Density of holes in Valence Band of Intrinsic Semiconductor (Derivation)

Density of holes in Valence Band of Intrinsic Semiconductor (Derivation)

We know that if an electron is transferred from valence band to conduction band, a hole is created in valence band.

Let dp be the number of holes per unit volume in valence band between the energy E and E + dE.

dp = Z (E) (1 − F (E)) dE .................(1)

where Z (E) dE → Density of states in the energy range E and E+ dE.

Since F (E) is the probability of electron occupation 1 - F(E) is the probability of an unoccupied electron state, i.e., (a) g probability of presence of hole.

Since E is very small when compared to E_F, in valence band (E – E_F) is a negative quantity. Therefore, is very small and it is neglected in the denominator term of eqn (2).

Density of states in valence band,

Here, m_h is the effective mass of the hole in valence band.

E_v, top of energy level in valence band is the potential energy of a hole at rest. Hence, (E_v - E) is the kinetic energy of the hole at level below E_v. So the term E in eqn (4) is replaced as (E_v - E).

The number of holes in valence band for the entire energy range is obtained by integrating eqn (6) between limits - ∞ to E_v.

To evaluate the integral in eqn (7), let us assume,

Substituting these values in eqn (7), we have

[-ve sign is omitted by interchanging the limits]

Using the gamma function, it is shown that

Substituting eqn (10) in eqn (9), we have

The equation (11) is the expression for the concentration of holes in valence band of intrinsic semiconductor.