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.

In Chapter 9, we discussed a quantum mechanical model that describes the flow of electrons coherently through a channel. All dissipative/phase-breaking processes were assumed to be limited to the contacts where they act to keep the electrons in local equilibrium. In practice, such processes are present in the channel as well and their role becomes increasingly significant as the channel length is increased. Indeed, prior to the advent of mesoscopic physics, the role of contactswas assumed to be minor and quantum transport theory was essentially focused on the effect of such processes. By contrast, we have taken a “bottom-up” view of the subject and now that we understand how to model a small coherent device, we are ready to discuss dissipative/phase-breaking processes.

Phase-breaking processes arise from the interaction of one electron with the surrounding bath of photons, phonons, and other electrons. Compared to the coherent processes that we have discussed so far, the essential difference is that phase-breaking processes involve a change in the “surroundings.” In coherent interactions, the background is rigid and the electron interacts elastically with it, somewhat like a ping pong ball bouncing off a truck. The motion of the truck is insignificant. In reality, the background is not quite as rigid as a truck and is set in “motion” by the passage of an electron and this excitation of the background is described in terms of phonons, photons, etc.