What exactly is a Moiré potential in 2D materials? Reading about 2D materials formed by transition metal dichalcogenides, I have seen that if you rotate one layer you form Moiré patterns. Apparently, these patterns produce a " Moiré potential that varies spatially". I'm having trouble understanding how this so-called Moiré potential is formed, what exactly it is, and how it affects the band structure. I would appreciate it if someone could give some information about this. Nothing too deep, just in order for me to get a grasp of it. Thank you.
 A: The simplest place to understand the moiré potential is actually in the untwisted heterobilayer transition metal dichalcogenides (TMD) systems. So no moiré pattern. In such a system, there are two key points to note: One is that, unlike graphene, there are actually anions and cations. The other is the band alignment makes one layer farther away from the Fermi level than the other. Imagine now you are a free electron moving in the layer that is closer to the Fermi level. Naturally, you want to hang out more near the anions of the other layer. This is what is producing the moiré potential.
There are two contributions to the value of the moiré potential: One is the virtual hopping between layers. The other is actual Coulomb interactions between layers. Both are interlayer effects, so they depend on the actual distances of the atoms in the two layers. In a small-angle moiré system, the interatom distance can be approximated and is what gives rise to the moiré length scale. In practice, it's very hard to make an accurate estimate of the moiré potential based on these considerations. So people just do DFT instead.
It's very important to understand that this potential is different from the interlayer tunneling in the twisted bilayer graphene (TBG) model. It may also be useful to look at the homobilayer model, turning off the interlayer tunneling, just to understand the potential is really just modulating the motions of free electrons in a given layer. So I hope now it's now clear how the potential affects the band structure: Like any periodic material when you have free electrons and a periodic structure, the bands fold and gaps are opened where there are degeneracies after the folding.
