Carrier Concentration in Intrinsic Semiconductors

CARRIER CONCENTRATION IN INTRINSIC SEMICONDUCTORS

Definition

The number of electrons in conduction band per unit volume of the material is called as electron concentration (n).

Similarly the number of holes in valence band per unit volume of the material is called hole concentration (p).

In general, the number of charge carriers per unit volume of the material is called carrier concentration. It is also known as density of charge carriers.

Density of Electrons in Conduction Band (Derivation)

The number of electrons per unit volume in conduction band for energy between E and E + dE is given by

dn = Z (E) F (E) dE .................(1)

where Z (E) dE - Density of states in energy between E and E + dE

F (E) - Probability of electron occupancy.

Number of electrons in conduction band for the entire range is calculated by integrating eqn (1) between energy E_c and +∞

E_C is energy corresponding to the bottom most level and +∞ is energy corresponding to the upper most level in conduction band. (Fig 3.5).

Density of states in conduction band between the energy range E and E + dE is given by dE is given by

The bottom edge of the conduction band (E) denotes the potential energy of an electron at rest. Therefore, (E-Ec) is the kinetic energy of conduction electron at higher energy levels.

Thus, in eqn (3), E is replaced as (E – E_C)

The probability of electron occupation is given by Fermi distribution function

Substituting eqns (4) and (5) in (2), we get

Since kT is very small and (E – E_F) is greater than kT, E - EF is very large compared to '1' Hence, '1' from the denominator of eqn (6) is neglected.

Now, eqn (6) becomes

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

Substituting above values in eqn (7), we have

Substituting eqn (9) in eqn (8), we have

Equation (10) is the expression for concentration of electrons in the conduction band of intrinsic semiconductor.