On the optical properties of carbon nanotubes--Part I. A general formula for the dynamical optical conductivity

TL;DR

This paper derives a general formula for the dynamical optical conductivity of single-walled carbon nanotubes, capturing their sharp absorption peaks by modeling their quasi-1D structure and electron interactions.

Contribution

It introduces a 1D effective many-body Hamiltonian for SWNTs and derives an asymptotic expansion for optical conductivity using linear response theory.

Findings

Sharp peaks in optical absorption spectrum explained

Asymptotic expansion relates conductivity to eigenvalues

Model captures quasi-1D electron interactions

Abstract

This paper is the first one of a series of two articles in which we revisit the optical properties of single-walled carbon nanotubes (SWNT). Produced by rolling up a graphene sheet, SWNT owe their intriguing properties to their cylindrical quasi-one-dimensional (quasi-1D) structure (the ratio length/radius is experimentally of order of 10^3). We model SWNT by circular cylinders of small diameters on the surface of which the conduction electron gas is confined by the electric field generated by the fixed carbon ions. The pair-interaction potential considered is the 3D Coulomb potential restricted to the cylinder. To reflect the quasi-1D structure, we introduce a 1D effective many-body Hamiltonian which is the starting-point of our analysis. To investigate the optical properties, we consider a perturbation by a uniform time-dependent electric field modeling an incident light beam along the longitudinal direction. By using Kubo's method, we derive within the linear response theory an asymptotic expansion in the low-temperature regime for the dynamical optical conductivity at fixed density of particles. The leading term only involves the eigenvalues and associated eigenfunctions of the (unperturbed) 1D effective many-body Hamiltonian, and allows us to account for the sharp peaks observed in the optical absorption spectrum of SWNT.