Download Steady-state and transient electron transport within the wide energy

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