Let's start w/ the basics. In the (non-relativistic) quantum world, we talk about the wavefunction \(\psi(\mathbf{r},t)\) of a system. The Schroedinger equation describes how the wavefunction evolves with time, and by solving it we can find the particular energy levels ("stationary states") for a given problem. The magnitude-squared of the spatial wavefunction, \(|\psi(\mathbf{r},t)|^2\) gives the probability of finding the particle in a particular place at a particular time.

The wavefunction a free particle with a well-defined momentum \(\mathbf{p}\) can be treated as a wave with a wavevector \(\mathbf{p}/\hbar \equiv \mathbf{k}\), and therefore a wavelength \(2 \pi \hbar/|\mathbf{p}|\). Higher momentum = shorter wavelength = the wavefunction has more closely spaced wiggles. The kinetic energy goes like \(p^{2}/2m\), as in classical nonrelativistic mechanics. (Note that the magnitude-squared of such a wave is constant as a function of spatial position. That is consistent with the uncertainty principle: Knowing the momentum precisely means that the position could be anything.)

Take a particle and stick it in an environment where the local potential energy varies periodically in space - ideally in a system so large that we can neglect boundary effects for now. The classic example of this is an electron in a crystalline solid. I've talked about this kind of spatial periodicity before. There are a couple of ways to think about this situation. We have replaced "continuous translational symmetry" (the environment of the particle is unchanged if we consider moving the particle anywhere) with "discrete translational symmetry" (now we have to move an integer number of lattice spacings to get back to the same environment for the particle). Mathematically, the single-particle stationary states can still be labeled by a parameter \(\mathbf{k}\), but they're Bloch waves rather than plane waves, and the energy \(E(\mathbf{k})\) is no longer necessarily proportional to \((\hbar k)^{2}\) all the time. Physically, when the naive spatial periodicity of the single-particle state matches up with the spatial periodicity (or some harmonic) of the lattice, it makes sense that there should be deviations from what we'd see with a free particle. The result is "band structure", ranges of energy densely filled with allowed single-particle states, separated by "band gaps", ranges of energy in which there is no way to make a single-particle state and still satisfy the Schroedinger equation with the spatially periodic potential energy.

The particular spatial periodicity of the lattice determines the form of \(E(\mathbf{k})\). For a hexagonal lattice like single-layer graphene, it turns out that there are two "Dirac points", and that near those special values of \(\mathbf{k}\), the form of \(E(k)\) looks like what is obeyed by photons in free space (!), with energy linearly dependent on \(k\).

The key point here: if we can tune the spatial periodicity of the potential arbitrarily, we can create interesting forms of \(E(\mathbf{k})\). That's really a neat idea. Carefully growing stacked multilayers of semiconductors along one direction has been used to create "minibands" for optoelectronic devices. Starting from a 2D surface state in copper, people have been able to put down patterns of CO molecules to create spatial periodicities in 2D, creating structures that look and act like graphene, or very recently even fractals. People have also tried doing this by patterning semiconductor structures, but it's very hard to get sufficient uniformity so that disorder isn't a problem.

Stacking graphene layers with some relative twist angle is a great way to create a 2D modulation with excellent uniformity over large areas (many many lattice spacings). This 2D modulation shows up because of the Moire pattern, which gives a spatially periodic coupling between the bands in each of the layers. By tweaking the relative angle, the spatial pattern can be tuned. By squishing on the bilayer, in principle the strength of the coupling can be tuned. This kind of 2D modulation should be possible in principle in twisted bilayers of all kinds of stackable materials.

The situation is even more interesting once we start thinking about electron-electron interactions.

Another way to think of bands: Start from the atomic energy levels of the individual, isolated constituent atoms. The electronic levels of each atom are sharply defined. All of the 4s orbitals, say, have the same energy. If you think about possible electronic states, the "band" made out of isolated (localized to individual atoms) 4s orbitals is very narrow in energy. If you built up some linear combination of those 4s orbitals that had a parameter \(\mathbf{k}\), the energy \(E(\mathbf{k})\) of that state would basically be independent of \(\mathbf{k}\). That is, the band would be "flat". Turn on hopping between atoms, and band broadens out in energy.

If we throw in a bunch of electrons and ask what is the lowest energy state of the many-electron system, we can often get away with mostly neglecting electron-electron interactions. Because of the Pauli Principle, we fill up the bands from the bottom up, and very often the (single-particle kinetic + lattice interactions) energy grows very rapidly, so much so that any electron-electron interactions are not very important. (That's what happens in the periodic table - as you go to atoms containing more and more electrons, the kinetic energy grows fast enough that e-e interactions don't really disrupt the basic hydrogen-like

*s*-*p*-*d*-*f*orbital structure of energy levels.)
In the twisted bilayers, it is possible to end up with some bands that are very flat - so flat that the typical electron-electron interaction energy is comparable or large compared to the bandwidth. In these flat band situations, electron-electron interactions can end up being very important in determining the collective many-body state of the electrons. That appears to be what people are seeing in the experiments mentioned previously.

The bottom line: Twisted stacking is a great, robust way to create a lateral spatially modulated potential, and therefore (within particular geometric limits) a "designer" band structure. The resulting bands can be very flat, so that electron-electron interaction effects (apparently) can lead to remarkable many-body responses, like the onset of superconductivity or magnetism.

## 3 comments:

Is there a reason that the same concept hasn't been applied to manipulate other (arguably more interesting) material systems, like transition metal dichalcogenides and other van der Waals bonded systems? It seems like they should show even more exotic behavior at these extreme conditions.

Anon, I assume there are people working on that right now. Right now graphene is probably the cleanest (in terms of least amount of impurity or vacancy disorder) and certainly has the best understood band structure, so it's a natural choice.

@Anonymous: There are already published studies of twisted TMD bilayers, see for example these 4 papers in Nature from February this year:

1. https://doi.org/10.1038%2Fs41586-019-0957-1

2. https://doi.org/10.1038%2Fs41586-019-0975-z

3. https://doi.org/10.1038%2Fs41586-019-0976-y

4. https://doi.org/10.1038%2Fs41586-019-0986-9

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