COMSOL 6.4 - Piezoresistive Pressure Sensor

Piezoresistive pressure sensors were some of the first MEMS devices to be commercialized. Compared to capacitive pressure sensors, they are simpler to integrate with electronics, their response is more linear, and they are inherently shielded from RF noise. They do, however, usually require more power during operation, and the fundamental noise limits of the sensor are higher than their capacitive counterparts. Historically, piezoresistive devices have been dominant in the pressure sensor market.

This example considers the design of the MPX100 series pressure sensors originally produced by the semiconductor products division of Motorola Inc. (now Freescale Semiconductor Inc.). Although the sensor is no longer in production, a detailed analysis of its design is given in Ref. 1, and an archived data sheet is available from Freescale Semiconductor Inc. (Ref. 2).

Model Definition

The model consists square membrane with side 1 mm and thickness 20 μm, supported around its edges by region 0.1mm wide, which is intended to represent the remainder of the wafer. The supporting region is fixed on its underside (representing a connection to the thicker handle of the device die). Near to one edge of the membrane an X-shaped piezoresistor (or Xducer™)¹ and part of its associated interconnects are visible. The geometry is shown in Figure 1.

The piezoresistor is assumed to have a uniform p-type dopant density of 1.32×1019 cm−3 and a thickness of 400 nm. The interconnects are assumed to have the same thickness but a dopant density of 1.45×1020 cm−3. Only a part of the interconnects is included in the geometry, since their conductivity is sufficiently high that they do not contribute to the voltage output of the device (in practice the interconnects would also be thicker in addition to having a higher conductivity but this also has little effect on the solution).

The edges of the die are aligned with the {110} directions of the silicon. The die edges are also aligned with the global X- and Y-axes in the COMSOL Multiphysics model. The piezoresistor is oriented at 45° to the die edge, and so lies in the [100] direction of the crystal. In the COMSOL Multiphysics model, a coordinate system rotated 45° about the global Z-axis is added to define the orientation of the crystal.

Figure 1: Left: Model geometry. Right: Detail showing the piezoresistor geometry.

Device Physics and Equations

The conductivity of the Xducer™ sensor changes when the membrane in its vicinity is subject to an applied stress. This effect is known as the piezoresistance effect and is usually associated with semiconducting materials. In semiconductors, piezoresistance results from the strain-induced alteration of the material’s band structure, and the associated changes in carrier mobility and number density. The relation between the electric field, E, and the current, J, within a piezoresistor is:

(1)

where ρ is the resistivity and Δρ is the induced change in the resistivity. In the general case both ρ and Δρ are rank 2 tensors (matrices). The change in resistance is related to the stress, σ, by the constitutive relationship:

(2)

where Π is the piezoresistance tensor (SI units: Pa−1Ωm), a material property. Note that the definition of Π in COMSOL Multiphysics includes the resistivity in each element of the tensor, rather than having a scalar multiple outside of Π (which is possible only for materials with isotropic conductivity). Π is in this case a rank-4 tensor; however, it can be represented as a matrix if the resistivity and stress are converted to vectors within a reduced subscript notation. Within the Voigt notation used by COMSOL Multiphysics for this purpose, Equation 2 becomes:

(3)

The Δρ vector computed from Equation 3 is assembled into matrix form in the following manner in Equation 1:

(4)

Silicon has cubic symmetry, and as a result the Π matrix can be described in terms of three independent constants in the following manner:

For p-type silicon the Π44 constant is two orders of magnitude larger than either the Π11 or the Π12 coefficients. The Π66 element (which is equal in magnitude to the Π44 element) couples the σxy shear stress, with the Δρxy off-diagonal term in the change in resistivity matrix. In turn, Δρxy couples a current in the x direction to an induced electric field in the y direction (and vice versa). This is the principle of the Xducer™ transducer. An applied voltage (typically 3 V; see Ref. 2) across the [100] oriented arm of the X produces a current (typically 6 mA; see Ref. 2) down this arm. Shear stresses are present in the Xducer™ as a result of the pressure induced deformation of the diaphragm in which it is implanted. Through the piezoresistance effect, these shear stresses cause an electric field or potential gradient transverse to the direction of current flow, in the [010] arm of the X. Across the width of the transducer, the potential gradient sums up to produce an induced voltage difference between the [010] arms of the X. According to the device data sheet, under normal operating conditions a 60 mV potential difference is generated from a 100 kPa applied pressure with a 3 V applied bias (Ref. 2).

The situation is complicated somewhat by the detailed current distribution within the device, since the voltage sensing elements increase the width of the current carrying silicon wire locally, leading to a “short circuit” effect (Ref. 3) or a spreading out of the current into the sense arms of the X.

The interfaces available in the MEMS Module solve Equation 3 and an inverse form of Equation 4, together with the equations of structural mechanics. In this model the coupling is used to model the structural equations on the domain level and to solve the electrical equations on a thin layer coincident with a boundary in the model geometry.

Results and Discussion

Figure 2 shows the displacement of the diaphragm as a result of a 100 kPa pressure difference. At the center of the diaphragm the displacement is 1.2 μm. A simple isotropic model for the deform displacement given in Ref. 1 predicts an order of magnitude value of 4 μm (assuming a Young’s modulus of 170 GPa and a Poisson’s ratio of 0.06). The agreement is reasonable considering the limitations of the analytic model, which is derived by a crude variational guess. A more accurate value for the shear stress in local coordinates at the midpoint of the diaphragm edge is given in Ref. 1 as:

where P is the applied pressure, L is the length of the diaphragm edge, and H is the diaphragm thickness. This equation predicts the magnitude of the local shear stress to be 35 MPa, in good agreement with the minimum value shown in Figure 3, which is also 35 MPa. Theoretically the shear stress should be maximal at the midpoint of the edge of the diaphragm. Figure 4 shows the shear stress along the edge in the model. This shows a maximum magnitude at the center of each of the two edges along which the plot is made, but the value of this maximum is less than the maximum stress in the model, in part due to the boundary conditions employed on the three dimensional diaphragm. The model: piezoresistive_pressure_sensor_shell.mph shows better agreement with the theoretical maximum shear stress along this edge.

Figure 2: Diaphragm displacement as a result of a 100 kPa applied pressure.

Figure 3: Shear stress, shown in the local coordinate system of the piezoresistor (rotated 45° about the z-axis of the global system). The shear stress is has its highest magnitude close to the piezoresistor with a value of approximately -35 MPa.

Figure 4: Plot of the local shear stress along two edges of the diaphragm.

The output of the model during normal operation shows good agreement with the manufacturer’s data sheet, given that the device dimensions and doping levels have been guessed. With an applied bias of 3 V a typical operating current of 5.9 mA is obtained (compare the current quoted in Ref. 2 of 6 mA). The model produces an output voltage of 52 mV, similar to the actual device output of 60 mV quoted in Ref. 2. The detailed current and voltage distribution within the Xducer™ is shown in Figure 5. There is clear evidence of the current flow “spreading out” into the sense electrodes (which are narrower), a phenomenon described in Ref. 3 as the “short circuit” effect. The asymmetry in the potential, which is induced by the piezoresistive effect, is also apparent in the figure.