Concentration of holes in valence band of p- type semiconductors [Derivation]

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

Density of ionised acceptors = N_ɑ NF(E_ɑ) …. (1)

N_ɑ - the number of acceptor atoms per unit volume.

E_ɑ - acceptor energy level

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

The eqn (1) becomes, density of ionised acceptors

e ^{(Ea – EF)/kT} is a large quantity and thus '1' from the denominator of R.H.S. of eqn (2) is neglected.

Now, the eqn (2) is modified as,

Density of ionised acceptors = Nɑe ^{(EF – Ea)/kT}

At equilibrium,

where ΔE = E_ɑ - E_U is the ionisation energy of acceptors.

Results

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

• 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.

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

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