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J Mater Sci: Mater Electron (2014) 25:4675–4713 DOI 10.1007/s10854-014-2226-2 REVIEW Steady-state and transient electron transport within the wide energy gap compound semiconductors gallium nitride and zinc oxide: an updated and critical review Walid A. Hadi · Michael S. Shur · Stephen K. O’Leary Received: 20 May 2014 / Accepted: 29 July 2014 / Published online: 12 September 2014 © Springer Science+Business Media New York 2014 Abstract The wide energy gap compound semiconductors, gallium nitride and zinc oxide, are widely 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 wide energy gap compound 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 analyzes of the electron transport within the wide energy gap compound semiconductors, gallium nitride and zinc oxide. 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, zincblende gallium nitride, and wurtzite zinc oxide. The electron transport that occurs within bulk zinc-blende gallium arsenide will also be considered, albeit primarily for benchmarking purposes. Most of our discussion will focus on results obtained from our ensemble semi-classical threevalley Monte Carlo simulations of the electron transport within these materials, our results conforming with stateof-the-art wide energy gap compound semiconductor orthodoxy. A brief tutorial on the Monte Carlo electron transport simulation approach, this approach being used to generate the results presented herein, 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. The wurtzite gallium nitride and zinc-blende gallium arsenide results, being presented in a previous review article of ours (O’Leary et al. in J Mater Sci Mater Electron 17:87, 2006), are also presented herein for the sake of completeness. 1 Introduction This paper is dedicated to the memory of our friend, mentor, and co-author, Professor Lester F. Eastman, of Cornell University, who passed away in 2013. Note to Reader Some of the results presented herein, and portions of the text, are borrowed from our previous review article; see O’Leary et al. [62]. This overlap in results and text is meant to make this particular review article as self-contained and complete as possible. W. A. Hadi · S. K. O’Leary (&) School of Engineering, The University of British Columbia, 3333 University Way, Kelowna, BC V1V 1V7, Canada e-mail: [email protected] M. S. Shur Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA Wide energy gap semiconductors, i.e., semiconductors with energy gaps wider than those associated with the more conventional semiconductors, crystalline silicon (c-Si) and crystalline gallium arsenide (c-GaAs) [1], offer considerable promise for novel electronic and optoelectronic device applications [2–9]. Owing to the fact that wider energy gap semiconductors tend to possess higher polar optical phonon energies, the saturation electron drift velocities exhibited by these materials tend to be higher [10–13]. In addition, the dielectric constants, both static and high-frequency, associated with the wider energy gap semiconductors tend to be smaller than those associated with the more conventional semiconductors [14, 15]. Both of these factors favor improved electron device performance [16–18]. An 123 4676 additional benefit of the wide energy gap semiconductors is their great tolerance to high applied electric field strengths, making them ideal for high-power device applications, the breakdown field of a semiconductor material increasing with the magnitude of its energy gap [19, 20]. Finally, the high thermal conductivities associated with these materials further adds to their allure [21, 22]. Two wide energy gap compound semiconductors currently attracting attention are gallium nitride (GaN) and zinc oxide (ZnO) [23–53].1 The wurtzite phase of GaN has a wide and direct energy gap; at room temperature, this energy gap is around 3.39 eV [54]. Wurtzite GaN also exhibits a high breakdown field [55, 56], elevated thermal conductivity [57, 58], and superb electron transport characteristics [59–62]. These attributes make GaN ideally suited for both electronic and optoelectronic device applications [63–78]. ZnO, while currently finding applications as a material for low-field thin-film transistor electron device structures [79] and as a potential material for transparent conducting electrodes [80], also possesses a direct energy gap [81, 82] with a magnitude that is very similar to that exhibited by GaN [83]. Thus, it might be expected that, with some further improvements in its material quality, ZnO may also be employed for some of the device roles currently implemented or envisaged for GaN. It is widely recognized that improvements in the design of GaN and ZnO-based devices may be achieved through a greater understanding of the underlying electron transport mechanisms. The scientific literature abounds with studies into the electron transport within both GaN and ZnO [3, 10, 59–62, 82, 84–130]. In this paper, we aim to provide some perspective on this large body of material by reviewing what is currently known about the electron transport within these wide energy gap compound semiconductors. This review article begins with an introduction to the general principles underlying electron transport within semiconductors. The use of Monte Carlo simulations, in the characterization of the electron transport within such 1 The earliest recorded studies on GaN, reported in the 1920s and 1930s, were performed on small crystals and powders [23]. Unfortunately, these materials were of insufficient quality for device applications. Thus, GaN remained a material of widely recognized but unrealized potential for many years. It was only when modern deposition approaches, such as molecular beam epitaxy and metalorganic chemical vapor deposition, were employed for the preparation of GaN that this material approached the levels of quality demanded of devices. Thus, intense interest into the material properties of GaN only really began in earnest in the early-1990s. While initial reports on the material properties of ZnO were made in the 1920s and 1930s, it was only much later that the quality of the material became sufficiently high that a diverse range of device applications could be considered. Accordingly, interest in the material properties of ZnO began in earnest in the early-2000s. Interest in both of these materials, and the device applications thus engendered, continues today. 123 J Mater Sci: Mater Electron (2014) 25:4675–4713 semiconductors, is then described. Then, a tabulation of the material and band structural parameters, corresponding to the semiconductors under investigation in this analysis, i.e., GaN and ZnO, is provided. A critical comparison with the material parameters corresponding to some other common compound semiconductors is also presented. Monte Carlo electron transport simulation results, corresponding to the materials under investigation in this analysis, are then featured. We conclude this review by presenting what is currently known about the electron transport within the wide energy gap compound semiconductors, GaN and ZnO, and how this understanding has evolved into its current form. Bulk electron transport within the materials under investigation, i.e., GaN and ZnO, will be the principal focus of this analysis. Results corresponding to GaAs will also be presented, albeit primarily for bench-marking purposes. In 2006, we reviewed what was then known about the electron transport within the III–V nitride semiconductors, GaN, indium nitride (InN), and aluminum nitride (AlN) [62]. In the present review, we focus on the wide energy gap compound semiconductors, GaN and ZnO. The GaN component of this analysis focuses on both the wurtzite and zinc-blende phases of this material; zincblende GaN, while less studied than its wurtzite counterpart, is presently being considered for a number of important device applications, as it offers certain advantages over the wurtzite phase of this material [131, 132]. The novelty in this particular review primarily resides in our zinc-blende GaN and wurtzite ZnO results. The wurtzite GaN and zinc-blende GaAs results are largely borrowed from our previously published 2006 review article [62]. The electron transport overview is also largely borrowed from this 2006 review article [62]. This overlap with our previously published review article [62] is introduced so that the present review article is reasonably selfcontained and complete. It should be noted, however, that the critical comparison between the material properties of GaN and ZnO with those associated with some other common compound semiconductors, which has been added to the electron transport overview section, constitutes an additional novel contribution to this review. The review of the electron transport literature, corresponding to the wide energy gap compound semiconductor, GaN, has been updated and revised, and is therefore another novel component to this analysis. Finally, the review of the scientific literature related to electron transport within ZnO, not being presented in our previous review article [62], represents an additional novel component to this analysis. This paper is organized in the following manner. In Sect. 2, the mechanisms underlying electron transport within semiconductors are introduced, and the use of Monte Carlo simulations, in the characterization of the J Mater Sci: Mater Electron (2014) 25:4675–4713 electron transport within semiconductors, is discussed. The material and band structural parameters, corresponding to wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are also featured in Sect. 2, these parameters being used in the subsequent analysis; a critical comparison between these material and band structural parameters, and those corresponding to some other common compound semiconductors, is also provided. Steady-state and transient electron transport results, obtained from our ensemble semi-classical threevalley Monte Carlo electron transport simulations within the wide energy gap compound semiconductors, GaN and ZnO, are then presented in Sect. 3. What is currently known about the electron transport of the wide energy gap compound semiconductors of interest, i.e., GaN and ZnO, and how this understanding has evolved into its current form, is then discussed in Sect. 4. Finally, the conclusions of this review are drawn in Sect. 5. 2 A primer on electron transport within semiconductors 2.1 Introduction Device performance is a primary consideration for a semiconductor device designer. The electron transport processes that occur within a semiconductor are known to play a critical role in determining this performance. Accordingly, understanding the electron transport that occurs within semiconductors, and developing tools whereby the mechanisms underlying it may be quantitatively characterized and related to semiconductor device performance, has been a high priority for semiconductor device designers for many years [133], and continues to be so [134, 135]. Through this understanding, improvements in semiconductor device performance have been achieved. Electrons within a semiconductor engage in continuous motion. In thermal equilibrium, this motion arises from the thermal energy that is present. Each electron undergoes a series of free-flights between scattering events. Scattering events occur owing to interactions between the transiting electrons and the lattice atoms, impurities, other electrons, and defects within the semiconductor. In thermal equilibrium, the electrons can move in all directions, without preference. For every electron moving in one direction, there is another electron moving in exactly the opposite direction. Accordingly, taken as an ensemble, the net electron current is nil. Through the application of an external electric field, thermal equilibrium no longer applies. Subject to this external electric field, each electron in the ensemble will experience a force in the opposite direction of the electric field. This force will act to modify the course of the free- 4677 flights that occur between scattering events. Given all of the factors at play in the transport of electrons between scattering events, for any individual electron, the impact of this external electric field may be negligible. Taken as an ensemble, however, and this external applied electric field will lead to a net electron current. Determining how the distribution of electrons within such an ensemble evolves with time under the action of such an external electric field is the fundamental issue at stake when one examines the electron transport within a semiconductor [133]. In this section, we provide a primer on the electron transport within semiconductors. We start-off by discussing how an ensemble of electrons may be treated as a continuum in phase-space. We then introduce the Boltzmann transport equation, and discuss how it may be used in order to examine, in the continuum limit, the evolution of an ensemble of electrons within phase-space under the action of an applied electric field. The principles underlying Monte Carlo simulation analyzes of electron transport within semiconductors are then presented, and an explanation, as to how this approach may be used in order to solve the Boltzmann transport equation, is then provided. We then introduce the ensemble semi-classical three-valley Monte Carlo simulation approach, used in our simulations of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO. The material and band structural parameter selections, corresponding to wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are then presented, a critical comparison with the material and band structural parameters corresponding to some other common compound semiconductors also being provided. Finally, the dependence of the scattering rates on the electron wave-vector, ~ k , corresponding to the key scattering processes, is provided for the cases of wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs. This chapter is organized in the following manner. In Sect. 2.2, the treatment of an electron ensemble as a continuum in phase-space, and the use of the Boltzmann transport equation, in order to determine the evolution of this ensemble under the action of an electric field, is presented. Then, in Sect. 2.3, the basic principles underlying Monte Carlo simulations of electron transport, and how such an approach yields a solution to the Boltzmann transport equation, is discussed. Details, related to the Monte Carlo algorithm employed, and various approximations that are often introduced, are then provided in Sect. 2.4. In Sect. 2.5, the material and band structural parameter selections, corresponding to wurtzite GaN, zincblende GaN, and wurtzite ZnO, are presented, a critical comparison with the material and band structural parameters corresponding to some other common compound semiconductors also being provided. The specifics of our ensemble semi-classical three-valley Monte Carlo 123 4678 J Mater Sci: Mater Electron (2014) 25:4675–4713 simulation approach to treating electron transport within a semiconductor are then featured in Sect. 2.6. Finally, the dependence of the scattering rates on the electron wavevector, ~ k , for the key scattering processes shaping the nature of the electron transport within a compound semiconductor, is provided for the key materials considered in this analysis, i.e., GaN, ZnO, and GaAs, in Sect. 2.7. 2.2 Electron transport in the continuum limit: the Boltzmann transport equation For the case of a bulk semiconductor, the individual nature of the electrons within an ensemble can be neglected. Instead, these electrons may be considered as being part of a continuum. In this continuum limit, the distribution of electrons may be characterizedthrough of the dis the specification tribution function, f ~; r~ k ; t , f ~; r~ k ; t DrDk representing the number of electrons within the infinitesimal volume DrDk around the point ~; r~ k in phase-space at time t, ~ r denoting the position and ~ k representing the electron wavevector; in this context, phase-space corresponds to the union of the real and electron wave-vector spaces. The Boltzmann transport equation provides a means of determining how this distribution function evolves with time under the action of an external electric field. The distribution function corresponding to an electron ensemble within a semiconductor evolves in time owing to three basic driving factors. There is the transfer of electrons in real-space, owing to the electron velocities that occur. There is the transfer of electrons in electron wave-vector space, these being related to the rate of change, with respect to time, of the electron wave-vector associated with the different electrons. Finally, there are the scattering processes that occur. Analytically, Shur [134] expresses the Boltzmann transport equation as of of _ ~ ~ rr f k rk f þ ; ¼ v ð1Þ ot ot scat where the terms following the equal sign in Eq. (1), from left to right, correspond to the three aforementioned driving factors, ~, v the electron velocity, being related to the electron band structure, ~ k ; see Eq. (4). That is, while the first term corresponds to the transfer of electrons in real-space, owing to the electron velocities that occur, the second term corresponds to the transfer of electrons in electron wavevector space, owing to the rate of change, with respect to time, of the electron wave-vectors associated with the different electrons, the final term corresponding to the contributions related to the different scattering processes that occur. Fundamentally, the Boltzmann transport equation is a continuity equation for the distribution function. Its solution provides for a complete characterization of the nature of the electron transport within a bulk 123 semiconductor. Further discussion on the Boltzmann equation is provided in the literature [136, 137]. Owing to its importance, a number of solutions to the Boltzmann transport equation have been devised over the years. Lowfield asymptotic analytical solutions include those of Chin et al. [86], Shur et al. [89], and Look et al. [138]. Higherfield approximate analytical solutions include those of Das and Ferry [3], Ferry [10], Conwell and Vassel [139], and Sandborn et al. [140]. Numerical techniques, that solve the Boltzmann transport equation directly, have also been developed. Unfortunately, the numerical computations that are demanded of such an approach are extremely intense, and typically, one must make approximations in order to allow for numerical tractability. These techniques are further discussed by Nag [133]. The electron transport within a semiconductor has both steady-state and transient components. Accordingly, the solution to the Boltzmann transport equation must take into account both aspects of the electron transport response. In considering steady-state electron transport, the state of the distribution function long after the application of the electric field, i.e., after all of the transients have been fully extinguished, is considered.2 In contrast, when considering transient electron transport, how the distribution function evolves in time is considered. Both the steady-state and transient components of electron transport, corresponding to the wide energy gap compound semiconductors, GaN and ZnO, are considered in this review article. As was mentioned previously, the Boltzmann transport equation applies to cases for which the corpuscular nature of the electrons within an ensemble may be neglected. For the case of simulating the electron transport within a bulk semiconductor, as in this analysis, the dimensions of the material are large and the treatment of the ensemble of electrons as a continuum is justifiable. When the dimensions are small, however, and quantum effects are significant, then the Boltzmann transport equation, and its continuum treatment of the electron ensemble, is incorrect. For such cases, quantum electron transport approaches must be employed instead for the treatment of the electron transport. Such approaches lie beyond the scope of the work reviewed here and are adequately discussed in the literature [141]. 2.3 Monte Carlo simulations of electron transport In a Monte Carlo simulation of electron transport, one employs a random number generator in order to simulate the random character of the electron transport within a 2 This requires that the electron ensemble has settled on a new equilibrium state. By an equilibrium state, however, we are not necessarily referring to thermal equilibrium, thermal equilibrium only being achieved in the absence of an applied electric field. J Mater Sci: Mater Electron (2014) 25:4675–4713 semiconductor. Instead of treating the ensemble of electrons as a continuum, as in the Boltzmann transport equation, one instead focuses on the electron transport of the individual electrons within the ensemble. Through tracking the motion of an individual electron for a long time, or through simulating the motion of a large number of electrons, a solution to the Boltzmann transport equation will emerge in the continuum limit. The accuracy of this solution will increase as the number of electrons in the simulation, or the length of the simulation time, is increased. This electron transport simulation approach is frequently employed in the simulation of the electron transport within semiconductors. With specific reference to the wide energy gap compound semiconductors being considered in this particular analysis, i.e., GaN and ZnO, Monte Carlo simulations of the electron transport are employed for the analysis of both materials. As was hinted at earlier, there are actually two broad categories of Monte Carlo simulation that can be employed in the analysis of electron transport: (1) single-particle, and (2) ensemble. In a single-particle Monte Carlo simulation, the transit of an individual electron is examined. Given sufficient time, the amount of time spent in any particular region of phase-space will be proportional to the distribution function, f ~; r~ k ; t , there. In contrast, for an ensemble Monte Carlo simulation, the transit of a large number of electrons is considered. In this approach, the number of electrons at any particular region in phase-space will be proportional to the distribution function, f ~; r~ k ; t , there. Assuming that the electron motion is ergodic, i.e., that the time averages correspond to the statistical averages, the resultant distribution functions should be identical [142]. Unfortunately, while single-particle Monte Carlo simulations are perfectly capable of resolving the steady-state electron transport, the resolution of the transient electron transport using this technique represents a challenge, variations in time being used as a proxy for ensemble variations in the single-particle approach. The ensemble Monte Carlo simulation technique, however, allows for the effective treatment of both the steady-state and the transient electron transport responses. Given that we are considering both aspects of the electron transport within the scope of this review, we focus on ensemble Monte Carlo simulations for the purposes of our analysis of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO. 2.4 The ensemble Monte Carlo simulation algorithm 2.4.1 Algorithm Within the framework of an ensemble Monte Carlo simulation of the electron transport within a semiconductor, the 4679 motion of a large number of electrons within the semiconductor, under the action of an applied electric field, is considered. Electrons transit through the semiconductor, accelerating under the action of the applied electric field, these accelerations being interrupted by scattering processes, these being related to the interaction of the transiting electrons with the thermal motion of the lattice, i.e., phonons, ionized impurities, lattice dislocations, and other electrons. Typically, each type of scattering is characterized by a scattering rate, the probability of a given “scattering event” occurring over an infinitesimal time interval being directly proportional to the product of its scattering rate and the duration of this time-interval. Clearly, for a Monte Carlo simulation, the accuracy of the results will be shaped, in large measure, by the selection of scattering rates employed. It is often the case that these scattering rates are determined at the outset of the simulation, 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 this analysis. The key elements of the Monte Carlo electron transport simulation algorithm are shown in Fig. 1. In the initialization phase, the initial scattering rate tables are determined. Each electron is assigned a specific point in phase-space, i.e., a value of ~ r and a value of ~ k , the initial distribution of electrons within phase-space being in accordance with Fermi-Dirac occupation statistics, i.e., the ensemble of electrons is initially assumed to be in thermal equilibrium. The evolution of this ensemble of electrons, under the action of an applied electric field, is the issue at stake in the study of the electron transport within a semiconductor [133]. We now consider the main body of the algorithm. In this phase, the motion of the electrons in the ensemble is divided into a number of small time-steps, Dt. From its initial point in phase-space, each electron is assumed to accelerate under the action of the applied electric field over the time-step, Dt, i.e., the electrons experience free-flight. 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, as is explained later. The time for each free-flight must be chosen carefully, and depends critically on the scattering rates at the beginning of the electron’s free-flight, as well as the scattering rates throughout its free-flight. Since the scattering rates change over the flight, the selection of the free-flight time is 123 4680 Fig. 1 A flowchart corresponding to our electron transport Monte Carlo simulation algorithm. A more detailed flowchart is shown in Appendix A of O’Leary et al. [62]. This figure is after that depicted in Figure 5 of O’Leary et al. [62] J Mater Sci: Mater Electron (2014) 25:4675–4713 Initialize Move each electron through one time−step Move electron to the end of the free−flight Determine scattering mechanism and wave−vector after scattering event Generate new free−flight time End of time−step reached? No Yes Calculate macroscopic quantities End of simulation reached? No Yes Output statistics complex. Methods used for generating the free-flight time have been extensively studied, and means of generating free-flight times are further detailed in Appendix A of O’Leary et al. [62]; Yorston [143] also provides a detailed discussion on this matter. 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 through the free-flight, a new free-flight time is chosen and the process repeats itself until that electron reaches the end of the current time-step. Once the electrons have gone through a time-step, the resultant electron distribution may be used to determine the electron transport properties of interest. The particular electron transport properties that we consider in our analysis include the electron drift velocity,3 the average electron energy, and the number of electrons in each valley. This process is repeated, time-step after time-step, until the entire simulation is complete. If the results are to be 3 By electron drift velocity, we are referring to the average electron velocity, determined by statistically averaging over the entire electron ensemble. 123 determined as a function of the applied electric field strength, periodic updates to the applied electric field strength selection are performed throughout the simulation, the time between updates being sufficient in order to ensure that steady-state is achieved before the next update to the field occurs. Once the simulation is complete, the results are sent to a file for the purposes of archiving, processing, and subsequent retrieval.4 2.4.2 The three-valley model approximation Transiting electrons in a semiconductor tend to congregate in the lower energy parts of the conduction band. A great simplification in the analysis may be achieved simply by focusing on the three lowest energy valleys of the conduction band rather than the entire band structure; these “valleys” actually correspond to the regions in ~ k -space that are in the immediate vicinity of the three lowest energy 4 The Monte Carlo approach to simulating the electron transport within semiconductors has been employed by many authors. A Monte Carlo electron transport simulation resource, with code included, may be found at https://nanohub.org/resources/moca. Further information about the Monte Carlo approach, beyond the electron transport context, may also be found at http://www.codeproject.com/Articles/ 767997/Parallelised-Monte-Carlo-Algorithms-sharp and http://www. codeproject.com/Articles/32654/Monte-Carlo-Simulation?q=Monte+ Carlo+code. J Mater Sci: Mater Electron (2014) 25:4675–4713 Valley 3 4681 Table 1 The material parameter selections, used for our simulations of the electron transport within wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are as specified below wurtzite GaN * m =me Valley 2 α=0 eV−1 m =m g=6 α=0 eV−1 * e g=1 2.1 eV 1.9 eV Valley 1 Parameter w-GaN Mass density (g/cm3 ) 6.15a Longitudinal sound velocity (cm/s) (see footnote 7) e α=0.189 eV−1 g=1 L−M Γ1 Γ2 Fig. 2 The three-valley model used to represent the conduction band electron band structure associated with wurtzite GaN for our Monte Carlo simulations of the electron transport within this material. The valley parameters, corresponding to wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are tabulated in Table 2; see Sect. 2.5.1. The degeneracy, g, associated with each valley is clearly indicated. This figure is after that depicted in Figure 1 of O’Leary et al. [62]. The online version of this figure is depicted in color conduction band minima.5 A further simplification may be achieved by adopting the Kane model in order to describe the form of each of these three valleys [144]. That is, in the vicinity of each conduction band valley minimum, the energy band is taken to be spherically symmetric, and of the form 2 k 2 h ¼ E ð1 þ aEÞ; 2m ð2Þ hk denoting the magnitude of the crystal momentum and E representing the electron energy, E ¼ 0 corresponding to the valley minimum, m being the effective mass of an electron at the valley minimum, and a being the non-parabolicity coefficient associated with that particular valley.6 Within 5 The conduction band minima may be degenerate, i.e., the same conduction band energy minima may be achieved at multiple points throughout the conduction band band structure. Valley 1 corresponds to those conduction band minima that are at the lowest energy. Valleys 2 and 3 correspond to those conduction band minima at the second most and third most lowest energy minima, respectively. 6 Albrecht et al. [82] generalize this relationship in order to include a second-order non-parabolocity coefficient that reduces to the traditional Kane model in the limit that this second-order non-parabolocity coefficient is set to zero. w-ZnO 6.15b 5.68c 6:56 10 5b 6:56 10 4:00 105 c Transverse sound velocity (cm/s) (see footnote 7) 2:68 105 a 2:68 105 b 2:70 105 c Acoustic deformation potential (eV) 8.3a 8.3b 3.83c Static dielectric constant 8.9a 8.9b 8.2c High-frequency dielectric constant 5.35a 5.35b 3.7c Effective mass (C1 valley) 0.20 me a 0.15 me b 0.17 me c Piezoelectric constant, e14 (C/cm2 ) (see footnote 8) 3:75 105 a 3:75 105 b 3:75 105 c Direct energy gap (eV) 3.39a 3.2b 3.4c Optical phonon energy (meV) 91.2a 91.2b 72.0c Inter-valley deformation potentials (eV/cm) (see footnote 9) 109 a 109 b 109 c Inter-valley phonon energies (meV) (see footnote 10) 91.2a 91.2b 72.0c * m =0.2 m zb-GaN 5a These parameter selections are drawn from O’Leary et al. [62], Foutz et al. [92], and Hadi et al. [126], for the cases of wurtzite GaN, zincblende GaN, and wurtzite ZnO, respectively; the parameters from Foutz et al. [92] have been slightly modified in order to more accurately reflect the reality of zinc-blende GaN a Parameters from O’Leary et al. [62] b Parameters slightly modified from Foutz et al. [92] c Parameters from Hadi et al. [126] the context of the Kane model, this non-parabolicity coefficient, 1 m 2 1 ; ð3Þ a¼ Eg me where Eg represents the corresponding energy gap [144]. The three-valley models, used to represent the conduction band structures associated with wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, for our Monte Carlo simulations of the electron transport within these materials, are described in Sect. 2.5 and detailed in Table 2. A representative three-valley model, corresponding to the specific case of wurtzite GaN, is depicted in Fig. 2, the degeneracy of each valley being clearly indicated. 123 4682 J Mater Sci: Mater Electron (2014) 25:4675–4713 Table 2 The band structural parameter selections, used for our Monte Carlo simulations of the electron transport within wurtzite GaN, zincblende GaN, and wurtzite ZnO, are as specified below w-GaN Valley number 1 2 3 Valley location C1 a C2 a L-Ma Valley degeneracy 1 1 6 Effective mass 0.2 me a me a me a Inter-valley energy separation (eV) 1.9a Energy gap (eV) zb-GaN w-ZnO 3.39 a 2.1a a 5.49a 5.29 Non-parabolicity (eV1 ) 0.189 0.0 Valley location Valley degeneracy C1 b 1b Xb 3b Effective mass 0.15 me b 0.40 me b Intervalley energy separation (eV) a b a 0.0a Lb 4b 0.60 me b 1.4 b 2.7b 4.6 b 5.9b Energy gap (eV) 3.2 Non-parabolicity (eV1 ) 0.226b 0.078b 0.027b Valley location C1 C2 L-Mc Valley degeneracy 1 c c c 0.17 me Intervalley energy separation (eV) Energy gap (eV) 3.4c Non-parabolicity (eV ) 0.66 6c 1 Effective mass 1 c c 0.42 me 4.4 c 0.70 me c 4.6c 7.8c c c 8.0c c 0.15 0.0c These parameter selections are drawn from O’Leary et al. [62], Foutz et al. [92], and Hadi et al. [126], for the cases of wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively; the parameters from Foutz et al. [92] have been slightly modified in order to more accurately reflect the reality of zinc-blende GaN a Parameters from O’Leary et al. [62] b Parameters modified from Foutz et al. [92] c Parameters from Hadi et al. [126] 2.4.3 The semi-classical approximation An electron wave-function associated with a semiconductor extends across the entire volume of the crystal. Accordingly, at any given instant, the electrons within a semiconductor are able to interact with all of the atoms and all of the other electrons that are present within it. That is, a given electron can simultaneously interact with a multitude of phonons, ionized impurities, lattice dislocations, and other electrons. Unfortunately, this perspective on electron transport is rather complex and does not provide much insight into the character of the electron transport within semiconductors. Thus, it is often the case that assumptions are introduced in order to render the analysis more tractable. The semi-classical treatment of the motion of electrons within a semiconductor is one of the most common simplifying assumptions that is introduced into analyzes of the electron transport within semiconductors. Within the framework of this assumption, each electron within the ensemble is treated as if it were a point particle. From a semi-classical perspective, an electron, with an electron wave-vector ~ k , has a velocity 123 1 ~ v ¼ rk ~ k ; ð4Þ h where ~ k denotes the corresponding electron band structure, i.e., the energy of the electron as a function of the electron wave-vector, ~ k [137]. Under the action of an applied electric field, ~ E , the rate of change of an electron’s momentum, ~ p ¼ h~ k , with respect to time, may be expressed as h ~ dk ~: ¼ qE dt ð5Þ Equations (4) and (5) define the semi-classical trajectory of this electron, assuming that the periodic potential associated with the underlying crystal is static. In reality, the thermal motion of the lattice, the presence of ionized imperfections and lattice dislocations, and interactions with the other electrons in the ensemble, result in the electron deviating from the path literally prescribed by Eqs. (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 aforementioned quantum mechanical notion of “scattering J Mater Sci: Mater Electron (2014) 25:4675–4713 events.” During a “scattering event,” the electron’s wavefunction abruptly changes. Quantum mechanics determines the probability of each type of “scattering event,” and dictates how to probabilistically determine the change in the electron 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. 2.4.4 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) defect scattering, i.e., related to lattice dislocations, and (3) carrier scattering [133]. For the wide energy gap compound semiconductors, GaN and ZnO, phonon scattering is the most important scattering mechanism, and it is featured prominently in our simulations of the electron transport within these materials. 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 charge. This mechanism is an important factor to consider in determining the electron transport within GaN and ZnO, and the effect of the doping concentration on the electron transport within these materials is featured prominently in our analysis. The final category of scattering mechanism, carrier scattering, or in our case, electron-electron scattering, has also been taken into account in some of 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, electron-electron scattering was not included in all of our simulations. Owing to their importance in determining the nature of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO, 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 4683 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 to as nonpolar 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 GaN and ZnO, 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 bonds within GaN and ZnO, 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 these wide energy gap compound 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 many compound semiconductors, and is known to be particularly important for the case of GaN; as will be seen later, intervalley transitions are not particularly important for the case of wurtzite ZnO, the large non-parabolicity associated with the lowest energy conduction band valley coupled with the wide conduction band inter-valley energy separation found within this material inhibiting the occurrence of inter-valley transitions. Inter-valley scattering is believed to be responsible for the negative differential mobility observed in the velocity-field characteristics associated with many compound semiconductors, such as GaN and GaAs. For the specific case of wurtzite ZnO, however, the large non-parabolicity of the lowest energy conduction band valley leads to a dramatic increase in the electron effective mass for applied electric field strengths approaching the peak field, and this leads to the observed negative differential mobility, i.e., the electrons become heavier and thus slower. A derivation of all of these scattering rates, as a function of the semiconductor parameters, can be found in the literature; see, for example, [133, 141, 145]. A formalism, which closely matches the form used in our ensemble semiclassical three-valley Monte Carlo simulations of electron transport, is found in Nag [133]. Many of the scattering rates that are employed for the purposes of our Monte Carlo simulations of the electron transport within the wide 123 4684 energy gap compound semiconductors, GaN and ZnO, are also explicitly tabulated in Appendix 22 of Shur [134]. Further discussion on the Monte Carlo simulation algorithm is amply provided in the scientific literature [62, 133–135, 144, 146–148]. 2.5 Parameter selections for wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs 2.5.1 Material and band structural parameter selections The material parameter selections, used for our simulations of the electron transport within wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are as specified in Table 1 [149– 151].7; 8; 9; 10 These parameter selections are drawn from O’Leary et al. [62], Foutz et al. [92], and Hadi et al. [126], for the cases of wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively. The band structural parameter selections, used for our Monte Carlo simulations of the electron transport within wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are as specified in Table 2. These parameter selections are also primarily drawn from O’Leary et al. [62], Foutz et al. [92], and Hadi et al. [126], for the cases of wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively [151, 152]11. The material and band structural parameters, corresponding to zinc-blende GaAs, are primarily drawn from Littlejohn et al. [151] and Blakemore [153]. 7 The longitudinal and transverse sound velocities are equal to sffiffiffiffiffi sffiffiffiffiffi Cl Ct and ; q q respectively, where Cl and Ct denote the respective elastic constants and q represents the density. 8 Piezoelectric scattering is treated using the well established zincblende scattering rates, and thus, a suitably transformed piezoelectric constant, e14 , must be selected. This may be achieved through the transformation suggested by Bykhovski et al. [149, 150]. The e14 value selected for wurtzite GaN is that suggested by Chin et al. [86]. The e14 values selected for zinc-blende GaN and wurtzite ZnO is that corresponding to wurtzite GaN. 9 All inter-valley deformation potentials are set to 109 eV/cm, following the approach of Gelmont et al. [85]. 10 We follow the approach of Bhapkar and Shur [90], and set the inter-valley phonon energies equal to the optical phonon energy, a relationship which holds approximately for the case of GaAs [151]. 11 The band structures are specified according to the three lowest energy conduction band valley minima, each minima corresponding to a valley, their locations in the band structures, the degeneracy of each valley, the effective mass of the electrons at each valley minimum, and the non-parabolicity coefficient corresponding to each valley being specified. 123 J Mater Sci: Mater Electron (2014) 25:4675–4713 2.5.2 A critical comparison with some other common compound semiconductors It is instructive to contrast the material parameters employed for our simulations of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO, with those corresponding to some other common compound semiconductors. The particular material parameters which we focus upon for the purposes of this critical comparative analysis are the electron effective mass, the polar optical phonon energy, the static dielectric constant, and the high-frequency dielectric constant, these parameters being known to play important roles in shaping the nature of the electron transport. The dependence of these parameters on the Eo energy gap at 300 K, as defined by Adachi [14], will be the focus of our analysis.12 The other common compound semiconductors considered in this analysis are aluminum arsenide (AlAs), aluminum phosphide (AlP), the zinc-blende phase of cadmium selenide (zb-CdSe), the wurtzite phase of cadmium sulphide (w-CdS), the zinc-blende phase of cadmium sulphide (zbCdS), gallium antimonide (GaSb), gallium phosphide (GaP), indium antimonide (InSb), indium arsenide (InAs), the wurtzite phase of indium nitride (w-InN), indium phosphide (InP), zinc selenide (ZnSe), and zinc telluride (ZnTe). While the energy gaps associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs are direct, some of the other common compound semiconductors considered in this critical comparative analysis possess indirect energy gaps, wherein the actual energy gap, Eg , i.e., the difference in energy between the minimum of the conduction band and the maximum of the valence band, does not coincide with the Eo energy gap. The material and band structural parameters, used for this critical comparative analysis, are drawn from Adachi [14] and Sze and Ng [154]. The material and band structural parameters, corresponding to these other common compound semiconductors, are tabulated in Table 3. A bar chart of the energy gaps, Eg , at 300 K, corresponding to a representative sampling of elemental and compound semiconductors, including some of the common compound semiconductors considered in this critical comparative analysis, is depicted in Fig. 3. In Fig. 4, we plot the electron effective mass associated with the lowest energy conduction band valley, expressed in units of the free electron mass, me, as a function of the 300 K Eo energy gap corresponding to the materials considered in this critical comparative analysis. It is noted that the electron effective mass monotonically increases in response to 12 For the case of direct-gap semiconductors, the Eo energy gap corresponds with the regular energy gap, Eg . For the case of indirectgap semiconductors, however, the Eo energy gap exceeds Eg . J Mater Sci: Mater Electron (2014) 25:4675–4713 4685 Table 3 The material and band structural parameters, corresponding to the other common compound semiconductors considered in this analysis Semi. Type Crys. B. Eg (eV) Eo (eV) m (me ) hxo (meV) s 1 InSb III–V zb D 0.17 0.17 0.013 23.65 17.2 15.3 InAs III–V zb D 0.359 0.359 0.024 29.93 14.3 11.6 w-InN III–V w Da 0.70a 0.70a 0.04a 73.0a 15.3a 8.4a GaSb III–V zb D 0.72 0.72 0.039 28.9 15.5 14.2 InP III–V zb D 1.35 1.35 0.07927 42.95 12.9 9.9 GaAs III–V zb D 1.43 1.43 0.067 35.3 12.90 10.86 AlSb III–V zb ID 1.615 2.27 0.14 42.16 11.21 9.88 zb-CdSe II–VI zb Db 1.675 1.675 0.119 26.2 9.6 6.2 AlAs III–V zb ID 2.15 3.01 0.124 49.8 10.06 8.16 GaP III–V zb ID 2.261 2.76 0.114 49.91 11.0 8.8 ZnTe II–VI zb D 2.27 2.27 0.117 26.0 9.4 6.9 zb-CdS II–VI zb Db 2.46 2.46 0.14 37.6 9.8 5.4 AlP III–V zb ID 2.48 3.91 0.220 62.12 9.6 7.4 w-CdS II–VI w Db 2.501 2.501 0.151c 37.7d 9.6e 5.35e ZnSe II–VI zb D 2.721 2.721 0.137 31.2 8.9 5.9 zb-GaN III–V zb Df 3.20f 3.20f 0.15f 91.2f 8.9f 5.35f w-GaN III–V w Df 3.39f 3.39f 0.20f 91.2f 8.9f 5.35f w f f f f f f ZnO II–VI D 3.40 3.40 0.17 72.0 3.7f 8.2 The material and band structural parameters, used for this critical comparative analysis, are mostly drawn from Adachi [14] and Sze and Ng [154]. The crystal types are either zinc-blende (zb) or wurtzite (w) for the case of these particular materials. The energy gaps all correspond to 300 K a Parameters from Hadi et al. [164] b The nature of the energy gaps associated with zinc-blende CdSe, zinc-blende CdS, and wurtzite CdS, was suggested by Sze and Ng [154] c The density-of-states effective mass value suggested by Adachi [14] for wurtzite CdS is employed d The polar optical phonon energy, determined at the E1 (LO) point, is adopted for the case of wurtzite CdS. The value suggested by Adachi [14] is employed e These values are obtained through averaging the parallel and perpendicular values supplied by Adachi [14] f Values from Tables 1 and 2 of this article increases in the Eo energy gap. The results of a linear leastsquares fit, depicted with the dashed line in Fig. 4, suggests that the electron effective mass essentially scales linearly with the Eo energy gap. Indeed, the deviations about this linear least-squares fit are noted to be relatively minor. The electron effective mass plays an important role in defining the low-field electron drift mobility, the higher this mass the lower the corresponding low-field electron drift mobility. This result suggests that the low-field electron drift mobility associated with a semiconductor will diminish as the Eo energy gap is increased. In Fig. 5, we plot the polar optical phonon energy, hxo , as a function of the 300 K Eo energy gap corresponding to the materials considered in this critical comparative analysis. We generally find that the polar optical phonon energy monotonically increases in response to increases in the Eo energy gap. In this case, however, there is a considerable amount of scatter about the trend; this scatter is sufficiently great that a linear least-squares fit, while indicative of a general trend, is not statistically significant. The polar optical phonon energy plays a critical role in determining the nature of the high-field electron transport. In particular, it is known that the high-field saturation electron drift velocity is, in large measure, determined by it [10]. This result suggests that semiconductors with wider Eo energy gaps favor higher saturation electron drift velocities. Hence, our interest in the wide energy gap compound semiconductors, GaN and ZnO. Finally, in Figs. 6 and 7, we plot the relative static and relative high-frequency dielectric constants as a function of the 300 K Eo energy gap corresponding to the materials considered in this critical comparative analysis. We note that these dielectric constants diminish as the Eo energy gap increases. We note, however, that the scatter is significant enough in order to render a linear least-squares fit of limited value. Diminished dielectric constants favor enhanced device performance, pointing once again to an advantage offered by wide energy gap compound semiconductors, such as GaN and ZnO. 2.6 Our Monte Carlo simulation approach For the purposes of this analysis of the electron transport within wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and 123 4686 J Mater Sci: Mater Electron (2014) 25:4675–4713 4.5 100 4 Polar optical phonon energy (meV) w−GaN direct−gap zb−GaN indirect−gap w−CdS 3.5 ZnO Energy Gap (eV) zb−CdS 3 ZnSe GaP AlP zb−CdSe 2.5 ZnTe AlSb GaAs 2 w−InN GaSb 1.5 AlAs BP InP Si 1 Ge 0.5 w−GaN AlP ZnSe ZnTe GaAs 0 1 2 AlSb zb−CdSe 0.1 ZnTe InP InSb ZnO w−InN InAs GaP 0.05 AlAs GaAs zb−GaN zb−CdS GaSb 0 0 1 2 3 3 w−CdS 4 15 Fig. 4 The electron effective mass associated with the lowest energy valley as a function of the Eo energy gap, at 300 K, for the wide energy gap compound semiconductors considered in this analysis and the other common compound semiconductors. The data for this plot is drawn from Table 3. The online version of this figure is depicted in color zinc-blende GaAs, we employ ensemble semi-classical three-valley Monte Carlo electron transport simulations. The scattering mechanisms considered are: (1) ionized impurity, (2) polar optical phonon, (3) piezoelectric, and (4) acoustic deformation potential. Inter-valley 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 zb−GaN GaP InP InAs AlAs AlSb CdSe AlP w−InN 10 GaAs III−V (direct−gap) 5 III−V (indirect−gap) ZnO ZnSe w−GaN ZnTe II−VI (direct−gap) 4 Eo energy gap (eV) 123 GaSb InSb Static relative dielectric constant e w−CdS GaSb 20 ZnSe 0.15 AlAs InAs 20 InSb AlP Fig. 5 The polar optical phonon energy as a function of the Eo energy gap, at 300 K, for the wide energy gap compound semiconductors considered in this analysis and the other common compound semiconductors. The data for this plot is drawn from Table 3. The online version of this figure is depicted in color w−CdS II−VI (direct−gap) 0.2 zb−GaN zb−CdSe 40 ZnO GaP Eo energy gap (eV) III−V (direct−gap) III−V (indirect−gap) AlSb InP w−InN 60 0 Fig. 3 The energy gap, Eg , at 300 K, for a number of elemental and compound semiconductors. The data for this plot is mostly drawn from Table 3. The online version of this figure is depicted in color Electron effective mass (m ) II−VI (direct−gap) 80 Semiconductor 0.25 zb−CdS III−V (indirect−gap) InAs InSb 0 w−GaN III−V (direct−gap) zb−CdS 0 0 1 2 3 4 Eo energy gap (eV) Fig. 6 The static relative dielectric constant as a function of the Eo energy gap, at 300 K, for the wide energy gap compound semiconductors considered in this analysis and the other common compound semiconductors. The data for this plot is drawn from Table 3. The online version of this figure is depicted in color simulations, the motion of ten-thousand electrons is considered. For our simulations, the crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 for all cases, unless otherwise specified. Electron degeneracy effects are accounted for by means of the rejection technique of Lugli and Ferry [155]. Electron screening is also accounted for following the Brooks-Herring method [156]. Further details of our approach are discussed in the literature [59, 61, 62, 85, 90, 92, 99, 100, 103, 157– 164]. High−frequency relative dielectric constant J Mater Sci: Mater Electron (2014) 25:4675–4713 4687 20 III−V (direct−gap) III−V (indirect−gap) InSb w−GaN GaSb 15 zb−GaN InAs GaAs AlAs AlP AlSb 10 GaP w−InN InP ZnTe CdSe ZnSe 5 zb−CdS 0 0 1 ZnO w−CdS II−VI (direct−gap) 2 3 4 Eo energy gap (eV) Fig. 7 The high-frequency relative dielectric constant as a function of the Eo energy gap, at 300 K, for the wide energy gap compound semiconductors considered in this analysis and the other common compound semiconductors. The data for this plot is drawn from Table 3. The online version of this figure is depicted in color 2.7 Scattering rates in GaN, ZnO, and GaAs In order to develop an appreciation for the role that the individual scattering mechanisms play in shaping the nature of the electron transport, it is instructive to contrast the dependence of the different scattering rates on the electron wave-vector, ~ k, for the different materials being considered in this analysis. In Figs. 8, 9, 10, and 11, we plot the various scattering rates as a function of the electron wave-vector, ~ k , for the cases of wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zincblende GaAs, respectively, these being the materials considered in this analysis. These are the scattering rates corresponding to the lowest energy valley in the conduction band, i.e., the C valley for the compound semiconductors considered in this review. The electrons in the upper valleys are found to experience similar scattering rates; each of the scattering rates considered in our simulations of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO, is described, in detail, by Nag [133]. 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 rates when the electron concentrations are high. 3 Steady-state and transient electron transport within GaN and ZnO 3.1 Introduction The current interest in the wide energy gap compound semiconductors, GaN and ZnO, 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 wide energy gap compound semiconductors, GaN and ZnO, has been the focus of intensive investigation for many the years. The literature abounds with studies on the steady-state and transient electron transport within these materials [3, 10, 59–62, 82, 84–130]. As a result of this intense flurry of research activity, novel wide energy gap compound semiconductor based devices are being deployed in commercial products today [8, 9, 22, 32, 38, 40, 41, 50]. Future developments in the wide energy gap compound semiconductor field will undoubtedly require an even deeper understanding of the electron transport mechanisms within these materials. In the previous section, we presented details of our semi-classical three-valley Monte Carlo simulation approach, that we employ for the analysis of the electron transport within the wide energy gap compound semiconductors, GaN and ZnO. 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 wide energy gap compound semiconductors under consideration in this analysis, i.e., GaN and ZnO, will be provided, and a comparison with the more conventional compound semiconductor, GaAs, will be presented. A comparison between the temperature dependence of the velocity-field characteristics associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs will then be presented. A similar analysis will then be performed for the doping dependence. Finally, the transient electron transport that occurs within the wide energy gap compound semiconductors under investigation in this analysis, i.e., GaN and ZnO, is characterized and compared with that corresponding to GaAs. This section is organized in the following manner. In Sects. 3.2, 3.3, and 3.4, the electron transport characteristics associated with wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, are presented and analyzed. For benchmarking purposes, in Sect. 3.5, an analogous analysis is performed for the case of zinc-blende GaAs, the electron transport characteristics associated with the wide energy gap compound semiconductors under consideration in this analysis, i.e., GaN and ZnO, being compared and contrasted with that corresponding to zinc-blende GaAs in Sect. 3.6. The sensitivity of the velocity-field characteristics associated with GaN and ZnO to variations in the crystal temperature will then be examined in Sect. 3.7, a comparison with that corresponding to zinc-blende GaAs 123 4688 J Mater Sci: Mater Electron (2014) 25:4675–4713 20 16 (a) 17 10 14 300 K −3 cm 12 3 10 8 2 6 4 4 2 0 zinc−blende GaN 300 K 16 1 Scattering Rate (1013 s−1) (a) 14 1 Scattering Rate (1013 s−1) 18 wurtzite GaN 12 17 10 10 8 2 6 2 5 2 4 6 8 0 10 0 2 Wave−vector (107 cm−1) 3 (b) (b) 8 wurtzite GaN 300 K 0.3 17 10 7 10 zinc−blende GaN 300 K s ) 8 12 −1 0.4 Scattering Rate (10 s ) 6 2.5 0.5 13 −1 4 Wave−vector (107 cm−1) 0.6 Scattering Rate (10 4 3 4 5 0 −3 cm −3 cm 0.2 6 17 10 7 cm−3 8 2 1.5 6 1 9 0.5 0.1 9 0 0 0 2 4 6 7 8 10 −1 Wave−vector (10 cm ) Fig. 8 The scattering rates for the lowest (C) valley as a function of the magnitude of the wave-vector for wurtzite GaN. The scattering mechanisms considered are: (1) ionized impurity (blue), (2) polar optical phonon emission (green), (3) inter-valley (1 ! 3) emission (red), (4) inter-valley (1 ! 2) emission (yellow), (5) acoustic deformation potential (magenta), (6) piezoelectric (blue), (7) polar optical phonon absorption (green), (8) inter-valley (1 ! 3) absorption (red), and (9) inter-valley (1 ! 2) absorption (yellow). The most important scattering rates are shown in a, i.e., scattering rates (1), (2), (3), (4), and (5), b depicting the other scattering rates, i.e., (6), (7), (8), and (9). This figure is after that depicted in Fig. 2 of O’Leary et al. [62]. The online version of this figure is depicted in color 0 2 4 6 7 8 10 −1 Wave−vector (10 cm ) Fig. 9 The scattering rates for the lowest (C) valley as a function of the magnitude of the wave-vector for zinc-blende GaN. The scattering mechanisms considered are: (1) ionized impurity (blue), (2) polar optical phonon emission (green), (3) inter-valley (1 ! 3) emission (red), (4) inter-valley (1 ! 2) emission (yellow), (5) acoustic deformation potential (magenta), (6) piezoelectric (blue), (7) polar optical phonon absorption (green), (8) inter-valley (1 ! 3) absorption (red), and (9) inter-valley (1 ! 2) absorption (yellow). The most important scattering rates are shown in a, i.e., scattering rates (1), (2), (3), (4), and (5), b depicting the other scattering rates, i.e., (6), (7), (8), and (9). The online version of this figure is depicted in color 3.2 Steady-state electron transport within wurtzite GaN also being presented. In Sect. 3.8, the sensitivity of the velocity-field characteristics associated with GaN and ZnO to variations in the doping concentration level will then be explored, a comparison with that corresponding to zincblende GaAs also being presented. Transient electron transport results are then featured in Sect. 3.9. Finally, the conclusions of this electron transport analysis are summarized in Sect. 3.10. 123 Our examination of results begins with wurtzite GaN. The velocity-field characteristic associated with this material is depicted in Fig. 12. This result is obtained through a steady-state Monte Carlo simulation of the electron transport within this material for the wurtzite GaN parameter selections specified in Tables 1 and 2, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . We note J Mater Sci: Mater Electron (2014) 25:4675–4713 20 4689 4 (a) 18 s ) 13 −1 wurtzite ZnO 14 2 Scattering Rate (10 Scattering Rate (1013 s−1) 16 1 300 K 12 10 17 10 −3 cm 8 6 3 4 5 2 0 3 3.5 4 zinc−blende GaAs 300 K 3 17 10 2 1.5 5 1 2 0.5 0 2 4 6 8 0 10 0 2 12 4 6 7 −1 10 20 (b) zinc−blende GaAs 7 Scattering Rate (1012 s−1) 18 10 8 −1 Wave−vector (10 cm ) Wave−vector (10 cm ) Scattering Rate (1012 s−1) (a) 1 4 7 wurtzite ZnO 8 300 K 17 10 6 −3 cm 9 4 6 2 300 K 16 8 9 17 10 14 −3 cm 12 10 8 6 4 8 (b) 7 2 0 cm−3 2.5 6 0 2 4 6 8 10 Wave−vector (107 cm−1) 0 0 2 4 6 8 10 Wave−vector (107 cm−1) Fig. 10 The scattering rates for the lowest (C) valley as a function of the magnitude of the wave-vector for wurtzite ZnO. The scattering mechanisms considered are: (1) ionized impurity (blue), (2) polar optical phonon emission (green), (3) inter-valley (1 ! 3) emission (red), (4) inter-valley (1 ! 2) emission (yellow), (5) acoustic deformation potential (magenta), (6) piezoelectric (blue), (7) polar optical phonon absorption (green), (8) inter-valley (1 ! 3) absorption (red), and (9) inter-valley (1 ! 2) absorption (yellow). The most important scattering rates are shown in a, i.e., (1), (2), (3), (4), and (5), b depicting the other scattering rates, i.e., (6), (7), (8), and (9). The online version of this figure is depicted in color Fig. 11 The scattering rates for the lowest (C) valley as a function of the magnitude of the wave-vector for zinc-blende GaAs. The scattering mechanisms considered are: (1) ionized impurity (blue), (2) polar optical phonon emission (green), (3) inter-valley (1 ! 3) emission (red), (4) inter-valley (1 ! 2) emission (yellow), (5) acoustic deformation potential (magenta), (6) piezoelectric (blue), (7) polar optical phonon absorption (green), (8) inter-valley (1 ! 3) absorption (red), and (9) inter-valley (1 ! 2) absorption (yellow). The most important scattering rates are shown in a, i.e., (1), (2), (3), (4), and (5), b depicting the other scattering rates, i.e., (6), (7), (8), and (9). The online version of this figure is depicted in color that initially the electron drift velocity monotonically increases with the applied electric field strength, reaching a maximum of about 2:9 107 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 saturating at about 1:4 107 cm/s for sufficiently high applied electric field strengths. We now focus on the results at low applied electric field strengths, i.e., applied electric field strengths less than 10 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, l, i.e., a linear low-field electron drift velocity dependence on the applied electric field strength, ~, applies in this regime; the negative sign comes vd ¼ lE about as the electron is negatively charged. Examining the distribution function for this regime, we find that it is very similar to the zero-field distribution function with a slight 123 4690 123 3 wurtzite GaN 300 K 2.5 Drift Velocity (10 7 cm/s) shift in the direction opposite of the applied electric field; this arises as electrons are negatively charged. 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, l, corresponding to the velocity-field characteristic depicted in Fig. 12, is around 850 cm2 /Vs. If we now examine the average electron energy as a function of the applied electric field strength, shown in Fig. 13, we see that there is a sudden increase at around 100 kV/cm; this result is obtained from the same steadystate wurtzite GaN Monte Carlo simulation of electron transport as that used to determine Fig. 12, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . In order to understand why this increase occurs, we note that the dominant energy loss mechanism for many compound semiconductors, including wurtzite GaN, is polar optical phonon scattering. When the applied electric field strength is less than 100 kV/cm, all of the energy that the electrons gain from the applied electric field is lost through polar optical phonon scattering. The other scattering mechanisms, i.e., ionized impurity scattering, piezoelectric scattering, and acoustic deformation potential scattering, do not remove energy from the electron ensemble, i.e., they are elastic scattering mechanisms. Beyond a certain critical applied electric field strength, however, and the polar optical phonon scattering mechanism can no longer remove all of the energy gained from the applied electric field. Other scattering mechanisms must start to play a role if the electron ensemble is to remain in equilibrium. The average electron energy increases until inter-valley scattering begins and an energy balance is re-established; the energy levels of the two lowest energy upper conduction band minima corresponding to this material, i.e., the two lowest energy conduction band upper-valleys, are depicted in Fig. 13. As the applied electric field strength is increased beyond 100 kV/cm, the average electron energy increases until a substantial fraction of the electrons have acquired enough energy in order to transfer into the upper valleys. In Fig. 14, we plot the occupancy of the valleys as a function of the applied electric field strength for the case of wurtzite GaN, this result being obtained from the same steady-state wurtzite GaN Monte Carlo simulation of electron transport as that used to determine Figs. 12 and 13, the motion of three-thousand electrons being considered for this analysis, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . As the effective mass of the electrons in the upper valleys is greater than that in the lowest valley, the electrons in the upper valleys will be slower. As more electrons transfer to the upper valleys, the electron drift velocity decreases. This accounts for the J Mater Sci: Mater Electron (2014) 25:4675–4713 17 10 2 −3 cm 140 kV/cm 1.5 1 0.5 0 0 200 400 600 800 1000 Electric Field (kV/cm) Fig. 12 The velocity-field characteristic associated with bulk wurtzite GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric field strengths it decreases until it saturates. The peak field, i.e., the applied electric field strength at which the maximum electron drift velocity occurs, 140 kV/cm, is clearly indicated with an arrow. This figure is after that depicted in Figure 6 of O’Leary et al. [62]. The online version of this figure is depicted in color negative differential mobility observed in the velocity-field characteristic depicted in Fig. 12. Finally, at sufficiently high applied electric field strengths, the number of electrons in each valley saturates. It can be shown that in the high-field limit, the number of electrons in each valley is proportional to the product of the density of states of that particular valley and the corresponding valley degeneracy. At this point, the electron drift velocity stops decreasing and achieves saturation. 3.3 Steady-state electron transport within zinc-blende GaN We continue our analysis with an examination of the steady-state electron transport within zinc-blende GaN. The velocity-field characteristic associated with this material is depicted in Fig. 15. This result is obtained through a steady-state Monte Carlo simulation of the electron transport within this material for the zinc-blende GaN parameter selections specified in Tables 1 and 2, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . We note that initially the electron drift velocity monotonically increases with the applied electric field strength, reaching a maximum of about 3:3 107 cm/s when the applied electric field strength is around 110 kV/cm. As with the case of wurtzite GaN, a linear regime of electron transport is observed, the low-field electron drift mobility, l, corresponding to the velocity-field characteristic depicted in Fig. 15, being about J Mater Sci: Mater Electron (2014) 25:4675–4713 2.5 4691 3000 valley 3 wurtzite GaN valley 2 1 wurtzite GaN 0.5 300 K 17 10 0 0 200 400 600 800 Number of Particles Electron Energy (eV) 1.5 100 kV/cm 300 K 1 2500 2 17 10 −3 cm 2000 2 100 kV/cm 1500 1000 3 500 cm−3 1000 Electric Field (kV/cm) 0 0 200 400 600 800 1000 Electric Field (kV/cm) Fig. 13 The average electron energy as a function of the applied electric field strength for bulk wurtzite GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Initially, the average electron energy remains low, only slightly higher than the thermal energy, 32 kb T , where kb denotes the Boltzmann 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 lowest and second lowest upper conduction band valley minima are depicted with the dashed lines. This figure is after that depicted in Figure 7 of O’Leary et al. [62]. The online version of this figure is depicted in color Fig. 14 The valley occupancy as a function of the applied electric field strength for the case of bulk wurtzite GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Soon after the average electron energy increases, i.e., at about 100 kV/cm, electrons begin to transfer to the upper valleys of the conduction band. There are three-thousand electrons employed for this simulation. The valleys are labeled 1, 2, and 3, in accordance with their energy minima, i.e., the lowest energy conduction band valley minimum corresponding to valley 1, the second lowest energy conduction band valley minimum corresponding to valley 2, the third lowest energy conduction band valley minimum corresponding to valley 3. This figure is after that depicted in Figure 8 of O’Leary et al. [62]. The online version of this figure is depicted in color 1250 cm2 /Vs. For applied electric fields strengths in excess of 110 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 saturating at about 2:1 107 cm/s for sufficiently high applied electric field strengths; it should be noted that this saturation occurs beyond the range of electric field strengths depicted in Fig. 15. If we examine the average electron energy as a function of the applied electric field strength, shown in Fig. 16, we see that there is a sudden increase at around 80 kV/cm; this result was obtained from the same steady-state zinc-blende GaN Monte Carlo simulation of electron transport as that used to determine Fig. 15, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . As with the case of wurtzite GaN, beyond a certain critical applied electric field strength, polar optical phonon scattering can no longer remove all of the energy gained from the applied electric field. The average electron energy increases until inter-valley scattering begins and an energy balance is re-established; the energy levels of the two lowest energy upper conduction band minima corresponding to this material, i.e., the two lowest energy conduction band upper valleys, are depicted in Fig. 16. In Fig. 17, we plot the occupancy of the valleys as a function of the applied electric field strength for the case of zinc-blende GaN, this result being obtained from the same steady-state zinc-blende GaN Monte Carlo simulation of electron transport as that used to determine Figs. 15 and 16, the motion of three-thousand electrons being considered for this steady-state electron transport analysis, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . This result is similar in character to that found for the case of wurtzite GaN. 3.4 Steady-state electron transport within wurtzite ZnO We now examine the steady-state electron transport within wurtzite ZnO. The velocity-field characteristic associated with this material is depicted in Fig. 18. This result is obtained through a steady-state Monte Carlo simulation of the electron transport within this material for the wurtzite ZnO parameter selections specified in Tables 1 and 2, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . We note that initially the electron drift velocity monotonically increases with the applied electric field strength, reaching a maximum of about 3:1 107 cm/s when the applied electric field strength is around 270 kV/cm. For applied electric fields 123 4692 J Mater Sci: Mater Electron (2014) 25:4675–4713 3.5 3.5 valley 3 zinc−blende GaN 3 300 K 1017 cm−3 Electron Energy (eV) 2.5 110 kV/cm 7 Drift Velocity (10 cm/s) 3 2 1.5 1 2 valley 2 1.5 1 zinc−blende GaN 80 kV/cm 300 K 0.5 0.5 0 2.5 0 100 200 300 400 0 500 1017 cm−3 0 100 200 Electric Field (kV/cm) strengths in excess of 270 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 saturating at about 2:2 107 cm/s for sufficiently high applied electric field strengths. As with the cases of wurtzite GaN and zinc-blende GaN, a linear regime of electron transport is observed for the case of wurtzite ZnO, the low-field electron drift mobility, l, corresponding to the velocity-field characteristic depicted in Fig. 18, being about 430 cm2 /Vs. If we examine the average electron energy as a function of the applied electric field strength, as shown in Fig. 19, we see that there is a sudden increase at around 200 kV/cm; this result is obtained from the same steady-state wurtzite ZnO Monte Carlo simulation of electron transport as that used to determine Fig. 18, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . For the case of this material, however, the increase in the average electron energy does not saturate once the average energy approaches the conduction band inter-valley energy separation. This is because the nonparabolicity of the lowest energy conduction band valley is extremely large and the conduction band inter-valley energy separation is extremely wide; in fact, the conduction band inter-valley energy separation is so wide that the lowest energy upper conduction band minimum is beyond the range of electron energies depicted in Fig. 19. This large non-parabolicity of the lowest energy conduction 123 400 500 Fig. 16 The average electron energy as a function of the applied electric field strength for bulk zinc-blende GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Initially, the average electron energy remains low, only slightly higher than the thermal energy, 32 kb T , where kb denotes the Boltzmann constant. At 80 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 lowest and second lowest upper conduction band valley minima are depicted with the dashed lines. The online version of this figure is depicted in color 3000 zinc−blende GaN 300 K 1 2500 Number of Particles Fig. 15 The velocity-field characteristic associated with bulk zincblende GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric field strengths it decreases until it saturates. The peak field, i.e., the applied electric field strength at which the maximum electron drift velocity occurs, 110 kV/cm, is clearly indicated with an arrow. The online version of this figure is depicted in color 300 Electric Field (kV/cm) 80 kV/cm 1017 cm−3 2 2000 1500 1000 500 0 3 0 100 200 300 400 500 Electric Field (kV/cm) Fig. 17 The valley occupancy as a function of the applied electric field strength for the case of bulk zinc-blende GaN for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Soon after the average electron energy increases, i.e., at about 80 kV/cm, electrons begin to transfer to the upper valleys of the conduction band. There are three-thousand electrons employed for this simulation. The valleys are labeled 1, 2, and 3, in accordance with their energy minima, i.e., the lowest energy conduction band valley minimum corresponds to valley 1, the second lowest energy conduction band valley minimum corresponding to valley 2, the third lowest energy conduction band valley minimum corresponding to valley 3. The online version of this figure is depicted in color 4693 3.5 3.5 3 3 2.5 270 kV/cm 2 1.5 1 wurtzite ZnO 300 K 0.5 17 10 0 0 200 400 600 800 Electron Energy (eV) Drift Velocity (10 7 cm/s) J Mater Sci: Mater Electron (2014) 25:4675–4713 2.5 2 1.5 200 kV/cm 1 wurtzite ZnO 300 K 0.5 17 −3 10 cm 1000 Electric Field (kV/cm) Fig. 18 The velocity-field characteristic associated with bulk wurtzite ZnO for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric field strengths it decreases until it saturates. The peak field, i.e., the applied electric field strength at which the maximum electron drift velocity occurs, 270 kV/cm, is clearly indicated with an arrow. This figure is after that depicted in Figure 1a of O’Leary et al. [129]. The online version of this figure is depicted in color band valley acts to dramatically enhance the effective mass of the higher energy electrons, and this is the primary factor responsible for the negative differential mobility exhibited by the velocity-field characteristic associated with wurtzite ZnO, as seen in Fig. 18. The combination of the large non-parabolicity of the lowest energy conduction band valley and the wide conduction band inter-valley energy separation limits the number of inter-valley transitions that can occur within wurtzite ZnO. In Fig. 20, we plot the occupancy of the valleys as a function of the applied electric field strength for the case of wurtzite ZnO, this result being obtained from the same steady-state wurtzite ZnO Monte Carlo simulation of electron transport as that used to determine Figs. 18 and 19, the motion of three-thousand electrons being considered for this steady-state electron transport analysis, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . It is seen that very little upper valley occupancy occurs, even for very high applied electric field strengths. 3.5 Steady-state electron transport within zinc-blende GaAs For bench-marking purposes, we now study the steadystate electron transport that occurs within zinc-blende GaAs. The velocity-field characteristic associated with this material is depicted in Fig. 21. This result is obtained through a steady-state Monte Carlo simulation of the 0 0 200 400 600 800 −3 cm 1000 Electric Field (kV/cm) Fig. 19 The average electron energy as a function of the applied electric field strength for bulk wurtzite ZnO for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Initially, the average electron energy remains low, only slightly higher than the thermal energy, 32 kb T , where kb denotes the Boltzmann constant. At 200 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 conduction band valley minima is beyond the scale depicted in this figure. This figure is after that depicted in Figure 1c of Hadi et al. [129]. The online version of this figure is depicted in color electron transport within this material for the zinc-blende GaAs parameter selections specified by Littlejohn et al. [151] and Blakemore [153], the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . We note that initially the electron drift velocity monotonically increases with the applied electric field strength, reaching a maximum of about 1:6 107 cm=s when the applied electric field strength is around 4 kV/cm. As with the cases of wurtzite GaN, zincblende GaN, and wurtzite ZnO, a linear regime of electron transport is observed for the case of zinc-blende GaAs, the low-field electron drift mobility, l, corresponding to the velocity-field characteristic depicted in Fig. 21, being about 5600 cm2 =Vs. For applied electric fields strengths in excess of 4 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 saturating at about 1:0 107 cm/s for sufficiently high applied electric field strengths. If we examine the average electron energy as a function of the applied electric field strength, shown in Fig. 22, we see that there is a sudden increase at around 2 kV/cm; this result is obtained from the same steady-state zinc-blende GaAs Monte Carlo simulation of electron transport as that 123 4694 J Mater Sci: Mater Electron (2014) 25:4675–4713 1.8 3000 zinc−blende GaAs 200 kV/cm 1 2000 300 K 1.4 10 1500 wurtzite ZnO 1000 300 K 2 3 17 10 500 −3 cm 17 −3 cm 1.2 7 Number of Particles 2500 Drift Velocity (10 cm/s) 1.6 4 kV/cm 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 Electric Field (kV/cm) Fig. 20 The valley occupancy as a function of the applied electric field strength for the case of bulk wurtzite ZnO for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Soon after the average electron energy increases, i.e., at about 200 kV/cm, small numbers of electrons begin to transfer to the upper valleys of the conduction band. There are three-thousand electrons employed for this simulation. The valleys are labeled 1, 2, and 3, in accordance with their energy minima, i.e., the lowest energy conduction band valley minimum corresponds to valley 1, the second lowest energy conduction band valley minimum corresponding to valley 2, the third lowest energy conduction band valley minimum corresponding to valley 3. This figure is after that depicted in Figure 1e of Hadi et al. [129]. The online version of this figure is depicted in color used to determine Fig. 21, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . As with the cases of wurtzite GaN and zincblende GaN, beyond a certain critical applied electric field strength, polar optical phonon scattering can no longer remove all of the energy gained from the applied electric field. The average electron energy increases until intervalley scattering begins and an energy balance is reestablished; the energy level of the lowest energy upper conduction band minimum corresponding to this material is depicted in Fig. 22, the second lowest energy conduction band minimum being beyond the range of energies depicted in Fig. 22. In Fig. 23, we plot the occupancy of the valleys as a function of the applied electric field strength for the case of zinc-blende GaAs, this result being obtained from the same steady-state zinc-blende GaAs Monte Carlo simulation of electron transport as that used to determine Figs. 21 and 22, the motion of three-thousand electrons being considered for this steady-state electron transport analysis, the crystal temperature being set to 300 K and the doping concentration being set to 1017 cm3 . This result is similar to that found for the cases of wurtzite GaN and zinc-blende GaN, i.e., a lot of upper valley occupancy occurs as the applied electric field strength is increased. 123 0 0 2 4 6 8 10 12 14 16 18 20 Electric Field (kV/cm) Fig. 21 The velocity-field characteristic associated with bulk zincblende GaAs for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric field strengths it decreases until it saturates. The peak field, i.e., the applied electric field strength at which the maximum electron drift velocity occurs, 4 kV/cm, is clearly indicated with an arrow. This figure is after that depicted in Figure 15 of O’Leary et al. [62]. The online version of this figure is depicted in color 3.6 Steady-state electron transport: a comparison of the wide energy gap compound semiconductors, GaN and ZnO, with GaAs In Fig. 24a, we contrast the velocity-field characteristics associated with the wide energy gap compound semiconductors under consideration in this analysis, i.e., wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, with that associated with zinc-blende GaAs. In all cases, we have set the crystal temperature to 300 K and the doping concentration to 1017 cm3 , the material and band structural parameters being as specified in Tables 1 and 2, respectively, i.e., these results are the same as those presented in Figs. 12, 15, 18, and 21, for the cases of wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs, respectively. We see that each of these wide energy gap compound semiconductors achieves a peak in its velocity-field characteristic. Zinc-blende GaN achieves the highest steady-state peak electron drift velocity, about 3:3 107 cm/s at an applied electric field strength of around 110 kV/cm. This contrasts with the case of wurtzite GaN, 2:9 107 cm/s at 140 kV/cm, and that of wurtzite ZnO, 3:1 107 cm/s at 270 kV/cm. For zinc-blende GaAs, the peak electron drift velocity, 1:6 107 cm/s, occurs at a much lower applied electric field strength than that for the wide energy gap compound semiconductors considered in this analysis, i.e., only 4 kV/cm. The peak and saturation electron drift velocities associated with these materials are depicted in Fig. 24b. The J Mater Sci: Mater Electron (2014) 25:4675–4713 4695 0.4 3000 zinc−blende GaAs valley 2 1 300 K 17 10 Number of Particles Electron Energy (eV) 2500 0.3 2 kV/cm 0.2 zinc−blende GaAs 0.1 2000 −3 cm 2 1500 1000 300 K 17 10 500 −3 cm 3 0 0 2 4 6 8 10 12 14 16 18 20 Electric Field (kV/cm) Fig. 22 The average electron energy as a function of the applied electric field strength for bulk zinc-blende GaAs for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Initially, the average electron energy remains low, only slightly higher than the thermal energy, 32 kb T , where kb denotes the Boltzmann constant. At 2 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 minimum corresponding to the lowest upper conduction band valley minimum is depicted with the dashed line. The other conduction band valley minima are beyond the scale depicted in the figure. This figure is after that depicted in Figure 16 of O’Leary et al. [62]. The online version of this figure is depicted in color peak electric fields, i.e., the electric field strengths at which these peak electron drift velocities occur, are depicted in Fig. 24c. 3.7 The sensitivity of the velocity-field characteristics associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs to variations in the crystal temperature The sensitivity of the velocity-field characteristics to variations in the crystal temperature is now explored. In Figs. 25, 26, and 27, the velocity-field characteristics associated with wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively, are presented for a number of different crystal temperatures; crystal temperatures between 100 to 700 K, in increments of 200 K, are considered in this analysis. The upper limit, 700 K, is chosen as this corresponds to the highest operating temperature which may be expected for AlGaN/GaN power devices. It is noted that crystal temperature variations do indeed play a significant role in shaping these velocity-field characteristics. Focusing on the peak electron drift velocity itself, for the specific case of wurtzite GaN, this velocity decreases from around 3:1 107 cm/s at 100 K to about 0 0 2 4 6 8 10 12 14 16 18 20 Electric Field (kV/cm) Fig. 23 The valley occupancy as a function of the applied electric field strength for the case of bulk zinc-blende GaAs for the crystal temperature set to 300 K and the doping concentration set to 1017 cm3 . Soon after the average electron energy increases, i.e., at about 2 kV/cm, electrons begin to transfer to the upper valleys of the conduction band. There are three-thousand electrons employed for this simulation. The valleys are labeled 1, 2, and 3, in accordance with their energy minima, i.e., the lowest energy conduction band valley minimum corresponds to valley 1, the second lowest energy conduction band valley minimum corresponding to valley 2, the third lowest energy conduction band valley minimum corresponding to valley 3. This figure is after that depicted in Figure 17 of O’Leary et al. [62]. The online version of this figure is depicted in color 2:2 107 cm/s at 700 K. The corresponding peak fields, i.e., the electric field strengths at which these peaks occur, are also found to vary with the crystal temperature, from around 130 kV/cm at 100 K to about 170 kV/cm at 700 K. Similar results are found for the other materials considered and the other electron transport metrics. To highlight the difference between the wide energy gap compound semiconductors, GaN and ZnO, with more conventional III–V compound semiconductors, such as GaAs, Monte Carlo simulations of the electron transport within zinc-blende GaAs have also been performed under the same conditions as the other materials. Figure 28 shows the results of these simulations. Clearly, the velocity-field characteristics associated with the wide energy gap compound semiconductors, GaN and ZnO, are less sensitive to variations in the crystal temperature than those associated with zinc-blende GaAs. A combination of scattering rates and occupancy issues account for the differences in behaviour, as has been explained by O’Leary et al. [62]. To quantify these dependencies further, the peak and saturation electron drift velocities associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zincblende GaAs are plotted as functions of the crystal temperature in Figs. 29a, 30a, 31a, and 32a, respectively, these results being determined from our steady-state Monte Carlo 123 4696 J Mater Sci: Mater Electron (2014) 25:4675–4713 8 10 b Fig. 24 a A comparison of the velocity-field characteristics associated wurtzite ZnO − 270 kV/cm (a) wurtzite GaN − 140 kV/cm Drift Velocity (cm/s) zinc−blende GaAs − 4 kV/cm 7 10 zinc−blende GaN − 110 kV/cm 6 10 0 10 1 2 10 3 10 10 Electric Field (kV/cm) Electron Velocity (107 cm/s) 3.5 peak (b) peak peak 3 2.5 saturation saturation 2 peak saturation 1.5 saturation 1 0.5 0 300 Peak Electric Field Strength (kV/cm) with the bulk wide energy gap compound semiconductors, wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, with that associated with bulk zinc-blende GaAs. The arrows correspond to the peak fields, i.e., the applied electric field strengths at which the peaks in the velocityfield characteristics occur, for each material considered. The crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 for all cases. The velocity-field characteristic beyond 400 kV/cm for the case of zinc-blende GaAs is not depicted owing to the fact that this material will be in breakdown over this range of electric field strengths. This plot is depicted on a logarithmic scale. This figure is after that depicted in Figure 18 of O’Leary et al. [62]. The online version of this figure is depicted in color. b A comparison of the peak and saturation electron drift velocities associated with the bulk wide energy gap compound semiconductors, wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, with that associated with bulk zinc-blende GaAs. The crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 for all cases. The online version of this figure is depicted in color. c A comparison of the peak electric field strengths, i.e., the electric field strengths at which the peak electron drift velocities occur, associated with the bulk wide energy gap compound semiconductors, wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, with that associated with bulk zinc-blende GaAs. The crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 for all cases. The online version of this figure is depicted in color w−GaN zb−GaN w−ZnO zb−GaAs (c) 250 3.8 The sensitivity of the velocity-field characteristics associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs to variations in the doping concentration 200 150 100 50 0 w−GaN zb−GaN w−ZnO zb−GaAs simulations of the electron transport within these materials. The corresponding low-field mobility dependencies on the crystal temperature, also determined from steady-state Monte Carlo simulations of the electron transport within these materials, are shown in Figs. 29b, 30b, 31b, and 32b, for the cases of wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs, respectively. For all materials, it is found that all of these electron transport 123 metrics diminish as the crystal temperature is increased. As may be seen through an inspection of Figs. 25, 26, 27, and 28, the peak and saturation electron drift velocities do not drop as much in GaN and ZnO as they do in GaAs in response to increases in the crystal temperature. The low-field electron drift mobilities in GaN and ZnO, however, are seen to fall quite rapidly with increased crystal temperature, this drop being particularly severe for temperatures at and below room temperature. This property will undoubtedly have an impact on high-power device performance. The doping concentration is a parameter which can be readily controlled in the fabrication of a semiconductor device. Understanding the effect of the doping concentration on the resultant electron transport characteristics is important. In Figs. 33, 34, and 35, the velocity-field characteristics associated with wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively, are presented for a number of different doping concentration levels; doping concentrations between 1017 and 1019 cm3 are considered, in decade increments; the 1016 cm3 results are imperceptibly distinct from the 1017 cm3 results for the cases wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, and thus, are not shown. It is noted that variations in the doping concentration do indeed play a significant role in shaping these velocity-field characteristics. Focusing on the peak electron J Mater Sci: Mater Electron (2014) 25:4675–4713 4697 3.5 3.5 100 K 17 3 10 2.5 100 K −3 cm 300 K 2 1.5 1 500 K 0.5 1017 cm−3 2.5 2 500 K 1.5 700 K 1 0.5 700 K 0 0 0 100 200 300 400 500 0 200 Fig. 25 The velocity-field characteristics associated with bulk wurtzite GaN for various crystal temperatures. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 19a of O’Leary et al. [62]. The online version of this figure is depicted in color 3.5 300 K 17 10 17 Drift Velocity (107 cm/s) 7 Drift Velocity (10 cm/s) 2.5 2 1.5 500 K 700 K 1 0.5 200 1000 zinc−blende GaAs −3 cm 2 100 800 2.5 3 0 600 Fig. 27 The velocity-field characteristics associated with bulk wurtzite ZnO for various crystal temperatures. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 1a of Hadi et al. [129]. The online version of this figure is depicted in color zinc−blende GaN 100 K 400 Electric Field (kV/cm) Electric Field (kV/cm) 0 wurtzite ZnO 300 K 3 Drift Velocity (10 7 cm/s) Drift Velocity (10 7 cm/s) wurtzite GaN 300 400 500 Electric Field (kV/cm) Fig. 26 The velocity-field characteristics associated with bulk zincblende GaN for various crystal temperatures. For all cases, we have assumed a doping concentration of 1017 cm3 . The online version of this figure is depicted in color drift velocity itself, for the specific case of wurtzite GaN, it is found that this velocity decreases from around 2:9 107 cm/s at 1017 cm3 to about 2:0 107 cm/s at 1019 cm3 . The corresponding peak fields, i.e., the applied electric field strengths at which the peak in the electron drift velocity occurs, are also found to vary with the doping concentration, albeit slightly, i.e., from around 140 kV/cm at 1017 cm3 to about 130 kV/cm at 1019 cm3 . Similar results are found for the other materials considered and for the other electron transport metrics. 10 100 K 1.5 −3 cm 300 K 500 K 1 700 K 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Electric Field (kV/cm) Fig. 28 The velocity-field characteristics associated with bulk zincblende GaAs for various crystal temperatures. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 19b of O’Leary et al. [62]. The online version of this figure is depicted in color To highlight the difference between the wide energy gap compound semiconductors, GaN and ZnO, with more conventional III–V compound semiconductors, such as GaAs, Monte Carlo simulations of the electron transport within zinc-blende GaAs have also been performed under the same conditions as the other materials; for the case of zinc-blende GaAs, as the 1016 cm3 result is visually distinct from the 1017 cm3 result, the 1016 cm3 result is also depicted. Figure 36 shows the results of these simulations. 123 4698 J Mater Sci: Mater Electron (2014) 25:4675–4713 3.5 4 (a) 3 17 10 3.5 −3 cm Drift Velocity (10 7 cm/s) Drift Velocity (107 cm/s) zinc−blende GaN wurtzite GaN 2.5 peak 2 1.5 1 saturation 0.5 17 10 cm−3 3 2.5 peak 2 1.5 saturation 1 0.5 (a) 0 0 100 200 300 400 500 600 0 700 0 100 200 Temperature (K) 300 400 500 600 700 Temperature (K) 2500 wurtzite GaN zinc−blende GaN 2500 1017 cm−3 17 10 Drift Mobility (cm /Vs) 2000 −3 cm 2000 2 2 Drift Mobility (cm /Vs) (b) 1500 1000 500 1500 1000 500 (b) 0 0 100 200 300 400 500 600 700 Temperature (K) Fig. 29 a The peak and saturation electron drift velocities associated with bulk wurtzite GaN as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite GaN. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 20a of O’Leary et al. [62]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk wurtzite GaN as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite GaN, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 20b of O’Leary et al. [62]. The online version of this figure is depicted in color Clearly, the velocity-field characteristics associated with the wide energy gap compound semiconductors, GaN and ZnO, are less sensitive to variations in the doping concentration than those associated with zinc-blende GaAs; in fact, for the case of 1019 cm3 doping, the peak in the velocity-field characteristic associated with zinc-blende GaAs completely disappears, the velocity-field characteristic associated with zinc-blende GaAs monotonically increasing with the applied electric field strength until 123 0 0 100 200 300 400 500 600 700 Temperature (K) Fig. 30 a The peak and saturation electron drift velocities associated with bulk zinc-blende GaN as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaN. For all cases, we have assumed a doping concentration of 1017 cm3 . The online version of this figure is depicted in color. b The low-field mobility associated with bulk zinc-blende GaN as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaN, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a doping concentration of 1017 cm3 . The online version of this figure is depicted in color. saturation is achieved for this particular case. A combination of scattering rates and occupancy issues account for the differences in behaviour, as has been explained by O’Leary et al. [62]. To quantify these dependencies further, the peak and saturation electron drift velocities associated with wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zincblende GaAs are plotted as functions of the doping concentration in Figs. 37a, 38a, 39a, and 40a, respectively, these results being determined from our steady-state Monte Carlo simulations of the electron transport within these materials. The corresponding low-field mobility dependencies on the J Mater Sci: Mater Electron (2014) 25:4675–4713 4699 2.5 4 wurtzite ZnO 17 10 cm 7 peak 2 1.5 saturation 1 0.5 0 zinc−blende GaAs peak 3 2.5 (a) −3 Drift Velocity (10 cm/s) Drift Velocity (10 7 cm/s) 3.5 1.5 1 0.5 saturation (a) 0 100 200 300 400 500 600 0 700 0 100 200 400 500 600 700 12000 1800 (b) wurtzite ZnO 1600 1017 cm−3 zinc−blende GaAs 17 10 10000 1400 Drift Mobility (cm2/Vs) 2 300 Temperature (K) Temperature (K) Drift Mobility (cm /Vs) 1017 cm−3 2 1200 1000 800 600 400 −3 cm 8000 6000 4000 2000 200 (b) 0 0 0 100 200 300 400 500 600 700 Temperature (K) Fig. 31 a The peak and saturation electron drift velocities associated with bulk wurtzite ZnO as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite ZnO. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 2b of Hadi et al. [129]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk wurtzite ZnO as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite ZnO, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a doping concentration of 1017 cm3 . The online version of this figure is depicted in color doping concentration, also determined from steady-state Monte Carlo simulations of the electron transport within these materials, are shown in Figs. 37b, 38b, 39b, and 40b, for the cases of wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs, respectively. For all materials, it is found that all of these electron transport metrics diminish as the doping concentration is increased. As may be seen through an inspection of Figs. 33, 34, 35, and 36, the peak and saturation electron drift velocities do not drop as much in GaN and ZnO as they do in GaAs in response to increases in the doping concentration. The low-field 0 100 200 300 400 500 600 700 Temperature (K) Fig. 32 a The peak and saturation electron drift velocities associated with bulk zinc-blende GaAs as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaAs. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 20c of O’Leary et al. [62]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk zinc-blende GaAs as a function of the crystal temperature. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaAs, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 20d of O’Leary et al. [62]. The online version of this figure is depicted in color electron drift mobilities associated with GaN and ZnO, however, are seen to fall quite rapidly with increased doping concentrations, this drop being particularly severe for doping concentrations at and below 1017 cm3 . 3.9 Transient electron transport Steady-state electron transport is the dominant electron transport mechanism in devices with larger dimensions. For devices with smaller dimensions, however, transient 123 4700 J Mater Sci: Mater Electron (2014) 25:4675–4713 3.5 3.5 10 Drift Velocity (10 7 cm/s) 3 1017 cm−3 wurtzite GaN −3 cm 2.5 18 10 2 −3 cm 19 10 −3 cm 1.5 1 100 200 300 400 500 18 2 10 −3 cm 1.5 19 10 cm−3 1 zinc−blende GaN 3.5 17 10 3 −3 cm 300 K 2.5 2 1018 cm−3 1.5 19 10 −3 cm 1 0.5 0 100 200 300 400 500 Electric Field (kV/cm) Fig. 34 The velocity-field characteristics associated with bulk zincblende GaN for various doping concentrations. For all cases, we have assumed a crystal temperature of 300 K. The 1016 cm3 doping concentration case is not shown in this plot as it is essentially indistinguishable from the 1017 cm3 case for this particular material. The online version of this figure is depicted in color electron transport must also be considered when evaluating device performance. Ruch [165] demonstrated, for both silicon and GaAs, that the transient electron drift velocity may exceed the corresponding steady-state electron drift velocity by a considerable margin for appropriate selections of the applied electric field strength. Shur and Eastman [166] explored the device implications of 123 0 200 400 600 800 1000 Electric Field (kV/cm) Fig. 33 The velocity-field characteristics associated with bulk wurtzite GaN for various doping concentrations. For all cases, we have assumed a crystal temperature of 300 K. The 1016 cm3 doping concentration case is not shown in this plot as it is essentially indistinguishable from the 1017 cm3 case for this particular material. This figure is after that depicted in Figure 23a of O’Leary et al. [62]. The online version is depicted in color Drift Velocity (10 7 cm/s) 2.5 0 0 Electric Field (kV/cm) 0 300 K 0.5 0.5 0 wurtzite ZnO 3 300 K Drift Velocity (107 cm/s) 17 Fig. 35 The velocity-field characteristics associated with bulk wurtzite ZnO for various doping concentrations. For all cases, we have assumed a crystal temperature of 300 K. The 1016 cm3 doping concentration case is not shown in this plot as it is essentially indistinguishable from the 1017 cm3 case for this particular material. This figure is after that depicted in Figure 2a of Hadi et al. [129]. The online version is depicted in color transient electron transport, and demonstrated that substantial improvements in the device performance may be achieved as a consequence. Heiblum et al. [167] made the first direct experimental observation of transient electron transport within GaAs. Since then, there have been a number of experimental investigations into the transient electron transport within III–V compound semiconductors; see, for example, [168–170]. Thus far, very little research has been invested into the study of the transient electron transport within the wide energy gap compound semiconductors, GaN and ZnO. In 1997, Foutz et al. [92] examined transient electron transport within the wurtzite and zinc-blende phases of GaN. In particular, they examined how electrons, initially in thermal equilibrium, respond to the sudden application of a constant electric field. In devices with dimensions greater than 0.2 lm, they found that steady-state electron transport is expected to dominate device performance. For devices with smaller dimensions, however, with the application of a sufficiently high electric field strength, they found that the transient electron drift velocity can considerably overshoot the corresponding steady-state electron drift velocity. This velocity overshoot was found to be comparable with that which occurs within GaAs. A subsequent analysis, reported by Foutz et al. [59] in 1999, contrasted the transient electron transport over a broad range of III–V nitride semiconductors. Following in the tradition of Foutz et al. [59, 92], we examine how an ensemble of electrons, initially in thermal J Mater Sci: Mater Electron (2014) 25:4675–4713 4701 3.5 (a) zinc−blende GaAs 2 10 Drift velocity (cm/s) 1016 cm−3 17 7 Drift Velocity (10 cm/s) 300 K 1.5 −3 cm 1 0.5 1017 cm−3 2.5 2 peak 1.5 1 0.5 18 10 −3 cm 19 10 0 wurtzite GaN 3 0 5 10 15 saturation 0 16 10 −3 cm 19 10 1500 (b) wurtzite GaN 1017 cm−3 1250 2 Drift Mobility (cm /Vs) equilibrium, respond to the application of a constant electric field for the cases of wurtzite GaN, zinc-blende GaN, and wurtzite ZnO. In particular, Figs. 41, 42, and 43 depict the transient electron drift velocity as a function of the distance displaced since the electric field was initially applied, for a number of electric field strength selections, for the cases of wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, respectively. For all cases, the crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 . These results are obtained from Monte Carlo electron transport simulations. For each transient electron transport simulation, the motion of ten-thousand electrons is considered. Focusing initially on the case of wurtzite GaN (see Fig. 41), we note that for the applied electric field strength selections 70 kV/cm and 140 kV/cm, that the electron drift velocity reaches steady-state very quickly, with little or no velocity overshoot. In contrast, for applied electric field strength selections above 140 kV/cm, significant velocity overshoot occurs. This result suggests that in wurtzite GaN, 140 kV/cm is a critical field for the onset of velocity overshoot effects. As was mentioned in Sects. 3.2 and 3.6, 140 kV/cm also corresponds to the peak in the velocityfield characteristic associated with wurtzite GaN; recall Figs. 12 and 24. Steady-state Monte Carlo simulations suggest that this is the point at which significant upper valley occupation begins to occur; recall Fig. 14. This tells us that the velocity overshoot that occurs within this material is related to the transfer of electrons to the upper valleys. A similar result is found for the case of zinc-blende 18 10 Doping concentration (cm−3 ) 20 Electric Field (kV/cm) Fig. 36 The velocity-field characteristics associated with bulk zincblende GaAs for various doping concentrations. For all cases, we have assumed a crystal temperature of 300 K. The 1016 cm3 doping concentration case is shown in this plot as it differs considerably from the 1017 cm3 case for this particular material. This figure is after that depicted in Figure 23b of O’Leary et al. [62]. The online version of this figure is depicted in color 17 10 1000 750 500 250 0 16 10 17 10 18 10 19 10 −3 Doping concentration (cm ) Fig. 37 a The peak and saturation electron drift velocities associated with bulk wurtzite GaN as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite GaN. For all cases, we have assumed a crystal temperature of 300 K. This figure is after that depicted in Figure 24a of O’Leary et al. [62]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk wurtzite GaN as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite GaN, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a crystal temperature of 300 K. This figure is after that depicted in Figure 24b of O’Leary et al. [62]. The online version of this figure is depicted in color GaN. For the case of wurtzite ZnO, however, the clearly observed velocity overshoot is related to the sudden increase in the electron effective mass that accompanies higher electron energies, i.e., increases in the electron effective mass quickly act to dampen the corresponding transient electron drift velocity. For bench-marking purposes, the case of zinc-blende GaAs is also considered; see Fig. 44. The critical fields found are 110 kV/cm for the case of zinc-blende GaN, 270 kV/cm for the case of wurtzite ZnO, and 4 kV/cm for the case of zinc-blende GaAs; see Figs. 42, 43, and 44, respectively. We note that for all cases 123 4702 J Mater Sci: Mater Electron (2014) 25:4675–4713 4 4 zinc−blende GaN (a) (a) wurtzite ZnO 1017 cm−3 300 K Drift Velocity (107 cm/s) Drift velocity (cm/s) 3.5 3 2.5 peak 2 1.5 1 saturation 3 peak 2 saturation 1 0.5 0 16 10 17 10 18 0 16 10 19 10 10 17 10 18 19 10 10 Doping Concentration (cm−3) Doping concentration (cm−3 ) 600 2000 300 k 2 1500 1250 1000 750 500 17 10 18 19 10 10 −3 Doping concentration (cm ) Fig. 38 a The peak and saturation electron drift velocities associated with bulk zinc-blende GaN as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaN. For all cases, we have assumed a crystal temperature of 300 K. The online version of this figure is depicted in color. b The low-field mobility associated with bulk zinc-blende GaN as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite GaN, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a crystal temperature of 300 K. The online version of this figure is depicted in color these critical electric field strengths are identical to the respective peak fields; recall Figs. 12, 15, 18, 21, and 24. We now compare the transient electron transport characteristics for the various materials. From Figs. 41, 42, 43, and 44, it is clear that certain materials exhibit higher peak overshoot velocities and longer overshoot relaxation times. It is not possible to fairly compare these different semiconductors by applying the same applied electric field strength for all of the materials, as the transient effects occur over such a disparate range of applied electric field strengths for each material. In order to facilitate such a comparison, we choose a field strength equal to twice the critical applied electric field strength for each material, i.e., 123 400 300 200 100 250 0 16 10 wurtzite ZnO 300 K 500 Drift Mobility (cm2/Vs) 1750 Drift Mobility (cm /Vs) (b) zinc−blende GaN (b) 0 16 10 17 10 18 19 10 10 −3 Doping Concentration (cm ) Fig. 39 a The peak and saturation electron drift velocities associated with bulk wurtzite ZnO as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite ZnO. For all cases, we have assumed a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 2b of Hadi et al. [129]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk wurtzite ZnO as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk wurtzite ZnO, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a crystal temperature of 300 K. The online version of this figure is depicted in color 280 kV/cm for the case of wurtzite GaN, 220 kV/cm for zincblende GaN, 540 kV/cm for wurtzite ZnO, and 8 kV/cm for zinc-blende GaAs. Figure 45a shows such a comparison of the velocity overshoot effects amongst the four materials considered in this analysis, i.e., wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs. The corresponding peak transient electron drift velocities are depicted in Fig. 45b. It is clear that among the wide energy gap compound semiconductors considered, zinc-blende GaN exhibits superior transient electron transport characteristics. In particular, zinc-blende GaN has the largest overshoot velocity and the distance over which this J Mater Sci: Mater Electron (2014) 25:4675–4713 4703 2 10 peak zinc−blende GaAs 8 1.5 7 Drift Velocity (10 cm/s) Drift velocity (cm/s) wurtzite GaN 9 1017 cm−3 1 0.5 saturation a) (a) 0 16 10 17 10 18 17 10 7 −3 cm 280 kV/cm 6 210 kV/cm 5 140 kV/cm 4 3 2 1 19 10 300 K 560 kV/cm 70 kV/cm 10 0 −3 Doping concentration (cm ) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Distance (μm) 8000 zinc−blende GaAs Drift Mobility (cm2/Vs) 7000 Fig. 41 The electron drift velocity as a function of the distance displaced since the application of the electric field, for various applied electric field strength selections, for the case of bulk wurtzite GaN. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 40a of O’Leary et al. [62]. The online version of this figure is depicted in color 1017 cm−3 6000 5000 4000 3000 2000 12 (b) zinc−blende GaN 300 K 10 17 10 18 10 19 10 Doping concentration (cm−3) Fig. 40 a The peak and saturation electron drift velocities associated with bulk zinc-blende GaAs as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaAs. For all cases, we have assumed a crystal temperature of 300 K. This figure is after that depicted in Figure 24c of O’Leary et al. [62]. The online version of this figure is depicted in color. b The low-field mobility associated with bulk zinc-blende GaAs as a function of the doping concentration. These results are determined from the results of Monte Carlo simulations of electron transport within bulk zinc-blende GaAs, i.e., the low-field slope of the velocity-field characteristic is determined. For all cases, we have assumed a crystal temperature of 300 K. This figure is after that depicted in Figure 24d of O’Leary et al. [62]. The online version of this figure is depicted in color overshoot occurs, 0.2 lm, is longer than in either wurtzite GaN and wurtzite ZnO. Zinc-blende GaAs is found to exhibit a longer overshoot relaxation distance, approximately 0.7 lm, but the electron drift velocity exhibited by zinc-blende GaAs is less than that of GaN and ZnO over most displacements. 3.10 Electron transport conclusions In this section, steady-state and transient electron transport results, corresponding to the wide energy gap compound 7 0 16 10 440 kV/cm 11 Drift Velocity (10 cm/s) 1000 9 17 10 8 cm−3 220 kV/cm 7 6 165 kV/cm 5 4 3 2 110 kV/cm 1 0 55 kV/cm 0 0.05 0.1 0.15 0.2 0.25 Distance (μm) Fig. 42 The electron drift velocity as a function of the distance displaced since the application of the electric field, for various applied electric field strength selections, for the case of bulk zinc-blende GaN. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . The online version of this figure is depicted in color semiconductors, GaN and ZnO, were presented, these results being obtained from our Monte Carlo simulations of electron transport within these materials. Steady-state electron transport was the dominant theme of our analysis. In order to aid in the understanding of these electron transport characteristics, a comparison was made between these results and those associated with zinc-blende GaAs. 123 4704 J Mater Sci: Mater Electron (2014) 25:4675–4713 8 1080 kV/cm 6 wurtzite ZnO zinc−blende GaAs 7 405 kV/cm 300 K 16 kV/cm 1017 cm−3 Drift Velocity (10 cm/s) 540 kV/cm 6 7 Drift Velocity (10 cm/s) 300 K 270 kV/cm 4 2 17 10 −3 cm 4 8 kV/cm 6 kV/cm 2 4 kV/cm 135 kV/cm 2 kV/cm 0 0 0.05 0.1 0.15 0.2 0.25 Distance (μm) 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Distance (μm) Fig. 43 The electron drift velocity as a function of the distance displaced since the application of the electric field, for various applied electric field strength selections, for the case of bulk wurtzite ZnO. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 5a of Hadi et al. [129]. The online version of this figure is depicted in color Fig. 44 The electron drift velocity as a function of the distance displaced since the application of the electric field, for various applied electric field strength selections, for the case of bulk zinc-blende GaAs. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . This figure is after that depicted in Figure 40d of O’Leary et al. [62]. The online version of this figure is depicted in color Our simulations showed that the wide energy gap compound semiconductors, GaN and ZnO, are less sensitive to variations in the crystal temperature and the doping concentration than that associated with zinc-blende GaAs. Finally, we presented some key transient electron transport results, these results indicating that the transient electron transport that occurs within zinc-blende GaN is the most pronounced of all of the materials under consideration in this review, i.e., wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs. have become available [37, 49, 50]. More wide energy gap compound semiconductor based device applications are currently under development, and these will undoubtedly become available in the near-term future. In this section, we present a brief overview of the GaN and ZnO electron transport field. We start with a survey, describing the evolution of the field. In particular, the sequence of critical developments that have occurred, contributing to our current understanding of the electron transport mechanisms within the wide energy gap compound semiconductors, GaN and ZnO, is chronicled. Then, some current literature is presented, particular emphasis being placed on the developments that have transpired over the past few years and how such developments continue to shape our understanding of the electron transport mechanisms within the wide energy gap compound semiconductors, GaN and ZnO. Finally, frontiers for further research and investigation are presented. This section is organized in the following manner. In Sect. 4.2, we present a brief survey, describing the evolution of the field. Then, in Sect. 4.3, the developments that have occurred over the past few years are highlighted. Finally, frontiers for further research and investigation are presented in Sect. 4.4. 4 Electron transport within the wide energy gap compound semiconductors, GaN and ZnO: a review 4.1 Introduction The first studies into the material properties of the wide energy gap compound semiconductors, GaN and ZnO, were reported on in the 1920s and 1930s [171–176]. Unfortunately, the materials available at the time, small crystals and powders, were of poor quality. Indeed, it was only in the late 1960s, when Maruska and Tietjen [54] employed chemical vapor deposition in order to fabricate GaN, that the electron device potential of the wide energy gap compound semiconductors became fully recognized. Since that time, however, dramatic improvements in the material quality of GaN and ZnO have been achieved. As a consequence, a number of commercial devices, which employ these wide energy gap compound semiconductors, 123 4.2 The evolution of the field The favorable electron transport characteristics of the wide energy gap compound semiconductors, GaN and ZnO, have been recognized for a long time now. As early as the J Mater Sci: Mater Electron (2014) 25:4675–4713 4705 separation, suggests a high saturation electron drift velocity for this material. As high-frequency electron device performance is, in large measure, determined by this saturation electron drift velocity [10], the recognition of this fact ignited enhanced interest in this material, and its wide energy gap compound semiconductor counterpart, ZnO.13 This enhanced interest, and the developments which have transpired as a result of it, are responsible for the wide energy gap compound semiconductor industry of today. Given that interest in the material properties of GaN preceded that of ZnO, we divide this chronology into two sections, the first one corresponding to GaN, the latter one corresponding to wurtzite ZnO. 8 (a) 300 K Drift Velocity (10 7 cm/s) 7 1017 cm−3 zinc−blende GaN 6 wurtzite GaN 5 4 3 2 wurtzite ZnO zinc−blende GaAs 1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Peak Transient Electron Drift Velocity (× 107 cm/s) Distance (μm) 8 4.2.1 Electron transport within GaN: a review (b) 6 4 2 0 w−GaN zb−GaN w−ZnO zb−GaAs Fig. 45 a A comparison of the velocity overshoot amongst the bulk wide energy gap compound semiconductors considered in this analysis, i.e., wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, and bulk GaAs. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . The applied electric field strengths chosen correspond to twice the critical applied electric field strength at which the peak in the steady-state velocity-field characteristic occurs (see Fig. 24), i.e., 280 kV/cm for the case of wurtzite GaN, 220 kV/cm for the case of zincblende GaN, 540 kV/cm for the case of wurtzite ZnO, and 8 kV/cm for the case of GaAs. This figure is after that depicted in Fig. 41 of O’Leary et al. [62]. The online version of this figure is depicted in color. b A comparison of the peak transient electron drift velocities associated with the bulk wide energy gap compound semiconductors considered in this analysis, i.e., wurtzite GaN, zinc-blende GaN, and wurtzite ZnO, with that associated with bulk zinc-blende GaAs. For all cases, we have assumed an initial zero-field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm3 . The applied electric field strengths chosen correspond to twice the critical applied electric field strength at which the peak in the steady-state velocity-field characteristic occurs (see Fig. 24), i.e., 280 kV/cm for the case of wurtzite GaN, 220 kV/cm for the case of zinc-blende GaN, 540 kV/cm for the case of wurtzite ZnO, and 8 kV/cm for the case of GaAs. The online version of this figure is depicted in color 1970s, Littlejohn et al. [84] pointed out that the large polar optical phonon energy characteristic of GaN, in conjunction with its wide conduction band inter-valley energy In 1975, Littlejohn et al. [84] were the first to report results obtained from semi-classical Monte Carlo simulations of the steady-state electron transport within wurtzite GaN. A one-valley model for the conduction band was adopted for the purposes of their analysis. Steady-state electron transport, for both parabolic and non-parabolic band structures, was considered in their analysis, non-parabolicity being treated through the application of the Kane model [144]. The primary focus of their investigation was the determination of the velocity-field characteristic associated with wurtzite GaN. All donors were assumed to be ionized and the free electron concentration was taken to be equal to the dopant concentration. The scattering mechanisms considered were: (1) ionized impurity, (2) polar optical phonon, (3) piezoelectric, and (4) acoustic deformation potential. For the case of the parabolic band, in the absence of ionized impurities, they found that the electron drift velocity monotonically increases with the applied electric field strength, saturating at a value of around 2:5 107 cm/s for the case of high applied electric field strengths. In contrast, for the case of the non-parabolic band, in the absence of ionized impurities, a region of negative differential mobility was found, the electron drift velocity achieving a maximum of about 2 107 cm/s at an applied electric field strength of around 100 kV/cm, further increases in the applied electric field strength resulting in a slight decrease in the corresponding electron drift velocity. The role of ionized impurity scattering was also investigated by Littlejohn et al. [84]. 13 Interest in the material properties of ZnO was ignited later than that associated with GaN, primarily on account of material quality considerations, i.e., high-quality GaN was prepared earlier, and a lack of familiarity with means of effectively handling II–VI compound semiconductors, many GaN processing techniques being borrowed directly from the GaAs case. 123 4706 In 1993, Gelmont et al. [85] reported on ensemble semiclassical two-valley Monte Carlo simulations of the electron transport within wurtzite GaN, this analysis improving upon the analysis of Littlejohn et al. [84] by incorporating inter-valley scattering into the simulations. They found that the negative differential mobility exhibited by wurtzite GaN is much more pronounced than that found by Littlejohn et al. [84], and that inter-valley transitions are responsible for this. For a doping concentration of 1017 cm3 , Gelmont et al. [85] demonstrated that the electron drift velocity achieves a peak value of about 2:8 107 cm/s at an applied electric field strength of around 140 kV/cm. The impact of inter-valley transitions on the electron distribution function was also determined and shown to be significant. The impact of doping and compensation on the velocity-field characteristic associated with bulk wurtzite GaN was also examined. Since these pioneering investigations, ensemble Monte Carlo simulations of the electron transport within GaN have been performed numerous times. In particular, in 1995, Kolnı́k et al. [87] reported on employing full-band Monte Carlo simulations of the electron transport within wurtzite GaN and zinc-blende GaN, finding that zincblende GaN exhibits a much higher low-field electron drift mobility than wurtzite GaN. The peak electron drift velocity corresponding to zinc-blende GaN was found to be only marginally greater than that exhibited by wurtzite GaN. Also in 1995, Mansour et al. [88] reported the use of semi-classical Monte Carlo simulations in order to determine how the crystal temperature influences the velocityfield characteristic associated with wurtzite GaN. In 1997, Bhapkar and Shur [90] reported on employing ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within bulk and confined wurtzite GaN. Their simulations demonstrated that the two-dimensional electron gas within a confined wurtzite GaN structure will exhibit a higher low-field electron drift mobility than bulk wurtzite GaN, by almost an order of magnitude, this being in agreement with experiment [177]. In 1998, Albrecht et al. [94] reported on employing ensemble semi-classical five-valley Monte Carlo simulations of the electron transport within wurtzite GaN, with the aim of determining elementary analytical expressions for a number of the electron transport metrics corresponding to wurtzite GaN, for the purposes of device modeling. The first known study of transient electron transport within GaN was that performed by Foutz et al. [92], reported in 1997. In this study, ensemble semi-classical three-valley Monte Carlo simulations were employed in order to determine how the electrons within wurtzite and zinc-blende GaN, initially in thermal equilibrium, respond to the sudden application of a constant electric field. The velocity overshoot which occurs within these materials was 123 J Mater Sci: Mater Electron (2014) 25:4675–4713 examined. It was found that the electron drift velocities that occur within the zinc-blende phase of GaN are slightly greater than those exhibited by the wurtzite phase owing to the slightly higher steady-state electron drift velocity exhibited by the zinc-blende phase of GaN. A comparison with the transient electron transport which occurs within GaAs was made. Using the results of this analysis, a determination of the minimum transit time, as a function of the distance displaced since the application of the applied electric field, was performed for all three materials considered in this study, i.e., wurtzite GaN, zinc-blende GaN, and zinc-blende GaAs. For distances in excess of 0.1 lm, both phases of GaN were shown to exhibit superior performance, i.e., reduced transit time, when contrasted with that associated with zinc-blende GaAs. A more general analysis, in which transient electron transport within wurtzite GaN was contrasted with that corresponding to two other III–V nitride semiconductors, AlN and InN, and zinc-blende GaAs, was then performed by Foutz et al. [59] and reported in 1999. As with their previous study, Foutz et al. [59] determined how electrons, initially in thermal equilibrium, respond to the sudden application of a constant electric field. For all the semiconductors considered, it was found that the electron drift velocity overshoot only occurs when the applied electric field strength exceeds a certain critical applied electric field strength unique to each material. The critical applied electric field strength was found to be 140 kV/cm for the case of wurtzite GaN, this corresponding to the peak field for this material, i.e., the electric field strength at which the peak electron drift velocity in the velocity-field characteristic associated with this material occurs; recall Fig. 12 of this article. This observation was also found to apply for the other semiconductors considered in their analysis. A comparison with the results of experiment was also performed. In addition to Monte Carlo simulations of the electron transport within these materials, a number of other types of electron transport studies have been performed. Reports on experimental measurements of the Hall mobility associated with wurtzite GaN are numerous, and include those made by Yoshida et al. [178] in 1983, Khan et al. [177] in 1995, Nakayama et al. [179] in 1996, and Hurni et al. [180] in 2012. Experimental measurements of the velocity-field characteristics and of the transient electron transport response of wurtzite GaN have been reported on by Wraback et al. in 2000 [11] and 2001 [12], respectively; further details, concerning these experimental measurements, were presented by Wraback et al. [13] in 2002. Theoretical investigations into the electron transport processes within these materials, in addition to the aforementioned Monte Carlo studies, are also numerous. In 1975, for example, Ferry [10] reported on the J Mater Sci: Mater Electron (2014) 25:4675–4713 determination of the velocity-field characteristic associated with wurtzite GaN using a displaced Maxwellian distribution function approach. For high applied electric field strengths, Ferry [10] found that the electron drift velocity associated with wurtzite GaN monotonically increases with the applied electric field strength, i.e., it does not exhibit a peak, reaching a value of about 2:5 107 cm/s at an applied electric field strength of 300 kV/cm. The device implications of this result were further explored by Das and Ferry [3]. In 1994, Chin et al. [86] reported on a detailed study of the dependence of the low-field electron drift mobilities associated with wurtzite GaN, and two other III–V nitride semiconductors, on the crystal temperature and the doping concentration. An analytical expression for the low-field electron drift mobility, l, determined using a variational principle, was employed for the purposes of this analysis. The results obtained were contrasted with those of experiment. Subsequent mobility studies were reported on in 1996 by Shur et al. [89] and in 1997 by Look et al. [138]. Then, in 1998, Weimann et al. [101] reported on a model for the determination of how the scattering of electrons by the threading dislocation lines within wurtzite GaN influences the low-field electron drift mobility. They demonstrated why the experimentally measured low-field electron drift mobility associated with this material is much lower than that predicted from Monte Carlo analyses, threading dislocations not being taken into account in the Monte Carlo simulations of the electron transport within wurtzite GaN. While the negative differential mobility exhibited by the velocity-field characteristics associated GaN is widely attributed to inter-valley transitions, and while direct experimental evidence confirming this has been presented [181], Krishnamurthy et al. [182] suggest that instead the inflection points in the bands, located in the vicinity of the C valley, are primarily responsible for the negative differential mobility exhibited by wurtzite GaN. The relative importance of these two mechanisms, i.e., inter-valley transitions and inflection point considerations, were evaluated by Krishnamurthy et al. [182], both for the case of wurtzite GaN and an alloy of GaN with another III–V nitride semiconductor, AlN. On the theoretical front, there have been a number of more recent developments. Hot-electron energy relaxation times within wurtzite GaN were studied by Matulionis et al. [183], and reported on in 2002. Bulutay et al. [184] studied the electron momentum and energy relaxation times within wurtzite GaN, and reported the results of their study in 2003. It is particularly interesting to note that the arguments of Bulutay et al. [184] add considerable credence to the earlier inflection point argument of Krishnamurthy et al. [182]. In 2004, Brazis and Raguotis [185] reported on the results of a Monte Carlo study 4707 involving additional phonon modes and a smaller conduction band inter-valley energy separation for the case of wurtzite GaN. Their results were found to be much closer to the experimental results of Wraback et al. [11] than those found previously. The influence of hot-phonons on the electron transport mechanisms within wurtzite GaN, an effect not considered in our simulations of the electron transport within these materials, i.e., we assumed steady-state phonon populations, has been the focus of considerable investigation. In particular, in 2004 itself, Gökden [106], Ridley et al. [107], and Silva and Nascimento [186], to name just three, presented results related to this research focus. These results suggest that hot-phonon effects play a role in influencing the nature of the electron transport within wurtzite GaN. In particular, Ridley et al. [107] point out that the saturation electron drift velocity and the peak field are both influenced by hot-phonon effects; it should be noted, however, that Ridley et al. [107] neglect conduction band inter-valley transitions in their analysis, their analysis challenging the conventional belief that the negative differential mobility exhibited by the velocity-field characteristics associated with wurtzite GaN is attributable to transitions into the upper energy conduction band valleys. More recent research into this topic was pursued by Martininez et al. [187] in 2006, Tas et al. [188] in 2007, Ramonas et al. [114] in 2007, and Matulionis et al. [189] in 2008. Research into the role that hot-phonons play in influencing the electron transport mechanisms within wurtzite GaN seems likely to continue into the foreseeable future. Research into how the electron transport within wurtzite GaN influences the performance of GaN-based devices is ongoing. In 2004, Matulionis and Liberis [190] reported on the role that hot-phonons play in determining the microwave noise within AlGaN/GaN channels. In 2005, Ramonas et al. [191] further developed this analysis, focusing on how hot-phonon effects influence power dissipation within AlGaN/GaN channels. The high-field electron transport within AlGaN/GaN heterostructures was examined and reported on in 2005 by Barker et al. [192] and Ardaravičius et al. [193]. A numerical simulation of the current-voltage characteristics of AlGaN/GaN high electron mobility transistors at high temperatures was performed by Chang et al. [194] and reported on in 2005. Other device modeling work, involving Monte Carlo simulations of the electron transport within wurtzite GaN, was reported on in 2005 by Yamakawa et al. [195] and Reklaitis and Reggiani [196], and many others since then. Finally, the determination of the electron drift velocity from experimental measurements of the unity gain cut-off frequency, fT , has been pursued by a number of researchers. The key challenge in these analyses is the deembedding of the parasitics from the experimental 123 4708 measurements so that the true intrinsic saturation electron drift velocity may be obtained. Following in the tradition of Eastman et al. [197], in 2005 Oxley and Uren [198] found a saturation electron drift velocity of about 1:1 107 cm/s for the case of wurtzite GaN. The role of self-heating was also probed by Oxley and Uren [198] and shown to be relatively insignificant. It should be noted, however, that a completely satisfactory explanation for the discrepancy between these experimental results and those of the Monte Carlo simulations has yet to be provided. 4.2.2 Electron transport within ZnO: a review In 1999, Albrecht et al. [82] were the first to report results obtained from semi-classical Monte Carlo simulations of the steady-state electron transport within wurtzite ZnO. A three-valley model for the conduction band was adopted for the purposes of their analysis. Steady-state electron transport, including consideration of the second-order nonparabolicity coefficient, was considered in their analysis, the non-parabolicity being treated through the application of a more general form of the Kane model than that considered by others; the second-order non-parabolicity, considered by Albrecht et al. [82], is not normally considered in the Kane model; see, for example, Fawcett et al. [144]. The primary focus of their investigation was the determination of the velocity-field characteristic associated with ZnO, and a determination as to how it varies with the crystal temperature, and how it contrasts with the case of wurtzite GaN. How the distribution function responds to the application of an electric field was also considered. The scattering mechanisms considered were: (1) polar optical phonon, (2) piezoelectric, and (3) acoustic deformation potential; ionized impurity scattering was not taken into account in the simulations of Albrecht et al. [82]. They found that for low applied electric field strengths that the electron drift velocity monotonically increases with the applied electric field strength, achieving a peak electron drift velocity of 3:2 107 cm/s at an applied electric field strength of 270 kV/cm at 300 K. Further increases in the electric field strength were found to lead to decreases in the resultant electron drift velocity, i.e., a region of negative differential mobility is observed. The saturation electron drift velocity could not be observed, as saturation occurs beyond the range of electric field strengths presented in the figures of Albrecht et al. [82]. Increases in the crystal temperature were noted to lead to a linearizing of the corresponding velocity-field characteristic and a concomitant decrease in the observed peak electron drift velocity. Increases in the applied electric field strength, beyond the peak field, were noted to have a dramatic effect on the electron distribution. Reasons for this were provided by Albrecht et al. [82]. 123 J Mater Sci: Mater Electron (2014) 25:4675–4713 In 2006, Guo et al. [109] reported on a full-band Monte Carlo simulation of the electron transport within wurtzite ZnO, this analysis improving upon the semi-classical threevalley analysis of Albrecht et al. [82]. Unlike the case of Albrecht et al. [82], ionized impurity scattering was taken into account in the Monte Carlo electron transport simulations of Guo et al. [109]. Using different material parameters than those employed by Albrecht et al. [82], Guo et al. [109] find a peak electron drift velocity of about 1:5 107 cm/s at an applied electric field strength of around 240 kV/cm when the crystal temperature is set to 300 K and the doping concentration is set to 1017 cm3 . The dependence of the velocity-field characteristic on the crystal temperature was also explored by Guo et al. [109], and the general trends observed were noted to be similar to those found by Albrecht et al. [82]. The role that the crystal temperature plays in shaping the dependence of the average electron energy on the applied electric field strength, and in determining the temperature dependence of the diffusion mobility, were further explored by Guo et al. [109]. Since these pioneering investigations, ensemble Monte Carlo simulations of the electron transport within ZnO have been performed by many researchers. In particular, in 2007, Bertazzi et al. [112] reported on Monte Carlo simulations of both the electron and hole transport within wurtzite ZnO. Impact ionization was considered in their analysis, something not being considered in the previous analyzes. The velocity-field characteristics, associated with both the electrons and the holes, were determined in this analysis, an anisotropy associated with the bands being taken into account; this anisotropy was shown to lead to only minor differences. In 2008, Furno et al. [118] explored the difference in results obtained using the full-band Monte Carlo approach with that obtained using the semi-analytical three-valley model. The differences are found to be quite minor. In 2010, Arabshahi et al. [122] contrasted the electron transport within wurtzite GaN with that that occurs in wurtzite silicon carbide and wurtzite ZnO. A detailed critical analysis of the differences between the electron transport within GaN and ZnO was also reported by O’Leary et al. [124] in 2010. O’Leary et al. [124] notes that conduction band inter-valley transitions do not occur in significant numbers for the case of wurtzite ZnO. Thus, it must be the pronounced non-parabolocity of the lowest energy conduction band valley associated with this material, and its impact on the effective mass, that leads to the observed negative differential mobility observed for the case of wurtzite ZnO. The first known studies of transient electron transport within ZnO were those performed by Arabshahi et al. [122] and O’Leary et al. [124]; both the reports from Arabshahi et al. [122] and O’Leary et al. [124] were published in 2010. In these studies, ensemble semi-classical three-valley J Mater Sci: Mater Electron (2014) 25:4675–4713 Monte Carlo simulations were employed in order to determine how the electrons within wurtzite ZnO, initially in thermal equilibrium, respond to the sudden application of a constant electric field. The velocity overshoot which occurs within these materials was examined and contrasted with that associated with wurtzite GaN. It was found that the electron drift velocities that occur within wurtzite ZnO are only slightly less than those exhibited within wurtzite GaN. This suggests that with some further improvements in the material quality, ZnO can also be considered for some of the high-field device applications currently being developed for GaN. 4.3 Recent developments Over the past few years, there have been a number of developments that have occurred that have further enriched our understanding of electron transport within wurtzite GaN. In 2011, Ilgaz et al. [199] studied the energy relaxation of hot electrons within AlGaN/GaN/GaN heterostructures. Then, in 2012, Naylor et al. [200] examined the steady-state and transient electron transport that occurs within bulk wurtzite GaN using an analytical bandstructure that more accurately reflects the nature of the actual band-structure. In 2012, Naylor et al. [201] also examined the electron transport that occurs within dilute GaNx As1x samples. In 2013, Bellotti et al. [202] employed a full-band model in order to determine the velocity-field characteristics associated with AlGaN alloys. Also in 2013, Dasgupta et al. [203] estimated the hot electron relaxation time associated with wurtzite GaN using a series of electrical measurements. In 2013, Zhang et al. [204] determined the hot-electron relaxation time within lattice-matched InAlN/AlN/GaN heterostructures. The potential for electron device structures was then explored. Clearly, the study of electron transport within wurtzite GaN remains an area of active inquiry. The electron transport that occurs within wurtzite ZnO has also been the focus of recent examination. In 2011, Hadi et al. [125] studied the sensitivity of the steady-state electron transport that occurs within wurtzite zinc oxide to variations in the non-parabolicity coefficient associated with the lowest energy conduction band valley. An ensemble semi-classical three-valley Monte Carlo simulation approach was used for the purposes of this analysis. It was found that for non-parabolicity coefficient selections beyond 0.4 eV1 , very few transitions to the upper energy conduction band valleys occur. This sensitivity to variations in the non-parabolicity coefficient was also found to have implications in terms of the form of the resultant velocity-field characteristics. Also in 2011, Hadi et al. [205] reviewed some recent results on the steady-state and transient electron transport 4709 that occurs within wurtzite ZnO. These results were also obtained using an ensemble semiclassical three-valley Monte Carlo simulation approach. It was shown that for electric field strengths in excess of 180 kV/cm, the steadystate electron drift velocity associated with wurtzite ZnO exceeds that associated with wurtzite GaN. The transient electron transport that occurs within wurtzite ZnO was studied by examining how electrons, initially in thermal equilibrium, respond to the sudden application of a constant electric field. These transient electron transport results demonstrated that for devices with dimensions smaller than 0:1 lm, GaN-based devices will offer the advantage, owing to their superior transient electron transport, while for devices with dimensions greater than 0:1 lm, ZnO-based devices will offer the advantage, owing to their superior high-field steady-state electron transport characteristics. Then, in 2012, Hadi et al. [127] employed a three-valley Monte Carlo simulation approach in order to probe the transient electron transport that occurs within wurtzite ZnO, wurtzite GaN, and zinc-blende GaAs. For the purposes of this analysis, Hadi et al. [127] followed the approach of O’Leary et al. [124], and studied how electrons, initially in thermal equilibrium, respond to the sudden application of a constant applied electric field. Through a determination of the dependence of the transient electron drift velocity on both the time elapsed since the onset of the applied electric field and the applied electric field strength, a complete characterization of the transient electron transport response of these materials is obtained. We then applied these results in order to estimate how the optimal cut-off frequency and the corresponding operating device voltage vary with the device length. These results clearly demonstrated the compelling advantage offered by wurtzite ZnO and wurtzite GaN, as opposed to zinc-blende GaAs, for electron devices operating in the terahertz frequency range if higher powers are required. Further analyzes of the steady-state and transient electron transport that occurs within wurtzite ZnO were performed by Hadi et al. [129, 163, 206–209]. While a general understanding of the nature of the electron transport processes within wurtzite ZnO has emerged, it is clearly not as well developed as for the case of GaN. Electron transport within bulk ZnO and ZnO-based electron devices is a focus of current scientific investigation and this will likely remain to be the case for many years to come. 4.4 Future prospectives It is clear that our understanding of the electron transport within the wide energy gap semiconductors, GaN and ZnO, is, at present at least, in a state of flux. A complete understanding of the electron transport mechanisms within 123 4710 these materials has yet to be achieved, and is the subject of intense current research. Most troubling is the discrepancy between the results of experiment and those of simulation. There are a two principle sources of uncertainty in our analysis of the electron transport mechanisms within these materials; (1) uncertainty in the material properties, and (2) uncertainty in the underlying physics. We discuss each of these subsequently. Uncertainty in the material parameters associated with the wide energy gap compound semiconductors, GaN and ZnO, remains a key source of ambiguity in the analysis of the electron transport with these materials [61]. Even for bulk wurtzite GaN, the most well studied of the III–V nitride semiconductors, uncertainty in the band structure remains an issue [91]. Given this uncertainty in the band structures associated with both phases of GaN, it is clear that new simulations of the electron transport within these materials will have to be performed once researchers have settled on appropriate band structures. We thus view the results presented in Sect. 3 as a baseline, the sensitivity analysis, presented in Section 3.11 of O’Leary et al. [62], providing some insight into how variations in the band structure will impact upon the results. Work on finalizing a set of band structural parameters, suitable for wurtzite GaN and its zinc-blende counterpart, and on performing the corresponding electron transport simulations, is ongoing. Given that wurtzite ZnO is far less studied than GaN, determining a set of band structural parameters, suitable for this particular material, remains an ongoing concern. Uncertainty in the underlying physics is also considerable. The source of the negative differential mobility, found in the velocity-field characteristics associated with the wide energy gap compound semiconductors, GaN and ZnO, remains a matter to be resolved. The presence of hotphonons within these materials, and how such phonons impact upon the electron transport mechanisms, remains another point of contention. It is clear that a deeper understanding of these electron transport mechanisms will have to be achieved in order for the next generation of wide energy gap compound semiconductor devices, based on GaN or ZnO, to be properly designed. J Mater Sci: Mater Electron (2014) 25:4675–4713 GaAs. Most of our discussion focused upon results obtained from our ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within these materials, our results conforming with state-of-the-art wide energy gap compound semiconductor orthodoxy. We began this review with the Boltzmann transport equation, this equation underlying most analyzes of the electron transport within semiconductors. A brief description of our ensemble semi-classical three-valley Monte Carlo simulation approach to solving the Boltzmann transport equation was then provided. The material and band structural parameters, corresponding to wurtzite GaN, zinc-blende GaN, wurtzite ZnO, and zinc-blende GaAs, were then presented. We then used these parameter selections, in conjunction with our ensemble semi-classical three-valley Monte Carlo simulation approach, in order to determine the nature of the steady-state and transient electron transport within the wide energy gap compound semiconductors, GaN and ZnO. Finally, we presented some recent developments on the electron transport within these materials, and pointed to fertile frontiers for further research and investigation. 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