Boundary conditions for phosphorene nanoribbons in the continuum approach

Abstract

We investigate the energy spectrum of single-layer black phosphorene nanoribbons (BPNs) by means of a low-energy expansion of a recently proposed tight-binding model that describes electron and hole bands close to the Fermi energy level. Using the continuum approach, we propose boundary conditions based on sublattice symmetries for BPNs with zigzag and armchair edges and show that our results for the energy spectra exhibit good agreement with those obtained by using the five-parameter tight-binding model. We also explore the behavior of the energy gap versus the nanoribbon width $W$. Our findings demonstrate that band gaps of armchair BPNs scale as $1/{W}^{2}$, while zigzag BPNs exhibit a $1/W$ tendency. We analyze the different possible combinations of the zigzag edges that result in twofold degenerate and nondegenerate edge states. Furthermore, we obtain expressions for the wave functions and discuss the limit of validity of such an analytical model.