The Fermi-Dirac distribution function is a fundamental concept in statistical mechanics that describes

• the probability of occupation of allowed energy states by electrons in a system at thermal equilibrium.
• It is particularly significant in
  • understanding the electrical behavior of semiconductors, as it governs the population of charge carriers (electrons and holes) in energy bands at different temperatures.

The Fermi-Dirac distribution function is mathematically represented as follows:

$$f\left(E\right)=\;\frac1{\left(1+e^{\left({\displaystyle\frac{\left(E-E_f\right)}{k_BT}}\right)}\right)}$$

Where:

• f(E) is the probability of occupation of an energy state E by an electron.
• E is the energy of the energy state being considered.
• E_f is the Fermi energy level, representing the highest energy level that electrons can occupy at absolute zero temperature.
• k is Boltzmann’s constant (approximately 8.617 x 10^-5 eV/K).
• T is the temperature in Kelvin.

At absolute zero temperature (T = 0 K),

• the Fermi-Dirac distribution function simplifies to a step function.
• All energy states below the Fermi energy level (E_f) are
  • fully occupied by electrons (f(E) = 1),
• while all states above E_f are
  • unoccupied (f(E) = 0).
• This situation corresponds to a fully filled valence band and an empty conduction band in semiconductors, making them insulators in this temperature regime.

As the temperature increases,

• the Fermi-Dirac distribution function smoothens out, allowing for the partial occupation of higher energy states.
• Some electrons gain thermal energy and transition from the valence band to the conduction band, resulting in the generation of free charge carriers (electrons and holes).
• The probability of occupation of energy states gradually decreases with increasing energy above the Fermi level.

The impact of the Fermi-Dirac distribution function on the electrical behavior of semiconductors at different temperatures is significant and has several key implications:

1.Temperature-dependent Carrier Concentration:

• The probability of occupation of energy states directly affects the concentration of charge carriers in the semiconductor material.
• At higher temperatures, more charge carriers are generated as more energy states become partially occupied.
• This increase in carrier concentration leads to a rise in electrical conductivity in semiconductors, enabling better current conduction.

2.Temperature Sensitivity:

• The Fermi-Dirac distribution function makes semiconductor electrical properties temperature-dependent.
• As temperature changes, the occupation probability of energy states changes, leading to variations in carrier concentration and electrical conductivity.
• This temperature sensitivity is essential for various semiconductor device applications, such as temperature sensors and thermistors.

3.Bandgap Reduction:

• At higher temperatures, the Fermi-Dirac distribution function causes a slight reduction in the effective bandgap of semiconductors.
• This reduction results in enhanced photon absorption and improved performance in optoelectronic devices like LEDs and solar cells.

4.Carrier Mobility:

• While the Fermi-Dirac distribution function primarily governs the occupation probability of energy states, it indirectly influences carrier mobility as well.
• Higher carrier concentrations at elevated temperatures can lead to increased scattering events, affecting the mobility of charge carriers in the material.

In summary, the Fermi-Dirac distribution function plays a critical role in describing the probability of occupation of allowed states in semiconductors at different temperatures.

It enables the understanding of carrier generation, carrier concentration, and electrical conductivity in these materials.

By capturing the complex behavior of electrons in energy bands, the Fermi-Dirac distribution function is a fundamental tool in the study and design of semiconductor devices, influencing their performance and operation across a wide range of temperature conditions.