Finite Temperature Lattice Properties of Graphene beyond the Quasiharmonic Approximation

The thermal and mechanical stability of graphene is important for many potential applications in nanotechnology. We calculate the temperature dependence of the lattice parameter, elastic properties, and heat capacity by means of atomistic Monte Carlo simulations that allow us to go beyond the quasiharmonic approximation. We predict an unusual, nonmonotonic, behavior of the lattice parameter with a minimum at $T\approx 900\mathrm{K}$ and of the shear modulus with a maximum at the same temperature. The Poisson ratio in graphene is found to be small $\approx 0.1$ in a broad temperature interval.

Figure 1

Temperature dependence of the lattice parameter $a$ (solid blue line) and nearest neighbor distance $R_{nn}$ (dashed red line). The scales of left ($a$) and right ($R_{nn}$) $y$ axes are related to each other by $\sqrt{3}$. At $T=0$, $a=2.4595\times {10}^{-10}\mathrm{m}$.

Figure 2

(a) Nearest neighbor radial distribution function $g(R_{nn})$ for the $N=8640$ sample at 300, 900, and 2300 K. The vertical lines indicate the length of double ($1.34\times {10}^{-10}\mathrm{m}$), conjugated ($1.42\times {10}^{-10}\mathrm{m}$), and single ($1.54\times {10}^{-10}\mathrm{m}$) bonds. (b) Standard deviation $\sigma (R_{nn})$ (red circles) and the best fit to $\sqrt{T}$ in the temperature range up to 500 K (solid blue line).

Figure 3

(a) 2D elastic moduli of graphene as a function of temperature: adiabatic bulk modulus $b_A$ (solid blue line with circles), isothermal bulk modulus $b_T$ (dashed blue line with squares), and shear modulus $\mu$ (solid red line with diamonds). (b) Adiabatic Young’s modulus $Y_A$ (solid blue line with circles) and isothermal $Y_T$ (dashed red line with squares).

Figure 4

Adiabatic ${\nu }_A$ and isothermal ${\nu }_T$ Poisson ratio of graphene as a function of temperature.

Figure 5

Temperature dependence of the molar heat capacity at constant volume $C_V$ (solid line) and fit to $C_V(T)=3R(1+T/E_0)$ (dashed line).