Band structure engineering : A simple way to optimize key features of one-dimensional energy bands

We determine the one-dimensional periodic potential required to produce an energy band structure in which the effective mass m at the bottom of the first miniband is minimized, and the peak group velocity within the miniband is maximized, subject to the constraint of maintaining a fixed specified energy gap 1 between the first and second bands. This problem is of considerable interest for the design of semiconductor superlattices and optical lattices that exhibit high carrier mobility combined with low interband Zener tunneling. We show that the problem is solved by a class of periodic potentials, known as 1-gap potentials, which, remarkably, support only two distinct energy bands separated by a single energy gap. We use the unique properties of 1-gap potentials to derive semi-analytical formulae that can be used to predict, ab initio, the potential profile required to generate any physically possible value of m or 1. Our results provide a simple, but powerful, band structure engineering tool that should facilitate the design of superlattice structures and optical lattices with optimized transport properties.