Mihail Nedjalkov - Stochastic Approaches to Electron Transport in Micro- and Nanostructures

More information about this series at http://www.springer.com/series/4960

This book is published under the imprint Birkhuser, http://www.birkhauser-science.com by the registered company Springer Nature Switzerland AG.

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Computational modeling is an important subject of microelectronics, which is the only alternative to expensive experiments for design and characterization of the basic integrated circuit elements, the semiconductor devices. Modeling comprises mathematical, physical, and electrical engineering approaches needed to compute the electrical behavior in these structures. The role of the computational approach is twofold: (1) to derive models describing the current transport processes in a given structure in terms of governing equations, initial and boundary conditions, and relevant physical quantities, and (2) to derive efficient numerical approaches for performing their evaluation. They are considered as the two sides of the problem, since physical comprehension is achieved on the expense of increased numerical complexity. Stochastic approaches are most widely used in the field in particular, because they reduce memory requirements on the expense of longer simulation time and avoid approximation procedures, requiring additional regularity.

The primary motivation of our book is to present the synergistic link between the development of mathematical models of current transport in semiconductor devices and the emergence of stochastic methods for their simulation. The book fills the gap between other monographs, which focus on the development of the theory and the physical aspects of the models [1, 2], their application [3, 4], and the purely theoretical Monte Carlo approaches for solving Fredholm integral equations [5]. Specific details about this book are given in the following.

The golden era of classical microelectronics is characterized by models based on the Boltzmann transport equation. Their physical transparency in terms of particles featured the widespread development of Monte Carlo methods in the field. At the beginning, almost 50 years ago, a variety of phenomenological algorithms, such as the Monte Carlo Single-Particle algorithm for stationary transport in the presence of boundary conditions, the Monte Carlo Ensemble algorithm relevant for transient processes with initial or boundary conditions, and Monte Carlo algorithms for small signal analysis, were derived using the probabilistic interpretation of the processes described. Thus, the stochastic method is viewed as a simulated experiment, which emulates the elementary processes in the electron evolution. The fact that these algorithms solve the transport equation was proved afterward. The inverse perspective, algorithms from the transport model, developed during the last 15 years of the twentieth century, gave rise to a universal approach based on the formal application of the numerical Monte Carlo theory on the integral form of the transport model. This is the Iteration Approach, which allows to unify the existing algorithms as particular cases as well as the derivation of novel algorithms with refined properties. These are the high precision algorithms based on backward evolution in time and algorithms with improved statistics based on event biasing, which stimulates the generation of rare events. As applied to the problem of self-consistent coupling with the Poisson equation, the approach gave rise to self-consistent event biasing and the concept for time-dependent particle weights. An important feature of the Iteration Approach is that the original model can be reformulated within the context of a Monte Carlo analysis in a way that allows for a novel improved model of the underlying physics.