Scaling properties of flexible membranes from atomistic simulations: Application to graphene

Structure and thermodynamics of crystalline membranes are characterized by the long-wavelength behavior of the normal-normal correlation function $G\left(q\right)$. We calculate $G\left(q\right)$ by Monte Carlo and molecular dynamics simulations for a quasiharmonic model potential and for a realistic potential for graphene. To access the long-wavelength limit for finite-size systems (up to $40000\text{atoms}$) we introduce a Monte Carlo sampling based on collective atomic moves (wave moves). We find a power-law behavior $G\left(q\right)\propto q^{-2+\eta }$ with the same exponent $\eta \approx 0.85$ for both potentials. This finding supports, from the microscopic side, the adequacy of the scaling theory of membranes in the continuum medium approach, even for an extremely rigid material such as graphene.

Figure 1

(Color online) Normal-normal correlation functions $G\left(q\right)/N$ calculated for a graphene system with $N=37888$ by ordinary MC simulations (red-dashed line), MD simulations and MC simulations with wave moves with the QH potential. The dashed lines show the asymptotic harmonic behavior with power laws $q^{-2}$ for large $q$ and the long-wavelength limit $q^{-\left(2-\eta \right)}$ with $\eta =0.85$.

Figure 2

(Color online) Normal-normal correlation functions $G\left(q\right)/N$ calculated for three systems with $N=12096$ ($L_x=177.08\text{Å}$, and $L_y=178.92\text{Å}$), $N=19504$ ($L_x=226.27\text{Å}$ and $L_y=225.78\text{Å}$), and $N=37888$ ($L_x=314.82\text{Å}$ and $L_y=315.24\text{Å}$) by MC simulations with wave moves with LCBOPII. The dashed lines show the asymptotic harmonic behavior with power laws $q^{-2}$ for large $q$ and the long-wavelength limit $q^{-\left(2-\eta \right)}$ with $\eta =0.85$. The dashed-dotted line is $G_a$ of Eq. (3) with the coefficients fixed by the asymptotic behavior. One can see that the crossover is much sharper in the simulations.

Figure 3

(a) Quadratic out-of-plane displacement $⟨h^2⟩$, averaged over all particles, as a function of the MC step for the same systems as in Fig. 2. The dashed horizontal lines denote $⟨⟨h^2⟩⟩$, the average $⟨h^2⟩$ over all MC steps except the first $3\times {10}^5$ steps of equilibration. (b) $⟨⟨h^2⟩⟩$ as a function of the average linear system size $\tilde{L}=\sqrt{L_x{L}_y}$ compared to the scaling law $⟨⟨h^2⟩⟩=C{L}^{2-\eta }$ with $C=0.00232$ and $\eta =0.85$ (dashed line). Both axis are in logarithmic scale.