The effects of non-uniform nanoscale deflections on capacitance in RF MEMS parallel-plate variable capacitors

This paper analyzes in detail the effect of nanoscale non-uniform deflections, caused by the residual stress phenomenon and/or applied actuation dc voltages, on the capacitance in RF MEMS parallel-plate variable capacitors in an effort to determine the severity of this effect on the ideal parallel-plate capacitance expression normally used. Closed-form capacitance expressions and integrals are given for second order, fourth order, elliptic paraboloidal and hyperbolic paraboloidal models of the bending behavior of the top plate caused by the residual stress and/or actuation. The theoretical analysis is then verified using the finite-element modeling method and results from both analyses exhibit excellent agreement. The equations obtained are then applied to a fabricated chip and other fabricated capacitors in the literature. It was found that anticipating the capacitance using the ideal parallel-plate formula can deviate from the capacitance incorporating the deflections by as much as 80%. The presented equations better anticipate the real capacitance of actual fabricated chips and can take into account deflections in the top plate even if they were in the order of a few nanometers.