Temperature dependent phonon frequency shift and structural stability of free-standing graphene: a spectral energy density analysis

Abstract

A spectral energy density based formalism is implemented to probe the temperature dependent frequency shift, linewidth, structural stability and coupling of normal modes of vibrations of free-standing graphene using a combination of lattice dynamics and molecular dynamics (MD). The in-plane lattice parameter shows a thermal contraction upto 1300 K and it expands thereafter. Frequency of the bending mode (ZA) becomes imaginary in the quasi-harmonic dispersion at higher temperatures, suggestive of a structural instability. However, the frequency of the ZA mode becomes real in the dispersion obtained from MD. Dynamical stability to the structure is restored by strong anharmonic coupling of phonon modes which is automatically incorporated in the MD simulations, whereas it is ignored in the quasi-harmonic dispersion. The mode resolved phonon spectra at Γ point show a blue-shift of degenerate longitudinal and transverse (LO/TO) optic modes. The blue-shift observed in canonical (NVT) and isobaric–isothermal (NPT) ensembles are more prominent than the shift predicted by quasi-harmonic approximation (QHA) due to the additional contribution from phonon–phonon coupling. The out-of-plane optic (ZO) mode frequencies are red-shifted in the QHA due to membrane-effect, whereas MD simulations show that the strong phonon–phonon coupling dominates the membrane effect leading to a blue-shift. The linewidth of LO/TO and ZO modes increases non-monotonically with temperature. The anharmonic coupling of normal modes at high symmetry points in the Brillouin zone is also discussed.