The Fermi energy level, often referred to simply as the Fermi level (E_f), is a critical concept in solid-state physics that plays a fundamental role in determining the electronic properties of materials, including semiconductors.

• It represents the highest energy level that electrons can occupy at absolute zero temperature (T = 0 K) when the material is in thermal equilibrium.
• The Fermi level separates filled electron energy states (below the Fermi level) from empty energy states (above the Fermi level) in a material’s electronic band structure.

In intrinsic semiconductors,

• which are pure semiconductors with no intentional doping, the Fermi level lies at the mid-point of the bandgap between the valence band (where electrons reside at lower energies) and the conduction band (where electrons are free to move and conduct electricity).
• At absolute zero, all energy states below the Fermi level are fully occupied, while all states above the Fermi level are empty.
• The Fermi level is a crucial reference point that affects the probability of electron occupation in energy states at any given temperature.
• The probability of occupation of an energy state E by an electron is described by the Fermi-Dirac distribution function, which depends on the difference between E and E_f, as well as the temperature T.

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

• At absolute zero temperature (T = 0 K), the Fermi-Dirac distribution function reduces to a step function, with
  • f(E) = 1 for E < E_f and
  • f(E) = 0 for E > E_f.
  • This corresponds to a fully filled valence band and an empty conduction band, as mentioned earlier.

As the temperature increases,

• the Fermi-Dirac distribution function smoothens out,
• allowing for partial occupation of higher energy states.
• More specifically, at finite temperatures, there is a non-zero probability for energy states both above and below the Fermi level to be occupied by electrons, but the probability decreases with increasing distance from the Fermi level.

Now, let’s discuss how the position of the Fermi level influences the electrical properties of doped and intrinsic semiconductors:

Intrinsic Semiconductors:

As mentioned earlier, in intrinsic semiconductors,

• the Fermi level is located at the middle of the bandgap.
• This means that the probability of electron occupation is nearly 0.5 for energy states near the center of the bandgap at room temperature.
• As temperature increases, more energy states become occupied by electrons, resulting in an increase in the concentration of charge carriers (electrons and holes).
• This leads to a rise in the electrical conductivity of the material.

Doped Semiconductors:

Doping is the intentional introduction of impurity atoms into the semiconductor lattice to alter its electrical properties.

• When dopants are added, the position of the Fermi level shifts.
• For N-type doping,
  • which introduces additional electrons,
  • the Fermi level moves closer to the conduction band.
  • As a result, the probability of occupation of energy states in the conduction band increases, leading to a higher concentration of free electrons and enhanced electrical conductivity.
• On the other hand, for P-type doping,
  • which introduces holes (absence of electrons),
  • the Fermi level moves closer to the valence band.
  • This results in an increased probability of hole occupation in the valence band,
  • leading to a higher concentration of holes and improved electrical conductivity.

In conclusion,

• the Fermi energy level is a critical reference point that influences the probability of electron occupation in semiconductors.
• Its position determines the concentration of charge carriers and thus significantly impacts the electrical properties of both doped and intrinsic semiconductors.
• By controlling the Fermi level through doping or temperature variation, engineers can tailor the electrical behavior of semiconductors to suit specific applications, making these materials indispensable in modern electronic devices and technology.