K points for monolayer graphene nanoribbons

[Luis M. Villamagua C.]

Dear everyone,

I have to optimize a monolayer graphene nanoribbon (i.e. 1nm wide and 3 nm long) structure before putting defects (like SW, vacancies, dislocations, etc.) on it. I was advices to carry out the k-point mesh optimization first (the correct choice of this mesh will apparently minimize the computing time and also get precise results -- correct me if I am wrong). For doing this, I found out that I should calculate the "total energy" of the system for different meshes and then see for which mesh the energy converges.  But with 3 number I can have 1000 different conbinations. What are the values I should start with, for example, 3x3x5, 5x5x3, 1x1x1, 1x1x5??? By the way, finding the "total energy" of the system in QuantumWise means to use the "total energy" tool calculator under analysis in the script?

Any advice will be well apreciated. I am very very new with this topic.

Thank in advance,
Luis

[Anders Blom]

Considering that it's a nanoribbon, you only need 1 kpoint in X and Y (assuming the direction of the ribbon is Z, which is best to choose in ATK). Thus you really only need to check 1x1xN with N=10,20,50,100. Either way will be relatively fast.

You can check how the total energy converges for DFT, but not for tight-binding. In many cases it can be better to check how other quantities change, like a particular band gap, forces or stress, density of states, or an effective mass - different quantities have different dependence on the k-point sampling.

[Luis M. Villamagua C.]

Sir,

I am trying to replicate the tutorial in the following link: http://quantumwise.com/documents/tutorials/latest/SiliconOptical/index.html/chap.Si.Band.html

They set the calculator mesh to (4x4x4) while for the DOS and the BandStructure to (15x15x15). I thought the mesh had to be the same in all cases.

Can you please clarify that to me??

Thanks in advance,
Luis

[Anders Blom]

What do you mean "in all cases"? For any system, of any shape, size or material? No, that is certainly not the case.

Or you mean, the calculator mesh and the DOS mesh? They can - and often should be - different; you are expanding different quantities is Fourier space. Other codes may not support it, but ATK does.

[Luis M. Villamagua C.]

Language barrier Sir  , What I meant was "the calculator mesh and the DOS mesh". I will try to read more about this to understad it properly.

I think I got the idea for the k-points btw. I have calculated the "Total Energy" of the nanoribbon (with a SW defect introduced in the midle of the nanoribbon) for different k-point meshes 1x1x5, 1x1x10, 1x1x20, 1x1x50, and 1x1x100 (attached image). Now I am calculating the total energies for meshes with Kc set from 10 to 20; I am doing this because I can see that the optimal mesh is in between this domain.

I have another question however ...  I tried to find the optimal mesh for the same nanoribbon but with no defect included. I mean, I created the ribbon with VNL's builder, and without doing anything else, I tested it with the same meshes as above. I calculated the total energy, but it increased (it didnt go down and then a little bit up to finally converge to a certain value of energy as in the nanoribbon with SW defect). Is there any logical reason for this?

Thanks in advance again,
Luis

[Anders Blom]

The total energy is not variational (in the sense of basis set completeness) with respect to the k-point sampling, so this often happens. The important thing is to find a high enough number that the variations are small. Also, going from odd to even can mean a big change (not necessarily to a lower energy) since one case (odd) involves the Gamma point and other doesn't.

[miguel]

Why we have to use DFT and not tight binding?

[miguel]

Does anyone knows why it is advised to choose Kc a multiple of 3?

[Anders Blom]

Does anyone knows why it is advised to choose Kc a multiple of 3?

Because the K point, where the Dirac cone is centered, is at 1/3 of the Brillouin zone.

[Anders Blom]

Why we have to use DFT and not tight binding?

Because in many tightbinding models there is no repulsive pair potential and hence no total energy.

[miguel]

Just for curiosity,

If I calculate the total energy with tight binding in quantumwise I will get an error, or will I get a non-variable value?? What about Extended Huckel???

[Anders Blom]

You will get some number, but it's not comparable between different configurations, meaning a configuration with smaller "total energy" is not necessarily closer to equilibrium. See http://quantumwise.com/documents/manuals/latest/ReferenceManual/index.html/chap.atkse.html#sect2.atkse.extended_huckel.total_energy for details.

[miguel]

Thank you very much Sir.