Revisiting phonon-phonon scattering in single-layer graphene

Understanding the mechanisms of thermal conduction in graphene is a longstanding research topic due to its high thermal conductivity. Studies based on the Peierls-Boltzmann transport equation (PBTE) have revealed many unique phonon transport properties in graphene, but most previous works considered only three-phonon scatterings and relied on interatomic force constants (IFCs) extracted at 0 K. Recently, the importance of four-phonon scattering in graphene was revealed by Feng and Ruan [Phys. Rev. B 97, 045202 (2018)]. In this paper, we explore the temperature-dependent IFCs with regard to phonon transport in graphene through our PBTE calculations. We demonstrate that the strength of four-phonon scatterings was severely overestimated by previous work that used the IFCs extracted at 0 K. By using IFCs at finite temperatures, we find that four-phonon scattering is weakened but still significantly reduced the thermal conductivity of graphene by around 50%, even at room temperature. Furthermore, in order to reproduce the prediction from molecular dynamics simulations, phonon frequency broadening has to be taken into account when determining the phonon scattering rates. Our study elucidates the phonon transport properties of graphene at finite temperatures and could be extended to other crystalline materials.

Figure 1

(a) Phonon dispersion of graphene along the high-symmetry lines, $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$. (b) Phonon dispersion curve of graphene for the ZA modes around the $\mathrm{\Gamma }$ point in the direction from $\mathrm{\Gamma }$ to M.

Figure 2

The computed classical thermal conductivity of graphene as a function of the number of atoms or q points. “3ph” means that only three-phonon scatterings are considered in the PBTE calculations, while “$3\mathrm{ph}+4\mathrm{ph}$” indicates both three-phonon and four-phonon scatterings are included. The EMD data from Fan et al. [9], Gill-Comeau and Lewis [10], and additional HNEMD results are also shown for comparison.

Figure 3

(a) Four-phonon and (b) three-phonon scattering rates of acoustic modes of graphene along $\mathrm{\Gamma }$-M direction.

Figure 4

Phonon scattering rates of ZA modes along the $\mathrm{\Gamma }$-M direction calculated by using different combinations of the interatomic force constants.

Figure 5

The calculated phonon scattering rates of the ZA modes along the $\mathrm{\Gamma }$-M direction. “3ph” and “4ph” in the legend present the three-phonon and four-phonon scattering rates, respectively. “3ph PBTE” and “$3\mathrm{ph}+4\mathrm{ph}\mathrm{PBTE}$” represent whether or not four-phonon scatterings are included in the PBTE calculations.

Figure 6

Classical thermal conductivity of graphene at 300 K as a function of sample size. The data from NEMD simulations in which the temperature gradient is determined by fitting the linear temperature profile in the sample are represented by open symbols: Qiu et al. [8], Xu et al. [6], and Fan et al. [7]. The results of NEMD simulations by Li et al. [69], where the temperature gradient is determined by the temperature difference of the two reservoirs, are shown as red diamonds.

Figure 7

Thermal conductivity of graphene as a function of temperature.

Figure 8

(a)–(e) Distribution (number of counts) of the thermal conductivity values from the HNEMD method. From (a) to (e), the numbers of atoms in the simulation cell are, respectively, 960, 3840, 8640, 24 000, and 96 000, the numbers of independent simulations (each with a production time of 18 ns in steady state) are 9, 8, 4, 2, and 1, and each thermal conductivity value is calculated from a time interval of 1.8, 1.6, 0.8, 0.4, and 0.2 ns. Therefore, there are 90 thermal conductivity values for each simulation cell size. (f) The averaged thermal conductivity, with the error bar calculated as the standard error of the distribution, as a function of the number of atoms in the simulation cell. One can see that in the HNEMD method, a larger system requires a shorter simulation time to achieve a given statistical accuracy. This is another advantage of the HNEMD method over the Green-Kubo method.