Electronic properties of graphyne-$N$ monolayer and its multilayer: even-odd effect and topological nodal line semimetalic phases

Abstract: We study the electronic structure and topological properties of monolayer and ABC-stacked multilayer of graphyne-$N$, which are a family of planar carbon sheets consisting of $sp$ and $sp_2$-bonding. By using the density-functional theory and the effective continuum model, we find a striking even-odd effect in the dependence of the band structure on $N$ (the number of carbon-carbon triple bonds between neighboring benzene rings). Specifically, even-$N$ graphyne monolayer has doubly-degenerate conduction and valence bands near the Fermi energy, and in its ABC multilayer, the band inversion of the doubly-degenerate bands leads to a nodal-line semimetal phase with non-trivial $\mathbb{Z}_2$ monopole charge. In contrast, odd-$N$ monolayer has singly-degenerate bands in separate valleys, and its ABC multilayer can have only $\mathbb{Z}_2$-trivial nodal lines. ABC graphynes with larger $N$ tend to be trivial insulators because of smaller interlayer coupling, while the external pressure induces a topological phase transition from the trivial phase to the nodal line semimetal phase.