Why does Fermi have a level?

Note

This is an old post that I started to write back in 2021, but never got around to extending and publishing it, hence why it may seem a little out of place with my more recent posts. I'm publishing in the current state with hopes to provide some more derivations in the future.

When you take your first solid-state physics¹ 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 ². 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.

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 state³.

Example of band structure and density of states forSiC (mp-8062)[1]

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 taken as the chemical potential of electrons in a periodic box at absolute zero temperature (Definition 4.1 in ref. [2]), representing the highest energy level occupied by electrons.

Warning

Its more correct to say the Fermi level is the energy that is half-way between the lowest and highest occupied electron state.

For brevity, the chemical potential is defined as:

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

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:

$$ \begin{equation} N = \int_{-\infty}^{E_F} g(E) \, dE \end{equation} $$

where $N$ is the total number of electrons, $E_F$ is the Fermi energy, and $g(E)$ is the density of states⁵. The Fermi energy $E_F$ is the energy that satisfies this equation.

Utility

The Fermi level serves as a crucial reference point, dividing occupied from unoccupied electronic states at zero temperature. It significantly influences the electronic, magnetic, and optical properties of materials. Understanding the band structure helps reveal how crystal symmetry affects electronic properties, while the density of states focuses on energy level distribution.

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 many good resources on this and a good start is Ashcroft and Mermin[3] or Simon[2].

Footnotes

---

1. Its probably "old-fashion" to use solid-state physics as this is considered a specific focus within condensed matter physics. ↩

2. 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. ↩

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

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

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

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] Simon, Steven H.. The Oxford Solid State Basics. United Kingdom, OUP Oxford, 2013.

[3] Ashcroft, Neil W., and Mermin, N. David. Solid State Physics. United States, Cengage Learning, 1976.

---

Reuse and Attribution