Mathematical Sciences: Classical and Quantum Fluid Transport Models for Semiconductor Devices

杰罗姆 研究半导体器件，专注于此类器件的流体动力学和量子流体动力学模型的模拟和数学分析。 经典研究强调微波和多组分模型以及摩擦的决定性作用，量子研究强调滞后和双稳态分析。 其基本原理是建立一种方法来处理经典结构中的热电子效应的各个方面，以及异质结构或化合物半导体中的隧穿和排斥效应。 用于模拟的算法是有限差分和有限元类型的冲击捕获算法。 多物种模型允许模拟电路涉及的古恩二极管，和它的三个谷电子载流子，由新的流体动力学模型。 这些相比，由厄巴纳组，通过Monte Carlo模拟玻尔兹曼方程的计算密集型的结果。 半导体器件的数学和计算机模型越来越需要结合两种类型的效应：（1）能量学，通常称为热电子效应，以及（2）小尺度或量子力学效应。 这个提议描述了两个重要的模型，第一个基于经典物理学，第二个基于量子物理学。 虽然基本物理定律已经被理解了至少75年，但将这些定律应用于半导体器件的努力，允许合理有效的建模和仿真，这对科学家，工程师和数学家来说是一个巨大的挑战。 经典建模的一个例子是古恩二极管，它被认为是至关重要的微波制造应用;这样的设备出现在汽车的电路，例如。 它的功能取决于电流和电压的微妙振荡，如果它们有任何用处，必须由计算机模型捕获。 首席研究员对这项工作可能导致的制造业部门的任何合作持开放态度。 量子研究的一个可能的应用是（由其他人）开发电路中的逻辑门，这些逻辑门不根据二元“关-开”定律运行，但允许多种状态的占用。 所描述的工作是高度协作和跨学科的，包括来自伊利诺伊大学厄巴纳分校的电气工程师。 来自亚利桑那州立大学、布朗大学和明尼苏达大学的计算数学家，以及来自柯朗研究所和斯坦福大学的数学分析师都参与了进来。 过去的工作成果定期在计算电子学研讨会上发表，并发表在电气工程和计算物理学杂志上。 这种做法仍在继续。

Jerome The investigator studies semiconductor devices, focusing on the simulation and mathematical analysis of the hydrodynamic and quantum hydrodynamic models for such devices. Microwave and multi-species models, and the decisive role of friction, are emphasized in the classical studies, and hysteresis, bistable-state analysis, are emphasized in the quantum studies. The rationale is to establish a way of dealing with various aspects of hot electron effects in classical structures, and the effects of tunneling and repulsion in heterogeneous structures, or compound semiconductors. The algorithms to be employed for the simulations are shock capturing algorithms of finite difference and finite element type. The multi-species models permit simulation of the circuit involving the Gunn diode, and its three valley electron carriers, by new hydrodynamic models. These are compared with the computationally intensive results obtained by the Urbana group, via Monte Carlo simulation of the Boltzmann equation. Mathematical and computer models of semiconductor devices have increasingly necessitated the incorporation of two types of effects: (1) energetics, often called the hot electron effect, and (2) small scale, or quantum mechanical, effects. This proposal describes two important models, the first based upon classical, and the second upon quantum, physics. Although the basic physical laws have been understood for at least seventy-five years, the efforts to apply these laws to semiconductor devices in a way that allows for reasonably efficient modeling and simulation represents an enormous challenge to scientists, engineers, and mathematicians. An example of classical modeling is the Gunn diode, which is acknowledged to be of critical importance in microwave manufacturing applications; such devices appear in the circuitry of the automobile, for example. Its function depends upon delicate oscillations of current and voltage, which must be captured by computer models if they are to be of any usefulness. The principal investigator is open to any collaboration in the manufacturing sector which might result from this work. A possible application of the quantum studies is the development (by others) of logic gates in circuits that do not operate according to binary "off-on" laws, but allow for multiple states of occupancy. The work described is highly collaborative and interdisciplinary, including electrical engineers from the University of Illinois at Urbana. Computational mathematicians from Arizona State and Brown Universities, and the University of Minnesota, and mathematical analysts from the Courant Institute and from Stanford University, are involved. The results of past work have been regularly presented at workshops on computational electronics, and published in journals of electrical engineering and computational physics. This practice is continued.