EE 147/247A Prof. PisterFall 2017

Homework 1Due Tuesday 9/5/2017 (9am Wednesday)

Questions:

Figure 1: Simple variable capacitor1. We will analyze a variable capacitor using CoventorWare. Start a new CoventorWare project and

open the Build Solid Model interface. To start your homework, create a new project, and then click on “Build Solid Model”.For the materials file, we will be using MUMPs.mpd, which you can find under:

<UserName>/Design_Files/Shared/MPD/MUMPs.mpdFor process, we will be using a predetermined two-mask SOI process called SOIMUMPS. You should be able to find this file here:

https://berkeley.box.com/v/EE147A-HW1Download the file from the link above and add it to your project devices folder.For the layout option, you will be constructing your own layout, so select “Create a new Layout from Process”.Note: If you look at the design rules for SOIMUMPS [1], you will find that the process has two options for the Silicon thickness you wafer:

From looking at the Process file, which you can open in Coventor, you can see that the Silicon thickness we are using for this homework is 10µm. Make sure to use this thickness for your calculations as well.

However, first some hand analysis:a. Calculate, by hand, the spring constant of the cantilever beam, the mass of the plate at the

end of the beam, and the resonant frequency of the structure, and the capacitance between the two fingers.

Solution: 5 pts. total: 1 pt. for effort, 1 pt. for correct spring constant, 1 pt. for correct mass, 1 pt. for correct resonant frequency, 1 pt. for correct capacitance

Beam dimensions:Lb= 100µmwb= 2µmtb= 10µm

Plate dimensions:Lp= 48µmwp= 48µmtp= 10µm

Young’s modulus E= 150GPa (based on Materials List for SOIMUMPs)Beam spring constant:

k beam=E t 3 w4 L3

k beam=(150 GPa)(2 µm )3(10 µm)

4(100 µm)3

k beam=3[ Nm ]

Mass of plate (ignore mass of beam):

mplate=Vρ

mplate=(48 µm ) ( 48 µm ) (10 µm )(2300[ kgm3 ])

m plate=53[ng]

However, if we take into account the etch holes:

mplateholes=( 48 µm−24 µm ) (48µm−24 µm ) (10 µm )(2300 [ kg

m3 ])m plateholes

=13.25[ng]

If we also add the mass of the finger, the total mass is:

mplatetotal=15 [ng]

Resonant frequency:

ω=√ kbeam

mplate

ω=√ 3 [ Nm ]

15 [ ng ]

ω=447∗1 03[ rads ]

ω=447 k [ rads ]

Capacitance between finger 1 and finger 2:

C=ε 0Ad

where A is the overlap area.

C=(8.85∗10−12[ F

m ])(37 µm ) (10 µm )

2 µm

C=1.63∗10−15 [F ]

C=1.63 f [F ]

b. How much will the end of Finger1 deflect with a 1uN load at the tip of the finger?

Solution: 1 pt. for approximately correct answerF=1µN

y plate=Fk

y plate=1∗1 0−6[ N ]

3[ Nm ]

y plate=0.33 µm with1 µN of force

c. How much will the tip of the plate deflect with 10V between the two fingers? You can assume the electrostatic force is a point load at the end of the beam (totally incorrect assumption, as we will see in a couple of weeks, but fine for now).

Solution: 2 pts. total; 1 pt. for approx. correct electric force, 1 pt. for approx. correct deflection

F el=12

ε0V 2 Ag2

F el=12 (8.85∗1 0−12 F

m )(10 V )2(37∗10−6 ) (10∗1 0−6 )

(2∗10−6 )2

F el=40 nN

y= Fk

y=13.3 nm

Now we will analyze the structure using CoventorWare:d. Using the Soi layer, create a layout of the variable capacitor from above. You can use the

SoiHole layer to create the holes in the mass. Important Final Layout Step: As the final step of the layout, use the layer Subtract to draw a rectangle around your layout. Then, select Modify-> Tools-> Boolean-> By LayerHere, you will subtract the Soi layer from the Subtract layer (or the rectangle you just drew). Make Input Layer 1 the Subtract layer, Input Layer 2 the Soi layer, and Output Layer will be NOT_SOI. Select “not” as the Operation. It should look like this:

e. Save your layout, and build the model once the layout is completed. Take a screenshot of your layout.

Solution: 1 pt. for screenshot of layout, as seen below

Layout before the Final subtract step:

Layout after the final subtract step:

*Ignore the white circle that appears on the bottom left of the plate. It’s just a rendering issue on the Layout Editor.

f. Build the solid model of the layout in CoventorWare. Take a screenshot of your solid model. Hide the substrate layer and add the SOI and oxide layers to the mesh model. Mesh the model using 5x5x5 Manhattan brick elements, and generate the mesh. Take a screenshot of the meshed model.Important Note: If you open the Preprocessor and only get a black screen, close CoventorWare. This is a rendering issue. To solve it, in your terminal enter:

export COV_NO_OPENGL=1

Solution: 2 pts. total; 1 pt. for screenshot of solid model, and 1 pt. for screenshot of meshed model, as seen below

Screenshot of solid model:

Screenshot of the meshed model:

Right click on the SOI option under Mesh Model, and select Properties. Under Properties-> Analysis Options, make sure Conductor is selected, and not Dielectric. Click OK.

Note: In the images displayed, the model is not rendering the mesh in order to allow for a clearer image.

Select and name the bottom face of each oxide part as Anchor.

Name the inner face of Finger1 Tip.

Name the variable capacitor conductor Finger1 (expand the Conductors/Dielectrics tab to see the different conductors available) ActuatorElectrode and name the anchored Finger2 GroundElectrode. Image examples are shown below.

ActuatorElectrode

GroundElectrode

Save your model.Begin a new MemElectro analysis. Put 1V between ActuatorElectrode and GroundElectrode to simulate charging the plate. Run the analysis and view the results.g. Open the capacitance matrix. Why is the capacitance different from your analytically

calculated capacitance?

Solution: 2 pts. total; 1 pt. for approx. capacitance value, 1 pt. for answer to question

Capacitance matrix:

The capacitance value we found in the matrix is: C=6.1fF. The calculated capacitance value is 1.63fF. The calculated capacitance is different from the analytical capacitance because Coventor takes into account the fringing fields of the fingers.

h. View the 3D results of the charge density. Zoom into the corners of the structure and take a screenshot. Why is the charge density greatest at the edges and corners?

Solution: 1 pt. for correct answer to question

A screenshot of the 3D results of the charge density can be seen below.

Zooming into the fingers, we observe the image below:

Notice that the corners are a darker color, closer to red than the rest of the plate, indicating a greater charge density in these areas. This higher charge density is due to the higher concentration of field lines at the corners.

Begin a new MemMech analysis. Under SurfaceBCs, put the fixall condition on patch Anchor and put a LoadPatchNodes condition on patch Tip. Define LoadPatchNodes with a vector LoadValue and -1 in Y. Units for this boundary condition are in uN and act on the entire face. Run the analysis.i. How much did the end of the plate deflect? Is there good agreement with your hand

calculation?

Solution: 2 pts.; 1 pt. for plate deflection value, 1 pt. for answering question

The resulting displacement matrix from this simulation can be seen below:

* Keep in mind that the displacements shown in the Coventor matrix results are already in microns.

The relevant displacement in this case is the minimum Node Y Displacement, which is about 1.53µm in the negative ‘y’ direction. The hand calculations resulted in a displacement of 0.33µm. The discrepancy comes from the fact that we were highly approximating the deflection in our hand calculations by assuming that the force was being applied at the point where the beam and the plate meet.

j. Open the deflected structure in the visualizer. Use the “Deform using displacements” option under the CoventorWare menu. Take a screenshot.

Solution: 1 pt. for screenshotBelow is the structure without deformation.

To observe the deflection, a zoom in of the fingers is shown below: without deformation on the left, and the deformed structure on the right.

Again, to observe the deflection more easily, another zoom in of the fingers is shown below: without deformation on the left, and the deformed structure on the right.

Begin a new MemMech analysis, this time change the physics in the first drop menu of the MemMech Settings window to “Modal (non-equilibrium).” Under SurfaceBCs place the fixall condition on Anchor. Calculate the first 5 mode shapes.k. Open the results and look at the modeDomain table. Record the frequency of each mode

shape. How does your calculated resonant frequency compare to the first mode? Why would it be different from your simple analysis?

Solution: 2 pts.; 1 pt. for frequency of first 5 mode shapes, 1 pt. for answering why they are different from the hand analysis

Below is the frequency of the first five mode shapes:

The calculate hand analysis resulted in a frequency of 447k rad/s, which is in the same order of magnitude as the simulated results. The simulated resonant frequency is given in Hz, so it would be equal to 186k rad/s. It is different from the simple analysis because Coventor takes into account other parameters such as the mass of the beam, which we ignored for the hand calculations.

l. In the 3D visualizer you can see the mode shapes and animate them using the Coventor > Mode Shapes menu option. Animate the mode shapes to see what is happening to the beam.

Begin a new CoSolveEM analysis. Under the DC_ConductorsBCs place 10V on ActuatorElectrode and 0V on GroundElectrode. Under SurfaceBCs place the fixall condition on Anchor. Run the analysis. It may take a while. You can increase the amount of memory the MemMech solver uses under the advanced options of the solver settings tab.m. How much did the tip of the plate deflect? Why is it different from your hand

calculation?

Solution: 2 pts. total; 1 pt. for tip deflection, 1 pt. for explanation on difference in results

In the z direction, the plate deflects about 73nm, which is about twice as much as the hand calculation. The difference between the calculated and the simulated displacements, results mostly from the fact that the hand calculations ignored fringing fields. Coventor also takes fringing fields into account, and so this almost doubles the electrostatic force, therefore doubling the deflection.

[1] SOIMUMPs design rules: http://www.memscapinc.com/__data/assets/pdf_file/0019/1774/SOIMUMPs.dr.v8.0.pdf