Periodic Hamiltonians and the Emergence of Band Structure: The Bloch Theorem and the Dirac Kronig–Penney Model

Crystals are defined by their atomic makeup and their crystal symmetry. The spatial symmetry leads to a periodic potential that localizes the electron or carrier at the lattice sites corresponding to the location of the atoms. But because of the periodicity of potential the particles wave function extends throughout the entire crystal, not localized at a specific atom. This delocalization of the wave function gives rise to the band structure in crystals such as those that are semiconductors. This chapter explores the physics that emerges in the wave function when the time independent Schrödinger equation is solved for a periodic potential.