3.2 Band

The band structure of a crystalline solid，that is，the energy-momentum（E-k）relationship，is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. The Bloch theorem，one of the most-important theorems basic to band structure，states that if a potential energy V（r）is periodic in the direct lattice space，then the solutions for the wave function ψ（r，k）of the Schrodinger equation

are of the form of a Bloch function

ψ（r，k）=exp（jk·r）Ub（r，k）） （3.4）

Here b is the band index，ψ（r，k）and Ub（r，k）are periodic in R of the direct lattice.

From the Bloch theorem one can also show that the energy E（k）is periodic in the reciprocal lattice. For a given band index，to label the energy uniquely，it is sufficient to use only k’s in a primitive cell of the reciprocal lattice. The standard convention is to use the Wigner-Seitz cell in the reciprocal lattice. This cell is the Brillouin zone or the first Brillouin zone. It is thus evident that we can reduce any momentum k in the reciprocal space to a point inside the Brillouin zone，where any energy state can be given a label in the reduced zone schemes.

For any semiconductor there is a forbidden energy range in which allowed states cannot exist. Energy regions or energy bands are permitted above and below this energy gap. The upper bands are called the conduction bands；the lower bands，the valence bands . The separation between the energy of the lowest conduction band and that of the highest valence band is called the bandgapor energy gap Eg，which is one of the most-important parameters in semiconductor physics.