Dear Joseph,
Thank you for your response. I should perhaps have been more specific; I'm not thinking of the metal as leads, but as a thin layer covering the semiconductor, such as is done in the Majorana devices. Something like the one that is shown on the about page taken from the unpublished work, except there they did not use different 'meshes' What I am ultimately interested in is solving the wavefunction in the entire system, so the metal and the semiconductor, which is why using equal lattice constants will end up being a problem for geometries of 2D or 3D. And for this self energy will probably not work. But thank you for the input, and making me realise I should refine my question.
I agree with you that even now I'm asking something too general, as there are probably different ways of tackling suck a problem, but I personally don't really see how this should be done. Perhaps every (for the sake of example) 4 metal sites should have hopping terms to every 1 semiconductor side, at the boundary, or something like that. Or maybe the boundary should be some interpolation, as the hopping will depend on t, which depends on a. It seems like a tough problem, but ultimately probably quite relevant. But as you point out this is a problem that might be discussed in literature, as it is a problem of finite difference methods in general I would think.
Kind regards,
Jonathan
Sent: Wednesday, July 12, 2017 at 3:03 AM
From: "Joseph Weston" <[email protected]>
To: "Jonathan Eaves" <[email protected]>, [email protected]
Subject: Re: [Kwant] Making heterostructures of different lattice constants
From: "Joseph Weston" <[email protected]>
To: "Jonathan Eaves" <[email protected]>, [email protected]
Subject: Re: [Kwant] Making heterostructures of different lattice constants
Hi Jonathan,
In the documentation it is written thjat the tight binding approximation is good for all quantum states with a wave length considerably larger than the lattice constant a. Now, if you are dealing with a system made out of a single material, then you can just choose a to be such that this is true, and make sure that the system is still solvable in a reasonable amount of time. But what if I am dealing with a heterostructure, in which one system has a much smaller wavelength? Say I put a metal on top of a semiconductor. Here you would need a much finer grid for the metal than for the semiconductor, but using that same grid for the semiconductor makes no sense computationally. Is this something that Kwant can deal with? Can I make a device that has a1 inside region 1 and a2 inside region 2? I would asssume the boundary layer is problematic, as you have different lattice constants, effective masses, and so on.
This is not something that Kwant will deal with for you; you will have to decide what the best way is to model your system. The choice will depend on a number of factors that will probably be pretty specific to exactly what behaviour you are trying to model. This is why stuff like this is not implemented directly in Kwant, and probably never will be.
That being said, there are probably members of the Kwant community who have addressed similar problems, and who could tell us what they did. I would imagine that the way to go would be to treat the leads as a self-energy, rather than directly constructing them as tight-binding models; this would mean that you would not have to finely "mesh" the lead. All of the information about what exactly happens at the interface is then in the self energy, so the problem is then just shifted to coming up with the right self energy. I would imagine that there is a reasonable literature on constructing such things, but I'm not an expert on this topic.
Happy Kwanting,
Joe
