APPLIED PHYSICS LETTERS 87, 222103 共2005兲
Steady-state and transient electron transport within bulk wurtzite indium
nitride: An updated semiclassical three-valley Monte Carlo simulation
analysis
Stephen K. O’Learya兲
Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
Brian E. Foutzb兲
School of Electrical Engineering, Cornell University, Ithaca, New York 14853
Michael S. Shur
Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy,
New York 12180-3590
Lester F. Eastman
School of Electrical Engineering, Cornell University, Ithaca, New York 14853
共Received 11 August 2005; accepted 12 October 2005; published online 21 November 2005兲
Recent experimentation, performed on bulk wurtzite InN, suggests that the energy gap, the effective
mass of the electrons in the lowest-energy valley, and the nonparabolicity coefficient of the
lowest-energy valley are not as originally believed for this material. Using a semiclassical
three-valley Monte Carlo simulation approach, we analyze the steady-state and transient electron
transport that occurs within bulk wurtzite InN using a revised set of material parameters, this revised
set of parameters taking into account this recently observed phenomenology. We find that the peak
electron drift velocity is considerably greater than that found previously. The impact that this revised
set of parameters has upon the transient electron transport is also found to be significant. © 2005
American Institute of Physics. 关DOI: 10.1063/1.2135876兴
For some time now, the III-V nitride semiconductors,
gallium nitride 共GaN兲, aluminum nitride 共AlN兲, and indium
nitride 共InN兲, have been recognized as promising materials
for novel electronic and optoelectronic device applications.
While initial interest in the III-V nitride semiconductors primarily centered upon GaN and AlN, in more recent years,
InN has also become a focus for attention.1,2 The present
interest in InN stems, in large measure, from its superb electron transport characteristics.3–7 In addition, while initial experimental results suggested a room-temperature energy gap
of 1.89 eV for the case of bulk wurtzite InN,8 more recent
experimental measurements have instead suggested an energy gap value between 0.7 and 1.0 eV for this material,9–11
this realization further fueling interest in InN.
Monte Carlo simulations of the steady-state electron
transport that occurs within bulk wurtzite InN were initially
performed by O’Leary et al.3 Using a semiclassical threevalley Monte Carlo simulation approach, they found that for
a crystal temperature of 300 K and a doping concentration of
1017 cm−3, InN exhibits an extremely high peak electron drift
velocity, about 4.3⫻ 107 cm/ s, and a saturation electron drift
velocity, about 2.5⫻ 107 cm/ s, which is comparable with
that which occurs within GaN. Similar InN steady-state
Monte Carlo simulation results were found by Bellotti et al.4
The transient electron transport that occurs within bulk
wurtzite InN was first explored through Monte Carlo simulations by Foutz et al.5 They found that InN exhibits the
highest overshoot electron drift velocity of the III-V nitride
a兲
Author to whom correspondence should be addressed; electronic mail:
stephen.oleary@uregina.ca
Present address: Cadence Design Systems, 6210 Old Dobbin Rd., Columbia, MD 21045.
b兲
semiconductors, and that the distance over which this overshoot occurs is greater than that of GaN and AlN.
The electron transport simulations of O’Leary et al.,3
Bellotti et al.,4 and Foutz et al.5 were performed assuming an
energy gap value of about 1.89 eV for the case of bulk
wurtzite InN. If the energy gap associated with InN is instead
between 0.7 and 1.0 eV, as suggested by recent experimental
results,9–11 then the nonparabolicity coefficient associated
with the lowest-energy valley, ␣, could be much greater than
initially anticipated. Recent experimental analyses performed
on InN, such as that of Arnaudov et al.,12 have also suggested that the effective mass of the electrons in the lowestenergy valley, m*, is much lower than initially anticipated.
These two factors play critical roles in influencing the nature
of the electron transport that occurs within the III-V nitride
semiconductors.13 Accordingly, a reanalysis of the steadystate and transient electron transport that occurs within bulk
wurtzite InN is called for. Despite this realization, many recent Monte Carlo analyses of the electron transport within
the III-V nitride semiconductors, such as that of Starikov et
al.,14 employ traditional bulk wurtzite InN parameter selections; it should be mentioned, however, that the Monte Carlo
simulations of the electron transport within InN, reported by
Liang et al.6 in 2004 and Tsen et al.7 in 2005, do employ
revised band structure parameters, but focus on the electron
transport solely within the lowest-energy valley. In this letter,
we study the electron transport within bulk wurtzite InN employing an ensemble semiclassical three-valley Monte Carlo
simulation approach, focusing on the determination of the
steady-state velocity-field characteristic associated with this
material as well as its transient electron transport response.
For the purposes of this analysis of the electron transport
within bulk wurtzite InN, we employ ensemble semiclassical
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222103-2
O’Leary et al.
three-valley Monte Carlo simulations. The scattering mechanisms considered are: 共1兲 ionized impurity, 共2兲 polar optical
phonon, 共3兲 piezoelectric, and 共4兲 acoustic deformation potential. Intervalley scattering is also considered, each valley
being assumed to be of the form employed by O’Leary et al.3
and Foutz et al.5 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 3000 electrons is examined, while for
our transient electron transport simulations, the motion of
10 000 electrons is considered. For our simulations, the crystal temperature is set to 300 K and the doping concentration
is set to 1017 cm−3 for all cases. Electron degeneracy effects
are accounted for by means of the rejection technique of
Lugli and Ferry.15 Electron screening is also accounted for
following the Brooks–Herring method.16 Further details of
our approach are presented in the literature.3,5,13,17–20
The material parameter selections, used for our simulations of the electron transport within bulk wurtzite InN, are
mostly those employed by Foutz et al.;5 it should be noted
that the InN parameters employed by Foutz et al.5 differ
slightly from those employed by O’Leary et al.,3 particularly
with respect to the upper valley effective masses. However,
following Tsen et al.,7 we set the energy gap, Eg, to 0.75 eV
and the effective mass associated with the electrons in the
lowest-energy valley, m*, to 0.045 me, where me denotes the
free electron mass. The nonparabolicity coefficient associated with the lowest-energy valley, ␣, is taken to range between 0.4 eV−1, as suggested by Tsen et al.,7 to 1.22 eV−1,
this upper limit corresponding to the value for ␣ obtained
through a direct application of the Kane model; Tsen et al.7
argue that in order for their electron transport simulation results to be consistent with their experimental results, ␣
should be much less than that obtained through a direct application of the Kane model. The recent band structural calculations of Carrier and Wei21 suggest, for the case of bulk
wurtzite InN, that the lowest point in the conduction band is
located at the center of the Brillouin zone, at the ⌫ point, the
first upper conduction-band valley minimum occurring at the
A point, 2.2 eV above the lowest point in the conduction
band, the second upper conduction-band valley minimum occurring at the ⌫ point, 2.8 eV above the lowest point in the
conduction band. For the purposes of this analysis, the band
structure of Carrier and Wei21 was adopted. As with Foutz
et al.,5 we assume that all of the upper valleys are completely
parabolic and ascribe an effective mass of me to all of the
electrons in these upper valleys.
In Fig. 1, we plot the velocity-field characteristic associated with bulk wurtzite InN for our revised parameter selections, for ␣ set to 0.4 eV−1. We find that initially the electron
drift velocity monotonically increases with the applied
electric-field strength, reaching a maximum of about 6.0
⫻ 107 cm/ s when the applied electric-field strength is around
22.5 kV/ cm; the applied electric-field strength at which
point the peak in the velocity-field characteristic occurs will
henceforth be referred to as the peak field. For applied
electric-field strengths in excess of 22.5 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. The low-field electron
drift mobility, ␮, is found to be around 10 000 cm2 / V s. In-
Appl. Phys. Lett. 87, 222103 共2005兲
FIG. 1. The velocity-field characteristic associated with bulk wurtzite InN.
For our revised parameter selections, we considered two selections for the
nonparabolicity coefficient, ␣ being set to 0.4 eV−1 and ␣ being set to
1.22 eV−1, these selections spanning over the range of values one might
expect for bulk wurtzite InN. We contrast these results with those obtained
using traditional bulk wurtzite InN parameter selections. For the sake of
comparison, a GaN velocity-field characteristic is also depicted. For all
cases, we have assumed a crystal temperature of 300 K and a doping concentration of 1017 cm−3. For each velocity-field characteristic, the peak field,
i.e., the applied electric-field strength at which point the maximum electron
drift velocity occurs, is indicated with an arrow.
creasing the nonparabolicity coefficient to 1.22 eV−1, we find
that the peak electron drift velocity is decreased slightly, to
about 5.9⫻ 107 cm/ s at a peak field of around 40 kV/ cm.
The low-field electron drift mobility, ␮, is essentially unmodified, however. These results contrast rather dramatically
with that associated with bulk wurtzite InN determined using
the traditional material parameter selections, the peak electron drift velocity, the saturation electron drift velocity, and
the low-field electron drift mobility being about 4.1
⫻ 107 cm/ s, 1.8⫻ 107 cm/ s, and 3400 cm2 / V s, respectively; there are some minor quantitative discrepancies with
the results of O’Leary et al.,3 as O’Leary et al.3 chose lighter
upper valley effective masses.
We now examine the transient electron transport that occurs within bulk wurtzite InN. In particular, following the
approach of Foutz et al.,5,17 we study how electrons, initially
in thermal equilibrium, respond to the sudden application of
a constant applied electric field. For our InN bulk parameter
selections, with the nonparabolicity coefficient set to
0.4 eV−1, in Fig. 2 we plot the electron drift velocity as a
function of the distance displaced since the electric field was
initially applied, for a number of applied electric-field
strength selections. We note that for the applied electric-field
strength selections 11.25 kV/ cm and 22.5 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 in excess of 22.5 kV/ cm,
significant velocity overshoot occurs. This result suggests
that in InN, for this particular selection of parameters, that
22.5 kV/ cm is a critical applied electric-field strength for the
onset of velocity overshoot effects. As was mentioned earlier,
22.5 kV/ cm also corresponds to the peak field in the
velocity-field characteristic associated with bulk wurtzite
InN; recall Fig. 1. This suggests that velocity overshoot is
related to the transfer of electrons to the upper valleys. Simi-
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222103-3
Appl. Phys. Lett. 87, 222103 共2005兲
O’Leary et al.
FIG. 2. 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 InN. For all cases, we have
assumed an initial zero-field electron distribution, a crystal temperature of
300 K, and a doping concentration of 1017 cm−3. For all cases, we employed
our revised parameter selections and set the nonparabolicity coefficient, ␣,
to 0.4 eV−1.
lar results were found for GaN, AlN, and gallium arsenide
共GaAs兲 by Foutz et al.5,17
We now compare the transient electron transport characteristics for the various materials. From Fig. 2, it is clear that
certain materials exhibit higher peak overshoot velocities and
longer overshoot relaxation times. It is not possible to fairly
compare these different semiconductors by applying the
same applied electric-field strength to all of the materials, as
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 peak field for each material, i.e., 280 kV/ cm for GaN,
130 kV/ cm for InN with the traditional parameter selections,
45 kV/ cm for InN with the revised parameter selections and
␣ being set to 0.4 eV−1, 80 kV/ cm for InN with the revised
parameter selections and ␣ being set to 1.22 eV−1, and
8 kV/ cm for GaAs. Figure 3 shows such a comparison of the
velocity overshoot effects amongst the materials considered
in our analysis, i.e., GaN, InN, and GaAs. It is clear that
among the three III-V compound semiconductors considered,
InN exhibits superior transient electron transport characteristics. In particular, InN has the largest overshoot electron drift
velocity, and the distance over which this overshoot occurs,
greater than 1 ␮m, is longer than in either GaN and GaAs.
In conclusion, we have studied steady-state and transient
electron transport within bulk wurtzite InN using an ensemble semi-classical three-valley Monte Carlo simulation
approach. We have found that the transport characteristics of
InN are superior to those of GaN as well as GaAs. In particular, InN is found to have an extremely high peak electron
drift velocity and favorable transient electron transport characteristics. This suggests that there may be distinct advantages offered by using InN in high-frequency centimeter and
millimeter wave devices. We hope that these new results will
stimulate further interest in this material.
The authors wish to thank the Office of Naval Research
for financial support. One of the authors 共S. K. O.兲 gratefully
acknowledges financial assistance from the Natural Sciences
and Engineering Research Council of Canada.
FIG. 3. A comparison of the velocity overshoot amongst the III-V nitride
semiconductors considered in this analysis and GaAs. The applied electricfield strength chosen corresponds to twice the critical applied electric field
strength at which the peak in the steady-state velocity-field characteristic
occurs 共recall Fig. 1兲, i.e., 280 kV/ cm for the case of GaN, 130 kV/ cm for
the case of InN with traditional parameter selections, 45 kV/ cm for the case
of InN with revised parameter selections and ␣ set to 0.4 eV−1, 80 kV/ cm
for the case of InN with revised parameter selections and ␣ set to 1.22 eV−1,
and 8 kV/ cm for the case of GaAs.
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