Thermal vibration of a double-layered graphene sheet with initial stress at low temperature

Abstract

Thermal fluctuations play an important role in the resonance properties of low dimensional structures as they serve as acoustic and vibration sensors of micro and nano machine. The quantum effects will be obvious during the thermal vibration of nanostructures because of the high natural frequency. Graphene has novel electronic properties and superior mechanical strength. The thermal vibration problems of graphene, which can be used as nano electronic components, have attracted a lot of research interest. The influence of quantum effects for the thermal vibration of single-layered graphene has been investigated. The dynamical behaviors of the double-layered graphene are more complicated due to the van der Waals force between two layers. However, few studies about the thermal vibration of double-layered rectangle graphene considering quantum effects have been found. To this end, an equivalent plate model with quantum effects of a double-layered graphene is established. The van der Waals interaction between two adjacent layers can be simplified as Winkler type spring because the thermal amplitude of graphene is small when the temperature is very low. The nonlocal effects and initial stress of the graphene are also taken into consideration. The natural frequency and the root of mean squared (RMS) amplitude of the double-layered graphene with different initial stress in different temperatures are calculated. The in-phase vibration mode and anti-phase vibration mode can be obtained from thermal vibration of the double-layered graphene. The natural frequency of the in-phase vibration of the double-layered graphene is the same as the natural frequency of the single-layered graphene with the same size. The natural frequency of the anti-phase vibration is larger than the in-phase vibration frequency. The RMS amplitude obtained by the plate model considering quantum effects is smaller than that obtained by the plate model together with the law of energy equipartition. The different of these two models becomes more obvious at the lower temperature. The RMS amplitude of the anti-phase vibration is smaller than the in-phase vibration RMS amplitude. The peak values of the RMS amplitude of the anti-phase vibration in different models are very close because the natural frequency of anti-phase vibration increases slowly with the increasing of model. The natural frequency of the in-phase vibration will increase with increasing of the initial stress, while the RMS amplitude of the in-phase vibration will decrease. However, the changes of the natural frequency and the RMS amplitude of the anti-phase vibration are very small when the initial stress changes. The initial stress has more significant influence for the in-phase vibration than the anti-phase vibration. The effect of zero-point energy for the thermal vibration of the double-layered graphene is also investigated. If the zero-point energy is taken into consideration to the mean modal energy, the RMS amplitude obtained by the plate model considering quantum effects is larger than that obtained by the plate model together with the law of energy equipartition in a wide range. The difference is obvious when the temperature is very low. Besides, the RMS amplitude of the graphene is not zero at the temperature of 0 K.