# Bands Aren’t Only For Crystalline Solids

If one goes through most textbooks on solid state physics such as Ashcroft and Mermin, one can easily forget that most of the solids in this world are not crystalline. If I look around my living room, I see a ceramic tea mug nearby a plastic pepper dispenser sitting on a wooden coffee table. In fact, it is very difficult to find something that we would call “crystalline” in the sense of solid state physics.

Because of this, one could almost be forgiven in thinking that bands are a property only of crystalline solids. That they are not, can be seen within a picture-based framework. As is usual on this blog, let’s start with the wavefunctions of the infinite square well and the two-well potential. Take a look below at the wavefunctions for the infinite well and then at the first four pairs of wavefunctions for the double well (the images are taken from here and here):

What you can already see forming within this simple picture is the notion of a “band”. Each “band” here only contains two energy levels, each of which can take two electrons when taking into consideration spin. If we generalize this picture, one can see that when going from two wells here to N wells, one will get energy levels per band.

However, there has been no explicit, although used above,  requirement that the wells be the same depth. It is quite easy to imagine that the potential wells look like the ones below. The analogue of the symmetric and anti-symmetric states for the E1 level are shown below as well:

Again, this can be generalized to N potential wells that vary in height from site to site for one to get a “band”. The necessary requirement for band formation is that the electrons be allowed to tunnel from one site to the other, i.e. for them “feel” the presence of the neighboring potential wells. While the notion of a Brillouin zone won’t exist and nor will Bragg scattering of the electrons (which leads to the opening up of the gaps at the Brillouin zone boundaries), the notion of a band will persist within a non-crystalline framework.

Because solid state physics textbooks often don’t mention amorphous solids or glasses, one can easily forget which properties of solids are and are not limited to those that are crystalline. We may not know how to mathematically apply them to glasses with random potentials very well, but many ideas used in the framework to describe crystalline solids are applicable when looking at amorphous solids as well.

### 3 responses to “Bands Aren’t Only For Crystalline Solids”

1. drie k.

But to me a band implies dispersion, i.e. changing energy with momentum. And I don’t think that is happening in amorphous solids; each well will have a slightly different depth, and therefore Pauli exclusion does not force dispersion upon hybridization.
So while there are indeed “bands” (regions in E) that have states where quasiparticles can live, these are flat (and fuzzy).

BTW, I’m missing your thoughtful posts – I gather as a postdoc you’re busy (potentially with applying for a well-deserved faculty position), but I enjoyed your posts. The (understandable) lack of their regular appearance is a pity.

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• Anshul Kogar

Thank you very much for your kind words, though I don’t know if they’re fully deserved! It has been tough, as I’ve gotten busier, to keep blogging regularly. It takes considerable effort to write a post of some value. With so much going on, it’s been difficult not only to blog, but to get ideas for interesting topics to blog about!

As for the science in this post, I would suggest that indeed there is some dispersion in amorphous solids. Consider a monatomic amorphous material. These would have the same depth of potential well, but not necessarily the same barrier length between atoms as they are unevenly spaced. Then you would very much get some dispersion. It would seem difficult to get a metallic glass without such a dispersion, see here for instance:

https://www.nature.com/articles/nature13617

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2. @Anshul Kogar I have been thinking about the same thing. Recently, I had a debate with a professor in my institute regarding this. They have published a paper claiming they found something they termed “Gaplike resonance” in disordered systems. Though they miss the point the only reason for any band like structure, which is closely related to “Anti-Crossing” is the coupling between the atoms. The periodic arrangement is not necessary for the gap. Yes periodicity is necessary for periodic band structure though.

Some papers which helped me understand are

https://aapt.scitation.org/doi/10.1119/1.3471177

https://aapt.scitation.org/doi/10.1119/1.5045332

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