.87) that is not present in the model for Ff ). Examination of eqn (4.81) shows that when y < 0 (as it is for the reflection case), then Teff 5 has a range of 0< Teff< 7’; where 7e—*0 as y —*0, and Teff —> T as y  —  —oo.  Therefore, the flux  July 12, 1995  81  density in the reflection case is characterised by a temperature that is always less than the lattice temperature 7; but unlike the enhancement case it remains positive under all conditions. Thus, the reflection case is identical to, and can be calculated by, the standard case but with a flux density characterised by 7ff instead of T (of course, the exp(-VbI’)&T) term must also be included). With 7ff instead of T used for the flux density in eqn (4.67), along with the exp(-Vb/4c1) term, the final model for the reflection case becomes (under the condition that b 17 < /pk5h(Up,eff)Up,effVt,eff  F  =  F  q  Nrat (Vbx  —  VBE)  ff) 8 th(U  Vteff  (UPeff + reff L r,eff)  Up,eff  (Upff) 3 qch  (ach  UffJ  —  (4.88)  Nrat(VbiVBE)  (Jmax(V’, 0) Ll;’  J  t,eff  frO”t,eff’  a r,eff  where I  Vb  FfrO  — —  FfsO  --j-. “  d  —  r,eff —  Finally, when Vb  3  Ich (U ,)Up,e,  I  q  t,ejj  (Vk/q) Sh(Upeff)  Vpk, then: Vbx— V—AE/q  Ff r  =  FfrOVteffe  Vt,eff  (4.89)  Eqns (4.88)-(4.89) present the analytic model for Fj,., which is basically the same as the model for Ff but with the flux density characterised by ,  The only potential issue (as concerns  error due to approximation) with eqn (4.88) (and eqn (4.78) as well) occurs at very low temperatures where tunneling is extremely large. Observation of Fig. 4.9 shows that for VBE  =  0.9V, the lower limit of integration is approaching the point at which the peak flux density occurs. However, when the temperature is reduced from 300K to 77K, then U,’, moves from 0.80 down to 0.086 (relative to  and the lower limit of integration ends up past the peak flux density.  When the peak flux density occurs outside of the region of integration, error will begin to occur with the model because the model is based upon a Taylor expansion about U. This potential error at low temperature is exacerbated in the calculation of Ff,. because Teff is even less than T (for an Al 00 As to GaAs flux, m,i Ga 7 3  =  me and mX,2 = O.O 92 O.O me, so that T= 81K when T 67  July 12, 1995  82  ______  300K). The solution to this problem is to perform the Taylor expansion of the integrand in eqn (4.87) for Ff,. and eqn (4.67) for Ff about the lower limit of integration; namely max( Vb, 0). Fortunately, in the course of solving the enhancement case, the desired expansion about Vb has already been performed. Eqn (4.84) is the expansion about Vb up to and including linear terms. If the second order terms are included, then using the transform preceding eqn (4.75) gives: r 1 ,, (4.90) 2 —th(Upeff+r)rbU—Jl—V—(U;—V;) 2 ch (U+r) Vb Jl—Vb 4 where the condition that V  >  0 is assured as this expansion is being used to solve the case where  V> U. Substituting eqn (4.90) into eqn (4.83), but using the limits of integration set out in  eqn (4.87), produces, after performing the integral over the Gaussian [81,#3.322.2]: V/q  Ffr  =  Ff  up,eff vt,ejj  e  ,  “  “°  ,  -.J1-Vb)  t,eff  e  r r,eff  —  rb  erf(—  L  r,eff  )  1  (4.91)  r,effj  where Ff,. 0 is defined in eqn (4.88), and (Treff is altered from its definition in eqn (4.88) to:  areff —  I UpeffVteff 4 (V/q) V Ji  Eqn (4.91) solves for Ff,. when U 0 <V used only when V  <  —  1, and is used instead of eqn (4.88). Eqn (4.88) is  <  U,’, (which is generally the case except under very low temperatures, or if  the heterojunction is such that AE is quite small). In a similar fashion, eqn (4.78) for the calculation of Fj 5 is further restricted to V Then, when U <V  <  U vTh =  U.  1 occurs, Fj is given by (after a simple extension from eqn (4.91)): V/q  5 Ff  <  Ffo  e  J  r  ,  ,  (rbVb -.J1-Vb)  e  r  [i  —  rb erf(_)]  (4.92)  where, in this case only: ar  I  =  I  uv V t  q(Vk/q)VJ1—V  and  rb  =  ach  1  1  J)  —  U°  Eqn (4.92), in concert with eqns (4.78)-(4.79) form the model for Ff . with an unrestricted place ment of the base barrier potential Vb, and the ability to model very low temperatures. Likewise, eqns (4.88)-(4.89) and (4.91) form the complete model for Fjr Finally, without any further exten sions, eqns (4.85)-(4.86) form the model for Ff e  Jtdyl2,1995  83  Before leaving this section a cautionary note regarding the numerical calculation of eqns (4.91) and (4.92) is in order. As V surpasses U by more that 3 areff (or ar), then the term 1  —  erf(x) (which is the complementary error function) rapidly approaches zero. One must ensure that the numerical code that generates erf(x) has the proper asymptotic form or else the result will be incorrectly forced to zero (i.e., 1  erf(x)  —  —>  by simply using the asymptotic form for 1  e_X/ (xJE)  —  ). Analytically, as areff (or a) —> 0, then  erf(x), eqn (4.91) is seen to become eqn (4.85) for  Fje, where the “—1” term in eqn (4.85) is dropped; this result is expected because under these con  ditions the linear Taylor expansion is sufficient.  4.5 The Effect of Emitter-Base SCR Control on I The previous section presented the analytic models for the calculation of the forward flux Ffl and included the mass boundary effects. The only assumption made in the development of the  models of the previous section was that the mass boundary be isotropic in terms of the transverse directed effective mass terms. In the event a material system is studied where this is not true, where such a system must posses an indirect bandgap because an anisotropic effective mass ten sor is required, then the models of the previous section can be used, but the final G integration must be performed using the general models of eqns (4.50)-(4.53) given at the end of Section 4.2. This section will connect the models of the previous section together to simulate an abrupt HBT where the CBS is responsible for current-limited-flow. This will provide insight into the models and allow for the effect of the mass boundary to be fully explored. Returning back to eqn (2.6) for a three-section device, the collector current density will be equal to J Let the simulated device be governed by the CBS in Section 1 (where the neutral base in Section 2  (‘2,  2  j  Fj base)’ and the collector in Section 3 (J  =  =  F CBS),  Ff coil). As  long as the demanded currents in the base and collector greatly exceed what the CBS can provide (i.e., Ff base and Ff coil>> Ff cBs) then if no significant recombination occurs throughout the base  and collector sections (i.e.,  Y2  =  1), eqn (2.6) produces: if Ffs+Ffe if Ff  cTjcBs  =  FfsFfrf  y=0 y > 0 ‘<O  .  (4.93)  where the multiplication of the electron flux by “-1” is not required due to the definition of July 12, 1995  84  It is very interesting to see that when the CBS is responsible for current-limited-flow, I will peer directly into the quantum mechanical nature of the CBS. Thus, the quantum mechanical effect of tunneling, including the effects of the mass barrier at the heterojunction itself, will be observable by simply measuring I. The simulated HBT will be based essentially on the following A1GaAs/GaAs HBT at 300K: emitter is Al 0 07 Ga base is GaAs; emitter doping ND 5x10 3 As; ; base doping NA 3 cm 17 ; emitter permittivity C 3 cm 19 1x10 1 is 1 l.98o; base permittivity £2 is l c-j; AE is 0.24eV; n, 9 . 2 25 3 c 6 2.25x10 ; 1 is 0.092m m m ; m 0 2 is 0.067m ; 0  -*  Nrat is 0.956; Vbj is l.671V; x(VBE=O) is  1 is 0.104; Vb is 649A; U is 0.488; U is 0.795; yis -0.373; lff is 81.5K; Up,effi5 1.80; U >0 when VBE < 1.43 1 V. Two other plausible devices are also considered for the reflection case; in order to make the comparisons direct, all parameters are identically maintained except m 2 is either lowered to  of m 1  ), or to 0 (= 0.046m  of m 1  ). The enhancement case typically does 0 (= 0.023m  not occur for electrons, but most certainly occurs for holes. Using the reciprocal relations to the ; 0.368mj. Changes to the effective density of states 0 reflection case gives m : 0.126m 2 ; 0.184m 0 due to the changing m 2 are not reflected into Vbj nor  Therefore, the simulations that are  about to be presented are contrived in terms of a physical analogue but as such allow for the most direct observation and comparison, regarding CBS transport, that is possible. Beginning with the reflection case, Fig. 4.10 plots Fj as well as Fj,.. using the analytic mod els of the previous section for the three m 2 cases of: 0.067m ; 0.046m 0 ; 0.023m 0 . At T equal to 0 300K as well as 200K, decreasing m 2 (and thus making y a larger negative number) results in an in crease to F! r Physically, as m 2 decreases, the mass barrier will demand a larger transfer of energy from U, 2 into U in order to conserve transverse momentum (see Fig. 4.5); thus, a larger number of particles will be reflected as they will not possess a sufficient amount of U, 2 energy to satisfy the momentum conservation requirements and enter the neutral base. Furthermore, as VBE is increased, Ffr begins to decrease and then decrease quite rapidly. The physical cause for this is the interplay between the base potential Vb and the mass barrier. As was just stated, the mass barrier moves ener . The point at which reflection occurs is when U, 2 2 gy from 2 U,, into Uj,  <  Vb. Obviously, as Vb is  made smaller, more energy can be removed from U,, 2 without encountering reflection. Since Vb de creases as VBE increases then F! r must decrease, relative to Ff, as VBE increases. The sudden de crease in Ff,. for VBE> 1 .4V corresponds to the point at which Vb goes below the reference poten  July 12, 1995  85  106  T=300K  io  Js  102 100  (a) Z  r;  2 m  =  0.023  10  =0.046 2 Fj;m Z  10 -6 108  0.8  ,_  10  =0.067 2 Fj;m .  •  0.9  •  •  •  •  •  •  1.0 1.4 1.5 1.1 1.2 1.3 Base-Emitter Voltage VBE (V) •  •  1.6  •  T=2OOK  r:046  Base-Emitter Voltage VBE (V) Fig. 4.10. Standard Flux Fj. and Reflection Flux Fj,. for an HBT with the parameters given near the start of this section. The only parameter being varied is the base side effective mass in . 2 The lines are obtained from the analytic models of eqns (4.78) (4.79) (4.92) for Ff and eqns (4.88) (4.89) (4.91) for Fj,.. while the solid dots are from the numerical calculation of eqn (4.67) for Fj and eqn (4.87) for Fjr (a) results for T = 300K. (b) results for T = 200K. July 12, 1995  86  tial energy E in the neutral emitter (Vb is <0 when VBE is> 1.431 V). Since the neutral emitter generates the flux that impinges upon the CBS, very few particles will have U, 2 reduced below zero by the mass barrier (unless the mass barrier is very strong due to a small 1 1m 2 m ) . Thus, once Vb decreases below zero, reflection will taper off quickly as there are essentially no more particles to reflect from the Vb barrier. Looking now at Fig. 4.11(b), as Tis reduced from 300K to 200K, there is an increase in Fj,. relative to Fj at low bias where Vb  >  0. The physical explanation for this fact is more complex.  First of all, any particle where U±,i is zero will be unaffected by the mass barrier because momen tum conservation is guaranteed when p is zero (see eqn (4.39)). This means that only particles where -‘yUj is comparable to, or larger than, U 1 will be affected by the mass barrier. Now, to tunnel through the potential barrier requires that the particle obtain a sufficient U 1 in order to pass through the CBS (on average an energy of UVpk is required). Any energy gained by U, 1 will do nothing to improve the particle’s chances of passing through the barrier; in fact it wiLl only serve to lower the particle’s availability because the occupancy decreases exponentially with any increase in total energy. Thus, the CBS preferentially picks out, from the random ensemble of par ticles impinging upon the barrier, those particles that possess a sufficiently high U 1 to pass through the barrier, while being blind to the amount of U 1 contained by each particle. Since U,’, decreases rapidly along with a decrease in 7 -YU±,i will become larger relative to U, 1 as T decreases, and the mass barrier will cause a larger reflection flux. Maintaining the focus upon Fig. 4.11, the effect of the mass boundary can be seen quite readily. In Fig. 4.11(a) the temperature is held constant and all three mass cases are presented. This clearly shows that as the mass barrier is strengthened by reducing m , the relative importance of 2 Ffr rapidly increases. Perhaps even more importantly, the effect that  has on the total flux F is  bias dependent. This shows that the mass barrier cannot be described by a simple multiplicative constant as has been suggested in the literature [51,79,82]. Another important feature that is clear ly brought out in both Figs. 4.11(a) and (b) is that for VBE> 1.43 (which corresponds to Vb <0), the effect of Fjr is negligible. As was discussed earlier, once Vb <0 there will be few particles left that can reflect from the potential barrier in the base. However, as the mass barrier is significantly strengthened to the point where m 1 is four times larger than m , the mass barrier is able to reflect 2 particles from Vb even when Vj, <0. These results clearly indicate that the position of Vj, is very  July 12, 1995  87  T=300K  08  0.6  (a)  0.4  0.%  10  ii  12  141516  13  Base-Emitter Voltage VBE (V) 0.25  0.20  T=IOOK  0.15  (b)  : :T00.  Base-Emitter Voltage VBE (V) Fig. 4.11. Relative importance Of Ff r to the total flux F (= Ff Fj) for an HBT with the same parameters as Fig. 4.10. The lines are obtained from the analytic models, while the solid dots are from numerical calculation. (a) results for T= 300K. (b) results form 2 = 0.067. Note: usable cur rents (i.e.,> 2 Acm begin at VBE> 1.OV for T= 200K, and VBE> 1.2V for T= lOOK. 8 l0 ) ,  —  July 12, 1995  88  0.6  •  05 T=100K  0.4 ,T=200K  (c)  :  0.3  •5N 4 . 2 O.%IHIO.ll.lI . 3 lI l  1.6  Base-Emitter Voltage VBE (V) Fig. 4.11. Continuation of Fig. 4.11 from the previous page. (c) results for m 2  106  .  •  •  •  •  •  •  •  •  =  0.046.  T=300K ,  i0 5 Fj 2 in F; m 2 = 0.023  =  10 0 10  2 m Ff; J, 2 Fr; m  =  0.046  =  0.067  ;l)  6 i0_  0.8  •  0.9  •  1.0  •  1.1  •  1.2  1.3  1.4  1.5  Base-Emitter Voltage VBE (V) Fig. 4.12. Standard Flux Fj and Reflection Flux Ff,. for an HBT with the same parameters as Fig. 4.10, but with AE reduced from 0.24eV down to 0.12eV. Note how reducing AE increases the relative importance of the reflecting potential barrier Vb by lowering Vbj (see Fig. 4.10(a)). July 12, 1995  89  important to the transport flux through the CBS. The conclusion is that during the design of the device it is beneficial to have a large AE so that Vj, is lowered, and the mass boundary will have a reduced effect. Finally, examination of Fig. 4.11(b) and (c) clearly demonstrates that lowering the temperature increases the relative importance of Ff,. in all cases. Obviously, the combination of lower temperatures and a stronger mass barriers produces the largest reflections. 00 The case of an Al Ga 7 3 As/GaAs HBT produces the rather fortuitous result that Vj, is be low zero right around the bias at which the device would routinely be operated. There are other material systems (like SiGe) and devices (HBTs with a smaller emitter Al content) where this is not the case. In these systems iXE is smaller so that Vb stands as a larger reflector. Fig. 4.12 shows what the effect of reducing AE from 0.24eV down to 0.12eV has on the transport flux. Under these conditions V , remains unchanged but Vbj is reduced by 0.12V to 1.551 V Therefore, rela 1 tively speaking, the mass barrier has a larger effect, and the effect occurs over a larger bias range. Reexamination of Figs. 4.10 and 4.11 show an excellent agreement between the analytic models of the previous section and the exact numerical calculation of eqns (4.67) and (4.87). These results clearly show that the approximations used to obtain the analytic models do not com promise the accuracy of the final results. This means that it is reasonable to look at the functional dependencies within these analytic models in order to obtain a deeper insight into the mechanisms by which transport occurs through the CBS. In the end, these analytic models will facilitate a full model for the HBT when other regions of the device (such as the neutral base, or the collector), are brought into the problem. Attention is now moved from the reflection to the enhancement case. As was stated at the start of this section, three cases will be considered for the enhancement case. In order to make comparisons with the reflection case simple, only m 2 is varied and it is chosen to be the reciprocal to the three reflection cases; namely 0.126m , 0.184m 0 , and 0.368m 0 . Fig. 4.13 is basically the 0 same as Fig. 4.10 (except that 7 is now positive under the case of enhancement), and plots Ffs, as well as Ffe. The same basic trends are observed for the enhancement case as were observed in the reflection case. In Fig. 4.13 at T equal to 300K as well as 200K, increasing m 2 (and thus increas ing y) results in an increase to Ff e Physically, as m 2 increases, the mass barrier will transfer more energy from Uj.i into U, 2 in order to conserve pj (see Fig. 4.5); thus, a larger number of parti cles will be moved from out of the base bandgap and into E to contribute to Fj. Furthermore, as July 12, 1995  90  ____  II.)  T=300K Ff s  1O  E  100. 2 = 0.368 Ffe; m  (a) 1O io-  84 Ffe;m2=O.l 10-6 26 Ffe;m2O.l a  a  •  a  •  a  •  a  •  •  •  10-8  0.8  1.0  0.9  1.1  1.2  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V) 10  •  •  •  •  •  T=200K :,  4 io 10  f1_\  E  1o  U) 10-2  2 = 0.368 Fje; m  4 io 10 =0.126 2 1e;m io1.0  a  1.1  •  a  1.2  •  a  1.3  a  •  1.4  1.5  •  a  1.6  Base-Emitter Voltage VBE (V) Fig. 4.13. Standard Flux Fj, and the Enhancement Flux Fice for an HBT with the parameters given near the start of this section. The only parameter being varied is the base side effective mass . The lines are obtained from the analytic models of eqns (4.78) (4.79) (4.92) for F 2 m 1 and eqns (4.85) (4.86) for Ffe, while the solid dots are from the numerical calculation of eqn (4.67) for Fj 5 and eqn (4.82) for Fje. (a) results for T = 300K. (b) results for T = 200K. July 12, 1995  91  VBE is increased, Fje begins to decrease and then decrease abruptly. The physical cause for this is  exactly the same as for the reflection case. As VBE increases Vb decreases so that fewer particles need to be helped over the barrier and Ffe decreases. In the event that Vb <0, every particle that makes it through the CBS must enter the base, since the enhancing mass barrier can only raise U, 2 and the minimum U, 2 is zero. Thus, once Vb decreases below zero, Ffe must abruptly vanish. Moving on to Fig. 4.14(b), as T is reduced from 300K to 200K, there is an increase in Fje relative to Ff.. The physical explanation for this fact is identical to the reflection case. Since U,’, decreases rapidly along with a decrease in 7 the particles will emerge from the CBS with a small er U, . As such, an increased number of particles will be available below Vb. With more particles 1 existing below Vb, the mass barrier may effect a larger transfer of particles from below to above the base barrier, and thus increase Fje as T is reduced. The most important difference to note between the enhancement and the reflection case is that a smaller increase occurs in Fje when compared to Fr,. for a similar increase in the strength of the mass barrier (which is affected by increasing or decreasing m 2 respectively). The reason for this arises purely because of the nature of enhancement and reflection. For the reflection case, as 2 becomes arbitrarily small ‘y — m  also have its U, 2  —> —cc,  shows that as 7—>  —cc  —oc•  With y —*  —cc,  every particle that hits the mass barrier will  leading to a total reflection of all carriers (examination of eqn (4.81)  then Teff> Tso that Ffr  —*  F and F —* 0). Thus, it is possible for the re  flecting mass barrier to become so effective that the transport flux is reduced to zero. For the en hancement case, there is a fixed ensemble of carriers launched from the neutral emitter towards the CBS that attempts to enter into the base. Once the CBS has removed its portion of the ensem ble, the enhancing mass barrier is left to increase U, 2 by removing energy from U±i. At the lim iting strength of the enhancing mass barrier (i.e., y = 1), the entire amount of Uj,i is transferred into U, 2 (see eqn (4.39)). Since the particles will have a one kT spread of energy in Uji, starting from Uj  =  0, the enhancing barrier will rapidly reach a limit by which it can no longer increase  Fje. Thus, the enhancing barrier will have a smaller effect on F than the reflecting barrier, and as  such will not experience the same increase in Ffe due to an increase in m 2 that Ff,. would realise for a similar decrease in m . 2 The differences just described between the reflecting and the enhancing case in the previous paragraph can also be understood from a graphical analysis of Figs. 4.6 and 4.7. For the enhanceJuly 12, 1995  92  ment case, there is a limit of y = 1. Looking at Fig. 4.6 for the integration in R , then obviously in 1 the limit when y  =  1, R will take on a fixed, non-vanishing shape with no possibility of an in  crease due to a change in the mass barrier. This leads to a maximum value for Ffe and thus F as well. For the reflection case of Fig. 4.7, there is a limit of y —>  —0o  When y —*  —00,  the region of  1 will be reduced to zero, and likewise, so will F This clearly shows that reflection integration R can produce a far larger effect upon F than enhancement can. Fig. 4.14 clearly demonstrates the effect of m , Vb and T upon Fje. Concentrating on Fig. 2 4.14(a), there is clearly an increase in Ffe as the strength of the mass barrier increases (i.e., as m 2 increases). However, looking back to Fig. 4.11(a) confirms that the enhancing case does indeed produce less of an effect than the reflecting case. Examination of Fig. 4.14(a) and (b) also shows that once Vj, is reduced below zero for VBE> 1.43V (Vb 1 changes with 7), Ffe  =  0 as there is no  longer a base barrier to surmount. Finally, Fig. 4.14(b) shows that reducing T increases F e in 1 much the same manner as for the reflecting case. Reexamination of Figs. 4.13 and 4.14 show an excellent agreement between the analytic models of the previous section and the exact numerical calculation of eqns (4.67) and (4.82). These results clearly show that the approximations used to obtain the analytic models do not com promise the accuracy of the final answer. It is important to keep in mind that under the condition where the CBS is responsible for current-limited-flow, then the results that have been displayed in this section are equal to J. Since for most abrupt HBTs the CBS is indeed responsible for limiting the current, then the mod elling of CBS transport becomes of paramount importance to the understanding of the device. With the analytic models presented in Section 4.4, and the general models of Sections 4.2 and 4.3, transport through complex structures like the CBS is now fully developed. Finally, it should be realised that the models of Sections 4.2 to 4.4 detennine the transport of charge through the entire EB SCR, and not just the CBS. Eqns (4.50)-(4.53) take into account any quantum mechanical effects, including transport via standard Drift-Diffusion (DD), without the need to appeal to high-energy phenomenological mobility models. By treating transport as a system of collision-less particles that originate from a thermal distribution, the problem of carrier heating and cooling, which needs to be included in DD models [83-85], is ameliorated. Thus, velocity overshoot, including carrier cooling as the electron surmounts Vbj, is modelled throughout the entire EB SCR. July 12, 1995  93  _  o.io  T=300K  .  (a) E  \  =0.368 2 m  0.06 =0.184 2 m  0.04  •  .  0.02 =0.l 1fl 6 2  0.00 0.8  •‘  0.9  •  I.  1.0  I  .1.  1.1  1.2  -  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V)  0.30 •“.-  0184 LZi2=  T=100K  (b)  : Base-Emitter Voltage VBE (V) Fig. 4.14. Relative importance of Fje to the total flux F ( Ff + Fje) for an HBT with the same parameters as Fig. 4.13. The lines are obtained from the analytic models, while the solid dots are from numerical calculation. (a) results for T = 300K. (b) results for m 2 = 0.184. Note: usable cur Acm begin at VBE> 1.OV for T= 200K, and VBE> 1.2V for T= lOOK. 8 10 ) rents (i.e.,> 2 July 12, 1995  94  4.6 Deviations from Maxwellian Forms and Non-Ballistic Effects This section will use the models of the previous sections in order to gain an understanding of the electron distribution that is injected into the neutral base from the emitter. With the neutral base width WB being pressed below  i000A, the truly ballistic device is being approached. In the regime  where the electron in transit through the neutral base suffers only a few collisions, then one cannot appeal to classical solutions that depend upon a thermalised distribution (i.e., drift-diffusion analy sis), nor can one avoid the effect of coffisions altogether and treat the ensemble ballistically throughout. In this in-between region, where collisions are important but do not dominate the transport characteristics, solution methods that solve the Boltzmann Transport Equation (BTE) must be used [42,43]. The issue with solving the BTE often hinges upon the shape of the particle ensemble distribution entering the neutral base. As there are less collisions within the base it be comes important to obtain the correct initial ensemble distribution. This section will provide a method to determine the correct ensemble distribution that enters the neutral base. Furthermore, the effect of collisions, or non-ballistic effects within the CBS will also be examined. It has long been recognized that the particle ensemble distribution entering the neutral base of abrupt HBTs is not Maxwellian [14,39-41]. A Maxwellian distribution is characterised by a Boltzmann distribution in energy, with a parabolic relationship between momentum (or k) and en ergy. Therefore, the Maxwellian distribution appears as a Gaussian distribution in k-space centred at k = 0 (see Fig. 4.15(a)). In the thennionic analysis of the EB heterojunction (i.e., no tunneling is considered through the CBS), one would have a Maxwellian distribution near the top of the CBS (see Fig. 4.2 at x = 0). Then, because of the abrupt potential drop beyond the CBS when going to wards the base, the Maxwellian distribution is pulled apart so that only the right-going half of the ensemble enters the base. This halved distribution is termed a hemi-Maxwellian (see Fig. 4.15(b)), and is identical to the full Maxwellian except that for k < 0 the distribution is zero (be cause the particles are only moving in the positive x-direction). Once the hemi-Maxwellian en semble has entered the neutral base, and if there have been no collisions from x = 0 to x = x, the distribution will no longer peak at k  =  with an increased energy of IXE  V, relative to E at x  —  0 with an energy of 0, but will be shifted towards larger k =  x. This shifted hemi-Maxwellian is  termed “hot” because it appears to look like a distribution that is characterised by a temperature which is higher that the lattice temperature T  July 12, 1995  95  1.0  0.7  0.5  (a)  .  i  0.2  z  0.0 -0.04 -0.02  -0.04 0.00  Normalised k±  -0.02  .  0.00  0,02  0.02  Normalised k  0.04 0.04  1.0  0.7  (b)  I  0.5  0.25  z  0.0 -0.04  / -0.02  -0.04 0.00  Normalised k±  ‘>:p.;  -0.02  —  I. 0.02  -  ----  0.00 0.02  Normalised k  0.04 0.04  Fig. 4.15. Ensemble particle distributions assuming a purely thermaliseci thermionic injection from the peak of the CBS in Fig. 4.2. (a) the initial Maxweffian distribution at x = 0. (b) the hemi Maxwellian distribution that is injected towards the neutral base (positive x-direction). k is norm alised to the length of the GaAs reciprocal lattice vector using an effective mass of 0.067. July 12, 1995  96  From the results of Fig. 4.15, and the arguments of the previous paragraph, the distribution entering the neutral base at x  = x, is clearly not Maxwellian. However, in terms of being able to  analyse the neutral base using drift-diffusion analysis, solutions based upon a hemi-Maxwellian distribution will differ from a full Maxwellian distribution by only a multiplicative constant. The issue of the hemi-Maxwellian being hot, however, will require that an energy-balancing scheme also be included by using the second moments of the BTE to arrive at hydro-dynamic drift-diffu sion analysis [16,17]. Many researchers who have studied transport within the EB SCR, or the neutral base, have relied on the assumption that the worst-case deviation from a Maxwellian would be a shifted or hot hemi-Maxwellian. This assumption is shown to be false when a structure like the CBS of Fig. 4.2 is present within the EB SCR. In fact, the distribution function entering the neutral base is appreciably distorted from either a Maxwellian, hemi-Maxwellian, or hot hemi-Maxwellian. Furthermore, the distortion to the ensemble distribution has a considerable bias dependence. Setting aside for the moment the issue of the mass barrier, which serves to distort the en semble distribution even further, tunneling through the CBS results in a profound change in the shape of the ensemble distribution. As was discussed in the explanation of Fig. 4.10, tunneling through the CBS preferentially picks out from the random Maxwellian ensemble of particles im pinging upon the barrier, those particles that possess a sufficiently high U 1 to pass through the barrier, while being blind to the amount of U±,i contained by each particle. Clearly, this will tend to focus the ensemble at x  = 0 towards higher U, 1 and destroy the circular symmetry that exists  between k and k± shown in Fig. 4.15(b) for the hemi-Maxwellian distribution. Finally, in moving from x = 0 to x = x, a number of particles will be reflected by the neutral base potential Vb which will clip off the distribution (much like a hemi-Maxwellian is cut from a Maxwellian) and result in a potentially hot ensemble entering the neutral base. Fig. 4.9 shows the ensemble distribution after an integration has occurred along the trans verse direction. The result, which was formally proven in Section 4.4, is essentially a Gaussian distribution versus U, . Since momentum p and wave vector k vary as the square root of U, 1 , the 1 ensemble distribution plotted in Fig. 4.9 wifi give a very distorted, non-Gaussian (i.e., non-Max wellian) shape when plotted against ki. Furthermore, Vb cuts the distribution off for particles where U, 2 (= U, 1  —  Vb because there is no mass barrier)  <  Vb. This results in a form that is indic  ative of, but distinctly different from, a hot hemi-Maxwellian (see Fig. 4.16). July 12, 1995  97  1.0  0.8 —  0.4  z  0.2  0.0 0.00  0.02  0.04  0.06  0.08  Normalised GaAs Wave Vector k, 8 cm ) 1 2 (1.11x10 Fig. 4.16. Ensemble distribution versus wave vector k 2 entering the neutral base (i.e., at x = x) (T= 300K). This is essentially a replot of Fig. 4.9 except when U, 2 < Vb the distribution is cut-off in and not displayed order to see the effect of the reflecting base potential. Also, Fig. 4.9 is a plot of the ensemble approaching the CBS from x = -x. Finally, k, 2 is normalised to the length of the reciprocal lattice vector (i.e., 2ir/a where a is the lattice constant).  Fig. 4.16 shows the distortion to the ensemble distribution along k, . At low bias, where Vb 2 is approaching Umax, the ensemble distribution is clipped very near the peak of the distribution, but, unlike a hemi-Maxwellian, not right at the peak. Further, the Gaussian form with respect to energy results in a very flat-topped and non-Gaussian form with respect to k. As the bias is in creased, Vj, recedes when compared to Um so that the distribution no longer has a clipped form. This results in a hot distribution that is asymmetric and which looks quite different from a shifted Maxwellian. Fig. 4.16 clearly shows the non-Maxwellian nature of the ensemble distribution en tering the neutral base. However, it does not show the distortion that occurs along k± (k±  =  k±,i  =  2 because of momentum conservation in eqn (4.36)). In order to see the full ensemble distribu k±, tion entering the neutral base  )), a three dimensional plot versus k,2 1 (= WCBS(Ux,1)fl(Ux,1 + Uj,  and k, , is displayed in Fig. 4.17. Observation of Fig. 4.17 clearly shows the non-Maxwellian or 2 non-hemi-Maxwellian shape of the electron ensemble distribution entering the neutral base at x = x. Furthermore, Fig. 4.17 also demonstrates that it would be a gross approximation to assume  July 12, 1995  98  1.00.  0.75—  0.50—  (a)  ri 0.25-  0.00/  -0.04 -0.02  ‘7  0.00  Normalised k±,z  I  0.06  0.02  Normalised k 2  0.04 0.10  1.00-  0.75—  0.50-  (b)  I  rj) E  0.25—  0  z  -0.04  ,  -0.02  zt±  ç\ 0 • 0 2  0.00  0.00  Normalised  0.02  ‘>)ç_ 0.04 0.08  0.04 0.06  Normalised k, 2  Fig. 4.17. Ensemble electron distribution entering the neutral base versus k (T=300K). The par ticle density is normalised to the peak of the distribution, and k is normalised to the length of the GaAs reciprocal lattice vector (= 1.1 lxi 08 cm 1)• (a) VBE = 1 .4V. (b) VBE = 0.9 V. Comparing (a) and (b) to Fig. 4.15 shows that these distributions are neither Maxwellian nor hemi-Maxwellian. July 12, 1995  99  that the shape of the ensemble distribution is invariant under a change in bias. These results clear ly indicate that the assumption of a hot Maxwellian or hemi-Maxwellian entering the neutral base in an abrupt HBT is erroneous. Figs. 4.16 and 4.17 have used the full HBT parameters that Section 4.5 has been based upon, except that the mass boundary has been neglected by setting m 2  . As was alluded to 1 m  earlier in this section, the mass boundary will have the effect of further distorting the ensemble distribution. Fig. 4.18 plots the electron ensemble distribution entering the base under the condi tion where m 2 = 0.023 (i.e., the reflecting case) to clearly observe the mass barrier effects. The ef fect of the reflecting mass barrier is to simultaneously pull the distribution towards lower k,2 and higher k± . Looking at Fig. 4.18(a) and comparing to Fig. 4.17(a) clearly shows the extension in 2 ; while careful observation of the constant k,2 line from the peak shows that the distribution is 2 k±, indeed being pulled and distorted towards lower k, . Comparison of Figs. 4.18(b) and 4.17(b) 2 clearly demonstrates the distortion due to the reflecting mass barrier upon the ensemble distribu tion. It is important to realise that, although the volume of the distribution is larger in Fig. 4.18 then in Fig. 4.17, there is an overall multiplicative factor of 0.25 (for this reflecting mass barrier) when computing the flux, leading to a net reduction in the total flux. Fig. 4.19 plots the electron ensemble entering the neutral base with an enhancing mass bar rier where m 2  =  0.368. The enhancing mass barrier distorts the distribution in exactly the opposite  fashion when compared to the reflecting mass barrier. The effect of the enhancing mass barrier is to simultaneously pull the distribution towards higher k, . Comparison of Fig. 2 , 1 2 and lower k 4.19(a) with Fig. 4.17(a) demonstrates that the distribution is certainly being pulled towards lower ; so much so that the distribution is starting to look Maxwellian. Closer examination of the 2 k±, contour lines in Fig. 4.19(a) shows the distortion that results from the extension in k, , which is a 2 clear deviation from a Maxwellian form. Further examination of Fig. 4.19(b) in comparison to Fig. 4.17(b) exemplifies the distortion to the ensemble due to the enhancing mass barrier. As sim ilarly occurred with the reflecting case, the volume in Fig. 4.19 appears smaller than the volume in Fig. 4.17. However, there is now a multiplicative constant of 4 (for this enhancing mass barrier) when computing the flux, leading to a net increase in the total flux. Figs. 4.16 through 4.19 clearly chronicle the effects that tunneling and the mass barrier have upon the electron ensemble distribution entering the neutral base. The one clear conclusion from  July 12, 1995  100  (a)  -0.04  0.75  -0.02  H  S  0.00  I  O.OOio  I  Normalised k, 2  1.0  : (b)  0.75  N  0.5  0.2  0.0 -0.04 -0.02  0.00 0.00  Normalised k±, 2  0.02  -  0.04  0.02  0.06 0.04 0.08  Normalised k, 2  Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier where m 2 = 0.023 450 and m 1 = 0.092. (a) VBE = 1.4V. The plot has been rotated relative to Fig. 4.17(a) to clearly display the distortion in the kx2 direction. (b) VBE = 0.9V. Again notice the extreme distortion compared to Fig. 4.17(b) for k 2 less than the peak. July 12, 1995  101  .  (a)  I 0.02  0.04 0.10  Normalised k, 2  .  (b) \  E 0  z 0.00  Normalised 0.04 0.08  Normalised k, 2  Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier where m 2 = 1 = 0.092 (the reciprocal to Fig. 4.18). (a) VBE = l.4V. The distribution looks Max 0.368 and in wellian but comparing to Fig. 4.17(a) shows it to be distorted towards larger kx,2. (b) VBE = 0.9V. Clearly k has been squashed relative to Fig. 4.17(b). 2 has been extended and July 12, 1995  102  the analysis of this section is that one cannot assume that the ensemble distribution entering the base has any resemblance to a Maxwellian or hemi-Maxwellian in either a normal or hot condi tion. Also, the change in the shape of the distribution over bias cannot be accounted for in a sim ple fashion (such as a constant multiplier). Further, it is the effect of tunneling that contributes most to the distortion of the ensemble distribution, with the mass barrier playing an important but generally subservient role. This fact returns us back to the starting comments of this chapter, i.e., that a failure to account for tunneling through the CBS can lead to considerable error in the analy sis of abrupt HBTs. In any event, the analytic models presented in this chapter can be used to con struct the correct electron ensemble distribution entering the neutral base. This correct neutral base ensemble distribution can then be used as a boundary condition in a subsequent BTE solu tion of the transport through the neutral base. The models presented in this chapter have assumed the condition of ballistic motion throughout the EB SCR. This assumption is relatively solid given that the EB SCR is generally quite narrow and as such is much smaller than the mean free path of the particle. Before going on to talk about the effects of non-ballistic motion throughout the EB SCR, it is important to pause for a moment to discuss the lower boundary of Vb used to calculate the flux through the CBS. Re examination of Figs. 4.1 and 4.2 show that Ff and Fr are calculated by assuming that a hemi-Max wellian distribution is launched into the EB SCR from both x = -x, and xi,, respectively. The final flux exiting the EB SCR is then determined by considering how tunneling through the CBS, as well as reflection by Vb and distortion due to the mass barrier, alters the course of the forward and reverse directed hemi-Maxwellians. To assume that a hemi-Maxwellian form exists at both x = and x, the distributions at these two points in space must be fully thermalised and characterised by the lattice temperature T This is a reasonable assumption given that x = -x,j and  are the de  pletion edges of the EB SCR, and as such are outside of where non-equilibrium effects would be gin to occur. It is for this reason that the flux is considered to be injected from x  =  -x, and  leading to the potential boundary of 17 b (which is equal to E at x = x) to enter the neutral base. The above argument corrects what Grinberg et al. [51] have suggested. In [51], the injection to the left is from x  =  0, not from x  =  x. The point x  =  0 is inside of the EB SCR and coincides  with the peak electric field. As such, the ensemble distribution at x =0 is expected to be at its larg est departure from equilibrium when compared to any other point within the EB SCR. Further-  July 12, 1995  103  more, to consider the point x  = 0 as the boundary condition, one would have to imagine that the  electron could ballistically tunnel a few hundred angstroms through the CBS and then suddenly  = 0, where it could then be carried into the neutral base by diffusion. Clearly, it is not reasonable to assume that x = 0 is the source of a thermalised Maxwellian distribution. thermalise at x  By adopting Grinberg’s proposals within [51], the lower limit of integration for the calcula tion of F would be reduced from Vb to Vb  -  VAt,  (=  Vpk  —  AE; see Fig. 4.2). The effect of this  change would be to increase F as the base potential has been lowered and will thus reflect fewer particles. For HBTs where the base doping is more than 30-fold larger than the emitter doping, then  will be very small and the error of adopting the proposals within [51] will be accordingly  small. However, as the doping of the EB junction becomes even slightly more symmetric, the er ror of using [511 will become increasingly large. Furthermore, as the temperature is reduced to the point where U,, occurs below Vb, there will be an exponential change to F for a linear change to Vt,. Thus, under low temperature conditions the methods contained within [51] for the inclusion of tunneling will be in error even for a highly asymmetric doping junction (see Fig. 4.20).  5.0  2.0  1.o  0.0 0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V) Fig. 4.20. Relative difference between the results obtained from the methods proposed in [51] to the model for F from this chapter. The device is based upon the same Al 0 07 Ga HET used in 3 As this section. Note how the reduction to Vb as proposed in [51] leads to an overestimation in the transport through the CBS, and therefore, to an overestimation of I. July 12, 1995  104  Finally, is it reasonable to consider ballistic motion throughout the entire EB SCR? Certain ly, to consider collisions to the particles while in the process of tunneling would be difficult. How ever, models that are similar to, but simpler than, the models presented in this chapter are able to explain the terminal characteristics of abrupt HBTs [22,25] because they include the effects of tunneling. Other more complex models, such as Monte Carlo simulation, which do not include the effect of tunneling, grossly underestimate I. Since ballistic motion is assumed in eqns (4.50)(4.53) when accommodating tunneling, and these models explain experimental findings, then ex perimental evidence tends to corroborate the assumption of ballistic motion throughout the EB SCR. For if there were even a moderate chance of only a single thermalising collision within the EB SCR, then the tunneling current would be drastically altered (any reduction or increase to the energy of the particle will cause a correspondingly rapid reduction or increase in the tunneling probability). Since experimental evidence does not support this, at most, there is a small probabil ity of a thermalising collision within the EB SCR. This justifies the assumption of ballistic motion throughout the EB SCR.  4.7 Conclusion To conclude this chapter, a summary of the past 40 years’ work in this area of electron trans port through a SCR is in order. The reason for this summary is to give due credit to all of the indi viduals who have made contributions, and to demonstrate how a large majority of this past work is disjoint from both the study of HBTs and itself. To begin with, Miller and Good [86] set out the re quirements for the WKB approximation to the Schrodinger equation in 1953, which formed the ba sis for the study by Murphy and Good [681 in 1956 of electron emission from metals into vacuum due to thermionic injection and tunneling (which they term field emission). [68] lead to the forma tion of the general charge transport model of eqn (4.2). The seminal work of Stratton [69] extends [68] by considering electron emission from semiconductors into vacuum, including the effect of a mass barrier based upon a spherical effective mass. The main concern in [68,69] is the incorpora tion of image force corrections which alter the tunneling potential and greatly increases the tunnel ing current. In [69], tunneling is only considered within the vacuum and not within the semiconductor, and does not consider the effect of a base barrier potential Vb (as Vb is far too nega tive to enter into the problem). Stratton and Padovani [75] apply [69] to Schottky barriers, and in clude tunneling within the semiconductor but still do not concern themselves with the effect of Vb. July 12, 1995  105  Also, [75] does not include the mass-barrier effect considered in [69]. In parallel to the work of [69,75], Christov independently repeats the work [70,73]. The work done in [69,70,73,75] is meant for the study of Schottky diodes, and is more concerned with surface effects (image force correc tion) than anything else. Furthermore, the potential profile being considered is linear and not the parabolic one found within the SCR; however, [69] does allude to the solution of an arbitrary po tential energy profile through the use of a Taylor expansion. The work up to this point forms the foundation for the study of Schottky diodes and band-offsets between metals and semiconductors. Crowell [76] derives the Richardson constant for a completely general effective mass tensor, but fails to rigorously derive the result by not presenting the relevant Jacobians. Instead, the work in [76] relies on simple arguments to obtain results that, while applicable to the study of pure ther mionic emission, are not clearly applicable when tunneling is considered. Crowell [77] continues the work in [76] in an effort to determine the correct effective mass to apply to a Schottky diode between two materials characterised by different effective masses. The work in [77], much Like that done in [76], is not mathematically rigorous, and as a result fails to obtain a vanishing trans port current under equilibrium conditions. Grinberg [82] solves this problem but only if thermionic emission is considered and not tunneling. The work of this chapter extends [82] by including tun neling and thermionic injection (eqn (4.60)) through a rigorous mathematical treatment. Finally, Crowell and Rideout [78] solve for tunneling through the parabolic potential barrier of the SCR, but do not include the effect of a mass barrier. They present the final transform (eqn (4.70)) used to evaluate the tunneling integral of eqn (4.67), but do not present its development (eqns (4.67)-(4.69)), nor do they provide for a spatially varying permittivity e or the effect of Vb. Eqns (4.50)-(4.53) derive for the first time charge transport through the EB SCR, including ther mionic emission and tunneling, between two semiconductors characterised by different effective mass tensors and e. Furthermore, the effect of Vb is properly included. The most important aspect of the work contained within this chapter is that for the first time all of the essential physical con structs of the EB junction within an abrupt HBT have been considered. The results of these con siderations are analytic models, based upon the solution of eqns (4.50)-(4.53), to simulate the transport of flux through the EB SCR. Since there were no special features of a specific material system employed within this chapter, the results of this chapter are applicable to any material sys tem. Finally, the developments presented here have focussed upon electron transport, but apply equally well to the transport of holes with basically little change to the models. July 12, 1995  106  CHAPTER 5 Recombination Currents  July 12, 1995  107  As was discussed in Chapter 3, one of the most important parameters of an HBT is the cur rent gain  [3. Whether one is designing Digital or Analogue circuits within an IC, an accurate un  derstanding of  13  is essential to the successful operation of the circuit. Chapter 4 dealt with the  calculation of transport through the CBS (in an npn device), which is often the determining factor for I in abrupt HBTs [18,25]. This chapter will finish off the model for I by using the general models of Chapter 2 to include the effect of neutral base transport along with transport through the CBS. More specifically to the calculation of [3, this chapter presents the physics underlying the creation of base current. Included in the analysis to follow is the interaction of ‘B with Ic that was alluded to in Chapter 2, and which occurs when transport through the CBS is responsible for cur rent-limited-flow (i.e., control of Ic). This chapter includes the modelling of four different components of the hole current that re sult in the base tenninal current. These components are: 1) Shockley-Read-Hall (SRH) recombi nation within the EB SCR; 2) Auger recombination within the EB SCR; 3) radiative recombination within the EB SCR; 4) neutral base recombination through all of the processes just detailed. The back injection of carriers (i.e., holes for the npn HBT being considered) from the base into the emitter is not accounted for because this back injection is effectively suppressed by the characteristics of the wide bandgap material that forms the emitter; however, inclusion of back injection is a trivial extension to the results that follow. Analytic models for the four previously mentioned recombination processes that are respon sible for the creation of ‘B will be presented. It is shown that these analytic expressions for the four base current components can be reduced to the familiar diode equations with two parameters -  namely the saturation current J and the injection index n. Even though the physical mechanisms  that control the base current in the presence of a heterojunction differ markedly from the homo junction case, one can still recover a simple diode model for the final representation. It is within this analysis that a surprising result regarding the injection index n is made. Standard theoretical calculations give a value of n  =  2 for the SRH current. However, it was found that n  =  2 applies  only in the limit of a wide, or symmetrically doped, EB SCR. For HBTs of interest, where the base doping is very high compared to the emitter doping (i.e., asymmetrically doped), a value of n =  1 is applicable under certain operating conditions.  July 12, 1995  108  _i  Most of the work that is to be presented in this chapter has been previously published by this author and Dr. D.L. Pulfrey [24]. Within the context of this published work, HBTs constructed within the AlGai.As material system were studied. The results of this chapter are general, how ever, and can be applied to other material systems as well. For the case of indirect material sys tems such as SiGei.., the only major change is that the radiative recombination rate is small enough to be ignored in comparison to SRH and Auger recombination.  5.1 Electron Quasi-Fermi Energy Splitting J\Ef The presence of an abrupt EB heterojunction in an npn HBT can lead to the splitting of the elec tron quasi-Fermi energy Epa. as first discussed by Penman and Feucht [50], and shown in Fig. 5.1. This splitting of E  (i.e.,  AE) has been alluded to in Chapter 2 and was found to be the driving  force for the transport current through the CBS (as was proven in Section 4.3, eqn (4.63)). Fig. 5.1 shows AE 1 and its position within the EB SCR.  results due to a departure from quasi-equi  librium, where the transport flux through the CBS is no longer a small perturbation to the forward and reverse equilibrium fluxes that are everywhere present within a semiconductor [50,18].  E(eV) E AE  t  ---  ‘—‘V 1 E -  qV  +  E -x  0  Fig. 5.1. Band diagram of the EB SCR showing the effect of the abrupt heterojunction on under an applied forward bias (reprint of Fig. 2.2). ji is the solution to the Poisson equation and is therefore continuous. Both the reference energy position and the intrinsic or mid-bandgap energy 1 are also shown. E Section 4.3 and eqn (4.63) clearly bring out AE, but do not locate the position of AE if it is indeed abrupt. Perlman and Feucht [50] have addressed the spatial variation of  and found  that in general AEj, occurs abruptly and coincidentally with the position of the EB Heterojunction (as is shown in Fig. 5.1). Finally, the hole quasi-Fermi energy Ej, has no discontinuity and is os July 12, 1995  109  tensibly constant throughout the EB SCR. The reason for the lack of a  within the EB SCR is  because transport through the neutral emitter and not the EB SCR dictates the back injection cur rent (this is proven at the end of Section 5.3 once the neutral emitter transport current is derived). Essentially, because the EB SCR is not responsible for the current-limited-flow of holes into the emitter, there is no iXEj, present within this region of the device. Traditionally, in the modelling of current transport in HBTs, zEp has been implicit in the calculation of the collector current density J and the neutral-base recombination current density NB  [20,51,87-89]. The calculation has proceeded via a balancing of  and NB against the com  bined thermionic/tunnel current ThT crossing through the CBS at the abrupt junction; i.e., ThT  =  NB +  (5.1)  AEfl.  J(  Further, it has been the usual practice when considering additional base current due to recombina tion in the EB SCR, to subsequently add this extra current SCR to the prior-calculated NB; i.e., =  (5.2)  ’NB(fn) SCR 1  Recently, Parikh and Lindholm [90] have emphasized that this calculation of B via direct super position is not strictly correct because the base-side component  SCR,B  of SCR should figure in  the original current-balancing equation which is used to compute AEp, and, subsequently, c, NB and scR,B; i.e., eqns (5.1) and (5.2) should be replaced by ThT  =  B,B  JscR,B+JNB+Jc =  SCR,B  —*AEffl  (5.3)  (AEffl) +JNB(IXEJfl)  (5.4)  where B,B is that portion of the base current arising from recombination in the metallurgical base (see Fig. 5.2). It can be appreciated that this more correct, self-consistent computation of SCR,B will only effect the base current if SCR,B is comparable to  NB  putation of  eqn (5.3)) in cases where  from the balancing equation  (i.e.,  and, furthermore, will only effect the com  f3 is low. To examine  these effects is one of the objectives of this chapter and, to ensure that their importance is not un derestimated, Auger and radiative recombination in the SCR have been considered, as well as the usual SRH recombination. The computation of  via eqn (5.3) can be done numerically, but an analytical solution  would be more insightful, and also very useful in HBT device modeffing because AE, and thus July 12, 1995  110  Jc N.B and SCR,.B could then all be computed directly from the physical properties of the device and the applied bias. Chapter 2 presented the analytic methods to determine both timate transport currents that produce J and  and the ul  Therefore, the second objective of this chapter is  to develop such an analytical expression for AE. A final aim is to show that the components of SCR,B’  even though they have an extra bias dependence through z\Ep, can be expressed as diode-  like equations. This fact should greatly facilitate the incorporation of these currents into a com plete, large-signal representation of the HBT, which may then be implemented in Circuit simula tors such as SPICE.  Fig. 5.2. Components of the collector (ic) and the base (SB) currents emphasising that ThT must equal the total of, J( + NB + SRflB + Aug,B + Rad,B when recombination due to Shockley Read-Hall (SRH), Auger (Aug) and radiative (Rad) processes is considered.  5.2 Modelling the Recombination Processes of HETs The “unique relationship”  [90]  between the collector current, the neutral-base current and  the base-side SCR recombination current comes about because all these currents depend upon the electron quasi-Fermi energy splitting at the heterojunction. As this splitting is greatest in the case of an abrupt heterojunction, we consider only this type of junction in this analysis. The junction is taken to be formed by an n-type Al 0 07 0a emitter and a p-type GaAs base (the same as the de 3 As vice in Section 4.5). To reduce the complexity of the algebra, without sacrificing much in the way July 12, 1995  111  ______________  of accuracy [90], the perinittivities and the effective densities of states have been taken as a con stant throughout the entire device.  5.2.1 SRH Recombination The recombination rate due to SRH recombination can be written as [90,911 n.  —  R  SRH  t  —  E -E fP) inh [cosh (Uf— E/kT) + b] “ 2kT  (55)  where: n(x) is the intrinsic carrier concentration, E(x) is the electron quasi-Fermi energy (see Fig. 5.1),  is the hole quasi-Fermi energy (assumed constant), =  , where t 0 Jt,  and t, are the hole and electron minority carrier lifetimes, respectively,  within the SCR, Uf= (Ep b  =  +  Efr)I2kT +  exp[(Ejj,  —  ln(tdt,),  Ep)/2kJ]•cosh[(Et  —  )fkT+ 1 E  ln(tolto)J,  where E is the energy level of the single recombination centre assumed in this work, and E(x) is the intrinsic Fermi energy. The latter has a discontinuity of AE 1 at the abrupt heterojunction (see Fig. 5.3), because the bandgap difference between the wide-bandgap emitter and the narrow bandgap base is generally not distributed evenly between the conduction and valence bands; i.e., 1 AE  AEG  =  +  AE  =  kT in  fl +  (5.6)  AE  where the subscripts p,n refer to the p-type base and the n-type emitter regions respectively. E 1 is related, therefore, to the electrostatic potential energy ji(x) via E (x)  =  x 0 N’ (x) IN(x)—AE x>0  (5.7)  Here we use the depletion approximation for if(x), namely: x0 iJ(x)  =  (5.8)  2  x>0  Julyl2,1995  112  where, using eqn (4.69) VP  —  Vp=q(lNrat)(Vbj_VBE),=’  =  VPk  I2eV  I2eVPk fl  =  —  where Nrat —  x=I  Al q N 2  CPNA eDNA + END  Vb.  kT =  =  (NAND  —liii q i  xP  —  xfl  —  AE  1+—  )  q  eDNA  ND  (5.9)  NA  kT =  —  £ND  (NAND  —mi q  1 AE 1+—  q  with VBE being the applied base-emitter voltage, Vb 1 the built-in potential, ND the emitter doping and NA the base doping. Eqn (5.9) has included the effects of a non-uniform permittivity for the time being.  -xn  0  Fig. 5.3. Energy Band diagram for the EB SCR of an HBT under equilibrium conditions. Notice the discontinuity of AE in the intrinsic energy E.  July 12, 1995  113  The SCR currents on each side of the heterojunction follow from  q$RSRHdx + qfRSRHdx  =  SRH  (5.10)  =SRH,B SRH,E  This equation can be solved using eqns (5.5)-(5.9), but the solution cannot be made analytic with simple transcendental functions. A closed-form solution demands that some approximation be made for W(x). Here we follow the linearisation procedure of Choo [92]; i.e., qi(x) Viinear(X) where x,  =  WBENrat,d, Xp  WBE(l  q(V—V) =  —  (x+x),—xxx  (5.11)  BE  Nrat,d), WBE = x, + x, and Nrat,d = NA! (NA  +  ND).  The linearisation of ji(x) in eqn (5.11) differs from that proposed by Parikh and Lindholm [90]. In [90], the linearisation is based upon a first order expansion of eqn (5.8) about the point where RSRH is a maximum. The problem with this type of expansion is the RSRH maximum must be well localised within the region of integration. If the Rpjj maximum is not within the region of integration (as it can be for reasonable operating biases), then the first order expansion proposed in [90] can lead to significant error. Eqn (5.11) alleviates this problem by appealing to the mean-val ue theorem to define the linearisation. In fact, as VBE approaches Vbj, eqn (5.11) becomes exact. Eqn (5.10) can now be evaluated using eqn (5.11) and x0  qV  Ef—Ef =  qBE—  fX>  (ND N 2kTlnL—J Ef+Ef  =  x0 —  2kT1n(.)  qV  AE x>0  to yield —  SRBB —  qnWB 2 te  (zo—z .r[q VBE_AEJn1 atan 2kT  51  +  (5.12) 1 q 2 W n SRFLE =  sinh  [ 2kT j atan  , 0 (Z,—Z +  1 July 12, 1995  114  with =  z 0 Z  zP 0 Z  —  —  =  q (VbI  —  VBE)/kT  ND  tpQ,fl 1  ND  tpQ 1  [ -— 4 -——exP[--  r  =  Jtpoxtnox  qV 2kT  —q—--—exp[—q  Nrat (“bi 2  —  VBE) + VBE1  2kT  J  (5.13)  ItpOpex P[qvBE_IXEffl  —  2kT  —  ND  ItpO,p  r  2qN (VbI  —  VBE) + qV + AEffl 2kT  —  1 2AE  where it is assumed that E and E 1 are coincident throughout the device [90], and, therefore, b from eqn (5.5) can be neglected for any reasonable operating conditions. Eqn (5.12) can be ob tained from eqn (5.10) by using integral 2.423 #9 in [81]. In all cases, the final subscript of p and  n refers to the p-type (base) and n-type (emitter) material regions respectively. Eqn (5.12) is equivalent to eqns (20) and (21) in Reference [90]. It is, perhaps, in a more ap pealing form as it can be readily seen to be an extension of the usual equation for SCR recombina tion in homojunctions. Also, the unique feature to HBTs, quasi-Fermi-energy splitting, is explicitly brought out by the presence of AEJk in the expression for SCR,B• Finally, the linearisation used to obtain eqn (5.11) results in the use of the doping ratio Nrat,d, and not the voltage ratio Nrat, within eqn (5.13). As was stated at the start of Section 5.2,  the effect of a non-uniform permittivity is quite small and can be neglected within the larger ap proximation of a linear NJ(x). For this reason, it is assumed that for all practical devices encoun tered that Nrat  Nrat,d; in fact, for the parameters used in Section 5.4, this is only a 0.4% error.  5.2.2 Auger Recombination As the doping concentrations increase, Auger recombination becomes an important consid eration. There are two Auger processes of interest [93]: 1) a conduction band electron recombines with a heavy-hole, transferring it to the light-hole band; 2) a hole recombines with a conduction band electron, and the energy is transferred to another conduction band electron. In the first case, the recombination rate is proportional to p n, while in the second it is proportional to pn 2 . When 2 Julyl2,1995  115  the equilibrium recombination rates are included, the total Auger recombination rate is: UAUg  where the constants A and  =  (An+Ap) (pn—n)  (5.14)  4 are the electron and hole Auger coefficients respectively.  Using the same techniques employed in arriving at eqn (5.5), the above equation can be re written as: 8 UAU  =  nexp(k?)AJAflAP.  [ZAUS+_]  [exp(  f119)  _i]  (5.15)  where ZAug  =  E +E -2E.1) AJA 2kT  The Auger recombination current is then given by  Aug  =  qfUAugdx+qJuAugdx Aug,B + Aug,E  (5.16)  which can be solved using eqns (5.15), (5.13), (5.11), (5.9), (5.7) and (5.6) to give: 2qnW Aug,B  exp  =  [qV—AEffl  [  j  kT  .  sinh  [qVBE—AEffl 2kT  [  0 (Z  2qn WBE Aug,E  =  exp  [  qV kT  sinh  [  qV 2kT]  —  Z) (A P nO p zz 0 + A, t ) 0 (5.17)  (4  —  Z) (A ntnnZnZn +A  Eqn (5.17) gives the Auger recombination currents that are generated from the base and the emit ter sides of the SCR.  5.2.3 Radiative Recombination For materials where there is a direct bandgap, it is important to consider direct band-to-band radiative recombination. The rate at which radiative recombination occurs will be proportional to the pn product [94]. When the equilibrium recombination rates are included, the total radiative re combination rate is: July 12, 1995  116  URad  =  B(pn—n)  (5.18)  where the constant B is the radiative recombination coefficient. The radiative recombination current is then given by  Rad  qfURaddX+qfURaddX  =  (5.19)  Rad,B + Rad,E  which can be solved using eqns (5.18), (5.9), (5.7) and (5.6) to give: Rad,B  Rad,E  =  qnpBpWBE(l_Nraj)[exp(  v r qBE hui,nBnwNrat[exP( 2  kT  qV-AE kT (5.20) —1  5.3 Current Balancing with the Neutral Region Transport Currents It is clear from Fig. 5.2 that the electron currents to the right (i.e., the base-side) of the het erojunction must equal the electron current due to the charge transport across the hetero-interface; i.e., (5.21)  JscR,B+JNB+Jc  ThT  where SCR,B  =  (5.22)  SRH,B + AugB + RadB•  The formulation given in eqns (5.21)-(5.22) was already treated in Section 2.2. Comparison of Fig. 5.2 with Fig. 2.3 shows an exact agreement. Therefore, the current balancing portrayed by eqns (5.21)-(5.22) can be solved using the models given in Section 2.2 if the various transport and recombination currents follow the general functional forms assumed in Chapter 2. ThT  is the transport current through the CBS that was solved for in Chapter 4. Eqn (4.63)  shows that the flux F through the CBS  (E  mr) has the functional form assumed in Chapter 2 (see  eqn (2.3) for J, i). This immediately allows the models of Chapter 4 to be used in concert with the models of Chapter 2 to solve for the collector and base terminal current densities J and spectively. Looking again at eqns (2.3) and (4.63) shows that  i2  1 =  Ff. and  =  B  re  cin3 AE. F  cludes both the thermionic emission and tunneling components involved in the transport over and through the CBS. Employing the formalisms of [51], Ff can be written as: July 12, 1995  117  Ff  =  y(VBE)  (kT) 4iqJm“ 1 m 1 Z  — — -_______ 1 J  2  ekT  (Vbx  VBE)  e  — -_______ VBE)  (VbZ  kT  qyuNe  kT  (5.23)  h  where 1) is the electron thermal velocity given by =  kT /eq 2itm,  (5.24)  and y(VBE) is the tunneling factor (this is not to be confused with the yin Chapter 4 used to char acterise the mass barrier). With y = 1, eqn (5.23) reduces to the thermionic injection current given by the last term in eqn (4.78). Essentially, yis given by FfIJth where th is the thermionic injection current and Fj= Fj CBS given in eqn (4.93). Failure to include yin eqn (5.23) will result in a severe overestimation of AEp 1 [18] (and an underestimation of the collector current). Finally, JI is the electrochemical potential relative to E formed by the doping ND within the neutral emitter. The approximate solution given in eqn (5.23) is strictly valid only if the emitter is non-degenerately doped. The neutral-base recombination current NB and the transport current through the neutral base Jc which must be used in eqn (5.21) follow from the standard, low-level injection solution to the continuity equation. Using the boundary condition that the driving potential at x = x,  —  start of the neutral base) is VBE  (i.e.,  the  AE (see Fig. 5.1), and for the case of a single heterojunction  structure operating in the forward active mode, the excess electron concentration near the collec tor is 11 (Wflb)  =  0, where Wb is the neutral base thickness relative to x  =  x,  then the expres  sions for these currents are IWflbN  2 —  NB —  NALflb  coshl— Lflb .  sinh  I—i  (W  Anb  csch  e  1  kT  2 qDn 1  1 — —  coshl— —i  NALflb  -L——j  and qDn,  qV — 8 zEf  (—)  [  .  sinh  qVEAEf  nb  qDn, —  Anb  csch  e  (W  ç-  (—-)  qV  —  1  ije  J  (5.25)  [e  nb  1] e  (5.26)  —  where D is the effective electron diffusivity in the base, and Lflb (= JDfltflb) is the electron minor ity carrier diffusion length in the base. Observation of eqns (5.25) and (5.26) show they possess the functional forms of  4  and J  respectively found in eqn (2.4) (i.e.,  4, = JNB(AEp=0)  July 12, 1995  118  and J  =  Jc(AEfij=0)). The approximate forms of eqns (5.25) and (5.26) introduce a negligible  error over almost all bias conditions given the magnitude of exp(qV/kT) compared to unity. The last remaining task before the models of Section 2.2 can be employed to solve eqns (5.21)-(5.22) is to ensure that Rad,B’ Aug,B’ and SRH,B have the same functional form as NB with Clearly,  respect to  Rad,B  in eqn (5.20) can be written in the same approximate form as NB  with respect to AEp. However, it is not clear that the same is true for  Aug,B  and  SRRB  in eqns  (5.17) and (5.12) respectively. In order to see Aug,B’ SRH,B’ and Ro4,B can be rewritten as:  )  Ar’  (Ti  \  Aug,B” BE’ ‘-fn) ‘  —  r cu Aug,Bk V BE’  (  e  kT  (5.27)  --  SRILB( BE’  fn)  ”BE’ 0 Rad,B  AE)  —  sB( BE’  )e  JRB( VBE,  0) e  kT  a plot of the error between the full and the approximate forms in eqn (5.27) is constructed. Fig. 5.4 plots the relative error between the right and left sides of eqn (5.27) for Aug,B’ SRI-1B’ and Rad,B with VBE fixed at 1 .OV. Fig. 5.4 shows that the error in using the approximate relations in eqn (5.27) is less than 10 parts per billion. With such a small error in using eqn (5.27), it is justified to state: J 3(VBE)  =  JNB(VBE, AE=0) + ‘ SRHBWBE’ 1  AE—0)  + JAug,B( VBE,  AEffl=0)  (5.28)  +  JRB(VBE, LEO).  Eqns (5.21)-(5.22) can now be solved using the models in Section 2.2. The transport current T through  the device (which is equal to the collector current) is given by eqn (2.7), with  from eqn (5.23), J =  1), and J  2  >>  from eqn (5.26),  =  (J  ,  =  J  3).  4  is given by eqn (5.28),  4  2  =  =  Fj’  0 (i.e.,  Y2  Using the above produces:  r-+--—i [  1 [J2,  .i2,3J  =  3 Ff(VBE)/y Jc(VBE,1Effl=O)if  <<J, J,i1”y 3  ’ 1 > 3 >’2, r  (5.29)  where y is given in eqn (2.7) as —  13  +4 3 3 J  and the  VBE  —  Jc+JNB+JsRH,B+JAug,B+JRacB jC  zEf=O  dependence has been omitted for clarity. Eqn (5.29) embodies the two different July 12, 1995  119  modes of operation that the HBT can function under; the first condition is where the CBS is re sponsible for current-limited-flow; while the second condition is the classic BJT regime of opera tion where the neutral base is responsible for current-limited-flow. Finally, the base terminal current can be solved directly by using eqn (2.9) to yield: B (VBE)  F+ SRIE =  Aug,E + RadE  JT(Y3  —  i).  (5.30)  Or, AE can be calculated by eqn (2.5) and substituted back into SCR,B of eqn (5.22) and NB• B is then given by the sum of all the hole currents (i.e., B SCR,B + NB + Aug,E + SRE + Rad,E) The beauty of eqn (5.30) is it solves for the base terminal current without the need to detennine the inner driving potential of iSE. However, if a detailed understanding of each component of the base terminal current is desired, then iXE must be solved for explicitly.  o  z:::: a  A ug,  SRHB  4  —6  .  0  50  100  150  200  250  300  Quasi-Fermi Energy Splitting AEf (mV) Fig. 5.4. Relative error between the approximate and exact forms given in eqn (5.27). The mate rial parameters are given in Section 5.4, and VBE is fixed at 1.OV Before leaving this section, it is important to verify that E, is indeed constant throughout the EB SCR. If there were a AEp,, present, it would have to be included in the emitter side hole current SCR,E just like  has been included in SCR,B• Essentially, the same current balancing  procedure given by eqns (5.21)-(5.22) needs to be performed regarding the transport of holes from July 12, 1995  120  the neutral base, through the EB SCR, and finally through the neutral emitter. The same models for the electron case can be applied to the hole case, but using the appropriate material parameters for a hole. Using the HBT parameters of Section 5.4, then the hole transport current through the , and the hole transport current through the neutral 2 EB SCR is 3.9x1Oexp(qV/k7)Acm emitter assuming a  3000A emitter cap at a doping of 10 3 is cm 20  2 exp(qV/kI)A 27 l.2x1W . cm  Clearly, the neutral emitter is the bottleneck to hole transport which validates the claim that is indeed zero through the EB SCR. This does not have to be the case, and a device can be imag ined where this is not true, leading to the requirement that hole transport be self-consistently solved with electron transport. It is quite interesting to realise that the valence band discontinuity AE does not limit the back injection of holes as the literature has lead the device community to  believe. The back injection of holes is ostensibly eliminated by the reduced number of minority holes due to a small n 1 that is characteristic of a wide bandgap material.  5.4 Full Model Results The values used for material parameters, unless otherwise stated, are: 5x10 NA: 19 ; 3 cm 1x10 Cbe: l2.9e; ; 3 cm ND: 17 to:  2Ons; AE: 0.24eV;  cm 3 4.21x10 ;  Ejr:  3 c 6 2.25x10 ; m  ll. E 9 rj; —*  to,p: Sns;  : 77.3 meV, Vb: 1.67 1V, 1 AE  ; 1 s 6 cm 32 Nrat,d: 0.952, Nrat: 0.956, x(VBE=l. 4 V): 271A, xp(VB=l. 4 V): 13.6A;A,,: 7.99x10 6s ; cm 31 5.75x1t1  31 cm 1.93x10 ; 1 s 6  B: s 3 c 7.82x1O’ ; 1 D: m2 30cm Wb: , 1 s  ; B: 1.29x10 1 s 6 cm 30 1.12x10 ; 1 s 3 cm 10  ioooA.  Results for the SCR currents are shown in Fig. 5.5. The slopes of the curves are not con stant, owing to the voltage dependence of WBE, but it is clear that all the base-side SCR recombi nation components have about the same ideality factor (n), and that this is considerably less than that of the emitter-side SCR recombination current (which is dominated by sRH,E)• Specifically, at VBE  =  thermore,  1.2 V, SCR,E = 1.90 and, adding all the base-side currents together, SCR,B =  1.14 at VBE = 1.2 V due to the effects of  =  1.19. Fur  These values are similar to  those reported elsewhere [90], and deserve further comment because SCR,B is so far removed from the “classical” value of n  =  2.  With reference to eqn (5.12) for the SRH current, because and  is so low and Nrat is  are both>> 1, leaving the atan term in eqn (5.12) to saturate at  1, Z,  t/2. The voltage depen July 12, 1995  121  dence of SCR,E is thus determined by the sinh qVI2kT term and n approaches 2. Contrarily, for SCR,B’  both Zk, and Z are generally  <<  1, so the atan term modulates the sinh term and reduces  the ideality factor from 2 towards 1.  100  10-2  4 io-  ‘ 1010  1012  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V) Fig. 5.5. Bias dependence of the SCR current from the emitter side, and the three components of the SCR current from the base side. Material parameters are taken from the start of Section 5.4. The width of the SCR on the base side of the heterojunction is much less than that on the emitter side, and this fact alone, via Nrat in the Z and Z , terms in eqn (5.12), would make 0 SCR,B <<SCR,E•  However, the much larger n 1 on the base side counterbalances this effect and al  lows the steeper-rising SCR,B current to exceed SCR,E beyond some forward bias. In the example shown in Fig. 5.5, this occurs around VBE = 1.45 V. This transfer from an n  2 slope to an n  1  slope in the SCR current does not occur in a homojunction device as there is no spatial change in 1 to inflate the current in the more highly-doped side of the junction. n In practical HBTs it is possible to imagine that the minority carrier lifetime in the highlydoped base will be less than that in the emitter. Indeed, photoluminescence measurements on maJuly 12, 1995  122  4x10 suggest that ‘c, 3 cm terial doped to 19 1  5Ops [951, and a value of 3Ops has been used to mod  el some experimental devices [90]. Fig. 5.6 shows that reducing the base-side t, to 5Ops causes SCR,.B  to exceed  at a bias of about 1.15 V. However, lest undue emphasis be placed upon  the significance of this change-over, note from Fig. 5.6 that SCR,B is always less than the quasineutral base recombination current NB• This indicates that, in practical devices, an observed change in base-current ideality factor from n  2 to n  1, will likely be due to a change from  Jc,gd0m1nated current to a JNB-dominated current. Only in circumstances where it is correct to attribute a much lower minority carrier lifetime to the base-side depletion region only, perhaps due to defects at the interface, can a situation be envisaged where SCR,B could dominate over NB’  and thus be responsible for the slope change to n  1, which is often seen experimentally. The  above point about the relative magnitudes of SCR,B and NB is an important one as it puts into practical perspective the theoretically-interesting fact that SCR,B has a different voltage depen dence to that of SCR,E  1  101  10-2  1  10-8  10-11  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V) Fig. 5.6. Gummel plot showing the importance of including the emitter- and base-SCR current components in the computation of the total base recombination current. Material parameters are from the start of Section 5.4 for two values of t,. July 12, 1995  123  While it is clear from the results of earlier work that AEp must be included in calculating SCR,B  [901, it is, perhaps, not evident how important it is to include SCR,B in the balancing equa  tion (5.21) to compute iXEp. Fig. 5.7 provides an answer for the material properties considered here. By not including SCR,B in eqn (5.21), yet using the subsequently-calculated tXE to eventu ally compute SCR,B’ leads to a result which is indistinguishable from that of the “full model”, where SCR,B is included in the balancing equation. This is a consequence of SCRB being much less than NB and J. However, also from Fig. 5.7, note that it is grossly incorrect to not include AEp in the calculation of SCR,B Because the electron quasi-Fermi energy splitting is so large for an abrupt junction [18], its omission leads to a large overestimation of SCR,B’ and, consequently, to a severe underestimation of the current gain. It is difficult to imagine a practical situation where it might be necessary to include SCR,B in the actual calculation of AE . A possible scenario is 1 one in which t, in the SCR is less than t, in the neutral base, perhaps due to interface defects, and that W is much larger than the usual  i000A.  The latter situation would reduce J, and the  former would increase SCR,B with respect to NB’ thus making SCRB become more prominent in eqn (5.21). The effect of these changes is shown in Fig. 5.8. Even though the gain has been re duced to a very low value, it appears that there is still no need to include SCR,B in the balancing equation. To summarise the results from the analysis of this section: it is necessary to include  in  the computation of SCR,B; but SCR,B need not be included in the balancing equation to estimate AEp; and SCR,B is not very important for devices based upon materials with the properties con sidered here, because SCR,B is usually less than either NB or  Of course, if parameters affect  ing Auger or radiative recombination in the SCR turn out to be greatly different than the values used here, then SCR,.B could become important One instance where SCR,B will definitely be larger than calculated here is in the case of HBTs which are compositionally graded at the base-emitter junction. The grading gives the junc tion a more homojunction-like character, so AE will be reduced, and SCR,B increased corre spondingly. However, because of the lower bandgap of the graded material in the emitter-side of the junction,  is increased and, therefore, SCR,E also. Thus it is not obvious whether SCR,B is any more important in graded-junction HBTs than it is in abrupt-junction HBTs. The results of  Parilch and Lindholm [90] suggest that SCR,E remains the dominant current. One situation in  July 12, 1995  124  3 o:x  Full model  2000 SCR,B  but  -  not in balancing eqn included  1000 not in balancing eqn /ndot SCR,B  w  .;  1.0  :i  Base-Emitter Voltage VBE (V) Fig. 5.7. Bias dependence of the current gain 13, showing the relative importance of including in the calculation of AE. Also shown is the dramatic error resulting from not including AE, in the calculation of SC.R,B SCR,B  10  8  in balancing eqn but AE included JSCR,BflOt  6-  4. Full model  2  SCR,B  7and 0.8  0.9  1.0  1.1  not in balancmg eqn not included 1.2  1.3  Base-Emitter Voltage VBE (V) Fig. 5.8. Bias dependence of the current gain 13 for the case of Wflb increased to 5000A and t, in the SCR reduced to 5 ps. Even in this extreme case there is little error in not including SCR,B in the balancing equation. July 12, 1995  125  which SCR,B could be increased without an associated increase in SCR,E is when recombination at the exposed base surface is important. Providing a reasonable expression for this surface re combination current were available, it could be added to the right-hand side of eqn (5.22) and used in the current balancing to compute AEj. However as can be deduced from Figs. 5.7 and  5.8, the inclusion of another component of SCR,B will only effect the estimate of AE if this new component is comparable in magnitude to  106  •  •  Wb  3 io c-’  100  Wflb  -  •  •  = lOOnm  = lOnm  -  .  ..:  -  SRH,E  3 io  Rec,B  ....•-‘ -  Wflb=lOnm ,..7  io-  io  ......  ...  7.....  10  io jRec,B Wab = •  1012  0.8  0.9  •  1.0  ////  w,=1oomn Wb=1Onm  lOl  lOOnm  108 •  1.1  •  1.2  1.0  •  1.3  -  1.4  1.4  1.2 •  -  1.6  •  1.5  1.6  Base-Emitter Voltage VBE (V) Fig. 59. Effect of changing the neutral base thickness Wflb when the CBS is responsible for cur rent-limited-flow. Lowering W,th leaves and sRgE unchanged, but results in the reduction to the base side recombination current Rec,B (= SC.R,B + J). Under high bias, where Rec,B domi nates, 13 increases with reductions in W,,,. While under low bias, where SRflE dominates, 13 is un altered by changes in W,,j,. Before leaving this section, it is interesting to see how current-limited-flow within the CBS leads to a mixing of the base and collector currents. For the HBT considered, the CBS is indeed responsible for current-limited-flow, so that  Jc  jCBS 1  Thus, if the neutral base transport curJuly 12, 1995  126  rent were increased by reducing Wflb,  Jc would remain unchanged because the CBS already rep  resent the bottleneck to charge transport through the device. However, the reduction to Wflb does have an effect on the device. Fig. 5.9 shows that the base-side components of the base terminal current are decreased by a reduction to Wflb. This decrease occurs due to a reduction of ‘y in eqn (5.29) because relatively speaking, a shorter neutral base will provide fewer occasions for recom bination. Therefore, opposite to what occurs in BJTs, the mixing of the collector and base currents due to current-balancing has coupled Wflb to the base instead of the collector current. Finally, for the sake of completeness, Fig. 5.10 replots the currents displayed thus far using the linearised W(x) from eqn (5.11) against the currents obtained with the full potential from the depletion approximation of eqn (5.8). As can be seen in Fig. 5.10, the error is indeed slight, and will be smaller than the uncertainty in the recombination parameters themselves. The linear NJ(x) is not required to solve the radiative recombination current, so there is no approximation used.  100  3 io106  1 01012  10-15  10-18  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  Base-Emitter Voltage VBE (V) Fig. 5.10. Comparison of the recombination currents when ji is given by the depletion approxi mation in eqn (5.8), and when it is given by the linearisation of eqn (5.11) (IEf is included).  July 12, 1995  127  5.5 Simple Analytic Diode Equations For the purpose of including the various SCR recombination current components in HBT device simulators, it would be convenient if a simple closed-form solution for iXEp existed. Fur ther, if the various current components could be expressed as diode-like equations, then their rep resentation in circuit simulators such as SPICE would be greatly facilitated. In this section, the approximations that need to be realised to effect these simplifications are discussed. The starting point for the reduction of eqns (5.12), (5.17) and (5.20) to diode-like expres sions is to examine the relative importance of the Z-terms which appear in the expressions for the SRH and Auger recombination currents. Fig. 5.11 shows the results from the full model calcula tions. From this figure, it appears reasonable to state that Z <<Z , 0  <<  1, 4>> 1  >>  Z, and generally  ZZ > 0 > 1. The Z-terms Z,, Z ,0 0 Z , , Z, are representative of the amount of recombination at x,, 0, 0 and xi,, (see Fig. 5.1), respectively. For the condition Zk <<Z, to remain valid, the depletion region on the base-side must not be vanishingly small. This can be ensured by having the doping density ratio NA/ND  30. Contrarily, there is a lower limit to the allowable value of NA/ND, below  which the recombination on the base-side of the depletion region becomes large and the inequali ty Z(),  <<  1 is violated. This limit is NA/ND  3. Therefore, keeping within the range 3  NA/ND  30, and following the usual practice of expressing WBE and €) by their equilibrium forms, eqns (5.12) and (5.17) reduce to  JsRH,BCs  riXE NDnP exp[ no,pni, ii 1Ifl  SRH,E  Cs  2t  qNVji  kT  JexpL  —  zXEf  kT  jqV  2kT  (5.31)  [qV—1XEffl  2  Aug,B  —  CSnI,PAP,PNAexp  L  1Aug,Esn4n,nNDP[  kT  q VBE kT  I  where C  =  Writing the radiative recombination currents in eqn (5.20) in similar form, gives July 12, 1995  128  ____  Rad,B  q CsVb 2 q VBE — IXEffl nipBp(l—Nrat)exp[ kT kT 1 V 5 qC  Rad,E  2  kT  ,vBnNratP  L  kT  Using these diode-like equations, along with the expressions for NB in eqn (5.25), (5.26) and  ThT =  in eqn  Ff in eqn (5.23) in the balancing equation of eqn (5.21), yields a convenient ex  pression for t.Ep; i.e., (VbZ VBE) kT  8 qV  —  e  kT  —  Recom (VBE) +  q’y’uNe  S,Recom +  qyuNe  +  e  (533)  —  kT  +  qDn / 0 W  e  where 8 qV I  (TI  ‘  Recom” “BE) Recom  — —  =  i  JS,SRBBe  nSRH,BkT  qV +.ISAU B 8 e  NDn. C ‘‘ e nO,p  Wflb, e  =  BO  =  Aug,BkT .i. I -I-.IsRaBe  “S,SRH,B + SAugB + SRa,B —  =  qV  Lflbtanh  1,  I \.  qN V 1 j  kT  +CsnPAPPNA+  n  qCV. flpBp(1Nrat) kT 1  b nb  The values for the saturation currents and ideality factors in eqn (5.33) can be found either through a statistical fitting method, or from the analytic diode equations in eqns (5.31)-(5.32). Note that the n factors appearing in eqn (5.33) are independent of AEp. Their values, based upon the diode forms in eqns (5.31)-(5.32) are: SRH,B  =  liNrat;  Aug,B  =  1;  RadB  =  1.  A comparison of the predictions of the diode forms in eqns (5.3 l)-(5.32) with results from the full expressions in eqns (5.l2),(5.17), and (5.20) is shown in Fig. 5.12. The agreement is very good, with the only discrepancies occurring at very high forward bias. As VBE approaches Vbj, the diminish ing depletion-region thickness becomes a factor in that the depletion approximation no longer holds. Thus, for values of VBE near Vbj, the voltage dependence of WBE needs to be included, and the as sumptions regarding the relative magnitudes of the Z functions re-addressed. July 12, 1995  129  106 . 4 i0  Zn  i02. N l0 10  1c  10  Zp  -.  106  ....-  1.0  1.2  1.4  I Vbl  1.6  Base-Emitter Voltage VBE (V) Fig. 5.11. Z-functions as computed from eqn (5.13) when using the material parameters from Section 5.4.  100  -  Exact Diode 102  ..  -  SCR,E  10  ,..  -  ..._/  -  101 .......-...  -  ......-  Rad,B  ......  108  101  10 1W  1012  -  Rad,B  SRH,B  •  0.8  0.9  Aug,B  1.0  •  &R,E  •  1.1  •  1.2  31.4 io1.3  •  1.4  1.5 1.5  -  1.6  •  1.6  Base-Emitter Voltage VBE (V) Fig. 5.12. Comparison of the full model and “diode-like” expressions for the SCR currents. The high-bias region of the figure is enlarged in the inset with SRH,B and Aug,B omitted for clarity. July 12, 1995  130  If, as found to be the case for the material parameters used here, it is not necessary to in clude  SCR,B  in the balancing equation, then the  and  Recom  S,Recom  terms can be omitted from  eqn (5.33). Finally, for most abrupt HBTs, the CBS represents the bottleneck to charge transport. In addition, for cases where f3>> 1, then T = Jm and eqns (5.29) and (5.33) can be further simpli fied to give (VbZ kT  qI1Ef =  DflnBo  q  —  VBE)  NratVbi+ (lNrat)VBE  kT  Wflb e’’D  Substituting this expression for  into the diode forms in eqns (5.31 )-(5.32), gives overall ide  ality factors for the base-side SRH, Auger and radiative currents of: lI( Nrat 2  —  1), liNrat and 1/  Nrat respectively (where the bias dependence of the tunneling factor y is not included). From this study of space-charge region recombination currents in a typical A1GaAs/GaAs HBT, it can be concluded that: 1. recombination currents in the base-side SCR are generally less than the neutral-base current and, therefore, need not be included in the current-balancing equation used to compute the qua si-Fermi energy splitting AEp, at the base-emitter junction; 2. however, when subsequently computing the base-side SCR currents, AE must be taken into account if the gain is not to be grossly underestimated; 3. the ideality factor for the base-side SCR currents is closer to 1, than to the normally-used value of 2; 4. a simple, yet acceptably-accurate analytical expression for AE can be derived; 5. the base-side SCR currents can be accurately represented by diode-like expressions, so facili tating their implementation in SPICE-style circuit simulators.  July 12, 1995  131  CHAPTER 6 The SiiGe HBT  July 12, 1995  132  The previous chapters have presented a collection of models for the calculation of the trans port and recombination currents in HBTs. Chapter 2 presented the generic models for current transport in an arbitrarily shaped device where there can be any number of sub regions within the defined regions of the emitter, base and collector. Chapter 4 presented the transport models for the movement of carriers through a forward biased pn-junction under the influence of a heterojunc tion. Chapter 5 presented the models for the recombination currents that occur both in the neutral regions of the device (specifically the base and the emitter), and the forward-biased EB SCR. Also included in Chapter 5 were models for the transport of charge through the neutral regions of the device. Finally, Chapter 3 presented the models for the calculation of the base transit time based upon an optimisation of either the base doping, or the base bandgap, or both. In all of the work presented thus far, no assumptions have been made that depended upon a specific attribute of a given material system. Thus, the models contained within this thesis are general, and may be ap plied to the study of an arbitrary HBT created within an arbitrary material system. Even though the models presented within this thesis are indeed applicable to any material system, whenever an analysis of a specific model was performed, the material system of AlGai..As was invariably chosen for the study. The reason for choosing the MGai..As material system is that current-day technologies for HBTs prefer this material system. The dominance of the MGai..As material system stems mainly from the fact that the lattice mismatch over the us able range of Al content (i.e., 0  x  0.45) is under 0.07% [61]. This nearly ideal lattice match al  lows for an arbitrary film thickness because there will be virtually no strain placed upon the lattice at the heterojunction. Coupled with the lattice-matched characteristic, the AlGai..As material system can also provide for large changes to the bandgap  (AEg)  [61]. However, compound semi  conductors like GaAs and AlAs have numerous undesirable features when it comes to manufac turing. The A1Gai..As material system lacks a usable native oxide, is a poor thermal conductor, cannot be pulled into wide ingots which results in small wafer diameters, is brittle, suffers from a high defect density, cannot employ ion implantation for bipolar devices, exposed surface layers have high recombination velocities, cannot be used in low-power applications because of the large Vbj inherent with large bandgaps, does not etch easily and generally lacks an abrupt end-point de tection for etching, and finally is expensive to manufacture. Given all of these manufacturing and electrical drawbacks, however, the lattice-matched attribute is important enough to make AlGai.As the preferred material system for the construction of HBTs. July 12, 1995  133  Essentially all of the manufacturing issues with regard to the AlGai..As material system are solved by using the Sii.Ge material system: save one issue. At issue with the Sii..Ge mate rial system is its large lattice mismatch. The Ge lattice is 4.2% larger than the Si lattice [96]. Even if the Ge content is constrained to be under 20% (i.e., 0  x  0.20) there would still be a 0.84%  lattice mismatch between a Si 0 02 Ge film and a Si substrate. The issue with a lattice mismatch of 8 around 1% is that to commensurately place an epitaxial film upon a given substrate would result in a strain within the film that would be large enough to tear the film apart [97-99]. If strain were allowed to tear the film and form dislocations, then deep states would form along the heterojunc tion interface which would greatly enhance recombination. Since the heterojunction will be formed in the middle of the EB SCR of the HBT, a plane of recombination centres at the hetero junction would result in an intolerably high base current; large enough to reduce  below 1.  There is no physical way to alter the bulk lattice constant of a material or alloy. However, if the epitaxial film is grown thin enough and at a low enough temperature, it will conform to the substrate [99]. Under such conditions, the epitaxial layer is said to be commensurately strained to fit the substrate, and the layer itself will be pseudomorphic [99]. Pseudomorphic films are thus strained in order to maintain the in-the-growth-plane crystalline structure of the substrate. The key to obtaining a pseudomorphic film is to ensure that the layer thickness is below the critical thick ness h [99]. However, in order to maintain a pseudomorphic film, and ensure that it does not re lax back to its bulk lattice constant, subsequent exposure of the layer to high temperature environments must be severely limited. In the past 5 years, great progress has been made at IBM in the quality of Sii.Gepseudomorphic films [31]. These developments have shown great poten tial regarding operating speeds [100-102], so much so that many other companies including the Japanese at NEC [1031 are developing SiGe IC processes. Through the recent successes regarding the high quality growth of pseudomorphic SiiGe films, the Sii..Ge material system is fast be coming a practical alternative for the manufacture of HBT-based ICs. In fact, with the massive in stalled base of Si-based IC manufacturing, coupled with the ability to integrate Sii..Ge films into the process, it is expected that Sii..Ge will rapidly displace MGai..As as the preferred material system for the manufacture of HBT-based ICs. This chapter will apply the general models obtained from the previous chapters to the study of HBTs based within the Sii..Ge material system. Due to the complex nature of Sii..Ge under  July 12, 1995  134  the influence of strain, a number of extensions to the work of previous chapters is necessary. Most importantly, due to the indirect nature of the Sii..Ge energy bands, there are six separate conduc tion band valleys [104] (compared to only one valley in a direct semiconductor such as GaAs). Each of these conduction band valleys will transport electrons. Since strain breaks the degeneracy of the six conduction band valleys, it will become important to consider electron transport within each valley separately. Once the needed extensions to the models of the previous chapters have been determined, a study of current-day SiGe HBTs can be performed. Furthermore, it will be shown that the use of strain can be turned into a tool for the HBT developer, instead of being seen only as a liability in terms of critical layer thickness.  6.1 The Effect of Strain on Sii..Ge The use of pure unstrained crystals of Ge in the formation of SiGe HBTs is possible, but due to the large lattice mismatch (—4%), would result in a high defect density at the heterointerface, severely degrading device performance. Furthermore, if only pure Si or Ge crystals were used in the formation of HBTs, there would be a considerable limitation imposed upon the ability to engi neer the bandgap within the HBT Instead, pseudomorphic Sii.Ge films, that are commensurate ly strained to become lattice matched to the substrate (which is pure Si in present day devices), are used. These pseudomorphic Sii..Ge layers will remain strained without relaxing as long as the layer thickness remains below the critical thickness h [97-99,105]. (For Si 030 grown on Ge 070  { lO0}  Si substrates the critical layer thickness is 600A, while for 045 Si it is only bOA). 055 Ge  Thus, unlike MGai..As, which is essentially lattice matched to GaAs and thus has no critical layer thickness, SiGe HBTs can have considerably less freedom in the choices for layer thickness es. The key to manufacturing SiGe HBTs is the commensurate growth of strained Sii..Ge lay ers to the underlying substrate. However, the strain in the plane of growth results in a distortion of the crystal structure that breaks the cubic symmetry and causes the crystal unit cell to become tet ragonal. With the breaking of the cubic symmetry comes a change to the dispersion relations for the energy of the Bloch electrons versus wave vector k. The most important effect of this symme try breaking is the relative change to the energy of the conduction band minima and the valence band maxima in k-space.  July 12, 1995  135  Constant energy surfaces near to the conduction band minima for pure unstrained Si and Ge are shown in Fig. 6.1. Looking at the case of Si, there are six separate but degenerate conduction band minima located along the (100) directions at the A point (which is 80% from the zone centre at F to the Brillouin zone edge at X). For alloys of Sii..Ge, these six minima are dependent both on the alloy content x and on the state of strain. Take, for an example, Sii..Ge grown on a Si sub strate with the direction of growth parallel to [001]. As x moves from 0 to 1, the Sii..Ge layer moves from an unstrained cubic structure to a compressively strained tetragonal structure [99,105]. As the strain decreases from zero (compression being negative strain), we find that the degeneracy of the six minima is lifted [105-108]. The two minima aligned to the normal of the in terface plane (i.e., parallel to the direction of growth) remain degenerate and are raised in energy, while the other four minima parallel to the interface plane also remain degenerate but are lowered in energy. For the case of Sii.Ge grown on a Ge substrate with the direction of growth still par allel to [001], the situation is reversed. As x moves from 1 to 0, the Sii..Ge layer moves from an unstrained cubic structure to an expanded, tensile-strained tetragonal structure. For this case of tensile strain, as the strain increases from zero, the two minima normal to the interface plane are lowered in energy, while the other four minima parallel to the interface plane are raised in energy. Thus, we find that there are now two types of A conduction band minima in a strained Sii..Ge film; those parallel (which will be termed E) and those perpendicular (which will be termed E) to the interface plane. Therefore, depending on the sign of the strain tensor (i.e., either compres sive or tensile), either the E or the E bands will form the ultimate conduction band. The valence band also suffers considerable change due to the symmetry breaking caused by strain. The valence band of pure, unstrained Si and Ge (or for that matter, all semiconductors), is composed of what should be three degenerate bands. These three bands are the light-hole (lh), heavy-hole (hh) and split-off bands (so). When the interaction of the electron’s internal angular momentum (spin), is coupled with its orbital angular momentum (termed spin-orbit coupling), the degeneracy of the so band is lifted [109,110]. The resultant interaction leaves the lh and hh bands degenerate with the so band maxima moved to a lower energy (see Fig. 6.2). The symmetry break ing caused by strain goes on to lift the degeneracy of the lh and hh bands. As in the conduction band, the valence band maxima is dependent both on the alloy content x and on the state of strain [105-108]. Returning to the case of Sii.Ge grown on a Si substrate, with the direction of growth parallel to [001], as x moves from 0 to 1 the Sii..Ge layer experiences an increasing compressive July 12, 1995  136  I/ \ 1.f1<H r L_ /\‘\.  ,..•‘  N /•......\ Si  \‘  \  ‘I  Fl  \\/7c  .1  /  \I )  \  x  F,.,,”  ‘\.  kL/ I  I\  ‘----—____\ ‘  II  /  Ge  L  Fig. 6.1. First Brillouin zone showing (in k-space) the constant energy surfaces near the bottom of the conduction band for Si and Ge. Also shown are the designations for the symmetry points and the degenerate bands E and E in strained Sii..Ge with the growth direction along [001]. July 12, 1995  137  strain. The result of compressive strain is an increase in the maxima of the hh band relative to the lh band, accompanied by a decrease in the maxima of the so band relative to the th band. Under tensile strain, however, the effect is reversed for the lh and hh bands (but not the so band). Return ing to the case of Sii..Ge grown on a Ge substrate, with the direction of growth still parallel to [001], as x moves from 1 to 0 the Sii.Ge layer experiences an increasing tensile strain. The re sult of tensile strain is an increase in the maxima of the th band relative to the hh band. However, there is still a decrease in the maxima of the so band relative to the hh band. Therefore, strain eliminates the degeneracies of all the valence bands, with the so band always moved to lower en ergies. However, depending on the sign of the strain tensor (i.e., either compressive or tensile), ei ther the lh or the hh band will form the ultimate valence band.  E hh band  so band  k  Fig. 6.2. Valence bands in unstrained Sii.Ge. The light hole (lh) and heavy hole (hh) bands re main degenerate for all values of Ge alloy composition x (only in the bulk state where there is no strain present). However, the split off (so) band maxima changes in energy with alloy content, where A(x) 0.044 + 0.246xeV. The previous paragraphs have outlined that the energy of the conduction band minima and the valence band maxima change under the effect of strain, while their position in k-space remains unaltered. However, it is also important to ascertain the effect of strain on the shape of the band in k-space, as this will set the effective mass which determines the velocity of the carrier and its probability for tunneling. Considering the valence band first, the effective mass for the lh and hh July 12, 1995  138  bands in pure unstrained Si and Ge are quite different. Therefore, as the Ge alloy content in the Sii.Ge layer changes, there must be a change to the shape of the band in k-space regardless of the strain state. To account for this varying shape of the lh and hh bands, a linear interpolation be tween the experimental values for the lh and hh masses in Si and Ge is used to arrive at the appro priate masses for the SiiGe layer [111]. It is further assumed that the effect of strain is negligible with regard to the shape of the band in k-space. This leads to: mhh  =  0.49  —  0.21x  (6 1)  —O.ll mffi=O.l x 6  where x is the Ge alloy content, and the masses are a fraction of the electron rest mass me (the Si and Ge hole masses are based upon [96]). The lh and hh effective masses are maintained separate ly instead of combining them into an effective density of states mass because under the influence of strain, the degeneracy breaking will result in a change to the effective density of states mass (see Section 6.3). For the conduction band, it is assumed that the conduction sub-bands E and E do not change shape with either a change in the Ge alloy content or the state of strain [107,112]. To first order in the strain tensor there must be a change to the effective mass for the electrons because the reciprocal lattice vector is being changed. However, this change will be relatively small as the maximum change to the reciprocal lattice vector is 4.2% over the entire range of Ge alloy content. As for the effect of the Ge alloy content, it is important to realise that Si and Ge (and therefore Sii..Ge) have conduction band minima at A and at L. The difference between Si and Ge is that the A minima form the ultimate conduction band in Si while the L minima form the ultimate con duction band in Ge. For Sii.Ge the A minima typically form the ultimate conduction band. However, if the Ge alloy content is high enough, then the ever-present L minima within the Sii.Ge alloy will form the ultimate conduction band [113]. It is therefore postulated [107,112] that the electron effective mass for the E and E bands are the same as that for the A minima in Si, while the effective masses for the L minima are the same as the Ge effective mass; i.e., [96];  m(A) (A) 1 m  =  0.19  =  m(L) (L) 1 m  =  0.98 0.082  =  1.64  (6.2)  Therefore, there is no change to the effective electron mass, for a given band, with either a change July 12, 1995  139  to the Ge alloy content or the state of strain. However, in similar fashion to the valence band, the effective density of states mass will change with strain depending on which band forms the ulti mate conduction band. The qualitative features that strain and Ge alloy content impart to the Sii..Ge layer have been presented. Using empirical deformation-potential theory [114-116], the quantitative features are now presented. The reason for using empirical deformation-potential theory, where the defor mation potentials are measured and not derived from first principles, is that current-day solid-state quantum mechanics is not sophisticated enough to predict the desired results with any reasonable tolerance (errors on the order of 1 eV are standard). To this end, the problem of including the strain state and the Ge alloy content is broken down into two independent problems. First of all, experimental measurements of the Sii.Ge bulk bandgap (i.e., unstrained) are performed over the entire range of 0  x  1 to produce the function Eg (x). Thus, Eg (x) contains all of the Ge alloy ef  fects. Then, empirical deformation-potential theory is used to determine the amount of degenera cy splitting that occurs within the sub-bands of the conduction and valence bands due to the addition of strain. Adding together Eg(x) with the results from deformation-potential theory pro duces the total change to the various bands within the Sii..Ge layer. Beginning with the calculation of Eg(x), in the seminal works of [113,117] the necessary exper imental measurements on the bandgap of bulk and strained Sii.,Ge have been performed. It has been found that for x < 0.85, the A minima form the ultimate conduction band minima in Sii..Ge. Howev er in the range 0.85 x  1, the L minima form the ultimate conduction band minima in bulk Sii..Ge.  Concentrating on the A minima alone, then using a quadratic fit to the data in [113] produces: Eg(X)  ( =  t  Eg,  1  2 x0.732 0.51446x+0.3h164x  EgsiO.l O 5 l  (6.3) 0.0813x  x>.0.732  where xis the Ge alloy content, ES 1 is the bulk Si bandgap, and all values are in eV. Eqn (6.3) gives the Sii..Ge bulk bandgap from the top of the valence band to the bottom of the A minima in the conduction band. Caution must be exercised when using eqn (6.3) for x > 0.85 as the L minima will form the ultimate conduction band in bulk Sii..Ge material. However, the strain imparted to the Sii..Ge layers used in HBTs is generally sufficient to reduce some of the A minima below the L minima even as x approaches 1 (i.e., pure Ge) [106]. For this reason it will be assumed that the L minima can be ignored. However, the energy of the L minima change much July 12, 1995  140  more rapidly for a given change to x than the A minima do. Thus, it would be possible to achieve larger band offsets using the L minima versus the A minima, or to achieve the same band offsets but with a smaller change in x (which would help address the critical layer thickness problem). The drawback to using the L minima is the substrate would have to be essentially Ge, and not Si, grown along (111). But, given the much higher mobilities in Ge versus Si, then SiGe HBTs based upon the L minima should outperform current SiGe HBTs based upon the A minima. Eg(X)  in eqn (6.3) solves the first problem of including the alloy effects into the conduction  and valence bands of Sii.Ge. The second problem of including the effect of strain is now ad dressed. Fig. 6.3 shows the effect of in-plane biaxial tension and compression. Fig. 6.3(a) shows the case where the substrate lattice constant as is larger than the alloy lattice constant aa. The commensurate growth of the alloy layer to the substrate forces the in-plane alloy lattice constant to match a . In so doing, a biaxial in-plane tension results in the pseudomorphic alloy film. In an 5 attempt to lower the energy contained within the film, the out-of-plane alloy lattice constant com presses below aa. The pseudomorphic alloy layer will then have a larger in-plane lattice constant when compared to the out-of-plane alloy lattice constant, leading to a tetragonal crystal instead of a cubic one. Contrarily, Fig. 6.3(b) shows the case where the substrate lattice constant a is small er than the alloy lattice constant aa. The commensurate growth of the alloy layer to the substrate forces the in-plane alloy lattice constant to match as. In so doing, a biaxial in-plane compression results in the pseudomorphic alloy film. In an attempt to lower the energy contained within the film, the out-of-plane alloy lattice constant expands past aa. The pseudomorphic alloy layer will now have a smaller in-plane lattice constant when compared to the out-of-plane alloy lattice con stant, which again leads to a tetragonal crystal instead of a cubic one. It is the fact that the pseudo morphic alloy layer has broken the cubic symmetry of the original lattice that leads to the changes in the conduction and the valence bands. The initial applied stress tensor to the alloy layer can be viewed as a uniaxial stress accompanied by a uniform hydrostatic pressure applied over the entire cell. If the in-plane interface is parallel to the x-y plane, with the direction of growth parallel to the z-direction, then the initial applied stress is [1081: 0  t  Applied stress  =  =  0  ti +  (6.4)  0 —t  July 12, 1995  141  where the growth is in the [001] direction, and a blank location in the tensor is zero. The first term on the right of eqn (6.4) is the hydrostatic pressure applied to the overall cell, while the second term is the uniaxial stress, of opposite direction to the biaxial stress, applied to the out-of-plane lattice constant. Therefore, the symmetry breaking of the alloy’s unit cell occurs along the direc tion of growth (i.e., the z-direction). Thus, any changes to the energies of the A conduction band minima will leave the A minima along [001] and the  [OOT] directions degenerate (i.e., E), as well  as the A minima along [010], [OTO], [100] and the [TOO] directions degenerate (i.e., E).  H  (a)  Haa  ‘Alloy’  WEww I  Alloy —  —  —  —  —  —  —  —  —  —  —  —  —  —  Substrate  -111.1  -a  (b) I I I  — ————  +  WWJEE Substrate  mm  —  —  —  —  —  —  —  —  —  —  —————  —Alloy  —  —  —  —  Substrate  —  —  —  —  —  —  —  —  —  —  III  -4-a --a Fig. 6.3. Commensurate growth of the SiiGe alloy layer to the Sii.Ge substrate, leading to a pseudomorphic alloy film. (a) the substrate lattice constant a 5 is larger than the alloy lattice constant aa. The resultant biaxial tension, which results from aa expanding to fit a , distorts the 5 out of plane alloy lattice constant by compressing it. (b) the substrate lattice constant as is smaller than the alloy lattice constant aa. The resultant biaxial compression, which results from aa com pressing to fit as, distorts the out of plane alloy lattice constant by expanding it. July 12, 1995  142  The final diagonal components e, e and e of the strain tensor, after the layer becomes pseudomorphic, are given by the relative difference between the final pseudomorphic lattice con stants and the initial bulk values [106,107,112,114]. Given that we are dealing with systems that are lattice matched to { 100 } substrates, and that the direction of growth is [001], then the strain tensor is: 3 a =  ]  [exx  aa  —  aa  =  asaa  (6.5)  ezz  1+v aa-as 3 a  exx+(i)  where v is the Poisson ratio (which is equal to 0.273 for Ge and 0.280 for Si [107], so on average is 0.277 for Sii..Ge). The lattice constants aa and as are obtained by a linear interpolation be tween the bulk lattice constants for Si and Ge giving: aa 3 a  5.43 5.43  = =  0.23xaA + 0.23x A 3 +  (6.6)  where xa is the Ge content in the alloy layer, and x is the Ge content in the substrate layer. In order to determine how E and E respond to strain it is instructive to define an average conduction band energy  .  The reason for defining  is that depending on the direction of strain,  either E or E will form the ultimate conduction band E; so using E as the reference would be come mathematically cumbersome. —  =  is given by the weighted average of E and E; i.e., 2 + 4 4E 2EC C6  =  Using deformation-potential theory, the change to =  where d and  (d +  ) 1: ë  =  (d +  2 + 4 2E C3EC (i.e.,  (6.7)  A) due to strain is [106,107,112,114]:  E) (e +  (6.8)  +  are the dilation and uniaxial deformation potentials respectively. Further the  change in energy for a specific A conduction band minima is given by: =  (6.9)  [d1+EUtàI}]:ë  where à is the unit vector parallel to the i ‘th A conduction band minima, and  { } denotes dyadic  product. For example, the change in the energy of the A conduction band minima along [100] is given by: July 12, 1995  143  100  :ë  +  =  = d(eYY+eZZ)  +  (d+U)eXX.  0 Finally, using eqns (6.9), (6.8) and (6.5) where and  as well as E and  =  then the energy difference between E  are given by:  1 E’°°  =  1 E°’°  1  =  1 —AE AE’°°  =  ({aa} —  1/3 0  =  e = e,  2  1/3  —  0  1  (6.10)  1/3) 1,_ = —e±  and E  =  1 E°°  } 1 — AE = ( {aà  =  0  1) :ë  1/3  —  0  =  —  1/3  1  :ë 1/3)  =  (—  (e  +  e)  2  (6.11)  +  2_ =  where  e±  —  e,  and  u(xa) =  9.16  +  6xaeV [106]. Eqns (6.10) and (6.11) give the change 2 O.  due to strain in the energy of the band minima for E and E respectively, relative to  .  Thus, E  and E are used both as a label and as a material parameter. Observation of eqns (6.10) and (6.11) confirm the general statements given earlier in the section regarding the changes to the conduction band due to strain. For compressive strain in the alloy layer; xa  >x 5  so that aa  >  a and eqn (6.5) has it that  eqns (6.10) and (6.11) have it that E <0 and E  >  e±  e  — e > 0.  Since  <  0, then  0, which confirms that under compression E  forms the ultimate conduction band. Contrarily, for tensile strain in the alloy layer, aa  >  as and eqn (6.5) has it that e± < 0. Then eqns (6.10) and (6.11) have it that E  Xa <x >  so that  0 and E  <  0, which confirms that under tension E forms the ultimate conduction band. With the changes to the conduction band due to strain determined, the valence band is now solved for. The designations for the hh, lh and so valence bands are based upon the valence band July 12, 1995  144  strain Hamiltonian [118]. To this end, it has been determined that the quantum numbers for total angular momentum J as well as magnetic moment (spin) m remain unchanged with the applica tion of strain. This leads to the hh band designation of  or I ; ±) for short; the lh  = 1 IJ= ; m  band designation of I ; ±); and the so band designation of I ; ±). The solution of the valence band strain Hamiltonian [118,107] produces:  I;±)  2 E 2 E  —  —  2  2  =  E  =  =  =  2  2 hh_  —  —  —  E  —  Du(xa)e±  u(Xa)  —  (e  —  e)  (E + A(Xa)) +  J9 (Er) 2 + A (Xa) 2  (E + A(Xa))  J9  —  —  2  (Er) + A (Xa) 2  —  (6.12)  2E!hA(Xa)  2E!bA(Xa)  where Du(Xa) is the valence band deformation potential equal to 3.15 + 4 l.l x aeV [106], and A(Xa) is the split off energy, defined in Fig. 6.2, which is equal to 0.044 + 0.246xaeV [107].  Using a similar technique to the one used for the solution of the conduction band, an aver age valence band energy  is defined and subsequently used as the reference point for all valence  band energies; i.e., —  =  Elth+ETh+E80 V  V  i  V  =  (6.13)  A(Xa).  It is interesting to note that , defined in eqn (6.13) is independent of the applied strain. Also, be cause  is not zero, the valence band energies in eqn (6.12) are not using , as their energy refer  ence (substituting eqns (6.11) and (6.10) into (6.7) gives  =  0, which shows that  is indeed the  energy reference for the conduction band). Observation of eqn (6.12) shows that under the condi tion of zero strain (i.e., e± erence for eqn (6.12) is not  0), then  E1 =  =  0, and E°  =  A(xa). Therefore, the energy ref  but the valence band edge of bulk Sii..Ge. The reason for using ,  will become obvious when the band offsets at a heterojunction are determined in Section 6.2. Eqns (6. 10)-(6. 12) determine the effect of uniaxial strain, due to the second term on the right-hand-side of eqn (6.4), on the conduction and valence bands of Sii.Ge. Eqn (6.3) deter mines the effect of the Ge alloy content. Finally, the effect of the hydrostatic force, due to the first term on the right-hand-side of eqn (6.4), is determined. The hydrostatic force results in either a net decrease or increase in the total volume of the crystal’s unit cell. A volume change in the unit cell will be accompanied by a change in the absolute energy of the conduction and valence bands. July 12, 1995  145  This net change in absolute energy is best determined by calculating the differential change to and  and i ,). Eqn (6.8) solves for 1  (i.e.,  and in a similar fashion A, = al: ë, where a  is another deformation potential that is characteristic of the material [105,107]. Put together, the hydrostatic change to the bulk Sii..Ge bandgap is:  /XEg  =  AE  —  AE  =  d-’3 (  —  a) 1: ë  =  (d +  —  a) (e +  +  e)  (6.14)  —a = 1.5— 0.l9xaeV [106].  where  Eqns (6.1)-(6.3), and (6.10)-(6.14) together determine the effect of Ge alloy content and strain on the conduction and valence bands. Specifically, the bandgap of a  Sii.Ge  alloy layer  commensurately strained to a Sii.Ge substrate is: Eg(Xa, x)  =  Eg(xa)  It must be remembered that for xa  >  + L\Eg +  min(E, E)  —  max(E, E).  (6.15)  0.85 it is possible for the L conduction band minima to be  come the ultimate conduction band. Therefore, the use of eqn (6.15) is valid for xa there is sufficient strain to ensure that the  ii  >  0.85 only if  minima, and not the L minima, still form the ultimate  conduction band. Fig. 6.4 plots the Sii.Ge bandgap for a variety of substrate cases. The most striking fea ture of Fig. 6.4 is the effect of strain on the bandgap. Comparing the bulk material bandgap to any of the other strained cases shows that the Ge alloy content of the Si ixpexa layer plays a far small er role than strain does in determining the bandgap. In fact, observation of the line for a pure Si substrate shows that Si 055 lattice matched to { 100} Si has a bandgap of 0.66eV, which is Ge 045 that of bulk Ge. The strange shape concerning the lines for material strained to substrates of , 025 Ge 075 Si  , 050 Ge i0 50  and Si-j 075 is due to the fact the material is shifting from a case Ge 25  of in-plane tension to compression. Take the example of a 5 OGeO substrate. When the Si0 O pseudomorphic layer has a Ge mole fraction in the range of 0  Xa  0.50, the layer is under in-  plane tension as the substrate has a larger lattice constant. Thus, as xa increases towards 0.50 the tension is decreasing and the bandgap will increase, with E forming the ultimate conduction band. When Xa  =  0.50 there is no strain and the bandgap will be given by the bulk value. Finally,  as xa increases past 0.50, the strain switches from in-plane tension to compression. When this change in the direction of the strain occurs, E forms the ultimate conduction band (this is why there is a corner in the plot, however, the E and E bandgaps continue on in a smooth fashion but do not form the ultimate bandgap). As xa increases past 0.50 the amount of in-plane compression July 12, 1995  146  continues to increase which reduces that bandgap once again. The essential feature of strain is that it always reduces the bandgap from the bulk unstrained value.  I  I  •  I  I  1.1 Bulk Material  10 0.9  on 25% Gb..,  .  \•..  0.8  -  on 50% Ge  .  0.7  ,.....,....•,,,,,,  on75%Ge  06 0.5 0.4 0.0  -  on GeX/ 2 on0%Ge  •  0.2  •  0.4  •  I  0.6  I  0.8  1.0  Germanium Mole Fraction Xa Fig. 6.4. Sii.Ge bandgap when grown commensurately to a variety of substrates oriented along (100). All values reflect the energy from the top of the valence band to the lowest A minima in the conduction band. The bulk material bandgap is for reference and is valid only forxa < 0.85; for Xa > 0.85 the bulk material line is not the ultimate bandgap but the bandgap to the A minima. Eqns (6.10)-(6.1l) give the conduction band energies of E and E relative to . Examina tion of eqns (6.10)-(6.11) shows that under zero strain, when e 1  =  0, E  =  =  0. Thus,  is the  position of the ultimate conduction band in the absence of strain. If the position of the unstrained conduction band is known, eqns (6.lO)-(6.11) will yield the offset to the conduction band due to any strain in the layer. Fig. 6.5 plots E and E relative to  using similar substrates as found in  Fig. 6.4. Observation of Fig. 6.5 shows the changes in E and E to be quite linear in terms of strain. Furthermore, whenever the pseudomorphic layer is under compression then E forms the conduction band, but when the layer is in tension then E forms the conduction band. Finally, E changes more rapidly than does E for a given increase in the amount of strain. July 12, 1995  147  0.50  •  •  •  on0%Ge 0.40  E  0.30-  2C E  on 25% Ge  on5O% Ge  020 L  0.10  on75%Ge -  0.00  -.-.....  -0.10 -0.20  ...-..  ...-....  .........-..-.  :.-•-  .--...-...  ...Sii75%Ge on5O% Ge on25%Ge  -0.30 0.0  •  0.2  I  0.4  •  I  0.6  on0%Ge I  0.8  1.0  Germanium Mole Fraction xa Fig. 6.5. E and E conduction band energies relative to the unstrained conduction band edge for SiiFe commensurately grown to a variety of substrates oriented along (100). The ulti mate conduction band edge will be formed by the band with the lowest energy. As was stated earlier, eqn (6.12) gives the energy offset of the hh, lh, and so bands relative to the unstrained valence band edge. Fig. 6.6 plots the hh and lh bands relative to the unstrained valence band using similar substrates as found in Figs. 6.4 and 6.5. The so band is not plotted be cause strain simply continues to lower the band peak even further, meaning that the so band will not be of any consequence regarding the transport of holes. Comparison of Fig. 6.6 with Fig. 6.5 shows that unlike the conduction bands, the valence bands respond in a non-linear fashion with re spect to an applied strain. Furthermore, there is not as large a change in the energy of the valence bands due to strain as there is in the conduction bands. Finally, whenever the Sii..Ge layer is under compression, then the hh band will form the ultimate conduction band; while under tension, the th band will form the ultimate conduction band. Finally, it is instructive to present a surface plot of constant energy in k-space, depicting the conduction bands in Sii..Ge under the influence of strain. Fig. 6.7 plots the surface of constant July 12, 1995  148  energy that envelopes the six A minima in Si 017 commensurately strained to (001) Si. Ge 083 Since the pseudomorphic Si 017 layer is under an in-plane compressive strain, then Fig. 6.5 Ge 083 shows that E will form the ultimate conduction band. The constant energy surface used in Fig. 6.7 is set at 209meV above the minimum in the E band. The energy separation between E and E for the case considered is 116 meV. As a result of the choice for the energy surface, the ellipses that represent  are reduced by 33% compared to the effipses that represent E. If a more realis  tic surface energy of 2kT (= 52meV at room temperature) were used instead of 200meV, then the E band would not be seen at all. This demonstrates the profound effect that strain imparts to the SiiGe layer.  0.20  0.15  0.10  0.05  0.00  -0.05  -0.10 0.0  0.2  0.4  0.6  0.8  1.0  Germanium Mole Fraction Xa Fig. 6.6. E and E’ valence band energies relative to the unstrained valence band edge for Sii..Fe commensurately grown to a variety of substrates oriented along (100). The ultimate va lence band edge will be formed by the band with the highest energy.  July 12, 1995  149  1.  1.00  1.00  k  Fig. 6.7. Constant energy surface plot depicting the E and E bands in Si 0 83 0 17 commensu Ge rately strained to (001) Si. The k-wave vectors are normalised to one-half the length of the recipro cal lattice vector. The constant energy surface is set at 209meV above the minimum in the E 4 band. The E band lies 116meV above the E band. The E ellipses have a longitudina1 extent o’f 0.8, while the E ellipses have a longitudinal extent that is 33% less than the E band, or 0.53. This section essentially presents a concise review of the relevant theories regarding the movement of the conduction and valence bands in Sii..Ge under the influence of strain. Further more, the most recent material parameters regarding deformation-potentials have been included. However, there is still considerable change occurring to the relevant material parameters of Sii..Ge at this time. As Sii..Ge becomes a more important material in mainstream commercial ICs, the need to ultimately obtain the relevant material parameters will force the solid-state com munity to finalise on the parameters. This process will most likely follow the course that July 12, 1995  150  MGaiAs took, in which a decade passed before the solid-state community settled on a firm set of material parameters. In any event, this section has clearly shown the profound effect that strain has on Sii..Ge; so much so that strain produces more of an effect on the bandgap than does the Ge alloy content.  6.2 Band Offsets in Sii..Ge Section 6.1 presented all of the relevant material parameters to describe the conduction and valence bands of a Sii..Ge alloy layer commensurately strained on top of a { 100) SiiGe substrate. This section will present the band offset models that predict the valence band and con duction band discontinuity at an abrupt heterojunction. Therefore, when the results of this section are combined with the results of Section 6.1, all of the relevant models for Sii.Ge regarding the position of the conduction and valence bands within a device can be determined. The seminal theoretical work on the band alignments between Sii..xex 1 and SiixGex (where the 1,r subscripts refer to the left and right films respectively), when commensurately strained on top of a { 100) Sii.Ge substrate, was done by Van de Walle and Martin [106,119,120]. They analysed a SiGe system in one dimension using a quantum mechanical model. To remove the issue of boundary conditions that would destroy the crystalline periodicity required to establish Bloch functions, they developed a supercell structure. The supercell structure had a unit lattice cell that was constructed of n Si atoms followed by n Ge atoms. By extending this unit supercell to infinity, though the Born-von Karman boundary conditions, Van de Walle and Martin were able to obtain the band offsets. In order to establish that the size of the supercell was large enough to ensure bulk material properties away from the heteroj unction, the band offsets were determined for a variety of n. Van de WaRe and Martin established that for n >5 the material was bulk-like away from the heterojunction. In fact, the shape of the Bloch electron’s wave function became bulk-like after moving only one lat tice constant away from the heterojunction. Therefore, Van de Walle and Martin concluded that the perturbing effect of the abrupt heterojunction was indeed localised to the space immediately sur rounding the interface. The main conclusion from the work of Van de Walle and Martin is that the average valence band offset A, between a pseudomorphic Si to Ge heterojunction, whether commensurately strained to either a { 100) Si or Ge substrate, is a constant of 0.54±0.04eV (where the Si er in energy that the Ge  is low  Numerous other individuals [12 1-124] have gone on to perform ex Julyl2,1995  151  perimental measurements of zVP with variations that are always lower than 0.54eV, and which are as low as 0.2eV. Recently, experimental measurements by Yu [125] have given E, = 0.49±0.13 eV. However, after performing an array of measurements on a variety of substrates (thereby changing the strain), it was found that A varied slightly with strain. The final results from Yu [125] were: AEv(Xai Xar)  where  Xa(, Xar  = (0.55  —  0.12x)  (6.16)  (Xai Xar),  and x refer to the Ge mole fraction in the left, right and substrate crystals respec  tively. Finally, Fig. 6.8 defines all of the energies and the offsets. At issue with eqn (6.16) is the considerable appeal to linear interpolation between material parameters for bulk Si and bulk Ge. To complicate things further, the material parameters that govern the conduction band and valence band movements due to strain have considerable variabil ity depending on which experimental method is used to obtain the results. At the moment there is no clear set of material parameters to use in order to determine the band offsets and movements within SiGe. The complexity of the SiGe system is quite high, however, it is essential that the ma terial science community finalise on a set of material parameters and models so that SiGe HBTs may be accurately simulated. Use of eqn (6.16) produces conduction band offsets AE that are far too large. Experimental measurements of AE [105,111,126,127] show that there should be no more than ±30meV of offset between Si ixapexa, and SiiGe grown on a pure Si { 100) substrate, where Xal and Xar can take on any value in the range of 0 to 1. Furthermore, recent measurements by Gan et. al. [128] have shown that AE should equal O. xaieV when xar = x =0. Use of eqn (6.16) produces 64 = 0.8OxaieV. By reducing LcP from 0.55eV back down to 0.49eV in eqn (6.16) produces: v(Xaiar)  = (0.49—0.12x)  (6.17)  (XaiXar).  Use of eqn (6.17) instead of eqn (6.16) reduces EiE to be no more than +48meV and -42meV (as compared to +30meV and -100meV), while also giving AE = O.7 xaieV. Finally, if Du(xa) in eqn 4 (6.12) is changed to 2.04  +  1.77xaeV [107] then IXE remains unchanged and AE = 0.68xaieV  The use of eqn (6.17) instead of eqn (6.16) is within the experimental error of the measurements in [125]. Further, eqn (6.17) when combined with Du(xa) = 2.04 + 1.77xaeV produces conduction and valence band offsets that match experimental observations closer than when the material val ues proposed within [125] are employed. Thus, there is no clear set of parameters as of yet for the July 12, 1995  152  modelling of the SiGe material system. However, the differences between the various models pre sented here is within 50meV. Therefore, in terms of the studies to be presented later on in this chapter, a small discrepancy of 50meV will simply cause a slight variation in the Ge alloy content of the various layers, but will not effect the ultimate function of the HBT.  E E  lh V  ,hh i—V  E Fig. 6.8. Conduction and valence band energies including all of the band offsets for a Sii3e 1 to a SiiGe heterojunction commensurately strained to a { l0O} Sii..Ge substrate. The des ignation of 1 and r refer to the left and right crystal respectively, where all A energies are referred to the crystal on the right. Eqn (6.17) provides the critical model that relates the band energies of two different SiGe crystals across an abrupt heterojunction. Once zVL, is known, then by using Fig. 6.8, all of the oth er relevant offsets can be determined by appealing to the models of Section 6.1. Using eqns (6.3), (6.1O)-(6.15), and (6.17) along with the aid of Fig. 6.8 yields: July 12, 1995  153  1 (A  +  =  A) + [max(E’ E’ 1) ,  —  —  max(E’ r E’ r)]  EE tAEv+(Al_A7) + AE1  A’) + (El_E.r)  .•  (6.18) AEc=AEv+(Al_A1) +  (E—E)  r E r)] ’ AE + [min(E’ E”) min(E’ 4 (El E’ ) = 1E + ,  =  —  —  lE (E r ) AE=LE+ 4  where the average bandgap Eg is equal to the bulk alloy bandgap given in eqn (6.3), plus the hy drostatic change to the bandgap AE 8 given in eqn (6.14). Therefore, eqn (6.18) provides all of the necessary information to calculate any of the band offsets within the SiGe material system. Although no one equation that forms the model of the SiGe material system is of a complex nature, the cumulative effect of each sub-model leads to a complex system as is evident from eqn (6.18). However, it is possible to arrive at a set of Taylor expansions for the models that govern the band movements within the conduction band. Unfortunately, the valence band models (i.e., eqn (6.12)) contain a square root dependence that proves impossible to approximate. Given the non linear nature of the strain tensor, is it is not possible to achieve a simple linear approximation for the conduction bands. By performing a multivariate Taylor expansion of the conduction band models in eqn (6.18), up to and including second order terms, yields:  AE  AE  0.1429  (Xar  0.1751  (Xar  0.1268  (Xar  —  —  —  —  Xai) x Xai) x  Xai) X  —  —  —  1 032789Xr + 0.O l55Xar + O.32985x 2 0.34084x,.  —  81Xar+ 3 4 0•  where all results are in eV, and E’ 2  =  —  —  O•0 X 2 5 22 ai  1 + O• 0.34281x Xai 4 8 433  1 0.3214lx + 0. 973Xar + 0.32338x 24  0.02723x + 0.0 836XaXs 4  —  0.01943x + 0.68368x  —  —  (6.19)  5070Xai 2 0.  0.68454Xa  E. Eqn (6.19) is accurate to within 1% of the full  model given in eqn (6.18) over the entire allowed range for Xal, Xa,, and x. The multivariate Tay lor expansions were centered around Xal = 0, Xar = 0.5, and x = 0.5. Thus, eqn (6.19) should strict ly be used with  Xal <Xar  however, if this is not true, then simply interchange  Xal  July 12, 1995  and  Xa,.  and 154  multiply the result by -1. If the interchange of variables is not performed for Xal > Xa,, then the er ror in eqn (6.19) will rise to 1.5%. Examination of eqn (6.19) provides insight into the conduction bands of SiGe. Considering 2  first, the two last linear terms in xa and x are the dominant terms. Therefore, to a crude ap  proximation, E’ 2  (x 684 O.  —  2 xa); which corrects the proposal of E’  O. X 6 a  by People [1051  and Pejcinovic [28] who considers only a Si substrate. Examination of the other models in eqn (6.19) shows a linear dependence upon the substrate Ge alloy content x. It is by no coincidence that the coefficient that governs the x dependence in AE is 0.1268, as compared to the coeffi cient of 0.12 in eqn (6.17). The largest portion of the substrate dependence in eqn (6.19) is due to the model for A Therefore, the material science community must determine for certain the ef fects of substrate strain, in order than SiGe devices can be developed where substrate strain is uti lised. Finally, the non-linear terms in eqn (6.19) stem mainly from the non-linear dependence that the bulk bandgap has on the Ge alloy content. In terms of the conduction band, Fig. 6.9 plots E and E to the left and right of a hetero junction under the proviso that the entire system is commensurately strained to a { 100) Sii..Ge substrate. The first thing to note is that EE is generally smaller than z\E, and is of such a nature that in going from the left to the right there is a downwards step. The reason for not classifying this as either a type I or II heterojunction is that the bandgap is not a monotonic function of strain, as is evidenced in Fig. 6.4. Thus, classification in terms of type I or II would require detailed knowledge of the strain state, which would destroy the simplicity of the type I or II designation. However, when going from a pure Si crystal to a Sii..Ge crystal there is always a small down wards AE. Contrarily, AE is in general quite large, much larger than  and is of an upwards  nature in going from a pure Si crystal to a Sii.Ge crystal. Most importantly, Fig. 6.9 clearly demonstrates that the character of the conduction band can change between E and E when crossing a heterojunction. Fig. 6.10 goes on to show that AE indeed has a complex nature when strain is brought into the picture. There are three distinct regions in Fig. 6.10: 1) when x 1 to E’ xar) then AE is governed by E 4. r  r  2) when Xal  <Xy <Xar  < (xai,  then AE is governed by E’ 1 to  3) when x> (xai, xar) then AE is governed by E’ 1 to E’ r To conclude this section AE is plotted in Fig. 6.11. The various parameters are identical to  the ones in Fig. 6.10. As with  AE also displays the same type of complex features which are  July 12, 1995  155  0.6  •  •  •  0.5 AE  4’  0.4  •  (a)  C  Substrate Germanium Mole Fraction x 0.5  •  •  •  0.3 0.2  E , 4 r  b  0.1  /E  \::T  02  10  04  Substrate Germanium Mole Fraction x Fig. 6.9. E and E conduction band minima to the left and right of an abrupt heterojunction when commensurately grown atop a { 100 } SiiGe substrate. All energies are relative to E on the right hand side of the heterojunction. (a) Xal 0, Xar = 0.15; (b) Xal 0, Xar = 0.30.  July 12, 1995  156  0.4  I  •  I  I  I  •  0.3  bE  0.2 E4,r  I  0.1  (c)  0.0  7•••••••••••  -0.1  \EZr  • •. . . . i  -0.2 -0.3 0.0  •  0.2  I  0.4  I  •  0.6  0.8  1.0  Substrate Germanium Mole Fraction x  0.3 1 E’ 0.2 0.1  I  0.0  (d) -0.1 -0.2  /  -0.3 -0.4 0.0  .l  .  0.2  0.4  0.6  0.8  1.0  Substrate Germanium Mole Fraction x Fig. 6.9. Continuation of Fig. 6.9 from the previous page. (c) = 0.60.  Xal  0, Xar = 0.45; (d)  July 12, 1995  Xal  =0,  Xar  157  0.2 0.1 0.0 -0.1 -0.2  (e)  -0.3  i.  -0.4  -0.5 -0.6-  0.2  0.4  zr. 0.6  1.0  0.8  Substrate Germanium Mole Fraction x Fig. 6.9. Continuation of Fig. 6.9 from the previous page. (e)  Xal  0, Xar  =  0.75.  toE  0.00  =  Xal +  0.2  -0.05— E2,1tOE2.r  -0.  0.00 0.40  0.75  0.60  Fig. 6.10. tE when xa,. = Xal +  Xal 1.00 0.80 0.20, afld Xal and x are varied. The right side is the reference. July 12, 1995  158  -0.05  “—;:  -0.10  0.00  -0.15  l.oo 0.80 Fig. 6.11. zXE. , when Xa,. = Xal + 0.20, and Xal and x are varied. The right side is the reference 1  due to the ultimate valence band changing from tinct regions in Fig. 6.11: 1) when x Xal <Xç <Xar lhl  byE  < (xai, xar)  hh  to  lh .  Just like Fig. 6.10, there are three dis  then AE is governed by E’ 1 to E’  then AE is governed by E’ 1 to E ” 1  r;  3) when x  > (xai, xar)  r;  2) when  then AE is governed  Ihr  toE  6.3 Electron Transport in Strained Sii..Ge Sections 6.1 and 6.2 present the necessary Sii..Ge material models to determine the overall band diagram, including offsets at abrupt heterojunctions, within any SiGe solid-state device. This section will focus on determining the transport models for electrons and holes within the Sii..Ge material system. Essentially, the models presented in all of the previous chapters are applicable to the study of SiGe-based devices. For example, Chapter 4 presented the EB SCR transport models July 12, 1995  159  which included the effects of tunneling and the mass barrier. Therefore, Chapter 4 can be applied to a SiGe device to determine if the EB SCR will generate current-limited-flow. However, care must be exercised in the application of Chapter 4, and indeed all of the other chapters, as there is a multi-band model for the Sii..Ge material system. This section will discuss and present the transport models for the multiband Sii.Ge material system. From the work in the previous two sections, it is clear that the conduction and valence bands are both broken down into two distinct sub-bands (the so valence band is ignored as it is always lower in energy than the lh and hh bands, especially under strain, and is of such a low carrier mass [96] that hole transport can be ignored). Unlike the AlGai..As material system, where the higher energy satellite band never forms the ultimate conduction band, E and E in the Sii.Ge materi al system can both form the ultimate conduction band. Thus, it is possible to have near equilibri um transport occur within both E and E at spatially separate points with the device; this is in contrast to the AlGai..As material system where transport in the satellite band need only be con sidered under extreme non-equilibrium injection conditions. Further, this multiband transport can also occur in the valence band of the Sii..Ge material system. Given the strange band offsets de picted in Figs. 6.9 to 6.11, it will be shown that transport within the Sii.Ge material system can offer a rich set of possibilities, both in terms of commercial HBT optimisation, and as a tool for research into the mechanics of transport within solids. Considering the valence band first of all, Fig. 6.2 shows that EL and E are degenerate un der the condition of no strain. More importantly, the maxima in both E! and E occur at the same point in k-space. Even under strain, the maxima in E and E’ remain coincident in their k space location. Therefore, there is very little issue regarding the conservation of crystal momen tum in moving between the lh and hh bands, if the mass barrier that would occur at the hetero junction for holes is neglected, then to a good approximation one need only consider the ultimate valence band in terms of hole transport. However, if the strain is small, so that the energy separa tion between E and E is less than —2k1 then transport within both bands needs to be consid ered. As is attested by eqn (6.1), the mass barrier for holes cannot be neglected as ‘y from eqn (4.80) is typically -2 but can be as small as -10. With y = -2, fully two-thirds of the current cross ing the mass barrier could be reflected, leading to a 3-fold reduction in the transport current. A 3fold reduction in the transport current would be equivalent to having an upwards step in energy of  July 12, 1995  160  28.5meV at room temperature. Therefore, when 11Ev is less than —2kT one must consider parallel transport within E and E!j. But, no matter how large or small AE is, the calculation of the va lence band effective density of states N must include both E and E’ due to the large difference in the lh and hh effective mass. The complexity of the valence band stems from the coincident k-space location of the band maxima for E and E. Examination of Fig. 6.7 shows that the Sii..Ge conduction band mini ma are not coincident in k-space. Thus, in order to move between any of the six A minima in Sii.Ge, crystal momentum must be conserved. There are two scattering processes that are re sponsible for intervalley scattering between the six conduction band A minima in Sii.Ge [129] (see Fig. 6.12); g scattering moves electrons between two bands that are along a common major k axis, such as the [001] and  [OOT] bands that form E; while f scattering moves electrons between  two bands that are not along a common major k-axis, such as the [100] and [010] bands within E. Given the proximity of the A minima to the Brillouin zone edge, an Umklapp process can eas ily take place, leading to g scattering, because of the relatively small k-space separation that must be conserved. On the other hand, f scattering involves a k-space conservation that is over one-half of the reciprocal lattice length. Therefore, it is found that f scattering rates are almost 10-fold low er than g scattering rates [129]. Tn terms of the E and E band groupings, g scattering will not re sult in movement between the E and E bands. Finally, for small distances, such as those that are typical of the EB SCR and neutral base width, f scattering is small enough to be ignored [108,130]. These two results regarding intervalley scattering allow the E and E bands to be treated independently, allowing for a large simplification as compared to the valence sub-bands. The arguments of the previous paragraph, justifying the independence of the E and E bands, must be considered in the light of an abrupt heterojunction. At an abrupt heterojunction, one would expect that a powerful Bragg plane could exist that would be perpendicular to the di rection of charge transport across the heterojunction. Such a powerful Bragg plane could enhance f scattering, leading to a coupling between the E and E bands. Consideration of the k-vector in volved in f scattering relative to the Bragg plane, shows the two are separated by 45°. With a 45° degree separation, it would not be expected that Bragg plane scattering at an abrupt heterojunction would lead to a significant increase in the f scattering rate [1081. Therefore, the independence of the E and E bands should be maintained even at an abrupt heterojunction. This leads to the for-  July 12, 1995  161  mation of a selection rule regarding transport in Sii..Ge that prohibits a mixing between the elec trons in E and E.  g scattering  4  E\,,  -K’  E  \I  YE2 f scattering gscattering  Fig. 6.12. Diagram of the A conduction band minima involved in f and g intervalley scattering. For clarity, only 1 of the 3 g scattering processes, and 1 of the 12 f scattering processes is shown. With the E and E bands treated independently of each other, the task of modelling elec tron transport within the Sii..Ge material system begins with calculation of the electron effective densities of states, N and N respectively. The density of states for band n is given by [131]: g(E)  =  n  f  —  E(k))  (6.20)  B.Z.  where n  =  2 or 4 in the case of Sii.Ge strained to a { 100) substrate, and B.Z. means Brillouin  zone. The pre-multiplying factor of n in eqn (6.20) results from the degeneracy of the E and E bands and the fortuitous designation where n is equal to 2 or 4. Then the effective density of states, assuming that the band-width is Eb and that Boltzmann statistics can be used, is equal to: Eb  N  =  E,,  E  JdE g(E)e’ =  Bt.  E(k)  E  6(E  —  .__ekT.  E(k))e’ =  (6.21)  Bt  July 12, 1995  162  Eqn (6.21) can be easily integrated with little error by assuming that the limit of integration  can be extended past the Brillouin zone to infinity; i.e.,  N  =  _!_e  h2k  1ik  00  1i2k  fdkxe2mu1cTfdkye2mh1cTfdkze2mtkT  3/2 =  2ne  (6.22)  where m  *  =  (m m 1 m)  1/3  The appearance of the term exp(-EIk7) in eqn (6.22) is due to the fact that the reference energy for the conduction sub bands is not located at the band minima, but at  One could have maintained  .  the reference energy at the band minima, but then N and N would have different energy referenc es and eqns (6.1O)-(6.11) could not be used directly within eqn (6.22). Furthermore, by employing a common energy reference of  ,  the total conduction band effective density of states is: 3/2(  N  =  1V+N  =  E  4 ( m’  (6.23)  .  2ich )  J  Finally, it is possible to reflect N from the energy reference of  back to the ultimate conduction  band minima by multiplying eqn (6.23) with exp(min(E, E)Ik7). The exact same methods used to determine  P4 and N can be applied to the calculation of  the hole effective density of states within the valence sub-bands, leading to:  N  =  2e  1 (mhhkT\ L21th  E  3/2  and N  )  =  2e’  3/2  (mlhkTN  L2ith  (6.24)  )  Then, owing to the different effective masses for the lh and hh, the total valence band effective density of states is given by: 3/2  N  =  N’ +N  kT  =  212ith  j  (mhh) e 2 ” 3 ’ + 3 (mlh) e 2 ” ’  .  (6.25)  )  In a similar fashion to the conduction band, the reference energy for the valence band is not locat ed at the ultimate valence band maxima, but at the location of the valence band maxima under the condition of no strain. To reflect N back to the energy of the ultimate valence band maxima, mul tiply eqn (6.25) by exp(-max(E, E)Ik1). July 12, 1995  163  Eqns (6.22)-(6.25) present the conduction and valence band effective density of states for the Sii.Ge material system. These equations represent an extension to the traditional definitions for effective density of states, necessitated by the complex band structure of Sii..Ge under the in fluence of symmetry-breaking strain. Finally, the electron and hole concentrations n and p respec tively are defined using eqns (6.25) and (6.23) in the usual non-degenerate manner, to yield: E  n  =  E  NceU’ and p  (6.26)  ‘  =  Ne  where Ef is the electron quasi-Fermi energy relative to  ,  and  is the hole quasi-Fermi ener  gy relative to the unstrained valence band maxima. After allowing for the fact that the conduction and valence band energy references are separated by g’ as is shown in Fig. 6.8, then:  n=pn=NNe 3 =  (  kT  \1tli  ‘  (m*  8 E kT  3/2  (  E  E  (  eICT + 2e’  ’ t E,  (mhh)  (6.27)  g  3/2ei; + 3/2e eu’, (mlh)  )  where the average bandgap  g  is equal to the bulk alloy bandgap given in eqn (6.3), plus the hy  drostatic change to the bandgap  Ag given  in eqn (6.14). Unlike eqns (6.22)-(6.26), n  given in  eqn (6.27) does not reference itself to an abstract energy reference, but is the standard definition for the intrinsic carrier concentration. With the effective density of states defined for the conduction and valence sub-bands in eqns (6.22)-(6.25), along with the carrier concentrations and n given in eqns (6.26)-(6.27), it is possible to define the built-in potential Vb 1 of apn-junction. Looking at Fig. 6.13, then clearly: Vbi  =  (Eje,j  —  (E  —  Eg,p)) +  (x x) —  =  ln(” ) ni,p c,n  +  (,  —  x).  (6.28)  Comparison of eqn (6.28) with eqn (4.69) shows, apart from the effect of a spatially varying effective density of states (which is neglected in eqn (4.69)), exact agreement if  —  =  Vbj  is the variation in the vacuum potential across the SCR extrapolated back to equilibrium condi tions. Thus, the electron affinities Xp and xn on the p- and n-sides of the junction are x  =  moves  and x = -x, respectively. If  and  are  evaluated at  spatially varying, then as a changing applied bias  and x,, Vb 1 will also vary with applied bias. It is well known that Anderson’s electron af  finity rule for the calculation of AE is not correct. However, at some distance far from the hetero Julyl2,1995  164  junction,  must become bulk-like. The question becomes how rapidly do Xp ifld Xn  and  return to their bulk values? The deviation of AE from  —  has been attributed to such things  as a complex rearrangement of charge surrounding the heterojunction. Thus, if this rearrangement of charge is abrupt, as is potentially suggested by Van de Walle and Martin [106, 119,120], then and  would definitely change over the width of the SCR; leading to an extra driving force for  the transport of charge than is not taken into account by any known theories. If this rapid variation in  and x turns out to be true, then Vbj will not be a constant as is given in eqn (4.69), but is in  stead given by eqn (6.28) with Xp and the desired information regarding  ,  being a function of position. Finally, Vbj contains all of  Xn’ and thus zSE. Therefore, if the pn-junction could be  driven up to and past Vbj, without resistive effects dominating the transport current, then informa tion regarding  ,  X and thus AE could be extracted. This possibility of operation near and past  Vbj will be considered in Section 6.4.  E (eV)  I  Unstrained E  Unstrained E  Fig. 6.13. Equilibrium band diagram of a np-junction, showing the relevant energies and poten tials. Ef is referenced to E while is referenced to E. Note that the Vacuum potential is con tinuous while E and E are not. July 12, 1995  165  Concentrating once again on the conduction band, the final models for electron transport can be determined. By the previous arguments, electron transport in the EB SCR and the neutral base can be modelled as two parallel conduction paths via E and E. It is further assumed, at least with the current-day knowledge of the Sii..Ge material system, that eqn (6.2) is correct, which precludes the formation of a mass barrier. Therefore, transport through the EB SCR would be given by the sum of E and E conduction solved by the standard transport model given by eqns (4.78)-(4.79) and (4.92). To this end, the correct parameters to use in the standard EB SCR transport model regarding E conduction are: m  =  1  =  mQX)  my,  (A) 1 m  1  4q2Jm imikT  =  m  1  =  m(Z)  1 .L  =  relative to  Jm 2 4tq k 1 m T  2 q  =  3 h  3 h  J2tm(A)kT  N (6.29)  While the correct parameters to use in the standard EB SCR transport model regarding E con duction are: m,  —  2 F fs0 —  =  (A) 1 m  m,  4irq J 2 m lmzl kT  =  m(z)  —  e  3 h  —  N 2 q —  1  m  =  m(A)  =  4itq J 2 m m 12 kTI N D N 3 h  ek  \  ‘1  —  —  relative to  N q 2 ‘N (A)kT 1 J2itm (6.30)  band degeneracy  E  J2tm ( 1 I)kT e  e kT 2 kT+  Examination of Fjo in eqns (6.29) and (6.30) reveals the exact context of parallel transport within  E and E. There are ND majority electrons that are distributed between the E and E bands, de pending upon the energy separation between the two. Since there are twice as many A minima in E as compared to E, there will be preferential transport within E, all other things being equal. July 12, 1995  166  Finally, the electrons within E and E move with a velocity that is proportional to the squareroot of l/mj and 1/rn 1 respectively. Therefore, neglecting the energy separation between E and E, the E band will carry 2Jmi/m  =  4.54 times the current compared to E. Furthermore, be  cause E has the light transverse mass parallel to the direction of transport, as compared to the heavy longitudinal mass for E, not only is the mobility higher [132] but the probability of tun neling through a given barrier will be much higher for E compared to E. Regarding transport within the neutral base, the independence between E and E can only be maintained if the neutral base width is small enough to preclude coupling via the f scattering process. For current-day SiGe HBTs the neutral base width Wfl, is under  ioooA and is rapidly ap  proaching 300A [10,26,28,100-103,133]. With such a small neutral basewidth it is reasonable to maintain  the separation between E and E used for the modelling of EB SCR transport. With E  and E treated independently, the neutral base transport current within either one of the E subbands is given by Kroemer [38] as: —  —.  NB  N(x) =  Wnb  qBE  fn  kT  [e  (6.31)  dx  -  n(x)  wherej = 2 or 4. It should be noted that eqn (6.31) is an extension of Kroemer’s work which was based upon Shockley boundary conditions. The reason for generalising the diffusion coefficient D, as was discussed in the previous paragraph, stems from the fact that the mobilities within the E and E bands will be different due to their highly anisotropic nature [132]. This leads to the conclusion that D  >  D because rn < rn . Further, each sub-band will have its own intrinsic carri 1  er concentration nb., which is determined in the same way as 1V, N and the total n to yield:  flj  NNe  kT  =  =‘  n  =  4 n  (6.32)  +  Finally, due to the independence of the E and E bands, a separate quasi-fermi energy must be present in order to account for the driving force within each sub-band. For this reason, there is to characterise transport within the E band, and AE to characterise transport within the E band. The final model for electron transport within the SiGe HBT is achieved using exactly the same methods employed in Section 5.3 for the derivation of eqn (5.29). Eqn (5.29) is based upon July 12, 1995  167  the general models of Section 2.2. Applying the models of Section 2.2 to the solution of transport within the conduction sub-bands yields for the E band:  =  1’ 40 [4L  (6.33)  where, based upon the findings of Chapter 5, the failure to include recombination effects specifi cally in the calculation of the total transport current To reiterate,  4 will produce an error that is of order 1/13.  4 is the EB SCR transport current solved by the standard transport model given by  eqns (4.78)-(4.79) and (4.92) with the pertinent parameters obtained from eqn (6.29). In a similar fashion, transport within the E band is: -1,  4=  + fs  (6.34) 1NB  where F 8 is once again the EB SCR transport current solved by the standard transport model giv en by eqns (4.78)-(4.79) and (4.92), but with the pertinent parameters obtained from eqn (6.30). Then, the total electron transport T through the HBT is given by the sum of  4 and 4.  To conclude this section, transport within the valence sub-bands is addressed. As was dis cussed earlier in this section, the coincident nature of E’ and E in terms of k-space location prohibits an independent treatment, such as was done for the conduction sub-bands, of the two va lence sub-bands. Fig. 6.2 clearly shows that the valence band of unstrained SiGe, and for that mat ter all semiconductors, is a multi-band system. To this end, transport within the unstrained valence band is determined by appealing to a single total effective mass that correctly produces the total valence band effective density of states. Then, by way of experimental measurement, a single mo bility is extracted to characterise the valence band as a whole. This method breaks down for the case of strained SiGe, as the degeneracy of E and E is lifted and the energy separation is de pendent upon the amount of strain present. This prohibits the use of a single effective mass and fixed mobility to characterise the valence band of strained SiGe. Yet, the valence sub-bands can not be treated independently for the purpose of determining charge transport, as was done for the conduction sub-bands. Essentially, the only way to solve transport within the strained SiGe valence band is to re sort to Monte Carlo simulation. However, as was pointed out at the start of Chapter 4, Monte CarJuly 12, 1995  168  lo simulators cannot presently model the non-local effects of tunneling. To this end, the following two assumptions are made: 1) hole transport within the EB SCR is considered ballistically due to the small width of the SCR, but the holes will always attempt to minimise their energy by moving to the highest sub-band; 2) due to the strong intervalley scattering that occurs between E and E, because of their coincidence in k-space, transport within the wider neutral regions of the  HBT is treated using a single equivalent valence band. The implication of the second assumption is straightforward; transport is treated in the stan dard single equivalent valence band approach. The only consideration that must be made in treat ing the valence band as a single valley is the mobility will change with strain. In a region where the Ge alloy content is not uniform the strain will change with position, which will move  and  E either closer or further apart in terms of energy. Since the lh mass is much smaller than the hh  mass, considerable change to the mobility of the material will occur as E and E move closer and further apart. This leads to a complex and spatially non-uniform mobility that is only due to the energy separation of the valence sub-bands. Other effects such as impurity and alloy scattering would also have to be considered. The implications of the first assumption are even more interesting than those of the second. For the purpose of tunneling, the lightest mass will produce the largest tunneling flux. But, con servation of transverse momentum must be ensured for a hole to change bands, which leads to the mass barrier results of Chapter 4. However, the hole will attempt to take the path of least resis tance by minimising its energy; it may either continue on in the sub-band it currently occupies, or change bands in an attempt to minimise its energy while taking into account the possible loss or gain due to the mass boundary effect. The complexity of transport within the SiGe valence band stems purely from the large difference in the lh and hh masses. If the lii and hh masses were the same, then transport would occur along the highest energy sub-band (in terms of electron ener gies), with a spatially varying N to consider. The model for the EB SCR in Chapter 4 is simple in that the heterojunction is abrupt; there by producing two regions, separated by a single mass barrier where the material parameters with in each region are a constant (see Fig. 4.2). As a result of this, the relative separation between E’ and E will not change, except at the mass boundary. Therefore, for the calculation of the EB SCR transport current for holes in the Sii..Ge material system: July 12, 1995  169  • initially consider the  and E bands independently, injecting a hemi-Maxwellian of holes  into the EB SCR, characterised by the individual mass of the band. • Using the standard flux model, given by eqns (4.78), (4.79) and (4.92), calculate the standard flux Ff (Fj does not include the mass barrier) using the appropriate mass from eqn (6.1), and q2  -  —  (N A)’ J21tmkT  where j is either hh or th. Within the standard flux model, the base barrier potential Vb longer the one from the originating band, but is given by the maximum of E and E in the neutral region (this is where the minimisation of hole energy enters the calculation). • If the mass barrier effects are not considered, then the problem ends here. But, the mass barrier can be quite large in the valence band, producing a potentially non-negligible effect. However, the mass barrier effects are only important if the aforementioned calculation of the standard flux has the holes changing between  Et1  and E. If the holes do change bands then eqns  (4.85)-(4.86) are used in the case of an enhancing mass barrier; where as eqn (4.87) is used for the reflecting mass barrier, but with the infinite upper limit of integration replaced with the Vb that is appropriate to the sub-band that injected the holes. The physical explanation of the valence band transport model is: holes ballistically travel through the EB SCR, perhaps tunneling through a Valence Band Spike (VBS), by way of indepen dent E’ and E bands. Upon reaching the mass barrier the holes attempt to occupy the lowest energy band, and do so by exchanging sub-bands, if necessary, while taking into account any loss es or gains due to the mass barrier. Depending upon the construction of the HBT, the emerging fluxes from the EB SCR, contained within E and E, will generally be characterised by differ ent driving forces of AE and AE respectively. However, due to the strong intervalley scatter ing that occurs between the valence sub-bands upon reaching the neutral region, a common quasi equilibrium condition of  will result for both E and E. Therefore, the final transport mod  el for holes is: 1 T,holes  where F and  F.h  =  1  (6.35)  [Fh+F+JT,TOl  are the full EB SCR transport models, and  T,utral  is the neutral region  transport current calculated by eqn (6.31) using n from eqn (6.27). July 12, 1995  170  6.4 The Accumulation Regime Beyond the Built-In Potential Chapter 4, and therefore Section 6.3, have both dealt with transport for an applied bias VBE that is less than the built-in potential Vb . For the case of a band diagram where there is a negative 1 step, as shown in Figs. 6.13 and 4.2, as VBE approaches Vbj, a current density of 2 A/cm will 6 —10 flow (this is based upon an emitter doping that is 18 ‘-.10 ) 3 cm . At a current density of 2 AIcm 6 lO , resistive effects will dominate the device and limit the internal forward bias to be much less than the external applied bias. For example, with an emitter area of 1 urn , there would be a current of 2 lOmA at a current density of 2 A/cm Even with an unrealistically low emitter contact resis 6 10 . tance of 502pm , there would be a 500mV drop to the external applied bias before it even 2 reached the junction. It is for this simple reason that observation of the device with a forward bias near, and certainly beyond, Vbj is not really experimentally possible. As is evidenced by the plot of tsE in Fig. 6.10, along with zE and AE shown in Fig. 6.9, there exists the possibility of constructing a positive-going potential step (see Fig. 6.14a) in the path of the electrons trying to surmount the potential barrier of the EB SCR. A positive potential step would force the electrons to surmount the entire barrier, because unlike the CBS there is no way to tunnel though the step. Therefore, if the step potential were as large as 240meV (i.e., AE =  -240meV), then by eqn (4.79) the charge flowing through the EB SCR at room temperature  would be reduced by a factor of exp(-240125.9)  10. Therefore, when VBE approaches Vbj, the  current density will have dropped to only 2 AIcm A current density of 2 lO . Afcm will certainly lO be observable, and would even allow for VBEtO exceed Vbj. Before going on to present a physical demonstration of operation beyond Vbj, the transport theory for this domain of operation is first developed. When VBE is exactly equal to Vbj, and if the resistive effects are negligible, then the band diagram will be flat except at abrupt heterojunctions or regions of spatially non-uniform Ge alloy content (see Fig. 6.l4b). For this reason, the point at which VBE is exactly equal to Vbj is termed flat-band (in much the same manner as the flat-band condition in MOSFETs). At flat-band there will be no space charge present. As VBE is increased past Vbj (see Fig. 6. 14c), an accumulation region of mobile electrons on the n-side, as well as mo bile holes on the p-side, of the heterojunction will begin to form (as has been the case throughout, a coincident hetero- and metallurgical-junction is assumed). This is contrast to the standard case where VBE  <  Vbj, and a depletion region forms where the space charge is composed of immobile July 12, 1995  171  _________ _  ion cores from the dopant atoms. For this reason, operation past Vb 1 is termed the accumulation regime. Finally, as VBE is increased further, the accumulation of charge will proceed exponential ly, with a net reduction to the potential step, and therefore, a continued exponential increase in the EB SCR transport current.  (a)  EE  n-side  E  (b)  VBE=Vbj  Ej_ \ 1 .. E,  mobile electrons -  -  -  -  (c)  \•  mobile holes —i  {,—Efh  V  Fig. 6.14. Band diagram for a np-junction with a positive step potential (i.e., AE <0). (a) equi librium; (b) flat-band where VBE = Vb; (c) accumulation region where VBE> Vbj. July 12, 1995  172  A reasonable first approximation to the complex accumulation regime begins by assuming that for operation just beyond Vb 1 the accumulation layer is non-degenerate. Based upon this as sumption, and neglecting the effects of a non-uniform e, the Poisson equation in one dimension becomes 2  9_ (ND  qi 2 d  —  ND e  kT  on the n-side (6.36)  =  2 dx  where  and  2 q  (N  —  NAekT) on the p-side  are in terms of electron energy (i.e., the negative of potential energy). Eqn (6.36)  is solved on the n-side (the p-side solution can be obtained directly from the n-side solution by symmetry arguments) to yield the following implicit transcendental function:  e  =  , 2 d+A  (6.37)  ND[(fl+Al)ekT+kT]  where A 1 and A 2 are arbitrary constants. There is no way to reduce eqn (6.37) down to a function of simple transcendental functions, nor will it be possible to invert the result. However, it is rea sonable to assume that the charge in the accumulation layer will overwhelm the background dopant ion potential. With this assumption eqn (6.36) is recast to: 2 2,,  — — 9 NDe 8  dx  2  kT  on the n-side (6.38)  kT  on the p-side  whose n-side solution is ?2  x  =  IJi ±4_$  e’  (6.39)  dW+A 1 . 2  ND AiecT+kT  It is interesting to note that the only difference between the approximate solution of eqn (6.39) and the full solution of eqn (6.37) is the extra term containing ‘qi, in the denominator of the radi July 12, 1995  173  cal. This linear term in ‘qi,. 1 produces the asymptotic solution to the underlying depletion space charge. Since eqn (6.39) assumes that the depletion space charge is negligible in comparison to the accumulation charge, the linear term in  qI,j  is lost.  The solution of eqn (6.39) begins with the determination of A . If the neutral assumption is 1 employed at  -x,  (the boundary to the accumulation region), then because the doping ND is a con  stant the electric field will vanish. Since the electric field is given by (lIq)diIdx, then talcing the derivative of eqn (6.39) with respect to N’,. inverting it, and setting it equal to zero with x  -x,  yields A 1  =  -kT With A 1  =  =  0 at  -k7 eqn (6.39) is solved using the change of variables ‘tin  y  =  2e’—l  to produce:  =  ±  asin  ekT 2 (  1  —  +A 2  e  sin  =  ]+ 2 A  .  (6.40)  q42N Finally, applying the energy reference of Nn = 0 at x  -x, to eqn (6.40), and choosing the positive  x-direction produces: (x+x’  e  =  (6.41)  cos l 2 1 ,)I ha  where 1 IekT  1 a =  By appealing to the symmetry of the problem, the p-side solution of eqn (6.38) is: ‘lip  e  =  (x—x 2I cos I ) 1 2a  (6.42)  where 1 a =  It is important to realise that ‘qi, is set equal to 0 at x = -x,, and  is set equal to 0 at x = x. How  ever, the form of the Poisson equation requires that when qI,j joins up with tion  (i.e.,  at x  =  at the heterojunc  0), the joint be analytic up to first derivatives. Given that we are solving for  accumulation and not depletion, then continuity of  and Nip requires that: July 12, 1995  174  ____________________________________  VBE— VbI  N’(O)  —  v(O)  =  2kT  q (VBE— Vbx)  =  cos  (2ai  )COS  1 (2a  pJ  (6.43)  Further, continuity of the electric field requires that:  (  =  —  dx  X,  “1  —----tan 1 a J 1 k2a  dx  =  I  “1.  —--tan ) 1 2a ap  (6.44)  It is a straightforward task involving considerable bookkeeping to solve simultaneously eqns (6.43) and (6.44) for x, and x. Eqns (6.43) and (6.44) form a quadratic equation involving the squared cosines of  and xJ2ai,. Choosing the positive roots of the solution for eqns  (6.43) and (6.44) yields:  x,  =  1 acos 2a  II4NANDeAVT+ (NA—ND) +NA—ND 2  I Al  (6.45) (IJ4NANDeET+ (ND—NA) +ND—NA 2 1 acosi I xp =2a vBE T NDe 2  where .A VBE  VBE  —  Vbj,  and VT = kTIq.  The accumulation regime solution of eqn (6.45) is certainly much more complex than eqn (5.9) for the depletion regime. However, the accumulation regime shares many similarities with the  depletion regime. In fact, when VBE is within the immediate neighbourhood of Vbj (i.e., small IWBE),  then a Taylor expansion of eqn (6.45) about the point AVBE  =  0 yields exactly the same  equations for x and x, that is obtained from the depletion regime. Further examination of eqn (6.45), however, shows that as AVBE increases, x and x, quickly saturate at a constant value of ltal,n  and ltal,p respectively. This saturation of the SCR width is a feature of the rapid accumulation  of mobile charge that screens out the applied bias with essentially no further increase to the extent of the SCR. This result is also the point at which the assumption of a non-degenerate accumulation layer will fail; so care must be exercised in the absolute application of eqn (6.45) for large IWBE. A useful metric from the depletion regime was the ratio of x,. to the total SCR width x, + Due to the complex nature of x and  in the accumulation regime, this same metric will not be a  simple constant. However, by appealing to a Taylor expansion about AVBE =0, and the asymptotic limit for large AVBE, it is found that: July 12, 1995  175  N  NN(N-N)  NA + ND  =  +  N rat  BE  (ND +  —  BE  —  hzee,D  VBE>VkfleeD  where 2 3VT(ND+NA)  —  knee,D —  (JNAND + NA) (JNAND + ND)  In a similar fashion, the metric for the splitting of AVBE between the n- and p-sides of the junction yields: N Vrat  =  IN’fl(°)I q  NN(N-N)  NA + ND +  VBE  —  VBE  Vee  (ND + NA) 3  BE  1  (6.47) A  where VT (ND + NA) 2 Ve,V_  NN AD  Nrat in eqn (6.46), as was stated a few paragraphs earlier, shares many of the same features as Nrat in eqn (5.9) under the depletion regime. Now, V.at in the depletion regime is exactly the same as Nrat, owing to the spatial uniformity of the space charge due to the immobile dopant ions. However, under the accumulation regime, l1 iat in eqn (6.47) starts out the same as Nrat, but due to the mobile nature of the accumulation space charge, quickly results in an equal portioning of the excess applied potential AVBE between the n- and p-sides of the junction. Therefore, the potential distribution in the accumulation regime differs markedly from what is found in the depletion re gime. Finally, Fig. 6.15 plots Njj and at 11 in both exact and approximate form, as well as x, and x, in order to gain a familiarity with the accumulation regime. Eqns (6.46) and (6.47) provide very useful tools for the solution of charge transport within the accumulation regime. Fig. 6. 14c shows that within the accumulation regime, the positive step potential has produced a CBS; but unlike the negative step potential within the depletion regime (see Fig. 4.2), the CBS now appears on the other side of the heterojunction. Taking the standard HBT case where NA  >>  ND, then x <<x, and for small AVBE one also finds ‘qi(O) <<ji,(O). These July 12, 1995  176  two findings mean that the CBS within the accumulation regime will be very narrow, and very weak in terms of a potential to be tunneled through. Strictly speaking, the transport current through the CBS in the accumulation regime requires that the general transport model of eqns (4.51) and (4.53) be solved using WCBS  =  1, and WN obtained from eqn (4.6) with the accumula  tion potential of eqn (6.42). However with the parameters used in Fig. 6.15, when AVBE  =  l2OmV, then the CBS stands only 28meV tall, and 17A wide at the base. Clearly, this small CBS will allow a significant current to pass though it. In any event, the largest that the CBS barrier could be, by assuming WN =0, would be an energy of IAEI CBS barrier could be, assuming that WN 6.16). Therefore, with Vat  =  and the smallest that the  —  1, would be an energy of  IAEI  —  qzV (see Fig.  1 for small IVBE (given the typical HBT doping), then the upper and  lower bounds for the effect of the CBS will be fairly close together.  1.00  0.90 rii  0.80  0.70  0.60 ZS.VBE (V)  0.50 0.00  I  0.10  0.20  0.30  0.40  Excess Applied Potential A VBE (V) Fig. 6.15. The exact and approximate forms for Nrat and Vrat from eqns (6.46)-(6.47). The ma terial parameters are: ND: 5x10 ; NA: 1x10 3 cm 17 ; e: 12.0. 3 cm 19 One of the essential results of Chapter 4 was that the peak emission flux density occurred at a fixed energy relative to the height of the CBS. This result occurred only because of the parabolic nature of the potential profile within the depletion regime. Given the fairly simple model presentJuly 12, 1995  177  ed for the accumulation regime, where degenerate effects have not been accounted for, there is lit tle point in solving the general transport models of Chapter 4. Instead, based on the arguments of the previous paragraph, it seems reasonable to characterise the accumulation CBS by an effective energy height. Finally, only thermionic emission over this effective CBS will be considered. Giv en the result from Chapter 4 that was mentioned at the start of this paragraph, the effective height of the CBS is given by: ECBS  =  =  where 0  Um  —  AEI  qzV + q (1  —qAV(l  —  —  UmaxIVBE  (6.48)  Umax+ UmaxVraj)  1, and Um will be taken as a phenomenological constant. Strictly, based upon  the analysis of Chapter 4, Umar will have a temperature dependence. However, as a first approxi mation, U can be taken as a constant independent of temperature. Then, the transport current under the accumulation regime is simpiy given by the thermionic term from eqn (4.79) as: ECBS  FfS  =  e 1 FfSO V  (6.49)  where both sub-bands within the valence and conduction bands need to be considered in the case of the Sii..Ge material system.  E  -\  1 reference ‘qc,  1 Iv(0)I = qzVV,Fig. 6.16. Diagram of the CBS that forms under the accumulation regime. Only the conduction band is shown, but a similar structure can occur in the valence band. Note: this is for one sub-band. July 12, 1995  178  6.5 Conventional and Novel Sii..Ge IIBT Structures The Sii..Ge material system represents a further step on the road to bandgap engineering. Unlike the AlGai..As material system, the Sii.Ge material system allows one to essentially manipulate IXEg and iXE (and thereby AE) independently. This independence between AEg and AE is achieved through two independent parameters: 1) the Ge mole fraction xa in the pseudo  morphic strained alloy layer; 2) the amount of compressive or tensile strain applied to the pseudo morphic alloy layer by the substrate (i.e., the substrate Ge mole fraction x ). The addition of strain 5 is the key to the rich possibilities regarding baudgap Engineering offered by the SiiGe material system. Sections 6.1 through 6.4 have set out the various material models and transport models to study the flow of charge within a SiGe HBT. This section will apply the results of these previous sections to the study of current-day SiGe HBTs structures, as well as some other novel structures. The study of highly strained pseudomorphic layers cannot be properly performed without consideration of the critical layer thickness h. As was stated early on in this chapter, the potential strain in the Sii.Ge material system can be quite large, owing to the 4.2% lattice mismatch be tween Si and Ge. As the in-plane strain is increased (see Fig. 6.3), the maximum thickness of the alloy layer decreases in an essentially exponential fashion. The determination of h has been the focus of numerous studies and controversies [97,99,105]. At present, there is still debate as to the exact model for h versus in-plane alloy strain, but the work of People [105] is at least a reason able reference point. In [105], the critical layer thickness is given as: h where h is in  =  1—v h 1 1 2 b (_)(_n(T)J i+v 20ic&  (6.50)  A, b = 4A (the magnitude of the Burger’s vector), v is the Poisson ratio from eqn  (6.5), aa is the unstrained (bulk) alloy lattice constant from eqn (6.6), andfis the alloy strain given by (aa  —  a)Ia (where a is the substrate lattice constant). Substituting all of these parameters into  eqn (6.50) gives: h  =  (5.43 + 2 0.23a” h 1.928 ln(). ,j 4 (5.43 +0.23aa)  (6.51)  Eqn (6.51) is an implicit phenomenological equation that People has fit to the best available data for h (see Fig. 6.17). Detailed information, such as what temperature and duration can a pseudo morphic layer tolerate before relaxing is still not conclusively known. July 12, 1995  179  101 0.0  0.2  0.4  0.6  0.8  1.0  Germanium Mole Fraction xa Fig. 6.17. Critical layer thickness for a Si xpexa layer on a { 100) Si substrate. If the substrate .  is SiiGe instead, then a good approximation is to find Ixa xj and use this on the above plot. -  Current-day SiGe HBTs, of which [100-1031 are examples, have all been based on a sub strate that is { 100) Si. The emitter and collector regions are pure Si, and the base is the only re gion made up of Sii..Ge The essential premise for this type of SiGe HBT stems directly from .  the early work of Kroemer [2,46,47] and Shockley [1] who called for a wide-bandgap emitter in jecting into a narrow-bandgap base. Within this physical construct, the Ge alloy content xa in the base is either fixed at some constant value, or a drift field is created in the base by increasing Xa as one proceeds from the emitter towards the base. Starting with a constant xa in the base of 0.2, then eqn (6.51) gives h  1550A. Because the  HBT is lattice matched to a pure Si substrate, all regions of the device except the base have E and E degenerate, as well as E and E’ degenerate. However, compressive strain in the base pro  duces E’ 2  -138meV, meaning that the ultimate conduction band in the base is E-like. Further,  compressive strain in the base makes the ultimate valence band E-1ike, with E’ lh  34meV  Fig. 6.18 presents the band diagram for the above device, with the relevant material parameters noted. Observation of Fig. 6.18 clearly shows that electron transport will occur via E. Since July 12, 1995  180  =  -100meV, while AE is 37meV, ostensibly all of the electrons contained by the E band in the  emitter (which is 33% of the total number of majority electrons) will be reflected by AE and not contribute to electron transport. Thus, if the EB SCR determines the transport current, then after including the different effective masses, I would be 18% less than expected from a simple exam ination of the device that does not account for the independence of E and E. However, if the neutral base region determines the transport current, then I would be larger than expected given that D is higher than the bulk value. In order to determine if it is the LB SCR or the neutral base that is responsible for current-limited-flow, the detailed construction of the device must be consid ered. For the devices in [100-102], where ND  >>  NA, then the neutral base is narrowly responsible  for current-limited-flow; although, inclusion of bandgap narrowing effects could lead to the EB SCR being responsible for current-limited-flow. However, for the devices in [10,1341, where ND NA, then depending on how bandgap narrowing in the base splits between E and E the EB SCR will be responsible for current-limited-flow; resulting in a much smaller increase to I than would be expected from neutral base transport considerations alone. This analysis of current-day SiGe HBTs shows that a failure to correctly model both E and E, including EB SCR limitations, could lead to an incorrect understanding of transport within the device.  2 • 0 Ge 08 Si  Si  Si  Ehl_  E’ E  Emitter  .  Base  Collector  AE  =  37meV  AE  =  —138meV  iXE  =  —100meV  AE  =  —104meV  IXE  =  37meV  AE  e  =  1120meV  fl e =  Eg,b  =  945meV  =  =  —138meV 3 c 9 6.94x10 m 3 cm 11 1.47x10  Without LB SCR limitations, E will transport 0.25% of the current in the neu tral base, leaving E to transport the re maining 99.75% of the current.  Fig. 6.18. Band diagram for an HBT with 20% Ge in the base, lattice matched to Si. The base is the reference. The effect of the LB and CB SCR potential is not shown for clarity. July 12, 1995  181  For SiGe HBTs, where the emitter and base are E-1ike, AE is too small to produce a CBS (see Fig. 6.10). Therefore, unlike AIGaAs HBTs, when the EB SCR limits the transport current in SiGe HBTs, then log I versus VBE will look identical to the case where the neutral base limits the transport current (i.e., the injection index will be unity). Thus, there will be no overt tell-tale sign in SiGe HBTs that the transport current is not being controlled by the neutral base. However, Ic will indeed be smaller than expected due to the EB SCR limitation, plus, the Early voltage should become theoretically infinite as basewidth modulation should no longer effect I [135]. The SiGe HBT where xa is varied across the base represents the device that has piqued the interests of the semiconductor community. By generating an aiding field in the base through a monotonically increasing  Xa  from the emitter to the base (and hence a decreasing Eg), an fT as  high as 113GHz has been obtained [1021. In order to achieve this remarkable metric the device was fabricated with as large a Axa in the base as possible; minimising the base transit time. To this end,  Xa  was 0 at the emitter and was linearly ramped up to 0.25 at the collector. The result is a  band diagram as depicted in Fig. 6. 19a. Since the neutral base closest to the emitter is pure Si, then one has essentially a homojunction for the EB SCR, and it is expected that the neutral base will limit the transport current (see Fig. 6. 19b). The base region, given the shape of the E and E bands, produces a demanded current that differs between the sub-bands by a factor of 8.3; i.e., the current in E will be 8.3-fold larger than E. This is not an overwhelming amount, which shows that 11% of the collector current is carried by the slower E band. In fact, using eqn (3.8) shows that tB for E is reduced 4.6-fold compared to tBo, while tB for E is reduced only 1.5-fold com pared to  ‘rho (where tBo is the ‘CB given in eqn (3.6)). Assuming that the final base transit time is  given by the average of the results from each band weighted with the relative current carried by the band, then the effective reduction to tB compared to tBO is (0.89/4.6 + 0.11/1.5)-i  =  3.8-fold.  If the two sub-bands were considered as one single band then tB would have been wrongly re duced 4.4-fold relative to  tBo, and I overestimated by  13%. In the above calculations the effect  of bandgap narrowing has not been accounted for. Inclusion of base bandgap narrowing could cause the EB SCR to limit the transport current (again, depending on how the bandgap narrowing splits between E and Er), which would greatly effect the current partitioning between the con duction sub-bands. Furthermore, the anisotropic nature of E and E has also not been accounted for, which would increase ‘CB even further given that  would be greatly reduced.  July 12, 1995  182  (a) E, E  Si  025 Ge 075 Si  Emitter Ehh, E ’ 1 1  .  Si  Base  I  lh  E  216meV  AE  =  0meV  =  44meV  AE  =  42meV  AE  =  216meV  AE  Eg e  =  1120meV  ‘e  b 8 Collector E  hh  E  —  =  904meV  = =  0meV 3 c 9 6.94x10 m 3 c 1 3.17x10’ m  Without EB SCR limitations, E will transport 11% of the current in the neutral base, leaving E to transport the remaining 89% of the current.  ‘i  (b) 108  •  •  E current withinthe EB SCR  1  6. E current within the neutral base  5 io  2/$  4.  ,...  >  / ,2’> “..  ‘C  ,-....-.  2 in  10 1  100 10-1  . .  /• <V /.. /..•.  :, / ‘./  10-2 0.6  •  0.7  E current within 2 the EB SCR  E current within the neutral• base 0.8  0.9  •  •  1.0  1.1  Base-Emitter Voltage VBE (V) Fig. 6.19. (a) Band diagram for an HBT with 25% linear grading of Ge in the base, lattice matched to Si. zS.E is from the 25% Ge point in the base to the emitter. Note: the base bandgap has a slightly parabolic nature due to the Ge alloy effects. (b) Transport currents through the various regions of the HBT, including the collector current. NA=5x10 , and 3 cm 20 , ND= 1x10 3 cm 18 Given E transport that within EB SCR the is not substantially larger than transport WB=700A. through the neutral base, I is subsequently 31% lower than expected from neutral base transport considerations alone. Thus, the neutral base is controlling I but the EB SCR does have an effect. July 12, 1995  183  The previous analysis of conventional SiGe HBT structures is not intended to be exhaustive, but it clearly demonstrates that the Sii..Ge material system cannot be characterised by an effec tive conduction band. In order to properly model a SiGe HBT, the rich nature of the E and E bands must be included via the models developed in Sections 6.1 to 6.4. Further, the assumption of Shockley boundary conditions (i.e., that the EB SCR is not responsible for current-limitedflow) can come under question in the design of SiGe HBTs. Finally, the importance of consider ing transport through the entire device becomes even more important when optimisation, or the extraction of material parameters, is sought after: for if the transport is being dictated by a region other than the one being considered, the result will be a an erroneous conclusion regarding either the correct path for optimisation or the material parameter being extracted. The main problem with the Sii..Ge material system is that the band offsets tend to be quite small because of the limits imposed on the Ge content by the critical layer thickness. For this rea son, it is still common to see ND >> NA in order to maintain a usable reduced, then NA must increase in order to offset a rapid decrease  f3. As the neutral base width is However, increasing NA  must be accompanied by an increase in ND or the gain will drop. With ND near the solid-solubility limit this is not really possible. Further, with NA and ND increasing, the EB capacitance will in crease, and a tunnel diode could form. The device in [134] attempted to solve this with a constant 22% Ge base content. By having a narrow bandgap in the base, the subsequent increase to Ib can be traded off for a higher base Gummel number. However, this precludes a graded base, as the EB heterojunction is required to maintain the gain, and the critical layer thickness will not allow for a higher Ge content (this is the alloy budget of Section 3.2). Therefore, in order to continue decreas ing the neutral basewidth without compromising fmax orf a way must be found to include higher Ge contents in the base. The answer to the problem of the previous paragraph is to lattice match the HBT to a Sii.Ge substrate, where x,> 0. Consider a  500A SigjgGe 02 emitter with a poly-Si cap, a base  graded from Si 025 at the emitter to Si Ge 075 0 04 Ge at the collector, all lattice matched to a 6 0 02 Si Ge collector and substrate (see Fig. 6.20). The base grading is started at 25% Ge instead of 8 20% in order to increase the transport current in E relative to E, thereby reducing the parasitic effect on tB found from the HBT in Fig. 6.19. Then, the 15% Ge base grading provides the aiding field to keep the base transit time small. However, unlike the HBT in Fig. 6.19, the optimum aug  July 12, 1995  184  mented-linear doping of Fig. 3.8 is used instead of the sub-optimum linear grading. The optimum base profile, due to the constant Ge regions near the emitter and the base, also increases the Early voltage and decreases the anomalous change to I due to the reverse Early voltage effect [11]. The  500A Si 0 02 Ge emitter next to the base ensures that the EB SCR will be free of dislocations that 8 will occur at the boundary to the poly-Si cap; plus it serves as an efficient source of E electrons. Finally, the poly-Si emitter cap provides stress relief to the system and a wide bandgap to kill the back injection of holes. With the wide bandgap of the poly-Si cap controlling the gain, NA can be significantly increased in order to increase f,, while ND can be decreased in order to decrease the EB SCR capacitance. The result is a 264-fold increase in I compared to a similar bulk Si de vice, with tB reduced 2.9-fold compared to  tBO.  These results are based upon the neutral base con  trolling I. As NA is increased to the point where bandgap narrowing becomes quite large, it is expected that the EB SCR will dictate I and limit the expected increase to  13.  bO.l W 6 Bj .,-l  C  0 75 Si 025 Ge  Sio.8oGeo.2o  040 1 Ge 060 Si  Emitter C  Base  WB 16 O.  0 80 Si 020 Ge Collector  c  =  2 AE  120meV  AEhhh1 v  —18meV  zE V  =  —34meV  i,b  9meV  E  =  3 cm 10 3.68x10  =  3 cm 11 5.05x10  e  =  990meV  b  =  876meV  ‘  =  120meV  tE  =  —34meV  E  Without EB SCR limi tations, E will trans port 4.5% of the current in the neutral base, leaving E to transport the remaining 95.5% of the current.  Fig. 6.20. Novel SiGe HBT based on a 20% Ge substrate. The incorporation of the optimum base grading provides the maximum reduction to ‘CB possible. The poly-Si emitter cap provides the wide bandgap necessary to control hole back injection, while lattice matching to a 20% Ge substrate allows a 40% Ge content in the base without being restricted by h. iXE is from the 40% Ge point in the base to the emitter. July 12, 1995  185  The operation of the novel transistor being proposed rests on two requirements: 1) that high quality SiiGe substrates can be formed; 2) that the poly-Si cap will indeed control the back injection of holes. The ability to grow high quality SiiGe substrates is currently an issue. At present, bulk epitaxial SiiGe layers on top of Si substrates have defect densities ranging from cm [311. This is too high to produce commercially yielding LSI ICs. However, 6 10 2 c 4 10 m to 2 given the infancy of epitaxially growing bulk SiiGe layers on Si, in time it is expected that the process will mature and the defect density will fall. The other option is to pull raw SiiGe in gots so that the starting wafer contains the desired substrate. In either case, for the study being presented here, it is sufficient to demonstrate the usefulness of using non-Si substrates in order to provide the impetus to grow low defect bulk Sii..Ge substrates on Si. The second question, re garding the efficacy of the poly-Si cap to control hole back injection, can only be answered by ex perimentation. However, recent work by Kondo et. al. [136,1371 for poly-Si to Si shows that the interface is not characterised by a high recombination velocity, and that the bandgap is, if any thing, larger than in bulk Si. Thus, n 1 in the poly-Si layer will be small compared to the n 1 in the base, controlling the back injection of holes and 3. Finally, the band alignment of the poly-Si layer to the Si 0 02 Ge emitter will only be an issue if the resulting AE is large enough to limit the elec 8 tron transport current through the entire device. Based upon Si lattice matched to , 02 G 8 Siij e AE should not exceed -90meV, which would not reduce the transport current given the high doping that would exist in the poly-Si layer. Therefore, it is expected that the poly-Si cap will control the hole back injection of the proposed SiGe HBT. This section concludes by examining an intriguing HBT structure that invokes all of the models of this chapter. Beginning with Fig. 6.9c for Xal strates where 0  3 x  =  0 and  Xar =  0.45, examination of sub  0.35 is very interesting. Let the left side be the emitter and the right side the  base. The emitter is under tensile strain so that the ultimate conduction band is E-Iike. Contrari ly, with the substrate range being considered, the base is under compressive strain and the ultimate conduction band is E-like. Just because the emitter conduction band is E does not preclude electrons from existing in E. In fact, given the band alignments for 0  x  0.35, more electrons  from E, rather than E, will be able to go from the emitter into the base. Essentially, the band with the lowest energy in both the emitter and the base will be the one that transports the current. With x  0.35, E will be responsible for current transport as E in the base is larger than E in  the emitter. July 12, 1995  186  2  1• E% Base 0 55 Si 045 Ge -E $E2  r—i——-— I I I  I I I  ç_)  C  \  •4444  Substrate Si 65 0 0 3e  4” s”  ;fSSd..  %%%%%t%.4CbS”  Emitter lr4nnl  o  i Vt)  io  V  4  C’  iM 44  :  4444  :1 —  b1  Collector Si 55 045 Ge  * ,,..nwt’*a”  Ehh v  e  AE  =  67meV  =  —237meV  =  —169meV  =  10 0 3 6.36 xl cm  =  o’i’  1i  2  9 E  e  =  932meV  =  908meV  cm  —3  —287meV  =  —168meV  =  =  —194meV  Fig. 6.21. Band diagram showing the conduction and valence sub-bands for an HBT where Xal = 0, Xar = 0.45, x = 0.35, NA= 1x10 , ND=5x10 3 cm 19 , and Wb=700A. 3 cm 17 Fig. 6.21 plots the band diagram, including SCR effects, for an HBT where Xal  0, Xar  =  0.45, x = 0.35, 3 cm 3 9 NA=lxlO’ , cm and wb=700A. As is the case for the HBT in 7 ND=5x10’ , Fig. 6.20, there is a high doped poly-Si cap on top of the emitter to provide stress relief and con trol the back injection of holes. What is interesting to note for the device in Fig. 6.21 is the emitter and base have essentially the same bandgap. Thus, there is no wide-gap emitter injecting into a narrow-gap base that is common to traditional HBT designs. Instead, the HBT is controlled by the band offsets and n 1 for the given sub-band within the neutral regions. Fig. 6.22 plots the EB SCR currents, the neutral base transport currents, and the final collector current that will occur within the device of Fig. 6.21. It is important to realise that Vbj device. For VBE  <  =  0.673 V due to the positive AE of this  V,j transport occurs via E through a small CBS, but with neutral base trans  port essentially controlling I. Thus, electron transport within the emitter is occurring in a band July 12, i995  187  that does not form the ultimate conduction band. Now, when I/BE> Vbj, the HBT is operating within the accumulation regime. Due to iXE = -169meV, EB SCR transport within E is reduced AIcm when VBE = Vbj. Furthermore, because AE lO to only 2  =  67meV, any increase in VBE past  Vbj will do nothing to increase the EB SCR current as there is no barrier to surmount, leaving only  the thermal movement of majority carriers to dictate the current. Thus, E transport is now con trolled by the EB SCR and not the neutral base. However, with the accumulation model of Section 6.4, E transport becomes the dominant path that controls I when VBE increases past V; lead ing to transport in the base that occurs within a band that does not form the ultimate conduction band. The final result is a very interesting log I versus VBE relationship that is due to the interac tion between the two conduction sub-bands.  106 1 1  c  1 102 101 100 10-1 10-2 1 0current within the neutral base 1 00.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Base-Emitter Voltage VBE (V) Fig. 6.22. Transport currents within the various regions of the HBT given in Fig. 6.21. The HBT of Fig. 6.21 may have some practical uses as a current source due to its flat I ver sus VBE relationship near V,j; however, it is probably more useful as a tool to investigate transport July 12, 1995  188  properties and band offsets within the Sii..Ge material system. Careful consideration of h for this HBT reveals some interesting results. Because the Ge content in the base is only 10% higher than in the substrate, then h  =  8054A. With such a large h it is conceivable that the base and in  trinsic collector regions could be formed without a heterojunction, thereby achieving an SHBT in stead of the DHBT common to SiGe devices. Furthermore, it is not unreasonable to imagine that the base and intrinsic collector could be formed in only 3000 to  5000A,  leaving considerable  room to the maximum h, which should help to increase the thermal budget for the base layer. The issue of DHBT devices is not a real concern in npn HBTs, due to the small AE, but would be of considerable appeal in making a pnp device. Finally, the result of a large h for the base and col lector regions is a significant lowering of the emitter h to 407 A. However, an  =  407A would  be wide enough to contain the emitter extent of the EB SCR. Therefore, the critical layer thick ness has been moved from out of the EB-SCR and into the neutral emitter, which will have less of an effect on device performance if dislocations due to strain relaxation occur. In conclusion to this chapter the following results regarding the Sii..Ge material system have been achieved: • A review of the literature, including the best material models, for the effect of strain on the conduction and valence sub-bands has been performed. • The band offset theory of Van de Walle and Martin, including the material models of Yu and Gan et. al., have been reviewed with the most consistent set of material parameters chosen to fit the experimental data available to date. To this end, a simple set of equations has been found to accurately describe the conduction band. • A theory regarding transport within the conduction E and E bands, and the valence E and E bands has been developed. The theory presented does not resort to an effective conduction and valence band, but considers carrier transport within both sub-bands. Included in this development is the full effective density of states and the intrinsic carrier concentration for all of the sub-bands. • A theory for the operation of an HBT past the built-in potential has been developed. • Finally, the models of this chapter, which are based upon the models of all the previous chapters, have been used to study current-day SiGe HBTs and a few other novel structures. The  July 12, 1995  189  most important result of this study is that the neutral base will no longer be the sole region controlling I as the neutral base width continues to shrink and the Ge grading in the base increases: the limitations of EB SCR transport must be considered. Furthennore, there is a significant error in both the calculation of charge flow and transit time by considering the subbands as a single effective conduction or valence band.  July 12, 1995  190  CHAPTER 7 Summary and Future Work  July 12, 1995  191  To begin with, Chapter 2 has presented a unique and general model (eqns (2.7) and (2.9)) for the simulation of HBTs. This model forms the framework for simulating charge transport within the entire HBT by providing a means to break the modelling effort into separate physical regions; each region characterised by its own unique physical transport process. Furthermore, the model presented in Chapter 2 allows for the existence of recombination sinks within each region; furthering the general nature of the model. Due to the abstract nature of eqns (2.7) and (2.9), it is possible to apply the model of Chapter 2 both to the microscopic transport of charge (i.e., to trans port over atomic distances), and to the macroscopic transport of charge (i.e., to transport over dis tances large enough to treat the electrons as a continuous flux, such as is done in drift-diffusion analysis). In so doing it may be possible to determine the point at which rapid spatial changes in the conduction or valence bands produce transport conditions that deviate from the models of drift and diffusion (such as can occur within an SCR, and certainly at the heterojunction where AE fonns). This may allow for a solution to a question posed by Dr. Mike Jackson of UBC as to the condition for which thermionic injection begins and drift-diffusion ends. However, the most logi cal extension to the work of Chapter 2 is to remove the restriction that Ep, (for an npn HBT) be a constant throughout the EB SCR. Chapter 3 presents some interesting ideas for optimising the metrics of an HBT by exploit ing the concept of current-limited flow outside of the neutral base. It would be a reasonable exten sion to the ideas of Chapter 3 to simulate and measure a number of HBT designs that exploit the optimisations that have been alluded to. Chapter 3 has also gone on to determine the simultaneous optimisation of the base bandgap and the base doping profiles for the minimisation of  tB.  This  work has, however, neglected the effect of a non-constant mobility with respect to doping varia tions. Numerical work [63] has shown that the optimum profiles which include the full pfl(NA) do not appear to be too complex, and certainly have a shape that is expected from consideration of the functional form of .I(N) itself. Therefore, it is expected that the analytic optimum profile, shown in Fig. 3.9, for the minimisation of tB could be extended to include either the full J.Ifl(NA) or a judicious approximation to it. Chapter 4 derives the model of charge transport with the EB SCR, including the effects of tunneling and momentum conservation across a mass boundary. To this end, the general models of eqns (4.50)-(4.53) were presented. Chapter 4 goes on to derive analytic approximation to eqns  July 12, 1995  192  (4.50)-(4.53). However, for the purpose of deriving analytic results, the mass boundary is consid ered in an isotropic fashion, but with the effective mass maintained as a diagonal tensor and not a simple scalar. Thus, a logical extension to the analytic work of Chapter 4 is to remove the assump tion of an isotropic mass boundary and resolve eqns (4.50)-(4.53) in an analytic form. Other extensions to the work of Chapter 4 are certainly alluded to in Section 4.6. By plotting the ensemble electron density entering the neutral base of the HBT, it was clear that the distribu tion could not be considered as a Maxwellian or even a hemi-Maxwellian. These distortions from a hemi-Maxwellian form are due to the effect of tunneling though the CBS. Since accurate simu lation of transport through a narrow base (in terms of mean free path [43]) demands a full solution to the BTE, then a way must be found to incorporate the non-local effect of tunneling into the BTE. A possible extension to the work of Chapter 4 is to connect the EB SCR transport models of the chapter to a BTE solver for the neutral base; thereby allowing for the inclusion of tunneling within the BTE via a hybrid model. The modelling of charge transport in Chapter 4, due to tunneling through the CBS contained within the EB SCR, is formulated upon ballistic considerations. It is common to consider tunnel ing electrons in a ballistic fashion, if for no other reason than to simplify the calculation of the tunneling probabilities. This position of neglecting thermalising collisions of the electron while undergoing tunneling is often substantiated on the grounds that tunneling distances are generally less than 100 or  200A, and are therefore significantly less than the mean free path. However, if  any collisions did occur while the electron is in the midst of tunneling, then the tunneling proba bility would be essentially reduced to zero. Thus, a potential extension to the work of Chapter 4 is to consider non-ballistic tunneling. The ultimate outcome of such non-ballistic tunneling consid erations would be the development of a Monte Carlo simulator that can incorporate non-local ef fects (i.e., tunneling). A final extension to the work of Chapter 4 can be found by careful observation of Fig. 4.9 and eqn (4.74). Im occurs at  which for a fixed temperature is a constant. Furthermore, the  flux density cLy. is fairly well centred about U, and will become even more localised as the temperature is reduced. Therefore, the tunneling current through the CBS can be thought of as oc curring at an energy of qU(V  —  VBE)Nrat relative to the conduction band minimum in the  emitter. Now, the tunneling current is very sensitive to the forward-directed effective mass, which  July 12, 1995  193  is dependent upon the full nature of the dispersion relation E(k). Then, with the CBS responsible for controlling I, by measuring I the tunneling current through the CBS can be determined. Fi nally, by extracting the effective mass through a matching of the measured I to the tunneling models of Chapter 4, it should be possible to infer E(k). Therefore, it should be possible to ex tend the work of Chapter 4 by developing an electrical spectroscopy method for the determination of E(k). Chapter 5 presents the models for the recombination currents that occur within both the EB SCR and the neutral base. Specifically, the need to balance the total current entering a region with the net current leaving plus any charge that has recombined within the region, is considered. This leads to a mixing of the base and collector currents of an HBT. The result of this mixing is a new connection between the physical construction of the HBT and it’s terminal characteristics. Regard ing future work, the basis for all of the recombination models (SRH, Auger, and radiative) used within Chapter 5 is essentially drift-diffusion. By the arguments of Chapter 4, drift-diffusion anal ysis is not applicable within the EB SCR. Therefore, combined with the extension being proposed for Chapter 4 (regarding integration with the BTE), the recombination currents should be recom puted from a particle scattering cross-section point of view. This would place the calculation of the recombination currents on par with the quantum mechanical view of a tunneling electron. Chapter 6 reviews the various material models that are required to understand the composi tion of the conduction and valence bands within pseudomorphically strained Sii.Ge. Further, the band offset models for the determination of IXE and AE at an abrupt heterojunction are also pre sented. Using these material models, transport models which include the two conduction subbands E, E and the two valence sub-bands E, E, are developed. It is shown that the multiband nature of strained Sii..Ge must be considered, even in present-day HBTs, lest considerable error regarding both the quantitative and qualitative aspects of charge transport be made. Regard ing future work, it is imperative that a final and consistent set of material parameters for Sii..Ge be obtained. Without a firm understanding of the material parameters, it is impossible to accurate ly determine the transport current. With this in mind, Chapter 6 presents a number of novel HBT structures, including a study of some present-day HBTs. In order to ascertain the validity of the models developed within Chapter 6, these SiGe HBTs should be fabricated and tested against these theories  July 12, 1995  194  Finally, Chapter 6 only considers substrates aligned to (100). However, there could be con siderable performance gains for growth along (111). Traditionally, BJTs have used (111) aligned substrates because epitaxial growth is the fastest for this orientation. (100) aligned substrates have come about because of the need to minimise surface states at the SiJSiO 2 interface in MOSFETs. One of the most interesting features of strained Sii..Ge is the possibility of only having charge transport occur parallel to the small transverse mass for electrons. The anisotropic nature of Si produces a 5-fold difference between the transverse and longitudinal mass for electrons. Thus, a significant improvement to tunneling and mobility can be had if the electrons predominantly move with the transverse mass. This would be further increased by using the (111) conduction bands instead of the (100) bands. In fact, the (111) bands have a 20-fold difference between the transverse and longitudinal mass for electrons, with the transverse mass near that of GaAs. There fore, a logical extension to the work of Chapter 6 would be the development of (111) aligned transport models. Finally, with the ability to set a large effective mass band at an arbitrary energy above a light effective mass band, it should in theory be possible to produce negative differential mobility, in terms of t versus electric field, within strained Sii.Ge; leading to the possibility of devices, such as Gunn diodes, which can only be presently made in materials such as GaAs. Therefore, a further extension to the work of Chapter 6 is to investigate the feasibility of generat ing and utilising strained Sii..Ge films that produce negative differential mobility versus electric field. As a final parting comment regarding future work, it is clear that with the rapid progress continuing in the development of ICs, device dimensions will continue to shrink at an exponential rate. Obviously, this will take devices down into the atomic realm where distances cover only 10 Angstroms and not a thousand. Even with present-day devices, where relevant dimensions are 500 to  ioooA,  quantum mechanical effects are important (as can be seen from the consideration of  tunneling in Chapter 4). As dimensions reduce to  ioA, clearly, classical mechanics will have no  part. For this reason, work on hydrodynamic models, which are really only a second order pertur bative solution of the BTE (drift-diffusion being the zero-th and first), will have very limited use fulness. Instead, a “full” quantum mechanical model will be required. But then what is meant by a “full” model? With relevant dimensions of 1 oA, it will not even be possible to utilise Bloch’s the orem because there will truly be no dimension over which the crystal can be considered as bulk. 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Tamaki, “Hetro-Emitter-Like Characteristics of Phosphorous Doped Polysilicon Emitter Transistors Part I: Band Structure in the Polysilicon Emitter Obtained from Electrical Measurements”, IEEE Trans. Electron Dev., vol. 42, no. 3, 419426, 1995. -  [137] M. Kondo, T. Kobayashi, Y Tamaki, “Hetro-Emitter-Like Characteristics of Phosphorous Doped Polysilicon Emitter Transistors Part II: Band Deformation Due to Residual Stress in the Polysilicon Emitter”, IEEE Trans. Electron Dev., vol. 42, no. 3, 427-435, 1995. -  July 12, 1995  205  Appendix A Ramped NAB(x) to Minimise tB The proof of eqn (3.17) begins by solving eqn (3.10) for ‘tB using the doping profile depicted in Fig. 3.4. To this end, it is seen that the doping profile of Fig. 3.4 is actually a subset of the profile depicted in Fig. 3.3 with h 1 =0 and h 2  =  h. Using the symbolic math tool MACSYMA©,  eqn (3.10) yields the following result for tB based upon the distribution presented in Fig. 3.3: 1og(x—h11og(I  d5 ()  hl-h2  Ne e integrate(\neAu,x,h2, 1);  (c6)  (1_h2)Ne  (d6)  U integrate(d5,x,hl,h2);  (c7)  Is U  -  1 zero or nonzero? nonzero; M1og(1  (d7) Ne  h2  —  (h2—hl)e  hi  log(U)  -  2-1  h2log(U) h2—hl  log(U)  integrateneAu,x1);  (cS)  Ne (1_x)  (d8)  U integrate(d5,xch2);  (c9)  Is U 1 zero or nonzero? nonzero; -  (  (d9)  I  Nel  hllog(U)  (h2_ hi) e  h2—hl  —  1og(t x  h11og(t  h2—hl  h2-hl  (h2_ hi) e  log(U)  —  h2 log(U) h2—hl  log(U)  (dO) integrate(\ne ,x,x,hl); (dlO)  Ne (hl_x)  Eqn (d5) is the exponential doping profile for h 1  x  , and it ensures that there are no jump 2 h  discontinuities at the break points h 1 and h 2 between the exponential doping profile and the re gions of constant doping. Then, eqns (d6)-(dlO) collect together the various sub-integrals required to solve eqn (3.10). It should be noted that the doping at x  =  0 is Ne, at x  =  1 is N, and that U  =  Ne/Nc. Using eqns (d5)-(dlO), eqn (3.10) produces: July 12, 1995  206  (cli)  tau = ratsimp(radcan(integrate(1Ane*flO,d7+d6),x,O,h1)+integrate(radcan(l/d5*(d9+d6)),x,h1,h2)+ integrate(\uAne*(d8),x,h2, 1));  Is U 1 zero or nonzero? nonzero; -  (  (dli)  f  [(h22_2h2+  l)U_hl  +Uh1 (h12u_2h1h2+2h1)1og2(  ÷((2h2_2h1)U_h1+(2h22+(_4h1_2)h2+2h12+2h1)U_h1  *log(U)+(_2h22+4h1h2_2h12)U_M+(2h22_4h1h2+2hl2)U_M  2 U _M 2 log ( U)  Eqn (dli) is the general model for tB from the optimum doping profile of Fig. 3.3.  Using the optimum equation for eqn (3.17) is obtained by setting h 1  =  ‘CR  given in eqn (dli), then the  0 and h 2  ‘CB  needed for the proof of  h; i.e.,  (c12) ev(dl 1,hl=O,h2=h); ‘dl2 ‘  _ (h ( _ u 2 2h)log(u)_2h u)+(2h 2h+1)u +2h u+2h log 2  2 U log (U)  Eqn (d12) can then be solved for the h that minimises ‘CB Differentiating eqn (d12) with respect to h, setting equal to zero, and solving for h produces:  (c13) ratsimp(diff(rhs(d12),h)); (d13)  (h_1)Ulog ( 2 U)+ (U+2h_ 1)log(U)_2hU+2h U log 2 (U)  (c14) solve(d13=O,h);  Eh.  [  (U)+2log(U)_2U+2 2 Ulog  (c16) ratsimp(radcan(ev(d12,d14)));  (d16)  2 log(U) —3 U 2U 2 + 4 U— 1 2  2  2  2U log (IJ)+4Ulog(U)_4U +4U  where (dl4) is the same as h in eqn (3.17), and eqn (d16) is the same as  ‘CR  in eqn (3.17) once the  July 12, 1995  207  factor of 1/2 is included within tBO. This completes the proof of eqn (3.17) for the ramped NAB(x) to minimise tB. It should be noted that the output displayed within this Appendix comes directly from MACSYMA©. As such, there is occasion to perform some intermediates steps that are not instructive to the proof but are more of a bookkeeping function for MACSYMA© itself. This is why some of the d-equations are missing. Finally, it can be shown that an intriguing symmetry exists in the ramped doping profile. If the profile is changed from that shown in Fig. 3.4 so that the exponential region follows the con stant doping region, then it is found that tB remains unchanged from what is given in eqn (3.17), and h  —*  1  —  h. Returning back to eqn (dli), the necessary change to the doping profile is accom  plished by setting h 1  =  h and h 2  =  1 in the optimum equation for tB given in eqn (dli); i.e.,  (c17) expand(ev(dl 1 ,hl=h,h2=1));  U 2 h  h 1—h  —  1 1—h  hU  log(U)  hU 2  (d17) —  h 1—h  —  1 1—h  log(U)  l—h  1—h  el_h +  2  2  1—h  log (U)  log (U)  h +  log(U)  2 h —  U 2 h  1  — log(U)  2  log (U)  —  1 1—h  (U) 2 log  +  2h +  h 1—h  1  2  log (U)  —  2  log (U)  2 U’’ h 2  +  (clS) substpart(xthru(map(radcan,piece)),d17,2);  (d18)  — U(hlog(U)+2(_h+2h_l)]+2(l_h)Ulog(U)÷2(h_h)log(U)+2(h_2h+ i) 2Ulog ( 2 U)  Eqn (d18) is the tB for the symmetric doping profile used to develop eqn (d12). As was done with eqn (d12), the optimum value for h is found by differentiation eqn (d18) with respect to h, setting equal to zero and solving; i.e., (c19) diff(rhs(dl 8),h)=O; (d19)  U (2 h log (U) 2  +  2 (2_ 2 h))_ 2U log(U) ÷ 2 (2 h — 1) log(U) 2Ulog ( 2 U)  +  2 (2 h— 2) —0 —  (c20) solve(d19,h); (d20)  rIh= [  (U+1)log(U)_2U+2 2  Ulog (U)+2log(U)_2U+2  July 12, 1995  208  Eqn (d20) is that h which renders eqn (d18) a minimum. Substituting (d20) back into (d18) yields the minimum tB; i.e., (c21) ratsimp(radcan(ev(d18,d20))); log(U)_3U 2 2U ÷ 4U_1  (d21)  (U) +4 U log(U) —4 U 2 2U 2 log 2 +4 U  Eqn (d21) is exactly the same as eqn (d16), showing that a symmetric change to the doping profile produces no change to the transit time. It can finally be shown that the symmetric change to the doping profile results in h  —>  1  —  h by adding together the h from eqn (d14) and (d20); i.e.,  (c26) rhs(first(d20))-i-d14; (U) 2 U log  (d26)  +  (1_ U) log(U)  +  Ulog ( 2 U) +2log(U’)_2U+2  (U + 1) log(U)_ 2 U + 2 Ulog ( 2 U)+2log(U)_2U÷2  (c27) ratsimp(combine(d26)); (d27)  1  Eqn (d27) proves that the symmetric change to the doping profile of Fig. 3.4 does indeed result in  h—> 1—h.  July 12, 1995  209  Appendix B Optimum NAB(x) to Minimise tB The proof of eqn (3.18) begins by solving eqn (3.10) for tB using the doping profile depict ed in Fig. 3.3. However, this task has already been accomplished in Appendix A as eqn (dli). Us ing eqn (dli) for the optimum  tB,  the pair h 1 and h 2 which minimise eqn (dli) is found. Using  the symbolic math tool MACSYMA©, the partial derivatives of eqn (dli) with respect to h 1 and are taken; i.e., (c29) ratsimp(diff(rhs(dl 1),hl));  ( 2 (h1U_h2+1)1og 2h1  +  (d29)  I4  +  u-hi  _u _h11og(u)  +(2h2_2h1)UM +(2h1_2h2)U_hl  ( 2 1og (c30) ratsinip(diff(rhs(dl l),h2));  (I  [(h2_l)u1_hlUMJlog2(u)  +[UM+(2h2_2h1—1)UM1og(U)+(2h1_2h2)  (d30)  h2  hi  *_M +(2h2_2hl)Ul_M  _hi  (u) 2 log  Eqns (d29) and (d30) present the simultaneous set of equations, once both are set equal to zero, that must be solved to determine the pair h 2 which minimise eqn (dii). Given the highly 1 and h non-linear form of these two equations it is not clear that an analytic solution is possible. There fore, before attempting to solve eqns (d29) and (d30), a numerical solution will be found so that a “feel” may be developed that will hopefully guide the steps to follow. Using MACSYMA©, a numerical Newton-Raphson solution to eqns (d29) and (d30) is found for three different cases of U; i.e.,  July 12, 1995  210  (c40) newton(ev([d29,c130],\u=3.9d0),[hl,h2],[O.25d0,O.75d0fl; C:\MACSYMA2\share\newton.fas being loaded. C:\MACSYMA2\matrix\bla_lu.fas being loaded. C:\MACSYMA2\matrix\blinalgLfas being loaded. (d40)  [hi  =  0.2975325725O992dQ h2 = o.70246742749006dq  (c41) d40[1]-i-d40[2]; (d41)  h2+hl=l.OdO  (c42) newton(ev([d29,d3O],\u5O.4dO),[hi,h2],[O.25dO,O.75dO]); (d42)  [hi  =  O.i689i9l7072612dQ h2= o.83io8o82927388dq  (c43) d42[l]+d42[2]; (d43)  h2+hi=1.OdO  (c44) newton(ev([d29,d30],\u=2000.4d0),[hl,h2],[O.25d0,O.75d0J); (d44)  [hi  =  O.i04i5470580558dQh2= O.895845294i9442dq  (c45) d44[1]+d44[2]; (d45)  h2+hi=l.OdO  The numerical results of eqns (d40)-(d45) indicate that h 1 =  +  =  1. In order to prove that h 1  1 is indeed a solution of eqns (d29) and (d30), the following is performed: substitute h 2  =  1  +  2 h  —  into both eqns (d29) and (d30); then, if the resulting eqns differ at most by a multiplicative constunt, then it is proven that h 1  +  2 h  =  1 is indeed a solution of eqns (d29) and (d30).  Using MACSYMA© to perform the above test yields: (c31) ev(d29,h2=l-hl);  (  1_2h1  +  (hlU+hi)log ( 2 U)  [(2 hi_2 (1_hi) +1)  u  1-2h1  —U  12M]log(U)  (d31) +(2hi_2(i_hl))U  U  1-2h1  +(2(i_hi)_2hi)U  1—2h1  ( 2 1og  July 12, 1995  211  (c32) ev(d30,h2=i-hl);  hi  II  (d32)  u  i-2M —  hi U  (U) 2 log  (l_hl)_i)U1_2M+U1_2hljlog(U) 2 + 1 +[(_2h  (2(1_hl)_2h1)U  hi i—2h1  U  +  (2h1_2(1_hl)) U  1—hi 1—2h1  (U) 2 log  ‘  (c33) ratsinip(combine(d31+d32)); 2h1  1  l t.lTT h 2 UI I—’  (d33)  +1  hu 2 I.lTT UI ‘.1  2hi 2hi—1  u (c34) radcan(expand(d33)); (d34)  0  Eqns (d31) and (d32), after substituting h 2  1  =  —  , are equal and opposite. Thus, these two equa 1 h  tions would differ by a multiplicative constant of “-1”. Eqns (d33) and (d34) prove that h 2  =  1  —  by showing the sum of eqns (d3 1) and (d32) vanishes. This result immediately asserts that there is only one independent equation to solve for. The solution for h 1 being:  (c35) distrib(expand(d3 1));  4 hi U  2hi i—2h1  1 1—2hi  —  u  log(U)  2hi 1—2h1  —  1 i—2hi  log(U)  —  4 hi U  1 log(U)  —  2h1 i2hl  1—2h1  —  (U) 2 log  +  1 (d35)  2U  i—2h1  l—2hi  —  2  —  4h1 —  log (U)  2  2  +  log (U)  +hiU  2  2h1 1—2hi  —  1 i—2h1  -i-hi  log (U) 1  2hi hhl 2 *U  1—2hi  (c36) map(radcan,d35); 1  4hi (d36)  (c37) (d37)  U log(U)  —  U log(U)  1 —  log(U)  4hi +  2  U log (U)  2 —  2  U log (U)  4h1 —  2  log (U)  2 +  2  log (U)  hi U  solve(d36=0,h 1);  rIhl= L  1 log(U)-i-2  July 12, 1995  212  Eqn (d37) proves eqn (3.18) for the optimum h , along with the result from eqn (d34) which 1 proves eqn (3.18) for the optimum h . Finally, using the optimum h 2 1 and h , the optimum 2  tB  is  found by substituting back into eqn (dli) found in Appendix A; i.e.,  (c38)  radcan(ev(dl 1,h2=1-hl)); 2h1-*-1  u 2h1—1  (  411  +L(h1_2h12)u211 (d38)  u+h1 log (bl ) 2 (U) 2 2h1-i-1 ((2h12_3h1+1)u÷4h12_2h1]Jlog(u)  ”’ 21 ÷u  2h1÷1  +u h 2 1 ((_4h12+4h1_1)u+4h12_4h1+  iJ  4h1  U  2h1—1  2  log(U)  (c39) radcan(ev(d38,d37)); 1  (d39)  Eqn (d39) is the same as  log(U) +2  tB  in eqn (3.18) once the factor of 1/2 is included within ‘CBO. This com  pletes the proof of eqn (3.18) for the optimum NAB(x) to minimise  tB.  July 12, 1995  213  

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