What is the Fermi-Dirac distribution function and how does it behave at different temperatures?

In one line

The Fermi-Dirac distribution function \(f(E) = 1/(1 + e^{(E - E_F)/kT})\) gives the probability that an electron energy state \(E\) is occupied at temperature \(T\).

Explained simply

Imagine seats in a stadium arranged by height, and fans fill from the bottom up, one per seat (electrons obey the Pauli exclusion principle). At absolute zero, every seat up to a certain row (the Fermi level) is filled and every seat above is empty. When it warms up, a few fans near that row jump to higher empty seats, blurring the boundary over a range of a few \(kT\).

How it works

1. Electrons are fermions: at most one per quantum state. Their occupancy follows \(f(E) = \dfrac{1}{1 + e^{(E - E_F)/kT}}\).
2. At T = 0 K: for \(E < E_F\), the exponent is \(-\infty\), so \(f = 1\) (all filled); for \(E > E_F\), \(f = 0\) (all empty). The graph is a sharp step.
3. At T > 0 K: at \(E = E_F\), \(f = 1/(1 + 1) = 1/2\). So the Fermi level is the energy at which the occupation probability is 50%.
4. States slightly below \(E_F\) have \(f\) a little less than 1, and states slightly above have \(f\) a little more than 0. The change happens over a few \(kT\) (about 0.026 eV at room temperature).
5. Probability that a state is empty (a hole): \(1 - f(E)\).
6. For \(E - E_F \gg kT\), \(f(E) \approx e^{-(E - E_F)/kT}\), the classical Boltzmann form.

Key formula

\(f(E) = \dfrac{1}{1 + e^{(E - E_F)/kT}}\). \(f(E)\): occupation probability (0 to 1), \(E\): energy of the state, \(E_F\): Fermi energy, \(k = 1.38\times10^{-23}\) J/K \(= 8.62\times10^{-5}\) eV/K, \(T\): temperature (K).

Quick example

At 300 K (\(kT \approx 0.0259\) eV), a state 0.1 eV above \(E_F\): \(f = 1/(1 + e^{3.86}) \approx 1/48.5 \approx 0.02\), so 2% occupied. A state 0.1 eV below \(E_F\) has \(f \approx 0.98\). For what energy is \(f = 0.01\)? \(e^{(E - E_F)/kT} = 99\), so \(E - E_F = kT\ln99 \approx 0.0259\times4.6 \approx 0.119\) eV.

Common mistakes

• Saying the Fermi level is the highest occupied level at all temperatures; that is true only at 0 K.
• Writing \(f(E_F) = 1\) at T > 0; it is exactly 1/2.
• Mixing eV and J for \(E - E_F\) and \(kT\).

How MU asks it

Asked as "Explain Fermi-Dirac distribution function and its variation with temperature" (5 marks) and numericals: "find the probability of occupation of a level ... eV above \(E_F\)" (3 marks). Write the function, analyse \(T = 0\) and \(T > 0\), draw \(f(E)\) versus \(E\) for both temperatures, and define the Fermi level.