Thermoelastic damping in micro- and nanomechanical beam resonators considering size effects

In this paper, we describe governing equations for modified coupled thermoelasticity in micro- and nanomechanical beam resonators, which can treat the effects of size by taking the relaxation time, the phonon mean-free path and the material length scale parameter into account. An analytical model of thermoelastic damping is derived using the complex-frequency approach. Numerical results of thermoelastic damping calculated using the proposed model are presented and compared to those calculated using the models proposed by Zener and Lifshitz and Roukes for a silicon thin beam. The results show that the nonlocal effect characterized by the length scale parameter is negligible even at sizes down to the phonon mean-free path. The size effects characterized by the relaxation time and the phonon mean-free path are significant for a micron-scale beam. Device miniaturization beyond the submicron scale will lead to increased energy dissipation due to thermoelastic damping considering size effects. The influence of size effects on thermoelastic damping can be suppressed by increasing aspect ratios. Finally, we present the range of geometry of a silicon beam resonator where the effects of size can be neglected by taking 10% as the permitted error bound.