Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction

Analysis of thermoelastic damping (TED) is an important component of the design of low-loss vacuum-operated micro- and nanomechanical resonators used in microelectromechanical systems (MEMS). The quasi-1-D theories developed by Zener in 1937, and subsequently improved by Lifshitz and Roukes in 2000, are now widely used in MEMS design. This paper presents an exact theory for TED with 2-D heat conduction that enables a detailed evaluation of the accuracy of the quasi-1-D theories. A Green's function method is used to solve the 2-D heat-conduction equation, and an expression for TED is derived in the form of an infinite series. The effects of beam geometry, length-to-thickness aspect ratio, natural frequency, flexural mode shapes, and structural boundary conditions on TED are investigated for the representative case of single-crystal silicon microbeam resonators. The errors in the exact quasi-1-D theory range from 2% to 80% depending upon the aspect ratio and the mode shape. Implications for the use of the quasi-1-D and 2-D theories in MEMS design are discussed.