Ab initio calculation of ideal strength and phonon instability of graphene under tension

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 $0\mathrm{K}$. 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 $E=1050\mathrm{GPa}$ and Poisson’s ratio $\nu =0.186$ 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 (${\epsilon }_{xx}=0.194$, ${\sigma }_{xx}=110\mathrm{GPa}$, ${\epsilon }_{yy}=-0.016$) and (${\epsilon }_{yy}=0.266$, ${\sigma }_{yy}=121\mathrm{GPa}$, ${\epsilon }_{xx}=-0.027$), 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.

Figure 1

(Color online) (a) Four-atom unit cell for stress-strain calculations. (b) Two-atom primitive cell for phonon calculations. (c) Primitive cell of the reciprocal lattice and the first Brillouin zone.

Figure 2

(Color online) DFPT calculated (a) phonon dispersion and (b) density of states of graphene sheet at zero stress. Note that the free bending wave ${\omega }^2=\kappa k^4$ near $\Gamma$ in (a) introduces a finite density of states at zero frequency in (b).

Figure 3

(Color online) The curves connected to the origin are the equivalent tensile stress ($d_0=3.34\mathrm{\AA })$ versus uniaxial strain in the $x$ and $y$ directions, respectively. The lines with initially negative slopes (scale labels to the right) are the finite-deformation Poisson’s ratios as functions of the uniaxial strain in the $x$ and $y$ directions, respectively. The red circles and triangles indicate the condition where peak stress could be attained for zigzag and armchair nanotubes, respectively.

Figure 4

(Color online) (a) Phonon dispersion and (b) density of states of graphene at ${\epsilon }_{xx}=0.18$. There is no soft mode yet. (c) Phonon dispersion at ${\epsilon }_{xx}=0.194$, ${\sigma }_{xx}=110\mathrm{GPa}$. (d) Blow-up of the unstable branch along $\mathbf{k}=q({\mathbf{b}}_1+{\mathbf{b}}_2)=k{\mathbf{e}}_x$. (e) Scan of the entire Brillouin zone at ${\epsilon }_{xx}=0.194$ to make sure that (d) is the first phonon instability. $⊛$ indicates imaginary frequency and $*$ indicates real frequency. (f) The unstable eigenvector corresponding to the soft mode at ${\epsilon }_{xx}=0.194$.

Figure 5

(Color online) (a) Phonon dispersion and (b) density of states of graphene at ${\epsilon }_{yy}=0.24$. There is no soft mode yet. (c) Phonon dispersion at ${\epsilon }_{yy}=0.266$, ${\sigma }_{yy}=121\mathrm{GPa}$. (d) Blow-up of the unstable branch along $\mathbf{k}=q({\mathbf{b}}_1-{\mathbf{b}}_2)=k{\mathbf{e}}_y$. (e) Scan of the entire Brillouin zone at ${\epsilon }_{yy}=0.266$ to make sure (d) is the first phonon instability. $⊛$ indicates imaginary frequency and $*$ indicates real frequency. (f) The unstable eigenvector corresponding to the soft mode at ${\epsilon }_{yy}=0.266$.