Nanoribbons in external electric fields

TL;DR

This paper analyzes how external electric fields affect the spectral properties of nanoribbons modeled by Schr"odinger operators, revealing the conditions under which flat bands persist or transform into spectral bands.

Contribution

It provides asymptotic descriptions of spectral band changes in nanoribbons under weak electric potentials, highlighting when flat bands remain or evolve into continuous spectrum bands.

Findings

Spectral bands are characterized for zero electric potential.

Weak electric fields cause predictable asymptotic shifts in spectral bands.

Conditions are identified for the persistence of flat bands under perturbation.

Abstract

We consider the Schr\"odinger operator on nanoribbons (tight-binding models) in an external electric potentials $V$. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the Schr\"odinger operator consists of two spectral bands and the flat band (i.e., the eigenvalue with infinite multiplicity) between them. If we switch on an weak electric potential $V\to 0$, then we determine the asymptotics of the spectral bands for small fields. In particular, we describe all potentials when the unperturbed eigenvalue remains the flat band and when one becomes the small band of the continuous spectrum.