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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 intheAlGai..AsandSii..GeMaterial 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 thesisin partial fulfillmentof therequirements for anadvanced degree atthe University of BritishColumbia, I agree thatthe Library shall makeit freely availablefor reference and study.I further agreethat permission forextensive copying of thisthesis for scholarlypurposes may begranted by the headof my departmentor by his or herrepresentatives. Itis understood that copyingor publication ofthis thesis for financialgain shall not beallowed without mywritten permission.(SiDepartment of&/c,ot/The University of BritishColumbiaVancouver, 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 toageneral material system, and are analytic. Extensions specifically required by the complexSii..Gematerial system are also developed. The optimum (to minimisebase 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 thatthetunneling 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 modelsdeveloped. 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 TablesvList of Figures viAcknowledgment xCHAPTER 1: Introduction 11.1 Modelling Details 41.2 Thesis Organisation7CHAPTER 2: A Multi-Regional Model for HBTs Leading toOptimisation by Current-Limited Flow92.1 Bandgap Engineering 102.2 Regional Decoupling and Current-Limited Flow122.3 Optimisation Through Current-Limited Flow18CHAPTER 3: Base Layer Decoupling and Optimisation213.1 Independent Optimisation ofCBE,andy 233.2 Reducing‘CEby Decoupling the Base fromIc273.3 Optimum Base Doping Profile to Minimise‘CB303.4 The Effect of a Non-Uniform n1 and D on the Optimum‘CE38CHAPTER 4: Transport Through the EB SCR434.1 Formulation of Charge Transport at the CBS454.2 Incorporation of Effective Mass Changes494.3 Calculation ofFrand a Unified Model for F674.4 Analytic CBS Transport Models704.4.1 Analytic Model for the Standard Flux 714.4.2 Analytic Model for the Enhancement FluxFfe 784.4.3 Analytic Model for the Reflection FluxFf 814.5 The Effect of Emitter-Base SCR Control on I844.6 Deviations from Maxwellian Forms and Non-Ballistic Effects954.7 Conclusion105July 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: TheSii..GeHBT1326.1 The Effect of Strain onSiiGe 1356.2 Band Offsets inSiiGe 1516.3 Electron Transport in StrainedSiiGe 1596.4 The Accumulation Regime Beyond the Built-In Potential 1716.5 Conventional and NovelSiiGeHBT Structures 179CHAPTER 7: Summary and Future Work 191References 197Appendix A: Ramped N(x) to MinimisetB206Appendix B: Optimum N(x) to MinimisetB210July 12, 1995 ivList of TablesTable 3.1:tBfor 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 HBT13Fig. 2.3. Hypothetical HBT structure showing the physical regions that govern chargetransport 14Fig. 2.4. The flowTthat results from a series connection of six pipes 16Fig. 2.5.Tfor 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 minimiseREandCBE27Fig. 3.3. Optimum doping profile N(x) obtaining by numerical minimisation33Fig. 3.4. The first trial function for N(x) inspired by the form suggestedby 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.tBusing N(x) from Fig. 3.3, where h1 1 — h2 and h2 is varied37Fig. 3.8. Optimum bandgap in the base to minimisetB40Fig. 3.9. The optimum stationary function y(x) which includes doping, bandgap,andbandgap reduction due to heavy doping for the minimisation of‘CB41Fig. 4.1. Abstract model of current flux within the region containing theCBS 46Fig. 4.2. Blow-up of theCBSfromFig. 3.1(b)49Fig. 4.3. Definitions of the cylindrical momentum space coordinatesfor the calculationof the Jacobian Transforms from k to U-space50Fig. 4.4. Domain of integration R1 for a uniformm*54Fig. 4.5. The effect that conservation ofphas uponU±,iandU±,2when a massboundary is placed at x =058Fig. 4.6. Domains of integration R1 and R2 for the enhancementcase 61Fig. 4.7. Domains of integration R1 and R2 for the reflectioncase 62Fig. 4.8. Collector current for an abrupt A1GaAs HBT with30% Al content in theemitter71Fig. 4.9. Flux density normalised to4max’for an Al03Ga07As/GaAs abrupt HBT75July 12, 1995 viFig. 4.10. Standard FluxFcand Reflection FluxFfrfor an HBT with the parametersgiven near the start of this section 86Fig. 4.11. Relative importance ofFfrto the total flux F for an HBT with the sameparameters as Fig. 4.10 88Fig. 4.12. Standard FluxFf.and Reflection FluxFffor an HBT with the sameparameters as Fig. 4.10, but withAEreduced from 0.24eV down to 0.12eV 89Fig. 4.13. Standard FluxFjand the Enhancement FluxFfefor an HBT with theparameters given near the start of this section 91Fig. 4.14. Relative importance ofFjeto 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 vectork,2entering the neutralbase 98Fig. 4.17. Ensemble electron distribution entering the neutral base versusk 99Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier101Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier102Fig. 4.20. Relative difference between the results obtained from the methods proposedin[511 to the modelfor F from this chapter 104Fig. 5.1. Band diagram of the EB SCR showing the effect of the abruptheterojunctionon under an applied forward bias (reprint of Fig. 2.2) 109Fig. 5.2. Components of the collector and the base currents emphasising thatThTmustequal the total of, +NB+SRN.B+Aug,B+Rad,B111Fig. 5.3. Energy Band diagram for the EB SCR of an HBT underequilibriumconditions 113Fig. 5.4. Relative error between the approximate and exact fonns givenin 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 ofincludingSCRBin the calculation of AE 125Fig. 5.8. Bias dependence of the current gain (3 for the case ofWflbincreased 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 5.4130Fig. 5.12. Comparison of the full model and “diode-like” expressions for the SCRcurrents130Fig. 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 unstrainedSii..Ge 138Fig. 6.3. Commensurate growth of theSii.Ge alloy layer to the Sii.Ge substrate,leading to a pseudomorphic alloy film142Fig. 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 forSii..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 forSiiGe commensurately grown to a variety of substrates orientedalong (100)149Fig. 6.7. Constant energy surface plot depicting the E and E bands inSi083Ge017commensurately strained to (001) Si150Fig. 6.8. Conduction and valence band energies including all of the band offsetsfor aSi1xaPexaito aSii.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.AEwhenXar = Xal+ 0.20, andXaland x are varied 158Fig. 6.11. AE whenXar = Xaj+ 0.20, andXaland x are varied 159Fig. 6.12. Diagram of the A conduction band minima involved in f andg intervalleyscattering162Fig. 6.13. Equilibrium band diagram of apn-junction, showing the relevant energies andpotentials165Fig. 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 forNratandYatfrom 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 aSii..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 HBTwhereXal= 0,Xar= 0.45, x = 0.35,NA=1x1019cm3,ND=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 whichIcould 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 presenceon my examiningcommittees helped to ensure that my final thesis was the best it could possiblybe.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 mea 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 forthe device that incorporates the various compromises between competing device metrics (suchas f3,fandRB)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 theemergenceof HBT-based Integrated Circuit (IC) processes. Finally, the models that are developedwithin thisthesis are in general free of any details specific to a single material system. However,given theimportance of theAlGai..AsandSii..Gematerial systems, these two systems willbe 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 hashad 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 processesachieve the maturity oftechnologies such as CMOS. Furthermore, with experimental results becoming moreprolific, andwith rapidly diminishing device dimensions, we are finding thatmuch 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, comesan increase inthe need for models that predict device operation. It is now possible to manufactureHBTs 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 ofthe crystal, theapplicability of models based upon classical continuous fields becomes questionable[15]. Thereis already general agreement that one must consider higher order moments beyondthe drift anddiffusion terms in the Boltzmann Transport Equation (BTE) in order to modeldeep submicron devices [16-17]. The BTE is based upon classical physical models that in generaldo 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 whenthese BJTmodels were developed without consideration of the physical processes thatgovern transportwithin an HBT. Presumably, the BJT model has enough degrees of freedomso that it can be manipulated to cover HBT operation. For example, one of the most common discrepanciesfoundwhen 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 theoreticallypredicted 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 thingsas why the collector and base injection indices differ between an HBT and a BJT [24,25].TheSii..Gematerial system has many unique physical considerations that othersystems,such as theAlGai..Asmaterial system, do not have to contend with. The uniqueattributes of theSii..Gematerial system are mostly due the effects of strain. Due to the large lattice mismatchbetween Si and Ge,Sii..Gefilms grown on top of Sii..Ge substrates (where x y) havea large degree of strain present within them if non-relaxed crystals with low defectdensity are to bemanufactured. The presence of strain breaks the cubic symmetry of the crystaland changes thebulk electrical properties [26-28] of the film. By varying the Ge alloy contentand the strain imparted to the SiGe film, it is possible to tailor both the bandgapand the offsets in the conductionand valence bands. Therefore, models specific to theSii..Gematerial system must be developedin order to understand charge transport within the complex band structurethat develops.Finally, the reason for focussing on theSii..GeandA1Gai.Asmaterial systems stemsfrom the maturity of AlGaAs devices, and the massive installedbase of Si-based IC technologiesthat would easily admit SiGe devices. From a manufacturing standpoint theAlGai..Asmaterialsystem 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 heterojunctionsbetween two materialsJuly 12, 1995 3characterised by different bandgaps. TheAlGai.Asmaterial system has essentially a fixed latflee constant over the entire range of Al mole fraction x. For this reason, theMGai.Asmaterialsystem is lattice matched and will admit an arbitrary heterojunction betweenAlGai..AsandAlGaiAs without developing a strain within one of the films. This lack of strain within theMGai.Asmaterial 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..Asmaterial system [29]. However, most solid-state devices are Si based[30]. With theadvancement of low-temperature Chemical Vapour Deposition (CVD) processing [31], theformation of high-quality commensurately strainedSii..Gefilms is becoming commercially viable.Therefore, given the manufacturing advantages of Si, it is expected that SiGe HBTs willshortlysurpass 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 emitterinto the base ofAlGai..Asnpn HBTs [18,24,25]. The research has centred around abrupt HBTs wherethe heterojunction between the wide-energy-gap emitter and the narrow-energy-gap baseis abrupt. In anabruptAlGai.AsHBT one finds the formation of a Conduction-Band Spike (CBS) between theemitter and the base (see Fig. 3.1). This spike, due ostensiblyto differences in the electron affinityof the materials used for the formation of the emitter and thebase, results in a large impediment tothe flow of electrons from the emitter into the base. In fact, if the CBSwere not taken into accountwhen modelling the HBT, the collector current would be overestimated by overthree orders ofmagnitude at room temperature (see Fig. 1.1). However, the modelling of chargetransportthrough the CBS cannot be based upon simple thermionicinjection 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 throughthe CBS can often be the limiting factor for the overall transport of charge within the HBT(i.e., the determination of the collector currentIc).This occurrence of current-limited flow outsideof the neutral base region willbe studied and exploited for device optimisation. Therefore, the modelling of therelevant physicalphenomena surrounding charge transport through the CBS, including tunnelingand conservationof transverse momentum across the heterojunction in a diagonal mass tensor, willbe investigated.July 12, 1995 41 W3,— LUI;,10C)io-90.8 1.0 1.2 1.4 1.6Base-Emitter VoltageVBE(V)Fig. 1.1. Collector current for an abrupt AIGaAs HBT with30% Al content in the emitter. Theemitter doping is 5x1017cm3,and the base doping is 1x1019cm3(see Section 4.5 for the complete device details). The top curve, where CBS limitations havebeen neglected, is arrived at byassuming Shockley boundary conditions and consideringonly neutral base transport.The possibility of regions other than the neutral base controlling I is intriguing.However,from a modelling perspective, the immediateconsequence of a multi-regional system controllingIcis the question of how to join these various regionstogether to form one cohesive transport model. Furthermore, the possibility exists that under multi-regionalcontrol ofI, older models, such asthose for the neutral base [38], which assume that onlythe specific region being studied controlsIcmay not longer be valid. It will be shown in Chapter 2 thatthere is a very simple prescriptionfor joining up all of the multi-regional transport modelsinto a complete transport model for the determination of I. It will be further demonstrated in Chapter6 that it is possible for two spatiallyseparate regions to control I simultaneously by having essentiallyidentical net-charge-transportcapacity through both regions; the ramificationof this is the inseparability of the two regions.With the general model of Chapter 2 providing theoverall method to link the various physical regions of the HBT together, then the problem of modelling chargetransport within the entireHBT is effectively decoupled into a set of models; onemodel for each relevant region. To this end,Chapter 3 investigates and develops models for the variousregions 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 ofthe base-transit time‘CD.Finally, the modelling of recombination events, which lead to the formationof thebase current‘B’is developed in Chapter 5 with the specific attributes ofa heterojunction included.These various regional models essentially form a toolbox for the study of charge transportwithinthe HBT, with the general transport model of Chapter 2 forming the blueprintfor 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 uponphenomenological mobility models (i.e., mobility models with an electric fielddependency). Instead,models that include the quantum mechanics of charge transport, which haveno appeal to saidphenomenological mobility models, are analytically solved for. However, the neutralbase chargetransport models are based upon drift-diffusion analysis. The reason forresorting to simpler drift-diffusion analysis for the neutral base is its been found that the neutral base oftendoes not represent the bottleneck to charge transport and thus does not dictate control overI ([25] and Fig.1.1). Nevertheless, as the neutral base thickness approaches and becomessmaller 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 electronsdo not follow exactly the simple models ofdrift-diffusion contained within the BTE [16,41]. Instead, a general solutionto the BTE is necessitated.In present-day HBTs, and even in some of the emerging high performance BJTs, theunderstanding of hot electrons can be essential to the accurate modellingof the device’s terminal characteristics [9,14]. The problem with general BTE solvers, suchas Monte Carlo simulation, is thatsome important QM effects cannot be modelled. The BTE is basedupon local potentials andtherefore cannot include some QM effects, such as tunneling, whichare inherently non-local. Aswas discussed and shown in Fig. 1.1, the failure to include tunneling resultsin 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 chargetransport through theCBS. Specifically, Section 4.6 will show that tunneling producesa 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 fortB,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 ofSii..Gefilms is dramatic. Chapter 6 reviews the various material models necessary for the description andstudy of the electrical characteristics of strainedSii.Ge.Specifically, once a review of the literature regarding theSii..Gematerial models is presented, a comparison to experimental results isperformed, and the most consistent set of material constants selected. The final resultis a complete set of models for the calculation of the bandgap including conductionand valence band offsets. Furthermore, strainedSii.Geresults in a two-band system both for the conduction and thevalence band. Chapter 6 uses theSii..Gematerial models and derives the necessary multi-bandcharge-transport models that are required to simulate SiGe HBTs. In fact, it is foundthat there is asubstantial error incurred by replacing the two-band system with a single effectiveband. Finally,the charge-transport models are applied to the study of present-day as wellas future SiGe HBTdesigns with some surprising results regarding operating voltages and critical layerthicknesses.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 Chapter2 is that itcan contain any number of physical regions to model the HBT, including sources and sinkswithineach region. Chapter 2 also introduces a method of optimisation through what istermed current-limited flow. Chapter 3 builds upon the ideas of Chapter 2by considering specific examples of device optimisation that can be performed within an HBT but not a BiT. The maindevelopment inChapter 3 is the solution for the optimum base bandgap and doping profile.Surprisingly, the optimum doping profile is not exponential, and the optimumbase 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 ofa spatiallynon-uniform effective mass, are developed. Finally, Chapter 4 goes on to showthe 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. Chapter5 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 modelsof Chapters 4and 5 for the simulation of SiGe HBTs. Models that include the effects of strain on theconductionand valence bands in theSii..Gematerial system are presented. Multi-band charge transportmodels, which include the material models of theSii..Gematerial system, are then developed.Finally, Chapter 6 brings all of the models developed within the chapter togetherfor the study ofnumerous present-day and future SiGe HBT designs.July 12, 1995 8CHAPTER 2A Multi-Regional Model for HBTs LeadingtoOptimisation 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 newone. Shockley alluded to the use of Bandgap Engineering in his BJT patent of 1948 [1], andKroemer first proposed the idea of using a wide-bandgap semiconductor for the emitter anda narrow-bandgapsemiconductor for the base in 1957 [2]. This junction betweentwo semiconductors with dissimilar bandgaps is a heterojunction, and leads to the creation of a Hetero-junctionBipolar Transistor(HBT). What makes the HBT of specific interest today, is that in 1957 it was notpossible 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 becomefeasible. Therefore,now is the time to fully explore the possibilities affordedby 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 ofthe 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 ofEJEis present from the construction of the device and is thereforeostensibly independent of the bias conditions (much the same as the electric field that isgenerated in the neutralbase due to a spatially varying doping is independent of bias). It is this manufactureddrivingforce, due to the spatial change in the bandgap and the band alignments, that givesrise to BandgapEngineering. It is possible to effect such a rapid change inEdEthat 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 inE/Esuch that the overall electric field is modified(such as adding a gradient toEdEthat aids in the transport of charge through the base) but is notoverwhelmed by the engineered field; or techniques that afford extremely rapid or abrupt changesinEIE,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 secondgroupof Bandgap Engineering techniques resulted in the idea of placing an abrupt downwards change inEto 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 resultsthat 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 comparedto BiTs. Thus, abruptBandgap Engineering may be the more promising road to follow in seekingto continue the evolution of BITs.Ballistic “launcher”didingField(N) (P) (N)Emitter Base CollectorEHole blockerFig. 2.1. The abrupt change ofEin the emitter-base junction “launches” electrons into the basewith a large kinetic energy. The gradual negative slope ofEin the base and the collector helps tospeed the electron through these regions. Finally, the abrupt change inEat 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 inE,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 fluxFrThe forward and reverse directed fluxes are[18,20,211:= qvn° andFr= qvn°= qnOekT (2.1)which producesAEfF=FfF=qn°l_ekT)=Fj(1_ekT),(2.2)where n0 is the electron concentration immediately to the left of the heterojunction,Ois 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 (whichcan include tunneling),and AE is the abrupt change in the electron quasi-Fermi-energy E. The reason for theappearance of the term /XEp in eqns (2.1) and (2.2) is due to the need for n° to surmount thebarrierAE.Therefore, the abrupt change inEgenerates the abrupt change inEqn (2.2) clearly shows that as F goes towards zero, thenso does In fact, if the conditions F << Fjand F<< Frare 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 FfandJuly 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 ofAEp,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 AEfr2occurs, resulting in a driving forceof V——in Region 3. Using the form given in eqn (2.2) for the transport current:O11f1kTn,l——eJ2 =2(V—E1)(1_ekT),Jfl,3 =3(V—AEffl1—iXEf2)’(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 asi°(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,f2jOe= n,2(2.4)2+J 3+43Then, equating J,,1 with J,,2 +J1,,2 gives:LEffl1 IEf1fn,2Ji(V)(1-e)=[J,(v)+4,(V)1e(1_ekT).Using eqn (2.4) in the above, and once again dropping the explicit dependence uponV, produces:iiE1,o— kT—n,lkn,2+n3+Jp25e— (J2+4)(3+43)+l(2+3÷43)The final transport current Jexiting the device is simply equal toJ,3.Substituting eqn (2.4) and(2.5) into J,,3 given in eqn (2.3) produces:JT(V)= Jfl3(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 currentTemanating fromthe device, and extends eqn (34) in [52]. It includes all of the recombinationeffects of Regions 2and 3, while allowing for a completely general relationship betweenthe applied bias and the forward directed flux Ff (where the fi(V) functions are F. The only stipulation placedupon the use ofeqn (2.6) is that fi(V— AEp) =J0(V)exp(—AEp/k7) (as willbe seen in later chapters, where eqn(2.6) is applied, this is exactly the functional fonn that results). Therefore, todetermine 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 defineJ1;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 throughany 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 examinationof 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 Regionj. Then, thetransported current through each successive region is not Jbut Now, the form of eqn(2.7) is exactly the same as that used for the calculation ofa connected series of conductors. Thisimmediately leads to the picture of a series of pipes through whicha currentJTmustpass (see Fig.2.4).Region 1 Region 2 Region 3 Region 4 Region 5 Region 6ITFig. 2.4. The flowTthat 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 andTwill accordingly be governed mostly by this region alone.July 12, 1995 16Looking at eqn (2.7) and letting J <<jkwherej k and k can range over all N, then Regionj will be responsible for the current-limited flow ofTand produce:JT(V)Ji(V)Iik(’(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 lose0kelectrons to recombination. Therefore, the current will be diminished byc’kineach region encountered, leaving a final current ofTexiting the device. This immediately leadsto eqn (2.8). Thus, in a device with say six regions, if Region 3 produces the limiting flow, thenT=Finally, looking once again at eqn (2.8), recombination events upstream of Regionj playnopart in the ultimate currentTThis 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 current4Adding all of the recombination events together gives:N N N+1F N N+1 N N+1J(V) ==flY= JT[YiIT [Ii=1 i=1 j=i+1 i=1 j=i+1 i=lj=i+1Then, after bringingyjinto 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, sinceYN÷11 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 withinthe device. Combining botheqns (2.7) and (2.9), the total electron and hole current entering and leavingthe device is known.As will almost always be the case, one region alone will dictatethe 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)-flx(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 functionalform,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 applyto any device.The important ramification is that one region alone will tend to determine theoverall transportthrough the entire device; creating a situation of current-limited flow. The key to achievinga situation of current-limited flow is the existence of a substantial AEp in one region. Finally,abruptBandgap Engineering techniques provide the capacity to createa situation of current-limited flowin any region of the device. In the next section and chapters to come, theconcept 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 beused 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 physicalprocess, willtend to dictate the transport current through the entire device. This sectionexamines how to intentionally design a specific region, through Bandgap Engineeringtechniques, to result in current-limited flow; thereby allowing for a decoupling ofTfrom 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 demandedjrthen one is free to optimise thatspecific region without affecting JFig. 2.5 shows the transport current that would result froma hypothetical three region device. For case (a), Region 3 controls J under low bias and Region 2 controlsunder high bias;while Region 1 plays no part at all. In case (b), the transport current inRegion 1 has been loweredso that Region 1, and neither Regions 2 or 3, controlsTunder all bias conditions. This demonstrates, in principle, the feasibility of engineering a specific region to be the source of current-limited flow, and thereby linkJTtothe 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,VAis to be maximised andRcminimised. However, increased collector doping decreases bothVAandRc,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 VoltageVBE(V)Fig. 2.5.Tfor 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 ina detrimental fashion, then one is free to optimise the desired metric. Current-limited flow provides thenecessary tool to decoupleTfrom all regions, and therefore all physical transport processes, saveJulyl2,1995 19one. Continuing on with the example of simultaneously optimisingVAandRc,ifTwere decoupled from the construction of the base and collector, say by making the emitter-baseSCR thesource of current-limited flow, thenVAwould no longer depend upon the collector doping; enabling the optimisation ofRcwithout affectingVA.With base-width modulation no longer an issue, in terms of the collector current and thereforeVA,it would be possible to increase the intrinsiccollector doping adjacent to the base and thereby reduceRc.A further optimisation, in terms ofthe base-collector capacitanceCBC,could also be had by placing a low-doped collector regionwithin the CB SCR (say at 1016cm3for 2000A) in order to setCBC,followed immediately by ahighly doped extrinsic collector to reduceRc.Optimisation of competing metrics is thus achievedby first identifying the coupling parameter; then, one other region that does not containthe coupling parameter is constructed (generally through abrupt Bandgap Engineering techniques)to bethe source of current-limited flow in order to provide for the controlof J- (i.e., the collector current).This chapter has provided a logical course to decouple otherwisecompeting metrics so theymay be simultaneously optimised. The tool for decoupling thecompeting metrics being the creation of current-limited flow outside of the region or regionsto be optimised. It is possible toachieve current-limited flow in any given region by resortingto abrupt Bandgap Engineering techniques. Thus, abrupt Bandgap Engineering providesthe necessary tool to further optimise BJTs.Finally, all the models for the various regions of the HBTare 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 thetransport 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 currentTdepended on thisregion alone; thereby decouplingTfrom all other regions of the device. OnceThas 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 thebase 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-resistanceRBJ,base-emitter capacitanceCBE,injection indexy(not to be confused with the y in Section 2.2 which is the recombination loss), Early voltageVA,base transittime‘CB,and the base-collector capacitanceCBCcould then be simultaneously optimised. The keyto the optimisation of these base metrics rests simply on the decoupling ofTfrom the base byconstructing one other region of the device in such a manner that it resultsin current-limited flow.This chapter takes the abstract concept of optimisation through current-limitedflow and applies it to the base region. The methods used to achieve the simultaneous optimisationof 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 methodsderived in Chapter 2.July 12, 1995 223.1 Independent Optimisation OfRB, 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 currentI divided by the basecurrentlB).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 EByis given by:“fl, B(3.1)wheren,Bis 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), inthe absence of neutral-base recombination, the gain is:I=!._•(3.2)‘‘“p,EThus, [3 is maximised asyis driven towards 1; meaning that is driven towards zero and/orn,Bis made as large as possible.In an npn BiT,n,Bis inversely proportional to the base Gummel numberG#B[54-56] given by:G#B=dx, (3.3)where D is the electron minority carrier diffusion coefficient,WBis 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,Eis inversely proportional to the emitter Gummel numberG#E[54-56] given by:G#E= i)dx, (3.4)where D is the hole minority carrier diffusion coefficient,WEis the neutral emitter width, and nis the emitter majority electron concentration(=emitter dopingNDEexcept under high-level injection). Thus, f3 is proportional toG#E/G#B.Now, the intrinsic base sheet-resistanceRBUis alsoinversely proportional toG#B[54]. However, unlike the case for [3, where it is desirable to reachamaximum,RBUis to be minimised in order to improve the high-frequency operation of the BJT.July 12, 1995 23SinceRBL]and13are both tied to the parameterG#B,and increasingG#BoptimisesRBUwhile deoptimising13,we realise these two metrics are competing and therefore cannot be simultaneouslyoptimised (at least in terms of the parameterG#B).As was discussed in Section 2.3, the key to optimising two otherwise competing metrics isto identify their common parameter (in this caseG#B)and remove its dependency from one of thetwo metrics. Continuing on with the case of optimisingRBJand13,it would appear possible to increaseG#Band thereby minimiseRBU,while also increasingG#Eand thereby maximise13.However,G#Ein a BJT cannot be increased becauseNDEis either at or very near its maximumphysical limit(1021cm3).Thus, without resorting to Bandgap Engineering techniques, the onlyavailable parameter isG#B,meaning that a compromise has to be made betweenRBUand13.Thiswas the motivation for the first HBT; to decouple 13 from its sole dependence uponG#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 exactlythe samepotential barrier ofVbj — VBE).Therefore, the ratio ofn,Btop,E (= 13)will be proportional to theratio of the available number of electrons in the emitter to the available numberof 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, thenpEwould be significantlyreduced and13increased (see Fig. 3.1(b)). Finally, if Bandgap Engineering wereemployed toachieve an initial 1000-fold increase in13(by reducingp,Ethrough a Bandgap EngineeredthenG#Bcould be increased 32-fold, thereby reducingRBD32-fold, while still leaving a net 32-fold increase in13.Thus, by creating a heterojunction at the EB metallurgical junction, it is possible to reducep,Ewithout increasingG#E.Then, the gains provided by a reducedpEare sharedbetween an increase in13and a decrease inRBIJ.The methods just described for the simultaneous optimisation ofRBand13demonstrate thepotential gains of abrupt Bandgap Engineering. However, the techniques describedabove did notfollow the exact prescription given in Section 2.3, and thus maintain a coupling betweenRBUand13.Instead of decoupling 13 fromG#B,another degree of freedom was added toG#E;namely theabrupt change oft1Ein the valence band at the EB junction. The dependence of13uponG#Bstillexists, butp,Eand 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 inthe emitterJuly 12, 1995 24i,E•However, since f3 still depends uponG#B,any change inG#Bdue to bias (such as the Earlyeffect [57], Kirk effect [58] or high level injection [56,59]), will still affect and generally degradeI.The reduction ofp,Ethrough simple abrupt Bandgap Engineering is thus seen as a good firststep, but falls short of the optimum case where f3 is decoupled fromG#Baltogether.To fully decouple 3 fromG#Bone looks at the spike inEat the EB junction shown in Fig.3.1(b). This Conduction-Band Spike (CBS) occurs in HBTs where thebase is made of GaAs andthe emitter is made ofAlGai.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 dropAEis sufficient to cause the CBS to be the region ofcurrent-limited flow (this will be fully discussed in Chapter 4). Thus,T (= n,Bin the absence ofsignificant neutral-base recombination) will be governed by the physical process of transportthrough the CBS, and not by the transport through the neutralbase. Furthermore, transportthrough the CBS has little dependence uponG4B(as long as the base doping is much larger thanthe emitter doping). Therefore,Tand thus are decoupled fromG#Bthrough the condition ofcurrent-limited flow at the CBS.The condition of current-limited flow in the region of the CBSfollows exactly the prescriptions of Section 2.3.n,Bhas now been decoupled fromG#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 seenby a holetrying to go from the base to the emitter is the same barrier seenby 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 leastAElarger than the barrier seen by anelectron trying to enter the base. Also note the formation of theConduction-Band Spike (CBS).July 12, 1995 25G#Bsuch 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 ofIXEand AK1,at the EB junction). With the collector current decoupled fromG#B, RBL]can be minimised by increasingG#Bthrough 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 capacitanceCBEcan also be minimised due to the conditionof current-limited flow at the CBS. The high-frequency performance of a BiT improves asCBEdecreases. MostnotablyfT(the frequency at whichf,under the conditions of an A.C. short circuitbetween emitter and collector, has dropped to unity) increases asCBEis reduced. SinceCBEisgiven by:qNNNABCBE= I,,whereNrat= N + N‘ . )I4kVbjVBE)AB DEthenNDEandNratneed to be minimised in order to reduceCBE.In a BJT, the need to maximisefforcesNDE>> N, meaning thatN is reduced in order to reduceCBE.Thus,CBEis connected toRBDas 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 thatRBUcan be optimised by increasing N. Finally,CBEis reduced in an abrupt HBT through the reduction ofNDE(for HBTs, N>> NDEso thatNrat1). The only limit to the reduction inNDEbeing the point atwhich a significant intrinsic emitter resistanceREbegins to occur (see Fig. 3.2).This section has presented the methods to simultaneously optimiseRB(J, CBE,and ‘y Optimisation of these metrics begins by decoupling fromyandCBEthe dependence upon N. Thisdecoupling is afforded by the creation of current-limited flow at the CBS. Withy andCBEdecoupled from N,RBis optimised by increasing N. Then,CBEis optimised by reducingNDE.Finally, the optimisation of y depends first of all uponp,E(which depends heavily on SCRrecombination [24]) and secondly uponn,B(which is governed by the flow ofTthrough theCBS); EB SCR recombination, which accounts for most ofp,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 considerableifNDE,1is 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 dopedNDE2extrinsic emitter is placed as a cap layer on top of the device, wherethe eventual contact layer isformed.3.2 ReducingtBby Decoupling the Base from I’Chapter 2 discussed the merits of Bandgap Engineering, where thenatural evolutionary pathof the BiT produces the HBT. Two Bandgap Engineeringtechniques were considered: techniquesthat created abrupt changes inE.JEleading to the creation of current-limited flow;and techniques that created gradual changes inEdEthat produced additional aiding fields forthe transport of charge. Then, Section 3.1 focussed uponthe benefits of current-limited flow producedbyan abrupt change ofiXEwithin the EB SCR. This section carries on withthe benefits to be derived from current-limited flow, but delves into thesecond group of Bandgap Engineeringtechniques - namely the creation of fields in the base toaid in the transport of charge through theregion.A major component of the total transit time fora BIT or HBT is still the neutral-base transittimetB.In the absence of any spatial variation to the bandgapor N, then under low level injection conditions, with the neutral base widthWBlarger than a few mean-freepaths , the base transit time is given by the standard equation:w(3.6)B2DtBcan be reduced from the value given in eqn (3.6), withoutreducingWB,by introducing an aiding field in the base (as is shown in Fig. 2.1). BJTs where an aidingfield has been placed in thebase are termed drift-base transistors [60]. This aiding field implies, foran npn BJT, a downwardsJuly 12, 1995 27Emitter-side SCR edgeslope toEin the neutral base. Before the creation of HBTs, a negative slope inEcould 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 reducetBbut 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 neutralbase), anaiding field can be created without the need to vary N [38]. Thus, by using Bandgap Engineering techniques to create a gradual down-slope toEin the base,tBcan be reduced without lowering N and compromising the Early voltage. Kroemer calculatedtBfor a non-uniform bandgapEg across the neutral base, and found [38]:w wIn.(x)p(z)=1__________dzdx, (3.7)Bp (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 - AEkT2(3.8)where F = AEgI(qWB), and AEg represents the difference between the bandgapat 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 intBthrough the addition of a graded bandgap in the base.July 12, 1995 28The reduction oftBthrough 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 createa gradedalloy in the base in order to provide for the downwards slope inE.In the case ofAlGai.As/GaAs HBTs, the bandgap is increased with an increase in the Al mole fraction x; while inSii..GeHBTs, the bandgap is decreased with an increase in the Ge mole fractionx. Now, inA1GaAs HBTs the Al mole fraction must remain below a maximum ofx = 0.45, for this is thepoint at which the material changes from a direct to an indirect bandgap [61]. Ina similar fashion,Sii..GeJSiHBTs 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 thata 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 formationofthe graded-base versus all the other bandgap-engineered regions of the device.Since the heterojunction in the EB SCR provides the most importantgains in terms of optimising the metrics of the device (namely decouplingy fromG#B),part of the total alloy budgetmust be allocated to its formation. In the case of AIGaAs HBTs, fully 66% of the maximumtotalalloy budget (Ax = 0.3 of a maximum 0.45) is spent in the formationof the EB heterojunction (Inreality, Ax < 0.45 is a maximum upper limit that is generally reduced to 0.30 for practicalapplications. With this reduced alloy budget, the EB heterojunction would consume the entirebudget). InSiGe HBTs, virtually the entire alloy budget of Ax < 0.2 is spent in the formationof the EB heterojunction. Therefore, irrespective of the material system used to form the HBT, little ifany ofthe alloy budget remains for the Engineered Bandgap in the base oncethe EB heterojunction hasbeen formed. This means there is little room to reduce‘CBthrough a manipulation of the bandgapwithin the base.The reduction of‘tBis 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 alloybudget;but drift-base transistors, based upon a non-uniform N(x), might become plausibleby the creation of an abrupt EB heterojunction. The reasons for abandoning drift-base transistorswere: 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 neutralbase 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 currentI from thebase, so that modulations toG#Bfrom changes toVCBno longer matter, provided punch-throughis avoided, of course. Following, once again, the prescriptions of Section 2.3, I isdecoupledfromG#Bby 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 insteadofthe neutral base region. With the two old problems associated with usinga non-uniform N(x)for the reduction oftBsolved, the optimum Nj(x) for the reduction oftBis investigated.3.3 Optimum Base Doping Profile to MinimisetBBandgap Engineering in the base is not really being considered in this section;however, itcan be included in the optimisation without any changes in the argumentsto follow (this includeseffects due to a manufactured change in the bandgap and changes to thebandgap due to heavydoping effects). Starting with eqn (3.7), then after substitutingp = N,tBbecomes:w_2WRn1 (x)NAB (z)= dzdx.(3.9)BNAB(x)D,(z)n(z)If D, is taken as some average constant, then eqn (3.9) is simplifiedeven further to become:w wj?n(x)?NAB(z)= =J Jdzdx. (3.10)BD,7ONAB(X)x nl(z)Eqn (3.10) provides the functional form oftBto be minimised. Using the calculus of variations,and searching for the weak variations in N(x)In (x), then the Euler-Lagrangecharacteristicequation 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 straightforwardand 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 intB.Within the confines of weak variations, a possible minimum could occur by admittingapiece-wise solution for N(x) composed of N sections whose form withineach 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 differentiationwith respectto the dependent variable x. In the case being considered, F = yly’. Then, using the exponentialsolution 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-pointx0 joining the two regions. If we let Region 1 joinwith Region 2, where the solution in Region 1 is A11e4Z1Xand the solution in Region 2 isA1,2eA2,2X,then the above equation requires that A2,1 = A2,= A2. Applying thefirst continuitycondition of eqn (3.15) at the point x = x0 produces:— == 1 1=A1 = A12= A1.ayY’2 A1 1Ae4° A1 2AeA2X0Thus, a piece-wise connection of exponentials is not admittedas a stationary solution for N(x).However, if the last equation is rewritten as A1 1AeA2X0A1 2AeA2XO,then as A2 —* 0 no restriction is placed on the values admitted forA1,and A1,2.This admittedsolution for y(x) is alsoa piece-wise and discontinuous set of constant solutions. As such, thissolution for y(x) tends towards a strong variation and care must be exercised in the absolute applicability ofthe 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, thenas 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 renderstBstationary 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 andtBwill 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)wheretBois thetBgiven by eqn (3.6). Clearly, as b —>—00,tB—*0. As a check, as b —>0,tBgiven 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 thattBis ostensibly reduced to zero.Therefore, the variational principles used to deduce that the optimum N(x) isa 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. Foreven thoughI is decoupled fromG#B, RBDstill depends onG#Band would become unreasonably large as b —*—00Eqn (3.10) is revisited, but this time‘CBis made stationary subject to boundary conditions uponN(x) at the emitter- and collector-sides of the neutral base. Since there appearsto be no simpleway of including these boundary conditions into the variational principles,a numerical minimisation was constructed [63]. The results of numerical attempts to rendertBstationary, subject to theboundary conditions placed upon N(x), produced a form that suggestsN(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 performedso far, where a constant was admitted as a solution to N(x). Even more convincing, the form being suggested fromthenumerical analysis is not a piece-wise connected set of exponentials (whichwas rejected as a possible stationary solution from the variational analysis), but isa piece-wise connection involvingconstant regions of doping, as is admissible from the variationalanalysis. 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) 5x1018cm3andN(xtWB) = 2x1016cmUsing the form for N(x) suggested from the numerical work, namely exponentialsseparated by regions of constant doping, analytic methods were employed to find the breakpoints between the exponentials and the constant regions that minimisedtB.Using the form of N(x)given in Fig. 3.4, then finding the break-point h that minimises‘CBgiven 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)](3.17)where‘CBOis still thetBgiven by eqn (3.6), h is normalised to the neutral base widthWB(andtherefore ranges from 0 at the emitter-side to 1 at the collector-side of the neutralbase), and U isthe doping ratio given byN(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 thatthe constant region occurs first followed by the exponential region. Eqn (3.17) representsthe solution ofthe simplest form of N(x) suggested from the numerical analysis.The process described above is repeated again, but this time with the optimumform (shownin Fig. 3.3) obtained from numerical analysis. Again, substituting thisform of N(x) into eqn(3.10) and minimising‘CBproduces, after considerable algebraic manipulation with the symbolicmathematics tool MACSYMA (see Appendix B):July 12, 1995 331 2h1l—h2= lnU+2andtB= tBolnU+2= tBo2hl,(3.18)where, h1 and h2 are nonnalised to the neutral base widthWB.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 thebase. It is very simple to provethattBgiven 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‘CBthen 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 formsuggested by Fig. 3.3.The process is continued by constructing more complexforms 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 whereh1 0 (h1 = 0 would give N(x) asshown in Fig. 3.3). Even though N(x) given by Fig.3.5(a) renderstBstationary, when comparedto the result obtained from eqn (3.18), it does not producethe absolute minimum value fortB.Infact, taking one final progression to using the N(x) as shownin Fig. 3.5(b), a stationary result isagain obtained, but it is larger still than the case shown inFig. 3.5(a) and therefore does not produce the absolute minimum value fortB.Therefore, eqn (3.18), with N(x) as shown in Fig.3.3,produces the absolute minimum intBsubject to the boundary conditions for the doping at theemitter- and collector-sides of the neutral base. Themost notable thing about the optimum form ofN(x), as shown in Fig. 3.3, is that it is not the pure exponentialthe device community has beenlead to believe is the optimum. This result answersthe problem posed in [64,65], where the authors used third order perturbation theoryto show that an exponential was indeed stationary but itdid not produce the absolute minimum fortB.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 shownbyFig. 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-wisecontinuous 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 thattBis minimised when:1 U+1h = andtB = tBO2U(3.19)Eqn (3.19) shows that a very simple jump discontinuity, or step, in the base doping proffleat exactly the half-way point in the neutral base, can reduce the base transit timeby a factor of two whencompared to the uniform base case(tBO).In fact, for any U 10, the full two-fold reduction intBis achieved. Still, for all relevant U,tBgiven 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-dopingcase shows thateven a very simple change to the base doping profile can produce a significantreduction in thetransit time through the neutral base. As for the technological objection thata perfect step-dopingprofile is impossible to create, any deviations from a step, say due to diffusion ofdopant during thethermal-cycle of the manufacturing process, will only tend to drive N(x)towards the optimumprofile and reducetBeven further: this result is obvious as a spreading of thestep-discontinuity in-creases the spatial extent of the aiding field and thereby decreases the transittime. Therefore, thestep-doping profile, although not as beneficial as the optimum doping profile, stillprovides for asignificant reduction oftB,but with very little complexity in terms of manufacturing.July 12, 1995 35Ix=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‘CBgiven 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 wasstated before, for the entire useful range of U (i.e., > 1),‘CBis minimised by the optimum doping proffle leading to eqn(3.18). However, for the range 1 U 7.389=e2,tBfrom the step-doping proffle is smaller thanthat from the pure exponential profile. Thus, not only have we foundout that the pure exponentialis not the optimum, we have also found that for small doping ratiosthe 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 optimumcase for the minimisation oftB.Thisresult shows that the optimum-doping case initially startsout looking much like the step-dopingcase, then as U increases, slowly transforms itself into the pure exponentialcase. Finally, for U =300, the optimum-doing case hash1 1 — h2 = 0.13 andtBis only 10% less when compared to thepure-exponential case; however, the optimum-dopingcase has a 49% larger Gummel number andthus a 49% smallerRBUwhen compared to the pure-exponential case. Clearly, the pure exponential case is not the optimum doping profile to use, either in terms of minimisingtBorRBU.Therefore, the optimum-doping case shown in Fig. 3.3 and governed byeqn (3.18) is the best base-doping profile to use in order to minimisetBwith the smallest impact onRB.July 12, 1995 36Table 3.1:tBfor 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 oftBO.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 0.550.5330 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 Ui03io41.00_;-- -,-‘\0.700.600.800.50Break Point h2Fig. 3.7.tBusingNAB(x)from Fig. 3.3, where h1 1 — h2 but h2 is varied asa parameter insteadof being given by eqn (3.18). h2 = 0.5 corresponds to the step-doping case, whileh2 = 1 corresponds to the pure exponential case. Finally, the line drawn on the surface is thetBthat 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 oftB.These two restrictions are: that the aiding field produced bythe 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 wouldbe atthe edge where high field effects begin to occur in heavy-doped GaAs. However,the second requirement thatDa(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) wouldstill 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 nextsection. Inany event,‘CBwill always be reduced by using a monotonicallydecreasing (from emitter towardsthe collector) non-uniform N(x). Therefore, if exact values and not generaltrends are required,then the optimum-doping case presented in this section must be applied with cautionif 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 conditionto theneutral base, in order to minimisetB.It was found that the optimum N(x) only depends on therelative doping ratio U, and not on the absolute doping givenby the boundary conditions. Furthermore, the optimum N(x) is not the pure exponential that the device communityhas 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 fromG#Bafforded by the creation of current-limitedflow at the CBS. Therefore, only by creating an abrupt HBT1 can the drift-baseBJT be manufactured without a significant reduction to the Early voltage.3.4 The Effect of a Non-Uniform n1 and D on the OptimumtBSection 3.3 derived the optimum base doping profile for the minimisationof‘CB.It wasfound that if the base doping was fixed at the emitter- and collector-sides ofthe neutral base, thenthe optimum N(x) was an augmented exponential shown in Fig. 3.3 and governedby eqn (3.18).1. It is possible to decouple IfromG#Bby 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 thebase if an EB heterojunction is to be formed in order to control f3. Therefore, the technique of current-limited flow isthe only practicalmethod to decouple I fromG#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 minimisingtB.Also, the effects of a non-uniform D(x) will be discussed.tBis 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 minimisestB.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 intothe 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 stillapplicable to the optimisation of n (x); for the stationary function y(x) has no dependenceon either N(x) or n1 (x).This immediately results in the optimum n (x) being given by the reciprocal toN(x) shown inFig. 3.3, with eqn (3.18) governing the placement of h1 and h2 and solving fortB.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 forEandEis ignored,then n (x) = n (x =0) exp (—AE8(x)/kT), where AEg(x) is now defined as the differenceinEg at x relative to Eg at the emitter-side of the neutral base. Since the optimumn (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 eithern (x) or N(x). Eqn (3.20)solves for the simultaneous optimisation of both n (x) and N(x). Thus, part ofthe 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 onG#BandRBD.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 givesUNAB(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 equaltoJ7.389 =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 minimisetB.The bandgap at the emitter-side of theneutral base (x= 0) is the reference point. U = n (xWB)/n (x :=0), where h1, h2 andtBaregiven 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 enoughto cause bandgap narrowing that couples n (x) to N(x). Since HBTs are characterisedby their very high base doping,bandgap narrowing effects need to be considered. Fortunately, the optimisation processthat renders y(x) stationary in eqn (3.20) does not dependupon the relationship between N(x) andn (x). Indeed, using eqns (3.21) and (3.18), the optimum y(x) has exactly thesame form as theoptimum N(x) shown in Fig. 3.3 (see Fig. 3.9). Therefore, with the optimumy(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 functionNB(x) for theactual N(x) is constructed by using h1 and h2 from y(x), and forcingNB(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 oftB,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 minimisestB.The break points h1 and h2,aswell as the transit timetBare 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, doesnot 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 presentedin thischapter, but as a whole cannot be construed as the same. However, current day BITs(and HBTs)are such thattBis an important but not dominant part of the total transit time (in the area of30%).July 12, 1995 41Therefore, more than a 2-fold reduction intBis really not warranted as the point of diminishingreturns would be surpassed. From the results presented earlier in this section,‘tBcan 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 reducetB,but only a full numerical optimisation will provide the true minimum [63].This section has found the optimum base profile for the minimisation oftBwhen 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 modelsandmethods 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 (suchas 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 currentTwill 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, thenTcould be constrained by the flux through this one feature alone. Fig. 3.1(b) showsthe general band diagram for HBTs built in theAlGai..Asmaterial system, where there is anabrupt heterojunction between the emitter and the base. The very nature of the sign ofAE,whencoupled to the fact that the emitter doping is much smaller that the base doping,produces a feature inEcalled the CB S [25]. The CBS can easily force the electrons to takea 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 forwarddirected flux, by four orders of magnitude at room temperature. A reductionby i04 in the forwarddirected flux will most certainly result in current-limited flow in the regioncontaining 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 isnot the only method oftransport through the region. Due to the quantum mechanical nature of the electron, andthe factthat the width of the CBS is typically of the same order as the de Brogliewavelength, the electroncould tunnel through the CBS instead of trying to increase its energy in orderto surmount the barrier. Since a reduction in the required energy to surmount the CBS leadsto 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 underestimateTby 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, thenI 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 withinthe 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,, andx 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+ek1where 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$d3kf2(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,Rris 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-xn0xpFig. 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 are2(2tY3electron 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 of1J;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 WU)= WU) = 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)- —hRf 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 andRwere equal, then thetwo integrals could be reduced to one integral with an integrand of (f1—f2)W(aUIa).One couldthen identify an Ff and F,. from this integrand (which strictly speaking is not the same as thatdefined in eqns (4.2) and (4.3), but for all practical situations is identical), giving:Ff $dpf1(U)W(U)L (4.4)hRfandFrij’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 andRrThe 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 FfandFdefined in eqns (4.4) and (4.5) begins by determining the transmission probabilityW(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 forW(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 ofW(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{_2f.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 massme,but to the moregeneral effective massm*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)whereVpkis 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) withWN(Ux)= 1 produces:xJ2mVkJl—U’+lWCBS(UX) = WcBs(U Vk)= ex[P(in( A[— AJi — uJ],(4.9)where U is normalised energy in terms ofVpk(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 whereVpkand x havenot yet been defined in terms of the material parameters and applied bias forthe EB SCR.WithW(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 determineU(p) and includethe effects of a non-uniform effective massm*that generally occurs at an abrupt heterojunction.July 12, 1995 48Once U(p) has been determined, the regions of integration Rf andRrare set out so that Ff andFrcan 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 massm*.This change inm*can either enhance or diminish theflux F in transit through the CBS when compared to the case wherem*is uniform throughout theregion. Failure to account for the change inm*can result in a significant error. Worse yet, this error is not simply a multiplicative constant as is stated by Grinberg [51], but hasa dependence onthe applied bias. Therefore, in solving for Ff andFusing eqns (4.4) and (4.5), the dispersion relationship U(p) needs to be determined in concert with the effects of a non-uniformm*.Concentrating on eqn (4.4) for Fj(the exact same results will apply to eqn (4.5) forFr),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 causea change of variables in the domain of integration fromp to U. To this end, the dispersion relationship willbe taken as parabolic, but left as a diagonal mass tensor to yield:U (p) = U(p) + U1(p)= ++£J’(4.10)wheremx,andmare 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 andP= psin€), (4.11)where eqn (4.10) has that:2 2pyPzandU.L=—+---—2m2m(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.2U2mPzPxpyFig. 4.3. Diagram showing the definitions of the cylindrical momentum space coordinates.At present, the non-uniformity ofm*has not been included, but it has also not been precluded. Setting aside the issues of a spatially varyingm*for the moment, the integration over p istransformed to U by the Jacobian:J(Px’PY’Pz —IUX,O,UL —apUxaeauao auap‘‘2apUxae(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 coordinatePxalone. From this realisation it immediately follows that:= 0and = 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 forU±and performing partial implicit differentiation with respect to Uj, gives:— p,— m — mpapmU± mzymz Py U±Then using eqn (4.11), which can be condensed and rewritten as = ptan2e, produces afterimplicit differentiation with respect toU±:— pyt (417)anFinally, substituting eqn (4.17) into (4.16), yields:1mmcos2e37)‘Z(4.18)aUPymcos2E)+m37sin2€)Pressing on and using p = ptan2e, but this time performing implicit differentiation with respect to (3, gives after some algebraic manipulation:ap37P cos9 apP1 1 Iapp= 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)mEi9mEi(3aem37paeFinally, substituting eqn (4.20) into (4.19), and using eqn (4.11) wherep, =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:immsin2eZ)‘Z• (4.23)Pmcos29+ 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+mysin9mcos9+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 =Xmm(4 27)U, 9, U)Pxmcos2€)+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 FjandFrMaintaining 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— 2qf uhJRL,u,e,u±)() (2i’m mm= _j•dOdUdUji_Z2Jfi(Ux+Ui..)W(Ux)_:ihR1 Pmcos O+msm 9 m2mm= —Jd9dUdU2 2fl(Ux+U±)WcBs(UX)WN(UX)(4.28)hR1mcosO+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-uniformm*has notbeen included. It is instructive to pause at this point and determine, under simplerconditions,WN(Ux)and thus the region of integration R1 before moving on to include the effect of a spatiallyvaryingm*.Withm,andmas 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 whenm*is non-constant). Re-examining Fig. 4.2 shows that in the region 0<x x,the potential profile that generatesWN(U)is of a strictly monotonically increasing nature (unlikethe CBS within the domain -x, x 0, which containsAE).Since we are considering a system inwhich their are no collisions that could either raise or lower the particle’s totalenergy, the particlemust emerge from the EB SCR with sufficient energy to enter into the neutralbase with an energythat is aboveE;else one would be admitting particle transport within the forbidden bandgap. Thisfact allows for a considerable simplification to the definition ofWN(UX);namely:(1 if U>VWN(U)= IX — b(4.29)k.0 ifUX<VbAlthough strictly speaking eqn (4.29) is not the full form forWN(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 forWN(Ux)is irrelevant, and eqn (4.29) suffices as it captures the essential feature ofWN(Ux).WithWN( U)defined in eqn (4.29), that last task to accomplish before Ff canbe solved forby eqn (4.28) is to determineR1.Re-examining Fig. 4.1, it is obvious that fora particle to enter theJuly 12, 1995 532itIdGmmm cos2O+msinG0Z=Immm cos2O+msinG0ZULEB 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 intoPx0. Furthennore, examination of eqn (4.12) showsthatp0 translates into U 0 (This is for the casewheremx> 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 UVbfrom eqn (4.29), then R1 will be as shown in Fig. 4.4.(a) (b)Ux/Ux+UEVbE000Fig. 4.4. Domain of integration R1 for a uniformm*.(a): case where the applied bias is such thatVb0; (b): case where the applied bias is such that Vj, 0. Note: Fig. 4.2 definesVb.UxEFfcan now be solved for by using eqns (4.28), (4.29), (4.9), (4.1), and the regionof integration R1 as shown in Fig. 4.4. Since R1 takes into accountWN(Ux),then solving eqn (4.28) yields:2itE E-U2mmFf =— J2 . 2fdU WCBS (U)fdU1f1(U + U1). (4.30)hmcose+msmO0Zmax(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 independentlyto yield:it/2It= 4Jmmtan’(StanqmJuly 12, 1995 54The above equation is evaluated by letting e approach itI2 from the left, giving:2itmmId9‘ Z= 2itjm m. (4.31)m cos29+m sin2eyz0Z3’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 energyU±which is random but evenlydistributed in all directions; every electron then passes through the CBSwith a probability oftransmission given byWCBSwhich 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., UVb),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 varyingm*into the transport current. The inclusion of a non-constantm*requires that the electron energy U(U + L1) be generalised to:U1 =U+Uand 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 atE(x=-x),so that using eqn (4.12) produces:1=andU1=+ (4.34)12m12mz1while2 2 2u—____VandU—‘2 Pz,2435X, 2— 2m2b J 2— 2m22mz2It is important to understand the exact meaning of eqns (4.33)-(4.35). To begin with, the energies U, U, andU±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 containingPx,2’it must also includethe offset potential energy ofVb.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, whileFrisbased 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 mechanicsdoes not allowone to consider momentum and position simultaneously. Eqn (4.3) must be interpreted withcare,because FfiS based upon a distribution in p-space located at x = -x,, whileFis based upon a distribution in p-space located at x = xi,. Essentially, due to the slow variation ofV(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 momentumpjcommutes 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) hasp=+p1)so that:py,1 = Py,2 p,andPz,i= Pz,2 = Pz(4.36)Since the potential energy V(x,y,z)(V(x)) does not vary in the transverse direction, thenP±,1P±,2cannot vary with x if collisions are prohibited. Using eqn (4.36) in eqns (4.34) and (4.35)shows that Uj1 andU±,2must 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= 2my1+ 2m1andUJ2= 2my2+ 2mz2Applying eqn (4.11) to the above yields, after a little algebraic manipulation:U±,1 m2mz2m 1cos2G+m 1sin2e_____= “‘ R(9) where R(G)= Z,(4.37)U±,2mylmz1mz2cos9+ my,2sin9Examination of eqn (4.37) shows the necessary condition that if m,1= m,i== mZ,2,thenUj1IU,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 differenceU±,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 IIx-xn0-x7oFig. 4.5. The effect that conservation ofphas upon U,1 andU±,2when a mass boundary isplaced at x = 0. Using m,1= m,i= m1 and m,2= m,2= m2 in eqn (4.37), thenU±,1IU±,2m21m1. (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 andUJ,2when 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— “ZR(€))=. (4.42)my2mz2my, 1mg,1—Finally, using this simplified form based on the functiony(the notation forywas initially set forth byChristov [70,73], but has been extended here to include anisotropic effects), eqn (4.37) becomes:ii1FT ‘ = 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 probabilityW(U),as defined in eqn (4.8), must be extended to:W(U) = WcBs(UXl)WN(UX2),(4.44)forWCBSis defined in Region 1 and thus depends upon U,1,whileWNis defined in Region 2and thus depends upon U,2.However, any function that depends upon total energy U (such as theFermi-Dirac distribution functionf(U))remains unaffected by the mass barrier due to the conservation of total energy set out in eqn (4.41). Therefore, eqn (4.29) forWNis rewritten as:11 if U >VWN(U2)x, 2 — b(445)X,‘0 ifU2<V,,The domain of integration R1,which is used for p- or U-space integrations performed atx =-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 mustpossessPx,10; or in terms of energy,U,i0. 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,2Vb,which results in a coupling between U,1 andU±,iwhen 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 R1:10,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.44.1and 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 fromU±,iinto 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,2and 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 into11x,2when 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 Ffin eqn (4.28).July 12, 1995 60E\,7Ux2+UJ.2=EU±2=R2\uX2oyE EFig. 4.6. Enhancement case where m1 <rn2 (i.e., y> 0 andy’ <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 thatVb0. Each domain of integration represents the ensemble of particles that contribute to Notice in R2 how the transfer ofenergy fromUjiinto U,2,due to the increasingm*in the direction of charge flow, leads to a focussing of the particles towards the direction of charge flow.+ U,1 =EVb-UXI(UL1yU2—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±,=ER20Ux2I ...........................................................Vb0EFig. 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, andx = x,, respectively: (a) theapplied bias is such that V,0; (b) the applied bias is such thatVb0. Each domain of integrationrepresents the ensemble of particles that contribute to Fp Notice in R1 how the removal of energyfrom U,,2 intoU±,2,due to the decreasingm*in the direction of charge flow, leads to a reflectionof the particles against the direction of charge flow. The reflection occurs because ofthe necessityfor particles to enter the base outside of the forbidden bandgap (i.e., U,2Vb, orpX20).EU,2U,1 + U,1 =EEE— Vb—UX1Y+ U = ER2Ux2Vb EVbVb-YE1—Y0(b)UJ-,1ZUx,1÷U±,1=EE-Vb1—YVb—UX1YU2—Y’Vb-YE1—yUx1EJuly 12, 1995 62Before eqn (4.28) is recast to include changes tom*(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—(U1’ L, 1 — U2 U2U2 9,u2)—2) — u1,1 1au2 au±,2Using eqns (4.40) and (4.43) produces:m 2m2u1=— m1mR(9)‘\x2J2J m 2m20“‘R(O)mylmz,1where R(9) is defined in eqn (4.37). Using eqn (4.37) yields, after substitution into the above:(Ui9U1,= (Ui U±1N= my,2mz2 mzlcos29+mylsinO(448)U2’,U±,2) 2’U2)my1m1m2cosO+ 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\U1,9,U±,)u2,u±,) u2,e,u±)Px,lmz2cosG+mysin9Examination 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 ofVb,which will not result in a deformation of the differential volumeelement. However, the termmi/p,iand notmx,2Ipx,2remains 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-uniformm*,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:2itEE— u,,, (4.50)2m1Ff= fde1cos2O+m 1sin2e$dUlWcBsx,l)J dU1,f(U+U±)0max(Vb,0)0max(Vb,O) E—U1+fdU1WCBS (U,1)Jd U11f1(u + U11)0Vb—UX1The termWN(Ux,2)is equal to 1 within the domain R1 and has been removed for clarity. However, ifWNdoes not have this simple form, then the fullWN(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:2itE E—U,2(4.51)2’m2Ff— Jd2 . 2JdU2WN (Ui,2)Jd U1,2f1(u2U12) WCBS (U, 1)h cos€)+ms1nE)o ‘ ‘yE 0Ux,2+ fdU2WN (U2)fdU1,2f1(U2+ U1,2)WCBS (U, )max(Vb,O)0In this caseWN (=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 whereWNdoes not have a simple form.However,WCBSremains nested within the third integral over U1,2 and cannot be easilyremoveddue to its dependence upon U1,which by way of eqn (4.40) is equal toU,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 andy< 0), then using eqn (4.28) with calculations based at x = -x and R1 defined in Fig. 4.7, produces:July 12, 1995 642itrE E—U,1(4.52)2m1 IFf_ SdOm1cos2+1 1sin29[mfdU1 WCBS(UX)jdU1f1(U+ U)0ax(Vb,O)0Vb- yE1—yE—U1— fdUWCBS (Ui,)fdU1f1(U1 + U1)max(Vb,O) Vb—UX,lAs was done with the enhancement case, the termWN(Ux,2) (=1 within the domain R1) has beenremoved for clarity. However, as is true for the enhancement case, ifWNdoes not have this simpleform, then the fullWN(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 overUj.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:2tE E—U,2(4.53)2m2Ff= i!Id0m2cosG+rn 2sine$dUxWN(Ux)jdUfl(UX+U±2)WcBs(UX1)oZJnax(Vb,0)00 E-U,2+ fdU2WN (U,2)$dU12f1(U,2+ U2)WCBS (U)min(Vb,O)—rAgain, as with the enhancement case, eqn (4.53) simplifies the problem of calculations involvinga complexWN,but at the expense of making calculations ofWCBSfar more complex. Basically, ifWNhas 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, ifWCBShas a simple form then use either eqn(4.51) or (4.53) for the calculation of Ff under enhancement or reflection respectively. Finally, ifbothWNandWCBShave 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 andUj,ito be identical in both equations and also equalto eqn (4.30) which is for a constantm*.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 fromUjiinto U,2,it is raised up intoEwithin the base to contribute to thetotal Ff As such, this current is termed the enhancement forward flux1enhanceFinally, 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 FfreflectSIt is also important to remember that when solving eqns (4.50)-(4.53),yandy’have a dependence upon e in general. Therefore, unlike eqn (4.30) (and thusFfstjJ)wherethe e integration can be treated as an independent multiplier to yield eqn (4.31), the calculation ofFjepice and1reflectwill 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 ofFrand 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 FfandFrespectively. 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 andFshare a dependency that is indicative of eqn (2.2), it is necessary to prove that the regions of integration for andFprovide 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)2m2F=-_ifdUfdUf,fd92 • 2ff(Uf)hm 2cos G+m 2sm 0000 0Z3’,where the superscriptfrefers 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 ofWCBS, WN,andf1to 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,2Vb,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 thep to UJacobian of eqn (4.27) in order to maintain the calculations at Xp. yields:002i(4.55)2mmFr = -_Jdu1fdUi,$de2z,2. 2W’)h m 2cos 0+m 2srn 0—000 03”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 forFrin 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 betweenFrandThe 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 thef-and r-functions istheir energy reference. SinceE(x=x) — E(x=-x) = Vb,and there is transverse symmetry, thenusing eqns (4.34) and (4.35):U1=U2—VandUI,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 reasony’and not y is used in the definition for U2,is because Regions 1 and 2 are interchanged for the calculation ofFrThis 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 mapsyinto y’. Since all of the functions used in eqns(4.54) and (4.55) are thermodynamically reversible (due to the fact the system iscollision-less),then a general functiongT(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= fdU1JdUI,domz,2cos2:::,sin2ef(U)Then, the only thing left to do before a direct comparison between eqn (4.58) forFrand 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 constantVb.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)2m2Fr= _fdUfdU.fde2., 2f(U)hm 2cos O+m 2sin ()0 0Z3’,Comparison of eqn (4.59) forFrand 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 =IdUf2 IdUf2dOm2mz2frh3J‘Jm2cosO+m 2sinO0 0‘[f{( U1’) h( U) — f( U”) h{( U”)] WBs(U1)W(U2)2it2’mm—i fifi“z’h3J ‘ .1 .1m2cosO+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 locatedat the bottom of the conduction band at x = -x,.Eqn (4.60) completes the proof that Fj andFrshare a dependency that is indicative of eqn(2.2). It also validates the modified definitions for Ff andFrgiven by eqns (4.4) and (4.5) respectively. Eqn (4.60) is brought into exact agreement with eqn (2.2) when thefiandf2distributionfunctions of eqn (4.1) are given by the Boltzmann approximation, leading to:July 12, 1995 69U -f1(U) =_______kT1+ekT(4.61)U - j.t2f2((1)=—2ekT1+ekTEqn (4.61), under the Boltzmann approximation, produces:U I’i____IS.Efl,f{( U) — f( U) = e e’ — ekTf(( U) (i — ekT)(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 — eJ.(4.63)Thus, the transport flux through the CBS has exactly the same fonn as eqn (2.2). This will allowthemodels 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 Ffthrougha complex two region system with an abrupt mass barrier in-between. The models also allow foran anisotropic effective mass tensorm*.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 modelof 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 VoltageVBE(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 1x1019cm3.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> 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 formofWNin 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 ofFjepi(or F!e for short), and aunique term for the reflection case of Fjreflect(orFjfor short). These terms, using eqns (4.50)and (4.52) are:July 12, 1995 712itE E-U12m1Ff=fd91cos29+msin2eIdUx,1WCBSx,1)fdU(U+U±,),(4.64)o“max(Vb,O)02it max(Vb,O) E—U1m 1m1Pe=dO2 2JdUX1WCBS(UXl)JdU±f(U+U±,),(4.65)hmz9+ m, 0o 0Vb—UX1Vb - yE2it E—U,12m1Ffr= 2+ .201dU1 WCBS(UX)JdU1f1(U+ U ).(4.66)o max(Vb,0) V,,—U,1yThe derivation of the analytic models begins withFj. Fjis 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) forFjbegins 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 = andm =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 1kT1+ekTFf=ZfdUWCBS (Ui)lnE —max(Vb,O)1+ekTThe integrand above becomes vanishingly small (at an exponential rate) for large U,, allowing fora simplification by letting E— 00to produce:/4itqjm 1m 1kT( -____Ff=$dUXWcBs(UX)ln,1+ekTmax(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)ekT(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 inWCBSfrom eqn (4.9) and making a changeof variables from absolute energy U to normalised energy U (where U= U/Vpk,andVpkis 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+12x2, x2—iX == (2and 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:ixfl./2mXlVPk y VPkFf5= 4itqJmlmz,1kT‘ke’IdU 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 whereWCBS= 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) andVpk(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 I2e2V xNDVb= x =q2NXP=qq2;= =(4.69)L2NAkTINANDN AEwhere N = and V. = — In I I +ratNA+C1ND q I2)qVbjis the built-in potential of the junction, n2 is the intrinsic carrier concentration in Region 2,NDis the emitter doping,NAis the base doping, e is the permittivity of the respective region, andVBEis the forward bias across the EB junction. The doping ratioNratdiffers slightly from that ineqn (3.5) due to a nonuniform e. Concentrating on the caseVb<Vpkthen using eqn (4.69) withineqn (4.68), along withIND= U + r where U= 5.VJeimgives: (4.70)________N(Vb.VBE) rFj= 4qJm1rnkTk;fdU erag(ch2(Up+r)PJ(4.71)hmax (VNrat (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 worthwhileto note that eqn(4.71), and the transform used to obtain it, follows that of Crowell and Rideout[781 usedin 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 1in1kTVpk;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 maximumcIoccurs 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 containingr 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 1.4Ir2rsh3(U +r)2— th(L1,+ r) =—P0 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—>0othen 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—oooccur 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 1mlkTVk —- Umax=y, z,Pe’e‘at U = ch2(U). (4.74)hpVBE=1.4VVBE=O.9VV, whenVBE= O.9V-V’, when1. 0.2 0.4 0.6 0.8Normalised Energy U(Vpk)Fig. 4.9. Flux densityfs’normalised tOmax’for anAl1j3Ga07As/GaAs abrupt HBT at twodifferent forward biases. The material parameters, the same as in Fig. 4.8, are: emitter dopingND5x1017cm3;base dopingNA1x1019cm3;emitter permittivityElll.9cj;AEis 0.24eV; n2 is2.25x106cmVbìis 1.67 1 V;mx,lis 0.091m0;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?maxoccurs 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; asNDincreases, or e decreases, the width x of the CBS decreases and U,’, becomes smaller, showing that tunneling is increasing; asm,idecreases 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)(_ch3(U+r))=ch2(U+r)pdrdUch3(U+r))2sh(U+r)and then eqn (4.71) becomes (under the condition thatVb<VPk):N,.,1(VbZ — VBE) Nrat (Vbx — VBE)F— 4itqJmlmzlkTVPk1ddC 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 issuewith the solution ofeqn (4.75) is thatr(C)cannot be determined in closed form. IfC(r)were invertible then eqn (4.75)could potentially be solved analytically. Observation of Fig. 4.9 shows thatI, the integrand ofeqn (4.71), has the form of a Gaussian. Indeed,‘jis extremely symmetric and suggests that aTaylor series expansion about U (i.e., r = 0) forC(r)is a potentially good approximation. Performing a Taylor expansion ofC(r)about r = 0 up to second order produces:sh(U)dCsh(U)C=—r2—th(U)= ——2rch (Ui)drch (Ui)Finally, substituting the above approximate equation forC(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)eufdreUVch3(U)jdrrch3(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 thatVb<Vpk)(see also eqn (4.92) for low temperature considerations):Nrat(Vb VBE)th(U)Ff5=FjsoPPPteu[erf()+(4.78)(ach(Jmax (vi, 0)— UNra: (i VBE)+erf +F VeafsOtror approximately asJuly 12, 1995 77Nrat (Vb1 — VBE)th(U)Nrai (Vbx — V8)btv sh(U )U V - 11 -________z’p p Pttp j’ TI I‘fs1fsOI mIfsOvtqch(U,,)where214iq im m kT—F— ‘v y,i z,i kTh3Finally, under the condition where Vj,Vk:VbI—V8—AE/qFf = FfSQVeV,(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 termFje.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!ecannot be performed independently. Further, eqns (4.42) and (4.37) show thatyhas a complex dependence upon € that would most likely cause the final integration over 0, forthe calculation ofFfe,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,1andmZ,2 = ammz,1.With this approximation, then using eqns(4.42) and (4.37) it is found that:m 1m1(a m 1cos2O+a m 1sin20”iy(0) = 1— ‘z m z m“= 1 —. (4.80)amy,lmz,1m,1cos20+ m 1sin20,) amEqn (4.80) reducesy(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 ofFjewill 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) Uy— i4tqim m kT —— ‘v y, i z,i kT 7K1,kTY— h3UxVYCBSkxi0Examination 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)wheream = =(4.81)m1mz,lTeffis then the effective temperature of the flux density. Under the enhancement case y> 0 andthusam>1, leading toTejf<0. With eqn (4.81) substituted into the equation preceding it, then:Vbmax(Vb,O) U4iq.jm 1m 1kT — --i- -Ffe= 3z,e”eTfdUXWCBS(UX)eeff(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 negativetemperatureTeffis 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 forFj,but the fact thatT?ff< 0must be accounted for. Population inversion, when combined with the fact thatWCBSalso 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 ofFj,will solve eqn(4.82) forFjewhen 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 ofmax(Vb,0). Eqn (4.82) is solved by returning to eqn (4.71) andintroducing lffinto all relevant equations to yield (under the condition thatVb<!‘Vbmax(V,,O)V/qrFje= 4qJm1m1kTee_VpkfdU-th(Up+r))(4.83)whereINkTp,eff— 2V1jj4emt,effqThe primes on the energies still denote normalisation with respect toVpk.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(Ueff+ r), anddl— 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 V2_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 thatVb< 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 whichWCBS= 1). Finally forVb <Vb — VPk,J1—max(VI,O)(‘VPkrbmax(Vb,O)F — FUp,jcVt,effqUV qUpffV8ff— 1fee e 1ebwhile forVb5 --__i_F —F VykT1 qV484fe fsOt,effeeAs a final check on the validity of the model forFje(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, whenVb0, the upper limit ofintegration in eqn (4.82) is zero and the integral itself vanishes. For the case whereVb> 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,thenU(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 forFjepresented in eqns (4.85)-(4.86), attention is finally focussedupon the solution of the reflection term1’JrEqn (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 ofFjce,the 9 integration to calculateFfrcannot be performed independently. To simplifythis problem, as was done withFfe,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 of1jrIn fact, using the same basic steps from eqns (4.80) to (4.82) will also solve forFjrThe 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:IIllV004irq im 1m 1kT — --- . ---— v ‘ ,‘ kT ‘1’-’I 4T1 WI (TI ‘ ffLfr3I ‘‘x “CBSk”x)”hJmax(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.Fjeon 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 fromVb< U,1<00. The form of the integral forandFrmust 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 forFfeand 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 ofexp(-VbI’1c7)that occurs in eqn (4.87) that is not present in the modelfor Ff5). Examination of eqn (4.81) shows that wheny< 0 (as it is for the reflection case), thenTeffhas a range of 0<Teff<7’; where 7e—*0 as y —*0, andTeff—> T asy——oo.Therefore, the fluxJuly 12, 1995 81density in the reflection case is characterised by a temperature that is always less than the latticetemperature7;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, theexp(-VbI’)&T)term must also be included).With 7ff instead of T used for the flux density in eqn (4.67), along with theexp(-Vb/4c1)term, the final model for the reflection case becomes (under the condition that17b<Nrat (Vbx — VBE)th(U8ff)F = F/pk5h(Up,eff)Up,effVt,effVteff Up,effreff(UPeff+ (4.88)q qch3(Upff) L r,eff)(ach(Jmax(V’, 0)— UffJNrat(VbiVBE)Ll;’ Jt,effafrO”t,eff’r,effwhereVb I3--j-. Ich (U ,)UF — F“d— Ip,e, t,ejjfrO — fsO r,eff— q (Vk/q)Sh(Upeff)Finally, whenVb Vpk,then:Vbx—V—AE/qFfr= FfrOVteffeVt,eff(4.89)Eqns (4.88)-(4.89) present the analytic model for Fj,., which is basically the same as themodel forFf,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 forVBE =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,. becauseTeffis even less than T(for an Al03Ga07As to GaAs flux,m,i = O.O92meandmX,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) forFfabout the lower limit of integration; namely max(Vb,0).Fortunately, in the course of solving the enhancement case, the desired expansion aboutVbhas already been performed. Eqn (4.84) is the expansion aboutVbup 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)rrb1Ffr= Ffp,eff t,ejje“°t,effer,eff— erf(—)(4.91)r,eff L r,effjwhere Ff,.0is defined in eqn (4.88), and(Treffis altered from its definition in eqn (4.88) to:IUpeffVteffareff— 4(V/q) V Ji —Eqn (4.91) solves for Ff,. whenU0<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 thatAEis 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,Fjis given by (after a simple extension from eqn (4.91)):V/q ,, rU vThJ(rbVb -.J1-Vb)rbFf5= Ffoe er[i — erf(_)] (4.92)where, in this case only:Iuva= IV t______andrb= ach1 1— Urq(Vk/q)VJ1—VJ)°Eqn (4.92), in concert with eqns (4.78)-(4.79) form the model for Ff.with an unrestricted placement of the base barrier potentialVb,and the ability to model very low temperatures. Likewise,eqns (4.88)-(4.89) and (4.91) form the complete model forFjrFinally, 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 3areff(orar),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, asareff(ora)—> 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 onIThe 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, 2Fjbase)’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., Ffbaseand 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+Ffeify> 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 dopingND5x1017cm3;base dopingNA1x1019cm3;emitter permittivity C1 is 1l.98o;base permittivity£2isl2.9c-j;AEis 0.24eV; n,252.25x106cm m1 is 0.092m0;m2 is 0.067m0;-* Nratis 0.956;Vbjis l.671V;x(VBE=O)is649A; U is 0.488; U is 0.795; yis -0.373; lffis 81.5K;Up,effi51.80; U1is 0.104;Vbis>0 whenVBE< 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(=0.046m0),or to of m1(=0.023m0).The enhancement case typically doesnot occur for electrons, but most certainly occurs for holes. Using the reciprocal relations to thereflection case gives m2: 0.126m0;0.184m0;0.368mj. Changes to the effective density of statesdue to the changing m2 are not reflected intoVbjnor 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 plotsFjas well as Fj,.. using the analytic models of the previous section for the three m2 cases of: 0.067m0;0.046m0;0.023m0.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,2energy to satisfy themomentum conservation requirements and enter the neutral base. Furthermore, asVBEis increased,Ffrbegins to decrease and then decrease quite rapidly. The physical cause for this is the interplaybetween the base potentialVband 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, asVbismade smaller, more energy can be removed fromU,,2without encountering reflection. SinceVbdecreases asVBEincreases then F!rmust decrease, relative toFf,asVBEincreases. The sudden decrease in Ff,. forVBE>1 .4V corresponds to the point at whichVbgoes below the reference potenJuly 12, 1995 85106ioT=300K102Js(a)100r;m2 = 0.023Z 10Fj;m2=0.046-6Z 10Fj;m2=0.067108. • • • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter VoltageVBE(V),_10• • •T=2OOKr:046Base-Emitter VoltageVBE(V)Fig. 4.10. Standard FluxFj.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)forFjand eqn (4.87) forFjr(a) results for T = 300K. (b) results for T = 200K.July 12, 1995 86tial energyEin the neutral emitter(Vbis <0 whenVBEis> 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 smallm21m1).Thus, onceVbdecreases below zero, reflection will taper off quickly as there are essentially no more particlesto reflect from theVbbarrier.Looking now at Fig. 4.11(b), as Tis reduced from 300K to 200K, there is an increase in Fj,.relative toFjat low bias whereVb> 0. The physical explanation for this fact is more complex.First of all, any particle whereU±,iis zero will be unaffected by the mass barrier because momentum conservation is guaranteed whenpis 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 ofUVpkis 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±,iwill 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 ofFfrrapidly 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 forVBE>1.43 (which corresponds toVb<0),the effect ofFjris negligible. As was discussed earlier, onceVb<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 fromVbeven when Vj, <0. These results clearly indicate that the position of Vj, is veryJuly 12, 1995 8708T=300K0.6(a)0.40.%10 ii 12 13 141516Base-Emitter VoltageVBE(V) VoltageVBE(V)Fig. 4.11. Relative importance Of Ffr 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 atVBE>1.OV for T= 200K, andVBE>1.2V for T= lOOK.July 12, 1995 880.6 •050.4 T=100K,T=200K(c) 0.3:O.%IHIO.ll.lI2.lI3.l4•5N1.6Base-Emitter VoltageVBE(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 VoltageVBE(V)Fig. 4.12. Standard FluxFjand Reflection Flux Ff,. for an HBT with the same parameters asFig. 4.10, but withAEreduced from 0.24eV down to 0.12eV. Note how reducingAEincreasesthe relative importance of the reflecting potential barrierVbby loweringVbj(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 largeAEso 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 systemsiXEis smaller so thatVbstands as a larger reflector. Fig. 4.12 showswhat the effect of reducingAEfrom 0.24eV down to 0.12eV has on the transport flux. Underthese conditions V1, remains unchanged butVbjis 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 0.126m0,0.184m0,and 0.368m0.Fig. 4.13 is basically thesame as Fig. 4.10 (except that7is now positive under the case of enhancement), and plotsFfs,aswell asFfe.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 FfePhysically, as m2 increases, the mass barrier will transfer moreenergy fromUj.iinto U,2 in order to conservepj (see Fig. 4.5); thus, a larger number of particles will be moved from out of the base bandgap and intoEto 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 VoltageVBE(V)10• • • • •T=200K:, io410E 1of1_\U)10-2Fje;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 VoltageVBE(V)Fig. 4.13. Standard FluxFj,and the Enhancement FluxFicefor 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) forF1and eqns(4.85) (4.86) forFfe,while the solid dots are from the numerical calculation of eqn (4.67) for Fj5and eqn (4.82) forFje.(a) results for T = 300K. (b) results for T = 200K.July 12, 1995 91VBEis increased,Fjebegins to decrease and then decrease abruptly. The physical cause for this isexactly the same as for the reflection case. AsVBEincreasesVbdecreases so that fewer particlesneed to be helped over the barrier andFfedecreases. In the event thatVb<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, onceVbdecreases below zero,Ffemust abruptly vanish.Moving on to Fig. 4.14(b), as T is reduced from 300K to 200K, there is an increase inFjerelative 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 belowVb.With more particlesexisting belowVb,the mass barrier may effect a larger transfer of particles from below to abovethe base barrier, and thus increaseFjeas T is reduced.The most important difference to note between the enhancement and the reflection case isthat a smaller increase occurs inFjewhen 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•Withy—*—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—>—ccthenTeff>Tso thatFfr—* 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,2by removing energy fromU±i.At the limiting strength of the enhancing mass barrier (i.e.,y= 1), the entire amount ofUj,iis transferredinto U,2 (see eqn (4.39)). Since the particles will have a one kT spread of energy inUji,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 inFfedue 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 ofy= 1. Looking at Fig. 4.6 for the integration in R1,then obviously inthe limit wheny= 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 forFfeand thus F aswell. For the reflection case of Fig. 4.7, there is a limit of y —>—0oWheny—*—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,Vband T uponFje.Concentrating on Fig.4.14(a), there is clearly an increase inFfeas 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 forVBE>1.43V(Vb1changes with 7),Ffe= 0 as there is nolonger a base barrier to surmount. Finally, Fig. 4.14(b) shows that reducing T increasesF1einmuch 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 surmountsVbj,is modelled throughout the entire EB SCR.July 12, 1995 93o.ioT=300K\m2=0.368.0.06(a)Em2=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 VoltageVBE(V)0.30•“.- LZi2=0184_T=100K(b):Base-Emitter VoltageVBE(V)Fig. 4.14. Relative importance ofFjeto 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 atVBE>1.OV for T= 200K, andVBE>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 basewidthWBbeing 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 provideamethod 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 characterisedby 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 goingtowards the base, the Maxwellian distribution is pulled apart so that only the right-going halfof 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 fork< 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 fromx = 0 to x = x, thedistribution will no longer peak at k = 0 with an energy of 0, but willbe shifted towards largerkwith an increased energy ofIXE— V, relative toEat x = x. This shifted hemi-Maxwellian istermed “hot” because it appears to look like a distributionthat 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 of0. (a)0.20.0-0.04-0.04-0.02. -±0,020.02 Normalisedk0.040.04I(b)z1.‘>:p.; — - ----- -0.02I.±0.040.04NormalisedkJuly 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 ofU±,icontained by each particle. Clearly, this will tendto focus the ensemble at x = 0 towards higher U,1 and destroy the circular symmetry that existsbetweenkandk±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 potentialVbwhichwill 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 againstki.Furthermore,Vbcuts the distribution off for particleswhere U,2(=U,1— Vbbecause 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 Vectork,2(1.11x108cm1)Fig. 4.16. Ensemble distribution versus wave vector k2 entering the neutralbase (i.e., at x = x)(T= 300K). This is essentially a replot of Fig. 4.9 except whenU,2< Vbthe 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,2is 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 distributionalongk,2.At low bias, whereVbis approachingUmax,the ensemble distribution is clipped verynear the peak of the distribution,but, unlike a hemi-Maxwellian, not right at the peak. Further, the Gaussian formwith respect toenergy results in a very flat-topped and non-Gaussian formwith respect to k. As the bias is increased, Vj, recedes when compared toUmso that the distribution no longer has a clipped form.This results in a hot distribution that is asymmetric and whichlooks quite different from a shiftedMaxwellian. Fig. 4.16 clearly shows the non-Maxwelliannature of the ensemble distribution entering the neutral base. However, it does not show thedistortion that occurs alongk± (k± = k±,i =k±,2because of momentum conservation in eqn (4.36)). In orderto see the full ensemble distribution entering the neutral base(= WCBS(Ux,1)fl(Ux,1+ Uj,1)), a three dimensional plot versusk,2and k,2,is displayed in Fig. 4.17. Observation of Fig. 4.17 clearly shows thenon-Maxwellian ornon-hemi-Maxwellian shape of the electron ensemble distribution entering theneutral base at x =x. Furthermore, Fig. 4.17 also demonstrates that it would bea gross approximation to assumeJuly 12, 1995 980.75—0.50--0.04,zt±-0.02ç\0•‘>)ç_0.06Normalisedk,20.04 0.081.00-(a).0.75—0.50—ri0.25-0.00-/-0.04-0.02 ‘7Normalisedk±,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 lxi08cm1)•(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 clearlyindicate 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 lowerk,2andhigherk±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 constantk,2line from the peak shows that the distribution isindeed being pulled and distorted towards lowerk,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 neutralbase 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 ofthe enhancing mass barrier isto simultaneously pull the distribution towards higherk,2and lower k1,2.Comparison of Fig.4.19(a) with Fig. 4.17(a) demonstrates that the distribution is certainlybeing 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 theextension ink,2,which is aclear deviation from a Maxwellian form. Further examination ofFig. 4.19(b) in comparison toFig. 4.17(b) exemplifies the distortion to the ensembledue to the enhancing mass barrier. As similarly occurred with the reflecting case, the volume in Fig. 4.19 appears smallerthan the volumein Fig. 4.17. However, there is now a multiplicative constant of 4 (for this enhancingmass 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 tunnelingand the mass barrier haveupon the electron ensemble distribution entering the neutral base. The one clear conclusionfromJuly 12, 1995 1001.0:0.750.50.20.0-0.04-0.02-±,,2S0.00HI(b)N0.00Normalisedk,2Fig. 4.18. Replot of Fig. 4.17 but this time including a reflecting mass barrier wherem2 = 0.023and m1 = 0.092. (a)VBE= 1.4V. The plot has been rotated450relative to Fig. 4.17(a) to clearlydisplay the distortion in thekx2direction. (b)VBE= 0.9V. Again notice the extreme distortioncompared to Fig. 4.17(b) fork2less than the peak.July 12, 1995 101.(a)I0.02.(b)\E0z0.00Normalisedk,2Fig. 4.19. Replot of Fig. 4.17 but this time including an enhancing mass barrier wherem2 =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 largerkx,2.(b)VBE= 0.9V.Clearlyk2has been extended and has been squashed relative to Fig. 4.17(b).Normalisedk,20.040.10Normalised0.040.08July 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 ofVbused to calculate the flux through the CBS. Reexamination of Figs. 4.1 and 4.2 show that FfandFrare 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 byVband distortion due to the mass barrier, alters the course of the forward andreverse directed hemi-Maxwellians. To assume that a hemi-Maxwellian form existsat 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 wouldbegin to occur. It is for this reason that the flux is considered to be injected from x = -x, andleading to the potential boundary of17b(which is equal toEat 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 expectedto 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 fromVbtoVb-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 becomeincreasingly large. Furthermore, as the temperature is reduced to thepoint where U,, occurs belowVb,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 VoltageVBE(V)Fig. 4.20. Relative difference between the results obtained fromthe methods proposed in [51] tothe model for F from this chapter. The device is based uponthe same Al03Ga07As HET used inthis section. Note how the reduction toVbas proposed in [51] leads to an overestimation in thetransport through the CBS, and therefore, to an overestimationofI.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 isa 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 thispast 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[681in 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 ofamass 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 vacuumand not within thesemiconductor, and does not consider the effect of a base barrier potentialVb(asVbis 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 ofVb.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 ofVb.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 bydifferent effectivemass tensors and e. Furthermore, the effect ofVbis properly included. The most important aspectof the work contained within this chapter is that for the first time all of the essentialphysical constructs of the EB junction within an abrupt HBT have been considered. The results of these considerations are analytic models, based upon the solution ofeqns (4.50)-(4.53), to simulate thetransport of flux through the EB SCR. Since there were no special features ofa specific materialsystem employed within this chapter, the results of this chapter are applicable to any materialsystem. 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 106CHAPTER5Recombination CurrentsJuly 12, 1995 107As was discussed in Chapter 3, one of the most important parameters of an HBT isthe current gain [3. Whether one is designing Digital or Analogue circuits within an IC, anaccurate understanding of13is essential to the successful operation of the circuit. Chapter 4 dealtwith thecalculation of transport through the CBS (in an npn device), which is often the determiningfactorfor I in abrupt HBTs [18,25]. This chapter will finish off the model for Iby using the generalmodels of Chapter 2 to include the effect of neutral base transport along with transportthroughthe 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‘BwithIcthat wasalluded to in Chapter 2, and which occurs when transport through the CBSis responsible for current-limited-flow (i.e., control ofIc).This chapter includes the modelling of four different components ofthe 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 EBSCR; 3) radiativerecombination within the EB SCR; 4) neutral base recombination throughall of the processes justdetailed. The back injection of carriers (i.e., holes for the npn HBT beingconsidered) from thebase into the emitter is not accounted for because this back injection is effectivelysuppressed 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 recombinationprocesses that are responsible for the creation of‘Bwill be presented. It is shown that these analytic expressionsfor thefour base current components can be reduced to the familiar diode equationswith two parameters- namely the saturation currentJ and the injection indexn. Even though the physical mechanismsthat control the base current in the presence of a heterojunction differ markedlyfrom the homojunction case, one can still recover a simple diode model forthe final representation. It is withinthis analysis that a surprising result regarding the injection indexn is made. Standard theoreticalcalculations give a value of n = 2 for the SRH current. However, it wasfound that n = 2 appliesonly in the limit of a wide, or symmetrically doped, EB SCR. ForHBTs of interest, where thebase doping is very high compared to the emitter doping (i.e., asymmetricallydoped), 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 publishedby thisauthor and Dr. D.L. Pulfrey [24]. Within the context of this published work,HBTs constructedwithin theAlGai.Asmaterial system were studied. The results of this chapter are general, however, and can be applied to other material systems as well. For thecase of indirect material systems such asSiGei..,the only major change is that the radiative recombination rateis 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 canlead 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 2and was found to be the drivingforce for the transport current through the CBS (as was proven in Section4.3, eqn (4.63)). Fig. 5.1shows AE1 and its position within the EB SCR. resultsdue to a departure from quasi-equilibrium, where the transport flux through the CBS is no longera small perturbation to the forwardand reverse equilibrium fluxes that are everywhere present withina semiconductor [50,18].EE(eV)AEt---‘—‘VE1EqV+_i-x 0Fig. 5.1. Band diagram of the EB SCR showing the effectof the abrupt heterojunction onunder an applied forward bias (reprint of Fig. 2.2). ji isthe solution to the Poisson equation and istherefore continuous. Both the referenceenergy 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 addressedthe spatial variation of and foundthat in general AEj, occurs abruptly and coincidentally withthe position of the EB Heterojunction(as is shown in Fig. 5.1). Finally, the hole quasi-Fermi energy Ej, has no discontinuityand 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 andNBagainst the combined thermionic/tunnel currentThTcrossing through the CBS at the abrupt junction; i.e.,ThT = NB +J( AEfl. (5.1)Further, it has been the usual practice when considering additionalbase current due to recombination in the EB SCR, to subsequently add this extra currentSCRto the prior-calculatedNB;i.e.,=1’NB(fn) SCR(5.2)Recently, Parikh and Lindholm [90] have emphasized that this calculation ofBvia direct superposition is not strictly correct because the base-side componentSCR,BofSCRshould figure inthe original current-balancing equation which is used to compute AEp,and, subsequently,c, NBandscR,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)whereB,Bis that portion of the base current arising from recombination inthe metallurgical base(see Fig. 5.2).It can be appreciated that this more correct, self-consistentcomputation ofSCR,Bwill onlyeffect the base current ifSCR,Bis comparable toNBand, furthermore, will only effect the computation of from the balancing equation (i.e., eqn (5.3)) incases 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 havebeen 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 modeffingbecause AE, and thusJuly 12, 1995 110Jc N.BandSCR,.Bcould 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 intoa 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 thatThTmustequal the total of, J( +NB+SRflB+Aug,B+Rad,Bwhen 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, theneutral-base current andthe base-side SCR recombination current comes about because all these currents depend upontheelectron quasi-Fermi energy splitting at the heterojunction. As this splittingis 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 Al030a07As emitter anda p-type GaAs base (the same as the device in Section 4.5). To reduce the complexity of the algebra, without sacrificing muchin 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—______________inhfP)(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,whereEis 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 AE1at 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 conductionand valence bands; i.e.,AEGflAE1=+ AE= kT in+ AE(5.6)where the subscripts p,n refer to the p-type base and the n-type emitterregions respectively. E1 isrelated, therefore, to the electrostatic potential energy ji(x) viaE (x) =IN’(x) x 0(5.7)N(x)—AE x>0Here we use the depletion approximation forif(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 AEkT(NANDAE1Vb.= —liii 1+— = —mi 1+—qi)q q qwithVBEbeing the applied base-emitter voltage,Vb1the built-in potential,NDthe emitter dopingandNAthe base doping. Eqn (5.9) has included the effects of a non-uniformpermittivity for thetime being.Fig. 5.3. Energy Band diagram for the EB SCR of an HBT under equilibriumconditions. Noticethe discontinuity of AE in the intrinsic energy E.I2eVPkfl= Al q2N—CPNAwhereNrat— 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 analyticwithsimple transcendental functions. A closed-form solution demands that some approximationbemade forW(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,XpWBE(l — Nrat,d), WBE= x, + x, andNrat,d = NA! (NA+ND).The linearisation of ji(x) in eqn (5.11) differs from that proposed by Parikhand Lindholm[90]. In [90], the linearisation is based upon a first order expansion of eqn (5.8) about the pointwhereRSRHis a maximum. The problem with this type of expansion is theRSRHmaximum mustbe well localised within the region of integration. If the Rpjj maximum is notwithin the region ofintegration (as it can be for reasonable operating biases), then the first orderexpansion proposed in[90] can lead to significant error. Eqn (5.11) alleviates this problem by appealingto the mean-value theorem to define the linearisation. In fact, asVBEapproachesVbj,eqn (5.11) becomes exact.Eqn (5.10) can now be evaluated using eqn (5.11) andEf—Ef=qV x0qBE—fX>(NDN2kTlnL—Jx0Ef+Ef = —2kT1n(.) qV AE x>0to yield— 2qnWB.rVBE_AEJn1(zo—zSRBB— te51[q2kTatan+(5.12)2qn1W (Z,—Z0,SRFLE=sinh[2kTjatan+ 1July 12, 1995 114with= q(VbI — VBE)/kT= Jtpoxtnoxz— ND1tpQ,fl[qV—4--——exP[--2kTND1tpQr2Nrat(“bi— VBE)+VBE1Z0 = —q—--—exp[—q2kTJ (5.13)z—ItpOpexqvBE_IXEfflP—P[2kTND ItpO,pr2qN(VbI — VBE)+ qV + AEffl — 2AE1Z02kTwhere it is assumed thatEand 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 allcases, 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 usualequation for SCR recombination in homojunctions. Also, the unique feature to HBTs, quasi-Fermi-energysplitting, isexplicitly brought out by the presence ofAEJkin the expression forSCR,B•Finally, the linearisation used to obtain eqn (5.11) results inthe use of the doping ratioNrat,d,and not the voltage ratioNrat,within eqn (5.13). As was stated at the start of Section5.2,the effect of a non-uniform permittivity is quite smalland can be neglected within the largerapproximation of a linearNJ(x).For this reason, it is assumed that for all practicaldevices encountered thatNrat Nrat,d;in fact, for the parameters used in Section 5.4, thisis only a 0.4% error.5.2.2 Auger RecombinationAs the doping concentrations increase, Auger recombination becomesan 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 conductionband electron. In the first case,the recombination rate is proportional top2n, while inthe 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 and4are 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[kTjsinh[2kT(Z0 — Z) (A PnO pzz0+ A,t0)(5.17)2qnWBEqV qVAug,E=exp[ kTsinh[ 2kT](4— Z) (AntnnZnZn+AEqn (5.17) gives the Auger recombination currents that are generatedfrom 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 considerdirect band-to-bandradiative recombination. The rate at which radiative recombinationoccurs will be proportional tothe pn product [94]. When the equilibrium recombination rates are included, the totalradiative 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 rqBERad,Ehui,nBnwNrat[exP(kT—15.3 Current Balancing with the Neutral Region TransportCurrentsIt 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 acrossthe hetero-interface;i.e.,ThTJscR,B+JNB+Jc(5.21)whereSCR,B = SRH,B+AugB+RadB•(5.22)The formulation given in eqns (5.21)-(5.22) was already treatedin Section 2.2. Comparison ofFig. 5.2 with Fig. 2.3 shows an exact agreement. Therefore, thecurrent balancing portrayed byeqns (5.21)-(5.22) can be solved using the models given in Section2.2 if the various transport andrecombination currents follow the general functional forms assumed inChapter 2.ThTis the transport current through the CBS that was solved for in Chapter4. Eqn (4.63)shows that the flux F through the CBS (Emr)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 concertwith themodels of Chapter 2 to solve for the collector and base terminal current densitiesJ andBrespectively. Looking again at eqns (2.3) and (4.63) shows thati2 1= Ff. and = AE. F3cin-cludes both the thermionic emission and tunneling components involved inthe transport over andthrough the CBS. Employing the formalisms of [51], Ff can be writtenas:July 12, 1995 1172J1(Vbx — VBE) (VbZ — VBE)4iqJm 1m 1(kT) — -_______ -_______Ff= y(VBE)“ZekTekTqyuNekT(5.23)hwhere 1) is the electron thermal velocity given by= /kT(5.24)eq 2itm,andy(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 currentgivenby the last term in eqn (4.78). Essentially, yis given byFfIJthwherethis the thermionic injectioncurrent and Fj= FjCBSgiven 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,JIis theelectrochemical potential relative toEformed by the dopingNDwithin the neutral emitter. Theapproximate solution given in eqn (5.23) is strictly valid only if the emitter isnon-degeneratelydoped.The neutral-base recombination currentNBand the transport current through the neutralbaseJcwhich must be used in eqn (5.21) follow from the standard, low-levelinjection solution tothe continuity equation. Using the boundary condition that thedriving potential at x = x, (i.e., thestart of the neutral base) isVBE— AE (see Fig. 5.1), and for the case of a singleheterojunctionstructure operating in the forward active mode, the excess electronconcentration near the collector is 11(Wflb)= 0, whereWbis the neutral base thickness relative tox = x, then the expressions for these currents areIWflbN2coshl— I—iqV8—zEf12coshl— —iqV1—LflbkT1qDn1NB— NALflb .(We—— NALflb .(We — ijesinh-L——jsinhç-J(5.25)andqDn,csch(—)[ qVEAEf—qDn,csch(—-)[e—1]e (5.26)Anb nb Anbnbwhere D is the effective electron diffusivity in the base,andLflb (=JDfltflb)is the electron minority carrier diffusion length in the base. Observation ofeqns (5.25) and (5.26) show they possessthe functional forms of4and 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 thatRad,B’ Aug,B’andSRH,Bhave the same functional form asNBwithrespect to Clearly,Rad,Bin eqn (5.20) can be written in the same approximate formasNBwith respect to AEp. However, it is not clear that the same is true forAug,BandSRRBin eqns(5.17) and (5.12) respectively. In order to seeAug,B’ SRH,B’andRo4,Bcan be rewritten as:) (Ti Ar’ \ r cu ( kTAug,B”‘BE’ ‘-fn) — Aug,BkVBE’e-- (5.27)SRILB( BE’ fn) sB( BE’)eRad,B0”BE’AE)— JRB(VBE,0) ekTa 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) forAug,B’ SRI-1B’andRad,BwithVBEfixed at 1 .OV. Fig. 5.4 shows that the error inusing 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 isjustified to state:J3(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 themodels in Section 2.2. The transport currentTthrough the device (which is equal to the collector current) is givenby eqn (2.7), with = Fj’from eqn (5.23), J=from eqn (5.26),4is given by eqn (5.28),42= 0 (i.e.,Y2= 1), and J2>> (J,J3).Using the above produces:= r-+--—i= [Ff(VBE)/y3J,i1”y3<<J,(5.29)[J2,1.i2,3JJc(VBE,1Effl=O)if1’r3>>’2,whereyis given in eqn (2.7) as— 3+4— Jc+JNB+JsRH,B+JAug,B+JRacB13jJ3 CzEf=Oand theVBEdependence has been omitted for clarity. Eqn (5.29) embodies thetwo 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 intoSCR,Bof eqn (5.22) andNB• 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 withoutthe need todetennine the inner driving potential of iSE. However, if a detailed understandingof eachcomponent of the base terminal current is desired, then iXE mustbe 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 formsgiven in eqn (5.27). The material parameters are given in Section 5.4, andVBEis fixed at 1.OVBefore leaving this section, it is important to verify that E, is indeed constantthroughoutthe EB SCR. If there were a AEp,, present, it would have to be included inthe emitter side holecurrentSCR,Ejust like has been included inSCR,B•Essentially, the same current balancingprocedure given by eqns (5.21)-(5.22) needs to be performed regarding the transportof 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 throughtheEB SCR is 3.9x1Oexp(qV/k7)Acm2,and the hole transport current through the neutralemitter assuming a 3000A emitter cap at a doping of 1020cm3is l.2x1W27exp(qV/kI)Acm2.Clearly, 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 canbe imagined where this is not true, leading to the requirement that hole transportbe self-consistentlysolved with electron transport. It is quite interesting to realise that the valence band discontinuityAEdoes not limit the back injection of holes as the literature has lead the device communitytobelieve. The back injection of holes is ostensibly eliminated by the reduced numberof 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:1x1019cm3;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,,: 7.99x1032cm6s1;5.75x1t131cm6s;1.93x1031 cm6s1; 1.12x1030cm6s1;B: 1.29x1010cm3B: 7.82x1O’1cm3s;D:30cm2s1,Wb:ioooA.Results for the SCR currents are shown in Fig. 5.5. The slopes of thecurves are not constant, owing to the voltage dependence ofWBE,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 dominatedbysRH,E)•Specifically,atVBE= 1.2 V,SCR,E= 1.90 and, adding all the base-side currents together,SCR,B= 1.19. Furthermore, = 1.14 atVBE= 1.2 V due to the effects of These values aresimilar tothose reported elsewhere [90], and deserve further comment becauseSCR,Bis so far removedfrom the “classical” value of n = 2.With reference to eqn (5.12) for the SRH current, because is so lowandNratis 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 VoltageVBE(V)Fig. 5.5. Bias dependence of the SCR current from the emitter side, and thethree components ofthe SCR current from the base side. Material parametersare taken from the start of Section 5.4.The width of the SCR on the base side of the heterojunction is muchless than that on theemitter side, and this fact alone, viaNratin the Z and Z0, terms in eqn (5.12), wouldmakeSCR,B <<SCR,E•However, the much larger n1 on the base side counterbalances this effectand allows the steeper-risingSCR,Bcurrent to exceedSCR,Ebeyond some forward bias. In the exampleshown in Fig. 5.5, this occurs aroundVBE= 1.45 V. This transfer from an n 2 slope to ann 1slope in the SCR current does not occur in a homojunctiondevice 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 inthe highly-doped base will be less than that in the emitter. Indeed, photoluminescencemeasurements on ma-dence ofSCR,Eis thus determined by the sinh qVI2kT term and n approaches 2. Contrarily, forSCR,B’bothZk,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-sidet, to 5Ops causesSCR,.Bto 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 thatSCR,Bis always less than the quasi-neutral base recombination currentNB•This indicates that, in practical devices, an observedchange in base-current ideality factor from n 2 to n 1, will likely be dueto a change fromJc,gd0m1nated current to aJNB-dominatedcurrent. Only in circumstances where it is correct toattribute a much lower minority carrier lifetime to the base-side depletion regiononly, perhapsdue to defects at the interface, can a situation be envisaged whereSCR,Bcould dominate overNB’and thus be responsible for the slope change to n 1, which is often seenexperimentally. Theabove point about the relative magnitudes ofSCR,BandNBis an important one as it puts intopractical perspective the theoretically-interesting fact thatSCR,Bhas a different voltage dependence to that ofSCR,E110-810-110.8 1.1 1.2 1.3 1.4 1.5Base-Emitter VoltageVBE(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. Materialparameters 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 includedin calculatingSCR,B[901,it is, perhaps, not evident how important it isto includeSCR,Bin the balancing equation (5.21) to compute iXEp. Fig. 5.7 provides an answerfor the material properties consideredhere. By not includingSCR,Bin eqn (5.21), yet using the subsequently-calculated tXEto eventually computeSCR,B’leads to a result which is indistinguishable from that ofthe “full model”,whereSCR,Bis included in the balancing equation. This isa consequence ofSCRBbeing muchless thanNBand J. However, also from Fig. 5.7, note that it is grosslyincorrect to not includeAEp in the calculation ofSCR,BBecause the electron quasi-Fermi energy splitting isso large foran abrupt junction [18], its omission leads to a large overestimation ofSCR,B’and, consequently,to a severe underestimation of the current gain. It is difficultto imagine a practical situation whereit might be necessary to includeSCR,Bin the actual calculation of AE1.A possible scenarioisone in which t, in the SCR is less than t, in the neutralbase, perhaps due to interface defects, andthat W is much larger than the usual i000A. The lattersituation would reduce J, and theformer would increaseSCR,Bwith respect toNB’thus makingSCRBbecome more prominent ineqn (5.21). The effect of these changes is shown inFig. 5.8. Even though the gain has been reduced to a very low value, it appears that thereis still no need to includeSCR,Bin the balancingequation.To summarise the results from the analysis of this section: itis necessary to include inthe computation ofSCR,B;butSCR,Bneed not be included in the balancingequation to estimateAEp; andSCR,Bis not very important for devices basedupon materials with the properties considered here, becauseSCR,Bis usually less than eitherNBor Of course, if parameters affecting Auger or radiative recombination in the SCR turnout to be greatly different than the valuesused here, thenSCR,.Bcould become importantOne instance whereSCR,Bwill definitely be larger than calculated here is inthe 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, andSCR,Bincreased correspondingly. However, because of the lower bandgapof the graded material in the emitter-sideofthe junction, is increased and, therefore,SCR,Ealso. Thus it is not obvious whetherSCR,Bisany more important in graded-junction HBTs than it is in abrupt-junctionHBTs. The results ofParilch and Lindholm [90] suggest thatSCR,Eremains the dominant current. One situation inJuly 12, 1995 1243o:xFull model2000 -SCR,Bnot in balancing eqnbut included1000SCR,Bnot in balancing eqnw.; 1.0 :i/ndotBase-Emitter VoltageVBE(V)Fig. 5.7. Bias dependence of the current gain13,showing the relative importance of includingSCR,Bin the calculation of AE. Also shown is the dramatic error resultingfrom not includingAE, in the calculation ofSC.R,B108JSCR,BflOtin balancing eqnbut AE included6-4.Full model2SCR,Bnot in balancmg eqn7and not included0.8 0.9 1.0 1.1 1.21.3Base-Emitter VoltageVBE(V)Fig. 5.8. Bias dependence of the current gain 13 for the case ofWflbincreased to 5000A and t, inthe SCR reduced to 5 ps. Even in this extreme case there is little error in not includingSCR,Binthe balancing equation.July 12, 1995 125whichSCR,Bcould be increased without an associated increase inSCR,Eis 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 ofSCR,Bwill 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- ......,..7io ... -7.....io10////w,=1oomn -jlOlWb=1OnmRec,BWab= lOOnm1081.0 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 VoltageVBE(V)Fig. 59. Effect of changing the neutralbase thicknessWflbwhen the CBS is responsible for current-limited-flow. LoweringW,thleaves andsRgEunchanged, but results in the reductiontothe base side recombination currentRec,B(=SC.R,B+ J). Under high bias, whereRec,Bdominates,13increases with reductions in W,,,. While under lowbias, whereSRflEdominates,13is unaltered by changes in W,,j,.Before leaving this section, it is interesting to see howcurrent-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 thatJc 1jCBSThus, if the neutral base transport cur-July 12, 1995126100io-31061 0-10-1510-180.8rent were increased by reducingWflb,Jcwould remain unchanged because the CBS already represent the bottleneck to charge transport through the device. However,the reduction toWflbdoeshave an effect on the device. Fig. 5.9 shows that the base-side componentsof the base terminalcurrent are decreased by a reduction toWflb.This decrease occurs due to a reduction of ‘y in eqn(5.29) because relatively speaking, a shorter neutralbase 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 coupledWflbto the base instead of the collector current.Finally, for the sake of completeness, Fig. 5.10 replots thecurrents displayed thus far usingthe linearisedW(x)from eqn (5.11) against the currents obtained withthe full potential from thedepletion approximation of eqn (5.8). As can be seen in Fig. 5.10, the erroris indeed slight, andwill be smaller than the uncertainty in the recombination parametersthemselves. The linearNJ(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 VoltageVBE(V)Fig. 5.10. Comparison of the recombination currents when ji is givenby 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 componentsin HBTdevice simulators, it would be convenient if a simple closed-form solution foriXEpexisted. Further, if the various current components could be expressed as diode-likeequations, 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) todiode-like expressions is to examine the relative importance of the Z-terms which appear in the expressionsfor theSRH and Auger recombination currents. Fig. 5.11 showsthe 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 representativeof the amount of recombination at x,,0, 0 and xi,, (see Fig. 5.1), respectively. For the conditionZk<<Z, to remain valid, the depletionregion on the base-side must not be vanishingly small. This canbe ensured by having the dopingdensity ratioNA/ND30. Contrarily, there is a lower limit to the allowable value ofNA/ND,belowwhich the recombination on the base-side of the depletion regionbecomes large and the inequality Z(), << 1 is violated. This limit isNA/ND3. Therefore, keeping within the range 3NA/ND30, and following the usual practice of expressingWBEand €) by their equilibrium forms, eqns(5.12) and (5.17) reduce toNDnPriXE — qNVji — zXEfJsRH,BCsexp[kTJexpLkTno,pni,ii1IfljqVSRH,ECs2t 2kT(5.31)2[qV—1XEfflAug,BCSnI,PAP,PNAexpLkTqVBE1Aug,Esn4n,nNDP[kTIwhereC =Writing the radiative recombination currents in eqn (5.20) insimilar form, givesJuly 12, 1995 128q CsVb2qVBE— IXEfflRad,BkTnipBp(l—Nrat)exp[kTqC5V12____Rad,EkT,vBnNratPLkTUsing these diode-like equations, along with the expressions forNBin eqn (5.25), in eqn(5.26) andThT= Ffin 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/WewhereqV8 qV qVI (TI ‘ — inSRH,BkT Aug,BkT.i. IRecom” “BE)— JS,SRBBe +.ISAU8Be -I-.IsRaBeRecom = “S,SRH,B+SAugB+SRa,B— qN1VjNDn.qCV.=C ‘‘ ekT+CsnPAPPNA+kT1flpBp(1Nrat)nO,p 1, nbWflb,e= LflbtanhI\. nbBO =The values for the saturation currents and idealityfactors in eqn (5.33) can be found eitherthrough a statistical fitting method, or from the analytic diode equationsin eqns (5.31)-(5.32).Note that the n factors appearing in eqn (5.33) are independentof 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 diodeforms 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.AsVBEapproachesVbj,the diminishing depletion-region thickness becomes a factor in that the depletion approximationno longer holds.Thus, for values ofVBEnearVbj,the voltage dependence ofWBEneeds to be included, and the assumptions regarding the relative magnitudes of the Z functions re-addressed.July 12, 1995 129106i04.Zni02.Nl0101-.c 10....- Zp106I1.0 1.2 1.4 1.6VblBase-Emitter VoltageVBE(V)Fig. 5.11. Z-functions as computed from eqn (5.13) when using the materialparameters fromSection 5.4.100 -ExactDiode102 - .. ,.. -SCR,E ..._/ -10 - .......-...101 ......- -Rad,BRad,B ......1081011W10 SRH,B&R,E -io-3Aug,B 1.41.5 1.61012• • • • • •0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Base-Emitter VoltageVBE(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 withSRH,BandAug,Bomitted for clarity.July 12, 1995 130If, as found to be the case for the material parameters used here, it is not necessary to includeSCR,Bin the balancing equation, then theRecomandS,Recomterms 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, thenT = Jmand eqns (5.29) and (5.33) can be further simplified to give(VbZ — VBE)kTqI1EfNratVbi+ (lNrat)VBE= DflnBoqkTWflbe’’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),liNratand 1/Nratrespectively (where the bias dependence of the tunneling factoryis 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 thanthe neutral-base currentand, therefore, need not be included in the current-balancing equationused to compute the quasi-Fermi energy splitting AEp, at the base-emitter junction;2. however, when subsequently computing the base-side SCR currents, AEmust 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 thenormally-used valueof 2;4. a simple, yet acceptably-accurate analytical expression for AEcan 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 6TheSiiGeHBTJuly 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 forthemovement 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 throughthe neutral regions of thedevice. Finally, Chapter 3 presented the models for the calculation of the base transit timebasedupon an optimisation of either the base doping, or the base bandgap, or both. In allof the workpresented thus far, no assumptions have been made that depended upona specific attribute of agiven material system. Thus, the models contained within this thesisare 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 thesisare indeed applicable to any materialsystem, whenever an analysis of a specific model was performed, the materialsystem ofAlGai..Aswas invariably chosen for the study. The reason for choosing theMGai..Asmaterialsystem is that current-day technologies for HBTs prefer this materialsystem. The dominance oftheMGai..Asmaterial system stems mainly from the fact that the lattice mismatchover the usable range of Al content (i.e., 0 x 0.45) is under0.07% [61]. This nearly ideal lattice match allows for an arbitrary film thickness because there will be virtually nostrain placed upon the latticeat the heterojunction. Coupled with the lattice-matched characteristic,theAlGai..Asmaterialsystem can also provide for large changes to the bandgap (AEg) [61]. However,compound semiconductors like GaAs and AlAs have numerous undesirable featureswhen it comes to manufacturing. TheA1Gai..Asmaterial system lacks a usable native oxide, is a poor thermalconductor,cannot be pulled into wide ingots which results in small wafer diameters,is brittle, suffers from ahigh defect density, cannot employ ion implantation for bipolardevices, exposed surface layershave high recombination velocities, cannot be used in low-powerapplications because of the largeVbjinherent with large bandgaps, does not etch easily and generally lacksan abrupt end-point detection for etching, and finally is expensive to manufacture. Given all ofthese manufacturing andelectrical drawbacks, however, the lattice-matched attribute isimportant enough to makeAlGai.Asthe preferred material system for the construction of HBTs.July 12, 1995 133Essentially all of the manufacturing issues with regard to theAlGai..Asmaterial systemare solved by using theSii.Gematerial system: save one issue. At issue with theSii..Gematerial system is its large lattice mismatch. The Ge lattice is 4.2% larger thanthe Si lattice [96]. Evenif the Ge content is constrained to be under 20% (i.e., 0 x 0.20) there would stillbe a 0.84%lattice mismatch between a Si08Ge02 film and a Si substrate. The issue witha lattice mismatch ofaround 1% is that to commensurately place an epitaxial film upon a given substrate wouldresultin 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 formalong the heterojunction interface which would greatly enhance recombination. Since theheterojunction will beformed in the middle of the EB SCR of the HBT, a plane of recombinationcentres at the heterojunction would result in an intolerably high base current; large enoughto reduce below 1.There is no physical way to alter the bulk lattice constant of a materialor 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 issaid 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 structureof the substrate. The keyto obtaining a pseudomorphic film is to ensure that the layer thickness isbelow the critical thicknessh[99]. However, in order to maintain a pseudomorphic film, and ensurethat it does not relax back to its bulk lattice constant, subsequent exposureof the layer to high temperatureenvironments must be severely limited. In thepast 5 years, great progress has been made at IBMin the quality ofSii.Gepseudomorphic films[31]. These developments have shown greatpotential regarding operating speeds [100-102],so much so that many other companies including theJapanese at NEC [1031 are developing SiGe IC processes. Through the recentsuccesses regardingthe high quality growth of pseudomorphicSiiGefilms, theSii..Gematerial system is fast becoming a practical alternative for the manufacture of HBT-basedICs. In fact, with the massive installed base of Si-based IC manufacturing, coupled withthe ability to integrateSii..Gefilms intothe process, it is expected thatSii..Gewill rapidly displaceMGai..Asas the preferred materialsystem for the manufacture of HBT-based ICs.This chapter will apply the general models obtained fromthe previous chapters to the studyof HBTs based within theSii..Gematerial system. Due to the complex nature ofSii..GeunderJuly 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 theSii..Geenergy 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 onSii..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 orGe crystals were used inthe formation of HBTs, there would be a considerable limitation imposed upon the abilityto engineer the bandgap within the HBT Instead, pseudomorphicSii.Gefilms, that are commensuratelystrained to become lattice matched to the substrate (which is pure Si in presentday devices), areused. These pseudomorphicSii..Gelayers will remain strained without relaxing as long as thelayer thickness remains below the critical thicknessh[97-99,105]. (ForSi070Ge030 grown on{lO0} Si substrates the critical layer thickness is 600A, while forSi045Ge055it is only bOA).Thus, unlikeMGai..As,which is essentially lattice matched toGaAs and thus has no criticallayer thickness, SiGe HBTs can have considerably less freedom in the choicesfor layer thicknesses.The key to manufacturing SiGe HBTs is the commensurate growth of strainedSii..Gelayers to the underlying substrate. However, the strain in the plane of growthresults in a distortion ofthe crystal structure that breaks the cubic symmetry and causes the crystal unit cell to becometetragonal. With the breaking of the cubic symmetry comes a changeto 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 andGeare shown in Fig. 6.1. Looking at the case of Si, there are six separatebut degenerate conductionband minima located along the (100) directions at the A point (which is 80% fromthe zone centreat F to the Brillouin zone edge at X). For alloys ofSii..Ge,these six minima are dependent bothon the alloy content x and on the state of strain. Take, for an example,Sii..Gegrown on a Si substrate with the direction of growth parallel to [001]. As x moves from0 to 1, theSii..Gelayermoves 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 alignedto the normal of the interface plane (i.e., parallel to the direction of growth) remain degenerateand are raised in energy,while the other four minima parallel to the interface plane also remain degeneratebut are loweredin energy. For the case ofSii.Gegrown on a Ge substrate with the direction of growth stillparallel to [001], the situation is reversed. As x moves from 1 to0, theSii..Gelayer moves from anunstrained cubic structure to an expanded, tensile-strained tetragonalstructure. For this case oftensile strain, as the strain increases from zero, thetwo minima normal to the interface plane arelowered in energy, while the other four minima parallelto the interface plane are raised in energy.Thus, we find that there are now two types of A conductionband minima in a strainedSii..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 ultimateconduction band.The valence band also suffers considerable change dueto the symmetry breaking caused bystrain. The valence band of pure, unstrained Siand Ge (or for that matter, all semiconductors), iscomposed of what should be three degenerate bands. These threebands are the light-hole (lh),heavy-hole (hh) and split-off bands (so). When the interactionof the electron’s internal angularmomentum (spin), is coupled with its orbitalangular momentum (termed spin-orbit coupling), thedegeneracy of the so band is lifted [109,110]. The resultant interactionleaves 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 andhh bands. As in the conductionband, the valence band maxima is dependent both on the alloy content x and onthe state of strain[105-108]. Returning to the case ofSii.Gegrown on a Si substrate, with the direction of growthparallel to [001], as x moves from 0 to 1 theSii..Gelayer experiences an increasing compressiveJuly 12, 1995 136I\1.f1<HL_ N//\‘\.r,..•‘ /•......\Si\‘ ‘----—____\ I\‘ Fl ‘I\\ \Ix\\/7c).1/‘\.F,.,,”kL//GeIIILFig. 6.1. First Brillouin zone showing (in k-space) the constant energy surfaces nearthe bottomof the conduction band for Si and Ge. Also shown are the designations for thesymmetry pointsand the degenerate bands E and E in strainedSii..Gewith 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 ofSii..Gegrown on a Ge substrate, with the direction of growth still parallel to[001], as x moves from 1 to 0 theSii.Gelayer 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 movedto 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 unstrainedSii.Ge.The light hole (lh) and heavy hole (hh) bands remain degenerate for all values of Ge alloy composition x (only inthe bulk state where there is nostrain present). However, the split off (so) band maxima changes in energywith alloy content,where A(x) 0.044 + 0.246xeV.The previous paragraphs have outlined that the energy of the conductionband minima andthe valence band maxima change under the effect of strain, while their position ink-space remainsunaltered. However, it is also important to ascertain the effect ofstrain on the shape of the band ink-space, as this will set the effective mass which determines the velocityof the carrier and itsprobability for tunneling. Considering the valence band first, the effective mass forthe 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.Gelayer 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 theSiiGelayer [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.llxwhere x is the Ge alloy content, and the masses are a fraction of the electron rest massme(the Siand Ge hole masses are based upon [96]). The lh and hh effective masses are maintained separatelyinstead 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 Edo 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 forthe electrons because thereciprocal lattice vector is being changed. However, this change willbe relatively small as themaximum change to the reciprocal lattice vector is 4.2% overthe entire range of Ge alloy content.As for the effect of the Ge alloy content, it is important to realise that Siand Ge (and thereforeSii..Ge)have conduction band minima at A and at L. The differencebetween Si and Ge is thatthe A minima form the ultimate conduction band in Si while the L minimaform the ultimate conduction band in Ge. ForSii.Gethe A minima typically form theultimate conduction band.However, if the Ge alloy content is high enough, then the ever-presentL minima within theSii.Gealloy 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 sameas that for the A minima inSi, while the effective masses for the L minima are the sameas 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, fora 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 theSii..Gelayer 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 theSii.Gebulk 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 theGe alloy effects. Then, empirical deformation-potential theory is used to determine the amountof degeneracy splitting that occurs within the sub-bands of the conduction and valence bandsdue to theaddition of strain. Adding together Eg(x) with the results from deformation-potentialtheory produces the total change to the various bands within theSii..Gelayer.Beginning with the calculation of Eg(x), in the seminal works of [113,117]the necessary experimental measurements on the bandgap of bulk and strainedSii.,Gehave been performed. It has beenfound that for x < 0.85, the A minima form the ultimate conductionband minima inSii..Ge.However in the range 0.85 x 1, the L minima form the ultimateconduction band minima in bulkSii..Ge.Concentrating on the A minima alone, then usinga quadratic fit to the data in [113] produces:(E 0.51446x+0.3h164x2x0.732Eg(X)=g, 1(6.3)tEgsiO.l5Ol0.0813x x>.0.732where xis the Ge alloy content,ES1is the bulk Si bandgap, and all values are in eV.Eqn (6.3) gives theSii..Gebulk bandgap from the top of the valenceband to the bottom ofthe A minima in the conduction band. Caution must be exercised when usingeqn (6.3) for x > 0.85as the L minima will form the ultimate conduction band in bulkSii..Gematerial. However, thestrain imparted to theSii..Gelayers used in HBTs is generally sufficient to reducesome 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 Lminima 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 ofSii.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 constantasis larger than the alloy lattice constantaa.Thecommensurate growth of the alloy layer to the substrate forces the in-plane alloylattice 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 latticeconstant compresses belowaa.The pseudomorphic alloy layer will then have a larger in-plane lattice constantwhen compared to the out-of-plane alloy lattice constant, leadingto a tetragonal crystal instead ofa cubic one. Contrarily, Fig. 6.3(b) shows the case where the substrate lattice constantais smaller than the alloy lattice constantaa.The commensurate growth of the alloy layer to the substrateforces the in-plane alloy lattice constantto matchas.In so doing, a biaxial in-plane compressionresults in the pseudomorphic alloy film. In an attemptto lower the energy contained within thefilm, the out-of-plane alloy lattice constant expandspastaa.The pseudomorphic alloy layer willnow have a smaller in-plane lattice constant when comparedto the out-of-plane alloy lattice constant, which again leads to a tetragonal crystal instead ofa cubic one. It is the fact that the pseudomorphic alloy layer has broken the cubic symmetry of the original latticethat leads to the changesin the conduction and the valence bands.The initial applied stress tensor to the alloy layer can be viewedas a uniaxial stress accompaniedby a uniform hydrostatic pressure applied over the entire cell. If the in-plane interfaceis parallel to thex-y plane, with the direction of growth parallel to thez-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, whilethe secondterm is the uniaxial stress, of opposite direction to the biaxial stress, appliedto the out-of-planelattice constant. Therefore, the symmetry breaking of the alloy’s unit cell occurs alongthe direction of growth (i.e., the z-direction). Thus, any changesto 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 theSiiGe alloy layer to theSii.Gesubstrate, leadingto a pseudomorphic alloy film. (a) the substrate lattice constanta5 is larger than the alloy latticeconstantaa.The resultant biaxial tension, which results fromaaexpanding to fit a5, distorts theout of plane alloy lattice constant by compressing it. (b) the substratelattice constantasis smallerthan the alloy lattice constantaa.The resultant biaxial compression, whichresults fromaacompressing to fitas,distorts the out of plane alloy lattice constant by expandingit.HHaa(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 systemsthat arelattice matched to{100}substrates, and that the direction of growth is [001], then the strain tensor is:a3— aa= [exx]= aaasaa(6.5)ezz1+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 forSii..Ge).The lattice constantsaaandasare 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)wherexais the Ge content in the alloy layer, and x is the Ge contentin the substrate layer.In order to determine how E and E respondto strain it is instructive to define an averageconduction band energy.The reason for defining is that depending onthe direction of strain,either E or E will form the ultimate conduction bandE;so usingEas the reference would become mathematically cumbersome. is given by the weightedaverage of E and E; i.e.,— 4E4+2E2 2E4+E2= C6C = 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)wheredand are the dilation and uniaxial deformationpotentials respectively. Further thechange in energy for a specific A conduction band minima is givenby:= [d1+EUtàI}]:ë (6.9)where à is the unit vector parallel to the i ‘th A conductionband minima, and{ }denotes dyadicproduct. For example, the change in the energy of the Aconduction 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, thenthe energy difference between Eand as well as E and are given by:= E’°°1 = E°’°1 = AE’°°1—AE= ({aa}—1 1/32 1=0 — 1/3(6.10)0 1/3)1,_= —e±andE = E°°1 =— AE =({aà1} — 1) :ë0 1/32=0 — 1/3: ë=(—(e + e)+(6.11)1 1/3)2_=wheree±— e, andu(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 Erespectively, 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 thegeneral statements given earlier in thesection regarding the changes to the conductionband due to strain. For compressive strain inthealloy layer;xa> x5 so thataa> a and eqn (6.5) has it thate±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 tensilestrain in the alloy layer,Xa<x so thataa < asand eqn (6.5) has it thate±< 0. Then eqns (6.10) and (6.11) have it that E > 0and E <0, which confirms that under tension E forms the ultimateconduction 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 valencebands are based upon the valence bandJuly 12, 1995 144strain Hamiltonian [118]. To this end, it has been determinedthat the quantum numbers for totalangular momentum J as well as magnetic moment (spin) m remainunchanged with the application of strain. This leads to the hh band designation ofIJ= ;m1= orI; ±)for short; the lhband designation ofI ;±); and the so band designation ofI ; ±).The solution of the valenceband strain Hamiltonian [118,107] produces:I;±)hh_22— 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)whereDu(Xa)is the valence band deformation potential equalto 3.15+ l.l4xaeV[106], andA(Xa)is the split off energy, defined in Fig. 6.2, which isequal to 0.044+ 0.246xaeV[107].Using a similar technique to the one used forthe solution of the conduction band, an average valence band energy is defined and subsequently used asthe reference point for all valenceband energies; i.e.,— Elth+ETh+E80i= V V V= A(Xa).(6.13)It is interesting to note that,defined in eqn (6.13) is independent ofthe applied strain. Also, because is not zero, the valence band energies in eqn (6.12) are notusing,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 conductionband). Observation of eqn (6.12) shows that underthe condition of zero strain (i.e.,e±0), thenE1= = 0, and E°= A(xa).Therefore, the energy reference for eqn (6.12) is not but the valence bandedge of bulkSii..Ge.The reason for using,will become obvious when the band offsets ata heterojunction are determined in Section6.2.Eqns (6. 10)-(6. 12) determine the effect ofuniaxial strain, due to the second termon theright-hand-side of eqn (6.4), on the conductionand valence bands ofSii.Ge.Eqn (6.3) determines the effect of the Ge alloy content. Finally, the effectof the hydrostatic force, due to the firstterm on the right-hand-side of eqn (6.4), is determined.The hydrostatic force results in eitheranet decrease or increase in the total volume of the crystal’s unitcell. A volume change in the unitcell will be accompanied by a change in the absoluteenergy of the conduction and valencebands.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 bulkSii..Gebandgap 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 aSii.Ge alloy layercommensurately strained to aSii.Gesubstrate is:Eg(Xa, x) = Eg(xa)+ L\Eg + min(E, E) — max(E, E). (6.15)It must be remembered that forxa> 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 forxa> 0.85 only ifthere is sufficient strain to ensure that the ii minima, and not the L minima, still form theultimateconduction band.Fig. 6.4 plots theSii.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 materialbandgap to anyof the other strained cases shows that the Ge alloy content of the Siixpexalayer plays a far smaller role than strain does in determining the bandgap. In fact, observation of the line fora pure Sisubstrate shows thatSi045Ge055 lattice matchedto{100} Si has a bandgap of 0.66eV, which isthat of bulk Ge. The strange shape concerning the lines for material strainedto substrates ofSi075Ge025,i050Ge050,and Si-j25Ge075 is due to the fact the material is shifting froma caseof in-plane tension to compression. Take the example of aSi05OGeOOsubstrate. When thepseudomorphic layer has a Ge mole fraction in the range of 0Xa0.50, the layer is under in-plane tension as the substrate has a larger lattice constant. Thus,asxaincreases towards 0.50 thetension is decreasing and the bandgap will increase, withE forming the ultimate conductionband. WhenXa= 0.50 there is no strain and the bandgap will be given by the bulk value. Finally,asxaincreases past 0.50, the strain switches from in-plane tensionto compression. When thischange in the direction of the strain occurs, E formsthe ultimate conduction band (this is whythere is a corner in the plot, however, the E and E bandgaps continue on ina smooth fashion butdo not form the ultimate bandgap). Asxaincreases 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 0Bulk Material -0.9on 25% Gb.., .0.8\•.. -on 50% Ge .0.7 ,.....,....•,,,,,,06on75%GeonGeX/20.5on0%Ge0.4• • I • I0.0 0.2 0.4 0.60.8 1.0Germanium Mole FractionXaFig. 6.4.Sii.Ge bandgap when grown commensurately to a variety of substrates orientedalong (100). All values reflect the energy from the top ofthe valence band to the lowest A minimain the conduction band. The bulk material bandgap is for reference and isvalid onlyforxa< 0.85;forXa> 0.85 the bulk material line is not the ultimate bandgap but the bandgapto the A minima.Eqns (6.10)-(6.1l) give the conduction band energiesof 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. Ifthe position of the unstrainedconduction band is known, eqns (6.lO)-(6.11) will yield the offsetto the conduction band due toany strain in the layer. Fig. 6.5 plots E and E relativeto using similar substrates as found inFig. 6.4. Observation of Fig. 6.5 shows the changes inE and E to be quite linear in terms ofstrain. Furthermore, whenever the pseudomorphic layeris under compression then E forms theconduction band, but when the layer is in tension then E forms theconduction band. Finally, Echanges more rapidly than does E for a given increase in the amountof strain.July 12, 1995 1470.50 • • •on0%Ge0.40 EE2on 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 FractionxaFig. 6.5. E and E conduction band energies relative to the unstrained conduction bandedgefor 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, andso bands relativeto the unstrained valence band edge. Fig. 6.6 plots the hh and lh bands relative tothe unstrainedvalence band using similar substrates as found in Figs. 6.4 and 6.5. Theso band is not plotted because strain simply continues to lower the band peak even further, meaningthat the so band willnot be of any consequence regarding the transport of holes. Comparisonof Fig. 6.6 with Fig. 6.5shows that unlike the conduction bands, the valence bands respond ina non-linear fashion with respect to an applied strain. Furthermore, there is not as large a change inthe energy of the valencebands due to strain as there is in the conduction bands. Finally, whenevertheSii..Ge layer isunder compression, then the hh band will form the ultimate conductionband; while under tension,the th band will form the ultimate conduction band.Finally, it is instructive to present a surface plot ofconstant energy in k-space, depicting theconduction bands inSii..Geunder the influence of strain. Fig. 6.7 plots the surfaceof constantJuly 12, 1995 148energy that envelopes the six A minima in Si083Ge017 commensurately strained to (001) Si.Since the pseudomorphicSi083Ge017 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, thentheE band would not be seen at all. This demonstrates the profound effect that strain impartsto theSiiGelayer.0.050.00-0.05-0.100.0 0.2 0.4 0.60.8 1.0Germanium Mole FractionXaFig. 6.6. E and E’ valence band energies relativeto 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 withthe highest energy. 12, 1995 1491.Fig. 6.7. Constant energy surface plot depicting the E and E bands in Si083Ge017commensurately 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 theE4band. 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 inSii..Geunder 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..Geat this time. AsSii..Gebecomes 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 that1.001.00kJuly 12, 1995 150MGaiAstook, 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 onSii..Ge;so much so that strain produces more of an effect on the bandgap than doestheGe alloy content.6.2 Band Offsets inSii..GeSection 6.1 presented all of the relevant material parameters to describe the conductionandvalence bands of aSii..Ge alloy layer commensurately strained on top of a{100)SiiGesubstrate. This section will present the band offset models that predict the valenceband and conduction band discontinuity at an abrupt heterojunction. Therefore, when the resultsof this sectionare combined with the results of Section 6.1, all of the relevant models forSii.Geregarding theposition of the conduction and valence bands within a device canbe determined.The seminal theoretical work on the band alignments betweenSii..xex1and SiixGex(where the 1,r subscripts refer to the left and right films respectively), whencommensurately 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 periodicityrequired to establish Blochfunctions, they developed a supercell structure. The supercell structure hada unit lattice cell that wasconstructed of n Si atoms followed by n Ge atoms. By extendingthis unit supercell to infinity, thoughthe Born-von Karman boundary conditions, Van de Walleand Martin were able to obtain the bandoffsets. In order to establish that the size of the supercellwas large enough to ensure bulk materialproperties away from the heterojunction, the band offsetswere determined for a variety of n. Van deWaRe and Martin established that for n >5 the material wasbulk-like away from the heterojunction.In fact, the shape of the Bloch electron’s wave function becamebulk-like after moving only one lattice constant away from the heterojunction. Therefore, Vande Walle and Martin concluded that theperturbing effect of the abrupt heterojunction was indeedlocalised to the space immediately surrounding the interface.The main conclusion from the work of Van de Walleand Martin is that the average valenceband offset A, between a pseudomorphic Sito Ge heterojunction, whether commensuratelystrained to either a{100) Si or Ge substrate, is a constant of0.54±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 lowerthan 0.54eV, and which areas low as 0.2eV. Recently, experimental measurements by Yu [125] have givenE, =0.49±0.13 eV. However, after performing an array of measurementson a variety of substrates(thereby changing the strain), it was found that A varied slightly with strain. The finalresultsfrom Yu [125] were:AEv(Xai Xar)= (0.55 —0.12x) (Xai Xar),(6.16)whereXa(, Xarand x refer to the Ge mole fraction inthe 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 movementsdue to strain have considerable variability depending on which experimental method is usedto obtain the results. At the moment there isno clear set of material parameters to use in order to determine theband 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 materialparameters and models so that SiGe HBTsmay be accurately simulated.Use of eqn (6.16) produces conduction band offsetsAEthat are far too large. Experimentalmeasurements ofAE[105,111,126,127] show that there should be no morethan ±30meV ofoffset between Siixapexa,andSiiGe grown on a pure Si{100) substrate, whereXalandXarcan take on any value in the range of 0 to 1. Furthermore, recentmeasurements by Gan et. al.[128] have shown thatAEshould equalO.64xaieVwhenxar =x =0. Use of eqn (6.16) produces= 0.8OxaieV.By reducingLcPfrom 0.55eV back down to 0.49eV ineqn (6.16) produces:v(Xaiar) = (0.49—0.12x) (XaiXar). (6.17)Use of eqn (6.17) instead of eqn (6.16) reducesEiEto be no more than +48meV and -42meV (ascompared to +30meV and -100meV), whilealso givingAE = O.74xaieV.Finally, ifDu(xa)in eqn(6.12) is changed to 2.04+ 1.77xaeV[107] thenIXEremains unchanged andAE = 0.68xaieVThe use of eqn (6.17) instead of eqn (6.16) is within theexperimental error of the measurementsin [125]. Further, eqn (6.17) when combined withDu(xa)= 2.04+ 1.77xaeVproduces conductionand valence band offsets that match experimental observations closerthan when the material values proposed within [125] are employed. Thus, there is no clear set ofparameters as of yet for theJuly 12, 1995 152modelling of the SiGe material system. However, the differences between the variousmodels presented here is within 50meV. Therefore, in terms of the studiesto be presented later on in thischapter, a small discrepancy of 50meV will simply cause a slight variation in theGe alloy contentof the various layers, but will not effect the ultimate function ofthe HBT.Fig. 6.8. Conduction and valence band energiesincluding all of the band offsets for aSii3e1to aSiiGeheterojunction commensurately strained toa{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 relatesthe band energies of two different SiGecrystals across an abrupt heterojunction. OncezVL, is known, then by using Fig. 6.8, all of the other relevant offsets can be determined by appealingto the models of Section 6.1. Using eqns(6.3),(6.1O)-(6.15), and (6.17) along with the aid of Fig. 6.8yields:EElhV,hhi—VEJuly 12, 1995 153= +(A1— A)+ [max(E’,E’1)— max(E’rE’r)]EE tAEv+(Al_A7) +AE1 .•A’)+ (El_E.r)(6.18)AEc=AEv+(Al_A1)+ (E—E)= AE+ [min(E’,E”) — min(E’rE4’r)]= 1E+(El— E’)AE=LE+(E4lEr)where the average bandgap Eg is equal to the bulk alloy bandgap given ineqn (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 SiGematerial system.Although no one equation that forms the model of the SiGe material systemis of a complexnature, the cumulative effect of each sub-model leadsto a complex system as is evident from eqn(6.18). However, it is possible to arrive at a set of Taylor expansions for themodels that govern theband movements within the conduction band. Unfortunately, the valence bandmodels (i.e., eqn(6.12)) contain a square root dependence that proves impossibleto approximate. Given the nonlinear nature of the strain tensor, is it is not possibleto achieve a simple linear approximation forthe conduction bands. By performing a multivariate Taylorexpansion of the conduction bandmodels in eqn (6.18), up to and including second order terms, yields:AE0.1429(Xar — Xai) x — 032789Xr + 0.O2l55Xar+ O.32985x1— O•02252Xai0.1751(Xar — Xai)x — 0.34084x,.—0•4381Xar+0.34281x1+O•43384Xai(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 towithin 1% of the fullmodel given in eqn (6.18) over the entire allowed range forXal, Xa,,and x. The multivariate Taylor expansions were centered aroundXal= 0,Xar= 0.5, and x = 0.5. Thus, eqn (6.19) should strictly be used withXal <Xarhowever, if this is not true, then simply interchangeXalandXa,.andJuly 12, 1995 154multiply the result by -1. If the interchange of variables is not performed forXal>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.Considering2first, the two last linear terms inxaand x are the dominant terms. Therefore, to a crudeapproximation, E’2O.684(x— xa);which corrects the proposal of E’2O.6Xaby People [1051and Pejcinovic [28] who considers only a Si substrate. Examination of theother models in eqn(6.19) shows a linear dependence upon the substrate Ge alloy contentx. It is by no coincidencethat the coefficient that governs the x dependence in AE is 0.1268, as comparedto 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 determinefor certain the effects of substrate strain, in order than SiGe devices canbe developed where substrate strain is utilised. Finally, the non-linear terms in eqn (6.19) stem mainly fromthe non-linear dependence thatthe bulk bandgap has on the Ge alloy content.In terms of the conduction band, Fig. 6.9 plots E and Eto the left and right of a heterojunction under the proviso that the entire system is commensuratelystrained to a{100)Sii..Gesubstrate. The first thing to note is that EE is generallysmaller than z\E, and is of such a naturethat in going from the left to the right there isa downwards step. The reason for not classifyingthis as either a type I or II heterojunction is that the bandgap is nota monotonic function of strain,as is evidenced in Fig. 6.4. Thus, classification in terms oftype I or II would require detailedknowledge of the strain state, which would destroythe simplicity of the type I or II designation.However, when going from a pure Si crystal to aSii..Gecrystal there is always a small downwards AE. Contrarily, AE is in general quite large, muchlarger than and is of an upwardsnature in going from a pure Si crystal to aSii.Gecrystal. Most importantly, Fig. 6.9 clearlydemonstrates that the character of the conduction bandcan change between E and E whencrossing a heterojunction. Fig. 6.10 goes onto show thatAEindeed has a complex nature whenstrain is brought into the picture. There are three distinctregions in Fig. 6.10: 1) when x <(xai,xar)thenAEis governed by E1 to E’r2) whenXal<Xy<XarthenAEis governed by E’1 to4. r3) when x>(xai, xar)thenAEis governed by E’1 to E’rTo conclude this sectionAEis plotted in Fig. 6.11. The various parameters are identicaltothe ones in Fig. 6.10. As withAEalso displays the same type of complex featureswhich areJuly 12, 1995 1550.6 • • •0.54’AE0.4• (a)CSubstrate Germanium Mole Fractionx0.5• • •0.30.2E4,rb0.102 04\::T/E10Substrate Germanium Mole Fraction xFig. 6.9. E and E conduction band minimato 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)Xal0,Xar= 0.15; (b)Xal0,Xar= 0.30.July 12, 1995 156Substrate Germanium Mole FractionxFig. 6.9. Continuation of Fig. 6.9 from theprevious page. (c)Xal0,Xar= 0.45; (d)Xal=0,Xar= 0.60.I • I I • IbEE4,r7•••••••••••\EZr•••........i• I • ISubstrate Germanium Mole Fraction xII0.2 0.8 1.0(c)(d)E’1E4,r./.l0.2 0.4 0.6 0.81.0July 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)Xal0,Xar= 0.75.-—toE= Xal+ 0.2E2,1tOE2.r0.750.000.400.601.000.80Fig. 6.10.tEwhenxa,. = Xal+ 0.20, afldXaland 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)thenAEis governed by E’1 to E’r;2) whenXal<Xç<XarthenAEis governed by E’1 to E1”r;3) when x >(xai, xar)thenAEis governedlhl IhrbyEtoE6.3 Electron Transport in StrainedSii..GeSections 6.1 and 6.2 present the necessarySii..Gematerial 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 theSii..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“—;:-0.10-0.150.00l.oo0.80Fig. 6.11. zXE.1,whenXa,. = Xal+ 0.20, andXaland x are varied. The right side is the referenceJuly 12, 1995 159which included the effects of tunneling and the mass barrier. Therefore, Chapter 4can 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 theSii..Gematerial system. This section will discuss and present thetransport models for the multibandSii.Gematerial system.From the work in the previous two sections, it is clear that the conduction and valencebandsare both broken down into two distinct sub-bands (theso valence band is ignored as it is alwayslower in energy than the lh and hh bands, especially under strain, and is of such alow carrier mass[96] that hole transport can be ignored). Unlike theAlGai..Asmaterial system, where the higherenergy satellite band never forms the ultimate conductionband, E and E in theSii.Gematerial system can both form the ultimate conduction band. Thus, it is possibleto have near equilibrium transport occur within both E and Eat spatially separate points with the device; this is incontrast to theAlGai..Asmaterial system where transport in the satellite band needonly be considered under extreme non-equilibrium injection conditions. Further,this multiband transport canalso occur in the valence band of theSii..Gematerial system. Given the strange band offsetsdepicted in Figs. 6.9 to 6.11, it will be shown that transport withintheSii.Gematerial system canoffer a rich set of possibilities, both in terms of commercialHBT optimisation, and as a tool forresearch into the mechanics of transport within solids.Considering the valence band first of all, Fig. 6.2 showsthatELand E are degenerate under the condition of no strain. More importantly, themaxima in bothE! and E occur at thesame point in k-space. Even under strain, the maximain E and E’ remain coincident in theirkspace location. Therefore, there is very little issue regardingthe conservation of crystal momentum in moving between the lh and hh bands, if themass barrier that would occurat the heterojunction for holes is neglected, then to a good approximationone need only consider the ultimatevalence band in terms of hole transport.However, if the strain is small, so that the energyseparation between E and E is less than —2k1 thentransport within both bands needs tobe considered. As is attested by eqn (6.1), the mass barrierfor holes cannot be neglectedas ‘y from eqn(4.80) is typically -2 but can be as small as -10. Withy= -2, fully two-thirds of the currentcrossing the mass barrier could be reflected, leading to a 3-foldreduction in the transport current. A3-fold reduction in the transport current would be equivalentto having an upwards step in energy ofJuly 12, 1995 16028.5meV at room temperature. Therefore, when11Evis less than —2kT one must consider paralleltransport within E and E!j. But, no matter how large or smallAEis, the calculation of the valence band effective density of statesNmust 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 locationof the bandmaxima for E and E. Examination of Fig.6.7 shows that theSii..Geconduction band minima are not coincident in k-space. Thus, in order to move betweenany of the six A minima inSii.Ge,crystal momentum must be conserved. There are twoscattering processes that are responsible for intervalley scattering between the six conductionband A minima inSii.Ge[129](see Fig. 6.12); g scattering moves electrons betweentwo bands that are along a common major kaxis, such as the [001] and [OOT] bands that form E; whilef scattering moves electrons betweentwo bands that are not along a common major k-axis, suchas the [100] and [010] bands withinE. Given the proximity of the A minima to the Brillouin zoneedge, an Umklapp process can easily take place, leading tog scattering, because of the relatively small k-space separation that mustbe conserved. On the other hand, f scattering involvesa k-space conservation that is over one-halfof the reciprocal lattice length. Therefore, it is foundthat f scattering rates are almost 10-fold lower thang scattering rates [129]. Tn terms of the E and E band groupings,g scattering will not result in movement between the E and Ebands. Finally, for small distances, suchas those that aretypical of the EB SCR and neutral base width, fscattering is small enough tobe ignored[108,130]. These two results regarding intervalley scatteringallow the E and E bandsto betreated independently, allowing for a large simplificationas compared to the valence sub-bands.The arguments of the previous paragraph, justifyingthe independence of the E and Ebands, must be considered in the light of an abruptheterojunction. At an abrupt heterojunction,one would expect that a powerful Bragg planecould exist that would be perpendicularto the direction of charge transport across the heterojunction.Such a powerful Bragg plane couldenhancef scattering, leading to a coupling between the E andE bands. Consideration of the k-vectorinvolved in f scattering relative to the Braggplane, shows the two are separatedby 45°. With a 45°degree separation, it would not be expected that Braggplane scattering at an abrupt heterojunctionwould lead to a significant increasein the f scattering rate[1081. Therefore, the independence ofthe E and E bands should be maintainedeven at an abrupt heterojunction. This leadsto the for-July 12, 1995 161mation of a selection rule regarding transport inSii..Gethat 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 andg intervalley scattering.For clarity, only 1 of the 3 g scattering processes, and 1 of the 12 f scattering processesis shown.With the E and E bands treated independently of each other,the task of modelling electron transport within theSii..Gematerial system begins with calculation of the electron effectivedensities of states, N and N respectively. The density of states for band n is givenby [131]:g(E) = nf— E(k)) (6.20)B.Z.where n = 2 or 4 in the case ofSii.Gestrained to a{100) substrate, and B.Z. means Brillouinzone. The pre-multiplying factor of n in eqn (6.20) results fromthe degeneracy of the E and Ebands and the fortuitous designation where n is equalto 2 or 4. Then the effective density ofstates, assuming that the band-width isEband that Boltzmann statistics can be used, is equalto:EbEE,,EE(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 1i2k3/2N= _!_e fdkxe2mu1cTfdkye2mh1cTfdkze2mtkT= 2ne (6.22)where*1/3m= (m1mm)The appearance of the term exp(-EIk7) in eqn (6.22) isdue 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 energyreferences 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 reflectNfrom the energy reference of backto the ultimate conductionband minima by multiplying eqn (6.23) with exp(min(E, E)Ik7).The exact same methods used to determine P4 and Ncan be applied to the calculation ofthe hole effective density of states within the valencesub-bands, leading to:3/2E3/2N = 2e(mhhkT\1and N = 2e’(mlhkTN(6.24)L21th)L2ith)Then, owing to the different effective masses for the lhand hh, the total valence band effectivedensity of states is given by:3/2N=N’ +N= 21kT(mhh)3”2e’+(mlh)3”2e’. (6.25)2ithj)In a similar fashion to the conduction band, the referenceenergy for the valence band is notlocated at the ultimate valence band maxima, but at the locationof the valence band maxima under thecondition of no strain. To reflectNback to the energy of the ultimate valenceband 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 fortheSii.Gematerial system. These equations represent an extension to the traditional definitionsfor effective density of states, necessitated by the complex band structure ofSii..Geunder 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 byg’as is shown in Fig. 6.8, then:E8n=pn=NNekT3(E E(E,t’g(6.27)= (kT‘ (m*3/2eICT+ 2e’(mhh)3/2ei;+(mlh)3/2eeu’,\1tli)where the average bandgapgis 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 standarddefinitionfor 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 potentialVb1of 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 effectof 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 extrapolatedback to equilibrium conditions. Thus, the electron affinitiesXpandxnon the p- and n-sides of the junction are evaluatedatx = and x = -x, respectively. If and are spatially varying, thenas a changing applied biasmoves and x,,Vb1will also vary with applied bias. It is well known that Anderson’selectron affinity rule for the calculation ofAEis not correct. However, at some distance far fromthe heteroJulyl2,1995 164junction, and must become bulk-like. The question becomes how rapidlydoXpifldXnreturn to their bulk values? The deviation ofAEfrom—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 drivingforce forthe transport of charge than is not taken into accountby any known theories. If this rapid variationin andxturns out to be true, thenVbjwill not be a constant as is given in eqn (4.69), butis instead given by eqn (6.28) withXpand being a function of position. Finally,Vbjcontains all ofthe desired information regarding,Xn’and thuszSE.Therefore, if the pn-junction could bedriven up to and pastVbj,without resistive effects dominating the transport current, then information regarding,Xand thusAEcould be extracted. This possibility of operation near andpastVbjwill be considered in Section 6.4.E (eV)IFig. 6.13. Equilibrium band diagram ofa np-junction, showing the relevant energies andpotentials. Ef is referenced toEwhile is referenced toE.Note that the Vacuum potential is continuous whileEandEare not.UnstrainedEUnstrainedEJuly 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 theSii..Gematerial system, that eqn (6.2) is correct,which precludes the formation of a mass barrier. Therefore, transport throughthe EB SCR wouldbe given by the sum of E and E conduction solved by the standard transport model givenbyeqns (4.78)-(4.79) and (4.92).To this end, the correct parameters to use in the standard EB SCR transport modelregardingE conduction are:m1=mQX) my,1m1(A)m1= m(Z) .L1 = relative to= 4q2JmimikT= 4tq2Jm1mkT= q2h3 h3J2tm(A)kTN(6.29)While the correct parameters to use in the standard EBSCR transport model regarding E conduction are:m,= m1(A) m,1= m(z) m = m(A) = relativetoF2— 4irq2JmlmzlkT — 4itq2Jm1mkTI2ND‘1— q2‘NNfs0— h3e— h3N— J2itm1(A)kT\(6.30)q2N ekband degeneracy— J2tm1(I)kTEekT+2kTExamination ofFjoin eqns (6.29) and (6.30) reveals the exact contextof parallel transport withinE and E. There areNDmajority electrons that are distributed betweenthe 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 preferentialtransport within E, all other things being equal.July 12, 1995 166Finally, the electrons within E and E move with a velocity that is proportionalto the square-root of l/mj and 1/rn1 respectively. Therefore, neglecting the energyseparation between E andE, the E band will carry2Jmi/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 comparedto E.Regarding transport within the neutral base, the independence between Eand E can onlybe maintained if the neutral base width is small enough to preclude coupling viathe f scatteringprocess. For current-day SiGe HBTs the neutral base widthWfl, is under ioooA and is rapidly approaching 300A [10,26,28,100-103,133]. With sucha small neutral basewidth it is reasonable tomaintain the separation between E and E used forthe modelling of EB SCR transport. With Eand E treated independently, the neutral base transport currentwithin either one of the E sub-bands is given by Kroemer [38] as:——. qBE fnNB=N(x)dx[ekT-(6.31)Wnbn(x)wherej = 2 or 4. It should be noted that eqn (6.31) is anextension of Kroemer’s work which wasbased upon Shockley boundary conditions. The reason for generalisingthe 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 anisotropicnature [132]. This leads to theconclusion that D > D becausern< rn1.Further, each sub-band will have its own intrinsic carrier concentration nb., which is determined in the sameway as 1V, N and the total n to yield:fljNNekT= =‘ n = n4 +(6.32)Finally, due to the independence of the E and Ebands, a separate quasi-fermi energy mustbepresent 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 characterisetransport within theE band.The final model for electron transport within the SiGe HBT isachieved 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 recombinationeffects specifically in the calculation of the total transport current4will produce an error that is of order1/13.To reiterate,4is the EB SCR transport current solved by the standard transport modelgiven byeqns (4.78)-(4.79) and (4.92) with the pertinent parameters obtained from eqn (6.29). Ina similarfashion, transport within the E band is:4=+-1,(6.34)fs 1NBwhere F8 is once again the EB SCR transport current solvedby the standard transport model given by eqns (4.78)-(4.79) and (4.92), but with the pertinent parameters obtainedfrom eqn (6.30).Then, the total electron transportTthrough the HBT is given by the sum of4and4.To conclude this section, transport within the valence sub-bandsis addressed. As was discussed earlier in this section, the coincident nature of E’ and E in termsof k-space locationprohibits an independent treatment, such as was done forthe conduction sub-bands, of the two valence sub-bands. Fig. 6.2 clearly shows that the valence band ofunstrained 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 effectivemass that correctly produces the totalvalence band effective density of states. Then, by way ofexperimental measurement, a single mobility is extracted to characterise the valenceband as a whole. This method breaks down for thecase of strained SiGe, as the degeneracy of Eand E is lifted and the energy separation is dependent upon the amount of strain present. Thisprohibits 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 determiningcharge transport, as was done for theconduction sub-bands.Essentially, the only way to solve transport withinthe strained SiGe valence band is to resort to Monte Carlo simulation. However, as was pointedout 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 ballisticallydue 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 neutralregions of theHBT is treated using a single equivalent valence band.The implication of the second assumption is straightforward; transport is treatedin the standard single equivalent valence band approach. The only consideration thatmust 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, whichwill move andE either closer or further apart in terms of energy. Since the lhmass is much smaller than the hhmass, considerable change to the mobility of the material willoccur as E and E move closerand further apart. This leads to a complex and spatiallynon-uniform mobility that is only due tothe energy separation of the valence sub-bands. Other effects such as impurityand alloy scatteringwould also have to be considered.The implications of the first assumption are even more interesting thanthose of the second.For the purpose of tunneling, the lightest mass will producethe largest tunneling flux. But, conservation of transverse momentum mustbe ensured for a hole to change bands, which leadsto themass barrier results of Chapter 4. However, the holewill attempt to take the path of least resistance by minimising its energy; it may either continueon in the sub-band it currently occupies, orchange bands in an attempt to minimise its energy whiletaking into account the possible loss orgain due to the mass boundary effect. The complexity of transportwithin 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 energysub-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 thatthe heterojunction is abrupt; thereby producing two regions, separated by a single mass barrier wherethe material parameters within each region are a constant (see Fig. 4.2). As a result ofthis, the relative separation between E’and E will not change, except at the mass boundary. Therefore, for thecalculation of the EBSCR transport current for holes in theSii..Gematerial 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— J21tmkTA)’where j is either hh or th. Within the standard flux model, the base barrier potentialVblonger 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 calculationof the standardflux has the holes changing betweenEt1and 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 integrationreplaced with theVbthat 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 BandSpike (VBS), by way of independent E’ and E bands. Upon reaching the mass barrierthe 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 ofthe HBT, the emergingfluxes from the EB SCR, contained within E and E, willgenerally be characterised by different driving forces of AE and AE respectively. However, dueto the strong intervalley scattering that occurs between the valence sub-bands upon reaching theneutral region, a common quasiequilibrium condition of will result for both Eand E. Therefore, the final transport model for holes is:1 1T,holes= [Fh+F+JT,TOl(6.35)where F andF.hare the full EB SCR transport models, andT,utralis 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 appliedbiasVBEthat is less than the built-in potentialVb1.For the case of a band diagram where there isa negativestep, as shown in Figs. 6.13 and 4.2, asVBEapproachesVbj,a current density of —106A/cm2willflow (this is based upon an emitter doping that is ‘-.1018cm3).Ata current density oflO6AIcm2,resistive effects will dominate the device and limit the internal forward bias tobe much less thanthe external applied bias. For example, with an emitter area of 1urn2,there would be a current oflOmA at a current density of 106A/cm2.Even with an unrealistically lowemitter contact resistance of 502pm2,there would be a 500mV drop to the external applied biasbefore it evenreached the junction. It is for this simple reason that observation of the device witha forward biasnear, and certainly beyond,Vbjis not really experimentally possible.As is evidenced by the plot oftsEin Fig. 6.10, along with zE and AE shown in Fig.6.9,there exists the possibility of constructing a positive-goingpotential step (see Fig. 6.14a) in thepath of the electrons trying to surmount the potential barrier of the EBSCR. A positive potentialstep would force the electrons to surmount the entire barrier, because unlikethe CBS there is noway to tunnel though the step. Therefore, if the step potential were as largeas 240meV (i.e.,AE= -240meV), then by eqn (4.79) the charge flowing throughthe EB SCR at room temperaturewould be reduced by a factor of exp(-240125.9) 10.Therefore, whenVBEapproachesVbj,thecurrent density will have dropped to only lO2AIcm.A currentdensity oflO2Afcm will certainlybe observable, and would even allow forVBEtOexceedVbj.Before going on to present a physical demonstrationof operation beyondVbj,the transporttheory for this domain of operation is first developed.WhenVBEis exactly equal toVbj,and if theresistive effects are negligible, then the band diagramwill be flat except at abrupt heterojunctionsor regions of spatially non-uniform Ge alloy content(see Fig. 6.l4b). For this reason, the point atwhichVBEis exactly equal toVbjis termed flat-band (in much the same manneras the flat-bandcondition in MOSFETs). At flat-band there will be nospace charge present. AsVBEis increasedpastVbj(see Fig. 6. 14c), an accumulation region of mobileelectrons on the n-side, as well as mobile holes on the p-side, of the heterojunction will beginto form (as has been the case throughout,a coincident hetero- and metallurgical-junction is assumed). This is contrastto the standard casewhereVBE<Vbj,and a depletion region forms where thespace charge is composed of immobileJuly 12, 1995 171ion cores from the dopant atoms. For this reason, operation pastVb1is termed the accumulationregime. Finally, asVBEis 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 —iVFig. 6.14. Band diagram for a np-junction with a positivestep potential (i.e.,AE<0). (a) equilibrium; (b) flat-band whereVBE = Vb;(c) accumulation region whereVBE> Vbj.July 12, 1995 172A reasonable first approximation to the complex accumulation regime begins by assumingthat for operation just beyondVb1the accumulation layer is non-degenerate. Based upon this assumption, and neglecting the effects of a non-uniform e, the Poisson equation inone dimensionbecomes29_(ND— N ekTon the n-sided2qiD= (6.36)dx2q2(N — NAekT) on the p-sidewhere and are in terms of electron energy (i.e., the negative of potentialenergy). Eqn (6.36)is solved on the n-side (the p-side solution can be obtained directly from then-side solution bysymmetry arguments) to yield the following implicit transcendental function:=ed+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 possibleto invert the result. However, it is reasonable to assume that the charge in the accumulation layer will overwhelmthe backgrounddopant ion potential. With this assumption eqn (6.36) is recast to:22,,—9—NDekTon the n-side8(6.38)dx2kTon the p-sidewhose n-side solution is?2IJie’x = ±4_$1dW+A2. (6.39)NDAiecT+kTIt is interesting to note that the only differencebetween 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 dopingNDis 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 toN’,.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 ofNn= 0 at x -x, to eqn (6.40), and choosing the positivex-direction produces:(x+x’e= cos2lI(6.41)ha1,)where1 IekTa1=By appealing to the symmetry of the problem, thep-side solution of eqn (6.38) is:‘lip(x—xe = cos2I I (6.42)2a1)wherea1=It is important to realise that ‘qi, is set equal to 0 at x = -x,, andis set equal to 0 at x = x. However, the form of the Poisson equation requires that when qI,j joinsup with at the heterojunction (i.e., at x = 0), the joint be analytic up to first derivatives. Given that we are solvingforaccumulation and not depletion, then continuity of andNiprequires that:July 12, 1995 174VBE— VbIN’(O)— v(O) = q (VBE— Vbx)2kT= cos(2ai)COS(2a1pJ(6.43)Further, continuity of the electric field requires that:—=—----tan(X,“1= —--tanI “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 involvingthesquared cosines of and xJ2ai,. Choosing the positive roots of the solution foreqns(6.43) and (6.44) yields:II4NANDeAVT+(NA—ND)2+NA—NDx, = 2a1 acosI Al____________________________________(6.45)(IJ4NANDeET+(ND—NA)2+ND—NAx =2a1 acosiIvp2NDeBE Twhere .AVBE VBE — Vbj,andVT= kTIq.The accumulation regime solution of eqn (6.45) is certainly muchmore complex than eqn(5.9) for the depletion regime. However, the accumulation regime sharesmany similarities with thedepletion regime. In fact, whenVBEis within the immediate neighbourhood ofVbj(i.e., smallIWBE),then a Taylor expansion of eqn (6.45) about the pointAVBE= 0 yields exactly the sameequations for x and x, that is obtained from the depletionregime. Further examination of eqn(6.45), however, shows that asAVBEincreases, x and x, quickly saturate at a constantvalue ofltal,nandltal,prespectively. This saturation of the SCR width is a feature ofthe rapid accumulationof mobile charge that screens out the applied bias with essentiallyno further increase to the extentof the SCR. This result is also the point at which the assumptionof a non-degenerate accumulationlayer will fail; so care must be exercised in the absolute application ofeqn (6.45) for largeIWBE.A useful metric from the depletion regime was the ratioof 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 expansionaboutAVBE =0,and the asymptoticlimit for largeAVBE,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 ofAVBEbetween the n- and p-sides of the junctionyields:N NN(N-N)Vrat=IN’fl(°)INA + ND+ (ND + NA)3VBE — VBE Vee(6.47)qBE1AwhereVT (ND+NA)2Ve,V_NNADNratin eqn (6.46), as was stated a few paragraphsearlier, shares many of the same featuresasNratin eqn (5.9) under the depletion regime. Now,V.atin the depletion regime is exactly thesame asNrat,owing to the spatial uniformity of the space chargedue to the immobile dopant ions.However, under the accumulation regime,1liatin eqn (6.47) starts out the same asNrat,but due tothe mobile nature of the accumulation space charge,quickly results in an equal portioning of theexcess applied potentialAVBEbetween the n- and p-sides of the junction.Therefore, the potentialdistribution in the accumulation regime differs markedlyfrom what is found in the depletion regime. Finally, Fig. 6.15 plots Njj and11atin both exact and approximate form,as well as x, andx, in order to gain a familiarity with the accumulationregime.Eqns (6.46) and (6.47) provide very useful tools forthe solution of charge transport withinthe accumulation regime. Fig. 6. 14c shows that withinthe accumulation regime, the positivesteppotential has produced a CBS; but unlike the negativestep potential within the depletion regime(see Fig. 4.2), the CBS now appears on the other side ofthe heterojunction. Taking the standardHBT case whereNA >> ND,then x <<x, and for smallAVBEone 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 usingWCBS= 1, andWNobtained from eqn (4.6) with the accumulation potential of eqn (6.42). However with the parameters used in Fig. 6.15, whenAVBE =l2OmV, then the CBS stands only 28meV tall, and 17A wideat 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 assumingWN=0, would be an energy ofIAEI— and the smallest that theCBS barrier could be, assuming thatWN= 1, would be an energy ofIAEI— qzV (see Fig.6.16). Therefore, withVat1 for smallIVBE(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 forNratandVratfrom eqns (6.46)-(6.47). The material parameters are:ND:5x1017cm3;NA:1x1019cm3;e: 12.0.One of the essential results of Chapter 4 was that the peak emission flux densityoccurred 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 fairlysimple 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 effectiveheightof the CBS is given by:ECBS =— qzV + q (1— UmaxIVBE= AEI—qAV(l— Umax+ UmaxVraj)(6.48)where 0Um1, andUmwill be taken as a phenomenological constant. Strictly, based uponthe analysis of Chapter 4,Umarwill have a temperature dependence. However, as a first approximation, U can be taken as a constant independent of temperature. Then, thetransport currentunder the accumulation regime is simpiy given by the thermionic term fromeqn (4.79) as:ECBSFfS = FfSOV1e (6.49)where both sub-bands within the valence and conductionbands need to be considered in the caseof theSii..Gematerial system.EFig. 6.16. Diagram of the CBS that forms underthe accumulation regime. Only the conductionband is shown, but a similar structure can occur in the valenceband. Note: this is for one sub-band.-\‘qc,1 referenceIv(0)I= qzVV,-1July 12, 1995 1786.5 Conventional and NovelSii..GeIIBT StructuresTheSii..Gematerial system represents a further step on the road to bandgap engineering.Unlike theAlGai..Asmaterial system, theSii.Gematerial system allows one to essentiallymanipulate IXEg andiXE(and therebyAE)independently. This independence between AEg andAEis achieved through two independent parameters: 1) the Ge mole fractionxain 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 theSiiGematerialsystem. 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 thicknessh.As was stated early on in this chapter, the potentialstrain in theSii.Gematerial 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 ofthealloy layer decreases in an essentially exponential fashion. The determination ofhhas been thefocus of numerous studies and controversies [97,99,105]. At present, there isstill debate as to theexact model forhversus in-plane alloy strain, but the work of People [105] is at least a reasonable reference point. In [105], the critical layer thickness is givenas:1—v1 b2 1 hh= i+v20ic&(_)(_n(T)J(6.50)wherehis in A, b = 4A (the magnitude of the Burger’s vector),v is the Poisson ratio from eqn(6.5),aais the unstrained (bulk) alloy lattice constant from eqn(6.6), andfis the alloy strain givenby(aa — a)Ia(whereais the substrate lattice constant). Substituting all of theseparameters intoeqn (6.50) gives:1.928(5.43 +0.23a”2 hh=(5.43+0.23aa),jln(-4-). (6.51)Eqn (6.51) is an implicit phenomenological equation that Peoplehas fit to the best available dataforh(see Fig. 6.17). Detailed information, such as what temperature and duration cana pseudomorphic layer tolerate before relaxing is still not conclusively known.July 12, 1995 1791010.0 0.2 0.4 0.60.8 1.0Germanium Mole FractionxaFig. 6.17. Critical layer thickness for a Si.xpexalayer on a{100) Si substrate. If the substrateisSiiGe instead, then a good approximation is to findIxa- 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 ofSii..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 fora wide-bandgap emitter injecting into a narrow-bandgap base. Within this physical construct,the Ge alloy contentxain thebase is either fixed at some constant value, or a drift field is created inthe base by increasingXaasone proceeds from the emitter towards the base.Starting with a constantxain the base of 0.2, then eqn (6.51)givesh1550A. Because theHBT is lattice matched to a pure Si substrate, all regionsof 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 inthe base is E-like. Further,compressive strain in the base makes the ultimate valenceband E-1ike, with E’lh34meVFig. 6.18 presents the band diagram for the above device, withthe relevant material parametersnoted. Observation of Fig. 6.18 clearly shows thatelectron transport will occur via E. SinceJuly 12, 1995 180= -100meV, while AE is 37meV, ostensibly all of the electrons containedby the E band in theemitter (which is 33% of the total number of majority electrons) will be reflectedby AE and notcontribute to electron transport. Thus, if the EB SCR determines the transport current,then afterincluding the different effective masses, Iwould be 18% less than expected froma 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, thenI wouldbe larger than expected giventhat D is higher than the bulk value. In order to determine if it is the LB SCR orthe neutral basethat is responsible for current-limited-flow, the detailed construction ofthe device must be considered. For the devices in [100-102], whereND >> NA,then the neutral base is narrowly responsiblefor current-limited-flow; although, inclusion of bandgap narrowingeffects could lead to the EBSCR being responsible for current-limited-flow. However, forthe devices in [10,1341, whereNDNA,then depending on how bandgap narrowing in the base splitsbetweenEandEthe EB SCRwill be responsible for current-limited-flow; resulting in a much smaller increaseto I than wouldbe expected from neutral base transport considerations alone. This analysis ofcurrent-day SiGeHBTs shows that a failure to correctly modelboth E and E, including EB SCR limitations,could lead to an incorrect understanding of transport withinthe device.AE = 37meV AE= —138meViXE = —100meV AE= —104meVIXE= 37meVAE= —138meVSi Si08Ge0•2Sie= 1120meV fle= 6.94x109cm3Ehl_Eg,b= 945meV = 1.47x1011cm3Without LB SCR limitations, EwillE’ Etransport 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. Thebase isthe reference. The effect of the LB and CB SCR potential is notshown for clarity.July 12, 1995 181For SiGe HBTs, where the emitter and base are E-1ike,AEis 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 logIversusVBEwill 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-talesignin SiGe HBTs that the transport current is not being controlled by the neutralbase. However,Icwill indeed be smaller than expected due to the EB SCR limitation, plus, the Earlyvoltage shouldbecome theoretically infinite as basewidth modulation shouldno longer effectI [135].The SiGe HBT wherexais varied across the base represents the device that has piqued theinterests of the semiconductor community. By generating an aiding field in thebase through amonotonically increasingXafrom the emitter to the base (and hence a decreasing Eg), anfTashigh as 113GHz has been obtained[1021. In order to achieve this remarkable metric the devicewas fabricated with as large aAxain the base as possible; minimising the base transittime. To thisend,Xawas 0 at the emitter and was linearly ramped up to 0.25 at the collector. Theresult 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). Thebase region, given the shape of the E and Ebands, produces a demanded current that differs between the sub-bandsby a factor of 8.3; i.e., thecurrent in E will be 8.3-fold larger than E. This is notan overwhelming amount, which showsthat 11% of the collector current is carriedby the slower E band. In fact, using eqn (3.8) showsthattBfor E is reduced 4.6-fold compared totBo,whiletBfor E is reduced only 1.5-fold compared to‘rho(wheretBois the‘CBgiven in eqn (3.6)). Assuming that the final basetransit time isgiven by the average of the results from each bandweighted with the relative current carried bythe band, then the effective reduction totBcompared totBOis (0.89/4.6+ 0.11/1.5)-i= 3.8-fold.If the two sub-bands were considered as one singleband thentBwould have been wrongly reduced 4.4-fold relative totBo,and I overestimated by 13%. In the above calculationsthe effectof bandgap narrowing has not been accounted for. Inclusionof base bandgap narrowing couldcause the EB SCR to limit the transport current (again, depending onhow the bandgap narrowingsplits betweenEand Er), which would greatly effect the current partitioningbetween the conduction sub-bands. Furthermore, the anisotropic nature of E andE has also not been accountedfor, which would increase‘CBeven further given that would be greatly reduced.July 12, 1995 182(a)E, E = 216meVAE = 0meV= 44meV AE = 42meVAE= 216meVAE= 0meVSiSi075Ge025. Ege= 1120meV‘e= 6.94x109cm3Emitter SiBasehhCollector E8b904meV 3.17x10’1cm3Ehh,E11’ —Elh IWithout 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• •1E current within-/the EB SCR>,2’> “..6.io5E current withinthe neutral base4.2/$‘C,... ,-....-.in21 . /•<VE2 current within10/../..•.the EB SCR100.10-1:, E current within/‘./the neutral base10-2• • • •0.6 0.7 0.8 0.9 1.01.1Base-Emitter VoltageVBE(V)Fig. 6.19. (a) Band diagram for an HBTwith 25% linear grading of Ge inthe base, latticematched to Si. zS.E is from the 25% Ge point inthe base to the emitter. Note: the basebandgap hasa slightly parabolic nature due to the Ge alloy effects. (b) Transportcurrents through the variousregions of the HBT, including the collectorcurrent.NA=5x1018cm3,ND= 1x1020cm3,andWB=700A.Given that E transport within the EB SCR is not substantiallylarger than transportthrough the neutral base, I is subsequently 31% lowerthan expected from neutral basetransportconsiderations alone. Thus, the neutralbase is controllingI but the EB SCR does have aneffect.July 12, 1995 183The previous analysis of conventional SiGe HBT structures is not intended to be exhaustive,but it clearly demonstrates that theSii..Gematerial 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 dictatedby a regionother than the one being considered, the result will be a an erroneous conclusion regardingeitherthe correct path for optimisation or the material parameter being extracted.The main problem with theSii..Gematerial 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 seeND >> NAin order to maintain a usable f3. As the neutral base width isreduced, thenNAmust increase in order to offset a rapid decrease However,increasingNAmust be accompanied by an increase inNDor the gain will drop. WithNDnear the solid-solubilitylimit this is not really possible. Further, withNAandNDincreasing, the EB capacitance will increase, and a tunnel diode could form. The device in [134] attempted tosolve this with a constant22% Ge base content. By having a narrow bandgap in thebase, the subsequent increase toIbcanbe traded off for a higher base Gummel number. However, this precludes a gradedbase, as the EBheterojunction is required to maintain the gain, and the critical layer thicknesswill 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 compromisingfmaxorf a way must be found to include higherGe contents in the base.The answer to the problem of the previous paragraph is to lattice matchthe HBT to aSii.Gesubstrate, where x,> 0. Consider a 500A SigjgGe02emitter witha poly-Si cap, a basegraded from Si075Ge025 at the emitter to Si06Ge04at the collector, all lattice matched to aSi08Ge02 collector and substrate (see Fig. 6.20). Thebase grading is started at 25% Ge instead of20% in order to increase the transport current in E relative to E, thereby reducingthe parasiticeffect ontBfound from the HBT in Fig. 6.19. Then, the 15% Ge base gradingprovides 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 toI 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,NAcan besignificantly increased in order to increasef,,whileNDcan be decreased in order to decreasethe EB SCR capacitance. The result is a 264-fold increase inI compared to a similar bulk Si device, withtBreduced 2.9-fold compared totBO.These results are based upon the neutral base controlling I. AsNAis increased to the point where bandgap narrowing becomes quite large, it isexpected that the EB SCR will dictate I and limit the expected increase to13..,-lCCWithout EB SCR limitations, E will trans= 120meVAEhhh1= —34meVport 4.5% of the currentc vin the neutral base,AE2 —18meV zE 9meV Ee= 990meVleaving E to transportV‘ the remaining 95.5% of= 120meVtE= —34meVEb= 876meV the current.Fig. 6.20. Novel SiGe HBT based on a 20% Ge substrate. The incorporation ofthe optimumbase grading provides the maximum reduction to‘CBpossible. 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 byh.iXE is from the 40%Ge point in the base to the emitter.Sio.8oGeo.2oEmitterbO.l6WBjSi075Ge025Si060Ge0401Si080Ge020CollectorO.16WBBase= 3.68x1010cm3i,b= 5.05x1011cm3July 12, 1995 185The operation of the novel transistor being proposed rests on two requirements: 1) that highqualitySiiGesubstrates can be formed; 2) that the poly-Si cap will indeed control the backinjection of holes. The ability to grow high qualitySiiGe substrates is currently an issue. Atpresent, bulk epitaxialSiiGe 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 bulkSiiGe layers on Si,in time it is expected that theprocess will mature and the defect density will fall. The other option is to pull rawSiiGeingots so that the starting wafer contains the desired substrate. In either case, for thestudy beingpresented here, it is sufficient to demonstrate the usefulness of using non-Sisubstrates in order toprovide the impetus to grow low defect bulkSii..Ge substrates on Si. The second question, regarding the efficacy of the poly-Si cap to control hole back injection, can onlybe 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 thebandgap is, if anything, larger than in bulk Si. Thus, n1 in the poly-Si layer will be small comparedto the n1 in thebase, controlling the back injection of holes and 3. Finally, the band alignment ofthe poly-Si layerto the Si08Ge02 emitter will only be an issue if the resultingAEis large enough to limit the electron transport current through the entire device. Based upon Si lattice matchedto Siij8Ge02,AEshould not exceed -90meV, which would not reduce thetransport current given the high dopingthat would exist in the poly-Si layer. Therefore, it is expected that the poly-Sicap will control thehole back injection of the proposed SiGe HBT.This section concludes by examining an intriguing HBT structure thatinvokes all of themodels of this chapter. Beginning with Fig. 6.9c forXal= 0 andXar= 0.45, examination of substrates where 0 x3 0.35 is very interesting. Letthe left side be the emitter and the right side thebase. The emitter is under tensile strain so that the ultimate conduction bandis E-Iike. Contrarily, with the substrate range being considered, thebase is under compressive strain and the ultimateconduction band is E-like. Just because the emitter conductionband is E does not precludeelectrons from existing in E. In fact, given the band alignments for 0x 0.35, more electronsfrom E, rather than E, will be able to go from the emitter into thebase. Essentially, the bandwith the lowest energy in both the emitter and the base will be the one thattransports the current.With x 0.35, E will be responsible for current transportas E in the base is larger than E inthe emitter.July 12, 1995 18621•E%BaseSi055Ge045r—i——-— $E2-E\I I CI I•4444SubstrateI I 4”ç_)%%%%%t%.4CbS”s”;fSSd.. Si0653e0Emitter__4 Vlr4nnl C’o i Vt) io iM:44:14444— *Collectorb1,,..nwt’*a”Ehhv Si55Ge045e= 6.36xl010cm32 1i —3= 67meV= o’i’ cm= —287meVAE = —237meVE9e= 932meV= —168meV= —169meV = 908meV= —194meVFig. 6.21. Band diagram showing the conduction andvalence sub-bands for an HBT whereXal =0,Xar= 0.45, x = 0.35,NA=1x1019cm3,ND=5x1017cm3,andWb=700A.Fig. 6.21 plots the band diagram, including SCR effects, for anHBT whereXal0,Xar =0.45, x = 0.35,NA=lxlO’9cm3,ND=5x10’7cm3,andwb=700A.As is the case for the HBT inFig. 6.20, there is a high doped poly-Si cap ontop of the emitter to provide stress relief and control the back injection of holes. What is interestingto note for the device in Fig. 6.21 is the emitterand base have essentially the same bandgap. Thus, thereis no wide-gap emitter injecting into anarrow-gap base that is common to traditional HBTdesigns. Instead, the HBT is controlledby theband offsets and n1 for the given sub-band within the neutralregions. Fig. 6.22 plots the EB SCRcurrents, the neutral base transport currents, and the final collectorcurrent that will occur withinthe device of Fig. 6.21. It is important to realise thatVbj= 0.673 V due to the positiveAEof thisdevice. ForVBE< V,j transport occurs via E through a small CBS,but with neutral base transport essentially controlling I. Thus, electron transportwithin the emitter is occurring ina bandJuly 12, i995 187that does not form the ultimate conduction band. Now, whenI/BE> Vbj,the HBT is operatingwithin the accumulation regime. Due toiXE= -169meV, EB SCR transport within E is reducedto only lO2AIcm whenVBE = Vbj.Furthermore, because AE = 67meV, any increase inVBEpastVbjwill do nothing to increase the EB SCR current as there is no barrier to surmount, leavingonlythe thermal movement of majority carriers to dictate the current. Thus, E transport is nowcontrolled by the EB SCR and not the neutral base. However, with the accumulation model ofSection6.4, E transport becomes the dominant path that controls I whenVBEincreases pastV;leading to transport in the base that occurs within a band that does not formthe ultimate conductionband. The final result is a very interesting log IversusVBErelationship 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 VoltageVBE(V)Fig. 6.22. Transport currents within the various regions of the HBT givenin Fig. 6.21.The HBT of Fig. 6.21 may have some practical uses asa current source due to its flatIversusVBErelationship near V,j; however, it is probably more useful as a tool to investigatetransportc10611110210110010-110-21 0-1 0-0.30.9 1.0July 12, 1995 188properties and band offsets within theSii..Gematerial system. Careful consideration ofhforthis HBT reveals some interesting results. Because the Ge content in the base is only 10% higherthan in the substrate, thenh= 8054A. With such a largehit 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 3000to 5000A, leaving considerableroom to the maximumh,which should help to increase the thermal budget for thebase layer. Theissue of DHBT devices is not a real concern in npn HBTs, due to the smallAE,but would be ofconsiderable appeal in making a pnp device. Finally, the result of a largehfor the base and collector regions is a significant lowering of the emitterhto 407 A. However, an = 407A wouldbe wide enough to contain the emitter extent of the EB SCR. Therefore, thecritical layer thickness has been moved from out of the EB-SCR and into the neutral emitter, whichwill have less ofan effect on device performance if dislocations due to strain relaxation occur.In conclusion to this chapter the following results regarding theSii..Gematerial systemhave been achieved:• A review of the literature, including the best material models, for theeffect of strain on theconduction and valence sub-bands has been performed.• The band offset theory of Van de Walle and Martin, includingthe material models of Yu andGan et. al., have been reviewed with the most consistent set of material parameterschosen 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 Ebands, and the valence E andE bands has been developed. The theory presenteddoes 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 statesand the intrinsic carrier concentration for allof the sub-bands.• A theory for the operation of an HBT past the built-in potentialhas been developed.• Finally, the models of this chapter, which are based uponthe models of all the previouschapters, have been used to study current-day SiGe HBTs anda 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 isasignificant error in both the calculation of charge flow and transit timeby 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 simulatingcharge transportwithin the entire HBT by providing a means to breakthe 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 recombinationsinks within each region;furthering the general nature of the model. Due to the abstract nature ofeqns (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 pointat which rapid spatial changes inthe conduction or valence bands produce transport conditions that deviate from the modelsof driftand diffusion (such as can occur within an SCR, and certainly atthe heterojunction whereAEfonns). This may allow for a solution to a question posedby Dr. Mike Jackson of UBC as to thecondition for which thermionic injection begins and drift-diffusionends. However, the most logical extension to the work of Chapter 2 is to remove the restriction thatEp, (for an npn HBT) be aconstant throughout the EB SCR.Chapter 3 presents some interesting ideas for optimising the metricsof 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 measurea number of HBT designs that exploit theoptimisations that have been alluded to. Chapter 3 has alsogone on to determine the simultaneousoptimisation of the base bandgap and the base dopingprofiles for the minimisation oftB.Thiswork has, however, neglected the effect ofa non-constant mobility with respect to doping variations. Numerical work [63] has shown that the optimum profileswhich include the fullpfl(NA)donot appear to be too complex, and certainly havea shape that is expected from consideration ofthe functional form of .I(N) itself. Therefore, it is expected thatthe analytic optimum profile,shown in Fig. 3.9, for the minimisation oftBcould be extended to include either the fullJ.Ifl(NA)ora judicious approximation to it.Chapter 4 derives the model of charge transport with the EB SCR, includingthe effects oftunneling and momentum conservation across a mass boundary. Tothis end, the general models ofeqns (4.50)-(4.53) were presented. Chapter 4 goes on to deriveanalytic approximation to eqnsJuly 12, 1995 192(4.50)-(4.53). However, for the purpose of deriving analytic results, the mass boundaryis considered in an isotropic fashion, but with the effective mass maintained as a diagonaltensor and not asimple scalar. Thus, a logical extension to the analytic work of Chapter 4 isto 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 Section4.6. By plottingthe ensemble electron density entering the neutral base of the HBT, it was clearthat the distribution could not be considered as a Maxwellian or evena 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 freepath [43]) demands a full solutionto the BTE, then a way must be found to incorporate the non-local effectof tunneling into theBTE. A possible extension to the work of Chapter 4 isto connect the EB SCR transport models ofthe chapter to a BTE solver for the neutral base; therebyallowing for the inclusion of tunnelingwithin the BTE via a hybrid model.The modelling of charge transport in Chapter 4, due to tunneling throughthe 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 reasonthan to simplify the calculation of thetunneling probabilities. This position of neglecting thermalisingcollisions of the electron whileundergoing tunneling is often substantiated onthe grounds that tunneling distances are generallyless than 100 or 200A, and are therefore significantlyless than the mean free path. However, ifany collisions did occur while the electron is in themidst of tunneling, then the tunneling probability would be essentially reduced to zero. Thus, a potentialextension to the work of Chapter 4 isto consider non-ballistic tunneling. The ultimate outcome ofsuch non-ballistic tunneling considerations would be the development of a Monte Carlo simulatorthat can incorporate non-local effects (i.e., tunneling).A final extension to the work of Chapter 4 can be foundby careful observation of Fig. 4.9and eqn (4.74). Im occurs at which for a fixed temperature isa constant. Furthermore, theflux densitycLy.is fairly well centred about U, and willbecome even more localised as thetemperature is reduced. Therefore, the tunneling currentthrough the CBS can be thought of as occurring at an energy of qU(V— VBE)Nratrelative to the conduction band minimumin theemitter. Now, the tunneling current is very sensitive to the forward-directedeffective mass, whichJuly 12, 1995 193is dependent upon the full nature of the dispersion relationE(k).Then, with the CBS responsiblefor controlling I, by measuring Ithe tunneling current through the CBS canbe determined. Finally, by extracting the effective mass through a matching ofthe measured I to the tunnelingmodels of Chapter 4, it should be possible to inferE(k).Therefore, it should be possible to extend the work of Chapter 4 by developing an electricalspectroscopy method for the determinationofE(k).Chapter 5 presents the models for the recombination currents that occur withinboth the EBSCR and the neutral base. Specifically, the need to balancethe 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 thismixing is a newconnection between the physical construction of the HBTand 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 thearguments of Chapter 4, drift-diffusion analysis is not applicable within the EB SCR. Therefore, combined with the extensionbeing proposedfor Chapter 4 (regarding integration with the BTE),the recombination currents should be recomputed from a particle scattering cross-section point of view. Thiswould place the calculation of therecombination currents on par with the quantum mechanicalview of a tunneling electron.Chapter 6 reviews the various material models that arerequired to understand the composition of the conduction and valence bands withinpseudomorphically strainedSii.Ge.Further, theband offset models for the determination ofIXEandAEat an abrupt heterojunction are also presented. Using these material models, transport models whichinclude the two conduction sub-bands E, E and the two valence sub-bands E, E, are developed. It is shownthat the multi-band nature of strainedSii..Gemust be considered, even in present-day HBTs, lestconsiderableerror regarding both the quantitativeand qualitative aspects of charge transport be made. Regarding future work, it is imperative that a final and consistentset of material parameters forSii..Gebe obtained. Without a firm understanding of the material parameters, it isimpossible to accuratelydetermine the transport current. With this in mind, Chapter6 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 shouldbe fabricated and tested againstthese theoriesJuly 12, 1995 194Finally, Chapter 6 only considers substrates alignedto (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 surfacestates at the SiJSiO2interface in MOSFETs.One of the most interesting features of strainedSii..Geis the possibility of only having chargetransport occur parallel to the small transverse mass for electrons. The anisotropicnature of Siproduces a 5-fold difference between the transverse and longitudinalmass for electrons. Thus, asignificant improvement to tunneling and mobility can be had if theelectrons predominantlymove with the transverse mass. This would be further increasedby using the (111) conductionbands instead of the (100) bands. In fact, the (111) bands have a 20-folddifference between thetransverse and longitudinal mass for electrons, with the transversemass near that of GaAs. Therefore, a logical extension to the work of Chapter 6 wouldbe the development of (111) alignedtransport models. Finally, with the ability to set a largeeffective mass band at an arbitrary energyabove a light effective mass band, it should in theorybe possible to produce negative differentialmobility, in terms of t versus electric field, within strainedSii.Ge;leading to the possibility ofdevices, such as Gunn diodes, which can only be presentlymade in materials such as GaAs.Therefore, a further extension to the work of Chapter6 is to investigate the feasibility of generating and utilising strainedSii..Gefilms that produce negative differential mobilityversus electricfield.As a final parting comment regarding future work, it is clearthat with the rapid progresscontinuing in the development of ICs, devicedimensions will continue to shrinkat an exponentialrate. Obviously, this will take devices down intothe atomic realm where distances cover only 10Angstroms and not a thousand. Even withpresent-day devices, where relevant dimensionsare 500to ioooA, quantum mechanical effects are important(as can be seen from the consideration oftunneling in Chapter 4). As dimensions reduceto 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 thezero-th and first), will have very limitedusefulness. Instead, a “full” quantum mechanical modelwill be required. But then whatis meant by a“full” model? With relevant dimensions of 1 oA, itwill not even be possible to utiliseBloch’s theorem because there will truly be no dimensionover which the crystal can be consideredas bulk.Furthermore, considering only the conduction electronsin a quantum mechanical fashion,and notJuly 12, 1995 195the core electrons, will not be acceptable at ioA. 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Lett., vol. 14, no. 6, 292-294, May 1993.[136] M. Kondo, T. Kobayashi, Y. Tamaki, “Hetro-Emitter-Like Characteristicsof PhosphorousDoped Polysilicon Emitter Transistors - Part I: Band Structure in the PolysiliconEmitterObtained from Electrical Measurements”, IEEE Trans. Electron Dev., vol. 42, no.3, 419-426, 1995.[137] M. Kondo, T. Kobayashi, Y Tamaki, “Hetro-Emitter-Like Characteristicsof PhosphorousDoped Polysilicon Emitter Transistors - Part II: Band Deformation Due to ResidualStressin the Polysilicon Emitter”, IEEE Trans. Electron Dev., vol. 42, no. 3, 427-435,1995.July 12, 1995 205Appendix ARampedNAB(x)to MinimisetBThe proof of eqn (3.17) begins by solving eqn (3.10) for‘tBusing the doping profile depictedin Fig. 3.4. To this end, it is seen that the doping profile of Fig. 3.4 is actuallya subset of theprofile depicted in Fig. 3.3 with h1 =0 and h2 = h. Using the symbolic mathtool MACSYMA©,eqn (3.10) yields the following result fortBbased 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)e2-1 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) eh2—hl h2—hl(h2_ hi) eh2-hl h2—hlNellog(U) log(U)(dO) integrate(\ne ,x,x,hl);(dlO)Ne (hl_x)Eqn (d5) is the exponential doping profile for h1x h2, and it ensures that there are no jumpdiscontinuities at the break points h1 and h2 between the exponential dopingprofile and the regions of constant doping. Then, eqns (d6)-(dlO) collect together the varioussub-integrals requiredto solve eqn (3.10). It should be noted that the doping at x = 0 isNe,at x = 1 isN,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_Mlog2(U)Eqn (dli) is the general model fortBfrom the optimum doping profile of Fig. 3.3.Using the optimum equation for‘CRgiven in eqn (dli), then the‘CBneeded 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(h2_2h+1)ulog(u)+(2hu+2h_2h)log(u)_2hu+2h‘22 U log (U)Eqn (d12) can then be solved for the h that minimises‘CBDifferentiating eqn (d12) with respect toh, setting equal to zero, and solving for h produces:(c13) ratsimp(diff(rhs(d12),h));(d13)(h_1)Ulog2(U)+ (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 sameas‘CRin eqn (3.17) once theJuly 12, 1995 207factor of 1/2 is included withintBO.This completes the proof of eqn (3.17) for the rampedNAB(x)to minimisetB.It should be noted that the output displayed within this Appendix comes directlyfrom MACSYMA©. As such, there is occasion to performsome 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 dopingprofile. Ifthe profile is changed from that shown in Fig. 3.4 so that the exponential regionfollows the constant doping region, then it is found thattBremains unchanged from what is given in eqn (3.17),and h —* 1 — h. Returning back to eqn (dli), the necessary changeto the doping profile is accomplished by setting h1 = h and h2 = 1 in the optimum equationfortBgiven in eqn (dli); i.e.,(c17) expand(ev(dl 1 ,hl=h,h2=1));h 1 h 1h 1h2U1—h — 1—hhU1—h — 1—hh 1 h2U1—h — 1—hlog(U) log(U) — log(U) + log(U)+ log2(U)(d17)2hUl—h 1—hel_h1—hh2 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)2Ulog2(U)Eqn (d18) is thetBfor the symmetric doping profile used to develop eqn (d12). Aswas done witheqn (d12), the optimum value for h is found by differentiationeqn (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)—02Ulog2(U) —(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 minimumtB;i.e.,(c21) ratsimp(radcan(ev(d18,d20)));2U2log(U)_3U÷4U_1(d21)2 U2log2(U) +4 U log(U) —4 U2 +4 UEqn (d21) is exactly the same as eqn (d16), showingthat a symmetric change to the dopingprofile produces no change to the transit time. Itcan finally be shown that the symmetric changeto the doping profile results in h —> 1 — h by adding together theh 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 Ulog2(U)+2log(U)_2U÷2(c27) ratsimp(combine(d26));(d27)1Eqn (d27) proves that the symmetric changeto the doping profile of Fig. 3.4 does indeed resultinh—> 1—h.July 12, 1995 209Appendix BOptimumNAB(x)to MinimisetBThe proof of eqn (3.18) begins by solving eqn (3.10) fortBusing the doping profile depicted in Fig. 3.3. However, this task has already been accomplished in Appendix Aas eqn (dli). Using eqn (dli) for the optimumtB,the pair h1 and h2 which minimise eqn (dli) is found. Usingthe symbolic math tool MACSYMA©, the partial derivatives of eqn (dli) withrespect 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_hilog2(u)Eqns (d29) and (d30) present the simultaneous set of equations, once both areset 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 thatan analytic solution is possible. Therefore, before attempting to solve eqns (d29) and (d30), a numericalsolution will be found so that a“feel” may be developed that will hopefully guide thesteps to follow.Using MACSYMA©, a numerical Newton-Raphson solutionto 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) indicatethat 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 eqnsdiffer at most by a multiplicativecon-stunt, then it is proven that h1 + h2 = 1 is indeed a solutionof eqns (d29) and (d30).Using MACSYMA© to perform the above testyields:(c31) ev(d29,h2=l-hl);(1_2h1(hlU+hi)log2(U)(d31)+ [(2 hi_2 (1_hi) +1) u1-2h1—U12M]log(U)+(2hi_2(i_hl))U1-2h1+(2(i_hi)_2hi)U1—2h1U 1og2(July 12, 1995 211(c32) ev(d30,h2=i-hl);hi ui-2M— hi U log2(U)I+[(_2h1+2(l_hl)_i)U1_2M+U1_2hljlog(U)(d32)Ihi 1—hi(2(1_hl)_2h1)Ui—2h1+ (2h1_2(1_hl)) U1—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 immediatelyasserts that there isonly one independent equation to solve for. Thesolution for h1 being:(c35) distrib(expand(d3 1));2hi 1 2hi 12h14 hi Ui—2h1 — 1—2hiu1—2h1 — i—2hi1 4 hi Ui2hl — 1—2h1log(U)1— log(U) — log(U)+ log2(U)(d35)i—2h1 — l—2hi2h1 — 12U 4h1 21—2hi i—2h1— 2 — 2 + 2+hiU -i-hilog (U) log (U) log (U)2hi 1*U2hhl1—2hi(c36) map(radcan,d35);4hi 1 1 4hi 2 4h1 2hi(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=Llog(U)-i-2July 12, 1995 212Eqn (d37) proves eqn (3.18) for the optimumh1,along with the result from eqn (d34) whichproves eqn (3.18) for the optimum h2. Finally, using the optimumh1 and h2, the optimumtBisfound 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 astBin eqn (3.18) once the factor of 1/2 is includedwithin‘CBO.This completes the proof of eqn (3.18) for the optimumNAB(x)to minimisetB.July 12, 1995 213


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