Steady-State and Transient Electron Transport Within the III V Nitride Semiconductors, GaN, AlN, and InN: A Review

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1 J Mater Sci: Mater Electron (2006) 17: DOI /s REVIEW Steady-State and Transient Electron Transport Within the III V Nitride Semiconductors, GaN, AlN, and InN: A Review Stephen K. O Leary Brian E. Foutz Michael S. Shur Lester F. Eastman Received: 15 July 2005 / Accepted: 26 July 2005 C Science + Business Media, Inc Abstract The III V nitride semiconductors, gallium nitride, aluminum nitride, and indium nitride, have, for some time now, been recognized as promising materials for novel electronic and optoelectronic device applications. As informed device design requires a firm grasp of the material properties of the underlying electronic materials, the electron transport that occurs within these III V nitride semiconductors has been the focus of considerable study over the years. In an effort to provide some perspective on this rapidly evolving field, in this paper we review analyses of the electron transport within the III V nitride semiconductors, gallium nitride, aluminum nitride, and indium nitride. In particular, we discuss the evolution of the field, compare and contrast results determined by different researchers, and survey the current literature. In order to narrow the scope of this review, we will primarily focus on the electron transport within bulk wurtzite gallium nitride, aluminum nitride, and indium nitride, for this analysis. Most of our discussion will focus on results obtained from our ensemble semi-classical three-valley Monte Author to whom correspondence should be addressed. Present address: Cadence Design Systems, 6210 Old Dobbin Lane, Columbia, Maryland 21045, USA. S. K. O Leary Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada, S4S 0A2 stephen.oleary@uregina.ca B. E. Foutz Lester F. Eastman School of Electrical Engineering, Cornell University, Ithaca, New York 14853, USA M. S. Shur Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York , USA Carlo simulations of the electron transport within these materials, our results conforming with state-of-the-art III V nitride semiconductor orthodoxy. A brief tutorial on the Monte Carlo approach will also be featured. Steady-state and transient electron transport results are presented. We conclude our discussion by presenting some recent developments on the electron transport within these materials. 1. Introduction The III V nitride semiconductors, gallium nitride (GaN), aluminum nitride (AlN), and indium nitride (InN), have, for some time now, been recognized as promising materials for novel electronic and optoelectronic device applications [1 9]. In terms of electronics, their wide energy gaps, large breakdown fields, high thermal conductivities, and favorable electron transport characteristics, make GaN, AlN, and InN, and alloys of these materials, ideally suited for novel high-power and high-frequency electron device applications. On the optoelectronics front, the direct nature of the energy gaps associated with GaN, AlN, and InN, make this family of materials, and its alloys, well suited for novel optoelectronic device applications in the visible and ultraviolet frequency range. While initial efforts to study these materials were hindered by growth difficulties, recent improvements in the material quality have made possible the realization of a number of III V nitride semiconductor based electronic [10 16] and optoelectronic [17 25] devices. These developments have fueled considerable interest in the III V nitride semiconductors, GaN, AlN, and InN. In order to analyze and improve the design of III V nitride semiconductor based devices, an understanding of the electron transport which occurs within these materials is necessary. Electron transport within bulk GaN, AlN, and InN has been extensively examined over the years [26 45].

2 88 J Mater Sci: Mater Electron (2006) 17: Unfortunately, uncertainty in the material parameters associated with GaN, AlN, and InN remains a key source of ambiguity in the analysis of the electron transport within these materials [45]. In addition, some recent experimental [46] and theoretical [47] developments have cast doubt upon the validity of widely accepted notions upon which our understanding of the electron transport mechanisms within the III V nitride semiconductors, GaN, AlN, and InN, has evolved. Further confounding matters is the sheer volume of research activity being performed on the electron transport within these materials, this presenting the researcher with a dizzying array of seemingly disparate approaches and results. Clearly, at this critical juncture at least, our understanding of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, remains in a state of flux. In order to provide some perspective on this rapidly evolving field, we aim to review analyses of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, within this paper. We start with a brief tutorial on the electron transport mechanisms within semiconductors, and on how the Monte Carlo approach may be used in order to probe such mechanisms. Then, focusing on the III V nitride semiconductors under investigation in this analysis, i.e., GaN, AlN, and InN, we present results obtained from ensemble semiclassical three-valley steady-state and transient Monte Carlo simulations of the electron transport within these materials, these results conforming with state-of-the-art III V nitride semiconductor orthodoxy. We conclude this review with a discussion on the evolution of the field and a survey of the current literature. In order to narrow the scope of this review, we will primarily focus on the electron transport within bulk wurtzite GaN, AlN, and InN for the purposes of this analysis. We hope that researchers in the field will find this review useful and informative. For our brief tutorial on the electron transport mechanisms within semiconductors, we begin with an introduction to the Boltzmann transport equation, this equation underlying most analyses of the electron transport within semiconductors. Then, the general principles underlying the ensemble semiclassical three-valley Monte Carlo simulation approach, that we employ in order to solve the Boltzmann transport equation, are presented. We conclude the tutorial by presenting the material parameters corresponding to bulk wurtzite GaN, AlN, and InN. We then use these material parameter selections and our ensemble semi-classical three-valley Monte Carlo simulation approach to determine the nature of the steady-state and transient electron transport within the III V nitride semiconductors. Finally, we present some recent developments on the electron transport within these materials. This paper is organized in the following manner. In Section 2, we present our tutorial on the electron transport mechanisms within semiconductors. In particular, the Boltzmann transport equation and our ensemble semi-classical three-valley Monte Carlo simulation approach, that we employ in order to solve this equation for the III V nitride semiconductors, GaN, AlN, and InN, are presented. The material parameters, corresponding to bulk wurtzite GaN, AlN, and InN, are also presented in the tutorial featured in Section 2. Then, in Section 3, using results obtained from our ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within the III V nitride semiconductors, we study the nature of the steady-state electron transport that occurs within these materials. Transient electron transport within the III V nitride semiconductors is also discussed in Section 3. A review of the III V nitride semiconductor electron transport literature, in which the evolution of the field is discussed and a survey of the current literature is presented, is then featured in Section 4. Finally, conclusions are provided in Section Electron Transport Within Semiconductors and The Monte Carlo Simulation Approach: A Tutorial 2.1. Introduction The electrons within a semiconductor are in a perpetual state of motion. In the absence of an applied electric field, this motion arises as a result of the thermal energy which is present, and is referred to as thermal motion. From the perspective of an individual electron, thermal motion may be viewed as a series of trajectories interrupted by a series of random scattering events. Scattering may arise as a result of interactions with the lattice atoms, impurities, other electrons, and defects. As these interactions lead to electron trajectories in all possible directions, i.e., there is no preferred direction, while individual electrons will move from one location to another, taken as an ensemble, assuming that the electrons are in thermal equilibrium, the overall electron distribution will remain static. Accordingly, no net current flow occurs. With the application of an applied electric field, E, each electron in the ensemble will experience a force, q E, q denoting the electron charge. While this force may have a negligible impact upon the motion of any given individual electron, taken as an ensemble, the application of such a force will lead to a net aggregate motion of the electron distribution. Accordingly, a net current flow will occur, and the overall electron ensemble will no longer be in thermal equilibrium. This movement of the electron ensemble in response to an applied electric field, in essence, represents the fundamental issue at stake when we study the electron transport within a semiconductor. In this chapter, we provide a brief tutorial on the issues at stake in our analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN. We begin our analysis with an introduction to the Boltzmann

3 J Mater Sci: Mater Electron (2006) 17: transport equation, this equation describing how the electron distribution function evolves under the action of an applied electric field, this equation underlying the electron transport within bulk semiconductors. We then introduce the Monte Carlo simulation approach to solving the Boltzmann transport equation, focusing on the ensemble semi-classical threevalley Monte Carlo simulation approach used in our own simulations of the electron transport within the III V nitride semiconductors. Finally, we present the material parameters corresponding to bulk wurtzite GaN, AlN, and InN. This chapter is organized in the following manner. In Section 2.2, the Boltzmann transport equation is introduced. Then, in Section 2.3, a brief discussion on the ensemble semiclassical three-valley Monte Carlo simulation approach to solving this Boltzmann transport equation is presented. Finally, in Section 2.4, our material parameter selections, corresponding to bulk wurtzite GaN, AlN, and InN, are presented The Boltzmann transport equation An electron ensemble may be characterized by its distribution function, f ( r, p, t), where r denotes the position, p represents the momentum, and t indicates time. The response of this distribution function to an applied electric field, E, is the issue at stake when one investigates the electron transport within a semiconductor. When the dimensions of the semiconductor are large, and quantum effects are negligible, the ensemble of electrons may be treated as a continuum, i.e., the corpuscular nature of the individual electrons within the ensemble, and the attendant complications which arise, may be neglected. In such a circumstance, the evolution of the distribution function, f ( r, p, t), may be determined using the Boltzmann transport equation. In contrast, when the dimensions of the semiconductor are small, and quantum effects are significant, then the Boltzmann transport equation, and its continuum description of the electron ensemble, is no longer valid. In such a case, it is necessary to adopt quantum transport methods in order to study the electron transport within the semiconductor [48]. For the purposes of this analysis, we will focus on the electron transport within bulk semiconductors, i.e., semiconductors of sufficient dimensions so that the Boltzmann transport equation is valid. Ashcroft and Mermin [49] demonstrated that this equation may be expressed as f t = p p f r r f + f t. (1) scat The first term on the right-hand side of Equation (1) represents the change in the distribution function due to external forces applied on the system. The second term on the righthand side of Equation (1) accounts for the electron diffusion which occurs. The final term on the right-hand side of Equation (1) describes the effects of scattering. Owing to its fundamental importance in the analysis of the electron transport within semiconductors, a number of techniques have been developed over the years in order to solve the Boltzmann transport equation. Approximate solutions to the Boltzmann transport equation, such as the displaced Maxwellian distribution function approach of Ferry [27] and Das and Ferry [28] and the non-stationary charge transport analysis of Sandborn et al. [50], have proven useful. Low-field approximate solutions have also proven elementary and insightful [30, 33, 51]. A number of these techniques have been applied to the analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN [27, 28, 30, 33, 51, 52]. Alternatively, more sophisticated techniques have been developed, these solving the Boltzmann transport equation directly. These techniques, while allowing for a rigorous solution of the Boltzmann transport equation, are rather involved, and require intense numerical analysis. They are further discussed by Nag [53]. For studies of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, the most common approach to solving the Boltzmann transport equation, by far, has been the ensemble semi-classical Monte Carlo simulation approach. In terms of the III V nitride semiconductors, using the Monte Carlo simulation approach, the electron transport within GaN has been studied the most extensively [26, 29, 31, 32, 34, 35, 40, 42, 45], AlN [37, 38, 42] and InN [36, 41, 42, 44] less so. The Monte Carlo simulation approach has also been used to study the electron transport within the two-dimensional electron gas of the AlGaN/GaN interface which occurs in high electron mobility AlGaN/GaN field-effect transistors [54, 55]. At this point, it should be noted that the complete solution of the Boltzmann transport equation requires a resolution of both steady-state and transient responses. Steadystate electron transport refers to the electron transport that occurs long after the application of an applied electric field, i.e., once the electron ensemble has settled to a new equilibrium state [56]. As the distribution function is difficult to quantitatively visualize, in the analysis of steady-state electron transport, researchers typically study the dependence of the electron drift velocity [57] on the applied electric field strength, i.e., they determine the velocity-field characteristic. Transient electron transport, by way of contrast, refers to the transport that occurs while the electron ensemble is evolving into its new equilibrium state. Typically, it is characterized by studying the dependence of the electron drift velocity on the time elapsed, or the distance displaced, since the electric field was initially applied. Both steady-state and transient electron transport within the III V nitride semiconductors, GaN, AlN, and InN, are reviewed within this paper.

4 90 J Mater Sci: Mater Electron (2006) 17: The ensemble semi-classical Monte Carlo simulation approach In the study of the electron transport within a semiconductor, the Monte Carlo approach is often used in order to solve the Boltzmann transport equation. In this approach, the motion of electrons within a semiconductor, under the action of an applied electric field, is simulated. The acceleration of each electron in the applied electric field, and the presence of scattering, are both taken into account in these simulations. The scattering events that an individual electron experiences are selected randomly, the probability of each such event being selected in proportion to the scattering rate corresponding to that particular event. Through such an analysis, one hopes to be able to estimate the resultant distribution function, f ( r, p, t). In simulating the electron transport within a semiconductor, there are a variety of different Monte Carlo approaches that researchers have adopted over the years. Most of these approaches may be classified as being either singleparticle Monte Carlo simulation approaches or ensemble Monte Carlo simulation approaches. In a single-particle Monte Carlo approach, one simulates the motion of a single electron, tracking its wave-vector for a sufficiently long time so that, in steady-state conditions, this wave-vector sweeps through all of phase space, the amount of time spent in any particular place in phase space being a proportionate predictor for the distribution function there. Ergodicity is implicitly assumed, i.e., it is assumed that time-averages are equal to ensemble-averages [58]. In an ensemble Monte Carlo simulation of the electron transport within a semiconductor, the motion of a large number of electrons, under the action of an applied electric field, is studied. The evolution over time of this distribution of electrons is interpreted as being indicative of the corresponding distribution function, the density of electrons at any point in phase space being a proportionate predictor for the distribution function there. Assuming that there are enough electrons used in the simulation, the law of large numbers dictates that the results will indeed correspond to those determined through an exact evaluation of the distribution function, f ( r, p, t). This approach allows for the ready analysis of both steady-state and transient electron transport. We have adopted an ensemble Monte Carlo simulation approach for the purposes of our analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN. Before describing the algorithm used for our Monte Carlo simulations, we first provide a brief overview of key modeling considerations. In particular, we present the three-valley model that is used to represent the conduction band electron band structure. Then, we discuss our semi-classical description for the motion of the electrons within this electron band structure. The interactions of the electrons with the semiconductor lattice, through the various scattering mechanisms, are then elaborated upon. Finally, after the basic physics of the electron transport within the III V nitride semiconductors has been introduced, a flow chart, describing the mechanics of our own particular Monte Carlo simulation approach, is presented The three-valley electron band structure model We restrict our attention to the analysis of the electron transport within the conduction band. In the absence of an applied electric field, electrons tend to occupy the lowest energy levels of the conduction band. When an electric field is applied, the average electron energy increases. Typically, however, only the lowest parts of the conduction band contain a significant fraction of the electron population. This allows for a considerable simplification in the analysis to be made. Instead of including the entire electron band structure for the conduction band, only the lowest valleys need be represented. The Monte Carlo simulation approach, used for our simulations of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, uses a three-valley model for the conduction band electron band structure, representing the three lowest energy minima of the conduction band. Within the framework of this three-valley model, the nonparabolicity of each valley is treated through the application of the Kane model, the energy band corresponding to each valley being assumed to be spherical and of the form 2 k 2 = E(1 + αe), (2) 2m where k denotes the magnitude of the crystal momentum, E represents the electron energy, E = 0 corresponding to the band minimum, m is the effective mass of the electrons in the valley, and the non-parabolicity coefficient, α,isgivenby α = 1 ( ) 2 1 m, (3) E g m e where m e and E g denote the free electron mass and the energy gap, respectively [59]. A schematic illustration of the threevalley model representing the conduction band electron band structure associated with bulk wurtzite GaN, used for the purposes of our ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within this material, is depicted in Fig. 1. Values for the valley parameters corresponding to bulk wurtzite GaN, AlN, and InN, used for the purposes of our simulations, are tabulated in Section The semi-classical motion of particles Electrons in a periodic potential possess wave-functions that can be distributed over volumes which are substantially larger

5 J Mater Sci: Mater Electron (2006) 17: Valley 3 m * =m e α=0 ev 1 GaN Valley 2 m * =m e α=0 ev ev 1.9 ev Valley 1 m * =0.2 m e α=0.189 ev 1 Fig. 1 The three-valley model used to represent the conduction band electron band structure associated with bulk wurtzite GaN for our Monte Carlo simulations of the electron transport within this material. The valley parameters, corresponding to bulk wurtzite GaN, AlN, and InN, are tabulated in Section 2.4 than the single unit cell. The electron is thus capable of interacting with many different components of the crystal simultaneously. It can interact with different phonons and different crystal impurities all at once. This picture, however, is too complex to handle directly, and several approximations are usually made in order to render the analysis tractable. One approximation that is commonly made is that the electrons behave as if they were point particles, whose motion, in response to an applied electric field, is well behaved and deterministic. The velocity of each electron may thus be expressed as v g = 1 kɛ( k), (4) where ɛ( k) denotes the electron band structure, i.e., the energy of the electron as a function of the electron wave-vector, k [60]. In addition, the rate-of-change of the electron s wavevector with time is proportional to the force that the electron experiences from the applied electric field, E, i.e., d k dt = q E, (5) where q denotes the electron charge. Equations (4) and (5) collectively determine the motion of an electron, assuming that the periodic potential associated with the underlying crystal is static. In reality, the thermal motion of the lattice, imperfections, and interactions with the other electrons in the ensemble, result in the electron deviating from the path literally prescribed by Equations (4) and (5). Although an individual electron s interaction with the lattice is very complex, the description is simplified considerably through the use of the quantum mechanical notion of scattering events. During a scattering event, the electron s wave-function abruptly changes. Quantum mechanics determines the probability of each type of scattering event, and dictates how to probabilistically determine the change in the wave-vector after each such event. With this information, the behavior of an ensemble of electrons may be simulated, this behavior being expected to closely approximate the electron transport within a real semiconductor. The probability of scattering is introduced into the Monte Carlo simulation approach through a determination of the scattering rates corresponding to the different scattering processes Scattering processes The scattering rate corresponding to a particular interaction refers to the expected number of scattering events of that particular interaction taking place per unit time. Quantum mechanics determines the scattering rates for the different processes based on the physics of the interaction. In general, scattering processes within semiconductors can be classified into three basic types; (1) phonon scattering, (2) carrier scattering, and (3) defect scattering [53]. For the III V nitride semiconductors, GaN, AlN, and InN, phonon scattering is the most important scattering mechanism, and it is featured prominently in our simulations of the electron transport within these materials. Carrier scattering, or in our case, electron-electron scattering, has also been taken into account in our simulations. It should be noted, however, that as this scattering mechanism leads to very little change in the results with a substantial increase in the running time, in an effort to determine our results as expeditiously as possible, electronelectron scattering was not included in our simulations. The final category of scattering mechanism, defect scattering, refers to the scattering of electrons due to the imperfections within the crystal. Throughout this work, it is assumed that donor impurities are the only defects present. These defects, when ionized, scatter electrons through their positive charge. This mechanism is an important factor in determining the electron transport within the III V nitride semiconductors, and the effect of the doping concentration on the electron transport within these materials is considered in our analysis. Owing to their importance in determining the nature of the electron transport within the III V nitride semiconductors, it is instructive to discuss the different types of phonon scattering mechanisms. Phonons naturally divide themselves into two distinctive types, optical phonons and acoustic phonons. Optical phonons are the phonons which cause the atoms of the unit cell to vibrate in opposite directions. For acoustic phonons, however, the atoms vibrate together, but the wavelength of the vibration occurs over many unit cells. Typically, the energy of the optical phonons is greater than that of the acoustic phonons. For each type of phonon, two types of interaction occur with the electrons. First, the deformations in the lattice, which arise from the interaction of the lattice with the phonons, changes the energy levels of the electrons, causing transitions to occur. This type of interaction is referred

6 92 J Mater Sci: Mater Electron (2006) 17: to as non-polar optical phonon scattering for the case of optical phonons and acoustic deformation potential scattering for the case of acoustic phonons. In polar semiconductors, such as the III V nitride semiconductors, the deformations which arise also induce localized electric fields. These electric fields also interact with the electrons, causing them to scatter. For the case of optical phonons, the interaction of the electrons with these localized electric fields is referred to as polar optical phonon scattering. For acoustic phonons, however, this mechanism is referred to as piezoelectric scattering. Owing to the extremely polar nature of the nitride bonds within the III V nitride semiconductors, GaN, AlN, and InN, it turns out that polar optical phonon scattering is very important for these materials. It will be shown that this mechanism alone determines many of the key properties of the electron transport within the III V nitride semiconductors. When the energy of an electron within a valley increases beyond the energy minima of the other valleys, it is also possible for the electrons to scatter from one valley to another. This type of scattering is referred to as inter-valley scattering. It is an important scattering mechanism for the III V compound semiconductors in general, and for the III V nitride semiconductors, GaN, AlN, and InN, in particular. Inter-valley scattering is believed to be responsible for the negative differential mobility observed in the velocity-field characteristics associated with these materials A derivation of all of these scattering rates, as a function of the semiconductor parameters, can be found in the literature; see, for example, [53, 61, 62]. A formalism, which closely matches the form used in our ensemble semi-classical three-valley Monte Carlo simulations of electron transport, is found in Nag [53]. Many of the scattering rates that are employed for the purposes of our Monte Carlo simulations of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, are also explicitly tabulated in Appendix 22 of Shur [63] Our Monte Carlo simulation approach For the purposes of our analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, we employ ensemble semi-classical three-valley Monte Carlo simulations. The scattering mechanisms considered are (1) ionized impurity, (2) polar optical phonon, (3) piezoelectric, and (4) acoustic deformation potential. Intervalley scattering is also considered. We assume that all donors are ionized and that the free electron concentration is equal to the dopant concentration. For our steady-state electron transport simulations, the motion of three thousand electrons is examined, while for our transient electron transport simulations, the motion of ten thousand electrons is considered. For our simulations, the crystal temperature is set to 300 K and the Fig. 2 The scattering rates for the lowest (Ɣ) valley as a function of the wave-vector for bulk wurtzite GaN. The scattering mechanisms are: (1) ionized impurity, (2) polar optical phonon emission, (3) inter-valley (1 3) emission, (4) inter-valley (1 2) emission, (5) acoustic deformation potential, (6) piezoelectric, (7) polar optical phonon absorption, (8) inter-valley (1 3) absorption, and (9) inter-valley (1 2) absorption. The most important scattering mechanisms are shown in Fig. 2(a), Fig. 2(b) depicting the other scattering mechanisms doping concentration is set to cm 3 for all cases, unless otherwise specified. Electron degeneracy effects are accounted for by means of the rejection technique of Lugli and Ferry [64]. Electron screening is also accounted for following the Brooks-Herring method [65]. Further details of our approach are discussed in the literature [29, 34 37, 42, 45]. Figs. 2 through 4 plot the scattering rates corresponding to the various scattering mechanisms as a function of the electron wave-vector, k, for the III V nitride semiconductors considered in this analysis, i.e., GaN, AlN, and InN. These are the rates corresponding to the lowest energy valley in the conduction band, i.e., the Ɣ valley for the III V nitride semiconductors under investigation in this review. The upper valleys have similar scattering rates. Each of the scattering mechanisms

7 J Mater Sci: Mater Electron (2006) 17: included in our simulations of the electron transport within the III V nitride semiconductors is described, in detail, by Nag [53]. For the ionized impurity, polar optical phonon, and piezoelectric scattering mechanisms, screening effects are taken into account. These screening effects tend to lower the scattering rate when the electron concentration is high The Monte Carlo algorithm Now that the fundamentals of electron transport within a semiconductor have been presented, a brief description of our ensemble semi-classical three-valley Monte Carlo algorithm will be provided. This description will be qualitative in nature. Further quantitative details are presented in Appendix A. For the purposes of our analysis, we employ an ensemble Monte Carlo approach. This approach simulates the transport of many electrons simultaneously. Often, in such an approach, the scattering rates are calculated once at the beginning of the program and remain fixed. However, more sophisticated techniques have been developed which depend upon the properties of the current electron distribution. These scattering rate formulas can be implemented using a self-consistent ensemble technique. This technique recalculates the scattering rate table at regular intervals throughout the simulation as the electron distribution evolves. This self-consistent ensemble Monte Carlo technique is the method employed for the purposes of our analysis. The essence of our Monte Carlo simulation algorithm, used to simulate the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, is as depicted in the flow chart shown in Fig. 5. During the initialization phase of our simulations, the initial scattering rate tables are computed. The initial electron distribution is set by assigning each electron a distinct wave-vector. The distribution of wave-vectors is chosen using Fermi-Dirac occupation statistics. As was mentioned earlier, the motion of three thousand electrons is studied for each steady-state electron transport simulation, while the motion of ten thousand electrons is considered for each transient electron transport simulation, these selections allowing us to achieve sufficient statistics. Next, the main body of the algorithm begins. In this phase, each electron moves through a series of time-steps, each timestep being of duration t. This is accomplished by moving the electron through a free-flight. During this free-flight, the electron experiences no scattering events, and its motion through the conduction band is determined semi-classically, i.e., as suggested by Equations (4) and (5). The time for each free-flight must be chosen carefully, and depends critically on the scattering rate at the beginning of the electron s flight, as well as the scattering rate throughout its free-flight. Since the scattering rate changes over the flight, the selection of Fig. 3 The scattering rates for the lowest (Ɣ) valley as a function of the wave-vector for bulk wurtzite AlN. The scattering mechanisms are: (1) ionized impurity, (2) polar optical phonon emission, (3) inter-valley (1 2) emission, (4) inter-valley (1 3) emission, (5) acoustic deformation potential, (6) piezoelectric, (7) polar optical phonon absorption, (8) inter-valley (1 2) absorption, and (9) inter-valley (1 3) absorption. The most important scattering mechanisms are shown in Fig. 3(a), Fig. 3(b) depicting the other scattering mechanisms. Note that piezoelectric scattering is more pronounced in bulk wurtzite AlN than in either bulk wurtzite GaN (see Fig. 2) or bulk wurtzite InN (see Fig. 4) the free-flight time is complex. Methods used for generating the free-flight time have been extensively studied, and the algorithm employed for our simulations is further detailed in Appendix A. At the end of each free-flight, the electron experiences a scattering event. The scattering event is chosen randomly, in proportion to the scattering rate for each mechanism. Finally, a new wave-vector for the electron is chosen, based on conservation of momentum and conservation of energy considerations, as well as the angular distribution function corresponding to that particular scattering mechanism. After the electron has moved, a new free-flight time is chosen

8 94 J Mater Sci: Mater Electron (2006) 17: Fig. 5 A flowchart corresponding to our Monte Carlo algorithm. A more detailed flowchart is shown in Appendix A end of the simulation, the accumulated statistics are sent to a file for the purposes of archiving, processing, and subsequent retrieval. Fig. 4 The scattering rates for the lowest (Ɣ) valley as a function of the wave-vector for bulk wurtzite InN. The scattering mechanisms are: (1) ionized impurity, (2) polar optical phonon emission, (3) inter-valley (1 2) emission, (4) inter-valley (1 3) emission, (5) acoustic deformation potential, (6) polar optical phonon absorption, (7) piezoelectric, (8) inter-valley (1 2) absorption, and (9) inter-valley (1 3) absorption. The most important scattering mechanisms are shown in Fig. 4(a), Fig. 4(b) depicting the other scattering mechanisms and the process repeats itself until that electron reaches the end of the current time-step. After all of the electrons have been moved through the time-step, macroscopic quantities are extracted from the resultant electron distribution. The relevant macroscopic quantities include the electron drift velocity, the average electron energy, and the number of electrons in each valley. The entire process repeats itself, time-step after time-step, until the end of the simulation is reached. When statistics are to be calculated as a function of the applied electric field strength, the applied electric field strength is also periodically updated throughout the simulation; it should be noted, however, that steady-state equilibrium must be achieved before the next update to the applied electric field strength occurs. At the 2.4. Parameter selections for bulk wurtzite GaN, AlN, and InN The material parameter selections, used for our simulations of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, are tabulated in Table 1 [3, 30, 33, 37, 66 78]. Most of these parameters are from Chin et al. [30], although we did select some values from other references [3, 33, 37, 42, 68, 69, 73 75]; these parameter selections are the same as those employed by Foutz et al. [42]. While the band structures corresponding to bulk wurtzite GaN, AlN, and InN, are still not agreed upon, for the purposes of this analysis, the band structures of Lambrecht and Segall [79] are adopted. For the case of bulk wurtzite GaN, the analysis of Lambrecht and Segall [79] suggests that the lowest point in the conduction band is located at the center of the Brillouin zone, at the Ɣ point, the first upper conduction band valley minimum also occurring at the Ɣ point, 1.9 ev above the lowest point in the conduction band, the second upper conduction band valley minima occurring along the symmetry lines between the L and M points, 2.1 ev above the lowest point in the conduction band; see Table 2. For the case of bulk wurtzite AlN, the analysis of Lambrecht and Segall [79] suggests that the lowest point in the conduction band is located

9 J Mater Sci: Mater Electron (2006) 17: Table 1 The material parameter selections corresponding to bulk wurtzite GaN, AlN, and InN. Most of these parameter selections are from Chin et al. [30]; the source of the other parameter selections is explicitly indicated in the table. This selection of parameters is the same as that employed by Foutz et al. [42] Parameter GaN AlN InN Mass density (g/cm 3 ) Longitudinal sound velocity (cm/s) [66] Transverse sound velocity (cm/s) [66] Acoustic deformation potential (ev) Static dielectric constant 8.9 [33] High-frequency dielectric constant [33] Effective mass (Ɣ 1 valley) [67] 0.20 m e [33] 0.48 m e 0.11 m e [3, 68, 69] Piezoelectric constant, e 14 (C/cm 2 ) [70, 71, 72] [37] Direct energy gap (ev) 3.39 [73] 6.2 [74] 1.89 [75] Optical phonon energy (mev) Intervalley deformation potentials (ev/cm) [76] Intervalley phonon energies (mev) [77] Table 2 The valley parameter selections corresponding to bulk wurtzite GaN, AlN, and InN. These parameter selections are from the band structural calculations of Lambrecht and Segall [79]. This selection of parameters is the same as that employed by Foutz et al. [42] Valley number GaN Valley location Ɣ 1 Ɣ 2 L-M Valley degeneracy Effective mass 0.2 m e m e m e Intervalley energy separation (ev) Energy gap (ev) Non-parabolicity (ev 1 ) AlN Valley location Ɣ 1 L-M K Valley degeneracy Effective mass 0.48 m e m e m e Intervalley energy separation (ev) Energy gap (ev) Non-parabolicity (ev 1 ) InN Valley location Ɣ 1 A Ɣ 2 Valley degeneracy Effective mass 0.11 m e m e m e Intervalley energy separation (ev) Energy gap (ev) Non-parabolicity (ev 1 ) at the center of the Brillouin zone, at the Ɣ point, the first upper conduction band valley minima occurring along the symmetry lines between the L and M points, 0.7 ev above the lowest point in the conduction band, the second upper conduction band valley minima occurring at the K points, 1 ev above the lowest point in the conduction band; see Table 2. For the case of bulk wurtzite InN, the analysis of Lambrecht and Segall [79] suggests that the lowest point in the conduction band is located at the center of the Brillouin zone, at the Ɣ point, the first upper conduction band valley minimum occurring at the A point, 2.2 ev above the lowest point in the conduction band, the second upper conduction band valley minimum occurring at the Ɣ point, 2.6 ev above the lowest point in the conduction band; see Table 2. We ascribe an effective mass equal to the free electron mass, m e, to all of the upper conduction band valleys. Thus, from Equation (3), it follows that the non-parabolicity coefficient, α, corresponding to each upper conduction band valley is zero, i.e., the upper conduction band valleys are completely parabolic. For our simulations of the electron transport within gallium arsenide (GaAs), the material parameters employed are from Littlejohn et al. [78] and Blakemore [80]. It should be noted that the energy gap associated with InN has been the subject of some controversy since The pioneering experimental results of Tansley and Foley [75], reported in 1986, suggested that InN has an energy gap of 1.89 ev. This value, or values similar to it [81], have been used extensively in Monte Carlo simulations of the electron transport within this material since that time [36, 41, 42, 44]; typically, the influence of the energy gap on the electron transport occurs through its impact on the non-parabolicity coefficient, α, i.e., through Equation (3), and on the effective mass associated with the lowest energy valley, m ; see, for example, Fig. 1 of Chin et al. [30]. In 2002, Davydov et al. [82], Wu et al. [83], and Matsuoka et al. [84] presented experimental evidence which instead suggests a considerably

10 96 J Mater Sci: Mater Electron (2006) 17: smaller energy gap for InN, around 0.7 ev. Most recently, within the context of an experimental study on the electron transport within InN, Tsen et al. [85] suggested an energy gap of 0.75 ev and an effective mass of m e for this material. As this new value for the energy gap associated with InN is still under active investigation [86], for the purposes of our present Monte Carlo simulations of the electron transport within this material, we adopt the traditional Tansley and Foley [75] energy gap value. The sensitivity of the velocity-field characteristic associated with bulk wurtzite GaN to variations in the non-parabolicity coefficient, α, and the effective mass associated with the lowest energy valley, m, will be explored, in detail, in Section 3, InN being expected to exhibit a similar behavior. The band structure associated with bulk wurtzite GaN has also been the focus of some controversy. In particular, Brazel et al. [87] employed ballistic electron emission microscopy measurements in order to demonstrate that the first upper conduction band valley occurs only 340 mev above the lowest point in the conduction band. This contrasts rather dramatically with more traditional results, such as the calculation of Lambrecht and Segall [79], which instead suggest that the first upper conduction band valley minimum within this material occurs about 2 ev above the lowest point in the conduction band. Clearly, this will have a significant impact upon the results. While the results of Brazel et al. [87] were reported in 1997, most bulk wurtzite GaN electron transport simulations have adopted the more traditional intervalley energy separation of about 2 ev. Accordingly, we have adopted the more traditional intervalley energy separation for the purposes of our present analysis. The sensitivity of the velocity-field characteristic associated with bulk wurtzite GaN to variations in the intervalley energy separation will be explored, in detail, in Section Steady-State and Transient Electron Transport Within Bulk Wurtzite GaN, AlN, and InN 3.1. Introduction The current interest in the III V nitride semiconductors, GaN, AlN, and InN, is primarily being fueled by the tremendous potential of these materials for novel electronic and optoelectronic device applications. With the recognition that informed electronic and optoelectronic device design requires a firm understanding of the nature of the electron transport within these materials, electron transport within the III V nitride semiconductors has been the focus of intensive investigation over the years. The literature abounds with studies on the steady-state and transient electron transport within these materials [26 47, 51, 52, 54, 55]. As a result of this intense flurry of research activity, novel III V nitride semiconductor based devices are starting to be deployed in commercial products today. Future developments in the III V nitride semiconductor field will undoubtably require an even deeper understanding of the electron transport mechanisms within these materials. In the previous section, we presented details of our semiclassical three-valley Monte Carlo simulation approach, that we employ for the analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN. In this section, a collection of steady-state and transient electron transport results, obtained from these Monte Carlo simulations, is presented. Initially, an overview of our steady-state electron transport results, corresponding to the three III V nitride semiconductors under consideration in this analysis, i.e., GaN, AlN, and InN, will be provided, and a comparison with the more conventional III V compound semiconductor, GaAs, will be presented. A comparison between the temperature dependence of the velocity-field characteristics associated with GaN and GaAs will then be presented, and our Monte Carlo results will be used in order to account for the differences in behavior. A similar analysis will be presented for the doping dependence. Next, detailed simulation results, in which the sensitivity of the velocity-field characteristics associated with AlN and InN to variations in the crystal temperature and the doping concentration is explored, will be presented. The sensitivity of the velocity-field characteristic associated with bulk wurtzite GaN to variations in the band structure will then be examined, this analysis providing us with some insight into the range of outcomes expected for these materials, this being a useful exercise, particularly for those III V nitride semiconductors which have, as yet, unresolved band structures, i.e., GaN and InN. Finally, the transient electron transport which occurs within the III V nitride semiconductors under investigation in this analysis, i.e., GaN, AlN, and InN, is determined and compared with that corresponding to GaAs. Our Monte Carlo results conform with state-of-the-art III V nitride semiconductor orthodoxy, although there have been some recent developments which have led to mild corrections to these results. These will be discussed in Section 4. This section is organized in the following manner. In Sections 3.2, 3.3, and 3.4, the velocity-field characteristics associated with GaN, AlN, and InN are presented and analyzed. For the purposes of comparison, in Section 3.5, an analogous analysis is performed for the case of GaAs, the velocity-field characteristics associated with the III V nitride semiconductors under consideration in this analysis, i.e., GaN, AlN, and InN, being compared and contrasted with that corresponding to GaAs in Section 3.6. The sensitivity of the velocity-field characteristic associated with GaN to variations in the crystal temperature will then be examined in Section 3.7, and a comparison with that corresponding to GaAs presented. In Section 3.8, the sensitivity of the velocity-field

11 J Mater Sci: Mater Electron (2006) 17: Fig. 6 The velocity-field characteristic associated with bulk wurtzite GaN. Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric field strengths it decreases until it saturates characteristic associated with GaN to variations in the doping concentration level will be explored, and a comparison with that corresponding to GaAs presented. The sensitivity of the velocity-field characteristics associated with AlN and InN to variations in the crystal temperature and the doping concentration will then be examined in Sections 3.9 and 3.10, respectively. The sensitivity of the velocity-field characteristic associated with bulk wurtzite GaN to variations in the band structure will then be examined in Section Our transient electron transport results are then presented in Section Finally, the conclusions of this electron transport analysis are summarized in Section Steady-state electron transport within bulk wurtzite GaN Our examination of results begins with bulk wurtzite GaN, the most commonly studied III V nitride semiconductor. The velocity-field characteristic associated with this material is depicted in Fig. 6. This result was obtained through a steadystate Monte Carlo simulation of the electron transport within this material for the GaN parameter selections specified in Tables 1 and 2, the crystal temperature being set to 300 K and the doping concentration being set to cm 3.We note that initially the electron drift velocity monotonically increases with the applied electric field strength, reaching a maximum of about cm/s when the applied electric field strength is around 140 kv/cm. For applied electric fields strengths in excess of 140 kv/cm, the electron drift velocity decreases in response to further increases in the applied electric field strength, i.e., a region of negative differential mobility is observed, the electron drift velocity eventually Fig. 7 The average electron energy as a function of the applied electric field strength for bulk wurtzite GaN. Initially, the average electron energy remains low, only slightly higher than the thermal energy, 3 2 k bt, where k b denotes Boltzmann s constant. At 100 kv/cm, however, the average electron energy increases dramatically. This increase is due to the fact that the polar optical phonon scattering mechanism can no longer absorb all of the energy gained from the applied electric field. The energy minima corresponding to the upper valleys are depicted with the dashed lines saturating at about cm/s for sufficiently high applied electric field strengths. By examining further the results of our Monte Carlo simulation, an understanding of this result becomes clear. First, we consider the results at low applied electric field strengths, i.e., applied electric field strengths less than 30 kv/cm. This is referred to as the linear regime of electron transport, as in this regime, the electron drift velocity is well characterized by the low-field electron drift mobility, μ, i.e., a linear low-field electron drift velocity dependence on the applied electric field strength, v d = μe, applies in this regime. Examining the distribution function for this regime, we find that it is very similar to the zero-field distribution function with a slight shift in the direction opposite of the applied electric field. In this regime, the average electron energy remains relatively low, with most of the energy gained from the applied electric field being transferred into the lattice through polar optical phonon scattering. We find that the low-field electron drift mobility, μ, corresponding to the velocity field characteristic depicted in Fig. 6, is around 850 cm 2 /Vs. If we examine the average electron energy as a function of the applied electric field strength, shown in Fig. 7, we see that there is a sudden increase at around 100 kv/cm; this result was obtained from the same steady-state GaN Monte Carlo simulation of electron transport as that used to determine Fig. 6. In order to understand why this increase occurs, we note that the dominant energy loss mechanism for many of the III V compound semiconductors, including bulk wurtzite