About the Number of k-points ?

[fan0221]

Hello everyone,
       I have some questions about  the Number of k-points in the Nanolanguage Scripter of the VNL. After i put a model in the Nanolanguage Scripter, there are two places that request me setting the parameters of the number of k-points  . One is in the Brillouin Zone Intergration of the Method, another is the Number of k-points (A),(B) in calculation the Current, Density of states, Transmission spectrum of the Analysis . Is there some difference between them? What's meaning of the k-points (A),(B) in the Analysis, it must be same with the parameters in the Brillouin Zone Intergration of the Method?
Thank you very much.

[Anders Blom]

The first setting, for the self-consistent calculation (SCF), is related to the band structure of the underlying electrode bulk crystal. To determine the Fermi level accurately and to sample the band structure properly, a certain number of k-points are needed. For a small unit cell (small in the XY directions that is), you need more k-points, perhaps up to 10x10 for good accuracy, while if you have a large unit cell you can perhaps even get by with a single point (1x1).

The second "analysis" settings determines the sampling of the 2D Brillouin zone when you calculate the transmission spectrum. Electrons incident from the elcetrode with different k-points have different probability of tunneling through the central region, and we must sum all contributions to obtain the total transmission at a certain energy. In this case it can be very hard to know up front how many points are needed, and one should really test by increasing them to see if the transmission spectrum changes.

The number of k-points do not have to be the same. If we take a well-known example of a FeMgO magnetic tunnel junction, an SCF sampling of 8x8 is sufficient for good results. However, the minority transmission spectrum for the parallel spin configuration requires upwards of 100x100 to reach a good value for the transmission. This is because there is large number of very narrow peaks in the transmission coefficients, which are related to the exact matching of the Fe band structure to the MgO energy levels.

Therefore, as we see in this example, the analysis k-points are related to the whole system, the transmission properties of the composite two-probe system, while the SCF k-points are only related to the electrode.

The only way to really find out how many k-points are needed to obtain converged results, is by running a sequence of calculation for different values. This goes for both the SCF and analysis k-points, although the transmission spectrum typically (but not always!) is less sensitive to the SCF k-points.

Increasing the number of k-points in the SCF does however improve the convergence, and if there are way too few k-points (e.g. using only Gamma point when one should have 5x5 instead) will shift the band structure and thus influence the results.

ATK is specifically parallelized over k-points, so when running on a cluster, it is not very expensive to increase the k-point sampling, if you run on many nodes. Memory usage does, however, increase with the number of k-points.

Of course, for one-dimensional systems like nanotubes, you always only need 1x1 k-points for both SCF and analysis. For graphene, which is periodic in two directions, you need N points in one direction and 1 point in the other (perpendicular to the sheet).

[privador]

My problem is same-selecting the bottom tab parameteres
For example graphene tutorial
Adition to the power spectrum there are other possible quantities.Electron density,Eigenvalues,Eigenstates etc

Could u do some example? for example simple graphene?

[Anders Blom]

This post is a bit detached, I may split it off later

Well, those other quantities are both simple and quite advanced.

If you want to calculate and visualize the electron density, for instance, just "click it over" to the list of "Selected quantities" (and make sure the specify a VNL file where you store the results), run the calculation, and then import the VNL file into the Nanoscope and plot it (right-click, Insert plot, etc). Let me know if you need a more detailed demo, and I'll do that

About the transmission eigenvalues etc, it takes a bit of knowledge of electron transport theory to fully make good use of them. I agree a tutorial on the matter would be needed, and I'm actually working on that.

[fan0221]

Thanks for all !

[serhan]

Hi Dr. Blom,

I'm a bit confused about the k-points also. 

1. In VNL, when we specify the number of k-points, are those numbers used for electrode calculation or scattering region calculation in ATK?
2. For example, if we use kx=1, ky=1 kz=100 or kx=1, ky=1, kz=10, what will the differerence of accuracy be? (Of course, it depends on the structure being simulated)
3. Does ATK take "kz=the given value" for the electrode calculation and "kz=1" for all scattering calculations?
4. In all of nanotube calculations, can we use kx=ky=1?

Cheers,
Serhan

[Anders Blom]

First of all, there is a lot of useful information on this point in this post.

To quickly recap via your specific questions:

1. The X/Y (actually, A/B) k-points are used both for the electrode and the SCF two-probe calculation with open boundary conditions, and the initial equivalent bulk calculation (if requested).

2. There will be a difference in accuracy which might prevent convergence. 10 is probably too low in all circumstances, while as many as 100 might not always be necessary. However, the electrodes are always the fastest part of the whole calculation, even with 100 k-points. (The VNL default of 500 is definitely too much in most situations, however!)

3. For the calculation with open boundary conditions (the real two-probe calculation) we don't use k-points at all in the Z direction, since there is no periodicity in this direction. This is the whole point of the two-probe method, actually otherwise we couldn't apply a bias across the system.

Alternatively, you can see it as infinitely many k-points in Z for the two-probe part, which hints at why you need so many kz-points for the electrode.

Also note that ATK uses kz=1 for the equivalent bulk calculation.

4. Yes, provided the system is correctly set up, that is the tube along Z and enough vacuum around it in X and Y.

[serhan]

Hi Dr. Blom,

Thanks again for the answers.

Regards,
Serhan

[hellboy]

Dear Sir,

I use K- points (1,1,100) for single wall CNT.

What will be the suitable number for a Double Wall CNT (DWCNT)?

[Anders Blom]

Same. K-points are related to translation symmetries, the first two numbers specifically to the periodicity of the structure in the X/Y plane, which is the same for any nanotube, single or double (i.e., there is no periodicity).

[hellboy]

I have a 5X5X5 Au electrodes cell and a (8, 0) CNT in the central region.

How many K-points are required in X-Y ?
What size of electrode unit cell is considered large so as to use 1X1 k-point in X-Y ?

best regards,

[Anders Blom]

If hope it's Au 5x5x3 or 5x5x6, since 5 layers in the electrode doesn't correspond to any sensible cleaving

The important role of the XY k-points is to establish a good band structure of the electrodes. So if you are really diligent, what you should do is compute the band structure of the electrode separately for an increasing number of k-points and see how it behaves. This can be done for a 1x1 cell, you just have to convert the k-points back (5x5 k-points for a 1x1 cell is 1x1 k-points for the 5x5 cell).

I don't think 5x5 is large enough to get an accurate results for only 1x1 k-points. For the basic band structure of gold, you need at least 9x9x9, so logically 3x3 seems reasonable for a 5x5 cell. (2x2 might be an option, but in my opinion it's generally better to include the Gamma point.)