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In this chapter a short overview will be given regarding the physics of semiconductors and some properties of the Boltzmann equation that is the starting point for describing the semiclassical transport of charge carriers.

In this chapter MEP is applied to close the moment equations for electrons in silicon semiconductors. In our model we consider the electrons distributed in the six X-valleys assumed as equivalent. The approximation given by Kane will be used as dispersion relation.

MEP can be used for solving the closure problem related to the moment systems associated to the electron transport equations. Here the case of 3D electron gas is considered. Lower dimensional electron gases will be treated in the next chapters.

To study the electron transport in a some tens of nanometers long channel of a metal oxide field effect transistor, in order to reduce the computational cost of simulations, it can be convenient to divide the electrons into a 2D and a 3D population. Near the silicon/oxide interface the two populations coexist, while in the remaining part of the device only the 3D component needs to be considered because quantum effects are negligible there. The major issue is the description of the scattering mechanisms between the 2D and the 3D electron populations, due to interactions of electrons with nonpolar optical phonons and interface modes. Here, we propose a rigorous treatment of these collisions based on an approach similar to that used in Fischetti and Laux (Phys Rev B 48:2244–2274, 1993), in the context of a Monte Carlo simulation. We also consider all the other main scatterings, which are those with acoustic phonons, surface roughness, and impurities.

We present a hierarchy of macroscopic models which generalize Cattaneo equation and can be used to describe heat conduction in semiconductor materials. In particular, from these models we derive a new formula for the lattice thermal conductivity which is a able to reproduce the results achieved by means of the celebrated Callaway formula and to give some insights into the dynamical thermal conductivity of semiconductor materials. The models make use of a set of macroscopic state variables for the acoustic phonons, that are moments of their occupation numbers. The evolution equations for these variables are obtained starting from the Boltzmann–Peierls transport equations, and are closed by means of the maximum entropy principle. All the main interactions of phonons among themselves, with isotopes and boundaries are taken into account. Numerical results are shown for the cases of silicon and germanium.

PurposeThis paper intends to present a hydrodynamical model which describes the hole motion in silicon and couples holes and electrons.Design/methodology/approachThe 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.FindingsThe 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/implicationsThe 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 implicationsThe 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/valueThe 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.

An exact closure is obtained of the 8-moment model for semiconductors based on the maximum entropy principle in the case of silicon semiconductors. The validity of the model is assessed, and comparisons with an approximate closure are presented and discussed.

A nanoscale double-gate MOSFET is simulated by using a subband model based on the maximum entropy principle (MEP).

We present the first macroscopical model for charge transport in compound semiconductors to make use of analytic ellipsoidal approximations for the energy dispersion relationships in the neighbours of the lowest minima of the conduction bands. The model considers the main scattering mechanisms charges undergo in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk 4H and 6H-SiC.

In this article we present a fluid dynamical model for the charge transport in semiconductors based on a recently found asymptotic solution of the semiconductor Boltzmann transport equation (BTE) for strong fields.

In this article, we present a theoretical foundation for tail electron hydrodynamical models (TEHM) in semiconductors.

In this paper we present a hydrodynamical model which, in principle, is able to describe charge transport in a generic compound semiconductor. The model makes use of an analytic approximation for the conduction bands. Energy dispersion relationships in the neighbors (valleys) of the lowest minima are, in fact, taken to be spherical, nonparabolic. The model considers the main scattering mechanisms in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk GaN and SiC.

A hydrodynamic subband model for semiconductors has been formulated in (Mascali and Romano, IL NUOVO CIMENTO 33C:155163, 2010) by closing the moment system derived from the Schrodinger-Poisson-Boltzmann equations on the basis of the maximum entropy principle (MEP). Explicit closure relations for the fluxes and the production terms are obtained taking into account scattering of electrons with acoustic and non-polar optical phonons, as well as surface scattering. Here a suitable numerical scheme is presented for the above model together with simulations of a nanoscale silicon diode.

In this paper we present exact closures of the 8-moment and the 9-moment models for the charge transport in silicon semiconductors based on the maximum entropy principle. The validity of these models is assessed by numerical simulations of an n-n-n device. The results are compared with those obtained from the numerical solution of the Boltzmann Transport Equation both by Monte Carlo method and directly by a finite difference scheme.

1. Kinetic model Semiconductors are characterized by a sizable energy gap between the va- lence and the conduction bands. The energy band structure of crystals can be obtained at the cost of intensive numerical calculations (and also semi-phenomenologically) by means of the quantum theory of solids. 1 The electrons, which mainly contribute to the charge transport, are those with energy near the lowest conduction band minima, each neighborhood being called valley. In silicon, which is the material we will deal with in this paper, there are six equivalent ellipsoidal valleys along the main crystallographic directions ¢ at about 85 % from the center of the flrst Brillouin zone, near the X points, which, for this reason, are termed as X-valleys. In the derivation of macroscopic models, usually, the energy in each valley is represented by analytical approximations. Among these, the most common one is the Kane dispersion relation, which describes the energy EA

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High-field phenomena in submicron electron devices cannot be described satisfactorily within the framework of the drift-diffusion models that do not include energy as a dynamical variable and are valid only in the quasi-stationary limit, while most hydrodynamical models suffer from serious theoretical drawbacks due to the ad hoc treatment of the closure problem [1]. Here we employ a moment approach, previously introduced in [2, 3] (see also [4] for a complete review) in which the closure procedure is based on the maximum entropy principle while the conduction bands are described by the Kane dispersion relation. The electrons in GaAs are considered as a mixture of two fluids, one representing the electrons in the Γ -valley and the other the electrons in the four equivalent L-valleys. The model comprises the balance equations of electron density, energy density, velocity and energy flux for both populations, coupled to the Poisson equation for the electric potential. We will give only a brief sketch of the model. For more details the interested reader is referred to [5]. One assumes that the conduction band is described in the neighborhood of each minimum (valley) by the Kane dispersion relation approximation

Consistent hydrodynamical models for electron transport in Si and GaAs semiconductors, free of any fitting parameter, have been formulated in (Cont. Mech. Thermodyn. 11 (1999) 307; Contemp. Mech. Thermodyn. 12 (1999) 31; Contemp. Mech. Thermodyn. 14 (2002) 405; COMPEL (to appear)) on the basis of the maximum entropy principle (MEP), by describing the valleys in the energy conduction band by means of the Kane dispersion relation. Explicit constitutive functions for fluxes and production terms appearing in the macroscopic balance equations of density, crystal momentum, energy and energy-flux have been obtained. Scatterings of electrons with polar (in the case of GaAs) and non-polar optical phonons, both for intervalley and intravalley interactions, and with acoustic phonons and impurities have been taken into account. Here we derive from the previous hydrodynamical models both low- and high-field mobilities. The results are compared with those given by the Caughey–Thomas formula and eventually the validity of the Einstein relation is investigated.

In this paper we present a theoretical foundation for tail electron hydrodynamical models (TEHM) in semiconductors with application to bulk silicon.

We present a review of recent developments in hydrodynamical modeling of charge transport in semiconductors. We focus our attention on the models for Si and GaAs based on the maximum entropy principle which, in the framework of extended thermodynamics, leads to the definition of closed systems of moment equations starting from the Boltzmann transport equation for semiconductors.

A hydrodynamical model for electron transport in GaAs semiconductors, which is free of any fitting parameter, has been formulated in [1] on the basis of the maximum entropy principle, by including both г and L conduction band valleys described by the Kane dispersion relation.

A hydrodynamical model based on the maximum entropy principle is formulated for GaAs semiconductors. Explicit closure relations for the moment equations of the electron density, energy, velocity and energy-flux are obtained by using the Kane dispersion approximation for the description of the conduction bands. All the relevant scattering mechanisms are included: interaction of electrons with acoustic, polar and non-polar optical phonons, impurities. Application to the bulk case reveal s that the model describes with accuracy the effect of negative differential conductivity, typical of GaAs, as well as the velocity overshoot and saturation.

The semiclassical Boltzmann (BE) equation is the reference model for charge transport in semic~nductors.~-~ In fact, not only the results of simulations by other models are usually compared with BE simulations, but most of the existing models in microelectronics have been or can be derived from BE.ly4 In the past, calculus of variations has been extensively applied to transport theory, mainly to obtain estimates of physical quantities or to derive