Robust Modeling of Acoustic Phonon Transmission in Nanomechanical Structures

The transmission of acoustic phonons is an important element in the design and performance of nano-mechanical devices operating in the mesoscopic limit. Analytic expressions for the power transmission coefficient, T, exist only in the low-frequency (quasi-static) limit described by thin-plate elastic theory, and for well-defined elastic wave-guiding geometries. We compare two numerical techniques based on finite-element computations to determine the frequency dependence of T for arbitrary phonon scattering structures. Both methods take into account acoustic mode conversion to acoustic and optical modes. In one case, phase and amplitude of complex-valued reflected waves are determined and related to transmission through a Fresnel equation, while in the other the magnitude of the transmitted mechanical power is directly calculated. The numerical robustness of these methods is demonstrated by considering the transmission across an abrupt junction in a rectangular elastic beam, a well-known problem of considerable importance in mesoscopic device physics. The simulations presented extend the standard results for acoustic phonon transmission at an abrupt junction, and are in good agreement with analytic predictions from thin-plate elastic theory in the long-wavelength limit. More generally, the numerical methods developed provide an effective tool for calculating acoustic mode energy loss in nano-mechanical resonators through mode conversion and heat transfer in arbitrary mesoscopic structures.