Difficulty in interpreting electronic band structures in solid state physics

Question

My background is in ab initio calculations of atoms and molecules, and I consider myself to have a good understanding of the underlying quantum theory of electronic structure and electronic spectra when it comes to interpreting such calculations. I'm trying to understand some solid state physics now, and I have a hard time interpreting electronic band structures.

For example, let's take a plot from this paper:

I think I have an understanding on what the x axis entails when it comes to the paths in the reciprocal space (although I find it very hard to have an intuition about it), and the meaning of the energy is clear. I can also identify conducting and insulating materials by the absence/presence of the gap at 0 eV.

However, I am having difficulty in understanding the rest of the information content in the plot. Below I list out questions I have:

1. What is the meaning of the fact that there's a crossing around -8 eV at the L point?

2. What is that the gap between the two unoccupied levels varies heavily along the k point path?

3. Can I extract any interesting observation from the plot, apart from checking whether the material is conducting or not?

Answer 1

One important step is to know how the Fermi energy ($E_F$) was treated. Some authors like to shift the band diagram after making $E_F=0$ (as looks like it was the case in the reference you cited) but others (as my self) prefer to let the $E_F$ with the calculated value. As the Fermi energy is an indicator of electron population, the energy levels below $E_F$ represents occupied states whereas energy levels above $E_F$ represents empty states. The occupied states are called valence band and the empty ones, conduction bands.

Saying that, things we look for in the band diagram:

• Alignment of the botom of the conduction band with the top of the valence band. If both (bottom and top) occurs at the same k-point, the material has a direct band gap. If the bottom occurs at a k-point different than the top, then the material has an indirect band gap. This has implications in the optical properties of the material, for example.
• The value of the band gap at the $\Gamma$ point (the central point of the Brillouin zone) or at the correspondig k-point for indirect band gap materials. This can be used to characterize the material as metal, semimetal, semiconductor or insulator.
• The parabolicity of the band around the bottom/top region. This is used to calculate the efective mass of the carriers (electrons and holes). This is used in Semiconductor Physics.
• It is possible to do a band diagram colored accordingly with the contribution of each orbital or even of each atom. This is called fat band diagram (take a look at the discussion here.

The analysis of electronic band together with the Partial Density of States (PDOS) or Total Density of States (DOS) can be used in electronic band engineering: modifying the material until you got the intended electronic properties.

As transport and optical properties are mainly related with the electrons in the valence/conduction bands, in general, the analisys is focused on that bands. The same for the k-poins related with the band gap (the central $\Gamma$ point or the indirect k-point.