Symmetry-Adapted Phonon Analysis of Nanotubes

TL;DR

This paper introduces a symmetry-adapted method using the Objective Structures framework to efficiently analyze phonons in nanotubes, providing both computational advantages and clearer physical interpretation.

Contribution

The paper develops a general approach for phonon analysis in symmetric nanostructures using block-circulant Hessians and Fourier transforms, applied specifically to carbon nanotubes.

Findings

Efficient phonon spectra computation for nanotubes.

Comparison of spectra for different nanotube chiralities.

Successful application to density of states calculations.

Abstract

The characteristics of phonons, i.e. linearized normal modes of vibration, provide important insights into many aspects of crystals, e.g. stability and thermodynamics. In this paper, we use the Objective Structures framework to make concrete analogies between crystalline phonons and normal modes of vibration in non-crystalline but highly symmetric nanostructures. Our strategy is to use an intermediate linear transformation from real-space to an intermediate space in which the Hessian matrix of second derivatives is block-circulant. The block-circulant nature of the Hessian enables us to then follow the procedure to obtain phonons in crystals: namely, we use the Discrete Fourier Transform from this intermediate space to obtain a block-diagonal matrix that is readily diagonalizable. We formulate this for general Objective Structures and then apply it to study carbon nanotubes of various chiralities that are subjected to axial elongation and torsional deformation. We compare the phonon spectra computed in the Objective Framework with spectra computed for armchair and zigzag nanotubes. We also demonstrate the approach by computing the Density of States. In addition to the computational efficiency afforded by Objective Structures in providing the transformations to almost-diagonalize the Hessian, the framework provides an important conceptual simplification to interpret the phonon curves.