Carrier Concentration in n-Type Semiconductor

CARRIER CONCENTRATION IN n – TYPE SEMICONDUCTOR

The energy band structure of n-type semi- conductor is as shown in Fig. 2.16.

At 0 K, E_F will lie exactly between E_c and E_d, but even at low temperature some electrons may go from E_d to E_c. Let us assume that E_c-E_F > K_B T. Then the density of electrons in conduction band can be written as

…………(1)

Let N_d be the number of donor energy levels per cm³ (i.e.,) density of state Z (E_d) dE, which has energy E_d below the conduction band.

If some electrons are donated from donar energy level to conduction band say for example if two electrons goes to conduction band then two vacant sites (holes) will be created in E, levels as shown in Fig. 2.16.

Thus, in general we can write the density of holes in donar energy level as

N (E_d) dE=Z (E_d) dE (1 - F (E_d))

i.e., n_h = N_d (1-F (E_d) ……..(2)

we know

………(3)

Since E_F -E_d>>K_BT

(or) E_d - E_F <<K_BT

(or) E_d - E_F / K_BT <<1

(or) e^(E_d ^{− E}_F^)/K_B^T << 1

1 + e^(E_d ^{− E}_F^)/K_B^T≈ 1

Equation (3) becomes

1 − F (E_d) = e^(E_d ^{− E}_F^)/K_B^T .....(4)

Substituting equation (4) in (2)

n_h = N_d e^(E_d ^{− E}_F^)/K_B^T ……(5)

At Equilibrium condition

Number of electrons per unit volume in conduction band (i.e. electron density) = Number of holes per unit volume in donar energy level. (i.e., hole density)

Equating equation (1) and equation (5) we get

Taking log on both sides

………(6)

At 0K  i.e, When T=0, we can write the above equation as

EF=(E_c+E_d)/2 …...(7)

Equation (7) shows that, at 0K, E_F will lie exactly in the midway between E_cand E_d

Expression for Density of Electrons in Conduction Band in terms of N_d

Substituting equation (6) in (1) we get  

Here E_c-E_d=ΔE is called as Ionisation energy of donors (i.e.) ΔE represents the amount of energy required to transfer an electron from donar energy level (E_d) to conduction band (E_c).

We can write

……..(8)