Thermal Expansion and Deformation of Graphene

Taking into consideration short-atomic-range interactions and anharmonic effects, we calculate the thermal expansion coefficients, Gruneisen parameters, the elastic modulus of graphene varying with temperature and the phonon frequency. The anharmonic effects associated with the graphene deformation are also discussed. The results show that the value of thermal expansion coefficient is negative in the moderate temperature range, and it becomes positive when the temperature grows to be higher than a certain value. The change rate of elastic modulus with respect to temperature and pressure are calculated, and phonon frequencies are estimated. In the process of graphene thermal expansion, it is accompanied with the change of bond length and the rotation around the axis normal to the plane. Our results indicate that the effects due to the bond change are more significant than that of the rotation. We also show that if anharmonic effects are ignored, the thermal expansion coefficient and the Gruneisen parameters are zero, and the elastic modulus and the phonon frequency are constant. If anharmonic effects are considered up to the second term, these values will vary with temperature, and become closer to the experimental value. The higher the temperature is, the more significant the anharmonic effects become.