Concentration of Holes in Valence Band of p-type Semiconductors

CONCENTRATION OF HOLES IN VALENCE BAND OF P-TYPE SEMICONDUCTORS [Derivation]

In p-type semiconductor, acceptor energy level is just above valence band (fig. 3.10).

Density of holes per unit volume in valence band is given by

E_v → Energy corresponding to top most level of valence band.

Density of ionised acceptors = N_a F(E_a)

N-the number of acceptor atoms per unit volume.

E_a- acceptor energy level

Here, F(E_a) is probability for finding electron in acceptor energy level ie., ionised acceptor.

The eqn (1) becomes, density of ionised acceptors

 is a large quantity and thus '1' from the (a) e denominator of R.H.S. of eqn (2) is neglected.

Now, the eqn (2) is modified as,

Density of ionised acceptors 

At equilibrium,

Taking log on both sides in eqn (5), we have

Rearranging,

Substituting eqn of E_F from (7) in (1), we get

where ∆E = E_a – E_v is the ionisation energy of acceptors.

Results

i. Density of holes in valence band is proportional to the square root of acceptor concentration.

ii. At high temperature, we must take into account the intrinsic carrier concentration of semiconductor due to breaking of covalent bond along hole concentration produced by acceptor impurity.

iii. At very high temperature, intrinsic carrier concentration over takes holes due to acceptor concentration.

iv. i.e., At very high temperature, p-type semiconductor behaves like an intrinsic semiconductor and acceptor concentration becomes insignificant.