Wave Packet Dynamics in a Biased Finite-Length Superlattice

In a superlattice containing a finite number of periods, the allowed values of the Bloch wave number form a discrete set, and the dynamics of an electron through k-space under the influence of an external force is necessarily that of a superposition wave packet composed of multiple Bloch waves. The present paper investigates this dynamics for a particularly simple class of “kcompact” wave packets, in which the spread over different k-values is minimized, and which propagate with an essentially rigid shape through kspace. 1) Introduction This paper concerns itself with the question of how the electron dynamics in a biased superlattice changes relative to that in an infinitely long superlattice, as the number of cells is reduced. In the extreme limit of, say, one or two cells, it becomes of course meaningless to speak of a superlattice at all. But it is far less clear what happens for an only slightly larger number of cells, say, 10 cells. At what point does it become meaningful to speak of a superlattice? The central issue is to what extent the dynamics can be described in terms of a dynamics in k-space. A familiar textbook theorem in solid-state physics states that, under the influence of an applied force F, the (reduced) Bloch wave number k of an electron in a one-dimensional periodic potential changes with time according to “Newton’s Law in k-Space,”