Electronic properties of two-dimensional rectangular graphyne based on phenyl-like building blocks

A rectangular graphyne sheet is composed of units similar to phenyl rings that are linked by acetylenic chains, as in hexagonal $\gamma$-graphyne. This system is organized over a rectangular lattice similar to that of the recently synthesized biphenylene network. We investigate the stability of this sheet from different perspectives and study its electronic structure. Rectangular graphyne is a semiconducting system in its pristine form and features a pair of highly localized states. These characteristics are correlated with the structural anisotropy of the system, since its frontier states behave like quasi--1D states embedded in the 2D lattice. We further consider modified systems in which longer acetylenic links are introduced. We discuss how a strategic choice of the position of these longer bridges can lead to specific changes of the electronic structure of the rectangular graphyne sheet.