^{1,2}, Sun-Chul Lee

^{1}and Yong-Sung Kim

^{1,2,a)}

### Abstract

In this paper, we report a non-equilibrium molecular dynamics study on the length-dependent lattice thermal conductivity of graphene with lengths up to 16 μm at room temperature. In the molecular dynamics simulations, whether the non-equilibrium systems reach the steady states is rigorously investigated, and the times to reach the steady states are found to drastically increase with the lengths of graphene. From the ballistic to the diffusive regime, the lattice thermal conductivities are explicitly calculated and found to keep increasing in a wide range of lengths with finally showing a converging behavior at 16 μm. That obtained macroscopic value of the lattice thermal conductivity of graphene is 3200 W/mK.

This work was supported by Future-based Technology Development Program (Nano Fields) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant No. 2012-0009623) and the Strategic Supercomputing Support Program (No. KSC-2013-C2-007) from the KISTI.

I. INTRODUCTION

II. METHODS

III. RESULTS

A. Steady states

B. Time to reach steady state

C. Length-dependent lattice thermal conductivity

IV. CONCLUSION

### Key Topics

- Thermal conductivity
- 51.0
- Graphene
- 49.0
- Ballistic transport
- 22.0
- Heat transport
- 12.0
- Molecular dynamics
- 11.0

## Figures

Supercell structures in the NEMD simulations for calculating the lattice thermal conductivities of graphene. (a) A magnified view near both the ends of the supercells along the length direction for the supercell width m = 4. (b) Supercell structure for the graphene size of m = 4 and n = 140, which is the smallest in our simulations. The length L of the heat transport region is in this case 2.45 nm. (c) Schematic illustration of the graphene with the length of n = 64 000, which is the longest in our simulations. The length L of the heat transport region is in this case 15.65 μm.

Supercell structures in the NEMD simulations for calculating the lattice thermal conductivities of graphene. (a) A magnified view near both the ends of the supercells along the length direction for the supercell width m = 4. (b) Supercell structure for the graphene size of m = 4 and n = 140, which is the smallest in our simulations. The length L of the heat transport region is in this case 2.45 nm. (c) Schematic illustration of the graphene with the length of n = 64 000, which is the longest in our simulations. The length L of the heat transport region is in this case 15.65 μm.

(a) Calculated temperature distributions along the length of graphene (y-axis) as a function of time (x-axis) in the NEMD simulations for the graphene supercell size of m = 4 and n = 48 000. (b) Spatial-averaged temperature evolutions for the hot (Th ) and cold (Tc ) thermostat regions, and the heat transport region (at 300 K). (c) Temperature distributions along the length of graphene in the initial state (0 ps) of T = 300 K (black thin solid line), after 5 ns (green dotted line), 10 ns (blue dashed line), and in the steady state after 15 ns (magenta dotted-dashed line) and 20 ns (red thick solid line). (d) Cumulative energy per supercell width m that flows along the heat transport direction in graphene as a function of time. After reaching the steady state, the constant energy per time flows, as shown in (d) by the linearly fit (red dashed) line.

(a) Calculated temperature distributions along the length of graphene (y-axis) as a function of time (x-axis) in the NEMD simulations for the graphene supercell size of m = 4 and n = 48 000. (b) Spatial-averaged temperature evolutions for the hot (Th ) and cold (Tc ) thermostat regions, and the heat transport region (at 300 K). (c) Temperature distributions along the length of graphene in the initial state (0 ps) of T = 300 K (black thin solid line), after 5 ns (green dotted line), 10 ns (blue dashed line), and in the steady state after 15 ns (magenta dotted-dashed line) and 20 ns (red thick solid line). (d) Cumulative energy per supercell width m that flows along the heat transport direction in graphene as a function of time. After reaching the steady state, the constant energy per time flows, as shown in (d) by the linearly fit (red dashed) line.

Cumulative energies that flow along the heat transport direction in graphene per supercell width m as a function of time for the graphene supercells of m = 4 and (a) n = 1000, (b) 2000, (c) 4000, (d) 8000, (e) 16 000, (f)32 000, (g) 48 000, and (h) 64 000. After reaching the steady state, the constant energy per time flows as shown by the linearly fit (red dashed) lines. The estimated times to reach the steady states are indicated by the (blue) vertical lines in (a)–(h). (i) The times to reach the steady states (black filled circles) and our simulation times (red filled squares) are shown as a function of the length L of graphene.

Cumulative energies that flow along the heat transport direction in graphene per supercell width m as a function of time for the graphene supercells of m = 4 and (a) n = 1000, (b) 2000, (c) 4000, (d) 8000, (e) 16 000, (f)32 000, (g) 48 000, and (h) 64 000. After reaching the steady state, the constant energy per time flows as shown by the linearly fit (red dashed) lines. The estimated times to reach the steady states are indicated by the (blue) vertical lines in (a)–(h). (i) The times to reach the steady states (black filled circles) and our simulation times (red filled squares) are shown as a function of the length L of graphene.

(a) Calculated lattice thermal conductivities (κ) of graphene at room temperature as a function of the length L of graphene for the supercell widths of m = 4 (black filled circles) and m = 40 (green filled squares). The (green) solid line indicates the ballistic thermal transport limit, 30 and the (green) dot-dashed line is the macroscopic thermal conductivity obtained in this study. The experimental data given in Ref. 5 are indicated by the (red) cross symbols. The length-dependent thermal conductivities obtained from the higher-order heat transport equation 32 are plotted by the (blue) dashed line. (b) The inverse of the calculated lattice thermal conductivity ( ) of graphene as a function of the inverse of L (1/L).

(a) Calculated lattice thermal conductivities (κ) of graphene at room temperature as a function of the length L of graphene for the supercell widths of m = 4 (black filled circles) and m = 40 (green filled squares). The (green) solid line indicates the ballistic thermal transport limit, 30 and the (green) dot-dashed line is the macroscopic thermal conductivity obtained in this study. The experimental data given in Ref. 5 are indicated by the (red) cross symbols. The length-dependent thermal conductivities obtained from the higher-order heat transport equation 32 are plotted by the (blue) dashed line. (b) The inverse of the calculated lattice thermal conductivity ( ) of graphene as a function of the inverse of L (1/L).

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