HW#6-2023-Solutions

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University of Michigan *

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414

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Mechanical Engineering

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Dec 6, 2023

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1 EECS 414 Introduction to MEMS Fall 2023 Reading Assignments ● Class Handouts and Notes, “Materials”, and “Mechanical Structures” Homework #6 Solutions Total: 190 Points Handed Out: Thursday Oct. 10, 2023 Due: Thursday Oct. 19, 2023 @ 9 pm 1. If you evaporate metal on the glass at the elevated temperature, what do you expect the stress of the deposited metal film at room temperature? The thermal expansion coefficient of metal is higher than that of the glass substrate. 5 points a) Uniform tensile stress b) Uniform compressive stress c) More compressive stress at the interface d) More tensile stress at the interface e) No stress 2. What is the typical stress you expect in the LPCVD silicon oxide film? 5 points a) Uniform tensile stress b) Uniform compressive stress c) Tensile stress gradient with more compressive stress at silicon interface d) Compressive stress gradient with more tensile stress at silicon interface e) None of these 3. In a bridge structure, which part experiences the highest stress when deflected? (mark all that apply): 5 points a) The top surface of the bridge close to the left anchor b) The bottom surface of the bridge close to the right anchor c) The top surface of the bridge in the middle d) The bottom surface of the bridge in the middle e) All of the above The first two are the correct answers, with the c and d also very close. 4. Intrinsic stress in deposited thin-films can be caused by the following (circle all that apply): 5 points a) Process conditions and the specific morphology of the film b) Effect of Young’s modulus c) Thermal expansion coefficient difference with the substrate d) The thickness of the film

2 5. Elastic materials have the following specific feature (circle only one): 5 Points a) Their Young’s modulus is very high b) They can stretch and not break c) Their stress-strain relationship is nonlinear d) Their strain changes linearly until they break 6. A plastic wrap (Glad warp) is stretched over a glass bowl and sticks to the bowl over its entire perimeter. The bowl is heated up to a temperature without melting the plastic wrap. What happens to the plastic wrap? 5 Points a) It stays flat b) It droops down c) It bulges up 7. This problem deals with the micromachined silicon probe structure for neural recording, as we discussed in the course a while ago, whose cross section along its long axis is shown below: The silicon substrate is boron-doped and is 15μm thick and 2mm long. It does not have any intrinsic stress. It is coated with three layers of LPCVD dielectric films, as shown: 4000 Å of silicon oxide + 1000 Å of silicon nitride + 4000Å of silicon oxide The silicon oxide layer has an intrinsic stress of -300 MPa, while the silicon nitride has an intrinsic stress of +1000 MPa. a) What is the stress of this composite structure? 10 Points Use this equation: s total (t 1 +t 2 +t 3 )= s 1 t 1 + s 2 t 2 + s 3 t 3 So: 0.9 s total =-300*0.8+1000*0.1 s total = -156 MPa b) What should the thickness of the nitride layer be in order to leave this probe flat after it is released, the oxide layers have the same thickness as in part a? 10 Points SiO SiN SiO

3 s total (t 1 +t 2 +t 3 ) = s 1 t 1 + s 2 t 2 + s 3 t 3 0=-300*0.8+1000*t nitride t nitride =0.24μm 8. This problem deals with spring constant formulas in slide 59 of the Structures Lecture. Please show why spring constant equations shown by the two red arrows are approximately equivalent? 10 Points Solution Referring to the table on slide 81, 82 of the structures lecture, equations for displacement as a function of load placement for various boundary conditions are shown. Using this table, spring constants for each structure can be extracted. Generally, this is accomplished by evaluating the expression for displacement at the position of the load specified in your problem. In this case, the loads happen to be at the positions that generate maximal displacement so no algebra is required. Match the provided expression to Hooke’s law and extract the spring constant by examination. Take the expression for a fixed-fixed beam loaded at the center for example,

4 We have: y = -WL^3/(192EI) Noting that here force is written as W, we have Hooke’s law: F = -k y y By rewriting the first equation we have W = -(1/(L 3 /(192EI))y This gives that k y = (192EI)/L 3 Using this process, we can write the general spring constants for both y and z. Then use the lengths defined in the figure to align the spring constants with the two exact structures we are looking at. Note that in the second case, there are two identical springs in parallel so the general spring constant needs to be multiplied by a factor of two. Once the spring constants match those shown in the reference image, the solution is complete. It is clear to see that by evaluating the expressions in the figure we can obtain the same final expressions for both y and z spring constants. 9. This problem deals with the accelerometer structure which has two proof masses as shown below. Proof mass 2 is supported by four tethers (supports, springs), as shown, and proof mass 1 is supported by two springs. This sensor is designed to measure in-plane accelerations along two directions (x- and y-directions). Please assume that: • The six support beams all have the same length, width, and thickness: L=L 1 =L 2 , and W=W 1 =W 2 , and t=t 1 =t 2 Derive expressions for the effective spring constant when mass 2 moves in the x direction, and the spring constant when mass 1 moves in the y direction, as a function of all relevant device parameters mentioned above. 20 Points