Video Tutorial: Capacitive Pressure Sensor

The first step is to import some model parameters, which we’ll use to specify variables within the model. Click “Parameters” and then select the desired file. Here, you can see there are three imported parameters: “p0” is a pressure. This is the ambient pressure at which the device is operating and we have two temperatures: “T0”, the ambient temperature at which the device operates, and a reference temperature, “Tref”, which is the temperature at which the device is bonded to its package.

Next, the geometry is configured. For convenience, the geometry is imported from an external file. To do this from the Home tab, select “Import” and choose the desired file. Here, you can see we’ve imported a pre-made geometry, which is one quarter of the device we are modeling.

To simply the configuration of the physics interface and to aid with postprocessing of the results, it is convenient to now define some coupling components and some geometry selections. Two coupling components have been added. An Average operator, which operates on boundary 12. Boundary 12 corresponds to the underside of the silicon membrane.

So, here we have the silicon membrane, the cavity, and the silicon die. You can see, highlighted in blue, the underside of the silicon membrane. And an Integration operator, which integrates one point. This point is the point where we expect the maximum deflection. It’s right in the center of the underside of the membrane.

An Explicit selection has been used to select the cavity domain. So, this is that vacuum cavity between the silicon membrane and the grounded body of the sensor. A second explicit selection has been used to select all domains and, then, a Difference operator has been used to remove the cavity domain. Leaving a selection, which is everything except for the cavity. This will be useful when assigning the linear elastic material properties later.

Next, we add the physics settings to the model. These include the pressure forces acting on the sensor, the applied voltages, and other appropriate domain and boundary conditions.

Next, we can apply the symmetry boundary conditions. From the physics ribbon, we select the boundary conditions, and under structural, we choose “Symmetry”. For the selection, we simply choose one of the symmetry planes we created earlier. We then repeat the process and select the other symmetry plane we created earlier.

Next, we need to add a Prescribed Displacement to a point on the model. Through the Physics ribbon, we choose from the Points, and from Electromechanics, choose “Prescribed Displacement”. You’re going to apply this to point 44 only, which is this corner point here.

Next, we need to add a Boundary Load condition. This is used to apply the pressure to the device. From the Physics ribbon, we select “Boundaries”, and from Structural we choose “Boundary Load”. This is applied to boundary 13 only, which corresponds to the top of the deflecting membrane. We’re going to set it to take a Pressure as its Load type, and we’re going to set the value of the Pressure to “p0”.

Now, we specify the electrostatic boundary conditions required. From the Physics ribbon, we choose “Boundary”, and from Electrical, we choose “Terminal”. This Terminal boundary condition is used to apply a voltage to the underside of the silicon membrane. Select boundary number 12 again and we’re going to change the Terminal type to “Voltage” and the default Electrical potential of “1 Volt” is sufficient.

Next, we once again return to the boundary conditions and the Electrical section, and we select Ground. And this we want to apply to the top of the silicon die directly below the cavity. So, we choose the corresponding boundary and choose boundary 9. If we zoom in, we can see the selection of these electrical boundary conditions. So, you can see the voltage of “1 Volt” has been applied on the underside of the silicon membrane and the Ground has been applied on the other side of the cavity.

Now that the physics has been configured, it’s a suitable time to apply some materials to the model. From the Home ribbon, select “New Material”. You can see that the new material by default is applied to all domains and COMSOL knows which physical properties are required for the model to compute. The first material we shall rename “Silicon” and we’ll use it for the silicon part of the model. Enter some values appropriate for the silicon material.

Next, we add a second new material. This material is going to be added only to the cavity. Here, COMSOL knows that in this domain, all that’s required is the Relative permittivity, which we set to “1” to represent the vacuum.

Finally, we can use a built-in material to specify the steel domains. From the Home ribbon, we click “Add Material”, and from the Built-In library list, select “Steel”. Apply this to the steel base selection we made earlier. Notice how the materials override each other as they’re subsequently added. Initially, the silicon material was applied to all domains and now it’s overridden by the vacuum and the steel in the appropriate domains.

Next, we set up a structured mesh to solve the problem on. From the Mesh ribbon, we select “Edit” to allow us to edit the default physics-induced mesh sequence. Rather than the default Free Tetrahedral, the geometry here calls for a Mapped, Swept Mesh.

First, we disable the default, Free Tetrahedral. We would like our mesh to be densest around the region of interest, which is the membrane that we expect to deflect. To do this, we add a Size node and select “Custom” and then specify a Maximum element size of “50 micrometers”. Apply this to a boundary and we’re going to choose boundary 3 only, which you can see is this boundary on the bottom of the device, which marks the cross section of the membrane.

Now, we add a Mapped mesh to the lower surface of the model. Choose boundaries 3, 16, and 32, which make up this bottom surface of the model, and here you can see that the smaller Maximum size node has reduced the area of the surface elements in this region directly below the deflecting membrane, and then we have nice sweeping, radial mesh elements.

We will configure the study so that it sweeps over a range of applied pressures so that the response of the sensor can be assessed as a function of the ambient pressure. Within the study node, we find study Step 1: Stationary, and expand the Study Extensions section. Here we can check an Auxiliary sweep, and add the “p0” pressure variable. Choose an appropriate range.

Now that the auxiliary sweep is configured, from the Home ribbon, we may Compute.

The default plot shows the total displacement. It is clear that much of the structure is not displaced at all in this initial study. In order to quantify the displacement, we can use the two coupling components that were created earlier.

From the Home ribbon, we’re going to Add Plot Group and choose to add a 1D Plot Group. Now, we want to add a Global plot and first we can use the Average operator. So, this will take the average of “w”, which is the vertical displacement over the “aveop” boundary, which was the underside of the silicon membrane.

Now that we have added additional physics to the physics interface, we see that the Materials section requires additional parameters. Specifically, we need to enter the coefficient of thermal expansion for the silicon. Note that the steel, as it came from the Built-In library, does not require us to add the additional coefficient of thermal expansion as this was already available.

In order to include the additional effects of thermal expansion, without overwriting the initial solution, we add a second study to the model. From the Study ribbon, we select “Add Study” and we add a second stationary study. Once again, we need to configure the study to perform an Auxiliary sweep over the “p0”, pressure, parameter.

Enter the same range as before, and then we can Compute.

Once again, the default plot shows the surface displacement. Notice now that the entire structure is displaced as a result of the differing rates of thermal expansion between the steel base plate and the silicon device. To compare these results with the first study, we can add the new solutions to the same plot groups.

Duplicate this Global plot and then change which solution it gets its data from. Tidy up the graph a bit. So, the solid lines correspond to the initial study with no thermal effects, and the dotted lines show that with the thermal effects, there is an increased deflection for a given pressure. For this plot, I moved the legend out of the way and changed the line style for clarity.

I can do the same for the capacitance. I can duplicate the original plot, change the solution from which it gets its data, and tidy the plot up a bit. And here we have created our plot for the capacitance as a function of pressure. I’ve changed the line color, and made some minor changes to the data labels in the legend, and you can see now the response of the capacitor to pressure is very different now that the effects of thermal stress are included in the simulation.