Bloch bands

In this blog, I will summarize some basic knowledge of Bloch bands. Here, I will use the example of a particle in a periodic lattice potential to calculate the wave function of a single particle and investigate the band structure.

A particle in a periodic potential is described by the following Schr\”odinger equation

where is the Hamiltonian, is the energy of the -th band, and is the wave function of the -th band. The wave function can be expanded in the basis of Bloch states

where is a periodic function with the same periodicity as the lattice potential . The wave vector is defined in the first Brillouin zone , where is the lattice constant. The Bloch states satisfy the following equation