Band Theory of Solids

Band Theory of Solids:
To understand the origin of the formation of energy levels and energy bands (leading to a band gap) in a semiconductor, we need to analyze the periodic potentials present in a semiconductor. If we solve the periodic wave functions (Bloch functions), the result is that the energy levels are grouped in bands, separated by energy band gaps. The behavior of electrons at the bottom of such a band is similar to that of a free electron and the electrons are affected by the presence of the periodic potential. The effect of a periodic arrangement on the electron energy levels is shown as the energy levels of electrons in a carbon crystal with the atoms arranged in a diamond lattice. These energy levels are plotted as a function of the lattice constant, a. 

Isolated carbon atom has six electrons in 1s, 2s and 2p orbital in pairs. The energy of an electron occupying the 2s and 2p orbital is indicated in the figure. As the lattice constant is reduced, there is an overlap of the electron wave-functions of adjacent atoms. It leads to a splitting of the energy levels consistent with Pauli’s exclusion principle that allows each energy level to contain a maximum of two electrons. The splitting results in an energy band containing 2N states in the 2s band and 6N states in the 2p band, where N is the number of atoms in the crystal. The allowed energy levels are so close together that they appear continuous. A further reduction of the lattice constant causes the 2s and 2p energy bands to merge and split again into two bands containing 4N states each. At zero Kelvin, the lower band is completely filled with electrons and labeled as the valence band. The upper band is empty and labeled as the conduction band.

The electrons of an isolated atom are at their discrete energy levels.
 When two atoms come close,theirelectron wave functions molecular orbitals  overlap.. One of these orbitals has lower energy than the original atomic level and the other orbital has higher energy. Lower energy orbital promotes the bonding of the two atoms. The higher energy orbital is less stable. It opposes bonding if it is occupied, and so it is called antibonding orbital. In solids, a very large number of atoms are closely packed in small space (4.93x1022atoms/cc for Si). As a result, various atomic orbitals overlap and form molecular orbitals. Some molecular orbitals are so close in energy that they form energy bands. The bands in which free electrons are present, are called valence

splits their energy levels into two

bands. Those with antibonding molecular orbitals form the conduction band. 

The comprehensive energy band diagrams of semiconductors are complex but their main features are same as that of a diamond crystal. Most semiconducting crystals such as Ge, Si, GaAs are anisotropic – have different properties along different crystallographic directions. So the variation of their energy versus wave number along different axes are different. The energy bandgap of semiconductors decreases with increasing temperature. Increased interatomic spacing decreases the average potential seen by the electrons, which in turn reduces the energy bandgap.
Eg(T) = Eg(0) – α.T2 / (T+β); where α and β are parameters.