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A one-dimensional solution of the Boltzmann transport equation with application to the compact modeling… St.Denis, Anthony Robert 1999

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A ONE-DIMENSIONAL SOLUTION OF T H E BOLTZMANN TRANSPORT EQUATION WITH APPLICATION T O T H E COMPACT MODELING OF MODERN BIPOLAR TRANSISTORS By Anthony Robert St. Denis B. A . Sc. University of British Columbia, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May 1999 © Anthony Robert St. Denis, 1999  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department of Electrical and Computer Engineering The University of British Columbia 2356 Main Mall Vancouver, British Columbia Canada, V6T 1Z4  Date:  Abstract The role of device design in the development of semiconductor technologies is an important one. To effectively design devices, the device engineer needs to understand the physics of device operation. The models used to predict device behavior should therefore be cast in such a form that the physics is readily identifiable. This is true for both compact device models and the detailed models of transport from which they are derived. A considerable effort has been expended here to develop models exhibiting this feature. The fundamental starting point for the transport analysis in this work is a one-dimensional, time-independent form of the Boltzmann transport equation (BTE). There are three main contributions made herein regarding the use of the B T E for analyzing transport in semiconductor devices. First, a field-free solution, valid for arbitrary boundary conditions and quite general collision integrals, is developed and used to study transport in the base of bipolar transistors, including both homojunction and heterojunction devices. Secondly, a kinetic approach to transport is used to develop an analogous solution for field-dependent transport. The field-dependent solution is used to make a study of the nature of carrier mobility in a forward-biased semiconductor barrier. Thirdly, forms for the incoming collision integrals are developed which involve integration over a single angle only, which makes them particularly well suited for use in both the field-free andfield-dependentsolutions developed in this work. In the device design process, it is very useful to appeal to compact device models in order to do preliminary designs, and to aid in the understanding of results from more detailed transport analyses. Compact models are closed form expressions describing device behavior, usually based on approximations of detailed device equations. In this work, compact models are developed which approximate the solution of the B T E in modern, short-base, bipolar transistors. Two models for collector current are derived. The first of these is appropriate for homojunction, and graded-heterojunction, devices. The second is appropriate for abruptheterojunction devices. A third compact model is derived for the base transit time in abrupt ii  heterojunction devices. Throughout this work, an attempt is made to understand transport phenomena in terms of the electron distribution function. This leads to a more fundamental understanding of device operation than can be gained by applying concepts inherent to drift-diffusion analyses, such as drift and diffusion current densities and quasi-Fermi levels. Such an understanding is necessary for the design and analysis of future, and state of the art present, devices whose intended very high frequency of operation demands that certain critical dimensions be reduced below the limits within which traditional drift-diffusion analysis applies.  iii  Contents Abstract  ii  List of Tables  viii  List of Figures  xiii  Acknowledgments  xiv  1  Introduction 1.1  1.2 2  Semiconductor device design, simulation and modeling  1  Thesis outline  4  Transport in semiconductors and the Boltzmann transport equation 2.1  7  Electron transport in semiconductors: dispersion relation, effective mass and collisions  7  2.2  The Boltzmann transport equation  8  2.3  One-dimensional, time-independent transport  2.4 3  1  relation  10  Overview of the solution  11  Collision integrals and lifetimes  12  3.1  Screened ionized-impurity scattering  13  3.1.1  Transition rate for impurity scattering  13  3.1.2  Incoming collision integral for impurity scattering  13  3.1.3  Outgoing collision integral and lifetime for impurity scattering  16  iv  3.2  Phonon scattering 3.2.1  17  General considerations and transition rates for phonon scattering  3.2.2  17  Incoming collision integral for polar optical phonon scattering  3.2.3  19  Outgoing collision integral and lifetime for polar optical phonon scattering  23  Collision integrals for acoustic phonon scattering  24  4 Application to base transport in BJTs and graded-junction HBTs  26  3.2.4  4.1  4.2  4.3  Solution of the field-free, time-independent transport relation  27  4.1.1  Development of the general result  27  4.1.2  Physical interpretation of the field-free solution  29  4.1.3  Boundary conditions for BJTs and graded-junction HBTs  31  Solution for acoustic phonon scattering only  31  4.2.1  Finding fo(z,k),  33  4.2.2  Finding the currents and mean velocities  36  4.2.3  Results  37  n(z), and a relation between them  Solution for screened ionized impurities and polar optical phonons  49  4.3.1  Computationally more efficient form of the solution  50  4.3.2  Results  51  5 A compact model for the collector current in BJTs  56  5.1  Compact collector current models  57  5.2  A new compact model for collector current in BJTs  61  5.2.1  63  B T E results in the current-balancing approach  6 Application to base transport in abrupt-junction HBTs  68  6.1  Transport equation and boundary conditions  69  6.2  Physical interpretation  72  6.3  Results  73  v  7  Kinetic approach to transport  93  7.1  Alternate derivation of the field-free solution  94  7.2  Kinetic approach to field-dependent transport  96  7.2.1  Ballistic components  97  7.2.2  Collision components  100  7.3  7.4  8  9  The field-dependent solution in spherical coordinates  106  Solution in a constant electric field  109  7.4.1  110  Solution with isotropic and elastic scattering  Field-dependent results  113  8.1  Specification of the problem under study  114  8.2  The equilibrium picture  116  8.3  The forward-mode results  118  8.4  The reverse-mode results  122  8.5  Comparison with the D D E  126  8.6  A study of carrier mobility  131  8.6.1  The issue of mobility in a forward-biased barrier  131  8.6.2  Numerical results  133  8.6.3  The dependence of the mobility on f  136  8.6.4  Concluding remarks  139  A  Summary, conclusions and suggestions for future work  141  9.1  The Boltzmann equation  141  9.2  Compact models  144  9.3  Suggestions for future work  145  A Evaluating properties of the distribution function A.l  147  General procedure  147  A.2 The concentration  147  A.3 The current density  148  A.4 The mean z-directed velocity  148  A.5 The mean z-directed kinetic energy  148  vi  B  A. 6 The mean kinetic energy  149  Derivation of the 1-dimensional drift-diffusion equation  150  B. l  150  The time term  B.2 The diffusion term  151  B.3 The drift term  152  B.4 The collision terms  153  B.4.1  Isotropic and elastic scattering  B.5 The momentum balance equation  154 154  References  156  vii  List of Tables 4.1  The layer structure of the GaAs B J T studied in this work  5.1  The exit-velocity factors, a and b, and the diffusivity-correction factor a, for any B J T in the active mode of operation with VBE > 0.21/  6.1  50  65  The layer structure of the A l G a i _ A s abrupt-junction H B T studied in this x  x  work  73  viii  List of Figures 2.1  A simple illustration indicating the definition of 6  3.1  Coordinate systems used for evaluating the incoming collision integral.  3.2  Coordinate systems used for evaluating the outgoing collision integral  17  3.3  Momentum conservation in phonon absorption and emission  19  4.1  One-dimensional transistor diagram  28  4.2  A schematic of the means by which carriers can come to be in the forward-  4.3  10 . . .  going part of the distribution at some some point z in a field-free base.  . . .  The equilibrium electron distribution function, and the function f (z,k),  at  0  the middle of a field-free base in a homojunction device 4.4  The function fo(z,k),  30  38  for an active-mode operating point of V E = 0.814i,E B  and VBC = 0, at 4 different basewidths 4.5  14  39  The distribution function, for an active-mode operating point of VBE = 0.8Vt,i,E and VBC = 0, at the edges and middle of the base, for three different basewidths. 40  4.6  The bias dependence of the carrier concentration profile in a homojunction device with a basewidth of one scattering length  4.7  41  The basewidth dependence of the carrier concentration profile, for active mode operation, in a homojunction device  4.8  42  The basewidth dependence of the carrier concentrations at the edges of the quasi-neutral base, for active-mode operation, in a homojunction device.  4.9  . .  43  The bias dependence of the collector current density, for a homojunction device with a quasi-neutral basewidth equal to one scattering length  44  4.10 The basewidth dependence of the collector current density, for active-mode operation, in a homojunction device  45  ix  4.11 The mean z-directed velocity profile, in the active mode, for a homojunction device with a basewidth of one scattering length  46  4.12 The bias dependence of the mean z-directed velocity profiles of the forwardand backward- going components of the electron distribution  47  4.13 The basewidth dependence of the velocities of carriers exiting the quasineutral base, in the active region of device operation  48  4.14 The distribution function in the middle of the base of a BJT, for both the scattering models studied in this work  52  4.15 Comparison of the basewidth dependence of the carrier concentration profiles, in the active-mode, for the two scattering models used in this work  53  4.16 Comparison of the basewidth dependence of the collector current, in the active-mode, for the two scattering models used in this work  54  4.17 Comparison of the mean z-directed velocities, in the active-mode, for the two scattering models used in this work 5.1  55  One-dimensional transistor diagram illustrating the fluxes used in the currentbalancing approach  5.2  62  The coordinate dependence of the local diffusivity, normalized to the bulk diffusivity D , for two different basewidths 0  5.3  A comparison of the basewidth dependence of the active-mode current density, between a solution to the B T E and a new compact model  5.4  64  66  A comparison of the basewidth dependence of the active-mode electron concentrations at the base edges, between a solution to the B T E and new currentbalancing results  6.1  67  Conduction band diagram of an abrupt np heterojunction, illustrating the origin of some of the distribution function components referred to in the text.  6.2  Conduction band diagram of an abrupt np heterojunction, illustrating the energy levels referred to in the text  6.3  70  The distribution of carriers injected into the base from the emitter, of an abrupt-junction HBT, for four different applied biases  6.4  69  74  The distribution function, at the edges and middle of the base, for three different applied biases and a basewidth of one scattering length x  76  6.5  The active mode distribution function, at the middle of the base, for four different basewidths  6.6  77  The ballistic, reflected and collision components of the forward-going portion of the distribution function, for an HBT with basewidth equal to one scattering length, in the active-mode  6.7  79  The ^-dependence of the electron mean-free path in GaAs when the collision mechanisms are due to screened, ionized-impurities and polar optical phonons, as studied in this work  6.8  80  The carrier concentration profiles in an abrupt-junction HBT, at several different applied biases, for a basewidth of one scattering length  6.9  81  The carrier concentration profiles in an abrupt-junction HBT, for several different basewidths, for active-mode operation  82  6.10 The current density as a function of applied bias in an abrupt-junction HBT, for a basewidth of one scattering length  83  6.11 The basewidth dependence of the active-mode current density in an abruptjunction H B T  84  6.12 The anti-symmetric part of the distribution, at the emitter edge of the base, in an abrupt-junction HBT  85  6.13 The bias dependence of the mean z-directed velocity profile, in an abruptjunction HBT with a basewidth of one scattering length  86  6.14 The bias dependence of the mean z-directed velocity profile in the forwardgoing ensemble, for an abrupt-junction HBT with a basewidth of one scattering length  87  6.15 The bias dependence of the mean z-directed velocity profile in the backwardgoing ensemble, for an abrupt-junction HBT with a basewidth of one scattering length  89  6.16 The basewidth dependence of the mean z-directed velocity profile in the forward-going ensemble, for an abrupt-junction H B T in the active-mode.  . .  90  6.17 The basewidth dependence of the mean z-directed velocity profile in the backward-going ensemble, for an abrupt-junction HBT in the active-mode.  .  91  6.18 A comparison of the basewidth dependence of the base transit time between homojunction and abrupt-heterojunction bipolar transistors xi  92  7.1  An illustration used for the derivation of the collision component of the electron distribution  7.2  96  A schematic of the means by which ballistic carriers can come to be in the forward-going ensemble when an arbitrary potential energy profile is present.  7.3  98  A schematic of the means by which ballistic carriers can come to be in the backward-going ensemble when an arbitrary potential energy profile is present. 100  7.4  A schematic of the means by which carriers can come to be in the collision component of the forward-going ensemble when an arbitrary potential energy profile is present  7.5  101  A schematic of the means by which carriers can come to be in the collision component of the backward-going ensemble when an arbitrary potential energy profile is present  8.1  105  A schematic of the potential energy profile, and the boundary conditions for injected carriers, used to study the field-dependent solution to the transport equation  114  8.2  The projection of carrier trajectories onto the (z, /c )-plane  8.3  The function fo(z,k)  at equilibrium, for a field strength £ — 10 V/m  region of width W = 2 x Z 8.4  116  2  5  and a 117  sc  The total equilibrium distribution, along with its ballistic and collision components, at the middle of a region with field strength £ = 10 V/m and width 5  W = l  118  sc  8.5  The full forward-mode distribution, for a variety of injection levels, at the middle of a region with field strength £ = l0 V/m 5  8.6  sc  .  121  sc  The width dependence of the full forward-mode distribution, at the middle of a region with field strength £ = 10 V/m  122  5  8.8  The full reverse-mode distribution, for a variety of injection levels, at the middle of a region with field strength £ = 10 V/m and width W — 2x Z . 5  sc  8.9  119  The field-strength dependence of the full forward-mode distribution, at the middle of a region with width W = 2 x Z  8.7  and width W — 2 x Z .  .  123  The field-strength dependence of the full reverse-mode distribution, at the middle of a region of width W — 2 x l  sc  xii  125  8.10 The width dependence of the full reverse-mode distribution, at the middle of a region with field strength £ = 10 V/m  126  5  8.11 Dependence of the current on region width for a field of 10 V/m and forward5  mode flux of 10 times that at equilibrium  128  5  8.12 Dependence of the current on field for a region width of 2 x /  s c  and forward-  mode flux of 10 times that at equilibrium  129  5  8.13 The dependence of the mobility on field strength for a region width of W — 2x l  sc  for both the forward- and reverse- modes  133  8.14 The dependence of the mobility on region width for a field strength of 10 V/m. 134 5  8.15 The dependence of the mean kinetic energy on position for a region width of W — 2 x Z and field strength of 10 V/m, for both the forward- and reverse5  sc  modes  135  8.16 The antisymmetric distributions of the forward- and reverse- modes for a region width of W = 2 x l  137  sc  8.17 A schematic illustrating the energy dependence of the antisymmetric part of the distribution, at z = W, for the reverse-mode  xiii  138  Acknowledgments This work would not have been possible without the support, at every level, of my friend and supervisor, Professor David L. Pulfrey. The time spent under his supervision was most enjoyable and rewarding, and I truly hope that we have many more opportunities to work together in the future. If any part of his exemplary character has rubbed off on me, I know I am much better for it. The support and friendship of family, colleagues and friends have made the graduate student experience a very happy and satisfying part of my life. Many times people have opened their homes to me, and I shall carry the fond memories with me always. I would especially like to thank my parents, whose patience, love and generosity appear to be without end. Finally, I would like to thank all the members of my supervisory and examination committees. Their thoroughness and thoughtfulness are greatly appreciated. Special thanks are due to my external examiner, Professor Mark Lundstrom of Purdue University, for providing a careful and thorough review of my work, and for thought provoking and enlightening comments and questions.  xiv  Chapter 1 Introduction 1.1  Semiconductor device design, simulation and modeling  The purpose of this work is to extend the state of the art of semiconductor device simulation and modeling for device design. A key aspect of device design is to be able, given the material composition and geometric layout of a device structure, to predict the physical behavior of the device. This requires the device designer to understand why devices behave physically as they do, i.e., to have an understanding of the inner workings of device behavior not available from device measurement. To achieve this end, the device designer may have to undertake first-principles calculations based on the physics underlying device operation. On the other hand, at the designer's disposal may be a compact device model, based on closedform solutions to approximations of detailed device equations, which can be an invaluable aid in predicting performance trends and tradeoffs, as well as in understanding principles of device operation. Historically, semiconductor devices have been developed through experimental iteration. In modern device fabrication, however, excessive use of design iteration is very costly, and is an inefficient way of developing device technologies. The more such experimental iterations can be reduced by first checking proposed process changes against device simulation results, the more cost effective and timely will be the technological developments. However, if device designers are not confirming design strategies with appropriate simulation tools, they will not be able to properly focus fabrication efforts in the most likely direction for device optimization, and may even be following wrong design principles. Device simulation can be useful throughout the lifetime of a given technology. Early on,  1  2 it can be used to predict, quantitatively, the fundamental limits, and relative advantages or disadvantages, of competing technologies.  This can be an invaluable aid in assessing  which technologies should receive the larger investment of resources. Later, in developing prototype fabrication processes, device simulation can be used as a debugging aid. Finally, it can also be used to make suggestions for incrementally improving the device performance of an established technology. In the device simulation process itself, the detailed numerical simulations required to achieve physically accurate descriptions of device behavior may be time-consuming endeavors in themselves. In such instances, it may be desirable to appeal to a compact device model to do preliminary design work, and then confirm and fine-tune the designs with the detailed numerical simulations. Compact models are usually constructed from closed-form solutions to simplified device equations. Their simplified nature makes them ideal for providing insight into the principles of device operation, and often suggests strategies for improving device performance. Compact device models have another important use. They are used by circuit designers, in conjunction with overall circuit simulators, to design useful circuits. However, the circuit designer and the device designer don't necessarily have the same view of compact modeling. Consider the compact model for the current in an ideal diode: / = I e '' * v  v  Q  - 1,  (1.1)  where lo is the saturation current, V is the applied bias and Vth is the thermal voltage. The a  circuit designer is mainly concerned with finding a value for I that fits as closely as possible 0  the terminal behavior of the device, such as could be found from electrical measurement. The device designer, however, is more interested to know how I can be related to specific 0  material properties and device layout, such as junction area, material composition and doping profile. In this way, the device engineer can attempt to design devices with the electrical characteristics desired by the circuit designer. Traditionally, the analysis of semiconductor devices has been largely based on the solution  3 of the five "equations of state", also referred to as the Shockley equations: J„  =  qn riE + qD Vn  (1.2)  J  =  qfi pE-qDpVp  (1.3)  p  n  p  1  3YI  Si  n  =  ^.J„-/!  +  G  I =-^-J -« +G P  V-E  =  ? [(p - „ ) - ( J V - - A t f ) ] .  (1.4)  (1.5) (1.6)  Equations (1.2) and (1.3) are the electron and hole transport equations, also known as driftdiffusion equations(DDE); Equations (1.4) and (1.5) are the electron and hole continuity equations; and Equation (1.6) is Poisson's equation. Closed-form solutions of these equations, developed through the use of approximations, are widely used in the construction of compact models for the terminal characteristics of semiconductor devices. Perhaps the most striking trend in microelectronics has been the intense drive for ever increasing miniaturization. Device feature sizes in the 1960s were on the order of 20/xm. Today they are at the submicron level, and continuing to shrink. There are several very good motivations for pushing this trend, including more efficient use of wafer real-estate, larger scale integration, improved power consumption and increased device speed. This drive towards device miniaturization, however, calls into question the validity of the Shockley set of equations. Generally speaking, results from the Shockley set are questionable when carrier concentrations or electric fields change abruptly over distances comparable to a carrier mean-free path length[l]. Both of these situations can be expected to exist, for example, in modern bipolar transistors, due to both the scaling of device dimensions and the presence of high doping levels. In such devices, a more fundamental description of carrier transport is needed, and researchers have turned to the Boltzmann transport equation (BTE). The first four equations in the Shockley set can be derived by taking moments of this fundamental equation. The solution of the B T E , however, is no small task. The benchmark technique has been the Monte Carlo approach[2, 3], although it has its own set of problems: for example, it can be computationally intensive and suffers from statistical limitations. Consequently, alternative methods are a topic of current research, including the m-flux method of Das and Lundstrom[4, 5] wherein the potential profile is approximated as piecewise linear, a spherical harmonic ex-  4 pansion technique of arbitrarily high order[6, 7], a field-free approach employing isotropic and elastic scattering only [8], and an approach wherein only isotropic scattering is used and the potential profile is approximated as a series of steps[9]. In this work, compact models are sought which approximate device behavior as described by the B T E , rather than by the more traditional D D E analysis. Regarding the B T E , solutions are sought which satisfy three important characteristics.  Firstly, they are exact,  including no interpolation of potential profiles or truncation of equations, and include both anisotropic and inelastic scattering mechanisms. Second, the solutions are written in terms of components of the distribution function, and each of these components is shown to have a clear physical interpretation based on its origin: this aids in understanding certain features of device behavior. Thirdly, the numerical methods required are simple and straightforward to implement.  1.2  Thesis outline  Chapter 2 begins with a very brief discussion of the nature of carrier transport in semiconductors, including the specific forms for the dispersion relation, velocity and effective mass used throughout the remainder of this work. The Boltzmann transport equation is introduced, with a short discussion of the forms of the collision integrals. The general form of the B T E is then reduced to the one-dimensional, time-independent form which is the fundamental starting point for the transport analyses presented in subsequent chapters. The chapter ends with an overview of the solutions for both the field-free and field-dependent situations. In Chapter 3 the collision integrals, which are perhaps the most interesting and difficult aspect of the B T E , are considered in detail. The incoming and outgoing collision integrals are developed from standard expressions for the transition rates for the three scattering mechanisms considered in this thesis; namely, acoustic phonon scattering, polar optical phonon scattering, and screened, ionized-impurity scattering.  A novel feature of the work is the  demonstration that the incoming collision integrals for the screened, ionized-impurity and polar optical phonon mechanisms can be reduced to integrals over a single angle when the distribution function is of the one-dimensional form studied here. The incoming collision integral for acoustic phonons, and the lifetimes developed from the outgoing collision integrals for all three mechanisms, are the standard ones appearing in the literature.  5 Chapter 4 is concerned with the solution of the field-free B T E , and its application to base transport in homojunction, and graded heterojunction, bipolar transistors. A physical interpretation of the solution for the forward- and backward- going parts of the electron distribution is offered in terms of ballistic and scattered (or collision) components.  The  boundary conditions for carriers injected into the base region are taken from the theory of thermionic emission, and two different scattering models are considered. The first includes only the isotropic and elastic acoustic phonon mechanism, which is considered to be the dominant mechanism in low field, moderately doped silicon devices[10]. The second includes both polar optical phonons and screened ionized-impurities, which are considered to be the dominant mechanisms in gallium arsenide devices[10]. In Chapter 5 a compact collector current relation for modern, short base, bipolar transistors is developed which is consistent with the solution of the field-free, time-independent B T E . The mean velocities of carriers exiting the base, as determined from the B T E solution, are used to improve a recently proposed[ll] set of modified Schottky boundary conditions for thermionic emission at the junctions to the base. These junction currents are then employed, along with an expression for the diffusion current in the base, in a current balancing approach to develop an analytical expression for the collector current. The diffusivity used in the base transport model is not the usual bulk diffusivity, but rather a basewidth dependent value defined so as to make the current evaluated using the B T E consistent with a Fick's law form. The resulting compact expression provides a good illustration of how the physics of a detailed treatment can be distilled into a tractable, engineering model[l3]. In Chapter 6 the field-free, time-independent B T E is applied to base transport in abruptjunction HBTs. The presence of a conduction band spike at the emitter side of the base requires that the thermionic injection boundary condition be augmented to include both tunneling through, and reflection from, the spike. A physical interpretation of the resulting solution is offered in terms of ballistic, scattered and reflected components of the electron distribution. The base transport in this case is shown to be different than in the homojunction situation. Compact models for both the collector current and base transit time are suggested. In Chapter 7, an alternate derivation of the field-free, time-independent solution to the B T E is presented using a kinetic approach to transport. This kinetic approach, based on the concept of path integrals, is then applied to the field-dependent transport problem. The  6 key difference between the field-free and field-dependent situations is that in the latter an electron's wave vector becomes a function of position due to its acceleration by the electric field. The approach is first applied to the situation of a general potential energy profile and collision integrals. A physical interpretation of the resulting expressions is provided. The general result is then reduced to a situation in which the collision integrals are taken to include only acoustic phonon scattering, and the electric field to be constant. In Chapter 8, the field-dependent solution is explored in detail for the case of a constant electric field and acoustic phonon scattering. The boundary conditions are taken to be the same as in the field-free situation studied in Chapter 4. A comparison is made with the predictions of the DDE, and it is found that using these boundary conditions can lead to nonphysical predictions by the DDE. A study of the carrier mobility is undertaken in which the issue of the nature of mobility in a forward-biased barrier is addressed. The work of others on this topic is discussed and it is concluded that some of the contradictions therein can be explained by showing that the mobility is a function of only the antisymmetric part of the distribution. A summary and conclusions are presented in Chapter 9, along with suggestions for future work. There are two appendices: Appendix A describes the procedures for evaluating properties of distribution functions; Appendix B describes how to derive the one-dimensional drift-diffusion equation by taking a moment of the Boltzmann transport equation.  Chapter 2 Transport in semiconductors and the Boltzmann transport equation In this chapter, some of the key features of electron transport in semiconductors are introduced. The discussion is restricted to those concepts which are important to this work. The basic precepts of the semiclassical model are presented, leading to the introduction of the Boltzmann transport equation. The B T E is reduced to a one-dimensional, time-independent form, which is the fundamental starting point for the analysis contained in later chapters. Overviews of the solution to this equation are presented, for both the field-free and fielddependent situations, which are developed in detail later in the work.  2.1  Electron transport in semiconductors: dispersion relation, effective mass and collisions  A survey of reference texts[10, 14, 15, 16] covering the theory of semiconductor devices shows that this theory is founded on that of quantum and statistical mechanics. In this work, the important results for understanding the nature of electron transport in semiconductors are accepted as postulates.  Basically, the electrons are treated as a dilute gas existing in a  weak periodic potential arising from a host crystal's ion cores. The individual electrons are described by their wave vector k, and obey an energy dispersion relation, E[k). The electrons move unimpeded through a perfectly periodic array of ion cores with a velocity described by v(k) = lv- E(k), k  (2.1)  and react to external forces in accordance with Newton's second law if the electronic mass  7  8  is replaced by an effective mass prescribed by  -l  d E(k) 2  m  h  where h is the reduced Planck constant.  (2.2)  dkdk  The potential inside a semiconductor crystal is  not perfectly periodic, however, and as a consequence an electron's classical trajectory is occasionally interrupted by what are considered to be local and instantaneous collisions which change the electron's wave vector. These perturbations of the periodic potential can arise due to the thermal agitation of the crystal lattice, crystal defects, the presence of other carriers or the controlled introduction of dopants. The collisions are considered to be effective in keeping electrons close to the minima of the dispersion relation, which is considered to be parabolic in shape, so that carriers can be approximated as having a scalar effective mass; i.e., E{k) =  2 1.2  (2.3)  2m '  with a velocity prescribed by v(k)  hk  (2.4)  m  Throughout this work, when the concept of particle mass is referred to, it is always to the effective mass.  2.2  The Boltzmann transport equation  The equation which governs the semiclassical picture of transport is the Boltzmann transport equation (BTE): df(f,k,t) dt  df(f,k,t)~  r  + v • V / ( f , k\ t) + j • V -/(f, k\ t) f  fc  dt  (2.5) coll  This is a continuity equation for the six-dimensional phase space defined by (r, k). The distribution function /(f, k, t) is the probability of occupancy of a state with wave vector k at a position r and time t. The left hand side of the equation is a total convective derivative, where the second term corresponds to the change of / due to particle streaming and the third term corresponds to the change of / due to the presence of an external force, F, on the  9 carriers. The right hand side represents the total rate of change in the distribution function due to collisions. In detail the right hand side can be written in terms of collision integrals:  J coll  n  i  (2.6) where the distribution function arguments (f, t) have been suppressed for clarity, the superscript indices  identify bands (or valleys), the subscript index n refers to the n  th  scattering mechanism and S J(k,k') is the rate of transition for carriers occupying a state l  with wave vector k to an empty state with wave vector k!. The scattering mechanisms considered in this work are intravalley only, in which case the collision term can be written as: df(k) dt  "(2^ /  {/(^[i-/^')]^^')-/^')!!-/^)]^'^)}^',  (2.7)  coll  where the sum over n mechanisms is implied. The incoming and outgoing collision integrals are identified as: ^ —  (--out  C-  jf f(r, k, t)S(k, k') [l - /(r, k', t) dk'  (2.8)  J  (2.9)  /(f, k', t)S(k', k) [l - /(f, k, t) dk'.  In this work, nondegenerate conditions are taken to apply. However, the methods developed herein are not restricted to this situation. In the nondegenerate limit the collision integrals become: O C  =  i n  ~y;  ^f(r,k,t)S(k,k')dk'  =  f(r,k',t)S(k',k)dk:.  (2.10) (2.11)  The outgoing collision integral can now be written in terms of the carrier lifetime r(k): Cout  J  =  fif,k,t) r(k)  f{rXt)S(k,k')dk'  (2.12)  where  r(k)  (2  (2.13)  10 Using the forms for the collision integrals developed above, the Boltzmann transport equation can be written in the following form:  (2.14)  2.3  One-dimensional, time-independent transport relation  The main effort of this work is to solve a one-dimensional, time-independent form of (2.14). This equation can be written out in detail in the following manner: dt  h dk  h dk  x  h dk  v  dx  x z  y  dy  z  dz  r  (2.15)  The problem is reduced to one spatial dimension by considering a distribution which is inhomogeneous in the z-direction only. Further, any external forces are considered to exist in the z-direction only. In the time-independent situation, then, equation (2.15) reduces to:  z  dz  (2.16)  h dk  T  z  Since the distribution is assumed inhomogeneous in the z-direction only, a solution of the following form is sought: f(r,k) = f(z,k,9), where 9 is the angle between k and the z-axis (see Figure 2.1).  (2.17) With this form for the  k  Figure 2.1: A simple illustration indicating the definition of 6. distribution function, for the particular scattering mechanisms important to this work, it is shown later that the incoming collision integral can be reduced to the following form: C (z, k, 6) = ^ f(z, k 9') • g(k, 9,9') • dff\ Jo in  u  (2.18)  11  where ki — k for elastic scattering mechanisms, or ki = k ± Ak for inelastic scattering mechanisms. The one-dimensional, time-independent transport equation which is solved in the remainder of this work can now be written down as: df{z,k,9) Vz 7j-  2.4  F df{z,k,9) 1- -jr z  f(z,k,9) 1 -  r = J }{z,ki,9) • g{k,9,9) • d9 .  (2.19)  Overview of the solution  In solving the one-dimensional, time-independent form of the B T E developed above, the nature of the boundary conditions on the distribution function leads to the need to write the solution in terms of its forward- and backward- going parts. The result is a set of coupled, implicit, integral relations. The implicit nature arises due to the need to perform a spatial integration over the incoming collision integral. In the field-free situation, for example, the relations are O<0<TT/2:  f {z,k,9)  = /+(0, k, 9)e~ ' + e~ '  n/2<9<n:  f~{z, k, 9)  = f~{W, k, 9)e ~ '  +  z/v  w  T  z/v  z/v  T  T  f e ' * —dz' Jo Vz  + e~ >  z  zlv  T  /v  T  (2.20)  f e ' * —dz'. (2.21) Jw v z  /v  T  z  These two coupled, implicit, integral relations for the forward- and backward- going components of the distribution function are equivalent to the time-independent, field-free B T E . Their implicit nature suggests an iterative approach to obtaining their solution; e.g., for the forward-going component, f+(z, k, 9) = f (0, k, 9)e~ l * + e~ ' +  z  v  T  z/v  [' e ' * ^^-dz', Jo v  T  z  /v  T  (2.22)  z  where the subscript m refers to the mth iteration, and the incoming collision integral is evaluated by using results from the (m — l)th iteration: Cin.m-1 = f' fm-x{z, k, 9') • g{k, 9, 9') • dff'. Jo  (2.23)  Similarly, for the backward-going component, f~(z, k, 9) = f-(W, k, 9)e - > w  z/v  T  + e~ ^ z  T  f ' ' ^^dz'. JW z z  /v  T  e  (2.24)  V  Note that, because (2.20) and (2.21) are coupled by the incoming collision integral, they must be solved simultaneously. As will become apparent later in this work, the relations developed for the field-dependent situation, which are somewhat more complicated although similar in form, may be solved in the same iterative fashion described above.  Chapter 3 Collision integrals and lifetimes In this chapter, expressions for the carrier lifetimes and collision integrals are developed. Three different electron scattering mechanisms are considered: nonpolar acoustic phonon scattering, polar optical phonon scattering and screened, ionized-impurity scattering. The former is considered to be the dominant mechanism in moderately doped, low-field silicon, while the latter two are considered to be the dominant mechanisms in highly doped gallium arsenide.  The starting point for the work here is the transition rate, S(k,k'),  for which  standard expressions are taken. It is shown that, for the one-dimensional problem studied in this work, the incoming collision integral can be reduced to an integration over a single angle, resulting in new expressions for both screened ionized-impurity and polar optical phonon scattering. These forms for the nondegenerate incoming collision integrals are specifically tailored for use in the transport equation developed in Section 2.3.  The results for the  lifetimes, and the incoming collision integral for acoustic phonons, are the standard ones appearing in the literature.  12  13  3.1  Screened ionized-impurity scattering  In this section the collision integrals and lifetime for screened, ionized-impurity scattering are developed. The transition rate used here is that of the Brooks-Herring approach. This approach is considered to provide the best results when screening is important[17], as can be expected in the heavily doped, quasi-neutral GaAs regions to which this mechanism is applied in this work.  3.1.1  Transition rate for impurity scattering  The Brooks-Herring approach provides the following result for the transition rate due to a screened, ionized impurity[18]: S%  k>) = \  —  £  ^6{E  - Eg),  P  (3.1)  6* [ 2 F ( 1 - C O S % , ) + T4 where Lp is the screening length given by:  (3-2)  i = J*f, D  V qn where n is the density of carriers with polarity opposite to that of the ionized impurities, ke is the Boltzmann constant, T is the temperature, e is the permittivity, q is the electronic charge and E is the electron energy. k  3.1.2  Incoming collision integral for impurity scattering  Using (3.1) in (2.11) gives the following expression for the incoming collision integral due to screened, ionized impurities: cl(z,k,e)  =  -A^J ( ,k',e')-N -s\k',k)dk' f  1  f(z,k',e')6{E- -E )  imp  i f W J  imp  [2nq N  J =  z  A  w r  k  J ,  k  r ^  p^ J  0  2  ( 1  _  c o s  )  +  ?  ^i  2  fMMjEt-Et)  i m P  he*  ^  £l  k  [ if (l-cos^ -) + 2  2  f c  « .  u  m  ,  d  e  ,  d  k  ,  ^] (3.3)  where the integration overfc-spacehas been written in spherical coordinates. Note that the usual procedure[10, 18] of multiplying the transition rate for a single impurity by the density  14  Figure 3.1: Coordinate systems used for evaluating the incoming collision integral. The primed coordinate system is chosen to be the same as the unprimed coordinates. of impurities,  has been followed here as well. In order to carry out the integration,  iV p, i m  the primed coordinates are chosen to be aligned with the unprimed coordinates, as shown in Figure 3.1, so a relation for cos0£,£ is needed. From vector calculus:  C  0  S  ^  =  ^ F '  ( 3 > 4 )  and from the definition of the spherical coordinates: k — A; sin 0 cos < > / k = k sin 9 sin cj> x  y  k = k cos 9,  (3.5)  x  so that: cos  =  (sin 9 cos (f>, sin 9 sin <f>, cos 9) • (sin 9' cos ft, sin 9' sin <f>', cos 9')  =  sin 9 cos 4> sin 9' cos ft + sin 9 sin (j> sin 9' sin ft + cos 9 cos 9'  — sin 9 sin 9' [cos <j) cos ft + sin <j> sin 0 ' ] + cos 9 cos 0' =  sin9sin0'[cos(^ -(/>')] + cos9cos9'.  (3.6)  Now, since the distribution function has no 0-dependence (recall it has no (^-dependence by definition of the one-dimensional solution), it is allowable to align the coordinate systems such that (j) — 0. Then cos 0£,£ = sin 9 sin 9' cos ft + cos 9 cos 6'  (3.7)  15  and  (2TT)  3  27rg JV  f2lT  7T  C} =  f{z, k', 9')5 (E -  4  imp  k' sin&d<j>'d&dk' 2  s  l2"  he  2  >k' JO JO  + yV  2k' (1 — sin 9 sin 6' cos <j)' — cos 9 cos 9') 2  (3-8) It is useful to employ the scaling property of the delta function, x  ,  W  X  (3.9)  to write (3.10) and make the transformation X = k . Then 2  Cl  n  f(z, s/Y', 9')5 {X' - X) sin  p2lT  7T  a-imp  =  Jk'Jo Jo  ffdtjtdff  \[X>  2X> (1 - sin 9 sin 9' cos ft - cos 0 cos 9') + ^ DJ  7T  2  7T  ttimn  7  7  n  1  0  f (z,k, 9') sin 9'd<p'd9'  /»27r  2A; (1 - sin 6> sin 0' cos <j>' - cos 0 cos 6') + yV 2  0  /(z,/v,0')sin0'#'d0'  27T  2 '  (- sin 9 sin 0')2A; cos </>' + ( ! - cos0 cos 9')2k + yjr 2  2  (3.11) where _ q miVimp ~ 27re fc ' 4  i  m  p  2  (3.12)  3  and the sifting property of the delta function, rb / f(x)S(x-x )dx Ja 0  (3.13)  = f(x ), 0  has been used. Making use of the tabulated integral, r 2 v  Jo  dx  2na  (a-+ 6cosz)  2  (3.14)  (a -6 ) / ' 2  2  3  2  then gives: ni  _  Q  i-P  L  1  27T • [(1 - cosflcosfl'^ + L7, ] sin 2  2  W 3/2  *  1  Jo  [(1 — cos 0 cos 0') 2 A; + L^] 2  — [(sin#sin#')2A; ] 2  [(1 - cos 9 cosfl')2/v + L~ ] sin 9'd9' 2  r f(z,k,9') Jo  2  D  3/2  -  [(1 - cos 0 cos 9>)2A; + L~ f - [(sin0sin0')2A; ] 2  2  2  2  (3.15)  16 With reference to (2.18) it can be seen that aimp • * • [ ( ! - cos 9 cos 6')2k + L^ ] sin 9' 2  g {k,9,9') =  2  -,3/2'  imp  [(1 - cos0cos0')2/c + L ~ ] 2  2  2  (3.16)  [(sin0sin0')2& ] 2  Equation (3.15) is the new form for the incoming collision integral due to screened, ionizedimpurity scattering.  3.1.3  O u t g o i n g collision integral a n d lifetime for i m p u r i t y scattering  Following an analysis similar to that for the incoming collision integral, using (3.1) in (2.10) gives the following for the outgoing collision integral: CL  =  ^^JN .S\k,k')dk' imp  f{z,k,o)  r  r  (2TT)3  J ,J k  0  r^^Kq^p I  KEp - Eg)  k' sin 9'd<p'd9'dk',  ;  He  2  2 ^ ( l - c o s ^ , ) + 7V (3.17)  where the integration over &-space has been written in spherical coordinates. Now, if one chooses a coordinate system for the integration variables that has z' lined up along k as in Figure 3.2, then 9^, becomes the polar angle in these coordinates. Then f( ,k,9) z  (2TT)3  r JJ  0  r 7r rp2ir  - * - k' sinm'd9'dk' 2/c (l -cos0') + 4 Dl E  he  7O  ]  2  2  2  2  (3.18)  Following the same procedures as for evaluating the incoming collision integral then provides the following: q mk  2L%  2ne H  4/v + TV  A  CL  =  f(k)  2  3  2  (3.19)  Comparing with (2.12), the lifetime for screened, ionized-impurity scattering can now be identified as the well accepted result[18]: _27re h r = q mk 2  1  A  3  4fc2  + 4 2L  2 D  (3.20)  17 x  x  z y  Figure 3.2: Coordinate systems used for evaluating the outgoing collision integral. The primed coordinate system is rotated about the y-axis by 6 with respect to the unprimed coordinates.  3.2  Phonon scattering  In this section the collision integrals and lifetimes for phonon scattering are developed. In developing the lifetimes, the usual procedure of integrating over the phonon wave vector, 8, is followed. In developing the incoming collision integrals, however, it is more convenient, for the purposes of this work, to carry out the integration in A;-space, i.e., over the electron wave vector.  3.2.1  G e n e r a l considerations a n d transition rates for p h o n o n scattering  The transition rates for phonon absorption and emission can be written as[18]: S (k, k') = a  *\Kp\  p  n  pu(8)  6[E - E , + hw{B)) k  k  (3.21)  where (3.22)  k' = k + P  and S (k,k') e  =  poj(B)  6[E - E , - hu(p)} k  k  (3.23)  18 where k' = k-j$.  (3.24)  Kp is a parameter which depends on the phonon scattering mechanism under consideration, p is the material mass density, ns is the phonon occupation number and to (8) is the phonon dispersion relation. The usual practice of taking the phonon occupation number to be the equilibrium value is followed here, i.e., 1  n'/3  (3.25)  hw(p)/k T _ i •  5  e  B  The phonon dispersion relation for acoustic phonons is taken to be to (/?) = v,a,  (3.26)  where v is the velocity of sound, while that for optical phonons is taken to be s  oo (8) = oo .  (3.27)  0  This is the usual practice in the analysis of semiconductor devices. The phonon occupancy can thus be written as 1 hv l3/k T _ I  a p e  s  (3.28)  B  for acoustic phonons and "°P  (3.29)  hw /k T _ i  _ e  0  B  for optical phonons. Further, the electron-phonon collisions are taken to obey a conservation of crystal momentum rule, as shown in Figure 3.3, as well as a conservation of energy rule. Conservation of energy requires that, for absorption,  £ , = £; + M/?),  ( - °)  E- , = E- -hw(8).  (3.31)  £  3  £  3  and for emission, k  k  With reference to Figure (3.3), the law of cosines for conservation of crystal momentum can be written as: B = k' + k + 2k'k cos e . 2  2  2  w  (3.32)  19  Figure 3.3: Momentum conservation in phonon absorption and emission. For phonon absorption, k + (3 = k , and for phonon emission k — (3 — k'. 1  3.2.2  I n c o m i n g collision integral for polar optical p h o n o n scattering  In developing the incoming collision integral for phonon scattering, the processes of absorption and emission are treated separately, and then summed to find the total result, i.e., Cm =  (3.33)  With reference to (2.11) one can write: Cl  =  Cl  =  j  S\k\k)f(z,k!,ff)dk'  (3.34)  j^y ^S (k',k)f(z,k',e')dk'.  (3.35)  e  3  Swapping k and k! in (3.21), substituting into (3.34) and going over into spherical coordinates gives "n = 7^- (Any p J  /  C  3  J  0  /  0  Jo  -^^f{z,k\e')5[E ,-E u {[)) k  + hto{(3)]k' sme'd j>'de'dk . (3.36) 2  k  l  (  Similarly, swapping k and k' in (3.23), substituting into (3.35) and going over into spherical coordinates gives d  = - i - - / (In)* p Jo  /  /  Jo Jo  \ ' u (P)  '-f(z,k\e')8[E ,  - E  k  - hw(S3)]k' m6'd4>'de'dk'. 2  k  S  (3.37)  For polar optical phonon scattering the parameter Kp is taken to be[18] e  2  ™  =  qup 2  L  7^~W  2  _  rq u 2  t  2 oP  ~ ~W~"  where r = ei,/(e e), CL is the contribution of the lattice to the permittivity and e  00  contribution of the electrons (e = e + too). L  ( 3  -  3 8 )  is the  20 The usual practice for evaluating the incoming collision integrals due to phonon scattering is to note that the conservation of crystal momentum requires a one-to-one correspondence between k' and 3, and then to integrate over 3 rather than A? [18]. However, for the purposes of this work, it is more convenient to integrate over k!. As was the case for the incoming collision integral due to impurity scattering, a primed coordinate system aligned with the unprimed coordinates is chosen (see Figure 3.1).  Furthermore, the unprimed coordinate  system is once again chosen so that A; has a ^-component of zero. For polar optical phonon absorption, using (3.38), (3.29) and (3.27) in (3.36) gives 1  £i „' ° a  P  P  /*7r  poo  /  /*27T  =  7A3- /  /  =  T—^Trr^nop / 1.271") J  1  2  2  ^^n f(z,k\9')5{E -E ov  fOO  PIT  / J  0  kl  P2TT  / J  0  rl  M  0  U  '  p  + h o ]k' sm9'd<j d9'dk' 2  k  J  ,  0  )  Ql\  '  ;  5[E , -E k  + hw ]k' sin B'dftdQ'dk'. 2  k  0  (3.39) Using the scaling property, (3.9), the delta function can be written as 5[E , -E k  + hw ] = -^S[k' - k + —u ], 2  k  (3.40)  2  0  0  and using the conservation of crystal momentum relation for 3 , (3.32), gives 2  , a , p o C  -  1  P  P  =  (  2m [»  2  2  ^  -  W  W  o  r  f * f(z,k',9')5[k' -k 2  Jo Jo  2  k'  2 +  k*  + fuj )  2  0  u2  2k>kcos9 ,  +  .  2  "  kk  ™  e  d  *  d  (3.41)  6  d  k  Applying the transformation X — k , and using the sifting property of the delta func2  tion, (3.13): ia,pop  1  0  1  2  2m  ffl Jo  Jo 2m  2 Jo Jo 1  Jo  x - -fu  2  2m k? f  0  v  2  + X + 2^/X - -f  2  [* 2  u VX  f(z,kf,9')  0  cos6 , kk  . (3.42)  21 where  ^w .  kf = \lk 2  (3.43)  0  n  The choice of coordinates being used here allows (3.7) to be substituted for cos9 :, giving: kk  2m kf  1  ^a.pop  rn  flit  /(z,kf, 9') sin ffd^dff 2k - IfuJo + 2kfk(sin 6 sin 6' cos $ + cos 9 cos 9') 2  r q co n 2  e  0  op  2m kf r*7T /•27T  FS  f (z,kf, 9') sin 9'd<j)'d9' [2k - ?fu) + 2kfk cos6 cos 9'] + 2kfk sin 9 sin 9' cos <}>' 2  0  Jo Jo  (3.44)  Using the tabulated integral, p2n Jo  dx  2TT  + b cos x  a  (3.45)  y/a - 6 2  2  gives •7a,pop •'in  ^pop^op^  JO  f(z,kf,8')sin ffdff -l 1 / 2 ' (2k - 2 f w + 2Jfcf A; cos 9 cos 0') - 4/cf /v sin 9 sin 0' 2  2  2  2  2  2  0  (3.46) where r mq u e  a p o p  =  (3.47)  0  4nh  2  '  Note that conservation of energy requires that for k < y ^ f ^ o , C ^  p o p  = 0. That is, if an  electron is to absorb a phonon, it cannot end up with an energy less than that of the phonon it absorbs. Comparing with (2.18) it can be seen that ct o n pkf sin & P  flpopCM.*')  (2k - ?fu  2  0  k>  W  0  + 2kfk cos 9 cos 9') - ikfk  2  if  P  2  -i 1 / 2 ' sin 9 sin 9' 2  2  yW ,  0, otherwise.  0  (3.48)  22  Turning now to phonon emission, using (3.38), (3.29) and (3.27) in (3.37) gives C£P°P  =  - i - - / pJ  /  0  op  (2TT)  3  -^^(n + Pu OO /»7T r2TT  2  kl  k  0  (  0  r 11  i)  +  l)f(z,k',e')S[E -E -}kv }k' sine'd j>'de'dk'  o p  2  JO  0  TTr q'uj (n e  /  J  0  Jo  P  2  Jo Jo  (3.49)  Using the scaling property, (3.9), the delta function can be written as 6[E , - E k  - Ku ) = ^5[k'  2  k  0  - k -  (3.50)  ^u ],  2  0  and using the conservation of crystal momentum relation for (3 , (3.32), gives 2  r-e,pop in  _  u  eQ  2m r°° r  U  7rr  —  0  V op •  r* 2  l  fr>_\?.  (27T)  ~hFJo  k' + k + 2k'kcos6 , 2  Jo Jo  }  2  kk  (3.51) Applying the transformation X = k , and using the sifting property of the delta func2  tion, (3.13): e, o  Cl  P  P  _ _  ^ q uj  ,  2  e  0  (27T)3 ^ ° P  1  '  _  nr g u) 2  -^-(n €  2m  n  m  +  0  o p  1  H  2  X  'X'  2  ~2  cos 6 i kk  2m  +l)^x v  j  2 n  s  + fuj f(z^x  + fu, ,e')  2  x  2  0  0  sine'd<j>'de' 2 Jo Jo  7rr, q u T2 2  0  X  ,^2mkt  + ?f L0 VX  + ?f U + X + 2y/X 0  o  cos 6 , kk  f(z,klff)  f  sniff dtfdff,  2k + fcu + 2k*kcos6 2  2  0  kkl  (3.52)  where (3.53) Once again the choice of coordinates being used allows (3.7) to be substituted for cose >, kk  23  giving: --  2m k\  2  ~  C i n  (2TT)3  I N  °  P  +  r  v  2  2  2,r  I  JO  2  h  J  /-  I  8TT  I  X  f (z, k°, 9') sin 9'dftdff 0 7„2 i 2 m , 2/c + ?f w + 2fc?fc(sin 6 sin 0' cos ^ + cos 9 cos 0') 2  •/ 0  0  ' n 2  o p  2  f(z, k°, 9') sin 9'dcj)'d9'  77  [2k + yw,, + 2kfk cos 0 cos 0'] + 2kfk sin 0 sin 0' cos 2  (3.54) Using (3.45) and (3.47) then gives C°*  op  =  /(z,/v?,0')sin 9'd&  apapCnop + l ) * ? / ' Jo  -i 1/2 '  (2& + ?f w„ + 2jfc?& cos 0 cos 0') - 4A;f A; sin 0 sin 0' 2  2  2  2  2  2  (3.55) Comparing with (2.18) it can be seen that *,(*,»,«•)  =  •w.v. + i w - y  T  (  3 5  6  )  (2A: + ?f w + 2fc?Jfc cos 9 cos 0') - 4kfk sin 0 sin 0' 2  2  2  2  2  0  Equations (3.46) and (3.55) are the new forms for the incoming collision integrals due to scattering by polar optical phonons.  3.2.3  Outgoing collision integral and lifetime for polar optical phonon scattering  As mentioned earlier, due to the one-to-one correspondence between k' and 8, one is at liberty to integrate over 8 rather than A;'. This is the usual practice, and substituting (3.21) and (3.23) into (2.10) leads to the following results for the outgoing collision integral due to phonon scattering:  24 Using (3.27), (3.29), and (3.38), and employing a set of limits on 8 derived from the conservation of crystal momentum and conservation of energy relations, the following results for polar optical phonons are achieved:  Note that for E < frco , i-e., k  k < ^/2mco /h, C^° = 0. That is, a carrier must p  0  0  have an energy greater than the optical phonon in order for emission to take place. Further, in the limit that k goes to zero, it can be shown that C ^  O P  = 2a [e  -  hWolkBT  pop  l)- f(z,k,e)/^2rnuj /h. 1  0  Comparing with (2.12) allows the lifetime to be written as the well-accepted result[18]: 1  _  3.2.4  ,  1  / yjl + fk0 /E 0  + 1\  k  ( 1 + y/1 -  Hu> /E 0  k  (3.61)  Collision integrals for acoustic phonon scattering  For nonpolar acoustic phonon scattering the parameter Kp is taken to be[18] \Kp\ = 8 D , 2  2  (3.62)  2  where D is the so-called deformation potential. In the analysis of acoustic phonons, the usual practice is to make use of the following approximations: hoo <C k T and v ^> v . The B  s  transition rates (3.21) and (3.23) then become S (k, k') = S (k, k') = * f6[E e  a  - EH],  D2k  2  K  (3.63)  and the incoming collision integral can be written as 1  ^  =  <7T fi 2  2  - ^ - ( 2 . ^ ) — / m  r°°  Jo  f  2lX  /  9m  /  f(z,k',6') -^6\k' -k }k' ine'dk'de'dcj>' Z  poo PIT / / f(z,k',e')6[k' Jo Jo  2  2  n  - k )k' sm6'dk'dO'dcf)',  2  2  S  Jo Jo 2  2  (3.64)  where _ a p  mD k T 2  B  irpv h 2  3  (3.65)  25 Making the transformation X = k , one can then write: 2  j = =  f(z,^/X ,e)8[X' J  X]^j-dX'sme'de'  a k\ j f(z,k,e)sme'de' 2 Jo oi kfo{z,k), &p  &v  (3.66)  where /o(z, A:) can be viewed as the distribution of carriers according to wave vector magnitude. Following a similar procedure for the outgoing collision integral, one finds Ct  =  a* kf(z,k,6), P  (3.67)  from which the lifetime is readily identified as r  a p  = —!—.  (3.68)  These forms for the collision integrals are equivalent to the isotropic and elastic forms used in [8].  Chapter 4 Application to base transport in BJTs and graded-junction HBTs In this chapter, a solution to the field-free, time-independent, one-dimensional transport equation is developed. The resulting coupled, implicit, integral relations for the forwardand backward- going ensembles are distinguished from others appearing in the literature by the presence of general forms for both the incoming collision integral and the boundary conditions for the distribution function. A physical interpretation of the solution, in terms of the ballistic and scattered components, is presented. Thermionic emission boundary conditions are applied to describe the distributions of carriers injected into the base region of a BJT. The results are expected to apply equally well to graded-junction HBTs. Results are generated for two different scattering models. The first includes only the isotropic and elastic acoustic phonon mechanism, which is the dominant scattering process in low-field, moderately doped, silicon. In this situation, considerable analytic progress can be made, and the results are related to similar work done by others. Results are generated for the distribution function, current and carrier densities, and carrier mean velocities. In the second, more detailed, scattering model, the mechanisms are those appropriate for gallium arsenide; namely, screened ionized-impurities and polar optical phonons, where the incoming collision integrals are described by the new equations derived in the previous chapter. A comparison is made between the results for the two scattering models.  26  27  4.1  Solution o fthe transport  4.1.1  field-free,  time-independent  relation  Development o ft h e general result  In the field-free situation, the one-dimensional B T E developed in Section 2.3 reduces to the following equation:  f  l-  +  dz  = ^ .  (4.1)  v  VT  z  Z  Since this is a linear, first order, ordinary differential equation describing the coordinate dependence of the distribution function for each value of the parameters k and 9, one can make use of an integrating factor to find its solution. That is,  J + P(z)y = Q(z)  (4.2)  has the general solution y = e - Z W * J eS ^ Q{z)dz p  + ce~ S ^  dz  p  d z  ,  (4.3)  where the constant of integration in the integral of P(z) is taken to be zero. Comparing (4.1) with (4.2) indicates that P(z) =  and Q(z) = C, /v , so the general solution for  (V T)~ Z  1  n  z  f(z,k, 9) can be written as f(z, k, 9) = ce~ * + e-* ' f z/v  T  /v  e * —dz.  T  z/v  T  JO  (4.4)  z  v  This result, in principle, requires a single boundary condition on the distribution function in order to eliminate the constant c. However, such a boundary condition must be valid for arbitrary 9, and in general such a boundary condition is not known. If, on the other hand, the equation is solved over two separate domains of 9, in particular 0 < 9 < ir/2 and 7r/2 < 9 < TT, then the distributions of carriers injected into the region under consideration (e.g., the base of a bipolar transistor, see Figure 4.1) from either end become suitable boundary conditions. These are a physically consistent set of boundary conditions[8], and, indeed, it is the usual practice to use such a set in analyzing transport in semiconductor devices[18]. Specifically, one has for 0 < 9 < ir/2 f (z, k, 9) = e-* * + e~ +  +  c  ,v  T  zlVzT  f e ' * —dz Jo z z  v  T  V  (4.5)  28 0  W  1  1  EMITTER  •.:  Forward  ••. '-I  >  Backward  '•. '•  <  1 ••. ••  BASE  ' SCR.:  6  -*  [ S C R . COLLECTOR  e  1  Figure 4.1: One-dimensional transistor diagram illustrating the forward and backward directions as referred to in the text. The emitter-base and collector-base space charge regions (SCR) are denoted by the dotted lines. and for n/2 < 6 <  IT  f~(z, k, 9) = c~e- ' z/v  + e~ ^  T  z  T  [* e * —dz. z/v  T  J o  (4.6)  z  V  Note the ranges of 9 are such that the plus superscript indicates the forward-going component of the distribution, and the minus superscript indicates the backward-going component, where forward-going refers to motion from the emitter towards the collector (see Figure 4.1). This is a convention used throughout this work. A "+" superscript refers to a component of the distribution function, or property derived therefrom, with 0 < 9 < n/2 (i.e., the forward-going ensemble). Similarly, a "—" superscript refers to a component of the distribution function, or property derived therefrom, with TT/2 < 9 < n (i.e., the backward-going ensemble). To eliminate c , the boundary condition for the flux injected at the left (e.g., emitter) +  side of the region, f (0,k,9), +  is employed. Similarly, c~ is eliminated with the boundary  condition for the flux entering from the right (e.g., collector) side of the region, f~(W, k, 9). Evaluating (4.5) at z = 0 and equating to / (0, k, 9) yields +  c = / (0,M), +  (4-7)  +  while evaluating (4.6) at z — W and equating to f~(W,k,6) yields - = /-(W,k,9)e * w/V  c  T  - / JO  e > ^dz. zlv  T  z  v  (4.8)  29  Substituting (4.7) and (4.8) back into (4.5) and (4.6), respectively, and noting  —  f* + $Y i yields the following results: f (z, k, 9) = / ( 0 , k, 9)e- * +  +  z/v  T  + [' - - 'V * ~dz', {z  z  v  (4.9)  T  e  z  v  JO  f~{z, k, 9) = f~(W, k, 9)e - V ~~ + /* -l'-*'V ' —dz'. (w  z  v  T  v  (4.10)  T  e  Vz  JW  These two integral relations are equivalent to the one-dimensional, time-independent transport relation. With reference to (2.18) it can be seen that they are coupled by the incoming collision integral appearing in the second term on their right hand sides. They differ from those developed in [8] in three important respects. Firstly they contain the general incoming collision integral rather than a simplified form appropriate for acoustic deformation potential scattering only. Secondly, the result for the backward going ensemble contains an additional term which occurs because of a better account of the collector boundary condition. This leads to a considerable difference in the results at low bias which will become evident in the work that follows. Thirdly, the boundary conditions, / ( 0 , k, 9) and f~(W, k, 9), appear +  in their most general form. This allows the solution to be extended to account for situations such as injection at an abrupt heterojunction, as is thoroughly examined in Chapter 6.  4.1.2  Physical interpretation of the field-free solution  Consider the solution for the forward-going component of the electron distribution, Equation (4.9). It is possible to give a physical interpretation of the two terms appearing in this equation. Clearly, if one were to consider out-scattering, but no in-scattering, then what would remain in the ensemble injected from the emitter (i.e., f (0,k,9)) +  at some point z  would be those electrons which have traveled to z ballistically. Setting the incoming collision integral to zero in the equation above would thus tell one what portion of the forward-going ensemble corresponds to this ballistic component, i.e., f£ (z,k,9)  = f (0,k,9)e- / * . +  AL  z  v  (4.11)  T  This interpretation implies that the second term in (4.9) corresponds to those forward-going carriers at the point z which have undergone a collision event since being injected into the base; they are referred to here as the collision component of the distribution, i.e., f+ (z,k,9)= OL  r  /  JO  z  e^ - '^C (z',k,9)—. z  z  Hz'  in  z  V  (4.12)  30 ^  COLLECTOR^  BASE  EMITTER  Ballistic Component  -i=H  A VCollision Component W  zi  Figure 4.2: A schematic of the means by which carriers can come to be in the forward-going part of the distribution at some some point z in a field-free base. A carrier may travel to z ballistically after injection at the boundary, or, a carrier may suffer a collision into a forwardgoing state somewhere to the left of z, at z\ say, then travel to z without suffering additional collisions. Note that the collision referred to here is the most recent collision suffered by the carrier before reaching z, and that the carrier in general may have suffered many previous collisions since being injected into the base. These ideas are illustrated in Figure 4.2, which shows schematically the two ways by which a carrier can come to be in the forward-going ensemble at z. The carrier can travel to z directly after being injected at the left boundary without suffering a collision, or it can suffer a collision into a forward-going state and then travel to z without further collisions. Note that a carrier in the collision component may have undergone many collisions before the most recent one preceding its arrival at z. Similarly, for the backward-going component, it is evident that corresponding expressions can be written for those carriers which have reached z ballistically after being injected from the collector, and for those carriers in the backward-going ensemble that have reached z after having undergone a collision, i.e., W * . , °) = f'{W, k, e)eW-'V"k  (4.13)  r  and  f  z  f- (z,k,6)= ot  JW  E  -(*-^C (z',M) i n  dz' — * v  •  (4-14)  31  4.1.3  B o u n d a r y c o n d i t i o n s for B J T s a n d g r a d e d - j u n c t i o n  HBTs  As described in [8] for BJTs, the Bethe theory[19] of thermionic emission can be used to generate the boundary conditions, / ( 0 , k,0) and f~(W,k,0),  needed to solve (4.9) and (4.10).  +  The results are also directly applicable to HBTs constructed with sufficient compositional grading near the metallurgical junction. The results are: /  +  ° '  (  M  )  /-WM)  =  f-f  =  f}  (4-15)  E k / k B T  e - E t / k B r  .  (4-i6)  ^  ( )  where, n*  E  =  nEe  -i(y^- BE)/kBT  n*  c  =  N C E  v  417  -9(Vbi.c-v c)/k r B  (  B  41g  )  N is the effective density of states in the conduction band given by c  1 /2mk T\  3 / 2  B  '  N  =  i { i £ - )  •  ( 4  '  1 9 )  and nc are the electron concentrations in the quasi-neutral emitter and collector, respectively, V B E and V B C  a  r  e  the applied biases and Vbi.E and V ^ c are the junction built-in  potentials, at the emitter and collector respectively. Physically, these boundary conditions describe a situation in which the two junctions bounding the base region of a bipolar transistor are perfectly absorbing to carriers impinging on them from inside the base region, and inject into the base hemi-Maxwellian distributions with weights determined by their respective applied biases.  When (4.17) and (4.18) are  substituted into (4.9) and (4.10), the time-independent solution in the field-free base region is given by: f (z,k,6) +  =  !^ -^T -z/v r e  e  z  N =  $Le- >'*» el -'V ' N B  c  4.2  f  Z  Jo  c  f-(z,k,0)  +  T  w  v T  -(z-z')/v rCin  e  z  dzl)  (  4  2  Q  )  Vz  + [' e - ^ ' ^ ^ d z ' . Jw v  (4.21)  z  Solution for acoustic phonon scattering only  In this section, the solution (4.20,4.21) is examined for the situation when the collision integrals include only the isotropic, and elastic, acoustic phonon mechanism. This mechanism  32  is the dominant scattering process in low-field, moderately doped silicon[10]. In this case, considerable analytic progress can be achieved. The presentation here directly extends the work of Grinberg and Luryi[8] to include a better account of transport at the collector junction, and leads to results that are valid in all modes of transistor operation. Results are presented for the distribution function itself, the carrier concentration, current density and carrier mean velocities. Some of the results are needed in the development of an analytical model for the collector current in Chapter 5, while others are of general interest and are compared, in Section 4.3, to the results when a more detailed set of collision integrals is employed in Equations (4.20,4.21). The incoming collision integral and lifetime associated with acoustic phonon scattering were developed in Chapter 3: 1 C  =  in  (4.22)  a kf (z,k). ap  (4.23)  0  With reference to (2.4), the z-directed velocity component can be written as hk v = —cos0. m  (4.24)  z  Combining (4.24) with (4.22) gives the following result: V T — l cos 9, Z  (4.25)  sc  where the scattering length is given by l  =  sc  (4.26)  a m ap  Combining (4.24) with (4.23) gives: Cin  fo{z,k)  =  v  l cos 6  z  sc  Substituting (4.25) and (4.27) into (4.20,4.21) gives f+(z,k,0)  =  ^ -E /y T -z/i e N e  h  B  e  KeoB  +  c  f~(z,k,9)  =  c  k  e  Z  0  ^k -E /*BT (w-z)/i N e  f J  e  scC0S  +  -(z-z')/i cosefo(z',k) , l cos 9  e  sc  z  sc  [ J  Z  w  ~(z-z>)/i co ofo(z',k) l cos 9  e  sc  S  dzl  sc  When n* — 0, these equations reduce to those found in [8]. Note that, as pointed out c  therein, quasi-neutral base recombination can be accounted for phenomenologically in this  33 formulation. By taking the lifetime derived from the outgoing collision integral to include recombination in addition to the acoustic phonon mechanism, a total mean-free path length is derived from 1 1 1 — = - + - , Hot  where l  cp  Z  sc  , (4.30)  'cp  is the mean-free path between recombination events. In the solution (4.20,4.21),  would replace l  tot  l  when /  sc  sc  appears in an exponential (i.e., in those places it appears due  to the outgoing collision integral). In principle, (4.28,4.29) can be solved using the iterative approach outlined in Section 2.4. However, the simple form of the incoming collision integral appearing in (4.28,4.29) allows considerable analytic progress to be made, resulting in a much simpler set of iterative equations, as developed below.  4.2.1  F i n d i n g fo(z,k),  n(z),  and a relation between t h e m  To develop a relationship between fo(z,k) and n(z) the first step is to integrate (4.28,4.29) over 9 to find an expression for f (z,k),  where  0  h(z,k)  =  \J  =  \ I ' f (z,k,9)sin8d9 (z,k,9)sin9d9+\ + £ /f  f{z, k, 9) sin 9d9 w/2  r  f~(z, k, 9) sinOdO.  + +  (4.31)  T/2  Let  ' \ IT ' t-ir/2  /o+(z k)=  k e)  sinm  (4-32)  10  and fo(*,k) = \f 2  f - (z,k,0) sin 6<W.  (4.33)  JTT/2 T/2  Using (4.28) in (4.32) gives r  f+(z,k)  =  n/2  ± I j L - s * A B r r \-'/i-«*° 0 2 Jo /  sin  d0  +  2 ./n  =  f c  2JSJ  Jo  e  -/,,  M u  +  l" f  f\-^ l  J  n  f  2 Jo Jo  c o s 9  ^^dz'sined9 ke COS 9  ^-'- Mm . du  u  (4.34)  iz  l  sc  34 where u = cos 9 and the order of integration in the second term has been changed. The integrals over u in this expression belong to a class known as exponential integrals which are defined by: E {x)=  (4.35)  / V  n  Jo  The properties of the exponential integrals, including highly accurate polynomial approximations, can be found in [20]. Equation (4.34) can be rewritten in terms of the exponential integrals as: ~z — z'-  ' Z '  (4.36)  Isc  2 Jo  -^sc-  For f ~(z,k), using (4.29) in (4.33) gives 0  2 Nc  JTT/2  I  If Z  e  -(z-S)/i„cosefo{z',k) , dz  (  4  3  ?  )  L cos 9  /2 JW JTT/2 J W  c  In terms of the exponential integrals, this can be written as fo-(z,k) =  \ f e - ^ E  1  W-z 2  +  w  (4.38)  2  Substituting (4.36) and (4.38) into (4.31) gives the following result for W-z  Inl 2 N  f (z,k): 0  +  r  2'sc Jo  fo(z',k)dz',  (4.39)  which, when n* — 0, reduces to the result found in [8]. c  In principle, (4.39) could be solved for fo(z,k) in an iterative fashion. The result could then be used in (4.28,4.29) to evaluate the distribution function itself, from which all relevant transport properties could be evaluated. However, the procedure can be made even simpler, and more computationally efficient, by first evaluating the electron concentration. Instead of evaluating the total electron concentration directly from (4.39), the concentrations in the forward- and backward- going ensembles are evaluated first and their sum formed to get the total concentration. The motivation for this procedure will become apparent in the following chapter, where the mean velocities of the counter-directed electron ensembles are required.  35  Using the result from Appendix A, the concentrations in the forward- and backward- going components of the distribution can be written as: n (z)  =  +  — /  f+(z,k)k dk  (4.40)  fo( ,k)k dk.  (- )  2  1 c°° =  -T /  z  2  4  41  Substituting (4.36) into (4.40) gives, for the concentration of the forward-going component, z-z'-\ fo(z',k) , n (z) IT Jo 2N I* sc dz  +  2  C  Z — Z  — —Eo  2  Lt  sc  - -  '•'sc J O  J  ''SC -  Z  —Eo  2  z  J  /  f (z',k)dz' 0  (4.42)  n(z')dz'.  'sc Jo  Similarly, for the backward-going component, substituting (4.38) into (4.41) gives n(z)  =  \W  -E  -z  2  z — z n{z')dz'.  E  x  2^sc Jz  (4-43)  Summing (4.42) and (4.43) gives the total electron concentration as a function of position in the quasi-neutral base region: n(z) =  \E  w <  2  +  \E  2  r  w  i I*  n(z')dz'.  I*  '"sc Jo  (4.44)  This result, when n* = 0, reduces to the result found in [8]. The implicit nature of (4.44) c  allows an iterative solution to be achieved for n(z) directly. Further, note that comparing (4.44) with (4.39) shows that n(z) and f (z,k) must be related by 0  /o(z,*) =  ^ e - *  /  N  k  B  (4.45)  r  c  the same result as in [8], where a simpler view of the collector boundary was taken. This result is to be expected given the scattering model and boundary conditions considered in its development. Since the boundaries inject carriers distributed in energy in a Maxwellian form, and the scattering is elastic, the carriers must remain in this form, as (4.45) illustrates. The solution (4.28,4.29) can be rewritten using (4.45) in the following manner: f+( ,k,0) Z  -E /k T  e  =  ^  - ^ / k B T  e  e  - * / / . c C O B »  B  [' -(*-*')f «<*» n(z )dz COS 0 Jo l  c sc  " C . -E /k T k  B  (W-z)/l cos0 sc  N  c  a  ,  e  Nl  1 * c.  f-(z,k,0)  k  +  J  (4.46)  -E /k T  e  k  B  f- -' < {z  +  Nl c  e  sc  c  o  s  Jw  @ JW  z  )/ls  cos8  n(z')dz''.(4.47)  Once the concentration has been found using (4.44), these equations can be used to evaluate the distribution function and its properties.  36  4.2.2  F i n d i n g the currents a n d mean  velocities  Perhaps the most salient property of the electron distribution is the current density. Using the form (4.46,4.47) for the distribution function, and the definition of the current density (A.5), one finds for the forward-going component: gh r 2n m Jo  r Jo  k3  J (z) +  2  ak -E /k T -z/i cose N poo pic/2 -E /k T rz -q / e / / 2ir m Jo Jo NJ cos 6 J e  k  B  e  sc  coses[nddedk  h  c  k  B  2  SC  S^l^  f  - l -du z  2  E  h1 Jo  c  -q  pz  +I  n* E E  3  cose  poo  pl z ul  c sc  R  z  T  2  -q2v  z  0  / n(z') / ue- ' «*du 27r m N l Jo Jo =  e^ - '^ n(z')dz'  k*e- *l^ dk +  ul  ue  2TT m N Jo  +  f  / Jo  k 3  z — z n(z')dz'  dkdz'  Ek/kBT  e  (4.48)  Jo where the definition of the exponential integrals, (4.35), has been used, and 4c  VR  (4.49)  27rm  is the Richardson velocity. In a similar fashion, the current density in the backward-going component can be found to be J (z)  — -q2vi  -n* E c  3  'sc  Jz  L  n{z')dz'  (4.50)  Summing (4.48) and (4.50) gives the total current density as a function of position in the quasi-neutral base: z  J(z) — —q2v n* E t  E  3  W-zi  -n*E  3  I*  >w \z — z sign(z — z')n(z')dz' + fI E 2  Jo  (4.51)  When n* — 0, this is the same as in [8]. c  The mean z-directed electron velocity can be found from (4.44) and (4.51) by noting that (Vz)  A*) -qn(z)'  (4.52)  It is also interesting to find the mean velocities of the forward- and backward- going components. Using (4.42) and (4.48) for the forward-going ensemble, and (4.43) and (4.50) for the backward-going ensemble, these velocities are given by: J (z) ^ -qn (z)' +  (vt)  =  +  J~(z) -qn-(z)'  (4.53) (4.54)  37  4.2.3  Results  In this section the results of solving the field-free, time-independent transport relation are presented. The results are generated, assuming isotropic and elastic scattering only, by the methods described above. First, the nature of the distribution function is examined. Following this, some of the important properties which can be computed from the distribution, such as the concentration, current density and mean velocities, are presented. In the active mode of operation, the results for the concentration and current density are similar to those found elsewhere[8]. The distribution function at the middle of the base, under equilibrium conditions, is shown in the top half of Figure 4.3. This is a Maxwellian distribution. Note that there is no angular dependence. The distributions shown in this chapter have all been normalized to the peak value of the distribution of carriers injected into the base from the emitter, i.e., n /N . E  c  Information about the spatial variation of the distribution can be found by viewing  the distribution function at different positions throughout the base, or by viewing a plot of fo{z, k), shown at equilibrium in the bottom half of Figure 4.3. The plot of fo(z, k) has the angular information contained in the plot of f(z, k, 9) integrated out, but nevertheless gives a good indication of how the size of the distribution, and its energy dependence, vary with position. Figure 4.4 shows the active mode results, with VBE = 0.8Vt>i,E and VBC = 0, for fo(z,k) at four different basewidths.  All four plots show the same dependence on energy, which  is predicted by (4.45). Note, however, that the weight of fo(z,k) decreases from emitter to collector, and this trend becomes more pronounced as the basewidth increases.  Note  also that there is a subtle curvature in the iso-A; lines, which can be seen most clearly near the ends of the k = 0 line. This curvature is more pronounced at short basewidths, and the relation (4.45) suggests this feature should be seen in the carrier concentration as well. Another trend is that the weight of this function, at the emitter side of the base, decreases as the basewidth becomes shorter. This is because the backward-going component gets smaller as the basewidth shrinks, as fewer carriers are scattered out of the forward-going ensemble injected from the emitter. Figure 4.5 shows the active mode results, with VBE = 0.8V i,E and VBC = 0, for f(z, k, 9) at b  three different points in the base, and for three different basewidths. Note that the forward-  38  Figure 4.3: The top half of this figure shows the equilibrium distribution function, normalized to n /N , at the middle of the base, for a basewidth of W = Z . This is a Maxwellian distribution, as evidenced by the angular isotropy and exponential energy dependence. Also shown is the function f (z, k) at equilibrium, normalized to n /N , for the same basewidth. E  c  sc  0  E  c  39  z = 0.5 x l  Bl  200  k (/im ) 600 -1  z = 10 x L  k {ji'in ) l  k (/rm )  Figure 4.4: The function fo(z,k), normalized to n /N , for an active-mode operating point of VQ = 0.8 x Vbi.E and VBC = 0, at 4 different basewidths. E  c  E  going component (i.e., 0 < 9 < TT/2) at the emitter edge of the quasi-neutral base always exhibits the angularly isotropic form of a hemi-Maxwellian. This, of course, is the boundary condition imposed by (4.15). A similar statement can be made about the backward-going component (i.e., ir/2 < 9 < ir) at the collector edge of the base, however the weight of this component is not visible on the scale shown, because the results are for the active mode of operation. At the centre of the base, and for the components of the distribution exiting the base at either end, it can be seen that the distribution displays an angular anisotropy. This feature is most pronounced at short basewidths, while at long basewidths the distribution approaches the shape of a Maxwellian (c.f. Figure 4.3). Note that the angular nature of the distribution is such that the carriers tend to be focussed about the z-axis in the positive direction. This angular anisotropy will be used to explain some of the features of the mean velocities presented later on. Note also the relative sizes of the forward- and backward- going  40  Figure 4.5: The distribution function, normalized to n /N , for an active-mode operating point of VBE = 0.8 x V ,E and VBC = 0, at the edges and middle of the base, for three different basewidths. E  U  c  41  0.0  0.0  0.6  0.4  0.2  0.8  1.0  z/W  Figure 4.6: The bias dependence of the carrier concentration profile, normalized to n , for a quasi-neutral basewidth of one scattering length. E  components. Here it can be clearly seen how the backward-going component is reduced as the basewidth is reduced, as fewer carriers scatter out of the forward-going ensemble before reaching the collector. Turning now to the carrier concentration, the bias dependence of n(z) is shown in Figure 4.6. The concentration has been normalized to n . This graph nicely illustrates the E  importance of accounting for the flux injected into the base from the collector, which was neglected in [8]. If no flux is injected from the collector, the normalized concentration would be bias independent, maintaining the high bias form shown in the plot (this can be seen by dividing equation (4.44) by n  E  and letting n* go to zero). This leads to the nonphysical c  situation of finite current at equilibrium. However, once enough forward bias is applied at the emitter junction so that the emitter-injected carriers dominate transport, the normalized  42  Figure 4.7: The basewidth dependence of the carrier concentration profile, normalized to n , for an active-mode operating point of VBE = 0-8 x Vbi,E E  a n  d VBC = 0.  concentration exhibits the same bias independent form as predicted in [8], For a transistor operating in the active mode, the basewidth dependence of the normalized concentration is illustrated in Figure 4.7. Note the curvature of the profiles, and that this curvature is more pronounced as the basewidth becomes shorter. The curvature of these profiles, coupled with the fact that the current is constant across the base (recall recombination has been neglected), leads to some interesting results regarding the compact modeling of base transport in short base devices[13j. In the long base limit, it can be seen that the concentration at the emitter side of the base approaches n , while that at the collector side E  approaches zero.  43  1.0  W/l  sc  Figure 4.8: The basewidth dependence of the carrier concentrations at the edges of the quasi-neutral base, normalized to n , for an active-mode operating point of VBE = 0.8 x Vb and VBC = 0. E  i]E  It is also interesting to look more closely at the basewidth dependence of n(0) and n(W), as these will be used in the following chapter. The normalized results are shown in Figure 4.8. As the basewidth gets small, both n(0) and n(W) tend towards 0.5. This is to be expected, as 0.5 x n is the number of carriers injected from the emitter. In the case of longer basewidths, E  n(0) tends toward unity; i.e., the majority of the carriers are reflected back to the emitter by the base. At the collector edge, n{W) is therefore tending towards zero.  44  0.80  0.60 h  e »as 0.40 -  0.20  0.00 ' 0.0  1  1  0.2  •  1  1  •  1  0.4  0.6  1  1  0.8  1.0  v /v BE  bi  Figure 4.9: The bias dependence of the collector current density, normalized to —qvnn , with V c = 0, for a quasi-neutral basewidth equal to one scattering length. E  B  The current density, normalized to the ballistic limit, — qvnn , is shown as a function E  of bias in Figure 4.9. It is especially clear from this figure that quantities normalized to n  E  show negligible bias dependence so long as the bias applied to the emitter-base junction  exceeds that applied to the collector-base junction by about 0.2 volts.  45  1.0  )  1  •  0.1  •  •  1.0  •  1  10.0  w/i  sc  Figure 4.10: The basewidth dependence of the collector current density, normalized to —qv n , at an active-mode operating point of VBE = 0.8Vt>i,E and VBC — 0. R  E  The normalized current density as a function of basewidth is illustrated in Figure 4.10. At short basewidths the normalized value tends to unity, which is to be expected because this is the current contained in the hemi-Maxwellian injected from the emitter. As the basewidth increases the normalized current density falls off and approaches zero asymptotically as the base width gets very long.  46  z/W  Figure 4.11: The profile of the mean z-directed velocity, normalized to 2VR, for a basewidth of one scattering length, at an active-mode operating point of VBE = 0.8 x Vbi,E and VBC = 0. The mean velocity in the z-direction, as a function of position, is shown for active-region operation in Figure 4.11. The velocity has been normalized to 2VR, the mean velocity of a hemi-Maxwellian distribution. The features of the mean velocity are readily predicted from the results for the carrier concentration and current density using equation (4.52). However, it is interesting to examine the behaviour of the mean velocities of the forwardand backward- going components. These are shown as a function of position, for several different biases, in Figure 4.12. Note how the mean speed of the forward-going ensemble is elevated above its boundary value of 2VR, while that for the backward-going ensemble is correspondingly reduced. This is a direct consequence of the angular anisotropy of the nonequilibrium distribution, as shown in Figure 4.5. The forward-going component is peaked near 0 = 0, leading to the increase in (v+), while the backward-going component is minimized  47  Figure 4.12: The bias dependence of the mean z-directed velocity profiles of the forwardand backward- going components of the distribution, normalized to 2VR, for a basewidth of one scattering length (VBC = 0). Note that the horizontal axis at 1.00 divides the vertical axis into separate scales for the forward- and backward- going ensembles. near 6 = IT, leading to the decrease in (v~). It is also interesting to look at the mean velocities of carriers exiting the base (recall the boundary conditions require that the mean velocities of carriers injected into the base be the hemi-Maxwellian value of 2VR),  as these results will  be used in the following chapter. Figure 4.13 illustrates these results. Note that at very short basewidths (v+(W)) approaches the value of (v*(0)), i.e., 2VR,  suggesting that the  injected distribution maintains its form throughout the base. For basewidths longer than one or two scattering lengths, (v^W)) reaches a value of about 1.15 x 2v . R  The mean  velocity of the backward-going component at the emitter edge gets lower and lower as the basewidth shrinks. This is because of the proximity of the emitter edge to the absorbing collector boundary, which tends to cool the backward-going ensemble in the active-mode.  48  10.0 w/k  Figure 4.13: The basewidth dependence of the normalized, z-directed velocities of carriers exiting the quasi-neutral base in the active region of operation, with VBE = 0.8Vbi,E and VBC = 0.  As the basewidth gets longer, (v~(0)) approaches the hemi-Maxwellian value of 2VR. The value of 1.15 x 2VR for the velocity of carriers entering the collector has been reported before, both for a long, field-free base with a perfectly absorbing collector[21], and for a short base, with a field present, and a partially absorbing collector[22]. In [21] it was suggested, without explanation, that it may be due to an angular dependence of the distribution near an absorbing boundary. This has now been confirmed as described above. One can understand how this angular anisotropy comes about by considering the nature of the streaming term in the B T E . The rate of change of the distribution due to particle streaming can be written as [23] dl  'df  dt  - —v cos 9 stream  dz'  (4.55)  49 For a device in the active mode of operation, | £ is negative.  Thus, carriers moving in  the +z-direction are continually streaming into states with 9 close to zero faster than into states with 6 close to 7 r / 2 . In other words, the forward-going ensemble becomes increasingly focussed about the z-axis. Steady-state is brought about by scattering, which in general tends to move a system toward equilibrium, and in this instance there is a net scattering out of states with 9 close to zero. This follows from the use of (3.67) and (3.66) for the collision integrals: out-scattering depends on 9 via f(z, k, 9), so there is more scattering out of states with 9 nearer to zero; in-scattering is isotropic as it depends on the function fo{z, k), which is a measure of the electron concentration n(z). Similarly, considering equation (4.55) for the case of the backward-going ensemble, one finds that there is a net streaming out of this component, and that carriers stream out of states with 9 nearer to TT more rapidly than they stream out of states with 9 nearer to 7r/2. This leads to a situation where the number of backward-going carriers in states with 9 near TT is depressed. Scattering out of these states is correspondingly depresssed, so the steady-state is brought about by a net scattering of carriers into these states. Near the collector boundary, where the low carrier concentration results in correspondingly low (equilibrium-restoring) scattering, the streaming described by (4.55) takes the system to its maximum departure from equilibrium. This leads to the maximum departure of the mean velocities from their equilibrium values of 2VR, as shown in Figure 4.12 (note that (v~) only goes to 2VR precisely at z = W because of the imposition of that value as a boundary condition on the flux associated with n ) . c  4.3  Solution for screened ionized impurities and polar optical phonons  In this section the field-free, time-independent transport relation, (4.20,4.21), is solved using collision integrals appropriate for screened ionized-impurities and polar optical phonons. These are considered to be the dominant scattering mechanisms in GaAs[10]. The layer structure of the device to be studied is shown in Table 4.1. While it may seem impractical to have a higher doping in the base than in the emitter of a homojunction device, the emphasis here is on electron transport in the base region, and the result can be expected to apply to practical heterojunction devices in which a graded emitter-base junction is present. Furthermore, such homojunction devices are of interest and have been used, along with  50 Doping 5 x 10 cm1 x 10 cm1 x 10 cm  Type n-type p-type n-type  Layer emitter base collector  17  3  19  a  ie  _a  Table 4.1: The layer structure of the GaAs BJT studied in this work. collector current models such as those discussed in Chapter 5, for characterizing heavily doped material[24]. The equations to be solved are (4.20) and (4.21). The incoming collision integral is taken to be  C (z, k, 9) = Cl(z, k, 9) + C?r (z, k, 9) + C*r (z, k, 9) P  P  in  (4.56)  where the individual mechanisms are as prescribed in (3.15), (3.46) and (3.55). The lifetime is taken to be 1 _ 1_ — r  r  i  1 T  (4.57)  P°P '  where the lifetimes for the individual mechanisms are as in (3.20) and (3.61).  4.3.1  C o m p u t a t i o n a l l y m o r e efficient f o r m o f t h e s o l u t i o n  As opposed to the case where the collision integral only included acoustic phonons, these equations do not lend themselves to the kind of analytic manipulation undertaken earlier. However, they can be rewritten in a form which makes an iterative solution more computationally efficient. The idea is to rewrite the collision terms so as to remove a redundancy in the integration over the spatial variable. Consider the physical interpretation given earlier, where the solution was written as the sum of ballistic and collision components. Following along the same lines, if the solution for /  +  is known at some point, z — Az say, the solution at z can be written as f {z,k,9) +  =  f (z+  Az,k,9)e- > Az/V  T  + f  e ~ ' ^—dz'. {z  z  )/v  Jz-Az  (4.58)  z  v  The first term in the equation above represents those carriers that have traveled from z-Az to z without suffering a collision while the second term represents those that have reached z after colliding into the forward-going ensemble somewhere between z-Az  and z. Rewriting  51 the first term gives  f {z,k,9)  = f {0,k,9)e- ^  +  +  z  + f+ {z - Az,k,e)e- ^  T  Az  + [*  T  OL  e^'^^dz',  Jz-Az  V  2  (4.59) from which it can be seen that the collision component at z can be written as /+  O L  (z,M)  fZ (z-Az,k,e)e- ' > +r  =  Az v  OL  T  e^'^^dz'.  Jz-Az  (4.60)  V  Z  Note that, when written in this manner, the integration from 0 to z — Az does not have to be reevaluated when calculating the solution at z, which is computationally more efficient, especially in regions where W is large. Similarly, the solution for the backward-going component can be written as  f-(z,k,6)  = f-{z + Az,k,6)e * + Az/v  T  ['  ~ 'V * —dz'  (z e  z  v  z  Jz+Az  = f-(W,k,e)e( - V * w  z  v  T  T  v  + fc {z + Az,k,9)e * Az/v  T  OL  + f  e '- ' ' ' ^dz', l  Jz+Az  z  )  v  T  z  v  (4.61) from which the collision component can be identified as / c ( * , k, 9) = / - (z + Az, k, 9)e ^ Az  0L  4.3.2  Results  c  OL  e ^ ^ d z ' .  + f Jz+Az  (4.62)  z  v  Figure 4.14 shows the active-mode form of the full distribution function in the middle of a base which is one scattering length in width. In this context, / is taken to be the equilibrium sc  value of the mean-free path length. The nonequilibrium value becomes, in general, bias- and position- dependent because of the inelastic scattering mechanism. However, in the case studied here the nonequilibrium value never strays very far from the equilibrium value of 46nm, which is in reasonable agreement with measured values in similarly doped material[25]. Figure 4.14 also shows, for the purposes of direct comparison, the distribution function for the case of acoustic phonon scattering, as considered earlier in this chapter. Not surprisingly, given the differences in the scattering models, the distributions shown in the figure exhibit some differences. Note that at low energies, when screened ionized-impurity and polar optical phonon scattering are considered, the carriers are much more evenly distributed between the  52  Figure 4.14: The normalized distribution function in the middle of a one scattering length base, at a forward-mode operating point of VBE = 0.8Vt>i,E and V c = 0. The top part is for the situation when only acoustic phonon scattering is used, the bottom part is for the situation when screened, ionized-impurity and polar optical phonon scattering are accounted for. B  53  1.0 screened ionized-impurities and polar optical phonons - acoustic phonons  0.8  0.6 W = 10.0 x l  sc  0.4  W = 1.0 x Z  sc  L- W = 0.5 x I  0.2  0.0  0.0  0.2  0.4  0.8  0.6  1.0  z/W  Figure 4.15: Comparison of the basewidth dependence of the carrier profiles, normalized to n , at an active-mode operating point of VBE = 0.8Vbi,E and VBC — 0, for the two scattering models used in this work. E  forward- and backward- going components. Further, the peak of the distribution no longer appears at k = 0. The carrier concentration profiles for basewidths of 0.5 x / , 1.0 x l s c  shown in Figure 4.15.  sc  and 10 x /  s c  are  Note that, despite the differences in the shape of the distribution  functions, their overall sizes, as represented by the carrier concentrations, are quite close. The small differences that do appear are manifest in more curvature for the case of scattering by screened ionized-impurities and polar optical phonons. The basewidth dependence of the normalized current density is shown in Figure 4.16. The curve corresponding to the more detailed scattering model can be seen to be a little bit lower. While the maximum difference occurs at about W — l , the currents can be seen to sc  be approaching each other in both the long and short base limits. This is to be expected  54  1.0  0.0 ' 0.1  •  •—•—  1.0  •—•  •—•—  10.0  w/i  sc  Figure 4.16: Comparison of the basewidth dependence of the collector current, normalized to —qvRn , at an active-mode operating point of VBE = 0.8Vbi,E and VBC = 0, for the two scattering models used in this work. E  as the ballistic limit current, — qvRn , is the same in both cases, and in the long base limit E  both distributions will be tending toward the same near-equilibrium form. The mean velocities of the forward- and backward- going components, for a basewidth of one scattering length, are compared in Figure 4.17.  Once again it can be seen that  the results for the two scattering models are in reasonable agreement, with those for the more detailed model showing a slightly greater departure from the equilibrium values. Note that the differences in the mean velocities can be predicted from comparing the relevant parts of the distribution functions shown in Figure 4.14. The forward-going ensemble in the detailed scattering case can seen to be pushed out in energy in comparison to the simple scattering case, while the backward-going ensemble is pushed towards low energies in the same comparison.  55  0.2  0.0  0.2  0.4  0.6  0.8  1.0  z/W  Figure 4.17: Comparison of the mean z-directed velocities, for a basewidth of one scattering length, normalized to 2VR, at an active-mode operating point of VBE = 0.8Vbi,E and VBC = 0, for the two scattering models used in this work. Overall, the results for the two scattering models considered here have been shown to be quite similar. Perhaps this is not too surprising for homojunction injection into a field-free base, as the carriers are never given an opportunity to get very far from their Maxwellian form in energy. It should not be presumed, however, that the same conclusion can be drawn for situations in which the injected carriers are not of a Maxwellian form, or in a region in which a finite electric field is present.  Chapter 5 A compact m o d e l for the collector current i n B J T s In this chapter a new, compact, collector current relation for modern, short base, bipolar transistors is developed which is consistent with the solution of the field-free, timeindependent B T E . The mean velocities of carriers exiting the base, which were shown in Chapter 4 to deviate from the Maxwellian value of 2VR due to the angular anisotropy of the nonequilibrium electron distribution, are used to improve a set of modified Schottky boundary conditions for thermionic emission at the junctions to the base. These junction currents are then employed, along with an expression for the diffusion current in the base, in a current-balancing approach to develop an analytical expression for the collector current. The diffusivity used in the base transport model is the average of the spatially dependent diffusivity, which is required to keep the current constant in the presence of the non-linear carrier concentration profile found from the solution to the B T E . The presence, in the resulting analytic expression, of correction factors associated with both the mean velocities of carriers exiting the base and the basewidth-dependent effective diffusivity, distinguish this equation from others which have appeared in the literature. The chapter also briefly discusses the uses of such compact collector current expressions, and summarizes similar work done by others.  56  57  5.1  Compact collector current models  The classic example of a compact model for the collector current in bipolar transistors is the Moll-Ross relation[26]:  Equation (5.1) is derived (e.g., for npn transistors) by applying the diffusion equation, ^ dn  ,  J = 1D.&  ( 5  .  '  2 )  to the base region subject to Shockley boundary conditions at the junctions and assuming negligible recombination in the quasi-neutral base. That is, one solves J  c  —  - ~qV  0  (5.3)  subject to the boundary conditions nJDo  =  eVBE/Vth  N (W)  n  =  ^  e  V B c / K h  j  (5  .4)  where the coordinates 0 and W refer to the emitter and collector edges of the quasi-neutral base, respectively, N is the doping level in the base, D is the electron diffusivity in bulk 0  material and  is the intrinsic carrier concentration.  While (5.1) is remarkably simple in both form and derivation, it is nevertheless an equation of great utility. It can be used to provide insight into device operation by illustrating the dependence of collector current on device and material parameters, such as basewidth, doping level and bandgap. It forms the heart of the Gummel-Poon[27] and Ebers-Moll[28] bipolar transistor models which find application in circuit simulators such as SPICE[29). It has also been extensively used to examine the effect of doping level on bandgap[30, 31, 32, 33]. With the advent of transistors employing very narrow base regions, Equation (5.1) has been called into question on account of both the use of the diffusion equation (5.2) and the nature of the boundary conditions (5.4). However, the remarkable utility of (5.1) has motivated work on its modification or extension in efforts to maintain its viability. The result has been the introduction of modified diffusion equations to account for the transition from a diffusive regime to a ballistic regime as the basewidth is reduced, and the use of more  58 sophisticated boundary conditions to account for the finite capacity of the junctions to emit and collect carriers. Some authors[34] have used plausibility arguments to make their case, while others[8] have founded their results in solutions to the B T E . Recognizing that microscopic diffusion theory is not applicable to regions where carrier concentrations change appreciably over distances comparable to a scattering length, Persky[34] argued it is plausible that thermionic emission over a barrier of zero height should form a limit to diffusive current in such instances. To account for this concept of a thermionic saturation of the diffusion current, Persky proposed the following empirical modification to the diffusion equation: J =  gA»fi , dn 1 + Em. nvR  (5.5)  In this equation it is evident that qnv plays the role of a diffusion saturation current density R  that J cannot exceed. In the limit that the classical diffusion current, qDdn/dx, is much less than qnvn, (5.5) reduces to the classical diffusion equation (5.2). It is important to note that there is no physical justification for the form of (5.5), but rather it is the simplicity of (5.5) that motivates its use to account, in at least a crude way, for the diffusion saturation effect which is observed in narrow-base transistors [35]. In an attempt to analyze the consequences of very small basewidth on the applicability of the diffusion equation, Rohr et al. [36] derive a transport relation from a hypothetical collisionless-base transistor, which they consider to be a limiting case for diminishing basewidth. The result, J  c  = -qh v B  R  [e  KBE/y  ' h  ]  VBc/Vth e  ,  (5.6)  where fig is the equilibrium minority carrier concentration in the base, is independent of basewidth, in contrast with the inverse relationship predicted by (5.1). It should be noted that (5.6) can also be obtained by taking the saturation limit of (5.5), instead of the classical diffusion equation, in an analogous derivation to that of (5.1). Perhaps the most enlightening analysis of diffusion in a short base has been given by Grinberg and Luryi[8]. They develop a solution to the B T E similar to that given in Chapter 4.2, except they use a simpler collector boundary condition which, as discussed therein, gives incorrect results at low bias.  59 In order that they can compare their results with those predicted by the diffusion equation, Grinberg and Luryi define a current reduction factor, £, as the ratio between the exact J and Jdiffi where Jdiff = -q^rn* ,  (5.7)  E  i.e., the classical diffusion equation employing a Shockley boundary condition at the emitter and a zero-concentration boundary condition at the collector. From the resulting equation, Av qn* R  C  E  3v7~  =  ^ ^  where w — W/l , a highly accurate analytic approximation for £(w) is provided. sc  It is worth noting that (5.8) has been experimentally verified as giving the correct dependence of collector current on basewidth[35].  Additionally, (5.8) has been used to evaluate  the doping dependence of bandgap, and brings the results from electrical measurement[24] into excellent agreement with those from a detailed theoretical analysis[37]. Recognizing the near linearity of their results for the carrier concentration, Grinberg and Luryi suggested a modified diffusion equation, similar in form to that used by Persky:  J=  ^  ~  (5.9)  y2vjin dz J  Equation (5.9) reduces to Persky's equation when y = 1 and 2v - » v . Employing a least R  R  squares fit to their results, Grinberg and Luryi find y = 2.01, however the fit for y = 2 is quite accurate (see Figure 6 of [8]). It should be noted however, as pointed out by Persky[34], that equations of the form of (5.9) cannot simply be used to replace the diffusion term of a one-dimensional form of the drift-diffusion equation to achieve a new drift-diffusion-like transport relation. In some more recent work, Hansen[ll] has argued that the main features of Grinberg and Luryi's current calculation can be explained if the boundary conditions to the classical diffusion equation are suitably modified Schottky boundary conditions. The boundary conditions employed by Hansen are, J  =  -q2v [n -n(0)}  J  =  q2v [n* - n(W)\,  R  R  E  c  (5.10)  60 which serve to constrain the current to that dictated by the usual Schottky boundary conditions (e.g., J — —qn VR in the active mode) in the short-base limit, and in the long-base E  limit force the carrier concentrations at the base edges to the values required by Shockley boundary conditions, i.e., n  E  and n* . c  Use of (5.10) in the classical diffusion equation (5.3) gives for the collector current Jc = -qD  E  (5.11)  0  0  w  3w  from which the current correction factor can be seen to be £  H = rjV,  (5.12)  a much simpler form than that provided by Grinberg and Luryi (see Equation (35) of [8]). It should be pointed out that this result is equivalent to that obtained by Tanaka and Lundstrom[l2] when they applied McKelvey's so-called one-flux method to analyze transport in short-base devices. Surprisingly, (5.11) yields results that differ by no more than 3% from the results obtained by Grinberg and Luryi, even though the classical diffusion equation has been employed in its development, and, as can be inferred from Figure 3 of [11], the resulting carrier concentration gradient for short base devices can be significantly different than that predicted by the B T E . This good agreement for the collector current results can be shown to result from a fortuitous cancellation of effects associated with the use of both the classical diffusion equation and the boundary conditions (5.10)[13]. In summary, Grinberg and Luryi have shown through a solution to the B T E the need to modify the bulk diffusivity for short base devices, as previously suggested by Persky, and subsequently experimentally verified. Unfortunately, their analytic result for the collector current, (5.8), suffers in that the concentration profile and effective diffusivity are implicitly contained in an all-embracing correction factor, £(w). It is thus difficult to gain physical insight from (5.8), a key feature of the original Moll-Ross relation.  Hansen's approach,  and the one-flux method, while giving accurate results for the collector current, give a less accurate description of the carrier concentration profile, and, furthermore, hide some of the interesting features of short-base transport uncovered by solving the B T E , for example: that the carriers exiting the base have a mean z-directed velocity which deviates from 2VR and that the effective diffusivity deviates from D . In the next section, the principle of current 0  61 balancing, in conjunction with results from the B T E solution of Chapter 4, is used to develop a new, compact, collector current relation in which these concerns are alleviated.  5.2  A new compact model for collector current in BJTs  An interesting feature of the B T E solution for a thin base transistor is the near linearity of the resulting minority carrier profile (see Figures 4.6 and 4.7). Since the neglect of recombination implies a coordinate independent current, this profile motivates the use of a diffusion equation of the usual form, i.e., n(W)  n(0) -  J = -g/J  —  eff  .  (5.13)  In (5.13) D ff is not the usual bulk diffusivity Do but rather an effective diffusivity defined so e  as to make the expression valid. Identifying the choice of Z) ff that achieves this task is left e  for later in the discussion, however a base transport relation of the form (5.13) suggests that the principle of current-balancing can be employed to develop a collector current relation. To utilize the method of current balancing, Equation (5.13) is coupled to expressions for current at the base edges. In general these boundary values for the minority carrier current can be written as: J(0)  =  —q [n (0)v (0) — n~(0)v~(0)]  J(W)  =  -q[n (W)v (W)  +  (5.14)  +  +  -n-(W)v-{W)]  +  .  (5.15)  These expressions simply state that the current is the difference of two opposing fluxes, as illustrated in Figure 5.1. For the fluxes injected into the base, an appeal is made once again to the Bethe theory of thermionic emission[19], i.e., n (0)v (0)^ f-2v +  +  = nv  1  R  E  n-{W)v-{W) = ^-2v  R  (5.16)  R  = n* v . c  (5.17)  R  These boundary conditions for the fluxes injected into the base are consistent with those imposed to achieve the B T E solution of Chapter 4. For the fluxes exiting the base, it was found in Chapter 4, for active-mode operation, that v (W) +  was greater than 2v  R  and that v~(0) was less than 2v . Here, an allowance is R  62 f(W,k )  f(0,k )  z  z  A  A  J-{W)< \  [~=J>j+(o) ~*v>- ft-2:  BASE  Figure 5.1: One-dimensional transistor diagram, illustrating the fluxes at the edges of the quasi-neutral base, as referred to in the text. made for mean velocities which deviate from 2v by introducing two parameters a and b R  defined such that v (0) = a2v  R  +(W)  = b2v .  V  R  (5.18) (5.19)  Additionally, the concentrations of these exiting carriers can be found by noting that the following relations for the total concentration must hold: n(0) n(W)  n (0) + n-(0)  (5.20)  = n (W) + n-{W).  (5.21)  =  +  +  Using (5.16-5.21) in (5.14, 5.15) gives the following results for the total current at either end of the base: J(0) J(W)  =  r l + <  -q2v  = q2v  R  R  H-o«(0)]  (5.22)  - 6n(W)].  (5.23)  2 1+ 6 *  It is worth noting that when a = b = 1, these boundary conditions reduce to the modified Schottky set employed by Hansen.  63  Employing the principle of current balancing, which in this instance requires J(0) = J{W) = Jc, the following base transport relation is arrived at: D r f  i„i(i±f!>_ .(i±a] n  1  ^  2lVt/ U ^ bi H  This is the new collector current relation. It has an explicit dependence on the effective diffusivity and on the current boundary conditions. The parameters associated with these dependences, a, b and D ff, can be found from a solution to the B T E  such as that described  e  in Chapter 4. All  that remains to complete the discussion of (5.24) is to identify the D ff that valie  dates (5.13). If one considers active mode operation and defines a local diffusivity D(z) according to J = qD(z)^-,  (5.25)  then it is apparent from Figure 4.7 that the increase in the concentration gradient near the base edges requires a corresponding decrease in D(z) to maintain a constant current. If one compares (5.25) to (5.13), it can be seen that D~$  as defined in (5.13) is simply the average  value of D~ (z) as defined from (5.25). 1  In Figure 5.2, the local diffusivity is shown, normalized to the bulk diffusivity D , for 0  two normalized basewidths. It can be seen that for short base devices, D(z) is everywhere less than D , so that D s is reduced as the basewidth is reduced. For long base devices, D fi 0  e  e  approaches Do except at the very edges of the base, leading to a D s that approaches Do, e  as expected, for the long-base limit.  5.2.1  BTE  results i n the current-balancing approach  In this section results are presented for the parameters a, b and D f[, which are needed in e  order for (5.24) to be useable. These values are obtained from the B T E  solution developed  in Section 4.2, where only acoustic phonons are considered in the collision integrals (recall the inclusion of more detailed scattering mechanisms made little difference to the transport properties for BJTs). The  exit velocities, v~(0) and v (W), for the left- and right- going ensembles were shown +  in Figure 4.13, from which the appropriate values for a and b can be selected for any particular normalized basewidth w = W/l . Values are tabulated in Table 5.1. These values are not sc  64  Figure 5.2: The coordinate dependence of the local diffusivity, normalized to the bulk diffusivity Do, for two different basewidths. material specific, and depend only on the basewidth and applied bias. The bias dependence is related to the relative magnitudes of the fluxes injected into either end of the base. In the case studied by Grinberg and Luryi [8] of zero flux injected at the collector, there is no bias dependence. In the case studied here, the accounting of carriers injected from the collector leads to the bias dependence of a and b suggested in Figure 4.12. For operation in the active mode, a value of VBE greater than 0.1 to 0.2 volts is required for the carriers injected from the emitter to overwhelmingly dominate in the transport process, i.e., for a and b to change from their equilibrium values of unity to the bias independent values appropriate to the perfectly absorbing and noninjecting collector situation considered in [8]. The material- and near bias- independence of a and b mean that Table 5.1 should prove useful for any B J T in the usual operating range of the active mode.  65  w  a(w)  b(w)  a(w)  0.483  1.00  0.865  1.144  0.801  0.922  1.153  0.883  0.945  1.154  0.918 0.938  Ul  a{w)  b(w)  a(w)  0.10  0.594  1.067  0.20  0.677  1.093  0.571  2.00  0.30  0.727  1.108  0.628  3.00  0.40  0.763  1.119  0.670  4.00  0.958  1.155  0.50  0.790  1.126  0.703  5.00  0.966  1.155  0.950  0.60  0.811  1.132  0.730  6.00  0.971  1.155  0.958  0.70  0.828  1.136  0.752  7.00  0.975  1.155  0.964  0.80  0.842  8.00  0.978  1.155  0.855  1.139 1.142  0.771  0.90  0.787  9.00  0.981  1.155  0.969 0.972  1.00  0.865  1.144  0.801  10.00  0.983  1.155  0.975  Table 5.1: The exit-velocity factors, a and b, and the diffusivity-correction factor a, for any B J T in the active mode of operation with VBE > 0 . 2 V \ Also presented in Table 5.1 is a diffusivity correction factor, a, defined from £>  eff  = aD .  (5.26)  0  The value of alpha is determined from the average value of the inverse local diffusivity defined by (5.25), as discussed above, with the carrier concentration and current density provided by the solution of the B T E . Armed with results for a, b, and 7J ff from Table 5.1 the current predicted from (5.24) e  can now be compared to that from the full B T E solution. By way of example, consideration is given to the active mode of operation with VBE = 1-0V and VBC = 0 . 0 V . In Figure 5.3, it can be seen that (5.24) provides near exact agreement with the B T E solution. This is to be expected, as the effective diffusivity has been defined to give exact agreement under the bias conditions for which a, b and a have been determined, and, as discussed earlier, these parameters have negligible bias dependence in the forward active mode. As well as providing a relation for the collector current, the current-balancing approach provides results for the carrier concentrations at the base edges. Using (5.24) in (5.22,5.23) gives n(0)  _  n  E  n(W) n  E  where S = 2vRW/D . efl  _  1+ a  5 + 1/6  2a  5 + 1/o+l/b.  1+ a  1  2ab  S + l/a + l/b\ '  (5.27) (5.28)  For the active-mode biases given above, these values are compared  with the B T E results in Figure 5.4. The excellent agreement shown confirms that the  66  Figure 5.3: A comparison of the basewidth dependence of the active-mode current density, between a solution to the B T E and the new compact model, Equation (5.24). principle of current balancing, when approached as in this work, can provide results for both carrier concentration and current which are consistent with a solution to the B T E .  67  Figure 5.4: A comparison of the basewidth dependence of the active-mode electron concentrations at the base edges, between a solution to the B T E and the new current-balancing results, Equations (5.27,5.28).  Chapter 6 A p p l i c a t i o n t o base transport i n abrupt-junction H B T s In this chapter the time-independent, field-free solution to the Boltzmann equation is extended to describe base transport in abrupt-heterojunction bipolar transistors. The thermionic emission boundary condition for carriers injected from the emitter into the base is augmented to account for tunneling through, and reflection from, a conduction-band spike at the metallurgical interface of the abrupt junction. These additional features of the junction lead to a different bias dependence of the transport than that occurring for thermionic emission alone. A physical interpretation of the resulting, new, expression for the forward-going component of the electron distribution in the base of abrupt-junction HBTs is presented. Results are generated for an A l G a i _ A s device operating in the forward mode, and include details of x  x  the distribution function and its components, the carrier concentration and current density, and the mean velocities of carriers. A key result is the near-ballistic level of the collector current, even at basewidths where the transport is clearly not ballistic; a compact model for this form of JQ is presented. Throughout, the results are compared to those obtained for the homojunction devices studied in Chapter 4. The basewidth dependence of the base transit time is evaluated, and a compact, analytical expression is provided which is in good agreement with the results obtained by solving the B T E .  68  69  BASE  EMITTER  thermionic components tunneling components  Figure 6.1: Conduction band diagram of an abrupt np heterojunction, illustrating the origin of some of the distribution function components referred to in the text.  6.1  Transport equation and boundary conditions  The transport equation forfield-free,steady-state applications was developed in Section 4.1.1, and, when written in terms of its forward- and backward- going components, was found to be: f (z, k, 9) = /+(0, k, 9)e~ ^ + f +  z  f~(z, k, 9) = f-(W, k, 6)e^ ~ ^ w  z  T  (6.1)  e-^- '^ ^dz',  T  2  + f  T  -(*-* )/".r^n ,  e  Jw  v  d 2  i  (6.2)  z  To develop the boundary conditions, / ( 0 , k, 9) and f~(W, k, 9), for these equations, proper +  consideration must be given to the processes governing transport at the base-emitter and base-collector junctions. In this chapter, the device structure under study consists of an abrupt-heterojunction emitter and a homojunction collector. At the base-collector junction, then, the same thermionic boundary condition used in Section 4.1.3 can be applied; i.e., f-(W,k,9) To develop an expression for f (0,k,9), +  ^  -E /k T k  B  (6-3)  consider the conduction band diagram for an  abrupt np heterojunction, as shown schematically in Figure 6.1. The presence of the conduction band spike at the metallurgical interface causes transport at the junction to be different from that in a homojunction, where the current is strictly due to thermionic emission over  70  BASE  EMITTER  a 0  -z, •n  Figure 6.2: Conduction band diagram of an abrupt np heterojunction, illustrating the energy levels referred to in the text. the junction barrier. Specifically, the spike is thin enough that carriers can tunnel through it, and such carriers can form an appreciable component of the total current injected from the emitter into the base. Furthermore, some of the carriers in the base with trajectories towards the emitter, that would otherwise continue into the emitter, will be reflected back into the base by the potential barrier formed by the spike. As a result, thermionic boundary conditions such as (6.3), which were used to analyze homojunction injection, are not appropriate for analyzing injection from abrupt heterojunctions. For the abrupt heterojunction, the forward-going component of the electron distribution at the quasi-neutral base edge is written as: / ( 0 , k, 6) = /+ (0,fe,9) + /f[ (0,fe,9). +  TE  In (6.4) /TTE(0'^>^) * * s  n e  FL  (6.4)  P * comprising carriers that have tunneled through, or been ar  thermionically injected over, the conduction band spike, and / R ( 0 ,fe,0) is the part comFL  prising carriers that have been reflected back into the base by the conduction band spike. These components of the distribution are shown schematically in Figure 6.1. The tunneling-thermionic-emission component, fxr (0, k, 6), can be found, with reference E  to Figure 6.2, from /+ (o,fe) = / (-z ,fe')r (fe) +  TE  n  E  (6.5)  71  where 7E(A;) is the so-called barrier transparency, representing the probability that an electron with wave vector k' in the emitter will tunnel through, or be thermionically injected over, the barrier and enter the quasi-neutral base with wave vector A;. Considering that the distribution in the quasi-neutral emitter can be represented as a Maxwellian, one can write h  2  f {-z ,k') +  n  = ^  e  x  P  2mkeT (fix "I" y " i "  (6.6)  z)  k  k  Conservation of energy requires 2m  % + k? + k? = kl + kl + k* +  j£E  (6.7)  bt  so (6.6) can be written as f (-z ,k') +  n  =  n  2m  E  -%  exp  " hk " n E b ^ e x p \ k B r 1J exp 2mksT 2  2  (6.8)  E  =  where E is defined in Figure 6.2. b  Further, the carrier effective mass is taken to be the  same on both sides of the barrier, so the transverse components of crystal momentum are conserved[38, 39] and the barrier transparency becomes a function of k only, i.e. Tn(k) = z  7E(A;,0). TO evaluate the barrier transparency, the W K B approximation gives[40]  exp|-|jT  y/2m  (E -E')dzY e  (6.9)  where the range of integration is defined in Figure 6.2, E is the conduction band energy c  and E' is the z-directed kinetic energy of the carrier in the quasi-neutral emitter. Using z  the depletion approximation to evaluate the conduction band profile in the emitter-base junction, one finds (6.10) where E cos 6 + E 2  k  V  =  b  E,k  (6.11)  P  and E  pk  is the height of the conduction band spike (see Figure 6.2). For z-directed energies  above the top of the conduction band spike, 7i;(A;,0) is taken to be unity, i.e., all such carriers are thermionically injected over the barrier. The definition of n ensures that there is  72 no tunneling for energies less than E . However, if E + E < 0 (as could be the case if the b  z  b  level of the conduction band in the quasi-neutral base is lower than that in the quasi-neutral emitter) then 7E(A;,0) is taken to be zero. With / (—z ,A?) taken from (6.8) and T (k,9) +  N  E  as described above, Equation (6.5) can be written as: E  n /TTE(°>M) = ]y- P E  hk 2  h  exp  e x  kT  2  2mk Tl  (6.12)  %(k,9).  B  R  To evaluate the component of the distribution reflected back into the base from the conduction band spike, for z-directed energies less than AE , where AE is defined in n  n  Figure 6.1, consideration must be given to both tunneling into the emitter and specular reflection; (6.13)  /RFL(0, k, 9) = f~ (0, k, TT - 9) [1 - T (k, 9)} E  Note that, because the WKB approximation involves only an integration over barrier energies above that of the carrier's z-directed energy, the barrier transparency is the same as that for carriers injected from emitter to base. Evaluating /~(0, k,ir — 9) from (6.2) one finds / " (0, Jfc, TT J  v  ' '  -  9) = f~{W,  — e)e ' w  k,Ti  *W +  vco  '  T  J  f  e  f-{W,k,ir-9)e-  dz  UCOS(TT-0)  -z'/vCOS(0)TC\n{z',k,7T  + / ( Jo  w/vcosWT  J) ,  «V«cosfr-fl)TCm(^M  w  (6.14)  - 9)  vcos(0)  Using (6.4), (6.12) and (6.14) in (6.1) and (6.3) in (6.2) gives the new set of coupled integral equations which need to be solved tofindthe distribution function in the quasi-neutral base of abrupt-junction HBTs: f (z,k,9)  = ^exp  +  \  E b  hk 2  1exp .  2  -j  2mk TJ 1 - k rJ z r-(z-z')11 exp L VT I B  —z exp \-V T  B  Z  +  dz' +  tints',  Z  PW  -E /k T  p  k  B  -W/v  +  -(z-z')/v rCin{z  c  -dz' I - T E ( M )  z  Jo  1 (6-15) exp — V T1 Z  -(z-z')/v,T in{z',k,9) C  -E /k T (W-z)/v k  B  e  c z  (6.16)  >w  6.2  Physical interpretation  Considerfirstthe result for the forward-going part of the distribution, Equation (6.15), from which it can be seen that there are now three terms in the solution, as opposed to the two  73  terms arising in the B J T case studied in Chapter 4. This difference arises because of the need to include reflection from the conduction-band spike, as well as injection from the emitter, into the boundary condition at the emitter edge of the quasi-neutral base. The first term in (6.15) is the part of the forward-going distribution that has traveled to z ballistically after injection from the emitter, which, as in the B J T case, is referred to in this work as fQ (z,k,9). AL  The second term in (6.15) represents carriers that have undergone a collision  event, into a forward-going state, at some point to the left of z. Note that this term is of identical form to the collision component of the forward-going part of the B J T solution, and as in that case, is referred to in this work as fc (z, OL  k, 9).  The third term in (6.15) represents carriers that have been reflected back into the base by the conduction-band spike, and then traveled to z without suffering a collision. This component of the forward-going ensemble is referred to in this work as f£ (z, FL  k, 9). These  carriers originate from the backward-going ensemble at z = 0, and thus include carriers which have suffered a collision since being injected into the base region, or have traveled ballistically since injection from the collector.  However, it has not been found useful to  distinguish the latter as a separate ballistic component of the forward-going distribution, especially for the forward mode of operation studied here. The solution for the backward-going component, Equation (6.16), is identical to that found in the study of homojunction devices in Chapter 4, and therefore has the same physical interpretation as described in Section 4.1.2.  6.3  Results  In this section some results for base transport in abrupt-junction HBTs are presented. The device studied is of the material composition detailed in Table 6.1. Equations (6.15,6.16) are Layer emitter base collector  Type n-type p-type n-type  Doping 5 x 10 cm1 x 10 cm1 x 10 cm17  3  19  a  16  3  Al Composition 0.3 0.0 0.0  Table 6.1: The layer structure of the A ^ G a ^ A s abrupt-junction H B T studied in this work. solved in the same iterative fashion as described earlier, with the incoming collision integrals and lifetimes determined from considering both screened ionized-impurities and polar optical  74  Figure 6.3: The distribution of carriers injected into the base from the emitter, of an abruptjunction HBT, normalized to n /N , for four different applied biases. E  c  phonons, as described in Section 4.3. In Figure 6.3 the distribution of electrons injected from the emitter into the base,  T / TE>  is shown for several applied biases. These plots, and indeed all of the distribution function plots of this chapter, have been normalized to the peak value of the distribution of carriers injected over the barrier seen looking from emitter to base, i.e., n /N , E  4 = n e- * . E  E  /kBT  c  where (6.17)  This is analogous to the normalization used in the B J T results of Chapter 4. However, in the B J T case, this scheme led to normalized distributions with a peak value of unity, while for abrupt-junction HBTs the peak value can be greater than unity, due to tunneling through the conduction band spike. In fact, the peak value of the injected distribution can be interpreted as a measure of the relative importance of tunneling in relation to thermionic emission. As the figure illustrates, tunneling is the dominant transport mechanism at the  75  junction for all the biases considered. With reference to Figure 6.3, there are three major differences between the forms of the injected distributions of the B J T and the abrupt-junction HBT. First of all, the carriers injected into the base of an abrupt-junction H B T exhibit an angular dependence not evident in the B J T case, where the injected carriers were of a hemi-Maxwellian form. The second point to note is that the carriers are more energetic than in the hemi-Maxwellian of the B J T case. Thirdly, the form of the (normalized) injected distribution is bias dependent, again in contrast to the B J T case, with the angular anisotropy and energy of the carriers increasing with applied bias. In fact at high biases it can be seen that there is little similarity between the injected distributions of the B J T and HBT. This bias dependence leads to a bias dependence of the full distribution and its properties, which is not found in the B J T case, where the normalized quantities exhibited no bias dependence in the active mode of operation. The angular anisotropy and energetic nature of the injected distribution arise due to the presence of the conduction band offset at the metallurgical interface of the abrupt junction, as illustrated in Figures 6.1 and 6.2. Carriers thermionically injected over the barrier gain additional longitudinal kinetic energy as they enter the base, which serves to focus the distribution about the z-axis. This is also true for carriers tunneling through the barrier. Furthermore, the barrier is more transparent to carriers with large z-directed components of kinetic energy. In Figure 6.4 the full distribution is shown at three different positions, and three applied biases, for a device with basewidth equal to one scattering length, / . Recall, as discussed in sc  Section 4.3, for the scattering mechanisms involved here, that l  sc  refers to the mean-free path  of the equilibrium distribution, the value of which is 46nm for the GaAs bases studied in this work. Note that the forward-going component at the emitter edge is significantly larger than the injected components of Figure 6.3. This is because the forward-going component at the boundary includes those carriers that are reflected back into the base from the conduction band spike. Note also that the definite peak that occurs in the injected distribution at high bias is still readily evident in the full distribution throughout the entirety of the base. This will have a noticeable effect on some of the properties of the distribution to be discussed later on. The basewidth dependence of the distribution function is illustrated in Figure 6.5, where  76  o  o  A  o I o  o  O  • * As  /  1  II  A ^  . . . . . . T o in o in o m Q ) m w o N in w  a.  •'  1 CD  \  18 m  1X1  'in  1 «•-.  O  O O O O O  v"  ^  ;  * <H>  *K  / o  T  <o in  CO CM •»- OD  o  o  o  AS  _  CD  A  °  Sco II  \^  4  W CO /co *~ / >  ^  Wf  -  T  o m o in o m ac in cu o r-- in CM  (O m  **-» O O O O O O <tt> to in co CM y-  TT  CO CJ ^ c©  o  o  O II M  o in o in o m ao m CM o h- m CM  yl x ^0 =  !q  ^ O O O O O O C Z D CD in CO CM i-  !q  /l x 90 =  a a  **-» CD  /l  !q  m  Tt-  co  CM  /l x 8-0 =  <a>  A  3a  Figure 6.4: The distribution function, normalized to n /N , at the edges and middle of the base, for three different applied biases and a basewidth of one scattering length. E  c  77  Figure 6.5: The active mode distribution function, normalized to n /N , at the middle of the base, for four different basewidths. The applied biases are VBE — 0.8Vbi,E and VBC = 0. E  c  the high bias (VBE = 0.8Vb and VBC = 0) distribution is shown at the middle of the base i]E  for four different basewidths. Because the bias, and hence the normalizations, is the same in these plots, the weights of these distributions are directly comparable. Note that the size increases considerably with increasing basewidth. This is in contrast to the B J T case, where the normalized concentration in the middle of the base is basewidth independent (see Figure 4.7). The increase seen here is due to the nature of the reflecting boundary at the emitter edge. The longer the basewidth, the larger is the chance of a carrier being scattered by the base back to the emitter edge. However, when a conduction band spike is present most of these carriers are, in turn, reflected back into the base rather than being absorbed by the emitter. This causes the carrier concentration to build up with increasing basewidth in abrupt-junction devices, as evidenced in the figure. Another interesting feature evident in Figure 6.5 is the relative importance of the injected distribution. Comparing with Figure 6.3, for short base devices, the high energy peak in  78 the full distribution indicates that the injected distribution is quite important. Conversely, for longer base devices, the high energy peak in the full distribution is reduced to a small part of the entire ensemble. This occurs because at long basewidths much more scattering takes place, which tends to drive the distribution towards a Maxwellian form. Further, the carriers that are reflected from the emitter boundary are even more thermalized than the basewidth alone would suggest, due to the extra path length they endure. One of the key features of the method employed here is the ability to readily divide the distribution up into its constituent parts as described earlier. Figure 6.6 examines this for the forward-going component at a basewidth of one scattering length in the active mode of operation. In the top row the ballistic component, comprised of carriers which have not suffered a collision since being injected from the emitter, is shown at the edges and the middle of the base. At z = 0, this is simply the component injected from the emitter,  /^TE-  As these carriers travel along the base, their number is reduced by collisions, as shown in the figure. However, at this basewidth there is still a significant number of these carriers which reaches the collector without having suffered a collision. The second row of Figure 6.6 shows the reflected component, /RFLC- ' k, 0). Note that the 2  size of this component decreases much faster than the ballistic component. This is due to the nature of the energy dependence of the carrier mean-free path length, as illustrated in Figure 6.7. Because the reflected component of the distribution has been quite thermalized from collisions, the shorter mean-free path length seen by its lower energy carriers causes a more rapid decline in the size of this component. The bottom row of Figure 6.6 shows the collision component of the forward-going ensemble, / C O L ( ' k, 0), comprising all carriers that have scattered into forward-going states to Z  the left of z. Clearly, at the emitter boundary this component should be zero, as the figure illustrates. What might not be so readily apparent is why this component of the distribution is smaller at the collector edge than it is at the middle of the base. The answer to this can be seen by examining the second term in Equation (6.15), which corresponds to this component of the distribution. The exponential term weights the integrand such that it is largest near the point z. The incoming collision integral, which can be seen from (2.11) to be closely related to the distribution function, has a higher value near the centre of the base in accordance with the relative size of the distribution function there, and so the collision component is correspondingly higher in the middle of the base than at the collector edge.  79  Figure 6.6: The ballistic, reflected and collision components of the forward-going portion of the distribution function, normalized to n /N , for an H B T with basewidth equal to one scattering length in the active-mode of operation with VBE = 0.8Vbi and VBC = 0. E  c  >E  80  200  k (/mi 1)  Figure 6.7: The A>dependence of the electron mean-free path in GaAs when the collision mechanisms are due to screened, ionized-impurities and polar optical phonons, as studied in this work. Turning now to the properties derived from the distribution function, consider the bias dependence of the normalized carrier concentration shown in Figure 6.8. The normalization here is to n | , i.e., twice the number of carriers thermionically injected into the base from the emitter.  This is analogous to using n  E  as the normalization for the B J T results in  Chapter 4. It is interesting to note that at low biases the shape of the curves is very close to the results of the homojunction device shown in Figure 4.6. At high biases, however, the curves are different than those for BJTs.  Not surprisingly, the differences in shape  occur mainly on the emitter side of the base, and can be attributed to the differences in the boundary conditions of the two devices, i.e., to the tunneling and reflection occurring in the abrupt-heterojunction device. Further, the curves continue to be bias dependent even in the forward active mode, in contrast to the BJT results. This is a consequence of the bias  81  Figure 6.8: The normalized carrier concentration profiles in an abrupt-junction HBT, at several different applied biases (VBC = 0), for a basewidth of one scattering length. The abscissas are position in angstroms and the ordinates are n(z)/n . E  dependence of the injected distribution evidenced in Figure 6.3. In regards to the weight of the normalized concentration, the general trend of decreasing weight with bias is followed in both devices, however the relative changes are much larger in the case of the HBT. The basewidth dependence of the normalized carrier concentration in abrupt-junction HBTs is illustrated in Figure 6.9 for an active mode bias of VBE = 0 . 8 ^ ^ and VBC = 0. The general trend of increasing size with basewidth, shown for the BJT case in Figure 4.7, is followed here as well. However, the relative change is much greater in the case of the HBT. Notice also that at short basewidths, e.g., W = 0.5 x Z , the H B T curves do not flatten sc  out as much as the BJT curves. This is a consequence of the tendency of carriers to pile up at the conduction band spike, as discussed earlier. As regards the shape of the curves, as the basewidth gets large the H B T curves start to resemble the B J T results except near the emitter edge. This is not surprising, as the farther the carriers can progress into a region,  82  Figure 6.9: The normalized carrier concentration profiles in an abrupt-junction HBT, for several different basewidths, at an active-mode bias of VBE = 0.814; and VBC = 0. The abscissas are z/W and the ordinates are n(z)/n . E  the more the collision mechanisms will drive the system toward quasi-equilibrium, while near the emitter edge the effect of the boundary is always apparent. Turning now to the current density, the bias dependence of the normalized value is shown in Figure 6.10.  The normalization here is to the current thermionically injected  from the emitter over the emitter-base junction, — qvnn . E  The corresponding results for  the homojunction device are shown in Figure 4.9. At low bias, the H B T exhibits the same increasing trend as the BJT. However, the HBT curve fails to level off in the forward active mode as does the BJT. In fact the HBT curve falls off considerably as the bias is increased. This is a consequence of the decreased relative importance of tunneling at higher biases, as discussed earlier. In the flat-band situation which would occur at VBE = VU,E, there would be no tunneling, and the curve in Figure 6.10 would reach its minimum value. Note also that the relative size of the HBT curve is much greater than that of the BJT. This is, once  83  Figure 6.10: The normalized current density as a function of applied bias in an abruptjunction HBT, for a basewidth of one scattering length. VBC = 0. again, due to the ability of carriers to tunnel through the conduction band spike. The basewidth dependence of the normalized current density for the HBT, operating in the active mode at VBE = 0-8Vbi,E and VBC = 0, is illustrated in Figure 6.11, along with the corresponding result for the B J T from Chapter 4. In both cases, the normalization is to the ballistic-limit current in the respective device. In the HBT, this includes not only the current thermionically injected over the barrier, but also the current tunneling through the conduction band spike, i.e., the total current contained in f^ . TE  At short basewidths, the  HBT can be seen to approach its ballistic limit much more closely than the BJT. In fact, for basewidths less than a scattering length, the HBT operates within about three percent of its limiting value. This trend is not too surprising, because of the highly focussed and energetic nature of the HBT's injected distribution at high bias, as shown in Figure 6.3.  84  0.0 0.1  10.0  1.0 w/i  sc  Figure 6.11: The normalized current density in an abrupt-junction HBT, at an active mode operating point of VBE = 0.8Vbi,E, as a function of basewidth. Considering now the results of Figure 6.11 for long basewidth, note that the decrease in collector current with increasing basewidth is far less pronounced for the HBT. In fact, the H B T collector current changes by only about ten percent as the basewidth changes from a tenth of a scattering length to ten scattering lengths, in comparison to almost eighty percent for the BJT. This is a surprising result, especially given the basewidth dependence of the shape of the distribution function. As illustrated in Figure 6.5, the energetic and focussed nature of the injected ensemble is washed out as the basewidth increases, and by W = 4 x Z  sc  the distribution is very much resemblant of a near-equilibrium form. To gain insight into why the collector current fails to fall off as rapidly for the H B T as the BJT, consider the anti-symmetric part of the distribution function, defined as f (z,k,0) = f (z,k,9)+  A  f-(z,k,ir-0).  (6.18)  85  Figure 6.12: The anti-symmetric part of the distribution, at the emitter edge of a base with W = 4 x l , for an abrupt-junction HBT at an active mode operating point of VBE = 0.8Vbi and VBC = 0. sc  This is the part of the distribution which carries the current, and is shown, at the emitter edge of a base with W = 4 x Z , in Figure 6.12. Note that at the emitter edge, /A(0, k, 6) sc  very closely resembles the injected distribution, / ^ T E '  a n c  ^  s  o  the current carried by the total  distribution is very close to the ballistic limit value carried by / ^ T E - The reason fA(0,k,6) so closely resembles / T  T E  can be seen from the form of the component reflected from the  conduction band spike, Equation (6.13). Because the reflection is specular, / ~ and / ^  F L  cancel each other out at z = 0, except for the small part of / ~ which tunnels through, or is thermionically injected over, the conduction band spike into the emitter. Only when / " becomes large enough that / ~ x 7E is comparable to / T T E  w u  ^ the collector current in  abrupt-junction HBTs show the same type of decline as in BJTs. The near-ballistic nature of the collector current in abrupt-junction HBTs, for a wide range of basewidths, suggests the following compact model for the collector current in these devices: J  where  r4  TE  and v^  TF  c  = -gn^TE^TTE)  ( 6  1 9  )  are the carrier concentration and mean z-directed velocity of the  86  0.0  0.0  0.2  0.4  0.6  0.8  1.0  z/W  Figure 6.13: The mean z-directed velocity profile in an abrupt-junction H B T , normalized to 2VR, as a function of applied bias with VBC = 0. The basewidth is one scattering length. injected distribution. Furthermore, an analytic approximation to (6.19), in diode-like form, has been developed by Searles et al. [41], which should prove useful in modeling the collector current in HBTs for circuit simulation applications. Consider now the electron velocity in abrupt-junction HBTs. The bias dependence of the mean z-directed velocity of the electron distribution is shown in Figure 6.13. Note that the basic shape of these curves is similar to the corresponding BJT curve shown in Figure 4.11. However, in the active mode of operation there would be no bias dependence in the B J T evidenced by the results for the normalized carrier concentration and current shown in Figures 4.6 and 4.9. The HBT, on the other hand, exhibits a bias dependence throughout the forward active mode of transistor operation. This is due to the bias dependence of the injected distribution at the abrupt-heterojunction which is not present in the homojunction  87  2.5 - 0.05 x V 0.1 x V - 0.2 x Vbi  bi  bi  - 0.4 - 0.6 -© 0.7 -o 0.8  2.0  xV x Vbi xV x Vi bi  bi  b  1.5  1.0  0.0  0.2  0.6  0.4  0.8  1.0  z/W  Figure 6.14: The bias dependence of the mean z-directed velocity profile in the forward-going ensemble for an abrupt-junction HBT with a basewidth of one scattering length. VBC = 0. case. Perhaps the most interesting results from the discussion of BJTs in Chapter 4 were those for the mean z-directed velocities of the forward- and backward- going components of the electron distribution. The bias dependence of the mean z-directed velocity in the forwardgoing component of an abrupt-junction HBT, with a basewidth of one scattering length, is illustrated in Figure 6.14. The corresponding results for a homojunction device are given in the top half of Figure 4.12. As has been the trend, the bias dependence of (u+) continues well into the active region for the HBT, as opposed to the BJT results, and in fact the bias dependence of the HBT case is strongest in the higher bias region. At low biases, the HBT curves are very similar in form to the B J T curves, although the values at the emitter edge are slightly higher than the hemi-Maxwellian value of 2VR exhibited by the BJT.  continues  88 to get considerably higher at the emitter edge as the bias is increased. This is due to the bias dependence of the injected distribution as illustrated in Figure 6.3, where it can be seen that the carriers are enhanced in energy and become increasingly more focussed about the z-axis as the bias is increased, leading to the increase in mean z-directed velocity seen here. At the collector edge,  shows the same tendency to increase that appeared in the B J T case,  and since both devices have homojunction collectors the reasons for this tendency are the same in both instances. At higher biases, away from the boundaries of the base, the mean z-directed velocity of the HBT can be seen to be decreasing. This is due to the tendency of the collision mechanisms to to drive the system back towards the equilibrium value of 2VR. This feature is not seen in the BJT because the value at the emitter edge is already pinned at 2VR because of the hemi-Maxwellian boundary condition imposed in that case. Figure 6.15 shows the bias dependence of the mean z-directed velocity in the backwardgoing ensemble for an abrupt-junction H B T with a basewidth of one scattering length. The corresponding results for the homojunction device are shown in the bottom half of Figure 4.12. Once again, the low bias profiles are very similar for the two devices, with the H B T continuing to show a bias dependence throughout the active regime, which is not seen in the B J T results. Near the collector edge of the base, the H B T profiles exhibit the same drop off seen in the B J T case. This not surprising given that both devices have the same collector structure. However, at the emitter edge, instead of reaching a final value of about 0.9 x 2VR, as in the BJT case, the mean z-directed velocity of the backward-going component of the H B T starts to increase with applied bias. This is because the backward-going carriers near the emitter originate largely from collisions out of the forward-going ensemble, which is elevated in energy, as discussed earlier. The basewidth dependence of the mean z-directed velocity in the forward-going ensemble is illustrated in Figure 6.16 for an active mode bias of VBE = 0.8Vbi and VBC = 0. For very short basewidths, (vf) can be seen to be quite high throughout the base. This again is due to the energetic and focussed nature of the injected distribution as shown in Figure 6.3. As the basewidth gets shorter, there is less and less scattering out of this ballistic component and consequently it forms a larger and larger part of the total forward-going distribution, leading to the high velocities exhibited in the figure. Conversely, as the basewidth increases there is more scattering and the collision component becomes the dominant part of the forward-going ensemble, leading to a  that is near to 2VR over most of the base, with the slight rises  89  Figure 6.15: The bias dependence of the mean z-directed velocity profile in the backwardgoing ensemble for an abrupt-junction H B T with a basewidth of one scattering length. VBC = 0. near the boundaries. The basewidth dependence of the mean z-directed velocity in the backward-going ensemble is illustrated in Figure 6.17 for an active mode bias of VBE = 0.8Vt,i and VBC — 0. As is to be expected, the profiles drop off near the collector boundary. For shorter basewidths, the value at the emitter edge is elevated above 2v for the same reasons surrounding the discusR  sion of Figure 6.15 above. As the basewidth gets longer, however, the collision mechanisms become effective in pushing the value back towards 2v . R  The enhanced carrier velocities of the short base abrupt-junction HBTs, discussed above, suggest that these devices may exhibit a reduced base transit time, TB, in comparison to  90  '  i  1  1  1  1/4 x / l/2x/ 3/4 x l 1x / 2x / o 4 x l • 6x / 0 10 X /  sc  s c  sc  s c  s c  G  sc  •  s c  <5  s c  2.0  ©•——e—  1.0,  ^  I  0.0  ,  —  3  *  I  0.2  e —  e  B r -S  ;  1 ^  ,  ;  B —— ©—  I  - — I  -B  i  r°  7*  19  0.4  © ^  0.6  &  Z H ^ - ^ ^-1  0.8  1  1  1.0  z/W  Figure 6.16: The basewidth dependence of the mean z-directed velocity profile in the forwardgoing ensemble for an abrupt-junction H B T at an active-mode bias of VBE = 0.8Vbi and VBC = 0. homojunction devices. The base transit time is calculated from  T B  =  -q • / T Jo  n{z)dz.  (6.20)  C  The basewidth dependence of T for both devices is shown in Figure 6.18. The H B T transit B  time can be seen to become significantly reduced with respect to that of the B J T as the basewidth is reduced to the order of one scattering length and beyond. Also shown for reference is an analytic expression for the base transit time in BJTs,  ° = 2D + *r  T  R  -  (6 21)  The close agreement between the B T E and the analytical result has been reported before for BJTs[42]. For small basewidths, this expression reduces to the ballistic limit,  while  91  Figure 6.17: The basewidth dependence of the mean z-directed velocity profile in the backward-going ensemble for an abrupt-junction HBT at an active-mode bias of VBE = 0.8Vt,i and VBC = 0. for long bases it approaches the long-standing classical expression,  It is proposed here  to modify (6.21) to account for the fact that the ballistic limit result is different in HBTs with respect to BJTs. The result, W TB = ^+  W  2  2L>  +  n  r  y  (6.22)  VTT (VBE) E  where « T ( V B E ) is the mean z-directed velocity of the injected distribution, / r r , is also T E  E  shown in the figure. It can be seen that the new expression, (6.22), is in good agreement with the results of the B T E solution. Note that the bias dependence of v^  TE  suggests that  the base transit time can be further reduced from that shown in the figure by operating at higher applied biases. It is noteworthy that for HBTs the reduction in base transit time below that for BJTs is  92  Figure 6.18: The base transit time as a function of basewidth, for both homojunction and abrupt-heterojunction devices at an operating point of VBE = 0.8Vbi,E and VBC = 0. not predicted by D D E simulators. Furthermore, it is an important feature for device design because of the tradeoff between increasing base resistance and decreasing base transit time as the basewidth is reduced. Finally, it should be noted that experimental measurement of the energetic nature of the electron distribution in abrupt-junction HBTs is a topic of current interest [43].  Chapter 7 K i n e t i c approach t o transport In this chapter a kinetic approach to transport is developed.  It is used first to provide  an alternate derivation of the field-free solution, and then it is applied to field-dependent transport. An expression for the probability that a carrier travels a distance z before collision, P(z), is derived from the concept of path integrals. This is then used to derive expressions for the ballistic components of the counter-directed electron ensembles. To derive the collision components, similar arguments are applied to the carriers that reach z after having suffered a scattering event. The analysis is straightforward in the field-free case, but in the fielddependent case it is complicated by the fact that a carrier's wave vector becomes a function of position. The development is carried out in terms of the z-component of wave vector, k , and then transformed to the preferred variables k and 6. The general result is then z  simplified to the situation where a constant electric field exists, and then further simplified by considering only isotropic and elastic scattering mechanisms.  93  94  7.1  Alternate derivation of the field-free solution  It is interesting, and instructive, to note that given the physical interpretation of the terms in the solution to the field-free, steady-state transport relation, the solution can be derived in another way. The procedure is to evaluate the different ways a carrier can come to be at some point z. Recognizing that the solution can be written as the sum of ballistic and scattered components, one has for the forward-going ensemble, f (z, k, 9) = / +  + A L  ( z , k, 6) + /  + 0 I  > , k, 6).  (7.1)  The two components can now be derived separately as follows. To derive the ballistic component note that f£ (z,k,6) AL  = f (0,k,e)P(z), +  (7.2)  where P(z) is the probability that a carrier travels to z without suffering a collision. Now the collision rate, or probability per unit time that a carrier suffers a collision, has been previously determined, i.e., — = collision rate. r  (7.3)  This means that — = probability of a collision between t and t + dt. r  (7.4)  Now, dz  (7.5)  and between t and t + dt a carrier moves between z and z + dz; thus, substituting for dt from (7.5) in (7.4) yields: dz  — probability of a collision between z and z + dz.  (7.6)  Further, the following equation must hold: P(z + dz) = P(z) 1 -  dz  (7.7)  which simply states that [the probability a carrier travels a distance z + dz without collision] equals [the probability that this carrier travels a distance z without collision] times [the  95 probability it travels between z and z + dz without collision]. Expanding Equation (7.7) gives P(z) + ^dz dz  = P(z)-P(z)  —, n  (7.8)  or  P  = - i i .  (7.9)  VT Z  Solving (7.9) and noting P(0) = 1 gives the following result: (7.10)  P(z) = exp  JO  z.  v  T  In the field-free situation, the carrier's wave vector, and thus v and r, does not change with z  position as the carrier moves ballistically in the base. Therefore, Equation (7.10) can be written in this instance as P(z) = e-* > . /v  (7.11)  T  Substitution of (7.11) into (7.2) yields the following result for the ballistic component of the forward-going ensemble: f£ (z,k,6) AL  = f (0,k,e)e-*^.  (7.12)  +  Turning now to the collision component of the forward-going ensemble, and referring to Figure 7.1, if one knows how many carriers have scattered into /  +  at z', then the arguments  surrounding the derivation of the ballistic component suggests that the number of these which reach z (without suffering another collision) is obtained by multiplying by exp ^—f , ^ z  •  The total collision component of the forward-going ensemble would then be obtained by integrating this product from zero to z. Now the rate at which carriers scatter into states in the distribution is simply the incoming collision integral, C . Multiplying this by dt gives: m  C;„(z')dt — number of carriers which scatter into / in the time dt.  (7.13)  Substituting for dt from (7.5), provides that: dz' , C i n ( s ' ) — = number of carriers which have scattered into / and remain within dz' of z'. (7.14)  96  z'  z  Figure 7.1: An illustration used for the derivation of the collision component of the electron distribution. Multiplying by the exponential and integrating over z' then gives: (7.15) In the field-free situation, the integral in the exponential can be evaluated to give: (7.16) Substituting (7.16) and (7.12) into (7.1) then yields the total forward-going component of the distribution, and the result is identical to that developed by starting from the field-free transport relation. The ballistic and collision components of the backward-going ensemble can be developed in an analogous fashion and, of course, yield results identical to those obtained by solving the field-free transport relation.  7.2  Kinetic approach to field-dependent transport  The kinetic approach used in the alternate derivation of the field-free solution can also be applied to the field-dependent problem. The electron distribution is once again partitioned into its forward- and backward- going components in order that the same boundary conditions, i.e., the distributions of carriers injected into the region at its boundaries, can be applied. The procedure is then the same as before: construct each of the counter-directed components by evaluating the different ways by which carriers can come to be in the distribution at some point z; i.e., either by traveling ballistically from the injecting boundaries, or by first suffering a collision before reaching z.  97 Before continuing, there are two points that need to be made. Firstly, it is convenient to consider the distribution function arguments to be {z, k , k±} rather than {z, k,6}, as this z  makes the following discussion more understandable. Once the results have been obtained, however, it is a simple matter to convert the variables back if it is so desired. Secondly, it must be recognized that the wave vector of a carrier becomes a function of position in the field-dependent case. In an arbitrary potential energy profile, V(z), the kinetic energy of a carrier varies between the points z and z' according to  assuming such a trajectory is energetically permissible. Since only one-dimensional potential profiles are considered, the perpendicular component of wave vector, k±, is conserved and therefore  *SM v +  2m  M  =*£W v . +  W  (7  . ) 18  2m  Furthermore, the z-component of wave vector can be written in the following fashion: 9m kt(z') = +^kl(z) + —[V{z)-V{z')} (7.19) K(z')  =  ~)jk*{z) + ^[V(z)-V{z')]  1  (7.20)  where the plus superscript identifies carriers that are moving to the right (forwards), and the minus superscript identifies those moving to the left (backwards).  7.2.1  Ballistic components  Consider first the forward-going component. Figure 7.2 shows the ways in which a forwardgoing carrier can reach the point z ballistically. Note that there are three significant differences between thefield-dependentandfield-freecases. Firstly, carriers injected from the boundary at W can be turned from the backward-going component into the forward-going component by the potential. This coupling of the forward- and backward- going components did not exist in thefield-freecase, where the collision integrals provided the only coupling between the counter-directed parts of the distribution. Secondly, not all carriers injected into the region (from either end) are necessarily going to reach the point z, as carriers can  98 V A  —  ^  1 0  Zpk,l  ^  jI  •  Cl  1  z  —t  *pk,2  Figure 7.2: A schematic of the means by which ballistic carriers can come to be in the forward-going ensemble, at some point z, when an arbitrary potential energy profile is present. The arrows represent the total z-directed energy of an injected carrier, and the distance between the arrows and the potential energy profile represents the z-directed kinetic energy of the carrier. be turned back toward the boundaries by the potential. Thirdly, the potential profile allows carriers which reach z to have a different kinetic energy than they possessed at their injecting boundary. The fact that carriers injected from either end of the region can come to be in the ballistic component of the forward-going ensemble at z suggests writing this component as the sum of two parts, /BAL( » Z  where f£  AL  ^J-)  =  /BAL,I( ' Z  k  z,  k±) +  /BAL,2( ' z i 2  k  kx),  is the part which consists of carriers injected at z = 0 and JQ  (7-21)  l  1 S AL  which consists of carriers injected at z = W. k\ < ^r[V(z k,i) — V(z)], where p  zero and z. Thus / g  A L x  V(z k,i) P  2  As suggested in Figure 7.2, f£  AL1  the P ^ ar  = 0 for  corresponds to the peak in the potential between  is comprised of carriers with k\ > ^ [ V ^ Z p k . i ) — V'(z)]. Relating this  k to the z-component of wave vector, k fl, at the boundary, one has, after evaluating (7.19) z  Z  at z' = 0: kz,o = \jkl + ~[V{z)~Vm  .  (7.22)  The ballistic component at z is then related to the boundary value by multiplying by P(z) as given by Equation (7.10), with the recognition that v and r are, in this instance, functions z  99 of position. The part of the ballistic component of the forward-going ensemble injected at 2 = 0 can thus be written as / B A L , l ( ^  A  :  - ^ ) = /  +  (  ' ^ 0 . ^ )  0  e  X  ~J  P  (kt(Z))T(k(S))  o  Vf  Xkl> -^[V{z )-V{z)]  <- )  2  7  pKl  23  = 0, otherwise, where k+(£) is found by substituting £ for z' in (7.19) and fc(f) = y/k+ {£) 2  Turning now to / B where  V^Zpk^)  A L ] 2  + k\ .  ' note that Figure 7.2 suggests that for k\ < ^[Viz^)  is the peak in the potential between W and z,  /BAL2  =  — V(z)},  0- Further, for  those carriers injected at W with enough z-directed kinetic energy to surmount the largest potential barrier between 0 and W, boundary. For ^r[V(.Zpk,2)  -  /BAL,2  V(z)] < k  2  = 0 as these carriers are absorbed at the left <  ^[Vmax -  V(z)], where  = V(z k i)  the maximum height of the potential profile (V  p  max  corresponds to  in Figure 7.2), the carriers  t  which reach z ballistically after being injected at z = W must have been injected with a z-component of wave vector described by substituting W for z' in Equation (7.20), i.e., k.,w^-^Jk  2  + ~[V(z)-V(W)].  (7.24)  The number that reach z after being turned around by the potential is then obtained by multiplying by the probability a carrier travels from W to £i, and then back to z, without suffering a collision. The arguments surrounding the derivation of Equations (7.10) and (7.23) suggest this probability is found from  w »,(*,-(0M*(0)  +  L «.(* (OM*(0)JJ' +  ( 7 - 2 5 )  where k~(£) is found by substituting £ for z' in (7.20). The part of the ballistic component of the forward-going ensemble injected at z = W can thus be written as / B A L ^ . **I ±-) k  = /  W  *,w,  *U.)  k  exp  \Jw »,(*r(OW*(0)  9m  4 «,(fc+(0)r(*(0)  9ro  if f?[V(z , ) - V{z)] < k < ^ [ V 2  pk  2  z  m a x  - V{z)]  = 0, otherwise. (7.26) The point Ci is found by setting z' to Ci in (7.18) and finding the point where k id) = 0; z  i.e., by inverting V(Ci) = ^  + Viz).  (7.27)  100  C2  2pk,2  Figure 7.3: A schematic of the means by which ballistic carriers can come to be in the backward-going ensemble, at some point z, when an arbitrary potential energy profile is present. From now on, as a matter of notational convenience, it will be taken to be understood that V+T  = v,{k?(Z))T{k{Q)  and  ^(MOM^O)-  =  V~T  The ballistic components of the backward-going ensemble can be evaluated using a similar procedure. With reference to Figure 7.3, the result for carriers injected from the left can be written as rC,2  /BAL,I(^. » ±) = / ( ° > **< , >> ±.) p [- ( j k  k  +  k  e x  o  ^  if | > ( * p k , i ) - V(z)] <kl<  + y | ^  —  c2  m  a  j  {  -  V(z)}  (7.28)  = 0, otherwise, while for those injected from the right one has /BAL,2( > 2:  -L) =  Kw,  k  ±)  exp  k  W z u  2m, if^>^[V>  p k  , )-y(z)]  (7.29)  2  = 0, otherwise.  7.2.2  Collision components  Consider first the forward-going component. Figure 7.4 shows the ways in which a carrier can come to be in the forward-going ensemble at the point z after suffering a collision. As  101  Zpk.l Cl  Ca  Cb pk,2 z  Figure 7.4: A schematic of the means by which carriers can come to be in the collision component of the forward-going ensemble, at some point z, when an arbitrary potential energy profile is present. indicated in the figure, there are three different ways in which this can come about, depending on the z-directed kinetic energy of the carrier after the collision: 1. A carrier can suffer a collision to the left of z, into a forward-going state with enough z-directed kinetic energy to surmount the maximum potential barrier between 0 and z. 2. A carrier can suffer a collision into a backward-going state at some point to the right of 2pk,i)  o  r  into a forward-going state between z k,i and z, with a z-directed kinetic p  energy such that ^ [ V ( z , ) - V(z)} < k]{z) < * f [ V > pk  2  3. A carrier can suffer a collision between z  pk)1  p M  ) - V(z)}.  and z k,2, into a state with a z-directed P  kinetic energy less than V(z k,2)> and thereby be caught in a potential well around z. P  The total collision component of the forward-going ensemble is found by summing the contributions of each of the mechanisms. The same basic procedure used in the field-free situation is followed here as well; i.e., find the number of carriers that have collided into the relevant component of the distribution at some point in space, multiply by the probability that these carriers reach z, and integrate this product over the appropriate region of space. Note that the main differences between the field-free andfield-dependentsituations are the number of  102 ways a carrier can come to be in the collision component of the forward-going ensemble and that, as discussed above, a carrier's wave vector becomes a function of position. Consider carriers of the first type enumerated above, which scatter into the forward-going ensemble at some point z' to the left of z. First note that if k < ^[V(z k,i) - V(z)], where 2  z  p  V{zpk i) corresponds to the peak in the potential between zero and z, then /COL.I ~ 0- ^> t  on the other hand, k > ^[V(zp^i) - V(z)], then the incoming collision integral and the 2  z  z-directed velocity need to be evaluated at a wave vector with z-component defined from Equation (7.19). Then, C (z',kl(z')) in  dz' v (kt(z'))  = number of carriers which have scattered into /  +  and remain  z  within dz' of z', and would have a z-component of wave vector equal to k if they were to reach z, z  (7.30) and C (z',kt(z'))in  dz'  exp  v (k+(z')) z  iL'  f Jz'  = number of such carriers that reach z.  i.  (7.31)  v T  Integrating from zero to z then gives C (z',fc (z')) +  in  /COL,I( > k i k±) — j Z  (* (*0) +  Z  Jo  v  .  exp  Jz' V+T  dz',  2m, Xk >^[V(z )-V(z))  (7.32)  2  z  pK1  = 0, otherwise. Consider now carriers of the second type enumerated above. The carrier scatters at z' into a state with wave vector such that  |>CSPM) -  V(z)] < k {z) < -^[V(z ) - Viz)], 2  2  z  pK1  (7.33)  and thereby has its motion restricted, as indicated in Figure 7.4, to Ci on the left. If the carrier scatters into a forward-going state between £i and z, then the carrier can travel directly to z. If, on the other hand, the carrier scatters into a backward-going state, then the path it follows after the collision is from z' to Ci in the backward direction, and then from Ci to z in the forward direction. Furthermore, to get the total number of carriers of this type one needs to integrate carriers that collide into backward-going states from W to Ci,  103 and those that collide into forward-going states from Ci to z. Following the same procedures used above gives /+  (  ^C (z',K(z')) f C (z',K(z'))  u h \  r  in  / /-ci  dz' +  in  C-M,kt(z'))  exp  (kt(z')) 9m  (7.34)  V+T  Jz'  9m  if ^[V(z )  - V(z)] < kl(z) < ^[V(z )  pK2  - V(z)}  pktl  — 0, otherwise. From now on, as a matter of notational convenience, it will be taken to be understood that vt(z') = v (kt(z')) and v;(z') = v (k;(z')). z  z  Now consider the third type of carrier enumerated above. These carriers are scattered into states without enough z-directed kinetic energy to escape a potential well around z. The contributions from carriers scattering into both forward- and backward- going states need to be summed up over the region of space from ( to ((,. Further, note that once a carrier a  reaches z in the forward direction, it will continue to travel back and forth in the potential well (until it undergoes another collision), and so there are, in principle, an infinite number of paths the carrier can take to arrive at z in the forward direction, all of which need to be accounted for. For example, the contribution of carriers which scatter into forward-going states between Ca and z can be written as  COL,Caz( > Z  zM  k  =  C (z',kt(z'))  f  in  *C (z',kl(z')) in  f  tW  V  'Ca  J<a  i:^C (z',fc+(z')) s: C (z',kt(z')) in  'Ca  V+(Z>)  in  vt(z')  exp  .  exp - / Jz'  exp exp  -  „  •CUz',kt(z')) vt(z')  — v  dz' + exp - f  t \ T  r di]  L /  — dz' + zT  V  r di  dz' +  —— exp - 2 d>  Jz'  - f 00  Vj  Jz'  Jz'  L  V+T\  -^-1 V+T\ r z  - 1 . J  J  dz' + zT  V  exp [-3 / -^-1 di'  [  exp  J  dz' +  V T\ Z  di_  (7.35)  104 where  VT  J  J  Z  V+T  z  V-T  J  Cb  J  C a  V+T  (7.36)  describes one round trip inside the potential well. Similar expressions can be written for the contribution of carriers that scatter into forward-going states between z and (j,, and the contribution of carriers that scatter into backward-going states between  and (, . The total contribution of these carriers can then a  be written as z^exp  ~f  —  n  <«C (z',k (z')) in  7  exp  r C (z',k?(z')) in  \Jz>  J  V-T  Q  dz'+  V+T  (7.37)  dz'+  exp  r& c (z',kt(z')) in  exp V+(z'') 2m iikl{z)<^[V{z )-V{z)\ Jz  V+T  \Jz>  J  < b  V-T  J  C  O  V+T  dz'j  w  = 0, otherwise, where z corresponds to the position of the top of the potential well (e.g., z — z w  w  pk]2  for the  potential energy profile of Figure 7.4). The overall collision component of the forward-going part of the distribution is then /COL( ' 2  z,  k  k±) —  kl) + / c O L , 2 ( ' 2  /cOL,l( > z  k  z,  k±) + fcOL,s( i  ^J-)"  Z  (7.38)  For the collision component of the backward-going ensemble, consider Figure 7.5. As in the forward-going case there are three ways by which a carrier can come to be in the backward-going ensemble at z after suffering a collision. Firstly, for carriers that come from the right and arrive at z with k (z) > ^ [ V ( z 2  z  ,  u  u  ^  pki2  ) — V(z)] one can write  f C (z',K(z'))  \  Z  in  ( ['  Xkl(z)> -^[V(z^ )-V(z)} 2  2  0, otherwise.  #  dz' (7.39)  Figure 7.5: A schematic of the means by which carriers can come to be in the collision component of the backward-going ensemble, at some point z, when an arbitrary potential energy profile is present. Secondly, for carriers that arrive at z in the backward-going ensemble with ^ [ V ^ Z p ^ i ) — V(z)\ < k (z) < | ? [ V > , ) - V(z)], one can write 2  pk  2  + r *L  <2 / c O L , 2 ( ' kg, k±_) — / Z  e  JO  U  X  P  L  Z\^)  / ' W  ~t V  \-JZ-  • * , - ( * ' ) )  4  z( ')  V  2m h  if ^ W * 2  L  Z  k  V  z  dz' +  T  \ - ( [ dz' ' *+  \Jz>  .  P  J<2  T  (7.40)  z  v  ,  2m  , i ) - V(z)] < kHz) < -^[V(z , )  - V{z)\  pk 2  = 0, otherwise. Thirdly, for carriers that are bound by a potential well around z after a collision, one can write  <QC- {z',kt{z')) exp m  rC (z',k;(z')) in  2m  exp  iik (z)<^[V(z )-V(z)} 2  z  0, otherwise.  w  \Jz> V+T 4  '<6  d£  exp  <°C (z',k-(z>)) in  .  \Jz>  V-T  v rj\ z  (7.41)  dz'+  J  C a  V+T  4  V-T  J \  J  106 The overall collision component of the backward-going part of the distribution is then / C O L ( ' zi k±) = / C O L , I ( ' kz, k±_) + /COL,2( > k , kj_) + / C O L , 3 ( ' kzi k±). Z  k  Z  2  (7-42)  Z  z  7.3  Thefield-dependentsolution in spherical coordinates  In this section thefield-dependentsolution is written down in terms of {z, k,6} instead of {z, k , k±}. The spatial dependence of wave vector magnitude was determined earlier to be z  as in Equation (7.17), which can be written as Hz') = /k (z) + ^[V(z)-V(z%  (7.43)  2  ]  The spatial dependence of the angle between k and the z-axis can be determined from 9 = cos  ^ ,  -1  (7.44)  k  and use of (7.19) and (7.20) gives ° (z')  e-(z>) 9  {  Z  lk*(z)c B*6(z) + ,r^;:; ^ k (z) +  ^  £[V(z)-V(z')}  2  l_m^^)^^)-v^))\  cos-  -  )  z,  n ygi[V(z)-V{z')]  0  = cos-  +  [  V  k*(z)  +  £.[V(z)-V(z')]  (746)  •  [  7  A  b  )  Let  h  2  ko =  \lk (z) + —[V(z)-V(0)]  (7.47)  2  lk (z)cos e(z) + 2  e  0 = cos-  1  £ [V(z)-V(0)]  2  [ f  k (z) + 2  t  ;  7.^ £[V(z)-V(0)]  :™  ^  and k a  =  \jk {z) + ^[V{z)-V(W)\  (7.49)  2  -if  l (z)cos e(z) + £;[V(z)-V(W)}\ k2  2  r  7  ,  m  Then the distribution function components developed in the previous section can be written as in the following equations.  /BAL^>M)  = / (0,A;oA)exp +  L  - ( • § - , Jo z. u  T  iik cos 9> -^[V(z , )-V(z)} 2  2  2  pk 1  = 0, otherwise,  ( 7  -  5 1 )  107  /BAL  2( > Z  ,  k  0  M  ) = / (W> if j£[VM  e x  P  - V(z)] < fc cos 0 < ^ \ V 2  - V(z)\  2  W  (7.52)  0, otherwise,  Qf ^ + £ ^ &  > °) = / ( ° > o, *o) exp [-  k  +  fc  if ^ [ V ( z  ) - V(z)] < /c cos 0 < ^ [ V 2  p k l l  - V(z)]  2  m a x  (7.53)  = 0, otherwise, di  /BAL,2( > >0) = f ( > w, Ow) exp 2 fc  w  k  2m, if^ cos e>^[l/(z 2  2  w pki2  VZT  (7.54)  )-l/(z)]  = 0, otherwise,  exp v+(z>) 2m, iik cos e>^[V(z )-V(z)} W  COL,lV > iV) — \ Jo Z  K  2  dz', Jz'  i,  v  T  (7.55)  2  pkil  — 0, otherwise,  , tcoifl&W)-J  /*  +  w  C (z',k(z'),6-(z')) exp v  I  dz' +  in  C (z',k(z'),6+(z')) in  vt(z')  Jci  \Jz>  { z l )  exp  V-T  _ r %dz'  . Jz'  J  0  V+T  (7.56)  Vf  if | > ( * p Zpk,2 k ) - V(z)] < A; cos 0 < -^[V(z ) - V{z)\ 2  | 2  — 0, otherwise,  2  2  pktl  108  {  d£  I  °°  u  x  C (z',k(z'),e-(z')) in  ft  ( f _^L CA  exp  ".-M  A.  exp <"C (z',A;(z'),0 (z')) exp v+(z>) 2m itk cos e<^[V(z )-V(z)} nr  f  +  iL  dz'+ (7.57)  dz'+  +  in  2  V+T J  \Jz'  J  V-T  a  <a  V+T  2  w  • 0, otherwise,  C (z',k(z'),6-(z')) in  C O L , I ( ' k,0) — 2  exp Jw 2m, iik cos e>^[V(z )-V(z)} W 2  dz'  VZT  (7.58)  2  pK2  0, otherwise,  r  <zho\  f ^(z',k(z'),e+(z'))  <2  <2  '  C (z',k(z'),9-(z')) in  exp  L C2  ^yCspw) -  if  exp  iL + f i£ V+T  d£  L C2  dz' +  Z  U  V-  dz' +  (7.59)  V(z)] < k cos 6<-^[v(zpk,2)- V{z)] 2  2  2  = 0, otherwise,  ln=0  L  J  & Ctaiy, A(^),  e  rc iz',kiz'),e-jz')) in  7  v~iz')  C6  j^C iz',kiz'),e-iz>)) in  if k cos 9< 2  2  = 0, otherwise.  2m —  x  [_ ( P i L +  f A  dz'+  'a exp  '_([ i L + /•ft' i L f i L Ca  exp  [V(z )-V(z)] w  (7.60)  dz'+ C6  +  dz'}  109  7.4  Solution in a constant electric field  As an example of the methods described above, a solution for the case of electron transport in a uniform electric field is developed. The electric potential, V, of a z-directed electric field, £ , can be found from (7.61) With £ constant the solution of this equation i (7.62)  V(z) = -£z + V(0), and multiplying by the charge on an electron, —q, gives the potential energy profile V(z) =q£z  (7.63)  + V{0).  In what follows the electric field is taken to be negative, so the field tends to push electrons from left to right, i.e., the direction referred to as forward in this thesis. This linear and negatively sloped potential profile provides that z k,i = 0, and that z k,2 always occurs at p  p  z. Using the equations for the various components of the distribution listed in the previous section, the solution can be written as  if k cos 6 > az 2  2  f (W,k ,6 )ex\o w  w  -  +  ^c {z',k{z'),e-{z')) m  wv  exp  v-(z')  dz' +  -  (7.64)  if k cos 9 < az 2  2  * C- {z',k(z'),6-{z')) exp m  v  7( ')  v  z  (7.65)  110  where 2m  a k(z') =  (7.66) (7.67)  s/k - a(z - z') 2  Ik cos 9 - a(z - z') 2  9 (z') =  cos'  k  Vk ^  9o  cos -l  ±  (7.68)  k — a(z — z') 2  2  0  2  1  (7.69)  az k cos 6 — az 2  2  (7.70)  k — az 2  kw  yjk + a(W - z)  Ow =  cos  (7.71)  2  k cos 9 + a(W - z) 2  (-  -1  z —  (7.72)  k + a(W - z) 2  k cos 9 2  Ci  2  2  (7.73)  Note that the solution (7.64,7.65) can also be written as f {z, k, 9) +  =  / ( 0 , fc ,0 ) exp [- ( I o  II+  +  0  Q  V+T  r CUz',k(z'),9+(z')) C (z',k(z'), in  Jo  v+(z>  dz'  exp  if k cos 9 > az 2  =  2  +  /~(Ci,*sinfl,7r/2)exp VJT  rc (z',k(z'),9 (z'))  di  +  m  4  v+(z> )  exp  (7.74)  if k cos 9 < az 2  f~(z,k,9)  =  2  f~{W,k ,9 )exp w  +  w  W  VZT  CUz',k(z'),9-(z'))  f  vi(z ) 1  exp  Jw  7.4.1  dz'  \Jz>  dz', V  Z  T  (7.75)  S o l u t i o n w i t h isotropic a n d elastic scattering  Consider now the situation in which only isotropic and elastic scattering is accounted for, such as would be the case when only acoustic phonon scattering is important. The lifetime and incoming collision integral for such a situation were found earlier to be  r = Cm =  1  (7.76)  cx^pk  a kf (z,k), ap  0  (7.77)  Ill where h{z,k) =-J^f(z,k,9)de. l  (7.78)  This lifetime leads to a mean free path length of l  sc  =  h VT  a  a p  (7.79)  m'  Then C (z',k(z'),e (z'))  _  ±  m  vf(z')  f (z',k(z')) 0  l cos 6 (z')' ±  (7.80)  sc  Furthermore, d£ Zsc  COS  ^(e)'  (7.81)  where 9 (£) is found by substituting £ for z' in (7.45) or (7.46), which gives ±  /  vfr  I  ±1 lsc  \J /  k +  a(£-z)  2  d£. k cos 9 + a(i - z) 2  2  (7.82)  Let \±(z')  =  (7.83)  l cos 9 (z') ±  sc  and (7.84)  J Xl Then, using the tabulated integrals dx px + q , J (ax + b) (px + q) aq - bp dx — — 1 ax + b 2a J ^/(ax + b)(px + q)  I  A/(ax + b) (px + q) ^aq — bp 2 a  2a  y/ap  In (^\Ja(px + q) + y/p(ax + b)j (7.85)  112  one has that - ! - J£; |cos0| - y/[k cos 0 la { 2  2  2  - az][k - az] + 2  sc  k{l + |cos0|)  k sin 0 In 2  ±1  I  2  - {k \cos0| - y/[k cos 0 a 2  2  A+ ,  2  2  (7.88)  sin0  - J\k cos 0 + a{W - z)][k + a(W - z)] + 2  2  2  fc(l + |cos0|)  k sin 0 In 2  (7.87)  2  (l + |cos0|)" |  sc  2  (7.86)  ^/fe + a{z' -z) + yjk cos 0 + a(z' - z)  2  - lk \cos0l a(  }  2  2  -^<{fc |cos0| + A; sin In la (  A  2  + a(z' - z)][k + a(z' - z)] +  2  2  2  Jt(l + |cos0|)  k sin 0 In 2  2  sjk — az + \/A; cos 0 — az 2  2  (7.89)  yjk + a(W -z) + ^/k cos 0 + a(W - z) 2  2  2  Then the solution (7.74,7.75) can be written for isotropic and elastic scattering as / + ( * , £ , 0)  =  / (O,fc ,0o)exp[-A+ ]+ +  o  z  [  X (z')f (z',k(z'))exp[-A+ ]dz'  Z  +  0  Jo  iZ  if k cos 6 > az 2  =  2  /-(Ci,fcsin0,7r/2)exp[-A+ J+ f i  X (z')f (z' +  0  )  k{z')) exp[-A+ ]dz' z  J^i  k cos 6 < az 2  f-{z,k,9)  =  (7.90)  2  f-(W,k ,9 )ex.p[-A ]+ w  w  Wz  f \-{z')f {z',k(z')) exp[-A;, ]dz'. 0  Jw  (7.91) These equations were solved numerically using an iterative procedure analogous to that used in the field-free solutions described earlier, and the results appear in the following chapter.  Chapter 8 Field-dependent results This chapter presents a study of field-dependent transport. The study is restricted to the situation in which the electric field is uniform and the collision integrals are those describing isotropic and elastic scattering mechanisms, such as acoustic phonon scattering. A detailed specification of the problem is presented first, followed by an examination of the equilibrium picture. Two modes of nonequilibrium transport are then analyzed in terms of dependence on bias, field strength and region width. The result for current density is then compared with results predicted by the drift-diffusion equation (DDE). It is shown that, using the same boundary conditions as in the field-free situation studied earlier, the D D E can lead to nonphysical predictions. Taking a particularfield-dependencefor the mobility, however, is shown to alleviate this problem. The mobility, as evaluated directly from the distribution function, is studied, and the issue of the nature of mobility in a forward-biased semiconductor junction is examined. The work of others on this topic is addressed, and some of the contradictions therein are explained by showing that the mobility depends only on the antisymmetric part of the distribution. Under the scattering model considered here, it is shown that the mobility is independent of the mode of device operation in a constant electric field.  113  114 f (k ) +  z  Figure 8.1: A schematic of the potential energy profile, and the boundary conditions for injected carriers, used to study the field-dependent solution to the transport equation.  8.1  Specification of the problem under study  The solution of the time-independent transport equation, including the effect of a constant electric field and collision integrals appropriate for isotropic and elastic scattering, has been developed in the previous chapter (7.90,7.91). In this study, the boundary conditions on the injected fluxes, f (0,k,9) +  and f~(W,k,9),  are considered to be hemi-Maxwellians, as was  the case for the homojunction devices studied earlier. The situation is shown schematically in Figure 8.1. The results here can be considered to apply equally well to the base region of a bipolar transistor or to a Mott barrier. Mott barriers are barriers in which the field is constant, while in the base region of a bipolar transistor a field can exist due to a varying doping profile or to compositional variance in the case of compound semiconductors. Two modes of operation are studied. The first, referred to here as the forward-mode, describes the situation in which the net flux exists in the same direction as the force on the carriers, i.e., the net flux is moving down the potential energy gradient. The second, referred to here as the reverse-mode, describes the situation in which the net flux exists in a direction opposite to that of the force on the carriers, i.e., the net flux is moving up the potential energy gradient. Finally, only low and moderate field strengths are examined, since the restriction on the collision integrals to elastic scattering cannot describe the velocity saturation effect occurring at high electric fields.  115 To represent the forward-mode situation, the flux injected from the left is increased beyond its equilibrium value, while that injected from the right is maintained at its equilibrium value. The reverse-mode is obtained by leaving the flux injected from the left at its equilibrium value while increasing that injected from the right. At equilibrium n(0) = fi and E  n{W) = n , and the following relation is obtained: c  n* = n e- ^K  (8.1)  £W  c  E  Employing the hemi-Maxwellian boundary conditions allows the solution (7.90,7.91) to be written as: f {z,k,9)  =  +  i -'( e  N  f c 2  -  a z  rA ( ')/o(2^(/))e- ^d ' Jo  le- t+ A  +  A  Z  c  Z  if k cos 9 > az 2  =  2  /-(Ci,fcsin0,7r/2)e- ci." + / ' A  X (z')fo(z',k(z'))e-^'-dz' +  if k cos 9 < az 2  f-{z,k,6)  =  (8.2)  2  ^ e - ^ + ^ - ^ e - ^ + [' A Jw  \-(z')f (z ,k{z'))e- ''.'dz', ,  A  0  c  (8.3) where rj = tl /2mk T 2  B  and the other parameters are as defined in Section 7.4.1.  Notice that in writing the solution, the six-dimensional phase space has been divided up into three distinct regions, which will be denoted as R , R and R . In terms of the x  2  3  backward- and forward- going components, the backward-going component will be referred to as R i , while the forward-going component will be referred to as R if k cos 9 > az, or 2  2  2  R  if k cos 9 < az. Note that, with reference to (7.66), the condition k cos 9 > az can 2  3  2  2  2  be written as E — q\£\z, i.e., it refers to z-directed kinetic energies at least as great as the z  potential energy difference between 0 and z. It is interesting to plot the projection of a carrier's possible phase space trajectories onto the (z, k )-p\&ne, as illustrated in Figure 8.2. Also shown in the figure are the possible ways z  in which a carrier can come to be in the electron distribution at some point in space, which were discussed in detail in Chapter 7. This diagram is useful in the interpretation of the results which follow. It can be appreciated immediately, for example, that ballistic carriers injected from the right that have been "turned around" by the potential gradient appear in R3  116  Figure 8.2: The projection of carrier trajectories onto the (z, fc )-plane. Also shown are the possible ways in which a carrier can come to be in the electron distribution at the point in space marked by the vertical dashed line. z  8.2  The equilibrium picture  The equilibrium picture itself is worth some study, as certain ideas that are needed to understand the results for the forward- and reverse- modes are relevant here as well. The function / , in a region with £ = l0 V/m and W = 2 x / , is shown in Figure 8.3. This 5  0  sc  plot exhibits the expected energy dependence and increasing weight with position, and has been normalized to the peak value of / at z = 0. In the rest of this chapter, the distribution function plots for forward-mode results have been normalized to the peak value of / at z = 0, while those for reverse-mode results have been normalized to the peak value of / at z = W. The total equilibrium distribution, along with its ballistic and collision components, is shown in Figure 8.4.  The total distribution is, of course, an appropriately weighted  Maxwellian. The ballistic and collision components, however, exhibit a number of interesting features. Consider first the ballistic component injected from the right. For the backward-  117  Figure 8.3: The function / (z, k) at equilibrium, for a field strength £ = 10 V/m and a region of width W = 2 x l . 5  0  sc  going part of this component (i.e., 90 < 6 < 180), notice how as 6 gets closer to 90, the distribution falls off. This is because such carriers see an effectively longer path than those directed more closely along the z-axis. For the forward-going part of this component (i.e., 0 < 6 < 90), notice that the component vanishes for k cos 6 > az (i.e., in R ) . This is 2  2  2  because ballistic carriers that have been injected from the right and subsequently turned into forward-going carriers by the field cannot have a z-component of wave vector such that k > az (see the trajectory corresponding to 2  m  /BAL2  Fig  u r e  8.2). Ballistic carriers with  a z-directed component of wave vector such that k > az must appear in the component 2  injected from the left, as shown in the lower left of Figure 8.4. Note that for carriers with k < az this component correspondingly disappears, and for angles closer to 90, the longer 2  effective path length once again causes the distribution to fall off more rapidly than for angles closer to 0. Consider now the collision component. Notice how there appears to be a depression of the forward-going part of this component where k cos 6 > az (i.e. in R ) . This occurs 2  2  2  because, as consideration of Figure 8.2 indicates, there is an effectively shorter path over which carriers can collide into appropriate states to end up in R at z, than to end up in R 3 . 2  118  BALLISTIC COMPONENT  TOTAL DISTRIBUTION  INJECTED FROM THE RIGHT  180  COLLISION COMPONENT  BALLISTIC COMPONENT INJECTED FROM THE LEFT  180  180  k (/im ) 1  Figure 8.4: The total equilibrium distribution, along with its ballistic and collision components, at the middle of a region with field strength £ — 10 V/m and width W — l . 5  sc  At equilibrium this is of course balanced by the ballistic part in R to yield the Maxwellian 2  shape of the equilibrium distribution. This depression gets less pronounced if the distribution is examined further along the region, and more pronouced the closer to the left hand edge the distribution is examined.  8.3  The forward-mode results  In this section the forward-mode results for the electron distribution are studied. Recall that the forward-mode is achieved by maintaining an equilibrium flux injected from the right while increasing the flux injected from the left beyond the equilibrium value. Consider first the bias dependence of the distribution. The results for a range of injection levels, at the halfway point of a region with W = 2 x Z and £ = 10 V/m, are shown in Figure 8.5. The first 5  sc  thing to notice is that after a certain level of injection is reached the shape of the normalized  119  Figure 8.5: The full forward-mode distribution, for a variety of injection levels, at the middle of a region with field strength S = 10 V/m and width W = 2 x / . 5  sc  120 distribution no longer changes with increasing injection. In the situation described here this occurs at a level somewhere between 20 and 100 times the equilibrium injection level. This behavior is similar to that found earlier for the case of a field-free region subject to the same boundary conditions on injected carriers. This behavior would not be expected to carry over into the situation of an abrupt heterojunction device consisting of a graded base region, as the shape of the injected carriers has both a bias and an angular dependence as discussed in Chapter 6. Consider now the higher level injection results, i.e., those for an injection level at least 100 times that of the equilibrium level. Note that the distribution vanishes for k  2  < az.  This is because at these injection levels the carriers injected from the left overwhelmingly dominate those injected from the right. Further, carriers injected from the left must have k > az because they pick up energy from the electric field as they move across the region. 2  This is also a consequence of the elastic scattering model applied in this instance. If carriers were allowed to give up energy in collision events, the distribution would not be expected to vanish at these low energies. Similarly, the nonvanishing distribution appearing at k  2  < az in the results for lower  levels of injection must then describe part of the electron ensemble injected from the right. It is interesting to note the angular isotropy of this part of the distribution. More will be said about this later on when the reverse-mode results are discussed. The dependence of the forward-mode distribution on field strength, at the middle of a region with W = 2 x Z , is shown in Figure 8.6. The level of injection is taken to be sc  n = 10 x n . Not unexpectedly, the low field result for £ = 10 V/m is very similar to the 5  E  3  E  field-free distribution shown in Figure 4.5, except when k is very small. For very small k, the difference between this and the field-free result is due to the forward-mode distribution tending to vanish for k  2  < az as discussed above. At £ = 10 V/m the displacement of 4  the distribution along the /c-axis has become noticeable, and this trend continues to become more pronounced as the field strength is further increased. This is simply a manifestation of the potential energy difference between z — 0 and z = W/2 increasing with increasing field strength. Another interesting trend is that the distribution becomes more closely focused about the z-axis as the field strength is increased. This is because it is only k that is increased z  more by the higher fields, while k± remains unchanged by the field. Finally, note the size  121  Figure 8.6: The field strength dependence of the full forward-mode distribution, at the middle of a region with width W — 2 x l and an injection level of n = 10 x n . 5  sc  E  E  of the distribution is decreasing with increasing field strength. This is because as the field increases the speed of the carriers, it must also rarefy their density in order to maintain a constant current density across the region. The dependence of the distribution on region width is shown in Figure 8.7. The results are taken at the middle of the region. Here the injection level is again n  = 10 x n and 5  E  E  the field strength is taken to be £ — 10 V/m. At very short region widths, the distribution 5  is concentrated in R . This is because the dominant part is the ballistic component injected 2  from the left. As the region width gets larger, carriers travel a larger distance to reach the middle of the region. They thereby undergo more collisions, leading to the decrease in the importance of the ballistic component and corresponding increase in the collision component. The result is the trend to a more angularly uniform shape of the distribution as the region width is increased. Another feature evident in the figure is the trend toward increased displacement along  122  W = l.0x L  W = 0.5 x L  180  180  k {nm ) l  V7 = 4.0 x Z  W = 2.0 x Z,  s  8  180  180  k (f«n  )  k (firri ) 1  Figure 8.7: The width dependence of the full forward-mode distribution, at the middle of a region with field strength £ = 10 V/m and an injection level of n = 10 x n . 5  5  E  E  the A;-axis as the region width is increased. This is again due to the carriers traveling a longer distance to reach the halfway point as the width is increased, leading to a larger increase in the kinetic energy derived from the electric field.  8.4  The reverse-mode results  Consider the bias dependence of the reverse-mode results, shown for a range of injection levels, at the halfway point of a region with W = 2 x /  and £ = 10 V/m, in Figure 8.8. 5  s c  Note that the change in the shape of the distribution is much more subtle than in the forward-mode. However, a similar aspect to the forward-mode is that above a certain level of injection, the normalized distribution becomes bias independent. One of the main features of the reverse-mode is that for k  2  < az, the distribution is  angularly isotropic. In fact, it has a Maxwellian energy dependence as well. This is somewhat surprising at first. However, consideration of the equilibrium situation suggests this is to  123  k (urn )  k (fim *)  1  Figure 8.8: The full reverse-mode distribution, for a variety of injection levels, at the middle of a region with field strength S = 10 V/m and width W = 2 x l . 5  sc  124 be expected. From the discussion of the forward-mode it is known that these carriers must have been injected from the right. At equilibrium, we know that these carriers must form a Maxwellian. Note that the only difference here is that the size of the distribution injected from the right has been increased, but it still maintains its hemi-Maxwellian form, so the same shape is achieved for carriers with k < az regardless of injection level. This explains 2  the low bias, forward-mode results for k < az as well. Finally, note that if inelastic scattering 2  mechanisms were to be included, this trend may not continue to hold since carriers injected from the left could come to be in states with k < az. 2  For k > az, notice that there is a depression in the distribution in R . This depression 2  2  becomes more pronounced as the bias is increased. This is because as the injection from the right is increased, the ballistic component in R (i.e., that injected from the left) becomes a 2  negligible part of the overall distribution. In R\ and R , although it may not be readily apparent by visual inspection, the distri3  bution is slightly lower than it would be for a truly Maxwellian form. This is because of the "missing" ballistic carriers in R effectively lowering the collision components in these 2  regions by lowering fo(z, k). The dependence of the reverse-mode distribution on field strength, at the middle of a region with W = 2 x l , is shown in Figure 8.9. sc  The level of injection is taken to be  n* — 10 x n* . As in the case of the forward-mode, the low field result for £ — 10 V/m is 5  c  3  c  very similar to the field-free distribution shown in Figure 4.5, except when k is very small. For very small k, the difference between this and the field-free result is due to the reversemode distribution tending to a Maxwellian for k < az as discussed above. At £ = 10 V/m 2  5  the Maxwellian form at low k is clearly evident, and further increasing the field strength leads to a situation where the distribution function is Maxwellian up to larger values of k. This is just a result of the potential energy difference between z = 0 and z — W/2 increasing with increasing field strength. Finally, note the size of the distribution is decreasing with increasing field strength. This is because as the field strength increases, fewer carriers injected from the right will have enough z-directed kinetic energy to reach the middle of the region. The dependence of the reverse-mode distribution on region width is shown in Figure 8.10. The results are taken at the middle of the region. Here the injection level is again n* — c  10 x n 5  and the field strength is taken to be £ = 10 V/m. 5  c  At very short region widths,  there is a clear depression of the distribution in R . As discussed above, this is a consequence 2  125  Figure 8.9: The field strength dependence of the full reverse-mode distribution, at the middle of a region of width W — 2 x / and an injection level of n* = 10 x n* . 5  s c  c  c  of the negligible number of carriers injected from the left in the reverse-mode, leading to a negligible ballistic component in R . As the region width gets larger, carriers travel a larger 2  distance to reach the middle of the region. They thereby undergo more collisions, leading to the decrease in the importance of ballistic components and corresponding increase in the importance of collision components, and the depression seen at short widths gradually disappears. Another feature of the width dependence of the reverse-mode is that as the width increases, the Maxwellian portion of the distribution exists up to higher energies. This is due to the fact that as the region width increases, so does the potential energy difference between the left edge and the middle of the region, which in turn determines the maximum energy of the Maxwellian portion of the reverse-mode distribution, as discussed earlier.  126  Figure 8.10: The width dependence of the full reverse-mode distribution, at the middle of a region with field strength £ = 10 V/m and an injection level of n* = 10 x n . 5  5  c  8.5  c  Comparison with the D D E  In this section results for the current density are presented. Both the field dependence and width dependence are examined, and compared to results obtained from the drift-diffusion equation. The expression for the D D E current is developed using the boundary conditions proposed by Hansen, (5.10), which are consistent with the hemi-Maxwellian boundary conditions applied to the B T E as discussed in Chapter 5. Consider the drift-diffusion equation for electrons, J = qnn£ + qD—,  (8.4)  which can be written as (8.5)  127  where e = ±S.  (8.6)  Equation (8.5) has solution n(x) = ce-  +  ex  (8.7)  or, in terms of the carrier concentrations at the boundaries, n(x)  =  n(0)e~ + - £ - ( 1 - e~ ) que  n(x)  =  n(W)e  £X  (8.8)  £X  eH/  e-  £I  + ^-(l-e  £ , y  e-  £ I  ).  (8.9)  Equating (8.8) and (8.9) provides „  n(V7)e^-n(0)  The boundary conditions for the current density, (5.10), can be written as „(0)  =  ni  n(W)  =  n' --^-,  +  ^ -  (8.11) (8.12)  c  and substituting into (8.10) gives the desired result:  This result can be rewritten in a form specific to base transport in bipolar transistors by noting that n(0)e  nE  (8.14)  VBE/Vth  n(W)e ' >, VBC  (8.15)  Va  and using (8.1): J  = -  ^  1  +  e  sw_^  {  1  _  e  £  W  y  (- ) 8  16  Equation (8.16) is reciprocal, and it was found that the B T E results for the current density are reciprocal as well. The results presented here are thus restricted to the forward mode.  128  0.0 0.1  10.0  1.0 W/l  sl  Figure 8.11: Dependence of the current on region width for a field of 10 V/m and forwardmode flux of 10 times that at equilibrium. 5  5  The width dependence of the forward-mode current density, calculated from the B T E with an injection level of n  = 10 x n 5  E  and an electric field of —10 V/m is shown in 5  E  Figure 8.11. Also shown are the results from Equation (8.16), where the mobility has been taken to be its equilibrium value, \x = no, where for the isotropic and elastic scattering studied here it can be shown that  Mo = l^lsc  (8-17)  The results for the D D E curve can be seen to overestimate those of the B T E curve. This is similar to the situation in the  field-free  to the corresponding result for the  shown in [11]. In comparing the B T E curve  field-free  shown in Figure 4.10, it can be seen  that the field-free result is slightly lower. However, the shapes of the B T E curves for both field-free andfield-dependenttransport are the same.  129  1.2 o BTE  G  1.0  o- - - -o DDE, n = no • • DDE, H = fi(£)  0.8  0.6  0.4 A  0.2  0.0  10°  10°  10  10  H  |f |  (V/m)  Figure 8.12: Dependence of the current on field for a region width of 2 x l and forwardmode flux of 10 times that at equilibrium. The field dependence of the mobility is given by Equation (8.22). sc  5  The field strength dependence of the forward mode current density, calculated from the B T E for a width of 2 x Z and an injection level of n = 10 x n 5  sc  E  E  is shown in Figure 8.12.  Also shown in the figure are the predictions of (8.16) with two different mobility models. The first mobility models assumes the low-field bulk mobility, / i , applies while the second 0  takes the mobility to be a field-dependent parameter, the form of which is given in (8.22) and is discussed shortly. Note that when /x is used, the D D E predicts currents higher than 0  the ballistic limit current for high field strengths. This is a consequence of the nature of the boundary conditions. Recall from Chapter 5 that the boundary condition at z = 0 can be  130  written as J  =  -q2v [n (0)  -n~(0)]  =  -q2v {ny2-n-(0)},  +  R  (8.18) (8.19)  R  or 3  -qn v E  R  = 1 - ^ , n  (8.20)  E  from which it can be seen that n~ (0) must be negative for the current to exceed the ballistic limit of — qn v . This is, of course, a nonphysical situation which does not arise when more E  R  traditional boundary conditions are applied to the DDE. This problem can be alleviated by employing a field-dependent value for the mobility. Consider a forward-mode situation where £ is negative and of large enough magnitude that e  eW  <C 1, and n 3> n* e . Then eW  E  c  Equation (8.16) reduces to ~  = —!: 2  .  (8.21)  H\£\  Note that if the mobility is restricted such that f i \ £ \ < 2v , then the current cannot exceed R  its ballistic limit. This suggests writing the mobility as KS) =  ^ r .  (8.22)  2v  R  In the low field limit, then, /x reduces to Ho, while for large fields /i is restricted to  2v /£. R  This model for the mobility was suggested in [44], although for different reasons than those presented here (see the following section for more details). In fact, it is the usual practice in D D E calculations to employ afield-dependentmobility. In Figure 8.12, the current predicted from (8.16) using the field dependent mobility of (8.22), as opposed to Ho, can be seen to more closely follow those of the B T E solution, especially at higher field strengths. In the forward-mode results studied here, it is quite reasonable to employ a mobility which is reduced as the field strength is increased. Because the carriers are heated by the field, it is expected that the carrier scattering rate will be increased, thereby leading to a reduced mobility (see Appendix B). In the reverse-mode, however, the situation is cloudy. This is because the net flow of carriers is against the field, i.e., the field tends to be cooling  131 the carriers, and it might be expected that the appropriate value of mobility to use in this instance is the bulk, low-field mobility, / i . More will be said about this issue in the following 0  section.  8.6  A study of carrier mobility  8.6.1  T h e issue o f m o b i l i t y i n a forward-biased  barrier  As alluded to in the previous section, there exists a long-standing issue over the nature of mobility in a forward-biased barrier (i.e., a barrier in which the net flux moves carriers up a potential energy gradient) [44, 45, 46, 47, 48, 49, 50, 51, 52]. In a reverse-biased barrier, in which the net flux moves carriers down a potential energy gradient, it is well understood that the electron distribution is heated, and this leads to a reduction in carrier mobility. In a forward-biased barrier, however, the electron distribution is cooled, and it therefore seems reasonable that the carrier mobility should not be retarded from its equilibrium value, and may even be enhanced. At equilibrium, of course, there are as many carriers being heated as being cooled so that the bulk, low field (i.e., equilibrium) value for mobility, fio, is obtained. Historically, this issue may not have had much practical importance in the analysis of bipolar transistors, since their operation was determined largely by transport through a low-field base region and a reverse-biased base-collector junction. In modern transistors, however, device scaling and bandgap engineering have led to optimized devices in which transport through the forward-biased emitter-base junction may be an important factor in limiting device performance. Consequently, the issue of the nature of carrier mobility in such regions has become one of considerable importance. A number of researchers have worked on this problem, and a wide range of contradictory results have been put forth. Stratton[45] argues, by coupling an approximate set of energy and momentum balance equations, that the mobility takes the form M — Mo -^r  ,  (8-23)  where T is the temperature of the lattice and T is the temperature of the electron distriL  e  bution. In such a model the mobility may be a function of position, and can be enhanced or degraded from (io, depending on the form of T . e  Taking a similar approach, but using a different set of approximations, Goldberg[48]  132  argues that H= 1  2  d£  (8.24)  - r° d~z Ti  where r is an energy relaxation time. This form for the mobility leads to the surprising e  result that the mobility may be enhanced, retarded, negative or infinite, depending on the second term in the denominator. By considering only the conduction (or drift) current and a constant built-in field, it is argued by Gunn[46] that, in a barrier carrying current, the mobility is that of a bulk semiconductor subjected to the same electric field. That is, the mobility is retarded regardless of the direction of the current. This view is supported by Thomas and Lin[49], except they note that if the barrier region becomes very short, then \i will start tending toward ^oIn [44] it is argued by Tanaka and Lundstrom that the mobility in a forward-biased Mott barrier should be of the form  „(£) =  where  £  c  —  2VR/no,  (8.25)  in order that the result for the current from a one-flux treatment (shown  by Vaidyanathan and Pulfrey[53] to be essentially equivalent to the DDE) agrees with that from thermionic emission theory. This is the same result discussed earlier, (8.22). In a numerical study based on a solution to the B T E , it is shown by Assad et al. [52] that in a forward-biased metal-semiconductor junction, (8.25) exhibits most of the main features of the mobility as evaluated directly from the distribution function. However, a clear physical description of why this occurs, and where the reasoning of analyses such as that in [45] lead to incorrect results, has yet to appear in the literature. In this work, a detailed study of carrier mobility has been made for the simplest possible situation: a region of constant field in which isotropic and elastic scattering is taken to be the overwhelmingly dominant scattering mechanism. The restricted form for scattering would most closely describe moderately doped, low-field silicon. The mobility is examined for its dependence on region width and field strength, for both the case when the net carrier flux is with the field (i.e., carriers are being heated) and when the net flux is moving against the field (i.e., carriers are being cooled). This, of course, corresponds to the study carried out  133  1.10 1.00 0.90 0.80 o  a.  S,  0.70  'EL  -a.  — £ -- 10 V/m, fm - - £ -- 10 V/m, fm 4  5  0.60  - £ = 10 V/m, o£-- 10 V/m, o£ = l0 V/m, a£ = 10 V/m, 4  0.50  5  6  0.40  i  0.2  0.0  ,  •CL,  fm rm rm rm  6  i  0.4  0.8  0.6  1.0  z/W  Figure 8.13: The dependence of the mobility on field strength for a region width of W = 2 x / for both the forward- and reverse- modes. The injected flux for each mode is 10 times its equilibrium flux. In the legend, fm indicates the forward-mode and rm indicates the reversemode. s c  5  earlier in this chapter. The results for the mobility derived from this study are presented in the next section.  8.6.2  N u m e r i c a l results  In this section results for the mobility, evaluated from the distribution functions found earlier in this chapter, are presented. The dependence of the mobility on field strength is shown for both modes of operation, in a region of width W — 2 x Z in Figure 8.13. In the forward sc  mode, n  = 10 x n , and for the reverse mode n* = 10 x n . At this point it needs to 5  E  5  E  c  c  be made clear that in this study, the forward-mode results correspond to the situation of a reverse-biased junction (i.e., carriers are heated by the field) and that the reverse-mode  134  1.10  1.00  S>  0.90 h  0.80  0.70 0.0  0.2  0.6  0.4  0.8  1.0  z/W  Figure 8.14: The dependence of the mobility on region width for a field strength of 10 V/m, in the forward-mode with an injected flux of 10 times that at equilibrium. 5  5  results correspond to the situation of a forward-biased junction (i.e., carriers are cooled by the field). The most important feature of Figure 8.13 is that the position- and field- dependence of the mobility are identical in both the forward- and reverse- modes. It is also interesting to note that near the left edge of the region, the mobility is slightly enhanced. Over most of the region, however, the mobility is reduced, with the reduction increasing as the field strength is increased. This supports the results of Gunn[46], where it is argued that the mobility is related to the field strength regardless of the mode of operation. The dependence of the mobility on region width, at a field strength of £ = 10 V/m, for 5  a forward-mode bias of n = 10 x n , is shown in Figure 8.14. The reverse mode results 5  E  E  are not shown, however, as they are the same. Note that as the region width gets small, the  135  Figure 8.15: The dependence of the mean kinetic energy on position for a region width of W = 2 x / and field strength of 10 V/m, for both the forward- and reverse- modes. The injected flux for each mode is 10 times its equilibrium flux. 5  s c  5  mobility tends toward the low field, bulk mobility, as argued by Thomas and Lin[49]. The results of Figures 8.13 and 8.14 are, however, in direct contradiction to Equation (8.23), which was proposed in [45].  This model for the mobility, despite the work  in [46] and [49], continued to find credibility in later efforts[47, 51] and has even been put into textbooks[54]. Equation (8.23) can be confirmed to give incorrect results by examining the mean kinetic energy of the electron ensemble, which is a direct measure of its temperature. The mean kinetic energy, u, for both forward- and reverse- modes, is illustrated in Figure 8.15. It can be seen that, as expected, the mean kinetic energy of the forward-mode has been enhanced while that for the reverse-mode has been reduced. This will lead (8.23) to incorrectly predict that in the forward-mode the mobility will be everywhere retarded and in the reverse-mode that it will be everywhere enhanced.  136 To understand why (8.23) leads to incorrect results, note that T is a property of the e  entire electron ensemble. The mobility however, as will be shown in the following section, is a function of only the antisymmetric part of the distribution. The antisymmetric part of the distribution may be only a small part of the total distribution and does not have to reflect the properties of the overall distribution, such as T . e  The antisymmetric part of the distribution is shown at two different positions, for both forward- and reverse- modes, in Figure 8.16. Note that because the antisymmetric part of the distribution for the forward-mode exists in the range of angles 0 < 9 < TT/2 and that for the reverse-mode in 7r/2 < 6 < TT, the results for both modes can be shown on the same plot. Note that the antisymmetric parts for the two modes are identical in terms of dependence on k and k. This leads to the mobilities being identical for the two modes. Furthermore, z  note that the antisymmetric parts get hotter for positions closer to the right hand side of the region. This leads to the decrease in the mobility as a function of position in Figures 8.13 and 8.14. It may seem strange that the antisymmetric part of the reverse-mode distribution should be hot when the effect of the field is to cool the carriers in this case. One can gain an intuitive feel as to why this comes about by considering Figure 8.17.  Recall that in the  reverse-mode the carriers injected from the left are negligible, so that only carriers injected from the right need to be considered. Because the scattering is elastic, then, the only carriers that can contribute to the net current must be injected from the right with energy greater than AE, where AE is defined in the figure. This is because lower energy carriers cannot reach the boundary at z = 0 and therefore are eventually returned to the injecting boundary at z = W. Consequently, at z = W, the antisymmetric part of the distribution must lie in the range of energies greater than AE.  8.6.3  T h e dependence of the m o b i l i t y o n JA  In this section it is shown that, for quite general scattering, the mobility depends only on the antisymmetric part of the distribution function. In Appendix B it is shown that  M = ^ « C » ,  (8-26)  137  f {0.1W,k,9) A  f (O.9W,k,0) A  Figure 8.16: The antisymmetric distributions of the forward- and reverse- modes, at two different positions in a region of width W = 2 x l . The injected flux for each mode is 10 times its equilibrium flux. Along the 0-axis, fm indicates the forward-mode and rm indicates the reverse-mode. 5  sc  138  f (W,E) A  f(W,E)  Figure 8.17: A schematic illustrating the energy dependence of the antisymmetric part of the distribution, at z = W, for the reverse-mode. Note that the antisymmetric part does not include the entire shaded area, but that the entire antisymmetric part is included in the range of energies defined by the shaded area. where  J k f{k)dk z  «r;»  (8.27)  7  L dt J coii  The approach will be to write the distribution function as the sum of its symmetric and antisymmetric parts, (8.28)  f(k) = fs{k) + f {k), A  and show that each of the terms arising in the numerator and denominator of (8.27) depend only on f . A  For the numerator:  Jk f(k)d k 3  z  = j k [f {k) + f (k))dk z  s  A  = J k f (k)dk + J k f (k)dk z  =  s  z  A  J k f (k)dk. z  (8.29)  A  The term involving f$ integrates to zero because its integrand is odd (k being odd and fs z  being even). For the denominator, the integrand is written in terms of its incoming and outgoing parts:  /  k,  df dt J coii  dk  "1  dt  out coll  dk.  (8.30)  139 For the outgoing term: 1 out  dt  coll  dk  (k) - I  r(k)  -I  r(k)  f. {k)dk + J  ^f (k)dk  ;  k  A  f (k)dk,  z  (8.31)  A  where the integration over the symmetric part is zero because fs and r(k) are even while k  z  is odd, so that the integrand is odd. For the incoming term: idt  dk  =  J k [y f(k')S(k', k)dk^ dk  =  Jk [J  =  Jk [J fs{k')S{k\k)dk'  =  Jk [J  z  f (k')S(k',k)dk' + J f (k')S(k',k)dk'  z  s  dk  A  dk + Jk [J z  f (k' )S(k', k)dk'dk A  z  z  f {k')S(k',k)dk'  dk,  (8.32)  A  where the term involving fs integrates to zero because the inner integral is an even function and k is odd. The inner integral is even for all the scattering mechanisms considered in this z  work (i.e., acoustic phonon, polar optical phonon and screened ionized impurity scattering), and is so for any mechanism for which the outgoing term can be written as in (2.12). Substituting for the various terms derived above gives the following result for the mobility in terms of the antisymmetric part of the distribution:  H=  I m J ^f (k)dkA  8.6.4  k f (k)dk z  A  (8.33)  jk [J z  f (k')S(k',k)dk']dk A  C o n c l u d i n g remarks  The results of this study of carrier mobility may be summarized in the following manner: 1. The forward-mode mobility is identical to the reverse-mode mobility, even though the overall energy of the forward-mode distribution is increased while that of the reversemode is decreased. 2. The antisymmetric part of the distribution has the same shape in both the forwardand reverse- modes and the mobility depends, for quite general scattering, only on this part of the distribution.  140 3. Expressions such as (8.23), while appropriate for forward-mode operation, can not correctly predict the mobility for reverse-mode operation (e.g., in a forward-biased semiconductor junction). 4. Although the mobility has been evaluated for isotropic and elastic scattering in a constant electric field, a conceptual framework has been developed which should prove useful in understanding results from studies involving more detailed scattering mechanisms or nonuniform electric fields.  Chapter 9 Summary, conclusions and suggestions for future work 9.1  The Boltzmann equation  A solution of the one-dimensional, time-independent, field-free Boltzmann transport equation, suitable for analyzing base transport in modern bipolar transistors was developed: (9.1) (9.2) This solution differs from others in the literature in that the boundary conditions for carriers injected into the region of interest and the incoming collision integrals are of a general form. Furthermore, it was shown that the general incoming collision integrals for screened, ionized impurities and polar optical phonons can be reduced to integrations over a single angle: (9.3) where g(k,6,6') depends on the scattering mechanism. Equations (9.1,9.2) were applied to analyze base transport in homojunction, graded-heterojunction and abrupt-heterojunction bipolar transistors. In the analysis of homojunction, and graded-heterojunction, devices, the boundary conditions were taken from the theory of thermionic emission: / (0,M) +  N  6  (9.4)  c  (9.5)  f-(W,k,0)  141  142 A physical interpretation of the solution was offered in terms of ballistic and scattered components of the counter-directed electron ensembles. Two separate scattering models were studied. The first included only the isotropic and elastic acoustic phonon mechanism, appropriate for the study of low field, moderately doped silicon. In this instance considerable analytic progress toward the solution can be achieved, a key result being an integral expression for the carrier concentration which is valid for any mode of transistor operation and which, when n* — 0, reduces to a similar result in the literature. c  In the second, more detailed scattering model, appropriate for gallium arsenide devices, both polar optical phonon and screened, ionized-impurity mechanisms were included. In this instance no analytic progress could be achieved. However, a more computationally efficient form of the solution was derived in which the collision terms were written as: /+ (z, k, 9) = OL  /coJz.M)  =  /+ (z - Az, k, 6)e- ^ A  T  OL  /coi> + A z , M ) e * A  M r  + f  Jz-Az  + f Jz+Az  e ^ ' ^ d z ' z  (9.6)  v  el'-'V^dz'.  (9.7)  z  v  These forms for the collision components do not require a reevaluation of the spatial integrals from 0 to z —Az for the forward-going ensemble, or from W to z + A z for the backward-going ensemble. In a comparison between the two scattering models, it was found that the distribution function results differed, most notably at low energies. Specifically, the carriers were more evenly distributed in angle at low energies for the more detailed model, and the peak value of the distribution was pushed out in energy as opposed to occurring at zero energy for the simple scattering model. However, the properties derived from the distribution, including the carrier and current densities, and carrier mean velocities, showed quite good agreement between the two models. In the analysis of abrupt-heterojunction devices, the thermionic emission boundary condition at the emitter edge of the base was augmented to include tunneling through, and reflection from, a conduction band spike appearing at the metallurgical interface of the abrupt junction. This led to the following form for the forward-going part of the distribu-  143  tion function: f (z,k,6)  E  h21.2 k .,  —z  2  h  = ^exp  +  exp  k r  lV T z  R  >/ 0 p  iV  -E /k T k  B  -W/V T  VT  dz' +  2  -(z-z')lv TC\n{z\ k, 7T  ,  Z  -(z-z'JlCin^.fc,*)  exp  +  c  0) J  z  d  l-T (k,9)  exp  E  Jo  c  —z  (9.8)  while that for the backward-going component was unchanged. In this analysis, only results for the more detailed scattering model were presented. It was found that the collector current in the abrupt-junction H B T falls off much less rapidly with basewidth than it does in the B J T or graded-junction HBT. The results for the carrier density and mean velocities were also different in the two cases. The base transit time in the abrupt-junction H B T was shown to offer an improvement over the other device structures when the basewidth was reduced to the order of a carrier scattering length and beyond. A physical interpretation of the terms in the field-free solution was developed.  This  interpretation suggested an alternate derivation of the field-free relations using a kinetic approach based on the concept of path integrals. This kinetic approach was then applied to find a set of iterative expression for the fielddependent transport problem. For a constant value of electric field, these expressions reduce to, f {z,k,6) +  =  / ((Uo,0o)exp  +  +  10  V+T  i;  c (z',Hz'),e (z'))  if k cos 6 > az 2  =  2  di  f-(W,k 9 )exp W)  f  w  ^c (z',k(z'),e-(z')) in  v+(z')  r di_  +4 exp  VzX*)  Jw  di_  +  m  i  v  T  \Jz>  exp  V+T  + VT Z  c (z',k(z'),e+(z')) in  v+(z>)  J  C l  V+T  dz' + dz'  exp  (9.9)  if k cos 9 < az 2  f~(z,k,B)  =  2  f~(W,k ,6 )exp w  w  dz  .  \JWV TJ_ Z  /  Jw  C (z',k(z'),6-(z')) in  vi [z>)  exp  Kf * .  \Jz> V  z  dz', (9.10)  144 which are quite similar in form to thefield-freesolution. However, the forward- and backwardgoing components of the distribution are now coupled by the electric field as well as by scattering, and the forward-going ensemble has been divided into two parts in phase-space. These equations were solved using acoustic phonon scattering only, and subject to the same hemi-Maxwellian boundary conditions applied to the field-free transport. Results for the distribution function were presented for two modes of operation: that in which the net flux of carriers moved down a potential gradient (referred to as the forward-mode), and that in which the net flux of carriers moved up a potential gradient (referred to as the reverse-mode). The forward-mode current density was compared to an analytical result derived by applying to the D D E a set of boundary conditions consistent with those applied to the B T E . This D D E approach is equivalent to the so-called one-flux approach, and it was shown that if the mobility is taken to be  this approach can lead to nonphysical results for the car-  rier concentration in the backward-going ensemble at the injecting boundary. However, it is also shown that this problem can be alleviated by applying a particular form for the field-dependent mobility which has been proposed in the literature. A detailed examination of the carrier mobility was carried out, and the issue of the nature of carrier mobility in a forward-biased semiconductor junction was considered. It was found that the carrier mobility depends only on the antisymmetric part of the distribution function and that, for the situation examined in this work, this part of the distribution does not depend on the polarity of the bias.  9.2  Compact models  The following compact model for the collector current density in homojunction, and graded heterojunction, bipolar transistors, consistent with the solution to B T E , was developed:  = = '"W i+W+l] •  J  where D ^ = aD , e  0  '  (9 n)  a, b and a are parameters determined from the B T E solution and  which, for active mode operation, depend only on the basewidth. In deriving this relation a set of expressions for the carrier concentrations at the quasi-neutral base edges was also found. A table was provided for the basewidth dependence of a, b and a which should make  145  these equations useful for determining the transport properties of any BJT operating in the forward, active mode. In abrupt-heterojunction devices, the following compact model for the collector current density was suggested: J = -gnTTE^TTE)  ( - ) 9  c  where n ^  T E  and  12  the concentration and mean z-directed velocity of the carriers  injected from the emitter into the base; either by thermionic injection over, or tunneling through, the conduction band spike at the metallurgical interface.  The result, although  equal to the ballistic limit current, was shown to be in reasonable agreement with the true current over the range of technologically important basewidths. For the base transit time in abrupt-heterojunction devices, the following model provides a conservative, yet accurate, estimate:  T  B  =  TTfT  2D  +  0  Note that v^  rE  + , UT^VBE) T /  V  (9-13)  has been given an explicit dependence on VBE in order to show that this  relation maintains a bias dependence even in the active mode of transistor operation.  9.3  Suggestions for future work  It is suggested that the scattering models be extended to include more mechanisms. The intervalley mechanism in gallium arsenide would be particularly interesting. This would require solving the transport equation simultaneously in each of the two valleys. However, this is necessary if one wants to study high-field transport in GaAs, where intervalley scattering becomes important, as do the transport properties of the electrons populating the upper valley. The abrupt-junction results can be extended to examine double heterostructure devices by employing the following for the boundary condition for the carriers injected from the collector: /-(W, k, 6) = f^ (W, E  k, 6) + f^ (W, k, 0), FL  (9.14)  146  where /  T  T  E  and /  f (z,k,6)  R F L  are analogous to f^  TE  The equations to be solved are then:  FL  + r(O,k,n-e)[l-T (k,e)}e- ^  = f^ (O,k,0)e-^  +  and /R -  T  z  rE  +  T  E  r -(z-z')/v,rCin(z',k,6) e  dzl  ^  ( g  JO z + f (W, k, TT - 0)[1 - %(k, 6)]e^ -^ ' v  f-(z, k, 6)  = fe (W, k, e)e^ -^ w  T  TE  +  w  Jw  v  r  v  z  where Tc(k,6) is the barrier transparency of the base collector junction. These equations could also be applied to the study of pseudo-heterojunction effects, where band offsets arise due to the doping dependence of bandgap in otherwise homojunction devices. Such effects have yet to be included in analyses of the doping dependence of bandgap from collector current measurements. The field-free solution can be extended to investigate the time dependence of base transport. Using the kinetic approach, but including the time dependence of the distribution function, the field-free solution becomes: f {z,k,9,t) +  =  f {0,k,9,t+  f )e- ' + z/v  T  =  f-(W,k,6,t-  e  (  9 1  7  )  v  )(~*)/^ w  f-(z,k,9,t)  r e - M / v ^ ^ ' ^ ' ^ T ) ^ z  Jo  + r  e  Jw  - M ) M r  C  |  " ^  f  c  ' ^ -  v  z  ~  \  z  '  .  (9.18)  These equations are not limited to small-signal analyses of the forward mode, and could be used to study the switching behaviour of modern, short-base bipolar transistors. Thefield-dependentanalysis used to study carrier mobility should be extended to include more detailed scattering mechanisms. Most important would be the inclusion of inelastic scattering, such as optical deformation potential scattering, in order to account for the velocity saturation effect at high electric fields. Further, a potential profile more closely approximating that in a semiconductor junction, such as that determined from the depletion approximation, should be used in order to account for effects due to a nonuniform electric field. Finally, an effort should be undertaken to formulate the field-dependent solution such that it can be efficiently solved consistently with Poisson's equation. This would involve writing routines to find the positions of any potential wells and then generating the required components of the forward- and backward- going ensembles as described in Chapter 7.  Appendix A Evaluating properties of the distribution function A.l  General procedure  The general procedure for finding properties of the distribution function is as follows. Let X(z, k, t) be any function that represents a property of an electron near z with wave vector near k at time t. The total amount of X at position z and time t is determined from the distribution function in the following manner: A[X(z, k, t)] = 2- j^-  j_ X(z, k, t)f(z, k, t)dk.  3  (A.l)  Note that this amounts to summing over all of /c-space the density of states times the probability occupancy times the value of the property. The mean value of X at position z and time t is determined from the distribution function in the following manner:  fx(z,k,t)f(z,k,t)dk (X(z, k, t)> =  W  J  k  n  n(z)  •  (A.2)  Note that this amounts to finding the total amount of the property and dividing by the number of electrons.  A.2  The concentration  The electron concentration is found from:  (A.3)  147  148 Going over into spherical coordinates l*oo POO  2  n(z)  pn PIT  p2ir p'ZTT  = — / / / f{z,k,6,(j))k sm6d(t)d9dk 471" Jo Jo Jo 2  i r°° r i r ir *  lk dk  2  = =  2  ^ j f f{z,k,6)sm0d6  i r°°  ~2 ft Jo  k dk 2  fo(z,k)k dk  (A.4)  2  Note that in the above, as throughout most of this work, / is not a function of <j>.  A.3  The current density  The z-directed electron current density is found from:  J(z)  = =  =  PIT  poo  n  J  3/ 1  p2n  J  J  v (k,6)f(z,k,6)k sm6d<t)d0dk  /  ~L J  2  z  ^ ' )^ ' ' '^  v  k  9  z  k  d  dek2dk  f°° f* hk / / —cos9f(z,k,6)sm6d0k dk 2  2vr Jo 2  l  h  J  Q  m  POO P%  / k f(z,k,6) cos 6 sin 6d6dk 2ir m Jo Jo 3  2  A.4  ^  sined<  (A.5)  The mean z-directed velocity  The mean z-directed velocity can be found directly by noting that:  („.W> = A.5  The mean z-directed kinetic energy  The mean z-directed kinetic energy is found from:  (A.6)  149  1  poo pn p2TV  c\  (u (z)) = -TTJTT^  /  z  (27T)-  JO  / u f(z,k,9)k sin9d<j>d9dk 2  z  JO  JO  2 r°° r (2TT) 7 JO  r n k f(z,k,6)k sin 9d(j)d9dk  0  OO  1  2  2n  3  /"7r  2  2  z  2m hk  fl'K 2  2  cos 0 / ( z , ik, 0)/c sin 9d(j>d9dk n(z (27r) )3 370 Jo Jo 2m O fc2 /-co PIT 1 2 ^—2n / / A: cos 9f(z,k, 9)k sin OdOdk n(z (2TT) 2m Jo Jo 2 7r Z" /'' 1 —2TT / A; / cos 9f(z,k, 9) sin 9d9dk n(z (2TT) 2m Jo Jo 2 /-CO /-7T 1 1 ti A 2 2  2  2  2  3  00  r  4  2  3  /  n(z 4n m Jo 2  A.6  k / f {z,k, 9) cos 9sin9d9dk. Jo  (A.7)  The mean kinetic energy  The mean kinetic energy is found from:  •I  (u(z)) =  n  POO PIT PZTT  uf(z,k,e)k sined<j>dedk n(z) (2n) Jo Jo Jo  — T - T - 7 — /  /  2  /  6  -f(z,k,6)k sin 9d<j>d9dk 1 2J r Jr Jf *K n(z)(2n) 2mk „ , „,, 2  2  3  0  0  2  2  2  Q  i 2 ti f°° r* ^T7TTT7^ / / k f(z,k, 9) sin 6d9dk n(z) (2TT) 2m JO JO 2 l 2 h f°° l fn , w „ —2TT2 / A; - / /(z, k, 9) sin 9d9dk n(z) (2TT) 2m J 2J 2  = =  2 7 r  4  3  4  3  0  1  1 Ti  = ^)V^L  2  f°°  ' -  k>h(z k)ik  0  -  (A  8)  Appendix B Derivation of the 1-dimensional drift-diffusion equation The Boltzmann transport equation can be written as  + * . -/(r, Z,t) + j - V -/(r, K t) = Vr  fc  df(f,k,t) dt  (B.l) J coll  To derive the drift-diffusion equation from the B T E , the method of moments is used. To apply the method, the B T E is multiplied by some function X(z, k, t) and then integrated over A;-space. If the B T E is satisfied then so will the resulting moment (or balance) equation. The procedure is applied term by term in the following development. The appropriate function to multiply by, in order to obtain the DDE, is X(z, k,t) = — qv . z  B.l  The time term  Multiplying the first term in the B T E by X and integrating gives: ^dk X(z,k,t)-  J  (B.2)  r  dt  Notice that  d i  {  x  f  )  -  x  m  +  dt>  f  (B.3)  or  X  dt~dt  { X f )  150  1  df  (B.4)  151 So, J X(zXt)j-dk  =  (Xf)-f^  J  d  =  ^Jxfdk-Jffdk  =  - [ n ( z ) (X(z, 11))) - n(z) ( - X ( z , k, t)),  (B.5)  where (A.2) has been used. With X(z, k,t) = -qv , dX/dt = 0 and z  J X(z,k,t)^-dk  =  i.[ (z)(-qv )] n  z  d_ [-qn{z,t)(v )} dt d_ J (z,t). dt z  (B.6)  z  B.2  The diffusion term  Multiplying the second term in the B T E by X and integrating gives: Jxv-Vfdk  = J  (B.7)  Xv ^dk a  where the summation convention has been used. Note that d  iXv f) dx.  J-  V n  a  { X  dx  f  )  *>dx  +  Xf  a  a  =  v fa  h vaX  dx  (B.8)  ——V X] dx ' dx 0  n  a  or t\  (v  9  Xv,  t  d  dx (Xv f) ~ V f— Q  a  v f  X  a  d  V  (B.9)  a  Xf-dx  a  So [ Xv -^-dk J dx a  a  =  d  / J  f  dx  v  (Xv f) a  a  dx d  dX - "of t  dv '. ~ f— dxr vt  x  a  1  d  k  c  A(n  W  (a))-n(z)(,|)-n( )(^) Z  (B.10)  152 where (A.2) has been used. Now, with hk v = — cos 9 m z  and for a position independent effective mass (i.e., a compositionally uniform material) one has that dv  a  0,  (B.ll)  ax= 0.  (B.12)  dx  Q  and since X(z,k, t) = —qv one also has z  dx  a  Further, since f(z, k, 9) (and therefore n(z)) varies in the z-direction only one has (n(z)(v X(z,k,t))) \ V / \ Q : V '  = —( {z)(-qv v )). ^  a  n  z  (B.13)  z  With the z-directed kinetic energy taken to be u = \mv = ^ 2 2m 2  2  z  K  (B.14) '  one has that 9  8x  -(n(z)(v X(z,k,t))) ° ' '  v a  v  /  x  a  -i£|-(n(z)(u,»  =  v  m Dz'  + _?£(«,> A„( ).  n(z)^(u ) m az 2q  z  m  dz  (B.15)  Z  So, y Xv-Vfdk  B.3  = ^ ( z ) ^ ( u ) + ^L( )^-n(z). 2  (B.16)  Uz  The drift term  Multiplying the third term in the B T E by A" and integrating gives:  j 4 ^  =J  d  T W j  x  k  (  B  1  7  ( B  .  »  where the summation convention has been used. Note that 8 dk  F K a  F  h'  d  h dk  J  a  _ h  dk  a  n  dk  a  1 8 )  153 assuming F does not depend on A:, or ,F Of Xh dk  F  a  d  a  h dk  a  (B.19)  H dk J  a  n  So J  F  a  h dk  dk =  d  h dkj*^  a  K/j-oo  h  f  a  /l  F  dk  H dk  dk  a  jdX(z,k,t)\  Q  (B.20)  -T \—dL—/ n{z)  since / goes to zero exponentially as k gets large, and where (A.2) has been used, With a force in the z-direction only we have F b  , ,/dX(z,k,t)\ /dX(z,k,t  an  - Tn " n  {  \  -— ) ~ \' \- ^ kdk a  vz )  F/dX{z,k,t ,/dX{z,k,t) \  b  Z/  z  = ~T \  dk  n{z)  /'  z  (B.21)  and with X(z,k, t) — —qv = —qhk /m one gets z  z  dX _ -qH dk  (B.22)  m '  z  and the mean value of a constant is just a constant so (B.23) m and  /  B.4  F X--V fdk  F = q^n{z). m  k  (B.24)  The collision terms  Multiplying the outgoing collision term in the B T E by X and integrating gives:  /  x\%]  dk.  (B.25)  L dt J coii  v  ;  With X(z, k, t) — —qv , the following expression is taken as a definition for the so-called z  momentum relaxation time, ((r^)): (B.26)  dk or -f(-qv )fdk  _  z  j(-qv ) z  §  dk Jcoll  - J k fdk z  Jk  dl dt  dk Jcoll  (B.27)  154  B.4.1  Isotropic a n d elastic scattering  If the scattering is restricted to the isotropic and elastic acoustic phonon mechanism, then r(k) =  (B.28)  and \dfl idti  in  \dfl idti  coll  =  \dfl idti  coll  out coll  (B.29)  & kf (z, k) - a kf(z, k, 0). ap  Q  ap  Multiplying the first term by k and integrating gives z  J a k kf (z, Ap  z  k)dk = 0.  0  (B.30)  This expression integrates to zero because the integrand is odd on account of k being odd z  and kf (z,k) being even. 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