The probability of occupation of allowed states in semiconductors is described by

• the Fermi-Dirac distribution function.

This function provides

• the probability that a particular energy state in a semiconductor is occupied by an electron at a given temperature.

The Fermi-Dirac distribution function is given by the following expression:

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

Where:

•  f(E) is the probability of occupation of the energy state E by an electron.
• E is the energy of the state being considered.
• E_f is the Fermi energy level, which represents 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.

• The Fermi-Dirac distribution function takes values between 0 and 1.
• At absolute zero temperature (T = 0 K), all energy states below the Fermi energy level (E_f) are completely occupied by electrons (f(E) = 1), while all states above E_f are unoccupied (f(E) = 0).
• As the temperature increases, some electrons gain thermal energy and move to higher energy states, leading to a broader distribution of occupied states.

In intrinsic semiconductors (pure semiconductors with no intentional doping),

• the Fermi level lies near the middle of the bandgap.
• As a result, at room temperature (T ≈ 300 K), the probability of occupation of energy states is close to 0.5 at the center of the bandgap.
• This implies that roughly half of the available energy states in the conduction band and valence band are occupied by electrons and holes, respectively.

In extrinsic semiconductors (doped semiconductors with intentional impurity atoms),

• the Fermi level shifts closer to either the conduction band (for N-type doping) or the valence band (for P-type doping).
• This shift results in a higher probability of occupation for energy states closer to the respective band edges.

The probability of occupation of allowed states in semiconductors is crucial in understanding the electrical behavior of these materials.

• It determines
  • the concentration of free charge carriers (electrons and holes) and
  • affects various electronic properties, such as
    • electrical conductivity,
    •  carrier mobility, and
    • carrier recombination rates.
• Understanding the Fermi-Dirac distribution function is essential for modeling and analyzing the behavior of semiconductor devices and optimizing their performance in practical applications.