The piezo-resistive coefficient π44 is reported for the case of single crystalline p-type silicon. By comparing the measured sensitivity of pressure sensors with the simulated sensitivity of these pressure sensors, we are able to extract π44 since this is the only free parameter in the simulation. A value of π44 = (108.3 ± 2.1) × 10−11 Pa−1 at a dopant concentration of (5.0 ± 4.5) × 1017 cm−3 was found, which is in good agreement with experimental literature data.

Topics
MEMS technology, Wheatstone bridge circuit, Circuit theorems, Sensors, Software engineering, Finite-element analysis, Chemical elements, Depth profiling techniques, Doping

Pressure sensors based on silicon Micro-Electro-Mechanical Systems (MEMS) are widely used in industry for numerous applications.1,2 A lot is known about MEMS pressure sensor design, manufacturing, and simulation.3–6 The measured signal of such silicon based pressure sensors is often based on the piezo-resistive properties of silicon.7 This means that an applied stress results in a change in the piezo-resistance of the silicon. Nevertheless, the underlying physical constants related to the piezo-resistivity in silicon, which have been obtained experimentally, have a rather large scatter.8 However, to predict the output of a pressure sensor as exact as possible, the values of the physical constants must be known precisely. Therefore, we report the piezo-resistive coefficient π44.

Mathematically, the piezo-resistivity can be described by a six-vector notation.9 The stress vector σλ is connected via the piezo-resistivity tensor πκλ to the change in resistivity vector Δρκ,

Δρκρ0=πκλσλ,
(1)

with ρ0 being the scalar resistivity without stress and κ, λ = 1, …, 6. Due to the cubic crystal structure of silicon, the piezo-resistivity tensor has only three independent components π11, π12, and π44, and the tensor is then

πκλ=π11π12π12000π12π11π12000π12π12π11000000π44000000π44000000π44.
(2)

In p-type silicon, the piezo-resistivity tensor component π44 is about two magnitudes larger than the other two components. Hence, the component π44 has the main impact on the resistivity change due to applied stress. Due to its importance, much effort8 has been undertaken in the past to obtain the component π44 by experiment as well as by theory. Usually, test structures8 or uniformly doped bars10 are prepared, which are stressed, e.g., by a four point bending fixture.8 The stress is calculated, and in consequence, π44 is calculated as well.

In this contribution, for the first time, the shear piezo-coefficient π44 is experimentally obtained by measuring the sensitivity of a pressure sensor. This sensitivity is simulated using the COMSOL Multiphysics® software package. The only free parameter in this simulation is the shear piezo-coefficient π44. This coefficient π44 is adapted in the simulation in a way that the measured sensitivity of the pressure sensor fits to the simulated sensitivity. In that way, the piezo-coefficient π44 is experimentally determined. The difference to previous studies is that a full featured pressure sensor is used with applied pressure to obtain the piezo-coefficient π44.

The schematic representation of the measured pressure sensor is shown in Fig. 1. Figure 1(a) shows a cross section of the pressure sensor, and Fig. 1(b) shows the top view of the pressure sensor. Black bars are the resistors. The pressure sensor consists of four resistors R1 to R4 placed on the edges of the membrane. The resistors themselves consist of boron doped bars with a size of 10 × 100 μm2 and a boron depth profile shown in Fig. 2. In Fig. 1(a), the cross section is shown for KOH etched membranes with tilted sidewalls.

FIG. 1.
FIG. 1. Schematic representation of the measured pressure sensor. (a) Cross section and (b) top view.
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Schematic representation of the measured pressure sensor. (a) Cross section and (b) top view.

FIG. 1.
FIG. 1. Schematic representation of the measured pressure sensor. (a) Cross section and (b) top view.
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Schematic representation of the measured pressure sensor. (a) Cross section and (b) top view.

Close modal
FIG. 2.
FIG. 2. Boron and phosphorous depth profiles (black and red lines) simulated by Atlas compared to the boron depth profile used for the COMSOL simulation of the resistors.
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Boron and phosphorous depth profiles (black and red lines) simulated by Atlas compared to the boron depth profile used for the COMSOL simulation of the resistors.

FIG. 2.
FIG. 2. Boron and phosphorous depth profiles (black and red lines) simulated by Atlas compared to the boron depth profile used for the COMSOL simulation of the resistors.
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Boron and phosphorous depth profiles (black and red lines) simulated by Atlas compared to the boron depth profile used for the COMSOL simulation of the resistors.

Close modal

The resistors are arranged in a Wheatstone bridge. The circuit diagram of this bridge is shown in Fig. 3. The output voltage UO is calculated by knowing the resistors R1 to R4 and the supply voltage US in the following way:

UO=USR1R1+R2−R3R3+R4.
(3)
FIG. 3.
FIG. 3. Circuit diagram of a Wheatstone bridge.
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Circuit diagram of a Wheatstone bridge.

FIG. 3.
FIG. 3. Circuit diagram of a Wheatstone bridge.
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Circuit diagram of a Wheatstone bridge.

Close modal

The pressure sensor is designed in a way that under pressure, R1 = R4 and R2 = R3 hold [see Fig. 1(b)]. It follows for the so-called sensitivity UO/US given in (mV/V),

UOUS=R1−R2R1+R2.
(4)

In the simulation, the resistance of the resistors R1 and R2 is obtained separately by applying Ohm’s law R1,2 = U1,2/I1,2. This is done without using a Wheatstone bridge. A constant voltage U1 = U2 = U is applied to R1 and R2, and the currents I1 and I2 are determined by COMSOL. Hence, we get

UOUS=I2−I1I1+I2.
(5)

The sensitivity UO/US is measured under applied pressure for the produced pressure sensors. Additionally, the currents I1 and I2 through the resistors are simulated by COMSOL under applied pressure.

In the following, details about the simulation are given. Figure 4 shows the geometry of the simulated pressure sensor with its two resistors placed on the edge of the membrane. The membrane size is 2 × 2 mm2. The geometry for the simulation follows the design of the produced pressure sensors. To save computation power, only the current through the resistors is simulated.

FIG. 4.
FIG. 4. Geometry of the simulated pressure sensor. Two simulated resistors R1 and R2 are clearly visible.
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Geometry of the simulated pressure sensor. Two simulated resistors R1 and R2 are clearly visible.

FIG. 4.
FIG. 4. Geometry of the simulated pressure sensor. Two simulated resistors R1 and R2 are clearly visible.
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Geometry of the simulated pressure sensor. Two simulated resistors R1 and R2 are clearly visible.

Close modal

The resistor itself consists of three bars, each 100 × 10 × 1 µm3 in size stacked one upon the other with different p-type doping. This is done to estimate the boron depth profile of the pressure sensor resistors. The depth profile of the simulated resistor is shown together with the boron depth profile in Fig. 2.

In Fig. 5, the mesh of the simulated pressure sensor is shown. The finer mesh in the region of the resistors is clearly visible.

FIG. 5.
FIG. 5. Mesh of the simulated pressure sensor.
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Mesh of the simulated pressure sensor.

FIG. 5.
FIG. 5. Mesh of the simulated pressure sensor.
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Mesh of the simulated pressure sensor.

Close modal

A pressure of 50 kPa is applied to the membrane from the backside. The sensitivity is directly measured on the assembled pressure sensor. Several pressure sensors with different membrane thicknesses are used. The measurements are done at room temperature.

In the simulation, a pressure of 50 kPa is applied to the membrane as well. The mechanical behavior as well as the electric current is solved by the simulation simultaneously. The elastic constants11,12 and the other piezo-coefficients10 used for the simulation are the standard COMSOL parameters (see Table I). The simulation is straightforward. Anisotropic mode must be switched on, and a rotated coordinate system (+45°) must be defined for the elements (resistors, etc.).

TABLE I.

Standard parameters used for the simulation.10–12 

Elastic constantsPiezo-coefficients
(GPa)(10−11 Pa−1)
C11 166 π11 6.6 
C12 64 π12 −1.1 
C44 80   
Elastic constantsPiezo-coefficients
(GPa)(10−11 Pa−1)
C11 166 π11 6.6 
C12 64 π12 −1.1 
C44 80   
View Large

The currents I1 and I2 through the two resistors are obtained by applying a voltage. Using Eq. (5), the sensitivity is obtained from the simulated currents.

The results of the mechanical part of the simulation can be displayed spatially resolved as the stress components σ1 = σxx and σ2 = σyy. In Fig. 6, the stress components σ1 and σ2 are depicted spatially resolved. The simulation is done for a 2 × 2 mm2 membrane with a thickness of 30 μm under a pressure of 50 kPa. It is clearly visible that the maximum stress exists in the region where the resistors are placed.

FIG. 6.
FIG. 6. Simulated stress components of a 2 × 2 × 0.03 mm3 membrane under a pressure of 50 kPa. (a) σ1 = σxx and (b) σ2 = σyy.
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Simulated stress components of a 2 × 2 × 0.03 mm3 membrane under a pressure of 50 kPa. (a) σ1 = σxx and (b) σ2 = σyy.

FIG. 6.
FIG. 6. Simulated stress components of a 2 × 2 × 0.03 mm3 membrane under a pressure of 50 kPa. (a) σ1 = σxx and (b) σ2 = σyy.
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Simulated stress components of a 2 × 2 × 0.03 mm3 membrane under a pressure of 50 kPa. (a) σ1 = σxx and (b) σ2 = σyy.

Close modal

The measured and simulated sensitivity is shown and compared in Fig. 7. The sensitivity is simulated for three different thicknesses of the membrane, 25, 30, and 35 μm. With decreasing membrane thickness, the sensitivity increases. The piezo-coefficient π44 is varied systematically from 104 × 10−11 to 114 × 10−11 Pa−1. With the increasing piezo-coefficient, the sensitivity increases as well.

FIG. 7.
FIG. 7. Simulated and measured sensitivity of a pressure sensor for different thicknesses of the membrane. In the simulation, the piezo-resistive coefficient π44 is varied systematically.
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Simulated and measured sensitivity of a pressure sensor for different thicknesses of the membrane. In the simulation, the piezo-resistive coefficient π44 is varied systematically.

FIG. 7.
FIG. 7. Simulated and measured sensitivity of a pressure sensor for different thicknesses of the membrane. In the simulation, the piezo-resistive coefficient π44 is varied systematically.
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Simulated and measured sensitivity of a pressure sensor for different thicknesses of the membrane. In the simulation, the piezo-resistive coefficient π44 is varied systematically.

Close modal

For the determination of the sensitivity as a function of the membrane thickness, three different groups of pressure sensors are used. Each group has a different membrane thickness. A membrane thickness of ∼26, ∼28, and ∼33 μm is used.

Obviously, the sensitivity is not a linear function of the membrane thickness t. The sensitivity is proportional to the inverse squared membrane thickness,13,14

UO/US∼1/t2.
(6)

Hence, the measured sensitivity as a function of the membrane thickness must be approximated by an inverse quadratic function. To estimate the measured sensitivity at a membrane thickness of 30 μm, an inverse quadratic function is fitted to the measured values. The result is depicted as the thick black line in Fig. 8. The thin black lines in Fig. 8 show the deviation of the measured sensitivity from the inverse quadratic fit function. In Fig. 8, the sensitivity for a membrane thickness of 30 μm is plotted as a function of the piezo-coefficient π44. The simulation results are the red squares connected with a red line.

FIG. 8.
FIG. 8. The sensitivity at a membrane thickness of 30 μm is plotted as a function of the piezo-coefficient π44. At the intersection of the measured value and the simulation, π44 is determined.
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The sensitivity at a membrane thickness of 30 μm is plotted as a function of the piezo-coefficient π44. At the intersection of the measured value and the simulation, π44 is determined.

FIG. 8.
FIG. 8. The sensitivity at a membrane thickness of 30 μm is plotted as a function of the piezo-coefficient π44. At the intersection of the measured value and the simulation, π44 is determined.
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The sensitivity at a membrane thickness of 30 μm is plotted as a function of the piezo-coefficient π44. At the intersection of the measured value and the simulation, π44 is determined.

Close modal

At the intersection of the measured and the simulated sensitivity, the piezo-coefficient π44 is determined. In this way, the piezo-coefficient π44 = (108.3 ± 2.1) × 10−11 Pa−1 is obtained. The error of the piezo-coefficient π44 is determined from the intersection of the simulated sensitivity with the thin black lines, which represent the error of the measured sensitivity.

To compare the piezo-coefficient π44 obtained in this work to experimental literature data, we follow the collection of literature data of Richter et al.8 These literature data from Refs. 10 and 15–19 are collected in Table II.

TABLE II.

Comparison of the determined piezo-coefficient π44 with literature data. Boldface denotes the piezo-coefficient obtained in this work.

ReferencesDopant density (×1018 cm−3)π44 (×10−11 Pa−1)
10  0.002 138.1 
15  0.02 93.1 ± 7.0 
16  0.03 113.5 ± 6.8 
This work 0.50 ± 0.45 108.3 ± 2.1 
17  0.8 105 ± 10.5 
18  1.5 87 ± 5.7 
19  3 111 
17  8.2 95 ± 9.5 
19  9 98 
19  50 78 
19  300 60 
19  500 48 
19  2000 35 
ReferencesDopant density (×1018 cm−3)π44 (×10−11 Pa−1)
10  0.002 138.1 
15  0.02 93.1 ± 7.0 
16  0.03 113.5 ± 6.8 
This work 0.50 ± 0.45 108.3 ± 2.1 
17  0.8 105 ± 10.5 
18  1.5 87 ± 5.7 
19  3 111 
17  8.2 95 ± 9.5 
19  9 98 
19  50 78 
19  300 60 
19  500 48 
19  2000 35 
View Large

The literature data collected in Table II together with the piezo-coefficient obtained in this work are plotted in Fig. 9. The piezo-coefficient π44 is plotted as a function of the dopant concentration of the resistor. A decrease in the piezo-coefficient with the increasing dopant concentration is visible.

FIG. 9.
FIG. 9. Graphical comparison of the determined piezo-coefficient π44 with literature data.
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Graphical comparison of the determined piezo-coefficient π44 with literature data.

FIG. 9.
FIG. 9. Graphical comparison of the determined piezo-coefficient π44 with literature data.
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Graphical comparison of the determined piezo-coefficient π44 with literature data.

Close modal

In our case, an average dopant concentration is used. We calculated the arithmetic average over the 3 µm thick resistor, which revealed an average boron concentration in the 3 µm of the resistor of (5.0 ± 4.5) × 1017 cm−3. This is, in fact, a rough approximation of the real boron diffusion profile. However, since the current flows through all three resistor bars, it is a valid simplifying assumption. The current density differs by a factor of 9 between the bar with the highest doping level and the bar with the lowest doping level. The error is taken as the standard deviation of the three used dopant concentrations within the resistor. This is a very rough estimate too. However, the model applied here with a boron diffusion profile does not allow to state a more concrete average concentration value. To get a more concrete concentration value, a step like boron profile would be necessary for example.

In Fig. 9, it is clearly visible that there is a scatter in the literature data of π44 in the dopant concentration region where we obtained the piezo-coefficient π44. Nevertheless, it can be stated that the piezo-coefficient π44 obtained in this work fits well to the reported values from the literature.

In summary, we performed a study to estimate the piezo-coefficient π44 in silicon. Basically, π44 was obtained by comparing an experiment with simulation. Therefore, pressure sensors with different membrane thicknesses were built and the sensitivity in a Wheatstone bridge arrangement was measured. This sensitivity was simulated using the simulation software COMSOL Multiphysics. Therefore, the full pressure sensor with two resistors was simulated. The current through the resistors was extracted under an applied pressure, and the sensitivity was calculated. The simulations were done under systematic variation of the piezo-coefficient π44. By comparing the sensitivity as a function of the piezo-coefficient π44 with the measured sensitivity, a value of π44 = (108.3 ± 2.1) × 10−11 Pa−1 at a dopant concentration of (5.0 ± 4.5) × 1017 cm−3 was found. This is in good agreement with the reported values from the literature. Based on these results, it is now possible to predict the sensitivity of a pressure sensor more precisely.

We acknowledge support for the publication costs from the Open Access Publication Fund of the Technische Universität Ilmenau.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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