In the free electron approximation, a Bloch state $|k\rangle$ is the linear superposition of free plane wave states $\sum_G C_G(k) |k+G\rangle$, where $G$ are the conjugate lattice. Since the coefficients $C_G$ varies with $k$, under an electric field the electron will not merely shift inside the same band, but will scatter to other bands. Is this effect significant in practice? Also, does the total scattering loss during one complete Bloch oscillation tend to zero as the applied electric field tends to zero?
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2 Answers
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The premise is not true. Under a uniform electric field in a perfect crystal, the electron moves within a single band, with k changing at a uniform rate. The coefficients $C_G$ change in accordance with the composition of the corresponding Bloch state. You can find a detailed proof in Kittel, "Quantum Theory of Solids".
See Kittel, "Quantum Theory of Solids", p190-193. One way to look at it is, transitions generally don't happen because of the adiabatic theorem. The conditions of the adiabatic theorem are supposedly "difficult to violate over an extended volume of crystal" in this context. (I have not tried to understand exactly what is required for a violation.)
There's another way to formulate the problem, in which you diagonalize a different Hamiltonian:
$H_{\mathbf{F}}=H_0-ie^{i\mathbf{k}\cdot\mathbf{x}}\mathbf{F}\cdot\nabla_\mathbf{k}e^{-i\mathbf{k}\cdot\mathbf{x}}$
Where F is the electric field and $H_0$ is the unperturbed Hamiltonian. It's not obvious from looking, but this Hamiltonian is susceptible to Bloch's theorem--it has bands like usual. The electric field exactly carries electrons within a single "band" of this modified Hamiltonian. I suppose you can use perturbation theory on the second term to find the scattering rates with respect to the ordinary bands of the crystal.
In practice, my experience involves semiconductor transport at not-especially-high fields. Scattering into another band by electric field alone (as opposed to defects, phonons, etc.) is not at all important in that context; I had never heard of it until now.
answered Nov 21, 2011 at 0:48
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$\begingroup$ The effect of an external field, during a time delta_t, is to change every free plane wave state (not Bloch state) k into k+delta_k, but doesn't change the pre-multiplying coefficients. Unless those coefficients are the same, you surely expect scattering into other bands, don't you? $\endgroup$– felixCommented Nov 21, 2011 at 1:04
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$\begingroup$ To clarify, I'm aware of the standard picture of electrons moving in the same band, but I'm saying that it has to be an approximation. $\endgroup$– felixCommented Nov 21, 2011 at 4:21
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$\begingroup$ I can see you have edited. Thanks for your answer! A professor I asked casually said that such scattering was a second order effect but didn't elucidate. I've voted you up but I'd wait until this question becomes clear to me before I'll accept the answer. $\endgroup$– felixCommented Nov 22, 2011 at 16:26
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According to a professor that I asked today, such an effect exists, and is called "Zener Tunneling".