### Fermi Level Calculator :

Enter the Fermi wave number in nm⁻¹, Fermi energy in eV, Fermi temperature in K, Fermi velocity in m/s, Fermi-Dirac distribution Energy in eV, and you get the Fermi-Dirac distribution temperature in Kelvin (K).

Fermi Level Formula:

Fermi level formula wavenumber kf_{(nm)} in nanometer is equal to the square of the 3π^{2} and multiply the number density n_{(em)} in electron meter.Hence the Fermi level formula can be written as

Fermi wavenumber or Fermi wave vector kf_{(nm)} =(3π^{2} * n_{(em)} )^{(1/3)}

Fermi velocity Vf = h* kF/m

Fermi energy Ef = h^{2} * KF^{2} / (2m)

Fermi temperature Tf = Ef /K

Where,

N = Number density

H = planck constant

M = 9.109 *10^{-31} kg mass of the electron

K = 1.38 *10^{-23} m^{2}-Kg /(s^{2}K) Boltzmann constant

The Fermi dirac distribution is the probability that na electron is in a special energy state. The Fermi function formula is

F(E) = 1/ (e^{[(E-Ef)/ (k*T)]} +1)

here,

E=energy of a particle in eV

T= temperature in K

### Example:1

Calculate the gold metal Fermi energy, Fermi velocity, Fermi temperature and Fermi wave number?

### Answer:

Given material is gold

Number density n =5.90*10^{28} electrons /m^{3}

Fermi wav vector kf =(3π^{2} * n )^{(1/3)}

Kf = (3π^{2} * 5.90*10^{28} )^{ (1/3)}

=12.043 1/nm

Fermi velocity vf =h* kF /m

Vf =6.626 *10^{-34} *12.043 /9.109 *10^{-31}

Vf = 8.76022 m/s

Fermi energy Ef =h^{2} *kf^{2} /(2 *m)

Ef =(6.626 *10^{-34})^{2} *12.043 1 ^{2} / (2* 9.109 *10^{-31})

Ef =2.89913 ev

Fermi temperature Tf =Ef /K

Tf =2.899 /1.38 *10^{-23}

=2.10072K

Therefore, the copper metal Fermi temperature is 2.10072K , Fermi energy 2.89913 ev,Fermi velocity is 8.76022 m/s and Fermi wave vector is 12.043 1/nm

### Example:2

Calculate the Fermi wave number of magnesium?

### Answer:

kf =(3π^{2} * n )^{(1/3)}

n= 8.61 *10^{28}electrons /m^{3}

kf =(3π^{2} * n )^{(1/3)}

kf =(3* π^{2} *8.61 *10^{28})^{(1/3)}

kf= 136607477m^{-1}