Created By : awnish

Reviewed By : Phani Ponnapalli

Last Updated : Apr 10, 2023

Use flexible Fermi Level Calculator to check the fermi parameters like fermi energy, fermi wave number, fermi temperature and fermi velocity details in a less amount of time. It just takes the number density of electrons and gives the accurate result easily & fastly.

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Number density of electrons

How to Calculate Fermi Energy?

Following are the steps to calulate the fermi energy, fermi velocity, fermi wave number and fermi temperature. Go through these steps and obtain the results in a less amount of time.

Fermi Wave Number

  • Get the number density of the electrons.
  • Multiply the number density with 3π²
  • Find 1/3 power of the product to check the fermi wave vector.

Fermi Energy

  • Get the square of the Planck's constant and fermi wave number.
  • Multiply those squares and divide it by twice the electron mass to check the fermi energy.

Fermi Velocity

  • Multiply the Planck's constant by the fermi wavenumber.
  • Divide the result by the mass of the electron to check the fermi velocity.

Fermi Temperature

  • Divide the fermi energy by the Boltzmann constant to get the fermi temperature.

What is the Fermi Level? | Fermi Level Formulas

The Fermi energy is used to obtain the maximum energy of an electron at absolute zero temperature. The Fermi level is a surface at absolute zero temperature where no electrons will have the energy to rise above the surface.

The formulas to calculate the Fermi parameters are along the lines.

Fermi Wavenumber or Fermi wave vector kf = 3π²n)(¹/Γéâ)

Fermi velocity Vf = h x kF/m

Fermi energy Ef = h² x kf²/(2m)

Fermi temperature Tf = Ef/k


n is the number density

h is Plack constant

m is the mass of the electron and m = 9.109 x 10-31 kg

k is the Boltzmann constant and k = 1.38 x 10-23 m² - kg/(s²K)

Fermi-Dirac distrubution

The Fermi Dirac distribution is the propability that na electron is in a special energy state. The fermi Function formula is

f(E) = 1 / (e[(E - Ef)/(k * T)] + 1)


E is the energy of a particle

T is the temperature

The number density details of various metals are as follows:

Metal Number density (*10^28 electrons/m³)
Cu (copper) 8.47
Ag (silver) 5.86
Au (gold) 5.90
Be (beryllium) 24.7
Mg (magnesium) 8.61
Ca (calcium) 4.61
Sr (strontium) 3.55
Ba (barium) 3.15
Nb (niobium) 5.56
Fe (iron) 17.0
Zn (zinc) 13.2
Cd (cadmium) 9.27
Al (aluminium) 18.1
Ga (gallium) 15.4
In (indium) 11.5
Sn (tin) 14.8
Pb (lead) 13.2


Question: Find the fermi energy, fermi velocity, fermi temperature and fermi wavenumber of copper?


Given material is copper

Number density n = 8.47 x 1028 electrons/m3

Fermi wave vector kf = (3π²n)(¹/Γéâ)

kf = (3π² x 8.47 x 1028)(¹/Γéâ)

= 13.586 1/nm

Fermi velocity Vf = h x kF/m

vf = 6.626 x 10-34 x 13.586/9.109 x 10-31

= 15,72,855 m/s

Fermi energy Ef = h² x kf²/(2m)

Ef = (6.626 x 10-34)² x 13.586²/(2 x 9.109 x 10-31)

= 7.003 eV

Fermi temperature Tf = Ef/k

tf = 7.003/1.38 x 10-23

= 81,612 K

Therefore, the copper metal fermi temperature is 81,612 K, fermi energy is 7.003 eV, fermi velocity is 15,72,855 m/s, and fermi wave vector is 13.586 1/nm.

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Frequently Asked Questions on Acceleration Calculator

1. What is the Fermi level?

Fermi level is the higher energy level where an electron obtains at zero temperature. It is the state between the conduction band and valence band because at zero temperature the electrons remain in the lowest energy.

2. What is the fermi level formula?

The Fermi level formulas to calculate the fermi parameters are given here.

  1. Fermi wave vector kf = (3π²n)(¹/Γéâ)
  2. Fermi velocity Vf = h x kF/m
  3. Fermi energy Ef = h² x kf²/(2m)
  4. Fermi temperature Tf = Ef/k

3. What are the applications of Fermi energy?

Some of the applications of Fermi energy are listed here.

  • It is used to describe metals, semiconductors and insulators.
  • It is applied in determining the electrical and thermal characteristics of solids.
  • It is also used in nuclear physics to understand the stability of white dwarfs.

4. How to use Fermi Level Calculator?

Fermi level Calculator is used to find the fermi parameters such as fermi wave number, Fermi energy, fermi temperature and Fermi velocity. You just need to provide material type and number density of electrons to obtain the result easily.