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/ ).

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

4/4

(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