Modern Theory of Solids

For a given Fermi energy level $E_F$, show that the probability of emptying energy level $KT$ below $E_F$ is equal to probability of occupying level $KT$ above $E_F$.

Find the temperature at which the probability of occupation of the energy state $0.75\ \mathrm{eV}$ above the Fermi energy is $30\%$.

Using image charge method, derive an expression of emission current density for Schottky effect.