Wim Magnus

IMEC, Silicon Process and Device Technology Division (SPDT), Quantum Device Modeling Group (QDM), Kapeldreef 75, Flanders, B-3001 Leuven, Belgium

E-mail address: wim.magnus@imec.be

Wim Schoenmaker

MAGWEL N.V., Kapeldreef 75, B-3001 Leuven, Belgium

E-mail address: wim.magnus@imec.be

Publisher Summary

This chapter provides an overview on electromagnetism. Electromagnetism, formulated in terms of the Maxwell equations and quantum mechanics, formulated in terms of the Schrödinger equation, constitutes the physical laws by which the bulk of natural experiences are described. Apart from the gravitational forces, nuclear forces and weak decay processes, the description of the physical facts starts with these underlying microscopic theories. The ambition of physicists, chemists and engineers, to provide tools for performing calculations, does not only boost progress in technology but also has a strong impact on the formulation of the equations that represent the physics knowledge and hence provides a deeper understanding of the underlying physics laws. With the advent of powerful computer resources, it has become feasible to extract information from these basic laws with unprecedented accuracy. In particular, the complexity of realistic systems manifests itself in the non-trivial boundary conditions, such that without computers, reliable calculations are beyond reach.

List of symbols

1 Preface

Electromagnetism, formulated in terms of the Maxwell equations, and quantum mechanics, formulated in terms of the Schrödinger equation, constitute the physical laws by which the bulk of natural experiences are described. Apart from the gravitational forces, nuclear forces and weak decay processes, the description of the physical facts starts with these underlying microscopic theories. However, knowledge of these basic laws is only the beginning of the process to apply these laws in realistic circumstances and to determine their quantitative consequences. With the advent of powerful computer resources, it has become feasible to extract information from these basic laws with unprecedented accuracy. In particular, the complexity of realistic systems manifests itself in the non-trivial boundary conditions, such that without computers, reliable calculation are beyond reach.

The ambition of physicists, chemists and engineers, to provide tools for performing calculations, does not only boost progress in technology but also has a strong impact on the formulation of the equations that represent the physics knowledge and hence provides a deeper understanding of the underlying physics laws. As such, computational physics has become a cornerstone of theoretical physics and we may say that without a computational recipe, a physics law is void or at least incomplete. Contrary to what is sometimes claimed, that after having found the unifying theory for gravitation and quantum theory, there is nothing left to investigate, we believe that physics has just started to flourish and there are wide fields of research waiting for exploration.

This volume is dedicated to the study of electrodynamic problems. The Maxwell equations appear in the form

     (1.1)

where Δ describes the near-by field variable correlation of the field that is induced by a source or field disturbance. Near-by correlations can be mathematically expressed by differential operators that probe changes going from one location the a neighboring one. It should be emphasized that “near-by” refers to space and time.

One could “easily” solve these equations by construction the inverse of the differential operator. Such an inverse is usually known as a Green function.

There are two main reasons that prevent a straightforward solution of the Maxwell equations. First of all, realistic structure boundaries may be very irregular, and therefore the corresponding boundary conditions cannot be implemented analytically. Secondly, the sources themselves may depend on the values of the fields and will turn the problem in a highly non-linear one, as may be seen from Eq. (1.1) that should be read as

     (1.2)

The bulk of this volume is dedicated to find solutions to equations of this kind. In particular, Chapters II, III, IV and V are dealing with above type of equations. A considerable amount of work deals with obtaining the details of the right-hand side of Eq. (1.2), namely how the source terms, being charges and currents depend in detail on the values of the field variables.

Whereas, the microscopic equation describe the physical processes in great detail, i.e., at every space–time point field and source variables are declared, it may be profitable to collect a whole bunch of these variables into a single basket and to declare for each basket a few representative variables as the appropriate values for the fields and the sources. This kind of reduction of parameters is the underlying strategy of circuit modeling. Here, the Maxwell equations are replaced by Kirchhoff's network equations. This is the starting point for Chapter VI.

The “basket” containing a large collection of fundamental degrees of freedom of field and source variables, should not be filled at random. Physical intuition suggests that we put together in one basket degrees of freedom that are “alike”. Field and source variables at near-by points are candidates for being grabbed together, since physical continuity implies that a all elements in the basket will have similar values.1

The baskets are not only useful for simplifying the continuous equations. They are vital to the discretization schemes. Since any computer has only a finite memory storage, the continuous or infinite collection of degrees of freedom must be mapped onto a finite subset. This may be accomplished by appropriately positioning and sizing of all the baskets. This procedure is named “grid generation” and the construction of a good grid is often of great importance to obtain accurate solutions.

After having mapped the continuous problem onto a finite grid one may establish a set of algebraic equations connecting the grid variables (basket representatives) and explicitly reflecting the non-linearity of the original differential equations. The solution of large systems of non-linear algebraic equations is based on Newton's iterative method. To find the solution of the set of non-linear equations F(x)=0, an initial guess is made: x=xinit=x0. Next the guess is (hopefully) improved by looking at the equation:

     (1.3)

where the matrix A is

     (1.4)

In particular, by assuming that the correction brings us close to the solution, i.e., , where , we obtain that

     (1.5)

Next we repeat this procedure, until convergence is reached. A series of vectors, xinit=x0,x1,x2,…,xn−1,xn=xfinal, is generated, such that |F(xfinal)|<ε, where ε is some prescribed error criterion. In each iteration a large linear matrix problem of the type A|x〉=|b〉 needs to be solved.

2 The microscopic Maxwell equations