# Basic Energy Band Theory

Band theory is essential in the study of solid state physics. The basic idea tends to center around two bands: the conduction and valence band (for reasons discussed later on). Between the two bands is a forbidden energy level (Energy gap) which depends on the resistivity or conductance of the material. In order to fully understand solid state devices such as transistors or solar cells, this must be discussed.

For a single atom, electrons occupy discrete energy levels called bands. When two atoms join together to form a diatomic element (such as Hydrogen), their orbitals overlap. The Pauli Exclusion Principle states that no two electrons can have the same quantum numbers. Now keep in mind that there are four types of quantum numbers. This means that when these two atoms combine the atomic orbitals must split to compensate so that no two electrons have the same energy. However for a macroscopic piece of a solid, the number of atoms is quite high (on the power of 10^22) and therefore the number of energy levels is also high. For this reason, adjacent energy levels are almost continuous, forming an energy band. The main bands under consideration are the valence (outermost band involved in chemical bonding) and conduction because the inner electron bands are so narrow. Band gaps or “forbidden zones” are leftover energy levels that are not covered by a band.

In order to apply band theory to a solid, the medium must be homogeneous or evenly distributed. The size of material must be considerable as well, which is not unreasonable considering the number of atoms in an appreciable piece of a solid. The assumption also must include that electrons do not interract with phonons or photons.

The “density of states” is a function that describes the number of states per unit volume, per unit energy. It is represented by a Probability Density function.

A Fermi-Dirac distribution function demonstrates the probability of a state of energy being filled with an electron. The probability is given below. The μ is generally expressed as EF which is the Fermi energy level or total chemical potential. kT is the familiar thermal energy which is the product of the Boltzmann constant and the temperature. From this equation it is clear that absolute zero temperature, the exponential term increases to infinity, causing the entire term to trend to zero. This leads to the conclusion that semiconductors behave as insulators at 0K.

The density of electrons can be calculated by multiplying this value with the density of states function and integrating over all energy.

Band-gap engineering is the process of changing a material’s band gap. This is usually done to semiconductors by changing the composition of alloys in the material.