boundary conditions

[yangzw1985]

Hellow,everyone!
      Can anyone tell me How boundary conditions should be choosing when solving poisson equation to perform a self-consistent calculation in a two probe system?
      Thank you !

[Anders Blom]

ATK solves the Poisson equation internally, with the relevant boundary conditions corresponding to how the problem is set up.

• Periodic boundary conditions apply in the Z direction if left and right electrodes are identical.
• A multi-grid method is used in the Z-direction if they are not.
• Periodic boundary conditions always apply in X and Y.

There is no real option for the user to "choose the boundary conditions", but perhaps that is not what your mean...?

[yangzw1985]

hi,Anders Blom,thanks a lot. Prhhaps I not make myself to be undersand. The question is that " if I have performed a self-consistent  I-V calculations, what are the bondary conditions chosen when solving poisson equation?".
how should I answer the question? Does it equal to how I determined the distance between the electrodes and insert nanowires along z directions? Thanks in advance!

[Anders Blom]

What you are looking for is thus, "between what points in the Z direction is the bias applied", is that right?

That is an important question, of course. The simple answer is that the bias is applied from the left end of the left electrode to the right end of the right electrode.

Let us schematically draw a two-probe system as L-1|L|C|R|R+1, where L/R are the left/right electrodes (and L-1/R+1 are identical copies of them) and C the central scattering region. The reason we include L-1 and R+1 is because ATK assumes that atoms in L (R) may only interact with atoms in L-1 (R+1), and therefore from ATK's internal perspective, the complete system calculated is actually the one written above (from L-1 to R+1).

The Poisson equation is solved in the region L|C|R.

If periodic boundary conditions are used, ATK uses an FFT method, and the bias can be taken out of the equation since it is represented by a linear additive term to the potential, and the second derivative of that is zero. The boundary conditions are then such that the electrostatic potential at the left boundary of the left electrode match the bulk-like potential in L-1, and correspondingly for the right boundary of the right electrode. The bias is then added to the solution as a linear ramp.

If the system is heterogeneous, for example if the left and right electrodes have different spin or if they are geometrically different, a real-space finite difference multi-grid method is used, and the bias is explicitly included in the left and right boundaries as a rigid shift of the corresponding bulk-potentials from L-1 and R+1 on the edges.

Thus, the "voltage drop" occurs over the length of the two electrodes plus the central region, as illustrated in the plots in e.g. this topic.

[yangzw1985]

    hi,Ander blom.I have performed I-V calculations of a li-znonanowire-li system and submitted a paper. The reviewer's technical questions are as follow:" Do the authors perform a self-consistent I-V calculation ? This should be stated in the manuscript. if yes, then what are the bondary conditions chosen when solving poisson equation?".
   what is the real mean of the reviewer? How should I answer the question? Thanks !

[Anders Blom]

Sure, the I-V calculation is self-consistent!

It seems to me the reviewer has not understood the methodology of the calculation. This is described in the methodology papers, and most people just refer to them instead of writing too many details about the calculation. Actually, you really need to include those references in the article; see http://quantumwise.com/publications/scientific-publications, the section "How to cite ATK".

In case you do, and the reviewer anyway want some specific details, I hope my earlier answer contains all the details you need (the boundary conditions are of the Dirichlet type; I didn't mention this term explicitly).

[yangzw1985]

Thanks very much, Anders blom. Now, I have another question, if I want to calcualate the DOS with applied voltage six. How should I set the voltage in the two electrodes.  0 left electrode and 6 right electrode; 6 left electrode and 0 right electrode; -3 left electrode and 3 right electrode; 3 left electrode and -3 right electrode. In my opinion, it should be work equally. Does the four methods have any difference ?

[Nordland]

In my experience, there is no difference in any way in how you apply the bias, so go ahead, and choice the one you want.
The only effect should mind, is that if you calculate the voltage drop, the voltage drop will reveal how you have set the voltage.

[Anders Blom]

Correct, there is no difference. ATK only uses the difference between the left and right voltages, not their absolute values.

However, I don't expect your calculation will converge with a bias of 6 V. ATK works well up to 1 or 2 V, but usually not beyond that. Some new algorithm are being designed for future releases that may allow for larger bias.

[yangzw1985]

HI, I try to answer the question, but I find its very difficult for me. can you give me more details on :What is the periodic boundary conditions of li-zno-li two probe system?

[Nordland]

The boundary condition of a twoprobe is always the same for all twoprobe:

In the z-axises the potential must match the electrode potential ( shifted by the voltage if there is a voltage applied), in the x and y axis there is periodic boundary conditions, such that the potential must be periodic in the x-axis and y-axis.

[Anders Blom]

For a real, infinite nanowire, this means that you must include enough vacuum around the wire in the XY plane to avoid residual electrostatic interactions between the repeated copies. In you have a Li surface from which tube sticks out of, the Li surface must be extended correspondingly.