Question

Band Theory

(A) Give a brief description of the formation of electron bands in crystals including reference to the atomic structure of the constituent atoms.

(B) Explain the following

i.Sodium, which has 2 atoms in a bcc (conventional cubic) unit cell, is a metal

ii.Calcium, which has 4 atoms in a fcc (conventional cubic) unit cell, is a metal

iii.Diamond, which has 8 atoms in a fcc (conventional cubic unit cell with a basis, is an electrical insulator, whereas silicon and germanium, which have similar structures, are semiconductors.

(C) Why is diamond transparent?

Answer 1

A)
Electrons are restricted to certain sets of discrete energy levels within atoms, with large gaps among them where no energy state is available for the electron to occupy. In solids they are also restricted and only allowed to certain energies. Instead of having such discrete energies, the available energy states form bands. This is very vital to the conduction process to make sure that electrons are present in the conduction band.
The discrete energy levels of the isolated atom will move to energy bands of solid due to the following reasons
Wave functions of electrons in neighboring atoms overlap in the solid thus it affects the potential energy
term as a result different energies are obtained and an electron is need not be localized at a particular atom.
An important parameter in the band theory is the Fermi level. It is the the top of the available electron energy
levels at low temperatures.

B) (i) The outer shell of metals have no more than three electrons. Sodium which has 2 atoms in BCC unit cell
It is a solid made up of ions with closed shells immersed in free electrons, which are free to move about
   the crystal under the influence of an electric field. Therefore strong attractive forces hold up the lattice together
   which makes it metal.
(ii) An FCC unit cell contains 4 atoms of calcium just count the atoms at the corners, faces and edges,
   here also the electrons are not restricted to move to a great extent under any influence such as electric field.
(iii) Diamond's structure is bcc with four of the corner atoms missing. Every atom in the tetrahedral structure has four nearest neighbors and it is this structure that is the basic building block of the diamond lattice.
This is its unique feature which does not allow electrical conductivity. Bonding forces arising from a
mechanical interaction between the shared electrons dominates.
For intrinsic semiconductors like silicon and germanium, the gap between valence and conduction
bands is less (no conduction occurs at 0 K) and at higher temperatures few number of electrons can
reach the conduction band.

C)  Diamond is not a conductor , the electrons that are in the valence band require a photon with at least 5.4 eV of energy to push an electron into the conduction band. In the process that photon would be absorbed because the band gap is very wide whereas visible light has energies of between 1.65 and 3.1 eV . That is the reason why
Diamonds are transparent.