BAND THEORY OF SOLIDS

BAND THEORY OF SOLIDS Consider that the Si or Ge crystal contains N atoms. Electrons of each atom will have discrete energies in different orbits. The electron energy will be same if all the atoms are isolated, i.e., separated from each other by a large distance. However, in a crystal, the atoms are close to each other ( to Å) and therefore the electrons interact with each other and also with the neighbouring atomic cores.

The overlap (or interaction) will be more felt by the electrons in the outermost orbit while the inner orbit or core electron energies may remain unaffected. Therefore, for understanding electron energies in Si or Ge crystal, we need to consider the changes in the energies of the electrons in the outermost orbit only. For Si, the outermost orbit is the third orbit (n = ), while for Ge it is the fourth orbit (n = ). The number of electrons in the outermost orbit is (2s and 2p electrons).

Hence, the total number of outer electrons in the crystal is 4N. The maximum possible number of outer electrons in the orbit is (2s + 6p electrons). So, out of the 4N electrons, 2N electrons are in the 2N s-states (orbital quantum number l = ) and 2N electrons are in the available 6N p-states. Obviously, some p-electron states are empty as shown in the extreme right of Figure.

This is the case of well separated or isolated atoms [region A of Figure]. Suppose these atoms start coming nearer to each other to form a solid. The energies of these electrons in the outermost orbit may change (both increase and decrease) due to the interaction between the electrons of different atoms. The 6N states for l = , which originally had identical energies in the isolated atoms, spread out and form an energy band [region B in Figure].

Similarly, the 2N states for l = , having identical energies in the isolated atoms, split into a second band (carefully see the region B of Figure) separated from the first one by an energy gap. At still smaller spacing, however, there comes a region in which the bands merge with each other. The lowest energy state that is a split from the upper atomic level appears to drop below the upper state that has come from the lower atomic level. In this region (region C in Figure), no energy gap exists where the upper and lower energy states get mixed.

Finally, if the distance between the atoms further decreases, the energy bands again split apart and are separated by an energy gap Eg (region D in Figure). The total number of available energy states 8N has been re-apportioned between the two bands (4N states each in the lower and upper energy bands). Here the significant point is that there are exactly as many states in the lower band (4N) as there are available valence electrons from the atoms (4N). Therefore, this band (called the valence band) is completely filled while the upper band is completely empty.

The upper band is called the conduction band.