2.1.3.2 Constant External Electric Field

For a uniform time-independent electric field the equation of motion in the quasi-momentum space (2.14) has the general solution

$$\vec{k}(t)=\vec{k}(0)+\frac{q\vec{E}}{\hbar}t.$$ (2.19)

This means that for a given time all wave vectors are shifted by the same vector. If an electron could travel in the quasi-momentum space between two collisions for distances longer than the size of the Brillouin zone, the static electric field would cause a time dependent current due to the periodicity of the average electron velocity in the quasi-momentum space. The fact that a static electric field can cause a time dependent electrical current cannot be explained from the classical point of view and represents the quantum mechanical property of the semiclassical model.

S. Smirnov: