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In Kittel's book chapter 8 there is a statement that says "The electron accelerates from k = 0 toward the zone boundary; when it reaches k = pi/a it reappears (as by an Umklapp process) at the zone boundary at the identical point - pi/a, " This statement is written under the definition of Bloch oscillations.

Having searched for other resources on the internet it seems that Bloch oscillations do not transfer momentum to the lattice, while Umklapp processes do. My question is, does the previous statement implicitly assume the presence of a phonon?

What decide if an electron once it reaches the BZ boundary will either experience Bloch oscillation or Umklapp process?

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  • $\begingroup$ Please state the book title, author, edition, chapter, page and possibly equation number(s). $\endgroup$
    – Tobias Fünke
    Commented Jul 8 at 21:00
  • $\begingroup$ "Introduction to solid state physics" Charles Kittel, chapter 8, page 217, 8th edition $\endgroup$
    – Gotaquestion
    Commented Jul 8 at 21:14

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An Umklapp process is any scattering event where the ingoing and outgoing momenta differ by a (nonzero) reciprocal lattice vector. This includes phonon-phonon, but also electron-phonon and electron-electron processes. Crystal momentum conservation exists because the Hamiltonian has a discrete translation symmetry: even in the limit that the lattice has infinite mass, you can still have Umklapp e-e scattering, so it doesn't need phonons. I think it's maybe not the best mental model to think of ph-ph Umklapp processes as "giving momentum to the lattice"—it really comes from the identification of the $k=0$ and $k=G$ points in $k$-space, and thus a rather different notion of momentum relative to the continuum case.

Bloch oscillation isn't a scattering process, so isn't quite correct to say that it's Umklapp, but the idea of folding back over the Brillouin zone is the same. Umklapp processes are when several excitations (e, ph, etc.) collide and one of their momenta is pushed over the edge of the BZ; Bloch oscillations are when a single electron is pushed there by an electric field. They're pretty hard to actually produce because the electron takes a long time to get all the way there, and in real materials it usually encounters an impurity first.

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  • $\begingroup$ Thanks, to make sure I understand it correctly, am I correct in saying that an Umklapp process will never occur unless two pseudo-particles interact? If that is the case, then the electric field will produce Bloch oscillations (assuming ideal lattice) when it reaches the BZ boundary unless it interacts with another pseudo-particle , is that right? $\endgroup$
    – Gotaquestion
    Commented Jul 9 at 17:52
  • $\begingroup$ Yes, Umklapp processes are scattering processes. Bloch oscillations strictly aren’t the fictitious “jumping back” in the BZ that happens when you hit the edge—they’re the position-space oscillation that happens when the momentum repeatedly wraps around. But you’ve got the idea. $\endgroup$
    – Rokas Veitas
    Commented Jul 10 at 5:53

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