(a) Schematic of a generic EM actuator with electrode having arbitrary geometry and support configurations and a (fractal) patterned bottom electrode . (b) Typical hysteretic - characteristics with the four defining characteristics marked—(1) Below pull-in state ( ), (2) Pull-in voltage ( ), (3) Post pull-in state ( ), and (4) Pull-out voltage ( ). (c) Geometry and support configurations of different geometry-classes being simulated to study scaling laws. The blue (horizontal) regions represent and the red (vertical) regions indicate a fixed support configuration. The gray (horizontal) regions represent Voltage is applied between and . The various geometry-classes are classified according to the electrostatic dimension of the system ( ) and the support configuration. Note that the carbon nanotube (CNT) can be subjected to either a fixed-fixed or a cantilever support configuration. (d) Typical shapes of at , at pull-in instability point ( ), post pull-in state ( ), and at pull-out instability point ( ) for fixed-fixed support configuration.
Verification of the scaling laws in Eqs. (3)–(6) , with simulation results from 100 randomly configured actuators. (a) Plot of vs. to verify Eq. (3) , and extraction of . (b) Plot of vs. , verifying Eq. (4) and determination of . (c) Extraction of . (d) Plot of vs. to verify Eq. (5) and extraction of . (e)Verification of scaling law in Eq. (6) and extraction of . (f) A table summarizing the values obtained for , , , and for the five GCs considered for this figure, with correlation coefficients ( ) of the fits.
Verification of scaling laws with simulation results from 100 randomly configured actuators with a fixed-fixed support configuration with fractal bottom electrode patterning. is varied between 1.2 and 1.8. Plots of (a) vs. (b) vs. , and (c) vs. to verify Eqs. (3)–(5) , respectively. (d) Verification of scaling law in Eq. (6) .
Basis for scaling relationships in the case of a fixed-fixed support configuration. (a) Top electrode shapes for 10 randomly sized fixed-fixed actuators at the point of pull-in instability ( ). (b) The beam shapes in (a) overlap perfectly in normalized dimensions and . (c) Beam shapes for a single fixed-fixed actuator during the post pull-in state. (d) The beam shapes in (c) overlap perfectly in normalized dimensions and . Inset: Variation of the length of the non-contact part of the beam ( ) with voltage ( plotted to verify Eq. (6) .
Analytical approximation for the scaling function into an analytically known function and a scaling parameter (Eq. (B3) . (a)Comparison between numerically obtained function and Eq. (B3) using a best-fit value of for the four geometry-classes with regular bottom electrode (1st column in Fig. 1(c) ) and a CNT electrode subjected to a fixed-fixed support configuration (3rd column in Fig. 1(c) ). (b) and (c) Summary of the values of obtained across various support configurations and electrostatic dimensions ( ) due to bottom electrode patterning. Note that the error bars obtained for are also drawn in (b), but are so small that they cannot be seen. (d) Plot of for all the four support configurations being studied. The normalized values of for the different support configurations overlap each other perfectly.
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