Can the Band Structure Diagram be Represented in Real Space Instead of k-Space?

[BeauGeste]

So band structure diagrams are always depicted in k-space (1st BZ -pi/a to pi/a).
Is it possible to show them in real space (1st WS cell -a/2 to a/2)?
It seems to me that this would be legal. Is it not done because it is not instructive?

 

[ZapperZ]

If you can do a 3D Fourier transform, you can get that. However, you can't call it "band dragram", because it won't be E vs. k anymore.

We don't do it because it tells us nothing useful.

Zz.

 

[BeauGeste]

I think it would be useful. It would show how the energy varies with distance from ions. It would show the periodic nature of the potential in real space (slightly less abstract). Is it less useful because an idea like band gap would not show up?
Also there's no reason you can't do it in 1-D too, right? Just a 1-D Fourier transform going from kx->x.

 

[Modey3]

BeauGeste,

My intuition tells me that if you took a single dispersion curve (the E vs k curves) at a given eigenvalue (say n = 1) and did a Fourier transform on the function you would get a function describing the total energy of a electron vs distance in the lattice for a given k. However, for a given n and k the energy of a electron is fixed. What I think you want to determine is the potential, kinetic energy components as a function of distance in the the lattice.

Remember due to periodicity, electrons on the same dispersion curve have k-values dependent on the dimensions of the lattice and the k-values of two electrons with the same spin cannot be the same.

Anyways, to satisfy yourself I would suggest taking a eigenfunction of the Kronig-Penney Hamiltonian at a value of k and determine the eigenvalue of that eigenfunction. Plot it as a function of distance. Then plot the kinetic/potential energy components as a function of distance. It will give information that you already should intuitively know. Thats why we don't use the E vs. X relation.

Best Regards
Modey3