Band structure calculation of BN monolayer

[Resear20]

Dear Users,

I calculated the Band structure of BN monolayer (hexagonal) for unit cell (2 atoms), I obtained the direct band gap at K point. after I calculated the band structure of BN (6 * 6)  monolayer (hexagonal) with (30 B and 30 N), I found the direct band gap at Gamma point. Please, could explain me why I obtain this difference?

thank you for your help 

[filipr]

In order to understand why the direct band gap at K in the primitive cell is not at the K point for the supercell you have to understand Bloch's theorem(https://en.wikipedia.org/wiki/Bloch%27s_theorem) and what the Brillouin zone (https://en.wikipedia.org/wiki/Brillouin_zone) actually is. The Brillouin zone for a supercell is not the same as for the primitive cell.

Let's consider a simple example in 1D. The 1D unitcell has length L. Then the k-points are given by k = t 2π/L, where t is the "fractional k-point", i.e. the coordinates inside the first Brillouin zone. The first Brillouin zone is defined as the coordinates of t for which -0.5 < t ≤ 0.5. If you have a fractional k-point outside this region it is wrapped back inside. Now if you consider an equivalent system, but where instead of using the primitive cell you consider a supercell of length 2L, then the k-points are given by k₂ = t₂ 2π/(2L). Likewise, the Brillouin zone of this system is defined as -0.5 < t₂ ≤ 0.5. You'll see that in terms of k, the Brillouin zone of the supercell is half the size of the primitive cell. Now the k-points of the states in the two are still the same. Let's consider the states at the K point of the Brillouin zone of the primitive cell. The K point is determined by t₁ = 0.5. We have k₁ = k₂, so 0.5 * 2π/L = k₂ = t₂ 2π/(2L) => t₂ = 0.5 * 2 = 1.0. So the K-point of the primitive cell corresponds to the t=1.0 fractional k-point of the supercell - this gets wrapped back to the Gamma point (t₂ = 0) in the first Brillouin zone of the supercell.

If you want to consider a band structure of a supercell as projected on the primitive cell you can use the effective band structure analysis tool (https://docs.quantumatk.com/tutorials/effective_band_structure/effective_band_structure.html).

Also, if any of this sound confusing I suggest you to revisit your basic solid state physics book and do some example calculations by hand.

[Resear20]

I am sorry but I don't agree with you.

I used the same space group P-6m2 for BN 1*1 and BN 6*6

I calculated the band structure using VASP code for the same structures. I obtained the VBM at K point for both structures.

[Petr Khomyakov]

It is not really about running it with different codes; this matter is fully geometrical related Brillouin zone folding when enlarging unit cell size.

If 1x1 BN refers to 1x1 primitive cell (not some other unit cell) of hexagonal BN, then K-point of the primitive Brillouin zone folds onto the Gamma-point of the 6x6 hexagonal BN Brillouin zone.

For example, 2x2 BN would have K-point folded on K-point, the same as for graphene, but 3x3 BN would have it on Gamma-point, meaning that 6x6 BN also has it folded on the Gamma-point.

I am guessing that there must be some issue with interpretation of the VASP band structure data.