Brillouin zone in bandstructure of graphene nanosheet

[sonal AGRAWAL]

Dear sir,
I have build a graphene nanosheet in ATK-VNL,  when I am calculating the band structure of nanosheet than the Brillouin zone point appear by default are only G, Z and bandstructure having bandgap of 0.02eV.  While In the hexagonal graphene sheet dirac cone with zero band gap is present at K point of brillouin zone but in the nanosheet which build from builder directly. I have explored the literature and have not found any paper which reported the electronic properties of retangular nanosheet.
1. Is there any difference in electronic properties of the hexagonal and unit cell or rectangular  shaped graphene sheet are they dependent on the supercell or i did something wrong.
2. brillouin zone point G, Z are for one dimensional material or it can also be used for Two dimensional material. I mean from the bandstructure how can we identify that this is the bandstructure of two-dimensional material or one dimensional material in case of unit cell.
For example graphene nanoribbon and nanosheet both have brillouin zoner point G, Z how can we differentiate both bandstructure.


Thanks in Advance

[Petr Khomyakov]

Please attach your python script and related log file.

[sonal AGRAWAL]

Thankyou for your reponse
Related log files and Python files are attached herewith please find the attachments.

[Petr Khomyakov]

The Graphene.py configuration has a rather small interlayer separation distance in C-direction, set it to 20 Ang, for example. The same separation distance should then be used for the nanosheet structure. Note that you have it in A-direction for the nanosheet, unlike the graphene sheet structure.

For the nanosheet, you may use, e.g., G, Z, G, Y, Z k-path. In the current definitions, G-Y is along the nanosheet, whereas G-Z is perpendicular to that. 

In principle, the electronic structure of graphene should not depend on the choice of its unit cell. Of course, the band structure plot will look somehow different because the Brilloiun zone depends on the unit cell choice, meaning that bands will be folded when going from the primitive cell of graphene to some unit cells. But these are still the same bands. So, no band gap should be seen in any band structure of pristine graphene.

I have also notice that you have used really few k-points  for SCF calculations, so I would suggest increasing it until convergence for total energy and band structure convergence is achieved, i.e., these 2 quantities do not virtually change when you further increase k-point sampling.

[sonal AGRAWAL]

Dear sir, Thankyou  for help

sir, I have increased the interlayer separation to  angstrong in both nanosheet and hexagonal sheet. After increasing separation, I have got exact bandgap for the hexagonal sheet, however in the case of nanosheet increasing separation doesn't give zero bandgap in nanosheet. I have calculated bandstruture for both G, Z  and G, Y BZ points.
here I am attaching the python file and bandstructure for G, Z, and G, Y Brillouin zone.
 in the G, Y direction I am getting the very high bandgap of 2.03ev. while in the G, z direction bandgap of 0.07ev
please help me out .

[Petr Khomyakov]

Did you check if increasing the k-point sampling resolves the issue?

k_point_sampling = MonkhorstPackGrid(
    nb=3,
    nc=3,
    )

Try using something like nb=12, nc=36 for the nanosheet system, instead of the coarse and highly-asymmetric k-grid, nb=3, nc=3, as set in the current script.

[sonal AGRAWAL]

Thanks for suggestion.
 
Iincreasing k point sampling improves the bandstructure but not exactly zero.
 I m getting bandgap of 0.07  in the G, Z direction, which is approximately zero, when I am selecting Full Brillouin zone points  G, Z, G, Y, Z than I am getting the same bandgap as G, Z however if I am selecting only G, Y then I m getting bandgap of 2.023eV, Which means it totally depends on the selection of Brillouin zone points.
1.  How can we identify which Brillouin Zone point should be select in the bandstructure?

 2. Nanosheet is periodic in both directions than why we are taking more K points in one direction than others.
 please clear my doubts.

Thanks & Regards
sonal

[Petr Khomyakov]

The best way to see whether your system has a band gap or not is to compute density of states. For graphene, you should use really many k-points to sample the Brillouin zone (for band structure and dos calculation) properly around the conical point, see Appnedix in Phys. Rev. B 87, 075414 – Published 7 February 2013. Note that one may have a band gap in some directions even for metallic systems, e.g., check the Fermi surface of bulk copper, which has a hole in one of the k-directions, even so copper is definitely a metal.

I was proposing an asymmetric k-grid because your original ribbon was defined roughly as a 3x1 rectangular cell. If you now use a nearly-square unit cell then it makes sense to adopt a symmetric k-grid. Typically, for a given crystal structure one chooses k-lines in-between k-symmetry points in the Brillouin zone, but then you have to set your unit cell type (Bravais lattice explicitly to allow QuantumATK to set them automatically). For a square lattice, G-X, Y, G is a natural route. So, just use a lot of k-point to eliminate the energy gap, which does not exist in pristine graphene, and in your calculations resulted from a coarse k-point sampling.

[sonal AGRAWAL]

Thanks for your help
This is really helpful for me