figure out the correct mesh for your Si unit cell and than divide it by 5 in each direction, as the reciprocal cell of the unit cell is 5 times larger in each direction than that of the supercell. e.g.: if your converged mesh for the Si unit cell is a 25x25x25 Monkhorst-Pack mesh, a 5x5x5 mesh is appropriate for the super cell. Of course you will have lass symmetry and thus a larger irreducible zone, increasing the relative number of points, but the k-point density is the same. write back if this is not clear, cheers, michael

Dear Joe,

In general, you have to try a particular k-point grid, compute the properties, then repeat with a finer k-point grid (i.e. more k-points). You keep trying finer and finer k-point grids until the properties you are interested in have all converged to the accuracy you care about.

Dr Wolloch is perfectly correct in his post that you can get an estimate from the k-point sampling required for the unit cell (scaled down by 5, because k-points sample reciprocal-space and you've scaled real-space up by a factor of 5). However, by creating a spherical cavity you've introduced surface states, and it is quite possible that these will require a finer k-point grid -- you will need to do the k-point convergence tests on the actual system you're interested in.

You will probably also need to think about whether the surfaces reconstruct, since the surface silicon will have only half the coordination (roughly) of the bulk silicon.

Hope that helps,

Phil Hasnip

according to Vasp tutorial, you can use less k point in the non periodic direction. for example if you add vacuum in z direction its safe to use n*m*1 grid. I hope I am right because I'm starting to do adsorption simulations based on that.

(page 28)

https://www.vasp.at/vasp-workshop/slides/k-points.pdf

If you have a direction in which the real system is not periodic, then the periodicity imposed on the system by the choice of plane-wave basis is not wanted. For the actual system, if you looked at the band-structure along the corresponding reciprocal direction the energies would not change at all, i.e. the bands would be completely flat. If the bands are flat, then you may as well only sample a single k-point in that direction, since you'll get the same energies from any other k-point.

In your simulation you can't do an infinite lattice vector, so of course there is some periodicity. This periodicity will cause the bands to have some dispersion, and they won't be completely flat -- however, this dispersion is actually a kind of error, it's an artefact of your finite lattice spacing, and would get less and less if you expanded your simulation cell in that direction. There is no point having more than one k-point in that direction, because having lots would just sample your error more precisely.

The question remains, which k-point coordinate should you choose for the aperiodic direction? You want to sample at the k-point coordinate which is "most typical", so somewhere in the middle of the Brillouin zone. If you think a bit about how the wavefunction in your cell will interfere with the one from its neighbour, you'll see that a k-point coordinate of 0 will lead to constructive interference (everything in phase; bad!) and a fractional k-point coordinate of 0.5 will lead to destructive interference (everything pi out of phase, i.e. one is minus the other; also bad!). What you want is the k-point coordinate which minimises the interference, and this is generally found to be 0.25 (i.e. the waves are pi/2 out of phase).

So if you have an orthorhombic cell and don't want periodicity in the z-direction, you would choose a k-point mesh Nkx x Nky x 1 with all the points having fractional coordinates of the form

k = (p/Nkx, q/Nky, 0.25)

where p and q are integers, as usual.

If you aren't sure how this works, I suggest you take a nice simple 2D system (e.g. graphene) with the material in the xy plane, set the lattice constant along z to 6 A and plot the band-structure just along kz. Then increase the lattice constant to 8 A and re-plot, then 10 A and 12 A. You'll see that as you increase the lattice constant in the z-direction the band minima move up in energy, the band maxima move down and the band energies in the middle stay pretty much constant. The "middle" is at 0.25.

Hope that helps,

Phil Hasnip

thanks for the explanation. so since im new in the field of solid state physics Im asking does that mean we should never use gamma point for non periodic calculations? is that the concept of shifting the k point mesh?

There is no scientific reason for choosing the gamma-point, the Baldereschi point would always be more accurate. However there is a computational reason why you might choose it: at the gamma-point (and a few other high-symmetry points) the wavefunctions can be made explicitly real, halving the memory requirements of the wavefunctions and pseudopotential projectors and speeding up conventional DFT calculations by up to 8 times compared to the same simulation cell with a general single k-point. However, a gamma-point calculation would need a significantly larger cell in order to get comparable accuracy to sampling at the Baldereschi point, so it isn't immediately obvious which would be quicker for a given accuracy.