Effect of support compliance and residual stress on the shape of doubly supported surface-micromachined beams

Doubly supported silicon-micromachined beams are increasingly used to study the mechanical properties of materials. Residual stresses in the beams and support compliance cause significant vertical deflections, which affect the performance of these micromachined devices. We present here both experimental results for doubly supported polysilicon surface-micromachined beams, and an elastic model of the devices that takes into account the compliance of the supports and the geometrical nonlinear dependence of the vertical deflections on the stress in the beam. An elastic one-dimensional model was used for the beams, and the response of the supports to forces and moments was obtained using finite-element method simulations. The model explains a previously observed gradual increase of the maximum vertical deflections of the beams with increasing length at a given constant residual stress and, in agreement with experimental observations, predicts two stable states for compressively stressed beams: one with the beam bent up, the other down. We introduce a critical biaxial residual stress sigma(rc), above which there are significant deflections of the beams. sigma(rc) can be used in practice to determine the maximum allowable compressive biaxial residual stress as a function of the beam length. The effect of variations of the compliance of the supports is reported.