Zigzag and armchair nanotubes in external fields

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.