Mode- And Direction-Dependent Akhiezer Damping in Single-Crystal Silicon Resonators

Abstract

In this work, we determine the intrinsic mechanical energy dissipation limit for single-crystal resonators due to anharmonic phonon-phonon scattering in the Akhiezer (Ωτ ≪ 1) regime and evaluate the quality factor limit for intrinsic silicon devices. The energy loss is derived using perturbation theory and the linearized Boltzmann transport equation for phonons, and includes the direction-and polarization-dependent mode-Grüneisen parameters in order to capture the strain-induced anharmonicity among phonon branches. Our new theoretical framework reveals that Akhiezer damping, previously thought to depend only on material properties, has a strong dependence on resonant mode shape. Critically, we show that shear mode devices do, in fact, experience Akhiezer damping, which is in direct contrast to the result produced when applying existing models. Our expression satisfies the pressing need for a reliable analytical model that can predict the Akhiezer damping limits for modern resonant microelectromechanical systems, where precise manufacturing techniques and accurate finite-element methods can be used to select particular vibrational mode shapes and crystal orientations.