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Fermi Energy, Fermi Level and Fermions

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What is Fermi Energy?

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.

FAQs on Fermi Energy, Fermi Level and Fermions

1. What is Fermi energy?

Fermi energy is a fundamental concept in quantum mechanics that describes the maximum kinetic energy an electron (or any other fermion) in a system possesses at the temperature of absolute zero (0 Kelvin). At this temperature, particles occupy the lowest available energy states. Due to the Pauli Exclusion Principle, which states no two fermions can occupy the same quantum state, electrons fill up energy levels from the bottom up. The energy of the highest occupied state at 0 K is the Fermi energy.

2. What is the key difference between Fermi energy and the Fermi level?

While often used interchangeably, there is a subtle but important difference:

  • Fermi Energy (EF): This specifically refers to the highest occupied energy level at absolute zero (0 K). It is a fixed value for a given material.
  • Fermi Level: This is a more general term representing the energy level at which there is a 50% probability of finding an electron at any temperature above absolute zero. The Fermi level's position can shift with changes in temperature or with the introduction of impurities (doping) in a material like a semiconductor.

At 0 K, the Fermi level is equal to the Fermi energy.

3. What are fermions, and what principle governs their behaviour?

Fermions are a class of subatomic particles characterized by having a half-integer spin (e.g., 1/2, 3/2). The most common examples relevant to electronics are electrons, protons, and neutrons. The behaviour of fermions is governed by the Pauli Exclusion Principle. This principle states that no two identical fermions can occupy the same quantum state simultaneously within a quantum system. This is the fundamental reason electrons in an atom fill up orbitals in a structured way, leading to the concept of Fermi energy in solids.

4. How is the Fermi energy of a metal calculated?

The Fermi energy (EF) for a metal at 0 K can be calculated using the following formula, which is derived from the free electron model:

EF = (h2 / 8π2m) * (3π2N/V)2/3

Where:

  • h is Planck's constant.
  • m is the mass of an electron.
  • N/V is the number of free electrons per unit volume (electron density).

This formula shows that Fermi energy depends directly on the concentration of free electrons in the metal.

5. What are some important applications or implications of Fermi energy?

The concept of Fermi energy is crucial in several areas of modern physics and engineering. Its main implications include:

  • Understanding Solids: It is essential for the band theory of solids, which explains why some materials are metals (conductors), some are insulators, and others are semiconductors.
  • Semiconductor Devices: The position of the Fermi level is critical in determining the electrical properties of semiconductors and is a key parameter in designing devices like diodes, transistors, and integrated circuits.
  • Astrophysics: Fermi energy helps explain the stability of compact stellar objects like white dwarf stars, where the immense gravitational collapse is counteracted by electron degeneracy pressure, a direct consequence of the Pauli Exclusion Principle and Fermi energy.

6. Why is the Pauli Exclusion Principle essential for the existence of Fermi energy?

The Pauli Exclusion Principle is the very foundation of Fermi energy. Without it, all electrons in a solid at absolute zero would simply fall into the lowest possible energy state. However, because the principle forbids multiple electrons from occupying the same quantum state, they are forced to fill up a sequence of discrete energy levels, one after another. The energy of the 'last' electron added to the system at 0 K defines the Fermi energy. Therefore, Fermi energy is a direct and necessary consequence of fermions obeying this exclusion principle.

7. How does the concept of Fermi level differ for metals, insulators, and semiconductors?

The position of the Fermi level within the energy band structure is what defines the electrical properties of a material:

  • In Metals: The Fermi level lies within the conduction band. This means there are plenty of available energy states for electrons to move into, allowing for high electrical conductivity.
  • In Insulators: The Fermi level is located in the middle of a large forbidden energy gap between the valence band (full) and the conduction band (empty). A significant amount of energy is required to excite an electron into the conduction band, making them poor conductors.
  • In Semiconductors: The Fermi level also lies within a forbidden energy gap, but this gap is much smaller than in insulators. For an intrinsic (pure) semiconductor, it lies in the middle of the gap. Thermal energy at room temperature is sufficient to excite some electrons into the conduction band, allowing for moderate conductivity.

8. Why does the Fermi level shift with doping in a semiconductor?

Doping involves intentionally adding impurities to a semiconductor to alter its electrical properties. This directly affects the electron and hole concentration, causing the Fermi level to shift:

  • N-type Doping: Adding a pentavalent impurity (e.g., Phosphorus to Silicon) creates an excess of free electrons. This increases the concentration of charge carriers in the conduction band, causing the Fermi level to shift upwards, closer to the conduction band.
  • P-type Doping: Adding a trivalent impurity (e.g., Boron to Silicon) creates an excess of holes (vacancies for electrons). This increases the concentration of available states in the valence band, causing the Fermi level to shift downwards, closer to the valence band.

This ability to control the Fermi level's position is the basis for creating p-n junctions and modern electronic devices.

9. What is the relationship between Fermi energy, Fermi temperature, and the Fermi-Dirac distribution?

These three concepts are deeply interconnected:

  • Fermi Energy (EF): The maximum electron energy at 0 K.
  • Fermi-Dirac Distribution (f(E)): A statistical function that gives the probability that a given energy state 'E' will be occupied by an electron at a certain temperature 'T'. At 0 K, this function is 1 for energies below EF and 0 for energies above it.
  • Fermi Temperature (TF): A theoretical temperature defined by the relation TF = EF / kB, where kB is the Boltzmann constant. It is not a physical temperature of the material but a threshold. For temperatures much lower than TF (T << TF), quantum effects dominate. For temperatures much higher than TF, electrons begin to behave more like a classical gas.