Inconsistency in CalculateCurrent for different number of integration points

[cwolowiec]

Hi,

I'm calculating current for a two probe system with graphene nano-ribbon electrodes.

I have converged sc files at .1V,.2V,.3V,.4V,.5V,.6V,.7V,.8V,.9V,1.0V.

At each bias, I have been calculating current using a different number of integration points over the energy. The default number of points is 100 and I also calculated current for 200, 400, 600, 800, and 1000 points as a check for consistency.

At lower bias, there is consistency across current calculations for different number of energy points. At higher bias, I'm getting some inconsistency. For example, at .8V I'm getting the following currents for different number of energy points:

100 Points
2.70322906734e-08 A

200 Points
6.15809652354e-09 A

400 Points
2.0376602094e-08 A

600 Points
2.67889424829e-08 A

800 Points
1.86633465855e-08 A

1000 Points
1.57990714177e-08 A


Furthermore, at a bias of .9V the currents have dropped a whole two orders of magnitude compared to .8V as seen from this data below:

100 Points
3.93849284671e-11 A

200 Points
3.90495735902e-10 A

400 Points
1.24586672635e-10 A

600 Points
2.48097024468e-10 A

800 Points
1.6369171909e-10 A

1000 Points
2.21587823888e-10 A

For lower bias up until a bias of .5V the I-V curve was monotonic increasing and there was more consistency in currents at a single bias. Clearly, at .9V the currents have dropped and are also inconsistent with the number of points.

Is there some explanation for this behavior? In another post, I recall reading that artificial scattering could occur if the electrode cell is not long enough in the z direction. Could artificial scattering be happening here at higher bias levels and not at lower bias levels? Note that, my electrode cells are 4 atoms thick (4.9 angstroms) and I have included this same number of atoms (thickness) for the screening layers on each electrode.

Any comments are appreciated,

Chris

[Nordland]

Hey Chris.

In order to be able to understand this behavior, it would be a good idea to see the transmission spectrum,
can you perhaps calculate the transmission spectrum for 0.8 V calculation in an energy range -2 eV to 2 eV using 1000 points.

Perhaps this will gives us the reason for this behavior. The current is related to the integral of the transmission spectrum in the interval
in roughly -V/2 to V/2, and hence more points might make this integral more precise.

So for 0.1 Volt you will have a point density of roughly 0.001 eV in your transmission spectrum, and this is properly fine as the transmission spectrum in most system do not vary that fast for low biases. Therefore when you got up a 1000 points, the integral remains the same.

For the 0.8 V calculation and 0.9 V calculation the point density is 8 time as rough, and therefore you will need at least 8 times as many points as 0.1 V calculation in order to get the same resolution. This combined with the fact that high bias calculation often has a rapid varying transmission spectrum might be the origin of your issue.

On a second note I have also seen a system with no artifical scattering for low bias, suddenly having artifical scattering at higher biases, which could be removed again, by adding more screenings layers.

Looking forward to see the transmission spectrum.

[Anders Blom]

I'd have to see things in more detail to be sure, but my guess is that your system is not conducting at all, and hence you are just computing "zero current" at various levels of accuracy.

You really should plot the transmission spectrum and check the values there. Most likely they are very close to zero. Is your ribbon zigzag? An armchair ribbon is semiconducting, so then the transmission and current will quite naturally be zero.

4 layers is very little, I'd go with 15-20 (in total) in the central region (cf. the graphene tutorial, Fig 12), and even if it apparently works to have 4 layers in the electrode, I'd actually rather use 6 or 8 for safety.

There may be some other artificial scattering going on, so check the geometry (distance between layers) carefully.

[cwolowiec]

Great! Thanks. All images are in .png format.

Attached are configuration files and one bias transmission spectrums.
The configuration consists of two zigzag edge GNR electrodes with a 5 angstrom vacuum gap in the middle. About 2.3 angstroms above the plane of the ribbon is a DNA molecule. I should mention that without the molecule, the current across the gap is on the order of 10^-14...which is much smaller than the previously reported currents with the molecule.

Hopefully this clears up some of previously raised issues..

Thanks again for your input,

Chris

[Forum Administrator]

The pictures - finally!

This system is really cool, btw!

We had big problems getting the pictures onto this post. Not because they were too big in file size, but probably because they had too high resolution (over 3000x3000 pixels). So the lesson is, don't make the pictures high-resolution, screen-resolution is fine, we're not going to print them in magazines

[cwolowiec]

Great!

To respond to Nordland's comments on the number of points and resolution of the transmission spectrum:

The two transmission plots contain 200 points between -.5eV and .5 eV and again 200 points between -5 eV and 5eV. I'll recalculated using the prescribed rules for resolution and integration at various bias.

Thanks,

chris

[Anders Blom]

First of all, the current is very, very small. I agree, it's not zero, but it's extremely small still, plus that it has some sharp features. So it might be difficult to integrate it very accurately.

As for how the current depends on the number of points, it's kind of irrelevant. You just need "enough" points that it saturates. Clearly that's difficult in your case, but again, the current is so small that I doubt one can consider the computed result physical.

The integration delta matters too, of course, as Nordland points out. Perhaps looking at that will stabilize the results a bit. But I would really consider moving the molecule just a little bit closer to the surface

[Anders Blom]

Another thing: what effect are you actually looking for here? The current-voltage characteristic is perhaps less relevant, while you have already proved one important point: in the presence of the molecule, the current changes by 2 orders of magnitude. To me, that's more important than the value of the current itself, or how it depends on bias.

But of course you should investigate all aspects.

[serhan]

Hi,

The topic on this thread made me panic    since I am planning to investigate the effects of high-bias on the resistivity of the nanostructures. As a conclusion, to guarantee the error in current calculation integral remains lower, which values would be good for number of points and delta? (Of course, in exchange of computer time  )

Cheers,
Serhan

[Anders Blom]

No need to panic, I hope

The calculation of the current depends on certain parameters. The point density is only one; the current is strongly dependent on k-point sampling (unless it's a 1D system), and there is also an indirect dependence on the mesh cut-off, basis set, and even geometrical parameters like vacuum gaps.

It's a natural part in a serious investigation to ensure that the results are converged in all these parameters in a systematic way. At the same time, "real" qualitative effects (such as, is the structure conducting or not; is this band gap larger than that, etc) are usually much less influenced, since the errors usually tend in the same way and therefore cancel.

One should not, in general, expect that atomic-scale simulations of this kind give absolutely correct results. There are too many unknown parameters (the precise geometrical structure being the primary one) to expect perferct agreement with experiments. Therefore, it's important to keep in mind to look for trends and qualitative effects rather than quantitative, and in the end also to use your sense of physics and corroborate the evidence with some real insight into the mechanisms behind the observed phenomenon.

All this rambling aside, now for a concrete answer You should never compute the current blindly. Always study the transmission spectrum carefully first. If it has strong peaks, you will need more points, if it's flat you need fewer. However, it is clear that in many cases the spectrum has both features: regions with relatively slowly varying T(E), and some peaks overlaid on this background. This is in the nature of resonant tunneling, in a sense.

Therefore, a really attractive approach would be to develop some kind of adaptive integration for the current, by which a larger number of points are used where T(E) varies strongly, and fewer where it is flat. This can be done using NanoLanguage, there is no need to go into the ATK core to implement such functionality. I'm going to have a look at it, but it might take some time...

[Nordland]

To Chris:

Your transmission spectrum looks like I thought they would, and therefore the is no reason to be worried about the results, part from the fact you will need to have a lot of energies points. If we for a second assumes that the transmission spectrum dont change under bias ( as it does ) we can try to understand the behavior of your results. The transmission spectrum is dominated by almost singular peaks, and hence transport is only allowed for certain incomming energies. Therefore if the system has to lead a current we need to get these peaks into actions. Therefore based on a quick judgement of the transmission spectrum, you will need to go to 5-6 voltages bias before you will start to see a real current build up, it will properly have the IV curve of a typical diode. The current is still very small even for 1 V.
The reason you need a lot of energies points to get in converged in current for the high bias, is that in order to get a good sampling of the peaks you need a high point density, or else we might overlook the the important peaks/features.
For the low bias calculation is the energy point density is still quite high, and the peak at the fermi level ( E = 0) is therefore quite good sampled and hence you will get quick convergence of the current.

So to sum up: The system you are studying, is dominated "singular" peaks and hence you need a really good sampling in order to get the correct resolution, and therefore you will need to increase the number of energy points as you increase biases.

As a note: It is not always need to have alot of energy points as the transmission spectrum might be more smooth - I have attached a transmission spectrum for graphene, and as you will see, it is quite smooth and not dominated by peaks or similar.

[Nordland]

Hi,

The topic on this thread made me panic    since I am planning to investigate the effects of high-bias on the resistivity of the nanostructures. As a conclusion, to guarantee the error in current calculation integral remains lower, which values would be good for number of points and delta? (Of course, in exchange of computer time  )

Cheers,
Serhan

Hey Serhan.

Panic is never good

If you ask me, this is always a matter of personally approach to the problem. When I am starting up calculations on a new set of systems, and in particular a system on which I wish to understand the current behavior for high biases etc, my first goal is to obtain a well converged transmission spectrum in terms of kpoints etc. From this information I observe the nature of the transmission spectrum and try to judge what kind of parameters I am going to need to get good results.

If I get a transmission spectrum like the one shown by Chris, I would except to get a IV curve of a diode, and hence I would know that I would need a lot of energies points in order to get a good current, making the calculation more costly. However as I always know from the transmission spectrum that the current would rise significant before the voltages gets quite high I might choose an non uniform sampling of the voltages:
V  = [0, 1, 2, 2.5, 3, 3.5, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0]

However if I have a transmission spectrum of that of a semi-metal, like the one I showed before I know that the energy point density only need to be high for the small voltages, and when it comes to the number of voltages needed to calculate a good IV curve, I would choose many small voltages, and few high:
V = [0.0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5, 1.0, 2.0, 4.0, 8.0]

I hope this can help you.

[cwolowiec]

Nordland,

Thanks for your comments regarding resolution and integration of T(E) to get the current. Before it seemed like guesswork. Now I have a systematic way to approach the subject. Having said that, I integrated the current at .9V using 9000 points and unfortuanately the order of magnitude did not change. At 1000 points, I = 2.2E-10. At 9000 points, I=5.9E-10.

The current for this system starts to saturate near .3V but is pretty monotonic and at least of the same order of magnitude through .8V. But at .9V and 1.0V, there is a huge drop in current. You mentioned the possibility of artificial scattering at higher bias. Can this be fixed by increasing the size of the electrode cell and/or increasing the screening layers in the central region or are things sufficient as is? I'd like to avoid adding atoms which as you know will increase calculation time.

Also, you mentioned the high bias Transmission spectrum should be no different than the low bias Transmission spectrum. I will look at the spectrums at higher bias and compare with the same at lower bias. If there are noticeable differences, which might explain the huge drop in current at .9V and 1.0V, could I assume that such differences are due to artificial scattering?

Chris

[Nordland]

Hey Chris.

I think we should try to pin down what are the goals and subsidiary goal of this study, and I will try to give my ideas on what these goals might be, and what route I would take, if I was you:

IV curve: The current behaves not perfectly clear, but in all respect of a device the current is zero for all the voltages, that are calculated so far, and features that I want to study, will not show be present before the voltages is significant higher, and hence I would go directly to higher voltages, and try to see if the current looks reasonable at these voltages.

Underlying mechanics: If the main goal is not to get the IV curve, but rather understand the physical behavior of this system, I would do the following things. First of all compare the transmission spectrum at 0.7 V with the one at 0.8 V - what are the differences?
Are the differences a result of the ill-convergence? What does a transmission spectrum at 0.75 V look like? Is there a certain voltage where the transmission spectrum changes fundemental shape? If it does, what are the reason of this value? Perhaps a molecular level
of the DNA molecule?
Alternative I would select a voltage for which the transmission spectrum is nice fx. 0.3 V, and then I would compare this transmission spectrum with one where I had doubled the number of atoms in the electrode, and the number of screening layers? Is there any difference?
And If I have accces to the newest ATK 2008.10 ( and used the constraint DensityMatrix), I would calculate the voltage drop for 0.3V for the original system and the extended system using the script I have posted here (link). Then I would check that the voltage drop looks as except - they should be somewhat similar or else the screening approximation has collapsed. If they agree, then I would do the same for 0.8 Volt? Still similar or radical different?

On a final note: If you have not done already, you should try to use ATK 2008.10 with the new constraint DensityMatrix as it is superior to bias calculation over the two old constraints...

[cwolowiec]

Nordland,

My overall objective is to find qualitative differences in the tunneling current for different types of DNA molecules placed above the GNR. I agree the current is small at 10-9 A but this is significantly larger than the current across the GNR without a molecule, 10^-14. Some ways I could increase the current further (keeping bias constant)  are to move the molecule closer to the ribbon and also to place the molecule over the edge since this is where the conduction in the ribbon occurs.

In the present system, regarding calculations at higher voltages. I'm guessing my choice of higher voltage should depend on the peaks and profile in the Transmission Spectrum. For example, if I see a peak at energy 4.1 eV, I'm guessing I should I run a calculation at bias 4.1V?

Regarding electrodes and screening layers: I'll follow your advice and recalculate at low bias and high bias to see if I can find any differences. I don't have version .10 yet, but I'll  see if I can get it installed on our cluster.

Thanks again and I appreciate your helpful comments.

chris