%0 Journal Article
%T Nonlinear Closure Relations for Electron Transport in Hydrodynamical Models
%A A. Salhoumi
%J Journal of Thermodynamics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/915297
%X Closure relations problem of hydrodynamical models in semiconductors is considered by expressing third- and fourth-order closure relations for the moments of the distribution function in terms of second-order Lagrange multipliers using a generalized Maxwell-Boltzmann distribution function within information theory. Calculation results are commented and compared with others to justify the accuracy of the approach developed in this paper. The comparison involves, in the first part with good agreements, the closure relations results obtained within extended thermodynamics which were checked by means of Monte Carlo simulations, in the second part, the results obtained by Grad's method which expands the distribution function up to fourth-order in Hermite polynomials. It is seen that the latter method cannot give any restriction on closure relations for higher-order moments, within the same conditions proposed in our approach. The important role of Lagrange multipliers for the determination of all closure relations is asserted. 1. Introduction The analysis of transport in small semiconductor devices is essential for the optimization of their functioning. Such transport could in principle be described by means of Boltzmann transport equation (BTE) for charge carriers. However, in small devices the electric fields are extremely large, and therefore nonlinear effects are unavoidable [1, 2] which leads to insurmountable difficulties to obtain solutions. Notwithstanding this, BTE contains more information than needed in practical applications. It is common in practice to consider only the lowest-order moments of the distribution function, which are directly related to density, charge flux, kinetic energy, heat flux, and so on. These variables are measured and controlled. This kind of approach is called a hydrodynamical approach [3¨C5]. The basic model, in which the various steps and approximations are derived and discussed in detail, is due to Blotekjaer [4]. So as to close the set of balance equations considered by Blotekjaer, one assumes that higher-order moments have the value appropriate for a displaced Maxwellian. A slightly different model has been suggested by H£¿nsch and Miura-Mattausch [6]. In their model, the distribution function is expanded in Legendre polynomials and only the first two terms in the expansion are retained. Only the five balance equations for particle number, momentum, and energy are considered; then the closure is accomplished by means of the Wiedemann-Franz law for heat flux. Both models [4, 6] are then further simplified in order to
%U http://www.hindawi.com/journals/jther/2013/915297/