Green's functions for the evaluation of anchor losses in mems

The issue of dissipation has a peculiar importance in micro-electro-mechanical-structures (MEMS). Among the sources of damping that affect their performance, the most relevant are [1]: thermoelastic coupling, air damping, intrinsic material losses, electrical loading due to electrode routing, anchor losses. Moreover, recent experimental results indicate the presence of additional temperature dependent dissipation mechanisms which are not yet fully understood (see e.g. [2, 12]). In a resonating structure the quality factor Q is defined as: Q = 2πW/ΔW (1) where ΔW and W are the energy lost per cycle and the maximum value of energy stored in the resonator, respectively. According to eq. (1), the magnitude of Q ultimately depends on the level of energy loss (or damping) in a resonator. The focus of the present contribution is set on anchor losses and the impact they have in the presence of axial loads. Anchor losses are due to the scattering of elastic waves from the resonator into the substrate. Since the latter is typically much larger than the resonator itself, it is assumed that all the elastic energy entering the substrate through the anchors is eventually dissipated. The semi-analytical evaluation of anchor losses has been addressed in several papers with different levels of accuracy [3, 6]. These contributions consider a resonator resting on elastic half-spaces and assume a weak coupling, in the sense that the mechanical mode, as well as the mechanical actions transmitted to the substrate, are those of a rigidly clamped resonator. The displacements and rotations induced in the half-space are provided by suitable Green's functions. Photiadis, Judge et al. [7] studied analytically the case of a 3D cantilever beam attached either to a semi-infinite space or to a semi-infinite plate of finite thickness. Their results are based on the semi-exact Green's functions established in [4]. More recently Wilson-Rae et al. [9, 10] generalized all these approaches using the involved framework of radiation tunnelling in photonics. Unfortunately, these contributions provide estimates of quality factors that differ quantitatively. In this paper we revisit the procedure of [7], which rests on simple mechanical principles, but starting from the exact Green's functions for the half space studied by Pak [14]. Through a careful analysis utilizing the theory of residues and inspired by the work of Achenbach [15], we show that the results obtained coincide exactly with those of [9], but for the case of torsion.