Fermi Energy or Fermi Level –

When the filling of electrons is undertaken in the energy bands, clearly the lowest energy level gets filled first. But there will be many more allowed energy levels left vacant as shown in fig below.

1) The Fermi energy or Fermi-level is defined as – The energy of highest occupied level at $0^{\circ}$ absolute is called the Fermi-energy & the level is called the Fermi level $E_F$.

2) All the energy levels above the Fermi-level at T = $0^{\circ}k$ are empty & those lying below are completely filled.

3) $E_F$ is taken as the datum or reference w.r.t. which other energy levels are compared.

Fermi-Dirac Function –

Fermi-Dirac function is given by,

$f(E)=\dfrac {1}{1+exp \frac {(E-E_{F})}{kt}}\\$ where $F(E)=$ probability that a particular energy level $E$ is occupied by electron $E_F=$ Fermi energy $k=$ Boltzmann constant $T=$ temperature in $^{\circ}k$ To show that for an intrinsic semiconductor, the Fermi-level lies half way between conduction & valency band. 1) At any temperature \(T\gt0^{\circ}k\), we know from semiconductor physics that, $n_{e}=N_{C}\ e^{\frac {-(E_{C}-E_{F})}{kT}}$ & $n_{v}=N_{V} \ e^{\frac {-(E_{F}-E_{V})}{kT}}$ where $n_e=$ numbers of electrons in conduction band $n_v=$ numbers of holes in valence band $N_C=$ Effective density of state in conduction band $N_V =$ Effective density of state in valency band. 2) For the best approx,, $N_C=N_V $ & For intrinsic semiconductor, $n_e=n_v$ $\therefore N_{C}\ e^{\frac {-(E_{C}-E_{F})}{kT}} = N_{V}\ e^{\frac {-(E_{F}-E_{V})}{kT}}$ $\therefore e^{\frac {-[E_{C}-E_{F}-E_{F}+E_{V}]}{kT}}=1$ $e^{-[E_{C}+E_{V}-2E_{F}]}=1$ 3) Taking logarithm on both side, $\dfrac {-(E_{C}+E_{V}-2E_{F})}{kT}=0$ $\therefore E_{F}=\dfrac {E_{C}+E_{V}}{2}$

Hence, the intrinsic semiconductor Fermi-level always remains at middle between the forbidden energy gap.