 Solution

For an electron wavepacket in one isolated band under a uniform electric field,Because $ϵ(k+2\pi /a)=ϵ(k)$, acceleration through successive Brillouin zones makes the velocity periodic rather than increasing indefinitely. For $ϵ={ϵ}_0-2J\cos (ka)$,for a positive field along the chosen axis. Integration givesThe excursion amplitude of a Bloch-oscillating wavepacket is $2|J|/(e|E|)$ about its mean and peak-to-peak extent $4|J|/(e|E|)$. The turning of velocity corresponds semiclassically to Bragg reflection near a zone boundary.

Coherent Bloch oscillations require scattering/dephasing times long compared with $T_B$ and a field weak enough to avoid substantial interband tunnelling. Ordinary metals usually scatter too rapidly; semiconductor superlattices and cold atoms in periodic optical potentials provide cleaner realizations. A completely filled band has canceling velocities, so its net current does not oscillate simply because individual wavepackets can.^[ref]