# Trapped modes in zigzag graphene nanoribbons

## TL;DR

This paper investigates the existence of trapped electronic modes in zigzag graphene nanoribbons using a Dirac model, identifying conditions for their occurrence near threshold energies and their dependence on potential size.

## Contribution

It introduces a mathematical framework based on the continuous Dirac model and augmented scattering matrix to derive conditions for trapped modes in graphene nanoribbons.

## Key findings

- Trapped modes can occur near threshold energies slightly below them.
- The multiplicity of trapped modes does not exceed one.
- Small potentials do not produce trapped modes outside the threshold.

## Abstract

We study a scattering on an ultra-low potential in zigzag graphene nanoribbon. Using mathematical framework based on the continuous Dirac model and augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies where the continuous spectrum changes its multiplicity and show that the trapped modes may appear for energies slightly less than a threshold and its multiplicity does not exceeds one. We prove that trapped modes do not appear outside the threshold, provided the potential is sufficiently small.