Introduction The understanding of the behavior of electrons in solids is one of the keys to understanding materials. The electron theory of solids is capable of explaining the optical, magnetic, thermal, as well as the electrical properties of materials. In other words, the electron theory provides important fundamentals for a technology which is often considered […]

# Electronic Properties of Materials

## The Wave-Particle Duality (Fundamentals of Electron Theory)

This topic is mainly concerned with the interactions of electrons with matter. Thus, the question "What is an electron?" is quite in order. Now, to our knowledge, nobody has so far seen an electron, even by using the most sophisticated equipment. We experience merely the actions of electrons, e.g., on a cathode-ray television screen or […]

## The Schrodinger Equation (Fundamentals of Electron Theory)

We shall now make use of the conceptual ideas which we introduced in the previous topic, i.e., we shall cast, in mathematical form, the description of an electron as a wave, as suggested by Schrodinger in 1926. All "derivations" of the Schrodinger equation start in one way or another from certain assumptions, which cause the […]

## Solution of the Schrodinger Equation for Four Specific Problems (Fundamentals of Electron Theory) Part 1

Free Electrons At first we solve the Schrodinger equation for a simple but, nevertheless, very important case. We consider electrons which propagate freely, i.e., in a potential-free space in the positive x-direction. In other words, it is assumed that no "wall," i.e., no potential barrier (V), restricts the propagation of the electron wave. The potential […]

## Solution of the Schrodinger Equation for Four Specific Problems (Fundamentals of Electron Theory) Part 2

Finite Potential Barrier (Tunnel Effect) Let us assume that a free electron, propagating in the positive x-direction, encounters a potential barrier whose potential energy, V0, ("height" of the barrier) is larger than the total energy, E, of the electron, but is still finite (Fig. 4.6). For this case we have to write two Schrodinger equations, […]

## Solution of the Schrodinger Equation for Four Specific Problems (Fundamentals of Electron Theory) Part 3

Electron in a Periodic Field of a Crystal (The Solid State) In the preceding sections we became acquainted with some special cases, namely, the completely free electron and the electron which is confined to a potential well. The goal of this section is to study the behavior of an electron in a crystal. We will […]

## Energy Bands in Crystals (Fundamentals of Electron Theory) Part 1

One-Dimensional Zone Schemes We are now in a position to make additional important statements which contribute considerably to the understanding of the properties of crystals. For this we plot the energy versus the momentum of the electrons, or, because of (4.8), versus the wave vector, k. As before, we first discuss the one-dimensional case. The […]

## Energy Bands in Crystals (Fundamentals of Electron Theory) Part 2

Wigner-Seitz Cells Crystals have symmetrical properties. Therefore, a crystal can be described as an accumulation of "unit cells." In general, the smaller such a unit cell, i.e., the fewer atoms it contains, the simpler its description. The smallest possible cell is called a "primitive unit cell." Frequently, however, a larger, nonprimitive unit cell is used, […]

## Energy Bands in Crystals (Fundamentals of Electron Theory) Part 3

Band Structures for Some Metals and Semiconductors Those readers who have skipped Sections 5.3 through 5.6 need to familiarize themselves with the (three-dimensional) first Brillouin zone for the face centered cubic (fcc) crystal structure (Fig. 5.19). The [100], the [110], and the [111] directions in k-space are indicated by the letters G — X, G […]

## Electrons in a Crystal (Fundamentals of Electron Theory) Part 1

Fermi Energy and Fermi Surface The Fermi energy, EF, is an important part of an electron band diagram. Many of the electronic properties of materials, such as optical, electrical, or magnetic properties, are related to the location of EF within a band. The Fermi energy is often defined as the "highest energy that the electrons […]