The 35 reference contexts in paper A. Borzdov V., V. Borzdov M., N. Dorozhkin N., А. Борздов В., В. Борздов М., Н. Дорожкин Н. (2016) “ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ЭЛЕКТРИЧЕСКИХ ХАРАКТЕРИСТИК ГЛУБОКОСУБМИКРОННОГО МОП-ТРАНЗИСТОРА СО СТРУКТУРОЙ «КРЕМНИЙ НА ИЗОЛЯТОРЕ» // NUMERICAL SIMULATION OF ELECTRIC CHARACTERISTICS OF DEEP SUBMICRON SILICON-ON-INSULATOR MOS TRANSISTOR” / spz:neicon:pimi:y:2016:i:2:p:161-168

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    DOI: 10.21122/2220-9506-2016-7-2-161-168 162 Introduction Silicon-on-insulator (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 short-channel effects in comparison with common MOSFETs
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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 single-photon 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 single-photon 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 short-channel 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 cross-section 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 so-called particle technique
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    . The Monte Carlo procedure is two-dimensional in real space and three-dimensional in momentum space. The latter is caused by the fact that state-of-the-art 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 state-of-the-art 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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    . 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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    . 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 three-dimensional.
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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 three-dimensional. 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 Si-SiO2 interfaces. For three-dimensional 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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    . 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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    . The transport is taken into account in the band of heavy and light holes, and in the split-off 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 split-off band. The scattering of holes by acoustic and optical phonons and by ionized impurity are involved in the model
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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 164 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 full-band
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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 full-band 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 full-band 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 full-band 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 full-band approach in
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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 split-off 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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    . 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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    . 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 full-band model
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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 full-band calculations. The aim of current study was to compare the influence of the choice between soft threshold Keldysh
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    and full-band [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 full-band calculations. The aim of current study was to compare the influence of the choice between soft threshold Keldysh [27] and full-band
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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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    . 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 – full-band model
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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 – full-band model [25], and dotted curves – Keldysh model
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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 – Current-voltage characteristics of the SOI MOSFET: solid curves – impact ionization process is not taken into account, dashed curves – full-band
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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 – Current-voltage characteristics of the SOI MOSFET: solid curves – impact ionization process is not taken into account, dashed curves – full-band [25] and dotted curves – Keldysh
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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 electron-hole 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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    , 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 electron-hole 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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    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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    and shortly discussed earlier as the most convenient from the computational point of view among others based on full-band 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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