Write the Fermi–Dirac distribution function and explain the terms involved in it. (asked in MU Dec 2024; also June 2025 KT)

Answer: \(f(E)=\dfrac{1}{1+e^{(E-E_F)/kT}}\), the probability that an energy state \(E\) is occupied by an electron at temperature \(T\).

\[f(E)=\frac{1}{1+\exp\left(\dfrac{E-E_F}{kT}\right)}\]

Terms:

• \(f(E)\): probability that an available energy state \(E\) is occupied by an electron (a number between 0 and 1).
• \(E\): energy of the state under consideration.
• \(E_F\): Fermi energy (Fermi level), the reference level of the distribution.
• \(k\): Boltzmann constant \(=1.38\times10^{-23}\) J/K; \(T\): absolute temperature in kelvin.

Meaning:

• At \(T=0\) K: \(f(E)=1\) for \(E<E_F\) (the exponent is \(-\infty\)) and \(f(E)=0\) for \(E>E_F\) (the exponent is \(+\infty\)). So all states below the Fermi level are completely filled and all states above it are empty; \(E_F\) is the highest filled level at 0 K.
• At \(T>0\) K: for \(E=E_F\), \(f(E)=\frac1{1+e^0}=\frac12\). So the Fermi level is the energy level whose probability of occupation is 50% at any temperature above 0 K. Some states just below \(E_F\) become empty and some just above it become occupied.