FFT/FFT2D/Multigrid

[Dipankar Saha]

For Device: we use MultiGrid...taking Dirichlet boundary condition in the C-direction or, the transport direction. No ambiguity.

But, if there is no gate....we can take FFT2D.  Now In that case, if the channel is a nano-ribbon, then  what will be the boundary conds. for A and B directions??

Similarly, for bulk...we take FFT. But, using that when we obtain the band struct. of a 2D material (say, graphene)....then what is the boundary condition in A direction (thickness) ??

[Anders Blom]

FFT = Fast Fourier transform, which means periodic boundary conditions. And that's what you see in the Script Generator too.

[Dipankar Saha]

Dr. A. Blom

Thanks a lot for your reply....

[Dipankar Saha]

But, if there is no gate....we can take FFT2D.  Now In that case, if the channel is a nano-ribbon, then  what will be the boundary conds. for A and B directions??

I just wanted to alter this old question.... a little bit..., simply for the sake of better clarity......
____________
1)  Say, for the two port device.... which is made of a 2-D channel (it's a sheet ...meaning the structure is periodic along the width)... 'C' is the transport direction...and 'B' is the width.... (obviously 'A' is one layer thin or thick)..../ As there is no gate ..... What should be the boundary conditions?? ___ Is it.... A, B, C --> Neumann,  Neumann, Dirichlet...?
More importantly,
then how that is different from the 1-D case..... where I made the struct. such that...in 'B' direction the channel is  not continuous  (prior to applying the  A, B, C --> Neumann,  Neumann, Dirichlet boundary conditions...)...?

2) Next, considering a diff. case___ say a device with the gate___ / What should be  boundary cond.  along the direction 'A'....? Now, here is the gate...where I have applied a vertical E-filed...!!
I find various suggestions (looking at the previously discussed topics) for such a case....though they are not essentially the same...!!!

http://quantumwise.com/forum/index.php?topic=1512.msg7502#msg7502

http://quantumwise.com/forum/index.php?topic=1597.0

Regards_
Dipankar

[Jess Wellendorff]

1) the difference between Dirichlet and Neumann boundary conditions is this:
- Dirichlet: specifies the value of the electrostatic potential at the boundary.
- Neumann: specifies the derivative of the electrostatic potential at the boundary (but the potential itself may take on any value, inclu. zero).

Neumann boundary conditions are therefore very useful when there is an applied bias along the B direction (when you have gates).
Dirichlet is appropriate along the transport direction (C).

If you have a graphene sheet and use the FFT2D poisson solver, the boundary conditions along A and B are periodic. This is just fine as long as the vacuum to the edges along A is sufficiently large.

2) When inserting a gate above the graphene, you have to choose which sort of gate you want to model. Is the gate very thin with vacuum above it, or is it essentially infinitely thick in comparison to the device? In the first case, Dirichlet boundary conditions along B is appropriate, in the second case Neumann would be better because it allows the electrostatic potential to drop of smoothly into the "infinite" gate metal.

[Dipankar Saha]

Thanks a lot Jess...for your answer...!!! 
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If you have a graphene sheet and use the FFT2D poisson solver, the boundary conditions along A and B are periodic. This is just fine as long as the vacuum to the edges along A is sufficiently large.

But, what if the vacuum to the edges along the thickness (along A) had not been set to a large value (as well as, I am using the periodic B.C. ) ? ?
Or, alternatively...it's better to ask......say, along  B ...I made my struct. periodic, and that represents the width of the sheet..... Besides, say there are three physical width values w1, w2, w3 .... which I am dealing with..../ Now, how the calculations will vary for w1, w2, w3....when in all those cases I'm considering the periodicity along B ??

[Jess Wellendorff]

If there is vacuum above the gate, the important questions is this: Are you trying to simulate a gate of this exact size with lots of vacuum above it, or is the gate really supposed to be much thicker but you are just using a thinner one in order to save computational time? In the first instance, the vacuum region should be no less than 5 Angstrom, perhaps more, and boundary conditions could be Periodic or Dirichlet. If the second, Neumann would be more appropriate.

[Dipankar Saha]

Jess Wellendroff,

Okay.....  / It's actually the first one.... thanks again.....

____________________

However,  you did not say anything about the following _

Or, alternatively...it's better to ask......say, along  B ...I made my struct. periodic, and that represents the width of the sheet..... Besides, say there are three physical width values w1, w2, w3 .... which I am dealing with..../ Now, how the calculations will vary for w1, w2, w3....when in all those cases I'm considering the periodicity along B ??