Analysis of frequency shifts due to thermoelastic coupling in flexural-mode micromechanical and nanomechanical resonators

Understanding the effects of thermoelastic coupling on the natural frequency of micro- and nanoscale resonators is essential for the design of frequency-sensitive microelectromechanical systems (MEMS). This paper presents an exact two-dimensional analysis of frequency shifts due to thermoelastic coupling in a beam undergoing flexural vibrations. The coupled heat conduction equation is solved for the thermoelastic temperature field by considering two-dimensional (2-D) heat conduction along the length and thickness of the beam. Thermoelastic coupling is modeled into the equation of motion for flexural vibrations through a temperature-dependent first moment of temperature distribution. The Galerkin technique is used to calculate the thermoelastically shifted frequencies. Detailed calculations are reported for the thermoelastic frequency shift in representative single crystal silicon and aluminum resonators over a full range of parameters. The effects of beam aspect ratio, structural boundary conditions, and mode number on the frequency shifts are discussed.