Calculation the transport properties of graphene

[Nordland]

9x9 is always better or the same as 3x3 in terms of accuracy and convergence - from case to case it might be enough with 3x3, and hence 9x9 will be a waste of time.

[Anders Blom]

Of course more k-points take longer time. For most system (esp. 3D bulk) you can always assume more k-points is more accurate, and a balance needs to be struck between computation time and accuracy, i.e. what is your acceptable level of inaccuracy for a lower k-point sampling than infinite. Infinite graphene is a bit unusual when it comes to k-point sampling, however, which of course is due to the special Fermi "surface" at the K point. Consider the following: below I list the top of the valence band and bottom of the conductance band at the K point for k-points up to (41x41).

Code

Nk	VB		CB
1	-2.83290807	-2.83266655
2	-0.09758413	-0.09734262
3	-0.00012076	0.00012076
4	0.00443931		0.00468083
5	0.02010983		0.02035134
6	-0.49142374	-0.49118223
7	-0.02140617	-0.02116465
8	-0.00251439	-0.00227288
9	-0.00012076	0.00012076
10	-1.78E-04		6.38E-05
11	-0.03717076	-0.03692925
12	-0.00330995	-0.00306843
13	-0.03310426	-0.03286275
14	-0.00071831	-0.0004768
15	-0.00012076	0.00012076
16	-2.31E-04		1.07E-05
17	-6.73E-05		1.74E-04
18	-0.000776		-0.00053449
19	-0.00167647	-0.00143496
20	-0.00037827	-0.00013676
21	-0.00012076	0.00012076
22	-2.00E-04		4.11E-05
23	-2.27E-04	1.48E-05
24	-0.00048527	-0.00024376
25	-0.00090307	-0.00066156
26	-2.62E-04		-2.07E-05
27	-0.00012076	0.00012076
28	-1.77E-04		6.41E-05
29	-2.40E-04		1.83E-06
30	-0.00035275	-0.00011124
31	-0.00058593	-0.00034442
32	-2.10E-04	3.17E-05
33	-0.00012076	0.00012076
34	-1.62E-04	7.91E-05
35	-2.26E-04	1.52E-05
36	-2.82E-04	-4.01E-05
37	-0.0004267	-0.00018519
38	-1.82E-04	5.95E-05
39	-0.00012076	0.00012076
40	-1.53E-04	8.89E-05
41	-2.10E-04	3.13E-05

This was computed with our Huckel model; since it's a symmetry-related result, the particular method has less of an influence on the conclusions, although in general the Huckel model is just as accurate (if not more, sometimes) for graphene, when using the Cerda basis set. The calculation of all results took 1 minute on my laptop, using the attached script, so there's no excuse for not doing a careful convergence study Now, if you look carefully, you will find that only for the following k-point samplings (Nk,Nk) will you have the two points straddling the Fermi level symmetrically:

Code

Nk = 3, 9, 15, 21, 27, 33, 39

So, quite simply: 3*an odd integer! Those are the ones you will want for infinite graphene in both 1D and 2D!

[mldavidhuang]

Thank you for your detail explanation, but I have some questions:

what is the parameter you calculate in the code? and 41 is not belong to the group of 3*odd, so the example you list can be consider a exception ?

And the symmetry of the parameters of VB and CB means for what? is the symmetry of the parameter you list can be a test of the right choice of the k point without doing the convergence study?

Another question is for other 2D material, For example, BN or monolayer MoS2, for these system how to determine the right k-samping? Is the rule still work?

[Anders Blom]

I'm afraid I don't quite understand your comments.

41x41x1 is just the largest k-point sampling I tried.

As mentioned in my previous post, VB and CB are the energies of the bottom of the conduction and top of the valence band at the K point.

The conclusions here are particular for graphene, but the method may be applied to other systems too.

[mldavidhuang]

Sorry for making my comments unclear.
It seems you use the symmetry of energies of VB and CB as the standard to determine the validity of the k-sampling. So can we determine the validity of the k-sampling of other materials in the same way, I mean, also through the symmetry of the VB and CB at K point?

[Anders Blom]

If the material is known to have a very special physics at the K point, like graphene has, then yes. Otherwise there is no reason to expect that.

[yongjunwinwin]

If the material has not special physics at these points, just take bulk Si for example, how the test the valid of k-sampling?

[Anders Blom]

Like you would converge anything: keep increasing the number of k-points until the quantity of interest doesn't change any more (within some acceptable accuracy, of course).

[zh]

If the material has not special physics at these points, just take bulk Si for example, how the test the valid of k-sampling?

Strongly recommend you to read this following paper:

Ann E Mattsson, Peter A Schultz, Michael P Desjarlais,
Thomas R Mattsson and Kevin Leung
Designing meaningful density functional theory calculations in materials science—a primer
Modelling Simul. Mater. Sci. Eng. 13 (2005) R1–R31
http://iopscience.iop.org/0965-0393/13/1/R01/

[mldavidhuang]

Thanks for your reference !

[esp]

I have create a simple GNR fet type device and have tried to follow the tutorial code .. i want to get the IV curve for a simple GNR ... code is below and attached image ..

It should be a 12,0 AGNR, gate voltage at 1V, electrodes at -0.1 and 0.1, .... IV curve is linear (only if I "symmetrize", otherwise it is a point) ... and i get flat conductance ... what am I doing wrong?

makeCfg() and doCalcs() are the only functions used for now ..

[Anders Blom]

A few things to consider. First of all, you don't get an IV curve from a single calculation, you have to run it at different bias, otherwise indeed it's just a point on the curve, not a curve.

Second, take care with your dielectric and metallic regions; they are sticking out of the box, which means they get wrapped around, not a good idea. Keep everything inside the box, atoms and what.

Finally, don't expect too much from a perfect 1D conductor. The IV curve can be trivially obtained from a simple band structure analysis, but to run a real finite-bias calculation on a perfect system is not a good idea. The reason is, that since it's perfect there should be no resistance (in the ballistic regime we are considering in ATK). But if there is no resistance, there is no point for the bias voltage to drop across the structure - a finite bias and a finite current means there must be finite resistance.

It would be better to do something like the simple but non-trivial system in the VNL tutorial, or the longer graphene tutorial. Or, if you want gates included, the graphene device tutorial.

[esp]

thank you

[esp]

I was unclear about the role of "the box" ... is there somewhere in the tutorials that explains this?

[Nordland]

Second, take care with your dielectric and metallic regions; they are sticking out of the box, which means they get wrapped around, not a good idea. Keep everything inside the box, atoms and what.

A small correction. Regions are not wrapped - this goes for both metallic and dielectric regions, but only the part inside the cell
is present in the calculation. Therefore your case the metallic region is disregard and you will not seen any effect of it.

As Anders explains one should be careful about deriving too much physics from a perfect metallic 1d systems, since the screening effect is almost infinite bad, and the effect of a voltage will be artificial since there is physical meaningful place for the voltage drop to be placed.