ABSTRACT
Over the past few years, more attention has been paid to an approach to light propagation in dense media based on a some what different way of looking at the solutions to Maxwell’s equations [1,2]. The different look is based on the realization that one can cast Maxwell’s equations as a mathematical linear operator and, thereby view the solutions in different terms. This is the familiar quantum mechanical formulation for which one defines the hamiltonian operator [3]. The hamiltonian operator acts on a fictitious set of functions called the wavefunctions, in the form
HC ¼ EC ð9:1Þ
H ¼ ðh=2pÞ 2
2m H2 þ VðxÞ ð9:2Þ
Here the periodic potential that the electron sees is contained in the V(x). The particular linear operator H represented by
Eq. (9.2) has a number of useful and important properties [2,3]. It is sufficient here to mention that, in general, the set of eigenfunctions H is orthogonal and complete, allowing one to express any state of the electron as an infinite linear combination of these unique functions. The concept is similar to being able to represent any vector in three spaces as a linear combination of the unit vectors in the x,y, and z directions. When Eqs. (9.1) and (9.2) are applied to electrons, say in a crystal where the electron experiences a periodic potential owing to the charged lattice, one then obtains a map of allowable bands of energy as a function of the wavevector k. For a simple illustrative example in one dimension, one writes the wavefunction in the form
CðxÞ ¼ uðxÞ expð jkxÞ ð9:3Þ Here the periodicity is contained in the property of u(x) such that u(xþa)¼u(x) where a is the period of the potential that the electron is in. The result of this formulation results in an energy (frequency) vs. k diagram as shown in Fig. 9.1. What one sees is the parabolic shape of the energy bands based on what one would expect for a free electron. [This corresponds to V(x)¼ 0 in Eq. (9.2), and u(x)¼ 1 in Eq. (9.3).] The symmetry of the structure allows one to draw the extended picture of the bands as shown. For example, the energies must be the same as k¼ 0 and all multiples 2p=a, where a is the period of the lattice. This will be true for any value of k within the range p=a < k < p=a which is termed the reduced Brillouin zone [3]. What is important to see, and forms the particular emphasis in what follows, is that in the presence of a nonzero potential, the bands split as shown. What this means is that there will be forbidden ranges of energies for the electron. These so-called bandgaps play the crucial role in all semiconductor devices. It is the judicious placing of impurities or defects in these materials that provide localized energy levels within the gap. It is the electrons and holes from these localized levels that are manipulated to produce the transistor.
