32. Electron Transport Within III-V Semiconductors

Transcription

1 Electron Transport Within III-V Tran Nitride Semiconductors StephenK.O Leary,PoppySiddiqua,WalidA.Hadi,BrianE.Foutz,MichaelS.Shur,LesterF.Eastman The III-V nitride semiconductors, gallium nitride, aluminum nitride, and indium nitride, have been recognized as promising materials for novel electronic and optoelectronic device applications for some time now. Since informed device design requires a firm grasp of the material properties of the underlying electronic materials, the electron transport that occurs within these III V nitride semiconductors has been the focus of considerable study over the years. In an effort to provide some perspective on this rapidly evolving field, in this paper we review analyses of the electron transport within these III V nitride semiconductors. In particular, we discuss the evolution of the field, compare and contrast results obtained by different researchers, and survey the more recent literature. In order to narrow the scope of this chapter, we will primarily focus on the electron transport within bulk wurtzite gallium nitride, aluminum nitride, and indium nitride for the purposes of this review. Most of our discussion will focus on results obtained from our ensemble semiclassical three-valley Monte Carlo simulations of the electron transport within these materials, our results conforming with state-of-the-art III V nitride semiconductor orthodoxy. 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 Electron Transport Within Semiconductors and the Monte Carlo Simulation Approach The Boltzmann Transport Equation Our Ensemble Semi-Classical Monte Carlo Simulation Approach Parameter Selections for Bulk Wurtzite GaN, AlN, and InN Steady-State and Transient Electron Transport Within Bulk Wurtzite GaN, AlN, and InN Steady-State Electron Transport Within Bulk Wurtzite GaN Steady-State Electron Transport: A Comparison of the III V Nitride Semiconductors with GaAs Influence of Temperature on the Electron Drift Velocities Within GaN and GaAs Influence of Doping on the Electron Drift Velocities Within GaN and GaAs Electron Transport in AlN Electron Transport in InN Transient Electron Transport Electron Transport: Conclusions Electron Transport Within III V Nitride Semiconductors: A Review Evolution of the Field Developments in the 21st Century Future Perspectives Conclusions References PartD 32 The III V nitride semiconductors, gallium nitride (GaN), aluminum nitride (AlN), and indium nitride (InN), have been known as promising materials for novel electronic and optoelectronic device applications for some time now [32.1 4]. In terms of electronics, their wide energy gaps, large breakdown fields, high thermal conductivities, and favorable electron transport characteristics, make GaN, AlN, and InN, and alloys of these materials, ideally suited for novel high-power and high-frequency electron device applications. On the optoelectronics front, the direct nature of the energy gaps associated with GaN, AlN, and InN, make this family of materials, and its alloys, well suited for novel optoelectronic device applications in the visible and ultraviolet frequency range. While initial efforts to study these materials were hindered by growth diffi- Springer International Publishing AG 217 S. Kasap, P. Capper (Eds.), Springer Handbook of Electronic and Photonic Materials, DOI 1.17/ _32

2 83 Part D Materials for Optoelectronics and Photonics PartD 32.1 culties, recent improvements in material quality have made the realization of a number of III V nitride semiconductor-based electronic [32.5 9] and optoelectronic [ ] devices possible. These developments have fuelled considerable interest in these III V nitride semiconductors. In order to analyze and improve the design of III V nitride semiconductor-based devices, an understanding of the electron transport that occurs within these materials is necessary. Electron transport within bulk GaN, AlN, and InN has been examined extensively over the years [ ]. Unfortunately, uncertainty in the material parameters associated with GaN, AlN, and InN remains a key source of ambiguity in the analysis of the electron transport within these materials [32.32]. In addition, some experimental [32.33] and theoretical [32.34] developments have cast doubt upon the validity of widely accepted notions upon which our understanding of the electron transport mechanisms within the III V nitride semiconductors, GaN, AlN, and InN, has evolved. Another confounding matter is the sheer volume of research activity being performed on the electron transport within these materials, presenting the researcher with a dizzying array of seemingly disparate approaches and results. Clearly, at this critical juncture at least, our understanding of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, remains in a state of flux. In order to provide some perspective on this rapidly evolving field, we aim to review analyses of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, within this chapter. In particular, we will discuss the evolution of the field and survey the more recent literature. In order to narrow the scope of this review, we will primarily focus on the electron transport within bulk wurtzite GaN, AlN, and InN for the purposes of this chapter. Most of our discussion will focus 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 III V nitride semiconductor orthodoxy. We hope that researchers in the field will find this review useful and informative. We begin our review with the Boltzmann transport equation, which underlies most analyses of the electron transport within semiconductors. The ensemble semiclassical three-valley Monte Carlo simulation approach that we employ in order to solve this Boltzmann transport equation is then discussed. The material parameters corresponding to bulk wurtzite GaN, AlN, and InN are then presented. We then use these material parameter selections and our ensemble semi-classical three-valley Monte Carlo simulation approach to determine the nature of the steady-state and transient electron transport within the III V nitride semiconductors. Finally, we present some developments on the electron transport within these materials. This chapter is organized in the following manner. In Sect. 32.1, we present the Boltzmann transport equation and our ensemble semi-classical three-valley Monte Carlo simulation approach that we employ in order to solve this equation for the III V nitride semiconductors, GaN, AlN, and InN. The material parameters, corresponding to bulk wurtzite GaN, AlN, and InN, are also presented in Sect. 32.1, these material parameters being updated for the specific case of wurtzite InN. Then, in Sect. 32.2, using results obtained from our ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within these III V nitride semiconductors, we study the nature of the steady-state electron transport that occurs within these materials. Transient electron transport within the III V nitride semiconductors is also discussed in Sect A review of the III V nitride semiconductor electron transport literature, in which the evolution of the field is discussed and a survey of the more recent literature is presented, is then featured in Sect Finally, conclusions are provided in Sect Electron Transport Within Semiconductors and the Monte Carlo Simulation Approach The electrons within a semiconductor are in a perpetual state of motion. In the absence of an applied electric field, this motion arises as a result of the thermal energy that is present, and is referred to as thermal motion. From the perspective of an individual electron, thermal motion may be viewed as a series of trajectories, interrupted by a series of random scattering events. Scattering may arise as a result of interactions with the lattice atoms, impurities, other electrons, and defects. As these interactions lead to electron trajectories in all possible directions, i. e., there is no preferred direction, while individual electrons will move from one location to another, when taken as an ensemble, and assuming that the electrons are in thermal equilibrium, the overall electron distribution will remain static. Accordingly, no net current flow occurs. With the application of an applied electric field E each electron in the ensemble will experience a force,

3 Electron Transport Within III-V Nitride Semiconductors 32.1 Electron Transport and Monte Carlo Simulation 831 qe, whereq denotes the magnitude of the electron charge. While this force may have a negligible impact upon the motion of any given individual electron, taken as an ensemble, the application of such a force will lead to a net aggregate motion of the electron distribution. Accordingly, a net current flow will occur, and the overall electron ensemble will no longer be in thermal equilibrium. This movement of the electron ensemble in response to an applied electric field, in essence, represents the fundamental issue at stake when we study the electron transport within a semiconductor. In this section, we provide a brief tutorial on the issues at stake in our analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN. We begin our analysis with an introduction to the Boltzmann transport equation. This equation describes how the electron distribution function evolves under the action of an applied electric field, and underlies the electron transport within bulk semiconductors. We then introduce the Monte Carlo simulation approach to solving this Boltzmann transport equation, focusing on the ensemble semi-classical three-valley Monte Carlo simulation approach used in our simulations of the electron transport within the III V nitride semiconductors. Finally, we present the material parameters corresponding to bulk wurtzite GaN, AlN, and InN. This section is organized in the following manner. In Sect , the Boltzmann transport equation is introduced. Then, in Sect , our ensemble semiclassical three-valley Monte Carlo simulation approach to solving this Boltzmann transport equation is presented. Finally, in Sect , our material parameter selections, corresponding to bulk wurtzite GaN, AlN, and InN, are presented The Boltzmann Transport Equation An electron ensemble may be characterized by its distribution function f.r; p; t/,wherer denotes the position, p represents the momentum, and t indicates time. The response of this distribution function to an applied electric field E is the issue at stake when one investigates the electron transport within a semiconductor. When the dimensions of the semiconductor are large, and quantum effects are negligible, the ensemble of electrons may be treated as a continuum, so the corpuscular nature of the individual electrons within the ensemble, and the attendant complications which arise, may be neglected. In such a circumstance, the evolution of the distribution function f.r; p; t/ may be determined using the Boltzmann transport equation. In contrast, when the dimensions of the semiconductor are small, and quantum effects are significant, then the Boltzmann transport equation, and its continuum description of the electron ensemble, is no longer valid. In such a case, it is necessary to adopt quantum transport methods in order to study the electron transport within the semiconductor [32.35]. For the purposes of this analysis, we will focus on the electron transport within bulk semiconductors, i. e., semiconductors of sufficient dimensions so that the Boltzmann transport equation is valid. Ashcroft and Mermin [32.36] demonstrated that this equation can be D Pp r pf Pr r r f ˇ : ˇscat The first term on the right-hand side of (32.1)represents the change in the distribution function due to external forces applied to the system. The second term on the right-hand side of (32.1) accounts for the electron diffusion which occurs. The final term on the right-hand side of (32.1) describes the effects of scattering. Owing to its fundamental importance in the analysis of the electron transport within semiconductors, a number of techniques have been developed over the years in order to solve the Boltzmann transport equation. Approximate solutions to the Boltzmann transport equation, such as the displaced Maxwellian distribution function approach of Ferry [32.14] anddas and Ferry [32.15] and the nonstationary charge transport analysis of Sandborn et al. [32.37], have proven useful. Low-field approximate solutions have also proven elementary and insightful [32.17, 2, 38]. A number of these techniques have been applied to the analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN [32.14, 15, 17, 2, 38, 39]. Alternatively, more sophisticated techniques have been developed which solve the Boltzmann transport equation directly. These techniques, while allowing for a rigorous solution of the Boltzmann transport equation, are rather involved, and require intense numerical analysis. They are further discussed by Nag [32.4]. For studies of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, by far the most common approach to solving the Boltzmann transport equation has been the ensemble semi-classical Monte Carlo simulation approach. Of the III V nitride semiconductors, the electron transport within GaN has been studied the most extensively using this ensemble Monte Carlo simulation approach [32.13, 16, 18, 19, 21, 22, 27, 29, 32], with AlN [32.24, 25, 29] and InN [32.23, 28, 29, 31] less so. The Monte Carlo simulation approach has also been used to study the electron transport within the two-dimensional electron gas of the AlGaN=GaN interface which occurs in high electron PartD 32.1

4 832 Part D Materials for Optoelectronics and Photonics PartD 32.1 mobility AlGaN=GaN field-effect transistors [32.41, 42]. At this point, it should be noted that the complete solution of the Boltzmann transport equation requires the resolution of both steady-state and transient responses. Steady-state electron transport refers to the electron transport that occurs long after the application of an applied electric field, i. e., once the electron ensemble has settled to a new equilibrium state (we are not necessarily referring to thermal equilibrium here, since thermal equilibrium is only achieved in the absence of an applied electric field). As the distribution function is difficult to visualize quantitatively, researchers typically study the dependence of the electron drift velocity (the average electron velocity determined by statistically averaging over the entire electron ensemble) on the applied electric field in the analysis of steady-state electron transport; in other words, they determine the velocity field characteristic. Transient electron transport, by way of contrast, refers to the transport that occurs while the electron ensemble is evolving into its new equilibrium state. Typically, it is characterized by studying the dependence of the electron drift velocity on the time elapsed, or the distance displaced, since the electric field was initially applied. Both steady-state and transient electron transport within the III V nitride semiconductors, GaN, AlN, and InN, are reviewed within this chapter Our Ensemble Semi-Classical Monte Carlo Simulation Approach For the purposes of our analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, we employ ensemble semi-classical Monte Carlo simulations. A three-valley model for the conduction band is employed. Nonparabolicity is considered in the lowest conduction band valley, this nonparabolicity being treated through the application of the Kane model [32.43]. In the Kane model, the energy band of the valley is assumed to be nonparabolic, spherical, and of the form 2 k 2 D E.1 C E/; (32.2) 2m where k denotes the magnitude of the crystal momentum, E represents the energy above the minimum, m is the effective mass, and the nonparabolicity coefficient is given by D m ; (32.3) E g m e where m e and E g denote the free electron mass and the energy gap, respectively [32.43]. The scattering mechanisms considered in our analysis are: 1. Ionized impurity 2. Polar optical phonon 3. Piezoelectric [32.44, 45] 4. Acoustic deformation potential. Intervalley scattering is also considered. Piezoelectric scattering is treated using the well established zinc blende scattering rates, and so a suitably transformed piezoelectric constant, e 14, must be selected. This may be achieved through the transformation suggested by Bykhovski et al. [32.44, 45]. We also assume that all donors are ionized and that the free electron concentration is equal to the dopant concentration. The motion of three thousand electrons is examined in our steadystate electron transport simulations, while the motion of ten thousand electrons is considered in our transient electron transport simulations. The crystal temperature is set to 3 K and the doping concentration is set to 1 17 cm 3 in all cases, unless otherwise specified. Electron degeneracy effects are accounted for by means of the rejection technique of Lugli and Ferry [32.46]. Electron screening is also accounted for following the Brooks Herring method [32.47]. Further details of our approach are discussed in the literature [32.16, 21 24, 29, 32, 48] Parameter Selections for Bulk Wurtzite GaN, AlN, and InN The material parameter selections, used for our simulations of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, are tabulated in Table These parameter selections are the same as those employed by Foutz et al. [32.29], with the exception of the case of InN, whose material parameters have been updated [32.49, 5]; following the lead of Polyakov et al. [32.5], the parameters corresponding to InN are as set by Hadi et al. [32.49]. While the band structures corresponding to bulk wurtzite GaN, AlN, and InN are still not agreed upon, the band structures of Lambrecht and Segall [32.51] are adopted for the purposes of this analysis except for the case of InN; for the case of InN, following the lead of Polyakov et al. [32.5], the band structure is as set by Hadi et al. [32.49]. For the case of bulk wurtzite GaN, the analysis of Lambrecht and Segall [32.51] suggests that the lowest point in the conduction band is located at the center of the Brillouin zone, at the point, the first upper conduction band valley minimum also occurring at the point, 1:9 ev above the lowest point

5 Electron Transport Within III-V Nitride Semiconductors 32.1 Electron Transport and Monte Carlo Simulation 833 Table 32.1 The material parameter selections corresponding to bulk wurtzite GaN, AlN, and InN. (These parameter selections are from Foutz et al. [32.29] andhadi et al. [32.49]) Parameter GaN AlN InN Mass density (g=cm 3 ) 6:15 3:23 6:81 Longitudinal sound velocity (cm=s) 6: : : Transverse sound velocity (cm=s) 2: : : Acoustic deformation potential (ev) 8:3 9:5 7:1 Static dielectric constant 8:9 8:5 15:3 High-frequency dielectric constant 5:35 4:77 8:4 Effective mass ( 1 valley) :2 m e :48 m e :4 m e Piezoelectric constant, e 14 (C=cm 2 ) 3: : : Direct energy gap (ev) 3:39 6:2 :7 Optical phonon energy (mev) 91:2 99:2 73: Intervalley deformation potentials (ev=cm) Intervalley phonon energies (mev) 91:2 99:2 73: Table 32.2 The band structure parameter selections corresponding to bulk wurtzite GaN, AlN, and InN. (These parameter selections are from Foutz et al. [32.29] and Hadi et al. [32.49]. These parameters were originally determined from the band structural calculations of Polyakov et al. [32.5] andlambrecht and Segall [32.51]) Valley number GaN Valley location 1 2 L M Valley degeneracy Effective mass :2 m e m e m e Intervalley energy separation (ev) 1:9 2:1 Energy gap (ev) 3:39 5:29 5:49 Nonparabolicity (ev 1 ) :189 : : AlN Valley location 1 L-M K Valley degeneracy Effective mass :48 m e m e m e Intervalley energy separation (ev) :7 1: Energy gap (ev) 6:2 6:9 7:2 Nonparabolicity (ev 1 ) :44 : : InN Valley location 1 2 L M Valley degeneracy Effective mass :4 m e :25 m e m e Intervalley energy separation (ev) 1:775 2:79 Energy gap (ev) :7 2:475 3:49 Nonparabolicity (ev 1 ) 1:43 : : PartD 32.1 in the conduction band, the second upper conduction band valley minima occurring along the symmetry lines between the L and M points, 2:1 ev above the lowest point in the conduction band; see Table The other materials considered in this analysis have suitably modified band structures, as described in Table We ascribe an effective mass equal to the free electron mass m e to all of the upper conduction band valleys, with the exception of the case of InN, the values suggested by Polyakov et al. [32.5] and Hadi et al. [32.49] being employed for the specific case of this material. The nonparabolicity coefficient corresponding to each upper conduction band valley is set to zero, so the upper conduction band valleys are assumed to be completely parabolic. For our simulations of the electron transport within gallium arsenide (GaAs), the material parameters employed are mostly from Littlejohn et al. [32.52], although it should be noted that the mass density, the energy gap, and the sound velocities are from Blakemore [32.53]. It should be noted that the energy gap associated with InN has been the subject of some controversy since 22. The pioneering experimental results of Tansley and Foley [32.54], reported in 1986, suggested that InN has an energy gap of 1:89 ev. This value has been used extensively in Monte Carlo simulations of the electron transport within this material since that time [32.23, 28, 29, 31]; typically, the influence of the energy gap on the electron transport occurs through its impact on the nonparabolicity coefficient. In 22,

6 834 Part D Materials for Optoelectronics and Photonics PartD 32.2 Davydov et al. [32.55], Wu et al. [32.56], and Matsuoka et al. [32.57], presented experimental evidence which instead suggests a considerably smaller energy gap for InN, around :7 ev. For the purposes of this analysis, the revised value for the InN energy gap is employed; the sensitivity of the velocity field characteristic associated with bulk wurtzite GaN to variations in the nonparabolicity coefficient has been explored, in detail, by O Leary et al. [32.32]. The band structure associated with bulk wurtzite GaN has also been the focus of some controversy. In particular, Brazel et al. [32.58] employed ballistic electron emission microscopy measurements in order to demonstrate that the first upper conduction band valley occurs only 34 mev above the lowest point in the conduction band for this material. This contrasts rather 32.2 Steady-State and Transient Electron Transport Within Bulk Wurtzite GaN, AlN, and InN dramatically with more traditional results, such as the calculation of Lambrecht and Segall [32.51], which instead suggest that the first upper conduction band valley minimum within wurtzite GaN occurs about 2 ev above the lowest point in the conduction band. Clearly, this will have a significant impact upon the results. While the results of Brazel et al. [32.58] were reported in 1997, electron transport simulations adopted the more traditional intervalley energy separation of about 2 ev until relatively recently. Accordingly, we have adopted the more traditional intervalley energy separation for the purposes of our present analysis. The sensitivity of the velocity field characteristic associated with bulk wurtzite GaN to variations in the intervalley energy separation has been explored, in detail, by O Leary et al. [32.32]. The current interest in theiii V nitride semiconductors, GaN, AlN, and InN, is primarily being fuelled by the tremendous potential of these materials for novel electronic and optoelectronic device applications. With the recognition that informed electronic and optoelectronic device design requires a firm understanding of the nature of the electron transport within these materials, electron transport within the III V nitride semiconductors has been the focus of intensive investigation over the years. The literature abounds with studies on steady-state and transient electron transport within these materials [ , 38, 39, 41, 42, 48]. As a result of this intense flurry of research activity, novel III V nitride semiconductor-based devices are starting to be deployed in today s commercial products. Future developments in the III V nitride 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 the Monte Carlo simulation approach that we employ for the analysis of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN. In this section, an overview of the steady-state and transient electron transport results we obtained from these Monte Carlo simulations is provided. In the first part of this section, we focus upon bulk wurtzite GaN. In particular, the velocity field characteristic associated with this material will be examined in detail. Then, an overview of our steady-state electron transport results, corresponding to the three III V nitride semiconductors under consideration in this analysis, i. e., GaN, AlN, and InN, will be given, and a comparison with the more conventional III V compound semiconductor, GaAs, will be presented. A comparison between the temperature dependence of the velocity field characteristics associated with GaN and GaAs will then be provided, and our Monte Carlo results will be used to account for the differences in behavior. A similar analysis will be presented for the doping dependence. Next, detailed simulation results for AlN and InN will be presented. Finally, the transient electron transport that occurs within the III V nitride semiconductors, GaN, AlN, and InN, is determined and compared with that in GaAs. This section is organized in the following manner. In Sect , the velocity field characteristic associated with bulk wurtzite GaN is presented and analyzed. Then, in Sect , the velocity-field characteristics associated with the III V nitride semiconductors under consideration in this analysis will be compared and contrasted with that of GaAs. The sensitivity of the velocity field characteristic associated with bulk wurtzite GaN to variations in the crystal temperature will then be examined in Sect , and a comparison with that corresponding to GaAs presented. In Sect , the sensitivity of the velocity field characteristic associated with bulk wurtzite GaN to variations in the doping concentration level will be explored, and a comparison with that corresponding to GaAs presented. The velocity field characteristics associated with AlN and InN will then be examined in

7 Electron Transport Within III-V Nitride Semiconductors 32.2 Steady-State and Transient Electron Transport 835 Electron velocity (1 7 cm/s) Electron energy (ev) GaN Fig The velocity field characteristic associated with bulk wurtzite GaN. Like many other compound semiconductors, the electron drift velocity reaches a peak, and at higher applied electric fields it decreases until it saturates Sect and Sect , respectively. Our transient electron transport analysis results are then presented in Sect Finally, the conclusions of this electron transport analysis are summarized in Sect Steady-State Electron Transport Within Bulk Wurtzite GaN Our examination of results begins with GaN, the most commonly studied III V nitride semiconductor. The velocity field characteristic associated with this material is presented in Fig This result was obtained through our Monte Carlo simulations of the electron transport within this material for the bulk wurtzite GaN parameter selections specified in Table 32.1 and Table 32.2; the crystal temperature was set to 3 K and the doping concentration to 1 17 cm 3. We see that for applied electric fields in excess of 14 kv=cm, the electron drift velocity decreases, eventually saturating at 1:4 1 7 cm=s for high applied electric fields. By examining the results of our Monte Carlo simulation further, an understanding of this result becomes clear. First, we discuss the results at low applied electric fields, i. e., applied electric fields of less than 3 kv=cm. This is referred to as the linear regime of electron transport as the electron drift velocity is well characterized by the low-field electron drift mobility in this regime, i. e., a linear low-field electron drift velocity dependence on the applied electric field, i. e., v d D E, applies in this regime. Examining the distribution function for this regime, we find that it is very similar to the zero-field distribution function with a slight shift in the Fig The average electron energy as a function of the applied electric field for bulk wurtzite GaN. Initially, the average electron energy remains low, only slightly higher 3 than the thermal energy, 2 k BT. At 1 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 direction opposite to the applied electric field. In this regime, the average electron energy remains relatively low, with most of the energy gained from the applied electric field being transferred into the lattice through polar optical phonon scattering. If we examine the average electron energy as a function of the applied electric field, shown in Fig. 32.2, we see that there is a sudden increase at around 1 kv=cm. In order to understand why this increase occurs, we note that the dominant energy loss mechanism for many of the III V compound semiconductors, including GaN, is polar optical phonon scattering. When the applied electric field is less than 1 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: they are elastic scattering mechanisms. However, beyond a certain critical applied electric field strength, 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 intervalley scattering begins and an energy balance is re-established. As the applied electric field is increased beyond 1kV=cm, the average electron energy increases un- PartD 32.2

8 836 Part D Materials for Optoelectronics and Photonics Number of particles 3 Drift velocity (cm/s) kv/cm 14 kv/cm kv/cm GaAs InN GaN AlN 45 kv/cm PartD Fig The valley occupancy as a function of the applied electric field for the case of bulk wurtzite GaN. Soon after the average electron energy increases, electrons begin to transfer to the upper valleys of the conduction band. Three thousand electrons were employed for this simulation. The valleys are labeled 1, 2, and 3, in accordance with their energy minima; the lowest energy valley is valley 1, the next higher energy valley is valley 2, and the highest energy valley is valley Fig A comparison of the velocity field characteristics associated with the III V nitride semiconductors, GaN, AlN, and InN, with that associated with GaAs. (After [32.29], with permission from AIP) Steady-State Electron Transport: A Comparison of the III V Nitride Semiconductors with GaAs til a substantial fraction of the electrons have acquired enough energy in order to transfer into the upper valleys. 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 (Fig. 32.3), the electron drift velocity decreases. This accounts for the negative differential mobility observed in the velocity field characteristic depicted in Fig Finally, at high applied electric fields, 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. Thus far, electron transport results corresponding to bulk wurtzite GaN have been presented and discussed qualitatively. It should be noted, however, that the same phenomenon that occurs in the velocity field characteristic associated with GaN also occurs for the other III V nitride semiconductors, AlN and InN. The importance of polar optical phonon scattering when determining the nature of the electron transport within the III V nitride semiconductors, GaN, AlN, and InN, will become even more apparent later, as it will be used to account for much of the electron transport behavior within these materials. Setting the crystal temperature to 3 K and the level of doping to 1 17 cm 3, the velocity field characteristics associated with the III V nitride semiconductors under consideration in this analysis GaN, AlN, and InN are contrasted with that of GaAs in Fig We see that each of these III V compound semiconductors achieves a peak in its velocity field characteristic. InN achieves the highest steady-state peak electron drift velocity, 5:6 1 7 cm=s at an applied electric field of 3 kv=cm. This contrasts with the case of GaN, 2:9 1 7 cm=s at 14 kv=cm, and that of AlN, 1:71 7 cm=s at 45 kv=cm. For GaAs, the peak electron drift velocity of 1:6 1 7 cm=s occurs at a much lower applied electric field thanthat forthe III V nitride semiconductors (only 4 kv=cm) Influence of Temperature on the Electron Drift Velocities Within GaN and GaAs The temperature dependence of the velocity field characteristic associated with bulk wurtzite GaN is now examined. Figure 32.5a shows how the velocity field characteristic associated with bulk wurtzite GaN varies as the crystal temperature is increased from 1 to 7 K, in increments of 2 K. The upper limit, 7 K, is chosen as it is the highest operating temperature that may be expected for AlGaN=GaN power devices. To

9 Electron Transport Within III-V Nitride Semiconductors 32.2 Steady-State and Transient Electron Transport 837 a) b) Drift velocity (1 7 cm/s) GaN Drift velocity (1 7 cm/s) GaAs K 5 K 3 K 7 K K 3 K 5 K 7 K Fig. 32.5a,b A comparison of the temperature dependence of the velocity field characteristics associated with (a) GaN and (b) GaAs. GaN maintains a higher electron drift velocity with increased temperatures than GaAs does PartD 32.2 a) b) Drift velocity (1 7 cm/s) Mobility (cm 2 /Vs) Drift velocity (1 7 cm/s) GaN Mobility (cm 2 /Vs) 12 GaAs Temperature (K) Temperature (K) Fig. 32.6a,b A comparison of the temperature dependence of the low-field electron drift mobility (solid lines), the peak electron drift velocity (diamonds), and the saturation electron drift velocity (solid points) for(a) GaN and (b) GaAs. The low-field electron drift mobility of GaN drops quickly with increasing temperature, but its peak and saturation electron drift velocities are less sensitive to increases in temperature than GaAs highlight the difference between the III V nitride semiconductors with more conventional III V compound semiconductors, such as GaAs, Monte Carlo simulations of the electron transport within GaAs have also been performed under the same conditions as GaN. Figure 32.5b shows the results of these simulations. Note that the electron drift velocity for the case of GaN is much less sensitive to changes in temperature than that associated with GaAs. To quantify this dependence further, the low-field electron drift mobility, the peak electron drift velocity, and the saturation electron drift velocity are plotted as a function of the crystal temperature in Fig. 32.6, these results being determined from our Monte Carlo simulations of the electron transport within these materials. For both GaN and GaAs, it is found that all of these electron transport metrics diminish as the crystal temperature is increased. As may be seen through an inspection of Fig. 32.5, the peak and saturation electron drift velocities do not drop as much in GaN as they do in GaAs in response to increases in the crystal temperature. The low-field electron drift mobility in GaN, however, is seen to fall quite rapidly with temperature, this drop being particularly severe for tem-

10 838 Part D Materials for Optoelectronics and Photonics a) Scattering rate (1 13 s 1 ) b) Scattering rate (1 12 s 1 ) 1 1 GaN 7 K GaAs 8 5 K 8 3 K 6 1 K 6 7 K 5 K 3 K K 2 2 PartD Fig. 32.7a,b A comparison of the polar optical phonon scattering rates as a function of the applied electric field strength for various crystal temperatures for (a) GaN and (b) GaAs. Polar optical phonon scattering is seen to increase much more quickly with temperature in GaAs a) Number of particles b) Number of particles 3 3 GaN 1 K GaAs K 1 7 K K Fig. 32.8a,b A comparison of the number of particles in the lowest energy valley of the conduction band, the valley, as a function of the applied electric field for various crystal temperatures, for the cases of (a) GaN and (b) GaAs. In GaAs, the electrons begin to occupy the upper valleys much more quickly, causing the electron drift velocity to drop as the crystal temperature is increased. Three thousand electrons were employed for these steady-state electron transport simulations peratures at and below room temperature. This property will likely have an impact on high-power device performance. Delving deeper into our Monte Carlo results yields clues into the reason for this variation in temperature dependence. First, we examine the polar optical phonon scattering rate as a function of the applied electric field strength. Figure 32.7 shows that the scattering rate only increases slightly with temperature for the case of GaN, from 6: s 1 at 1 K to 8: s 1 at 7 K, for high applied electric field strengths. Contrast this with the case of GaAs, where the rate increases from 4: 1 12 s 1 at 1 K to more than twice that amount at 7 K, 9: s 1, at high applied electric field strengths. This large increase in the polar optical phonon scattering rate for the case of GaAs is one reason for the large drop in the electron drift velocity with increasing temperature for the case of GaAs. A second reason for the variation in temperature dependence of the two materials is the occupancy of the upper valleys, shown in Fig In the case of GaN, the upper valleys begin to become occupied at roughly

11 Electron Transport Within III-V Nitride Semiconductors 32.2 Steady-State and Transient Electron Transport 839 a) Drift velocity (1 7 cm/s) b) Drift velocity (1 7 cm/s) cm 3 GaN 1 16 cm 3 GaAs cm cm cm cm cm Fig. 32.9a,b A comparison of the dependence of the velocity field characteristics associated with (a) GaN and (b) GaAs on the doping concentration. GaN maintains a higher electron drift velocity with increased doping levels than GaAs does PartD 32.2 a) Drift velocity (1 7 cm/s) Mobility (cm 2 /Vs) b) Drift velocity (1 7 cm/s) GaN Mobility (cm 2 /Vs) GaAs Doping concentration (cm 3 ) Doping concentration (cm 3 ) Fig. 32.1a,b A comparison of the low-field electron drift mobility (solid lines), the peak electron drift velocity (diamonds), and the saturation electron drift velocity (solid points) for(a) GaN and (b) GaAs as a function of the doping concentration. These parameters are more insensitive to increases in doping in GaN than in GaAs the same applied electric field strength, 1 kv=cm, independent of temperature. For the case of GaAs, however, the upper valleys are at a much lower energy than those in GaN. In particular, while the first upper conduction band valley minimum is 1:9 ev above the lowest point in the conduction band in GaN, the first upper conduction band valley is only 29 mev above the bottom of the conduction band in GaAs [32.53]. As the upper conduction band valleys are so close to the bottom of the conduction band for the case of GaAs, the thermal energy (at 7 K, k B T Š 6 mev) is enough in order to allow for a small fraction of the electrons to transfer into the upper valleys even before an electric field is applied. When electrons occupy the upper valleys, intervalley scattering and the upper valleys larger effective masses reduce the overall electron drift velocity. This is another reason why the velocity field characteristic associated with GaAs is more sensitive to variations in crystal temperature than that associated with GaN Influence of Doping on the Electron Drift Velocities Within GaN and GaAs One parameter that can be readily controlled during the fabrication of semiconductor devices is the doping concentration. Understanding the effect of doping on the resultant electron transport is also important.

12 84 Part D Materials for Optoelectronics and Photonics a) Scattering rate (1 13 s 1 ) b) Scattering rate (1 12 s 1 ) 1 1 GaN cm cm cm 3 4 GaAs 1 16 cm cm cm cm 3 PartD Fig a,b A comparison of the polar optical phonon scattering rates as a function of the applied electric field, for both (a) GaN and (b) GaAs, for various doping concentrations a) Number of particles b) Number of particles cm 3 GaN 1 18 cm cm cm cm GaAs Fig a,b A comparison of the number of particles in the lowest valley of the conduction band, the valley, as a function of the applied electric field, for both (a) GaN and (b) GaAs, for various doping concentration levels. Three thousand electrons were employed for these steady-state electron transport simulations In Fig. 32.9, the velocity field characteristic associated with GaN is presented for a number of different doping concentration levels. Once again, three important electron transport metrics are influenced by the doping concentration level: the low-field electron drift mobility, the peak electron drift velocity, and the saturation electron drift velocity; see Fig Our simulation results suggest that for doping concentrations of less than 1 17 cm 3, there is very little effect on the velocity field characteristic for the case of GaN. However, for doping concentrations above 1 17 cm 3,the peak electron drift velocity diminishes considerably, from 2:9 1 7 cm=s for the case of 1 17 cm 3 doping to 2: 1 7 cm=s for the case of 1 19 cm 3 doping. The saturation electron drift velocity within GaN is found to only decrease slightly in response to increases in the doping concentration. The effect of doping on the lowfield electron drift mobility is also shown. It is seen that this mobility drops significantly in response to increases in the doping concentration level, from 12 cm 2 =.Vs/ at 1 16 cm 3 doping to 4 cm 2 =.Vs/ at 1 19 cm 3 doping. As we did for temperature, we compare the sensitivity of the velocity-field characteristic associated with GaN to doping with that associated with GaAs. Figure 32.1 shows this comparison. For the case of GaAs, it is seen that the electron drift velocities decrease much more with increased doping than those associated with GaN. In particular, for the case of GaAs, the peak electron drift velocity decreases from

13 Electron Transport Within III-V Nitride Semiconductors 32.2 Steady-State and Transient Electron Transport 841 a) Drift velocity (1 7 cm/s) b) Drift velocity (1 7 cm/s) AlN 1 K Mobility (cm 2 /Vs) 4 AlN K 5 K 7 K Temperature (K) Fig The velocity field characteristic associated with AlN (a) for various crystal temperatures. The trends in the low-field mobility (solid line), the peak electron drift velocity (diamonds), and the saturation electron drift velocity (solid points), are also shown. AlN exhibits its peak electron drift velocity at very high applied electric fields. AlN has the lowest peak electron drift velocity and the lowest low-field electron drift mobility of the III V nitride semiconductors considered in this analysis (b) PartD :8 1 7 cm=s at1 16 cm 3 doping to :6 1 7 cm=s at 1 19 cm 3 doping. For GaAs, at the higher doping levels, the peak in the velocity-field characteristic disappears completely for sufficiently high doping concentrations. The saturation electron drift velocity decreases from 1: 1 7 cm=s at1 16 cm 3 doping to :61 7 cm=sat1 19 cm 3 doping. The low-field electron drift mobility also diminishes dramatically with increased doping, dropping from 78 cm 2 =.Vs/ at 1 16 cm 3 doping to 22 cm 2 =.Vs/ at 1 19 cm 3 doping; see Fig Once again, it is interesting to determine why the doping dependence in GaAs is so much more pronounced than it is in GaN. Again, we examine the polar optical phonon scattering rate and the occupancy of the upper valleys. Figure shows the polar optical phonon scattering rates as a function of the applied electric field, for both GaN and GaAs. In this case, however, due to screening effects, the rate drops when the doping concentration is increased. The decrease, however, is much more pronounced for the case of GaAs than for GaN. It is believed that this drop in the polar optical phonon scattering rate allows for upper valley occupancy to occur more quickly in GaAs rather than in GaN (Fig ). For GaN, electrons begin to occupy the upper valleys at roughly the same applied electric field strength, independent of the doping level. However, for the case of GaAs, the upper valleys are occupied more quickly with greater doping. When the upper valleys are occupied, the electron drift velocity decreases due to intervalley scattering and the larger effective mass of the electrons within the upper valleys Electron Transport in AlN AlN has the largest effective mass of the III V nitride semiconductors considered in this analysis. Accordingly, it is not surprising that this material exhibits the lowest electron drift velocity and the lowest low-field electron drift mobility. The sensitivity of the velocity field characteristic associated with AlN to variations in the crystal temperature may be examined by considering Fig As with the case of GaN, the velocity field characteristic associated with AlN is extremely robust to variations in the crystal temperature. In particular, its peak electron drift velocity, which is 1:81 7 cm=s at 1 K, only decreases to 1:21 7 cm=s at 7 K. Similarly, its saturation electron drift velocity, which is 1:5 1 7 cm=s at 1 K, only decreases to 1: 1 7 cm=s at 7 K. The low-field electron drift mobility associated with AlN also diminishes in response to increases in the crystal temperature, from 375 cm 2 =.Vs/ at 1 K to 4 cm 2 =.Vs/ at 7 K. The sensitivity of the velocity field characteristic associated with AlN to variations in the doping concentration may be examined by considering Fig It is noted that the variations in the velocity field characteristic associated with AlN in response to variations in the doping concentration are not as pronounced as those which occur in response to variations in the crystal temperature. Quantitatively, the peak electron drift velocity drops from 1:7 1 7 cm=s at1 17 cm 3 doping to 1:3 1 7 cm=s at1 19 cm 3 doping. Similarly, its saturation electron drift velocity drops from 1:4 1 7 cm=s at1 17 cm 3 doping to 1:2 1 7 cm=s

14 842 Part D Materials for Optoelectronics and Photonics a) Drift velocity (1 7 cm/s) b) Drift velocity (1 7 cm/s) cm cm 3 AlN Mobility (cm 2 /Vs) 25 AlN cm PartD Doping concentration (cm 3 ) Fig The velocity field characteristic associated with AlN for various doping concentrations (a). The trends in the low-field electron drift mobility (solid line), the peak electron drift velocity (diamonds), and the saturation electron drift velocity (solid points), are also shown (b) a) Drift velocity (1 7 cm/s) b) Drift velocity (1 7 cm/s) 6 6 InN 5 5 Mobility (cm 2 /Vs) 25 InN K 3 K 5 K 7 K Temperature (K) Fig The velocity field characteristic associated with InN for various crystal temperatures (a). The trends in the low-field electron drift mobility (solid line), the peak electron drift velocity (diamonds), and the saturation electron drift velocity (solid points), are also shown (b). InN has the highest peak electron drift velocity and the highest low-field electron drift mobility of the III V nitride semiconductors considered in this analysis at 1 19 cm 3 doping. The influence of doping on the low-field electron drift mobility associated with AlN is also observed to be not as pronounced as for the case of crystal temperature. Figure 32.14b shows that the low-field electron drift mobility associated with AlN decreases from 14 cm 2 =.Vs/ at 1 16 cm 3 doping to 1 cm 2 =.Vs/ at 1 19 cm 3 doping Electron Transport in InN InN has the smallest effective mass of the three III V nitride semiconductors considered in this analysis. Accordingly, it is not surprising that it exhibits the highest electron drift velocity and the highest low-field electron drift mobility. The sensitivity of the velocity-field characteristic associated with InN to variations in the crystal temperature may be examined by considering Fig As with the cases of GaN and AlN, the velocity field characteristic associated with InN is extremely robust to increases in the crystal temperature. In particular, its peak electron drift velocity, which is 6:1 7 cm=s at 1 K, only decreases to 4:21 7 cm=s at 7 K. Similarly, its saturation electron drift velocity, which is 1:5 1 7 cm=s at 1 K, only decreases to