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Prove that the Fermi level lies exactly at the centre of the forbidden energy gap in case of an intrinsic semiconductor. OR Derive an expression for Fermi level for an intrinsic semiconductor.
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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.