The k-points live in a 3D-space, and in the electronic structure calculation, you are sampling this space. For some systems you can get away with few sample points, some need more - typically metals. You need to select your k-points somewhat systematically, for example, by using a Monkhorst-Pack mesh or similar. This scheme also assigns weights to each k-point.

Your question seems to refer to the visualisation of the electronic structure, where you take different paths - but these on their own are not sufficient to sample the k-points.

As shown by Albert, the k-points distribute as a 3-dimensional mesh in the k-spce. According to my experience in the code VASP, to get a meaningful or accurate/reliable results at the DFT or other level,

a) you need to calculate the electron distribution in the crystal with a dense k-mesh self-consistently.

b)The obtained electron density (e.g. CHARCAR in VASP) can be used to calculate the eigenenergies of the electrons at the selected k-points, e.g. along the high symmetry lines as Brajesh used ) in the crystal non-self-consistently to get e.g. dispersion curves of electronic bandstructure.

Hi Brajesh,

actually for the phonon and Band structures (with Quantum espresso) you must manualy introduce the k-point for best resolution of the graph when you plot it.

so to generate the k-point and their weight you have tow ways:

1) I think that with Quantum espresso there is some routin called kpoint.x give you the high symetry k-points and their weights but first you must define your structure.

2) by using the Xcrysden , that is good beacause you can figure your Brillouin zone and you can chose the path of points and the interval between them.

Thank you a lot. Using a Monkhorst-Pack mess (6X6X6) I was able to get correct DoS which is almost similar to other reported values. But the problem I start facing when I try to plot electronic structure along different paths. I start getting uneven k- points in different paths which makes E-k diagram looks very bad and unable to understand anything out of that.

@ Bartok and Fang Please explain meaning by selecting the k points systematically.

@ Balot Can you please elaborate how by Xcrysden.

Hi,

for the DOS calculation the k-points depend of the size of your structure, as it very smal you must use very denser grid of k-points.

for Band calculation it is normal you had a bad graph , if you had define an automatic k-points, for good resolution yous must introduse all the path by your self as i said for you you can make it by Xcrysden by this way:

one you have an SCF output you open it with xcrysden and the you go into Tools>k-path ...

i hop that help you.

Thanks a lot. Let me try it out.

Dear Mr Tiwari,

All of above descriptions are true. But Physical character such as phonon and optic are very sensitive to K-mesh and you must have a dense k-point to obtaining correct data. usually, I use 12*12*12 for electronic (DOS & band structure), 36*36*36 for PHONON and OPTICAL properties.

@ Jalilian Thank you. If I understand correctly by using default k-points ( Monkhorst-Pack mess) of 12X12X12 mess I can get good electronic structure and other physical properties.

Dear Mr Tiwari,

12*12*12 k-mesh just is suitable for electronic properties. For PHONON and OPTIC you must choose a high value of k-mesh.

Thank u so much.. It means I need to have a high permanence computer.

Hi Brajesh,

I'm using a Monkhorst-Pack K-mesh (80X80X80) with LAMMPS.

Dear Dr Tiwali,

There are many good responses to your question already, so I won't repeat them here. Two points I would like to add:

1. The important quantity when discussing k-points is not the number, but their spacing in reciprocal-space.

The k-point mesh you need for a given sampling depends on the size of the Brillouin zone (effectively the reciprocal-space unit cell). If you have a large real-space simulation cell, then the reciprocal-space cell is small and you won't need a large k-point mesh. For very large simulations you may only need a single k-point.

Typical k-point spacings range from 0.03 x 2pi/A to 0.05 x 2pi/A, where the smaller separation will usually give excellent quality sampling of the Brillouin zone even for metallic systems, and the latter is rather coarse but would usually give reasonable qualitative results for semiconducting and insulating systems. Of course you need to test the convergence for any particular system, but I would usually investigate k-point meshes over this range, starting around 0.04 x 2pi/A.

2. The "weight" of the k-points describes the contribution of data at a single k-point to the data for the whole Brillouin zone. Naively you would expect a calculation with N k-points to have all the k-points weighted by 1/N, but the weights also account for the symmetries of your system. Most simulations have some symmetry, even if it's just time-reversal symmetry.

To take a simple example, imagine you have a single band sampled on a 1x1x3 k-point mesh centred on the origin; in theory you need to compute the solutions for the 3 k-points

E_1 = E(0,0,1/3)

E_2 = E(0,0,0)

E_3 = E(0,0,-1/3).

and the total energy of the state is

E=(E_1 + E_2 + E_3)/3 = E_1/3 + E_2/3 + E_3/3

We rewrite this more generally as E=sum_i E_i w_i, so in this case:

E = E_1.w_1 + E_2.w_2 + E_3.w_3

where w_i is the weight of k-point i -- in this case all the weights are 1/3.

Now consider the symmetry of the system. Time-reversal symmetry means that +k and -k are equivalent, so E(0,0,-1/3) = E(0,0,+1/3) and hence E_3 = E_1. Now we only have to compute:

E_1 = E(0,0,1/3)

E_2 = E(0,0,0)

but when we add the energies up to compute the total energy, we need to remember that E_1 represents two points in reciprocal-space, and E_2 still represents one. We do this by setting w_1=2/3 and leaving w_2=1/3, so that:

E = E_1.w_1 + E_2.w_2

which gives the same value of E as before. Using this weighting with the symmetries has allowed us to get the same quality of calculation using only 2 points explicitly instead of the original 3.

Additional symmetries may reduce the number of k-points further, and lead to further modifications to the k-point weights; however the k-point weights for a calculation with N k-points should all still be of the form a/N, where "a" is an integer that describes how many k-points in the original N-point set are equivalent to this particular k-point.

Dear Dr. Hasnip. Thank you so much. First time I got a very clear understanding of weight in sampling k-points. Anyway I am working on antiferromagneticLaCrO3 in Pnma space group. I think by XCrysden is fine to get this weight?

You should read the papers of Chadi and Cohen (Phys. Rev. B 8, 5747 (1973)), and after that the paper of Mokhirst and Pack (Phys. Rev. B 13, 5188 (1976)). There you will find the theory behind the k-point sampling of the Brillouin zone.

Thanks a lot @Paulo. These papers are really step by step clarification how to get k-points for the system. 

Kuntal Kabra

In Case of VASP calculation you nay know about your KPOINTS from your OUTCAR file.

Hi,

Step by step process of accurate (both total and projected) DOS and band structure calculations using VASP and some ways of plotting the band structure and DOS of our system is explained in a Youtube video.

The link is:

https://www.youtube.com/watch?v=RYziHtNFaPs

Hello,

This may be obvious but how is it possible to read out the bandgap from different phases.

For example if I wanted to calculate Si [111] vs Si[110].

I am trying to represent how different phases will dominate in different morphologies but I don't know how to make a structure that is Si [111] vs whatever the natural crystalline file is showing.

I saw some articles saying that it might be done by specific k-points

I also saw some stuff on cutting slabs. Overall I am just not sure.

Jordan Campbell do you mean "phases" or "faces"? In bulk silicon the electronic structure is completely periodic in both real- and reciprocal-space, so it doesn't change if you create a supercell, even if it is rotated with respect to the primitive cell. These would all be the same bulk phase of silicon though (usually the cubic diamond structure, unless you're studying its behaviour at high-pressures).

If you forced the band-structure calculation to start at the gamma-point (0,0,0) and go in only one particular direction, then the band-gap would often be different if you changed which direction you calculated the band-structure along, but this is simply an artefact of how you restricted the calculation to only a single line through the Brillouin zone. Cleaving a crystal in a particular direction does break the translational symmetry along that direction, so perhaps for a very thin film there might be some interest in looking at which k-point paths are no longer present, but for thicker films I don't see how this would be useful.

If you were to cleave the silicon so that you had a slab geometry, then the electronic structure of the surface would definitely depend on the orientation of the surface, i.e. which face was at the surface. Silicon's surface reconstructions are well known, but even if you fixed the atomic positions in the bulk locations or passivated the surfaces (e.g. with hydrogen) the electronic structure would be different cleaving along [110] and [111].

When considering which faces dominate the shape of a single-crystal, you usually try to compute the surface energies and use a Wulff construction (see, for example, https://en.wikipedia.org/wiki/Wulff_construction).

Sorry if I've misunderstood what you're interested in, I hope this has helped a little, at least.