Analysis and analytical modeling of static pull-in with application to MEMS-based voltage reference and process monitoring

The pull-in voltage of one- and two-degrees-of-freedom (DOF) structures has been symbolically and numerically analyzed with respect to drive mode dependence and hysteresis. Moreover, the time and temperature stability has been investigated and tested. Modeling results have been applied in the design of both folded-spring-suspended 1-DOF structures and single-side-clamped 2-DOF beams with a nominal pull-in voltage in the 5-10 V range and fabricated in an epi-poly process. Asymmetrically driven structures reveal pull-in close to the value predicted by the model (V-pi 1-DOF is 4.65 V analytically simulated and 4.56 V measured; V-pi 2-DOF is 9.24 V analytically simulated, 9.30 V in FEM and 9.34 V measured). Also the hysteresis is in close agreement (release voltage, V-r, 1-DOF is 1.41 V analytically simulated and 1.45 V measured; V-r 2-DOF is 9.17 V analytically simulated, 9.15 V in FEM and 9.27 V measured). In symmetrically operated devices the differences between the computed and measured V-pi and V-r are much larger and are due to process dependencies, which make these devices very suitable for process monitoring. The 2-DOF asymmetrically operated device is the most suitable for MEMS-based voltage reference. The stability in time is limited by charge build-up and calls for a 100-hour initial burn-in. Temperature dependence is -100 muV/K at V-pi approximate to 5 V, however, is calculable and thus can be corrected or compensated.