K points in Transmission Spectrum Analysis

[ruyam]

In the Transmission spectrum tool, we have an option to select k point sampling in transverse transport direction. I read in PRL 104,076807 (2010)(http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.104.076807) and in Physical Review B paper on Nickel Graphene Interface (by Dr. Stokbro et. al.) that to get a transmission spectrum of a Graphene sheet and not GNR, we increase the k-points in transmission spectrum so that effective width(W) will be W=w*k, where w is the width of the GNR and 'k' is the k-point chosen(Dr. Stokbro et. al. increased it to 501 to get an effective width of 2134 Ang).
 
Is my understanding correct?

[Anders Blom]

I (being of co-author of said paper) must admit I find the notion of "equivalent width" a bit misleading. The reason the k-point sampling is increased is quite simply to obtain convergence in the results, where an integral over k-space is approximated by a discrete sum over k-points. Actually you probably don't need that many k-points at all, provided you put them in the "correct" places - that is around the K point. But since we use a Monkhorst-Pack grid you have to use a high number of points in the whole zone just to get a lot of points around K. But most of the points don't help and just cost in computation time...

[ruyam]

Dear Anders Sir,

To recheck the k-point issue and its relation with effective width, I tested with a simple GNR FET and took some 20 points. I plotted the current vs. k-point sampling plot with the data and got a trend which is completely opposite to what I expected. According to quantum confinement effect, I expected the current to decrease as the k-point sampling is decreased (assuming that the effective width theory presented in previous post is true). From what I got I dont understand whether the way I am seeing it is wrong or I fail to understand the k-point sampling in the transmission spectrum tool.

I have a couple of questions regarding it:
1) How should I understand the k-point sampling and its effect in current changes.
2) So, how should I analyze a Graphene FET and NOT GNR FET ? Is choosing a periodic boundary condition in the Calculator(Poisson solver) enough to get results of a infinite graphene sheet in transverse transport direction ?

PFA my python file and the current vs. k-point sampling plot.

Thanks.

[Anders Blom]

I guess it just proves my point that the k-points are not a way to emulate a wider ribbon width.

[ruyam]

Thanks Anders Sir. 
I'll then use periodic boundary conditions to simulate an infinite graphene sheet.

[ruyam]

Dear Anders Sir,

To my previous query, I want to understand a few things regarding it?

1) If k-points are for convergence then why is the current so high for low k-point and then tend to stabilize as we increase them? How should we get its correct value of k-points to get the correct current value? Is it only by doing it for a some values of k-points and checking the trend? Is k-point sampling something analogous to meshing in TCAD?

2) Now, if I want to simulate, suppose a 100nm/150nm width Graphene based FET, then can I do that in ATK ? If, yes how and what parameter should I tune to get it?

[Anders Blom]

1) Sort of yes
2) You couldn't, and you wouldn't. Such a wide ribbon is infinite for most practical purposes - meaning the features that are relevant to compute in ATK

[ruyam]

How should I normalize the results then?
I take periodic boundary condition and have a width 'x' and get a contact resistance of say 'y ohm' for some bias. How should I normalize it in the unit of 'ohm-um' ? Should i multiply it with the width 'x' to normalize my result?

[Anders Blom]

Yes