Band Theory of Solids

The first quantum theory of electronic behavior in solid is Sommerfeld theory, which treat electrons as free gas, and do not include the underlying ionic field and the electron-electron interaction. Band theory is the next step, which take the ion field into consideration. The greatest success of band theory is its ability to explain the metal and insulator. Insulation corresponds to fully occupied band while metals are partially filled. These partially unoccupied bands can readily be occupied upon thermal excitation or application of an electric field.

In case of band theory, we apply so-called one-electron approximation, where the total wave function for the system is given by a combination of wave functions, each of which involves the coordinates of only one electron. The field seen by a given electron is assumed to be that of the fixed nuclei plus some average field produced by the charge distribution of all other electrons. The problem essentially involves a potential field which has the same periodicity as the crystal field. Motion of electrons in a periodic potential result in energy bands separated by forbidden regions.