I'm working on a data-driven project involving material properties and I'm wondering about the following:

I know that the notion of band gap and electrical conductivity are inversely related, and the former can help in classifying materials in metals, semiconductors and insulators. Nevertheless, I would like to know if there are cases where materials can exhibit a very low band gap yet not be conducting. I guess there are, as I expect conductivity depends on a variety of factors that won't include the band gap alone.

  • $\begingroup$ The notion of band gap and electrical conductivity are inversely related - could you give a source of this statement? Do you mean this is the case for only narrow band gap intrinsic semiconductor (and hence intrinsic conductivity) or for any materials (doped materials, wide band gap semiconductors, insulators, etc)? $\endgroup$ Feb 23 at 16:49
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    $\begingroup$ @AbdulMuhaymin: Kittel, C. Introduction to solid state physics, p220. 7th ed.; John Wiley & Sons, Inc., 1996. $\endgroup$
    – Camps
    Feb 23 at 16:57
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    $\begingroup$ @Camps Since I don't have that specific version in hand, I cannot check directly but I checked the 8th edition. In ch. 8, it is written that intrinsic conductivity and intrinsic carrier concentration are largely related with the gap. So I think (not sure) this is true for intrinsic semiconductors only. I am asking the source because I fail to see any straightforward inverse relationship between these two. For example, InAs has higher mobility and lower gap than GaAs which is consistent with the OPs notion, but PbS has even lower gap than InAs and yet its mobility is lower than GaAs. $\endgroup$ Feb 24 at 18:34
  • $\begingroup$ Please consider accepting my answer below so that the question is marked as answered! Thanks! $\endgroup$ Apr 3 at 19:07

1 Answer 1


There are such materials! See chp. 11 in Girvin-Yang which gives a much better discussion than I am providing here. You could probably also look up 'Anderson insulator' online or in any other solid-state text book for a similar explanation.

The Einstein relation for conductivity is $ \sigma \sim \rho D $ where $\sigma$ is conductivity, $\rho$ is the density of states at the Fermi level, and $D$ is the diffusion constant. $\rho$ basically tells you how many electrons there are that can conduct and $D$ tells you how easily these electrons can move.

If $\rho=0$, there are no electrons that can move, so there is no conductivity. This is the case for gapped materials like band insulators.

On the other hand, there are materials where $D=0$ while $\rho \neq 0$. The only example I can think of is an Anderson insulator. My understanding is tenuous, but basically in systems with strong disorder, it is possible for electrons to get trapped in deep potential wells. There are electrons in the system, i.e. $\rho \neq 0$, but the potential makes it very difficult for them to move and $D\rightarrow0$, i.e. $\sigma\rightarrow 0$ even though $\rho \neq 0$. I think the fact that the potential is disordered is important since a very strong lattice potential would lead to a different type of metal-insulator transition, namely a Mott transition, where the limit of isolated atoms is approached. Like I said, my understanding of Anderson insulators is poor, but hopefully this answer helps! Feel free to read more about Anderson insulators and explain them to me.

Cheers, Ty

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    $\begingroup$ Thank you, Ty. It seems to me that even though such cases do exist, they are quite uncommon. For instance, if we were to consider all currently experimentally reported organic compounds, what proportion do you think these materials could represent? $\endgroup$ Feb 27 at 9:24
  • $\begingroup$ Agreed, they are very uncommon. I assume that no currently reported organic compounds are Anderson insulators. $\endgroup$ Feb 27 at 16:31
  • $\begingroup$ Thank you, by the way, I meant INorganic ^^" $\endgroup$ Feb 28 at 17:09

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