Carbon nanotubes in an inhomogeneous transverse magnetic field: exactly solvable model

Abstract

A class of exactly solvable models describing carbon nanotubes in the presence of an external inhomogeneous magnetic field is considered. The framework of the continuum approximation is employed, where the motion of the charge carriers is governed by the Dirac-Weyl equation. The explicit solution of a particular example is provided. It is shown that these models possess nontrivial integrals of motion that establish N = 2 nonlinear supersymmetry in case of metallic and maximally semiconducting nanotubes. Remarkable stability of energy levels with respect to small fluctuations of longitudinal momentum is demonstrated.

A qualitative study of a nanotube model using an iterative Taylor method

“…Therefore, functions depending on φ have to show periodicity properties in terms of this angle. It has been proven that these solutions should be either periodic or anti-periodic 8 . This periodicity properties have to be inherited by the functions ϕ ± (x).…”

“…In here, we shall not intend to study equations (7) with boundary conditions (8) in their full generality. Due to the form of the problem under our consideration, we shall restrict ourselves to the linear case.…”

“…The given solution shows two energy bands with positive energy with one gap between them. The first energy band is finite while the second one is unbounded 8 . The purpose of this section is to analyze some aspects on the behavior of the energy bands when k z = 0 using an iterative Taylor method to be described next.…”

“…On the nanotube, a magnetic field B acts with a vector potential A = A φ n φ + A z n z , where n φ and n z are the unit vectors tangent to the circumference and to the longitudinal direction. The component A φ was assumed to vanish 8 , while A z is a function of the angle φ alone, A z ≡ A z (φ). Along the present article, we are keeping the same assumptions, were A φ and A z (φ) are given data.…”

Dirac electron in graphene under supersymmetry generated magnetic fields

“…A class of exactly solvable metallic and maximally semiconducting nanotubes in the presence of an external inhomogeneous magnetic field is shown to possess nontrivial integrals of motion, which establishes the N = 2 nonlinear supersymmetry [36]. The nonlinear hidden supersymmetry generated by local supercharges is also shown to appear naturally in the finite-gap and time-reversal invariant configurations of twisted nanotubes [38].…”