Beyond the semiclassical description of Bloch oscillations

Electrons moving in a tilted periodic potential perform a periodic motion, known as Bloch oscillation. Within a semiclassical description, the crystal momentum increases linearly with time until it reaches the boundary of the first Brillouin zone in reciprocal space. Then, it reenters the first Brillouin zone by the opposite edge. This periodic motion in reciprocal space is accompanied by an oscillation in real space. The angular frequency of the oscillations and their amplitude can be calculated within the semiclassical framework. Nevertheless, the semiclassical approach cannot explain the rich phenomenology of the Bloch oscillations, such as the breathing of the electronic wave packet. We present a simple description of the Bloch oscillations of tightly bound electrons in biased lattices at a basic level and calculate exactly the wavefunction as a function of time.