Zigzag nanoribbons in external electric and magnetic fields

TL;DR

This paper studies the spectral properties of zigzag nanoribbons under external magnetic and electric fields, revealing how these fields influence flat and non-flat bands and solving inverse spectral problems for small potentials.

Contribution

It provides a detailed analysis of how magnetic and electric fields affect the spectrum of zigzag nanoribbons, including asymptotics and conditions for flat band preservation or splitting.

Findings

Magnetic field alters the continuous spectrum but not the flat band.

Weak electric potential can split the flat band into a small spectral band.

Conditions are identified under which the flat band remains unchanged or splits.

Abstract

We consider the Schr\"odinger operators on zigzag nanoribbons (quasi-1D tight-binding models) in external magnetic fields and an electric potential $V$. The magnetic field is perpendicular to the plane of the ribbon and the electric field is perpendicular to the axis of the nanoribbon and the magnetic field. If the magnetic and electric fields are absent, then the spectrum of the Schr\"odinger (Laplace) operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on the magnetic field, then the spectrum of the magnetic Schr\"odinger operator consists of some non-flat bands and one flat band between them. Thus the magnetic field changes the continuous spectrum but does not the flat band. If we switch on a weak electric potential $V\to 0$, then there are two cases: (1) the flat band splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed flat band remains the flat band. We describe all potentials when the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Moreover, we solve inverse spectral problems for small potentials.