CHAPTER 2 ANALYTICAL MODELING FOR PULL-IN VOLTAGE OF CANTILEVER FOR WIDE RANGE OF DIMENSIONS

Transcription

1 34 CHAPTER ANALYTICAL MODELING FOR PULL-IN VOLTAGE OF CANTILEVER FOR WIDE RANGE OF DIMENSIONS.1 INTRODUCTION Electrostatic actuation is one of the methods to obtain actuation in MEMS. This method of actuation is very popular because of its simplicity, fast actuation rates, low power consumption, and use of silicon (Haluzan et al 1) as its material. This actuation method is common for driving MEMS devices because it is compatible with microfabrication. These actuators are available in the form of 1. Cantilever beams and. Fixed-fixed beams. Electrostatically actuated micro beams are widely used in MEMS. Considering the popularity of the cantilever structure, this Chapter investigates the pull-in voltage modeling of the same, for a wide range of dimensions with better accuracy. Electrostatic actuation-based devices are essentially simple capacitors composed of two parallel micro-plates, usually with a beam type arrangement. The device is designed such that one plate is free to move, while the other one remains fixed. When voltage is applied between the two plates, an electrostatic force is created between them, so the free plate will deflect towards the fixed plate. If the applied voltage is further increased the electrostatic force exceeds the elastic restoring force which is developed within the deformed plate, and makes sudden contact with the other plate.

2 35 This phenomenon is called as pull-in. The corresponding voltage is called pull-in voltage. In the electro-static MEMS actuators, the pull-in phenomenon is very significant for the design of devices because, in some applications such as microphones (Pedersen et al 1998) and pressure sensors (Ho et al & Sallese et al 1), it is essential to avoid the pull-in effect. Since the contact between the two plates makes a short circuit, the device cannot be operated unless there is a dielectric layer of separation. But in some applications like optical systems (Chik 1997), the applied voltage is purposely tuned or optimized, such that when the plate touches the substrate, it turns the switch on or off. Therefore, the realization of such devices, requires not only the availability of the fabrication technique, but also a thorough knowledge of the behavior of the device structure such that the pull-in effect between the two plates can be either avoided or utilized, depending upon the particular application. In order to meet the aforementioned requirements, the critical bias voltage at which the pull-in occurs, must be precisely computed well in advance so that the designer can ensure the operation of the device according to the specification requirements. Hence an accurate evaluation of the pull-in voltage is necessary and also essential in the design of electrostatically actuated MEMS devices. The pull-in problem of beams cannot be solved analytically. Numerical techniques using the Finite Element Analysis (FEA) are computationally expensive and are time-consuming. Therefore, closedform expressions are useful for designers as they provide some basic information regarding the pull-in voltage. Chowdhury et al (5) devised a closed-form model for pull-in voltage calculation, by considering Meijs (Meijs & Fokkema 1984) as a better capacitance model (Barke 1988). The

3 36 pull-in voltage for selective dimensions of the cantilever beam is computed and presented. But further investigation of a wide range of dimensions of the cantilever beam shows that the closed-form model used in the work (Chowdhury et al 5) alone is not sufficient. Different capacitance models are available in literature, (Chang 1976, Meijs & Fokkema 1984, Yuan & Trick 1984, Elmasry 198, Sakurai & Tamaru 1983, Palmer 193) accurate (Barke 1988). Therefore, the present work takes into consideration all other capacitance models available in literature (Meijs & Fokkema 1984, Yuan &Trick 1984, Elmasry 198, Sakurai & Tamaru 1983, Palmer 1937). Based on these capacitance models, closed-form pull-in voltage equations are obtained, and the pull-in voltage of the cantilever beam is computed for a wide range of dimensions. Further, the suitability of each model for the calculation of the pull-in voltage has also been investigated for a wide range of dimensions. A detailed comparative analysis is carried out, by comparing the pull-in voltages obtained from the closed-form models, with the simulation results and also with the experimental results.. SIGNIFICANCE OF THE FRINGING FIELD ON PULL-IN VOLTAGE In the micro cantilever beams, if the ratio of the width to the depth of the beam is small (i.e. w/h < 1.5, where w-width of the beam and h thickness of the beam) the fringing field capacitance increases the total capacitance by 1.5 to 3 times (Rabaey 1996). If w/d plate approximation does not work, and an error of upto % is reported in the literature (Chowdhury et al 5), where d is the gap between the substrate and the beam, as shown in Figure.1.

4 37 Figure.1 Cantilever beam separated from a fixed ground plane by a dielectric spacer The capacitance between the side walls of the beam and the substrate is called the fringing capacitance, which is too large to be neglected. Therefore, a simplified model that approximates capacitance as the sum of the parallel plate capacitance and the fringe field capacitance, has to be used, as shown in Figure.. Figure. (a) Fringing field capacitance (b) Model of fringing field capacitance (Rabaey 1996) Based on the above modeling many authors have developed empirical formulas to calculate the capacitance, as discussed in the previous section.

5 38.3 ANALYTICAL MODELING.3.1 Cantilever Beam Pull-in Voltage Model The lumped parameter model of the actuator is shown in Figure.3. The actuator is assumed to be in vacuum environment to ensure zero external mechanical loading of the top electrode. It is also assumed that any damping within the system, the equation of the motion of the movable plate, due to an electrostatic force F E can be expressed as d z M kz F dt E (.1) where F E - Electrostatic force k - Spring constant M - z - Mass Displacement Figure.3 Lumped parameter model of the parallel plate electrostatic actuator

6 39 The electrostatic attraction force of the plate can be found by differentiating the stored energy of the capacitor, with respect to the position of the movable plate, and is given as d F E 1 CV AV = dz d z (.) where A C d z A Area of the plates C Capacitance Permittivity of free space (.3) and the spring force (elastic restoring force) is represented as FM kz (.4) where F M - Mechanical elastic restoring force At static equilibrium FM F E If the electrostatic force is increased by increasing the applied voltage, and if that force is greater than the elastic restoring force, the equilibrium is lost, and the movable plate will collapse on the fixed ground plate. This phenomenon is known as the pull-in. substituting Equation (.) in.4, Equation (.5) is arrived at. kz AV = ( d ) z (.5) simplifying Equation (.5), Equation (.5a) is arrived.

7 4 kz AV d z d z (.5.a) kzd +kz -4 kd z = AV 3 (.5.b) is obtained. Differentiating Equation (.5.b) with respect to z, Equation (.5.c) 6 8 = kd kz kdz A V z v (.5.c) At equilibrium (or) at pull-in v z = kd 6kz 8 kd z = (.5.d) d 3z 4 d z = (or) 3z 4d z d (.5.e) Solving Equation (.5.e) z pi d 3 (.5.f) Substituting z pi in Equation (.5) Equation (.6) can be arrived at. V PI Equation (.6) as 3 8kd 7 A (.6) The spring constant of the movable plate is solved from k 7 8 AV PI 3 d (.7)

8 41 If the applied voltage is increased beyond the pull-in voltage, the resulting electrostatic force will overcome the elastic restoring force, and will cause the movable plate to collapse on the fixed ground plane and the capacitor will be short circuited. By expanding equation (.5), using a Taylor series approximation about a distance z as outlined by Ladabaum (1999), Equation (.8) is arrived, AV AV AV ( )( 1) F ( z z )..., E z z 3 z z ( d z) ( d z) ( d z) (.8) After simplification and rearrangement of the terms in Equation (.8), Equation (.9) has been derived. F E AV z z 1... ( d z) d z (.9) By substituting F E from Equation (.9) into Equation (.1), Equation (.1) is derived. M d z AV z z dt d z d z kz ( ) (.1) That is rearranging Equation (.1), Equation (.11) is arrived. d z AV z M k k z dt d z d z soft ( ) (.11) From Equation (.11) it is evident, that the electrostatic attraction force effectively modifies the spring constant of the movable plate, and the term within the parenthesis on the left-hand side of Equation (.11) represents the effective spring constant at a specific voltage. The amount of modification is termed as spring softening, and can be expressed as in Equation (.1).

9 4 k soft AV ( d z ) 3 (.1) For parallel-plate geometries, the electrostatic force is always uniform. But for cantilever beam geometry (Figure.1), the electrostatic force becomes increasingly non-uniform as the beam deforms, as in Figure.4. As a result, the tip of the cantilever will experience a higher attractive force, compared to the region closer to the fixed end. Following Osterberg (1995), an expression for a uniform pressure (P) causing a cantilever tip deflection of z can be derived as in Equation (.13). P kz wl 3 Eh 4 l 3 z (.13) Figure.4 Deformation of the beam due to electrostatic force where E is the plate modulus E / (1- ) for wide beams (w 5h). For narrow beams (w < 5h), E E (Osterberg 1995). A uniform linearised model of the electrostatic force can be obtained from Equation (.11) and Equation (.1), by linearizing the electrostatic force about the zero deflection point (z = ), as shown in Figure.5.

10 43 Figure.5 Linearization of the electrostatic force about the zero deflection point (z = ) were wlv d z soft dt d M k k z k soft wlv d 3 (.14) (.15) Rearranging Equation (.14) and neglecting the time-dependent term for a static case, the force equilibrium relation for any displacement z can be obtained. It is given in Equation (.16). wlv kz k softz d (.16) The right-hand side of (.16) is equal to approximate. The uniform and linear electrostatic force (F E -linear-uniform), and the left-hand side represents the elastic restoring force. The effective linearised uniform electrostatic pressure (P eff ) on the beam can be evaluated from Equation (.16) as Equation (.17). P eff FE linear uniform V ksoft z wl d wl (.17)

11 44 Substituting k soft from Equation (.15) into Equation (.17) and replacing z in Equation (.17) by the pull-in deflection z = 1/3d, the pull-in electrostatic pressure P PI-electrostatic can be evaluated as P PI electrostatic 5 V 6 PI d (.18) where V PI represents the pull-in voltage. In order to compensate for the error that arises due to neglecting the higher order terms in the Taylor series expansion, and the error due to linearization, a Compensation Factor (cf) has been determined by a trial and error method, while comparing the results with the Coventor Ware FEA model results. The compensation factor is applied to Equation (.18). Now, the compensated pull-in electrostatic pressure P PIelectrostatic is expressed as P cf V PI electrostatic 5 PI 6do (.19) By substituting the pull-in deflection, z = 1/3d, into Equation (.13), the elastic restoring pressure at pull-in is obtained. P PI elastic Eh d Eh d 3 l 3 9l (.) Since at the pull-in equilibrium, the electrostatic pressure is just counterbalanced by the elastic restoring pressure (P PI-electrostatic = P PI-elastic ), Equation (.19) and Equation (.) can now be solved simultaneously, to yield the final closed-form expression for the pull-in voltage V PI as

12 45 V PI 3.E h d 5 ( l ) cf d 6 4 (.1).3. Closed form Models of Pull-in Voltage from Capacitance Models In the previous section, the procedure to determine the closed-form model of the pull-in voltage is discussed. But the accuracy of the pull-in voltage computation can be improved, only by computing the capacitance of a parallel plate accurately. As reported in the literature, different capacitance models (Chang 1976, Meijs & Fokkema 1984, Yuan & Trick 1984, Elmasry 198, Sakurai & Tamaru 1983, Palmer 193) are considered for deriving the pull-in volatge models, and the same is to be checked for a wide range of dimensions of the cantilever strcuture..3.3 Capacitance Models Since the simple parallel plate model is not adequate for capacitance calculation these capacitance models are considered. These capacitance models are basically parallel-plate models, with fringing fields. These models are mainly based on approximated formulas of exact analytical solution, of the line to ground capacitance problem in a VLSI environment. It is reported that (Barke 1988) all these models are easy to use and fast to compute and also result in accurate and reliable capacitance values. Further it is reported that, numerical methods may also be used, but it may require expensive computations even for geometrically small problems. It is also emphasised that there is a strong need for simple and fast approximating formulas to calcualte the capacitance values. Amoung the capacitance models Chang 1976) is based on the technique of conformal transformation, and hence it is giving more

13 46 accurate capacitance values as comparing with exact analytical solutions. The other capacitance models (Meijs & Fokkema 1984, Yuan & Trick 1984, Elmasry 198, Sakurai & Tamaru 1983, Palmer 1937) are compared with the as it is computationally expensive, but the remaining models (Meijs & Fokkema 1984, Yuan & Trick 1984, Elmasry 198, Sakurai & Tamaru 1983, Palmer 1937) have been taken into consideration for deriving the pull-in voltage of cantilever beams using the procedure explained in section.3.1. All these capacitance models that are available in literature are presented elaborately in Appendix A..3.4 Pull-in Voltage Models Based on the above capacitance models the closed-form models of the pull-in voltages are derived, using the procedure given in Section.3., and whose final form is presented from Equation (.) through (.6). It is to be noted that the model 1 shown below has been already discussed in the work of Chowdhury et al (5). Here, as explained before, the compensation factor (cf) is applied to every model, and its value is the same irrespective of the change in dimensions. Pull-in voltage based on model 1 V.E h PI,1 4 where A h and cf.76 d l cf A d d w d w.5 (.)

14 47 Pull-in voltage based on model V 3.Eh d PI, 4 l cf B1 where cf.76 (.3) where B 5(w - h) 1 1d w 3 d 1dh w log sigma d h h 1 d d d d d h h 16 d 4 log sigma d h h h h h d h 1 h sigma w log sigma where sigma d h d d h h 1 1 Pull-in voltage based on model 3 V.E h PI,3 4 l cf C1 3 d (.4) where C d h w 5h log 5 h 5d 4 h h d h h(6d 5 h) 1 6d 3d 3d w 3 d h d h w 3d d h d h w where cf =.5 Pull-in voltage based on model 4 V =. E h PI,4 4 3 d l cf D 1 (.5)

15 48 where D h d w d where cf.8 Pull-in voltage based on model 5 V.E h PI,5 4 3 d l ( cf ) E 1 (.6) Where E 5 4 6d 3 dw 1 where cf.89.4 FEM SIMULATION USING COVENTORWARE CoventorWare is a software tool for designing and simulating MEMS and microfluidic devices. It provides two distinct modules: 1.ARCHITECT and the. DESIGNER and ANALYZER module. The former provides a system level approach to the MEMS design, and the latter provides the FEM-based approach. Both design flows require information about the fabrication process. The Process Editor and Material Properties Data Base are the two main sub-modules of the Designer and Analyzer module. Modeling in the Designer and Analyzer requires the knowledge of fabrication. The designer phase starts from creating a -D layout by the layout editor. The stack information is provided in the Process Editor. Then the Solid Modeller uses the layout and Process Editor details to automatically build a 3-D solid model. Then, in the preprocessor, the mesh has to be generated. Then in the analyzer phase, the model is solved using a suite of field solvers

16 49 which simulate the physical behavior of the devices using FEM (or) BEM (Boundary Element Method), or a combination of both..4.1 Design using CoventorWare Material Properties Database: The first step in creating a design is to enter the material properties associated with the fabrication process in the Material Properties Database (MPD). The materials available in the MPD are only accessible for simulations. Process Editor: Here the fabrication steps should be defined by the user. The fabrication sequence has to be created in steps, by selecting the prototype steps from a process library. Each step parameter has to be specified. Layout Editor: This is used to define all the masks required by the process file. There are many options for providing the layout file. One option is to draw the shapes that define each mask, using the layout editor. Another option is to draw the layout with the layout tools available in the GDS II (Graphic Database System), DXF (Drawing exchange Format) (or) CIF (Caltech Intermediate Graphic) file format. Mesh convergence Study: To conduct a mesh convergence study, different meshing elements have to be applied for the model. Then the accuracy of the result has to be checked for each case. By increasing the number of mesh elements further, if the accuracy is not varied much, then it is concluded that mesh convergence has been obtained. In this work about 1 mesh elements have been used. The Manhattan type element is used for meshing the cantilever. The Meshed cantilever beam is shown in Figure.6.

17 5 Figure.6 Meshed cantilever beam.4. Analysis using CoventorWare This simulation study uses the following solvers. 1. MemElectro,. MemMech and 3. CoSolveEM. MemElectro: This solver is used to solve the electrostatics problem. It uses BEM to solve the problem. This tool computes the capacitance and force between conductors. MemMech: displacement and stress. It can be set to run the thermal, thermo mechanical, electro thermal and electro thermo mechanical analyse, computing the effects of temperature on the model. CoSolveEM: This analysis is used to analyze the electrostatic actuation of the mechanical structure. MemElectro provides the electrostatic analysis and MemMech provides the mechanical analysis. CoSolveEM maintains the consistency between the two solutions; that is the mechanical deformation is correct for the applied electrostatic force. The most common application is the pull-in voltage determination. The other applications are the contact and

18 51 lift-off analysis, and electrostatic spring softening. The MemMech and MemElectro solver parameter has to be set, to setup the CoSolve Simulation..5 FABRICATION AND CHARACTERISATION OF CANTILEVER BEAMS To validate the analytical results, cantilever beams are fabricated and characterized for the pull-in voltage and resonant frequency. The resonant frequency of the cantilever beams is also obtained through simulation studies. Fabrication of cantilever beam has been carried out in CeNSE (Centre for Nano Science and Engineering), Indian Institute of Science, Bangalore through INUP facility (Indian Nanoelectronics User Program). The SOI wafer is used for fabrication. The SOI wafer specifications are given in Table.1. Table.1 SOI wafer specifications Sl.No Property Value 1. Size 1/4 th of 6" wafer. Type N 3. Resistivity 1-1 ohm-meter 4. Device layer Thickness 5±.5 µm 5. BOX (Buried Oxide) thickness µm 6. Handle wafer thickness 4±1 µm The cantilevers are fabricated, using the bulk micromachining technique on an N type SOI wafer. The mask which is used to fabricate the cantilever is shown in Figure.7. The fabrication of the cantilever beam is a simple one mask process, using a Positive PhotoResist (PPR) (AZ-514) for the UV lithography. The 1.4µm thick PhotoResist also acts as a mask layer, for etching the silicon. The Silicon is etched using the Deep Reactive Ion

19 5 Etching (DRIE) technology to form the patterns. The Sacrificial etching of oxide releases the cantilever with bond pads of x µm. The cantilever design length is varied from 5 µm to µm and the width is fixed at 5 µm in all cases. The fabrication flow for the device is shown in Figure.8. Figure.7 Mask design of the cantilevers The silicon wafer (a) is first cleaned with the piranha (3:1, H SO 4 : H O ) solution. Then the wafer is patterned (b) with a mask and photoresist AZ 514 (thickness of PPR 1.4 µm) with a dose of 3 mj/cm. Then, the Silicon is etched by (c) Deep Reactive Ion Etching at 1 º C, 75W power. For deposition, the pressure is 8 mtorr, and for etching 13 mtorr pressure is used. During deposition C 4 F 8 with flow rate of 8 sccm gas is used; for etching SF 6 with a flow rate of sccm is used; the power during deposition is 16 watts, and for etching watts is applied. The Silicon is dry etched at the etch rate of 4 µm/min. After this, the photo resist is removed by ashing, using oxygen gas for 3 minutes. To increase the conductivity of the cantilevers, phosphorous (d) is diffused at a pre-deposition temperature of 95 º C for 15 minutes, and drive-in temp of 15 º C for 3 minutes with POCl 3. After this, the PSG is removed by a quick dip in the solution of 1:4:3, HF:HNO 3 :DI water. The sheet resistance -cm, and after -cm. Finally, (e) the sacrificial oxide layer is etched in

20 53 buffered hydrofluoric acid and the cantilevers are kept in the Critical Point Drier (CPD) for 1 hour to release them properly. Figure.8 Fabrication flow of the cantilever In the following section, the fabrication of cantilever beams and microgripper is explained in detail with corresponding figures.

21 54 Figure.9 Wafer kept in piranha solution for cleaning (CeNSE Laboratory, IISc) UV Lithography Cleaning: Wafer is first cleaned with Acetone and IPA (Iso Propel Alcohol), and then dried with nitrogen gas. The wafer is kept on a heater for dehydration for min at a temperature of 5 º C. This is shown in Figure.9. Spincoat of Photoresist: First PR AZ 514 is poured and kept in a spin coater. Recipe for spin coating is 4 RPM for 4 sec. The thickness of PR after spin coating measured is 1.4 µm. This is shown in Figure.1. Prebake of PR: After spin coat the wafer is prebaked, for that it is kept on a heater for 11 º C for 5 sec.

22 55 Figure.1 Wafer placed on a spin coat (CeNSE Laboratory, IISc) Loading in EVG Mask aligner: The mask is loaded first, (Ref Figure.11) aligned then the sample is loaded. Separation distance is 3 µm. Energy of UV source supplied is 3mJ/cm. Figure.11 Loading the sample in a EVG mask aligner (CeNSE Laboratory, IISc)

23 56 Developing: The sample is unloaded from the Mask aligner and it is developed using AZ351B (1:4, AZ351B: DI water) for 4 sec. After developing it is rinsed with DI water. Then the sample is dried with N gas. Post Baking: In this final step the sample is kept on a heater with a temperature of 11 º C for 3 minutes and again dried with N. DRIE of Si: Silicon is dry etched with the etch rate of 4 µm/min. After this the photo resist is removed by ashing using oxygen gas for 3 minutes. This is shown in Figure.1. Figure.1 Loading the sample for DRIE (CeNSE Laboratory, IISc) To increase the conductivity of the cantilevers, phosphorous is diffused with pre-deposition temperature of 95 º C for 15 minutes, and drivein temp of 15 º C for 3 minutes with POCl 3. The whole process of Phosphorous diffusion is shown in Figure.13. After this the PSG is removed by quick dip in the solution of 1:4:3, HF:HNO 3 :DI water. The sheet -cm and -cm.

24 57 Figure.13 Loading the wafer for phosphorous diffusion (CeNSE Laboratory, IISc) Figure.14 Four Point probe station to measure sheet resistance (CeNSE Laboratory, IISc) Four point probe station which is shown in Figure.14 is used to measure the sheet resistance of the sample after phosphorous diffusion. The -cm -cm which is shown in Figure.15.

25 58 Figure.15 Measurement of sheet resistance (CeNSE Laboratory, IISc) The metal contacts are deposited with gold and chrome deposition. This is done with gold-chrome sputtering unit which is shown in Figure.16. The samples loaded for gold-chrome sputtering is shown in Figure.17. In Physical Vapour Deposition (PVD) coating is done using sputtering which is one of the most flexible methods. The coating material chrome and gold is inserted into the vacuum chamber as a cathode under the form of a metal plate. After the chamber has been emptied, the process gas is introduced. Figure.16 Sputtering unit (CeNSE Laboratory, IISc)

26 59 Figure.17 Samples loaded for sputtering of gold-chrome (CeNSE Laboratory, IISc) The sacrificial oxide layer is etched in buffered hydrofluoric acid and the cantilevers and microgrippers are kept in critical point drier for 1 hour to release it properly. Figure.18 and Figure.19 show the CPD (Critical Point Drier) and the sample loaded in the CPD. Figure.18 Critical point drier (CeNSE Laboratory, IISc)

27 6 The SEM images of the fabricated cantilevers are shown in Figure.. The dimensions of the fabricated cantilevers are given in Table.. Figure.19 Samples loaded in critical point drier for releasing (CeNSE Laboratory, IISc) Figure. SEM image of fabricated cantilever beams (CeNSE Laboratory, IISc)

28 61 Table. Dimensions of the fabricated cantilevers Length of cantilevers (l) Width (w) Thickness (h) Gap (d ) µm 5 µm 5 µm µm 16 µm 5 µm 5 µm µm 1 µm 5 µm 5 µm µm 5µm 5 µm 5 µm µm.6 RESULTS AND DISCUSSION.6.1 Pull-in Voltage Analysis of the Cantilever Beam The pull-in voltages of cantilever beam have been computed, using the models mentioned in section 3, for a wide range of dimensions. The specific ranges used for the calculation of the pull-in voltage are based on the paper (Barke 1988). Here, the values of d are selected as very low, to suit the parallel plate approximation (Chowdhury et al 5) as considered in the derivation of the closed-form models. The cantilever beam is modeled and analyzed for a wide range of dimensions using CoventorWare. Meshing is done based on the mesh convergence study. CoSolveEM is used to detect the pull-in voltage. The FEM model used in this study, is shown Figure.1.

29 6 Figure.1 FEM simulation model of the cantilever beam using Coventor Ware The pull-in voltage for different ranges is computed and presented in Tables.3 through Table.6. The values of the pull-in voltage are validated using the CoSolve results; wherein the CoSolve results have been already verified with the experimental results (Chowdhury et al 5). It is reported that the difference between the experimentally measured values and CoSolve results is.83%. Therefore, the authors Chowdhury et al (5), have used the CoSolve results as a bench mark. Though it is claimed, that the accuracy of the pull-in voltage obtained from the FEA based models is the best when compared with that of the closed-form models (Barke 1988), the former is time consuming as against the latter. The results of the pull-in voltage of the closed-form models are compared with those of CoSolve, and the deviation is computed as error. If the closed-form (Chowdhury et al 5); then that model can be considered as a better model. The better suited model amongst them is as shown from Table.3 through Table.6. Here, it is to be noted, that for each closed-form model, the compensation factor applied is unique across all dimensions, and it has taken care of the reduction of error substantially.

30 63 Table.3 Pull-in voltage and %Error comparison for w/d Sl. No. Dimension: h = 1.3µm, d v =.6, µm. w/d V PI, %Error V PI, %Error V PI, %Error V PI, %Error V PI, %Error Cosolve Table.3 shows the pull-in voltage and its % Error for w/d. In this range V PI, 1 error is within %, and less compared to that of the other models. Therefore, for the range of w/d PI, 1 is considered to be a better suited model.

31 64 Table.4 Pull-in voltage and % Error comparison for w/d 1 Sl. No. Dimension:h = 1.3µm, d v =.6, µm. w/d V PI, %Error V PI, %Error V PI, %Error V PI, %Error V PI, %Error Cosolve Table.4 shows the pull-in voltage and the % Error for the range of w/d range V PI, 4 error is within %, and less compared to that of the other models. Therefore, in the range of w/d V PI, 4 is considered to be a better suited model.

32 65 Table.5 Pull-in voltage and % Error comparison for Sl.No. Dimension: h =.75µm, d v =.6, GPa, l =1 µm. ( ) w V PI, %Error V PI, %Error V PI, %Error V PI, %Error V PI, %Error Cosolve Table.5 shows that V PI, 1. Therefore, if the thickness of the beam is equal to the gap between the the model V PI, 1 is considered to be a better suited model.

33 66 Table.6 Pull-in voltage and % Error comparison for h/d Sl. No. Dimension: w = 1.5µm, d µm. h/d V PI, %Error V PI, %Error V PI, %Error V PI, %Error V PI, %Error Cosolve In Table.6, the pull-in voltages and % Error for h/d given. In this range, the error of V PI, 5 is within %, and less compared with that of the other models. Therefore, for the beams with higher thickness (h), the model V PI, 5 is considered to be a better suited model. Moreover, Figures. and.3 shows, the pull-in voltage versus width of the beam, and the pull-in voltage versus height of the beam respectively, with respect to both the closed-form model results and the Coventor results. Here, the results are plotted for a wide range; viz, width varying from.5 µm to 5 µm and height varying from 1 µm to 1 µm. It

34 67 could be seen that the models 1, 4 and 5 are found to be closer to the Coventor results. This coincides with the results already presented in Tables.3 through.6. Therefore, it could be concluded that a particular model is better suited for a particular dimension, but not for the entire range. Chowdhury et al (5) presented a closed-form model for the computation of the pull-in voltage, based on Meijs capacitance model. Investigation shows that this model is suitable only for particular dimensions, but not for a wide range of variations in the dimension of the cantilever beam. Therefore, for the calculation of the pull-in voltage, an appropriate model has to be chosen for the appropriate dimensions of the cantilever beam. It is observed that the pull-in voltage values can be computed in a few micro-seconds, using the closed-form models as against the FEA model, which takes a relatively longer time duration. It is clear that the closed-form models can be used against the FEA model for speedier computation of the pull-in voltage, without sacrificing the accuracy. Figure. Pull-in voltage versus width of the cantilever beam

35 68 Figure.3 Pull-in voltage versus height of the cantilever beam To validate the analytical results, cantilever beams are fabricated and characterized for the pull-in voltage and resonant frequency. The pull-in voltage measured for the µm length cantilever beam is 58 volts. To find the resonant frequency of the cantilever beams, the MSA (Microsystem Analyzer) - Polytech MSA 5 is used, and the experimental set up is shown in Figure Resonant Frequency Analysis of the Cantilever Beam The modal analysis is used to understand the dynamic properties of the structure such as natural frequency (resonant frequency), damping and mode shapes. A cantilever is a continuous system, where its mass and elasticity are distributed all over its volume. Hence, there are infinite natural frequencies. And also, corresponding to every natural frequency, it has a particular shape of vibration, called the mode shape. The lowest natural frequency is called the fundamental or natural frequency, and the corresponding mode is called the fundamental mode or simply the first mode. In this study, silicon cantilevers are fabricated with four different lengths, and their resonant frequency is measured.

36 69 Figure.4 Experimental setup to measure the resonant frequency using the Micro System Analyzer (CeNSE Laboratory, IISc) Figure.5 Arrangement of the probes on the cantilever when measuring the resonant frequency (CeNSE Laboratory, IISc) The arrangement of the probes on the cantilever beam is shown in Figure.5. A periodic chirp signal of.5 volts is applied. The resonant

37 7 frequency which is obtained experimentally, is shown through Figures.6 to.9. Table.7 shows the experimental and simulated results of the resonant frequency and pull-in voltage for different lengths of cantilever beams. Figure.6 Resonant frequency of the µm cantilever Figure.7 Resonant frequency of the 16 µm cantilever

38 71 Figure.8 Resonant frequency of the 1 µm cantilever Figure.9 Resonant frequency of the 5 µm cantilever

39 7 Table.7 Resonant Frequency and Pull-in voltage of cantilever beams Length of cantilever Resonant Frequency (Coventor) (Simulation) Resonant Frequency (experimental) Pull-in Voltage (Coventor) Simulation) (volts) Pull-in Voltage (experimental) (volts) µm KHz KHz µm 68.8 KHz KHz µm KHz KHz µm.74 MHz.9 MHz As the length of the cantilever beam reduces, then the resonant frequency and pull-in voltage increases. This aspect has been experimentally validated as shown in Table.7. The resonant frequencies of the cantilever beams obtained from the simulation and from the experiment are closer. A slight deviation in the resonant frequencies of experimental and simulation values may be due to the residual stress presented in the fabricated cantilever. Due to the limitations in the measurement set up, the pull-in voltage of the cantilever beam of length µm alone could be experimentally measured. The pull-in voltage in this case is found to be 58 V experimentally, while it is V from the closed-form model. Table.8 shows the pull-in voltage results of all the three cases, analytical, FEM and experimental values. Here, the experimental pull-in voltage is higher compared to that of the simulated and analytical results. From Table.8., it could be observed that both V PI, and V PI,5 shows an error less than %. But V PI,5 provides comparatively a very low error value. Therefore it can be chosen as the better option.

40 73 Table.8 Pull-in voltage and % Error comparison with experimental results for h/d 15 Dimensions: width of the beam = 5µm, gap d GPa, Length l = µm, thickness h = 5 µm, h/d =.5 Model Pull-in voltage % Error Analytical and Experimental % Error Analytical and Simulation V PI, V PI, V PI, V PI, V PI, Cosolve Experimental Comparing the results presented in Tables.6 and Table.8 for the range of h/d PI,5 shows less error in both the cases. Thus, the closed-form models are validated with the experimental results, atleast for one case that is h/d Thus, this section presented the closed-form models for the computation of the pull-in votage of the cantilever beams. For the accurate evaluation of the pull-in voltage, the fringe field effects have been considered and with the help of different capacitance models respective pull-in voltage models are obtained. The simulation and analytical results are compared with the experimental results. The pull-in voltage and resonant frequency are calculated and verified, through simulation and experimental methods. The results clearly show that a particular closed-form model has to be chosen for a particular dimension of the cantilever beam based electro static actuator.

41 74.7 CONCLUSION In this Chapter, five closed-form models for the calculation of the pull-in voltages, have been derived from different capacitance models with fringe fields. These models are validated, by comparing its results with the CoSolve FEA results, for a wide range of dimensions. Further, the experimental result of the pull-in voltage is compared with both the simulation and closed-form model results. The closed-form model results coincide with the experimental results but the error percentage is slightly higher than %. The closed-form models for the computation of the pull-in voltage obtained in this Chapter, and the Meijs (Meijs & Fokkema 1984) capacitance model, are found to be useful in modeling the electrostatically actuated single finger comb drive and microgripper which are discussed in Chapters 3 and 4 respectively.