Fermi Energy, Fermi Level and Fermions

Energy in quantum mechanics is a complicated affair and is different from how energy is perceived in terms of light energy, thermal energy, etc. in Physics. An example of such a form of energy is Fermi energy. Let’s understand more about Fermi energy and topics related to it, in this article.

The concept of Fermi energy derives its name from the 20th-century Italian physicist, Enrico Fermi, who defined a measure of energy to understand the point at which the electrons in a solid are at their most loosely bonded. Some solids are tighter and are more strongly held together than others, and the difference between the extent to which each one is held together is called the Fermi level. If one were to understand it generally, Fermi level is the highest kinetic energy that can be achieved by an electron at an absolute zero temperature.

More specifically, at absolute zero temperature, the highest kinetic energy that an electron can achieve, i.e. the difference in energy between the highest and lowest states of fermions in a quantum system, is called Fermi energy.

How to Calculate Fermi Energy?

To put a number to the Fermi energy in a system, every state that has equal energy should be organized into sets. These sets shall be in an increasing order, where single particles are added continuously and state with the lowest energy are gradually filled up. The energy in the highest state is how one arrives at the Fermi energy quantity.

Here is a simpler way to understand it. The particles are cooled down to absolute zero or at zero Kelvin, and this essentially removes all potential energy from the metal. But when the energy is removed, electrons still continue to move around freely. The ones that move fastest in terms of velocity are moving at Fermi energy.

What is a Fermi Level?

There is a minor difference between Fermi level and Fermi energy, even though in most instances they are used as one and the same. The Fermi level refers to the highest energy level an electron experiences at 0 Kelvin. This "level" is the Fermi level. But the Fermi level would be different from Fermi energy in temperatures that are not absolute. This is why both the terms are necessary.

What are Fermions?

Based on the Fermi-Dirac statistics, when a group of subatomic particles have angular momentum that is odd and have half-integral movement (\[ \frac{1}{2} or \frac{3}{2} spin \]), each of these particles In the system is known as fermions. Leptons, baryons, nuclei of odd mass number, are some examples of particle classes that are classified as fermions by the definition above. Here, it is important to discuss Pauli's Exclusion Principle. According to it, one quantum state cannot be occupied by more than one particle of this form and type.

Boltzmann Constant

Named after the Austrian physicist Ludwig Boltzmann, the Boltzmann Constant is a physics constant that is found in basic as well as quantum physics in Statistical Mechanics. The constant is symbolized as “k” and has the value of \[1.380649 \times 10^{-23}\] joules per kelvin (K). The constant is relevant to Fermi levels as the constant k allows one to calculate the amount of energy or heat generated in correspondence with random thermal motions of a particle in a substance.

Bonus Information - Band Theory

The Band Theory is an inseparable fundamental theory in solid-state electronics, as it helps a student note the electrical and thermal nature of various solids. The Theory organizes electrons and their behavior in specific states into bands, which represents a range of allowed energies. Every brand has a specific level at which it conducts, and this Fermi level at each band is an important part of the theory and has various practical applications.

Did You Know?

The Band of Solid Theory helps to explain the quantum state that an electron attains inside a solid metal. Each molecule is made up of different discrete energy levels and how an electron function within a molecule is described appropriately in this theory. Fermi level is an important part of this theory which states about the top of the available electron energy levels at low temperatures. The position of the Fermi level for the conduction band is a vital parameter in determining the electrical properties.