The period of the Bloch oscillations. Consider an electron that is subjected to an electric field. The electric field exerts a force F=-qE on the electron. Assume that the electron is initially not in motion, i.e., k = 0. Upon application of the electric field, the k value of the electron increases from 0 to π/a. At this value of k, Bragg reflection occurs, and the electron assumes a k value of –π/a. Then, the electron is again accelerated to k = π/a. At this point, the electron again undergoes Bragg reflection, and the cycle starts from the beginning. The process described above is called the Bloch oscillation of the electron in an energy band of the solid-state crystal.

(a) Show that the period of the Bloch oscillation is given by τ =2πh qEa, where a is the periodicity of a one-dimensional atomic chain.

(b) Calculate the period of the Bloch oscillations for a = 4 Å and E = 1250 V.cm^-1. Compare the period of the Bloch oscillations with a typical inelastic scattering times. What conclusions do you draw from the comparison? Are the Bragg reflections important scattering events for the movement of electrons in a crystal? Typical inelastic scattering times are 10-¹¹s for low fields and 10^-13 s for high fields.