Strong discontinuities for the 2-D MEP hydrodynamical model of charge transport in semiconductors

Abstract

Abstract.For the balance equations of charge transport in semiconductors based on the maximum entropy principle (see Anile and Romano (1999) and Romano (2000)), we derive and study the Rankine-Hugoniot jump conditions. Introduction.In the hierarchy of macroscopic models of charge transport in semiconductors beyond the drift-diffusion equations [3,4,5] and the energy-transport models [6,7,8], one finds the hydrodynamical models that are obtained from an infinite set of moment equations of the Boltzmann Transport Equation by a suitable truncation procedure (see [9,10] for a complete review on the mathematical modelling for semiconductors).It is well known that moment systems require a closure assumption in order to yield a closed system of evolution equations. In [1, 2] the balance equations for density, momentum, energy, and energy flux of electrons have been considered for silicon and, by using the maximum entropy principle (MEP) for the closure, explicit constitutive relations for the stress tensor, the flux of energy flux tensor, and production terms have been obtained both in the parabolic and the Kane dispersion relation approximations for the energy conduction bands. The model fits into Extended Thermodynamics [10,11,12] and Levermore's theory of moments [13] and does not contain any fitting parameters. The validity of the constitutive equations has been checked in [14,15] and the general mathematical properties have been studied in [16], where it has been shown that the evolution equations form a hyperbolic system in the physically relevant region of the field space. Applications of the model have been presented in [15,16] for 1-D problems and in [17] for a 2-D simulation of a silicon MESFET (metal semiconductor field effect transistor). At last, the mathematical aspects of the model related to the stability of the equilibrium state for a 2-D silicon MESFET have been investigated in [18].In the present paper, we continue the study of the system of balance equations proposed in [1,2]. We are interested in piecewise smooth solutions of this system with smooth parts separated by a surface of strong discontinuity. Using a standard approach