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In the ideal case, an electron in a lattice under the influece of a static force will undergo bloch oscillations.

A simple hamiltonian for this system would be:

H=H

_{°}+Fx and V(x+d)=V(x)

If I used the kronig-Penney Model would I be able to derive the period of oscillation for the electron or would I not see that at all?

I'm asking because we have two hamiltonians; one for the potential well and on for the free space between wells. So to properly model the lattice we have to use a transfer hamiltonian to couple the potential wells to each other. Otherwise the states are not orthogonal to each other (I'm not sure why this matters I read it online). Possibly because the basic bloch functions are not localized and they need to be for oscillation to be observed? We also lose the translational invariance when we add the extra force.

Can someone offer me some insight into how the oscillations arise in a periodic lattice?

I've also come across the Wannier-Stark Ladder when looking for solutions of the hamiltonian. It seems to me the most common way to model an electron in a lattice with an electric field. Can anyone direct me to an introductory derivation of this or anything that will help me understand it and apply it?

Also I see a lot about the tight binding model for the electron, any good sources for reading up on that?

Thanks a bunch for the help, my textbooks really don't help much with this. I've mostly been reading publications that basically assume I already know a lot about bloch oscillations.