Joseph W. Jerome

Solution of the hydrodynamic device model using high-order nonoscillatory shock capturing algorithms

Simulation results for the hydrodynamic model are presented for an n/sup +/-n-n/sup +/ diode by use of shock-capturing numerical algorithms applied to the transient model with subsequent passage to the steady state. The numerical method is first order in time, but of high spatial order in regions of smoothness. Implementation typically requires a few thousand time steps. These algorithms, termed essentially nonoscillatory, have been successfully applied in other contexts to model the flow in gas dynamics, magnetohydrodynamics, and other physical situations involving the conservation laws of fluid mechanics. The presented semiconductor simulations reveal temporal and spatial velocity overshot, as well as overshoot relative to an electric field induced by the Poisson equation. Shocks are observed in the transient simulations for certain low-temperature parameter regimes. >

Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study

Poisson--Nernst--Planck (PNP) systems are considered in the case of vanishing permanent charge. A detailed case study, based on natural categories described by system boundary conditions and flux, is carried out via simulation and singular perturbation analysis. Our results confirm the rich structure inherent in these systems. A natural quantity, the quotient of the Debye and characteristic length scales, serves as the singular perturbation parameter. The regions of validity are carefully analyzed by critical comparisons and contrasts between the simulation and the perturbation solution, which can be represented in closed form.

Compressible Euler-Maxwell equations

Abstract The Euler-Maxwell equations as a hydrodynamic model of charge transport of semiconductors in an electromagnetic field are studied. The global approximate solutions to the initial-boundary value problem are constructed by the fractional Godunov scheme. The uniform hound and H −1 compactness are proved. The approximate solutions are shown convergent by weak convergence methods. Then, with some new estimates due to the presence of electromagnetic fields, the existence of a global weak solution to the initial-boundary value problem is established for arbitrarily large initial data in L∞

Energy Models for One-Carrier Transport in Semiconductor Devices

Moment models of carrier transport, derived from the Boltzmann equation, have made possible the simulation of certain key effects through such realistic assumptions as energy dependent mobility functions. This type of global dependence permits the observation of velocity overshoot in the vicinity of device junctions, not discerned via classical drift-diffusion models, which are primarily local in nature. It has been found that a critical role is played in the hydrodynamic model by the heat conduction term. When ignored, the overshoot is inappropriately damped. When the standard choice of the Wiedemann-Franz law is made for the conductivity, spurious overshoot is observed. Agreement with Monte-Carlo simulation in this regime has required empirical modification of this law, as observed by IBM researchers, or nonstandard choices. In this paper, simulations of the hydrodynamic model in one and two dimensions, as well as simulations of a newly developed energy model, the RT model, will be presented. The RT model, intermediate between the hydrodynamic and drift-diffusion model, was developed at the University of Illinois to eliminate the parabolic energy band and Maxwellian distribution assumptions, and to reduce the spurious overshoot with physically consistent assumptions. The algorithms employed for both models are the essentially non-oscillatory shock capturing algorithms, developed at UCLA during the last decade. Some mathematical results will be presented, and contrasted with the highly developed state of the drift-diffusion model.

Qualitative properties of steady-state Poisson-Nernst-Planck systems: mathematical study

We examine qualitative properties of solutions of self-consistent Poisson--Nernst--Planck systems, including uniqueness. In the case of vanishing permanent charge, the predominant case studied, our results unveil a rich structure inherent in these systems, one that is determined by the boundary conditions and the signs of the oppositely charged carrier fluxes. A particularly significant special case, that of simple boundary conditions, is shown to lead to uniqueness and to a complete characterization. This case underlies the more complicated cases studied later. A contraction mapping principle is included for completeness and allows for an arbitrary permanent charge distribution.

Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation

In this paper we introduce a new method for numerically solving the equations of the hydrodynamic model for semiconductor devices in two space dimensions. The method combines a standard mixed finite element method, used to obtain directly an approximation to the electric field, with the so-called Runge-Kutta Discontinuous Galerkin (RKDG) method, originally devised for numerically solving multi-dimensional hyperbolic systems of conservation laws, which is applied here to the convective part of the equations. Numerical simulations showing the performance of the new method are displayed, and the results compared with those obtained by using Essentially Nonoscillatory (ENO) finite difference schemes. From the perspective of device modeling, these methods are robust, since they are capable of encompassing broad parameter ranges, including those for which shock formation is possible. The simulations presented here are for Gallium Arsenide at room temperature, but we have tested them much more generally with considerable success.

Analytical approaches to charge transport in a moving medium

ABSTRACT We consider electrodiffusion in an incompressible electrolyte medium which is in motion. The Cauchy problem is governed by a coupled Navier-Stokes/Poisson–Nernst–Planck system. We prove the existence of a unique smooth local solution for smooth initial data, with nonnegativity preserved for the ion concentrations. We make use of semigroup ideas, originally introduced by T. Kato in the 1970s for quasi-linear hyperbolic systems. The time interval is invariant under the inviscid limit to the Euler/Poisson–Nernst–Planck system.

A finite element approximation theory for the drift diffusion semiconductor model

Two-sided estimates are derived for the approximation of solutions to the drift-diffusion steady-state semiconductor device system which are identified with fixed points of Gummel’s solution map. The approximations are defined in terms of fixed points of numerical finite element discretization maps. By use of a calculus developed by Krasnosel’skii and his coworkers, it is possible both to locate approximations near fixed points in an “a priori” manner, as well as fixed points near approximations in an “a posteriors” manner. These results thus establish a nonlinear approximation theory, in the energy norm, with rate keyed to what is possible in a standard linear theory. This analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation.

On a steady-state quantum hydrodynamic model for semiconductors

A third order quantum perturbation of the stress tensor, and a relaxation approximation to represent averaged collisions, are employed as perturbations of the isentropic model for a collisionless plasma. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. As formulated, the model is a reduced version of the quantum hydrodynamic model for semiconductors. Existence is demonstrated for the model, which is shown to be equivalent to a non-standard integro-differential equation. An unusual boundary condition, with the important physical interpretation of specifying the quantum potential at the (current) inflow boundary, is identified as essential for the theory.

Device benchmark comparisons via kinetic, hydrodynamic, and high-hield models

Abstract This paper describes benchmark comparisons for a GaAs n + −n−n + diode. A global kinetic model is simulated, and compared with various realizations of the hydrodynamic model, depending on mobility calibration. Finally, the channel region alone is simulated, with interior boundary conditions derived from the kinetic model, by use of the high-field (augmented drift-diffusion) model.