Formula for the Absorption Coefficient for Multi-Wall Nanotubes

TL;DR

This paper develops a formalism to calculate the absorption spectrum of coaxial nanotubes, focusing on plasmon excitations and their dependence on tube radii, providing insights into their optical properties.

Contribution

It introduces a self-consistent field theory for plasmon-induced absorption in multi-wall nanotubes, including numerical calculations of peak positions and mode splitting.

Findings

Peak positions depend on inner and outer tube radii.

Higher plasmon energies correspond to larger peaks and oscillator strengths.

Coupled modes are split by Coulomb interaction and vary with tube dimensions.

Abstract

We present a formalism for calculating the absorption coefficient of a pair of coaxial tubules. A spatially nonlocal, dynamical self-consistent field theory is obtained by calculating the electrostatic potential produced by the charge density fluctuations as well as the external electric field. There are peaks in the absorption spectrum arising from plasma excitations corresponding either to plasmon or particle-hole modes. In this paper, we numerically calculate the plasmon contribution to the absorption spectrum when an external electric field is applied. The number of peaks depends on the radius of the inner as well as outer tubule. The height of each peak is determined by the plasmon wavelength and energy. For a chosen wave number, the most energetic plasmon has the highest peak corresponding to the largest oscillator strength of the excited modes. Some of the low-frequency plasmon modes have such weak coupling to an external electric field that they are not seen on the same scale as the modes with larger energy of excitation. We plot the peak positions of the plasmon excitations for a pair of coaxial tubules. The coupled modes on the two tubules are split by the Coulomb interaction. The energies of the two highest plasmon branches increase with the radius of the outer tubule. On the contrary, the lowest modes decrease in energy as this radius is increased. No effects due to inter-tubule hopping are included in these calculations.