written 3.9 years ago by |
It can be shown for intrinsic semiconductors that fermi energy level lies midway between conduction band and valence band. The proof is as follows:
At any temperature (T>0K)
ne = number of electrons in conduction band
nv = number of electrons in valence band
We have, by definition of intrinsic semiconductors:
The number of electrons in conduction band is given by
ne=Nce−(EC−EF)KT
The number of electrons in valence band is given by
nv=Nve−(EF−EV)KT
Where,
Nc = effective density of states in conduction band
NV = effective density of states in valence band
For best approximation, we consider Nc = NV
For intrinsic semiconductors, nc = nv
∴Nce−(EC−EF)KT=Nve−(EF−EV)KT
NVNC=e−(EC−EF)KTe−(EF−EV)KT
NVNC=e−(EC−EF−EF+EV)KT
NVNC=e−(EC−2EF+EV)KT
As NC = NV = 1;
we get,
e−(EC−2EF+EV)KT=1
Taking log on both sides, we get,
−(EC−2EF+EV)KT=0
∴EC+EV=2EF
∴EC+EV2=EF
Thus, the Fermi level lies exactly at the centre of the forbidden energy gap in case of an intrinsic semiconductor.