Fermi–Dirac Distribution Function: Definition, Formula, Graph & Fermi Energy
Quick Exam Notes
- Definition: The probability that an energy state is occupied by an electron at temperature T.
- Formula: \( f(E) = \frac{1}{1 + e^{(E - E_f)/kT}} \)
- Semiconductors:
- Intrinsic → \(E_f\) at mid-gap
- n-type → \(E_f\) shifts toward conduction band
- p-type → \(E_f\) shifts toward valence band
- Graph: At T = 0 K, all states below Ef are filled; above Ef are empty. At T > 0 K, the curve smoothens.
- Applications: Explains electrical conductivity, heat capacity of metals and carrier concentration in semiconductors
Fermi level
It is the highest energy state occupied by a free electron at absolute temperature.
Fermi energy
It is the maximum energy possessed by a free electron at absolute temperature.
Figure 1: Fermi energy and Fermi level diagram illustrating electron occupancy in a semiconductor.
Fermi level in intrinsic and extrinsic semiconductor
Figure 2: Fermi level in intrinsic and extrinsic semiconductors.In intrinsic semiconductors, it lies near the middle of the band gap, while in n-type it shifts toward the conduction band and in p-type it shifts toward the valence band.
Fermi-Dirac Distribution
Electrons are fermions which have half integral spin. They obey Fermi Dirac statistics. It is denoted as f(E) and is given by
\[ f(E) = \frac{1}{1+\exp \left ( \frac{E-E_F}{kT} \right )} \]The above equation represents the Fermi Dirac (FD) statistics which tells us the probability of occupation of a given energy state as a function of a temperature. Lets discuss the following cases to understand the significance of FD function.
Figure 3: Fermi–Dirac distribution function graph. At 0 K, states below the Fermi energy are fully occupied and above are empty. At higher temperatures, the probability curve smoothens around the Fermi level.
At temperature T = 0 K,
\[ f(E) = \begin{cases} 1 & \text{if } E < E_f, \\ 0 & \text{if } E > E_f, \end{cases} \]This means at 0 kelvin all the energy states below the Fermi energy are occupied and all energy state above Ef are unoccupied. At 0 kelvin f(E) is just a delta function.. Also at any temperature T, for E = Ef ; f(E) = 1/2.
if we increase the temperature some of the states above Ef will get populated. If we further increase the temperature (T2), f(E) will increase. However, irrespective of temperature, probability at Fermi energy Ef will always be half.
Summary
The Fermi–Dirac distribution gives the probability that an electron occupies an energy level at temperature T. At T=0 K, all states below the Fermi energy are filled and above are empty. At T > 0 K, the curve smoothens near the Fermi level. This concept explains electron behavior in metals and semiconductors, helping to understand carrier concentration, conductivity, and device operation.
MCQs on Fermi–Dirac Distribution
-
The Fermi–Dirac distribution is used to describe:
- a) Bosons
- b) Fermions
- c) Classical particles
- d) Photons
Answer
b) Fermions
-
The Fermi–Dirac probability function is:
- a) \( f(E) = e^{-(E/kT)} \)
- b) \( f(E) = \frac{1}{1 + e^{(E - E_f)/kT}} \)
- c) \( f(E) = \frac{1}{e^{(E - E_f)/kT}} \)
- d) \( f(E) = \frac{1}{1 - e^{(E - E_f)/kT}} \)
Answer
b) \( f(E) = \frac{1}{1 + e^{(E - E_f)/kT}} \)
-
At absolute zero (T = 0 K), the Fermi–Dirac distribution shows:
- a) All states are partially filled
- b) All states are empty
- c) States below Ef are filled and above Ef are empty
- d) Random filling of states
Answer
c) States below Ef are filled and above Ef are empty
-
The Fermi energy is defined as:
- a) Highest occupied level at T = 0 K
- b) Lowest energy state available
- c) Average electron energy
- d) Conduction band minimum
Answer
a) Highest occupied level at T = 0 K
-
Which of the following fields relies heavily on Fermi–Dirac statistics?
- a) Semiconductor physics
- b) Classical thermodynamics
- c) Newtonian mechanics
- d) Optics
Answer
a) Semiconductor physics