Simulation of hot carriers in semiconductor devices

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (p. 109-113).by Khalid Rahmat.Ph.D

Simulation of hot carriers in semiconductor devices

Includes bibliographical references (p. 109-113).Supported by the U.S. Navy. N00174-93-C-0035Khalid Rahmat

Angular adaptivity with spherical harmonics for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses Pn and filtered Pn expansions, allowing for different expansion orders across space/energy. Our spatial discretisation is specifically designed to use less memory than competing DG schemes and also gives us direct access to the amount of stabilisation applied at each node. For filtered Pn expansions, we then use our adaptive process in combination with this net amount of stabilisation to compute a spatially dependent filter strength that does not depend on a priori spatial information. This applies heavy filtering only where discontinuities are present, allowing the filtered Pn expansion to retain high-order convergence where possible. Regular and goal-based error metrics are shown and both the adapted Pn and adapted filtered Pn methods show significant reductions in DOFs and runtime. The adapted filtered Pn with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with P0 discretisations in problems with heavy advection.Comment: arXiv admin note: text overlap with arXiv:1901.0492

Synergies between Numerical Methods for Kinetic Equations and Neural Networks

The overarching theme of this work is the efficient computation of large-scale systems. Here we deal with two types of mathematical challenges, which are quite different at first glance but offer similar opportunities and challenges upon closer examination. Physical descriptions of phenomena and their mathematical modeling are performed on diverse scales, ranging from nano-scale interactions of single atoms to the macroscopic dynamics of the earth\u27s atmosphere. We consider such systems of interacting particles and explore methods to simulate them efficiently and accurately, with a focus on the kinetic and macroscopic description of interacting particle systems. Macroscopic governing equations describe the time evolution of a system in time and space, whereas the more fine-grained kinetic description additionally takes the particle velocity into account. The study of discretizing kinetic equations that depend on space, time, and velocity variables is a challenge due to the need to preserve physical solution bounds, e.g. positivity, avoiding spurious artifacts and computational efficiency. In the pursuit of overcoming the challenge of computability in both kinetic and multi-scale modeling, a wide variety of approximative methods have been established in the realm of reduced order and surrogate modeling, and model compression. For kinetic models, this may manifest in hybrid numerical solvers, that switch between macroscopic and mesoscopic simulation, asymptotic preserving schemes, that bridge the gap between both physical resolution levels, or surrogate models that operate on a kinetic level but replace computationally heavy operations of the simulation by fast approximations. Thus, for the simulation of kinetic and multi-scale systems with a high spatial resolution and long temporal horizon, the quote by Paul Dirac is as relevant as it was almost a century ago. The first goal of the dissertation is therefore the development of acceleration strategies for kinetic discretization methods, that preserve the structure of their governing equations. Particularly, we investigate the use of convex neural networks, to accelerate the minimal entropy closure method. Further, we develop a neural network-based hybrid solver for multi-scale systems, where kinetic and macroscopic methods are chosen based on local flow conditions. Furthermore, we deal with the compression and efficient computation of neural networks. In the meantime, neural networks are successfully used in different forms in countless scientific works and technical systems, with well-known applications in image recognition, and computer-aided language translation, but also as surrogate models for numerical mathematics. Although the first neural networks were already presented in the 1950s, the scientific discipline has enjoyed increasing popularity mainly during the last 15 years, since only now sufficient computing capacity is available. Remarkably, the increasing availability of computing resources is accompanied by a hunger for larger models, fueled by the common conception of machine learning practitioners and researchers that more trainable parameters equal higher performance and better generalization capabilities. The increase in model size exceeds the growth of available computing resources by orders of magnitude. Since $2012$, the computational resources used in the largest neural network models doubled every $3.4$ months\footnote{\url{https://openai.com/blog/ai-and-compute/}}, opposed to Moore\u27s Law that proposes a $2$-year doubling period in available computing power. To some extent, Dirac\u27s statement also applies to the recent computational challenges in the machine-learning community. The desire to evaluate and train on resource-limited devices sparked interest in model compression, where neural networks are sparsified or factorized, typically after training. The second goal of this dissertation is thus a low-rank method, originating from numerical methods for kinetic equations, to compress neural networks already during training by low-rank factorization. This dissertation thus considers synergies between kinetic models, neural networks, and numerical methods in both disciplines to develop time-, memory- and energy-efficient computational methods for both research areas

Multidisciplinary benchmarks of a conservative spectral solver for the nonlinear Boltzmann equation

The Boltzmann equation describes the evolution of the phase-space probability distribution of classical particles under binary collisions. Approximations to it underlie the basis for several scholarly fields, including aerodynamics and plasma physics. While these approximations are appropriate in their respective domains, they can be violated in niche but diverse applications which require direct numerical solution of the original nonlinear Boltzmann equation. An expanded implementation of the Galerkin-Petrov conservative spectral algorithm is employed to study a wide variety of physical problems. Enabled by distributed precomputation, solutions of the spatially homogeneous Boltzmann equation can be achieved in seconds on modern personal hardware, while spatially-inhomogeneous problems are solvable in minutes. Several benchmarks against both analytic theoretical predictions and comparisons to other Boltzmann solvers are presented in the context of several domains including weakly ionized plasma, gaseous fluids, and atomic-plasma interaction.Comment: 17 pages, 5 figures, 1 tabl

무한한 상관길이를 가지는 위스너 수송 방정식의 결정론적 해

학위논문(박사) -- 서울대학교대학원 : 공과대학 전기·정보공학부, 2022. 8. 최우영.We propose a new discrete formulation of the Wigner transport equation (WTE) with infinite correlation length of potentials. Since the maximum correlation length is not limited to a finite value, there is no uncertainty in the simulation results, and Wigner-Weyl transformation is unitary in our expression. For general and efficient simulation, the WTE is solved self-consistently with the Poisson equation through the finite volume method and the fully coupled Newton-Raphson scheme. By applying the proposed model to resonant tunneling diodes and double gate MOSFET, transient and steady-state simulation results including scattering effects are shown.본 연구에서는 무한한 상관 길이를 가지는 위그너 수송 방정식의 새로운 수치해석적 풀이법을 제시하였다. 최대 상관 길이가 한정된 값으로 제한되지 않기 때문에, 시뮬레이션 결과에 불확실성이 발생하지 않으며, 제안된 표현법에서는 Wigner-Weyl 변환이 unitary하다. 일반적이고 효율적인 시뮬레이션을 위해, 위그너 수송 방정식을 푸아송 방정식과 유한 체적법과 뉴턴-랩슨 방식을 통해 self-consistent하게 풀었다. 제안된 모델을 resonant tunneling diode와 double gate MOSFET에 적용하여, 산란효과를 고려한 동적 그리고 정적 시뮬레이션 결과를 보여주었다.Chapter 1 Introduction 1 1-1 Various models for device simulation 1 1-2 Numerical problems in solving WTE 5 Chapter 2 Simulation methods 10 2-1 WTE with infinite correlation length 10 2-2 Numerical Methods 13 2-3 Multi-dimensional Simulation Methods 23 Chapter 3 Simulation methods 26 2-1 Simulation results according to the correlation length 26 2-2 Simulation for resonant tunneling diode 30 2-3 Simulation for double gate MOSFET 51 Chapter 4 Conclusion 70 Appendix 72 A-1 Numerical integration method of the nonlocal potential terms 72 A-2 2D electron density and electric potential results 75 A-3 Wigner function for each subband 78 References 85 Abstract 92박

Trinity: A Unified Treatment of Turbulence, Transport, and Heating in Magnetized Plasmas

To faithfully simulate ITER and other modern fusion devices, one must resolve electron and ion fluctuation scales in a five-dimensional phase space and time. Simultaneously, one must account for the interaction of this turbulence with the slow evolution of the large-scale plasma profiles. Because of the enormous range of scales involved and the high dimensionality of the problem, resolved first-principles global simulations are very challenging using conventional (brute force) techniques. In this thesis, the problem of resolving turbulence is addressed by developing velocity space resolution diagnostics and an adaptive collisionality that allow for the confident simulation of velocity space dynamics using the approximate minimal necessary dissipation. With regard to the wide range of scales, a new approach has been developed in which turbulence calculations from multiple gyrokinetic flux tube simulations are coupled together using transport equations to obtain self-consistent, steady-state background profiles and corresponding turbulent fluxes and heating. This approach is embodied in a new code, Trinity, which is capable of evolving equilibrium profiles for multiple species, including electromagnetic effects and realistic magnetic geometry, at a fraction of the cost of conventional global simulations. Furthermore, an advanced model physical collision operator for gyrokinetics has been derived and implemented, allowing for the study of collisional turbulent heating, which has not been extensively studied. To demonstrate the utility of the coupled flux tube approach, preliminary results from Trinity simulations of the core of an ITER plasma are presented.Comment: 187 pages, 53 figures, Ph.D. thesis in physics at University of Maryland, single-space versio

Numerical methods for drift-diffusion models

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions