Zigzag and armchair nanotubes in external fields

Abstract

We consider the Schrödinger 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 $|\mB|\to {163}({\pi2}-{πkN}+πs)\tan {\pi2N}, k=1,2,..,N, s\in \Z$, then some spectral band for zigzag nanotube shrinks into a flat band and the corresponding asymptotics are determined.

29 pages, 6 figures

References in corpus (5)

• Spectral estimates for periodic Jacobi matrices
• Electron properties of carbon nanotubes in a periodic potential
• Effective masses for zigzag nanotubes in magnetic fields
• Inverse problem for the discrete 1D Schrödinger operator with small periodic potentials
• Magnetic Schrödinger operators on armchair nanotubes