Electron transport within the wurtzite and zinc-blende phases of gallium nitride and indium nitride

Abstract

Wide energy gap semiconductors are broadly recognized as promising materials for novel electronic and opto-electronic device applications. As informed device design requires a firm grasp on the material properties of the underlying electronic materials, the electron transport that occurs within the wide energy gap semiconductors has been the focus of considerable study over the years. In an effort to provide some perspective on this rapidly evolving and burgeoning field of research, we review analyzes of the electron transport within some wide energy gap semiconductors of current interest in this paper. In order to narrow the scope of this review, we will primarily focus on the electron transport that occurs within the wurtzite and zinc-blende phases of gallium nitride and indium nitride in this review, these materials being of great current interest to the wide energy gap semiconductor community; indium nitride, while not a wide energy gap semiconductor in of itself, is included as it is often alloyed with other wide energy gap semiconductors, the resultant alloy often being a wide energy gap semiconductor itself. The electron transport that occurs within zinc-blende gallium arsenide will also be considered, albeit primarily for bench-marking purposes. 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 wide energy gap semiconductor orthodoxy. A brief tutorial on the Monte Carlo electron transport simulation approach, this approach being used to generate the results presented herein, will also be provided. Steady-state and transient electron transport results are presented. The evolution of the field, a survey of the current literature, and some applications for the results presented herein, will also be featured. We conclude our review by presenting some recent developments on the electron transport within these materials. This review is the latest in a series of reviews that have been published on the electron transport processes that occur within the class of wide energy semiconductor materials. The results and references have been updated to include the latest developments in this rapidly evolving field of study.

Appendices

Appendix: Further details related to our Monte Carlo algorithm

We now provide further details of the semi-classical Monte Carlo algorithm employed for the purposes of our simulations of the electron transport within the III–V nitride semiconductors, GaN and InN. Initially, we overview a more detailed flow chart corresponding to our approach. Then, details of some of the trickier parts of our Monte Carlo algorithm will be discussed. The generation of the free-flight time will then be covered and then the selection of the scattering event (after a free-flight) will be described.

1.1 Flow chart for our Monte Carlo algorithm

A more detailed flow chart for our Monte Carlo algorithm is shown in Fig. 49. This flow chart provides a detailed description of how the dynamics of the electrons are handled, as well as how the statistics are kept during the simulation.

Fig. 49

(Copyright permission was obtained from Springer)

A more complete flowchart for our Monte Carlo algorithm used for simulating electron transport within the III–V nitride semiconductors, GaN and InN. This figure is the same as that depicted in Fig. A1 of O’Leary et al. [129]

Fig. 50

(Copyright permission was obtained from Springer)

The scattering mechanism selection process. This figure has been modified from Fig. A2 of O’Leary et al. [129]. The online version of this figure is depicted in color

When the simulation initializes, it reads the input file and sets the simulation parameters. Next, the initial electron distribution is determined. During this stage, each electron in the simulation is given an initial wave-vector in accordance with a Maxwell-Boltzmann distribution. At the same time, a rejection technique is used in order to ensure that the number of electrons in any given region of $\vec {k}$-space never exceeds the Fermi-Dirac limit. This technique provides a close approximation to an initial Fermi-Dirac distribution.

Next, the electric field is set and the scattering rate tables are initialized. The time-step is set to zero and then a loop is entered which moves each particle through free-flights and scattering events until the end of the time-step is reached. After all of the particles are moved, macroscopic quantities, such as the electron drift velocity, are calculated over the distribution and stored in temporary arrays. At the end of the simulation, the accumulated statistics are output to a file. In the next sections, details of some of these steps are provided.

1.2 Generation of the free-flight times

The electron energy and its wave-vector, $\vec {k}$, determine the probability that this electron will scatter by means of any of the aforementioned scattering processes. In between each scattering event, the electron’s motion is determined through semi-classical physics, i.e., Eqs. (4) and (5). The amount of time between each scattering event is determined statistically, based on the total scattering rate,

$$\begin{aligned} \lambda \left( \vec {k}\right) = \sum _i \lambda _i \left( \vec {k}\right) , \end{aligned}$$

(9)

which is just the sum of the individual scattering rates corresponding to each scattering mechanism. The statistically determined time between scattering events is known as the free-flight time, $t_{f}$.

Generating a proper distribution of free-flight times is essential in order to obtain correct simulation results. A number of methods, used for the purposes of generating these free-flight times, have already been studied in detail [167]. A derivation of the algorithm used in our simulations of the electron transport within the III–V nitride semiconductors, GaN and InN, will be provided here.

We first note that the probability distribution, $P\left( t\right)$, for the free-flight time, of length t, is just the probability that an electron survives without a collision to time t multiplied by the probability of a collision within a small interval, dt, around t. The probability of a collision within dt of t is simply the product of the scattering rate at time t and dt. The first part of the distribution, the probability that the electron survives to time t without a collision, can be found by assuming that the scattering processes are Poisson in nature. For a Poisson process, the probability of no scattering event for any interval, $\delta t = t_2 - t_1$, is $P\left( N = 0\right) = \exp \left( -\lambda \delta t\right)$. If the scattering rate is constant, this would be the distribution we require. However, the scattering rate changes with time as the electron drifts under the action of the applied electric field. To take into account the fact that the scattering rates change with time, we divide the interval, $\left[ 0, t\right]$, into i small intervals. The probability, $p_i$, that no scattering event occurs, in interval i, is

$$\begin{aligned} p_i = \exp \left( -\lambda _i \delta t\right) , \end{aligned}$$

(10)

where $\lambda _i$ is the scattering rate during interval i and $\delta t$ is the duration of interval i. The probability that no scattering event occurs in any of the i intervals, 0 through N, is the product of the probabilities for each interval, i.e.,

$$\begin{aligned} p\left( t\right)= & {} \prod \nolimits _{i = 0}^N \exp \left[ -\lambda _i \delta t\right] , \nonumber \\= & {} \exp \left( -\sum \nolimits _{i = 0}^N \lambda _i \delta t\right) . \end{aligned}$$

(11)

Letting the intervals become very small, i.e., $\delta t \rightarrow dt$, the sum of Eq. (11) reduces to an integral, i.e.,

$$\begin{aligned} p\left( t\right) = \exp \left[ -\int _0^t \lambda \left( \vec {k}\left( t'\right) \right) dt'\right] . \end{aligned}$$

(12)

The free-flight time distribution then becomes the scattering rate multipled by $p\left( t\right)$, i.e.,

$$\begin{aligned} P\left( t\right) = \lambda \left( \vec {k}\left( t\right) \right) \exp \left[ -\int _0^t \lambda \left( \vec {k}\left( t'\right) \right) dt'\right] . \end{aligned}$$

(13)

In order to generate random free-flight times, with a given $P\left( t\right)$, we apply a direct method [76]. In particular, we select a random number, r, with a uniform distribution between [0, 1], and set it equal to the integrated probability distribution function, i.e.,

$$\begin{aligned} r = \int _0^tP\left( t'\right) dt'. \end{aligned}$$

(14)

Substituting Eq. (13) into Eq. (14), and solving the integral, yields

$$\begin{aligned} r = 1 - \exp \left[ -\int _0^{t} \lambda \left( \vec {k}\left( t'\right) \right) dt'\right] . \end{aligned}$$

(15)

Thus, we conclude that

$$\begin{aligned} -\ln \left( 1-r\right) = \int _0^{t} \lambda \left( \vec {k}\left( t'\right) \right) dt'. \end{aligned}$$

(16)

A time, t, must be found which satisfies the above equation for the random number, r.

One difficulty in evaluating the integral over $\lambda$ is that it is a complicated function of t. This problem can be overcome by introducing an artificial scattering mechanism, known as the self-scattering mechanism, $\lambda _0\left( \vec {k}\right)$. This new mechanism makes the total scattering rate constant over some interval of time, i.e.,

$$\begin{aligned} \Gamma = \lambda _0\left( \vec {k}\right) + \lambda \left( \vec {k}\right) . \end{aligned}$$

(17)

Yorston [167] discusses several algorithms for generating the free-flight times using this self-scattering concept. One of the the most efficient algorithms, and the one employed in our Monte Carlo simulations of the electron transport within the III–V nitride semiconductors, GaN and InN, is the constant-time method. In this method, a fixed time, $t_{\text{inc}}$, is chosen, and the integral in Eq. (16) is carried out over intervals of length $t_{\text{inc}}$. In each interval, a self-scattering mechanism, $\lambda _0\left( \vec {k}\right)$, is added in order to make the total scattering rate constant and greater than $\lambda \left( \vec {k}\right)$ during the $t_{\text{inc}}$ interval. Fig. 50 illustrates this algorithm. The free-flight time is chosen when the total integral satisfies Eq. (16). At that time, $\lambda _0\left( \vec {k}\right)$ and each $\lambda _i\left( \vec {k}\right)$ are used to determine the choice of scattering event.

In the case a self-scattering mechanism is chosen, special treatment is necessary. The integral for the next free-flight time must continue where the previous one left off. In the example shown in Fig. 50, the integral from t to $4 t_{\text{inc}}$ is first used, then that from $4 t_{\text{inc}}$ to $5 t_{\text{inc}}$ is used, and so on.

1.3 Choice of scattering event

Once the electron finishes its free-flight, it scatters. The choice of the scattering event is also made with a random number. This time, the probability that a particular scattering event is selected is directly proportional to the scattering rates corresponding to that particular mechanism. A random number, r, uniformly distributed between [0, 1], is chosen, and the scattering mechanism, i, which satisfies

$$\begin{aligned} S_i< r < S_{i+1}, \end{aligned}$$

(18)

where

$$\begin{aligned} S_i = \frac{\sum _{j=0}^i \lambda _j\left( \vec {k}\right) }{\Gamma }, \end{aligned}$$

(19)

is selected, where

$$\begin{aligned} \Gamma = \sum _{i} \lambda _i\left( \vec {k}\right) . \end{aligned}$$

(20)

Once the scattering mechanism is selected, the final wave-vector of the electron must be chosen. This selection must, of course, obey conservation of energy. With this requirement, there exists a sphere in $\vec {k}$-space into which the electron is allowed to scatter. Therefore, by determining the angle (azimuthal and polar) from the electron’s original direction, we may uniquely select the final wave-vector for the electron, and at the same time select the phonon with which the electron is scattering, in order to obey conservation of momentum considerations. For all the scattering mechanisms selected in our Monte Carlo approach, the selection of the azimuthal angle is done with a uniform distribution, i.e., there is no preference in terms of the azimuthal angle. However, many of the scattering mechanisms have a preference with the polar angle. For each of the scattering mechanisms in the Monte Carlo approach, the dependence of the scattering rate with the polar angle is known, i.e.,

$$\begin{aligned} \lambda _i\left( \vec {k}\right) = \int _0^{2\pi }P_i\left( \theta , \vec {k}\right) d\theta . \end{aligned}$$

(21)

There are three different techniques available for converting random numbers with a uniform distribution into one with an arbitrary distribution. These are the direct, rejection, and combined techniques, which are all described by Jacoboni and Lugli [76]. For most of the scattering mechanisms used in our Monte Carlo approach, the rejection technique is used to determine the polar angles. However, some of the most important mechanisms are handled differently. For polar optical phonon and piezoelectric scattering, a combined technique is used. For ionized impurity scattering at low energies, when non-parabolicity can be ignored, the direct technique is used. In other cases, the rejection technique is used, except when the distribution is highly peaked, in which case a combined technique is used.

The simulation continues, moving the electron through each time-step until a special time-step is reached, known as the collection time. After this special time-step, the macroscopic averages, which are stored in temporary arrays, are averaged and stored in permanent arrays. Each average is simply the average over all of the electrons in the simulation. For example, for the electron drift velocity,

$$\begin{aligned} \bar{v}\left( t\right) = \frac{\Sigma v_i\left( t\right) }{N}, \end{aligned}$$

(22)

where N denotes the total number of electrons. After each collection time, the scattering rates tables are also recalculated. This occurs because some of the scattering rates, i.e., polar optical phonon, ionized impurity, and piezoelectric, are a function of the electron temperature, which changes throughout the simulation. If the simulation requires that the applied electric field strength to be updated, then it is updated after every fourth collection time (this number can be adjusted). The average from that fourth collection time is assumed to be in steady-state and is associated with the electric field during that interval. At the end of the simulation, the quantities stored in the permanent arrays are written to an output file.

1.4 Monte Carlo codes available on the internet

A variety of Monte Carlo codes, for the purposes of simulating the steady-state and transient electron transport within bulk semiconductors, are available on the internet. For example, SDemon is available at: https://www.nanohub.org/simulation_tools/sdemon_tool_information

For a description of the theoretical basis of SDemon and its implementation, please consult, M. A. Stettler, “Monte Carlo Studies of Electron Transport in Silicon Bipolar Transistors,” MSEE Thesis, Purdue University, West Lafayette, Indianna, December 1990 and the SDemon User’s Manual. The program is written in Fortran 77. The program author was M. A. Stettler of Purdue University.

Note to Reader

Many of the results presented herein, and portions of the text, are borrowed from our previous review articles, i.e., “Steady-state and transient electron transport within the III–V nitride semiconductors, GaN, AlN, and InN: a review,” which was published in the Journal of Materials Science: Materials in Electronics in 2006 [129], “Steady-state and transient electron transport within the wide energy gap compound semiconductors gallium nitride and zinc oxide: an updated and critical review,” which was published in the Journal of Materials Science: Materials in Electronics in 2014 [142], and “A 2015 perspective on the nature of the steady-state and transient electron transport within the wurtzite phases of gallium nitride, indium nitride, and zinc oxide: a critical and retrospective review,” which was published in the Journal of Materials Science: Materials in Electronics in 2015 [143]. Copyright permission was obtained from Springer. Some results and portions of the text were also borrowed from three other articles of ours previously published in the Journal of Applied Physics, namely, “Transient electron transport in wurtzite GaN, InN, and AlN” [120], “The steady-state and transient electron transport within bulk zinc-blende indium nitride: the impact of crystal temperature and doping concentration variations” [155], and “The sensitivity of the electron transport within bulk zinc-blende gallium nitride to variations in the crystal temperature, the doping concentration, and the non-parabolicity coefficient associated with the lowest energy conduction band valley” [156]. Copyright permission was obtained from the American Institute of Physics.