Fermi – Dirac Distribution Function

By
Last updated:
  • It is an expression for the distribution of electrons among the energy levels as a function of temperature.
  • The probability of finding an electron in a particular energy state of energy E is given by,
\mathrm{F}(\mathrm{E})=\frac{1}{1+\exp \left(\frac{\mathrm{E}-\mathrm{E}_{\mathrm{F}}}{\mathrm{KT}}\right)}

Where, EF  Fermi energy (highest energy level of an electron)

K – Boltzmann’s constant 

T – Absolute temperature 

Effect of Temperature on Fermi Function

Case 1: At T=0 K and E< EF

\begin{array}{ll} \mathrm{F}(\mathrm{E})=\frac{1}{1+\exp (-\infty)}=\frac{1}{1+0} & \left\lvert\, \begin{array}{l} \mathrm{e}^{0}=1 \\ \mathrm{e}^{-\infty}=0 \\ \mathrm{e}=\infty \end{array}\right. \\ \mathrm{F}(\mathrm{E})=1=100 \% & \end{array}
  • It means that 100% chance to find the particle [electron]
  • At zero Kelvin, all energy states below EF are occupied by electrons.

Case 2: At T=0 K and E>EF

\begin{aligned} & F(E)=\frac{1}{1+\exp (-\infty)}=\frac{1}{1+\infty}=\frac{1}{\infty}=0 \\ & F(E)=0=0 \% \end{aligned}
  • It means that 0% chance to find the particle [electron]
  • At zero Kelvin, all energy states above EF are empty.

Case 3: At T>0 K and E=EF

\begin{aligned} & F(E)=\frac{1}{1+\exp (0)}=\frac{1}{1+1}=\frac{1}{2}=0.5 \\ & F(E)=0.5=50 \% \end{aligned}
  • It means that 50% chance to find the particle [electron]
  • At zero Kelvin, energy states above EF are empty and below EF are filled.

fermi

Fig 1.1

Where EF is Fermi energy or Fermi Level.

Fermi level : The Fermi level is the highest reference energy level of a particle at absolute zero.

Importance : It is the reference energy level which separates the filled energy levels and vacant energy levels.

Fermi energy [EF ] : It is the maximum energy of the quantum state corresponding to Fermi energy level at absolute zero.

Importance : Fermi energy determines the energy of the particle at any temperature.

Variation of Fermi function with temperature

The Fermi function varies with respect to the temperature. At 0 K all the energy states below EF are filled and all those above EF are empty.
Fermi Graph
Fig 1.2

When the temperature is increased, the electron takes an energy KT and hence the Fermi function falls to zero.

Read More Topics
Energy distribution of electrons in metals
Three types of optical fibre
Fibre – Optic sensor working principle
For Feedback - techactive6@gmail.com

Leave a Comment