Curl and Effective Height of Micromachined Bi-Layer Cantilevers Under Differing Residual Stresses

In silicon surface-micromachining, for example PolyMUMPstrade, it is common to include a metal layer in addition to the doped polysilicon layers. However, the residual stresses of any metal layers do not necessarily match the residual stresses within the polysilicon layers, leading to non-uniform residual stresses within structures. This is most visible in cantilevers, which exhibit significant curvature. This can be either beneficial or detrimental, depending on the application. In either case, it is important to be able to predict the radius of curvature for the cantilever, as well as the final height of the tip. While the importance of this phenomenon has been recognized for some time, the equations available in the literature only reflect specific cases. For example, they consider layers of equal thickness, or non-uniform stresses with a uniform material. Unfortunately, this does not cover all cases, including the particularly important case of PolyMUMPstrade. In this paper, we develop, from minimum energy considerations, a completely general equation for predicting both (1) the uniform strain of a cantilever and (2) the radius of curvature of a cantilever. These equations apply to an arbitrary number of layers, each with arbitrary material parameters and residual stress profiles. These results are checked against finite element simulations