Microelectromechanical Systems HW2

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EE-446 MEMS (Microelectromechanical Systems), Spring 2018, Homework #2. 1. (25’) A poly-silicon surface-micromachined comb accelerometer device structure is shown in Figure 1. The device is similar as ADXL150 accelerometer developed by Analog Devices Inc. which has been widely used for automobile airbag application. Given the design parameters as follow: Beam width Wb=2μm, beam length Lb=200μm, beam thickness tb=2μm, Mass width Wm=80μm, mass length Lm=800μm, mass thickness tm=2μm. Finger width Wf=6μm, finger length Lf=180μm, finger thickness tf=2μm. Assume there are totally 40 comb finger groups (20 in top and 20 in bottom), i.e., Nf=40. (In order for simplification, only 20 comb finger groups are shown in diagram.) The static capacitance gap (when acceleration a=0) between each movable finger and its left (right) fixed finger is d0=2μm. The density of poly-Si: ρ=2.33×103kg/m3, Young’s modulus of poly-Si: E=1.70×1011Pa. Assume all the eight folded-beam sections have the same dimensions (Wb, Lb and tb). Ignore the deformation of the connectors and the movable mass. Use small deflection approximation.

Figure 1. Structure diagram of poly-Si surface-micromachined comb accelerometer

(1). Briefly explain the working principle of this accelerometer device. (2). Calculate the total sensitive mass Ms of the accelerometer, which consists of the movable mass, plus the mass of 40 movable fingers.

X

Y

Acceleration a

anchor

connector

folded beam

left fixed fingers

right fixed fingers

movable fingers

Lb

Wb

Wm Lm mass

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(3). Use equations to find the spring constant kb1 of one section of a folded-beam. (4). Calculate the spring constant of each folded beam. (5). Derive the total spring constant ktot of the whole device. (6). Derive the displacement sensitivity of the whole device. That is, the lateral displacement Sdx in response to unit gravity acceleration input (1g) in X direction. (7). Movable comb fingers constitute differential capacitance C1 (C2) with left (right) fixed fingers. Calculate the static differential capacitance of the device C1=C2=C0=? (8). Find out the resonant frequency of the device for the working mode (vibration along X direction). 2.(15’) A bulk-micromachined Si MEMS piston micromirror is shown in Figure 2. The micromirror is supported by two beams connected to anchors. Aluminum bottom electrode was pre-deposited on substrate right underneath the mirror. Both the Al bottom electrode and the mirror have the same size and they are perfectly aligned to each other. A DC driving voltage Vd is applied between the mirror and Al electrode to activate the mirror to move perpendicular to substrate. It can be used to modulate the phase of incident light. Given the width and length of the square micromirror as: W=L=120µm, initial static capacitance gap (when Vd=0) d0=6µm, total spring constant of both beams: Ktot=0.16N/m, In order to achieve micromirror displacement as Δx=1.5µm, what is the required driving voltage Vd=? What is the pull- down threshold voltage Vth=? What is the maximum controllable displacement of the mirror without pull-down effect?

Figure 2. An electrostatic actuated MEMS piston micromirror

3. (20’) A surface-micromachined poly-silicon comb resonator device is shown in Figure 3. Electrostatic push-pull driving is used to activate the device. Driving voltages V1 and V2 are applied to the left and right fixed comb fingers separately. Assume N is the number of left (right) movable driving fingers, t is device thickness, ε is dielectric constant of air (ε=8.85×10-12F/m), d is capacitance gap, V1 and V2 are driving voltages. If N=40, t=2µm, d=2µm, and we apply the driving voltages as: )cos(12221 tVVV  , )cos(12222 tVVV  1). Calculate the electrostatic forces F1 and F2 experienced by the central mass toward left and right. 2). What is the total driving force Ftot=F1-F2? Is Ftot a constant force or periodic force? Will the central mass vibrate? 3). Will the movable mass move along vertical (Y) direction? Why? 4). Assume for each segment of beam, beam width Wb=4μm, beam length Lb=200μm, beam thickness tb=2μm. Young’s modulus of poly-Si is E=170GPa. What is the spring constant kb1 of each segment of beam? What is the spring constant kbf of one folded-beam? What is the spring constant ktot of the whole resonator? 6). What is the maximum displacement of the movable fingers under above push-pull driving? 7). Assume the total mass of the movable part is Ms=10μg, what is the resonant frequency f0 of the comb resonator?

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Figure 3. Electrostatic push-pull driving of a MEMS comb resonator.

4.(25’) A surface-micromachined MEMS aluminum torsional micromirror is shown in Figure 4. The aluminum micromirror is supported by two beams connected to the anchors on substrate. There are left and right bottom fixed electrodes below the mirror surface. The torsional micromirror is activated by applying driving voltage V between the mirror and one of its two bottom driving electrodes. Given the shear modulus of aluminum as: G=26GPa (1GPa=109Pa), dielectric constant of air: ε=8.85×10-12F/m. The design parameters of the torsional micromirror are shown in Table 1.

Table 1. Design parameters of the aluminum torsional micromirror Design Parameters Values

Mirror width (a) 60μm

Mirror length (L) 60μm

Thickness of mirror and the beams (t) 0.4μm

Torsional beam width (Wb) 2μm

Length of one section of torsional beam (Lb) 40μm

Inner distance between two electrodes (a1) 6μm

Outer distance between two electrodes (a2) 56μm

Gap between mirror and substrate (h) 4μm

(a). 3D view

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(b). cross sectional view

Figure 4. Schematic diagram of the cross-section of the micromirror 1). Find the inertial momentum (It) of the torsional beam. 2). Find the torsional stiffness of one section of torsional beam (St1). 3). There are two sections of torsional beams in this device. Are they connected in parallel or in series? Find the total torsional stiffness of both torsional beam sections (St_tot). 4). What is the nominal maximum allowed torsional angle θmax=? (in unit of degree) 5). In order to achieve a torsional angle of θ=2º, what is the required driving voltage Vt applied between the micromirror and bottom left driving electrode? (Hint: first find normalized rotation angle max/ ). 6). Ignore a1 (i.e. assume α=0), calculate the snap-down angle θsnap of the micromirror. What is the corresponding snap-down voltage (i.e. the maximum driving voltage without snap-down effect) Vmax=? Can the micromirror be driven to remain equilibrium at a torsional angle θ where θsnap<θ< θmax? Why? 5. (15’) A (100) silicon wafer has initial native oxide layer of 0.05µm (thickness). Assume one hour dry oxidation at 1100oC is followed by 6 hours wet oxidations at 1100oC for this Si wafer. Ignore the effect of initial rapid growth regime, use Deal-Grove model to calculate oxide thickness for each step (dry oxidation and wet oxidation) of this Si wafer. What is the total thickness of Si material consumed in the surface due to thermal oxidation in both steps? For (100) Si wafer at T=1100 oC, the following data is given: dry oxidation: A=0.1396µm, B=0.0236µm2/hr, wet oxidation: A=0.1827µm, B=0.5289µm2/hr. Due on 02/27/2018, Tuesday in class.

a2

h

a1

a

substrate

bottom electrodes

mirror