Solid State Communications 150 (2010) 2182–2185
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Steady-state and transient electron transport within bulk wurtzite zinc oxide
Stephen K. O’Leary a,∗ , Brian E. Foutz b , Michael S. Shur c , Lester F. Eastman d
a
School of Engineering, The University of British Columbia, Okanagan, 3333 University Way, Kelowna, British Columbia, Canada V1V 1V7
b
608 Nettle Court, Charlottesville, VA 22903, USA
Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
c
d
School of Electrical Engineering, Cornell University, Ithaca, NY 14853, USA
article
info
Article history:
Received 22 June 2010
Received in revised form
30 August 2010
Accepted 31 August 2010
by E.Y. Andrei
Available online 8 September 2010
Keywords:
A. Zinc oxide
B. Gallium nitride
C. Semiconductors
D. Electron transport
abstract
We study the steady-state and transient electron transport that occurs within bulk wurtzite zinc oxide
using an ensemble semi-classical three-valley Monte Carlo simulation approach. We find that for electric
field strengths in excess of 180 kV/cm, the steady-state electron drift velocity associated with bulk
wurtzite zinc oxide exceeds that associated with bulk wurtzite gallium nitride. We also present evidence
to suggest that the negative differential mobility exhibited by the velocity-field characteristic associated
with bulk wurtzite zinc oxide is not related to transitions to the upper valleys. The transient electron
transport that occurs within bulk wurtzite zinc oxide is studied by examining how electrons, initially in
thermal equilibrium, respond to the sudden application of a constant electric field. From these transient
electron transport results, we conclude that for devices with dimensions smaller than 0.1 µm, gallium
nitride based devices will offer the advantage, owing to their superior transient electron transport, while
for devices with dimensions greater than 0.1 µm, zinc oxide based devices will offer the advantage, owing
to their superior high-field steady-state electron transport.
© 2010 Elsevier Ltd. All rights reserved.
Zinc oxide (ZnO) is a direct-gap II–VI semiconductor that
has recently become the focus of considerable interest [1–4].
Traditionally, this material was prepared either as a powder or
in oriented polycrystalline form [5]. Unfortunately, the defective
nature of these samples limited the range of applications that could
be considered [6–8]. In recent years, however, dramatic advances
in ZnO deposition have been achieved, and high-quality bulk
wurtzite ZnO crystals have been fabricated [9]. This has greatly
expanded the range of opportunities for ZnO in electronic and
optoelectronic device applications. The wide energy gap of ZnO,
around 3.4 eV at room temperature [10], is very similar to that
of gallium nitride (GaN), suggesting that ZnO is also a suitable
candidate for high temperature and high power electron device
applications. Its availability in bulk wurtzite crystalline form, and
the fact that it is almost exactly lattice matched with GaN, make it
possible to use ZnO as a substrate in GaN epitaxy [11]. Adding to its
allure, ZnO is amenable to wet chemical etching [12], possesses a
large exciton binding energy [13], and is able to robustly function
in extreme radiation conditions [14]. These favorable material
properties suggest that ZnO will remain a focus of attention for
many years to come.
With its wide energy gap, large polar optical phonon energy,
and large intervalley energy separation, ZnO is expected to ex-
∗
Corresponding author. Tel.: +1 250 807 8091; fax: +1 250 807 9850.
E-mail address: [email protected] (S.K. O’Leary).
0038-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2010.08.033
hibit favorable electron transport characteristics [15]. A number of
studies of the electron transport that occurs within ZnO have been
reported over the years. Experimental measurements of the mobility of this material were reported by Hutson [16] in 1957 and
Look et al. [9] in 1998. Then, in 1999, Albrecht et al. [5] reported
on Monte Carlo simulations of the steady-state electron transport
that occurs within bulk wurtzite ZnO. Further Monte Carlo analyses of the steady-state electron transport that occurs within bulk
wurtzite ZnO have been reported by Guo et al. [17] in 2006, by
Bertazzi et al. [18] in 2007, and by Furno et al. [19] in 2008. Unfortunately, these Monte Carlo results have yet to produce a satisfactory consensus, there being large differences observed between
the obtained velocity-field characteristics. Variations in the material and band structural parameter selections employed for these
Monte Carlo simulations are responsible for the differences in the
obtained results.
In this communication, we examine the steady-state and transient electron transport that occurs within bulk wurtzite ZnO. In
particular, we focus on how the electron transport that occurs
within this II–VI semiconductor differs from that which occurs
within bulk wurtzite GaN, another wide energy gap semiconductor currently being employed for a number of electronic and optoelectronic device applications. We start with an examination of
the steady-state electron transport, initially focusing on the dependence of the electron drift velocity on the applied electric field
strength. The dependence of the occupancy of the lowest energy
S.K. O’Leary et al. / Solid State Communications 150 (2010) 2182–2185
conduction band valley on the applied electric field strength is also
probed. We then focus on the transient electron transport that
occurs within this material, examining how electrons, initially in
thermal equilibrium, respond to the sudden application of a constant electric field. From these transient electron transport results,
we draw conclusions about the performance of ZnO and GaN based
devices.
For the purposes of this analysis of the electron transport within
bulk wurtzite ZnO, we employ an ensemble semi-classical threevalley Monte Carlo simulation approach. The scattering mechanisms considered are (1) ionized impurity, (2) polar optical
phonon, (3) piezoelectric, and (4) acoustic deformation potential. Intervalley scattering is also considered. The non-parabolicity
of each valley is treated through the application of the Kane
model [20,21]. 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 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 cm−3 for all cases.
Electron degeneracy effects are accounted for by means of the rejection technique of Lugli and Ferry [22]. Electron screening is also
accounted for following the Brooks–Herring method [23]. Further
details of our Monte Carlo simulation approach are presented in
the literature [24–33].
Most of the material parameter selections, used for our
simulations of the electron transport within bulk wurtzite ZnO, are
from Albrecht et al. [5]. Unfortunately, as Albrecht et al. [5] did not
provide an exhaustive list of all of the material parameters needed
for our Monte Carlo simulations, some of the material parameters
employed by us are drawn from other sources in the literature, or
through a direct fit with the results of Albrecht et al. [5], i.e., we
tweaked the material parameters until the resultant velocity-field
characteristic corresponded with that found by Albrecht et al. [5].
With the exception of the second order nonparabolicity factor,
which we neglect in our analysis [20,21], we employ the same
three-valley band model as that employed by Albrecht et al. [5].
The lowest energy conduction band valley electron effective mass
selection, 0.17 me , where me denotes the free electron mass,
while smaller than the selections of Adachi [10] (0.234 me ), Guo
et al. [17] (0.54 me ), and Furno et al. [19] (0.21 me perpendicular
and 0.23 me parallel), is in keeping with the relation between the
electron effective mass and the direct energy gap found in other
III–V and II–VI semiconductors, as can be seen in Fig. 1; given the
considerable overlap in authorship between Bertazzi et al. [18] and
Furno et al. [19], it will be assumed that the choice of electron
effective mass is similar. As with Albrecht et al. [5], anisotropy in
the bands is neglected, the simulations of Bertazzi et al. [18] and
Furno et al. [19] demonstrating that accounting for this anisotropy
only leads to a slight correction.
In Fig. 2a, we plot the velocity-field characteristic associated
with bulk wurtzite ZnO for our parameter selections. The crystal
temperature is set to 300 K and the doping concentration is set
to 1017 cm−3 for the purposes of this simulation. We find 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; 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 fields 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. The low-field
2183
Fig. 1. The electron effective mass associated with the lowest energy valley in the
conduction band as a function of the direct energy gap for the case of a number of
III–V and II–VI direct-gap and indirect-gap semiconductors. The values depicted in
this plot are mostly from Adachi [10] except for the value corresponding to InN,
which is drawn from O’Leary et al. [30], the value corresponding to GaN, which is
drawn from O’Leary et al. [31], and a few of the ZnO values, as indicated in the plot
itself. A linear least squares fit corresponding to this data is depicted with the dashed
line. The nature of the energy gap, i.e., whether it is direct or indirect, is specifically
indicated. For the case of GaN, the wurtzite phase is considered. For the purposes of
our simulations, we follow Albrecht et al. [5] and set the electron effective mass to
0.17 me .
Fig. 2a. The electron drift velocity as a function of the applied electric field
strength, for the cases of bulk wurtzite ZnO and bulk wurtzite GaN. The peak fields
corresponding to each material, i.e., 270 kV/cm for the case of bulk wurtzite ZnO
and 140 kV/cm for the case of bulk wurtzite GaN, are indicated. The transition
field, beyond which the steady-state electron drift velocity associated with bulk
wurtzite ZnO exceeds that associated with bulk wurtzite GaN, i.e., 180 kV/cm, is
also indicated. For both cases, we assume a crystal temperature of 300 K and a
doping concentration of 1017 cm−3 . A similar plot is depicted in Figure 5 of Albrecht
et al. [5].
electron drift mobility is found to be around 430 cm2 /Vs for the
case of bulk wurtzite ZnO [34–37].
In Fig. 2a, we also plot the velocity-field characteristic associated with bulk wurtzite GaN, this velocity-field characteristic being determined using the same bulk wurtzite GaN material and
band structural parameter selections as that employed by O’Leary
et al. [31]. As before, the crystal temperature is set to 300 K and the
doping concentration is set to 1017 cm−3 for the purposes of this
simulation. While the low-field electron drift mobility associated
with bulk wurtzite GaN, around 850 cm2 /Vs, is greater than that
associated with bulk wurtzite ZnO, its peak electron drift velocity, about 2.9 × 107 cm/s, is less than that found for bulk wurtzite
ZnO. The fact that the peak field associated with bulk wurtzite GaN,
2184
S.K. O’Leary et al. / Solid State Communications 150 (2010) 2182–2185
Fig. 2b. The occupancy of the lowest energy conduction band valley, Γ1 , as a
function of the applied electric field strength, for the cases of bulk wurtzite ZnO
and bulk wurtzite GaN. The total number of electrons considered in each simulation
is three thousand. The results depicted in this plot are determined from the same
simulations used to determine the results depicted in Fig. 2a. The peak field
corresponding to each material is indicated. For both cases, we assume a crystal
temperature of 300 K and a doping concentration of 1017 cm−3 .
around 140 kV/cm, is less than that associated with bulk wurtzite
ZnO, suggests that ZnO offers some distinct advantages for use in
high-field electron device applications. Indeed, as can be seen in
Fig. 2a, for electric fields in excess of 180 kV/cm, the steady-state
electron drift velocity associated with bulk wurtzite ZnO exceeds
that associated with bulk wurtzite GaN.
In Fig. 2b, we plot the occupancy of the lowest energy conduction band valley as a function of the applied electric field strength
for the cases of bulk wurtzite ZnO and bulk wurtzite GaN; for both
materials, the lowest energy conduction band valley corresponds
to the central Γ valley, which we refer to as the Γ1 valley. As before,
the crystal temperature is set to 300 K and the doping concentration is set to 1017 cm−3 for the purposes of these simulations, the
results depicted in Fig. 2b being obtained using the same Monte
Carlo stimulations as those used to determine Fig. 2a. We note that
while the onset of transitions to the upper valleys occurs at applied electric field strength selections lower than the peak field,
i.e., 140 kV/cm, for the case of bulk wurtzite GaN, for the case of
bulk wurtzite ZnO very few upper valley transitions occur, even
for applied electric field strength selections well in excess of the
peak field. A detailed analysis demonstrates that even for an applied electric field strength of 1000 kV/cm, the lowest energy conduction band valley of bulk wurtzite ZnO remains 92% populated.
The large intervalley energy separation exhibited by bulk wurtzite
ZnO, 4.4 eV between the lowest energy conduction band valley
and the first upper conduction band valley, is the primary factor
responsible for this difference [38]; for the case of bulk wurtzite
GaN, O’Leary et al. [31] assert that this energy difference is 1.9 eV.
Clearly, the peak exhibited in the velocity-field characteristic associated with bulk wurtzite ZnO is not related to upper valley
transitions. Instead, as was noted by Albrecht et al. [5], the large
non-parabolicity coefficient associated with bulk wurtzite ZnO is
leading to the observed negative differential mobility exhibited by
bulk wurtzite ZnO [39].
We now examine the transient electron transport that occurs
within bulk wurtzite ZnO. In particular, following the approach of
Foutz et al. [24,27], we study how electrons, initially in thermal
equilibrium, respond to the sudden application of a constant applied electric field. For our ZnO bulk parameter selections, in Fig. 3
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
Fig. 3. 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 assume a crystal temperature of
300 K and a doping concentration of 1017 cm−3 .
the applied electric field strength selections 135 and 270 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 270 kV/cm, significant velocity
overshoot occurs. This result suggests that for bulk wurtzite ZnO,
for this particular selection of parameters, that around 270 kV/cm
is a critical applied electric field strength for the onset of velocity overshoot effects. As was mentioned earlier, around 270 kV/cm
also corresponds to the peak field in the velocity-field characteristic associated with bulk wurtzite ZnO; recall Fig. 2a. Similar results were found for GaN, and other III–V semiconductors, by Foutz
et al. [27,24].
We now compare the transient electron transport characteristics corresponding to bulk wurtzite ZnO and bulk wurtzite GaN.
From the earlier analysis of Foutz et al. [24,27], 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 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 peak field for each material, i.e., 540 kV/cm for the case of bulk wurtzite ZnO and 280
kV/cm for the case of bulk wurtzite GaN. Fig. 4 shows such a comparison of the velocity overshoot effects for ZnO and GaN. We find
that the peak transient electron drift velocity associated with bulk
wurtzite ZnO, around 5.6 × 107 cm/s, is less than that associated
with bulk wurtzite GaN, about 6.7 × 107 cm/s, and that the velocity overshoot exhibited by both materials occurs over 0.1 µm. This
suggests that for devices with dimensions smaller than 0.1 µm,
GaN based devices will offer the advantage, owing to their superior transient electron transport. In contrast, for devices with dimensions greater than 0.1 µm, ZnO based devices will offer the
advantage, owing to their superior high-field steady-state electron
transport.
In conclusion, we have studied the steady-state and transient
electron transport that occurs within bulk wurtzite ZnO using
an ensemble semi-classical three-valley Monte Carlo simulation
approach. We found that for electric field strengths in excess of
180 kV/cm, that the steady-state electron drift velocity associated
with bulk wurtzite ZnO exceeds that associated with bulk wurtzite
GaN. We also presented evidence to suggest that the negative
differential mobility exhibited by the velocity-field characteristic
associated with bulk wurtzite ZnO is not related to transitions
to the upper valleys. The transient electron transport that occurs
S.K. O’Leary et al. / Solid State Communications 150 (2010) 2182–2185
2185
[19] E. Furno, F. Bertazzi, M. Goano, G. Ghione, E. Bellotti, Solid-State Electron. 52
(2008) 1796.
[20] In the Kane model, the energy bands are assumed to be non-parabolic,
spherical, and of the form
h̄2 (⃗
k − k⃗o )2
2m∗
= E (1 + α E ),
where ⃗
k denotes the crystal momentum wave-vector, k⃗o represents the
location of the valley minimum, E is the energy with respect to this valley
minimum, i.e., E = 0 corresponds to the valley minimum, m∗ is the effective
mass, and α is the non-parabolicity coefficient. Typically, one sets
α=
Fig. 4. A comparison of the velocity overshoot for bulk wurtzite ZnO and bulk
wurtzite GaN. The applied electric field strength selections correspond to twice
the peak field for each case, i.e., 540 kV/cm for the case of bulk wurtzite ZnO and
280 kV/cm for the case of bulk wurtzite GaN. For both cases, we assume a crystal
temperature of 300 K and a doping concentration of 1017 cm−3 . The plots crossover
where they intersect.
within bulk wurtzite ZnO was studied by examining how electrons,
initially in thermal equilibrium, respond to the sudden application
of a constant electric field. From these transient electron transport
results, we concluded that for devices with dimensions smaller
than 0.1 µm, GaN based devices will offer the advantage, owing to
their superior transient electron transport, while for devices with
dimensions greater than 0.1 µm, ZnO based devices will offer the
advantage, owing to their superior high-field steady-state electron
transport. We hope that these results will stimulate further interest
in this material.
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
Acknowledgements
[34]
The authors wish to thank Dr. Kai Liu for his assistance in translating an article written in a foreign language. Financial support
from the Office of Naval Research is acknowledged. One of the authors (S.K.O.) gratefully acknowledges financial assistance from the
Natural Sciences and Engineering Research Council of Canada.
[35]
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The steady-state results of Albrecht et al. [5] differ slightly from those
found by us. While we aimed to faithfully reproduce the results of Albrecht
et al. [5], uncertainty as to the exact values of the material and band structural
parameters probably accounts for most of these slight differences. The reasons
for the differences in the value of the low-field electron drift mobility remain
unknown, however; the velocity-field characteristics of Albrecht et al. [5]
suggest a low-field electron drift mobility of 300 cm2 /Vs while we find
430 cm2 /Vs.
A number of experimental measurements of the low-field electron mobility
associated with ZnO have been reported over the years. An initial determination of the mobility associated with crystalline ZnO was reported by Hutson [16] in 1957, a Hall mobility of 180 cm2 /Vs at 300 K being found for
doping concentrations of the order of 1017 cm−3 ; crystalline ZnO needles
were considered in the analysis of Hutson [16]. In 1998, Look et al. [9] determined a Hall mobility of 205 cm2 /Vs at 300 K for doping concentrations
of the order of 1017 cm−3 , their measurements being performed on large diameter ZnO crystals. Makino et al. [36] find much higher Hall mobility values,
440 cm2 /Vs at 300 K, for epitaxial layers of ZnO, this exceeding the bulk theoretical limit corresponding to crystalline ZnO; Makino et al. [36] find this limit
to be 430 cm2 /Vs at 300 K. We suspect that this enhancement in the mobility may be due to the presence of a two-dimensional electron gas; enhancement in the low-field mobility associated with ZnO due to the presence of a
two-dimensional electron gas was also found by Tampo et al. [37] in 2006.
While there is some variation in the obtained experimental low-field mobility
results, differences in the nature of the materials are believed to account for
most of these variations. Differences between these experimental results and
the results of our simulation may be accounted for if one considers the polaron
mass enhancement, the quality of the crystals, the Hall coefficient, and other
factors. Finally, it is interesting to note that the bulk theoretical limit corresponding to crystalline ZnO found by Makino et al. [36], 430 cm2 /Vs at 300 K,
is in agreement with the results of our simulations.
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87 (2005) 022101.
H. Tampo, H. Shibata, K. Matsubara, A. Yamada, P. Fons, S. Niki, M. Yamagata,
H. Kanie, Appl. Phys. Lett. 89 (2006) 132113.
The large intervalley energy separation, i.e., larger than the energy gap itself,
seems unusual. An intervalley energy separation that exceeds the energy gap
is also found for the case of indium nitride [30].
This has yet to be widely recognized. Guo et al. [17], for example, attribute
the peak observed in their velocity-field characteristic associated with bulk
wurtzite ZnO as being due to transitions to the upper valleys.