What is the meaning of the (A x B) when talking about CDWs?

[kevmeister]

TL;DR Summary

For example the 2x2 CDW, 3x3 CDW, etc.
It seems to me like this is one of those things that is so trivial that no one ever explains it.

What is the A x B labeling referring to when discussing charge density waves? For example the 2x2 CDW structure in TiTe2 (link). Is it

I have searched through a fair amount of literature (from Grüner to more recent experimental studies) and it doesn't seem to be explained anywhere that I've looked. I'm probably way overthinking it but just wanted some clarification.
Maybe this post will come up in a google search for some future physicists when they have the same question!

 

[kevmeister]

I may have just answered this via chatGPT:

The "AxB" notation in the context of charge density waves (CDWs) typically refers to the modulation pattern or superlattice structure formed by the CDW in a crystalline material. In CDWs, there is a periodic modulation of the electron density or charge density in a material, resulting in a spatially varying charge distribution.

The "AxB" notation represents the periodicity of the CDW modulation along two perpendicular crystal lattice directions. The "A" and "B" refer to the periodicity of the modulation along the respective crystallographic axes. For example, in the case of a "2x2" CDW, the modulation pattern repeats every 2 lattice units along one crystallographic axis (A direction) and every 2 lattice units along another perpendicular axis (B direction). This means that the CDW creates a superlattice with a periodicity of 2 lattice units in both the A and B directions.

The CDW modulation pattern or superlattice structure can have different periodicities depending on the material and the underlying crystal structure. The "AxB" notation is commonly used to describe the spatial arrangement of the CDW modulation and provides important information about the CDW properties and behavior in a given material.

So, it's basically however many latice vectors make up one side of the CDW "lattice", i.e., superlattice?

 

[TeethWhitener]

Yes, the notation is superlattice notation. CDW's generally cause a periodic distortion in the lattice such that the unit cell needs to be enlarged. For a 2x2 superlattice, the new unit cell including the CDW is 2 of the old unit cells in each dimension (note that AxB implies a 2D material. A 3D CDW superlattice would be notated AxBxC).

I'd be wary of using ChatGPT to "answer" questions like this right now. It's really just a super duper autocomplete. It got the answer mostly right this time, or at least it seems that you were able to divine the right answer from it, but even so, note that it still makes a subtle mistake: the lattice directions don't have to be perpendicular (e.g., a hexagonal lattice). They only have to be linearly independent.

 

[kevmeister]

Thanks for the reply! When this label is present does that automatically signify that the CDW is commensurate with the underlying lattice, i.e., modulated by an integer multiple of lattice vectors?

Sidenote:
I also caught the subtleties in the GPT response, definitely far from perfect! It's pretty obvious when my students use it for HWs. I mainly use it to get started on a code.