Fermi Energy - Definition, Formula, FAQs

Fermi energy may be a concept in quantum physics usually pertaining to the energy difference between the very best and lowest occupied single-particle states during a quantum system of non-interacting fermions at temperature. In this article, we will discuss, Fermi energy and Fermi level.

What is Fermi?

A unit of length adequate to 10⁻¹⁵ metre (one femtometre), utilized in atomic (nuclear) physics. It’s almost like the diameter of a proton. The meaning of Fermi is “Above”.

The Fermi is a unit of measurement of length. The Fermi function determines the probability that an energy level (E) is crammed with an electron when the fabric we are working with is under equilibrium conditions.

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What is Fermi level (Electrochemical potential)?

At absolute zero temperature, the highest level occupied by an electron is called the Fermi level . When the transfer is complete, the Fermi levels of the 2 materials are equal.

What is Fermi energy?

Definition: Named after the Physicist by name Fermi. At absolute zero temperature, the energy occupied by an electron is called the Fermi energy. It is often defined as:

Fermi energy is a concept in quantum mechanics usually pertaining to the energy of the greatest occupied quantum state during a system of fermions at absolute 0 temperatures.

Formula: The equation for Fermi energy is given by:

Where, N= number of particles

m₀ = rest mass of each fermion

V = volume of the system, h = reduced Planck constant

Using this definition of Fermi energy, various related quantities can be useful.

The Fermi temperature is defined as

T_F= E_F/k_B

Where,

k_B = Boltzman constant

E_F = Fermi energy

The Fermi temperature is often thought of because the temperatures at which thermal effects are like quantum effects related to Fermi statistics. The Fermi temperature for metal may be a few orders of magnitude above temperature.

Other quantities defined during this context are Fermi momentum

And Fermi velocity:

v_F= p_F/m₀

These quantities are sequentially the momentum and group velocity of a fermion at the Fermi surface.

The Fermi momentum can also be represented as

Where

k_F = Fermi wave vector, is the radius of Fermi sphere

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Calculating Fermi Energy

The lowest possible Fermi energy of a system is calculated by grouping the states with equal energy into sets and order according to the increasing energy level. On successive addition of particles with time and 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 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 sometimes used interchangeably to refer to one another. Although, both the terms are equal at one temperature, they're different at other temperatures.

Fermi energy is applied in finding the electrical and thermal characteristics of the solids. It’s one of the important concepts in superconductor physics and quantum physics. It’s utilized in semiconductors and insulators.

What is Fermi energy level?

The probability of occupation of energy levels in the valence band and conduction band is named Fermi level.

In intrinsic or pure semiconductor, the amount of holes in the valence band is adequate to the amount of electrons within the conduction band. Hence, the probability of occupation of energy levels in conduction band and valence band are same. Hence, the Fermi level for the intrinsic semiconductor lies under the middle of the forbidden band.

Metals

Under the electron model, the electrons during a metal are often considered to make a Fermi gas. The number density N/V of conduction electrons in metals ranges between 10²⁸ and 10²⁹ electrons/m³. It is responsible for the Fermi energy of the order of 2 to 10 electron volts.

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What is the Fermi level in semiconductors?

At absolute zero temperature, the highest energy level occupied by an electron in the conduction band is called the Fermi energy.

Electrons settle into rock bottom available energy states at temperature and build a "Fermi sea" of electron energy states. The Fermi Level (with Fermi energy E_f) is that the “surface” of this ocean where electrons won't have enough energy to rise above the surface. It’s the energy state which is settled by the very best electron orbital at 0 Kelvin (absolute 0 temperature) and a parameter of the Fermi-Dirac distribution:

Where,

T= absolute temperature

k = Boltzmann’s constant

The above function represents the probability f (E) of an electron to occupy a state with energy E. For electrical conduction to take place, the distribution function has to be non-zero in the conduction band. In solids, the probability of occupancy with the energy level decides whether a material is a conductor, semiconductor, or insulator.

There’s a function between the valence and transfer (conduction) band called the energy gap; the higher the energy gap, the more energy is needed to transfer the electron from the valence band to the conduction band. At the Fermi level (when E=E_f), the probability simplifies to ½ and thus E_f lies halfway between the valence and conduction band, or within the middle of the energy gap (E_gap/2).

For semiconductors, electrons occupying the transfer (conduction) band level mean the capability of conducting electricity with energy. At higher temperatures, more electrons can bridge the energy gap and contribute to conduction.

The Fermi level is usually positioned somewhere under the band gap for any semiconductor material when it is intrinsic. The doped/ Fermi level for extrinsic semiconductors depends on the doping concentration and the kind of dopant used (p or n-type).

For conduction to occur, the Fermi level lies above the valence band since the density of states (DOS) is greater there. For p-type semiconductors, since the free obtainable electrons are delocalised in the products, the Fermi level might drop down to the valence band since the DOS accordingly decreases. Just the opposite occurs in case of n-type semiconductors; because of maximum free electrons, DOS around the transfer (conduction) band goes up and Fermi level increases. Position of Fermi level in intrinsic semiconductor lies in the middle of the forbidden gap.

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