Assessment of a High Resolution Centered Scheme for the Solution of Hydrodynamical Semiconductor Equations

Abstract

Hydrodynamical models are suitable to describe carrier transport in submicron semiconductor devices. These models have the form of nonlinear systems of hyperbolic conservation laws with source terms, coupled with Poisson's equation. In this article we examine the suitability of a high resolution centered numerical scheme for the solution of the hyperbolic part of these extended models, in one space dimension. Because of the lack of physically significant exact analytical solutions, the method is assessed against a benchmark for the system of compressible, unsteady Euler equations with source terms, which has an exact solution; the latter is shown to be nearly identical to the numerical one. The method is then used to solve the extended hydrodynamical model (EM) based on the maximum entropy closure recently introduced by Anile, Romano, and Russo, simulating a ballistic diode n + − n − n + , which models a metal oxide semiconductor field effect transistor (MOSFET) channel. Results are presented for the reduced-and full-equation EM formulation at steady state, for an initially discontinuous electron density at the junctions. Transient results show the evolution of highly nonlinear waves emanating from the neighborhood of the junctions. Introduction.Enhanced functional integration in modern electron devices requires an accurate modeling of energy transport in semiconductors in order to describe high-field phenomena such as hot-electron propagation, impact ionization, and heat generation in the bulk material. Furthermore, when using compound semiconductors for high frequency applications, usually one deals with multivalley band structures and in these cases the transfer of carriers from one valley to the other must also be modeled. The standard drift-diffusion models cannot cope with highfield phenomena because they do not comprise energy as a dynamical variable. Also they do not incorporate dynamical transfer of carriers from one valley to the other and this renders them ill-suited for simulating time dependent high frequency phenomena. Therefore, generalizations of the drift-diffusion equations have been sought which would incorporate energy as a dynamical variable and which also could treat time dependent high frequency phenomena. Because of their mathematical similarity to the equations of compressible fluid flow, these models are called hydrodynamical models. Semiconductor hydrodynamical models are obtained from the infinite hierarchy of moment equations of the semiclassical Boltzmann transport equation (BTE) by a suitable truncation procedure. This requires making suitable assumptions on (i) choosing the appropriate moments, (ii) closing the hierarchy of moment equations by finding appropriate expressions for the N + 1 order moment in terms of the previous

Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations

“…Anile and coworkers (Anile et al 2000) applied the central scheme SLIC in the context of the time-dependent hydrodynamical semiconductor equations, obtaining satisfactory results for a system whose characteristic information is not available. In the context of special and general relativistic MHD Koide et al (1996Koide et al ( , 1998) applied a second-order central scheme with nonlinear dissipation developed by Davis (1984) to the study of relativistic extragalactic jets and black hole accretion.…”

Numerical problems in semiconductor simulation using the hydrodynamic model: a second-order finite difference scheme

“…Concerning this latter point, we notice that the same argument could be used to explain the considerable loss of accuracy on the steady electron current at the drain junction that is experienced by Anile and coworkers [3][4][5], where an extended hydrodynamic model taking into account the first five moments of the Boltzmann equations is solved by the Nessyahu-Tadmor scheme: in this case the slope limiters act directly on the primitive variables U, and, looking at the results presented in the above works, at least two functions in the set U, the electron energy and the heat flux, exhibit points of local extrema at the end of the channel at steady state.…”

“…In semiconductor device simulation, high-order shock capturing algorithms have been applied in [17,19], where the HD model is solved by a third-order ENO finite difference approximation, and in [3][4][5], where the second-order Nessyahu-Tadmor finite difference scheme is used for the solution of an improved hydrodynamic model including viscous terms in the momentum-energy equations. In the aforementioned papers, numerical simulations of the one-dimensional n þ À n À n þ diode are presented to illustrate the performance of the discretization method.…”

A Finite Difference Scheme Solving the Boltzmann–Poisson System for Semiconductor Devices

“…We used a V bias = 1V and the same constants as in [1]. For the discretization of the variables t, z, w, µ we used the following values.…”