Anharmonic renormalization of the dispersion of flexural modes in graphene using atomistic calculations

We investigate through an atomistic method the effects of anharmonicity on the dispersion of flexural modes of graphene. Using a calculation based on ensemble averages of correlations among displacements and forces, we calculate the temperature-dependent frequencies for a semiempirical potential for graphene. We find that the dispersion relation of the flexural modes of graphene is renormalized by anharmonic coupling to other modes. Our calculations confirm that the anharmonic continuum results of Mariani and von Oppen [Phys. Rev. Lett. 100, 076801 (2008)] hold in detail for small wave number and at low temperatures. We examine the deviation from the continuum result outside of that range.

Figure 1

Dispersion of the flexural modes, calculated using the moments method at two temperatures $(T=0\text{K}$ and $1200\text{K})$. The points are the results of our calculations, and the lines are simple power-law fits. These results are for the modified Tersoff potential. The dispersion curves for several intermediate temperatures have been calculated and are smooth intermediates to these curves, so they have been omitted from this plot for the sake of presentation.

Figure 2

Plot of $({\omega }^4/k^6)$ vs $k^2$ at various temperatures. Points are the results of the present calculation. The lines are simple, one-parameter fits to the values for the lowest five $k$ vectors. The single parameter for each temperature is the vertical offset, the slope for all being determined by the $T=0$ K calculations. Error bars are statistical, as discussed in the text.

Figure 3

Plot of $({\omega }^4-{\omega }_0^4)/k^6$ vs $k^2$ at low temperatures. This is a replotting of the calculations from Fig. 2 to emphasize the temperature dependence. Points are the results of the present calculations, and the horizontal lines are simple fits to the values of the lowest five $k$ vectors. Error bars are statistical, as discussed in the text.

Figure 4

Plot of $k_c^2$ vs $T$, showing the linear behavior at low temperatures predicted by Mariani and von Oppen [10]. The points are our calculations, and the line is a one-parameter linear fit to the lower temperature calculations.

Figure 5

Plot of $\text{ln}\left(k_c^2\right)$ vs $\text{ln}\left(T\right)$, along with a linear fit to the lower temperature calculations. The continuum theory predicts a slope of 1 [10].