Cyclic- and helical-symmetry-adapted phonon formalism within density functional perturbation theory

TL;DR

The paper addresses efficient first-principles phonon calculations in nanostructures with cyclic and/or helical symmetry by developing a symmetry-adapted DFPT framework. It constructs a symmetry-adapted dynamical matrix and acoustic sum rules within a real-space, high-order finite-difference formulation, and stabilizes the Sternheimer equations for robust phonon responses. Implemented in Cyclix-DFT/M-SPARC, the method is validated against plane-wave results for carbon nanotubes, delivering accurate phonon spectra and enabling extraction of elastic moduli and phonon-scaling laws for ring modes and radial breathing modes. The approach promises substantial computational savings for bending and torsion problems and lays a foundation for studying electron–phonon interactions in cyclic/helical nanostructures.

Abstract

We present a first-principles framework for the calculation of phonons in nanostructures with cyclic and/or helical symmetry. In particular, we derive a cyclic- and helical-symmetry-adapted representation of the dynamical matrix at arbitrary phonon wavevectors within a variationally formulated, symmetry-adapted density functional perturbation theory framework. In so doing, we also derive the acoustic sum rules for cylindrical geometries, which include a rigid-body rotational mode in addition to the three translational modes. We implement the cyclic- and helical-symmetry-adapted formalism within a high-order finite-difference discretization. Using carbon nanotubes as representative systems, we demonstrate the accuracy of the framework through excellent agreement with periodic plane-wave results. We further apply the framework to compute the Young's and shear moduli of carbon nanotubes, as well as the scaling laws governing the dependence of ring and radial breathing mode phonon frequencies on nanotube diameter. The elastic moduli are found to be in agreement with previous density functional theory and experimental results, while the phonon scaling laws show qualitative agreement with previous atomistic simulations.

Figures (7)

• Figure 1: Illustration of nanostructures exhibiting (a) cyclic, (b) helical, and (c) combined cyclic and helical symmetries. Fundamental atoms are indicated by plus markers, while their cyclic and helical images are shown in magenta and purple, respectively.
• Figure 2: Illustration of (a) zigzag, (b) armchair, and (c) chiral single-walled carbon nanotubes. The two fundamental carbon atoms are shown in black and brown, with their cyclic and helical images depicted in violet and green, respectively.
• Figure 3: Comparison of the phonon frequencies computed using the cyclic- and helical-symmetry-adapted framework and the planewave code ABINIT.
• Figure 4: Cyclic- and helical-symmetry-adapted phonon band structure for the (16,0) carbon nanotube at $\nu_\mathbf{q} = 0$.
• Figure 5: Atomic displacements corresponding to the phonon modes of interest that are common to the carbon nanotubes studied, illustrated using the $(16,0)$ nanotube as a representative example. Arrows indicate the directions of atomic perturbations, and their lengths denote the corresponding magnitudes. The torsional mode corresponds to a zero-frequency rigid-body motion. The four ring modes are characterized by the $\nu$-fold symmetry inherent in their vibrational patterns. RBM denotes the radial breathing mode.