Chapter 2 Modeling of Scratch Drive Actuator
2.3 Dynamic analysis of SDA
To explore the dynamic behavior of SDA including friction effect, a mass-spring-damper model of single degree of freedom (SDOF) is proposed as an approximation. The equation of motion and its normalized form are given as Eq. (2-25) and Eq. (2-26)
The coordinate x and its first and second derivatives of with respect to time are the displacement, velocity and acceleration of SDA, respectively. The lump-mass m defines the total mass of SDA array including the ring around SDA. The coefficient k defines the spring constant of the suspended tether spring. The damper may contain one or two types of friction; the first is viscous damping of coefficient c of contact surface between main beam and insulated layer.
The second one is the Coulomb friction of coefficient µ accounting for the stick-slip phenomenon in contact surfaces. The force FN is the normal force between the contact area of main beam and insulated layer. In MEMS, the body force such as gravity force is often much less than surface traction force as the electrostatic force. So the normal force is simplified to contain only the electrostatic force in the contact area. The force FO is the amplitude of driving force which is the resultant horizontal component of the electrostatic force applied along the main beam of SDA, the force along bushing is ignored here. The driving voltage is in sinusoidal wave sin(ωt) of angular velocity ω. However, it should be noted that the electrostatic force is an attractive force between contact surfaces, no matter the charge on main beam is positive or negative. Therefore, the driving force is expressed as the absolute value of driving voltage waveform, i.e., abs(sin(ωt)) multiplying FO, which is the motive force that makes SDA scratch forward. The friction terms in Eq. (2-25) may be and rearranged as following
26)
In Eq. (2-27), sign(~) is the sign function result of the operand in parenthesis; the gain (c/m) is defined as a regulation factor that modifies viscous friction coefficient. The offset (µFN/m) is the dry friction force that should be overcome in motion. To simplify the derivation, the static and dynamic dry friction coefficients are assumed to be same as µ.
When the Coulomb friction is activated, the analysis of displacement (total travel distance) response is too complex to derive an exact and compact analytical solution. To solve this type of nonlinear vibration problems, some methods have been developed such as perturbation method, averaging method, multiple scales method and direct separation of motions [26-30].
Instead of these methods, direct numerical simulation method as Simulink in Matlab has been adopted to investigate the dynamic response in a straight forward way to simulate the dynamical behavior of SDA.
To evaluate the fatigue behavior of SDA, the information of stress bounds are needed. Eq.
(2-21) may calculate the maximum and minimum stresses as following
The equivalent stress σR, the fully reverse stress of same fatigue life [31], is expressed as
The minimum moment Mmin at specific input, though not easily be derived, lies between zero for fully discharge mode and the maximum moment at Vp in constantly contact mode.
27)
Figure 2-1. Structures and elements of SDA. (a) Support beams are free at contact part.
(b) contact part is constrained to slide without rotation [24].
substrate
dielectrics rail
main plate
bushing support beam W
L
Wp
Lp
t
h
(a)
(b)
Figure 2-2. Five states in SDA operation at different input voltages: (a) initial, zero input; (b) pull-in, voltage Vpi; (c) priming, voltage Vpr; (d) post-priming, voltage Vpp; (e) one-step forward after discharge.
Figure 2-3. Coordinate system and electrostatic force density in post-priming configuration. The electrostatic force density in contact region is constant. In non-contact region, the electrostatic force density follows Petersen model [8].
Figure 2-4. Step size of SDA operated in fully-discharge mode. (a) Initial state. (b) The main plate rotates around bushing paw and causes lateral backward movement
∆x1. (c) The curvature shortening effect causes the bushing top lateral movement
∆x2 backward more. Also, the bushing paw displaces ∆x3 forward. (d) SDA is discharged fully with one net step size ∆x =∆x3−∆x1−∆x2.
Figure 2-5. Step size of SDA operated in constantly-contact mode. (a) Initial state at voltage VPP. (b) Partly discharge to voltage VP , main plate springs back. (c) SDA is again charged to VPP, the step size ∆x =∆x(VPP)−∆x(VP).
Chapter 3
Performance Enhancement of Scratch Drive Actuator with Modified Flexible Joint
3.1 Concept design of low voltage Scratch Drive Actuator (LVSDA)
The concept design of the proposed LVSDA is shown in Figure 3-1, where two narrow beams as flexible joint are placed between the main plate and scratch plate. The operation procedures of LVSDA and conventional SDA are illustrated in Figure 3-2. They are both at rest initially with zero input in Figure 3-2(a). When the input increases to snap-through voltage, the free edge of main plate of LVSDA or SDA will touch the dielectrics (Figure 3-2(b)). At the same input voltage, LVSDA will snap through more due to the smaller flexural rigidity to generate a larger bushing lateral displacement, as shown in Figure 3-2(b). At an even higher input voltage, the main plate will contact with the dielectrics more area and further push the bushing forward, as Figure 3-2(c). After discharging, LVSDA or SDA will bounce back to complete one cycle with one forward step size ∆x, as Figure 3-2(d).
In order to fabricate and test the proposed LVSDA, the geometric parameters in LVSDA are defined in Figure 3-3(a). The design of testing device is illustrated in Figure 3-3(b), where a tether spring consisted of four box springs links four LVSDAs to the contact electrode. The comparison between the proposed LVSDA and conventional SDA is based on the same device size. It means all LVSDAs and SDAs here have the same total plate length L and plate width W, which are fixed as 80 µm and 65 µm, respectively.
L will remain to be 80 µm, as well as the plate length of SDA. The thickness t of plate, spring, and support beams, as well as the bushing height, of LVSDA and SDA is all set at 2 µm.
Each support beam has length Lp= 25 µm and width Wp= 3 µm. Different flexible joint
dimensions and locations are designed to investigate their effects on performance. A shorter or wider flexible joint will provide a more rigid joint. Two different flexible joint widths Wj are designed as 6 µm and 12 µm. The designed flexible joint length Lj includes 15 µm and 20 µm.
The designed scratch plate length Ls includes 10 µm, 15 µm, and 20 µm. A shorter Ls means the flexible joint is closer to the bushing. Due to different combinations on scratch plate length and flexible joint length, the corresponding main plate length Lm of LVSDA includes 50 µm, 45 µm and 40 µm to keep the total plate length as 80 µm. The designed LVSDA dimensions are denoted as LV-Ls-Lj-Lm-Wj. For example, the test device expressed as LV-10-20-50-12 has following dimensions: Ls=10 µm, Lj=20 µm, Lm=50 µm, and Wj=12 µm. Eight types of LVSDA used in this work are listed in Table 3-1.
The spring constant of one box spring can be expressed as [24]:
k1=EtWr3
/(Lr3
), (3-1)
where E is the Young’s modulus of spring material, t and Wr are the thickness and width of spring beam, respectively, and Lr is the half length of box spring. For the tether spring composed of 4 box springs, the equivalent spring constant becomes
k = EtWr3/(4Lr3), (3-2) The output force of LVSDA or SDA can be determined from the result of spring constant multiplied by the deflection of spring.
3.2 Qualitative Analysis of LVSDA
In order to analyze the deflection of LVSDA under given input, a detailed free body diagram for input between snap-through and priming is plotted in Figure 3-4. The equivalent electrostatic force is labeled as FE. To satisfy this demand, the flexural rigidity EI should
decrease or the moment should increase. As the increasing of moment needs increasing the input which violates the purpose of low-voltage, the reasonable way is to reduce the flexural rigidity EI of the device. By decreasing EI, the LVSDA can be deflected to provide certain force at lower voltage than SDA. In other words, the electrostatic force to resist the mechanical force can be reduced.
3.3 Finite element analysis of LVSDA
The finite element analysis is realized by SolidWorks® and CosmosWorks®. The behavior of LVSDA has been analyzed in detail and will be compared with the experimental results. For stress simulations, since plastic deformation may happen in flexible joint at high driving voltage, the finite element analysis (FEA) in this work are performed by elasto-plastic model in nonlinear static analysis package of CosmosWorks. According to the material properties of micro electroplated nickel [26-29], the Young’s modulus E is selected as 171 GPa, yielding stress is 323 MPa and tangent modulus in plastic region is assigned as 17.1 MPa. The fatigue stress is 195 MPa, and the ultimate stress is 560 MPa. Parameters used in FEA are listed in Table 3-2. The solid modeling process is first to sketch the main body of LVSDA and substrate separately, combining them as assembly with suitable conditions. Then apply material properties to each part; assign constraints according to requirements; apply electrostatic pressure and external load, the pressure is specific for every input voltage. Assign contact conditions on faces that may contact each other during loading, the no-penetration condition is the most important setting to make sure the LVSDA will not penetrate into the ground layer.
The meshing size of element often uses default value or even finer value; the results always converge at about 1.0e-3 level. The friction coefficient between contact surfaces is set as 0.2 in all cases.
The electrostatic force on scratch plate, flexible joint, support beams and bushing are neglected. The left edge of main plate is set to be pinned with the dielectric layer and no penetration is allowed. The electrostatic pressure for a given input voltage is approximated by the parallel capacitor model for each strip of 2.5 µm along the main plate. Then stress in the test structure and the contact length between main plate surface and dielectric layer at different input voltages can be simulated. In simulations, the input voltage starts from 40 volts to 120 volts with increment of 10 volts, since all measured threshold voltages are at least 40 volts. A typical simulated result from the CosmosWorks is shown in Figure 3-5 for device LV-20-20-40-6 at 70 volts. It is found that the maximum stress happens around flexible joint corner next to the main plate and exceeds the yielding stress (Figure 3-5(a)). The contact length between the main plate and dielectric layer is about 20 µm (Figure 3-5(b)). The fully-discharge shape is shown in Figure 3-5(c), in which permanent vertical displacement due to plastic deformation will cause degradation of output force level for same input when yielding begins.
Figure 3-1. Concept design of low-voltage Scratch Drive Actuator (LVSDA) with flexible joint.
Figure 3-2. Operation procedures of conventional SDA and LVSDA: (a) at rest initially; (b) snap-through ;(c) contact more at higher input voltage; (d) discharge and complete one step size ahead.
Figure 3-3. Design layout of micro-electroplated nickel SDA. (a) Definition of LVSDA geometric parameters. In another word, the sum of main plate length Lm, flexible joint length Lj and scratch plate length. (b) Four LVSDAs with tether spring connected to the contact electrode.
Figure 3-4. Qualitative analysis on flexible joint effect. (a) Equivalent electrostatic force is FE . (b) Moment M along device similar to SDA. (c) ( M/I ) ratio changes hugely along flexible joint in LVSDA. So, the flexible joint reduces flexural rigidity, increases bending stress, induces more deflection and makes LVSDA easier output at lower voltage.
Figure 3-5. Finite Element Analysis (FEA) results of LVSDA with Ls=20, Lj=20, Lm=40 and Wj=6 µm ( LV-20-20-40-6) at 70 V. (a) Stress distribution, where the maximum von-Mises stress happens at the flexible joint corners close to the main plate. (b) Side view at maximum loading, contact length is about 20 µm. (c) Side view after loading is fully-relax, contact length is about 2 µm. The permanent vertical deflection UY shows plastic deformation at left side on flexible joint that will affect the LVSDA output performance hereafter.
Table 3-1. Notation of LVSDA testing types
Notation
LV-Ls-Lj-Lm-Wj
Ls (µm)
Scratch Plate Length
Lj (µm)
Flexible Joint Length
Lm (µm)
Main Plate Length
Wj (µm)
Flexible Joint Width
LV-10-20-50-6 10 20 50 6
LV-10-20-50-12 10 20 50 12
LV-15-15-50-6 15 15 50 6
LV-15-15-50-12 15 15 50 12
LV-15-20-45-6 15 20 45 6
LV-15-20-45-12 15 20 45 12
LV-20-20-40-6 20 20 40 6
LV-20-20-40-12 20 20 40 12
Table 3-2. Simulation parameters for FEA
Parameters magnitude Parameters magnitude
Total main plate length, L, µm 80 Support beam width, Wp, µm 3 Scratch plate length, Ls, µm 10, 15, 20 Plate thickness for all, t, µm 2 Flexible joint length, Lj, µm 15, 20 Bushing height, h, µm 2 Main plate length, Lm, µm 40, 45, 50 Bushing width, Wb, µm 3 Main plate width, W , µm 65 Box spring length, Lr, µm 110 Flexible joint width, Wj, µm 6, 12 Spring beam width, Wr, µm 3 Support beam length, Lp, µm 25 Young’s modulus of nickel , E, 171 Spring thickness, same as t, 2 Shear modulus, G, GPa 69 Tether spring constant, 1.98 Yielding stress, σy, MPa 323
Tangent modulus, σt, MPa 17.1 Element size, µm 2.5
Total nodes 26220 Total elements 15525
Note: The structure material, microelectroplated nickel, is defined as isotropic hardening of von-Mises plasticity once yielding. The solver use sparse matrix of Newton-Raphson method based on force control algorithm.
Chapter 4 Fabrication and Results
In order to verify the validity of the proposed models of SDA and LVSDA, a two-mask micro-electroplated nickel surface-micromachining process is chosen to construct Ni SDA and LVSDA array structures. The test structure is composed of four SDAs or LVSDAs connected to a suspended spring which is anchored to the contact electrode.
4.1 Fabrication process
The fabrication process is summarized as following: (a) Starting from a 4-in. RCA-clean (100) wafer, a 6000Å thick LPCVD silicon nitride is grown in furnace as the dielectric layer (Figure 4-1(a)). (b) First patterning process: coating 2µm photoresist FH6400 as sacrificial layer, 90℃ soft bake 10 minutes, hydration reaction 20 minutes, then creating the pattern of bushing and contact electrode by the first mask (Figure 4-1(b)). (c) Sputtering process: 200Å thick Ti and 1500Å thick Cu are sputtered sequentially as the adhesive layer and seed layer, respectively (Figure 4-1(c)). (d) Second patterning process: coating 5µm thick photoresist AZ9260, then creating the pattern of electroplating mold by the second mask (Figure 4-1(d)).
(e) Electroplating nickel process: electroplating the Ni test structure with Watt bath with current density 10 mA/cm2 over 10 minutes to form 2µm thick Ni (Figure 4-1(e)). (f) Release process: using Acetone to remove electroplating mold AZ9260, then removing Cu seed layer by CR-7T solution about 20 seconds. Ti adhesive layer is then removed by BOE solution about 10 seconds, then the removal of FH6400 sacrificial layer by Acetone for 30 minutes. By immersing in IPA solution and vibrating about 20 seconds for releasing, then drying at 60℃, a fully suspended test Ni structure can be obtained as in Figure 4-1(f). Typical fabricated results are shown in Figure 4-2, including (a) the SDA array, (b) the LVSDA array, (c) close-view of
bushing, and (d) close-view of LVSDA. The lithography parameters are listed I Table 4-1;
while the electroplating parameters are listed in Table 4-2 for reference.
The experimental equipments for loading test include an optical microscope mounted with a CCD camera, PC with image process software, chip position table, function generator, high-voltage power amplifier and probe station. The test chip is fixed on the position table by vacuum chuck. Two probes are adjusted to touch the contact electrode of SDA array and substrate. The test signal is generated from the function generator, amplified and calibrated by the high-voltage power amplifier in sinusoidal waveform of 500 Hz. This input is composed of a DC offset at 30 volts often and AC voltage depending on the control command. For example, when the driving signal needs to vary in the range of 30 to 120 volts, the chosen DC offset is 30 volts and AC amplitude is 90 volts in peak to peak definition. The motion of SDA or LVSDA array is recorded by the CCD in computer and then analyzed by using image process software.
Typical measurement set-up and driving test are shown in Figure 4-3.
Figure 4-1. Fabrication process of micro-electroplated nickel SDA. (a) LPCVD Si3N4, (b) bushing and fixture patterning, (c) Cu/Ti seed layer, (d) device structure patterning, (e) nickel electroplating, (f) release.
(a)
(b)
(c)
(d)
(e)
(f)
Table 4-1. Lithography parameters
Photoresist FH6400 AZ9260
Thickness 2.1 µm 5 µm
Spinning 500 rpm (10 sec) 1200 rpm (25 sec)
1000 rpm (10 sec) 5000 rpm (30 sec)
Waiting 5 min 10min
Soft Bake 2 min 10 min
Hydration 5 min > 20 min
Exposure 3 sec (+20%) (46mW/cm2) 6 sec (+20%) (46mW/cm2) Develop 30 sec (AZ-400k) 2 min 30 sec (AZ-400k)
Fix 1 min (DI Water) 1 min (DI Water)
Hard Bake 10 min X
Table 4-2. Electroplating parameters
Mask Mask #2
Metal Nickel
Area 3.526 cm2
Current Density 10 mA/cm2 Electroplating Rate 0.2 µm/min
Figure 4-3. Driving test of released devices. (a) Measurement set-up, and (b) driving test of SDA array.
Figure 4-2. SEM pictures of fabricated results. (a) Top view of SDA array; (b) Top view of LVSDA array; (c) Side view of bushing and main plate; (d) Close up of LVSDA.
4.2 Analytical and testing results of SDA
In this subsection, the simulated and numerical results of SDA model developed in chapter 2 will be presented and compared with testing results of Ni SDA of 80 µm long and 65µm wide.
Some discussions will be made on the key factors of SDA performance.
4.2.1 Static analytical results of SDA
Numerical codes on Matlab software are developed to perform the deflection analysis on the first proposed model based on parameters listed in Table 1. The non-contact length ln is calculated first from Eq. (2-16) at each specified input voltage from 40 to 120 volts with increment in 10 volts. The results are plotted in Figure 4-4 where the non-contact length in this work is shorter than that of Kazuaki’s work. As the simulation of electrostatic force in this work is better than Kazuaki’s work, the chosen size of SDA may be smaller and save the chip size.
The priming voltage Vpr of the device is calculated from Eq. (2-17) as shown in Figure 4-5. All models, including this work, Linderman’s and Kazuaki’s works, how that shorter plate needs higher priming voltage to maintain the contact between main plate and dielectrics. This work still shows smaller priming voltage than that of the other two models for same plate length. The step size for different control method out of Eq. (2-23) and Eq. (2-24) are plotted in Figure 4-6.
The step sizes in fully-discharged mode of this work and Kazuaki’s work have been shown in Figure 4-6(a). As the main plate tends to turn over at the end of discharge, the SDA operation is not stable enough. Consequently, constantly-contact mode is a stable way for SDA operation as shown in Figure 2-5. The SDA is driven between two input levels which are all equal to or larger than priming voltage, and the main plate always keeps contact with dielectrics during charge and discharge cycles. The step size in this mode is the step size at maximum input minus the step size at minimum input. Figure 4-6(b) shows the results when priming voltage is the minimum.
In Figure 4-7(a), the first reported bending moment diagrams of SDA are shown based on Eq. (2-20). The bending moment is zero over the entire contact part and at the starting point of non-contact part. Then it increases rapidly to maximum; after that, it goes down to the local minimum at the right end. The value of maximum moment and where it happens depends on the input level. From the trajectory of maximum moment Mm, it is obvious that Mm becomes higher and shifts rightward as input increases. The maximum static bending stress σm along main plate for different input is plotted in Figure 4-7(b). This maximum stress, calculated by Eq. (2-21) as from 182 to 495 MPa for input voltages from 40 volts to 120 volts, also locates at same place of maximum moment for same input voltage.
Some mechanical properties of micro-fabricated Nickel useful in this work are adopted from literature; the Young’s modulus of Nickel is chosen as 171 MPa; the yield strength is chosen as 323 MPa [32, 33]; the ultimate strength is chosen as 560 MPa [34, 38]. As the fatigue strength, [29] reported a value of 195 MPa for one million life cycles and constructed the S-N curve of Nickel under cyclic reverse loading test. For the driving test in continuously charge/discharge, the suitable failure mode is fatigue instead of yielding stress for static load.
This topic will be discussed in subsection 4.2.2.
4.2.2 Dynamic analytical results of SDA
As some devices fail at higher voltage, it is proper to adopt the fatigue design theory [30] for cyclic loading to explore the cause. For a fully reversal loading with zero average stress, the stress amplitude σR and the device life cycles N is related by the Basquin’s equation,
where the coefficients A and B can be determined from the fatigue test data. For a non-fully
where the coefficients A and B can be determined from the fatigue test data. For a non-fully