Commensurate supercells for rotated graphene layers¶
In the Stash, select the newly added item and click “Copy” to make an identical copy of the graphite structure.
Select the copied structure and delete the uppermost layer of carbon atoms highlighted in the picture below.
Click on. Drag the 2-layer structure and drop it in the first slot, and the 1-layer structure in the second, as shown in the figure below.
Click the button “Select Surface Cells…”, and in the window that opens up click the button “Set Matching Parameters”.
In the Set Matching Parameters widget that opens up, set the range of rotation angles to be searched from 20° to 24°, in increment of 1°, and click “OK”.
The structure with the lowest number of atoms is normally selected by default in the 2D plot on the left-hand bottom side of the Select Surface Cells widget (the red spot in the figure below). In this case this is also the structure with the lowest strain (in fact, no strain at all, i.e. it’s a perfectly commensurate match). This is indeed the geometry you want, with the two lattices rotated by 21.79° one with respect to each other. Click “Apply” to select this cell.
In the 3D window in the Builder you will now see a preview of the structure. You can rotate it, and if you would prefer to change some parameters you can still do it (for instance, you can add more layers of graphite to the “substrate” by clicking the plus button on the left under the first slot). Otherwise press “Create” to finalize the geometry.
Additional rotated structures¶
For more ideas on commensurate structures, see e.g. Ref. . All the structures reported in the paper can easily be built using the procedure shown above. However, as noted in the article, the number of atoms becomes very large in many cases. The simple cases shown in Figure 7 in Ref.  are quite manageable - note that in these structures there are only 2 layers rotated with respect to each other, not 3 as above. Both A-A and A-B stacking sequences can be used, so the easiest way is actually still to build the structure as above, and then remove either the middle or bottom layers, and adjust the layer separation.
Note that in these systems it will be necessary to increase the parameters nmax and mmax too in the Set Matching Parameters widget. For instance, to find the 5.1° supercell shown in Figure 7a in Ref. , you need to set them to 14. Notice that in this case the default suggestion in the Select Surface Cells widget may not be the structure with zero strain, if there are strained structures with a smaller number of atoms. So make sure to always select a structure with no strain (there may be a few, naturally take the one with the smallest number of atoms), by clicking the blue dots in the plot (the active choice is indicated by the red dot as shown above).
Here below you can find a few examples of the rotated bilayer graphene structures that can be created using QuantumATK and the procedure explained above.