About the charge option

[F. Fuchs]

Hello everyone,

I want to clarify myself about the meaning of the charge option for electrodes (ATK version 12.8.2).
It is clear that the Fermi level gets shifted when additional charge is added.

To estimate the influence of the electrode charge in a device configuration, I looked at one single (7,0) CNT (bulk configuration) and varied the charge (periodic boundary conditions along the CNT, Neumann boundary conditions perpendicular to the CNT (as this is needed for the device configuration)).
Afterwards, I subtracted the Fermi level from the resulting band structures (to see the influence on the band structure and the Fermi level separately).
 
The result is attached.
You can see the energy values at the Gamma point versus different amounts of additional charge. The red curve is the Fermi level, the black curves correspond to the band structure. The inset shows the band structure of the (7,0) CNT for 0.0 additional charge.

The movement of the Fermi level relative to the bands is clear. However, there is also a shift of the whole band structure in addition, which I don't understand yet (shouldn't the black curves be constant?).

Can someone explain this behavior?
I would also be interested in further information on how exactly the additional charge is added to the system.

Thanks in advance!

[Anders Blom]

You're basically subtracting the Fermi level twice. The absolute Fermi level is meaningless, so we always set the relative Fermi level to zero for the band structure, so it's already subtracted when you extract the band positions at the Gamma point. If you subtract it again, you will get the effect you see, but it's not relevant.

[F. Fuchs]

I agree that only the position of the bands relative to the Fermi level should be important.

However, I observe the following:

When I look at the same system and increase the size of the unit cell, the slope of the curves in the previous picture increases. This means that the shift of the band structure values due to charge addition is increased by using bigger cells (only the values are shifted, the look of the band structure is not influenced). The Fermi level changes as a result, too.
By looking at the Fermi level versus the system size for different doping levels, I obtain the attached picture ("fermi_dependency.png"). Only for big cells, the Fermi level is independent of the cell size.

The observed effect is interesting for us because it correlates with the observed shift of the transfer characteristic of our CNTFET (see http://quantumwise.com/forum/index.php?topic=2670.msg12714#msg12714). The charged CNT is used as electrodes for the transistor.

We think that the observed effect (shift of the transfer characteristic) comes from electrostatics inside the cell, because the additional charge increases the electrostatic effects inside the cell. Therefore, a bigger cell is needed so that the manipulation of the field by the Neumann boundaries is small.

We know what to do against this effect (using small amounts of charge and big cells), but we don't understand the effect yet.
Maybe someone can enlighten us.

[Anders Blom]

I think the fundamental issue is that your system is charged, so yes, as you say, there is a contribution of the boundary condition. Actually when computing systems with a net charge, one should really use the multipole boundary condition, but on the other hand you don't really want a charged system - a doped structure is not charged, it just has an additional amount of free charge, but it is always balanced by bound local charges of the opposite sign.

In ATK 13.8 it is possible to compensate the free charge by such "compensation charges", and this neutralizes the system and should - I hope - stabilize your energies.

[F. Fuchs]

Thank you for the information!

One final question (related to ATK 12.8.2, since ATK 13.8 is unfortunately not available yet):
When I add additional charge to the electrodes, does this also increase the net charge of the central region?
To put it in another way: Can the additional charge of the electrodes flow into the the central region?

(I would think so because the system is an open system. However, this would make the additional charge to an unsuitable method for simulating a system with doped electrodes. But obviously it works and it is also used for the SiFET tutorial.)

[Anders Blom]

To some extent it does due to the shift of the Fermi level, but this charge cannot localize in the central region, only pass through it in steady-state. So, put crudely, the charge in the electrodes might increase the current but doesn't modify the potential landscape.

I don't think these points make the method unsuitable, however the lack of compensation charge means it's an approximation, and therefore of course we added the compensation charge in later versions of ATK. Very importantly also, the compensation charge can be used in the central region to create a doping effect there as well, allowing not just p-i-n or n-i-n studies, but also p-n or n-p-n, etc.

[F. Fuchs]

Thank you for the information!

We now tried to surround the electrodes by a metallic region in order to shield the electrostatics from the boundaries (I basically added a cylindrical gate to the electrodes. This is easy because we have carbon nanotubes as electrodes). This seems to work pretty good.
However, we are now wondering if it is necessary to add a metallic region to the electrode copies inside the central region as well.
What do you think?

[Anders Blom]

There is nothing that prohibits that, and basically it's just a question of how long you want the gate to be. It may, possibly, be harder to converge the calculation with a gate on the electrode copy, but apart from that the choice is up to you, I think.