How Do Moiré Patterns Affect Graphene's Electronic Properties?

[Jufa]

Homework Statement

I have to do a presentation on Moiré patterns. The problem I'm struggling with is to connect its history with its applications. I fail to understand the formalism of the subject in order to demonstrate how the moiré patterns lead to applications such as superconductivity. Anyone can help?

Relevant Equations

No relevant equations.

No relevant equations.

 

[berkeman]

Can you post links to the reading you've done so far? What kind of "practical applictions" do you think there are for these patterns?

 

[TeethWhitener]

I fail to understand the formalism of the subject in order to demonstrate how the moiré patterns lead to applications such as superconductivity. Anyone can help?

Are you talking about the discovery of superconductivity in twisted bilayer graphene by Pablo Jarillo-Herrero's group at MIT? It made the science news rounds a few years ago. The phenomenon is quite straightforward to understand if you know basic solid state physics, but pretty much impossible to understand without that. What is your level of physics knowledge?

 

[Jufa]

Are you talking about the discovery of superconductivity in twisted bilayer graphene by Pablo Jarillo-Herrero's group at MIT? It made the science news rounds a few years ago. The phenomenon is quite straightforward to understand if you know basic solid state physics, but pretty much impossible to understand without that. What is your level of physics knowledge?

Currently I'm taking the third course of Physics in Barcelona. The presentation is for the subject "Photonics".
I haven't studied solid state physics.

 

[berkeman]

I did a simple Google search for Applications of Moiré patterns, and got lots of good hits. Many/most of them involved straightforward math at first glance.

 

[Jufa]

Can you post links to the reading you've done so far? What kind of "practical applictions" do you think there are for these patterns?

https://www.quantamagazine.org/when-magic-is-seen-in-twisted-graphene-thats-a-moire-20190620/

This first link reffers to how the magic angle produces a macroscopic hexagonal geometry. Here I understand that this macroscopical structure gives electrons freedom to move straightforward without encountering obstacles which lead to superconductivity at not supercool temperatures. Am I right?

https://www.pnas.org/content/108/30/12233
This second link is an attempt to modelling this phenomenon but I fail to understand the hamiltonian that is used. Is there a simpler way to approach the problem? It would be enough to understand how the dependence of the Hamiltonian on the rotated angle gives rise to a magic angle that makes it easy for the electron to tunnel from one layer to the other.

 

[TeethWhitener]

It’s going to be very difficult to explain any of this without a basic knowledge of band theory.

A picture will help:

The easiest way to think about it is in two steps:
1) Start by assuming the two graphene layers don’t interact. Then the Brillouin zone of the system will be two hexagons twisted relative to one another in reciprocal space by the same angle that the graphene layers are twisted in real space (Figure b above). The band structure of the total system will feature the Dirac cones twisted relative to one another, and the level crossings that occur due to the twisting of the cones will have an angle-dependent energy (Figure c, left diagram).
2) Add in interaction between the two layers. This turns the Dirac cone level crossings into avoided crossings, with a characteristic energy that is more or less constant (Figure c, right diagram).

At a certain “magic” angle, the interlayer interaction energy from step 2 and the angle dependent energy from step 1 will be equal. This causes the extremely flat avoided crossing bands to land right at the Fermi level (Figure d), leading to an increase in the density of states by several orders of magnitude. BCS tells us that increasing the density of states at the Fermi level increases the superconducting critical temperature.