# What is the electrical behavior of the material with this band gap?

Presently I am doing DFT calculations to analyse the electronic structure of a material. I got the band structure as shown in the figure. I would like to know the electrical behavior of this material. The Fermi Energy is shifted to 0 eV.  • +1. Welcome to our new community and thank you for contributing your question here! We hope to see much more of you in the future!!! You might have to be more specific about what you want, for example by the phrase "nature of the material", but it's a good start at a first question! May 20, 2021 at 17:04
• In order to give you an answer, we need to know where the Fermi level is. It was translated to zero ($E-E_{Fermi}$)?
– Camps
May 20, 2021 at 17:39
• Thanks for helping me remove the ambeguity in the question. The question has been edited to clarify these omissions. May 20, 2021 at 18:13
• Your question indicates that there should be a band gap, but there is none. This is a metal. May 20, 2021 at 18:39
• You only consider a single avoided crossing visible in your band structure and see that as a gap because it is near the Fermi energy. But there actually is at least one band at every energy in your band structure, even at the energies relating to this avoided crossing. There you find two other bands. You have to consider the whole range of k points. May 21, 2021 at 8:14

The short answer is that this is a metal because of the following reasons:

1. An insulator or semiconductor features a bandgap between the occupied and the unoccupied states. For a temperature of 0K the Fermi energy discriminates between these two different occupations. This means the bandgap has to be around the Fermi level. A bandgap means that there are no states within a certain energy region. In the example from the question there is at least one band (or state) at every energy in the plotted energy window. There may be avoided crossings between two bands, but if there are other bands covering the energy region of this avoided crossing, e.g., at other k points, you have no bandgap. As a remark I would also like to add that these states do not necessarily have to show up in the band structure: Band structures cover the states along a high-symmetry path in the Brillouin zone (BZ). The band edges do not necessarily have to be found on this path. They can also be at different coordinates within the BZ. Nevertheless most of the time the band structure already gives a very good view on the electronic structure.

2. The avoided crossings in the example are probably artifacts of the visualization coming from connecting the i-th calculated states between the considered k points by lines. Such a visualization is sensitive to the order of the states in the underlying data files, which typically does not reflect physics. I suggest to plot band structure visualizations only with points at the calculated energy eigenvalues. Lines between the points may be misleading.

In a broader answer one may abstract from the example and consider a material with a tiny band gap at the Fermi level. Of course, there are several types of materials that come near such a situation. I would like to discuss this from the point of view of semiconductor physics:

In semiconductors you have a larger bandgap and you can dope your material with certain atoms to place states within the bandgap. Typically these dopands are chosen such that these states are found near one of the band edges. In such situations there are temperatures T with a $$k_\mathrm{B}T$$ that is at least in the same ballpark as the energy difference between the dopant states and the nearest band edge but considerably smaller than the band gap. This allows an ionization of the dopant atoms and the electrons (or holes) related to the dopant states become free charge carriers. This is practical because you can control the concentration of charge carriers within a large temperature window by the amount of doping. Within the Drude model the conductivity of the material $$\sigma_0$$ is connected to this charge carrier concentration $$n$$ by

$$\begin{equation} \sigma_0 = \frac{n q^2 \tau}{m}, \end{equation}$$

where $$q$$ is the charge of the carrier, $$m$$ is its effective mass, and $$\tau$$ its average traveling time between collisions.

You have a different situation in which there are no dopant atoms but you have a very small bandgap. At a vanishing temperature such a material has no free charge carriers and therefore an insulating behavior. For finite temperatures $$k_\mathrm{B}T$$ is at least in the same ballpark as the bandgap. If you increase the temperature starting with very small T, this will also increase the concentration of free charge carriers and with this the conductivity of the material. This is a different behavior than what you expect in metals. In metals you would expect a minimal resistivity at tiny T and an increase of the resistivity with temperature because the electron-phonon scattering would also increase. Of course, you would also have this mechanism in your situation but it would not dominate the behavior. At much higher temperatures you would observe a metal-like behavior.

Let me finally mention that there are variations of the tiny bandgap situation that may be of interest when thinking about the behavior of such a system. For example you might have Dirac cones at the Fermi level. This means no bandgap but a certain dispersion of the bands. Another case may be the the presence of Weyl points. I will not dive into the physics of such cases.

EDIT: An additional thought: Depending on the XC functional DFT calculations typically underestimate the bandgap and sometimes predict metals even though the material actually is an insulator. Maybe the example shows such a case.