Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields
The electron distribution function and transport characteristics of hot electrons in GaN semiconductor are calculated by the Monte Carlo method. We studied the electron transport at temperatures of 10, 77, and 300 K under low and moderate electric fields. We found that, at low temperatures and lo...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
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irk-123456789-1183292017-05-30T03:05:27Z Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields Syngaivska, G.I. Korotyeyev, V.V. The electron distribution function and transport characteristics of hot electrons in GaN semiconductor are calculated by the Monte Carlo method. We studied the electron transport at temperatures of 10, 77, and 300 K under low and moderate electric fields. We found that, at low temperatures and low electric fields (a few hundreds of V/cm), the second “ohmic” region is to be observed on the I-V characteristic. In this case, the mean energy is very slowly dependent on the field. The streaming effect can occur in bulk GaN with low electron concentration (<10¹⁶ cm⁻³) at low temperatures and electric fields of a few kV/cm. 2007 Article Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields / G.I. Syngaivska, V.V.Korotyeyev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 54-59. — Бібліогр.: 16 назв. — англ. 1560-8034 PACS 02.70.Uu, 72.10.-d http://dspace.nbuv.gov.ua/handle/123456789/118329 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The electron distribution function and transport characteristics of hot electrons
in GaN semiconductor are calculated by the Monte Carlo method. We studied the
electron transport at temperatures of 10, 77, and 300 K under low and moderate electric
fields. We found that, at low temperatures and low electric fields (a few hundreds of
V/cm), the second “ohmic” region is to be observed on the I-V characteristic. In this case,
the mean energy is very slowly dependent on the field. The streaming effect can occur in
bulk GaN with low electron concentration (<10¹⁶ cm⁻³) at low temperatures and electric
fields of a few kV/cm. |
| format |
Article |
| author |
Syngaivska, G.I. Korotyeyev, V.V. |
| spellingShingle |
Syngaivska, G.I. Korotyeyev, V.V. Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Syngaivska, G.I. Korotyeyev, V.V. |
| author_sort |
Syngaivska, G.I. |
| title |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields |
| title_short |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields |
| title_full |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields |
| title_fullStr |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields |
| title_full_unstemmed |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields |
| title_sort |
monte carlo simulation of hot electron effects in compensated gan semiconductor at moderate electricfields |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118329 |
| citation_txt |
Monte Carlo Simulation of hot electron effects in compensated GaN semiconductor at moderate electricfields / G.I. Syngaivska, V.V.Korotyeyev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 54-59. — Бібліогр.: 16 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT syngaivskagi montecarlosimulationofhotelectroneffectsincompensatedgansemiconductoratmoderateelectricfields AT korotyeyevvv montecarlosimulationofhotelectroneffectsincompensatedgansemiconductoratmoderateelectricfields |
| first_indexed |
2025-07-08T13:49:32Z |
| last_indexed |
2025-07-08T13:49:32Z |
| _version_ |
1837086881726070784 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
54
PACS 02.70.Uu, 72.10.-d
Monte Carlo simulation of hot electron effects
in compensated GaN semiconductor at moderate electric fields
G.I. Syngayivska, V.V. Korotyeyev
V. Lashkaryov Institute for Semiconductor Physics, Department of Theoretical Physics
41, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: singg@ukr.net, koroteev@ukr.net
Abstract. The electron distribution function and transport characteristics of hot electrons
in GaN semiconductor are calculated by the Monte Carlo method. We studied the
electron transport at temperatures of 10, 77, and 300 K under low and moderate electric
fields. We found that, at low temperatures and low electric fields (a few hundreds of
V/cm), the second “ohmic” region is to be observed on the I-V characteristic. In this case,
the mean energy is very slowly dependent on the field. The streaming effect can occur in
bulk GaN with low electron concentration (<1016 cm–3) at low temperatures and electric
fields of a few kV/cm.
Keywords: Monte Carlo method, hot electrons, electron transport.
Manuscript received 16.10.07; accepted for publication 19.12.07; published online 31.01.08.
1. Introduction
In recent years, a new class of semiconductors – the
group-III nitrides – is actively investigated due to their
possible usage in both power electronics and opto-
electronics. Several materials such as InN, GaN, and
AlN are included to this group of semiconductors. The
investigation of kinetic properties of these materials and
especially those of GaN is carried out in wide ranges of
electric fields and temperatures. A number of works has
been done to investigate kinetic properties of electrons in
bulk GaN at high electric fields under room or higher
temperatures. In most of these investigations, relatively
high electron concentrations have usually considered.
For example, in [1-4], electron kinetic properties have
been studied at electric fields of 100-600 kV/cm and
electron concentrations of 1016–1018 cm–3. It is noted that
GaN has become of great interest for the use in many
high-power semiconductor devices [5, 6].
The group-III nitride semiconductors, in
particularly GaN, differ from other semiconductor АIIIВV
compounds well studied at the present moment. The
nitrides have specific material characteristics: the large
optical phonon energy εLO , the large energy splitting
between the Γ-minimum and the lowest satellite valleys
in the conduction band ∆ε , and strong electron-polar-
optical-phonon coupling (characterized by the Fröhlich
constant α). For GaN, these quantities are equal to
εLO ≈ 0.092 eV, ∆ε ≈ 1.4 – 1.8 eV [3], and α = 0.41,
respectively. These material properties allow one to
assume the existence of new electron phenomena at a
moderate electric field which are hardly to be observable
in other known polar semiconductors such as GaAs. The
one of such effects is the streaming regime induced by
optical phonon emission. In this regime, the electron
motion is quasiperiodic in the energy space with
alternating quasiballistic acceleration and emission of a
polar optical phonon. The streaming regime can be
realized at low lattice temperatures (10 – 77 K) and low
electron concentrations (1014–1015 cm–3) [7, 8].
The main purpose of this work is to study
characteristic properties of the electron transport in GaN
at moderate electric fields by the Monte Carlo method.
This numerical method is one of the most effective tools
to study the electron transport in semiconductors. It
allows one to take into account the band structure and all
actual scattering mechanisms and is well described in the
literature, in particular in [9-11].
2. Subject of investigation
We studied the transport characteristics of hot electrons
in a bulk GaN of the cubic modification. Applied electric
fields are assumed to be steady. The compensated GaN
samples with low electron concentrations (1014–1015 cm–3)
are considered. The ionized impurity concentration is of
1016 cm–3. All material parameters used in the simulation
are taken from [12].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
55
Fig. 1. Total scattering rate as a function of the electron energy
in compensated GaN at 10, 77, and 300 K.
The following scattering mechanisms are taken into
account: the scattering by acoustic phonon, polar optical
phonon, and ionized impurities. The large intervalley
separation in group-III-nitride semiconductors allows us
to neglect the intervalley scattering in the considered
range of electric fields. The electron band is assumed
parabolic for the simulation of the electron transport.
Importantly, the electron concentration is chosen low to
neglect the electron-electron scattering.
Under these assumptions, the obtained dependences
of the scattering rate on the electron energy at different
lattice temperatures are shown in Fig. 1. In the “passive”
energy region ( LOLO ωεε h=< ), the ionized impurity
scattering dominates at 10 and 77 K. In the “active”
energy region (ε > εLO ), the polar optical-phonon
emission is always the dominant scattering mechanism.
For 300 K, two scattering mechanisms, i.e. the ionized
impurity scattering and the polar optical-phonon
absorption, are essential in the “passive” energy region.
3. Electron distribution function
In this section, we demonstrate an analytical method of
solving the Boltzmann equation in case of weak electric
fields and compare the analytical solution with a
numerical one obtained by the Monte Carlo method.
3.1. Analytical model
The momentum electron distribution function )( pf
r
contains the full information about electron properties
and is traditionally the main subject of the research. The
distribution function is used to calculate the mean kinetic
characteristics. Moreover, the knowledge of )( pf
r
allows us to give the full qualitative picture of kinetic
phenomena taking place in the semiconductor.
The electron distribution function )( pf
r
in a homo-
geneous steady electric field E
r
can be determined from
the Boltzmann transport equation
col
)(
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
t
f
pd
pdf
Ee v
rr
. (1)
Here
col
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
t
f
is the collision integral. Under
certain conditions, the integro-differential equation (1)
can be reduced to a chain of differential equations by
using the Legendre polynomial expansion of the
distribution function [13]. In other words, the
distribution function is expanded as
∑=
k
kk Pfpf )θ(cos)ε()(
r
, (2)
where fk (ε) are coefficients of expansion of the
distribution function, Pk (cos θ) are the Legendre
polynomials, and θ is the angle between the electron
momentum p
r
and the electric field E
r
.
Coefficients of the expansion of the electron
distribution fk (ε) can be used to determine different
mean kinetic characteristics. For example, in order to
calculate the mean energy 〈ε〉 and the drift velocity vdr,
we must find two first coefficients f0(ε) and f1(ε):
∫∫ εεπ=ε=〉ε〈 dppfpdpf 2
0
3 )(4)( rr , (3)
∫∫
π
== dppfvpdpfvv x
2
1
3
dr )ε(
3
4)(
rr
. (4)
The determination of the coefficients f0 (ε) and f1(ε)
is the main task of the analytical theory.
3.2. Case of weak fields
At small electric fields and low lattice temperatures, the
electron distribution is formed by the elastic and
quasielastic scattering mechanisms. The electron
distribution function calculated by the Monte Carlo
method in the momentum space is shown in Fig. 2, a.
Here, px and py are the components of the momentum,
and px is parallel to the electric field. On the contour plot
(Fig. 2, b), we show the contour lines representing the
lines of constant values of the distribution function
from 0.1 to 0.9 with a step of 0.1. Two first coefficients
of the Legendre polynomial expansion of the distribution
function are shown on Fig. 2, c. It is clearly seen that the
coefficient f0(ε) is essentially larger than f1(ε) anywhere
in the energy region. This means that the electron distri-
bution function is quasiisotropic at this electric field.
Thus, at weak fields, we can use the diffusion
approximation which assumes that | f1(ε)| << | f0(ε)| and
f2 = … = fk = 0. In the diffusion approximation, Eq. (1)
can be written as
.
)(
,)(
)(
3
10
0opac
1
2
2 p
f
dp
df
eEfLL
dp
fpd
p
eE
τ
−=+=
(5)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
56
a
b
c
Fig. 2. Electron distribution function (a), contour plot (b), and
coefficients of expansion (2) of the distribution function (c) for
GaN at 10 K and E = 100 V/cm, LOLO ε2mp = .
In (5), Lac and Lop are the collision operators for
acoustic phonons (quasielastic interaction) and polar
optical phonons (inelastic interaction); Lac is a diffe-
rential second-order operator, the effect of Lop is reduced
to an algebraic expression in finite differences [13]; and
)(/1)(/1)(/1 impac ppp τ+τ=τ , where )(ac pτ , )(imp pτ
are the momentum relaxation times on acoustic phonons
and ionized impurities, respectively. Solving (5), we can
find the analytical expression for f0(ε), f1(ε) and then
will calculate the mean kinetic characteristics.
3.3. Comparison of numerical and analytical results
It is appropriate now to compare the mean kinetic cha-
racteristics obtained analytically with those calculated by
the Monte Carlo method. The results of calculations are
shown in Fig. 3, where the mean electron energy
(Fig. 3, a), the electron drift velocity (Fig. 3, b), and the
longitudinal D|| and transverse D⊥ diffusion coefficients
(Fig. 3, c) are presented as functions of the electric field.
The calculations are carried out by assuming a compen-
sated GaN at 10 K. It is clear that the analytical solution
coincides with the numerical one in a range of low
electric fields. With increase in the applied field, the
distribution function takes a more anisotropic form. The
diffusion approximation is disturbed, and one can see a
weak divergence between curves 1 and 2 in Fig. 3.
a
b
c
Fig. 3. Mean electron energy (a), drift velocity (b), and
diffusion coefficient (c) as functions of the electric field: 1 –
the Monte Carlo method; 2 – the analytical solution. D0 is the
diffusion coefficient in the equilibrium state.
V
d
r/
V
L
O
<
ε>
/ε
L
O
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
57
a
b
c
Fig. 5. Mean electron energy (a) and drift velocity (b) and
(c) as functions of the electric field for GaN (Ni = 1016 cm–3,
Ne = 1014 cm–3).
3.4. Case of strong fields
A strongly anisotropic electron distribution is formed in
the strong applied field. The formation of the anisotropic
electron distribution is characteristic of the streaming
phenomenon. This anisotropic electron distribution
function calculated by the Monte Carlo method at an
electric field of 4 kV/cm is shown in Fig. 4, a. Here, the
x-component of the momentum, xp , is parallel to the
electric field. Contour lines on the plot shown in
Fig. 4, b are not spherical as those in Fig. 2, b, but they
are elongated along the applied field. Three first
coefficients of expansion (2) of the electron distribution
function are shown in Fig. 4, c. In the streaming regime,
these moments are quantities of the same order. This is
strongly different from the case of low electric fields
when the spherically symmetric part 0f entirely domi-
nates (Fig. 2, c).
a
b
c
Fig. 4. Electron distribution function (a), contour plot (b), and
coefficients of expansion (2) of the distribution function (c) for
GaN at 10 K and E = 4 kV/cm.
4. Mean kinetic characteristics of electrons
In this section, we describe some specific properties of
the mean kinetic characteristics. At first, we consider a
case of weak electric fields. Then we will discuss a
peculiarity of the electron motion in the streaming
regime and analyze the kinetic properties of electrons
under strong electric fields.
4.1. Case of weak fields
At low temperatures (from 10 to 77 K) and fields less than
50 V/cm, the mean energy grows rapidly with increase in
the field (see Fig. 5, a). In this range of temperatures and
fields, electrons interact mainly with acoustic phonons and
ionized impurities. So the energy and momentum
relaxations occur on acoustic phonons and ionized
impurities, respectively. These scattering mechanisms
cannot restrict the growth of the mean energy as an
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
58
applied field increases. In larger electric fields, the role of
the electron interaction with polar optical phonons
(emission phonons) becomes more important. This
scattering mechanism suppresses a growth of the mean
energy. As a result, the saturation of the mean electron
energy is observed at a field of 100 V/cm and more.
At fields less than 50 V/cm, the dependence of the
drift velocity on the electric field has two unusual
properties. At heating fields and temperatures from 10 to
77 K, we observe the anomalous growth of mobility with
temperature (see Fig. 5, b). This mobility growth is due
to a decrease of the role of the ionized impurity
scattering with increase in the lattice temperature in the
Brooks-Herring model. For temperatures from 77 K and
higher, the role of electron-phonon scattering mecha-
nisms (acoustic and polar optic phonons) becomes more
important, and the electron mobility decreases with
increase in the temperature.
At electric fields of 100 V/cm and higher (Fig. 5, c)
where the quasisaturation of the mean energy takes place,
we observe the so-called second “ohmic” region for the
drift velocity [14, 15]. In case of the second “ohmic”
region, the main mechanism of electron energy relaxation
is the emission of polar optical phonons. The electron
momentum relaxation comes from ionized impurities.
Electron mobility is estimated as s)/(Vcm105.1 24 ⋅⋅ for
10 K and s)/(Vcm101.1 24 ⋅⋅ for 77 K on this part of the
I-V characteristic. For comparison, for the electron
mobility on the first “ohmic” region, we have the
following estimations: s)/(Vcm104 23 ⋅⋅ for 10 K and
s)/(Vcm105.8 23 ⋅⋅ for 77 K.
For the lattice temperature equal to 300 K,
drastically another situation occurs. At 300 K, the
inelastic scattering mechanism – the polar optical-
phonon absorption – is essential in the “passive” energy
region in addition to the elastic scattering mechanism
(the ionized impurity scattering). As a result, the mean
energy depends weakly on the applied field and the I-V
characteristic is quasiohmic in the wide range of electric
fields with the mobility estimated to be
s)/(Vcm108.1 23 ⋅⋅ .
4.2. Case of strong fields
In higher electric fields (a few kV/cm), we can observe
another hot electron effect. The low lattice temperature,
a low electron concentration, and the domination of the
polar optical-phonon emission over other scattering
mechanisms provide conditions for the appearance of the
streaming effect at some range of steady electric fields.
In the streaming regime, the electron motion
becomes almost periodic [7, 15]. During one period T,
an electron accelerates quasiballistically until it reaches
the critical velocity mv /ε2 LOLO = . The electron then
loses its energy by emitting an optical phonon and starts
the next period of acceleration. The scheme of this
motion is shown in Fig. 6.
Fig. 6. Sketch of changing the electron velocity with time in
the streaming regime.
The streaming regime is characterized by the
saturation of both the mean energy and the drift velocity.
These quantities as functions of the electric field are
shown in Fig. 7. At fields of 2–10 kV/cm, it can see the
region of the saturation of the mean energy and the drift
velocity. At this region of the I-V characteristic, the drift
velocity is equal to LO5.0~ v . At the lattice temperature
equal to 300 K, the streaming is not possible in the
considered range of electric fields. This is clearly seen
from the last figure.
a
b
Fig. 7. Mean electron energy (a) and drift velocity (b) as
functions of the electric field for GaN.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 54-59.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
59
5. Conclusions
We have studied the distribution function and properties
of the electron transport in group-III nitride
semiconductors at moderate electric fields. In bulk GaN,
at low temperatures in the wide range of heating electric
fields, the electron distribution function is quasiisotropic,
and the second “ohmic” region on the I-V characteristic
is observed. Simultaneously, the saturation of the mean
electron energy is observed. As the electric field
increases, the electron distribution takes on the spindle
shape, and the electrons demonstrate the streaming
regime. In compensated GaN, particularly, the streaming
effect occurs at 10 K and 77 K at fields of 2–10 kV/cm.
The novel transport effects studied in this paper can
find applications to the construction of devices. The
existence of an extended quasisaturation region for the
mean electron energy on a growth of the drift velocity
corresponds to the steady noise level of hot electrons and
the increase of the signal/noise ratio. This can be used in
the development of new semiconductor devices, in
particular, of sensitive photodetectors. The peculiar
features of the electron transport in the streaming regime
can be used in the design of tunable sources of tera-
Hertz radiation [16].
Acknowledgement
The authors would like to express their gratitude to
Professor V.A. Kochelap for his valuable discussions on
various aspects of this work.
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