Analysis of a silicon piezoresistive pressure sensor

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Milton Thornton
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Analysis of a silicon piezoresistive pressure sensor This lab uses the general purpose finite element solver COMSOL to determine the stress in the resistors in a silicon piezoresistive pressure

1 Analysis of a silicon piezoresistive pressure sensor This lab uses the general purpose finite element solver COMSOL to determine the stress in the resistors in a silicon piezoresistive pressure sensor and hence determine the sensitivity of the device. The pre-lab will use the approximate simple analytic solution for the problem discussed in class to obtain an initial estimate for the sensitivity. In your final lab conclusion, be sure to compare the numerical solution to the approximate solution. If the numerical solution is exact, how accurate is the approximate solution? The basic pressure sensor structure we will study is shown in plan view (IC layout view) in Fig. 1 below. The sensor consists of four p-type resistors implanted/diffused into a lightly doped n-type substrate. The substrate is etched away to leave a thin silicon plate (membrane) supporting the resistors. The top and bottom resistors are transverse resistors: the current flow is effectively perpendicular to the stress. The left and right resistors are longitudinal resistors: the current flow is effectively parallel to the stress. The resistors are laid out so that their centers are all the same distance from the edge of the plate. Figure 1 Pressure sensor layout Figure 2 Pressure sensor side profile The following parameters describe the sensor: L = 2mm = side length of plate a = 120µm = spacing from resistor center to plate edge b = (10 + last digit of student number)µm = width of resistor c = 200µm = length of resistor H = 20µm =thickness of plate R S = 250 Ω/ = sheet resistance of resistors The resistors are diffused into the top surface of the plate with a depth of 1µm

2 1. Pre-Lab Calculations: In class we showed that for a long thin beam of thickness H and length L, the displacement w(x) of the beam in response to applied pressure P is given by: w( x) x a a2x a3x a4 where 2 L P a LP EH EH 2EH a P Here we take x=0 at the edge of the beam. We will use this result as an approximation to the behavior of the square plate, noting that we expect the plate to be stiffer than the beam (and hence deflect less for given applied pressure). x will now be the distance from the edge of the plate, measured from the center of the plate edge normal to the edge. 1. What is the maximum displacement of the plate for P = 100kPa (roughly one atmosphere)? Use this pressure in all the following calculations. Also use E = 160GPa (Young s modulus for Si), π 11 = 7x10-11 Pa -1, π 12 = -1x10-11 Pa -1, π 44 = 138x10-11 Pa The stress σ at the surface of the plate is given by EH ( x) 2 2 w 2 x Using this result, find an expression for σ(x). Sketch σ versus x. 3. Where is σ maximum? (Hint: be sure to consider the value of σ at the edges of the plate) 4. Compute σ at the center of the resistors. 5. Compute the resistor value R before pressure is applied. 6. Estimate the relative resistance change R/R for the two kinds of resistors. Use the value for σ at the resistor center. 7. Re-draw Fig.1 and show how the resistors should be connected to produce a Wheatstone bridge with maximum output sensitivity. 8. If the bias voltage applied across the bridge is 5V compute the sensitivity S (sometimes expressed as V/V/Pa) of the bridge output to pressure. 9. How much electrical power is dissipated in the sensor for 5V bias? 10. If the bridge output is loaded with a typical DVM with an input impedance of 10M, by how much does the bridge output voltage change compared to an infinite load impedance?

Select a 3D space dimension and click the right arrow. 4.

3 2. Numerical Simulation: 1. Start COMSOL 4.X. 2. The default window should open including the Model Wizard. 3. Select a 3D space dimension and click the right arrow. 4. Under Add Physics, choose Structural Mechanics Solid Mechanics, click the + symbol to add the physical model and the press the right arrow.

To begin drawing your structure, expand Model 1 from the drop down list. Right-click on Geometry 1 and select block.

4 5. From the list of studies choose Stationary, and then click on the checkered flag. 6. Your Model Builder will now be populated. Bear in mind that COMSOL 4 processes the model building and simulation sequentially as it is entered in the drop down list. To begin drawing your structure, expand Model 1 from the drop down list. Right-click on Geometry 1 and select block. In the Block window you can now enter the dimensions and position of your new object. Note that you can choose a centered or corner based orientation for your objects position.

5 7. Draw a Si die composed of a plate 2000x2000x20 μm. The top of the plate should be at z = 300 μm. The plate should be broken up into 4 (Or just two to save memory) segments as shown below. Make sure to have at least one line passing through the center and reaching both edges. You have to do this so that you can later plot along that line. Remember to save your work often. 8. Now draw a rectangular frame around the plate which is 300 μm wide and 300 μm thick. There are a number of ways to achieve this so use whichever method works best for you. For example, you could draw each side individually, or use one large block and subtract a smaller block from it by using Boolean Operations. Feel free to rename the blocks to more meaningful names. If you sized everything correctly, the bottom of the frame should be at z = 0.

6 9. Now you need to define the resistor volume for later calculation of the stress in the resistors. Draw 4 resistors of length c width b and thickness 1 μm such that they are diffused in the top plate surface. You ll need to figure out where to put them. 10. The plate, resistors and frame are one solid piece of material. Use Boolean Operations to create one composite object of all the blocks. Right-click Geometry 1 and select Boolean Operations Union. Select all your objects (Ctrl-a) and press the + symbol to add them to the union. 11. Your geometry should now be complete. It is now necessary to define the material of the structure. Right-click Materials in the model builder context window and select Material Browser. Find and select silicon. Note the values for the material properties, then click the + symbol to add the material to your model. Be sure that all your objects are part of the Input objects list. If they aren t, add them or simply select all domains.

8 12. Now we need a piezoresistive coefficient. Right-click Global Definitions from the Model Builder context window and select Parameters. Enter the name pi_44 and an expression of 138.1e-11[m^2/N]. The description is optional. Later we will also need the volume of the resistor to be defined, so add another parameter called resvolume with your resistor volume in cubic meters ( eg: 2e-15[m^3] ). 13. Now we must add all the domains to the physical model. Left-click Solid Mechanics from the Model Builder context window. Under Domains choose Selection All domains.

14. The edges of the plate need to be constrained. Right-click Solid Mechanics and select Fixed Constraint. Add the bottom of your support structure to the selection list. 15.

9 14. The edges of the plate need to be constrained. Right-click Solid Mechanics and select Fixed Constraint. Add the bottom of your support structure to the selection list. 15. Now we need to define the load. Right click Solid Mechanics and select Boundary Load. Select and add the top plate of your structure to the selection list. Add a force of Fz = -1 N/m 2.

10 The problem is now defined by the geometry and boundary conditions. To solve the problem we need to break the structure into blocks or elements, with as many as possible in regions of interest or rapidly changing solutions. 3-D problems, particularly with high aspect ratio features will create extremely large meshes. If you mesh the structure using the default settings, you may get an impractically large number of elements! 16. Now create a mesh. Left-click Mesh 1 and choose a mesh element size of coarser. Right-click Mesh 1 and select Build All to build the Mesh. You can try using smaller element sizes to get a more accurate result if your computer has enough memory for the computation. There are also several tricks we can use to try and reduce the size of the problem. One simple one is to scale the z dimension of our flat plate. The smallest dimension in the problem will typically be used to set the element size. Other tricks include using the symmetry of the problem so you only have to simulate ½ or ¼ of the plate. If you know something about the solution you may be able to select more appropriate element shapes, such as plate or shell elements. We ll leave these techniques for FEA experts for now. 17. With the structure, boundary conditions and mesh in place, we are ready to solve the problem. Right-click Study 1 and select Compute to solve. 18. To plot the displacement (or stress) across the membrane, right-click Results and add a 1D Plot Group. Set up the axes manually using known values and uncheck Preserve aspect ratio. Right-click 1D Plot Group 3 and add a Line Graph. Select the line which you wish to plot over (this is why we made the plate in 4 pieces. For the Y-Axis, choose the appropriate expression eg: displacement (solid.disp) or stress (solid.sx or solid.sy). For the X-axis choose the coordinate over which you wish to plot. You can add other 1D Plot Groups to create additional 1D plots. What is the maximum displacement of the plate?

12 You ll need to be able to export and save your plots. In order to accomplish this, you need to select the Plot Group you re interested in (eg: 1D Plot Group 3) and select Add Image to Report. Now expand the Report tree in the Model Builder context window. Left-click the image you are interested in and specify a filename, file type and location. While still under the Report tree, right click the image you re interested in and select Export. An image of the type you requested will be created at the specified location. Be sure to export all the plots listed in the Submission Checklist.

13 19. Right click on derived values and select Volume integration. Add one of your resistors to the selection list. For the expression, use pi_44/2*(solid.sx-solid.sy)/resvolume. Right-click Derived Values and choose Clear and Evaluate All to compute your answer. Compare these values to the ones obtained in the pre-lab. Calculate the overall sensitivity for the bridge. 20. You can move object by adjusting the geometry settings. Simply change the parameters, re-mesh, re-compute, re-evaluate and re-plot. Can you find a location for the transverse stressed resistors (parallel to the plate edge) that will balance the bridge? Justify selection of a location, then move your resistors and re-simulate. Submit the new values for R/R for the longitudinal and transverse resistors. Submission Checklist All pre-lab calculations Resistor coordinates Cross section plot of total displacement Cross section plot of normal stress sx Cross section plot of normal stress sy Values for ΔR/R for all 4 resistors, and calculation of the sensitivity A discussion of the differences between this solution and the pre-lab New locations for the resistors and re-calculated ΔR/R and sensitivity, as well as justification for your choice of location your COMSOL model file to amiles@doe.carleton.ca.

ANSYS Workbench Guide

ANSYS Workbench Guide Introduction This document serves as a step-by-step guide for conducting a Finite Element Analysis (FEA) using ANSYS Workbench. It will cover the use of the simulation package through