Orazio Muscato, Wolfgang Wagner and Vincenza Di Stefano
– The purpose of this paper is to deal with the self-heating of semiconductor nano-devices.
Abstract
Purpose
The purpose of this paper is to deal with the self-heating of semiconductor nano-devices.
Design/methodology/approach
Transport in silicon semiconductor devices can be described using the Drift-Diffusion model, and Direct Simulation Monte Carlo (MC) of the Boltzmann Transport Equation.
Findings
A new estimator of the heat generation rate to be used in MC simulations has been found.
Originality/value
The new estimator for the heat generation rate has better approximation properties due to reduced statistical fluctuations.
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Orazio Muscato and Vincenza Di Stefano
The purpose of this paper is to set up a consistent off‐equilibrium thermodynamic theory to deal with the self‐heating of electronic nano‐devices.
Abstract
Purpose
The purpose of this paper is to set up a consistent off‐equilibrium thermodynamic theory to deal with the self‐heating of electronic nano‐devices.
Design/methodology/approach
From the Bloch‐Boltzmann‐Peierls kinetic equations for the coupled system formed by electrons and phonons, an extended hydrodynamic model (HM) has been obtained on the basis of the maximum entropy principle. An electrothermal Monte Carlo (ETMC) simulator has been developed to check the above thermodynamic model.
Findings
A 1D n+−n−n+ silicon diode has been simulated by using the extended HM and the ETMC simulator, confirming the general behaviour.
Research limitations/implications
The paper's analysis is limited to the 1D case. Future researches will also consider 2D realistic devices.
Originality/value
The non‐equilibrium character of electrons and phonons has been taken into account. In previous works, this methodology was used only for equilibrium phonons.
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Hydrodynamic‐like models are commonly used for describing carrier transport in semiconductor devices. One major problem of this formulation is how to model the production terms…
Abstract
Hydrodynamic‐like models are commonly used for describing carrier transport in semiconductor devices. One major problem of this formulation is how to model the production terms. In this paper the relaxation‐time approximation and the moments expansion of the production terms are checked with Monte Carlo simulations for a one dimensional n+ – n – n+ silicon diode in the spherical parabolic band approximation.
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A. Majorana, O. Muscato and C. Milazzo
Time‐depending solutions to the Boltzmann‐Poisson system in one spatial dimension and three‐dimensional velocity space are obtained by using a recent finite difference numerical…
Abstract
Time‐depending solutions to the Boltzmann‐Poisson system in one spatial dimension and three‐dimensional velocity space are obtained by using a recent finite difference numerical scheme. The collision operator of the Boltzmann equation models the scattering processes between electrons and phonons assumed in thermal equilibrium. The numerical solutions for bulk silicon and for a one‐dimensional n+‐n‐n+ silicon diode are compared with the Monte Carlo simulation. Further comparisons with the experimental data are shown.
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Marco Coco and Giovanni Nastasi
The purpose of this paper is to simulate charge transport in monolayer graphene on a substrate made of hexagonal boron nitride (h-BN). This choice is motivated by the fact that…
Abstract
Purpose
The purpose of this paper is to simulate charge transport in monolayer graphene on a substrate made of hexagonal boron nitride (h-BN). This choice is motivated by the fact that h-BN is one of the most promising substrates on account of the reduced degradation of the velocity due to the remote impurities.
Design/methodology/approach
The semiclassical Boltzmann equations for electrons in the monolayer graphene are numerically solved by an approach based on a discontinuous Galerkin (DG) method. Both the conduction and valence bands are included, and the inter-band scatterings are taken into account as well.
Findings
The importance of the inter-band scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the inter-band scatterings produces sizable variations in the average values, like the current density, at zero Fermi energy, whereas, as expected, the effect of the inter-band scattering becomes negligible by increasing the absolute value of the Fermi energy.
Research limitations/implications
The correct evaluation of the influence of the inter-band scatterings on the electronic performances is deeply important not only from a theoretical point of view but also for the applications. In particular, it will be shown that the time necessary to reach the steady state is greatly affected by the inter-band scatterings, with not negligible consequences on the switching on/off processes of realistic devices. As a limitation of the present work, the proposed approach refers to the spatially homogeneous case. For the simulation of electron devices, non-homogenous numerical solutions are required. This last case will be tackled in a forthcoming paper.
Originality/value
As observed in Majorana et al. (2019), the use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the inter-band scatterings, is computationally very expensive because the valence band is highly populated and a huge number of particles is needed. Even by simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electron-hole pairs. The DG approach, used in this paper, does not suffer from the previous drawbacks and requires a reasonable computing effort.
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A.M. Blokhin and R.S. Bushmanov
The purpose of this paper is to reply to the following question: do there exist piecewise smooth solutions to the 2D MEP hydrodynamical model of charge transport in semiconductors…
Abstract
Purpose
The purpose of this paper is to reply to the following question: do there exist piecewise smooth solutions to the 2D MEP hydrodynamical model of charge transport in semiconductors with smooth parts separated by a surface of strong discontinuity?
Design/methodology/approach
A standard approach is used to obtain jump conditions for the balance equations under consideration.
Findings
For the balance equations of charge transport in semiconductors based on the maximum entropy principle Rankine‐Hugoniot jump conditions were derived and studied. Considering the important case of planar discontinuity, the authors discuss the legitimacy of the introduction of surface charge and surface current in the Rankine‐Hugoniot jump conditions.
Research limitations/implications
The jump conditions are derived for the balance equations written for the case of the parabolic approximation of energy bands. However, it is possible also to perform the analysis of corresponding jump conditions for the case of Kane dispersion relation approximation.
Originality/value
The paper presents derivation and study of Rankine‐Hugoniot jump conditions for the 2D MEP hydrodynamical model of charge transport in semiconductors.
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A.M. Blokhin and A.A. Iohrdanidy
A mathematical gas dynamic model for semiconductor devices is numerically analysed. The well‐known ballistic diode problem is taken as an example.
Abstract
A mathematical gas dynamic model for semiconductor devices is numerically analysed. The well‐known ballistic diode problem is taken as an example.
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Orazio Muscato and Wolfgang Wagner
To provide an accurate analysis of the systematic error introduced by the constant time technique free flight mechanism, due to the choice of the time step and number particles.
Abstract
Purpose
To provide an accurate analysis of the systematic error introduced by the constant time technique free flight mechanism, due to the choice of the time step and number particles.
Design/methodology/approach
A homogeneous (bulk) silicon semiconductor is studied by using direct simulation Monte Carlo (DSMC).
Findings
The systematic error turns out to be of the first order with respect to the time step. The efficiency of the method is tackled.
Research limitations/implications
The analysis is limited to the bulk case. Future researches will consider non homogeneous devices
Originality/value
An accurate analysis of an “old” free flight mechanism has been performed, and its limits have been stated.
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Luca Ballestra and Fausto Saleri
In this paper, we solve by a finite difference upwinded method an extended hydrodynamic model for semiconductors, with viscous terms in the momentum equation. In particular, we…
Abstract
In this paper, we solve by a finite difference upwinded method an extended hydrodynamic model for semiconductors, with viscous terms in the momentum equation. In particular, we consider the simulation of a one‐dimensional n+‐n ‐n+ diode, whose solution exhibits at low temperatures strong discontinuities, and investigate the effect of the momentum viscosity on the shock waves. Numerical experiments, performed also on a two‐dimensional test case, demonstrate that the numerical scheme, working on non‐uniform grids, is suitable to describe solutions with strong variations in time and space. Well‐posedness for the boundary conditions is discussed, and a linear stability estimate is established for the one‐dimensional n+‐n ‐n+ diode benchmark problem.
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Giovanni Mascali and Vittorio Romano
This paper intends to present a hydrodynamical model which describes the hole motion in silicon and couples holes and electrons.
Abstract
Purpose
This paper intends to present a hydrodynamical model which describes the hole motion in silicon and couples holes and electrons.
Design/methodology/approach
The model is based on the moment method and the closure of the system of moment equations is obtained by using the maximum entropy principle (hereafter MEP). The heavy, light and split‐off valence bands are considered. The first two are described by taking into account their warped shape, while for the split‐off band a parabolic approximation is used.
Findings
The model for holes is coupled with an analogous one for electrons, so obtaining a complete description of charge transport in silicon. Numerical simulations are performed both for bulk silicon and a p‐n junction.
Research limitations/implications
The model uses a linear approximation of the maximum entropy distribution in order to close the system of moment equations. Furthermore, the non‐parabolicity of the heavy and light bands is neglected. This implies an approximation on the high field results. This issue is under current investigation.
Practical implications
The paper improves the previous hydrodynamical models on holes and furnishes a complete model which couples electrons and holes. It can be useful in simulations of bipolar devices.
Originality/value
The results of the paper are new since a better approximation of the band structure is used and a description of both electron and hole behavior is present, therefore the results are of a certain relevance for the theory of charge transport in semiconductors.