Strongly buckled square micromachined membranes

Abstract

The buckling of compressively prestressed square membranes with built-in edges is investigated experimentally and analyzed theoretically. The buckling depends weakly on Poisson's ratio and essentially is a function of the reduced prestrain, /spl epsi/~/sub 0/=/spl epsi//sub 0/a/sup 2//h/sup 2/, where /spl epsi//sub 0/ is the physical prestrain, a is the width, and h is the thickness of the membrane. As /spl epsi/~/sub 0/ becomes increasingly negative, the membrane undergoes two symmetry breaking buckling transitions. Beyond the first transition occurring at /spl epsi/~/sub cr1/, the buckling profile has all the reflection and rotation symmetries of a square. The reflection symmetries are lost through a second instability transition at /spl epsi/~/sub cr2/. The bifurcation points, /spl epsi/~/sub cr1/ and /spl epsi/~/sub cr2/, and buckling profiles were calculated using analytical energy minimization and nonlinear finite-element simulation. Both methods agree. The buckling of micromachined plasma-enhanced chemical vapor deposition silicon nitride membranes on a silicon wafer is interpreted in terms of the theoretical results. Good matching between measured and calculated buckling profiles is found. The extracted strain values are consistent irrespective of the size and buckling mode of the membranes. From the average strain across the wafer /spl epsi//sub 0/=-3.50/spl times/10/sup -4/ and complementary wafer curvature measurements, a Young's modulus of 130 GPa is deduced. Methods for the straightforward extraction of /spl epsi//sub 0/ from experimental center deflections of buckled square membranes are described.