A Mathematical Walk into the Paradox of Bloch Oscillations

Abstract

We mathematically describe the apparently paradoxical phenomenon that the electric current in a semiconductor can flow because of collisions, and not despite them. A model of charge transport in a one-dimensional semiconductor crystal is considered, where each electron follows the periodic Hamiltonian trajectories, determined by the semiconductor band structure, and undergoes non-elastic collisions with a phonon bath. Starting from the detailed phase-space model, a closed system of ODEs is obtained for averaged quantities. Such a simplified model is nevertheless capable of describing transient Bloch oscillations, their damping and the consequent onset of a steady current flow, which is in good agreement with the available experimental data.

Keywords:

• Bloch oscillations
• semiconductors
• collisions
• transport
• diffusion

Acknowledgment

The author wishes to thank his team of students at the XI Modelling Week of Universidad Complutense de Madrid, Antonio Luna, Carlos Domingo, Jaime Oliver, Jared M. Field, Rahil Sachak-Patwa and Sergio Montoro, where the first nucleus of this work originated.