Scaling laws for the band gap and optical response of phosphorene nanoribbons

We report the electronic structure and optical absorption spectra of monolayer black phosphorus (phosphorene) nanoribbons (PNRs) via first-principles simulations. The band gap of PNRs is strongly enhanced by quantum confinement. However, differently orientated PNRs exhibit distinct scaling laws for the band gap vs the ribbon width $w$. The band gaps of armchair PNRs scale as $1/{w}^{2}$, while zigzag PNRs exhibit a $1/w$ behavior. These distinct scaling laws reflect a significant implication of the band dispersion of phosphorene: electrons and holes behave as nonrelativistic particles along the zigzag direction but resemble relativistic particles along the armchair direction. This unexpected merging of nonrelativistic and relativistic properties in a single material may produce novel electrical and magnetotransport properties of few-layer black phosphorus and its ribbon structures. Finally, the respective PNRs host electrons and holes with markedly different effective masses and optical absorption spectra, which are suitable for a wide range of applications.