Capacitive Pressure Sensor

Capacitive pressure sensors are gaining market share over their piezoresistive counterparts since they consume less power, are usually less temperature sensitive and have a lower fundamental noise floor. This model performs an analysis of a hypothetical sensor design discussed in Ref. 1, using the electromechanics interface. The effect of a rather poor choice of packaging solution on the performance of the sensor is also considered. The results emphasize the importance of considering packaging in the MEMS design process.

Model Definition

The model geometry is shown in Figure 1. The pressure sensor is part of a silicon die that has been bonded to a metal plate at 70 °C. Since the geometry is symmetric, only a single quadrant of the geometry needs to be included in the model, and it is possible to use symmetry boundary condition.

Figure 1: The model geometry. Left: The symmetric device geometry, with one quadrant highlighted in blue, showing the symmetry planes. Right: In COMSOL only the highlighted quadrant is modeled, and the symmetry boundary condition is used on the cross section walls.

A detailed 2D section through the functional part of the device is shown in Figure 2. A thin membrane is held at a fixed potential of 1 V. The membrane is separated from a ground plane chamber sealed under high vacuum. The sides of the chamber are insulating to prevent a connection between the membrane and the ground plane (for simplicity the insulating layer is not modeled explicitly in the COMSOL model; this approximation has little effect on the results of the study.).

Figure 2: Cross section through the device showing the capacitor. The vertical axis has been expanded to emphasize the gap.

When the pressure outside of the sealed chamber changes, the pressure difference causes the membrane to deflect. The thickness of the sealed chamber now varies across the membrane and its capacitance to ground therefore changes. This capacitance is then monitored by an interfacing circuit, such as the switched capacitor amplifier circuit discussed in Ref. 1.

Thermal stresses are introduced into the structure as a result of the thermal conductivity mismatch between the silicon die and the metal plate, and the elevated temperature used for the bonding process. These stresses change the deformation of the diaphragm in response to applied pressures and alter the response of the sensor. In addition, because the stresses are temperature dependent, they introduce an undesired temperature dependence to the device output.

Initially the sensor is analyzed in the case where there are no packaging stresses. Then the effect of the packaging stress is considered. First, the device response at fixed temperature is evaluated with the additional packaging stress. Finally the temperature dependence of the device response at a fixed applied pressure is assessed.

Results and Discussion

Figure 3 shows the deformation of the membrane when a pressure of 25 kPa is applied to it, in the absence of packaging stresses. Figure 4 shows the potential on a plane located between the plates. The deformation of the membrane is of the form expected, and results in a nonuniform potential between the plates.

Figure 3: Quadrant deflection when the pressure difference across the membrane is 25 kPa. As expected the deflection is greatest in the center of the membrane.

Figure 4: Electric potential in the sealed chamber, plotted on a slice between the two plates of the capacitor. The potential has become nonuniform as a result of the pressure-induced deformation of the diaphragm.

Figure 5: Maximum and mean displacement of the membrane as a function of the applied pressure.

Figure 6: Capacitance of the membrane as a function of applied pressure, both with and without the packaging stresses. The linearized zero pressure capacitance variation, taken from Ref. 1, is also shown for comparison.

Figure 5 shows the mean and maximum displacements of the membrane as a function of applied pressure. At an applied pressure of 10 kPa the diaphragm displacement in the center is 0.89 μm. The average displacement of the diaphragm is 0.27 μm. These values are in good agreement with the approximate model given in Ref. 1 (maximum displacement 0.93 μm, average displacement 0.27 μm).

Figure 6 shows that the capacitance of the device increases nonlinearly with applied pressure. The gradient of the curve plotted is a measure of the sensitivity of the sensor. At zero applied pressure the sensitivity of the model (1/4 of the whole sensor) is 7.3·10−6 pF/Pa (compare to the value of 6.5·10−6 pF/Pa given in Ref. 1). The device sensitivity is therefore 29·10−6 pF/Pa (compare to 26·10−6 pF/Pa. calculated in Ref. 1). Assuming the interfacing electronics use the switched capacitor amplifier circuit presented in Ref. 1 this corresponds to a sensor transfer function of 29 μV/Pa (compared to 26 μV/Pa from Ref. 1). Using a smaller pressure step to produce the plot improves the agreement leading to a response at the origin of 6.7·10−6 pF/Pa (27·10−6 pF/Pa for the device, corresponding to 27 μV/Pa). The response is nonlinear, so that at 20 kPa the model output is 14.3·10−6 pF/Pa (device output 57 pF/Pa or 57 μV/Pa)). This nonlinear response adds to the complexity of designing the interfacing circuitry. Note that, for comparison with these figures, the circuitry proposed in Ref. 1, has a noise floor corresponding to a capacitance of 17·10−6 pF, or 0.6 Pa at zero applied pressure (assuming an average of 100 consecutive measurements). This resolution is approximately four times the fundamental sensitivity of the device imposed by mechanical noise from thermal fluctuations.

Next the response of the device is considered when packaging stresses are present in the model. For this part of the discussion it is assumed that the device is operated at 20°C and that the system was stress and displacement free at the bonding temperature (70°C). Figure 7 shows the displacement of the structure at the room temperature operating point, with an applied pressure of 25 kPa. The membrane displacement at its center is shown in Figure 5. The complex interaction between the thermal stresses and the stresses introduced as a result of the applied pressure has resulted in both an initial offset displacement and an increased dependence of the displacement on the pressure.

Figure 7: The displacement of the structure due to an applied pressure of 25 kPa when packaging stresses are also included in the model. Displacements are shown at the operating temperature of 20 °C, and are assumed to be zero at the die bonding temperature of 70 °C.

Figure 8: Temperature dependence of the capacitance of the packaged device. The capacitance varies with temperature as a result of temperature induced changes in the packaging stress within the diaphragm.

The response of the device with the additional packaging stresses is shown in Figure 6. At zero applied pressure the sensitivity of the COMSOL model has increased from 6.5·10−6 pF/Pa to 10·10−6 pF/Pa (40·10−6 pF/Pa for the entire device). The effect is even more pronounced at a pressure of 20 kPa, where the model that includes thermal stresses shows a pressure sensitivity of 25·10−6 pF/Pa (100 pF/Pa for the entire device) compared to the unstressed value of 14.3·10−6 pF/Pa. The sensitivity of the device to pressure has almost doubled. While this effect might seem desirable, an unwanted dependence on temperature has been introduced into the device response. Since the thermal stresses are temperature dependent, the response of the device is also now temperature dependent. The final study in the model assesses this issue.

Figure 8 shows the capacitance of the device, with an applied pressure of 20 kPa, as the temperature is varied. The temperature sensitivity of the model response is given by the gradient of this curve, approximately 3.5·10−3 pF/K (14·10−3 pF/K for the whole device). With a pressure sensitivity of 25·10−6 pF/Pa at 20 kPa (for a single quadrant of the device) this corresponds to an equivalent pressure of 140 Pa/K in the sensor output. Compared to the unstressed performance of the sensor (0.6 Pa with the circuit proposed in Ref. 1) this number is very large. The model shows the importance of carefully considering the packaging in the MEMS design process.

Reference

1. V. Kaajakari, Practical MEMS, Small Gear Publishing, Las Vegas, 2009.

MEMS_Module/Sensors/capacitive_pressure_sensor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

For convenience, the device geometry is inserted from an existing file. You can read the instructions for creating the geometry in the Appendix — Geometry Instructions.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file capacitive_pressure_sensor_geom_sequence.mph.

In the toolbar, click

Add parameters to the model. These will be used subsequently to perform parametric studies.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
p0	20[kPa]	20000 Pa	Pressure
T0	20[degC]	293.15 K	Operating temperature
Tref	70[degC]	343.15 K	Die Bonding temperature

SI units or their multiples, such as Pa and kPa, as well as non-SI units, such as degrees Celsius can be entered in the COMSOL Desktop enclosed by square brackets.

Next, add a nonlocal coupling to compute a derived global quantity from the model. These couplings can be convenient for processing results and COMSOL’s solvers can also use them during the solution process, for example to include integral quantities in the equation system. Here, a nonlocal average coupling is added so that the average displacement of the diaphragm can be computed and a point integration is used to make available the displacement of the center point of the diaphragm.

Definitions

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 12 only.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Point 4 only.

Next, define selections to simplify the setup of materials and physics.

Steel Base

In the toolbar, click

In the window for , locate the section.

In the text field, type -100[um].

Locate the section. From the list, choose .

In the text field, type Steel Base.

Cavity

In the toolbar, click

Select Domain 3 only.

In the window for , type Cavity in the text field.

All domains

In the toolbar, click

In the window for , locate the section.

Select the check box.

In the text field, type All domains.

Solid

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

In the window for , type Solid in the text field.

Electrostatics

In the toolbar, click

Select Domains 3 and 4 only.

In the window for , type Electrostatics in the text field.

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Apply the structural symmetry boundary condition on the symmetry boundaries.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Symmetry 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Note that the electrical symmetry boundary condition (the feature) is applied by default.

The motion of the structure is constrained in most directions by the structural symmetry boundary conditions. However, the whole device can still slide up and down the z-axis. Apply a point constraint to prevent this.

Prescribed Displacement 1

In the toolbar, click

and choose .

Select Point 44 only.

In the window for , locate the section.

Select the check box.

Apply a to represent the pressure acting on the surface of the diaphragm.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 13 only.

In the window for , locate the section.

From the list, choose .

In the p text field, type p0.

Moving Mesh

Deforming Domain 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Symmetry/Roller 1

In the window, click .

Select Boundaries 7 and 8 only.

Doing this allows the mesh to move in the z direction.

Add and features to the model to apply boundary conditions for the electrostatics parts of the problem.

The default feature was set to use solid material type. Add one more feature to represent the nonsolid (void) domain.

Electrostatics (es)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Charge Conservation 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

With the assumption that the silicon membrane is a good conductor, use the Domain Terminal feature to set a bias voltage on the domain. Note: The Domain Terminal feature will be very handy for a conducting domain with a complex shape and many exterior boundaries - instead of selecting all the boundaries to set up the Ground, Terminal, or Electric Potential boundary condition, we only need to select the domain to specify the Domain Terminal with the same effect. In addition, the computation load is reduced, because the electrostatic degrees of freedom within the Domain Terminal do not need to be solved for.

Terminal 1

In the toolbar, click

and choose .

Select Domain 4 only.

In the window for , locate the section.

From the list, choose .

The default value of 1 V is fine in this instance.

Ground 1

In the toolbar, click

and choose .

Select Boundary 9 only.

The pressure sensor consists of a silicon die with an enclosed cavity held at a low pressure. The pressure sensor is bonded onto a cylindrical steel plate during the packaging process. COMSOL includes a with many predefined material properties. This model uses a predefined material for the steel plate, but sets up the silicon as a user-defined material with isotropic material parameters to allow comparison with Ref. 1. The cavity also needs ‘material’ properties (to define the relative permittivity) and a user defined material is used to set the relative permittivity to 1 in this region.

Materials

Silicon

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	170[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.06	1	Young’s modulus and Poisson’s ratio
Density	rho	2330	kg/m³	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	11.7	1	Basic

In the text field, type Silicon.

By default, the silicon is in all domains. Some of these selections will be overridden as other materials are added.

Vacuum

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Locate the section. From the list, choose .

In the text field, type Vacuum.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Steel AISI 4340 (mat3)

In the window for , locate the section.

From the list, choose .

Next set up a structured mesh to solve the problem on.

Mesh 1

Mapped 1

In the toolbar, click

and choose .

Select Boundaries 3, 16, and 32 only.

Size 1

Right-click and choose .

In the window for , locate the section.

In the list, select .

Click

Select Boundaries 3 and 32 only.

In the list, select .

Click

Select Boundary 3 only.

Locate the section. Click the button.

Locate the section.

Select the check box. In the associated text field, type 50[um].

Click

Swept 1

In the toolbar, click

In the window for , click

Set up a study that sweeps over a range of applied pressures, so that the response of the sensor can be assessed.

Study 1

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Click

The continuation parameter p0 (Pressure) is added by default. This is the correct parameter to sweep over.

Click

In the dialog box, type 0 in the text field.

In the text field, type 5000.

In the text field, type 25000.

Click .

In the toolbar, click

Results

Displacement (solid)

Much of the structure is not displaced in this initial study. To facilitate results analysis, add a selection to the solution. This will ensure that only the domains of interest are displayed in the plots.

Study 1/Solution 1 (sol1)

In the window, expand the node, then click .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Displacement (solid)

Click the

button in the toolbar.

The plot now shows the displacement of the diaphragm only, which, as expected, is maximum in the center of the sensor.

Next, plot the electric potential in an xy-orientated plane between the sensor diaphragm and the ground plane.

Streamline Multislice 1

In the window, expand the node.

Right-click and choose .

Multislice 1

In the window, under click .

In the window for , locate the section.

Find the subsection. From the list, choose .

In the text field, type 0.

Find the subsection. From the list, choose .

In the text field, type 0.

Find the subsection. In the text field, type -0.0023[mm].

Click to expand the section. Select the check box.

Selection 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Due to the deformation of the diaphragm the potential is nonuniformly distributed in the plane.

Next, plot the deformation of the diaphragm as a function of the pressure differential across it. Include both average and maximum displacements.

1D Plot Group 4

In the toolbar, click

and choose .

Global 1

Right-click and choose .

Use the point integration and surface average couplings defined earlier to evaluate the displacement at the midpoint of the membrane and the average displacement.

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
intop1(w)	um	Maximum Displacement
aveop1(w)	um	Average Displacement

Diaphragm Displacement vs. Pressure

In the window, click .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Diaphragm displacement.

Locate the section.

Select the check box. In the associated text field, type Pressure (Pa).

Select the check box. In the associated text field, type Displacement (\mu m).

Locate the section. From the list, choose .

In the text field, type Diaphragm Displacement vs. Pressure.

In the toolbar, click

At an applied pressure of 10 kPa the diaphragm displacement in the center is 0.89 um. The average displacement of the diaphragm is 0.27 υm. These values are in good agreement with the approximate model given in Ref. 1 (maximum displacement 0.93 υm, average displacement 0.27 υm).

Now plot the sensor capacitance as a function of the applied pressure. If the switched capacitor amplifier described in Ref. 1 is used to produce the output, the sensor output or transfer function is directly proportional to the change in capacitance.

1D Plot Group 5

In the toolbar, click

and choose .

Global 1

Right-click and choose .

Since the boundary condition was used for the underside of the diaphragm, COMSOL automatically computes its capacitance with respect to ground. The value of the capacitance is available as a variable in results analysis.

In the window for , click in the upper-right corner of the section. From the menu, choose .

Next, compare the computed capacitance with the small-displacement, linearized analytic expression derived in Ref. 1.

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
es.C11	pF	Capacitance
0.738[pF](1+8.87e-6[1/Pa]p0)	pF	Linearized Analytic Capacitance

Model Capacitance vs. Pressure

In the window, click .

In the window for , locate the section.

From the list, choose .

In the text area, type Model Capacitance vs. Pressure.

Locate the section.

Select the check box. In the associated text field, type Pressure (Pa).

Select the check box. In the associated text field, type Capacitance (pF).

Locate the section. From the list, choose .

In the text field, type Model Capacitance vs. Pressure.

In the toolbar, click

Next, add thermal expansion to the model to assess the effects of packaging stresses on the device performance.

Solid Mechanics (solid)

Linear Elastic Material 1

In the window, under click .

Thermal Expansion 1

In the toolbar, click

and choose .

The model temperature should be set to the previously defined room temperature parameter, T0.

In the window for , locate the section.

From the T list, choose . In the associated text field, type T0.

The reference temperature indicates the temperature at which the structure had no thermal strains. In this case, set it to the previously defined parameter, Tref, which represents the temperature at which the silicon die was bonded to the metal carrier plate.

Click

for .

Global Definitions

Default Model Inputs

In the window, under click .

In the window for , locate the section.

Find the subsection. In the text field, type Tref.

Materials

Silicon (mat1)

COMSOL shows a warning in the material properties settings to indicate a missing property.

In the window, under click .

In the table, add a value for the thermal expansivity of silicon to the appropriate row:

Add a new study to compute the system response including thermal expansivity effects.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 2

In the window, click .

In the window for , locate the section.

Clear the check box.

Step 1: Stationary

In the window, under click .

In the window for , locate the section.

Select the check box.

Click

The continuation parameter p0 (Differential Pressure) is added by default. This is the correct parameter to sweep over.

Click

In the dialog box, type 0 in the text field.

In the text field, type 5000.

In the text field, type 25000.

Click .

In the toolbar, click

Create a mirrored dataset to visualize a cross section of the device.

Results

Mirror 3D 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Displacement (solid) 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Notice that the entire structure is now displaced at room temperature as a result of thermal expansion.

Now look at the effect of the thermal stress on the response of the sensor.

Add an additional node to the previously defined plot. This separate node can point to a different dataset, enabling a plot of the displacement of the thermally stressed device alongside the unstressed plot.

Global 2