<i>Ab initio</i>calculation of ideal strength and phonon instability of graphene under tension

TLDR

Graphene‑based sp²‑carbon nanostructures such as carbon nanotubes and nanofibers can fail near their ideal strengths because of their extremely small dimensions. The study calculates graphene phonon spectra under uniaxial tension to identify the strain at which phonon instability first occurs, thereby determining the defect‑free crystal’s strength at 0 K. Density functional perturbation theory is used to compute phonon spectra while applying uniaxial tensile strain along the x (nearest‑neighbor) and y (second nearest‑neighbor) directions, corresponding to zigzag and armchair nanotube deformations. The calculated Young’s modulus (1050 GPa) and Poisson’s ratio (0.186) agree with prior work, and phonon instabilities appear at εxx = 0.194 (σxx = 110 GPa) and εyy = 0.266 (σyy = 121 GPa), indicating longitudinal elastic waves that imply brittle cleavage fracture at low temperatures, while a phonon band gap emerges under high strain, offering a spectroscopic signature.

Abstract

Graphene-based $s{p}^{2}$-carbon nanostructures such as carbon nanotubes and nanofibers can fail near their ideal strengths due to their exceedingly small dimensions. We have calculated the phonon spectra of graphene as a function of uniaxial tension by density functional perturbation theory to assess the first occurrence of phonon instability on the strain path, which controls the strength of a defect-free crystal at $0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Uniaxial tensile strain is applied in the $x$ (nearest-neighbor) and $y$ (second nearest-neighbor) directions, related to tensile deformation of zigzag and armchair nanotubes, respectively. The Young's modulus $E=1050\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and Poisson's ratio $\ensuremath{\nu}=0.186$ from our small-strain results are in good agreement with previous calculations. We find that in both $x$ and $y$ uniaxial tensions, phonon instabilities occur near the center of the Brillouin zone, at (${\ensuremath{\epsilon}}_{xx}=0.194$, ${\ensuremath{\sigma}}_{xx}=110\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, ${\ensuremath{\epsilon}}_{yy}=\ensuremath{-}0.016$) and (${\ensuremath{\epsilon}}_{yy}=0.266$, ${\ensuremath{\sigma}}_{yy}=121\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, ${\ensuremath{\epsilon}}_{xx}=\ensuremath{-}0.027$), respectively. Both soft phonons are longitudinal elastic waves in the pulling direction, suggesting that brittle cleavage fracture may be an inherent behavior of graphene and carbon nanotubes at low temperatures. We also predict that a phonon band gap will appear in highly stretched graphene, which could be a useful spectroscopic signature for highly stressed carbon nanotubes.