The piezo-resistive coefficient *π*_{44} 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 *π*_{44} since this is the only free parameter in the simulation. A value of *π*_{44} = (108.3 ± 2.1) × 10^{−11} Pa^{−1} at a dopant concentration of (5.0 ± 4.5) × 10^{17} cm^{−3} was found, which is in good agreement with experimental literature data.

## I. INTRODUCTION

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 *π*_{44}.

Mathematically, the piezo-resistivity can be described by a six-vector notation.^{9} The stress vector *σ*_{λ} is connected via the piezo-resistivity tensor *π*_{κλ} to the change in resistivity vector Δ*ρ*_{κ},

with *ρ*_{0} 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 *π*_{11}, *π*_{12}, and *π*_{44}, and the tensor is then

In *p*-type silicon, the piezo-resistivity tensor component *π*_{44} is about two magnitudes larger than the other two components. Hence, the component *π*_{44} has the main impact on the resistivity change due to applied stress. Due to its importance, much effort^{8} has been undertaken in the past to obtain the component *π*_{44} by experiment as well as by theory. Usually, test structures^{8} or uniformly doped bars^{10} are prepared, which are stressed, e.g., by a four point bending fixture.^{8} The stress is calculated, and in consequence, *π*_{44} is calculated as well.

In this contribution, for the first time, the shear piezo-coefficient *π*_{44} 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 *π*_{44}. This coefficient *π*_{44} 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 *π*_{44} 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 *π*_{44}.

## II. EXPERIMENT

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 *R*_{1} to *R*_{4} placed on the edges of the membrane. The resistors themselves consist of boron doped bars with a size of 10 × 100 *μ*m^{2} 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.

The resistors are arranged in a Wheatstone bridge. The circuit diagram of this bridge is shown in Fig. 3. The output voltage *U*_{O} is calculated by knowing the resistors *R*_{1} to *R*_{4} and the supply voltage *U*_{S} in the following way:

The pressure sensor is designed in a way that under pressure, *R*_{1} = *R*_{4} and *R*_{2} = *R*_{3} hold [see Fig. 1(b)]. It follows for the so-called sensitivity *U*_{O}/*U*_{S} given in (mV/V),

In the simulation, the resistance of the resistors *R*_{1} and *R*_{2} is obtained separately by applying Ohm’s law *R*_{1,2} = *U*_{1,2}/*I*_{1,2}. This is done without using a Wheatstone bridge. A constant voltage *U*_{1} = *U*_{2} = *U* is applied to *R*_{1} and *R*_{2}, and the currents *I*_{1} and *I*_{2} are determined by COMSOL. Hence, we get

The sensitivity *U*_{O}/*U*_{S} is measured under applied pressure for the produced pressure sensors. Additionally, the currents *I*_{1} and *I*_{2} 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 mm^{2}. 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.

The resistor itself consists of three bars, each 100 × 10 × 1 *µ*m^{3} 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.

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 constants^{11,12} and the other piezo-coefficients^{10} 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.).

Elastic constants . | Piezo-coefficients . | ||
---|---|---|---|

(GPa) . | (10^{−11} Pa^{−1})
. | ||

C_{11} | 166 | π_{11} | 6.6 |

C_{12} | 64 | π_{12} | −1.1 |

C_{44} | 80 |

Elastic constants . | Piezo-coefficients . | ||
---|---|---|---|

(GPa) . | (10^{−11} Pa^{−1})
. | ||

C_{11} | 166 | π_{11} | 6.6 |

C_{12} | 64 | π_{12} | −1.1 |

C_{44} | 80 |

The currents *I*_{1} and *I*_{2} through the two resistors are obtained by applying a voltage. Using Eq. (5), the sensitivity is obtained from the simulated currents.

## III. RESULTS

The results of the mechanical part of the simulation can be displayed spatially resolved as the stress components *σ*_{1} = *σ*_{xx} and *σ*_{2} = *σ*_{yy}. In Fig. 6, the stress components *σ*_{1} and *σ*_{2} are depicted spatially resolved. The simulation is done for a 2 × 2 mm^{2} membrane with a thickness of 30 *μ*m under a pressure of 50 kPa. It is clearly visible that the maximum stress exists in the region where the resistors are placed.

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 *μ*m. With decreasing membrane thickness, the sensitivity increases. The piezo-coefficient *π*_{44} is varied systematically from 104 × 10^{−11} to 114 × 10^{−11} Pa^{−1}. With the increasing piezo-coefficient, the sensitivity increases as well.

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 *μ*m 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}

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 *μ*m, 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 *μ*m is plotted as a function of the piezo-coefficient *π*_{44}. The simulation results are the red squares connected with a red line.

At the intersection of the measured and the simulated sensitivity, the piezo-coefficient *π*_{44} is determined. In this way, the piezo-coefficient *π*_{44} = (108.3 ± 2.1) × 10^{−11} Pa^{−1} is obtained. The error of the piezo-coefficient *π*_{44} 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 *π*_{44} 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.

References . | Dopant density (×10^{18} cm^{−3})
. | π_{44} (×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 |

The literature data collected in Table II together with the piezo-coefficient obtained in this work are plotted in Fig. 9. The piezo-coefficient *π*_{44} 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.

In our case, an average dopant concentration is used. We calculated the arithmetic average over the 3 *µ*m thick resistor, which revealed an average boron concentration in the 3 *µ*m of the resistor of (5.0 ± 4.5) × 10^{17} 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 *π*_{44} in the dopant concentration region where we obtained the piezo-coefficient *π*_{44}. Nevertheless, it can be stated that the piezo-coefficient *π*_{44} obtained in this work fits well to the reported values from the literature.

## IV. CONCLUSIONS

In summary, we performed a study to estimate the piezo-coefficient *π*_{44} in silicon. Basically, *π*_{44} 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 *π*_{44}. By comparing the sensitivity as a function of the piezo-coefficient *π*_{44} with the measured sensitivity, a value of *π*_{44} = (108.3 ± 2.1) × 10^{−11} Pa^{−1} at a dopant concentration of (5.0 ± 4.5) × 10^{17} 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.

## ACKNOWLEDGMENTS

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

## DATA AVAILABILITY

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