I have obtained a band structure plot (see image below) for a crystalline material. The Fermi level is set at the zero energy level. Based on the band structure, I am uncertain whether the material can be classified as metallic or semi-metallic.

Could you please provide your opinion on the nature of the material? Do the valence and conduction bands touch at specific points or along specific lines below or above the Fermi level for a bandstructure to be considered semi-metallic? Does the small opening in Figure 2 along the Gamma-X line mean a band gap opening, or something else?

Any insights or explanations regarding the conductivity properties would be greatly appreciated.

UPDATE: The band structure belongs to the same material, Figure 3 and is a close-up on the Gamma-X line shown in Figure 1, while Figure 4 is a close-up on Figure 2 case.

Figure 1. enter image description here

Figure 2. enter image description here

Figure 3. enter image description here

Figure 4. enter image description here


1 Answer 1


(Edited after discussion with the asker and seeing the zoomed-in band structures. A commenter suggested adding Addison of hybrid functionals as a correction to the band gap underestimation of [semilocal] DFT.)

Your material—both in Fig. 1/3 without and in Fig. 2/4 with spin-orbit coupling—is a semimetal. The inset regions make it clear: there is a positive direct gap everywhere, though it's pretty close along the $\Gamma \to X$ line. However, there is a small region of the Brillouin zone where valence bands are above the Fermi energy, and another where conduction bands pass below it. This is exactly how Wikipedia characterizes semimetals, as "negative indirect-gap semiconductors": the conduction band minimum is below the valence band maximum, but somewhere else in $\mathbf{k}$-space.

If you can do a density of states calculation, having negligible density at $E_F$ will support the semimetal conclusion. That said, the direct gap is so small that I wouldn't be surprised if the material has pretty strongly metallic character anyway.*

As an aside, topological semimetals seem to be more in the spotlight lately, and their definition is somewhat different: They have valence and conduction bands that meet at isolated points on the Fermi surface. See, for instance, the top answer to this question and Assumption 2.5 in the arXiv preprint it mentions. These include Weyl and Dirac semimetals; a famous example is graphene, with its gapless points linear in $\mathbf{k}$.

* Although your question tag suggests your calculation uses DFT. If you're using a standard (semilocal) functional, it almost certainly underestimates the band gap, so the true behavior is likely to be "less metallic" than your computation predicts. More expensive methods like DFT+$U$, Koopmans–compliant functionals, hybrid functionals including a fraction of Hartree–Fock exchange, and $GW$ are likely to improve the quantitative band structure.

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    $\begingroup$ Other than the methods you mentioned, using hybrid functional also might improve the band gap (if there is any) or open a noticeable band gap. $\endgroup$ Commented Nov 15, 2023 at 18:50
  • $\begingroup$ @elutionary Yes the band structure is for the same material, the only difference is that in Figure 2,spin-orbit coupling was taken into consideration. I have updated the question to include a close-up on the line between gamma and X as you can see in Figure 3 and Figure 4. According to what I understand from your answer, both cases are showing a semi-metallic characteristic? Besides, for a topoligical semi-metal as you mention in the definition, does the meeting point for valence and conduction bands mean that they touch each other at a single point?Could this definition be applied to my case? $\endgroup$ Commented Nov 16, 2023 at 0:24
  • $\begingroup$ @JaafarMehrez Ahh, your update is helpful. You certainly don't have a topological semimetal, as the two valence bands (the 'leftmost' ones, that go up a little then sharply down along the Γ -> X path) don't actually intersect the two conduction bands. I am kind of surprised, though, that the Fermi energy is below the top of the "valence-like" bands. Assuming you got that from the scf calculation, though, it should be trustworthy—it might predict that you have a metal after all, with or without SOC (since one or two valence bands are not fully occupied). This is a tough case... $\endgroup$
    – elutionary
    Commented Nov 16, 2023 at 15:26
  • $\begingroup$ @elutionary Thanks for your reply, hm I am kinda confused now. According to this wiki here en.wikipedia.org/wiki/Semimetal, in a semi-metal the valence and conduction bands are crossing the fermi level and overlapping without intersection (You can check the figures in that wiki), which is similar to the case I got here.. so doesn't that make the band I have semi-metallic? $\endgroup$ Commented Nov 17, 2023 at 0:22
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    $\begingroup$ @elutionary I think we both agree on that it is a semimetal, if you don't mind, would you like to update your answer so that other people could make use of that in the future, thanks and best regards ~ $\endgroup$ Commented Nov 18, 2023 at 2:04

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