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Let's consider a system of free electrons moving in a one dimensional lattice with dispersion $\varepsilon(k) = -2t\cos{ka}$, ($a$ is the lattice spacing and $t$ the hopping amplitude). Let's now superimpose a uniform force $F$ to the system, so that the Hamiltonian reads $$ H = -\frac{\hbar^2}{2m}\nabla^2 + V(x) - Fx, $$ $V(x)$ being the periodic lattice potential. I know that the external force induces Bloch oscillations of the electrons, which are spatial oscillations of amplitude $\propto t/F$ and frequency $\propto F$.

However, I would naively expect that, at least in the limit $F\gg t$, the electrons should not oscillate, but instead they should follow the foce and accelerate uniformly. This idea is suggested by the fact that we typically refer to the system with $F=0$ as a "metal", so an external force should drag the electrons. In other words, I find the Bloch oscillation very counter-intuitive; moreover it seems like this model does not explain the expected conduction properties of electrons.

Can anyone tell me how the intuitive classical behavior of an acceleration resulting from the external force is recovered in this formalism?

Qmechanic
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Matteo
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1 Answers1

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Imagine your potential is an infinite series of square wells. Now apply your $-Fx$. You could increase the $F$ until the the top right corner of your well is near the height of or is below the bottom left corner. Obviously, a wave packet is going to "slide down the bumpy hill" and accelerate.

This has the problem of severely breaking the translation periodicity, and Bloch's theorem would no longer apply, and you wouldn't have what we consider to be metals. So, this is not quite what you were asking for.

Also, I suspect you desire billiard ball physics when this is wave mechanics.

Dr. Nate
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