Question

Use the band structure diagram for each element below to explain:

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

b. Calcium, which has four atoms in a fcc (conventional cubic) unit cell, is a metal;

c. Diamond, which has eight 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. Why is diamond transparent?

Answer 1

to distinguish between metals and non metals on the basis of band theory, we consider and energy band filled with electrons upto a certain value. It is the effective number of free electrons present in this band that gives information about the conductivity of the element.

For a lattice of parameter L, the total effective number of free electrons in an energy band is given by:

....(1)

where m is the mass of an electron, and dE/dk is the energy curve.

Now Lattice parameter L for

(a) Sodium is L_Na  = 4.2906 Å

(b) Calcium is L_Ca = 5.5884 Å

(c) Diamond is L_Dia = 3.57 Å

and Ge, Si is L_GeSi = 5.5 Å .

After putting values for L, m, , in (1), We get:

N_eff (Na) = 2.91 x 10³⁹ . dE/dk,

N_eff(Ca) = 3.7911 x 10³⁹ dE/dk,

N_eff (Dia) = 2.42 x 10³⁹ dE/dk,

N_eff (Ge,Si) = 3.73 x 10³⁹ dE/dk

ThedE/dk curve for Metals (Na, Ca):

The electrons partly occupy the conduction band and partly the valence band, which makes these elements of metallic character.

The dE/dk curve for Diamond:

The eletrons completely occupy the valence band and conduction band is completely empty, making it an insulator at 0k.

The dE/dk curve for semiconductors (Ge,Si):

At temperatures greater than 0k, even the perfect insulators act as conductors.

WHY IS DIAMOND TRANSPARENT:

The band gap of diamonds is very wide, i.e. a radiation of energy 6 eV is needed to make diamond electrons jump to conduction band, and visible light has energy between 1.65 eV to 3.1 eV only. therefore diamond doesnt absorb visible light, which makes it transparent.

While, graphite has electrons in the conduction band already, and these electrons will absorb radiation of any energy, making it look black.