# Fermi Energy

In Physics, the concept of energy is tricky because it has different meanings depending on the context. For example, in atoms and molecules, energy comes in different forms such as light energy, electrical energy, heat energy, etc. In quantum mechanics, it gets even trickier. In this branch of physics, scientists rely on concepts like Fermi energy which refers to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature.

## What is Fermi Energy?

Named after the physicist, Enrico Fermi, a Fermi level is the measure of the energy of least tightly held electrons within a solid. It is important in determining the thermal and electrical properties of solids. The value of the Fermi level at absolute zero temperature (−273.15 °C) is known as the Fermi energy. It is also the maximum kinetic energy an electron can attain at 0K. Fermi energy is a constant for each solid.

### Calculating Fermi Energy

To determine the lowest possible Fermi energy of a system, we first group the states with equal energy into sets and arrange them in an increasing order of energy. We then add particles one at a time, successively filling up the unoccupied quantum states with the lowest energy.

When all the particles are arranged accordingly, the energy of the highest occupied state is the Fermi energy. In Spite of the extraction of all possible energy from a metal by cooling it to near absolute zero temperature (0 Kelvin), the electrons in the metal still move around. The fastest ones move at a velocity corresponding to a kinetic energy equal to the Fermi energy.

The Fermi level and Fermi energy are usually confusing terms and often used interchangeably to refer each other. Although, both the terms are equal at absolute zero temperature, different at other temperature.

Fermi energy is applied in determining the electrical and thermal characteristics of the solids. It is one of the important concepts in superconductor physics and quantum mechanics. It is used in semiconductors and insulators.

### Fermi Level

The highest energy level that an electron can occupy at the absolute zero temperature is known as the Fermi Level. The Fermi level lies between the valence band and conduction band because at absolute zero temperature the electrons are all in the lowest energy state. Due to lack of sufficient energy at 0 Kelvin, the Fermi level can be considered as the sea of fermions (or electrons) above which no electrons exist. The Fermi level changes as the solid is warmed and as electrons are added to or withdrawn from the solid.

### Fermi Energy Level

The reason for the existence of this energy level is due to Pauli’s exclusion principle which states two fermions cannot occupy that same quantum state. So, if a system has more than one fermions, each fermion has a different set of magnetic quantum numbers associated with it.

The Fermi Temperature can be defined as the energy of Fermi level divided by the Boltzmann’s constant. It is also the temperature at which the energy of the electron is equal to the Fermi energy. It is the measure of the electrons in the lower states of energy in metal.
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## Applications of Fermi Energy

It is one of the important concepts in quantum mechanics and condensed matter physics. Some fermi energy applications are given in the points below.

• It is used in semiconductors and insulators.
• It is used to describe insulators, metals, and semiconductor.
• Fermi energy is applied in determining the electrical and thermal characteristics of the solids.
• It is also important in nuclear physics to understand the stability of white dwarfs. White dwarfs are stars that have mass comparable to the Sun but have about a hundredth of its radius.

### Value of Fermi energy for different elements

 Element Fermi Energy eV Li 4.74 K 2.12 Na 3.24 Cs 1.59 Rb 1.85 Ag 5.49 Cu 7.00 Be 14.3 Au 5.53 Ca 4.69 Mg 7.08 Ba 3.64 Sr 3.93 Fe 11.1 Nb 5.32 Zn 9.47 Mn 10.9 Hg 7.13 Cd 7.47 Al 11.7 Ga 10.4 In 8.63 Tl 8.15 Sn 10.2 Pb 9.47 Bi 9.90 Sb 10.9

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## Fermi Energy Important Questions

Q1. For intrinsic semiconductor, where does the Fermi energy level lie?

Ans: It lies close to valence band.

Q2. For p-type of extrinsic semiconductor, where does the Fermi energy level lie?

Ans: It lies close to conduction band.

Q3. For n-type of extrinsic semiconductor, where does the Fermi energy level lie?

Ans: It lies close to conduction band.

Q4. On what does the Fermi level of extrinsic semiconductor depend on?

Ans: Fermi level of extrinsic semiconductor depend on:

• Temperature
• Concentration of impurities
• Donor element

Q5. How to calculate the position of Fermi level for p-type material?

Ans: $E_{F}=E_{V}+kTln\frac{N_{D}}{N_{A}}$

Q6. Determine the movement of the position of Fermi energy level when the temperature increases.

Ans: As the temperature increases, the Fermi energy level tends to move towards the centre of the energy gap.

Q7. What is Quasi-fermi energy level?

Ans: Quasi-fermi energy level is defined as the change in the level of Fermi level as the charge carriers are added excessively to the semiconductor.

Q8. What happens to the Fermi energy level when p-type and n-type semiconductors are joined together?

Ans: Fermi energy level decreases for n-type and for p-type it increases until equilibrium is obtained.

Q9. What happens to the Fermi energy level when forward bias is applied to the PN junction diode?

Ans: When forward bias is applied, for n-type the Fermi energy level increases and for p-type the Fermi energy level it decreases.

Q10. What happens to the Fermi energy level when reverse bias is applied to the PN junction diode?

Ans: When reverse bias is applied, for n-type the Fermi energy level decreases and for p-type it will increase.

#### Practise This Question

The magnitude of the focal length of a diverging less is 20 cm. What is the power of this lens?