HW#6-2023-Solutions

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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,

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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

6 Solution: 10. Please provide the equation for the maximum deflection of a bridge (a beam supported at two ends) of length L, and uniform rectangular cross-section of thickness h and width w, as a function of a load F applied uniformly across the bridge width in the direction of the beam thickness at the very center of the bridge, as a function of beam dimensions and material properties. 10 Points Solution : This is basically the equation for a standard bridge as the load is applied across the width of the beam (not across the length of the beam). From Roark&Young equations, we see that: 11. Please provide the equation representing the relationship between Pressure ( P ), and deflection ( w c ) in the center of a square diaphragm with side length of a and thickness h , made from one uniform material. Please specify any other parameters used in this relationship. Also show the maximum stress built into the diaphragm when it is deflected in the center. 10 Points Solution : The equation representing relation between Pressure ( P ) and deflection ( w c ) at the center of square diaphragm with side length 2 a is given by, where, E : Youngs Modulus; v : Poisson’s Ratio To find similar relation for diaphragm length a can be found out by replacing a by a/2 in the above equation, ࠵? = ࠵? 1 − ࠵? ! 2 " ℎ " ࠵? " [4.20 ࠵? # ℎ + 1.58 ࠵? # $ ℎ $ ] Maximum stress can be given by the equation,

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7 ࠵? %&’ = ࠵? 1 − ࠵? ! 2 ! ℎ ! ࠵? ! [5.02 ࠵? # ℎ + 3.77 ࠵? # ! ℎ ! + 0.638 ࠵? # $ ℎ $ ] 12. In the figure below, the top view of a cantilever beam is shown. The beam is made from silicon with a uniform thickness of 500μm. A single comb finger is attached to the center of the cantilever beam as shown and is used to apply a force in the direction shown on the cantilever. The bottom of the cantilever is attached to a fixed support. The top of the beam is free to move except when it touches a mechanical stop that is 1μm away from its tip as shown. The other critical dimensions for the beam are shown in the figure. For the rest of this problem assume the following: - Young’s Modulus for all Silicon = 190 GPa - Silicon thickness for all areas = 500μm a) Calculate the spring constant of the cantilever beam in the x-direction before the tip touches the mechanical stop. 10 points

8 Solution: For this part note that we have a cantilever beam where the force is applied in the center and we want to find the deflection at the tip. We use the equations given in the tables at the end of Structures slides. This particular case is shown below. Now we can calculate the force to deflection relationship from the above: ࠵? ( = −࠵? 6࠵?࠵? (2࠵? $ − 3࠵? ! ࠵? + ࠵? $ ), ࠵?ℎ࠵?࠵?࠵? ࠵? = ࠵? 2 ࠵? ( = −࠵? 6࠵?࠵? A 5࠵? $ 8 B = −࠵? 6࠵? ࠵?ℎ $ 12 A 5࠵? $ 8 B = −࠵?. 5࠵? $ 4࠵?࠵?ℎ $ ࠵? ( = −࠵?. 5࠵? $ 4࠵?࠵?ℎ $ = −࠵?. 5(2 × 10 )$ ) $ 4 × 190 × 10 * × 500 × 10 )+ × (2 × 10 )+ ) $ −࠵? ࠵? ( = 0.076 ࠵? ࠵? , ࠵?࠵?࠵? 1µ࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?ℎ࠵? ࠵?࠵?࠵?࠵?࠵? ࠵?࠵? 0.076µ࠵? Note also that if we assume that the finger where the force is applied remains straight and does not bend, then we have to use a different formula for the bending of this cantilever beam. In this case, we can simply assume we have a cantilever beam with a fixed end and a guided end (where the force is applied through the finger) and its length being equal to 1000μm. In this case we have to use the cantilever beam with the guided end formula which is: ࠵? ! = −࠵? 12࠵?࠵? (࠵? " ) → ࠵? = 12࠵?࠵? ࠵? " = ࠵?࠵?ℎ " ࠵? " = 0.76 ࠵?/࠵? As you can see, the spring constant in this case is much larger, and so the beam will likely bend as in the first case shown above unless the finger is really forced to be straight, which will not be the case. b) Calculate the spring constant of the beam after its tip touches the mechanical stop. 10 points Solution: For this part you can assume that when the beam deflects, its tip is always resting on the mechanical stop and does not move down.

9 For this part, the equation for deflection-force is different because now the tip of the cantilever is supported by the stop and so we need to use the following relationship obtained from the formula table at the end of the “Structures” slides: Now we can calculate the force to deflection relationship from the above: ࠵? %&’ = −࠵? 6࠵?࠵? ࠵?(࠵? − ࠵?) ! ( ࠵? 2࠵? + ࠵? ) , ! , ࠵?ℎ࠵?࠵?࠵? ࠵? = ࠵? 2 > 0.414࠵? ࠵? %&’ = −࠵?࠵? $ 6࠵?࠵? ࠵?(࠵? − ࠵?) ! ( ࠵? 2࠵? + ࠵? ) , ! ࠵? %&’ = −࠵? 6࠵?࠵? A ࠵? $ 8 B T 1 5 = −࠵? 6࠵? ࠵?ℎ $ 12 A ࠵? $ 8 B T 1 5 = −1.18࠵? → −࠵? = 0.85࠵? %&’ Therefore, the spring constant is k= 0.85 N/m 13. This problem deals with spring constant of the three mechanical structures below. All views are from the top, and all support beams have a uniform thickness of h, and width of w. The length of all the relevant support beams is shown. The mass is a square in the middle and you can assume that it is rigid. The black regions show where beams are fixed to the substrate. a) Show the expression for the spring constant of this structure when moving in the z direction, as a function of device dimensions and material properties. 10 Points Solution: We assume the mass moves uniformly and flatly up and down. There is one folded beam on the left side, but this one is a little different since on the left there are really two folded beams in L L x y z-axis out of plane

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10 parallel and the spring constant of each is 6EI/L 3 , and 2 cantilever guided beams, 2x6EI/L 3 , in parallel so the total spring constant is: k z = 42EI/L 3 b) Show the expression for the spring constant of this structure when moving in the z direction, as a function of device dimensions and material properties. 10 Points Solution: There are four cantilever guided beams, 48EI/L 3 or one bridge and two cantilever guided beams. c) Show the expression for the spring constant of this structure when moving in the z direction, as a function of device dimensions and material properties. 10 Points Solution: Each beam is like a cantilever beam with two guided ends. Two cantilever guided beams with length of L and two with length of 2L, all in parallel. k z = 24EI/L 3 +24EI/8L 3 = 27EI/L 3 14. This problem deals with the structure whose top view is shown below. It consists of a moveable finger supported by a folded beam as shown. The moveable finger has a thickness of 5μm, and the folded beam sections are 2μm wide, and 5μm thick. The fixed finger also has a thickness of 5μm. A force F is applied as shown to move the structure in the x direction. Calculate the spring constant of this structure in the x direction. - Young’s modulus for all materials is 170 GPa - All other critical dimensions are as shown. 20 Points L L x y z-axis out of plane 2L 2L x y z-axis out of plane

11 Solution: This is a folded beam, but only with one half of the beam compared to what we have seen in class. I spent almost one whole lecture on this topic a few weeks ago when there was some confusion about the derivation of the spring constant for folded beams. I gave you all this problem to see whether people are listening to the lectures or follow the slides that I produce to help understand some of these topics. This exact spring was also used in previous year exams. The spring constant of the whole structure in the x direction is given by: ࠵? ’ = ࠵? ’ ∆࠵? But the spring constant of each half of either the top or the bottom folded beam is given by (see the figure below): ࠵? -.#/012 = ࠵? ’ /2 ∆࠵?/2 = ࠵? ’ ∆࠵? But this is a guided beam whose spring constant we showed in class and is given as: ࠵? -.#/012 = ࠵? ’ ∆࠵? = 12࠵?࠵? ࠵? $ The spring constant of the whole structure is then: ࠵? ’ = ࠵? ’ ∆࠵? = 12࠵?࠵? ࠵? $ = ࠵?ℎ࠵? $ ࠵? $

12 Using this expression, and the values provided (h=5μm, w=2μm, L=500μm, and E=170GPa), the spring constant in the x direction is then calculated to be: k x = 0.0544 N/m 15. This figure was shown in slide #40 of Mechanical structures slide. Please briefly explain what the three figures mean? For example, what do the x-axis and y-axis in these plots represent? Why is plot “After bending” is as shown, and is there any stress in the film, if not please briefly explain why not? 20 Points ∆ x/2 ∆ x F x /2 Force on each of the top and bottom parts of the folded beam Deflection of each half of one of the folded beams F x /2 Force on each half of one of the folded beams Moved position of the folded beam original position of the folded beam

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13 Solution: These three represent the same cantilever beam at successive points in time. The plots describe the internal stresses in the beam as a function of position across the thickness of the beam. They tell us magnitude and direction of stress (tensile or compressive) as a function of height. In the first image, the beam is supported by the sacrificial layer. The beam is in compression. The top face of the beam is under the greatest compressive stress and the bottom face of the beam has the lowest compressive stress. In the second image, the beam has been released. The compressive offset is relieved by the expansion of the bar. However, the stress gradient now remains. Now, the top face of the beam is under compressive stress and the bottom face of the beam is under tensile stress. In the third image, the beam has relieved all internal stresses by bending. The beam is under zero stress throughout its volume.