# Optical Matrix Elements of Single-Walled Carbon Nanotubes for Longitudinal Polarization of Light

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This Demonstration shows the dependence of the value of the optical matrix element for light polarized parallel to the nanotube axis over the Brillouin zone (BZ) of a single-walled carbon nanotube (SWNT). A general analytical expression for for any SWNT chirality and unit cell unrolled in 2D has been obtained in the literature (see details) from the tight-binding electronic structure of graphene. The profile of is plotted over the 2D graphene BZ as a function of the graphene wavevectors . The optical selection rules obey the azimuthal and axial quantum number conservation, hence only vertical optical transitions between electronic bands with the same quantum number and at the same are allowed. The profile of over the nanotube 1D BZ is obtained by applying the zone-folding method to the 2D analytical function and evaluating this expression along the allowed SWNT cutting lines, which are line sections of the 2D BZ and which also provide the SWNT electronic bands. The 1D plots so obtained are then superimposed to show how the matrix element changes value and sign over each cutting line of index . The plots shown in this Demonstration show that the properties of are closely related to the SWNT geometry and determine the profile of the optical absorption spectrum for a given chirality.

Contributed by: Jessica Alfonsi (University of Padova, Italy) (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The calculations in this Demonstration have been reproduced from the equations in the following reference:

J. Jiang, R. Saito, A. Grueneis, G. Dresselhaus, and M. S. Dresselhaus, "Optical Absorption Matrix Elements in Single-Wall Carbon Nanotubes," *Carbon* 42(15), 2004 pp. 3169–3176.

Other related references:

A. Zarifi and T. G. Pedersen, "Analytic Approach to the Linear Susceptibility of Zigzag Carbon Nanotubes," *Physical Review B* 74(15), 155434, 2006.

E. Malic, M. Hirtschulz, F. Milde, A. Knorr, and S. Reich, "Analytical Approach to Optical Absorption in Carbon Nanotubes," *Physical Review B* 74(19), 195431, 2006.

J. Alfonsi and M. Meneghetti, "Small Crystal Approach for the Electronic Properties of Double-Wall Carbon Nanotubes," *New Journal of Physics* 11, 043002, 2009.

J. Alfonsi, "Small Crystal Models for the Electronic Properties of Carbon Nanotubes", Ph.D. thesis, University of Padova, 2009, Chapters 2-4 and references therein.