Question

Explain the concept of fermi level. Prove that the Fermi level exactly at the centre of the Forbidden energy gap in intrinsic semiconductor.

Answer 1

FERMI LEVEL
Fermi level is not an allowed energy level it is an imaginary reference level used to specify other energy levels. Fermi level is defined as the highest filled energy level in any solid at absolute zero temperature.
Hence, at absolute zero temperature all energy levels below E_F are empty for which the

probability of occupancy can be written from Fermi-Dirac distribution function.
At any temperature T>0K in 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.

`N = n_c + n_v `

`f(E_c) = 1/(1+e (E_c-E_F)/(kT))`………………………(1)

`f(E_v) = 1/(1+e(-(E_c-E_F))/(kT)` ………………………(2)

`n_V = Nf(E_V) = N/(1+e((E_c - E_F)/(kT))`

`N= N/(1+e((-E_c - E_F))/(kT)) + N/(1+e((E_c - E_F)/(kT))`

`1= 1/(1+e(-(E_c - E_F))/(kT)) + 1/(1+e((E_c - E_F)/(kT))`

`1=((2+e^((E_c -E_F)/kT) +e ^(-(E_c-E_F)/kT)))/([1+e(E_c-E)/(kT)][1+e(-E_c+E_F)/(kT)]`

Solving above equation using cross multiplication method.

`e^((E_c-2E_F+E_V)/(KT) = 1`

`(E_c-2E_F+E_V)/(KT)=0`

`E_F = (E_C+E_V)/2`

Hence it is proved that fermi energy level in intrinsic semiconductor is at the Centre of forbidden energy gap.