Laminated composites have been used extensively as printed circuit board (PCB) material in electronic packaging. Determination of the macroscopic properties of a PCB made of a laminated composite is a key step in designing a new PCB or selecting a PCB tailored for a specific packaging application. A laminated composite PCB consists of a number of laminae where each lamina itself is a composite made of discontinuous filler or continuous filler (straight fibers or woven fabrics). A lamination model is used to predict the overall properties of a laminated composite for a given set of properties of the laminae. It is assumed that the properties of each lamina are already obtained by experiment or predicted by a composite model such as the Eshelby model. Here, we apply the classical lamination theory only when a thin plate composed of laminae is subjected to thermomechanical loading and/or an electric field with the aim of finding the deflection of the plate and stress field within the laminae. The case of thermomechanical loading is important for the analysis of PCBs while that of an electric field relates to a piezoelectric laminated plate. Similarly, the behavior of electromagnetic waves in laminated composites provides key quantitative information in designing switchable windows for radomes and surface plasmon resonance sensors. In the following, we shall discuss first the classical lamination theory focusing on (i) mechanical behavior under hygro-thermomechanical loading, then extend it to (ii) the problem of a piezoelectric laminated composite subjected to an electric field, followed by (iii) stress analysis in a thin-coating–substrate system, and (iv) electromagnetic wave propagation in a layered medium.

What is an electronic composite?

“Composite” is a well-accepted word, generally referring to structural components with enhanced mechanical performance. There are a number of textbooks and review articles on these types of composites (e.g., Kelly, 1973; Tsai and Hahn, 1980; Hull, 1981; Chawla, 1987; Clyne and Withers, 1993; Daniel and Ishai, 1994; Gibson, 1994; Hull and Clyne, 1996).

Historically, composites with enhanced mechanical performance have been in existence from ancient-Egyptian time, c. 2000 BC, when bricks were made of mud, soil and straw (Exodus, Chapter 5, verse 7). Structural composites are designed primarily to enhance the mechanical properties of a matrix material by introducing reinforcement; the primary mechanical properties to be enhanced are strength, stiffness, and fracture resistance.

Normally, a composite consists of a matrix material and one kind of filler, but sometimes more than one kind of filler can be used, forming a “hybrid composite”. Depending on the matrix material, one can group composites into three basic types: polymer matrix composites (PMCs), metal matrix composites (MMCs), and ceramic matrix composites (CMCs). Among these, PMCs are the most popular type for electronic composites due to their low processing temperatures and associated cost-effectiveness.

An “electronic composite” is defined as a composite that is composed of at least two different materials and whose function is primarily to exhibit electromagnetic, thermal, and/or mechanical behavior while maintaining structural integrity.

Percolation model based on base lattices

A percolation model was proposed first by Broadbent and Hammersley (1957). Since then, there have been a large number of studies of percolation problems, and two books (Stauffer and Aharony, 1991; Sahini, 1994). A percolation model (Shante and Kirkpatrick, 1971; Essam, 1972; 1980) is composed of two simple elements, a site and a bond, which are interconnected. In most cases there is a base lattice underlying sites and bonds. The key variable in percolation problems is the probability p of occupied bonds or sites. The probability that each bond or site is occupied or empty is entirely independent of whether its neighbors are occupied or empty. When p = 1, the model becomes exactly the same as the base lattice and none of the elements is empty.

There are two types of percolation problem. If the occupancy of bonds is governed by a stochastic mechanism, the problem is called a bond percolation problem, whereas if that of the sites is so governed, the problem is called a site percolation problem. Figures 6.1, 6.2, and 6.3 show examples of these problems. A two-dimensional (2D) square lattice is taken as a base lattice, as shown in Fig. 6.1. Figure 6.2 shows a bond percolation problem of this 2D base lattice, where the probability of bonds pb = 0.5. Figure 6.3 show a site percolation problem of the same base lattice, where the probability of sites ps = 0.5.

The purpose of this book is to convey the conceptual framework that underlies the microscopic or atomistic theory of matter, emphasizing those aspects that relate to electronic properties, especially current flow. Even a hundred years ago the atomistic viewpoint was somewhat controversial and many renowned scientists of the day questioned the utility of postulating entities called atoms that no one could see. What no one anticipatedwas that by the end of the twentieth century, scientistswould actually be “seeing” and taking pictures of atoms and even building “nanostructures” engineered on a nanometer length scale. The properties of such nanostructures cannot be modeled in terms of macroscopic concepts like mobility or diffusion. What we need is an atomic or microscopic viewpoint and that is what this book is about.

The microscopic theory of matterwas largely developed in the course of the twentieth century following the advent of quantum mechanics and is gradually becoming an integral part of engineering disciplines, as we acquire the ability to engineer materials and devices on an atomic scale. It is finding use in such diverse areas as predicting the structure of new materials, their electrical and mechanical properties, and the rates of chemical reactions, to name just a few applications. In this book, however, I will focus on the flow of current through a nanostructure when a voltage is applied across it. This is a problem of great practical significance as electronic devices like transistors get downscaled to atomic dimensions.

In Chapter 1, we saw that current flow typically involves a channel connected to two contacts that are out of equilibrium with each other, having two distinct electrochemical potentials. One contact keeps filling up the channel while the other keeps emptying it causing a net current to flow from one contact to the other. In the next chapter we will take up a quantum treatment of this problem. My purpose in this chapter is to set the stage by introducing a few key concepts using a simpler example: a channel connected to just one contact as shown in Fig. 8.1.

Since there is only one contact, the channel simply comes to equilibrium with it and there is no current flow under steady-state conditions. As such this problem does not involve the additional complexities associated with multiple contacts and nonequilibrium conditions. This allows us to concentrate on a different physics that arises simply from connecting the channel to a large contact: the set of discrete levels broadens into a continuous density of states as shown on the right-hand side of Fig. 8.1.

In Chapter 1 I introduced this broadening without any formal justification, pointing out the need to include it in order to get the correct value for the conductance. My objective in this chapter is to provide a quantum mechanical treatment whereby the broadening will arise naturally along with the “uncertainty” relation γ = ħ/τ connecting it to the escape rate 1/τ for an electron from the channel into the contact.

As we move from the hydrogen atom (one electron only) to multi-electron atoms, we are immediately faced with the issue of electron–electron interactions, which is at the heart of almost all the unsolved problems in our field. In this chapter I will explain (1) the self-consistent field (SCF) procedure (Section 3.1), which provides an approximate way to include electron–electron interactions into the Schrödinger equation, (2) the interpretation of the energy levels obtained from this so-called “one-electron” Schrödinger equation (Section 3.2), and (3) the energetic considerations underlying the process by which atoms “bond” to form molecules (Section 3.3). Finally, a supplementary section elaborates on the concepts of Section 3.2 for interested readers (Section 3.4).

The self-consistent field (SCF) procedure

One of the first successes of quantum theory after the interpretation of the hydrogen atom was to explain the periodic table of atoms by combining the energy levels obtained from the Schrödinger equation with the Pauli exclusion principle requiring that each level be occupied by no more than one electron. The energy eigenvalues of the Schrödinger equation for each value of l starting from l = 0 (see Eq. (2.3.8)) are numbered with integer values of n starting from n = l + 1. For any (n, l) there are (2l + 1) levels with distinct angular wavefunctions (labeled with another index m), all of which have the same energy.