Gap opening in two-dimensional periodic systems

TL;DR

This paper introduces a new method for controlling spectral gaps in two-dimensional periodic systems using differential operators and narrow potential walls, allowing precise gap opening and width control.

Contribution

The authors develop a novel approach to open and control spectral gaps in 2D periodic systems, with gaps occurring at internal Brillouin zone points, expanding the understanding of spectral manipulation.

Findings

Gaps can be opened around specific dispersion curve points.

Gap widths can be controlled via the perturbation parameter.

Edges of the gaps are attained at internal Brillouin zone points.

Abstract

We present a new method of gap control in two-dimensional periodic systems with the perturbation consisting of a second-order differential operator and a family of narrow potential `walls' separating the period cells in on direction. We show that under appropriate assumptions one can open gaps around points determined by dispersion curves of the associated `waveguide' system, in general any finite number of them, and to control their widths in terms of the perturbation parameter. Moreover, the distinctive feature of those gaps is that their edge values are attained by the corresponding band functions at internal points of the Brillouin zone.