Energy Models for One-Carrier Transport in Semiconductor Devices

Moment models of carrier transport, derived from the Boltzmann equation, 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 was 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 required empirical modification of this law, or nonstandard choices. Simulations of the hydrodynamic model in one and two dimensions, as well as simulations of a newly developed energy model, the RT model, are presented. The RT model, intermediate between the hydrodynamic and drift-diffusion model, was developed 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. Some mathematical results are presented and contrasted with the highly developed state of the drift-diffusion model

Numerical experiments on the accuracy of ENO and modified ENO schemes

Further numerical experiments are made assessing an accuracy degeneracy phenomena. A modified essentially non-oscillatory (ENO) scheme is proposed, which recovers the correct order of accuracy for all the test problems with smooth initial conditions and gives comparable results with the original ENO schemes for discontinuous problems

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

Semiannual final report, 1 October 1991 - 31 March 1992

A summary of research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period 1 Oct. 1991 through 31 Mar. 1992 is presented

Nonlinear dynamics and numerical uncertainties in CFD

The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching, approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with spurious behavior observed in CFD computations

Freedrop Testing and CFD Simulation of Ice Models from a Cavity into Supersonic Flow

Weapon release at supersonic speeds from an internal bay is highly advantageous. For this reason, both experimental and numerical methods were used to investigate store separation from a cavity (L=D=4.5) into Mach 2.94 flow. The experiment used a piezoresistive pressure transducer, Schlieren and high-speed photography for data acquisition. The computational solution used the OVERFLOW solver. A sphere and a Mk-82, scaled to 1:20, were formed using frozen tap water. The sphere model was freedrop tested experimentally and computationally, while the sub-scale store shaped model was freedrop tested experimentally. The total pressure was varied to alter the dynamic response of the model. Computed spectra of pressure signals were in reasonable agreement with those measured experimentally, while the trajectory and dynamics of the CFD sphere release closely matched the experiment. Two sawtooth spoiler devices were tested for effectiveness at high Mach numbers. Pressure measurements showed a detuning of the Rossiter tones but with an increase in the broadband levels. Furthermore, spoiler testing demonstrated the capability to enhance store separation. Mk-82 shapes were also tested, which proved that the experimental process can be used with representations of actual stores. Reducing the test pressure conditions to sub-atmospheric levels allowed sub-scale models to be accurately scaled in mass and moment of inertia using heavy Mach scaling laws

Dynamics of Numerics & Spurious Behaviors in CFD Computations

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos

Numerical Simulations of Shock and Rarefaction Waves Interacting With Interfaces in Compressible Multiphase Flows

Developing a highly accurate numerical framework to study multiphase mixing in high speed flows containing shear layers, shocks, and strong accelerations is critical to many scientific and engineering endeavors. These flows occur across a wide range of scales: from tiny bubbles in human tissue to massive stars collapsing. The lack of understanding of these flows has impeded the success of many engineering applications, our comprehension of astrophysical and planetary formation processes, and the development of biomedical technologies. Controlling mixing between different fluids is central to achieving fusion energy, where mixing is undesirable, and supersonic combustion, where enhanced mixing is important. Iron, found throughout the universe and a necessary component for life, is dispersed through the mixing processes of a dying star. Non-invasive treatments using ultrasound to induce bubble collapse in tissue are being developed to destroy tumors or deliver genes to specific cells. Laboratory experiments of these flows are challenging because the initial conditions and material properties are difficult to control, modern diagnostics are unable to resolve the flow dynamics and conditions, and experiments of these flows are expensive. Numerical simulations can circumvent these difficulties and, therefore, have become a necessary component of any scientific challenge. Advances in the three fields of numerical methods, high performance computing, and multiphase flow modeling are presented: (i) novel numerical methods to capture accurately the multiphase nature of the problem; (ii) modern high performance computing paradigms to resolve the disparate time and length scales of the physical processes; (iii) new insights and models of the dynamics of multiphase flows, including mixing through hydrodynamic instabilities. These studies have direct applications to engineering and biomedical fields such as fuel injection problems, plasma deposition, cancer treatments, and turbomachinery.PhDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133458/1/marchdf_1.pd