The Fermi Energy

When you take your first solid-state physics or electronic materials class, you will be regularly confronted with band structure diagrams. At first, you'll be like "what in the hell" as there is so much information packed into a single plot. You have the x-axis, which contains Greek letters that indicate high-symmetry points, and then the paths between them indicate directions. Moreover, you have to remember this is in reciprocal space of the crystal structure, so you're dealing with wave-vectors ¹. The y-axis isn't as frustrating, for each k-point represents the allowable energies.

Now you may ask, "How do I use this kind of plot to extract information?" Yeah, that's not trivial, and understanding it is an objective of a solid-state physics course. However, in brief, you have things like phase and group velocity, which correspond to derivatives, i.e., $\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}}$, and tell you about how electrons conduct.

One particular aspect of band structure I want to cover is the Fermi energy level. The Fermi level plays a crucial role as it helps in understanding the behavior of electrons in a material. We typically use it to denote whether a material is a conductor, semiconductor, or insulator.

What is an Electronic Band Structure?

An electronic band structure is a representation of the allowed energy levels of electrons in a solid material. It is obtained by solving the quantum mechanical problem for electrons moving in a periodic potential, which represents the regular array of atoms in a crystal. The solutions to the Schrödinger equation for this periodic potential yield the energy bands, which are key features in the electronic band structure. In the band structure, the energy of each state (or eigen-energy) is plotted against the wavevector (or k-point), which is a measure of the wavelength or crystal momentum of the state².

Example of banstructure and density of states for SiC (mp-8062)

Fermi Level as a Reference

To compare the electronic energy levels, we need a reference, and this is where the Fermi level comes in. The Fermi level is essentially the chemical potential of electrons. It separates occupied and unoccupied electronic states at absolute zero temperature³. The Fermi level is defined as:

$$\begin{array}{}\text{(1)}& \mu ={\left(\frac{\partial U}{\partial N}\right)}_{V,S}\end{array}$$

where $\mu$ is the chemical potential (or Fermi level), $U$ is the internal energy, $V$ is the volume, $S$ is the entropy, and $N$ is the number of particles.

Fermi Level and Conductivity

In metals, the Fermi level often lies within an energy band, leading to partially filled bands. This allows for easy movement of electrons within the band and thus good electrical conductivity. In semiconductors and insulators, the Fermi level lies in a band gap, with all states below it fully occupied (forming the valence band) and all states above it empty (forming the conduction band).

Calculating the Fermi Level

The Fermi energy is calculated based on the number of electron states and the number of electrons in the system. It can be obtained from the following equation:

$$\begin{array}{}\text{(2)}& N={\int }_{-\mathrm{\infty }}^{E_F}g(E)dE\end{array}$$

where $N$ is the total number of electrons, $E_F$ is the Fermi energy, and $g(E)$ is the density of states⁴. The Fermi energy $E_F$ is the energy that satisfies this equation.

Utility

In essence, the Fermi level provides a dividing line between the states that are occupied and those that are unoccupied. It plays a pivotal role in determining the electronic, magnetic, and optical properties of solids. When you want to know how symmetry of the crystal structure impacts the electronic behavior, you look at the band structure. If the interest is just in how the energy levels stack up, you can focus on the density of states. In both cases, the Fermi energy tells us about the material.

The next step would be to classify a material as metals, insulators, semiconductors, etc. based on the bandstructure. I'm not going to cover this in detail here, but there are many good resources on this and a good start is Ashcroft and Mermin [2].

References

[1] Materials Data on SiC by Materials Project. LBNL Materials Project; Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States), 2020. https://doi.org/10.17188/1282015.

[2] Ashcroft, Neil W., and Mermin, N. David. Solid State Physics. United States, Cengage Learning, 2022.

Footnotes

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1. You can think of the wavevector as a type of momentum representation of the electronic wavefunction. This is sometimes called the crystal momentum as well. It provides information about how the electronic state is moving in space. ↩

2. The wavevector is also a measure of the phase change of the electron wavefunction as it moves across the crystal. ↩

3. At non-zero temperatures, the Fermi-Dirac distribution comes into play, affecting the occupation of states. ↩

4. This equation is often solved numerically, especially for complex materials. ↩

Reuse and Attribution

Bringuier, S., The Fermi Energy, Dirac's Student, (2023). Retrieved from https://www.diracs-student.blog/2023/08/the-fermi-energy.html.

  @misc{Bringuier_17AUG2023,
  title        = {The Fermi Energy},
  author       = {Bringuier, Stefan},
  year         = 2023,
  month        = aug,
  url          = {https://www.diracs-student.blog/2023/08/}# 
                 {the-fermi-energy.html},
  note         = {Accessed: 2025-04-16},
  howpublished = {Dirac's Student [Blog]},
  }