Decay and double-decay properties of edge bands of phosphorene ribbons

Phosphorene (a mono-layer of black phosphorus) recently spurred much attention due to its potential for application. We notice there are two types of zigzag edge and also two types of armchair edge for phosphorene lattice. We study the winding number of various types of edge of phosphorene ribbons and conclude that, besides on the typical zigzag edge, the flat zero-energy edge band can be found in the ribbon of another nontypical armchair edge. The localization of these edge bands is investigated analytically. We find every single edge state of the nontypical armchair edge decays to the bulk at two different decay rates.