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DOI: 10.21122/22209506201672161168
162
Introduction
Silicononinsulator (SOI) technology in micro and nanoelectronics has gained a great interest
in the last decades. Deep submicron SOI MOSFETs
are regarded as promising elements for modern integrated circuits in different electronic applications
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[1, 2]
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. Among the advantages of submicron SOI
MOSFETs, in comparison with common «bulk»
MOSFETs, are the lower power dissipation and
increased operation speed, lower leakage currents,
and higher radiation hardness.
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Among the advantages of submicron SOI
MOSFETs, in comparison with common «bulk»
MOSFETs, are the lower power dissipation and
increased operation speed, lower leakage currents,
and higher radiation hardness. Deep submicron SOI
MOSFETs are less vulnerable to shortchannel effects in comparison with common MOSFETs
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[3]
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.
Results of recent investigations show that very
promising is the use of submicron SOI MOSFETs
as different sensors and detectors. For instance, the
possibility to use SOI MOSFETs as electric field
sensors was proposed in [4].
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Results of recent investigations show that very
promising is the use of submicron SOI MOSFETs
as different sensors and detectors. For instance, the
possibility to use SOI MOSFETs as electric field
sensors was proposed in
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[4]
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. Also recently the possibility to use deep submicron SOI MOSFETs as
unique singlephoton detectors at room temperature
was demonstrated in [5, 6].
Today the development of modern devices of
micro and nanoelectronics, including various sensor devices, can not be done without computer simulation of their characteristics.
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For instance, the
possibility to use SOI MOSFETs as electric field
sensors was proposed in [4]. Also recently the possibility to use deep submicron SOI MOSFETs as
unique singlephoton detectors at room temperature
was demonstrated in
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[5, 6]
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.
Today the development of modern devices of
micro and nanoelectronics, including various sensor devices, can not be done without computer simulation of their characteristics. Thereupon it must be
noted that ensemble Monte Carlo method has been
widely used as a powerful tool for simulation of carrier transport phenomena in different semiconductor
devices.
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Thereupon it must be
noted that ensemble Monte Carlo method has been
widely used as a powerful tool for simulation of carrier transport phenomena in different semiconductor
devices. By means of Monte Carlo simulation static,
dynamic and noise characteristics of submicron SOI
MOSFETs have been calculated
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[7–10]
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. One of the
advantages of the method is the possibility of incorporation of rather sophisticated models describing different physical processes into the simulation procedure. Ensemble Monte Carlo simulation thus is one
of the most promising methods for the simulation of
deep submicron SOI MOSFETs, which allows account of all necessary mechanisms of carrier scattering.
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Ensemble Monte Carlo simulation thus is one
of the most promising methods for the simulation of
deep submicron SOI MOSFETs, which allows account of all necessary mechanisms of carrier scattering.
The simulation procedure also enables inclusion of
semiconductor band structure calculations and account of quantum effects as well
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[10–14]
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.
It is known that inclusion of effects related to
impact ionization becomes very important in numerical simulations of shortchannel MOSFETs. This is
caused by the fact that in such MOSFETs electric field
strengths are high enough to make impact ionization
rate be comparable or even higher than other dominant scattering mechanisms.
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Outlines of ensemble Monte Carlo transport
simulation
The crosssection of the SOI MOSFET structure under consideration is presented in Figure 1.
The simulated structure is a fully depleted single gate
SOI MOSFET with the conducting silicon channel
laying between gate and buried oxides
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[8, 15, 16]
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.
The device dimensions denoted in the Figure 1 are
as follows: the source, gate and drain lengths are
LS = LG = LD = 100 nm, channel thickness Wc = 30 nm,
the thickness of buried oxide layer is Wb = 100 nm,
and the thickness of the silicon substrate layer
Wsub = 200 nm.
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After every time step Δt the
163
Poisson equation with appropriate boundary conditions is solved in order to update the electrostatic
potential. The calculation of free carrier charge density within the simulation dimensions is performed
using socalled particle technique
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[17]
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. The Monte
Carlo procedure is twodimensional in real space and
threedimensional in momentum space. The latter is
caused by the fact that stateoftheart planar technology implies that the device width in the dimension perpendicular to the figure plane (see Figure 1)
is much higher than its length L = LS +LG +LD and
depth W = Wc +Wb +Wsub.
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latter is
caused by the fact that stateoftheart planar technology implies that the device width in the dimension perpendicular to the figure plane (see Figure 1)
is much higher than its length L = LS +LG +LD and
depth W = Wc +Wb +Wsub. The time step Δt is chosen
to be 1 fs. A general description of the Monte Carlo
simulation approach may be found elsewhere
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[18]
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.
It is considered that the contacts of the drain, the
source, and the substrate are ideal ohmic contacts. The
metallic gate is assumed to be aluminum. Ideal ohmic
contact model implies that a contact is in thermal equilibrium though the current is flowing through it.
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The
latter means that the contact injects particles to provide charge neutrality in a small region of semiconductor near the contact edge. We suppose that injected
particles have Maxwellian distribution and also we
use the injection model which takes into account that
particles are not injected simultaneously
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[19]
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. Particles reaching the contact from inside the simulation
domain leave the device freely.
It must be mentioned that in present work we
neglect size quantization effects and consider electron and hole gases as purely threedimensional.
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It must be mentioned that in present work we
neglect size quantization effects and consider electron and hole gases as purely threedimensional. Such
approximation must be reasonable for considered
channel width
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[8, 15]
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. Another problem, which arises while simulating charge carrier transport in SOI
MOSFETs, is the treatment of carrier scattering by
SiSiO2 interfaces. For threedimensional electron
and hole gases the scattering of charge carriers by the
interfaces is usually regarded as the combination of
diffusive and specular reflections of particles from
the interface.
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Electron transport in the conduction band of silicon is simulated in valleys X and L, with account of
the nonparabolicity effect. The intravalley and intervalley electron scattering by phonons, scattering by
the ionized impurity, plasmons, and impact ionization process are taken into account
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[18, 20]
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.
It is known that the band structure of silicon in
valley X can be represented by three pairs of equivalent valleys, the isoenergetic surfaces of which in k
space are ellipsoids of a revolution with the axes of
symmetry oriented along crystallographic directions of
the type (100).
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In this study, the hole transport is simulated similarly
to the electron transport in the effective mass
approximation allowing for the nonparabolicity and
anisotropy of the dispersion relation in the valence
band. To do that we follow the work by RodriguezBolivar et al.
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[21]
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. The transport is taken into account
in the band of heavy and light holes, and in the splitoff
band. The scattering of holes by acoustic and optical
phonons and by ionized impurity are involved in the
model [22, 23].
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The transport is taken into account
in the band of heavy and light holes, and in the splitoff
band. The scattering of holes by acoustic and optical
phonons and by ionized impurity are involved in the
model
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[22, 23]
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. Dispersion relations for holes can be
written in the form:
(2)
(3)
(4)
(5)
EE
k
m
i
ii
().1
2
22
1
3
+=
=
α∑
Ek
k
m
LLAgEE()(,)(),;=+()≥
22
20
10θφχ
Ek
k
m
soEE
so
()sososo(),;=+≥
22
2
χ∆∆
g
B
A
C
A
(,)(sinsincoscossin).θφθφφθθ=++
2
2
2
2
42222
Ek
k
m
HHAgEE()(,)(),;=−()≥
22
20
10θφχ
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In equalities (2)–(5), indices «H», «L», and «so»
tering rate of electrons by ph
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The most common values for these parameters
for silicon are Eth = 1.2 eV and P = 0.38 for so called
«soft» threshold model and Eth = 1.8 eV and P = 100
for so called «hard» threshold model
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[27]
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. Briefly
the difference between these two kinds of Keldysh
models can be described as follows. In the hard
threshold model it is assumed that during the impact
ionization event the rules of energy and momentum
conservation must be fulfilled.
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Due to this fact restriction associated with
the momentum conservation may be neglected. Previously, the comparison of soft and hard threshold
models was done while simulating electrical characteristics and effective threshold energy in deep submicron MOSFET in
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[28]
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. Also in the framework of
Keldysh model some aspects of impact ionization
effective threshold energy in deep submicron silicon MOSFETs were investigated in [16, 29]. In this
study we will use the parameters of the soft threshold
as by now it is supposed that impact ionization process is more likely to occur within the soft threshold
model and estimations based on fullband
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17709
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Previously, the comparison of soft and hard threshold
models was done while simulating electrical characteristics and effective threshold energy in deep submicron MOSFET in [28]. Also in the framework of
Keldysh model some aspects of impact ionization
effective threshold energy in deep submicron silicon MOSFETs were investigated in
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[16, 29]
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. In this
study we will use the parameters of the soft threshold
as by now it is supposed that impact ionization process is more likely to occur within the soft threshold
model and estimations based on fullband calculations indicate this.
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The most common
situation for Keldysh model is the definition of particle final states after scattering via the assumption
that near threshold the group velocities of the final
particles are equal and for spherical parabolic bands
all wave vectors are collinear
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[27]
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.
By now more sophisticated models of impact
ionization process based on fullband calculations
have been developed [25, 30–32]. These types of
models usually have no fitting parameters, but their
implementation is restricted by the complexity of
scattering rate calculation and definition of the particle final states which result in too much computational effort.
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most common
situation for Keldysh model is the definition of particle final states after scattering via the assumption
that near threshold the group velocities of the final
particles are equal and for spherical parabolic bands
all wave vectors are collinear [27].
By now more sophisticated models of impact
ionization process based on fullband calculations
have been developed
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[25, 30–32]
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. These types of
models usually have no fitting parameters, but their
implementation is restricted by the complexity of
scattering rate calculation and definition of the particle final states which result in too much computational effort.
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These types of
models usually have no fitting parameters, but their
implementation is restricted by the complexity of
scattering rate calculation and definition of the particle final states which result in too much computational effort. Basing on the fullband approach in
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[32]
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the expression for impact ionization scattering
rate for silicon was derived in a rather simple fitted
form:
(7)
where electron energy E is in eV. Moreover, the calculation revealed that the average energy of secondary
generated particles depends linearly on the primary
electron energy after the scattering event.
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4.22; B = –0.78;
C = 4.80; θ and j are the angles in a spherical
coordinate system in the space of wave vectors; m0 is
the free electron mass; mso is the hole effective mass
in the splitoff band; Δso = 0.044 eV; and χ are the
functions that describe the nonparabolicity of the
dispersion relation in the valence band, the form and
approximation of which are presented in
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[21]
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.
Impact ionization process simulation
Impact ionization is a threshold process
[24– 26]. In a simple case, threshold energy Eth can
be determined using the energy and momentum
conservation laws and minimization of the energy of
final particles.
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values of threshold
energies are possible and it may be concluded
that the effective (or average) threshold energy of
charge carriers depend on electric field strength.
The effective threshold energy can be defined as
the energy corresponding to maximum value of
the product of impact ionization cross section and
electron distribution function
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[25]
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. When simulating
the electric properties of bulk silicon and silicon
MOSFETs by the Monte Carlo method, in order
to calculate the dependence of impact ionization
scattering rate WII(E) with specified threshold energy
Eth on energy E, many authors currently use Keldysh
formula [24, 27]:
(6)
where P is a parameter and Wph(
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When simulating
the electric properties of bulk silicon and silicon
MOSFETs by the Monte Carlo method, in order
to calculate the dependence of impact ionization
scattering rate WII(E) with specified threshold energy
Eth on energy E, many authors currently use Keldysh
formula
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[24, 27]
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:
(6)
where P is a parameter and Wph(Eth) is the total scat165
sible to define carrier states after the impact ionizaa rapid rise of current density in the channel for
VD > 1.5 V.
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scattering rate WII(E) with specified threshold energy
Eth on energy E, many authors currently use Keldysh
formula [24, 27]:
(6)
where P is a parameter and Wph(Eth) is the total scat165
sible to define carrier states after the impact ionizaa rapid rise of current density in the channel for
VD > 1.5 V. While the fullband model
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[25]
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gives
a rather moderate avalanche multiplication in the
channel under considered simulation conditions. The
latter proves that the use of more rigorous models
based on the calculation of realistic silicon band
structure may be crucial for calculation of submicron
SOI MOSFET characteristics.
tion event.
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In our opinion the procedure is the most
suitable for application in Monte Carlo simulations
among other approaches based on fullband calculations. The aim of current study was to compare the
influence of the choice between soft threshold Keldysh
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[27]
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and fullband [25] electron impact ionization models on the calculation of the SOI MOSFET
characteristics and determine the device operation
modes when impact ionization starts to make sufficient influence on the channel current.
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In our opinion the procedure is the most
suitable for application in Monte Carlo simulations
among other approaches based on fullband calculations. The aim of current study was to compare the
influence of the choice between soft threshold Keldysh [27] and fullband
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[25]
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electron impact ionization models on the calculation of the SOI MOSFET
characteristics and determine the device operation
modes when impact ionization starts to make sufficient influence on the channel current.
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models on the calculation of the SOI MOSFET
characteristics and determine the device operation
modes when impact ionization starts to make sufficient influence on the channel current. In current
study we regard only impact ionization by electrons
since they are the main charge carriers in the SOI
MOSFET. Also the threshold energy for holes is high
enough (1,49 eV)
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[33]
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.
Results and discussion
The current–voltage (I–V) characteristics of
the simulated SOI MOSFET both with and without
account of the impact ionization process are presented
in Figure 2.
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Figure 3 – Electron drift velocity (a) and average kinetic
energy (b) along the transistor channel at VD = 2,5 V and
VG = 1,5 V: solid curves correspond to the case when
impact ionization process is neglected, dashed curves –
fullband model
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[25]
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, and dotted curves – Keldysh
model [27]
In the Figures 3 and 4 the results of calculated
electron drift velocities and average energy versus
the distance along the transistor channel with the use
of both Keldysh and full band models are presented.
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Figure 3 – Electron drift velocity (a) and average kinetic
energy (b) along the transistor channel at VD = 2,5 V and
VG = 1,5 V: solid curves correspond to the case when
impact ionization process is neglected, dashed curves –
fullband model [25], and dotted curves – Keldysh
model
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[27]
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In the Figures 3 and 4 the results of calculated
electron drift velocities and average energy versus
the distance along the transistor channel with the use
of both Keldysh and full band models are presented.
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At the same time
the difference in current densities (see Figure 2) for
given models is already significant. The latter may be
Figure 2 – Currentvoltage characteristics of the SOI
MOSFET: solid curves – impact ionization process is not
taken into account, dashed curves – fullband
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[25]
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and
dotted curves – Keldysh [27] model of impact ionization
Analysis of Figure 2 shows that the linear
region of the I–V characteristics for the transistor
corresponds to the drain voltage range 0 ≤ VD ≤ 0.5 V.
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The latter may be
Figure 2 – Currentvoltage characteristics of the SOI
MOSFET: solid curves – impact ionization process is not
taken into account, dashed curves – fullband [25] and
dotted curves – Keldysh
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[27]
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model of impact ionization
Analysis of Figure 2 shows that the linear
region of the I–V characteristics for the transistor
corresponds to the drain voltage range 0 ≤ VD ≤ 0.5 V.
The saturation region occurs at voltages VD > 0.5 V,
up to approximately 1.5 V.
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For
a given transistor structure Keldysh model gives
166
referred to much higher electronhole pair generation
Figure 4 – Electron drift velocity (a) and average
kinetic energy (b) along the transistor channel at VD = 3.5 V
and VG = 1.5 V: solid curves correspond to the case when
impact ionization process is neglected, dashed curves – fullband model
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[25]
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, and dotted curves – Keldysh model [27]
It should be mentioned here that according to our
simulation the scattering rates calculated by all fullband approaches, presented in [25, 30–32] give close
values of the drain current.
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structure Keldysh model gives
166
referred to much higher electronhole pair generation
Figure 4 – Electron drift velocity (a) and average
kinetic energy (b) along the transistor channel at VD = 3.5 V
and VG = 1.5 V: solid curves correspond to the case when
impact ionization process is neglected, dashed curves – fullband model [25], and dotted curves – Keldysh model
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[27]
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It should be mentioned here that according to our
simulation the scattering rates calculated by all fullband approaches, presented in [25, 30–32] give close
values of the drain current. So the most convenient
may be the use of equation (7) for calculation of
impact ionization scattering rate as it has the same
simplicity as Keldysh formula (6).
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a) and average
kinetic energy (b) along the transistor channel at VD = 3.5 V
and VG = 1.5 V: solid curves correspond to the case when
impact ionization process is neglected, dashed curves – fullband model [25], and dotted curves – Keldysh model [27]
It should be mentioned here that according to our
simulation the scattering rates calculated by all fullband approaches, presented in
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[25, 30–32]
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give close
values of the drain current. So the most convenient
may be the use of equation (7) for calculation of
impact ionization scattering rate as it has the same
simplicity as Keldysh formula (6).
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So the most convenient
may be the use of equation (7) for calculation of
impact ionization scattering rate as it has the same
simplicity as Keldysh formula (6). For the definition
of the final states we chose the procedure proposed
in
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[32]
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and shortly discussed earlier as the most
convenient from the computational point of view
among others based on fullband approach.
Conclusion
In this study electric characteristics of a deep
submicron SOI MOSFET with 100 nm channel
length have been simulated by means of ensemble
Monte Carlo method.
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