written 8.3 years ago by | • modified 8.3 years ago |
Let,
$n_e$ be the number of electrons in the semiconductor band.
$n_v$ be the number of holes in the valence band.
At any temperature, T>0K
$n_e = N_c.e^{-(E_c - E_F)/kT}$ and
$n_v = N_v.e^{-(E_F-E_c)/kT}$
Where, $N_c$ is the effective density of states in the conduction band.
$N_v$ is the effective density of states in the valence band.
For best approximation, $N_c = N_v$
For an intrinsic semiconductor, $n_c = n_v$
$N_c . e^{-(E_c - E_F)/kT} = N_v . e^{-(E_F - E_c)/kT}$
$\frac{e^{-(E_c - E_F)/kT}}{e^{-(E_F - E_c)/kT}} = \frac{N_v}{N_c}$
$e^{-(E_c + E_V + 2E_F)/kT} = 1 \ \ \ ( N_c = N_v)$
Taking ln on both sides,
$\frac{-(E_C + E_V - 2E_F)}{kT} = 0$
$E_C = \frac{E_C + E_V}{2}$
Thus, Fermi level in an intrinsic semiconductor lies at the centre of the forbidden gap.