Boundary conditions for phosphorene nanoribbons in the continuum approach

TL;DR

This paper develops a continuum model with boundary conditions for phosphorene nanoribbons, accurately predicting their energy spectra and band gap scaling, aligning well with tight-binding results.

Contribution

It introduces boundary conditions based on sublattice symmetries for phosphorene nanoribbons, enabling analytical calculation of energy spectra and gap behavior.

Findings

Armchair BPN band gap scales as 1/W^2

Zigzag BPN band gap scales as 1/W

Good agreement with tight-binding model

Abstract

We investigate the energy spectrum of single layer black phosphorene nanoribbons (BPN) 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 BPN 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 behaviour of the energy gap versus the nanoribbon width $W$. Our findings demonstrate that band gap of armchair BPNs scale as $1/W^2$, while zigzag BPNs exhibit a $1/W$ tendency. We analyse the different possible combinations of the zigzag edges that result two-fold degenerate and non-degenerate edge states. Furthermore, we obtain expressions for the wave functions and discuss the limit of validity of such analytical model.