Question

Show that for intrinsic semiconductors of the Fermi level lies midway between the conduction band and the valence band .With the help of diagram explain effect of impurity concentration on Fermi level of N type semiconductor.

Answer 1

• At any temperature T > OK is an intrinsic semiconductor, a number of electrons are
found in the conduction band and the rest of the valence electrons are left behind in
the valence band.

• Let there be nc number of electrons in the conduction band and nv number of
electrons in the valence band.

Hence, the total number of electrons in the intrinsic semiconductor is
N=nc +n ................................. (7)

At T = 0 K all N electrons occupy energy states in the valence band.

• Out of these total N number of valence electrons Amy NC number of electrons cam
reach the conduction band.
The probability of occupancy of an energy level in the conduction band can be written
from equation (7) as

` f (EC)= 1/(1+e ^(Ec Et)//kT)`

where Ec is the potential energy of a rest electron in conduction band.

• Here, Ec is the minimum energy required for the electron to reach the bottom level of
the conduction band. The extra energy is converted to its kinetic energy with which it
moves freely in the conduction band at any energy level.

• Hence, the number if electrons found in the conduction band is

nc= Nf(Ec} = `N/(7 +e(Ec-EtJ JkT)` ........................ {2}

• Similarly, any nv number if valence electrons from the total of N electrons can bring
left behind in the valence band.

• The probability of occupancy of a level in the valence band is given by

f(Ev} = `1/(7 +e·(Et-Ev) /kT)`......................... {3}

• Hence, the number of electrons in the valence band can bring written as

nV=NF{Ev} =`N/(7 +e {Ef-Ev) /kT)....................... (4}`

• Substituting equations (2} and ( 4} in {7 ), it is found that.

` N = N/(1 +e(Ec-Et) JkT) + N/(7 +e {Et-Ev) / kT)`

`[1 + e ^(Ec -Et )//kt]` `[1+e^((- Et-Ev)//kT)]= 2 + e ^(-(Et-Ev)//kT)+e^((Ec-Et )//Kt)`

`1 + e ^((Ec -Et )//kt)` `+e^((- Et-Ev)//kT)= 2 + e ^(-m(Et-Ev)//kT)+e^((Ec-Et )//Kt)`

`e^((Ec-2Ef+Ev)//kT)=1`

`Ec -2EF + Ev//kT = 0`

`Ec+Ev= 2EF`

`(EF = Ec+Ev)/2`

Thus the Fermi energy /eve/lies in the middle of the forbidden energy gap in an intrinsic
semiconductor.

Vsristlon of Fermi Level with Impurity concentrstlon:

• At low impurity concentration the impurity atoms do not interact with each
other. Hence, the extrinsic carriers have their own discrete energy levels. 
• With the increase in impurity concentration the interaction of the impurity atoms start
and the Fermi level varies in the following way.
• As the impurity atoms interacts the donor electron are shared by the neighbouring
atoms.

• This results in splitting of the donor level and formation of the donor band below the
conduction band. With the increase in impurity concentration the width of the band
increases. At one stage it overlaps with the conduction band. As the donor band widens the forbidden gap decreases.ln the process the Fermi level shifts upwards
and finally enters the conduction band as shown: