3

If we apply an electric field to a 1D lattice so that the quasimomentum increases as $$\langle\dot q\rangle=eE$$

what happens when we reach the limit $\frac \pi a$? Does the quasimomentum cycle round to $-\frac \pi a$ or does it jump to the higher Brillouin zone?

Cameron
  • 1,179

1 Answers1

4

It cannot jump to a higher Brillouin zone, since the Brillouin zones are just equivalent descriptions of the toroidal momentum space of a system with a discrete translation symmetry. In a way the choice of the Brillouin zone is a choice of coordinates. So if we keep using the same choice of coordinates then the momentum wraps around from $\pi/a$ to $-\pi/a$.

This effect is known as Bloch oscillation: The velocity of the electron (and therefore the position) oscillates. In other words in a perfect metal with infinite extent a constant voltage causes an alternating current.

Note, however, that a boundary of the metal scatters the electrons and thereby inhibits the effects. Further, electron-electron interactions and phonon-electron interactions, impurites, and, if these are eradicated, Frenkel pairs mean that it requires very low temperatures and high electric fields to reach the domain where Bloch oscillations actually occur, as soon as there are strong enough scattering effects, the electron momentum will not wrap around but scattering will limit the acquired momentum.

Sebastian Riese
  • 11,682
  • So would we have faster oscillations of the wavefunction when we have say $k=3\pi$ since we have $e^{ikr}$ in the wavefunction? – Cameron Jun 04 '18 at 19:23
  • I do not understand this question. What is oscillating is $k$ itself (since it moves through the Brillouin zone which is periodic). – Sebastian Riese Jun 04 '18 at 19:35
  • In my question I say that the k is increasing due to an electric field. So when $k=\pi/a$ does it then jump back to $-\pi/a$? Is there some energy lost somehow? If we apply an electric field constantly surely the momentum should keep increasing, which to me feels like energy is being lost somewhere – Cameron Jun 04 '18 at 19:38
  • Could you give a reference about Fraenkel pairs? I can't seem to find anything on the web about them. – untreated_paramediensis_karnik Jun 04 '18 at 20:00
  • I might messed up the name, it's Frenkel pair – I'll edit to fix it, sorry about that. The bottom line is, that even a pure solid has impurities at non-zero temperatur, since some atoms jump from their place in the lattice (leaving a vacancy) to a location between lattice points (giving a interstitial atom). – Sebastian Riese Jun 04 '18 at 23:51
  • 1
    @James No energy is lost since the electron does not escape to infinity, but oscillates, so it does not extract an arbitrary amount of energy from the electric field. The momentum, on the other hand, is no longer conserved in the lattice: Only the crystal momentum (which is momentum up to inverse lattice vectors) is conserved. – Sebastian Riese Jun 04 '18 at 23:57