In order to study quantum effects in a two-dimensional crystal lattice of a free-standing monolayer graphene, we have performed both path-integral Monte Carlo (PIMC) and classical Monte Carlo (MC) simulations for temperatures up to 2000 K. The REBO potential is used for the interatomic interaction. The total energy, interatomic distance, root-mean-square displacement of the atom vibrations, and the free energy of the graphene layer are calculated. The obtained lattice vibrational energy per atom from the classical MC simulation is very close to the energy of a three-dimensional harmonic oscillator $3k_{B}T$. The PIMC simulation shows that quantum effects due to zero-point vibrations are significant for temperatures $T<1000$ K. The quantum contribution to the lattice vibrational energy becomes larger than that of the classical lattice for $T<400$ K. The lattice expansion due to the zero-point motion causes an increase of 0.53% in the lattice parameter. A minimum in the lattice parameter appears at $T\simeq 500$ K. Quantum effects on the atomic vibration amplitude of the graphene lattice and its free energy are investigated.

• Received 13 May 2015
• Revised 2 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.195416