In this course, only semiconductors that have the diamond or zinc sulfide structures, (see figure 21.19 on page 761 of the text) which feature tetrahedral bonding geometries, will be covered. Since all of the bonds in these tetrahedral solids are derived from sp³ hybrid orbitals, the methane molecule provides a good starting point for describing their band structure. A simplified molecular orbital diagram for CH₄ is shown in Figure 1.

On the left of this figure is shown a carbon atom that has been hybridized to yield four equivalent sp³ orbitals, each directed towards a different comer of a tetrahedron. On the right are the 1s valence orbitals of the four hydrogen atoms. These combine with the four sp³ orbitals to form four bonding orbitals that are stabilized in energy relative to the atomic orbitals of carbon and hydrogen; and four antibonding orbitals. (Note: each bonding orbital points in a different direction, but all four of them have the same energy).

Four sigma C-H bonds are formed by placing two spin-paired electrons (one from carbon and one from hydrogen) in each bonding orbital. As a result, the bonding orbitals are completely filled, but the antibonding orbitals are completely empty.

The situation in diamond is analogous to that for methane, except that the orbitals on the right hand side of the diagram are now the sp³ hybrid orbitals of the four adjacent carbon atoms that bond to the central atom. However, in the solid, one important new feature arises.  Weaker secondary interactions now occur between the central carbon atom and carbon atoms farther away from it than the four adjacent atoms to which it is directly bound. These weak interactions cause the energy levels of the collection of bonding and antibonding orbitals to be broadened. As a result, relatively narrow bands of bonding and the antibonding orbitals are observed.
The band consisting of bonding orbitals is called the valence band, and the  band consisting of antibonding orbitals is called the conduction band.
The difference in energy between them is called the band gap, E_g.

The magnitude of the band gap depends primarily on the same factors that determine the separation in energy between the bonding and antibonding orbitals in an isolated molecule; namely, the factors that determine the strength of the bonds between adjacent atoms. Bond strengths of homonuclear diatomic molecules are determined primarily by the size of the combining atoms (or bond lengths).  However, because polar covalent bonds tend to be stronger than purely covalent bonds, bond strengths are also affected by the difference in electronegativities of the combining atoms.

The band width (the spread of energies within the valence or conduction bands) is determined by the strength of the weak interactions between non nearest-neighbor atoms (i.e. those outside of the immediate tetrahedron).

In the lowest energy state of the system (which is occupied at very low temperatures), the valence band is completely filled with electrons, and the conduction band is completely empty (see methane above).

Depending on the magnitude of their band gap, tetrahedral solids are somewhat arbitrarily characterized as being either insulators (E_g> 3eV = 290 kJ/mol) or semiconductors (E_g < 3eV). Diamond, which has a band gap of 5.5eV, is a good insulator, while silicon and germanium are semiconductors. The band gaps of several other tetrahedral solids are listed in Table 1. Two other important properties of each solid; namely, the edge length of the cubic unit cell, which is related to atomic size, and the strength of the bonds between adjacent atoms are also listed in the table. The relationship that exists between the band gap, the bond strength and the size of the bonding atoms is nicely illustrated.
 
 

Over the last quarter of a century, chemists have been very inventive in creating new tetrahedral solids that have band gaps that differ from those of the four pure elements shown in Table 1.  One example of such a solid is gallium arsenide, the material created by combining the Group III element gallium, which immediately precedes germanium in the periodic table, with the Group V element arsenic, which immediately follows it.  Gallium arsenide, GaAs, has the zinc sulfide structure (see page 761 of the text), which is identical to the "diamond" structure exhibited by germanium, except that the germanium atoms are replace by alternate gallium and arsenic atoms.  Since one half of the germanium atoms are replaced by Ga atoms having one fewer valence electron, and the other half are replaced by As having one additional valence electron, the total number of valence electrons is preserved. Therefore, this new isoelectronic species has the same tetrahedral bonding structure as germanium.  Also since Ga, Ge, and As are next to one another in the periodic table, all three elements have similar atomic radii.  As a result, the unit cell length of GaAs (565 pm) is almost identical to that of germanium.  However, the band gaps of the two materials are quite different. Because GaAs has much stronger polar covalent bonds, it has the larger band gap (1.42 eV versus 0.66 eV for germanium).