Christian Ringhofer - US grants

This research will consider mathematical and numerical aspects of models describing semiconductor transport in novel device structures. Two aspects of the problems will be considered in tandem. First, increasing miniaturization results in effects which, while they can still be described by the Basic Semiconductor Equations (BSE), are intrinsically three- dimensional in nature. To address this problem, this research will develop and analyze an iterative algorithm for solution of the nonlinear BSE using a decoupling procedure based on asymptotic analysis. Secondly, miniaturization leads to physical phenomena, such as "hot electrons," which are not described by the BSE. This research will thus also consider the development and use of new model equations. This research could have a direct impact on the design of semiconductor devices and aid in the development of advanced technology.

Proposal # 0204543
PI: Dieter H. Armbruster
Institution: Arizona State Univsity
Title: Dynamics and Control in Semiconductor Manufacturing Lines

ABSTRACT

It is proposed to apply nonlinear dynamics methods from dynamical systems theory and fluid dynamics to three problems in industrial mathematics: 1) The organization of production lines with flexible workers. A dynamical systems model will be derived and analyzed. The goal is to determine whether chaos or other instabilities are present, to find out whether the production line is still self-organizing, and to determine the resulting throughput. 2) A switched arrival system
for three parallel machines has been shown to be chaotic. Increasingly realistic production scenarios will be added to this model, and the resulting dynamics will be determined. Networks of such systems will be studied to determine whether synchronization or partial synchronization can occur. Such systems are hybrid dynamical systems that require new tools to study their dependence on changing parameters. Such tools will be developed as extensions of standard dynamical systems methods. 3) A hierarchy of fluid-dynamical models to describe the flow of products through factories based on conservation laws will be developed. These models are based on the methods of gas dynamics and fluid averaging and promise to allow fast and accurate simulations for supply networks.

This project is based on an ongoing collaboration between researchers from Mathematics, Industrial Engineering, and Intel Corporation. We expect that our results have a significant impact on the understanding of the relationship between policies and management decisions and the resulting performance of factories and whole business networks. The three major areas of inquiry will be: i) The management concept of bucket brigades: The organizational principle of bucket brigades will be studied for workers with different skill levels along the production line as well as changing skill levels due to learning. ii) There are theoretical results stating that certain work allocation rules for parallel machines will lead to chaotic behavior. These results are based on highly abstracted models. We will study more realistic models and larger networks of production machines. iii) Supply chain modeling and simulation: We will derive models that allow fast scalable simulations of production flows in a supply chain. The long-term goal is to optimize production across the whole supply chain, an intermediate goal is to generate simulation tools that allow to explore business questions and to pose "what if" questions on these simulations.

The purpose of the proposed research effort is to develop methodology which
allows simulation of effects playing an important role in nano-scale
devices, within the framework of existing device simulators. The important
underlying mechanisms are, on one hand, quantum effects such as tunneling
and quantization, and, on the other hand, many-body effects, electron-ion
interactions and single-dopant effects. The goal of the project is to
include these effects into existing models and codes in such a way, that
their influence on device performance is accurately represented, while
simulations can be made at a computational cost comparable to that of
existing particle-based device simulators. The basic approach to achieve
this goal consists of the use of novel forms of effective quantum potentials
and quantum versions of existing particle-based approaches to simulate
short-range many-body interactions. Comparison will be made with
experimental data provided by the Nanostructures Research Group at ASU, and
Intel and Motorola. The calibrated tools can help device designers in
fabricating optimal device structures with a reduced cost. The educational
effort of this project will be closely tied to the proposed research
activities. This will be accomplished via specially developed
interdisciplinary courses in a classical as well as a distance learning
format, and through the direct participation of students in the research
project on the graduate as well as the undergraduate level. Course
materials will be developed for this purpose and will be placed on the
Computational Electronics Hub that is being developed as part of an existing
NSF sponsored project.

Production flow is a strongly multiscale phenomenon: Networks of factories and distribution centers building a supply chain represent the large flow - long timescale networks, whereas the networks of individual machines and operators typically represent the local fast scales. Much progress has been made to characterize the individual local production unit: Discrete event and agent based models are routinely developed to study the dynamics of flows through such networks. However, due to the stochastic nature of the processes involved and due to the complexity of the networks, such simulations are prohibitively expensive to maintain and are not equipped well to answer questions on the behavior of the networks as a whole. The goal of this proposal is to generate the mathematical foundation for a link between the local and fast time scales and the global long-term time scales in complex production networks. Based on this link we will derive continuum simulation models of network production flows based on continuum approximations for product as well as production stages, leading to partial differential equation models related to traffic flow models. Comparisons between the continuum models and large-scale discrete event simulations will be performed. As an additional tool linking small-scale simulation to large-scale simulation we will also explore an equation-free modeling approach to these production systems.

There is a huge need for fast and accurate simulations of production flows in factories and even more so at the enterprise level or at the whole supply chain level allowing the user to vary policies and business scenarios and ask "what if" questions. This is typically true for example in semiconductor factories or car manufacturing. Currently, these simulations suffer from the fact that the complexity of the systems, the number of parts involved and the stochastic nature of production does not allow the simulations to be performed as part of a decision tool. Instead, simulation scenarios are typically done off-line and the results discussed in planning meetings. Answering a new question typically requires a detailed off-line recalculation. Our approach has the potential to develop the fundamental models to describe such flows through a scalable and fast simulation and discuss their validity and limitations. Overall we are developing the theory and the tools for a much better strategy-planning and business-evaluation environment.

The retina is part of the central nervous system and an ideal region to study information processing in the brain. It is accessible, well documented, and studied by researchers spanning the clinical, experimental, and theoretical sciences. Image processing in the retina begins in the outer plexiform layer, where bipolar, horizontal, and photoreceptor cells interact. The goal of this research is to mathematically model, in detail, the subcircuits of the outer plexiform layer, capturing spatio-temporal dynamics on two spatial scales: the scale of an individual synapse and the scale of the receptive field. The availability of electrophysiological, anatomical, and molecular biological data provides a rare opportunity to develop a complex multiscale model for the system. Mathematical modeling will be used to discover the functional impact of various circuitry elements, much the way an experimentalist does with selective pharmacological agents, except that not all the agents that are needed actually exist. A primary objective is to explore two competing hypotheses for explaining synaptic feedback effects that are observed experimentally in the outer plexiform layer of the retina. Specifically, the project will investigate if feedback effects in the cone photoreceptor's synapse are driven by electrical or chemical mechanisms, or both. New insights into the understanding of retinal circuitry from this project will impact research in health and medicine, including biomedical engineering, image processing, visual psychophysics, and pharmacology. The modeling and computational techniques resulting from this project will provide efficient and innovative approaches for other problem areas in applied mathematics and engineering.

Kinetic equations play a central role in many areas of mathematical physics, from micro- and nano-physics to continuum mechanics. They are an indispensable tool in the mathematical description of applications in physical and social sciences, from semi-conductors, polymers and plasma to traffic networking and swarming. The ultimate goal of this proposal is to develop novel analytical and numerical methods based on kinetic descriptions of complex phenomena with multiple scales and with a wide range of applications. The objective is to achieve a better understanding of problems which are in the forefront of current research and to contribute to the solution of long-standing problems by synergetic collaboration of theory, modeling and numerics. To this end, this Focus Research Group (FRG) will provide a platform, led by leading researchers from Universities of Maryland, Brown, Iowa State, Wisconsin-Madison, Arizona State, Austin-Texas and Toulouse, France, who will merge their expertise in the construction, analysis and implementation of kinetic descriptions for a selected suite of problems with crossing scales from quantum and micro scales to the macro scales. Topics to be discussed include kinetic descriptions of microscopic and quantum phenomena, and kinetic descriptions of macroscopic phenomena. As a recent novel example for the kinetic methodology we will use kinetic descriptions to study hyperbolic flows for complex supply chains. The theoretical and modeling aspects of this research program, on both microscopic and macroscopic scales, will be integrated with kinetic-based numerical methods for capturing ``smaller scales phenomena".

The rationale behind this proposal is a timely effort to address several important issues in modern applied mathematics. Kinetic theories are not new. Yet, there have been many major developments in kinetic modeling, kinetic theories and related numerical methods, with the potential for a considerable impact on emerging new fields in physical and social sciences. The proposed effort will significantly strengthen the leading role that the US researchers can play in pursuing cutting-edge research and training a new generation of applied mathematicians in this important field. We expect this project to contribute to the development of scientific workforce by advanced training for doctoral and postdoctoral researchers and by providing a platform for interdisciplinary interactions with researchers from related disciplines. Internal and external interactions will be maintained through synergetic collaborations which will come to fruition during the three annual workshops to be held in Maryland (Year 1), France and Brown (Year 2), and Wisconsin (Year 3). International meetings will be held as part of a series of interdisciplinary workshops organized by the Center for Scientific Computation and Mathematical Modeling (CSCAMM) at the University of Maryland. Project researchers will collaborate with the DOE Center for Multiscale Plasma Dynamics in CSCAMM, the DOE Ames Laboratory at Iowa State University, and the Institute for Computational Engineering and Sciences (ICES) at UT Austin.