The concept of energy is difficult in physics since its meaning varies depending on the situation. For instance, energy can take on several forms in atoms and molecules, such as heat, electricity, light, and so forth. It gets considerably more complicated in quantum mechanics. Within this area of physics, scientists rely on ideas such as the Fermi energy, which is the energy of the highest-occupied quantum state at zero temperature in a system of fermions.
The phrase “Fermi energy” is frequently employed to denote the Fermi level, which is closely linked to but distinct from Fermi energy (also termed electrochemical potential). At least in the sense that they are used in this article, there are a few significant distinctions between the Fermi level and Fermi energy:
While the Fermi level is defined at any temperature, the Fermi energy is only defined at absolute zero. Whereas the Fermi level is the overall energy level made up of both kinetic and potential energy, the Fermi energy is an energy differential that typically corresponds to kinetic energy.
What is Fermi Energy?
A Fermi level, so named in honour of physicist Enrico Fermi, is a measurement of the energy of the least securely bound electrons in a material. Determining the thermal and electrical characteristics of solids is crucial. It is characterized by The energy difference between the greatest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature is known as the Fermi energy in quantum mechanics.
The Fermi energy is the value of the Fermi level at -273.15 °C, which is the absolute zero temperature. At 0K, it is also the highest possible kinetic energy that an electron may have. Every solid has a constant Fermi energy.
Fermi Level
The Fermi Level is the maximum energy level an electron may occupy at zero degrees Celsius. The Fermi level lies between the valence band and the conduction band because electrons are at their lowest energy state at absolute zero temperature.
The Fermi level can be thought of as a hypothetical energy level of an electron in band structure theory, which is used in solid state physics to evaluate the energy levels in a solid. At thermodynamic equilibrium, this energy level would have a 50% probability of being occupied at any given time.
Fermi Energy Level
Pauli’s exclusion principle, which asserts that two fermions cannot occupy the same quantum state, is responsible for the creation of this energy level. Since each fermion has a unique set of magnetic quantum numbers, a system including several fermions will have distinct sets of magnetic quantum numbers for each fermion.
The Fermi temperature is defined as the energy of the Fermi level divided by Boltzmann’s constant. The temperature of the electron is often referred to as the temperature at which its energy equals the Fermi energy. Measurements are made of the electrons in metal in lower energy states.
Calculation of Fermi Energy
The states with equal energy are first grouped into sets and arranged in ascending order of energy to get the lowest feasible Fermi energy of a system. The empty quantum states with the lowest energy are then gradually filled in by adding particles one at a time.
The energy of the highest occupied state, when all the particles are aligned appropriately, is known as the Fermi energy. Even after cooling metal to almost absolute zero (0 Kelvin), which removes all available energy from it, electrons in the metal continue to travel. The fastest ones travel at a speed that is equivalent to the Fermi energy in kinetic energy.
Energy Level
Fermi Energy Formula
EF = ℏ2 /2mo (3π2N / V)2/3
mo = Rest mass
ℏ = reduced planck constant.
N = number of Fermions
V = Volume of system
we know the volume of the system, the rest mass, the number of occupied spaces, we can use the formula to determine the Fermi energy. It’s important to remember that the Fermi energy is independent temperature of material . Also, the variation in the number of fermions per unit volume is the only reason we get varied values of Fermi energy for different metals.
Steps to Calculate The Fermi Energy
We have the following steps to calculate the Fermi energy shown below
- Find out the Fermion’s rest mass value.
- For the system you are considering, find the number density N/V or N and V independently.
- Find the values of the relevant constants. This is the decreased Planck’s constant.
- Fill in the formula with the values and constants, then run the analysis.
Rest mass of fermions electron is mo = 9.1 x 10-31
Number density of copper is N/V = 8.5 x 10 28
Reduced planck constant h = 1.04 x 10-34
EF = (1.01 x 10-34 )2 (3π2 × 8.5 ×1028)2/3 / 2× 9.1 × 10−31
On calculating the following equation the EF = 1.1214 × 1018 J
To get value in electron volt divide EF with 1.6 x 10-19 so we get new EF = 7.0 eV
Applications of Fermi Energy
In quantum mechanics, it is a crucial idea. Here are a few applications of Fermi energy.
- It is applicable to semiconductors and insulators.
- It talks about semiconductors, metals, and insulators.
- Fermi energy can be used to determine the thermal and electrical properties of the solids.
- An essential component of nuclear physics is the understanding of white dwarf stability. White dwarfs are stars with a radius equal to that of the Sun but a fraction of its mass.
Conclusion
We can determine the electron density, the number of holes in the material, and the relative fraction of holes and density with regard to temperature thanks to the fermi energy levels. Understanding how electrical energy flows through semiconductor materials is essential to comprehending the material’s characteristics. The separation between the valence and conduction bands in metallic conductors is quite small. As a result, it will be simple for the partially filled valence electrons to migrate toward the conduction band. Conversely, a large band gap exists in semiconductors and insulators between the valence and conduction bands. The restricted band gaps contain their fermi energy levels.
As a result, semiconductors become partially conductive within a specific temperature range, while insulators lack a conduction band. Because of their lengthy residence times in the high energy states and large thermal excitations, semiconductors become excellent electrical conductors at high temperatures. The two main markers of any solid material’s electrical characteristics are its band gap and fermi energy levels.
Fermi Energy – FAQs
Define Fermi Energy
In a quantum system of non-interacting fermions, the energy difference between the highest and lowest occupied single-particle states is known as the Fermi energy.
Why Fermi Energy is important ?
The Fermi level, named for the physicist who initially proposed it, is a measurement of the energy of the least securely confined electrons within a solid. Determining the electrical and thermal characteristics of solids is crucial.
In case of insulator is Fermi level exist ?
Yes fermi level exist at T = 0k and maximum electron find at this temperature
Define Fermi Paradox?
The argument over the likelihood and extent of sentient life existing outside of Earth in the universe is known as the Fermi conundrum. There isn’t any proof that it was ever there. Scientist Enrico Fermi was the first to look into the Fermi paradox while investigating the possibility of extraterrestrial civilization. There are worlds similar to our own that may have evolved sentient life, and these advanced entities may have acquired the ability to travel between galaxies.
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