Bandgap Extremization: Some Exact Results

TL;DR

This paper introduces a variational method to optimize bandgaps in one-dimensional systems under specific constraints, providing exact solutions for potential configurations that maximize the bandgap.

Contribution

It develops a novel variational approach for bandgap maximization with exact solutions for constrained potentials, applicable across dimensions.

Findings

Maximum bandgap achieved by potential mixtures at bounds.

Exact extremal potential configurations for fixed moments.

Potential application to photonic bandgap engineering.

Abstract

We present here a variational method for maximizing the bandgap in a one-dimensional system where the potential is subject to given constraints. Two specific examples are studied in detail. In the first, we show that if the potential is constrained to lie between two values, the largest bandgap is obtained by a mixture of the highest and lowest potential - an exact result valid in any dimension. The second example fixes the first and second moments of the potential and seeks to extremize the bandgap. An exact result is obtained. Finally, we indicate how our techniques may be applied to photonic bandgaps.