Electron Transport Within III-V Nitride Semiconductors

  • Stephen K. O’Leary
  • Poppy Siddiqua
  • Walid A. Hadi
  • Brian E. Foutz
  • Michael S. Shur
  • Lester F. Eastman
Part of the Springer Handbooks book series (SPRINGERHAND)

Abstract

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 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. 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 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, 32.2, 32.3, 32.4]. 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 difficulties, recent improvements in material quality have made the realization of a number of III–V nitride semiconductor-based electronic [32.5, 32.6, 32.7, 32.8, 32.9] and optoelectronic [32.10, 32.11, 32.12, 32.9] 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 [32.13, 32.14, 32.15, 32.16, 32.17, 32.18, 32.19, 32.20, 32.21, 32.22, 32.23, 32.24, 32.25, 32.26, 32.27, 32.28, 32.29, 32.30, 32.31, 32.32]. 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 semi-classical 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. 32.2. 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. 32.3. Finally, conclusions are provided in Sect. 32.4.

32.1 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, −qE, where q 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. 32.1.1, the Boltzmann transport equation is introduced. Then, in Sect. 32.1.2, our ensemble semi-classical three-valley Monte Carlo simulation approach to solving this Boltzmann transport equation is presented. Finally, in Sect. 32.1.3, our material parameter selections, corresponding to bulk wurtzite GaN, AlN, and InN, are presented.

32.1.1 The Boltzmann Transport Equation

An electron ensemble may be characterized by its distribution function \(f(\boldsymbol{r},\boldsymbol{p},t)\), where r denotes the position, p represents the momentum, and t indicates time. The response of this distribution function to an applied electric field E is the issue at stake when one investigates the electron transport within a semiconductor. When the dimensions of the semiconductor are large, and quantum effects are negligible, the ensemble of electrons may be treated as a continuum, 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 \(f(\boldsymbol{r},\boldsymbol{p},t)\) 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 expressed as
$$\frac{\partial f}{\partial t}=-\boldsymbol{\dot{p}}\cdot\nabla_{\mathrm{p}}f-\boldsymbol{\dot{r}}\cdot\nabla_{\mathrm{r}}f+\left.\frac{\partial f}{\partial t}\right|_{\mathrm{scat}}\;.$$
(32.1)
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] and Das 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, 32.20, 32.38]. 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, 32.15, 32.17, 32.20, 32.38, 32.39]. 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 [32.40].

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, 32.16, 32.18, 32.19, 32.21, 32.22, 32.27, 32.29, 32.32], with AlN [32.24, 32.25, 32.29] and InN [32.23, 32.28, 32.29, 32.31] 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 mobility AlGaN ∕ GaN field-effect transistors [32.41, 32.42].

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.

32.1.2 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 Γ valley is assumed to be nonparabolic, spherical, and of the form
$$\begin{aligned}\displaystyle\frac{\hbar^{2}k^{2}}{2{m}^{*}}=E\left(1+\alpha E\right),\end{aligned}$$
(32.2)
where ℏk denotes the magnitude of the crystal momentum, E represents the energy above the minimum, m* is the effective mass, and the nonparabolicity coefficient α is given by
$$\begin{aligned}\displaystyle\alpha=\frac{1}{E_{\mathrm{g}}}\left(1-\frac{{m}^{*}}{{m}_{\mathrm{e}}}\right)^{2}\;,\end{aligned}$$
(32.3)
where me and Eg denote the free electron mass and the energy gap, respectively [32.43].
The scattering mechanisms considered in our analysis are:
  1. 1.

    Ionized impurity

     
  2. 2.

    Polar optical phonon

     
  3. 3.

    Piezoelectric [32.44, 32.45]

     
  4. 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, e14, must be selected. This may be achieved through the transformation suggested by Bykhovski et al. [32.44, 32.45]. 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 steady-state electron transport simulations, while the motion of ten thousand electrons is considered in our transient electron transport simulations. The crystal temperature is set to 300 K and the doping concentration is set to 1017 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, 32.21, 32.22, 32.23, 32.24, 32.29, 32.32, 32.48].

32.1.3 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 32.1. 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, 32.50]; following the lead of Polyakov et al. [32.50], 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. [32.50], 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 Γ point, the first upper conduction band valley minimum also occurring at the Γ point, 1.9 eV above the lowest point in the conduction band, the second upper conduction band valley minima occurring along the symmetry lines between the L and M points, 2.1 eV above the lowest point in the conduction band; see Table 32.2. The other materials considered in this analysis have suitably modified band structures, as described in Table 32.2. We ascribe an effective mass equal to the free electron mass me to all of the upper conduction band valleys, with the exception of the case of InN, the values suggested by Polyakov et al. [32.50] and Hadi et al. [32.49] being employed for the specific case of this material. The nonparabolicity coefficient α 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].
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] and Hadi et al. [32.49])

Parameter

GaN

AlN

InN

Mass density (g ∕ cm3)

6.15

3.23

6.81

Longitudinal sound velocity (cm ∕ s)

6.56 × 105

9.06 × 105

6.24 × 105

Transverse sound velocity (cm ∕ s)

2.68 × 105

3.70 × 105

2.55 × 105

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 (Γ1 valley)

0.20 me

0.48 me

0.04 me

Piezoelectric constant, e14 (C ∕ cm2)

\(\mathrm{3.75\times 10^{-5}}\)

\(\mathrm{9.2\times 10^{-5}}\)

\(\mathrm{3.75\times 10^{-5}}\)

Direct energy gap (eV)

3.39

6.2

0.7

Optical phonon energy (meV)

91.2

99.2

73.0

Intervalley deformation potentials (eV ∕ cm)

109

109

109

Intervalley phonon energies (meV)

91.2

99.2

73.0

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. [32.50] and Lambrecht and Segall [32.51])

  

Valley number

1

2

3

GaN

Valley location

Γ 1

Γ 2

L–M

Valley degeneracy

1

1

6

Effective mass

0.2 me

m e

m e

Intervalley energy separation (eV)

1.9

2.1

Energy gap (eV)

3.39

5.29

5.49

Nonparabolicity (eV−1)

0.189

0.0

0.0

AlN

Valley location

Γ 1

L-M

K

Valley degeneracy

1

6

2

Effective mass

0.48 me

m e

m e

Intervalley energy separation (eV)

0.7

1.0

Energy gap (eV)

6.2

6.9

7.2

Nonparabolicity (eV−1)

0.044

0.0

0.0

InN

Valley location

Γ 1

Γ 2

L–M

Valley degeneracy

1

1

6

Effective mass

0.04 me

0.25 me

m e

Intervalley energy separation (eV)

1.775

2.709

Energy gap (eV)

0.7

2.475

3.409

Nonparabolicity (eV−1)

1.43

0.0

0.0

It should be noted that the energy gap associated with InN has been the subject of some controversy since 2002. 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, 32.28, 32.29, 32.31]; typically, the influence of the energy gap on the electron transport occurs through its impact on the nonparabolicity coefficient α. In 2002, 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 0.7 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 α 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 340 meV above the lowest point in the conduction band for this material. This contrasts rather 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].

32.2 Steady-State and Transient Electron Transport Within Bulk Wurtzite GaN, AlN, and InN

The current interest in the III–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 [32.13, 32.14, 32.15, 32.16, 32.17, 32.18, 32.19, 32.20, 32.21, 32.22, 32.23, 32.24, 32.25, 32.26, 32.27, 32.28, 32.29, 32.30, 32.31, 32.32, 32.33, 32.34, 32.38, 32.39, 32.41, 32.42, 32.48]. 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. 32.2.1, the velocity–field characteristic associated with bulk wurtzite GaN is presented and analyzed. Then, in Sect. 32.2.2, 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. 32.2.3, and a comparison with that corresponding to GaAs presented. In Sect. 32.2.4, 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 Sect. 32.2.5 and Sect. 32.2.6, respectively. Our transient electron transport analysis results are then presented in Sect. 32.2.7. Finally, the conclusions of this electron transport analysis are summarized in Sect. 32.2.8.

32.2.1 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. 32.1 . 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 300 K and the doping concentration to 1017 cm−3. We see that for applied electric fields in excess of 140 kV ∕ cm, the electron drift velocity decreases, eventually saturating at \({\mathrm{1.4\times 10^{7}}}\,{\mathrm{cm/s}}\) for high applied electric fields. By examining the results of our Monte Carlo simulation further, an understanding of this result becomes clear.
Fig. 32.1

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

First, we discuss the results at low applied electric fields, i. e., applied electric fields of less than 30 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 μ in this regime, i. e., a linear low-field electron drift velocity dependence on the applied electric field, i. e., vd = μE, applies in this regime. Examining the distribution function for this regime, we find that it is very similar to the zero-field distribution function with a slight shift in the direction opposite 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 100 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 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: 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.
Fig. 32.2

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 than the thermal energy, \(\frac{3}{2}k_{\mathrm{B}}T\). 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

As the applied electric field 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. 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. 32.1.
Fig. 32.3

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 3

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.

32.2.2 Steady-State Electron Transport: A Comparison of the III–V Nitride Semiconductors with GaAs

Setting the crystal temperature to 300 K and the level of doping to 1017 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. 32.4. 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, \({\mathrm{5.6\times 10^{7}}}\,{\mathrm{cm/s}}\) at an applied electric field of 30 kV ∕ cm. This contrasts with the case of GaN, \({\mathrm{2.9\times 10^{7}}}\,{\mathrm{cm/s}}\) at 140 kV ∕ cm, and that of AlN, \({\mathrm{1.7\times 10^{7}}}\,{\mathrm{cm/s}}\) at 450 kV ∕ cm. For GaAs, the peak electron drift velocity of \({\mathrm{1.6\times 10^{7}}}\,{\mathrm{cm/s}}\) occurs at a much lower applied electric field than that for the III–V nitride semiconductors (only 4 kV ∕ cm).
Fig. 32.4

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)

32.2.3 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 100 to 700 K, in increments of 200 K. The upper limit, 700 K, is chosen as it is the highest operating temperature that may be expected for AlGaN ∕ GaN power devices. To 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 32.5 b 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.
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

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 temperatures at and below room temperature. This property will likely have an impact on high-power device performance.
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

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 \({\mathrm{6.7\times 10^{13}}}\,{\mathrm{s^{-1}}}\) at 100 K to \({\mathrm{8.6\times 10^{13}}}\,{\mathrm{s^{-1}}}\) at 700 K, for high applied electric field strengths. Contrast this with the case of GaAs, where the rate increases from \({\mathrm{4.0\times 10^{12}}}\,{\mathrm{s^{-1}}}\) at 100 K to more than twice that amount at 700 K, \({\mathrm{9.2\times 10^{12}}}\,{\mathrm{s^{-1}}}\), 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.
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 second reason for the variation in temperature dependence of the two materials is the occupancy of the upper valleys, shown in Fig. 32.8. In the case of GaN, the upper valleys begin to become occupied at roughly the same applied electric field strength, 100 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 290 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 700 K, kBT ≅ 60 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.
Fig. 32.8a,b

A comparison of the number of particles in the lowest energy valley of the conduction band, the Γ 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

32.2.4 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. 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. 32.10. Our simulation results suggest that for doping concentrations of less than 1017 cm−3, there is very little effect on the velocity–field characteristic for the case of GaN. However, for doping concentrations above 1017 cm−3, the peak electron drift velocity diminishes considerably, from \({\mathrm{2.9\times 10^{7}}}\,{\mathrm{cm/s}}\) for the case of 1017 cm−3 doping to \({\mathrm{2.0\times 10^{7}}}\,{\mathrm{cm/s}}\) for the case of 1019 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 low-field electron drift mobility is also shown. It is seen that this mobility drops significantly in response to increases in the doping concentration level, from 1200 cm2 ∕  ( V s )  at 1016 cm−3 doping to 400 cm2 ∕  ( V s )  at 1019 cm−3 doping.
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

Fig. 32.10a,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

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 32.10a,b 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 \({\mathrm{1.8\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1016 cm−3 doping to \({\mathrm{0.6\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1019 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 \({\mathrm{1.0\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1016 cm−3 doping to \({\mathrm{0.6\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1019 cm−3 doping. The low-field electron drift mobility also diminishes dramatically with increased doping, dropping from 7800 cm2 ∕  ( V s )  at 1016 cm−3 doping to 2200 cm2 ∕  ( V s )  at 1019 cm−3 doping; see Fig. 32.10a,b.

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 32.11a,b 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. 32.12a,b). 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.
Fig. 32.11a,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

Fig. 32.12a,b

A comparison of the number of particles in the lowest valley of the conduction band, the Γ 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

32.2.5 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. 32.13. 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 \({\mathrm{1.8\times 10^{7}}}\,{\mathrm{cm/s}}\) at 100 K, only decreases to \({\mathrm{1.2\times 10^{7}}}\,{\mathrm{cm/s}}\) at 700 K. Similarly, its saturation electron drift velocity, which is \({\mathrm{1.5\times 10^{7}}}\,{\mathrm{cm/s}}\) at 100 K, only decreases to \({\mathrm{1.0\times 10^{7}}}\,{\mathrm{cm/s}}\) at 700 K. The low-field electron drift mobility associated with AlN also diminishes in response to increases in the crystal temperature, from 375 cm2 ∕  ( V s )  at 100 K to 40 cm2 ∕  ( V s )  at 700 K.
Fig. 32.13

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)

The sensitivity of the velocity–field characteristic associated with AlN to variations in the doping concentration may be examined by considering Fig. 32.14. 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 \({\mathrm{1.7\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1017 cm−3 doping to \({\mathrm{1.3\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1019 cm−3 doping. Similarly, its saturation electron drift velocity drops from \({\mathrm{1.4\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1017 cm−3 doping to \({\mathrm{1.2\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1019 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 140 cm2 ∕  ( V s )  at 1016 cm−3 doping to 100 cm2 ∕  ( V s )  at 1019 cm−3 doping.
Fig. 32.14

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)

32.2.6 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. 32.15. 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 \({\mathrm{6.0\times 10^{7}}}\,{\mathrm{cm/s}}\) at 100 K, only decreases to \({\mathrm{4.2\times 10^{7}}}\,{\mathrm{cm/s}}\) at 700 K. Similarly, its saturation electron drift velocity, which is \({\mathrm{1.5\times 10^{7}}}\,{\mathrm{cm/s}}\) at 100 K, only decreases to \({\mathrm{1.0\times 10^{7}}}\,{\mathrm{cm/s}}\) at 700 K. The low-field electron drift mobility associated with InN also diminishes in response to increases in the crystal temperature, from about 22000 cm2 ∕  ( V s )  at 100 K to below 2300 cm2 ∕  ( V s )  at 700 K.
Fig. 32.15

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

The sensitivity of the velocity–field characteristic associated with InN to variations in the doping concentration may be examined by considering Fig. 32.16. These results suggest a similar robustness to the doping concentration for the case of InN. In particular, it is noted that for doping concentrations below 1017 cm−3, the velocity–field characteristic associated with InN exhibits very little dependence on the doping concentration. When the doping concentration is increased above 1017 cm−3, however, the peak electron drift velocity diminishes. Quantitatively, the peak electron drift velocity decreases from \({\mathrm{5.6\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1017 cm−3 doping to \({\mathrm{4.2\times 10^{7}}}\,{\mathrm{cm/s}}\) at \({\mathrm{5\times 10^{18}}}\,{\mathrm{cm^{-3}}}\) doping; a numerical instability occurs for the case of InN at large doping concentrations, owing to its small electron effective mass, and this prevented the determination of a 1019 cm−3 doping concentration result. The saturation electron drift velocity only drops slightly, however, from \({\mathrm{1.4\times 10^{7}}}\,{\mathrm{cm/s}}\) at 1017 cm−3 doping to \({\mathrm{1.3\times 10^{7}}}\,{\mathrm{cm/s}}\) at \({\mathrm{5\times 10^{18}}}\,{\mathrm{cm^{-3}}}\) doping. The low-field electron drift mobility, however, drops significantly with doping, from 12000 cm2 ∕  ( V s )  at 1016 cm−3 doping to 4900 cm2 ∕  ( V s )  at \({\mathrm{5\times 10^{18}}}\,{\mathrm{cm^{-3}}}\) doping.
Fig. 32.16

The velocity–field characteristic associated with InN 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)

32.2.7 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 electron transport must also be considered when evaluating device performance. Ruch [32.59] 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. Shur and Eastman [32.60] explored the device implications of transient electron transport, and demonstrated that substantial improvements in the device performance can be achieved as a consequence. Heiblum et al. [32.61] 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 the III–V compound semiconductors [32.62, 32.63, 32.64].

Thus far, very little research has been invested into the study of transient electron transport within the III–V nitride semiconductors, GaN, AlN, and InN. Foutz et al. [32.21] examined transient electron transport within both 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 μm, they found that steady-state electron transport is expected to dominate device performance. For devices with smaller dimensions, however, upon the application of a sufficiently high electric field, 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.

Foutz et al. [32.29] performed a subsequent analysis in which the transient electron transport within all of the III–V nitride semiconductors under consideration in this analysis were compared with that which occurs within GaAs. In particular, following the approach of Foutz et al. [32.21], they examined how electrons, initially in thermal equilibrium, respond to the sudden application of a constant electric field. A key result of this study, presented in Fig. 32.17a-d, plots the transient electron drift velocity as a function of the distance displaced since the electric field was initially applied for a number of applied electric field strengths and for each of the materials considered in this analysis. The InN results have been updated owing to the revisions in the material parameter selections corresponding to this particular material; see Figure 32.17a-dc.
Fig. 32.17a–d

The electron drift velocity as a function of the distance displaced since the application of the electric field for various applied electric field strengths, for the cases of (a) GaN, (b) AlN, (c) InN, and (d) GaAs. In all cases, we have assumed an initial zero field electron distribution, a crystal temperature of 300 K, and a doping concentration of 1017 cm−3. (After [32.29] with permission, copyright AIP)

Focusing initially on the case of GaN (Fig. 32.17a-da), we note that the electron drift velocity for the applied electric field strengths 70 kV ∕ cm and 140 kV ∕ cm reaches steady-state very quickly, with little or no velocity overshoot. In contrast, for applied electric field strengths above 140 kV ∕ cm, significant velocity overshoot occurs. This result suggests that in GaN, 140 kV ∕ cm is a critical field for the onset of velocity overshoot effects. As mentioned in Sect. 32.2.2, 140 kV ∕ cm also corresponds to the peak in the velocity-field characteristic associated with GaN; recall Fig. 32.4. Steady-state Monte Carlo simulations suggest that this is the point at which significant upper valley occupation begins to occur; recall Fig. 32.3. This suggests that velocity overshoot is related to the transfer of electrons to the upper valleys. Similar results are found for the other III–V nitride semiconductors, AlN and InN, and GaAs; see Figs. 32.17a-db–d.

We now compare the transient electron transport characteristics for the materials. From Fig. 32.17a-d, 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 to each of the materials, as the transient effects occur over such a disparate range of applied electric field strengths in 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. Figure 32.18 shows a comparison of the velocity overshoot effects amongst the four materials considered in this analysis, i. e., GaN, AlN, InN, and GaAs. It is clear that among the three III–V nitride semiconductors considered, InN exhibits superior transient electron transport characteristics. In particular, InN has the largest overshoot velocity and the distance over which this overshoot occurs, 0.6 μm, is longer than in either GaN or AlN. GaAs exhibits a longer overshoot relaxation distance, approximately 0.7 μm, but the electron drift velocity exhibited by InN is greater than that of GaAs for all distances.
Fig. 32.18

A comparison of the velocity overshoot amongst the III–V nitride semiconductors and GaAs. The applied electric field strength chosen corresponds to twice the critical applied electric field strength at which the peak in the steady-state velocity–field characteristic occurs (Fig. 32.4), i. e., 280 kV ∕ cm for the case of GaN, 900 kV ∕ cm for the case of AlN, 60 kV ∕ cm for the case of InN, and 8 kV ∕ cm for the case of GaAs. (After [32.29] with permission, copyright AIP)

32.2.8 Electron Transport: Conclusions

In this section, steady-state and transient electron transport results, corresponding to the III–V nitride semiconductors, GaN, AlN, and InN, were presented, these results being obtained from our Monte Carlo simulations of the 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 GaN and GaAs. Our simulations showed that GaN is more robust to variations in crystal temperature and doping concentration than GaAs, and an analysis of our Monte Carlo simulation results showed that polar optical phonon scattering plays the dominant role in accounting for these differences in behavior. This analysis was also performed for the other III–V nitride semiconductors considered in this analysis – AlN and InN – and similar results were obtained. Finally, we presented some key transient electron transport results. These results indicated that the transient electron transport that occurs within InN is the most pronounced of all of the materials under consideration in this review (GaN, AlN, InN, and GaAs).

32.3 Electron Transport Within III–V Nitride Semiconductors: A Review

Pioneering investigations into the material properties of the III–V nitride semiconductors, GaN, AlN, and InN, were performed during the earlier half of the 20th Century [32.65, 32.66, 32.67]. The III–V nitride semiconductor materials available at the time, small crystals and powders, were of poor quality, and completely unsuitable for device applications. Thus, it was not until the late 1960s, when Maruska and Tietjen [32.68] employed chemical vapor deposition to fabricate GaN, that interest in the III–V nitride semiconductors experienced a renaissance. Since that time, interest in the III–V nitride semiconductors has been growing, the material properties of these semiconductors improving considerably over the years. As a result of this research effort, there are currently a number of commercial devices available that employ the III–V nitride semiconductors. More III–V nitride semiconductor-based device applications are currently under development, and these should become available in the near future.

In this section, we present a brief overview of the III–V nitride semiconductor electron transport field. We start with a survey describing the evolution of the field. In particular, the sequence of critical developments that have occurred that contribute to our current understanding of the electron transport mechanisms within the III–V nitride semiconductors, GaN, AlN, and InN, is chronicled. Then, some of the more recent literature related to the electron transport within the III–V nitride semiconductors, GaN, AlN, and InN, is presented, with particular emphasis being placed on the most recent developments in the field that have occurred in the 21st Century, and how such developments have shaped our understanding of the electron transport mechanisms within these materials. Finally, frontiers for further research and investigation are presented.

This section is organized in the following manner. In Sect. 32.3.1, we present a brief survey describing the evolution of the field. Then, in Sect. 32.3.2, the developments that have occurred in the 21st Century are discussed. Finally, frontiers for further research and investigation are presented in Sect. 32.3.3.

32.3.1 Evolution of the Field

The favorable electron transport characteristics of the III–V nitride semiconductors, GaN, AlN, and InN, have long been recognized. As early as the 1970s, Littlejohn et al. [32.13] pointed out that the large polar optical phonon energy characteristic of GaN, in conjunction with its large intervalley energy separation, suggests a high saturation electron drift velocity for this material. As high-frequency electron device performance is, to a large degree, determined by this saturation electron drift velocity [32.14], the recognition of this fact ignited enhanced interest in this material and its III–V nitride semiconductor compatriots, AlN and InN. This enhanced interest, and the developments which have transpired as a result of it, are responsible for the III–V nitride semiconductor industry of today.

In 1975, Littlejohn et al. [32.13] were the first to report results obtained from semi-classical Monte Carlo simulations of the steady-state electron transport within bulk wurtzite GaN. A one-valley model for the conduction band was adopted in their analysis. Steady-state electron transport, for both parabolic and nonparabolic band structures, was considered in their analysis, nonparabolicity being treated through the application of the Kane model [32.43]. The primary focus of their investigation was the determination of the velocity-field characteristic associated with 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. 1.

    Ionized impurity

     
  2. 2.

    Polar optical phonon

     
  3. 3.

    Piezoelectric

     
  4. 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 about \({\mathrm{2.5\times 10^{7}}}\,{\mathrm{cm/s}}\) for the case of high applied electric fields. In contrast, for the case of the nonparabolic band, and in the absence of ionized impurities, a region of negative differential mobility was found, the electron drift velocity achieving a maximum of about \({\mathrm{2\times 10^{7}}}\,{\mathrm{cm/s}}\) at an applied electric field strength of about 100 kV ∕ cm, with 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. [32.13].

In 1993, Gelmont et al. [32.16] reported on ensemble semi-classical two-valley Monte Carlo simulations of the electron transport within bulk wurtzite GaN, this analysis improving upon the analysis of Littlejohn et al. [32.13] by incorporating intervalley scattering into the simulations. They found that the negative differential mobility found in bulk wurtzite GaN is much more pronounced than that found by Littlejohn et al. [32.13], and that intervalley transitions are responsible for this. For a doping concentration of 1017 cm−3, Gelmont et al. [32.16] demonstrated that the electron drift velocity achieves a peak value of about \({\mathrm{2.8\times 10^{7}}}\,{\mathrm{cm/s}}\) at an applied electric field of about 140 kV ∕ cm. The impact of intervalley 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 Mansour et al. [32.18] reported the use of such an approach in order to determine how the crystal temperature influences the velocity-field characteristic associated with bulk wurtzite GaN. Later that year, Kolník et al. [32.19] reported on employing full-band Monte Carlo simulations of the electron transport within bulk wurtzite GaN and bulk zinc blende GaN, finding that bulk zinc blende GaN exhibits a much higher low-field electron drift mobility than bulk wurtzite GaN. The peak electron drift velocity corresponding to bulk zinc blende GaN was found to be only marginally greater than that exhibited by bulk wurtzite GaN, however. In 1997, Bhapkar and Shur [32.22] 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. In 1998, Albrecht et al. [32.27] reported on employing ensemble semi-classical five-valley Monte Carlo simulations of the electron transport within bulk wurtzite GaN, with the aim of determining elementary analytical expressions for a number of electron transport metrics corresponding to bulk wurtzite GaN, for the purposes of device modeling.

Electron transport within the other III–V nitride semiconductors, AlN and InN, has also been studied using ensemble semi-classical Monte Carlo simulations of the electron transport. In particular, by employing ensemble semi-classical three-valley Monte Carlo simulations, the velocity–field characteristic associated with bulk wurtzite AlN was studied and reported by O’Leary et al. [32.24] in 1998. They found that AlN exhibits the lowest peak and saturation electron drift velocities of the III–V nitride semiconductors considered in this analysis. Similar simulations of the electron transport within bulk wurtzite AlN were also reported by Albrecht et al. [32.25] in 1998. The results of O’Leary et al. [32.24] and Albrecht et al. [32.25] were found to be quite similar. The first known simulation of the electron transport within bulk wurtzite InN was the semi-classical three-valley Monte Carlo simulation of O’Leary et al. [32.23], reported in 1998. InN was demonstrated to have the highest peak and saturation electron drift velocities of the III–V nitride semiconductors. The subsequent ensemble full-band Monte Carlo simulations of Bellotti et al. [32.28], reported in 1999, produced results similar to those of O’Leary et al. [32.23].

The first known study of transient electron transport within the III–V nitride semiconductors was that performed by Foutz et al. [32.21], 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 that occurs within these materials was 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 the smaller effective mass of the electrons within the zinc blende phase of this material. A comparison with the transient electron transport that occurs within GaAs was made. Using the results from 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: wurtzite GaN, zinc blende GaN, and GaAs. For distances in excess of 0.1 μm, both phases of GaN were shown to exhibit superior performance (reduced transit time) when contrasted with that associated with GaAs.

A more general analysis, in which transient electron transport within GaN, AlN, and InN was studied, was performed by Foutz et al. [32.29], and reported in 1999. As with their previous study, Foutz et al. [32.29] determined how electrons, initially in thermal equilibrium, respond to the sudden application of a constant electric field. For GaN, AlN, InN, and GaAs, it was found that the electron drift velocity overshoot only occurs when the applied electric field exceeds a certain critical applied electric field strength unique to each material. The critical applied electric field strengths, 140 kV ∕ cm for the case of wurtzite GaN, 450 kV ∕ cm for the case of AlN, 65 kV ∕ cm for the case of InN (this was the critical applied electric field strength corresponding to our previous InN parameter selections [32.29]), and 4 kV ∕ cm for the case of GaAs, were shown to correspond to the peak electron drift velocity in the velocity-field characteristic associated with each of these materials; recall Fig. 32.4. It was found that InN exhibits the highest peak overshoot velocity, and that this overshoot lasts over prolonged distances compared with AlN, InN, and GaAs. A comparison with the results of experiment was 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. In 1975, for example, Ferry [32.14] reported on the determination of the velocity-field characteristic associated with wurtzite GaN using a displaced Maxwellian distribution function approach. For high applied electric fields, Ferry [32.14] found that the electron drift velocity associated with GaN monotonically increases with the applied electric field strength (it does not saturate), reaching a value of about \({\mathrm{2.5\times 10^{7}}}\,{\mathrm{cm/s}}\) at an applied electric field strength of 300 kV ∕ cm. The device implications of this result were further explored by Das and Ferry [32.15]. In 1994, Chin et al. [32.17] reported on a detailed study of the dependence of the low-field electron drift mobilities associated with the III–V nitride semiconductors, GaN, AlN, and InN, on crystal temperature and doping concentration. An analytical expression for the low-field electron drift mobility μ determined using a variational principle, was used for the purposes of this analysis. The results obtained were contrasted with those from experiment. A subsequent mobility study was reported in 1997 by Look et al. [32.38]. Then, in 1998, Weimann et al. [32.26] reported on a model for determining how the scattering of electrons by the threading dislocations within bulk wurtzite GaN influence 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 were not taken into account in the Monte Carlo simulations of the electron transport within the III–V nitride semiconductors, GaN, AlN, and InN.

While the negative differential mobility exhibited by the velocity-field characteristics associated with the III–V nitride semiconductors, GaN, AlN, and InN, is widely attributed to intervalley transitions, and while direct experimental evidence confirming this has been presented [32.69], Krishnamurthy et al. [32.34] suggest that the inflection points in the bands located in the vicinity of the Γ valley are primarily responsible for the negative differential mobility exhibited by wurtzite GaN instead. The relative importance of these two mechanisms (intervalley transitions and inflection point considerations) were evaluated by Krishnamurthy et al. [32.34], for both bulk wurtzite GaN and an AlGaN alloy.

32.3.2 Developments in the 21st Century

Thus far, the electron transport results that have been presented have focused on developments in our understanding of the nature of the electron transport within the III–V nitride semiconductors, GaN, AlN, and InN that were acquired in the latter part of the 20th Century. There have been a number of interesting more recent developments in the study of the electron transport within the III–V nitride semiconductors which have influenced the direction of thought in this field. On the experimental front, in 2000 Wraback et al. [32.33] reported on the use of a femtosecond optically detected time-of-flight experimental technique in order to experimentally determine the velocity–field characteristic associated with bulk wurtzite GaN. They found that the peak electron drift velocity, \({\mathrm{1.9\times 10^{7}}}\,{\mathrm{cm/s}}\), is achieved at an applied electric field strength of 225 kV ∕ cm. No discernible negative differential mobility was observed. Wraback et al. [32.33] suggested that the large defect density characteristic of the GaN samples they employed, which were not taken into account in Monte Carlo simulations of the electron transport within this material, accounts for the difference between this experimental result and that obtained using simulation. They also suggested that decreasing the intervalley energy separation from about 2 eV to 340 meV, as suggested by the experimental results of Brazel et al. [32.58], may also account for these observations.

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 de-embedding of the parasitics from the experimental measurements so that the true intrinsic saturation electron drift velocity may be obtained. Eastman et al. [32.70] present experimental evidence which suggests that the saturation electron drift velocity within bulk wurtzite GaN is about \({\mathrm{1.2\times 10^{7}}}{-}{\mathrm{1.3\times 10^{7}}}\,{\mathrm{cm/s}}\). A more recent report, by Oxley and Uren [32.71], suggests a value of \({\mathrm{1.1\times 10^{7}}}\,{\mathrm{cm/s}}\). The role of self-heating was also probed by Oxley and Uren [32.71] and shown to be relatively insignificant. A completely satisfactory explanation for the discrepancy between these results and those from the Monte Carlo simulations has yet to be provided.

Wraback et al. [32.72] performed a subsequent study on the transient electron transport within wurtzite GaN. In particular, using their femtosecond optically detected time-of-flight experimental technique in order to experimentally determine the velocity overshoot that occurs within bulk wurtzite GaN, they observed substantial velocity overshoot within this material. In particular, a peak transient electron drift velocity of 7.25 × 107cm/s was observed within the first 200 fs after photoexcitation for an applied electric field strength of 320 kV ∕ cm. These experimental results were shown to be consistent with the theoretical predictions of Foutz et al. [32.29].

On the theoretical front, there have been a number of recent developments. In 2001, O’Leary et al. [32.30] presented an elementary, one-dimensional analytical model for the electron transport within the III–V compound semiconductors, and applied it to the cases of wurtzite GaN and GaAs. The predictions of this analytical model were compared with those of Monte Carlo simulations and were found to be in satisfactory agreement. Hot-electron energy relaxation times within the III–V nitride semiconductors were studied by Matulionis et al. [32.73] and reported in 2002. Bulutay et al. [32.74] studied the electron momentum and energy relaxation times within the III–V nitride semiconductors and reported the results of this study in 2003. It is particularly interesting to note that their arguments add considerable credence to the earlier inflection point argument of Krishnamurthy et al. [32.34]. In 2004, Brazis and Raguotis [32.75] reported on the results of a Monte Carlo study involving additional phonon modes and a smaller intervalley energy separation for bulk wurtzite GaN. Their results were found to be much closer to the experimental results of Wraback et al. [32.33] than those found previously.

The influence of hot-phonons on the electron transport mechanisms within the III–V nitride semiconductors, GaN, AlN, and InN, has been the focus of considerable investigation. In particular, in 2004 Silva and Nascimento [32.76], Gökden [32.77], and Ridley et al. [32.78], to name just three, presented results related to this research focus. These results suggest that hot-phonon effects play a significant role in influencing the nature of the electron transport within the III–V nitride semiconductors, GaN, AlN, and InN. In particular, Ridley et al. [32.78] point out that the saturation electron drift velocity and the applied electric field strength at which the peak in the velocity–field characteristic occurs are both influenced by hot-phonon effects. The role that hot-phonons play in influencing device performance was studied by Matulionis and Liberis [32.79]. Research into the role that hot-phonons play in influencing the electron transport mechanisms within the III–V nitride semiconductors, GaN, AlN, and InN, seems likely to continue into the foreseeable future.

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 GaN. In 2011, Ilgaz et al. [32.80] studied the energy relaxation of hot electrons within AlGaN/GaN/GaN heterostructures. Then, in 2012, Naylor et al. [32.81] examined the steady-state and transient electron transport that occurs within bulk wurtzite GaN using an analytical band structure that more accurately reflects the nature of the actual bandstructure. In 2012, Naylor et al. [32.82] also examined the electron transport that occurs within dilute GaN x As1−x samples. In 2013, Bellotti et al. [32.83] employed a full-band model in order to determine the velocity-field characteristics associated with AlGaN alloys. Also in 2013, Dasgupta et al. [32.84] estimated the hot electron relaxation time associated with wurtzite GaN using a series of electrical measurements. In 2013, Zhang et al. [32.85] 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 same situation applies for the cases of AlN and InN, and alloys of these materials [32.50, 32.86, 32.87, 32.88, 32.89, 32.90, 32.91, 32.92, 32.93, 32.94, 32.95]. Further information, regarding our current understanding of the nature of the electron transport within these materials, may be found in the recently published review articles of Hadi et al. [32.96] and Siddiqua et al. [32.97].

32.3.3 Future Perspectives

It is clear that our understanding of the electron transport within the III–V nitride semiconductors, GaN, AlN, and InN, is, at present at least, in a state of flux. A complete understanding of the electron transport mechanisms within 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 principal 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 III–V nitride semiconductors, GaN, AlN, and InN, remains a key source of ambiguity in the analysis of the electron transport with these materials [32.32]. Even for bulk wurtzite GaN, the most studied of the III–V nitride semiconductors considered in this analysis, uncertainty in the band structure remains an issue [32.58]. The energy gap associated with InN and the effective mass associated with this material continue to fuel debate; see, for example, Davydov et al. [32.55], Wu et al. [32.56], and Matsuoka et al. [32.57]. Variations in the experimentally determined energy gap associated with InN, observed from sample to sample, further confound matters. Shubina et al. [32.98], for example, suggested that nonstoichiometry within InN may be responsible for these variations in the energy gap. Further research will have to be performed in order to confirm this. Given this uncertainty in the band structures associated with the III–V nitride semiconductors, it is clear that new simulations of the electron transport will have to be performed once researchers have settled on the appropriate band structures. We thus view our present results as a baseline, the sensitivity analysis of O’Leary et al. [32.32] providing some insights into how variations in the band structures will impact upon the results.

Uncertainty in the underlying physics is considerable. The source of the negative differential mobility remains a matter to be resolved. The presence of hot-phonons within these materials, and how such phonons impact upon the electron transport mechanisms within these materials, 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 III–V nitride semiconductor-based devices to be properly designed.

32.4 Conclusions

In this chapter, we reviewed analyses of the electron transport within the III–V nitride semiconductors GaN, AlN, and InN. In particular, we have discussed the evolution of the field, surveyed the more recent literature, and presented frontiers for further investigation and analysis. In order to narrow the scope of this review, we focused on the electron transport within bulk wurtzite GaN, AlN, and InN for the purposes of this paper. 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 III–V nitride semiconductor orthodoxy.

We began our review with the Boltzmann transport equation, since this equation underlies most analyses 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 parameters, corresponding to bulk wurtzite GaN, AlN, and InN, were then presented. We then used 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 presented some recent developments on the electron transport within these materials, and pointed to fertile frontiers for further research and investigation.

Notes

Acknowledgements

The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada. The work at Rensselaer Polytechnic Institute (M. S. Shur) was supported by the US Army Cooperative Research Agreement (Project Monitor Dr. Meredith Reed).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Stephen K. O’Leary
    • 1
  • Poppy Siddiqua
    • 1
  • Walid A. Hadi
    • 2
  • Brian E. Foutz
    • 3
  • Michael S. Shur
    • 4
  • Lester F. Eastman
    • 5
  1. 1.School of EngineeringUniversity of British ColumbiaKelownaCanada
  2. 2.Dept. of Electrical and Computer EngineeringUniversity of WindsorWindsorCanada
  3. 3.CharlottesvilleUSA
  4. 4.Dept. of Physics, Applied Physics, and AstronomyRensselaer Polytechnic InstituteNew YorkUSA
  5. 5.Dept. of Electrical and Computer EngineeringCornell UniversityIthacaUSA

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