Fermi Level

The Fermi energy (E_F) refers to the highest energy state of an electron at absolute zero temperature in a solid material. It represents the energy level at which the probability of finding an electron is 50%. In other words, at E_F, the occupation probability of electron states is exactly one-half. This energy level is significant because, according to the Pauli exclusion principle, no two electrons in a system can occupy the same quantum state simultaneously. The Fermi energy serves as a reference point for understanding the distribution of electrons in a material, particularly in determining the behavior of electrons near the top of the valence band or the bottom of the conduction band.

On the other hand, the Fermi level is a concept derived from the Fermi-Dirac distribution function, which describes the statistical distribution of electrons in a system at thermal equilibrium. Unlike the Fermi energy, which is a property of a material independent of temperature, the Fermi level depends on temperature and external factors such as doping or applied electric fields. The Fermi level represents the energy level at which the probability of finding an electron is equal to its probability at absolute zero temperature but with the effects of temperature and other external factors taken into account. It is an energy reference point that defines the equilibrium between the probability of finding electrons in occupied and unoccupied states.

The Fermi level (E_F) can be defined in terms of the Fermi-Dirac distribution function as:

\[ f(E) = \frac{1}{1 + e^{\frac{E – E_F}{k_B T}}} \]

Where:

– \( f(E) \) is the Fermi-Dirac distribution function, representing the probability of finding an electron with energy \( E \).

– \( E \) is the energy of the electron.

– \( E_F \) is the Fermi level.

– \( k_B \) is Boltzmann’s constant.

– \( T \) is the temperature in Kelvin.

At thermal equilibrium, the Fermi level represents the energy level at which \( f(E) \) is 0.5 (50% probability).

In metals, the Fermi level lies within the conduction band, which is the band of energy levels that electrons can freely move through. Since the Fermi level is within this band, metals exhibit high electrical conductivity, as electrons are readily available for conduction. At absolute zero, all states up to the Fermi level are filled with electrons, and those above it are empty, creating a sharp boundary between filled and empty states.

In semiconductors, the position of the Fermi level depends on the doping level or the presence of impurities. In an intrinsic semiconductor (undoped), the Fermi level lies close to the middle of the band gap, the energy range between the valence band (where electrons are bound to atoms) and the conduction band. When doped with impurities, the Fermi level shifts either towards the conduction band (for n-type doping, which introduces extra electrons) or towards the valence band (for p-type doping, which introduces electron “holes” or vacancies in the electron structure). This movement of the Fermi level affects the conductivity of the semiconductor, allowing it to be used in electronic devices such as transistors and diodes.

In insulators, the Fermi level lies within the band gap, far from the energy bands where electrons can freely move. As a result, insulators have very low electrical conductivity since there are no available states for electrons to conduct through at the Fermi level. Instead, electrons are tightly bound to their respective atoms, requiring a significant amount of energy to transition into the conduction band.