HW#10-2023

© Do not reproduce or post without Permission of Author Khalil Najafi 1 EECS 414 Introduction to MEMS Fall 2023 Reading Assignments ● Class Handouts and Notes, and “Piezoresistive sensors” ● COMSOL tutorials Homework #10 Total: 220 Points Handed Out: Thursday Nov. 16, 2023 Due: Tuesday Nov 28, 2023 @ 9 pm Note: this is due on Tuesday not Thursday 1. For p-type (100) plane silicon wafer, piezoresistance coefficients becomes maximum in: 5 Points a) [100] direction b) [110] direction c) [111] direction d) [010] direction e) [211] direction f) None of the above 2. In (100) plane silicon, in which orientation is Young’s Modulus maximum? 5 Points a) [100] direction b) [110] direction c) [111] direction d) [010] direction e) [211] direction f) None of the above 3. In a p-type silicon piezoresistive sensor, as you increase the temperature from 25ºC to 100°C, how would its sensitivity change? 5 Points a) Increase b) Decrease c) Almost same d) None of the above 4. The stress on the top surface of a deflected square diaphragm (circle all that apply): 5 points a) Is always tensile b) Is always compressive c) Is always zero d) All the above e) None of the above

© Do not reproduce or post without Permission of Author Khalil Najafi 2 5. The large piezoresistance effect in se miconductors is due to: (circle all that apply) 5 points a) The large change in device dimensions b) The high Young’s modulus c) The Poisson’s effect d) Changes in the band-gap and energy bands e) Joule heating f) The fact that semiconductors are more lightly doped than metals 6. Why are piezoresistors in a Wheatstone bridge aligned along the <110> direction on an n- type silicon wafer? Provide a brief and precise answer. 5 points 7. In piezoresistive pressure sensors with a square diaphragm, four piezoresistors are usually used along the four sides of the square. Briefly explain why this is done instead of using two resistors along only two sides. 5 points 8. This problem deals with the piezoresistive pressure sensor whose structure is shown below. The pressure sensor is to be fabricated on a (100) silicon wafer. One p-type piezoresistor with a resistivity of 7.8 Ω -cm is fabricated on a 3mm square n-type diaphragm of thickness 5μm. The length of the piezoresistor is also approximately 3mm. The diaphragm edge is aligned along the <110> flat. For the rest of this problem assume the following: - Density of Si = 2330 Kg/m 3 - Young's Modulus of Si : 190 GPa (all directions) - Poisson’s ratio = 0.1 (in all directions) - The resistivity of p-type resistor = 7.8 Ω -cm

© Do not reproduce or post without Permission of Author Khalil Najafi 4 10. This problem deals with the force sensor structure whose top view is shown below. The sensor consists of a suspended cross structure that is made of n-type silicon and has a uniform thickness of 5μm. It is supported at four anchor points, as shown. This sensor supports three p-type piezoresistors as shown. The resistors are connected in series and the total resistance R is measured. Note that the three piezoresistors, R 1 , R 2 , and R 3 , are rectangular and have equal dimensions, and have a nominal value of R 0 . A force F (not shown) is applied in the center of the structure and vertical to it (that is perpendicular to the plane of the paper). This applied force deflects the structure and creates stress in the different regions of this structure. Note that the piezoresistors are electrically isolated from the rest of the silicon beam. The resistance R is measured as output. The beam is made of silicon and you can assume that silicon has Young’s modulus E, Poisson ratio n , and assume all piezoresistors have longitudinal piezoresistance coefficient of p L and transverse piezoresistance coefficient of p T . Part a: Find and expression for the approximate spring constant, k, of this structure as a function of the dimensions of the structure (Length=L, Width=W, and Thickness=h), and any relevant material properties? 10 Points Part b: Find an expression for the force sensitivity of this structure. Sensitivity is defined as ࠵? = ∆"/" ∆$ . Your expression should be defined as a function of the longitudinal and transverse piezoresistance coefficients ( p L and p T ), the dimensions of the structure (Length=L, Width=W, and Thickness=h), and any relevant material properties. 20 Points L W R1 R2 R3 R Support Anchors

© Do not reproduce or post without Permission of Author Khalil Najafi 5 11. The following shows six different designs (layouts) of piezoresistive pressure sensors. The sensors utilize four piezoresistors (R1-R4) that are connected in a standard Wheatstone bridge configuration (connections are not shown). All electrical connections to resistors are to their short side. The side length of the square diaphragm used in these sensors is equal to a. Assume the piezoresistor length is much smaller than the diaphragm side length a. Assume that all piezoresistors are p-type, with resistivity of 10 Ω cm, and are twice as long as they are wide. You are supposed to identify the THREE (3) designs that have higher pressure sensitivity among the six? Justify and explain your answer. Please CIRCLE the three highest sensitivity designs. Note that I do not need you to calculate each exact sensitivity. Just find the three with the highest sensitivity based on qualitative or semi-quantitative analysis. 20 points 12. This problem deals with the piezoresistive pressure sensor shown below, fabricated in (100) silicon. A p-type silicon piezoresistor with a resistivity of 7.8 Ω -cm and the shape as shown in the figure is placed on the edge of the diaphragm. Note the resistor has three sections with the length of each section as shown. All three sections have the same width w. You can assume that the diaphragm side dimension is much larger than the length L of the piezoresistor. Calculate the total relative resistance change ∆ R/R of this pressure sensor. Note that you are supposed to obtain a numerical answer for this part as a function of the longitudinal stress s l . All the information you need, such as relevant piezoresistor coefficients, etc., can be obtained from various sources you have available, like your textbook, and my lecture notes. Make sure that you show your work. 30 points Diaphragm Side Length=a P-Type Piezoresistors (A) (B) (C) (D) (E) (F) R1 R2 R3 R4 <110> 3a/5 3a/5

© Do not reproduce or post without Permission of Author Khalil Najafi 7 (a) Calculate the longitudinal piezoresistive coefficient and transverse piezoresistive coefficient of the n-type piezoresitors at room temperature. 10 points (b) Find the accelerometer sensitivity for three resistors, ∆ R1/R1/ ∆ a, ∆ R2/R2/ ∆ a, and ∆ R3/R3/ ∆ a, respectively, to acceleration in y direction at room temperature. 10 points (c) Find the accelerometer sensitivity for three resistors, ∆ R1/R1/ ∆ a, ∆ R2/R2/ ∆ a, and ∆ R3/R3/ ∆ a, respectively, to acceleration in z direction at room temperature. 20 points (d) When you operate this sensor at -25 o C, what will be ∆ R1/R1/ ∆ a? 10 points