Introduction to the Quantum Theory of Solids

In the last chapter, use applied quantum mechanics and Schrodinger's wave equation
to determine the behavior of electrons in the presence of various potential
functions. We found that one important characteristic of an electron bound to an
atom or bound within a finite space is that the electron can take on only discrete values
of energy; that is, the energies are quantized. We also discussed the Pauli exclusion
principle, which stated that only one electron is allowed to occupy any given
quantum state. In this chapter, we will generalize these concepts to the electron in a
crystal lattice.
One of our goals is to determine the electrical properties of a semiconductor material,
which we will then use to develop the current-voltage characteristics of semiconductor
devices. Toward this end, we have two tasks in this chapter: to determine
the properties of electrons in a crystal lattice, and to determine the statistical characteristics
of the very large number of electrons in a crystal.
To start, we will expand the concept of discrete allowed electron energies that
occur in a single atom to a band of allowed electron energies in a single-crystal solid.
First we will qualitatively discuss the feasibility of the allowed energy bands in a
crystal and then we will develop a more rigorous mathematical derivation of this theory
using Schrodinger's wave equation. This energy band theory is a basic principle
of semiconductor material physics and can also be used to explain differences in
electrical characteristics between metals, insulators, and semiconductors.
Since current in a solid is due to the net flow of charge, it is important to determine
the response of an electron in the crystal to an applied external force, such as an
electric field. The movement of an electron in a lattice is different than that of an electron
in free space. We will develop a concept allowing us to relate the quantum mechanical
behavior of electrons in a crystal to classical Newtonian mechanics. This analysis leads to a parameter called the electron effective mass. As part of this development,
we will find that we can define a new particle in a semiconductor called a
hole. he motion of both electrons and holes gives rise to currents in a semiconductor.
Because the number of electrons in a semiconductor is very large, it is impossible
lo follow the motion of each individual particle. We will develop the statistical
behavior of electrons in a crystal, noting that the Pauli exclusion principle is an important
factor in determining the statistical law the electrons must follow. The resulting
probability function will determine the distribution of electrons among the available
energy states. when we develop the theory of the semiconductor in
equilibrium

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