^{1}

^{*}

^{2}

^{1}

^{3}

^{3}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Gap asymmetry in differential capacitors is the primary source of the zero bias output of force-balanced micro accelerometers. It is also used to evaluate the applicability of differential structures in MEMS manufacturing. Therefore, determining the asymmetry level has considerable significance for the design of MEMS devices. This paper proposes an experimental-theoretical method for predicting gap asymmetry in differential sensing capacitors of micro accelerometers. The method involves three processes: first, bi-directional measurement, which can sharply reduce the influence of the feedback circuit on bias output, is proposed. Experiments are then carried out on a centrifuge to obtain the input and output data of an accelerometer. Second, the analytical input-output relationship of the accelerometer with gap asymmetry and circuit error is theoretically derived. Finally, the prediction methodology combines the measurement results and analytical derivation to identify the asymmetric error of 30 accelerometers fabricated by DRIE. Results indicate that the level of asymmetry induced by fabrication uncertainty is about ±5 × 10^{−2}, and that the absolute error is about ±0.2 μm under a 4 μm gap.

The development of micro-electro-mechanical-systems (MEMS) usually involves the fabrication of small structures with relative errors larger than that observed in traditional fabrication technology. Therefore, evaluating the level of structural error and its influence in micro-fabrication has significance in the design of new devices and improvement of processes. Given the immaturity and diversity of micro fabrication techniques, researchers and institutions adopt evaluation methods with focus and specificities particular to such approaches. For example, Cigada used the electrical method to measure the dynamic behaviors of a MEMS gyroscope with fabrication error [

The rest of the paper is structured as follows: Section 2 presents the centrifuge tests conducted in this study, and the observations of zero bias in accelerometers fabricated by DRIE. Section 3 discusses the prediction process, including the analytical derivation and reverse calibration method. Finally, Section 4 provides the results of the proposed experimental–theoretical method.

Micro accelerometers are fabricated using DRIE bulk silicon technology and silicon bonding technology to provide high sensitivity and large signal output. These components have been proven successful in MEMS devices. The structure of the micro accelerometer used in this study is shown in

“Bi-” refers to two opposite ways of loading bias input voltage on the electrodes of the capacitors. For the positive direction, the top electrode shown in

The data are depicted in

Identifying the effect of asymmetric error on the accelerometer requires deriving and analyzing the operation principle of the sensor. The diagram of a typical differential accelerometer structure is shown in

The gap asymmetry caused by the micro fabrication is considered first. Its equivalent value is represented by the average value, which equals the ratio of the sum of the capacitor gaps and gap number (_{1} and _{2} are the average values of the top and bottom electrode gaps, respectively. _{1} ≠ _{2} reflects the asymmetry from the fabrication. When the bias voltages are applied to the fixed plates of the capacitors, the mass block moves to a position where the mechanical force equals the electrostatic force under closed-loop control.

Assuming that the sensing circuit is of an ideal state, the instantaneous feedback voltage on the block subjected to bias voltage can be expressed as:
_{a}_{1} and _{2} are the total capacitance values of the top and bottom capacitors, respectively.

When a voltage is applied, the mass block moves because of the asymmetry induced by the electrostatic force produced by the top and bottom capacitors. The movement deforms the supporting beams. The resultant force of the mass block can therefore be expressed as:
_{s}_{e}_{1} and _{e}_{2} denote the electrostatic forces formed by two groups of capacitors, which can be expressed as follows:
_{0} denotes the vacuum permittivity, _{d}_{f}

The balance position of the mass block is where the resultant force is zero, _{r}

To simplify, we set one average value _{1} = _{0}, and the other average value _{2} = _{0} + _{f}_{0} is a constant, _{out} is the output voltage, and _{0}.

When the frequency of the driving voltage is considerably larger than the natural frequency of the mass block, the electrostatic forces can be calculated using the effective value of the voltage. According to the law of action and reaction, when the external acceleration is applied to the accelerometer, inertial force _{a}_{r}_{a}_{a}_{a}_{d}_{e}_{s}_{e}_{0}_{d}^{2}_{0}^{−3}.

Therefore, the input acceleration can be expressed as a function of output voltage:
_{s}_{g}

We conclude that the presence of _{g}_{b}_{b}

According to the preceding derivation, the input acceleration can be expressed as:
_{b}_{b}_{d}

The output voltage is:

Therefore, the input can be expressed in terms of the output as follows:

We conclude that the existence of a circuit deviation also leads to bias output voltage because of the nonzero constant term _{c}_{b}

The influence of the structure and circuit error has been analyzed under deep feedback control. Gain value

The conclusion derived from _{out} = 0 to the original point. It is also independent of the parameters of the closed-loop control circuit. Second, the second term on the right indicates how the asymmetric error influences the linear constant of the calibration function.

Reverse calibration pertains to input acceleration calibrated as a function of output voltage. According to the analytical procedure, formulating the input function of the output is more direct and clear than formulating the output function of the input. Thus, reverse calibration is introduced to study the gap asymmetric error resulting from micro fabrication. The results of the reverse calibration based on the least-squares method are:
_{outp}_{outn}_{p}_{n}

The theoretical–experimental prediction is established on the analytical and calibration functions. The ratio of constant term to linear term coefficient is selected in predicting the error, and can be expressed as:

The conclusion derived from

The reverse calibration result of a selected sensor leads to the coefficient ratio:

Therefore, the predicted relative error that arises from the tested accelerometer is about 4 × 10^{−2}, and the absolute error is about 0.16 μm at an average gap of 4 μm. Under common conditions of engineering design, 30 sensors are tested and predicted. The results are shown in ^{−2}, and that of the absolute error is about ±0.2 μm under a 4 μm gap. For verification, those sensors are investigated by optical microscopy and SEM, from which the finger images are shown in

A fabrication error prediction technique based on a mature MEMS device is presented in this paper. The combined analytical and reverse calibration methods enable direct and clear gap error prediction. Reverse calibration is suitable for deriving the relationship, and the analytical relationship provides the guidelines for rational calibration. In addition, the absence of a quadric term improves prediction precision.

The ratio of the prediction coefficients eliminates the need to calculate the values of parameters, which are difficult to determine. The prediction indicates that the range of relative asymmetric error is about ±5 × 10^{−2}, and that the absolute error is about ±0.2 μm under a 4 μm gap. These results coincide with the values measured by optical microscopy and SEM.

This research was partially supported by National Natural Science Foundation of China (Grant No. 51175437), Fundamental Research Funds for the Central Universities (Grant No. ZYGX2011J085). The authors would like to acknowledge Zhang Fengtian, Shi Zhigui and Du Lianming for their assistance in fabrication and experiments.

SEM of the sensor's structure.

Differential sensor.

Data of experiments on the selected sensor.

Diagram of an accelerometer.

Simplified structure with average gaps.

The test results in two directions.

Relative error of calibration.

Asymmetric error

The prediction results of thirty sensors.

The gap measurements using microscopy. (

The measurement results of thirty sensors.

Parameters of the accelerometer system.

DC voltage (_{d} |
4.5 |

Coefficient (_{0}) |
0.4 |

Ratio of Stiffness (R) | 0.1906 |

Initial gap designed (μm) | 4 |