Pressure Sensor COMSOL

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Capacitive Pressure SensorSOLVED WITH COMSOL MULTIPHYSICS 3.5a

®

pressure_sensor.book Monday, December 1, 2008 1:23 PM

pressure_sensor.book Page 1 Monday, December 1, 2008 1:23 PM

Capa c i t i v e P r e s s u r e S e n s o r

Introduction

This example examines the operation of a capacitive MEMS pressure sensor, treating both 2D and 3D model versions.

The operating principle of a capacitive pressure sensor is to measure the change in capacitance between two electrodes when a change in pressure displaces one of the electrodes, located on a thin diaphragm. The diaphragm separates a reference compartment kept at vacuum pressure and a pressurized compartment; see Figure 1.

Figure 1: One quarter of the pressure sensor (the two vertical cross-sectional planes are symmetry planes). The vacuum compartment looks like a small ashtray with a thin diaphragm at its bottom. Between this diaphragm and the fixed base is a very thin pressurized compartment.

At the bottom of the pressurized compartment is a fixed base (with one electrode), while the diaphragm (with a counterelectrode) is located at its top. As the pressure changes, the diaphragm that separates the two compartments is displaced, and the

Vacuum

Dividing diaphragm

Pressurized

Base with

with electrode

compartment

counterelectrode

compartment

C A P A C I T I V E P R E S S U R E S E N S O R | 1


change in separation between the two electrodes results in a corresponding change in the capacitance.

Although the deformation of the sensor is primarily caused by the applied pressure, any initial stresses in the material also affect the deformation. Therefore, the manufacturing process and the selected materials directly influence sensor operation. For example, in some structures the membrane and cavities are engraved onto silicon and sealed with layers of glass. Because the materials are bonded together at a high temperature, cooling them down to the sensor’s normal operating temperature produces undesirable stresses in the material that affect device performance.

The sensor in this example measures static pressures of a magnitude from zero to atmospheric pressure. The model first computes the initial stresses from the manufacturing process; then it accounts for the structure’s mechanical deformation resulting from an applied pressure. It finally calculates the sensor’s capacitance for the deformed shape: the 2D model calculates the capacitance from a computed electric field, whereas the 3D model simply integrates infinitesimal capacitance contributions over the electrode boundary.

The 2D model describes a geometry that differs from that of a true 3D sensor (and of the 3D model). The deforming diaphragm is rectangular and fixed at all boundaries. When viewed in 2D, however, a bridge-like structure, fixed only at the two edges, results. Figure 2 illustrates the model’s deformed geometry in 2D.



Figure 2: 2D view of a pressure sensor.

Model Definition

M O D E L G E O M E T R Y

The pressure sensor consists of a silicon structure that includes a micrometer-thick diaphragm situated between two glass layers. Figure 3 shows the geometry and the dimensions are given in Table 1. In addition, two 1 mm2 rectangular plates at the pressurized compartment’s top (applied potential) and bottom (grounded) form the electrodes.

TABLE 1: DEVICE COMPONENT DIMENSIONS AND MATERIALS

PROPERTY TOP AND BOTTOM LAYERS

MIDDLE LAYER VACUUM COMPARTMENT

PRESSURIZED COMPARTMENT

Shape Rectangular Rectangle with engraved cavities

Symmetric trapezoid

Rectangular

Width/Length

2.5 mm 2.5 mm diaphr.: 1.5 mm

top: 1.9 mm bottom: 1.5 mm

1.5 mm



Figure 3: Materials used in the structural analysis of the capacitive pressure sensor in 3D.

S T R E S S A N D D E F O R M A T I O N

During manufacturing, the sensor is bonded together in a vacuum and at a high temperature before it is cooled down. Therefore, during this process no external forces act on the sensor’s boundaries, but internal stresses appear because the two materials have different coefficients of thermal expansion. This process also produces a vacuum in the upper cavity that serves as the reference pressure.

During regular operation, the sensor is fixed to a solid surface, and ambient pressure pushes on all outer boundaries. The temperature also changes, which produces extra stresses due to thermal expansion.

For a linear elastic material, the stress-strain relationship—taking into account initial stress, σ0, initial strain, ε0, and thermal strain, εth—is

Height 0.5 mm 0.5 mm diaphr.: 20 μm

0.475 mm 5 μm

Material Glass, HOYA, SD-2

Silicon Vacuum Air

TABLE 1: DEVICE COMPONENT DIMENSIONS AND MATERIALS

PROPERTY TOP AND BOTTOM LAYERS

MIDDLE LAYER VACUUM COMPARTMENT

PRESSURIZED COMPARTMENT

Glass

Silicon

Glass

Symmetry planes



where D is the elasticity tensor, and the 6-dimensional vectors σ and ε give the normal and shear values of the stresses and strains.

Initially only thermal expansion is active. It is given by

where αvec are the coefficients of thermal expansion, T is the ambient temperature, and Tref is the reference temperature. The manufacturing stage produces the initial stress for normal operation, where further thermal expansion takes place. This model assumes that the sensor is close to its initial geometry after manufacturing, so that the initial strain equals zero. Furthermore, you solve the first application mode using the small deformation assumption but allow large deformations for the second one.

C A P A C I T A N C E

To compute the sensor’s capacitance, the 2D model solves for the electric field in the deformed geometry (or frame), which is defined by the Moving Mesh (ALE) application mode. Using a port boundary condition, the capacitance is obtained from the energy of the electric field from the equation

where U is the potential difference between the plates (U = 1 V for the port boundary condition) and We is the electric energy density. The area Ωd corresponds to the narrow air gap in the sensor.

The 3D model uses another method for the capacitance calculation, namely integrating over the surface of the capacitor according to

σ Dεel D ε εth– ε0–( ) σ0+= =

εth

εx

εy

εz

γxy

γyz

γxz

αvec T Tref–( )= =

C 2

U2------- We Ωdd

Ωd

∫=

C ε 1h---dA∫=



where h denotes the local distance across the capacitor and ε is the permittivity of air. This calculation rests on the assumption that the lower glass block does not deform much, so that the local separation only depends on the initial distance and the diaphragm deformation. In the model this expression is further multiplied by 4 to get the full capacitance of the model.

Results and Discussion

Figure 4 shows the results from the 2D model after the bonding phase, where bonding took place at 400 °C and the sensor is then cooled down to 22 °C. In the image the x-axis and y-axis have different scales, and the structural deformation is scaled by a factor 20.

It appears that the membrane slightly pulls toward the larger cavity even though there are no applied loads. Stresses appear near the boundaries of the different materials and in the silicon membrane, which is narrower than other parts of the sensor. The maximum appears at the lower left corner of the smaller cavity.

Figure 4: Initial stresses of the materials in the pressure sensor.



Figure 5 shows the results from the 3D model when the sensor is in operation: it is exposed to a pressure of one atmosphere at 15 °C. The largest stress in the diaphragm appears near the position where the diaphragm connects the surrounding material.

Figure 6 shows the results from the 2D mode at the same conditions. The figure is arbitrarily scaled and is focused on the left half of the lower cavity. The diaphragm deforms toward the vacuum with maximum deformation in the middle. Maximum stresses appear at the upper corners of the lower cavity where the membrane attaches to the silicon boundaries.

The streamlines show the electric field in the lower cavity. The lines are vertical between the two electrodes. Some field lines appear outside of the electrode region, but the field strength is very small there (dark blue color).



Figure 5: Sensor deformation and von Mises Stress (MPa) when exposed to ambient pressure.

Figure 6: Sensor deformation, stresses (von Mises Stress in MPa; left colorbar) and electric field (electric field strength in kV/m; right colorbar) when exposed to ambient pressure.



Figure 7 show the capacitance values computed from the electric field for four conditions as described in Table 1. Conditions A and B have the same ambient temperature but a higher bonding temperature. In Condition C all temperature values are the same so no stresses or deformations arise from thermal effects. For Condition D, thermal deformation does result from the ambient temperature.

The capacitance for the full sensor modeled in the 3D model is shown in Figure 8. This capacitance corresponds to condition A in Table 1.

You can also compare the computed capacitance values to those for a plate capacitor with a plate size of 1 mm times unity and a gap of 5 μm when not deformed. The capacitance for this plate capacitor is C = ε0A/d = 1.771·10−9 F; the corresponding value from the COMSOL Multiphysics model is 1.7760763·10−9 F computed with settings for Condition C and zero pressure.

TABLE 1: TEMPERATURE CONDITIONS FOR COMPUTING THE CAPACITANCE VALUES

TEMPERATURE A B C D

High bonding (°C) 400 200 22 22

Low bonding (°C) 22 22 22 22

Ambient (°C) 15 15 22 -18



Figure 7: Computed capacitance vs. ambient pressure for different temperature conditions from the 2D model: Condition A, triangles; Condition B, squares; Condition C, +; Condition D, *.



Figure 8: Capacitance of the sensor as the function of the operating pressure from the 3D model.

Modeling in COMSOL Multiphysics

In COMSOL Multiphysics you solve the 2D problem using four application modes: two Plane Strain application modes, one Moving Mesh (ALE) application mode, and one Electrostatics application mode. The latter two are defined in a frame to allow the mesh to move.

In the 2D model you solve for the mesh movement and the electric field only in the small air gap where the ambient pressure is applied to the sensor. The mesh movement follows the structural deformation on the model’s parts.

The solution process takes place in four steps:

1 The first Plane Strain application mode represents the sensor’s fabrication, and it computes the initial stresses that result from thermal expansion. This is solved with the static solver.

2 The second Plane Strain application mode solves the deformation and stresses that result when the sensor is exposed to ambient temperature and pressure. It uses the



initial stresses and deformation from the first Plane Strain application mode. This is solved with a parametric solver for different values of ambient pressure.

3 Using a parametric solver, you solve the ALE mesh for each ambient pressure.

4 Finally, you solve the electric field for each ambient pressure using a parametric solver.

You solve 3D problem using only two Solid, Stress-Strain application modes. Similarly, for the 2D model, you must use a large deformation analysis for the second structural application mode.

The solution process of the 3D model takes place in two steps, which are the same as the first two steps for the 2D model, but using the Solid, Stress-Strain application mode.

The following table contains descriptions and default values for constants that define the different manufacturing and ambient conditions. If you want to conduct a static analysis to solve for single solutions, you can define a constant Pambient; it is not needed for the parametric analysis.

CONSTANT VALUE DESCRIPTION

TbondingH 400 oC High bonding temperature

TbondingL 22 oC Low bonding temperature

Tambient 15 oC Ambient temperature

Pambient 101 kPa Ambient pressure


Pressure Sensor COMSOL

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