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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

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A Theoretical Toolbox for the Simulation and Design of HBTs Constructed inthe AlGai..As and Sii..Ge Material SystemsbyShawn Searles, P.Eng.B.Sc.E.E., The University of Manitoba, 1987M.Eng., Carleton University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 12, 1995© Shawn Searles, 1995In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(SiDepartment of &/c,ot/The University of British ColumbiaVancouver, CanadaDate / 795AbstractA theoretical toolbox for the simulation of Heterojunction Bipolar Transistors (HBTs), including the effects of tunneling, recombination, and the optimum non-linear base proffle (for theminimisation of the base transit time), is developed. The models developed are applicable to ageneral material system, and are analytic. Extensions specifically required by the complexSii..Ge material system are also developed. The optimum (to minimise base transit time) basedoping is found to be non-exponential, and the optimum base bandgap grading is not linear. Ageneral transport model for HBTs, including recombination processes, is developed that accountsfor the complex nature of charge transport throughout the entire device. Unique methods for optimising HBT metrics, which cannot be employed for Bipolar Junction Transistors (BJTs), are alsopresented. 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 developed. The implications of having different electron effective masses in the two sides of the heterojunction, leading to what is termed a mass boundary, is fully explored. It is found that thetunneling of electrons within the emitter-base SCR leads to a non-Maxwellian minority-particleensemble distribution entering the neutral base. Finally, transport within SiGe HBTs is considered, with all of the relevant material models presented and multi-band transport models developed. This treatment leads to a variety of interesting conclusions regarding the operation ofpresent-day SiGe HBTs and possible future designs.July 12, 1995 iiTable of ContentsAbstract .iiTable of Contents iiiList of Tables vList of Figures viAcknowledgment xCHAPTER 1: Introduction 11.1 Modelling Details 41.2 Thesis Organisation 7CHAPTER 2: A Multi-Regional Model for HBTs Leading toOptimisation by Current-Limited Flow 92.1 Bandgap Engineering 102.2 Regional Decoupling and Current-Limited Flow 122.3 Optimisation Through Current-Limited Flow 18CHAPTER 3: Base Layer Decoupling and Optimisation 213.1 Independent Optimisation of CBE, andy 233.2 Reducing ‘CE by Decoupling the Base from Ic 273.3 Optimum Base Doping Profile to Minimise‘CB 303.4 The Effect of a Non-Uniform n1 and D on the Optimum‘CE 38CHAPTER 4: Transport Through the EB SCR 434.1 Formulation of Charge Transport at the CBS 454.2 Incorporation of Effective Mass Changes 494.3 Calculation of Fr and a Unified Model for F 674.4 Analytic CBS Transport Models 704.4.1 Analytic Model for the Standard Flux 714.4.2 Analytic Model for the Enhancement Flux Ffe 784.4.3 Analytic Model for the Reflection Flux Ff 814.5 The Effect of Emitter-Base SCR Control on I 844.6 Deviations from Maxwellian Forms and Non-Ballistic Effects 954.7 Conclusion 105July 12, 1995 iiiCHAPTER 5: Recombination Currents .1075.1 Electron Quasi-Fermi Energy Splitting 1Ep, 1095.2 Modelling the Recombination Processes of HBTs 1115.2.1 SRH Recombination 1125.2.2 Auger Recombination 1155.2.3 Radiative Recombination 1165.3 Current Balancing with the Neutral Region Transport Currents 1175.4 Full Model Results 1215.5 Simple Analytic Diode Equations 128CHAPTER 6: The Sii..GeHBT 1326.1 The Effect of Strain on SiiGe 1356.2 Band Offsets in SiiGe 1516.3 Electron Transport in Strained SiiGe 1596.4 The Accumulation Regime Beyond the Built-In Potential 1716.5 Conventional and Novel SiiGe HBT Structures 179CHAPTER 7: Summary and Future Work 191References 197Appendix A: Ramped N(x) to Minimise tB 206Appendix B: Optimum N(x) to Minimise tB 210July 12, 1995 ivList of TablesTable 3.1: tB for the four doping cases: Optimum, Ramp, Step, and Exponential 37July 12, 1995 vList of FiguresFig. 1.1. Collector current for an abrupt A1GaAsHBT.5Fig. 2.1. Band diagram of an HBT including a graded-base bandgap 11Fig. 2.2. Band diagram of the emitter-base junction within an abrupt HBT 13Fig. 2.3. Hypothetical HBT structure showing the physical regions that govern chargetransport 14Fig. 2.4. The flow T that results from a series connection of six pipes 16Fig. 2.5. T for a three region HBT in the absence of recombination 19Fig. 3.1. Band diagram of both a homojunction BJT and an HBT 25Fig. 3.2. Emitter cap layer design to minimise RE and CBE 27Fig. 3.3. Optimum doping profile N(x) obtaining by numerical minimisation 33Fig. 3.4. The first trial function for N(x) inspired by the form suggested by Fig. 3.3 34Fig. 3.5. The second and third trial functions for N(x) 35Fig. 3.6. Step-doping proffle for N(x) 36Fig. 3.7. tB using N(x) from Fig. 3.3, where h1 1 — h2 and h2 is varied 37Fig. 3.8. Optimum bandgap in the base to minimise tB 40Fig. 3.9. The optimum stationary function y(x) which includes doping, bandgap, andbandgap reduction due to heavy doping for the minimisation of ‘CB 41Fig. 4.1. Abstract model of current flux within the region containing the CBS 46Fig. 4.2. Blow-up of theCBSfromFig. 3.1(b) 49Fig. 4.3. Definitions of the cylindrical momentum space coordinates for the calculationof the Jacobian Transforms from k to U-space 50Fig. 4.4. Domain of integration R1 for a uniform m* 54Fig. 4.5. The effect that conservation ofp has upon U±,i and U±,2 when a massboundary is placed at x =0 58Fig. 4.6. Domains of integration R1 and R2 for the enhancement case 61Fig. 4.7. Domains of integration R1 and R2 for the reflection case 62Fig. 4.8. Collector current for an abrupt A1GaAs HBT with 30% Al content in theemitter 71Fig. 4.9. Flux density normalised to4max’ for an Al03Ga07As/GaAs abrupt HBT 75July 12, 1995 viFig. 4.10. Standard Flux Fc and Reflection Flux Ffr for an HBT with the parametersgiven near the start of this section 86Fig. 4.11. Relative importance of Ffr to the total flux F for an HBT with the sameparameters as Fig. 4.10 88Fig. 4.12. Standard Flux Ff.and Reflection Flux Ff for an HBT with the sameparameters as Fig. 4.10, but with AE reduced from 0.24eV down to 0.12eV 89Fig. 4.13. Standard Flux Fj and the Enhancement Flux Ffe for an HBT with theparameters given near the start of this section 91Fig. 4.14. Relative importance of Fje to the total flux F for an HBT with the sameparameters as Fig. 4.13 94Fig. 4.15. Ensemble particle distributions assuming a purely thermalised thermionicinjection from the peak of the CBS in Fig. 4.2 96Fig. 4.16. Integrated ensemble distribution versus wave vector k,2 entering the neutralbase 98Fig. 4.17. Ensemble electron distribution entering the neutral base versus k 99Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier 101Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier 102Fig. 4.20. Relative difference between the results obtained from the methods proposedin [511 to the model for F from this chapter 104Fig. 5.1. Band diagram of the EB SCR showing the effect of the abrupt heterojunctionon under an applied forward bias (reprint of Fig. 2.2) 109Fig. 5.2. Components of the collector and the base currents emphasising that ThT mustequal the total of, + NB + SRN.B + Aug,B + Rad,B 111Fig. 5.3. Energy Band diagram for the EB SCR of an HBT under equilibriumconditions 113Fig. 5.4. Relative error between the approximate and exact fonns given in eqn (5.27) 120Fig. 5.5. Bias dependence of the SCR current from the emitter side, and the threecomponents of the SCR current from the base side 122Fig. 5.6. Gummel plot showing the importance of including the emitter- and base-SCRcurrent components in the computation of the total base recombination current 123Fig. 5.7. Bias dependence of the current gain f3, showing the relative importance ofincluding SCRB in the calculation of AE 125Fig. 5.8. Bias dependence of the current gain (3 for the case of Wflb increased to 5000Aand t, in the SCR reduced to 5ps 125July 12, 1995 viiFig. 5.9. Effect of changing the neutral base thickness W,,, when the CBS isresponsible for current-limited-flow 126Fig. 5.10. Comparison of the recombination currents when qi is given by the depletionapproximation and when it is given by the linearisation of eqn (5.11) 127Fig. 5.11. Z-functions as computed from eqn (5.13) when using the material parametersfrom Section 5.4 130Fig. 5.12. Comparison of the full model and “diode-like” expressions for the SCRcurrents 130Fig. 6.1. First Briulouin zone showing (in k-space) the constant energy surfaces nearthe bottom of the conduction band for Si and Ge 137Fig. 6.2. Valence bands in unstrained Sii..Ge 138Fig. 6.3. Commensurate growth of the Sii.Ge alloy layer to the Sii.Ge substrate,leading to a pseudomorphic alloy film 142Fig. 6.4. SiiGe bandgap when grown commensurately to a variety of substratesoriented along (100) 147Fig. 6.5. E and E conduction band energies relative to the unstrained conductionband edge for Sii..Ge commensurately grown to a variety of substratesoriented along (100) 148Fig. 6.6. E and E’ valence band energies relative to the unstrained valence bandedge for SiiGe commensurately grown to a variety of substrates orientedalong (100) 149Fig. 6.7. Constant energy surface plot depicting the E and E bands inSi083Ge17commensurately strained to (001) Si 150Fig. 6.8. Conduction and valence band energies including all of the band offsets for aSi 1xaPexai to a Sii.Ge heterojunction commensurately strained to a { 100)SiiGe substrate 153Fig. 6.9. E and E conduction band minima to the left and right of an abruptheterojunction when commensurately grown atop a { 100} Sii.Ge substrate 156Fig. 6.10. AE when Xar = Xal + 0.20, and Xal and x are varied 158Fig. 6.11. AE when Xar = Xaj + 0.20, and Xal and x are varied 159Fig. 6.12. Diagram of the A conduction band minima involved in f and g intervalleyscattering 162Fig. 6.13. Equilibrium band diagram of apn-junction, showing the relevant energies andpotentials 165Fig. 6.14. Band diagram for a np-junction with a positive step potential (i.e., AE <0) 172July 12, 1995 viiiFig. 6.15. The exact and approximate forms for Nrat and Yat from eqns (6.46)-(6.47) 177Fig. 6.16. Diagram of the CBS that forms under the accumulation regime 178Fig. 6.17. Critical layer thickness for a Sii..Ge layer on a { 100) Si substrate 180Fig. 6.18. Band diagram for an HBT with 20% Ge in the base, lattice matched to Si 181Fig. 6.19. Band diagram and Transport currents for an HBT with 25% linear grading ofGe in the base, lattice matched to Si 183Fig. 6.20. Novel SiGe HBT based on a 20% Ge substrate 185Fig. 6.21. Band diagram showing the conduction and valence sub-bands for an HBTwhere Xal = 0, Xar = 0.45, x = 0.35, NA=1x1019cmND=5x1017cm3,andWb700A 187Fig. 6.22. Transport currents within the various regions of the HBT given in Fig. 6.21 188July 12, 1995 ixAcknowledgmentI would like to thank first of all, Professor Dave L. Pulfrey, my Ph.D. supervisor. I returnedfrom industry to obtain my Ph.D. because I was interested in performing research that probed intothe complex theories of solid-state device operation. Thanks to Dr. Pulfrey and his forthcomingguidance, I was able to navigate a steady course through the often turbulent waters of academicresearch, and attain the research goals I had planned to explore. Dr. Pulfrey provided constant encouragement to my work, offered valuable assistance, and provided me a learning experience thatI know will serve me for the rest of my life. Dr. Pulfrey, however, went even further in his contributions during the time I worked on my Ph.D. He allowed and encouraged me to pursue other lifeinterests, so that I can proudly say that my Ph.D. research was indeed a time that touched and enriched all aspects of my life. So to you Dr. Puifrey I can only offer in return my simple but sincerest thanks.I would also like to thank Professor Mike Jackson who provided me with many ideasthroughout my Ph.D. research. Without the presence of Dr. Jackson, my Ph.D. research would nothave been as interesting nor as fulfilling as is has been. I would also like to thank Professor TomTiedje, who offered me an excellent course in solid-state quantum mechanics; without which Icould not have performed the Ph.D. research that I have done. Dr. Tiedje, you have challenged meand as a result, provided me a fundamental base from which I will solve many questions yet tocome. I would also like to thank Dr. Jackson, Dr. Tiedje, Professor Nick Jaeger, Professor MattYedlin, Professor Jeff Young, and Professor Fred Lindholm, whose presence on my examiningcommittees 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 willingpartner in my Ph.D. research efforts. Your caring presence has provided me a reference point thatI could always count upon, no matter how hard things became during the course of my researchgoals.July 12, 1995 xCHAPTER 1IntroductionJuly 12, 1995 1The main objective of the Ph.D. research being presented in this thesis is the creation ofmodels that will foster a deeper understanding regarding the physics surrounding a Heterojunction Bipolar Transistor (HBT). To this end, physically based models for the transport of chargewithin an HBT will be developed. These physics-based models will allow for the simulation ofpresent-day HBT structures and novel structures for the future. By clearly identifying the relevantmechanisms by which charge transport takes place within the HBT, an optimum design for the device that incorporates the various compromises between competing device metrics (such as f3,fand RB) can be obtained. A further goal is to reduce all of the models developed within this thesisto 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 emergenceof HBT-based Integrated Circuit (IC) processes. Finally, the models that are developed within thisthesis are in general free of any details specific to a single material system. However, given theimportance of the AlGai..As and Sii..Ge material systems, these two systems will be extensively 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, over30 years ago, Kroemer developed much of the fundamental physics regarding the operation of theHBT [2]. However, it has not been until the last five years that industry has had the capability tomanufacture HBTs with suitable yields to be commercially viable [3-5]. Also, the material research is still continuing and has a long way to go before HBT processes achieve the maturity oftechnologies such as CMOS. Furthermore, with experimental results becoming more prolific, andwith rapidly diminishing device dimensions, we are finding that much of the physics laid downfor 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 inthe need for models that predict device operation. It is now possible to manufacture HBTs withactive basewidths approaching 100 A [10-121 and with features that change over distances of lessthan 10 A [13-14]. As device dimensions approach the atomic lattice spacing of the crystal, theapplicability of models based upon classical continuous fields becomes questionable [15]. Thereis already general agreement that one must consider higher order moments beyond the drift anddiffusion terms in the Boltzmann Transport Equation (BTE) in order to model deep submicron devices [16-17]. The BTE is based upon classical physical models that in general do not incorporateJuly 12, 1995 2quantum mechanical (QM) phenomena. It has been recognised that the correct modelling of tunneling, a QM effect, is of paramount importance to the correct prediction of HBT operation [18-21]. Thus, models of HBTs that incorporate QM phenomena are becoming increasingly importantin order to maintain accurate simulation of the HBT.The general relationship between the terminal currents and voltages of an HBT can still bepredicted today by models designed for Bipolar Junction Transistors (BJTs) [22]. However, it isnot always clear why we can continue to apply BJT models to HBT operation when these BJTmodels were developed without consideration of the physical processes that govern transportwithin an HBT. Presumably, the BJT model has enough degrees of freedom so that it can be manipulated to cover HBT operation. For example, one of the most common discrepancies foundwhen using BiT models for HBT simulation is that the injection indices (ideality factors) for thecollector and base terminal currents do not correspond to what is theoretically predicted for BJToperation [23]. Thus, in order to accurately predict HBT operation, and to further develop HBTprocesses so as to advance device operation, one needs to understand such things as why the collector 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 theSii..Ge material system are mostly due the effects of strain. Due to the large lattice mismatch between Si and Ge, Sii..Ge films grown on top of Sii..Ge substrates (where x y) have a large degree of strain present within them if non-relaxed crystals with low defect density are to bemanufactured. The presence of strain breaks the cubic symmetry of the crystal and changes thebulk electrical properties [26-28] of the film. By varying the Ge alloy content and the strain imparted to the SiGe film, it is possible to tailor both the bandgap and the offsets in the conductionand valence bands. Therefore, models specific to the Sii..Ge material system must be developedin 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 stemsfrom the maturity of AlGaAs devices, and the massive installed base of Si-based IC technologiesthat would easily admit SiGe devices. From a manufacturing standpoint the AlGai..As materialsystem offers no redeeming features when compared to Si, save one - the lack of strain. Obviously, the key to the operation of an HBT is the formation of heterojunctions between two materialsJuly 12, 1995 3characterised by different bandgaps. The AlGai.As material system has essentially a fixed latflee constant over the entire range of Al mole fraction x. For this reason, the MGai.As materialsystem is lattice matched and will admit an arbitrary heterojunction between AlGai..As andAlGaiAs without developing a strain within one of the films. This lack of strain within theMGai.As material system helps to ensure a defect-free heterointerface that greatly facilitatesthe manufacture of HBTs. For this reason, most commercially available HBTs are based in theAlGai ..As material system [29]. However, most solid-state devices are Si based [30]. With theadvancement of low-temperature Chemical Vapour Deposition (CVD) processing [31], the formation 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 shortlysurpass A1GaAs HBTs as the most prolific commercially available HBT [32-37].1.1 Modelling DetailsResearch has been conducted into the injection of electrons from the emitter into the base ofAlGai..As npn HBTs [18,24,25]. The research has centred around abrupt HBTs where the heterojunction between the wide-energy-gap emitter and the narrow-energy-gap base is abrupt. In anabrupt AlGai.As HBT one finds the formation of a Conduction-Band Spike (CBS) between theemitter and the base (see Fig. 3.1). This spike, due ostensibly to differences in the electron affinityof the materials used for the formation of the emitter and the base, results in a large impediment tothe flow of electrons from the emitter into the base. In fact, if the CBS were not taken into accountwhen modelling the HBT, the collector current would be overestimated by over three orders ofmagnitude at room temperature (see Fig. 1.1). However, the modelling of charge transportthrough the CBS cannot be based upon simple thermionic injection alone. Since the width of theCBS is typically less than boA near the top of the spike, the occurrence of a tunneling currentcannot be neglected. Finally, it will be shown that transport through the CBS can often be the limiting factor for the overall transport of charge within the HBT (i.e., the determination of the collector current Ic). This occurrence of current-limited flow outside of the neutral base region willbe studied and exploited for device optimisation. Therefore, the modelling of the relevant physicalphenomena surrounding charge transport through the CBS, including tunneling and conservationof transverse momentum across the heterojunction in a diagonal mass tensor, will be investigated.July 12, 1995 41 W3,— LUI;,10C)io-90.8 1.0 1.2 1.4 1.6Base-Emitter Voltage VBE (V)Fig. 1.1. Collector current for an abrupt AIGaAs HBT with 30% Al content in the emitter. Theemitter doping is 5x1017cm3,and the base doping is 1x109cm3(see Section 4.5 for the complete device details). The top curve, where CBS limitations have been neglected, is arrived at byassuming 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 controllingIc is the question of how to join these various regions together to form one cohesive transport model. Furthermore, the possibility exists that under multi-regional control ofI, older models, such asthose for the neutral base [38], which assume that only the specific region being studied controlsIc may not longer be valid. It will be shown in Chapter 2 that there is a very simple prescriptionfor joining up all of the multi-regional transport models into a complete transport model for the determination of I. It will be further demonstrated in Chapter 6 that it is possible for two spatiallyseparate regions to control I simultaneously by having essentially identical net-charge-transportcapacity 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 physical regions of the HBT together, then the problem of modelling charge transport within the entireHBT 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 siJuly 12, 1995 5multaneous optimisation of the base bandgap and doping profile (provisions are also made for theinclusion of bandgap narrowing due to heavy doping effects) for the minimisation of the base-transit time ‘CD. Finally, the modelling of recombination events, which lead to the formation of thebase 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 withinthe HBT, with the general transport model of Chapter 2 forming the blueprint for the ultimate operation of the device.The modelling efforts presented in this thesis regarding charge transport through the EBSCR are rigorous in that no appeal has been made to drift-diffusion analysis based upon phenomenological 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 saidphenomenological mobility models, are analytically solved for. However, the neutral base chargetransport models are based upon drift-diffusion analysis. The reason for resorting to simpler drift-diffusion analysis for the neutral base is its been found that the neutral base often does not represent 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 meanfree 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 ofdrift-diffusion contained within the BTE [16,41]. Instead, a general solution to the BTE is necessitated.In present-day HBTs, and even in some of the emerging high performance BJTs, the understanding of hot electrons can be essential to the accurate modelling of the device’s terminal characteristics [9,14]. The problem with general BTE solvers, such as Monte Carlo simulation, is thatsome important QM effects cannot be modelled. The BTE is based upon local potentials andtherefore cannot include some QM effects, such as tunneling, which are inherently non-local. Aswas discussed and shown in Fig. 1.1, the failure to include tunneling results in a gross error regarding the transport of charge through the HBT. Section 4.6 will address the issue of mergingclassical BTE solvers with the models developed in Chapter 4 for charge transport through theCBS. Specifically, Section 4.6 will show that tunneling produces a considerable distortion to theminority-particle ensemble distribution entering the neutral base (deviations that are far fromJuly 12, 1995 6Maxwellian or even hemi-Maxwellian). Finally, it should be noted that the use of drift-diffusionmodels in the neutral base will not produce gross errors like the failure to include tunnelingthrough the CBS. Instead, drift-diffusion models can be employed in the neutral base, but withcorrections 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. calculations, such as for tB, there would be an error.The final modelling effort of this thesis pertains directly to the design and simulation ofSiGe HBTs. As has been alluded to, the effect of strain on the electrical characteristics of Sii..Gefilms is dramatic. Chapter 6 reviews the various material models necessary for the description andstudy of the electrical characteristics of strained Sii.Ge. Specifically, once a review of the literature regarding the Sii..Ge material models is presented, a comparison to experimental results isperformed, and the most consistent set of material constants selected. The final result is a complete set of models for the calculation of the bandgap including conduction and valence band offsets. Furthermore, strained Sii.Ge results in a two-band system both for the conduction and thevalence band. Chapter 6 uses the Sii..Ge material models and derives the necessary multi-bandcharge-transport models that are required to simulate SiGe HBTs. In fact, it is found that there is asubstantial 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 HBTdesigns with some surprising results regarding operating voltages and critical layer thicknesses.1.2 Thesis OrganisationThis thesis is organised into five main chapters. Chapter 2 presents a general model for theHBT that is highly abstract in nature. The main tenet of the general model in Chapter 2 is that itcan contain any number of physical regions to model the HBT, including sources and sinks withineach region. Chapter 2 also introduces a method of optimisation through what is termed current-limited flow. Chapter 3 builds upon the ideas of Chapter 2 by considering specific examples of device optimisation that can be performed within an HBT but not a BiT. The main development inChapter 3 is the solution for the optimum base bandgap and doping profile. Surprisingly, the optimum doping profile is not exponential, and the optimum base bandgap is not linear. Chapter 4July 12, 1995 7moves on to develop the necessary models for charge transport within the emitter-base SCR. Specifically, models for the tunneling of electrons through the CBS, including the effect of a spatiallynon-uniform effective mass, are developed. Finally, Chapter 4 goes on to show the effect of tunneling on the emerging minority-carrier ensemble distribution entering the neutral base. Chapter 5rounds out the ideas presented in Chapter 2 by developing the necessary models for the recombination of minority carriers within the emitter-base SCR and the neutral base. Chapter 5 concludesby using the model of Chapter 2 to bring together the various regional models of Chapters 3through 5 for the simulation of an A1GaAs HBT. Chapter 6 builds upon the models of Chapters 4and 5 for the simulation of SiGe HBTs. Models that include the effects of strain on the conductionand valence bands in the Sii..Ge material system are presented. Multi-band charge transportmodels, 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 ofnumerous present-day and future SiGe HBT designs.July 12, 1995 8CHAPTER 2A Multi-Regional Model for HBTs Leading toOptimisation by Current-Limited FlowJuly 12, 1995 9Since the invention of the Bipolar Junction Transistor (BiT) in 1948 by Brattain, Bardeenand 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 continuous development of the design and manufacture of the BJT shows no sign of ending nor anyabating in the pace at which improvements are made. The question then, is what direction or directions will the course of BiT development take in the future?The latest innovation in the evolution of the BIT has been termed Bandgap Engineering byCapasso [45]. By altering the actual semiconductor within the active portion of the BIT, generallyby forming some sort of alloy, the shape of the bandgap can be altered to provide another force togovern the motion of electrons with the device. This idea, however, is not a new one. Shockley alluded to the use of Bandgap Engineering in his BJT patent of 1948 [1], and Kroemer first proposed the idea of using a wide-bandgap semiconductor for the emitter and a narrow-bandgapsemiconductor for the base in 1957 [2]. This junction between two semiconductors with dissimilar 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 manufacture HBTs due to the infancy of the art of semiconductor manufacture. It has only been in thelate 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 continued development of the BIT.2.1 Bandgap EngineeringThe force acting upon an electron/hole within a semiconductor is the sum of the electricfield due to any spatially varying charge, and the field of a spatially varying conduction/valenceband (EJE) [71. The electric field due to the spatially non-uniform charge is the standard forceresponsible for drift and it changes with applied bias. However, the effect of the field due to thevariation of EJE is present from the construction of the device and is therefore ostensibly independent of the bias conditions (much the same as the electric field that is generated in the neutralbase due to a spatially varying doping is independent of bias). It is this manufactured drivingforce, due to the spatial change in the bandgap and the band alignments, that gives rise to BandgapEngineering. It is possible to effect such a rapid change in EdE that the affects of the standardJuly 12, 1995 10electric field are negligible and unimportant. One can therefore expect to create HBTs with markedly different tenninal characteristics than those possible with BITs. Finally, and most importantly, the terminal characteristics of HBTs can have a completely different dependence upon thephysical 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 notoverwhelmed by the engineered field; or techniques that afford extremely rapid or abrupt changesin EIE, so much so that electron/hole transport no longer depends upon the electric field due tothe space-charge but is governed completely by the engineered bandgap.The first group of Bandgap Engineering techniques was applied to the newly emerging HBTin the form of an additional adding field in the base and the collector, in order to afford a morerapid transit of the electron/hole through the device [2,46,47]. Shortly thereafter, the second groupof Bandgap Engineering techniques resulted in the idea of placing an abrupt downwards change inE to provide a sudden increase in the kinetic energy to the electron as it entered the base (ballisticinjection; see Fig. 2.1) [12,14,48]. The aiding field in the base produced results that were expected; the ballistic launcher however, did not. In the end, it was the abrupt Bandgap Engineeringtechnique that provided the most unique results in HBTs when compared to BiTs. Thus, abruptBandgap Engineering may be the more promising road to follow in seeking to continue the evolution of BITs.Ballistic “launcher”didingField(N) (P) (N)Emitter Base CollectorE Hole blockerFig. 2.1. The abrupt change of E in the emitter-base junction “launches” electrons into the basewith a large kinetic energy. The gradual negative slope of E in the base and the collector helps tospeed the electron through these regions. Finally, the abrupt change in E at the emitter-base junction suppresses hole back-injection into the emitter.July 12, 1995 112.2 Regional Decoupling and Current-Limited FlowWithin any region of a solid-state device, charge flow or transport results in a spatial variationto 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 ofa continuous spatial change in Efi and arrive at the standard drift-diffusion transport equations. However, when the change in the conduction or valence bands is not small, as occurs in abrupt BandgapEngineering, then Ef does not vary in a continuous fashion but instead changes abruptly as well[50,7]. This abrupt change in Efis due to a departure from conditions of quasi-equilibrium, where thetransported current through the region is large in comparison to the equilibrium charge flows that result 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 toan abrupt change in E, as shown in Fig. 2.1. Fig. 2.2 shows what the abrupt emitter-base heterojunction would look like, including the effect of the potential energy variation due to the Space-Charge-Region (SCR). The transport flux F is then given by the forward directed flux Ff minusthe backward directed flux Fr The forward and reverse directed fluxes are [18,20,211:= qvn° and Fr = qvn° = qnOekT (2.1)which producesAEfF=FfF=qn°l_e kT)=Fj(1_e kT), (2.2)where n0 is the electron concentration immediately to the left of the heterojunction, O is theelectron concentration immediately to the right of the heterojunction that is capable of surmounting 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 appearance of the term /XEp in eqns (2.1) and (2.2) is due to the need for n° to surmount the barrierAE. Therefore, the abrupt change in E generates the abrupt change inEqn (2.2) clearly shows that as F goes towards zero, then so does In fact, if the conditions F << Fjand F << Fr are satisfied, then AEp 0. This is exactly what is meant by quasi-equilibrium; as long as the total transport current merely perturbs the equilibrium fluxes, the result willbe a vanishingly small Conversely, if the transport current is not small compared to Ff andJuly 12, 1995 12F,, 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 transport current to exceed the available forward directed flux.Fig. 2.2. Band diagram of the emitter-base junction showing the effect of the abrupt heterojunction on E7c,, under an applied forward bias. ji is the solution to the Poisson equation and is therefore continuous; however, the midgap energy E1 need not be.The condition of F Ff is termed current-limited flow, and is a manifestation wherebyquasi-equilibrium is grossly violated. The region in which current limiting has occurred respondsby generating as large a as necessary such as to reduce the demanded F to be no more thanFp Obviously, the transport current through the entire device will be governed by the region inwhich current limiting has occurred. Furthermore, the physical construction of the region limitingthe 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 thetotal transport current irrespective of any other physical portion of the device.To examine the effects of current limiting by a region, consider the hypothetical structureshown in Fig. 2.3. Fig. 2.3 shows three different but adjoining regions with a total applied bias ofV 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 resuits if holes are considered instead. To further generalise this picture consider a sink, or recombination 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. ThisJuly 12, 1995 13ElectronFlowHoleFlowFig. 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 recombination processes in Regions 2 and 3 that generate currents J,2 and J,3 respectively. Conservationof current forces J,,1 = J,,2 + J,2, and J,,2 = J,,3 + J,,,3. Note: Ep, is assumed to be a constant.procedure has been referred to as current balancing [51,52], but is generalised here to also allowfor sinks (and with a simple extension, sources as well). Thus, the sink causes the electron andhole currents emanating from the region to couple together as the total electron flux entering theregion must be conserved [24]. Now, the driving force in Region 1 is the full applied bias of VHowever, at the boundaries, one needs to consider a drop of AEp,x (where x = 1 or 2) through theregion. Thus, the driving force in Region 2 is not Vbut V— AEfi. Likewise, at the second boundary, another drop in the electron quasi-Fermi energy of AEfr2 occurs, resulting in a driving forceof V——in Region 3. Using the form given in eqn (2.2) for the transport current:O 11f kTn,l——eJ2 =2(V—E1)(1_e kT ),Jfl,3 = 3 (V—AEffl 1 —iXEf 2)’ (2.3)fn, 2J2 =2(V—AE1)(1_eJ,3 =4(V—AEffll_AEffl2.pJuly 12, 1995 14It is important to realise that the hole currents represent electrons that have recombined; hencetheir direction of flow as presented in Fig. 2.3 and their connection withIf theJ0(V— AEp) functions can be expressed as i°(V)exp(—AEp1/k7), then, equating J,,2with J,,3 + J,,,3 gives:LEf2 AEf,,2kT )= [3(+43(]e,which produces, after dropping the explicit dependence upon V,f2 jOe= n,2 (2.4)2+ J 3+4 3Then, equating J,,1 with J,,2 +J1,,2 gives:LEffl1 IEf1 fn,2Ji(V)(1-e)=[J,(v)+4,V)1e(1_e kT).Using eqn (2.4) in the above, and once again dropping the explicit dependence upon V, produces:iiE1 ,o— kT — n,lkn,2+ n3+Jp 25e— (J2+4)(3+43) +l(2+3÷43)The final transport current Jexiting the device is simply equal to J,3. Substituting eqn (2.4) and(2.5) into J,,3 given in eqn (2.3) produces:JT(V) = Jfl 3 (V) = 1 , (2.6)+ +1J1(V) J22(V) J3(V)where—‘2,2+4,2and—3+43— -‘2,2 —Eqn (2.6) provides a very simple form for the ultimate transport current T emanating fromthe device, and extends eqn (34) in [52]. It includes all of the recombination effects of Regions 2and 3, while allowing for a completely general relationship between the applied bias and the forward directed flux Ff (where the fi(V) functions are F. The only stipulation placed upon the use ofeqn (2.6) is that fi(V— AEp) =J0(V)exp(—AEp/k7) (as will be seen in later chapters, where eqn(2.6) is applied, this is exactly the functional fonn that results). Therefore, to determine the transJuly 12, 1995 15port current that results from coupling three regions together, it is sufficient to calculate the forwarddirected 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 given by eqn (2.6) used to define J1;Regions 4 and 5 then become Regions 2 and 3 in the analysisleading up to eqn (2.6). Finally, a recursive application of the above procedure gives:N+i= JJ ‘v.,(V)J (2.7)i=i j=i+iJj+Jjwherey4 1, and = ‘Jn, jEqn (2.7) is the general formula for the calculation of transport current through any multi-regional HBT (of which a BJT is a subset). The ramifications of eqn (2.7) are striking and generally lead to current-limited flow within a single region. An examination of eqn (2.7) begins withthe y4 functions, which are termed the recombination loss; y4, therefore, represents the additionalcurrent that must exist in order to satisfy the recombination events within Region j. Then, thetransported 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. Thisimmediately 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 ITFig. 2.4. The flow T that results from a series connection of six pipes (the flow entering only equalsthe flow leaving (= J) when there is no recombination in any of the regions). Obviously, the pipe inRegion 4 is the most restrictive and T will accordingly be governed mostly by this region alone.July 12, 1995 16Looking at eqn (2.7) and letting J <<j k wherej k and k can range over all N, then Regionj will be responsible for the current-limited flow of T and produce:JT(V)Ji(V) Ii k(’ (2.8)k=j+1where (= l/) is the transport efficiency of Regionj and expresses the fraction of the transportcurrent that is lost to recombination within the region. Eqn (2.8) is exactly the form expected fromthe arguments presented in Fig. 2.4. For if Region j is responsible for the current-limited flow, Twould equal J3 in the absence of recombination. However, each subsequent region downstreamwill lose 0k electrons to recombination. Therefore, the current will be diminished by c’k ineach region encountered, leaving a final current of T exiting the device. This immediately leadsto 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 nopart in the ultimate current T This is no surprise since all of the regions upstream of Regionj cansupply the demanded current within Region j. However, every region from 1 to N contributes tothe recombination current, and must be included in the calculation of the total hole current 4Adding all of the recombination events together gives:N N N+1 F N N+1 N N+1J(V) = = fl Y = JT[Yi IT [Ii=1 i=1 j=i+1 i=1 j=i+1 i=lj=i+1Then, after bringing yj into the multiplication and letting i = i’ — 1 in the second term:r N N+1 N+1N+1 1 1N+1 N N+1 N N+1J(V)= JTLH- HYij = JTLH,+H- flN+1i=lj=i i’=2j=i’ j=1 i=2j=i i’=2j=iFinally, since YN÷1 1 from eqn (2.7), and i’ is a dummy variable, the above reduces to:J(=JT(V)[fl(V)_11. (2.9)Eqn (2.9) provides for the total hole current generated within the device. Combining botheqns (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 tocurrent-limited flow. Then, eqn (2.7) can be replaced by its approximate form, eqn (2.8), to yieldafter substitution into eqn (2.9):July 12, 1995 17J(V) =JJ(V)[fl(k(v)- fl x(V)]. (2.10)k=1 k=j+1The results of this section are models for the total electron and hole currents entering andleaving 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 themodels 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 transportthrough the entire device; creating a situation of current-limited flow. The key to achieving a situation of current-limited flow is the existence of a substantial AEp in one region. Finally, abruptBandgap Engineering techniques provide the capacity to create a situation of current-limited flowin any region of the device. In the next section and chapters to come, the concept of current-limited 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 together all of the models for each of the relevant regions of an HBT.2.3 Optimisation Through Current-Limited FlowThe main conceptual result of the last section was that one region, or physical process, willtend to dictate the transport current through the entire device. This section examines how to intentionally design a specific region, through Bandgap Engineering techniques, to result in current-limited flow; thereby allowing for a decoupling of T from the physical transport processes in allother regions of the device. Finally, once J’ is decoupled from a specific region, by ensuring thattransport through the region is much larger than the demanded jr then one is free to optimise thatspecific region without affecting JFig. 2.5 shows the transport current that would result from a hypothetical three region device. 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 loweredso that Region 1, and neither Regions 2 or 3, controls T under all bias conditions. This demonstrates, in principle, the feasibility of engineering a specific region to be the source of current-limited flow, and thereby link JTto the physical process in that region alone.Julyl2,1995 18In order to see how optimisation can occur by engineering a specific region to be the sourceof current-limited flow, one begins by identifying the need for decoupling. Imagine there are twospecific metrics, say Early voltage (VA) and collector resistance (Rc), that are to be optimised. Ifthese two metrics are connected to one parameter, in this case collector doping, and the two metrics do not both move towards their optimum value with either an increase or decrease in the oneparameter, 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 VAand Rc, forcing a compromise between the two metrics to be made. If it were possible to decoupleeither of these metrics from the one parameter, then it would be possible (in terms of this one parameter only) to optimise both metrics. Therefore, decoupling the metrics from their commoncompeting parameter is the key to removing the compromise and achieving a true optimum.1i031O-10.8case(b) i2 11.0 1.2 1.4Base-Emitter Voltage VBE (V)Fig. 2.5. T for a three-region HBT in the absence of recombination. The solid lines represent themaximum regional currents .1°, while the dashed lines are J For case (a), Region 1 is never the limiting region; while for case (b), Region 1 is the source of current-limited flow.1.6At the heart of decoupling is the separation of the transport current from the physical process that is to be optimised. For if the transport current is not affected, or at least not in a detrimental fashion, then one is free to optimise the desired metric. Current-limited flow provides thenecessary tool to decouple T from all regions, and therefore all physical transport processes, saveJulyl2,1995 19one. Continuing on with the example of simultaneously optimising VA and Rc, if T were decoupled from the construction of the base and collector, say by making the emitter-base SCR thesource of current-limited flow, then VA would no longer depend upon the collector doping; enabling the optimisation of Rc without affecting VA. With base-width modulation no longer an issue, in terms of the collector current and therefore VA, it would be possible to increase the intrinsiccollector doping adjacent to the base and thereby reduce Rc. A further optimisation, in terms ofthe base-collector capacitance CBC, could also be had by placing a low-doped collector regionwithin the CB SCR (say at 106cm3for 2000A) in order to set CBC, followed immediately by ahighly doped extrinsic collector to reduce Rc. Optimisation of competing metrics is thus achievedby first identifying the coupling parameter; then, one other region that does not contain the coupling parameter is constructed (generally through abrupt Bandgap Engineering techniques) to bethe source of current-limited flow in order to provide for the control of J- (i.e., the collector current).This chapter has provided a logical course to decouple otherwise competing metrics so theymay be simultaneously optimised. The tool for decoupling the competing metrics being the creation of current-limited flow outside of the region or regions to be optimised. It is possible toachieve current-limited flow in any given region by resorting to abrupt Bandgap Engineering techniques. 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 througheqns (2.7) and (2.9) (or their approximate forms eqns (2.8) and (2.10)) for the calculation of thetotal electron and hole currents entering and leaving the device.July 12, 1995 20CHAPTER 3Base Layer Decoupling and OptimisationJuly 12, 1995 21Traditionally, the base region, or more specifically the neutral base region, has determined theoverall performance of the BJT. As such, the physical construction of the base is of paramount importance to the function of the BJT At issue with the base is the fact that there are basically two degrees of freedom within the base; namely the base doping profile N(x) and the neutral base widthWB (in an HBT a third parameter, namely the bandgap in the base Eg(x), is also available). Againstthese 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 simultaneously optimise all of the metrics. Thus, an inherent compromise is forced to exist betweenmany 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 transport current within an HBT. It was found that through abrupt Baudgap Engineering, it was possible to construct a specific region in such a fashion that the transport current T depended on thisregion alone; thereby decoupling T from all other regions of the device. Once T has been decoupled from all other regions of the device, save one, the task of independently optimising each region becomes trivial.The possibility of decoupling J- from the physical construction of the base promises toeliminate the interdependence that the base-controlled metrics have upon each other. Once thebase 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 intrinsic base sheet-resistance RBJ, base-emitter capacitance CBE, injection index y (not to be confused with the y in Section 2.2 which is the recombination loss), Early voltage VA, base transittime ‘CB, and the base-collector capacitance CBC could then be simultaneously optimised. The keyto the optimisation of these base metrics rests simply on the decoupling of T from the base byconstructing 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 applies it to the base region. The methods used to achieve the simultaneous optimisation of the baseregion metrics follow directly from the prescriptions of Chapter 2. Specifically, the base metricsRBD, CBE, ‘ VA, and ‘C13 are considered for optimisation. Finally, once the optimum models foreach of these metrics within the base region have been derived, they are linked together for thecalculation of the total electron and hole currents by the methods derived in Chapter 2.July 12, 1995 223.1 Independent Optimisation Of RB, CBE, AndIn the design of any transistor, the sufficient design criteria is to provide for a gain that isgreater 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 divided by the base currentlB). In current-day BITs, the manufactured materials are so pure, that forthe most part, the recombination of minority carriers being transported through the neutral baserepresents only a small fraction of the total‘B [53]. Therefore, [3 will depend on the injection efficiency y of the Emitter-Base (EB) junction. For an npn BJT the EB y is given by:“fl, B (3.1)where n,B is the electron transport current through the base, and is the hole current injectedinto the emitter (also known as hole back-injection). Using eqn (3.1), in the absence of neutral-base recombination, the gain is:I=!._• (3.2)‘‘“p,EThus, [3 is maximised as y is driven towards 1; meaning that is driven towards zero and/or n,Bis 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:G#B=dx, (3.3)where D is the electron minority carrier diffusion coefficient, WB is the neutral base width, andpis 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, (3.4)where D is the hole minority carrier diffusion coefficient, WE is the neutral emitter width, and nis the emitter majority electron concentration (= emitter doping NDE except under high-level injection). Thus, f3 is proportional to G#E/G#B. Now, the intrinsic base sheet-resistance RBU is alsoinversely proportional to G#B [54]. However, unlike the case for [3, where it is desirable to reach amaximum, RBU is to be minimised in order to improve the high-frequency operation of the BJT.July 12, 1995 23Since RBL] and 13 are both tied to the parameter G#B, and increasing G#B optimises RBU while deoptimising 13, we realise these two metrics are competing and therefore cannot be simultaneouslyoptimised (at least in terms of the parameter G#B).As was discussed in Section 2.3, the key to optimising two otherwise competing metrics isto identify their common parameter (in this case G#B) and remove its dependency from one of thetwo metrics. Continuing on with the case of optimising RBJ and 13, it would appear possible to increase G#B and thereby minimise RBU, while also increasing G#E and thereby maximise 13. However, G#E in a BJT cannot be increased because NDE is either at or very near its maximumphysical limit ( 1021 cm3). Thus, without resorting to Bandgap Engineering techniques, the onlyavailable parameter is G#B, meaning that a compromise has to be made between RBU and 13. Thiswas the motivation for the first HBT; to decouple 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 electron to enter the base as it is for a hole to enter the emitter (the two carriers see exactly the samepotential barrier of Vbj — VBE). Therefore, the ratio of n,B to p,E (= 13) will be proportional to theratio of the available number of electrons in the emitter to the available number of holes in thebase (= NDE/N — G#E/G#B). Now, if it were possible to alter the bandgap of the EB junction sothat the holes had to surmount a larger barrier than the electrons, then pE would be significantlyreduced and 13 increased (see Fig. 3.1(b)). Finally, if Bandgap Engineering were employed toachieve an initial 1000-fold increase in 13 (by reducing p,E through a Bandgap Engineeredthen G#B could be increased 32-fold, thereby reducing RBD 32-fold, while still leaving a net 32-fold increase in 13. Thus, by creating a heterojunction at the EB metallurgical junction, it is possible to reduce p,E without increasing G#E. Then, the gains provided by a reduced pE are sharedbetween an increase in 13 and a decrease in RBIJ.The methods just described for the simultaneous optimisation of RB and 13 demonstrate thepotential gains of abrupt Bandgap Engineering. However, the techniques described above did notfollow the exact prescription given in Section 2.3, and thus maintain a coupling between RBU and13. Instead of decoupling 13 from G#B, another degree of freedom was added to G#E; namely theabrupt change of t1E in the valence band at the EB junction. The dependence of 13 upon G#B stillexists, but p,E and thus (3, by the addition of a heterojunction within the EB SCR, now has another dependence of exp(-AEJk1) [2,46,47] through the intrinsic carrier concentration in the emitterJuly 12, 1995 24i,E• However, since f3 still depends upon G#B, any change in G#B due to bias (such as the Earlyeffect [57], Kirk effect [58] or high level injection [56,59]), will still affect and generally degradeI. The reduction of p,E through simple abrupt Bandgap Engineering is thus seen as a good firststep, but falls short of the optimum case where f3 is decoupled from G#B altogether.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 andthe emitter is made of AlGai.As [25]. The barrier to electrons entering the base lies somewherebetween q(V1— VBE) and q(V1— VBE)—tSE depending on the amount of tunneling through theCBS. In general, it is found that the drop AE is sufficient to cause the CBS to be the region ofcurrent-limited flow (this will be fully discussed in Chapter 4). Thus, T (= n,B in the absence ofsignificant neutral-base recombination) will be governed by the physical process of transportthrough the CBS, and not by the transport through the neutral base. Furthermore, transportthrough the CBS has little dependence upon G4B (as long as the base doping is much larger thanthe emitter doping). Therefore, T and thus are decoupled from G#B through the condition ofcurrent-limited flow at the CBS.The condition of current-limited flow in the region of the CBS follows exactly the prescriptions of Section 2.3. n,B has now been decoupled from G#B, meaning that processes connected to(a) BJT (b) HBTE (eV) E (eV)+Fig. 3.1. (a): Band diagram of a homojunction BJT. Clearly, the potential barrier seen by a holetrying to go from the base to the emitter is the same barrier seen by an electron trying to go fromthe emitter to the base. (b): Band diagram of an HBT. Through abrupt Bandgap Engineering, thebarrier seen by a hole trying to enter the emitter is a least AE larger than the barrier seen by anelectron trying to enter the base. Also note the formation of the Conduction-Band Spike (CBS).July 12, 1995 25G#B such as the Early effect, Kirk effect, and high-level injection, which degraded the collectorcurrent of BJTs, are no longer an issue for the abrupt HBT (the term abrupt refers to the abruptchange of IXE and AK1, at the EB junction). With the collector current decoupled from G#B, RBL]can be minimised by increasing G#B through an increase in N, while leaving and therefore ‘yunaffected.Before leaving this section to discuss the further optimisation of the base and collector, itshould be noted that the EB junction capacitance CBE can also be minimised due to the conditionof current-limited flow at the CBS. The high-frequency performance of a BiT improves as CBEdecreases. Most notablyfT (the frequency at which f, under the conditions of an A.C. short circuitbetween emitter and collector, has dropped to unity) increases as CBE is reduced. Since CBE isgiven by:qNN NABCBE= I,, where Nrat= N + N ‘ . )I4kVbjVBE) AB DEthen NDE and Nrat need to be minimised in order to reduce CBE. In a BJT, the need to maximise fforces NDE >> N, meaning thatN is reduced in order to reduce CBE. Thus, CBE is connected toRBD as well, and leads to another condition where only a compromise and not a true optimum canbe reached. CBS-limited flow in an abrupt HBT decouples 3 from N, so that RBU can be optimised 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 atwhich 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 Optimisation of these metrics begins by decoupling from y and CBE the dependence upon N. Thisdecoupling is afforded by the creation of current-limited flow at the CBS. With y and CBE decoupled from N, RB is optimised by increasing N. Then, CBE is optimised by reducing NDE. Finally, the optimisation of y depends first of all upon p,E (which depends heavily on SCRrecombination [24]) and secondly upon n,B (which is governed by the flow of T through theCBS); EB SCR recombination, which accounts for most of p,E’ is covered in Chapter 5, whilethe current within the CBS is covered in Chapter 4. The optimisation afforded by the abrupt HBTin comparison to the BJT is stunning, as none of the methods discussed in this section would havebeen applicable to a BJT because the gain of the transistor would have been reduced below unity.July 12, 1995 26Fig. 3.2. The resistance of the intrinsic emitter will become considerable if NDE,1 is reducedwithout bound. To minimise this parasitic resistance, the width of the intrinsic emitter is onlymade large enough to contain the emitter extent of the EB SCR. Then, a highly doped NDE2 extrinsic emitter is placed as a cap layer on top of the device, where the eventual contact layer isformed.3.2 Reducing tB by Decoupling the Base from I’Chapter 2 discussed the merits of Bandgap Engineering, where the natural evolutionary pathof the BiT produces the HBT. Two Bandgap Engineering techniques were considered: techniquesthat created abrupt changes in E.JE leading to the creation of current-limited flow; and techniques that created gradual changes in EdE that produced additional aiding fields for the transport of charge. Then, Section 3.1 focussed upon the benefits of current-limited flow produced byan abrupt change of iXE within the EB SCR. This section carries on with the benefits to be derived from current-limited flow, but delves into the second group of Bandgap Engineering techniques - namely the creation of fields in the base to aid in the transport of charge through theregion.A major component of the total transit time for a BIT or HBT is still the neutral-base transittime tB. In the absence of any spatial variation to the bandgap or N, then under low level injection conditions, with the neutral base width WB larger than a few mean-free paths , the base transit time is given by the standard equation:w(3.6)B 2DtB can be reduced from the value given in eqn (3.6), without reducing WB, by introducing an aiding field in the base (as is shown in Fig. 2.1). BJTs where an aiding field has been placed in thebase are termed drift-base transistors [60]. This aiding field implies, for an npn BJT, a downwardsJuly 12, 1995 27Emitter-side SCR edgeslope to E in the neutral base. Before the creation of HBTs, a negative slope in E could only beachieved by varying N from a high value near the emitter-side of the neutral base, to a low valuenear the collector-side of the neutral base [60]. This non-uniform N(x) would indeed reduce tBbut at the expense of having a low base doping nearest the collector; leading to a reduced magnitude of the Early voltage. Therefore, the drift-base transistor had a rather limited range of optimisation as the aiding field was coupled in a compromising fashion to the Early voltage. Add to thisthe fact that the optimum N(x) was an un-manufacturable exponential, then the optimum drift-base transistor was a good idea that was generally beyond the manufacturing capabilities of theday.Enter Bandgap Engineering once again. The issue with the drift-base transistor was the lowbase doping near the collector. By using a graded bandgap in the base (where the bandgap is largenear the emitter-side of the neutral base and small near the collector-side of the neutral base), anaiding field can be created without the need to vary N [38]. Thus, by using Bandgap Engineering techniques to create a gradual down-slope to E in the base, tB can be reduced without lowering N and compromising the Early voltage. Kroemer calculated tB for a non-uniform bandgapEg across the neutral base, and found [38]:w wIn.(x) p(z)=1__________dzdx, (3.7)B p (x) D, (z) n (z)where n1 is the intrinsic carrier concentration. The derivation in [38] which leads to eqn (3.7) isbased 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 linear grading of the bandgap in the base is used, such that n (x) = n (x= 0) exp (qFx/kT , eqn(3.7) gives:w2 - AE kT2 (3.8)where F = AEgI(qWB), and AEg represents the difference between the bandgap at the emitter-sideof the neutral base and the bandgap at the collector-side of the neutral base. As an example, if D= 30cm2s1,WB = 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 28The reduction of tB through a graded-base transistor is very attractive. When coupled to thefact that the Early voltage is not compromised, Bandgap Engineering in the base appears to holdnothing but gains. The only requirement of a graded-base transistor is the need to create a gradedalloy 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 inSii..Ge HBTs, the bandgap is decreased with an increase in the Ge mole fraction x. Now, inA1GaAs HBTs the Al mole fraction must remain below a maximum of x = 0.45, for this is thepoint 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 discussedfully in Chapter 6). Thus, an “alloy budget” exists in the HBT, meaning that a decision must bemade in the allocation of alloy mole fraction among the various regions of the HBT. Therefore, acompromise must be made in the amount of Bandgap Engineering allocated to the formation ofthe 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 optimising the metrics of the device (namely decoupling y from G#B), part of the total alloy budgetmust be allocated to its formation. In the case of AIGaAs HBTs, fully 66% of the maximum totalalloy budget (Ax = 0.3 of a maximum 0.45) is spent in the formation of the EB heterojunction (Inreality, Ax < 0.45 is a maximum upper limit that is generally reduced to 0.30 for practical applications. With this reduced alloy budget, the EB heterojunction would consume the entire budget). InSiGe HBTs, virtually the entire alloy budget of Ax < 0.2 is spent in the formation of the EB heterojunction. Therefore, irrespective of the material system used to form the HBT, little if any ofthe alloy budget remains for the Engineered Bandgap in the base once the EB heterojunction hasbeen formed. This means there is little room to reduce ‘CB through a manipulation of the bandgapwithin 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 creation of an abrupt EB heterojunction. The reasons for abandoning drift-base transistors were: itwas not possible to manufacture the steep doping profile in the base required to generate the aiding field; and the low base doping near the collector-side of the neutral base resulted in an intolerJuly 12, 1995 29ably low Early voltage. The first problem, namely the manufacture of the highly non-uniformN(x), is no longer an issue with advanced MBE and MOCVD growth techniques. The secondproblem, a decrease to the Early voltage, is solved by decoupling the collector current I from thebase, so that modulations to G#B from changes to VCB no longer matter, provided punch-throughis avoided, of course. Following, once again, the prescriptions of Section 2.3, I is decoupledfrom G#B by creating a situation of current-limited flow at the CBS formed by the EB heterojunction; thereby linking I to the physical transport mechanisms associated with the CBS instead ofthe 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 tBBandgap Engineering in the base is not really being considered in this section; however, itcan be included in the optimisation without any changes in the arguments to follow (this includeseffects due to a manufactured change in the bandgap and changes to the bandgap due to heavydoping effects). Starting with eqn (3.7), then after substituting p = N, tB becomes:w_2 WRn1 (x) NAB (z)= dzdx. (3.9)B NAB (x) D, (z) n (z)If D, is taken as some average constant, then eqn (3.9) is simplified even further to become:w wj ?n(x) ?NAB(z)= =J J dzdx. (3.10)B D,7ONAB(X) x 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 characteristicequation that minimises eqn (3.10) is:= C, (3.11)y dxwherewrNAB (z)y(x) = t n(z) dz, (3.12)and C is an arbitrary constant. The solution of eqn (3.11) is straightforward and yields:July 12, 1995 30y(x) =_A1eA2X (3.13)where A1 and A2 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 takingfor now that n(x) is constant, then differentiating eqn (3.12) and substituting in eqn (3.13) gives:N(x) = AiA2eA2) = ae (3.14)Eqn (3.14) gives the standard exponential solution [601 for the doping profile in the base thatleads to a minimum in tB.Within the confines of weak variations, a possible minimum could occur by admitting apiece-wise solution for N(x) composed of N sections whose form within each section is givenby eqn (3.14). The conditions of continuity at any break-point joining two regions being [62]:and F—y’ () be continuous, (3.15)where F is the integrand that is to be made stationary, and primes denote differentiation with respectto the dependent variable x. In the case being considered, F = yly’. Then, using the exponential solution for y(x) in eqn (3.13), and applying the second continuity condition of eqn (3.15) produces:F— — — 2which must be continuous at the break-point x0 joining the two regions. If we let Region 1 joinwith Region 2, where the solution in Region 1 is A1 1e4Z 1X and the solution in Region 2 isA1,2eA2,2X, then the above equation requires that A2,1 = A2, = A2. Applying the first continuitycondition of eqn (3.15) at the point x = x0 produces:— == 1 1=A1 = A1 2 = A1.ay Y’2 A1 1Ae4° A1 2A eA2X0Thus, a piece-wise connection of exponentials is not admitted as a stationary solution for N(x).However, if the last equation is rewritten as A1 1A eA2X0 A1 2A eA2XO, then as A2 —* 0 no restriction is placed on the values admitted forA1, and A1,2.This admitted solution for y(x) is alsoa piece-wise and discontinuous set of constant solutions. As such, this solution for y(x) tends towards a strong variation and care must be exercised in the absolute applicability of the weak variational principles used to obtain this result. With that cautionary note in mind, if the form ofA1,and A1,2 are carefully chosen to bea1/A2anda2IA respectively, then as A2 —* 0, N(x) also becomes a piece-wise and discontinuous set of constant solutions.July 12, 1995 31The weak variational principles used to find the N(x) that renders tB stationary are constructed in a such a manner that only y(x) be defined at the end points of the integration. SinceN(x) is given by y’, there is no simple way to specify the doping values at the end points of theintegration (namely the emitter and collector edges to the neutral base). Further examination ofeqn (3.14) shows that there are no bounds to the value of the constant b in the exponent of the exponential defining N(x). In fact, by letting b —> —00, an infinitely large aiding field can be created in the base and tB will be reduced to zero. To see this, eqn (3.14) is used in (3.10) to give:2(eb_b_ 1)tB = tBOb2(3.16)where tBo is the tB given by eqn (3.6). Clearly, as b —> —00, tB —*0. As a check, as b —>0, tB given by eqn (3.16) goes to ‘rho. Thus, no matter what N is forced to be at the emitter-side of theneutral 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 exponentialare based upon an unrestricted doping at the collector-side of the neutral baseIt is not reasonable to allow the doping at the collector-side of the neutral base to become arbitrarily small, even in the presence of current-limited flow at the CBS. For even though I is decoupled from G#B, RBD still depends on G#B and would become unreasonably large as b —* —00Eqn (3.10) is revisited, but this time ‘CB is made stationary subject to boundary conditions uponN(x) at the emitter- and collector-sides of the neutral base. Since there appears to be no simpleway of including these boundary conditions into the variational principles, a numerical minimisation was constructed [63]. The results of numerical attempts to render tB stationary, subject to theboundary conditions placed upon N(x), produced a form that suggests N(x) be exponential inthe middle of the base but have two constant regions attached on the ends (see Fig. 3.3). This result seems plausible in light of the variational analysis performed so far, where a constant was admitted as a solution to N(x). Even more convincing, the form being suggested from thenumerical analysis is not a piece-wise connected set of exponentials (which was rejected as a possible stationary solution from the variational analysis), but is a piece-wise connection involvingconstant regions of doping, as is admissible from the variational analysis. In any event, it is clearthat the boundary conditions placed upon N(x) cause the exponential solution from simple variational analysis to become non-stationary.July 12, 1995 32loW0.0 0.25 0.50 0.75 1.0Normalised Position in the Base x (WB)Fig. 3.3. Optimum doping profile N(x) obtaining by numerically minimisin eqn (3.10) withthe boundary conditions N(x=0) 5x1018cm3and N(xtWB) = 2x1016cmUsing the form for N(x) suggested from the numerical work, namely exponentials separated by regions of constant doping, analytic methods were employed to find the break points between the exponentials and the constant regions that minimised tB. Using the form of N(x)given in Fig. 3.4, then finding the break-point h that minimises ‘CB given by eqn (3.10) produces,after considerable algebraic manipulation with the symbolic mathematics tool MACSYMA (seeAppendix A):(U1nU+1—U)lnU U[U(2lnU—3)+4]—1(U1nU+2)lnU+2(l—U) tB_tBou[(Ulu+2)lu+2(1u)](3.17where ‘CBO is still the tB given by eqn (3.6), h is normalised to the neutral base width WB (andtherefore ranges from 0 at the emitter-side to 1 at the collector-side of the neutral base), and U isthe doping ratio given by N(x=0)IN(x=WB). The interesting thing to note about eqn (3.17) isthat 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 constant region occurs first followed by the exponential region. Eqn (3.17) represents the solution ofthe simplest form of N(x) suggested from the numerical analysis.The process described above is repeated again, but this time with the optimum form (shownin 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 symbolicmathematics tool MACSYMA (see Appendix B):July 12, 1995 331 2h1l—h2= lnU+2 and tB = tBolnU+2 = tBo2hl, (3.18)where, h1 and h2 are nonnalised to the neutral base width WB. Eqn (3.18) shows the beauty of thesymmetric form used for N(x); namely that the length of each of the constant regions is thesame, and the exponential region is perfectly centred within the base. It is very simple to provethat tB given by eqn (3.18) is always smaller then that given by eqn (3.17). Therefore, the form ofN(x) given in Fig. 3.3 produces a smaller ‘CB then the form given in Fig. 3.4.CC0.0 0.25 0.50 0.75 1.0Normalised 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 toN(x) given in Fig. 3.3. When eqn (3.10) is minimised using the N(x) given by the form shownin Fig. 3.5(a), it is possible to find a stationary result where h1 0 (h1 = 0 would give N(x) asshown in Fig. 3.3). Even though N(x) given by Fig. 3.5(a) renders tB stationary, when comparedto the result obtained from eqn (3.18), it does not produce the absolute minimum value for tB. Infact, taking one final progression to using the N(x) as shown in Fig. 3.5(b), a stationary result isagain obtained, but it is larger still than the case shown in Fig. 3.5(a) and therefore does not produce 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 theemitter- and collector-sides of the neutral base. The most notable thing about the optimum form ofN(x), as shown in Fig. 3.3, is that it is not the pure exponential the device community has beenlead to believe is the optimum. This result answers the problem posed in [64,65], where the authors used third order perturbation theory to show that an exponential was indeed stationary but itdid not produce the absolute minimum for tB.July 12, 1995 34(a) (b)0.0 0.25 0.50 0.75 1.0 0.0 0.25 0.50 0.75 1.0Normalised Position in the Base x (WB) 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 byFig. 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 surrounding eqn (3.15). Tn the proof that showed N(x) could not be constructed of piece-wise continuous exponentials, it was found that N(x) could be constructed of piece-wise discontinuousconstants. In the simplest case, if N(x) is constructed as shown in Fig. 3.6, then it is straight forward to show that tB is minimised when:1 U+1h = and tB = tBO 2U (3.19)Eqn (3.19) shows that a very simple jump discontinuity, or step, in the base doping proffle at exactly the half-way point in the neutral base, can reduce the base transit time by a factor of two whencompared to the uniform base case (tBO). In fact, for any U 10, the full two-fold reduction in tBis achieved. Still, for all relevant U, tB given by the step-doping case of eqn (3.19) is larger thanthat achieved by the optimum-doping case of eqn (3.18). However, the step-doping case shows thateven a very simple change to the base doping profile can produce a significant reduction in thetransit time through the neutral base. As for the technological objection that a perfect step-dopingprofile is impossible to create, any deviations from a step, say due to diffusion of dopant during thethermal-cycle of the manufacturing process, will only tend to drive N(x) towards the optimumprofile and reduce tB even further: this result is obvious as a spreading of the step-discontinuity in-creases the spatial extent of the aiding field and thereby decreases the transit time. Therefore, thestep-doping profile, although not as beneficial as the optimum doping profile, still provides for asignificant reduction of tB, but with very little complexity in terms of manufacturing.July 12, 1995 35I x=h-”0.0 0.25 0.50 0.75 1.0Normalised 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), withb = -mU), shows some interesting results (see Fig. 3.7). In all four cases as U —* 1, t(3— tBO: thisis required and acts as a check to the validity of the four models. As was stated before, for the entire useful range of U (i.e., > 1), ‘CB is minimised by the optimum doping proffle leading to eqn(3.18). However, for the range 1 U 7.389=e2,tB from the step-doping proffle is smaller thanthat from the pure exponential profile. Thus, not only have we found out that the pure exponentialis not the optimum, we have also found that for small doping ratios the step-doping profile is betterthan the exponential. An examination of Table 3.1 shows that as U becomes large, the pure exponential case and the ramped case both approach the optimum case for the minimisation of tB. Thisresult shows that the optimum-doping case initially starts out looking much like the step-dopingcase, then as U increases, slowly transforms itself into the pure exponential case. Finally, for U =300, the optimum-doing case hash1 1— h2 = 0.13 and tB is only 10% less when compared to thepure-exponential case; however, the optimum-doping case has a 49% larger Gummel number andthus a 49% smaller RBU when compared to the pure-exponential case. Clearly, the pure exponential case is not the optimum doping profile to use, either in terms of minimising tB or RBU. Therefore, the optimum-doping case shown in Fig. 3.3 and governed by eqn (3.18) is the best base-doping profile to use in order to minimise tB with the smallest impact on RB.July 12, 1995 36Table 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 Exponential3 0.65 0.69 0.67 0.727.389=e2 0.50 0.54 0.57 0.5710 0.46 0.50 0.55 0.5330 0.37 0.40 0.52 0.42100 0.30 0.32 0.51 0.34300 0.26 0.27 0.50 0.29-.Cz1murn’i0.75j—0.50—Exponential0.25—“a100 ‘2,101‘7,102Doping Ratio U i03io4 1.00_;-- -,-‘\ 0.70 0.600.800.50Break Point h2Fig. 3.7. tB using NAB(x) from Fig. 3.3, where h1 1 — h2 but h2 is varied as a parameter insteadof being given by eqn (3.18). h2 = 0.5 corresponds to the step-doping case, while h2 = 1 corresponds to the pure exponential case. Finally, the line drawn on the surface is the tB that resultsfrom the optimum-doping case given by eqn (3.18).July 12, 1995 37There are two major restrictions placed on the use of the optimum-doping case for the mmimisation of tB. These two restrictions are: that the aiding field produced by the non-uniformN(x) be small enough to neglect high field effects; and the variation in D(x) be small enough toignore. The first requirement is not terribly restrictive, for even with a base width of ioooA, and U= 100, the aiding field is l.7xlO4VIcm which is acceptable for heavy-doped Si and would be atthe edge where high field effects begin to occur in heavy-doped GaAs. However, the second requirement 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 significant variation with U in the range of 7 to 30. The issue of a non-uniform D(x), as well as variations in n(x) due to Bandgap Engineering and heavy doping, are considered in the next section. Inany event, ‘CB will always be reduced by using a monotonically decreasing (from emitter towardsthe 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 isconsiderable 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 theneutral base, in order to minimise tB. It was found that the optimum N(x) only depends on therelative doping ratio U, and not on the absolute doping given by the boundary conditions. Furthermore, the optimum N(x) is not the pure exponential that the device community has thought wasthe case, but is an augmented exponential as shown in Fig. 3.3. They key to applying the results ofthis section hinge on the decoupling of I from G#B afforded by the creation of current-limitedflow at the CBS. Therefore, only by creating an abrupt HBT1 can the drift-base BJT be manufactured without a significant reduction to the Early voltage.3.4 The Effect of a Non-Uniform n1 and D on the Optimum tBSection 3.3 derived the optimum base doping profile for the minimisation of ‘CB. It wasfound that if the base doping was fixed at the emitter- and collector-sides of the neutral base, thenthe 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 inSection 3.2, the alloy budget generally prohibits any significant Bandgap Engineering in the base if an EB heterojunction is to be formed in order to control f3. Therefore, the technique of current-limited flow is the only practicalmethod to decouple I from G#B.July 12, 1995 38Due the arguments presented in Section 3.2, Section 3.3 found the optimum N(x) without regard to the optimumn1(x). However, due to the heavy base doping that is characteristic of HBTs,bandgap narrowing will certainly cause variations to n1(x) when a non-uniform N(x) is present.This section will consider the joint optimisation of N(x) and n1(x) in terms of minimising tB.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 moment 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 produces after differentiation with respect to x:NAB(x)y(x)=— 2n (x)Using eqn (3.11), which is the O.D.E. that renders y(x) stationary, in the above equation yields:NAB (x)2 = —Cy (x). (3.20)n (x)At this point Section 3.3 lets n1(x) be a constant, which can then be absorbed into the arbitraryconstant C, to yield eqn (3.14). However, one could just as easily let N(x) be a constant andsolve for n (x). If this is done, then all of the results of Section 3.3 are still applicable to the optimisation of n (x); for the stationary function y(x) has no dependence on either N(x) or n1 (x).This immediately results in the optimum n (x) being given by the reciprocal to N(x) shown inFig. 3.3, with eqn (3.18) governing the placement of h1 and h2 and solving for tB. The onlychange is that U is now given by the ratio n (x = WB) /n (x =0) (the endpoints have been interchanged to keep U> 1). If the variation in the effective density of states for E and E is ignored,then n (x) = n (x =0) exp (—AE8x)/kT), where AEg(x) is now defined as the difference inEg at x relative to Eg at the emitter-side of the neutral base. Since the optimum n (x) is given bythe reciprocal to N(x) shown in Fig. 3.3, and given that Fig. 3.3 is a log plot, then LS.Eg(x) looksexactly like Fig. 3.3 but it would be linear and not log (see Fig. 3.8). Therefore, just like in the optimum doping case, the optimum bandgap-graded-base HBT is not purely linear, but is the augmented 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 fieldcan be created by a non-uniform N(x), and the rest of the aiding field can be created by a BandJulyl2,1995 39gap Engineered n (x). This realisation allows the burden of generating an aiding field to beshared between two physically different parameters. By using both n (x) and N(x), far less ofthe alloy budget needs to be used in order to generate n (x), and a smaller decrease in N(x) willnecessarily have a smaller impact on G#B and RBD. As an example, let ‘CB = O.StBo. This requiresthat U = 7.389=e2,whereNAB(x) n2(x)U= n(x) NAB(X) X=WB(3.21)Letting both N(x) and n (x) share equally in generating the aiding field gives UNAB (which isthe U for eqn (3.18)) equal to U,z (which is the U for n (x) shown in Fig. 3.8) which is equal toJ7.389 =e. Thus, the doping in the base as well as n (x) change by only 2.7-fold, meaning thatAEg is only lkT-kTlnU0.0 0.25 0.50 0.75 1.0Normalised Position in the Base x (WB)Fig. 3.8. Optimum bandgap in the base to minimise tB. The bandgap at the emitter-side of theneutral base (x= 0) is the reference point. U = n (x WB) /n (x :=0), where h1, h2 and tB aregiven by eqn (3.18).So far this section has only presented the case where N(x) and n (x) are treated independently of each other. This will not be the case whenN is large enough to cause bandgap narrowing 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 renders y(x) stationary in eqn (3.20) does not depend upon the relationship between N(x) andn (x). Indeed, using eqns (3.21) and (3.18), the optimum y(x) has exactly the same form as theoptimum 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 (AE8 (x), NAB (x)).July 12, 1995 40In general, the dependence that n has with respect to N will be too complex to allow fora closed-form analytic solution. In this case a possible solution process is to use an iterative approach where y(x) is first solved for using eqns (3.21) and (3.18). A trial function NB (x) for theactual N(x) is constructed by using h1 and h2 from y(x), and forcing NB (x) to take the form ofFig. 3.3 (while obeying the original doping boundary conditions). Finally, using eqn (3.20), a newN(x) is solved for using n (iE (x), NB (x)) and y(x). This process can be repeated until little change is observed in N(x). In the event that the convergence of this iterative method is tooslow, then higher-order numerical methods such as Newton-Raphson iteration could be used instead. Thus, it is a simple matter to include banclgap narrowing into the optimum base profile forthe minimisation of tB, for the stationary function y(x) that defines both N(x) and n (x) is independent of both these functions.NAB (x)n(x) x=W1,0.0 0.25 0.50 0.75 1.0Normalised Position in the Base x (WB)Fig. 3.9. The optimum stationary function y(x) that minimises tB. The break points h1 and h2, aswell 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 onthe 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 foundthus far. Strictly, to accomplish this minimisation, one must apply the methods of variational calculus to eqn (3.9) directly; which leads to an O.D.E. that is not soluble in terms of any know transcendental functions. The effect of a non-uniform D(x) is investigated numerically in [63] forlarge U, and the result is a solution that has elements of the stationary functions presented in thischapter, 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 41Therefore, more than a 2-fold reduction in tB is really not warranted as the point of diminishingreturns would be surpassed. From the results presented earlier in this section, ‘tB can be reduced 2-fold with only a 2.7-fold reduction in N(x) across the base when coupled with a AEg of lkTWith 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 reduce 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 thedoping and the bandgap have been constrained at the emitter- and collector-sides of the neutralbase. The optimum base profile has the form of an augmented exponential shown in Fig. 3.9, notthe long-established pure exponential [60] that has been mistakenly assumed. Further, the solutionpresented allows for the simultaneous optimisation of N(x) and n (x), and can also include theeffects of bandgap narrowing due to heavy doping. Perhaps the most interesting and startling result occurs by using both N(x) and n (x) to generate the aiding field in the base, thereby reducing the overall variation in each parameter across the neutral base. Finally, all of the models andmethods 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 orSiGe).July 12, 1995 42CHAPTER 4Transport Through the EB SCRJuly 12, 1995 43In BJTs it is customary to apply the Shockley boundary condition at both edges to the EBSCR in order to determine the quasi-Fermi levels [67]. The Shockley boundary conditions arebased upon the assumption that no matter what physical process is responsible for the movementof charge through the EB SCR, the total transport current T will be very small compared to theforward and reverse directed fluxes at any point within the region. This argument follows exactlythe development of Section 2.1. Applying eqn (2.2) under the conditions described in this paragraph leads to AEp 0. In fact, the Shocidey boundary conditions simply state that Ep1 andare constant across the EB SCR. These boundary conditions allow for an enormous simplificationbecause the exact details of the transport through the EB SCR no longer need to be understood orincluded 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 conduction and valence bands that reduce the forward directed flux. If one of these spikes is largeenough, then T could be constrained by the flux through this one feature alone. Fig. 3.1(b) showsthe general band diagram for HBTs built in the AlGai..As material system, where there is anabrupt heterojunction between the emitter and the base. The very nature of the sign of AE, whencoupled to the fact that the emitter doping is much smaller that the base doping, produces a feature in E called the CB S [25]. The CBS can easily force the electrons to take a path that requiresan increase in energy of nearly 240meV. To increase the electron energy by 240meV, with respectto a homojunction, would reduce the available number of electrons, and therefore the forward directed flux, by four orders of magnitude at room temperature. A reduction by i04 in the forwarddirected 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 transport current within the EB SCR.The thermionic injection of electrons over the top of the CBS is not the only method oftransport through the region. Due to the quantum mechanical nature of the electron, and the factthat the width of the CBS is typically of the same order as the de Broglie wavelength, the electroncould tunnel through the CBS instead of trying to increase its energy in order to surmount the barrier. Since a reduction in the required energy to surmount the CBS leads to an exponential increase in the forward directed flux, tunneling and therefore the quantum mechanical nature of theJulyl2,1995 44electron also needs to be considered when deriving the physical models for transport through theCBS. Failure to include this tunneling current will underestimate Tby up to two orders of magnitude [25] (see also Fig. 4.8). Therefore, no matter how powerful a model is used (such as MonteCarlo modelling), if tunneling is not accounted for through the CBS, the terminal characteristicsof the device will be greatly underestimated.All of the previous chapters have relied on the existence of current-limited flow in one region of the device that is separated from both the base and the collector. Specifically, the regionproviding current-limited flow occurred at the EB heterojunction where the CBS is formed. Sincethe transport current through the device leads to I, and because current-limited flow at the CBScontrols the transport current, then I is governed completely by the transport mechanisms of theCBS. Under the condition of CBS control, I has no dependence on the physical construction ofeither 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 toaccurately predict the terminal characteristics of the device. This chapter investigates and derivesmodels for the transport of charge through the region containing the CBS, including effects due totunneling and a varying effective mass.4.1 Formulation of Charge Transport at the CBSThe transport of charge through the region where the CBS is formed can be found by viewing the system as a set of forward and reverse directed fluxes (Ff and F,. respectively) entering theregion from opposite sides (see Fig. 4.1). If there is no source or sink of carriers within the regionconsidered, then just like eqn (2.2) F =J7c(-x)— Fr(Xp), where F is the transport flux, x, is thethickness of the SCR extending from the heterojunction into the emitter, and x is the thickness ofthe SCR extending from the heterojunction into the base. If at the points -x,, and x it is acceptableto state that the system is fully thermalised, based upon a local Fermi energy Ef, then the carrierdistribution with respect to total energy U is:f(U)= U—,.L’ (4.1)l+e k1where f is the Fermi-Dirac distribution function and p. is the electrochemical potential (which isusually termed the Fermi energy Ef). Using eqn (4.1) and the quantum mechanics of crystallineJuly 12, 1995 45solids, the transport flux through the region containing the CBS in the x-direction can be writtenin the standard form [68-70]:F = Ff— Fr=3j•dkf1( U) (1 — f2(U) ) WU)-- —2q3$d3kf(U) (1—f1(U) ) WU)-j- (4.2)(2i) Rrwhere W £I) and W1 U) are the forward and reverse directed transmission probabilities respectively, f1 is the Fermi-Dirac distribution at -x,, f2 is the Fermi-Dirac distribution at x, Rf is thevalid 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 0 xpFig. 4.1. Abstract model of current flux within the region containing the CBS. The EB heterojunction is centred at x = 0, with x being the excursion into the emitter (Region 1), and x, beingthe excursion into the base (Region 2). There is a flux Ff entering the region at x = -x, and anotherflux F,. entering from x = xi,. The net transport flux F is equal to Fj— F,. in the absence of any sinksor sources within the region.The interpretation of eqn (4.2) is straight forward in that: there are 2(2tY3 electron statesper unit volume in k-space (including spin degeneracy);f1(1-f2 (in the case of the forward directed flux) is the probability of an electron existing in Region 1 and being able to move to an emptystate in Region 2; W U) is the probability of the electron moving from -x, to with a forwarddirected 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 usingW L1) and W U) respectively. This allows for a non-reversible system to be studied, whereelectron collisions with the lattice (but not with other electrons) can be included. Strictly, if coffisions are considered that change the total energy U of the electron, and not simply its direction ink-space, then the vacancy probability 1—f2(U) (in the case of the forward directed flux) will notdepend on U, but will depend on the exit energy in Region 2. However, if any type of collision isI IJuly 12, 1995 46considered, then W( U) and W U) will be of an extremely complex nature and would require anumerical calculation of eqn (4.2) (this could be accomplished by a Monte-Carlo simulator usingnon-local mathematics; however, no such simulator exists at this time). As a result, eqn (4.2) issimplified by considering collision-less or ballistic transport throughout the entire region, leadingto W U) = W U) = W(U). With the assumption of ballistic transport throughout the regionfrom -x, to x, and converting from k to momentum p (= tik), eqn (4.2) yields:F = FfFr = jd3pfi(U) (1—f2(U) )W(U)-—h Rf px?Jd3pf2(U) (1—f1(U) )W(U)-1-. (4.3)h Rr 1)XExamining eqn (4.3) shows that if the regions of integration Rf and R were equal, then thetwo integrals could be reduced to one integral with an integrand of (f1—2)W(aUIa). One couldthen identify an Ff and F,. from this integrand (which strictly speaking is not the same as that defined in eqns (4.2) and (4.3), but for all practical situations is identical), giving:Ff $dpf1(U)W(U)L (4.4)h RfandFrij’d3Pf2(U)W(Ux)aP_ (4.5)h RrThe key to the definitions of eqns (4.4) and (4.5) is the equivalence of Rf and Rr The fact that thisis indeed true is proven later on in Section 4.3 once the effects of a non-uniform effective masshave been brought into the picture.The solution of Ffand F defined in eqns (4.4) and (4.5) begins by determining the transmission 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 Schrodingerequation, even for a potential obtained from the depletion approximation, is complex enough torequire a numerical solution. Failure to obtain an analytic form for W(U) would hide the rich interplay that exists between the final transport model for the CBS and the physical attributes suchas doping concentration, temperature, effective mass, electron affinity, and bias conditions. An approximate but analytic form is thus sought for the solution of W(U). To this end, one could apJuly 12, 1995 47peal to the asymptotic formalisms in the complex plane used by Landau and Lifshitz [71], or tothe JWKB method [72], to obtain:W(U) = exp [e {Pdx}] = exp [e {_2 f.Jv(x)— Udx}] (4.6)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 movewithout any quantum mechanical reflection. Eqn (4.6) presents a simple analytic solution forW(U), where the particle mass m is in general not equal to the electron mass me, but to the moregeneral 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 approximation results in a parabolic form for V(x), then one can write:V(x) VPk(1+) for-xxO, (4.7)where Vpk is the peak energy of the CBS, and the reference energy is at the bottom of the conduction band where x = -x,. Eqn (4.7) is appropriate to the case where the heterojunction and the metallurgical junction are coincident. The domain 0 x x, will be considered separately so thatW(U) may be separated into two functions; one for Region 1 (WCBS(Ux)) and another for Region2 (WN(Ux), where N stands for Notch), leading to:W(U) = WCBS(UX)WN(UX). (4.8)Using eqns (4.6)-(4.8) with WN(Ux) = 1 produces:xJ2mVk Jl—U’+lWCBS(UX) = WcBs(U Vk) = ex[ P (in ( A[ — AJi — u J], (4.9)where U is normalised energy in terms of Vpk (i.e., U = Ux/Vk). Eqn (4.9) forms the basic kernel for the transmission probability, and it is written in a most general form where Vpk and x havenot 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 regions of integration Rj and R,. are determined), Ff and F,. can be obtained once the energy dispersion relationship U(p) has been set out. The following section will determine U(p) and includethe effects of a non-uniform effective mass m* that generally occurs at an abrupt heterojunction.July 12, 1995 48Once U(p) has been determined, the regions of integration Rf and Rr are set out so that Ff and Frcan be solved for using eqns (4.4) and (4.5) in the next section.E(eV)t2-xn xpFig. 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 ChangesIn general, the two materials that form the abrupt heterojunction shown in Fig. 4.2 are characterised by a different effective mass m*. This change in m* can either enhance or diminish theflux F in transit through the CBS when compared to the case where m* is uniform throughout theregion. Failure to account for the change in m* can result in a significant error. Worse yet, this error is not simply a multiplicative constant as is stated by Grinberg [51], but has a dependence onthe applied bias. Therefore, in solving for Ff and F using eqns (4.4) and (4.5), the dispersion relationship 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 isrealised that the integration is being performed over p-space. As the entire integrand is dependentupon total energy U and x-directed energy U,, it would be beneficial to cause a change of variables in the domain of integration fromp to U. To this end, the dispersion relationship will be taken as parabolic, but left as a diagonal mass tensor to yield:U (p) = U(p) + U1(p) = ++ £J’ (4.10)where mx, and m are the effective masses for particles that have momenta ofPx py, and reJuly 12, 1995 49spectively. As before, U is the x-directed energy, and now, U is the transverse directed energy. Itis also important to realise that eqn (4.10) implicitly places the energy reference at the band extrema. A further simplification can be achieved by a change from cartesian momentum coordinates to cylindrical momentum coordinates. Since we are considering devices that behaveessentially as one-dimensional, symmetry dictates that the azimuth direction in the cylindricalsystem 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 2py Pzand U.L=—+---—2m 2m (4.12)Eqns (4.lO)-(4.12) together allow for the solution of Ff The only approximation being made isthat UQ,) can be adequately described within the parabolic approximation. However, the full masstensor has been retained (albeit in diagonal form) so that anisotropic materials such as Si, SiGeand strained semiconductors can be modelled with the results to follow in this chapter.2U 2mPzPxpyFig. 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 precluded. Setting aside the issues of a spatially varying m* for the moment, the integration over p istransformed to U by the Jacobian:J(Px’PY’Pz —IUX,O,UL —apUx aeau ao auap‘‘2 apUx ae(4.13)July 12, 1995 50The solution of the Jacobian in eqn (4.13) rests on the definitions in eqns (4.11) and (4.12). Looking at eqn (4.12) for the definition of U shows a dependence upon the canonical coordinate Pxalone. From this realisation it immediately follows that:= 0 and = 0. (4.14)Furthermore, eqn (4.12) also produces:3p m=—f. (4.15)auSo 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 thedefinition for U± and performing partial implicit differentiation with respect to Uj, gives:— p, — m — mpapm U± mz y mz Py U±Then using eqn (4.11), which can be condensed and rewritten as = p tan2e, produces afterimplicit differentiation with respect to U±:— pyt (417)anFinally, substituting eqn (4.17) into (4.16), yields:1 mmcos2e37 )‘ Z (4.18)aU Py mcos2E)+m37sin2€)Pressing on and using p = p tan2e, but this time performing implicit differentiation with respect to (3, gives after some algebraic manipulation:ap37 P cos9 ap P 1 1 I ap p= Py sin2O[cose— sine] = sine [cose — sine (4.19)Then, returning back to eqn (4.12) for Uj and performing implicit differentiation with respect toOyields:0= - =---- (420)mEi9 mEi(3 ae m37paeFinally, substituting eqn (4.20) into (4.19), and using eqn (4.11) where p, = pcosO/sin(3, produces:—— m37(421)—mcos2(3+ msin29July 12, 1995 51The 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:= 0. (4.22)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:i mmsin2eZ )‘ Z• (4.23)P mcos29+ msin2OThen, substituting eqn (4.21) into (4.20) produces:mz—=p . (4.24)‘ mcos2€ + msin2eFinally, 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:m0 0px(P, PyPz= 0_________________mmcos29/p (4 26)! U, € U±) mzcos29+mysin mcos9+msinG0pm mmsin2e/pmcos2€) + msin2€) mcos2€) + msin2€)Given the sparse nature of the matrix in eqn (4.26), the solution of the determinant quickly yields:(PX P = X mm (4 27)U, 9, U) Px mcos2€)+msinE)Eqn (4.27) is the Jacobian that allows the integral definitions in eqns (4.4) and (4.5) to be transformed from p to U. As will be seen shortly, this greatly facilitates the development of the modelsfor Fjand FrMaintaining 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 52F — 2q f uhJR L,u,e,u±) ( ) (2 i’m mm= _j•dOdUdUji_ Z 2 Jfi(Ux+Ui..)W(Ux)_:ih R1 Pmcos O+msm 9 m2 mm= —J d9dUdU 2 2 fl(Ux+U±)WcBs(UX)WN(UX) (4.28)h R1 mcos O+msin 0where R1 = Rj to reflect that F1 originates at -x, within Region 1. Eqn (4.28) is the full model fortransport through the CBS. However as was stated previously, the effect of a non-uniform m* has notbeen included. It is instructive to pause at this point and determine, under simpler conditions, WN(Ux)and thus the region of integration R1 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 (thisis 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 (unlikethe CBS within the domain -x, x 0, which contains AE). Since we are considering a system inwhich their are no collisions that could either raise or lower the particle’s total energy, the particlemust emerge from the EB SCR with sufficient energy to enter into the neutral base with an energythat is above E; else one would be admitting particle transport within the forbidden bandgap. Thisfact allows for a considerable simplification to the definition of WN(UX); namely:(1 if U>VWN(U) = I X — b (4.29)k.0 if UX<VbAlthough strictly speaking eqn (4.29) is not the full form for WN(Ux), it captures the ultimate result since any particle that enters the neutral base within the forbidden bandgap (i.e., U, < Vb) willwithin 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 irrelevant, 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 forby eqn (4.28) is to determineR1.Re-examining Fig. 4.1, it is obvious that for a particle to enter theJuly 12, 1995 532itI dGmmm cos2O+msin0 Z=Immm cos2O+msinG0 ZULEB SCR at x = -x, and contribute to Ffi it must possess a positive x-directed momentum. With theenergy reference shown in Fig. 4.2, a positive x-directed momentum translates into Px 0. Furthennore, examination of eqn (4.12) shows thatp 0 translates into U 0 (This is for the casewhere mx > 0 and therefore applies to electrons. To consider holes, it is best to use a negative holeenergy instead of a negative hole mass so that all of the results in this chapter may be applied directly.). If the requirement that U U1 + U E be imposed (where E is the bandwidth for U(p)),then together with U 0, and U Vb from eqn (4.29), then R1 will be as shown in Fig. 4.4.(a) (b)Ux/Ux+ UEVb E 000Fig. 4.4. Domain of integration R1 for a uniform m*. (a): case where the applied bias is such thatVb 0; (b): case where the applied bias is such that Vj, 0. Note: Fig. 4.2 defines Vb.UxEFfcan now be solved for by using eqns (4.28), (4.29), (4.9), (4.1), and the region of integration R1 as shown in Fig. 4.4. Since R1 takes into account WN(Ux), then solving eqn (4.28) yields:2it E E-U2 mmFf =— J 2 . 2 f dU WCBS (U) f dU1f1(U + U1). (4.30)h mcose+msmO0 Z max(Vb,O) 0Examination of eqn (4.30) reveals that the integral over E) has no dependence on the results of thesecond and third integrals. This allows the 0 integral to be performed independently to yield:it/2It= 4Jmmtan’ ( StanqmJuly 12, 1995 54The above equation is evaluated by letting e approach itI2 from the left, giving:2itmmI d9 ‘ Z = 2itjm m. (4.31)m cos29+m sin2e y z0 Z 3’Eqn (4.31) solves for the anisotropic effective mass tensor and is evaluated in such a manner thanall branch points of the inverse tangent are respected. Therefore, as long as one can assert that thesecond and third integrals of eqn (4.30) are indeed independent of (3, then one can substitute eqn(4.31) into (4.30) to obtain:E E-U4tqJm mFf=h3ZJdUW(U)fdUf(U+U). (4.32)max(V,0) 0Eqn (4.32), with the region of integration R1 as shown in Fig. 4.4, gives us a flavour for thetransport current through the CBS. The interpretation of eqn (4.32) yields: a thermalised ensembleof electrons at x = -x, (characterised by the distribution f1 with an electrochemical potential t ofE,1)is injected to the right, towards the CBS; each electron within the ensemble is characterisedby a forward-directed energy U and a transverse directed energy U± which is random but evenlydistributed in all directions; every electron then passes through the CBS with a probability oftransmission given by WCBS which is dependent upon U, alone; the transverse directed portion ofthe electron’s energy leads to a contribution given by the geometric mean of the two transverse effective masses; finally, only electrons that can enter the neutral base outside of the forbiddenbandgap (i.e., U Vb), and are within the bandwidth E of the conduction band, are allowed tocontribute to the transport current. Eqn (4.32) solves for Fjunder the condition that the effectivemass tensor is a constant throughout the CBS.Returning back to eqn (4.28), the main thrust of this section is continued; namely the incorporation of a spatially varying m* into the transport current. The inclusion of a non-constant m*requires that the electron energy U ( U + L1) be generalised to:U1 = U+U and U2 = (4.33)where energies with a subscript of 1 refer to transit within Region 1 (i.e., -x, x 0), while energies with a subscript of 2 refer to transit within Region 2 (i.e., 0 <x xv). The reason for the generalisation that leads to eqn (4.33) is that the spatial change in the effective mass tensor results ina mixing of the x-directed and transverse directed energies. Therefore, one cannot maintain a to-July 12, 1995 55tally separate view of U, and U1. Now, the energy reference continues to be located at E(x=-x),so that using eqn (4.12) produces:1=and U1 = + (4.34)1 2m 1 2mz 1while2 2 2u —____V and U— ‘2 Pz,2 435X, 2— 2m 2 b J 2 — 2m 2mz 2It is important to understand the exact meaning of eqns (4.33)-(4.35). To begin with, the energies U, U, and U± represent total energies within their respective regions. Band diagrams suchas those shown in Fig. 4.2 do not show the total energy U, but instead show only U. In the eventthat the system possesses transverse symmetry, then the potential energy is V(x,y,z) V(x). Whenthere is transverse symmetry, it is possible cast the full three dimensional problem into two decoupled one dimensional problems whose solution only depends upon U or U1 respectively. For thisreason, U,2 is not simply given by the kinetic energy term containing Px,2’ it must also includethe offset potential energy of Vb. Thus, eqn (4.35) gives the total energy U2 located at x = xi,,while eqn (4.34) gives the total energy U1 located at x = -x. The reason for defining the energiesat -x, and Xp being that Ff is based upon particles injected to the right from x = -x,j, while Fr isbased upon particles injected to the left from x = x. Furthermore, because the potential energyV(x,y,z) V(x) does not vary in the transverse direction, Uj and U.,2 do not contain an offsetpotential energy term.The cumbersome nature of the energy relations given by eqns (4.34) and (4.35) arise fromthe quantum mechanical nature of the problem. Looking back to eqn (4.3) shows the flux beingcalculated by an integration over p-space. Strictly speaking, quantum mechanics does not allowone 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 distribution in p-space located at x = xi,. Essentially, due to the slow variation of V(x) over the atomicdimensions, it is possible to cast the problem into quasi-classical form [15] where one can speakof distinct p-space distributions at largely separated positions in real space. Finally, because wetransform p into U, the same concerns for p-space apply to U-space as well.July 12, 1995 56Due 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 conservation ofp. Therefore, at the heterojunction separating Region 1 from Region 2 (i.e., at x 0),P±, 1 P±,2 (where eqn (4.11) has p = +p1)so that:py,1 = Py,2 p, and Pz,i = Pz,2 = Pz (4.36)Since the potential energy V(x,y,z) ( V(x)) does not vary in the transverse direction, then P±,1P±,2 cannot vary with x if collisions are prohibited. Using eqn (4.36) in eqns (4.34) and (4.35)shows that Uj1 and U±,2 must remain constants of the motion. Therefore, eqn (4.36) must holdequally 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 2 2 2UJ1= 2my 1 + 2m 1and UJ2= 2my 2 + 2mz 2Applying eqn (4.11) to the above yields, after a little algebraic manipulation:U±,1 m 2mz2 m 1cos2G+m 1sin2e_____= “‘ R(9) where R(G) = Z, (4.37)U±, 2 my lmz 1 mz2cos9+ my,2sin9Examination of eqn (4.37) shows the necessary condition that if m,1= m,i = = mZ,2, thenUj 1IU,2= 1. Eqn (4.37) represents the change in the transverse energy that must occur to conserve pj in the face of a spatially varying effective mass tensor. It is instructive at this point to reveal the full implications of eqn (4.37) upon the total energy within the system. Fig. 4.5 shows theeffect of eqn (4.37) when m1 = m,i = m1, and m,2 = mZ,2 = m2. When m1 <m2, thenAs will be described in the next paragraph, total energy must be conserved throughout Regions 1 and 2. Thus, when m1 <m2, the positive difference Uj1— Uj is transferred into U,2which leads to an enhancement in the forward directed flux. Conversely, when m1 > m2, then Ujj< Uj. Thus, when m1 > m2, the negative difference U±,i — U,2 is removed from U,2 whichleads 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 totalenergy must be conserved at the heterojunction separating Region 1 from Region 2 (i.e., x = 0).Thus,1 2Furthermore, since there are no collisions within the two regions, the above conservation require-July 12, 1995 57ment 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 heterojunctionseparating Region 1 from Region 2. It is important to remember that most of the proceeding arguments are based upon the conservation ofp. This conservation can only be asserted if the Hamiltonian of the entire system has translational symmetry along the transverse spatial dimension, ifthe heterojunction contains a corrugation or surface roughness, then one could not assert thatp isconserved. This would lead to a considerable increase in the complexity of the model that wouldnecessarily require a detailed view of the device at the atomic level.(a) (b)I I Ix-xn 0 -x7 oFig. 4.5. The effect that conservation ofp has upon U,1 and U±,2 when a mass boundary isplaced at x = 0. Using m,1 = m,i = m1 and m,2 = m,2 = m2 in eqn (4.37), then U±,1IU2 m21m1. (a): when m1 <m2, energy is removed from Uj1 and transferred to U,2 when moving fromthe left to the right; (b): when m1 > m2, energy is removed from U,2 and transferred to Uj whenmoving from the left to the right.With eqns (4.38), (4.37), (4.35) and (4.34), the effect of a spatially varying effective masstensor 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(U,2,U,2)in Region 2. This mapping is solved for by substituting eqn (4,37) into (4.38), giving:I I IJuly 12, 1995 58U2 =U1+(e)U, (4.39)andUx1 = U2+ y’(€)) U,2, (4.40)andU = U1 + U,1 = U2+ U1,2 (4.41)wherem 1m m2y(9) = 1 — “ ‘ R(9) and y’(9) = 1 — “ Z R(€)) = . (4.42)my 2mz 2 my, 1mg, 1 —Finally, using this simplified form based on the function y (the notation for y was initially set forth byChristov [70,73], but has been extended here to include anisotropic effects), eqn (4.37) becomes:ii 1FT ‘ = 1 = (4.43)J,2 ‘i’where the explicit dependence upon 9 has been dropped for simplification. Eqn (4.41) simply asserts the fact that a collision-less system is being considered, while eqns (4.39) and (4.40) represent the energy mapping that occurs when crossing the heterojunction at x = 0 from the left orfrom the right respectively.Returning back to eqn (4.28) for the calculation of Ffi the integral is being performed overU-space located at x = -x, with a domain of integration R1. Using the formalisms for passingthrough the heterojunction that were developed in eqns (4.39)-(4.43), it is important to realise thatthe transmission probability W(U), as defined in eqn (4.8), must be extended to:W(U) = WcBs(UXl)WN(UX2), (4.44)for WCBS is defined in Region 1 and thus depends upon U,1, while WN is defined in Region 2and thus depends upon U,2. However, any function that depends upon total energy U (such as theFermi-Dirac distribution function f(U)) remains unaffected by the mass barrier due to the conservation of total energy set out in eqn (4.41). Therefore, eqn (4.29) for WN is rewritten as:11 if U >VWN(U 2) x, 2 — b (445)X,‘0 if U2<V,,The domain of integration R1, which is used for p- or U-space integrations performed at x =-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 = -x, and contribute to Ffi it mustpossess Px,1 0; or in terms of energy, U,i 0. And, the requirement that U (= + U1,) E(where E is the bandwidth for U(p)) is still maintained. However, eqn (4.45) imposes the condiJuly 12, 1995 59tion that U,2 Vb, which results in a coupling between U,1 and U±,i when eqn (4.39) is used tomap from Region 2 into Region 1. Therefore, using the three boundary conditions set out in thisparagraph, along with eqn (4.39), yields the following boundary for R1:1 0,U1+U,E, (4.46)UX,l+YUI,l Vb.It is also possible to transform R1 (which is applicable to an integration carried out at x = -x) intoR2 (which is applicable to an integration carried out at x = x) by substituting eqns (4.40) and(4.41) into (4.46) to produce the following boundary for R2:U2+ yU1,2 0,U,2+ U1,2 B, (447)UX2Vb.When the effective mass tensor is uniform, then eqns (4.42) and (4.37) produce y = =0. Under these uniform conditions, then indeed eqn (4.46) produces the R1 as shown in Fig. 4.4. However,when y’ 0, R1 becomes distorted from that shown in Fig. 44.1 and y’ can take on any value inthe range —0o (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 m1 <m2where 0 <‘y 1 (and —00 < ‘y’ <0), and energy is transferred from U±,i into U2 which leads to an enhancement in the forward directed flux; secondly, when m1 > m2 where —oo <‘y <0 (and 0< y’ 1), and energy is removed fromUx,2 and transferred into which leads to a reduction in the forward directed flux. Fig. 4.6 showsR1 and R2 for the case where y> 0, while Fig. 4.7 shows R1 and R2 for the case where y <0. Examination of Fig. 4.6 shows a focussing ofR2 towards the direction of charge flow. This is due to the energy transfer into 11x,2 when passing through the heterojunction, leading to what is termed currentenhancement. Conversely, examination of Fig. 4.7 shows a reflection in R1 against the direction ofcharge flow. This is due to the energy removal from U,,2 past the heterojunction, leading to what istermed current reflection. The current reflection occurs because ultimately, no carrier may enter thebase within the forbidden bandgap (i.e., U,2 < Vb). As a result of Figs. 4.6 and 4.7, care must be exercised in applying the integration boundary R1 (or R2) to the solution of Ff in eqn (4.28).July 12, 1995 60E \,7Ux2+UJ.2=EU±2=R2\uX2o yE EFig. 4.6. Enhancement case where m1 <rn2 (i.e., y> 0 and y’ <0). Domains of integration R1and R2 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 integration represents the ensemble of particles that contribute to Notice in R2 how the transfer ofenergy from Uji into U,2, due to the increasing m* in the direction of charge flow, leads to a focussing of the particles towards the direction of charge flow.+ U,1 =EVb-UX I(UL1 yU 2—yp00(a)U,2E(b)E1—y’00 ‘Vb yE,U,1 + Uj = EU,;,2—y,July 12, 1995 61(a)*U,2E-VbEUJ_,2 =_____—YU,2+U±,=ER20 Ux2I ...........................................................Vb 0 EFig. 4.7. Reflection case where m1 > m2 (i.e., y < 0 and y’ > 0). Domains of integration R1 andR2 from eqns (4.46) and (4.47) for the calculation of Ff at x = -x, and x = x,, respectively: (a) theapplied bias is such that V, 0; (b) the applied bias is such that Vb 0. Each domain of integrationrepresents the ensemble of particles that contribute to Fp Notice in R1 how the removal of energyfrom U,,2 into U±,2, due to the decreasing m* in the direction of charge flow, leads to a reflectionof the particles against the direction of charge flow. The reflection occurs because of the necessityfor particles to enter the base outside of the forbidden bandgap (i.e., U,2 Vb, orpX2 0).EU,2U,1 + U,1 =EEE— Vb—UX1Y+ U = ER2Ux2Vb EVb Vb-YE1—Y0(b)UJ-,1ZUx,1÷U±,1=EE- Vb1—YVb—UX1YU 2—Y’Vb-YE1—yUx1EJuly 12, 1995 62Before eqn (4.28) is recast to include changes to m* (by including R1 from Figs. 4.6 and 4.7)it is instructive to calculate the Jacobian that transforms integrations performed within Region 1into those performed within Region 2. In other words, we wish to determine:au1 au11 , Uj—(U 1’ L, 1 — U2 U 2U2 9,u2)— 2) — u1, 1 1au2 au±,2Using eqns (4.40) and (4.43) produces:m 2m 2u1 =— m1mR(9)‘\x2J2J m 2m 20 “ ‘ R(O)my lmz, 1where R(9) is defined in eqn (4.37). Using eqn (4.37) yields, after substitution into the above:(Ui9 U1, = (Ui U±1N = my,2mz2 mzlcos9+mylsinO (448)U 2’ , U±, 2) 2’ U 2) my 1m 1 m2cosO+ my,2sinOFinally, by combining the above Jacobian for a change in variables from Region 1 to Region 2with the Jacobian given by eqn (4.27) for a change in variables from p to U (which in this case issubscripted to reflect calculations within Region 1), gives:U,1”—3(P,i’P,i’P,i”1— m1 my,2mz2 (449)1\U,9,U±,)u2,u±,) u2,e,u±) Px,lmz2cosG+ ysin9Examination 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 becausethe 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 volumeelement. However, the term mi/p,i and not mx,2Ipx,2 remains in eqn (4.49). The reason for thisdiscrepancy from perfect symmetry lies in the fact that is an ensemble of particles originatingat 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 isconserved 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 U/ap,1 (Px,1’mx,1) and notJuly 12, 1995 63The final transport model for Fp including the effects of a non-uniform m*, is presented. Forthe enhancement case (i.e., m1 <m2 andy> 0), then using eqn (4.28) with calculations based at x= -x, and R1 defined in Fig. 4.6, produces that:2it E E— u,,, (4.50)2 m1Ff= fde1cos2O+m 1sin2e$dUlWcBsx,l)J dU1f(+ ±0 max(Vb,0) 0max(Vb,O) E—U1+ f dU 1 WCBS (U, 1) J d U1 1 f1(u + U1 1)0 Vb—UX1The term WN(Ux,2) is equal to 1 within the domain R1 and has been removed for clarity. However, if WN does not have this simple form, then the full WN(Ux,2) = WN(Ux,1 + yU11)must remainin eqn (4.50), where the coupling of the canonical variables forces it to remain nested within thethird integral over U,1. If this is the case, it may be beneficial to calculate Ff at x = x. Using R2as defined in Fig. 4.6, along with eqns (4.48) and (4.28), produces:2it E E—U,2 (4.51)2’ m2Ff— J d 2 . 2 J dU 2 WN (Ui, 2) J d U1,2f1(u 2 U1 2) WCBS (U, 1)h cos€)+ms1nE)o ‘ ‘yE 0Ux,2+ f dU2WN (U 2) fdU1,2f1(U2+ U1, 2) WCBS (U, )max(Vb,O) 0In this case WN (= 1 within the domain R2) has been left in to show its general inclusion for thecalculation 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 U1,2 and cannot be easily removeddue to its dependence upon U1,which by way of eqn (4.40) is equal to U,2 + y’ U1,2.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 F1, one must multiplyby “-1”.Finally, for the reflection case (i.e., m1 > m and y < 0), then using eqn (4.28) with calculations based at x = -x and R1 defined in Fig. 4.7, produces:July 12, 1995 642it r E E—U,1 (4.52)2 m1 IFf_ SdOm 1cos2 +1 1sin29[m f dU1 WCBS(UX ) j dU1f1(U + U )0 ax(Vb,O) 0Vb - yE1—y E—U1— f dU WCBS (Ui, ) f dU1f1(U1 + U1)max(Vb,O) Vb—UX,lAs was done with the enhancement case, the term WN(Ux,2) (= 1 within the domain R1) has beenremoved for clarity. However, as is true for the enhancement case, if WN does not have this simpleform, then the full WN(Ux,2) = WN(Ux,1 + YU±,i) must remain in eqn (4.52), where the couplingof the canonical variables forces it to stay nested within the third integral over Uj.i. If this is thecase, it may be beneficial to calculate Fj at x = x. Using R2 as defined in Fig. 4.7 along with eqns(4.48) and (4.28), produces:2t E E—U,2 (4.53)2 m2Ff= i!Id0m 2cosG+rn 2sine$dUxWN(Ux)jdfl( X+U±2)WcBs(UX1o Z Jnax(Vb,0) 00 E-U,2+ f dU2WN (U, 2) $ dU12f1(U,2+ U 2) WCBS (U )min(Vb,O)—rAgain, as with the enhancement case, eqn (4.53) simplifies the problem of calculations involvinga complex WN, but at the expense of making calculations of WCBS far more complex. Basically, ifWN has a simple form then use either eqn (4.50) or (4.52) for the calculation of Ffunder enhancement 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, ifboth WN and WCBS have a complex form then little can be done to reduce the complexity of theproblem.Eqns (4.50)-(4.53) present a rigorous model, that includes the effect of quantum mechanicaltunneling, for the calculation of the forward flux entering a two region system with an abruptmass- and hetero-junction in-between. These equations solve, for the first time, the transport current within a complex region while allowing for an anisotropic media. As such, these equationsJuly 12, 1995 65represent 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 featuresfound within HBT structures which were not accounted for by the aforementioned authors in theirstudy of Schottky diodes. Furthermore, the models presented here overcome the problem encountered by Perlman and Feucht [79], who solved the same system but neglected tunneling. Due tothe neglect of tunneling, the models in [79] have an un-physical discontinuous change when themass boundary is placed coincidently with the potential boundary. It is important to be able tomodel transport through complex regions like the CBS, for in modem abrupt HBT structures thistransport 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 onematerial system. Therefore, the results of this section can be applied equally well to any materialsystem.In concluding this section it is important to mention some cautionary comments and shedsome 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 equalto eqn (4.30) which is for a constant m*. For this reason, this double integral is termed thestandard forward flux as this is the standard flux that would flow in the absence of themass barrier. The last double integral in eqn (4.50) represents an additional flux that wouldnormally have entered the base within the forbidden bandgap, but due to the mass boundarytransferring energy from Uji into U,2, it is raised up into E within the base to contribute to thetotal Ff As such, this current is termed the enhancement forward flux1enhance Finally, the lastdouble integral in eqn (4.52) represents a flux that would normally have entered the base withinE, but due to the mass boundary removing energy from U2, it is lowered into the forbiddenbandgap within the base and is lost from the total Fp As such this current is termed the reflectedforward flux FfreflectS It is also important to remember that when solving eqns (4.50)-(4.53), y andy’ have a dependence upon e in general. Therefore, unlike eqn (4.30) (and thus FfstjJ) wherethe e integration can be treated as an independent multiplier to yield eqn (4.31), the calculation ofFjepice and 1reflect will have y(9) and y’ () nested within the integrand, making for apotentially stiff problem to solve due to the complex nature of the 0 integral.July 12, 1995 664.3 Calculation of Fr and a Unified Model for FThe total transport flux F is equal to Ff— F,, as is given by eqn (4.2). The models of the previous section, given in eqns (4.50)-(4.53), concentrate on the calculation of Ff The reason for maintaining a focus upon Fj while neglecting F,, is that the two the fluxes are essentially identical, savefor a change in the electrochemical potential within the distribution functionsf1andf2used to determine Ffand F respectively. Furthennore, under the condition of current-limited-flow due to a givenregion, 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 indicative of eqn (2.2), it is necessary to prove that the regions of integration for and F provide forthe form given in eqn (2.2). The calculation of F,, and the ultimate proof that eqn (4.2) (and thus thetransport 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 effectof a mass boundary. Included within eqn (4.3) is the requirement that tunneling, or any other conduction process for that matter, that moves electrons from one state to another depend upon theprobability that the final state be unoccupied (= (1 — f) h). Using eqn (4.3) for Ffr eqn (4.44) forW, eqn (4.34) for U,1, and the Jacobian given by eqn (4.49) to move calculations to xi,,, yields:21r (454)2 m2F= -_ifdUfdUf,9 2 • 2 ff(Uf)h m 2cos G+m 2sm 000 0 0 Z 3’,where the superscriptf refers to functions that have their energy reference located at the bottom ofthe conduction band at x = -x. To arrive at the infinite extent for the region of integration it is onlynecessary to extend the definitions of WCBS, WN, andf1 to implicitly account for the fact that theflux density must be zero outside of the region R2 defined by eqn (4.47) (i.e., WCBS(Ux,1) 0when U,1 0, WN(Ux,2) 0 when U,2 Vb, andf1(U) 0 when U E). No loss to the generality of these function occurs as a result of this extension. Likewise for F,, but using only the p to UJacobian of eqn (4.27) in order to maintain the calculations at Xp. yields:00 2i (4.55)2 mmFr = -_Jdu1fdUi,$de 2z,2. 2 W’)h m 2cos 0+m 2srn 0—00 0 0 3”July 12, 1995 67where the superscript r refers to functions that have their energy reference located at the bottom ofthe conduction band at x = x. Note that in eqn (4.55) the subscripts referring to Regions 1 and 2have been interchanged to reflect the reverse direction of flow for Fr in comparison to Ff Therefore, both eqns (4.54) and (4.55) have been constructed so that the integration over U-space occurs at the point x = x. This will facilitate direct comparison between Fr andThe task that remains is to recast the r-superscripted functions of eqn (4.55) into thef-superscripted functions of eqn (4.54). The only difference that exists between the f- and r-functions istheir energy reference. Since E(x=x) — E(x=-x) = Vb, and there is transverse symmetry, thenusing eqns (4.34) and (4.35):U1 = U2—V and UI,i = U2 =.u’ = U1+U, = UVb.Finally, recasting eqns (4.39) and (4.40) into r andfform, gives:= U2+7’U, and U2 = U1+’y’U,.The reason y’ and not y is used in the definition for U2, is because Regions 1 and 2 are interchanged for the calculation of Fr This regional interchange maintains consistency with Section4.2 where the flux always originates in Region 1. With the interchange of Regions 1 and 2, all ofthe effective masses are also interchanged. Finally, observation of eqns (4.42) and (4.37) showsthat 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-superscripted). Using this functional translation, along with eqns (4.57) and (4.56) gives:WBs(Ul +y’Ui,1+ Vb) = WBs(U2+y’U2)wrf iii! ‘“CBS x,1”wifi’rir— wrf’rif ‘Vb)— VVNkLI2),f( u’ + Vb) = f( (if),f( U’) = f{( U’ + Vb) = f{( Ui).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 isequal to 1 — f, eqn (4.55) becomes:(4.56)(4.57)WBs(U2)= WBs(U2+Vb) == W(U) ==f;(U’) =July 12, 1995 6821t (4.58)Fr = fdU1Jd I,domz,c s2:::sin2ef(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 (U1,UI1) into(U2,U2). Examination of eqn (4.56) shows that the only difference between points in(U1,UI,1) space and points in (U2,U2)space is a constant Vb. Since the addition of a constant does not distort the differential volume element, the Jacobian is unity. This allows eqn(4.58), along with UI,1 = U,2,to immediately transform into:2it (459)2 m2Fr = _fdUfdU.e 2., 2 f(U)h m 2cos O+m 2sin ()0 0 Z 3’,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) ), whileFf deals with transport from Region 1 to Region 2 (i.e., f(( U) h( U”) ). Therefore, the transportflux is:_21tF = F — F = I dUf2 IdUf 2 dO m2mz2f r h3 J ‘ J m2cosO+m 2sinO0 0 ‘[f{( U1’) h( U) — f( U”) h{( U”) ] WBs(U1)W(U2)2it2’ mm—i fi f i “ z’h3 J ‘ .1 .1 m2cosO+m 2sinO—e 0 0[f{( U) — f( U”) I WBs(U1)W(U2) . (4.60)Thefsuperscripts have been included as a reminder that the energy reference is located at the bottom 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) respectively. Eqn (4.60) is brought into exact agreement with eqn (2.2) when the fi and f2 distributionfunctions of eqn (4.1) are given by the Boltzmann approximation, leading to:July 12, 1995 69U -f1(U) =_______kT1+e kT(4.61)U - j.t2f2((1)=—2e kT1+e kTEqn (4.61), under the Boltzmann approximation, produces:U I’i___IS.Efl,f{( U) — f( U) = e e’— e kT f(( U) (i — e kT) (4.62)where AE-I2• Since AEp is a constant with respect to the canonical variables defining theintegration in eqn (4.60), then substituting eqns (4.62) and (4.54) into (4.60) gives:iF = Ff— Fr = F1 — e J. (4.63)Thus, the transport flux through the CBS has exactly the same fonn as eqn (2.2). This will allow themodels of this chapter to be used with the results of Chapter 2. Eqn (4.63) also justifies the methodology 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 increases, so does F increase; when the system is at equilibrium (AE 0), the transport flux vanishes.4.4 Analytic CBS Transport ModelsSection 4.2 presented the general models for the calculation of the transport flux Ff througha complex two region system with an abrupt mass barrier in-between. The models also allow foran 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 obtaininganalytic models, and not simply resorting to numerical calculation, the rich interplay that existsbetween the physical attributes such as doping concentration, temperature, effective mass, electron affinity, and bias conditions, will be brought out for study in the final transport model of theCBS. The key component to all of the models presented in this chapter is the inclusion of the effects due to tunneling. Any model or simulator (such as the highly acclaimed Monte Carlo simuJuly 12, 1995 70lator) that fails to account for the vast increase in transport current through the CBS due totunneling, will be grossly inaccurate even if every conceivable scattering process and other driving force outside of tunneling is accounted for (see Fig. 4.8).106i03 - tunneling and thermionic emission100io-3 -10-6 - ...‘ no tunneling, only thermionic emissionio-9 I • I • I •0.8 1.0 1.2 1.4 1.6Base-Emitter Voltage VBE (V)Fig. 4.8. Collector current for an abrupt A1GaAs HBT with 30% Al content in the emitter. Theemitter doping is 5x1017cm3,and the base doping is 1x109cm3.Notice the large error that resuits if the tunneling current through the CBS is not accounted for. Also, the tunneling current hasa 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 FluxWith the result of eqn (4.63), the development returns to the main goal of this section; deriving analytic models for Ff from eqns (4.50)-(4.53). For the problems being considered, the formof WN in eqn (4.45) suggests that eqn (4.50) be used for the enhancement case (i.e., m1 <m2 and> 0), and eqn (4.52) be used for the reflection case (i.e., m1 > m2 and ‘y < 0). As was discussednear the very end of Section 4.2, eqns (4.50) and (4.52) share a common term called1Jstandard (orFj8 for short), plus a unique term for the enhancement case of Fjepi (or F!e for short), and aunique term for the reflection case of Fjreflect (or Fj for short). These terms, using eqns (4.50)and (4.52) are:July 12, 1995 712it E E-U12 m1Ff=fd91cos29+m sin2eIdUx,1WCBSx,1) f dU(+U±,),(4.64)o “ max(Vb,O) 02it max(Vb,O) E—U1m 1m 1Pe=dO 2 2 J dUX1WCBS(UXl) J dU±f(+ ,),(4.65)h mz 9+ m, 0o 0 Vb—UX1Vb - yE2it E—U,12 m1Ffr= 2 + .201 dU1 WCBS(UX ) J dU1f1(U + U ).(4.66)o max (Vb, 0) V,,—U,1yThe derivation of the analytic models begins with Fj. Fj is the most important term, and as itwiil 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 andUj contain no term with a dependence upon 0. This allows the 9 integral to be performed independently, 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:E_Ux-,.L14itqjm 1m 1kT 1+e kTFf=Z f dU WCBS (Ui) ln E —max(Vb,O) 1+e kTThe integrand above becomes vanishingly small (at an exponential rate) for large U,, allowing fora simplification by letting E — 00 to produce:/4itqjm 1m 1kT ( -____Ff=$dUXWcBs(UX)ln,1+e kTmax(Vb, 0)In general, even if the emitter is degenerately doped, the energies U at which the above integrandproduces significant contributions to F15 occurs at energies where U is a few kT larger than t1.This allows what is essentially an assertion of the Boltzmann approximation that leads to eqn(4.61), so that:July 12, 1995 72I004itqqm 1m 1kT — -—Ff=ekTJdUXWCBS(Ux)e kT (467)max(Vb, 0)Eqn (4.67) provides for the model of the standard flux, where the integrand multiplied by the leading constants is the standard flux density.Eqn (4.67) is now solved for by substituting in WCBS from eqn (4.9) and making a changeof variables from absolute energy U to normalised energy U (where U = U/Vpk, and Vpk is theheight of the CBS as defined in Fig. 4.2). Before performing these changes to eqn (4.67), the solution process is further facilitated by the following change of variables:,J1—U +1 2x 2 , x2—iX == ( 2 and Ji — = 2l+x x+lLettingx=e3’ = and J1—U =th(y),ch (y)where ch(y) is the hyperbolic cosine of y, and th(y) is the hyperbolic tangent of y. Using the aboveequations, along with the normalised energies from the start of the paragraph, yields for Vj, < VPk:i xfl./2mXlVPk y VPkFf5= 4itqJm lmz, 1 kT ‘ke’ I dU e— th(i))— kT ch2(y) (4.68)‘nax(V,0)pkkT+ ----eVpkwhere all energies, including V, are in terms of normalised energy (i.e., Vb = V ‘k). The lastterm inside of the square brackets is the thermionic injection term where WCBS = 1. In the eventthat V> 1 (i.e., Vb> k) then the CBS is at an energy too low to effect the transport current and:2i4tqm1m(kT) -F = e eUp 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 withoutconnection to the material parameters of the device (where the device is arbitrarily chosen as annpn HBT). However, using the depletion approximation gives [24,80]:July 12, 1995 73V 61NDVPk = Vp=q(l_Nraj)(Vbj_VBE)‘ = eNplc 2AI2elVk I2eV x NDVb = x =q2NXP=qq2;= = (4.69)L2NA kT INANDN AEwhere N = and V. = — In I I +rat NA+C1D q I 2 ) qVbj is the built-in potential of the junction, n2 is the intrinsic carrier concentration in Region 2,ND is the emitter doping, NA is the base doping, e is the permittivity of the respective region, andVBE is the forward bias across the EB junction. The doping ratio Nrat differs slightly from that ineqn (3.5) due to a nonuniform e. Concentrating on the case Vb < Vpk then using eqn (4.69) withineqn (4.68), along withI ND= U + r where U= 5.VJeimgives: (4.70)________N (Vb.VBE) rFj= 4qJm1rnkTk;fdU erag(ch2(Up+r)PJ (4.71)hmax (V Nrat (Vb — VBE)kT v+-—eVpkwhere V is the thermal voltage kTIq, and U = ch2(U + r). As will be shown shortly, eqn(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 thedevelopment of Schottky diodes. Furthermore, U, is the V1 normalised version of Efj4-J from [751.The standard forward flux density for a given energy U is:_______N (Vb.VBE) r= 4qm 1in 1 kT Vpk;erat(Ch2(U+ r)-th(u÷ r))(4.72)where the energy U (= ch2(U + r)) is defined in terms of r. The energy at which the maximumcI occurs can be found directly from eqn (4.72). In terms of the variable r, and given that exponentials are analytic functions, 4c will be at a maximum when the exponent containing r in eqnJuly 12, 1995 74(4.72) is at a maximum. To this end it is found that:I-r . -ThermionicInjectionRegimek I1.0 1.2 1.4I r 2rsh3(U +r)2 — th(L1,+ r) = — P 0 r = 0, —UP, ±00. (4.73)drch (U+r) ) ch (U+r)sh(y) is the hyperbolic sine of y. Examination of the definition for U, in terms of r, shows that r hasa range of-Un r < oo• Furthermore, when r = -U then U = 1, which coffesponds to the top of theCBS, and when r —>0o then U =0 (it should be noted that U < 1 deals with the tunneling of electrons through the CBS while U> 1 deals with thermionic injection over the CBS). The solutionsof r = -U,, and —oo occur due to the mapping used to define U, in terms of i and do not representthe absolute maximum that is being sought. Thus, the maximum occurs when r =0 and gives:th(U)4itqjm 1m lkTVk —- Umax= y, z, P e’e ‘ at U = ch2(U). (4.74)h pVBE= 1.4VVBE=O.9V, when VBE = O.9V-V’, when1.00.80.60.40.20.00.0 0.2 0.4 0.6 0.8Normalised Energy U (Vpk)Fig. 4.9. Flux density fs’ normalised tO max’ for anAl1j3Ga07s/GaAs abrupt HBT at twodifferent forward biases. The material parameters, the same as in Fig. 4.8, are: emitter doping ND5x1017cm3;base doping NA 1x109cm3;emitter permittivity El ll.9cj; AE is 0.24eV; n2 is2.25x106c Vbì is 1.67 1 V; mx,l is 0.091m;Tis 300K. Note that energies U <V would enterthe base within the forbidden bandgap, and although displayed here are reflected in reality.July 12, 1995 75As was also found in [78], eqn (4.74) presents a surprising result that the energy U atwhich the peak flux density Im occurs is independent of the applied bias. Therefore, relative tothe 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)) increases from 0 towards infinity, U moves from 1 towards zero, and tunneling becomes increasinglydominant over thermionic emission; as ND increases, or e decreases, the width x of the CBS decreases and U,’, becomes smaller, showing that tunneling is increasing; as m,i decreases theprobability 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 throughthe barrier then it is to obtain enough thermal energy to pass overtop of the CBS; finally, in thelimit as 11 goes to zero, the system should evolve to a state that is purely describable by classicalmechanics, 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 integration over . Using eqn (4.73), it is found that for:C = r —th(U +r), dr= (_2rsh(U+r)(_ch3U+ ))=ch2(U+r) p drdU ch3(U+r) ) 2sh(U+r)and then eqn (4.71) becomes (under the condition that Vb < VPk):N,.,1 (VbZ — VBE) Nrat (Vbx — VBE)F— 4itqJm lmzl kT VPk1d dC 1 + kT - V1h3e,r——e_V_eEqn (4.75) has had the limits of integration from eqn (4.71) temporarily removed for clarity. Atthis point no approximations have been introduced into the solution. At issue with the solution ofeqn (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 ofeqn (4.71), has the form of a Gaussian. Indeed,‘j is extremely symmetric and suggests that aTaylor series expansion about U (i.e., r = 0) for C(r) is a potentially good approximation. Performing a Taylor expansion of C(r) about r = 0 up to second order produces:sh(U) dC sh(U)C=—r2 —th(U)= ——2rch (Ui) dr ch (Ui)Finally, substituting the above approximate equation for C (r) back into the integral within eqnJuly 12, 1995 76(4.75) yields:Nrat (T7bz — VBE) Nrat (Vbx — VBE) th (Un) Nrat (Vb — VBE) sh (Un) r2fdr-e = —2sh(U)eu fdreUV ch3(U)j drr ch3(U) JThe above equation is simply the integration of a Gaussian, and results in an error-function solution. With the limits of integration from eqn (4.71) reintroduced, the solution of the above is:Nrat (Vb — VBE) th (Un) (ach (____________— u1sh(U)eri + erfJmax (V,, .0)) P), (4.76)ch (Un) awhere— /ch3(UpUpkT—VP, sh(U)Eqn (4.76) solves for the integral in eqn (4.75) and produces the analytic model for Ff,.thatis sought after. The complexity of eqn (4.76) stems mainly from the evaluation of the boundaryconditions. Fig. 4.9 shows and the boundaries of integration. As long as the majority of iscontained within the two boundaries, then the error functions will both approach 1, and eqn (4.76)can be approximated by:sh(U) Nra: (Vb1-VBE)th(U)2q ae . (4.77)ch3(U)Eqn (4.77) is the simplified model for the integral in eqn (4.75), but it still contains most of the important features regarding CBS transport. Thus, the final and approximate models for Ff5 arefound by substituting either eqn (4.76) or eqn (4.77) respectively, into the integral of eqn (4.75) toobtain (under the condition that Vb < Vpk) (see also eqn (4.92) for low temperature considerations):Nrat(Vb VBE) th(U)Ff5 = FjsoPPPte u [erf()+ (4.78)(ach(Jmax (vi, 0) — U Nra: (i VBE)+erf +F Vea fsOtror approximately asJuly 12, 1995 77Nrat (Vb1 — VBE) th(U) Nrai (Vbx — V8)btv sh(U )U V- 11 -________z’ p p Pt t p j’ TI I‘fs1fsO I mIfsOvtqch(U,,)where214iq im m kT—F— ‘v y,i z,i kTh3Finally, under the condition where Vj, Vk:VbI—V8—AE/qFf = FfSQVe V, (479)4.4.2 Analytic Model for the Enhancement Flux FfeWith the analytic model for Ff presented in eqns (4.78)-(4.79), attention is focussed uponthe solution of the enhancement term Fje. Examination of eqn (4.65) shows that the integrationover Ujj has a lower limit that includes y(G). Thus, unlike the solution for Fj5, the G integrationto calculate F!e cannot be performed independently. Further, eqns (4.42) and (4.37) show that yhas a complex dependence upon€that would most likely cause the final integration over 0, forthe calculation of Ffe, to become analytically intractable. To alleviate this complexity an approximation is made. So far, all of the models presented use a general mass tensor that is diagonal withrespect to the direction of transport. This general mass tensor formulation is maintained, but themass barrier will be confined to the study of an isotropic change in the transverse direction of themass 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:m 1m 1 (a m 1cos2O+a m 1sin20” iy(0) = 1 — ‘ z m z m “ = 1 — . (4.80)amy, lmz, 1 m,1cos20+ m 1sin20 ,) amEqn (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 followexactly the one for the calculation of Fj5 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:V max (Vb, 0) U y— i4tqim m kT —— ‘v y, i z, i kT 7K1,kT Y— h3U x VYCBSk xi0Examination of the above equation shows that T inside of the integral can be redefined with:July 12, 1995 78m2 m2Teff = T = Ty’ = T(1 —am) where am = = (4.81)m1 mz,lTeff is then the effective temperature of the flux density. Under the enhancement case y> 0 andthus am> 1, leading to Tejf< 0. With eqn (4.81) substituted into the equation preceding it, then:Vb max(Vb,O) U4iq.jm 1m 1kT — --i--Ffe= 3z,e”eTfdUXWCBS(UX)e eff (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 negativetemperature Teff is to cause an increase to the electron distribution as one proceeds to higher energies. 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 < 0must be accounted for. Population inversion, when combined with the fact that WCBS also increases with increased energy, means that the peak flux density will no longer occur at an energyof U,’ given in eqn (4.74), but will instead occur at the upper energy boundary allowed into theproblem.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 longermakes sense to use an expansion that is centred about U, as population inversion moves thepeak flux density to an energy of max(Vb, 0). Eqn (4.82) is solved by returning to eqn (4.71) andintroducing lffinto all relevant equations to yield (under the condition that Vb <!‘ Vb max(V,,O) V/q rFje= 4qJm1m1kTee_ Vpkf dU-th(Up+r))(4.83)whereIN kTp,eff— 2V1jj4em t,eff qThe primes on the energies still denote normalisation with respect to Vpk. Unlike the developmentthat took eqn (4.71) into (4.75), eqn (4.83) is expanded about V. Furthermore, the condition ofpopulation inversion causes the integrand in eqn (4.83) to become basically exponential in termsof U. Remembering from eqn (4.71) that U = ch2(U eff + r), and dl — U = th(Upeff + r),July 12, 1995 79then a Taylor expansion about max(Vb, 0) for the exponent inside of the integral of eqn (4.83), upto and including linear terms, is:V/q________V/qU V 2 _th(Up,eff+T)U (rU —Jl—max(V,0)), (4.84)p,eff t,effch (U+r) ) p,eff t,effwhererb= 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 theintegral 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 integralfrom 1 up to V (over which WCBS = 1). Finally for Vb <Vb — VPk,J1—max(VI,O) (‘ VPkrbmax(Vb,O)F — FUp,jcVt,eff qUV qUpffV8f— 1fe e e 1ebwhile for Vb5 --__i_F —F V ykT1 qV48 4fe fsO t,effe eAs a final check on the validity of the model for Fje (i.e., eqns (4.85)-(4.86)), observation ofeqn (4.65) and the region of integration in Fig. 4.6 shows that as ‘y — 0, Fj — 0. This occurs because when y =0 there is no mass barrier and F = Ff. Obviously, when Vb 0, the upper limit ofintegration in eqn (4.82) is zero and the integral itself vanishes. For the case where Vb > 0, examination of eqn (4.82) shows that the terms containing y are:U(1—y) —Vb‘kTeThe enhancement case is being considered, where 0 < y < 1. Furthermore, since the limits of integration have it that 0< U < Vb, then U(1— y) — Vb <—YUX <0. Therefore, the terms that makeupthe exponent of the above equation are always negative. Then, as 7 approaches 0 from the positiveside, the exponent goes to negative infinity and eqn (4.82) goes to zero. The exact same development 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 804.4.3 Analytic Model for the Reflection Flux Ff,.With the analytic model for Fje presented in eqns (4.85)-(4.86), attention is finally focussedupon the solution of the reflection term 1’Jr Eqn (4.66) is the general model for Ff,., and it alsocontains y within the Uj as well as the U,1 integrations. Therefore, as was the case with the solution of Fjce, the 9 integration to calculate Ffr cannot be performed independently. To simplifythis problem, as was done with Ffe, the mass barrier is assumed to consist of an isotropic changein the transverse direction of the mass tensor. This allows eqn (4.80) to be used in the solution of1jr In fact, using the same basic steps from eqns (4.80) to (4.82) will also solve for Fjr The onlychange that occurs is to the upper limit of integration over U,1,which will approach infinity as E—> 00 (this is because y < 0 for the reflection case). The final result is:I Ill V 004irq im 1m 1kT — --- . ---— v ‘ , ‘ kT ‘1’-’ I 4T1 WI (TI ‘ ffL fr 3 I ‘‘x “CBSk”x)”h Jmax (Vb, 0)Eqn (4.87) is identical to eqn (4.82) save the limits of integration. This fact occurs becauseof the symmetry of the problem being considered. As was stated before, Ff is the standard fluxthat would flow if there was no mass barrier at all. Fje on the other hand, is the flux of carriersthat would normally enter the base within the forbidden bandgap (i.e., U2 < Vb), but due to themass barrier, is raised up into the conduction band to contribute to the total flux; thus the integration is carried out from 0 < U,1 <Vb. Finally, Fj,. is the flux of carriers that would normally enterthe base within the conduction band (i.e., U,,2> Vb), but due to the mass barrier, is lowered downinto the forbidden bandgap to become reflected and take away from the total flux; thus the integration is carried out from Vb < U,1 <00. The form of the integral for and Fr must be thesame since the Jacobian transforms and the boundary conditions given in eqns (4.46)-(4.47) donot depend on the sign ofEven though the models for Ffe and Ff,. are ostensibly identical, their analytic solutions arenot. This occurs because in Ff,, Teffis positive (the same as for Ff5). In fact, eqn (4.87) is identicalto eqn (4.67) for Fj5, except the temperature of the flux density is no longer T but T (there isalso a constant multiplier of exp(-VbI’1c7) that occurs in eqn (4.87) that is not present in the modelfor Ff5). Examination of eqn (4.81) shows that when y < 0 (as it is for the reflection case), then Teffhas a range of 0< Teff< 7’; where 7e—*0 as y —*0, and Teff —> T as y — —oo. Therefore, the fluxJuly 12, 1995 81density in the reflection case is characterised by a temperature that is always less than the latticetemperature 7; but unlike the enhancement case it remains positive under all conditions. Thus, thereflection case is identical to, and can be calculated by, the standard case but with a flux densitycharacterised 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 17b <Nrat (Vbx — VBE) th(U8ff)F = F/pk5h(Up,eff)Up,effVt,eff Vteff Up,eff reff (UPeff + (4.88)q qch3(Upff) L r,eff)(ach(Jmax(V’, 0)— UffJ Nrat(VbiVBE)Ll;’ J t,effa frO”t,eff’r,effwhereVb I 3--j-. Ich (U ,)UF — F “ d — I p,e, t,ejjfrO — fsO r,eff— q (Vk/q) Sh(Upeff)Finally, when Vb Vpk, then:Vbx— V—AE/qFf r = FfrOVteffeVt,eff (4.89)Eqns (4.88)-(4.89) present the analytic model for Fj,., which is basically the same as themodel for Ff,but with the flux density characterised by The only potential issue (as concernserror due to approximation) with eqn (4.88) (and eqn (4.78) as well) occurs at very lowtemperatures 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 downto 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 occurwith the model because the model is based upon a Taylor expansion about U. This potentialerror at low temperature is exacerbated in the calculation of Ff,. because Teff is even less than T(for an Al03Ga07As to GaAs flux, m,i = O.O92me and mX,2 = O.O67me, so that T= 81K when TJuly 12, 1995 82300K). 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 Vbhas already been performed. Eqn (4.84) is the expansion about Vb up to and including linearterms. If the second order terms are included, then using the transform preceding eqn (4.75) gives:2r—th(Upeff+r)rbU—Jl—V—(U;—V;) 1 ,, (4.90)ch (U+r) 4Vb Jl—Vbwhere the condition that V > 0 is assured as this expansion is being used to solve the case whereV> U. Substituting eqn (4.90) into eqn (4.83), but using the limits of integration set out ineqn (4.87), produces, after performing the integral over the Gaussian [81,#3.322.2]:V/q , ,u v“-.J1-Vb) r rb 1Ffr = Ffp,eff t,ejje “° t,eff e r,eff — erf(— ) (4.91)r,eff L r,effjwhere Ff,.0 is defined in eqn (4.88), and (Treff is altered from its definition in eqn (4.88) to:I UpeffVteffareff— 4 (V/q) V Ji —Eqn (4.91) solves for Ff,. when U0 <V < 1, and is used instead of eqn (4.88). Eqn (4.88) isused only when V < U,’, (which is generally the case except under very low temperatures, or ifthe heterojunction is such that AE is quite small).In a similar fashion, eqn (4.78) for the calculation of Fj5 is further restricted to V < U.Then, when U <V < 1 occurs, Fj is given by (after a simple extension from eqn (4.91)):V/q , , rU vTh J (rbVb -.J1-Vb) rbFf5 = Ffo e e r [i — erf(_)] (4.92)where, in this case only:I uva = I V t______and rb = ach 1 1 — Ur q(Vk/q)VJ1—V J) °Eqn (4.92), in concert with eqns (4.78)-(4.79) form the model for Ff.with an unrestricted placement 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 extensions, eqns (4.85)-(4.86) form the model for FfeJtdyl2,1995 83Before 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 thatthe numerical code that generates erf(x) has the proper asymptotic form or else the result will beincorrectly forced to zero (i.e., 1 — erf(x) —> e_X/ (xJE) ). Analytically, as areff (or a) —> 0, thenby simply using the asymptotic form for 1 — erf(x), eqn (4.91) is seen to become eqn (4.85) forFje, where the “—1” term in eqn (4.85) is dropped; this result is expected because under these conditions the linear Taylor expansion is sufficient.4.5 The Effect of Emitter-Base SCR Control on IThe previous section presented the analytic models for the calculation of the forward fluxFfl and included the mass boundary effects. The only assumption made in the development of themodels of the previous section was that the mass boundary be isotropic in terms of the transversedirected 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 tensor is required, then the models of the previous section can be used, but the final G integrationmust 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 HBTwhere the CBS is responsible for current-limited-flow. This will provide insight into the modelsand 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 beequal to J Let the simulated device be governed by the CBS in Section 1 (where j = FCBS),the neutral base in Section 2 (‘2, 2 Fjbase)’ and the collector in Section 3 (J = Ffcoil). Aslong as the demanded currents in the base and collector greatly exceed what the CBS can provide(i.e., Ffbase and Ffcoil>> FfcBs) then if no significant recombination occurs throughout the baseand collector sections (i.e., Y2 = 1), eqn (2.6) produces:Ff if y=0cTjcBs = Ffs+Ffe if y > 0 . (4.93)FfsFfrf ‘<Owhere the multiplication of the electron flux by “-1” is not required due to the definition ofJuly 12, 1995 84It is very interesting to see that when the CBS is responsible for current-limited-flow, I will peerdirectly into the quantum mechanical nature of the CBS. Thus, the quantum mechanical effect oftunneling, including the effects of the mass barrier at the heterojunction itself, will be observableby simply measuring I.The simulated HBT will be based essentially on the following A1GaAs/GaAs HBT at 300K:emitter is Al03Ga07As; base is GaAs; emitter doping ND 5x1017cm3;base doping NA1x109cm3;emitter permittivity C1 is 1 l.98o; base permittivity £2 isl2.9c-j; AE is 0.24eV; n,252.25x106cm m1 is 0.092m;m2 is 0.067m;-* Nrat is 0.956; Vbj is l.671V; x(VBE=O) is649A; U is 0.488; U is 0.795; yis -0.373; lff is 81.5K; Up,effi5 1.80; U1is 0.104; Vb is>0 when VBE < 1.43 1 V. Two other plausible devices are also considered for the reflection case; inorder to make the comparisons direct, all parameters are identically maintained except m2 is eitherlowered to of m1 (= 0.046m), or to of m1 (= 0.023m). The enhancement case typically doesnot occur for electrons, but most certainly occurs for holes. Using the reciprocal relations to thereflection case gives m2: 0.126m;0.184m;0.368mj. Changes to the effective density of statesdue to the changing m2 are not reflected into Vbj nor Therefore, the simulations that areabout to be presented are contrived in terms of a physical analogue but as such allow for the mostdirect 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 models of the previous section for the three m2 cases of: 0.067m;0.046m;0.023m At T equal to300K as well as 200K, decreasing m2 (and thus making y a larger negative number) results in an increase to F!r Physically, as m2 decreases, the mass barrier will demand a larger transfer of energyfrom U,2 into U in order to conserve transverse momentum (see Fig. 4.5); thus, a larger numberof particles will be reflected as they will not possess a sufficient amount of U,2 energy to satisfy themomentum 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 interplaybetween the base potential Vb and the mass barrier. As was just stated, the mass barrier moves energy from U,,2 into Uj,2.The point at which reflection occurs is when U,2 < Vb. Obviously, as Vb ismade smaller, more energy can be removed from U,,2without encountering reflection. Since Vb decreases as VBE increases then F! r must decrease, relative to Ff, as VBE increases. The sudden decrease in Ff,. for VBE> 1 .4V corresponds to the point at which Vb goes below the reference potenJuly 12, 1995 85106io T=300K102 Js(a) 100 r; m2 = 0.023Z 10Fj;m2=0.046-6Z 10 Fj;m2=0.067108 . • • • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V),_ 10 • • •T=2OOKr:046Base-Emitter Voltage VBE (V)Fig. 4.10. Standard Flux Fj. and Reflection Flux Fj,. for an HBT with the parameters givennear the start of this section. The only parameter being varied is the base side effective mass in2.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 86tial energy E in the neutral emitter (Vb is <0 when VBE is> 1.431 V). Since the neutral emittergenerates the flux that impinges upon the CBS, very few particles will have U,2 reduced belowzero by the mass barrier (unless the mass barrier is very strong due to a smallm21m). Thus, onceVb decreases below zero, reflection will taper off quickly as there are essentially no more particlesto 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 momentum conservation is guaranteed when p is zero (see eqn (4.39)). This means that only particleswhere -‘yUj is comparable to, or larger than, U1 will be affected by the mass barrier. Now, totunnel through the potential barrier requires that the particle obtain a sufficient U1 in order topass through the CBS (on average an energy of UVpk is required). Any energy gained by U,1will do nothing to improve the particle’s chances of passing through the barrier; in fact it wiLl onlyserve to lower the particle’s availability because the occupancy decreases exponentially with anyincrease in total energy. Thus, the CBS preferentially picks out, from the random ensemble of particles impinging upon the barrier, those particles that possess a sufficiently high U1 to passthrough the barrier, while being blind to the amount of U1 contained by each particle. SinceU,’, decreases rapidly along with a decrease in 7-YU±,i will become larger relative to U,1 as Tdecreases, 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 quitereadily. In Fig. 4.11(a) the temperature is held constant and all three mass cases are presented. Thisclearly shows that as the mass barrier is strengthened by reducing m2, the relative importance ofFfr rapidly increases. Perhaps even more importantly, the effect that has on the total flux F isbias dependent. This shows that the mass barrier cannot be described by a simple multiplicativeconstant as has been suggested in the literature [51,79,82]. Another important feature that is clearly 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 leftthat can reflect from the potential barrier in the base. However, as the mass barrier is significantlystrengthened to the point where m1 is four times larger than m2, the mass barrier is able to reflectparticles from Vb even when Vj, <0. These results clearly indicate that the position of Vj, is veryJuly 12, 1995 8708 T=300K0.6(a)0.40.% 10 ii 12 13 141516Base-Emitter Voltage VBE (V)0.250.200.15 T=IOOK(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 sameparameters as Fig. 4.10. The lines are obtained from the analytic models, while the solid dots arefrom numerical calculation. (a) results for T= 300K. (b) results form2= 0.067. Note: usable currents (i.e.,> l08Acm2)begin at VBE> 1.OV for T= 200K, and VBE> 1.2V for T= lOOK.July 12, 1995 880.6 •050.4 T=100K,T=200K(c) 0.3:O.%IHIO.ll.lI2.l3l4•5N1.6Base-Emitter Voltage VBE (V)Fig. 4.11. Continuation of Fig. 4.11 from the previous page. (c) results for m2 = 0.046.106. • • • • • •T=300K, i0Fj5in2F; m2 = 0.0230= 1010Ff; m2 = 0.046J,Fr; m2 = 0.067;l)i0_6 • • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5Base-Emitter Voltage VBE (V)Fig. 4.12. Standard Flux Fj and Reflection Flux Ff,. for an HBT with the same parameters asFig. 4.10, but with AE reduced from 0.24eV down to 0.12eV. Note how reducing AE increasesthe relative importance of the reflecting potential barrier Vb by lowering Vbj (see Fig. 4.10(a)).July 12, 1995 89important to the transport flux through the CBS. The conclusion is that during the design of thedevice it is beneficial to have a large AE so that Vj, is lowered, and the mass boundary will have areduced effect. Finally, examination of Fig. 4.11(b) and (c) clearly demonstrates that lowering thetemperature increases the relative importance of Ff,. in all cases. Obviously, the combination oflower temperatures and a stronger mass barriers produces the largest reflections.The case of an Al03Ga07As/GaAs HBT produces the rather fortuitous result that Vj, is below zero right around the bias at which the device would routinely be operated. There are othermaterial systems (like SiGe) and devices (HBTs with a smaller emitter Al content) where this isnot the case. In these systems iXE is smaller so that Vb stands as a larger reflector. Fig. 4.12 showswhat the effect of reducing AE from 0.24eV down to 0.12eV has on the transport flux. Underthese conditions V1, remains unchanged but Vbj is reduced by 0.12V to 1.551 V Therefore, relatively 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 analyticmodels 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 compromise the accuracy of the final results. This means that it is reasonable to look at the functionaldependencies within these analytic models in order to obtain a deeper insight into the mechanismsby which transport occurs through the CBS. In the end, these analytic models will facilitate a fullmodel 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 thestart of this section, three cases will be considered for the enhancement case. In order to makecomparisons with the reflection case simple, only m2 is varied and it is chosen to be the reciprocalto the three reflection cases; namely 0.126m, 0.184m, and 0.368m Fig. 4.13 is basically thesame as Fig. 4.10 (except that 7 is now positive under the case of enhancement), and plots Ffs, aswell as Ffe. The same basic trends are observed for the enhancement case as were observed in thereflection case. In Fig. 4.13 at T equal to 300K as well as 200K, increasing m2 (and thus increasing y) results in an increase to Ffe Physically, as m2 increases, the mass barrier will transfer moreenergy from Uj.i into U,2 in order to conserve pj (see Fig. 4.5); thus, a larger number of particles will be moved from out of the base bandgap and into E to contribute to Fj. Furthermore, asJuly 12, 1995 90II.)T=300KFfs1OE 100.(a) Ffe; m2 = 0.3681Oio-Ffe;m2=O.l8410-6Ffe;m2O.l2610-8a • a • a • a • a • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)10 • • • • •T=200K:, io410E 1of1_\U)10-2 Fje; m2 = 0.368____io4101e;m2=0.126io- a • a • a • a • a1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 4.13. Standard Flux Fj, and the Enhancement Flux Fice for an HBT with the parametersgiven near the start of this section. The only parameter being varied is the base side effective massm2. The lines are obtained from the analytic models of eqns (4.78) (4.79) (4.92) for F1 and eqns(4.85) (4.86) for Ffe, while the solid dots are from the numerical calculation of eqn (4.67) for Fj5and eqn (4.82) for Fje. (a) results for T = 300K. (b) results for T = 200K.July 12, 1995 91VBE is increased, Fje begins to decrease and then decrease abruptly. The physical cause for this isexactly the same as for the reflection case. As VBE increases Vb decreases so that fewer particlesneed to be helped over the barrier and Ffe decreases. In the event that Vb <0, every particle thatmakes it through the CBS must enter the base, since the enhancing mass barrier can only raise U,2and 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 Fjerelative 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 smaller U,1.As such, an increased number of particles will be available below Vb. With more particlesexisting below Vb, the mass barrier may effect a larger transfer of particles from below to abovethe base barrier, and thus increase Fje as T is reduced.The most important difference to note between the enhancement and the reflection case isthat a smaller increase occurs in Fje when compared to Fr,. for a similar increase in the strength ofthe mass barrier (which is affected by increasing or decreasing m2 respectively). The reason forthis arises purely because of the nature of enhancement and reflection. For the reflection case, asm2 becomes arbitrarily small ‘y — —oc• With y —* —cc, every particle that hits the mass barrier willalso have its U,2 —> —cc, leading to a total reflection of all carriers (examination of eqn (4.81)shows that as 7—> —cc then Teff> Tso that Ffr —* F and F —* 0). Thus, it is possible for the reflecting mass barrier to become so effective that the transport flux is reduced to zero. For the enhancement case, there is a fixed ensemble of carriers launched from the neutral emitter towardsthe CBS that attempts to enter into the base. Once the CBS has removed its portion of the ensemble, the enhancing mass barrier is left to increase U,2 by removing energy from U±i. At the limiting strength of the enhancing mass barrier (i.e., y = 1), the entire amount of Uj,i is transferredinto U,2 (see eqn (4.39)). Since the particles will have a one kT spread of energy in Uji, startingfrom Uj = 0, the enhancing barrier will rapidly reach a limit by which it can no longer increaseFje. Thus, the enhancing barrier will have a smaller effect on F than the reflecting barrier, and assuch will not experience the same increase in Ffe due to an increase in m2 that Ff,. would realisefor a similar decrease in m2.The differences just described between the reflecting and the enhancing case in the previousparagraph can also be understood from a graphical analysis of Figs. 4.6 and 4.7. For the enhance-July 12, 1995 92ment case, there is a limit of y = 1. Looking at Fig. 4.6 for the integration in R1, then obviously inthe limit when y = 1, R will take on a fixed, non-vanishing shape with no possibility of an increase due to a change in the mass barrier. This leads to a maximum value for Ffe and thus F aswell. For the reflection case of Fig. 4.7, there is a limit of y —> —0o When y —* —00, the region ofintegration R1 will be reduced to zero, and likewise, so will F This clearly shows that reflectioncan produce a far larger effect upon F than enhancement can.Fig. 4.14 clearly demonstrates the effect of m2, Vb and T upon Fje. Concentrating on Fig.4.14(a), there is clearly an increase in Ffe as the strength of the mass barrier increases (i.e., as m2increases). However, looking back to Fig. 4.11(a) confirms that the enhancing case does indeedproduce less of an effect than the reflecting case. Examination of Fig. 4.14(a) and (b) also showsthat once Vj, is reduced below zero for VBE> 1.43V (Vb1 changes with 7), Ffe = 0 as there is nolonger a base barrier to surmount. Finally, Fig. 4.14(b) shows that reducing T increases F1e inmuch the same manner as for the reflecting case.Reexamination of Figs. 4.13 and 4.14 show an excellent agreement between the analyticmodels 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 compromise the accuracy of the final answer.It is important to keep in mind that under the condition where the CBS is responsible forcurrent-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 modelling 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 ofcharge through the entire EB SCR, and not just the CBS. Eqns (4.50)-(4.53) take into account anyquantum mechanical effects, including transport via standard Drift-Diffusion (DD), without the needto appeal to high-energy phenomenological mobility models. By treating transport as a system ofcollision-less particles that originate from a thermal distribution, the problem of carrier heating andcooling, 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 93o.io T=300K\ m2=0.368.0.06(a) E m2=0.1840.04 •.0.021fl2=0.l60.00 •‘ • I. I .1. -0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)0.30•“.- LZi2=0184_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 sameparameters as Fig. 4.13. The lines are obtained from the analytic models, while the solid dots arefrom numerical calculation. (a) results for T = 300K. (b) results for m2 = 0.184. Note: usable currents (i.e.,> 108Acm2)begin at VBE> 1.OV for T= 200K, and VBE> 1.2V for T= lOOK.July 12, 1995 944.6 Deviations from Maxwellian Forms and Non-Ballistic EffectsThis section will use the models of the previous sections in order to gain an understanding ofthe electron distribution that is injected into the neutral base from the emitter. With the neutral basewidth WB being pressed below i000A, the truly ballistic device is being approached. In the regimewhere the electron in transit through the neutral base suffers only a few collisions, then one cannotappeal to classical solutions that depend upon a thermalised distribution (i.e., drift-diffusion analysis), nor can one avoid the effect of coffisions altogether and treat the ensemble ballisticallythroughout. In this in-between region, where collisions are important but do not dominate thetransport 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 particleensemble distribution entering the neutral base. As there are less collisions within the base it becomes important to obtain the correct initial ensemble distribution. This section will provide amethod 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 baseof abrupt HBTs is not Maxwellian [14,39-41]. A Maxwellian distribution is characterised by aBoltzmann distribution in energy, with a parabolic relationship between momentum (or k) and energy. Therefore, the Maxwellian distribution appears as a Gaussian distribution in k-space centredat k = 0 (see Fig. 4.15(a)). In the thennionic analysis of the EB heterojunction (i.e., no tunneling isconsidered 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 towards the base, the Maxwellian distribution is pulled apart so that only the right-going half of theensemble 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 (because the particles are only moving in the positive x-direction). Once the hemi-Maxwellian ensemble has entered the neutral base, and if there have been no collisions from x = 0 to x = x, thedistribution will no longer peak at k = 0 with an energy of 0, but will be shifted towards larger kwith an increased energy of IXE — V, relative to E at x = x. This shifted hemi-Maxwellian istermed “hot” because it appears to look like a distribution that is characterised by a temperaturewhich is higher that the lattice temperature TJuly 12, 1995 95izFig. 4.15. Ensemble particle distributions assuming a purely thermaliseci thermionic injectionfrom the peak of the CBS in Fig. 4.2. (a) the initial Maxweffian distribution at x = 0. (b) the hemiMaxwellian distribution that is injected towards the neutral base (positive x-direction). k is normalised to the length of the GaAs reciprocal lattice vector using an effective mass of 0.067.1.00.70.5. (a)0.20.0-0.04-0.04-0.02. -0.020.000.00Normalised k± 0,020.02 Normalised k0.04 0.04I(b)z1.00.70.50.250.0-0.04/-0.02-0.040.00 ‘>:p.; — - ---- - -0.02I. 0.000.02 0.02Normalised k±0.04 0.04Normalised kJuly 12, 1995 96From the results of Fig. 4.15, and the arguments of the previous paragraph, the distributionentering the neutral base at x = x, is clearly not Maxwellian. However, in terms of being able toanalyse the neutral base using drift-diffusion analysis, solutions based upon a hemi-Maxwelliandistribution will differ from a full Maxwellian distribution by only a multiplicative constant. Theissue of the hemi-Maxwellian being hot, however, will require that an energy-balancing schemealso be included by using the second moments of the BTE to arrive at hydro-dynamic drift-diffusion analysis [16,17]. Many researchers who have studied transport within the EB SCR, or theneutral base, have relied on the assumption that the worst-case deviation from a Maxwellianwould be a shifted or hot hemi-Maxwellian. This assumption is shown to be false when a structurelike the CBS of Fig. 4.2 is present within the EB SCR. In fact, the distribution function enteringthe neutral base is appreciably distorted from either a Maxwellian, hemi-Maxwellian, or hothemi-Maxwellian. Furthermore, the distortion to the ensemble distribution has a considerable biasdependence.Setting aside for the moment the issue of the mass barrier, which serves to distort the ensemble distribution even further, tunneling through the CBS results in a profound change in theshape of the ensemble distribution. As was discussed in the explanation of Fig. 4.10, tunnelingthrough the CBS preferentially picks out from the random Maxwellian ensemble of particles impinging upon the barrier, those particles that possess a sufficiently high U1 to pass through thebarrier, while being blind to the amount of U±,i contained by each particle. Clearly, this will tendto focus the ensemble at x = 0 towards higher U,1 and destroy the circular symmetry that existsbetween k and k± shown in Fig. 4.15(b) for the hemi-Maxwellian distribution. Finally, in movingfrom x = 0 to x = x, a number of particles will be reflected by the neutral base potential Vb whichwill clip off the distribution (much like a hemi-Maxwellian is cut from a Maxwellian) and resultin a potentially hot ensemble entering the neutral base.Fig. 4.9 shows the ensemble distribution after an integration has occurred along the transverse direction. The result, which was formally proven in Section 4.4, is essentially a Gaussiandistribution versus U,1. Since momentum p and wave vector k vary as the square root of U,1, theensemble distribution plotted in Fig. 4.9 wifi give a very distorted, non-Gaussian (i.e., non-Maxwellian) shape when plotted against ki. Furthermore, Vb cuts the distribution off for particleswhere U,2 (= U,1 — Vb because there is no mass barrier) < Vb. This results in a form that is indicative of, but distinctly different from, a hot hemi-Maxwellian (see Fig. 4.16).July 12, 1995 971.00.8—0.40.2z0.00.00 0.02 0.04 0.06 0.08Normalised GaAs Wave Vector k,2 (1.11x108cm1)Fig. 4.16. Ensemble distribution versus wave vector k2 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-offand not displayed in order to see the effect of the reflecting base potential. Also, Fig. 4.9 is a plotof the ensemble approaching the CBS from x = -x. Finally, k,2 is normalised to the length of thereciprocal lattice vector (i.e., 2ir/a where a is the lattice constant).Fig. 4.16 shows the distortion to the ensemble distribution along k,2.At low bias, where Vbis 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 toenergy results in a very flat-topped and non-Gaussian form with respect to k. As the bias is increased, 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 shiftedMaxwellian. Fig. 4.16 clearly shows the non-Maxwellian nature of the ensemble distribution entering the neutral base. However, it does not show the distortion that occurs along k± (k± = k±,i =k±,2 because of momentum conservation in eqn (4.36)). In order to see the full ensemble distribution entering the neutral base (= WCBS(Ux,1)fl(Ux,1 + Uj,1)) a three dimensional plot versus k,2and k,2, is displayed in Fig. 4.17. Observation of Fig. 4.17 clearly shows the non-Maxwellian ornon-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 assumeJuly 12, 1995 980.75—0.50--0.04,zt±-0.02 ç\0•20.000.040.02 ‘>)ç_ 0.06 Normalised k,20.04 0.081.00-(a).0.75—0.50—ri0.25-0.00-/-0.04-0.02 ‘7Normalised k±,z(b)0.00I0.020.04 0.100.06Normalisedk21.00-0.25—rj)E0zNormalisedII0.00Fig. 4.17. Ensemble electron distribution entering the neutral base versus k (T=300K). The particle density is normalised to the peak of the distribution, and k is normalised to the length of theGaAs 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 99that the shape of the ensemble distribution is invariant under a change in bias. These results clearly indicate that the assumption of a hot Maxwellian or hemi-Maxwellian entering the neutral basein an abrupt HBT is erroneous.Figs. 4.16 and 4.17 have used the full HBT parameters that Section 4.5 has been basedupon, except that the mass boundary has been neglected by setting m2 m1. As was alluded toearlier in this section, the mass boundary will have the effect of further distorting the ensembledistribution. Fig. 4.18 plots the electron ensemble distribution entering the base under the condition where m2 = 0.023 (i.e., the reflecting case) to clearly observe the mass barrier effects. The effect of the reflecting mass barrier is to simultaneously pull the distribution towards lower k,2 andhigher k±2. Looking at Fig. 4.18(a) and comparing to Fig. 4.17(a) clearly shows the extension ink±,2;while careful observation of the constant k,2 line from the peak shows that the distribution isindeed being pulled and distorted towards lower k,2. Comparison of Figs. 4.18(b) and 4.17(b)clearly demonstrates the distortion due to the reflecting mass barrier upon the ensemble distribution. It is important to realise that, although the volume of the distribution is larger in Fig. 4.18then 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 barrier where m2 = 0.368. The enhancing mass barrier distorts the distribution in exactly the oppositefashion when compared to the reflecting mass barrier. The effect of the enhancing mass barrier isto simultaneously pull the distribution towards higher k,2 and lower k1,2. Comparison of Fig.4.19(a) with Fig. 4.17(a) demonstrates that the distribution is certainly being pulled towards lowerk±,2; so much so that the distribution is starting to look Maxwellian. Closer examination of thecontour lines in Fig. 4.19(a) shows the distortion that results from the extension in k,2, which is aclear deviation from a Maxwellian form. Further examination of Fig. 4.19(b) in comparison toFig. 4.17(b) exemplifies the distortion to the ensemble due to the enhancing mass barrier. As similarly occurred with the reflecting case, the volume in Fig. 4.19 appears smaller than the volumein 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 haveupon the electron ensemble distribution entering the neutral base. The one clear conclusion fromJuly 12, 1995 1001.0: 0.750.50.20.0-0.04-0.02- 0.020.000.04Normalised k±,2 0.02 0.060.04 0.08I-0.04-0.02(a)0.75O.OOioNormalised k,2S0.00HI(b)N0.00Normalised k,2Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier where m2 = 0.023and m1 = 0.092. (a) VBE = 1.4V. The plot has been rotated 450 relative to Fig. 4.17(a) to clearlydisplay the distortion in the kx2 direction. (b) VBE = 0.9V. Again notice the extreme distortioncompared to Fig. 4.17(b) for k2 less than the peak.July 12, 1995 101.(a)I0.02.(b)\E0z0.00Normalised k,2Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier where m2 =0.368 and in1 = 0.092 (the reciprocal to Fig. 4.18). (a) VBE = l.4V. The distribution looks Maxwellian but comparing to Fig. 4.17(a) shows it to be distorted towards larger kx,2. (b) VBE = 0.9V.Clearly k2 has been extended and has been squashed relative to Fig. 4.17(b).Normalised k,20.04 0.10Normalised0.04 0.08July 12, 1995 102the analysis of this section is that one cannot assume that the ensemble distribution entering thebase has any resemblance to a Maxwellian or hemi-Maxwellian in either a normal or hot condition. Also, the change in the shape of the distribution over bias cannot be accounted for in a simple fashion (such as a constant multiplier). Further, it is the effect of tunneling that contributesmost to the distortion of the ensemble distribution, with the mass barrier playing an important butgenerally 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 analysis of abrupt HBTs. In any event, the analytic models presented in this chapter can be used to construct the correct electron ensemble distribution entering the neutral base. This correct neutralbase ensemble distribution can then be used as a boundary condition in a subsequent BTE solution of the transport through the neutral base.The models presented in this chapter have assumed the condition of ballistic motionthroughout the EB SCR. This assumption is relatively solid given that the EB SCR is generallyquite narrow and as such is much smaller than the mean free path of the particle. Before going onto talk about the effects of non-ballistic motion throughout the EB SCR, it is important to pausefor a moment to discuss the lower boundary of Vb used to calculate the flux through the CBS. Reexamination of Figs. 4.1 and 4.2 show that Ffand Fr are calculated by assuming that a hemi-Maxwellian distribution is launched into the EB SCR from both x = -x, and xi,, respectively. The finalflux exiting the EB SCR is then determined by considering how tunneling through the CBS, aswell as reflection by Vb and distortion due to the mass barrier, alters the course of the forward andreverse 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 characterisedby the lattice temperature T This is a reasonable assumption given that x = -x,j and are the depletion edges of the EB SCR, and as such are outside of where non-equilibrium effects would begin to occur. It is for this reason that the flux is considered to be injected from x = -x, andleading to the potential boundary of 17b (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 injectionto the left is from x = 0, not from x = x. The point x = 0 is inside of the EB SCR and coincideswith the peak electric field. As such, the ensemble distribution at x =0 is expected to be at its largest departure from equilibrium when compared to any other point within the EB SCR. Further-July 12, 1995 103more, to consider the point x = 0 as the boundary condition, one would have to imagine that theelectron could ballistically tunnel a few hundred angstroms through the CBS and then suddenlythermalise at x = 0, where it could then be carried into the neutral base by diffusion. Clearly, it isnot reasonable to assume that x = 0 is the source of a thermalised Maxwellian distribution.By adopting Grinberg’s proposals within [51], the lower limit of integration for the calculation of F would be reduced from Vb to Vb - VAt, (= Vpk — AE; see Fig. 4.2). The effect of thischange would be to increase F as the base potential has been lowered and will thus reflect fewerparticles. 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 accordinglysmall. However, as the doping of the EB junction becomes even slightly more symmetric, the error of using [511 will become increasingly large. Furthermore, as the temperature is reduced to thepoint where U,, occurs below Vb, there will be an exponential change to F for a linear change toVt,. Thus, under low temperature conditions the methods contained within [51] for the inclusion oftunneling will be in error even for a highly asymmetric doping junction (see Fig. 4.20).5.02.01.o0.00.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 4.20. Relative difference between the results obtained from the methods proposed in [51] tothe model for F from this chapter. The device is based upon the same Al03Ga07As HET used inthis section. Note how the reduction to Vb as proposed in [51] leads to an overestimation in thetransport through the CBS, and therefore, to an overestimation of I.July 12, 1995 104Finally, is it reasonable to consider ballistic motion throughout the entire EB SCR? Certainly, to consider collisions to the particles while in the process of tunneling would be difficult. However, models that are similar to, but simpler than, the models presented in this chapter are able toexplain the terminal characteristics of abrupt HBTs [22,25] because they include the effects oftunneling. Other more complex models, such as Monte Carlo simulation, which do not include theeffect 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 experimental evidence tends to corroborate the assumption of ballistic motion throughout the EBSCR. For if there were even a moderate chance of only a single thermalising collision within theEB SCR, then the tunneling current would be drastically altered (any reduction or increase to theenergy of the particle will cause a correspondingly rapid reduction or increase in the tunnelingprobability). Since experimental evidence does not support this, at most, there is a small probability of a thermalising collision within the EB SCR. This justifies the assumption of ballistic motionthroughout the EB SCR.4.7 ConclusionTo conclude this chapter, a summary of the past 40 years’ work in this area of electron transport through a SCR is in order. The reason for this summary is to give due credit to all of the individuals who have made contributions, and to demonstrate how a large majority of this past work isdisjoint from both the study of HBTs and itself. To begin with, Miller and Good [86] set out the requirements for the WKB approximation to the Schrodinger equation in 1953, which formed the basis for the study by Murphy and Good [681 in 1956 of electron emission from metals into vacuumdue to thermionic injection and tunneling (which they term field emission). [68] lead to the formation 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 amass barrier based upon a spherical effective mass. The main concern in [68,69] is the incorporation of image force corrections which alter the tunneling potential and greatly increases the tunneling current. In [69], tunneling is only considered within the vacuum and not within thesemiconductor, and does not consider the effect of a base barrier potential Vb (as Vb is far too negative to enter into the problem). Stratton and Padovani [75] apply [69] to Schottky barriers, and include tunneling within the semiconductor but still do not concern themselves with the effect of Vb.July 12, 1995 105Also, [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 meantfor the study of Schottky diodes, and is more concerned with surface effects (image force correction) than anything else. Furthermore, the potential profile being considered is linear and not theparabolic one found within the SCR; however, [69] does allude to the solution of an arbitrary potential energy profile through the use of a Taylor expansion. The work up to this point forms thefoundation 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 workin [76] relies on simple arguments to obtain results that, while applicable to the study of pure thermionic emission, are not clearly applicable when tunneling is considered. Crowell [77] continuesthe work in [76] in an effort to determine the correct effective mass to apply to a Schottky diodebetween two materials characterised by different effective masses. The work in [77], much Likethat done in [76], is not mathematically rigorous, and as a result fails to obtain a vanishing transport current under equilibrium conditions. Grinberg [82] solves this problem but only if thermionicemission is considered and not tunneling. The work of this chapter extends [82] by including tunneling and thermionic injection (eqn (4.60)) through a rigorous mathematical treatment.Finally, Crowell and Rideout [78] solve for tunneling through the parabolic potential barrierof 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 thermionic emission and tunneling, between two semiconductors characterised by different effectivemass tensors and e. Furthermore, the effect of Vb is properly included. The most important aspectof the work contained within this chapter is that for the first time all of the essential physical constructs of the EB junction within an abrupt HBT have been considered. The results of these considerations are analytic models, based upon the solution of eqns (4.50)-(4.53), to simulate thetransport of flux through the EB SCR. Since there were no special features of a specific materialsystem employed within this chapter, the results of this chapter are applicable to any material system. Finally, the developments presented here have focussed upon electron transport, but applyequally well to the transport of holes with basically little change to the models.July 12, 1995 106CHAPTER 5Recombination CurrentsJuly 12, 1995 107As was discussed in Chapter 3, one of the most important parameters of an HBT is the current gain [3. Whether one is designing Digital or Analogue circuits within an IC, an accurate understanding of 13 is essential to the successful operation of the circuit. Chapter 4 dealt with thecalculation of transport through the CBS (in an npn device), which is often the determining factorfor I in abrupt HBTs [18,25]. This chapter will finish off the model for I by using the generalmodels of Chapter 2 to include the effect of neutral base transport along with transport throughthe CBS. More specifically to the calculation of [3, this chapter presents the physics underlying thecreation of base current. Included in the analysis to follow is the interaction of‘B with Ic that wasalluded to in Chapter 2, and which occurs when transport through the CBS is responsible for current-limited-flow (i.e., control of Ic).This chapter includes the modelling of four different components of the hole current that result in the base tenninal current. These components are: 1) Shockley-Read-Hall (SRH) recombination within the EB SCR; 2) Auger recombination within the EB SCR; 3) radiativerecombination within the EB SCR; 4) neutral base recombination through all of the processes justdetailed. The back injection of carriers (i.e., holes for the npn HBT being considered) from thebase into the emitter is not accounted for because this back injection is effectively suppressed bythe characteristics of the wide bandgap material that forms the emitter; however, inclusion of backinjection is a trivial extension to the results that follow.Analytic models for the four previously mentioned recombination processes that are responsible for the creation of‘B will be presented. It is shown that these analytic expressions for thefour 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 mechanismsthat control the base current in the presence of a heterojunction differ markedly from the homojunction case, one can still recover a simple diode model for the final representation. It is withinthis analysis that a surprising result regarding the injection index n is made. Standard theoreticalcalculations give a value of n = 2 for the SRH current. However, it was found that n = 2 appliesonly in the limit of a wide, or symmetrically doped, EB SCR. For HBTs of interest, where thebase 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 108Most of the work that is to be presented in this chapter has been previously published by thisauthor and Dr. D.L. Pulfrey [24]. Within the context of this published work, HBTs constructedwithin the AlGai.As material system were studied. The results of this chapter are general, however, and can be applied to other material systems as well. For the case of indirect material systems such as SiGei.., the only major change is that the radiative recombination rate is smallenough to be ignored in comparison to SRH and Auger recombination.5.1 Electron Quasi-Fermi Energy Splitting J\EfThe presence of an abrupt EB heterojunction in an npn HBT can lead to the splitting of the electron 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 drivingforce for the transport current through the CBS (as was proven in Section 4.3, eqn (4.63)). Fig. 5.1shows AE1 and its position within the EB SCR. results due to a departure from quasi-equilibrium, where the transport flux through the CBS is no longer a small perturbation to the forwardand reverse equilibrium fluxes that are everywhere present within a semiconductor [50,18].EE(eV)AE t---‘—‘VE1EqV+_i-x 0Fig. 5.1. Band diagram of the EB SCR showing the effect of the abrupt heterojunction onunder an applied forward bias (reprint of Fig. 2.2). ji is the solution to the Poisson equation and istherefore continuous. Both the reference energy position and the intrinsic or mid-bandgap energyE1 are also shown.Section 4.3 and eqn (4.63) clearly bring out AE, but do not locate the position of AE if itis indeed abrupt. Perlman and Feucht [50] have addressed the spatial variation of and foundthat 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 109tensibly constant throughout the EB SCR. The reason for the lack of a within the EB SCR isbecause transport through the neutral emitter and not the EB SCR dictates the back injection current (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 theemitter, 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 thecalculation of the collector current density J and the neutral-base recombination current densityNB [20,51,87-89]. The calculation has proceeded via a balancing of and NB against the combined thermionic/tunnel current ThT crossing through the CBS at the abrupt junction; i.e.,ThT = NB + J( AEfl. (5.1)Further, it has been the usual practice when considering additional base current due to recombination in the EB SCR, to subsequently add this extra current SCR to the prior-calculated NB; i.e.,=1’NB(fn) SCR (5.2)Recently, Parikh and Lindholm [90] have emphasized that this calculation of B via direct superposition is not strictly correct because the base-side component SCR,B of SCR should figure inthe original current-balancing equation which is used to compute AEp, and, subsequently, c, NBand scR,B; i.e., eqns (5.1) and (5.2) should be replaced byThT = JscR,B+JNB+Jc —*AEffl (5.3)B,B = SCR,B (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 onlyeffect the base current if SCR,B is comparable to NB and, furthermore, will only effect the computation of from the balancing equation (i.e., eqn (5.3)) in cases where f3 is low. To examinethese effects is one of the objectives of this chapter and, to ensure that their importance is not underestimated, Auger and radiative recombination in the SCR have been considered, as well as theusual SRH recombination.The computation of via eqn (5.3) can be done numerically, but an analytical solutionwould be more insightful, and also very useful in HBT device modeffing because AE, and thusJuly 12, 1995 110Jc N.B and SCR,.B could then all be computed directly from the physical properties of the deviceand the applied bias. Chapter 2 presented the analytic methods to determine both and the ultimate transport currents that produce J and Therefore, the second objective of this chapter isto develop such an analytical expression for AE. A final aim is to show that the components ofSCR,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 complete, large-signal representation of the HBT, which may then be implemented in Circuit simulators such as SPICE.Fig. 5.2. Components of the collector (ic) and the base (SB) currents emphasising that ThT mustequal the total of, J( + NB + SRflB + Aug,B + Rad,B when recombination due to ShockleyRead-Hall (SRH), Auger (Aug) and radiative (Rad) processes is considered.5.2 Modelling the Recombination Processes of HETsThe “unique relationship” [90] between the collector current, the neutral-base current andthe base-side SCR recombination current comes about because all these currents depend upon theelectron quasi-Fermi energy splitting at the heterojunction. As this splitting is greatest in the caseof an abrupt heterojunction, we consider only this type of junction in this analysis. The junction istaken to be formed by an n-type Al030a7As emitter and a p-type GaAs base (the same as the device in Section 4.5). To reduce the complexity of the algebra, without sacrificing much in the wayJuly 12, 1995 111of accuracy [90], the perinittivities and the effective densities of states have been taken as a constant throughout the entire device.5.2.1 SRH RecombinationThe recombination rate due to SRH recombination can be written as [90,911n. E -ER—______________inh fP) (55)SRH— t [cosh (Uf— E/kT) + b] “ 2kTwhere: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),= Jt,0t,where and t, are the hole and electron minority carrier lifetimes, respectively,within the SCR,Uf= (Ep + Efr)I2kT + ln(tdt,),b = exp[(Ejj,— Ep)/2kJ]•cosh[(Et —E1)fkT+ ln(tolto)J,where E is the energy level of the single recombination centre assumed in this work, and E(x) isthe intrinsic Fermi energy. The latter has a discontinuity of AE1 at the abrupt heterojunction (seeFig. 5.3), because the bandgap difference between the wide-bandgap emitter and the narrowbandgap base is generally not distributed evenly between the conduction and valence bands; i.e.,AEG flAE1 = + AE = kT in + AE (5.6)where the subscripts p,n refer to the p-type base and the n-type emitter regions respectively. E1 isrelated, therefore, to the electrostatic potential energy ji(x) viaE (x) = I N’ (x) x 0 (5.7)N(x)—AE x>0Here we use the depletion approximation for if(x), namely:x0iJ(x) =2(5.8)x>0Julyl2,1995 112where, using eqn (4.69)= Vp=q(lNrat)(Vbj_VBE),=’VP— £NDVPk— eDNA_______kT (NAND AE kT (NAND AE1Vb. = —liii 1+— = —mi 1+—q i ) q q qwith VBE being the applied base-emitter voltage, Vb1 the built-in potential, ND the emitter dopingand NA the base doping. Eqn (5.9) has included the effects of a non-uniform permittivity for thetime being.Fig. 5.3. Energy Band diagram for the EB SCR of an HBT under equilibrium conditions. Noticethe discontinuity of AE in the intrinsic energy E.I2eVPkfl= Al q2N— CPNAwhere Nrat— eDNA + ENDI2eVx=I =xP— NDxfl— NA (5.9)-xn 0July 12, 1995 113The SCR currents on each side of the heterojunction follow fromSRH = q$RSRHdx + qfRSRHdx=SRH,B SRH,E (5.10)This equation can be solved using eqns (5.5)-(5.9), but the solution cannot be made analytic withsimple transcendental functions. A closed-form solution demands that some approximation bemade for W(x). Here we follow the linearisation procedure of Choo [92]; i.e.,q(V—V)qi(x) Viinear(X)=(x+x),—xxx (5.11)BEwhere x, = WBENrat,d, Xp WBE(l— 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 pointwhere RSRH is a maximum. The problem with this type of expansion is the RSRH maximum mustbe well localised within the region of integration. If the Rpjj maximum is not within the region ofintegration (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-value 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) andEf—Ef=qV x0qBE— fX>(ND N2kTlnL—J x0Ef+Ef =—2kT1n(.) qV AE x>0to yield— 2qnWB.r VBE_AEJn1 (zo—zSRBB— te51 [q 2kT atan +(5.12)2qn1W (Z,—Z0,SRFLE=sinh [ 2kT j atan + 1July 12, 1995 114with= q (VbI— VBE)/kT= Jtpoxtnoxz— ND 1tpQ,fl [ qV—4--——exP[-- 2kTND 1tpQ r 2Nrat (“bi — VBE) + VBE1Z0 = —q—--—exp[—q 2kT J (5.13)z— ItpOpex qvBE_IXEfflP—P[ 2kTND ItpO,p r 2qN (VbI — VBE) + qV + AEffl — 2AE1Z02kTwhere it is assumed that E and E1 are coincident throughout the device [90], and, therefore, bfrom eqn (5.5) can be neglected for any reasonable operating conditions. Eqn (5.12) can be obtained from eqn (5.10) by using integral 2.423 #9 in [81]. In all cases, the final subscript ofp andn 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 appealing form as it can be readily seen to be an extension of the usual equation for SCR recombination in homojunctions. Also, the unique feature to HBTs, quasi-Fermi-energy splitting, isexplicitly 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 ratioNrat,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 approximation of a linear NJ(x). For this reason, it is assumed that for all practical devices encountered 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 RecombinationAs the doping concentrations increase, Auger recombination becomes an important consideration. There are two Auger processes of interest [93]: 1) a conduction band electron recombineswith a heavy-hole, transferring it to the light-hole band; 2) a hole recombines with a conductionband electron, and the energy is transferred to another conduction band electron. In the first case,the recombination rate is proportional top2n, while in the second it is proportional to pn2.WhenJulyl2,1995 115the equilibrium recombination rates are included, the total Auger recombination rate is:UAUg = (An+Ap) (pn—n) (5.14)where the constants A and 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 rewritten as:UAU8 = nexp(k?)AJAflAP. [ZAUS+_] [exp( f119) _i] (5.15)whereE +E -2E.ZAug= AJA 2kT1)The Auger recombination current is then given byAug = qfUAugdx+qJuAugdxAug,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 [qV—AEffl. [qVBE—AEfflAug,B=exp [ kT j sinh [ 2kT(Z0 — Z) (A PnO pzz0+ A,t0)(5.17)2qn WBE qV qVAug,E=exp [ kT sinh [ 2kT] (4 — Z) (A ntnnZnZn +AEqn (5.17) gives the Auger recombination currents that are generated from the base and the emitter sides of the SCR.5.2.3 Radiative RecombinationFor materials where there is a direct bandgap, it is important to consider direct band-to-bandradiative recombination. The rate at which radiative recombination occurs will be proportional tothe pn product [94]. When the equilibrium recombination rates are included, the total radiative recombination rate is:July 12, 1995 116URad = B(pn—n) (5.18)where the constant B is the radiative recombination coefficient.The radiative recombination current is then given byRad = qfURaddX+qfURaddXRad,B + Rad,E (5.19)which can be solved using eqns (5.18), (5.9), (5.7) and (5.6) to give:qV-AERad,B = qnpBpWBE(l_Nraj)[exp( kTv (5.20)2 r qBERad,E hui,nBnwNrat[exP( kT —15.3 Current Balancing with the Neutral Region Transport CurrentsIt is clear from Fig. 5.2 that the electron currents to the right (i.e., the base-side) of the heterojunction must equal the electron current due to the charge transport across the hetero-interface;i.e.,ThT JscR,B+JNB+Jc (5.21)whereSCR,B = SRH,B + AugB + RadB• (5.22)The formulation given in eqns (5.21)-(5.22) was already treated in Section 2.2. Comparison ofFig. 5.2 with Fig. 2.3 shows an exact agreement. Therefore, the current balancing portrayed byeqns (5.21)-(5.22) can be solved using the models given in Section 2.2 if the various transport andrecombination 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 (seeeqn (2.3) for J, i). This immediately allows the models of Chapter 4 to be used in concert with themodels of Chapter 2 to solve for the collector and base terminal current densities J and B respectively. Looking again at eqns (2.3) and (4.63) shows that i2 1 = Ff. and = AE. F3cin-cludes both the thermionic emission and tunneling components involved in the transport over andthrough the CBS. Employing the formalisms of [51], Ff can be written as:July 12, 1995 1172 J1 (Vbx — VBE) (VbZ — VBE)4iqJm 1m 1(kT) — -_______ -_______Ff= y(VBE) “ Z ekT e kT qyuNe kT (5.23)hwhere 1) is the electron thermal velocity given by= / kT (5.24)eq 2itm,and y(VBE) is the tunneling factor (this is not to be confused with the yin Chapter 4 used to characterise the mass barrier). With y = 1, eqn (5.23) reduces to the thermionic injection current givenby the last term in eqn (4.78). Essentially, yis given by FfIJth where th is the thermionic injectioncurrent and Fj= FjCBS given in eqn (4.93). Failure to include yin eqn (5.23) will result in a severeoverestimation of AEp1 [18] (and an underestimation of the collector current). Finally, JI is theelectrochemical potential relative to E formed by the doping ND within the neutral emitter. Theapproximate solution given in eqn (5.23) is strictly valid only if the emitter is non-degeneratelydoped.The neutral-base recombination current NB and the transport current through the neutralbase Jc which must be used in eqn (5.21) follow from the standard, low-level injection solution tothe continuity equation. Using the boundary condition that the driving potential at x = x, (i.e., thestart of the neutral base) is VBE — AE (see Fig. 5.1), and for the case of a single heterojunctionstructure operating in the forward active mode, the excess electron concentration near the collector is 11 (Wflb) = 0, where Wb is the neutral base thickness relative to x = x, then the expressions for these currents areIWflbN2 coshl— I—i qV8—zEf 1 2 coshl— —i qV 1— Lflb kT 1qDn1NB— NALflb . (We—— NALflb . (We — ijesinh-L——j sinh ç- J (5.25)andqDn,csch (—) [ qVEAEf — qDn, csch (—-)[e — 1]e (5.26)Anb nb Anb nbwhere D is the effective electron diffusivity in the base, and Lflb (= JDfltflb) is the electron minority carrier diffusion length in the base. Observation of eqns (5.25) and (5.26) show they possessthe functional forms of 4 and J respectively found in eqn (2.4) (i.e., 4, = JNB(AEp=0)July 12, 1995 118and J = Jc(AEfij=0)). The approximate forms of eqns (5.25) and (5.26) introduce a negligibleerror 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 withrespect to Clearly, Rad,B in eqn (5.20) can be written in the same approximate form as NBwith 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:) (Ti Ar’ \ r cu ( kTAug,B” ‘ BE’ ‘-fn) — Aug,Bk V BE’ e-- (5.27)SRILB( BE’ fn) sB( BE’ ) eRad,B0”BE’ AE) — JRB(VBE, 0) e kTa plot of the error between the full and the approximate forms in eqn (5.27) is constructed. Fig. 5.4plots the relative error between the right and left sides of eqn (5.27) for Aug,B’ SRI-1B’ and Rad,Bwith 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) +‘1SRHBWBE’ 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 currentT through the device (which is equal to the collector current) is given by eqn (2.7), with = Fj’from eqn (5.23), J = from eqn (5.26), 4 is given by eqn (5.28), 4 2 = 0 (i.e., Y2= 1), and J 2 >> (J , J 3). Using the above produces:= r-+--—i= [ Ff(VBE)/y3 J,i1”y3<<J, (5.29)[J2,1 .i2,3J Jc(VBE,1Effl=O)if 1’r3>>’2,where y is given in eqn (2.7) as— 3+4— Jc+JNB+JsRH,B+JAug,B+JRacB13 jJ3 C zEf=Oand the VBE dependence has been omitted for clarity. Eqn (5.29) embodies the two differentJuly 12, 1995 119modes of operation that the HBT can function under; the first condition is where the CBS is responsible for current-limited-flow; while the second condition is the classic BJT regime of operation 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) = JT(Y3F+ SRIE Aug,E + RadE— i). (5.30)Or, AE can be calculated by eqn (2.5) and substituted back into SCR,B of eqn (5.22) and NB• Bis 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 todetennine the inner driving potential of iSE. However, if a detailed understanding of eachcomponent of the base terminal current is desired, then iXE must be solved for explicitly.oz::::Aa ug,SRHB4—6 .0 50 100 150 200 250 300Quasi-Fermi Energy Splitting AEf (mV)Fig. 5.4. Relative error between the approximate and exact forms given in eqn (5.27). The material parameters are given in Section 5.4, and VBE is fixed at 1.OVBefore leaving this section, it is important to verify that E, is indeed constant throughoutthe EB SCR. If there were a AEp,, present, it would have to be included in the emitter side holecurrent SCR,E just like has been included in SCR,B• Essentially, the same current balancingprocedure given by eqns (5.21)-(5.22) needs to be performed regarding the transport of holes fromJuly 12, 1995 120the neutral base, through the EB SCR, and finally through the neutral emitter. The same modelsfor the electron case can be applied to the hole case, but using the appropriate material parametersfor a hole. Using the HBT parameters of Section 5.4, then the hole transport current through theEB SCR is 3.9x1Oexp(qV/k7)Acm2,and the hole transport current through the neutralemitter assuming a 3000A emitter cap at a doping of 102cm3is l.2x1W27exp(qV/kI)AcmClearly, the neutral emitter is the bottleneck to hole transport which validates the claim thatis indeed zero through the EB SCR. This does not have to be the case, and a device can be imagined where this is not true, leading to the requirement that hole transport be self-consistentlysolved with electron transport. It is quite interesting to realise that the valence band discontinuityAE does not limit the back injection of holes as the literature has lead the device community tobelieve. The back injection of holes is ostensibly eliminated by the reduced number of minorityholes due to a small n1 that is characteristic of a wide bandgap material.5.4 Full Model ResultsThe values used for material parameters, unless otherwise stated, are:ND: 5x1017cm3;NA: 1x109cm3;Cbe: l2.9e; Ejr: ll.9Erj; to,p: Sns;to: 2Ons; AE: 0.24eV; 4.21x10cm 2.25x106cm3—* AE1: 77.3 meV, Vb: 1.67 1V,Nrat,d: 0.952, Nrat: 0.956, x(VBE=l.4V): 271A, xp(VB=l.4V): 13.6A;A,,: 7.99x1032cm6s15.75x1t13cm6s; 1.93x1031 cm6s1; 1.12x1030cm6s;B: 1.29x100cm;B: 7.82x1O’1cs;D:30cm2s1,Wb: ioooA.Results for the SCR currents are shown in Fig. 5.5. The slopes of the curves are not constant, owing to the voltage dependence of WBE, but it is clear that all the base-side SCR recombination components have about the same ideality factor (n), and that this is considerably less thanthat of the emitter-side SCR recombination current (which is dominated by sRH,E)• Specifically,at VBE = 1.2 V, SCR,E = 1.90 and, adding all the base-side currents together, SCR,B = 1.19. Furthermore, = 1.14 at VBE = 1.2 V due to the effects of These values are similar tothose reported elsewhere [90], and deserve further comment because SCR,B is so far removedfrom the “classical” value of n = 2.With reference to eqn (5.12) for the SRH current, because is so low and Nrat is 1, Z,and are both>> 1, leaving the atan term in eqn (5.12) to saturate at t/2. The voltage depenJuly 12, 1995 121101010120.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 5.5. Bias dependence of the SCR current from the emitter side, and the three components ofthe 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 theemitter side, and this fact alone, via Nrat in the Z and Z0, terms in eqn (5.12), would makeSCR,B <<SCR,E• However, the much larger n1 on the base side counterbalances this effect and allows the steeper-rising SCR,B current to exceed SCR,E beyond some forward bias. In the exampleshown in Fig. 5.5, this occurs around VBE = 1.45 V. This transfer from an n 2 slope to an n 1slope in the SCR current does not occur in a homojunction device as there is no spatial change inn1 to inflate the current in the more highly-doped side of the junction.In practical HBTs it is possible to imagine that the minority carrier lifetime in the highly-doped base will be less than that in the emitter. Indeed, photoluminescence measurements on ma-dence of SCR,E is thus determined by the sinh qVI2kT term and n approaches 2. Contrarily, forSCR,B’ both Zk, and Z are generally << 1, so the atan term modulates the sinh term and reducesthe ideality factor from 2 towards 1.10010-2io-4‘July 12, 1995 122terial doped to 4x1019cm3suggest that ‘c,1 5Ops [951, and a value of 3Ops has been used to model some experimental devices [90]. Fig. 5.6 shows that reducing the base-side t, to 5Ops causesSCR,.B to exceed at a bias of about 1.15 V. However, lest undue emphasis be placed uponthe significance of this change-over, note from Fig. 5.6 that SCR,B is always less than the quasi-neutral base recombination current NB• This indicates that, in practical devices, an observedchange in base-current ideality factor from n 2 to n 1, will likely be due to a change fromJc,gd0m1nated current to a JNB-dominated current. Only in circumstances where it is correct toattribute a much lower minority carrier lifetime to the base-side depletion region only, perhapsdue to defects at the interface, can a situation be envisaged where SCR,B could dominate overNB’ and thus be responsible for the slope change to n 1, which is often seen experimentally. Theabove point about the relative magnitudes of SCR,B and NB is an important one as it puts intopractical perspective the theoretically-interesting fact that SCR,B has a different voltage dependence to that of SCR,E110-810-110.8 1.1 1.2 1.3 1.4 1.5Base-Emitter Voltage VBE (V)Fig. 5.6. Gummel plot showing the importance of including the emitter- and base-SCR currentcomponents in the computation of the total base recombination current. Material parameters arefrom the start of Section 5.4 for two values of t,.110110-20.9 1.0 1.6July 12, 1995 123While it is clear from the results of earlier work that AEp must be included in calculatingSCR,B [901, it is, perhaps, not evident how important it is to include SCR,B in the balancing equation (5.21) to compute iXEp. Fig. 5.7 provides an answer for the material properties consideredhere. By not including SCR,B in eqn (5.21), yet using the subsequently-calculated tXE to eventually 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 muchless than NB and J. However, also from Fig. 5.7, note that it is grossly incorrect to not includeAEp in the calculation of SCR,B Because the electron quasi-Fermi energy splitting is so large foran 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 whereit might be necessary to include SCR,B in the actual calculation of AE1. A possible scenario isone in which t, in the SCR is less than t, in the neutral base, perhaps due to interface defects, andthat W is much larger than the usual i000A. The latter situation would reduce J, and theformer would increase SCR,B with respect to NB’ thus making SCRB become more prominent ineqn (5.21). The effect of these changes is shown in Fig. 5.8. Even though the gain has been reduced to a very low value, it appears that there is still no need to include SCR,B in the balancingequation.To summarise the results from the analysis of this section: it is necessary to include inthe computation of SCR,B; but SCR,B need not be included in the balancing equation to estimateAEp; and SCR,B is not very important for devices based upon materials with the properties considered here, because SCR,B is usually less than either NB or Of course, if parameters affecting Auger or radiative recombination in the SCR turn out to be greatly different than the valuesused here, then SCR,.B could become importantOne instance where SCR,B will definitely be larger than calculated here is in the case ofHBTs which are compositionally graded at the base-emitter junction. The grading gives the junction a more homojunction-like character, so AE will be reduced, and SCR,B increased correspondingly. However, because of the lower bandgap of the graded material in the emitter-side ofthe junction, is increased and, therefore, SCR,E also. Thus it is not obvious whether SCR,B isany more important in graded-junction HBTs than it is in abrupt-junction HBTs. The results ofParilch and Lindholm [90] suggest that SCR,E remains the dominant current. One situation inJuly 12, 1995 1243o:xFull model2000-SCR,B not in balancing eqnbut included1000SCR,B not in balancing eqnw.; 1.0 :i/ndotBase-Emitter Voltage VBE (V)Fig. 5.7. Bias dependence of the current gain 13, showing the relative importance of includingSCR,B in the calculation of AE. Also shown is the dramatic error resulting from not includingAE, in the calculation of SC.R,B108 JSCR,BflOt in balancing eqnbut AE included6-4.Full model2 SCR,B not in balancmg eqn7and not included0.8 0.9 1.0 1.1 1.2 1.3Base-Emitter Voltage VBE (V)Fig. 5.8. Bias dependence of the current gain 13 for the case of Wflb increased to 5000A and t, inthe SCR reduced to 5 ps. Even in this extreme case there is little error in not including SCR,B inthe balancing equation.July 12, 1995 125which SCR,B could be increased without an associated increase in SCR,E is when recombinationat the exposed base surface is important. Providing a reasonable expression for this surface recombination current were available, it could be added to the right-hand side of eqn (5.22) andused in the current balancing to compute AEj. However as can be deduced from Figs. 5.7 and5.8, the inclusion of another component of SCR,B will only effect the estimate of AE if this newcomponent is comparable in magnitude to106 • • • •Wb = lOOnmio3 - Wflb = lOnm -c-’ 100 ...: -SRH,Eio3 - ....•-‘ Rec,BWflb=lOnmio- ...... ,..7 io ... -7.....io 10 //// w,=1oomn -j lOl Wb=1OnmRec,BWab = lOOnm 108 1.0 1.2 1.4 1.61012• • • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 59. Effect of changing the neutral base thickness Wflb when the CBS is responsible for current-limited-flow. Lowering W,th leaves and sRgE unchanged, but results in the reduction tothe base side recombination current Rec,B (= SC.R,B + J). Under high bias, where Rec,B dominates, 13 increases with reductions in W,,,. While under low bias, where SRflE dominates, 13 is unaltered by changes in W,,j,.Before leaving this section, it is interesting to see how current-limited-flow within the CBSleads to a mixing of the base and collector currents. For the HBT considered, the CBS is indeedresponsible for current-limited-flow, so that Jc 1jCBS Thus, if the neutral base transport cur-July 12, 1995 126100io-31061 0-10-1510-180.8rent were increased by reducing Wflb, Jc would remain unchanged because the CBS already represent the bottleneck to charge transport through the device. However, the reduction to Wflb doeshave an effect on the device. Fig. 5.9 shows that the base-side components of the base terminalcurrent 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 recombination. Therefore, opposite to what occurs in BJTs, the mixing of the collector and base currentsdue 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 usingthe linearised W(x) from eqn (5.11) against the currents obtained with the full potential from thedepletion approximation of eqn (5.8). As can be seen in Fig. 5.10, the error is indeed slight, andwill 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.10120.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 5.10. Comparison of the recombination currents when ji is given by the depletion approximation in eqn (5.8), and when it is given by the linearisation of eqn (5.11) (IEf is included).July 12, 1995 1275.5 Simple Analytic Diode EquationsFor the purpose of including the various SCR recombination current components in HBTdevice simulators, it would be convenient if a simple closed-form solution for iXEp existed. Further, if the various current components could be expressed as diode-like equations, then their representation in circuit simulators such as SPICE would be greatly facilitated. In this section, theapproximations 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 expressions is to examine the relative importance of the Z-terms which appear in the expressions for theSRH and Auger recombination currents. Fig. 5.11 shows the results from the full model calculations. From this figure, it appears reasonable to state that Z <<Z0, << 1, 4>> 1 >> Z, and generallyZZ0>> 1. The Z-terms Z,, Z0,Z0,, 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 depletionregion on the base-side must not be vanishingly small. This can be ensured by having the dopingdensity ratio NA/ND 30. Contrarily, there is a lower limit to the allowable value of NA/ND, belowwhich the recombination on the base-side of the depletion region becomes large and the inequality Z(), << 1 is violated. This limit is NA/ND 3. Therefore, keeping within the range 3 NA/ND30, and following the usual practice of expressing WBE and €) by their equilibrium forms, eqns(5.12) and (5.17) reduce toNDnP riXE — qNVji — zXEfJsRH,BCs exp[ kT JexpL kTno,pni, ii1Ifl jqVSRH,E Cs 2t 2kT (5.31)2 [qV—1XEfflAug,B CSnI,PAP,PNAexp L kTq VBE1Aug,Esn4n,nNDP[kT IwhereC =Writing the radiative recombination currents in eqn (5.20) in similar form, givesJuly 12, 1995 128q CsVb 2 q VBE — IXEfflRad,B kT nipBp(l—Nrat)exp[ kTqC5V1 2____Rad,E kT ,vBnNratP L kTUsing these diode-like equations, along with the expressions for NB in eqn (5.25), in eqn(5.26) and ThT = Ff in eqn (5.23) in the balancing equation of eqn (5.21), yields a convenient expression for t.Ep; i.e.,(VbZ— VBE) qV8kT — Recom (VBE) + q’y’uNekT+ e (533)e—S,Recom + qyuNekT+qDn0/W ewhereqV8 qV qVI (TI ‘ — i nSRH,BkT Aug,BkT .i. IRecom” “BE) — JS,SRBBe +.ISAU8Be -I-.IsRaBeRecom = “S,SRH,B + SAugB + SRa,B— qN1VjNDn. qCV.=C ‘‘ e kT +CsnPAPPNA+ kT1flpBp(1Nrat)nO,p 1, nbWflb, e = Lflbtanh I\. nbBO =The values for the saturation currents and ideality factors in eqn (5.33) can be found eitherthrough 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 uponthe 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 thefull 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 diminishing 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 assumptions regarding the relative magnitudes of the Z functions re-addressed.July 12, 1995 129106i04.Zni02.Nl0101-.c 10 ....- Zp106 I1.0 1.2 1.4 1.6 VblBase-Emitter Voltage VBE (V)Fig. 5.11. Z-functions as computed from eqn (5.13) when using the material parameters fromSection 5.4.100 -ExactDiode102 - .. ,..-SCR,E ..._/-10 - .......-...101 ......- -Rad,BRad,B ......1081011W0 SRH,B &R,E -io-3Aug,B 1.4 1.5 1.61012 • • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter Voltage VBE (V)Fig. 5.12. Comparison of the full model and “diode-like” expressions for the SCR currents. Thehigh-bias region of the figure is enlarged in the inset with SRH,B and Aug,B omitted for clarity.July 12, 1995 130If, as found to be the case for the material parameters used here, it is not necessary to include SCR,B in the balancing equation, then the Recom and S,Recom terms can be omitted fromeqn (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 simplified to give(VbZ — VBE)kTqI1Ef NratVbi+ (lNrat)VBE= DflnBo q kTWflb e’’DSubstituting this expression for into the diode forms in eqns (5.31 )-(5.32), gives overall ideality factors for the base-side SRH, Auger and radiative currents of: lI(2Nrat — 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/GaAsHBT, it can be concluded that:1. recombination currents in the base-side SCR are generally less than the neutral-base currentand, therefore, need not be included in the current-balancing equation used to compute the quasi-Fermi energy splitting AEp, at the base-emitter junction;2. however, when subsequently computing the base-side SCR currents, AE must be taken intoaccount 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 valueof 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 facilitating their implementation in SPICE-style circuit simulators.July 12, 1995 131CHAPTER 6The SiiGe HBTJuly 12, 1995 132The previous chapters have presented a collection of models for the calculation of the transport and recombination currents in HBTs. Chapter 2 presented the generic models for currenttransport in an arbitrarily shaped device where there can be any number of sub regions within thedefined regions of the emitter, base and collector. Chapter 4 presented the transport models for themovement of carriers through a forward biased pn-junction under the influence of a heterojunction. Chapter 5 presented the models for the recombination currents that occur both in the neutralregions of the device (specifically the base and the emitter), and the forward-biased EB SCR. Alsoincluded in Chapter 5 were models for the transport of charge through the neutral regions of thedevice. Finally, Chapter 3 presented the models for the calculation of the base transit time basedupon an optimisation of either the base doping, or the base bandgap, or both. In all of the workpresented thus far, no assumptions have been made that depended upon a specific attribute of agiven material system. Thus, the models contained within this thesis are general, and may be applied 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 materialsystem, whenever an analysis of a specific model was performed, the material system ofAlGai..As was invariably chosen for the study. The reason for choosing the MGai..As materialsystem is that current-day technologies for HBTs prefer this material system. The dominance ofthe MGai..As material system stems mainly from the fact that the lattice mismatch over the usable range of Al content (i.e., 0 x 0.45) is under 0.07% [61]. This nearly ideal lattice match allows for an arbitrary film thickness because there will be virtually no strain placed upon the latticeat the heterojunction. Coupled with the lattice-matched characteristic, the AlGai..As materialsystem can also provide for large changes to the bandgap (AEg) [61]. However, compound semiconductors like GaAs and AlAs have numerous undesirable features when it comes to manufacturing. 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 ahigh defect density, cannot employ ion implantation for bipolar devices, exposed surface layershave high recombination velocities, cannot be used in low-power applications because of the largeVbj inherent with large bandgaps, does not etch easily and generally lacks an abrupt end-point detection for etching, and finally is expensive to manufacture. Given all of these manufacturing andelectrical drawbacks, however, the lattice-matched attribute is important enough to makeAlGai.As the preferred material system for the construction of HBTs.July 12, 1995 133Essentially all of the manufacturing issues with regard to the AlGai..As material systemare solved by using the Sii.Ge material system: save one issue. At issue with the Sii..Ge material system is its large lattice mismatch. The Ge lattice is 4.2% larger than the Si lattice [96]. Evenif 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 Si08Ge02 film and a Si substrate. The issue with a lattice mismatch ofaround 1% is that to commensurately place an epitaxial film upon a given substrate would resultin a strain within the film that would be large enough to tear the film apart [97-99]. If strain wereallowed to tear the film and form dislocations, then deep states would form along the heterojunction interface which would greatly enhance recombination. Since the heterojunction will beformed in the middle of the EB SCR of the HBT, a plane of recombination centres at the heterojunction 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, ifthe epitaxial film is grown thin enough and at a low enough temperature, it will conform to thesubstrate [99]. Under such conditions, the epitaxial layer is said to be commensurately strained tofit the substrate, and the layer itself will be pseudomorphic [99]. Pseudomorphic films are thusstrained in order to maintain the in-the-growth-plane crystalline structure of the substrate. The keyto obtaining a pseudomorphic film is to ensure that the layer thickness is below the critical thickness h [99]. However, in order to maintain a pseudomorphic film, and ensure that it does not relax back to its bulk lattice constant, subsequent exposure of the layer to high temperatureenvironments must be severely limited. In the past 5 years, great progress has been made at IBMin the quality of Sii.Gepseudomorphic films [31]. These developments have shown great potential regarding operating speeds [100-102], so much so that many other companies including theJapanese at NEC [1031 are developing SiGe IC processes. Through the recent successes regardingthe high quality growth of pseudomorphic SiiGe films, the Sii..Ge material system is fast becoming a practical alternative for the manufacture of HBT-based ICs. In fact, with the massive installed base of Si-based IC manufacturing, coupled with the ability to integrate Sii..Ge films intothe process, it is expected that Sii..Ge will rapidly displace MGai..As as the preferred materialsystem for the manufacture of HBT-based ICs.This chapter will apply the general models obtained from the previous chapters to the studyof HBTs based within the Sii..Ge material system. Due to the complex nature of Sii..Ge underJuly 12, 1995 134the influence of strain, a number of extensions to the work of previous chapters is necessary. Mostimportantly, due to the indirect nature of the Sii..Ge energy bands, there are six separate conduction 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 degeneracyof the six conduction band valleys, it will become important to consider electron transport withineach valley separately. Once the needed extensions to the models of the previous chapters havebeen determined, a study of current-day SiGe HBTs can be performed. Furthermore, it will beshown that the use of strain can be turned into a tool for the HBT developer, instead of being seenonly as a liability in terms of critical layer thickness.6.1 The Effect of Strain on Sii..GeThe use of pure unstrained crystals of Ge in the formation of SiGe HBTs is possible, but dueto 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 inthe formation of HBTs, there would be a considerable limitation imposed upon the ability to engineer the bandgap within the HBT Instead, pseudomorphic Sii.Ge films, that are commensurately strained to become lattice matched to the substrate (which is pure Si in present day devices), areused. These pseudomorphic Sii..Ge layers will remain strained without relaxing as long as thelayer thickness remains below the critical thickness h [97-99,105]. (ForSi070Ge3 grown on{ lO0} Si substrates the critical layer thickness is 600A, while forSi045Ge5 it is only bOA).Thus, unlike MGai..As, which is essentially lattice matched to GaAs and thus has no criticallayer thickness, SiGe HBTs can have considerably less freedom in the choices for layer thicknesses.The key to manufacturing SiGe HBTs is the commensurate growth of strained Sii..Ge layers to the underlying substrate. However, the strain in the plane of growth results in a distortion ofthe crystal structure that breaks the cubic symmetry and causes the crystal unit cell to become tetragonal. With the breaking of the cubic symmetry comes a change to the dispersion relations forthe energy of the Bloch electrons versus wave vector k. The most important effect of this symmetry breaking is the relative change to the energy of the conduction band minima and the valenceband maxima in k-space.July 12, 1995 135Constant energy surfaces near to the conduction band minima for pure unstrained Si and Geare shown in Fig. 6.1. Looking at the case of Si, there are six separate but degenerate conductionband minima located along the (100) directions at the A point (which is 80% from the zone centreat F to the Brillouin zone edge at X). For alloys of Sii..Ge, these six minima are dependent bothon the alloy content x and on the state of strain. Take, for an example, 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 layermoves 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 thedegeneracy of the six minima is lifted [105-108]. The two minima aligned to the normal of the interface 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 loweredin energy. For the case of Sii.Ge grown on a Ge substrate with the direction of growth still parallel to [001], the situation is reversed. As x moves from 1 to 0, the Sii..Ge layer moves from anunstrained cubic structure to an expanded, tensile-strained tetragonal structure. For this case oftensile strain, as the strain increases from zero, the two minima normal to the interface plane arelowered 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..Gefilm; 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 compressive 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 bystrain. The valence band of pure, unstrained Si and Ge (or for that matter, all semiconductors), iscomposed 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 angularmomentum (spin), is coupled with its orbital angular momentum (termed spin-orbit coupling), thedegeneracy of the so band is lifted [109,110]. The resultant interaction leaves the lh and hh bandsdegenerate with the so band maxima moved to a lower energy (see Fig. 6.2). The symmetry breaking caused by strain goes on to lift the degeneracy of the lh and hh bands. As in the conductionband, 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 growthparallel to [001], as x moves from 0 to 1 the Sii..Ge layer experiences an increasing compressiveJuly 12, 1995 136I \ 1.f1<H L_ N/ /\‘\. r ,..•‘ /•......\Si\‘ ‘----—____\ I\‘ Fl ‘I \\ \I x\ /7c ).1/ ‘\. F,.,,”kL/ / GeI IILFig. 6.1. First Brillouin zone showing (in k-space) the constant energy surfaces near the bottomof the conduction band for Si and Ge. Also shown are the designations for the symmetry pointsand the degenerate bands E and E in strained Sii..Ge with the growth direction along [001].July 12, 1995 137strain. The result of compressive strain is an increase in the maxima of the hh band relative to thelh band, accompanied by a decrease in the maxima of the so band relative to the th band. Undertensile strain, however, the effect is reversed for the lh and hh bands (but not the so band). Returning 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 result 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, straineliminates the degeneracies of all the valence bands, with the so band always moved to lower energies. However, depending on the sign of the strain tensor (i.e., either compressive or tensile), either the lh or the hh band will form the ultimate valence band.kFig. 6.2. Valence bands in unstrained Sii.Ge. The light hole (lh) and heavy hole (hh) bands remain degenerate for all values of Ge alloy composition x (only in the bulk state where there is nostrain 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 andthe valence band maxima change under the effect of strain, while their position in k-space remainsunaltered. However, it is also important to ascertain the effect of strain on the shape of the band ink-space, as this will set the effective mass which determines the velocity of the carrier and itsprobability for tunneling. Considering the valence band first, the effective mass for the lh and hhEhh bandso bandJuly 12, 1995 138bands in pure unstrained Si and Ge are quite different. Therefore, as the Ge alloy content in theSii.Ge layer changes, there must be a change to the shape of the band in k-space regardless ofthe strain state. To account for this varying shape of the lh and hh bands, a linear interpolation between the experimental values for the lh and hh masses in Si and Ge is used to arrive at the appropriate masses for the SiiGe layer [111]. It is further assumed that the effect of strain isnegligible with regard to the shape of the band in k-space. This leads to:mhh = 0.49 — 0.21x (6 1)mffi=O.l6—O.lxwhere x is the Ge alloy content, and the masses are a fraction of the electron rest mass me (the Siand Ge hole masses are based upon [96]). The lh and hh effective masses are maintained separately instead of combining them into an effective density of states mass because under the influenceof 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 notchange shape with either a change in the Ge alloy content or the state of strain [107,112]. To firstorder in the strain tensor there must be a change to the effective mass for the electrons because thereciprocal lattice vector is being changed. However, this change will be relatively small as themaximum 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 thereforeSii..Ge) have conduction band minima at A and at L. The difference between Si and Ge is thatthe A minima form the ultimate conduction band in Si while the L minima form the ultimate conduction 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 theSii.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 inSi, while the effective masses for the L minima are the same as the Ge effective mass; i.e., [96];m(A) = 0.19m1(A) = 0.98 (6.2)m(L) = 0.082m1(L) = 1.64Therefore, there is no change to the effective electron mass, for a given band, with either a changeJuly 12, 1995 139to the Ge alloy content or the state of strain. However, in similar fashion to the valence band, theeffective density of states mass will change with strain depending on which band forms the ultimate conduction band.The qualitative features that strain and Ge alloy content impart to the Sii..Ge layer havebeen presented. Using empirical deformation-potential theory [114-116], the quantitative featuresare now presented. The reason for using empirical deformation-potential theory, where the deformation potentials are measured and not derived from first principles, is that current-day solid-statequantum mechanics is not sophisticated enough to predict the desired results with any reasonabletolerance (errors on the order of 1 eV are standard). To this end, the problem of including thestrain 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 theentire range of 0 x 1 to produce the function Eg (x). Thus, Eg (x) contains all of the Ge alloy effects. Then, empirical deformation-potential theory is used to determine the amount of degeneracy splitting that occurs within the sub-bands of the conduction and valence bands due to theaddition of strain. Adding together Eg(x) with the results from deformation-potential theory produces 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 experimental measurements on the bandgap of bulk and strained Sii.,Ge have been performed. It has beenfound that for x < 0.85, the A minima form the ultimate conduction band minima in Sii..Ge. However 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:( E 0.51446x+0.3h164x2 x0.732Eg(X)=g, 1 (6.3)t EgsiO.l5Ol 0.0813x x>.0.732where xis the Ge alloy content, ES1 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 ofthe A minima in the conduction band. Caution must be exercised when using eqn (6.3) for x > 0.85as the L minima will form the ultimate conduction band in bulk Sii..Ge material. However, thestrain imparted to the Sii..Ge layers used in HBTs is generally sufficient to reduce some of the Aminima below the L minima even as x approaches 1 (i.e., pure Ge) [106]. For this reason it will beassumed that the L minima can be ignored. However, the energy of the L minima change muchJuly 12, 1995 140more rapidly for a given change to x than the A minima do. Thus, it would be possible to achievelarger band offsets using the L minima versus the A minima, or to achieve the same band offsetsbut 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 basedupon 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 conductionand valence bands of Sii.Ge. The second problem of including the effect of strain is now addressed. Fig. 6.3 shows the effect of in-plane biaxial tension and compression. Fig. 6.3(a) showsthe case where the substrate lattice constant as is larger than the alloy lattice constant aa. Thecommensurate growth of the alloy layer to the substrate forces the in-plane alloy lattice constantto match a5. In so doing, a biaxial in-plane tension results in the pseudomorphic alloy film. In anattempt to lower the energy contained within the film, the out-of-plane alloy lattice constant compresses below aa. The pseudomorphic alloy layer will then have a larger in-plane lattice constantwhen compared to the out-of-plane alloy lattice constant, leading to a tetragonal crystal instead ofa cubic one. Contrarily, Fig. 6.3(b) shows the case where the substrate lattice constant a is smaller than the alloy lattice constant aa. The commensurate growth of the alloy layer to the substrateforces the in-plane alloy lattice constant to match as. In so doing, a biaxial in-plane compressionresults in the pseudomorphic alloy film. In an attempt to lower the energy contained within thefilm, the out-of-plane alloy lattice constant expands past aa. The pseudomorphic alloy layer willnow have a smaller in-plane lattice constant when compared to the out-of-plane alloy lattice constant, which again leads to a tetragonal crystal instead of a cubic one. It is the fact that the pseudomorphic alloy layer has broken the cubic symmetry of the original lattice that leads to the changesin the conduction and the valence bands.The initial applied stress tensor to the alloy layer can be viewed as a uniaxial stress accompaniedby a uniform hydrostatic pressure applied over the entire cell. If the in-plane interface is parallel to thex-y plane, with the direction of growth parallel to the z-direction, then the initial applied stress is [1081:t 0Applied stress== ti + 0 (6.4)0 —tJuly 12, 1995 141where the growth is in the [001] direction, and a blank location in the tensor is zero. The first termon the right of eqn (6.4) is the hydrostatic pressure applied to the overall cell, while the secondterm is the uniaxial stress, of opposite direction to the biaxial stress, applied to the out-of-planelattice constant. Therefore, the symmetry breaking of the alloy’s unit cell occurs along the direction of growth (i.e., the z-direction). Thus, any changes to the energies of the A conduction bandminima will leave the A minima along [001] and the [OOT] directions degenerate (i.e., E), as wellas the A minima along [010], [OTO], [100] and the [TOO] directions degenerate (i.e., E).‘Alloy’IWEww-111.1-a+WWJEE mmSubstrate— — — — — — — — — — — — — —SubstrateIIII I I————— —Alloy— — — — — — — — — — — — — —— — — —— Substrate — — — — —————— III--aFig. 6.3. Commensurate growth of the SiiGe alloy layer to the Sii.Ge substrate, leadingto a pseudomorphic alloy film. (a) the substrate lattice constant a5 is larger than the alloy latticeconstant aa. The resultant biaxial tension, which results from aa expanding to fit a5, distorts theout of plane alloy lattice constant by compressing it. (b) the substrate lattice constant as is smallerthan the alloy lattice constant aa. The resultant biaxial compression, which results from aa compressing to fit as, distorts the out of plane alloy lattice constant by expanding it.H Haa (a)Alloy(b)-4-aJuly 12, 1995 142The final diagonal components e, e and e of the strain tensor, after the layer becomespseudomorphic, are given by the relative difference between the final pseudomorphic lattice constants and the initial bulk values [106,107,112,114]. Given that we are dealing with systems that arelattice matched to { 100 } substrates, and that the direction of growth is [001], then the strain tensor is:a3— aa= [exx ] = aa asaa (6.5)ezz 1+v aa-asexx+(i)a3where v is the Poisson ratio (which is equal to 0.273 for Ge and 0.280 for Si [107], so on averageis 0.277 for Sii..Ge). The lattice constants aa and as are obtained by a linear interpolation between the bulk lattice constants for Si and Ge giving:aa = 5.43 + 0.23xaAa3 = 5.43 + 0.23x3A(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 averageconduction 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 become mathematically cumbersome. is given by the weighted average of E and E; i.e.,— 4E+2E2 2E4+= C6 C = C3 C (6.7)Using deformation-potential theory, the change to (i.e., A) due to strain is [106,107,112,114]:= (d + ) 1: ë = (d + E) (e + + (6.8)where d and are the dilation and uniaxial deformation potentials respectively. Further thechange in energy for a specific A conduction band minima is given by:= [d1+EUtàI}]:ë (6.9)where à is the unit vector parallel to the i ‘th A conduction band minima, and { } denotes dyadicproduct. For example, the change in the energy of the A conduction band minima along [100] isgiven by:July 12, 1995 143100=+ :ë= d(eYY+eZZ) + (d+U)eXX.0Finally, using eqns (6.9), (6.8) and (6.5) where e = e, then the energy difference between Eand as well as E and are given by:= E’°°1 = E°’°1 = AE’°°1—AE = ({aa} —1 1/3 2 1=0 — 1/3 (6.10)0 1/3)1,_= —e±andE = E°°1 =— AE = ( {aà1} — 1) :ë0 1/3 2=0 — 1/3 : ë=(— (e + e) + (6.11)1 1/3)2_=where e± — e, and u(xa) = 9.16 +O.26xaeV [106]. Eqns (6.10) and (6.11) give the changedue to strain in the energy of the band minima for E and E respectively, relative to.Thus, Eand 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 thesection regarding the changes to the conduction band due to strain. For compressive strain in thealloy layer; xa > x5 so that aa > a and eqn (6.5) has it that e± e — e > 0. Since > 0, theneqns (6.10) and (6.11) have it that E <0 and E > 0, which confirms that under compression Eforms the ultimate conduction band. Contrarily, for tensile strain in the alloy layer, Xa <x so thataa < as and eqn (6.5) has it that e± < 0. Then eqns (6.10) and (6.11) have it that E > 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 nowsolved for. The designations for the hh, lh and so valence bands are based upon the valence bandJuly 12, 1995 144strain Hamiltonian [118]. To this end, it has been determined that the quantum numbers for totalangular momentum J as well as magnetic moment (spin) m remain unchanged with the application of strain. This leads to the hh band designation of IJ= ; m1= or I ; ±) for short; the lhband designation of I ; ±); and the so band designation of I ; ±). The solution of the valenceband strain Hamiltonian [118,107] produces:I;±) hh_2 2— E — Du(xa)e± — u(Xa) (e — e)2E2 — 2 = E = — (E + A(Xa)) + J9 (Er) + A2(Xa) — 2E!hA(Xa) (6.12)2E2 — 2 = = — (E + A(Xa)) — J9 (Er) + A2(Xa) — 2E!bA(Xa)where Du(Xa) is the valence band deformation potential equal to 3.15 + l.l4xaeV [106], andA(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 average valence band energy is defined and subsequently used as the reference point for all valenceband energies; i.e.,— Elth+ETh+E80 i= V V V= A(Xa). (6.13)It is interesting to note that,defined in eqn (6.13) is independent of the applied strain. Also, because is not zero, the valence band energies in eqn (6.12) are not using,as their energy reference (substituting eqns (6.11) and (6.10) into (6.7) gives = 0, which shows that is indeed theenergy reference for the conduction band). Observation of eqn (6.12) shows that under the condition of zero strain (i.e., e± 0), then E1 = = 0, and E° = A(xa). Therefore, the energy reference for eqn (6.12) is not 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 theright-hand-side of eqn (6.4), on the conduction and valence bands of Sii.Ge. Eqn (6.3) determines the effect of the Ge alloy content. Finally, the effect of the hydrostatic force, due to the firstterm on the right-hand-side of eqn (6.4), is determined. The hydrostatic force results in either anet decrease or increase in the total volume of the crystal’s unit cell. A volume change in the unitcell will be accompanied by a change in the absolute energy of the conduction and valence bands.July 12, 1995 145This net change in absolute energy is best determined by calculating the differential change toand (i.e., andi1,). Eqn (6.8) solves for and in a similar fashion A, = al: ë, where ais another deformation potential that is characteristic of the material [105,107]. Put together, thehydrostatic change to the bulk Sii..Ge bandgap is:/XEg = AE — AE= (3d-’- — a) 1: ë = (d + — a) (e + + e) (6.14)where —a = 1.5— 0.l9xaeV [106].Eqns (6.1)-(6.3), and (6.10)-(6.14) together determine the effect of Ge alloy content andstrain on the conduction and valence bands. Specifically, the bandgap of a Sii.Ge alloy layercommensurately strained to a Sii.Ge substrate is:Eg(Xa, x) = Eg(xa) + L\Eg + min(E, E) — max(E, E). (6.15)It must be remembered that for xa > 0.85 it is possible for the L conduction band minima to become the ultimate conduction band. Therefore, the use of eqn (6.15) is valid for xa > 0.85 only ifthere is sufficient strain to ensure that the ii minima, and not the L minima, still form the ultimateconduction band.Fig. 6.4 plots the Sii.Ge bandgap for a variety of substrate cases. The most striking feature of Fig. 6.4 is the effect of strain on the bandgap. Comparing the bulk material bandgap to anyof the other strained cases shows that the Ge alloy content of the Si ixpexa layer plays a far smaller role than strain does in determining the bandgap. In fact, observation of the line for a pure Sisubstrate shows thatSi045Ge5 lattice matched to { 100} Si has a bandgap of 0.66eV, which isthat of bulk Ge. The strange shape concerning the lines for material strained to substrates ofSi075Ge2,i050Ge050,and Si-j25Ge075 is due to the fact the material is shifting from a caseof in-plane tension to compression. Take the example of a Si05OGeO substrate. When thepseudomorphic 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 thetension is decreasing and the bandgap will increase, with E forming the ultimate conductionband. 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 thischange in the direction of the strain occurs, E forms the ultimate conduction band (this is whythere is a corner in the plot, however, the E and E bandgaps continue on in a smooth fashion butdo not form the ultimate bandgap). As xa increases past 0.50 the amount of in-plane compressionJuly 12, 1995 146continues to increase which reduces that bandgap once again. The essential feature of strain is thatit always reduces the bandgap from the bulk unstrained value.I I I • I1.11 0 Bulk Material -0.9 on 25% Gb.., .0.8 \•.. -on 50% Ge .0.7 ,.....,....•,,,,,,06 on75%Geon GeX/20.5on0%Ge0.4 • • I • I0.0 0.2 0.4 0.6 0.8 1.0Germanium Mole Fraction XaFig. 6.4. Sii.Ge bandgap when grown commensurately to a variety of substrates orientedalong (100). All values reflect the energy from the top of the valence band to the lowest A minimain 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.Examination of eqns (6.10)-(6.11) shows that under zero strain, when e1 = 0, E = = 0. Thus, is theposition of the ultimate conduction band in the absence of strain. If the position of the unstrainedconduction band is known, eqns (6.lO)-(6.11) will yield the offset to the conduction band due toany strain in the layer. Fig. 6.5 plots E and E relative to using similar substrates as found inFig. 6.4. Observation of Fig. 6.5 shows the changes in E and E to be quite linear in terms ofstrain. Furthermore, whenever the pseudomorphic layer is under compression then E forms theconduction band, but when the layer is in tension then E forms the conduction band. Finally, Echanges more rapidly than does E for a given increase in the amount of strain.July 12, 1995 1470.50 • • •on0%Ge0.40 EE2 on 25% Ge0.30- Con5O% Ge020on75%GeL 0.10 -0.00 -.-..... ...-.. ...-....-0.10 .........-..-....Sii75%Ge-0.20 :.-•-.--...-... on5O% Geon25%Geon0%Ge-0.30 • I • I I0.0 0.2 0.4 0.6 0.8 1.0Germanium Mole Fraction xaFig. 6.5. E and E conduction band energies relative to the unstrained conduction band edgefor SiiFe commensurately grown to a variety of substrates oriented along (100). The ultimate 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 relativeto the unstrained valence band edge. Fig. 6.6 plots the hh and lh bands relative to the unstrainedvalence band using similar substrates as found in Figs. 6.4 and 6.5. The so band is not plotted because strain simply continues to lower the band peak even further, meaning that the so band willnot be of any consequence regarding the transport of holes. Comparison of Fig. 6.6 with Fig. 6.5shows that unlike the conduction bands, the valence bands respond in a non-linear fashion with respect to an applied strain. Furthermore, there is not as large a change in the energy of the valencebands due to strain as there is in the conduction bands. Finally, whenever the Sii..Ge layer isunder 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 theconduction bands in Sii..Ge under the influence of strain. Fig. 6.7 plots the surface of constantJuly 12, 1995 148energy that envelopes the six A minima in Si083Ge17 commensurately strained to (001) Si.Since the pseudomorphicSi083Ge17 layer is under an in-plane compressive strain, then Fig. 6.5shows 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 andE for the case considered is 116 meV. As a result of the choice for the energy surface, the ellipsesthat represent are reduced by 33% compared to the effipses that represent E. If a more realistic surface energy of 2kT (= 52meV at room temperature) were used instead of 200meV, then theE band would not be seen at all. This demonstrates the profound effect that strain imparts to theSiiGe layer.0.050.00-0.05-0.100.0 0.2 0.4 0.6 0.8 1.0Germanium Mole Fraction XaFig. 6.6. E and E’ valence band energies relative to the unstrained valence band edge forSii..Fe commensurately grown to a variety of substrates oriented along (100). The ultimate valence band edge will be formed by the band with the highest energy.0.200.150.10July 12, 1995 1491.Fig. 6.7. Constant energy surface plot depicting the E and E bands in Si083Ge0 17 commensurately strained to (001) Si. The k-wave vectors are normalised to one-half the length of the reciprocal lattice vector. The constant energy surface is set at 209meV above the minimum in the E4band. The E band lies 116meV above the E band. The E ellipses have a longitudina1 extent o’f0.8, 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 themovement of the conduction and valence bands in Sii..Ge under the influence of strain. Furthermore, the most recent material parameters regarding deformation-potentials have been included.However, there is still considerable change occurring to the relevant material parameters ofSii..Ge at this time. As Sii..Ge becomes a more important material in mainstream commercialICs, the need to ultimately obtain the relevant material parameters will force the solid-state community to finalise on the parameters. This process will most likely follow the course that1.00 1.00 kJuly 12, 1995 150MGaiAs took, in which a decade passed before the solid-state community settled on a firm setof material parameters. In any event, this section has clearly shown the profound effect that strainhas on Sii..Ge; so much so that strain produces more of an effect on the bandgap than does theGe alloy content.6.2 Band Offsets in Sii..GeSection 6.1 presented all of the relevant material parameters to describe the conduction andvalence bands of a Sii..Ge alloy layer commensurately strained on top of a { 100) SiiGesubstrate. This section will present the band offset models that predict the valence band and conduction band discontinuity at an abrupt heterojunction. Therefore, when the results of this sectionare combined with the results of Section 6.1, all of the relevant models for Sii.Ge regarding theposition of the conduction and valence bands within a device can be determined.The seminal theoretical work on the band alignments between Sii..xex1 and SiixGex(where the 1,r subscripts refer to the left and right films respectively), when commensurately strainedon top of a { 100) Sii.Ge substrate, was done by Van de Walle and Martin [106,119,120]. Theyanalysed a SiGe system in one dimension using a quantum mechanical model. To remove the issueof boundary conditions that would destroy the crystalline periodicity required to establish Blochfunctions, they developed a supercell structure. The supercell structure had a unit lattice cell that wasconstructed of n Si atoms followed by n Ge atoms. By extending this unit supercell to infinity, thoughthe Born-von Karman boundary conditions, Van de Walle and Martin were able to obtain the bandoffsets. In order to establish that the size of the supercell was large enough to ensure bulk materialproperties away from the heterojunction, the band offsets were determined for a variety of n. Van deWaRe 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 lattice constant away from the heterojunction. Therefore, Van de Walle and Martin concluded that theperturbing effect of the abrupt heterojunction was indeed localised to the space immediately surrounding the interface.The main conclusion from the work of Van de Walle and Martin is that the average valenceband offset A, between a pseudomorphic Si to Ge heterojunction, whether commensuratelystrained to either a { 100) Si or Ge substrate, is a constant of 0.54±0.04eV (where the Si is lower in energy that the Ge Numerous other individuals [12 1-124] have gone on to perform exJulyl2,1995 151perimental measurements of zVP with variations that are always lower than 0.54eV, and which areas 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 resultsfrom Yu [125] were:AEv(Xai Xar) = (0.55 — 0.12x) (Xai Xar), (6.16)where Xa(, Xar and x refer to the Ge mole fraction in the left, right and substrate crystals respectively. 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 materialparameters for bulk Si and bulk Ge. To complicate things further, the material parameters thatgovern the conduction band and valence band movements due to strain have considerable variability depending on which experimental method is used to obtain the results. At the moment there isno clear set of material parameters to use in order to determine the band offsets and movementswithin SiGe. The complexity of the SiGe system is quite high, however, it is essential that the material science community finalise on a set of material parameters and models so that SiGe HBTsmay be accurately simulated.Use of eqn (6.16) produces conduction band offsets AE that are far too large. Experimentalmeasurements of AE [105,111,126,127] show that there should be no more than ±30meV ofoffset between Si ixapexa, and SiiGe grown on a pure Si { 100) substrate, where Xal and Xarcan take on any value in the range of 0 to 1. Furthermore, recent measurements by Gan et. al.[128] have shown that AE should equalO.64xaieV when xar = x =0. Use of eqn (6.16) produces= 0.8OxaieV. By reducing LcP from 0.55eV back down to 0.49eV in eqn (6.16) produces:v(Xaiar) = (0.49—0.12x) (XaiXar). (6.17)Use of eqn (6.17) instead of eqn (6.16) reduces EiE to be no more than +48meV and -42meV (ascompared to +30meV and -100meV), while also giving AE = O.74xaieV. Finally, if Du(xa) in eqn(6.12) is changed to 2.04 + 1.77xaeV [107] then IXE remains unchanged and AE = 0.68xaieVThe use of eqn (6.17) instead of eqn (6.16) is within the experimental error of the measurementsin [125]. Further, eqn (6.17) when combined with Du(xa) = 2.04 + 1.77xaeV produces conductionand valence band offsets that match experimental observations closer than when the material values proposed within [125] are employed. Thus, there is no clear set of parameters as of yet for theJuly 12, 1995 152modelling of the SiGe material system. However, the differences between the various models presented here is within 50meV. Therefore, in terms of the studies to be presented later on in thischapter, a small discrepancy of 50meV will simply cause a slight variation in the Ge alloy contentof the various layers, but will not effect the ultimate function of the HBT.Fig. 6.8. Conduction and valence band energies including all of the band offsets for a Sii3e1to a SiiGe heterojunction commensurately strained to a { l0O} Sii..Ge substrate. The designation of 1 and r refer to the left and right crystal respectively, where all A energies are referredto the crystal on the right.Eqn (6.17) provides the critical model that relates the band energies of two different SiGecrystals across an abrupt heterojunction. Once zVL, is known, then by using Fig. 6.8, all of the other 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:EElhV,hhi—VEJuly 12, 1995 153= + (A1 — 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)= AE + [min(E’ , E”) — min(E’ r E4’ r)]= 1E + (El — E’ )AE=LE+ (E4lr)where the average bandgap Eg is equal to the bulk alloy bandgap given in eqn (6.3), plus the hydrostatic change to the bandgap AE8 given in eqn (6.14). Therefore, eqn (6.18) provides all of thenecessary 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 complexnature, 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 theband 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 nonlinear nature of the strain tensor, is it is not possible to achieve a simple linear approximation forthe conduction bands. By performing a multivariate Taylor expansion of the conduction bandmodels in eqn (6.18), up to and including second order terms, yields:AE 0.1429 (Xar — Xai) x — 032789Xr + 0.O2l55Xar + O.32985x1— O•0225Xai0.1751 (Xar — Xai) x — 0.34084x,.—0•4381Xar+ 0.34281x1+O•4338Xai (6.19)AE 0.1268 (Xar — Xai) X— 0.3214lx +0.24973Xar + 0.32338x1—0.25070Xai— 0.02723x +0.04836XaXs — 0.01943x + 0.68368x — 0.68454Xawhere all results are in eV, and E’2 =— E. Eqn (6.19) is accurate to within 1% of the fullmodel given in eqn (6.18) over the entire allowed range for Xal, Xa,, and x. The multivariate Taylor expansions were centered around Xal = 0, Xar = 0.5, and x = 0.5. Thus, eqn (6.19) should strictly be used with Xal <Xar however, if this is not true, then simply interchange Xal and Xa,. andJuly 12, 1995 154multiply the result by -1. If the interchange of variables is not performed for Xal > Xa,, then the error in eqn (6.19) will rise to 1.5%.Examination of eqn (6.19) provides insight into the conduction bands of SiGe. Considering2 first, the two last linear terms in xa and x are the dominant terms. Therefore, to a crude approximation, E’ 2 O.684(x— xa); which corrects the proposal of E’ 2 O.6Xa by People [1051and 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 coincidencethat the coefficient that governs the x dependence in AE is 0.1268, as compared to the coefficient of 0.12 in eqn (6.17). The largest portion of the substrate dependence in eqn (6.19) is due tothe model for A Therefore, the material science community must determine for certain the effects of substrate strain, in order than SiGe devices can be developed where substrate strain is utilised. Finally, the non-linear terms in eqn (6.19) stem mainly from the non-linear dependence thatthe 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 heterojunction under the proviso that the entire system is commensurately strained to a { 100) Sii..Gesubstrate. The first thing to note is that EE is generally smaller than z\E, and is of such a naturethat in going from the left to the right there is a downwards step. The reason for not classifyingthis 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 detailedknowledge 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 downwards AE. Contrarily, AE is in general quite large, much larger than and is of an upwardsnature in going from a pure Si crystal to a Sii.Ge crystal. Most importantly, Fig. 6.9 clearlydemonstrates that the character of the conduction band can change between E and E whencrossing a heterojunction. Fig. 6.10 goes on to show that AE indeed has a complex nature whenstrain is brought into the picture. There are three distinct regions in Fig. 6.10: 1) when x < (xai,xar) then AE is governed by E1 to E’ r 2) when Xal <Xy <Xar then AE is governed by E’1 to4. r 3) when x> (xai, xar) then AE is governed by E’1 to E’ rTo conclude this section AE is plotted in Fig. 6.11. The various parameters are identical tothe ones in Fig. 6.10. As with AE also displays the same type of complex features which areJuly 12, 1995 1550.6 • • •0.54’AE0.4• (a)CSubstrate Germanium Mole Fraction x0.5 • • •0.30.2E4,rb0.102 04\::T/E10Substrate Germanium Mole Fraction xFig. 6.9. E and E conduction band minima to the left and right of an abrupt heterojunctionwhen commensurately grown atop a { 100 } SiiGe substrate. All energies are relative to E onthe right hand side of the heterojunction. (a) Xal 0, Xar = 0.15; (b) Xal 0, Xar = 0.30.July 12, 1995 156Substrate Germanium Mole Fraction xFig. 6.9. Continuation of Fig. 6.9 from the previous page. (c) Xal 0, Xar = 0.45; (d) Xal =0, Xar= 0.60.I • I I • IbEE4,r7••••••••••• \EZr•••........i• I • ISubstrate Germanium Mole Fraction xII0.2 0.40.40.30.20.10.0-0.1-0.2-0.30.00.30.20.10.0-0.1-0.2-0.3-0.40.00.6 0.8 1.0(c)(d)E’1E4,r/.l0.2 0.4 0.6 0.8 1.0July 12, 1995 1570.2 0.4 0.6 0.8 1.0Substrate Germanium Mole Fraction xFig. 6.9. Continuation of Fig. 6.9 from the previous page. (e) Xal 0, Xar = 0.75.-0.0.20.10.0-0.1-0.2-0.3-0.4-0.5-0.6-i. zr.(e)0.00-0.05—toE= Xal + 0.2E2,1tOE2.r0.750.000.400.601.00 0.80Fig. 6.10. tE when xa,. = Xal + 0.20, afld Xal and x are varied. The right side is the reference.XalJuly 12, 1995 158hh lhdue to the ultimate valence band changing from to . Just like Fig. 6.10, there are three distinct regions in Fig. 6.11: 1) when x < (xai, xar) then AE is governed by E’1 to E’ r; 2) whenXal <Xç <Xar then AE is governed by E’1 to E1”r; 3) when x > (xai, xar) then AE is governedlhl IhrbyE toE6.3 Electron Transport in Strained Sii..GeSections 6.1 and 6.2 present the necessary Sii..Ge material models to determine the overallband diagram, including offsets at abrupt heterojunctions, within any SiGe solid-state device. Thissection will focus on determining the transport models for electrons and holes within the Sii..Gematerial system. Essentially, the models presented in all of the previous chapters are applicable tothe study of SiGe-based devices. For example, Chapter 4 presented the EB SCR transport models-0.05“—;: -0.10-0.15 0.00l.oo 0.80Fig. 6.11. zXE.1, when Xa,. = Xal + 0.20, and Xal and x are varied. The right side is the referenceJuly 12, 1995 159which included the effects of tunneling and the mass barrier. Therefore, Chapter 4 can be appliedto a SiGe device to determine if the EB SCR will generate current-limited-flow. However, caremust be exercised in the application of Chapter 4, and indeed all of the other chapters, as there is amulti-band model for the Sii..Ge material system. This section will discuss and present thetransport models for the multiband Sii.Ge material system.From the work in the previous two sections, it is clear that the conduction and valence bandsare both broken down into two distinct sub-bands (the so valence band is ignored as it is alwayslower 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 higherenergy satellite band never forms the ultimate conduction band, E and E in the Sii.Ge material system can both form the ultimate conduction band. Thus, it is possible to have near equilibrium transport occur within both E and E at spatially separate points with the device; this is incontrast to the AlGai..As material system where transport in the satellite band need only be considered under extreme non-equilibrium injection conditions. Further, this multiband transport canalso occur in the valence band of the Sii..Ge material system. Given the strange band offsets depicted in Figs. 6.9 to 6.11, it will be shown that transport within the Sii.Ge material system canoffer a rich set of possibilities, both in terms of commercial HBT optimisation, and as a tool forresearch into the mechanics of transport within solids.Considering the valence band first of all, Fig. 6.2 shows that EL and E are degenerate under the condition of no strain. More importantly, the maxima in both E! and E occur at thesame point in k-space. Even under strain, the maxima in E and E’ remain coincident in their kspace location. Therefore, there is very little issue regarding the conservation of crystal momentum in moving between the lh and hh bands, if the mass barrier that would occur at the heterojunction for holes is neglected, then to a good approximation one need only consider the ultimatevalence band in terms of hole transport. However, if the strain is small, so that the energy separation between E and E is less than —2k1 then transport within both bands needs to be considered. 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 crossing the mass barrier could be reflected, leading to a 3-fold reduction in the transport current. A 3-fold reduction in the transport current would be equivalent to having an upwards step in energy ofJuly 12, 1995 16028.5meV at room temperature. Therefore, when 11Ev is less than —2kT one must consider paralleltransport within E and E!j. But, no matter how large or small AE is, the calculation of the valence band effective density of states N must include both E and E’ due to the large differencein the lh and hh effective mass.The complexity of the valence band stems from the coincident k-space location of the bandmaxima for E and E. Examination of Fig. 6.7 shows that the Sii..Ge conduction band minima are not coincident in k-space. Thus, in order to move between any of the six A minima inSii.Ge, crystal momentum must be conserved. There are two scattering processes that are responsible 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 kaxis, such as the [001] and [OOT] bands that form E; while f scattering moves electrons betweentwo bands that are not along a common major k-axis, such as the [100] and [010] bands withinE. Given the proximity of the A minima to the Brillouin zone edge, an Umklapp process can easily take place, leading to g scattering, because of the relatively small k-space separation that mustbe conserved. On the other hand, f scattering involves a k-space conservation that is over one-halfof the reciprocal lattice length. Therefore, it is found that f scattering rates are almost 10-fold lower than g scattering rates [129]. Tn terms of the E and E band groupings, g scattering will not result in movement between the E and E bands. Finally, for small distances, such as those that aretypical 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 betreated 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 Ebands, 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 direction of charge transport across the heterojunction. Such a powerful Bragg plane could enhancef scattering, leading to a coupling between the E and E bands. Consideration of the k-vector involved 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 heterojunctionwould lead to a significant increase in the f scattering rate [1081. Therefore, the independence ofthe E and E bands should be maintained even at an abrupt heterojunction. This leads to the for-July 12, 1995 161mation of a selection rule regarding transport in Sii..Ge that prohibits a mixing between the electrons in E and E.g scattering4-K’ E\,,EYE2 \If scatteringgscatteringFig. 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 electron transport within the Sii..Ge material system begins with calculation of the electron effectivedensities 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 Brillouinzone. The pre-multiplying factor of n in eqn (6.20) results from the degeneracy of the E and Ebands and the fortuitous designation where n is equal to 2 or 4. Then the effective density ofstates, assuming that the band-width is Eb and that Boltzmann statistics can be used, is equal to:Eb E E,, E E(k)N = JdE g(E)e’= Bt. 6(E — E(k))e’ = Bt.__ekT. (6.21)July 12, 1995 162Eqn (6.21) can be easily integrated with little error by assuming that the limit of integrationcan be extended past the Brillouin zone to infinity; i.e.,00 1ik h2k 1i2k 3/2N = _!_e fdkxe2mu1cTfdkye2mh1cTfdkze2mtkT = 2ne (6.22)where* 1/3m = (m1m )The appearance of the term exp(-EIk7) in eqn (6.22) is due to the fact that the reference energy forthe conduction sub bands is not located at the band minima, but at.One could have maintainedthe reference energy at the band minima, but then N and N would have different energy references and eqns (6.1O)-(6.11) could not be used directly within eqn (6.22). Furthermore, by employinga common energy reference of,the total conduction band effective density of states is:3/2( EN = 1V+N=4(m’. (6.23)2ich ) JFinally, it is possible to reflect N from the energy reference of back to the ultimate conductionband 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 ofthe hole effective density of states within the valence sub-bands, leading to:3/2 E 3/2N = 2e(mhhkT\1and N = 2e’(mlhkTN (6.24)L21th ) L2ith )Then, owing to the different effective masses for the lh and hh, the total valence band effectivedensity of states is given by:3/2N = N’ +N = 21 kT (mhh)3”e’+ (mlh)3”2e’ . (6.25)2ith j )In a similar fashion to the conduction band, the reference energy for the valence band is not located at the ultimate valence band maxima, but at the location of the valence band maxima under thecondition of no strain. To reflect N back to the energy of the ultimate valence band maxima, multiply eqn (6.25) by exp(-max(E, E)Ik1).July 12, 1995 163Eqns (6.22)-(6.25) present the conduction and valence band effective density of states forthe Sii.Ge material system. These equations represent an extension to the traditional definitionsfor effective density of states, necessitated by the complex band structure of Sii..Ge under the influence of symmetry-breaking strain. Finally, the electron and hole concentrations n and p respectively are defined using eqns (6.25) and (6.23) in the usual non-degenerate manner, to yield:E En = NceU’ and p = Ne‘ (6.26)where Ef is the electron quasi-Fermi energy relative to,and is the hole quasi-Fermi energy relative to the unstrained valence band maxima. After allowing for the fact that the conductionand valence band energy references are separated by g’ as is shown in Fig. 6.8, then:E8n=pn=NNe kT3 ( E E ( E,t’ g (6.27)= ( kT ‘ (m* 3/2 eICT + 2e’ (mhh) 3/2ei; + (mlh) 3/2e eu’,\1tli )where the average bandgap g is equal to the bulk alloy bandgap given in eqn (6.3), plus the hydrostatic change to the bandgap Ag given in eqn (6.14). Unlike eqns (6.22)-(6.26), n given ineqn (6.27) does not reference itself to an abstract energy reference, but is the standard definitionfor the intrinsic carrier concentration.With the effective density of states defined for the conduction and valence sub-bands ineqns (6.22)-(6.25), along with the carrier concentrations and n given in eqns (6.26)-(6.27), it ispossible to define the built-in potential Vb1 of apn-junction. Looking at Fig. 6.13, then clearly:Vbi = (Eje,j — (E — Eg,p)) + (x— x) = ln(”)+ (,—x). (6.28)ni,p c,nComparison of eqn (6.28) with eqn (4.69) shows, apart from the effect of a spatially varyingeffective density of states (which is neglected in eqn (4.69)), exact agreement if— = Vbjis the variation in the vacuum potential across the SCR extrapolated back to equilibrium conditions. Thus, the electron affinities Xp and xn on the p- and n-sides of the junction are evaluated atx = and x = -x, respectively. If and are spatially varying, then as a changing applied biasmoves and x,, Vb1 will also vary with applied bias. It is well known that Anderson’s electron affinity rule for the calculation of AE is not correct. However, at some distance far from the heteroJulyl2,1995 164junction, and must become bulk-like. The question becomes how rapidly do Xp ifld Xnreturn to their bulk values? The deviation of AE from—has been attributed to such thingsas a complex rearrangement of charge surrounding the heterojunction. Thus, if this rearrangementof charge is abrupt, as is potentially suggested by Van de Walle and Martin [106, 119,120], thenand would definitely change over the width of the SCR; leading to an extra driving force forthe transport of charge than is not taken into account by any known theories. If this rapid variationin and x turns out to be true, then Vbj will not be a constant as is given in eqn (4.69), but is instead given by eqn (6.28) with Xp and being a function of position. Finally, Vbj contains all ofthe desired information regarding,Xn’ and thus zSE. Therefore, if the pn-junction could bedriven up to and past Vbj, without resistive effects dominating the transport current, then information regarding,X and thus AE could be extracted. This possibility of operation near and pastVbj will be considered in Section 6.4.E (eV)IFig. 6.13. Equilibrium band diagram of a np-junction, showing the relevant energies and potentials. Ef is referenced to E while is referenced to E. Note that the Vacuum potential is continuous while E and E are not.Unstrained EUnstrained EJuly 12, 1995 165Concentrating once again on the conduction band, the final models for electron transportcan be determined. By the previous arguments, electron transport in the EB SCR and the neutralbase can be modelled as two parallel conduction paths via E and E. It is further assumed, atleast 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 wouldbe given by the sum of E and E conduction solved by the standard transport model given byeqns (4.78)-(4.79) and (4.92).To this end, the correct parameters to use in the standard EB SCR transport model regardingE conduction are:m 1 = mQX) my, 1 m1(A) m 1 = m(Z) .L1 = relative to= 4q2Jm imikT = 4tq2Jm1mkT= q2h3 h3 J2tm(A)kT N(6.29)While the correct parameters to use in the standard EB SCR transport model regarding E conduction are:m, = m1(A) m, 1 = m(z) m = m(A) = relative toF2— 4irq2Jm lmzl kT — 4itq2Jm1m kTI2ND ‘1 — q2‘NNfs0— h3e— h3 N — J2itm1(A)kT\ (6.30)q2N ek band degeneracy— J2tm1(I)kT Ee kT+2 kTExamination of Fjo in eqns (6.29) and (6.30) reveals the exact context of parallel transport withinE and E. There are ND majority electrons that are distributed between the E and E bands, depending upon the energy separation between the two. Since there are twice as many A minima inE as compared to E, there will be preferential transport within E, all other things being equal.July 12, 1995 166Finally, the electrons within E and E move with a velocity that is proportional to the square-root of l/mj and 1/rn respectively. Therefore, neglecting the energy separation between E andE, the E band will carry 2Jmi/m = 4.54 times the current compared to E. Furthermore, because E has the light transverse mass parallel to the direction of transport, as compared to theheavy longitudinal mass for E, not only is the mobility higher [132] but the probability of tunneling 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 onlybe maintained if the neutral base width is small enough to preclude coupling via the f scatteringprocess. For current-day SiGe HBTs the neutral base width Wfl, is under ioooA and is rapidly approaching 300A [10,26,28,100-103,133]. With such a small neutral basewidth it is reasonable tomaintain the separation between E and E used for the modelling of EB SCR transport. With Eand E treated independently, the neutral base transport current within either one of the E sub-bands is given by Kroemer [38] as:——. qBE fnNB=N(x)dx[e kT-(6.31)Wnbn(x)wherej = 2 or 4. It should be noted that eqn (6.31) is an extension of Kroemer’s work which wasbased upon Shockley boundary conditions. The reason for generalising the diffusion coefficientD, as was discussed in the previous paragraph, stems from the fact that the mobilities within theE and E bands will be different due to their highly anisotropic nature [132]. This leads to theconclusion that D > D because rn < rn1. Further, each sub-band will have its own intrinsic carrier concentration nb., which is determined in the same way as 1V, N and the total n to yield:flj NNe kT = =‘ n = n4 + (6.32)Finally, due to the independence of the E and E bands, a separate quasi-fermi energy must bepresent in order to account for the driving force within each sub-band. For this reason, there isto characterise transport within the E band, and AE to characterise transport within theE band.The final model for electron transport within the SiGe HBT is achieved using exactly thesame methods employed in Section 5.3 for the derivation of eqn (5.29). Eqn (5.29) is based uponJuly 12, 1995 167the general models of Section 2.2. Applying the models of Section 2.2 to the solution of transportwithin the conduction sub-bands yields for the E band:= [4L401’ (6.33)where, based upon the findings of Chapter 5, the failure to include recombination effects specifically in the calculation of the total transport current 4 will produce an error that is of order 1/13.To reiterate, 4 is the EB SCR transport current solved by the standard transport model given byeqns (4.78)-(4.79) and (4.92) with the pertinent parameters obtained from eqn (6.29). In a similarfashion, transport within the E band is:4= +-1,(6.34)fs 1NBwhere F8 is once again the EB SCR transport current solved by the standard transport model given 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 discussed earlier in this section, the coincident nature of E’ and E in terms of k-space locationprohibits an independent treatment, such as was done for the conduction sub-bands, of the two valence sub-bands. Fig. 6.2 clearly shows that the valence band of unstrained SiGe, and for that matter all semiconductors, is a multi-band system. To this end, transport within the unstrained valenceband is determined by appealing to a single total effective mass that correctly produces the totalvalence band effective density of states. Then, by way of experimental measurement, a single mobility is extracted to characterise the valence band as a whole. This method breaks down for thecase of strained SiGe, as the degeneracy of E and E is lifted and the energy separation is dependent upon the amount of strain present. This prohibits the use of a single effective mass andfixed mobility to characterise the valence band of strained SiGe. Yet, the valence sub-bands cannot be treated independently for the purpose of determining charge transport, as was done for theconduction sub-bands.Essentially, the only way to solve transport within the strained SiGe valence band is to resort to Monte Carlo simulation. However, as was pointed out at the start of Chapter 4, Monte Car-July 12, 1995 168lo simulators cannot presently model the non-local effects of tunneling. To this end, the followingtwo assumptions are made: 1) hole transport within the EB SCR is considered ballistically due tothe small width of the SCR, but the holes will always attempt to minimise their energy by movingto the highest sub-band; 2) due to the strong intervalley scattering that occurs between E andE, because of their coincidence in k-space, transport within the wider neutral regions of theHBT is treated using a single equivalent valence band.The implication of the second assumption is straightforward; transport is treated in the standard single equivalent valence band approach. The only consideration that must be made in treating the valence band as a single valley is the mobility will change with strain. In a region wherethe Ge alloy content is not uniform the strain will change with position, which will move andE either closer or further apart in terms of energy. Since the lh mass is much smaller than the hhmass, considerable change to the mobility of the material will occur as E and E move closerand further apart. This leads to a complex and spatially non-uniform mobility that is only due tothe energy separation of the valence sub-bands. Other effects such as impurity and alloy scatteringwould 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, conservation of transverse momentum must be ensured for a hole to change bands, which leads to themass barrier results of Chapter 4. However, the hole will attempt to take the path of least resistance by minimising its energy; it may either continue on in the sub-band it currently occupies, orchange bands in an attempt to minimise its energy while taking into account the possible loss orgain due to the mass boundary effect. The complexity of transport within the SiGe valence bandstems purely from the large difference in the lh and hh masses. If the lii and hh masses were thesame, then transport would occur along the highest energy sub-band (in terms of electron energies), with a spatially varying N to consider.The model for the EB SCR in Chapter 4 is simple in that the heterojunction is abrupt; thereby producing two regions, separated by a single mass barrier where the material parameters within 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 EBSCR 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 holesinto 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 standardflux Ff (Fj does not include the mass barrier) using the appropriate mass from eqn (6.1), and- q2(N— J21tmkT A)’where j is either hh or th. Within the standard flux model, the base barrier potential Vblonger the one from the originating band, but is given by the maximum of E and E in theneutral 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 barriercan 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 standardflux 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 forthe reflecting mass barrier, but with the infinite upper limit of integration replaced with the Vbthat is appropriate to the sub-band that injected the holes.The physical explanation of the valence band transport model is: holes ballistically travelthrough the EB SCR, perhaps tunneling through a Valence Band Spike (VBS), by way of independent E’ and E bands. Upon reaching the mass barrier the holes attempt to occupy the lowestenergy band, and do so by exchanging sub-bands, if necessary, while taking into account any losses or gains due to the mass barrier. Depending upon the construction of the HBT, the emergingfluxes from the EB SCR, contained within E and E, will generally be characterised by different driving forces of AE and AE respectively. However, due to the strong intervalley scattering that occurs between the valence sub-bands upon reaching the neutral region, a common quasiequilibrium condition of will result for both E and E. Therefore, the final transport model for holes is:1 1T,holes= [Fh+F+JT,TOl (6.35)where F and F.h are the full EB SCR transport models, and T,utral is the neutral regiontransport current calculated by eqn (6.31) using n from eqn (6.27).July 12, 1995 1706.4 The Accumulation Regime Beyond the Built-In PotentialChapter 4, and therefore Section 6.3, have both dealt with transport for an applied bias VBEthat is less than the built-in potential Vb1. For the case of a band diagram where there is a negativestep, as shown in Figs. 6.13 and 4.2, as VBE approaches Vbj, a current density of —106A/cm2willflow (this is based upon an emitter doping that is ‘-.1018cm3).At a current density of lO6AIcm2,resistive effects will dominate the device and limit the internal forward bias to be much less thanthe external applied bias. For example, with an emitter area of 1 urn2, there would be a current oflOmA at a current density of 106A/cm2.Even with an unrealistically low emitter contact resistance of 502pm2,there would be a 500mV drop to the external applied bias before it evenreached the junction. It is for this simple reason that observation of the device with a forward biasnear, 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 thepath of the electrons trying to surmount the potential barrier of the EB SCR. A positive potentialstep would force the electrons to surmount the entire barrier, because unlike the CBS there is noway 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 temperaturewould be reduced by a factor of exp(-240125.9) 10. Therefore, when VBE approaches Vbj, thecurrent density will have dropped to only lO2AIcm.A current density of lO2Afcm will certainlybe observable, and would even allow for VBEtO exceed Vbj.Before going on to present a physical demonstration of operation beyond Vbj, the transporttheory for this domain of operation is first developed. When VBE is exactly equal to Vbj, and if theresistive effects are negligible, then the band diagram will be flat except at abrupt heterojunctionsor regions of spatially non-uniform Ge alloy content (see Fig. 6.l4b). For this reason, the point atwhich VBE is exactly equal to Vbj is termed flat-band (in much the same manner as the flat-bandcondition in MOSFETs). At flat-band there will be no space charge present. As VBE is increasedpast Vbj (see Fig. 6. 14c), an accumulation region of mobile electrons on the n-side, as well as mobile 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 casewhere VBE < Vbj, and a depletion region forms where the space charge is composed of immobileJuly 12, 1995 171ion cores from the dopant atoms. For this reason, operation past Vb1 is termed the accumulationregime. Finally, as VBE is increased further, the accumulation of charge will proceed exponentially, with a net reduction to the potential step, and therefore, a continued exponential increase in theEB SCR transport current.(a)_EEn-sideE(b) VBE=VbjEj_1\..E,mobile electrons____ ____- - --(c) \• {,—Efhmobile holes —i VFig. 6.14. Band diagram for a np-junction with a positive step potential (i.e., AE <0). (a) equilibrium; (b) flat-band where VBE = Vb; (c) accumulation region where VBE> Vbj.July 12, 1995 172A reasonable first approximation to the complex accumulation regime begins by assumingthat for operation just beyond Vb1 the accumulation layer is non-degenerate. Based upon this assumption, and neglecting the effects of a non-uniform e, the Poisson equation in one dimensionbecomes29_ (ND — N e kT on the n-sided2qi D= (6.36)dx2q2 (N — NAekT) on the p-sidewhere and 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 bysymmetry arguments) to yield the following implicit transcendental function:=e d+A2, (6.37)ND[(fl+Al)ekT+kT]where A1 and A2 are arbitrary constants. There is no way to reduce eqn (6.37) down to a functionof simple transcendental functions, nor will it be possible to invert the result. However, it is reasonable to assume that the charge in the accumulation layer will overwhelm the backgrounddopant ion potential. With this assumption eqn (6.36) is recast to:22,,—9NDe kT on the n-side8 (6.38)dx 2kT on the p-sidewhose n-side solution is?2IJi e’x = ±4_$ 1dW+A2. (6.39)ND AiecT+kTIt 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 radiJuly 12, 1995 173cal. This linear term in ‘qi,.1 produces the asymptotic solution to the underlying depletion spacecharge. Since eqn (6.39) assumes that the depletion space charge is negligible in comparison tothe accumulation charge, the linear term in qI,j is lost.The solution of eqn (6.39) begins with the determination of A1. If the neutral assumption isemployed at -x, (the boundary to the accumulation region), then because the doping ND is a constant the electric field will vanish. Since the electric field is given by (lIq)diIdx, then talcing thederivative of eqn (6.39) with respect to N’,. inverting it, and setting it equal to zero with = 0 atx -x, yields A1 = -kT With A1 = -k7 eqn (6.39) is solved using the change of variables‘tiny = 2e’—lto produce:= ± asin(2ekT — 1 + A2 e = sin A2]+. (6.40)q42NFinally, applying the energy reference of Nn = 0 at x -x, to eqn (6.40), and choosing the positivex-direction produces:(x+x’e = cos2l I (6.41)ha1,)where1 IekTa1=By appealing to the symmetry of the problem, the p-side solution of eqn (6.38) is:‘lip (x—xe = cos2 I I (6.42)2a1)wherea1=It is important to realise that ‘qi, is set equal to 0 at x = -x,, and is set equal to 0 at x = x. However, the form of the Poisson equation requires that when qI,j joins up with at the heterojunction (i.e., at x = 0), the joint be analytic up to first derivatives. Given that we are solving foraccumulation and not depletion, then continuity of and Nip requires that:July 12, 1995 174VBE— VbIN’(O) — v(O) = q (VBE— Vbx) 2kT = cos (2ai )COS (2a1 pJ (6.43)Further, continuity of the electric field requires that:—=—----tan ( X, “1 = —--tan I “1. (6.44)dx dx a1 k2a1J ap 2a1)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 thesquared cosines of and xJ2ai,. Choosing the positive roots of the solution for eqns(6.43) and (6.44) yields:II4NANDeAVT+ (NA—ND)2+ A NDx, = 2a1 acos I Al____________________________________(6.45)(IJ4NANDeET+ (ND—NA)2+ D NAx =2a1 acosi I vp2NDe BE Twhere .AVBE 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 thedepletion regime. In fact, when VBE is within the immediate neighbourhood of Vbj (i.e., smallIWBE), then a Taylor expansion of eqn (6.45) about the point AVBE = 0 yields exactly the sameequations 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 ofltal,n and ltal,p respectively. This saturation of the SCR width is a feature of the rapid accumulationof mobile charge that screens out the applied bias with essentially no further increase to the extentof the SCR. This result is also the point at which the assumption of a non-degenerate accumulationlayer 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 asimple constant. However, by appealing to a Taylor expansion about AVBE =0, and the asymptoticlimit for large AVBE, it is found that:July 12, 1995 175N NN(N-N)N=_____NA + ND+(ND +BE — BE — hzee,DratVBE>VkfleeDwhere— 3VT(ND+NA)2knee,D— (JNAND + NA) (JNAND + ND)In a similar fashion, the metric for the splitting of AVBE between the n- and p-sides of the junctionyields:N NN(N-N)Vrat =IN’fl(°)I NA + ND + (ND + NA)3 VBE — VBE Vee (6.47)q BE 1 AwhereVT (ND + NA)2Ve,V_ NNADNrat in eqn (6.46), as was stated a few paragraphs earlier, shares many of the same featuresas Nrat in eqn (5.9) under the depletion regime. Now, V.at in the depletion regime is exactly thesame as Nrat, owing to the spatial uniformity of the space charge due to the immobile dopant ions.However, under the accumulation regime, 1liat in eqn (6.47) starts out the same as Nrat, but due tothe mobile nature of the accumulation space charge, quickly results in an equal portioning of theexcess applied potential AVBE between the n- and p-sides of the junction. Therefore, the potentialdistribution in the accumulation regime differs markedly from what is found in the depletion regime. Finally, Fig. 6.15 plots Njj and 11at in both exact and approximate form, as well as x, andx, 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 withinthe accumulation regime. Fig. 6. 14c shows that within the accumulation regime, the positive steppotential 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 standardHBT case where NA >> ND, then x <<x, and for small AVBE one also finds ‘qi(O) <<ji,(O). TheseJuly 12, 1995 176two findings mean that the CBS within the accumulation regime will be very narrow, and veryweak in terms of a potential to be tunneled through. Strictly speaking, the transport currentthrough 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 accumulation 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 CBSwill allow a significant current to pass though it. In any event, the largest that the CBS barriercould be, by assuming WN =0, would be an energy of IAEI — and the smallest that theCBS barrier could be, assuming that WN = 1, would be an energy of IAEI — qzV (see Fig.6.16). Therefore, with Vat 1 for small IVBE (given the typical HBT doping), then the upper andlower bounds for the effect of the CBS will be fairly close together.1.000.90rii0.800.700.60ZS.VBE (V)0.50 I0.00 0.10 0.20 0.30 0.40Excess Applied Potential AVBE (V)Fig. 6.15. The exact and approximate forms for Nrat and Vrat from eqns (6.46)-(6.47). The material parameters are: ND: 5x1017cm3;NA: 1x109cm3;e: 12.0.One of the essential results of Chapter 4 was that the peak emission flux density occurred ata fixed energy relative to the height of the CBS. This result occurred only because of the parabolicnature of the potential profile within the depletion regime. Given the fairly simple model present-July 12, 1995 177ed for the accumulation regime, where degenerate effects have not been accounted for, there is little point in solving the general transport models of Chapter 4. Instead, based on the arguments ofthe previous paragraph, it seems reasonable to characterise the accumulation CBS by an effectiveenergy height. Finally, only thermionic emission over this effective CBS will be considered. Given the result from Chapter 4 that was mentioned at the start of this paragraph, the effective heightof the CBS is given by:ECBS = — qzV + q (1 — UmaxIVBE= AEI —qAV(l — Umax+ UmaxVraj)(6.48)where 0 Um 1, and Um will be taken as a phenomenological constant. Strictly, based uponthe analysis of Chapter 4, Umar will have a temperature dependence. However, as a first approximation, U can be taken as a constant independent of temperature. Then, the transport currentunder the accumulation regime is simpiy given by the thermionic term from eqn (4.79) as:ECBSFfS = FfSO V1e (6.49)where both sub-bands within the valence and conduction bands need to be considered in the caseof the Sii..Ge material system.EFig. 6.16. Diagram of the CBS that forms under the accumulation regime. Only the conductionband is shown, but a similar structure can occur in the valence band. Note: this is for one sub-band.-\‘qc,1 referenceIv(0)I = qzVV,-1July 12, 1995 1786.5 Conventional and Novel Sii..Ge IIBT StructuresThe 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 essentiallymanipulate IXEg and iXE (and thereby AE) independently. This independence between AEg andAE is achieved through two independent parameters: 1) the Ge mole fraction xa in the pseudomorphic strained alloy layer; 2) the amount of compressive or tensile strain applied to the pseudomorphic alloy layer by the substrate (i.e., the substrate Ge mole fraction x5). The addition of strainis the key to the rich possibilities regarding baudgap Engineering offered by the SiiGe materialsystem. Sections 6.1 through 6.4 have set out the various material models and transport models tostudy the flow of charge within a SiGe HBT. This section will apply the results of these previoussections 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 withoutconsideration of the critical layer thickness h. As was stated early on in this chapter, the potentialstrain in the Sii.Ge material system can be quite large, owing to the 4.2% lattice mismatch between Si and Ge. As the in-plane strain is increased (see Fig. 6.3), the maximum thickness of thealloy layer decreases in an essentially exponential fashion. The determination of h has been thefocus of numerous studies and controversies [97,99,105]. At present, there is still debate as to theexact model for h versus in-plane alloy strain, but the work of People [105] is at least a reasonable reference point. In [105], the critical layer thickness is given as:1—v 1 b2 1 hh= i+v 20ic& (_)(_n(T)J (6.50)where h is in 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 givenby (aa — a)Ia (where a is the substrate lattice constant). Substituting all of these parameters intoeqn (6.50) gives:1.928 (5.43 + 0.23a” hh=(5.43 +0.23aa) ,j ln(-4-). (6.51)Eqn (6.51) is an implicit phenomenological equation that People has fit to the best available datafor h (see Fig. 6.17). Detailed information, such as what temperature and duration can a pseudomorphic layer tolerate before relaxing is still not conclusively known.July 12, 1995 1791010.0 0.2 0.4 0.6 0.8 1.0Germanium Mole Fraction xaFig. 6.17. Critical layer thickness for a Si.xpexa layer on a { 100) Si substrate. If the substrateis 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 substrate that is { 100) Si. The emitter and collector regions are pure Si, and the base is the only region made up of Sii..Ge . The essential premise for this type of SiGe HBT stems directly fromthe early work of Kroemer [2,46,47] and Shockley [1] who called for a wide-bandgap emitter injecting into a narrow-bandgap base. Within this physical construct, the Ge alloy content xa in thebase is either fixed at some constant value, or a drift field is created in the base by increasing Xa asone 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 theHBT is lattice matched to a pure Si substrate, all regions of the device except the base have E andE degenerate, as well as E and E’ degenerate. However, compressive strain in the base produces 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 34meVFig. 6.18 presents the band diagram for the above device, with the relevant material parametersnoted. Observation of Fig. 6.18 clearly shows that electron transport will occur via E. SinceJuly 12, 1995 180= -100meV, while AE is 37meV, ostensibly all of the electrons contained by the E band in theemitter (which is 33% of the total number of majority electrons) will be reflected by AE and notcontribute to electron transport. Thus, if the EB SCR determines the transport current, then afterincluding the different effective masses, I would be 18% less than expected from a simple examination of the device that does not account for the independence of E and E. However, if theneutral base region determines the transport current, then I would be larger than expected giventhat D is higher than the bulk value. In order to determine if it is the LB SCR or the neutral basethat is responsible for current-limited-flow, the detailed construction of the device must be considered. For the devices in [100-102], where ND >> NA, then the neutral base is narrowly responsiblefor current-limited-flow; although, inclusion of bandgap narrowing effects could lead to the EBSCR being responsible for current-limited-flow. However, for the devices in [10,1341, where NDNA, then depending on how bandgap narrowing in the base splits between E and E the EB SCRwill be responsible for current-limited-flow; resulting in a much smaller increase to I than wouldbe expected from neutral base transport considerations alone. This analysis of current-day SiGeHBTs 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.AE = 37meV AE = —138meViXE = —100meV AE = —104meVIXE = 37meV AE = —138meVSi Si08Ge•2 Si e = 1120meV fl e = 6.94x10cm3Ehl_ Eg,b = 945meV = 1.47x101cm3Without LB SCR limitations, E willE’ E transport 0.25% of the current in the neutral base, leaving E to transport the reEmitter . Base Collector 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 isthe reference. The effect of the LB and CB SCR potential is not shown for clarity.July 12, 1995 181For 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 inSiGe HBTs, then logI versus VBE will look identical to the case where the neutral base limits thetransport current (i.e., the injection index will be unity). Thus, there will be no overt tell-tale signin SiGe HBTs that the transport current is not being controlled by the neutral base. However, Icwill indeed be smaller than expected due to the EB SCR limitation, plus, the Early voltage shouldbecome theoretically infinite as basewidth modulation should no longer effectI [135].The SiGe HBT where xa is varied across the base represents the device that has piqued theinterests of the semiconductor community. By generating an aiding field in the base through amonotonically increasing Xa from the emitter to the base (and hence a decreasing Eg), an fT ashigh as 113GHz has been obtained [1021. In order to achieve this remarkable metric the devicewas fabricated with as large a Axa in the base as possible; minimising the base transit time. To thisend, Xa was 0 at the emitter and was linearly ramped up to 0.25 at the collector. The result is aband 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 basewill limit the transport current (see Fig. 6. 19b). The base region, given the shape of the E and Ebands, produces a demanded current that differs between the sub-bands by a factor of 8.3; i.e., thecurrent in E will be 8.3-fold larger than E. This is not an overwhelming amount, which showsthat 11% of the collector current is carried by the slower E band. In fact, using eqn (3.8) showsthat tB for E is reduced 4.6-fold compared to tBo, while tB for E is reduced only 1.5-fold compared to ‘rho (where tBo is the ‘CB given in eqn (3.6)). Assuming that the final base transit time isgiven by the average of the results from each band weighted with the relative current carried bythe 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 reduced 4.4-fold relative to tBo, and I overestimated by 13%. In the above calculations the effectof bandgap narrowing has not been accounted for. Inclusion of base bandgap narrowing couldcause the EB SCR to limit the transport current (again, depending on how the bandgap narrowingsplits between E and Er), which would greatly effect the current partitioning between the conduction sub-bands. Furthermore, the anisotropic nature of E and E has also not been accountedfor, which would increase ‘CB even further given that would be greatly reduced.July 12, 1995 182(a)E, E = 216meV AE = 0meV= 44meV AE = 42meVAE = 216meV AE = 0meVSi Si075Ge2. Eg e = 1120meV ‘e = 6.94x10cm3Emitter SiBasehh Collector E8b 904meV 3.17x10’cmEhh, E11’ — Elh I Without EB SCR limitations, E willE‘i transport 11% of the current in the neutralbase, leaving E to transport the remaining89% of the current.(b)108• •1 E current within- /the EB SCR >,2’> “..6.io5 E current withinthe neutral base4. 2/$ ‘C,... ,-....-.in21 . /• <V E2 current within10 /.. /..•. the EB SCR100 .10-1 :, E current within/ ‘./ the neutral base10-2 • • • •0.6 0.7 0.8 0.9 1.0 1.1Base-Emitter Voltage VBE (V)Fig. 6.19. (a) Band diagram for an HBT with 25% linear grading of Ge in the base, latticematched to Si. zS.E is from the 25% Ge point in the base to the emitter. Note: the base bandgap hasa slightly parabolic nature due to the Ge alloy effects. (b) Transport currents through the variousregions of the HBT, including the collector current. NA=5x1018cm3,ND= 1x1020cm3,andWB=700A. Given that E transport within the EB SCR is not substantially larger than transportthrough the neutral base, I is subsequently 31% lower than expected from neutral base transportconsiderations alone. Thus, the neutral base is controlling I but the EB SCR does have an effect.July 12, 1995 183The 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 effective conduction band. In order to properly model a SiGe HBT, the rich nature of the E and Ebands must be included via the models developed in Sections 6.1 to 6.4. Further, the assumptionof Shockley boundary conditions (i.e., that the EB SCR is not responsible for current-limited-flow) can come under question in the design of SiGe HBTs. Finally, the importance of considering transport through the entire device becomes even more important when optimisation, or theextraction of material parameters, is sought after: for if the transport is being dictated by a regionother than the one being considered, the result will be a an erroneous conclusion regarding eitherthe 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 quitesmall because of the limits imposed on the Ge content by the critical layer thickness. For this reason, it is still common to see ND >> NA in order to maintain a usable f3. As the neutral base width isreduced, then NA must increase in order to offset a rapid decrease However, increasing NAmust be accompanied by an increase in ND or the gain will drop. With ND near the solid-solubilitylimit this is not really possible. Further, with NA and ND increasing, the EB capacitance will increase, and a tunnel diode could form. The device in [134] attempted to solve this with a constant22% Ge base content. By having a narrow bandgap in the base, the subsequent increase to Ib canbe traded off for a higher base Gummel number. However, this precludes a graded base, as the EBheterojunction is required to maintain the gain, and the critical layer thickness will not allow for ahigher Ge content (this is the alloy budget of Section 3.2). Therefore, in order to continue decreasing the neutral basewidth without compromisingfmax orf a way must be found to include higherGe contents in the base.The answer to the problem of the previous paragraph is to lattice match the HBT to aSii.Ge substrate, where x,> 0. Consider a 500A SigjgGe02emitter with a poly-Si cap, a basegraded from Si075Ge2 at the emitter to Si06Ge04 at the collector, all lattice matched to aSi08Ge02 collector and substrate (see Fig. 6.20). The base grading is started at 25% Ge instead of20% in order to increase the transport current in E relative to E, thereby reducing the parasiticeffect on tB found from the HBT in Fig. 6.19. Then, the 15% Ge base grading provides the aidingfield to keep the base transit time small. However, unlike the HBT in Fig. 6.19, the optimum augJuly 12, 1995 184mented-linear doping of Fig. 3.8 is used instead of the sub-optimum linear grading. The optimumbase profile, due to the constant Ge regions near the emitter and the base, also increases the Earlyvoltage and decreases the anomalous change to I due to the reverse Early voltage effect [11]. The500A Si08Ge02 emitter next to the base ensures that the EB SCR will be free of dislocations thatwill 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 theback injection of holes. With the wide bandgap of the poly-Si cap controlling the gain, NA can besignificantly increased in order to increasef,, while ND can be decreased in order to decreasethe EB SCR capacitance. The result is a 264-fold increase in I compared to a similar bulk Si device, with tB reduced 2.9-fold compared to tBO. These results are based upon the neutral base controlling I. As NA is increased to the point where bandgap narrowing becomes quite large, it isexpected that the EB SCR will dictate I and limit the expected increase to 13..,-lCCWithout EB SCR limitations, E will trans= 120meV AEhhh1 = —34meV port 4.5% of the currentc v in the neutral base,AE2 —18meV zE 9meV E e = 990meV leaving E to transportV‘ the remaining 95.5% of= 120meV tE = —34meV E b = 876meV the current.Fig. 6.20. Novel SiGe HBT based on a 20% Ge substrate. The incorporation of the optimumbase grading provides the maximum reduction to ‘CB possible. The poly-Si emitter cap providesthe wide bandgap necessary to control hole back injection, while lattice matching to a 20% Gesubstrate 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.Sio.8oGeo.2oEmitterbO.l6WBjSi075Ge025Si060Ge4 1Si080Ge020CollectorO.16WB Base= 3.68x1010cmi,b = 5.05x1011cm3July 12, 1995 185The operation of the novel transistor being proposed rests on two requirements: 1) that highquality SiiGe substrates can be formed; 2) that the poly-Si cap will indeed control the backinjection of holes. The ability to grow high quality SiiGe substrates is currently an issue. Atpresent, bulk epitaxial SiiGe layers on top of Si substrates have defect densities ranging from104cm2to 106cm2[311. This is too high to produce commercially yielding LSI ICs. However,given the infancy of epitaxially growing bulk SiiGe layers on Si, in time it is expected that theprocess will mature and the defect density will fall. The other option is to pull raw SiiGe ingots so that the starting wafer contains the desired substrate. In either case, for the study beingpresented here, it is sufficient to demonstrate the usefulness of using non-Si substrates in order toprovide the impetus to grow low defect bulk Sii..Ge substrates on Si. The second question, regarding the efficacy of the poly-Si cap to control hole back injection, can only be answered by experimentation. However, recent work by Kondo et. al. [136,1371 for poly-Si to Si shows that theinterface is not characterised by a high recombination velocity, and that the bandgap is, if anything, larger than in bulk Si. Thus, n1 in the poly-Si layer will be small compared to the n1 in thebase, controlling the back injection of holes and 3. Finally, the band alignment of the poly-Si layerto the Si08Ge02 emitter will only be an issue if the resulting AE is large enough to limit the electron transport current through the entire device. Based upon Si lattice matched to Siij8Ge02,AEshould not exceed -90meV, which would not reduce the transport current given the high dopingthat would exist in the poly-Si layer. Therefore, it is expected that the poly-Si cap will control thehole back injection of the proposed SiGe HBT.This section concludes by examining an intriguing HBT structure that invokes all of themodels of this chapter. Beginning with Fig. 6.9c for Xal = 0 and Xar = 0.45, examination of substrates where 0 x3 0.35 is very interesting. Let the left side be the emitter and the right side thebase. The emitter is under tensile strain so that the ultimate conduction band is E-Iike. Contrarily, with the substrate range being considered, the base is under compressive strain and the ultimateconduction band is E-like. Just because the emitter conduction band is E does not precludeelectrons from existing in E. In fact, given the band alignments for 0 x 0.35, more electronsfrom E, rather than E, will be able to go from the emitter into the base. Essentially, the bandwith 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 inthe emitter.July 12, 1995 18621•E%BaseSi055Ge045r—i——-— $E2 -E \I I CI I •4444 SubstrateI I 4”ç_) %%%%%t%.4CbS” s” ;fSSd.. Si0653e0Emitter__4 Vlr4nnl C’o i Vt) io iM: 44:1 4444— * Collectorb1 ,,..nwt’*a” Ehhv Si55Ge045e = 6.36xl010cm32 1i —3= 67meV = o’i’ cm= —287meVAE = —237meV E9 e = 932meV = —168meV= —169meV = 908meV = —194meVFig. 6.21. Band diagram showing the conduction and valence sub-bands for an HBT where Xal =0, Xar = 0.45, x = 0.35, NA=1x109cm3,ND=5x1017cm3,and Wb=700A.Fig. 6.21 plots the band diagram, including SCR effects, for an HBT where Xal 0, Xar =0.45, x = 0.35, NA=lxlO’9cmND=5x10’7cm3and wb=700A. As is the case for the HBT inFig. 6.20, there is a high doped poly-Si cap on top of the emitter to provide stress relief and control the back injection of holes. What is interesting to note for the device in Fig. 6.21 is the emitterand base have essentially the same bandgap. Thus, there is no wide-gap emitter injecting into anarrow-gap base that is common to traditional HBT designs. Instead, the HBT is controlled by theband offsets and n1 for the given sub-band within the neutral regions. Fig. 6.22 plots the EB SCRcurrents, the neutral base transport currents, and the final collector current that will occur withinthe device of Fig. 6.21. It is important to realise that Vbj = 0.673 V due to the positive AE of thisdevice. For VBE < V,j transport occurs via E through a small CBS, but with neutral base transport essentially controlling I. Thus, electron transport within the emitter is occurring in a bandJuly 12, i995 187that does not form the ultimate conduction band. Now, when I/BE> Vbj, the HBT is operatingwithin the accumulation regime. Due to iXE = -169meV, EB SCR transport within E is reducedto only lO2AIcm when VBE = Vbj. Furthermore, because AE = 67meV, any increase in VBE pastVbj will do nothing to increase the EB SCR current as there is no barrier to surmount, leaving onlythe thermal movement of majority carriers to dictate the current. Thus, E transport is now controlled by the EB SCR and not the neutral base. However, with the accumulation model of Section6.4, E transport becomes the dominant path that controls I when VBE increases past V; leading to transport in the base that occurs within a band that does not form the ultimate conductionband. The final result is a very interesting log I versus VBE relationship that is due to the interaction between the two conduction sub-bands.current withinthe neutral base0.4 0.5 0.6 0.7 0.8Base-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 flatI versus VBE relationship near V,j; however, it is probably more useful as a tool to investigate transportc10611110210110010-110-21 0-1 0-0.3 0.9 1.0July 12, 1995 188properties and band offsets within the Sii..Ge material system. Careful consideration of h forthis HBT reveals some interesting results. Because the Ge content in the base is only 10% higherthan in the substrate, then h = 8054A. With such a large h it is conceivable that the base and intrinsic collector regions could be formed without a heterojunction, thereby achieving an SHBT instead of the DHBT common to SiGe devices. Furthermore, it is not unreasonable to imagine thatthe base and intrinsic collector could be formed in only 3000 to 5000A, leaving considerableroom to the maximum h, which should help to increase the thermal budget for the base layer. Theissue of DHBT devices is not a real concern in npn HBTs, due to the small AE, but would be ofconsiderable appeal in making a pnp device. Finally, the result of a large h for the base and collector regions is a significant lowering of the emitter h to 407 A. However, an = 407A wouldbe wide enough to contain the emitter extent of the EB SCR. Therefore, the critical layer thickness has been moved from out of the EB-SCR and into the neutral emitter, which will have less ofan effect on device performance if dislocations due to strain relaxation occur.In conclusion to this chapter the following results regarding the Sii..Ge material systemhave been achieved:• A review of the literature, including the best material models, for the effect of strain on theconduction and valence sub-bands has been performed.• The band offset theory of Van de Walle and Martin, including the material models of Yu andGan et. al., have been reviewed with the most consistent set of material parameters chosen to fitthe experimental data available to date. To this end, a simple set of equations has been found toaccurately describe the conduction band.• A theory regarding transport within the conduction E and E bands, and the valence E andE bands has been developed. The theory presented does not resort to an effective conductionand valence band, but considers carrier transport within both sub-bands. Included in thisdevelopment is the full effective density of states and the intrinsic carrier concentration for allof 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 previouschapters, have been used to study current-day SiGe HBTs and a few other novel structures. TheJuly 12, 1995 189most important result of this study is that the neutral base will no longer be the sole regioncontrolling I as the neutral base width continues to shrink and the Ge grading in the baseincreases: the limitations of EB SCR transport must be considered. Furthennore, there is asignificant error in both the calculation of charge flow and transit time by considering the sub-bands as a single effective conduction or valence band.July 12, 1995 190CHAPTER 7Summary and Future WorkJuly 12, 1995 191To 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 transportwithin the entire HBT by providing a means to break the modelling effort into separate physicalregions; each region characterised by its own unique physical transport process. Furthermore, themodel 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 ispossible to apply the model of Chapter 2 both to the microscopic transport of charge (i.e., to transport over atomic distances), and to the macroscopic transport of charge (i.e., to transport over distances large enough to treat the electrons as a continuous flux, such as is done in drift-diffusionanalysis). In so doing it may be possible to determine the point at which rapid spatial changes inthe conduction or valence bands produce transport conditions that deviate from the models of driftand diffusion (such as can occur within an SCR, and certainly at the heterojunction where AEfonns). This may allow for a solution to a question posed by Dr. Mike Jackson of UBC as to thecondition for which thermionic injection begins and drift-diffusion ends. However, the most logical extension to the work of Chapter 2 is to remove the restriction that Ep, (for an npn HBT) be aconstant throughout the EB SCR.Chapter 3 presents some interesting ideas for optimising the metrics of an HBT by exploiting the concept of current-limited flow outside of the neutral base. It would be a reasonable extension to the ideas of Chapter 3 to simulate and measure a number of HBT designs that exploit theoptimisations that have been alluded to. Chapter 3 has also gone on to determine the simultaneousoptimisation of the base bandgap and the base doping profiles for the minimisation of tB. Thiswork has, however, neglected the effect of a non-constant mobility with respect to doping variations. Numerical work [63] has shown that the optimum profiles which include the full pfl(NA) donot appear to be too complex, and certainly have a shape that is expected from consideration ofthe 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) ora judicious approximation to it.Chapter 4 derives the model of charge transport with the EB SCR, including the effects oftunneling and momentum conservation across a mass boundary. To this end, the general models ofeqns (4.50)-(4.53) were presented. Chapter 4 goes on to derive analytic approximation to eqnsJuly 12, 1995 192(4.50)-(4.53). However, for the purpose of deriving analytic results, the mass boundary is considered in an isotropic fashion, but with the effective mass maintained as a diagonal tensor and not asimple scalar. Thus, a logical extension to the analytic work of Chapter 4 is to remove the assumption 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 plottingthe ensemble electron density entering the neutral base of the HBT, it was clear that the distribution could not be considered as a Maxwellian or even a hemi-Maxwellian. These distortions froma hemi-Maxwellian form are due to the effect of tunneling though the CBS. Since accurate simulation of transport through a narrow base (in terms of mean free path [43]) demands a full solutionto the BTE, then a way must be found to incorporate the non-local effect of tunneling into theBTE. A possible extension to the work of Chapter 4 is to connect the EB SCR transport models ofthe chapter to a BTE solver for the neutral base; thereby allowing for the inclusion of tunnelingwithin the BTE via a hybrid model.The modelling of charge transport in Chapter 4, due to tunneling through the CBS containedwithin the EB SCR, is formulated upon ballistic considerations. It is common to consider tunneling electrons in a ballistic fashion, if for no other reason than to simplify the calculation of thetunneling probabilities. This position of neglecting thermalising collisions of the electron whileundergoing tunneling is often substantiated on the grounds that tunneling distances are generallyless than 100 or 200A, and are therefore significantly less than the mean free path. However, ifany collisions did occur while the electron is in the midst of tunneling, then the tunneling probability would be essentially reduced to zero. Thus, a potential extension to the work of Chapter 4 isto consider non-ballistic tunneling. The ultimate outcome of such non-ballistic tunneling considerations would be the development of a Monte Carlo simulator that can incorporate non-local effects (i.e., tunneling).A final extension to the work of Chapter 4 can be found by careful observation of Fig. 4.9and eqn (4.74). Im occurs at which for a fixed temperature is a constant. Furthermore, theflux density cLy. is fairly well centred about U, and will become even more localised as thetemperature is reduced. Therefore, the tunneling current through the CBS can be thought of as occurring at an energy of qU(V— VBE)Nrat relative to the conduction band minimum in theemitter. Now, the tunneling current is very sensitive to the forward-directed effective mass, whichJuly 12, 1995 193is dependent upon the full nature of the dispersion relation E(k). Then, with the CBS responsiblefor controlling I, by measuring I the tunneling current through the CBS can be determined. Finally, by extracting the effective mass through a matching of the measured I to the tunnelingmodels of Chapter 4, it should be possible to infer E(k). Therefore, it should be possible to extend the work of Chapter 4 by developing an electrical spectroscopy method for the determinationof E(k).Chapter 5 presents the models for the recombination currents that occur within both the EBSCR and the neutral base. Specifically, the need to balance the total current entering a region withthe net current leaving plus any charge that has recombined within the region, is considered. Thisleads to a mixing of the base and collector currents of an HBT. The result of this mixing is a newconnection between the physical construction of the HBT and it’s terminal characteristics. Regarding future work, the basis for all of the recombination models (SRH, Auger, and radiative) usedwithin Chapter 5 is essentially drift-diffusion. By the arguments of Chapter 4, drift-diffusion analysis is not applicable within the EB SCR. Therefore, combined with the extension being proposedfor Chapter 4 (regarding integration with the BTE), the recombination currents should be recomputed from a particle scattering cross-section point of view. This would place the calculation of therecombination 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 composition of the conduction and valence bands within pseudomorphically strained Sii.Ge. Further, theband offset models for the determination of IXE and AE at an abrupt heterojunction are also presented. Using these material models, transport models which include the two conduction sub-bands E, E and the two valence sub-bands E, E, are developed. It is shown that the multi-band nature of strained Sii..Ge must be considered, even in present-day HBTs, lest considerableerror regarding both the quantitative and qualitative aspects of charge transport be made. Regarding future work, it is imperative that a final and consistent set of material parameters for Sii..Gebe obtained. Without a firm understanding of the material parameters, it is impossible to accurately determine the transport current. With this in mind, Chapter 6 presents a number of novel HBTstructures, including a study of some present-day HBTs. In order to ascertain the validity of themodels developed within Chapter 6, these SiGe HBTs should be fabricated and tested againstthese theoriesJuly 12, 1995 194Finally, Chapter 6 only considers substrates aligned to (100). However, there could be considerable performance gains for growth along (111). Traditionally, BJTs have used (111) alignedsubstrates because epitaxial growth is the fastest for this orientation. (100) aligned substrates havecome about because of the need to minimise surface states at the SiJSiO2 interface in MOSFETs.One of the most interesting features of strained Sii..Ge is the possibility of only having chargetransport occur parallel to the small transverse mass for electrons. The anisotropic nature of Siproduces a 5-fold difference between the transverse and longitudinal mass for electrons. Thus, asignificant improvement to tunneling and mobility can be had if the electrons predominantlymove with the transverse mass. This would be further increased by using the (111) conductionbands instead of the (100) bands. In fact, the (111) bands have a 20-fold difference between thetransverse and longitudinal mass for electrons, with the transverse mass near that of GaAs. Therefore, a logical extension to the work of Chapter 6 would be the development of (111) alignedtransport models. Finally, with the ability to set a large effective mass band at an arbitrary energyabove a light effective mass band, it should in theory be possible to produce negative differentialmobility, in terms of t versus electric field, within strained Sii.Ge; leading to the possibility ofdevices, 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 generating and utilising strained Sii..Ge films that produce negative differential mobility versus electricfield.As a final parting comment regarding future work, it is clear that with the rapid progresscontinuing in the development of ICs, device dimensions will continue to shrink at an exponentialrate. Obviously, this will take devices down into the atomic realm where distances cover only 10Angstroms and not a thousand. Even with present-day devices, where relevant dimensions are 500to ioooA, quantum mechanical effects are important (as can be seen from the consideration oftunneling in Chapter 4). As dimensions reduce to ioA, clearly, classical mechanics will have nopart. For this reason, work on hydrodynamic models, which are really only a second order perturbative solution of the BTE (drift-diffusion being the zero-th and first), will have very limited usefulness. 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 theorem because there will truly be no dimension over which the crystal can be considered as bulk.Furthermore, considering only the conduction electrons in a quantum mechanical fashion, and notJuly 12, 1995 195the core electrons, will not be acceptable at ioA. 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Electron Dev., vol. 42, no. 3, 427-435, 1995.July 12, 1995 205Appendix ARamped NAB(x) to Minimise tBThe proof of eqn (3.17) begins by solving eqn (3.10) for ‘tB using the doping profile depictedin Fig. 3.4. To this end, it is seen that the doping profile of Fig. 3.4 is actually a subset of theprofile depicted in Fig. 3.3 with h1 =0 and h2 = 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:d51og(x—h11og(I() hl-h2Ne e(c6) integrate(\neAu,x,h2, 1);(d6) (1_h2)NeU(c7) integrate(d5,x,hl,h2);Is U - 1 zero or nonzero?nonzero;M1og(1- h2log(U)(d7) (h2—hl)e 2-1 h2—hlNelog(U) log(U)(cS) integrateneAu,x1);(d8) Ne (1_x)U(c9) integrate(d5,xch2);Is U - 1 zero or nonzero?nonzero;( hllog(U) — 1og(t x h11og(t — h2 log(U)(d9) I (h2_ hi) e h2—hl h2—hl (h2_ hi) e h2-hl h2—hlNellog(U) log(U)(dO) integrate(\ne ,x,x,hl);(dlO) Ne (hl_x)Eqn (d5) is the exponential doping profile for h1 x h2, and it ensures that there are no jumpdiscontinuities at the break points h1 and h2 between the exponential doping profile and the regions of constant doping. Then, eqns (d6)-(dlO) collect together the various sub-integrals requiredto 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:h2 — hiJuly 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;( [(h22_2h2+ l)U_hl +Uh1 (h12u_2h1h2+2h1)1og2((dli) f ÷((2h2_2h1)U_h1+(2h22+(_4h1_2)h2+2h12+2h1)U_h1*log(U)+(_2h22+4h1h2_2h12)U_M+(2h22_4h1h2+2hl2)U_M2U_M log2(U)Eqn (dli) is the general model for tB from the optimum doping profile of Fig. 3.3.Using the optimum equation for ‘CR given in eqn (dli), then the ‘CB needed for the proof ofeqn (3.17) is obtained by setting h1 = 0 and h2 h; i.e.,(c12) ev(dl 1,hl=O,h2=h);‘dl2(h_2h+1)ulogu)+(2h +2hh)log( )_2h‘22 U log (U)Eqn (d12) can then be solved for the h that minimises‘CB Differentiating eqn (d12) with respect toh, setting equal to zero, and solving for h produces:(c13) ratsimp(diff(rhs(d12),h));(d13) (h_1)Ulog2U)+ (U+2h_ 1)log(U)_2hU+2hU log2(U)(c14) solve(d13=O,h);Eh.[ Ulog2(U)+2log(U)_2U+2(c16) ratsimp(radcan(ev(d12,d14)));2 U2 log(U) —3 U2 + 4 U— 1(d16)2 2 22U log (IJ)+4Ulog(U)_4U +4Uwhere (dl4) is the same as h in eqn (3.17), and eqn (d16) is the same as‘CR in eqn (3.17) once theJuly 12, 1995 207factor 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 directlyfrom MACSYMA©. As such, there is occasion to perform some intermediates steps that are notinstructive to the proof but are more of a bookkeeping function for MACSYMA© itself. This iswhy some of the d-equations are missing.Finally, it can be shown that an intriguing symmetry exists in the ramped doping profile. Ifthe profile is changed from that shown in Fig. 3.4 so that the exponential region follows the constant 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 accomplished by setting h1 = h and h2 = 1 in the optimum equation for tB given in eqn (dli); i.e.,(c17) expand(ev(dl 1 ,hl=h,h2=1));h 1 h 1 h 1h2U 1—h — 1—h hU 1—h — 1—h h 1 h2U 1—h — 1—hlog(U) log(U) — log(U) + log(U) + log2(U)(d17) 2hU l—h 1—h el_h 1—h h2 2h 1— 2 + 2 — 2 + 2 — 2log (U) log (U) log (U) log (U) log (U)h2U’’+ 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( )Eqn (d18) is the tB for the symmetric doping profile used to develop eqn (d12). As was done witheqn (d12), the optimum value for h is found by differentiation eqn (d18) with respect to h, settingequal to zero and solving; i.e.,(c19) diff(rhs(dl 8),h)=O;(d19) U (2 h log2(U) + 2 (2_ 2 h))_ 2U log(U) ÷ 2 (2 h — 1) log(U) + 2 (2 h— 2)—02Ulog( ) —(c20) solve(d19,h);r (U+1)log(U)_2U+2(d20) Ih= 2[ Ulog (U)+2log(U)_2U+2July 12, 1995 208Eqn (d20) is that h which renders eqn (d18) a minimum. Substituting (d20) back into (d18) yieldsthe minimum tB; i.e.,(c21) ratsimp(radcan(ev(d18,d20)));2Ulog( )_3U÷4 1(d21)2 U2 log2(U) +4 U log(U) —4 U2 +4 UEqn (d21) is exactly the same as eqn (d16), showing that a symmetric change to the dopingprofile produces no change to the transit time. It can finally be shown that the symmetric changeto 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;(d26) U log2(U) + (1_ U) log(U) + (U + 1) log(U)_ 2 U + 2Ulog2(U) +2log(U’)_2U+2 Ulog(U)+2log(U)_2U÷2(c27) ratsimp(combine(d26));(d27) 1Eqn (d27) proves that the symmetric change to the doping profile of Fig. 3.4 does indeed result inh—> 1—h.July 12, 1995 209Appendix BOptimum NAB(x) to Minimise tBThe proof of eqn (3.18) begins by solving eqn (3.10) for tB using the doping profile depicted in Fig. 3.3. However, this task has already been accomplished in Appendix A as eqn (dli). Using eqn (dli) for the optimum tB, the pair h1 and h2 which minimise eqn (dli) is found. Usingthe symbolic math tool MACSYMA©, the partial derivatives of eqn (dli) with respect to h1 andare taken; i.e.,(c29) ratsimp(diff(rhs(dl 1),hl));(h1U_h2+1)1og2(d29)+ 2h1 + u-hi _u _h11og(u)I4 +(2h2_2h1)UM +(2h1_2h2)U_hl1og2((c30) ratsinip(diff(rhs(dl l),h2));( [(h2_l)u1_hlUMJlog2(u)I +[UM+(2h2_2h1—1)UM1og(U)+(2h1_2h2)(d30) h2 hi*_M +(2h2_2hl)Ul_M_hi log2(u)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 h1 and h2 which minimise eqn (dii). Given the highlynon-linear form of these two equations it is not clear that an analytic solution is possible. Therefore, 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) isfound 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.OdOThe numerical results of eqns (d40)-(d45) indicate that h1 + = 1. In order to prove that h1 + h2= 1 is indeed a solution of eqns (d29) and (d30), the following is performed: substitute h2 = 1 —into both eqns (d29) and (d30); then, if the resulting eqns differ at most by a multiplicative con-stunt, then it is proven that h1 + h2 = 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)(d31)+ [(2 hi_2 (1_hi) +1) u 1-2h1 —U 12M]log(U)+(2hi_2(i_hl))U 1-2h1 +(2(i_hi)_2hi)U 1—2h1U 1og2(July 12, 1995 211(c32) ev(d30,h2=i-hl);hi u i-2M— hi U log2(U)I +[(_2h1(l_hl)_i)U1_2M+U1_2hljlog(U)(d32) I hi 1—hi(2(1_hl)_2h1)U i—2h1 + (2h1_2(1_hl)) U 1—2h1U‘ log2(U)(c33) ratsinip(combine(d31+d32));2h1 1+1t.lTT2hl I.lTT2hu(d33) UI I—’ UI ‘.12hiu2hi—1(c34) radcan(expand(d33));(d34) 0Eqns (d31) and (d32), after substituting h2 = 1— h1, are equal and opposite. Thus, these two equations would differ by a multiplicative constant of “-1”. Eqns (d33) and (d34) prove that h2 = 1—by showing the sum of eqns (d3 1) and (d32) vanishes. This result immediately asserts that there isonly one independent equation to solve for. The solution for h1 being:(c35) distrib(expand(d3 1));2hi 1 2hi 1 2h14 hi U i—2h1 — 1—2hi u 1—2h1 — i—2hi 1 4 hi U i2hl — 1—2h1log(U)1— log(U) — log(U) + log2(U)(d35) i—2h1— l—2hi 2h1 — 12U 4h1 2 1—2hi i—2h1—2—2 + 2 +hiU -i-hilog (U) log (U) log (U)2hi 1*U2hhl 1—2hi(c36) map(radcan,d35);4hi 1 1 4hi 2 4h1 2 hi(d36) — — + 2—2 — 2 + 2U log(U) U log(U) log(U) U log (U) U log (U) log (U) log (U) U(c37) solve(d36=0,h 1);r 1(d37) Ihl=L log(U)-i-2July 12, 1995 212Eqn (d37) proves eqn (3.18) for the optimum h1, along with the result from eqn (d34) whichproves eqn (3.18) for the optimum h2. Finally, using the optimum h1 and h2, the optimum tB isfound by substituting back into eqn (dli) found in Appendix A; i.e.,(c38) radcan(ev(dl 1,h2=1-hl));2h1-*-1u2h1—1 (bl2u+h1)log2(U)( 411 2h1-i-1+L(h1_2h12)u211 ÷u21”’ ((2h12_3h1+1)u÷4h12_2h1]Jlog(u)(d38) 2h1÷1+u2h1 ((_4h12+4h1_1)u+4h12_4h1+ iJ4h12h1—1 2U log(U)(c39) radcan(ev(d38,d37));1(d39)log(U) +2Eqn (d39) is the same as tB in eqn (3.18) once the factor of 1/2 is included within ‘CBO. This completes the proof of eqn (3.18) for the optimum NAB(x) to minimise tB.July 12, 1995 213

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