Learn how to calculate the Fermi level and many more fundamental quantities of solids state physics with our Fermi level calculator. We will keep the physics and the math simple and understandable, trying to help you understand the basics of particle statistics and statistical mechanics.

Keep reading our article: you will learn what a fermion is, why we use the Fermi-Dirac distribution to describe them, and what are the Fermi parameters. We will explain to you the Fermi level, the equation for the Fermi energy, temperature, the other parameters, and much more!

Introduction to the Fermi-Dirac distribution: what is the Fermi level?

Every portion of matter in the known universe is made by only two types of particles: bosons and fermions.

Bosons like to huddle: if we cool down a bosonic system down to the absolute zero, all the particles will end in the lowest possible energy level in a phenomenon we call Bose-Einstein condensation.

Fermions, on the other hand, are more introvert, and they occupy energy levels alone (or in pairs, if you ignore the spin, for example).

🙋 Enrico Fermi was an Italian scientist considered one of the most influential figures in modern quantum and nuclear physics development. His contributions spaced from fundamental physics to the development of atomic energy for peaceful and military purposes.

These vastly different behaviors are reflected in the distributions of these two types of particles. In a Bose-Einstein condensate, all particles have the same energy, while in a Fermi system, the energy of particles increases. If we are at absolute zero, all levels are tightly occupied up to a specific energy, the Fermi energy, with a corresponding energetic level, called the Fermi level.

Calculate the Fermi parameters: the formula for the fermi energy, velocity, and much more

The fundamental quantity we use in calculating the Fermi level, energy, and the other Fermi parameters is the electron number density.

You can find the tabulated values of this quantity in many solid state physics and online. The tool number density calculator can help you, too.

Once you know this value, you can proceed to calculate the various Fermi parameter:

  • Firstly, we will calculate the Fermi wave vector;
  • Its value will appear in the formula for the Fermi energy;
  • With a different equation than the fermi energy, we can calculate the Fermi velocity;
  • Eventually, the Fermi energy allows calculating the Fermi temperature;
  • The Fermi level requires a deeper analysis.

To calculate the Fermi wave vector, the maximum energy of a fermion at the absolute zero, we apply the following formula:

kF=(3π2n)13\bold{k}_{\text{F}} = \left(3\cdot\pi^2\cdot n\right)^{\frac{1}{3}}

The Fermi wave vector is the radius of the sphere of occupied states in the k-space.

🙋 Wavevector? Isn't an electron a particle? Not exactly! Visit our De Broglie wavelength calculator to learn why you are probably only half correct!

The formula for the Fermi energy uses the relationship between energy and momentum in particle physics:

εF ⁣= ⁣(kF)22m ⁣= ⁣22m(3π2n)23\varepsilon_{\text{F}} \!= \!\frac{\left(\hbar\cdot k_{\text{F}}\right)^2}{2\cdot m} \!=\! \frac{\hbar^2}{2\cdot m}\left(3\cdot\pi^2\cdot n\right)^{\frac{2}{3}}

The Fermi energy is the energy of the highest occupied energy level at the absolute zero. Do you know that even if momentum at a particle level has a different meaning and definition than the one we use every day, the law of conservation of momentum still holds? It's a cornerstone of our universe!

With a simple formula we can calculate the Fermi velocity from the Fermi wave vector:

vF=kFm\bold{v}_{\text{F}} = \frac{\hbar\cdot\bold{k}_{\text{F}}}{m}

The Fermi temperature is an essential concept in particle physics: it's the temperature at which quantum effects fade, leaving room to thermal effects. We compute it using another famous (or infamous, if you're studying physics) relationship:

TF=εFkBT_{\text{F}} = \frac{\varepsilon_{\text{F}}}{k_{\text{B}}}

In these formulas, we meet two fundamental constants of Nature:

  • \hbar, pronounced "hbar": the reduced Planck's constant. It has value h/2πh/2\pi and is crucial in almost every aspect of quantum physics; and
  • kBk_{\text{B}}, the Boltzmann's constant, a cornerstone of thermodynamics, and, as we've seen, a unit of scale for energies and temperatures.

What is the Fermi level: calculations for the statistical occupation of energy levels

The parameters above are all independent of the temperature. In fact, they are best defined when the temperature is 0 K0\ \text{K}. There is another quantity, however, that requires the temperature to acquire a meaning: the Fermi level. To understand what is the Fermi level and how to calculate it, we need to introduce the Fermi-Dirac distribution. The Fermi-Dirac distribution is a probability function that describes the occupation of energy levels. It has the recognizable form:

f(εi)=1eεiεFkBT+1f(\varepsilon_i) = \frac{1}{e^{\frac{\varepsilon_i - \varepsilon_{\text{F}}}{k_{\text{B}}\cdot T}} + 1}

This equation has result bounds between 11 if the level with energy εi\varepsilon_i is occupied and 00 if the corresponding level is free. The values are probability of occupation.

The Fermi-Dirac distribution has a box shape for zero temperatures, with a sharp step at εi=εF\varepsilon_i = \varepsilon_{\text{F}}. If the temperature increases above absolute zero, we can't use the Fermi energy in the equation anymore, and we better substitute it with the Fermi level. The Fermi level, μ(T)\mu(T), is the energy of the energy level with occupation probability P=0.5P=0.5 at a given temperature and, for reasons deeply connected to statistical mechanics, corresponds to the chemical potential. To calculate the Fermi level, we use the following formula:

μ(T)(kBT)2εF+εF\mu(T) \approx \frac{(k_{\text{B}}\cdot T)^2}{\varepsilon_{\text{F}}} + \varepsilon_{\text{F}}

🙋 The Fermi level and the Fermi energy corresponds only at the absolute zero!

Applications of the equation for the Fermi energy: the case of aluminum

Take aluminum, a nice and light metal with widespread applications in quantum technologies and kitchenware (for all the possible different reasons). The number density of electrons in aluminum is n=18.1×1028n = 18.1\times10^{28} electrons per cube meter. From this value, we can compute or ask our Fermi level calculator to do this for us, the values of the various Fermi parameters. We then find:

  • Fermi wave vector: kF=17.5 nm1\bold{k}_{\text{F}} = 17.5\ \text{nm}^{-1};
  • Fermi energy: εF=11.67 eV\varepsilon_{\text{F}} = 11.67\ \text{eV};
  • Fermi velocity: vF=2025911 m/s\bold{v}_{\text{F}} = 2025911\ \text{m}/\text{s}; and
  • Fermi temperature: TF=135,400 KT_{\text{F}} = 135,400\ \text{K}.

These parameters vary greatly for different metals. Feel free to explore them in our tool!

🙋 We expressed the Fermi energy in electronvolts. Learn what is this measurement unit at our volt to electronvolt calculator and how to convert between energy units at our handy energy converter!

Davide Borchia
Material properties
Material
Select...
Number density of electrons
* 10²⁸ electrons/
cu yd
Fermi parameters
Fermi wave number
nm
⁻¹
Fermi energy
eV
Fermi temperature
K
Fermi velocity
ft/s
Fermi - Dirac distribution
Energy
eV
Temperature
K
Fermi Function
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