Carrier Concentration in n-type Semiconductors

CARRIER CONCENTRATION IN n-TYPE SEMICONDUCTORS [Derivation]

The energy band diagram of n-type semiconductor is shown in figure 3.8. In n-type semiconductor, the donor level is just below conduction band.

Density of electrons per unit volume in conduction band is given by

Since, F(E_d) is the probability for finding electron in donor energy level (unionised donor), therefore 1 - F(E_d) is the probability for finding ionised donors.

 E_d represents the donor energy level and N_d denotes donor concentration i.e., the number of donor atoms per unit volume of the material.

 is very small in eqn (3) when compared to '1'. Hence, it is neglected.

At equilibrium, the density of electron in conduction band is equal to the density of ionised donors.

Equating (1) and (4), we get

Taking log on both sides, we have

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

Rearranging the expression (9), we get

where ∆E = E_C - E_d is the ionisation energy of the donor. i.e., ∆E denotes the amount of energy required to transfer an electron from donor energy level E_d to conduction band E_C.

Results

i. The density of electrons in conduction band is proportional to the square root of donor concentration. The equation (11) is valid only at low temperatures.

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

iii. At very high temperatures, intrinsic carrier concentration which is generated thermally due to breaking of covalent bond over takes electrons due to donor impurity.

iv. That is, at very high temperature, n-type semiconductor behaves like intrinsic semiconductor and donor concentration becomes insignificant.

Fermi level

Fermi level gives the probability of finding an electron at a given energy value. If Fermi level lies exactly at the middle of the two levels, then the probability of finding an electron is half, e.g., as in an intrinsic semiconductor.

In extrinsic semiconductor, Fermi level strongly depends on temperature as well as the nature of doping and doping concentration.

The Fermi level is little below conduction band in n-type semiconductor and it is just above valence band in p-type semiconductor..