Zigzag and armchair nanotubes in external fields

TL;DR

This paper analyzes the spectral properties of zigzag and armchair nanotubes under external magnetic and electric fields, revealing band structures, eigenvalues, and asymptotic behaviors for different potential strengths.

Contribution

It provides a detailed spectral analysis of nanotubes in external fields, including spectral band descriptions and asymptotics, using a reduction to Jacobi operators.

Findings

Spectral bands and eigenvalues are fully described.

Asymptotics of spectral bands are determined for small and large potentials.

Spectral behavior as a function of magnetic field strength is characterized.

Abstract

We consider the Schr\"odinger operator on the zigzag and armchair nanotubes (tight-binding models) in a uniform magnetic field $\mB$ and in an external periodic electric potential. The magnetic and electric fields are parallel to the axis of the nanotube. We show that this operator is unitarily equivalent to the finite orthogonal sum of Jacobi operators. We describe all spectral bands and all eigenvalues (with infinite multiplicity, i.e., flat bands). Moreover, we determine the asymptotics of the spectral bands both for small and large potentials. We describe the spectrum as a function of $|\mB|$. For example, if $|\mB|\to {163}({\pi2}-{\pi kN}+\pi s)\tan {\pi2N}, k=1,2,..,N, s\in \Z$, then some spectral band for zigzag nanotube shrinks into a flat band and the corresponding asymptotics are determined.