Band structure of twisted bilayer graphene

[SSSarangi]

Hello,

I am trying to calculate the band structure of twisted bilayer graphene (tBLG) at arbitrary twist angles using Pybinding.

As a first step, I am using the built-in generators to prepare the model. To maintain periodicity of the moiré structure, my approach is to construct a rectangular domain with x and y dimensions corresponding to the commensurate unit cell. I obtain these values from external tools such as twister and atomsk.

Following the tutorials, I generate the model, add translational symmetry, and then calculate the band structure. However, no matter what twist angle I choose, I always obtain the same band structure.

Could you please let me know what I might be doing wrong? I highly appreciate your help in the matter. Also, I would also like to understand the reason of not obtaining the correct band structure of bilayer graphene when the twist angle is specified as 0.
#--------------
model = pb.Model(
two_graphene_monolayers(),
twist_layers(theta=21.78), pb.rectangle(x = 0.6508548, y = 1.1273136),
pb.translational_symmetry()
)

plt.figure(figsize=(6.5, 6.5))

solver = pb.solver.lapack(model)
bands = solver.calc_bands(Gamma, K1, M, Gamma)
bands.plot(point_labels=[r'$\Gamma$', 'K', 'M', r'$\Gamma$'])
plt.title(r"$\theta$ = 21.78 $\degree$")

[BertJorissen]

This is indeed an interesting problem.
I'm actually working on a general kind of implementation that allows the connection of any twisted bilayer, like TMD or even hBN.
There are several issues you need to consider:

• making a commensurate supercell
• generating interlayer hoppings
• allowing for the interlayer hoppings to connect over the boundaries of the supercell
• considereing the orbital characteristics of the twisted layer (as graphene is pz, you can neglect this one)
• applying the correct fuctional dependence on the cosine factors (Slater-Koster) and length (Exponential decay)
• calculating the moire Brillouin-zone
• calculating the low energy bands (APRACK as LAPACK takes too long)
• connecting the interesting bands; disentangling the bands
• and a lot of other things

There are some 'internal' scripts used by Robin Smeyers (https://pubs.rsc.org/en/content/articlelanding/2023/nr/d2nr07171a/unauth), Misa Andelkovic (https://www.science.org/doi/full/10.1126/sciadv.aay8897) and me that can calculate these bands.
As I'm currently quite busy, I don't have the time to clean these up for you but you can have a look at Misa's thesis (https://cmt.uantwerpen.be/ua/thesis_misa_andelkovic.pdf) where he explains this in appendix C.
Feel free to ask more questions, but I can't guarantee that I can always supply you with the needed scripts.
I hope this pushes your research in the right direction ;)

[SSSarangi]

Hey Bert,

Thank you very much for your response and suggestions. I will go through the documents and try to find a way to make this work for my research. I will reach out in case of more questions.

Regards,
Soumya

[SSSarangi]

Thank you for your earlier suggestions. I went through the recommended thesis and successfully constructed the tBLG systems based on the codes provided in Appendix C. The system creation worked well for various twist angles, and the resulting structures appear to have the correct commensurate shapes and number of atoms.

However, I am currently facing difficulty in adding the hopping positions to the bottom layer (in the add_one_atomic_layer function). The function returns only the atomic positions for the bottom layer, and my attempts to create two separate lattices within a single model have not worked as intended.

From my understanding, correctly defining the hopping sites is essential for obtaining an accurate band structure, as pb.translational_symmetry() does not seem to function properly otherwise. The thesis also refers to a GitHub repository containing a working example of the tBLG case, but I was unable to locate it.

I would greatly appreciate any guidance or clarification you could provide regarding the correct way to include the hopping terms for the bottom layer or a reference to the working example, if available.

Thank you very much for your time and help.

Kind regards,
Soumya

[BertJorissen]

Hi Soumya
the issue with the Lattice and Model is unfortunatly a key part of how the Hamiltonians are set up in Pybinding.
Currently, you can only have one Lattice per Model. This means that you have to create a gaint Lattice for the whole supercell (both layers + connections).
I have a script for this, but it is not yet at a stage that I can share this with you as it uses a lot of 'hacks' to make things work.
If you manage to set up the whole model like Misa did, you will still struggle with connecting the periodic structures. It is better to extract the atomic positions and connect each hopping seperatly.

The GitHub pages Misa refered to are the code_tutorials on his page.
If you have patience, you can also try this online in MyBinder (though, it can be this is broken).

I'll see wether I can create a minimal example for your studies with a tBLG cell.

Either way, you still have to create the commensurate cell. This is explained in i.e.

• https://doi.org/10.1038/s42254-025-00818-4
• https://doi.org/10.1038/s41578-020-0214-0 (see also https://ciqm.harvard.edu/uploads/2/3/3/4/23349210/carrfang2020.pdf and box 1)
• https://doi.org/10.1063/5.0023984
• https://doi.org/10.21468/SciPostPhys.14.3.053
• https://doi.org/10.1021/acs.jpcc.6b01496

[SSSarangi]

Hi Bert,

Thank you for your suggestions. I am in fact working on the code to bring both the layers together and make them part of same lattice but I have not been successful yet. I think I need to understand the lattice.add_hoppings function better. And thank you for sharing the link to the tutorials and papers. The tutorial is actually available in the main page of pybinding and I have taken inspiration from that tutorial as well. Please let me know if you can come up with an example for band structure of tBLG. It would be a great help for people working in this area. Thank you again for developing Pybinding.

Thanks,
Soumya