Proceedings of E240, Winter 2015 DD SHIN, RN Haksar, BJ Galligan and J Baek

WAFER-ENCAPSULATED DIFFERENTIAL RESONANT ACCELEROMETER WITH PASSIVE TEMPERATURE COMPENSATION SCHEME

Dongsuk D. Shin, Ravi N. Haksar, Benjamin J. Galligan and Jeesu Baek Department of Mechanical Engineering, Stanford University

Stanford, California

Abstract. This paper presents the design of a resonant accelerometer with a passive temperature-compensation scheme for use in an inertial measurement unit (IMU). This is achieved by shape tuning a pair of resonant beams to yield a similar temperature coefficient of frequency (TCf) for differential temperature compensation, while separating the two resonant frequencies to increase the dynamic range. Preliminary design analysis suggests a sensitivity of 375 Hz/g, dynamic range of ±50 g, and matching TCfs within 0.06 ppm difference. From these results the temperature effects are reduced by a factor of more than 500.

Background. In the last decade, microelectromechanical systems (MEMS) research has spurred the development of microsensors for almost every possible sensing modality. One emerging area of interest is the inertial measurement unit (IMU), a combination of accelerometers, gyroscopes, and magnetometers used to determine the position, orientation, and velocity of a moving object. With the increased demand for small, inexpensive, and high-performance MEMS IMUs, there have been efforts to co-fabricate the required structures—accelerometers, gyroscopes, and often resonator clocks for timing references—and wafer-level package them on a single die. This approach, however, presents technical challenges: high-performance gyroscopes and clock resonators require a high quality factor (Q), whereas typical capacitive accelerometers (where a seismic mass displacement due to acceleration is sensed via parallel plate capacitance) generally require lower Q to achieve critical damping.

Moreover, traditional capacitive accelerometers are limited with regards to miniaturization and performance. First, because they are capacitive displacement transducers, a large proof mass translation is required for acceptable sensitivity to small accelerations. Second, parallel plate capacitors are susceptible to pull-in instability, which limits the minimum gap size and thus requires a larger proof mass [1]. These limitations in gap and proof mass size pose severe constraints on sensor’s dynamic range, high-shock survivability, and displacement sensitivity.

To overcome these issues and achieve single-die integration, resonant sensing can be used instead of capacitive sensing. In the design shown in Fig. 1, a pair of resonant beams is affixed to a proof mass, such that external acceleration will generate axial force on the beams. This will cause frequency shifts, used as signals, in the two resonators. The design’s symmetry provides a differential response to linear acceleration and intrinsically compensates for off-axis acceleration and temperature changes. While resonant sensing introduces a different set of challenges associated with frequency measurement, because it is not strictly limited by gap size, it presents some key upsides: high sensitivity, large dynamic range, immunity to pull-in instability, and high-shock survivability. Furthermore, it requires high Q and hence low-damping techniques, making it an ideal candidate for IMU integration on a single die.

Figure 1. SEM micrograph of Christensen et. al’s device and inset showing close up of sensing geometry.

A major issue with implementing the resonant accelerometer design lies in the resonant structure’s high sensitivity to temperature fluctuations. The two beams were originally designed to be identical in order to differentially read the acceleration signal and cancel out the temperature effects. However, while resonant beams exhibit the same temperature coefficients of frequency (TCf), the two modes synchronize near their resonant frequencies; this prevents the tracking of two separate peaks and thereby the ability to track inertial forces through differential frequency. To ensure that the two modes do not interact with each other, the two beams’ resonant frequencies should be far apart for some desired dynamic range (roughly ±50g), while having the same TCfs to cancel out temperature effects.

Therefore, the goal of the project is to design a resonant accelerometer with a passive temperature-compensation scheme to overcome this temperature sensitivity issue, which will enable further development of resonant accelerometers in many applications and markets, such as consumer devices. The device

Proceedings of E240, Winter 2015 DD SHIN, RN Haksar, BJ Galligan and J Baek

will be fabricated using a wafer-scale encapsulation process. The performance of the accelerometer should have a high sensitivity (>300 Hz/g), low temperature sensitivity (<0.01 ppm/˚C), large dynamic range (>50 g), high shock survivability (>10000 g), and low noise (<0.1 mg/√Hz)).

High performance IMUs are critical for accurate measurement of an object’s movement and rotation in 3D space. In particular, a temperature-compensated MEMS accelerometer would be useful for navigation in small to medium-sized robotic vehicles that either operate autonomously or provide feedback for human operator inputs. For this type of application, low temperature sensitivity is necessary to prevent excessive drift. This type of navigation, called dead reckoning, has historically been limited to larger vehicles that have the space for high performance, large-footprint electromechanical devices. Reducing the size without performance loss is critical for smaller vehicles and the development of better autonomous robotic systems.

Previous efforts to compensate for temperature effects have included building additional structures [2] to reduce thermal stress on resonant beams, as well as adding separate DETFs on the same die for active compensation [3]. As shown in the design space in Fig. 2 [2]-[11], there is a distinct tradeoff between the sensor performance—namely sensitivity—and the temperature effects. In addition, direct incorporation of temperature compensation scheme into the accelerometer design has not been explored in the past. This work proposes to design a passive temperature compensation scheme into Christensen et. al’s previous device [4] via shape tuning. The two beams will exhibit similar TCf behavior while maintaining a sufficient frequency split and the performance of the device in terms of sensitivity will be minimally affected in the compensated system.

Figure 2. Comparison of performance of previous work to the current and proposed designs. Previous efforts have been unable to effectively compensate for temperature effects while maintaining high sensitivity.

Design. The first part of the redesign of the current work involves modifying the dimensions of one of the two resonant beams. The two beams support the mass, which rotates about a fixed point when an external acceleration is applied. The change in beam dimensions modifies the beam stiffness, which is related to the mechanical resonant frequency (see Analysis section for more details). By modifying only one of the beams, the beams will have different resonant frequencies, increasing the dynamic range of the device. The original beams have a thickness of 40 µm, a length of 160 µm, and a width of 3.1 µm. The modified beam will have an overall width of 3 µm with a thicker section of 3.1 µm. This section is centered on the beam and has a length of 20 µm.

The next part of the redesign is to make sure that the TCf of the modified beam still closely matches the other unmodified resonant beam. Since the dimensions of one of the beams were changed, a mismatch in two TCfs is created, causing the sensitivity fluctuation over temperature. To change the TCf to be near that of the other beam, the center was thickened. The relationship between beam geometry and TCf change is described in the Analysis section, which shows how this change is feasible. Fig. 3 shows the redesign of the current work.

Figure 3. Proposed resonant accelerometer design with inset that shows the modified resonant beam geometry. The red section denotes the increased beam width.

The preliminary tests of an existing device shows the effect of difference in TCf of two beams on the device sensitivity. In the original design, fabrication errors produced differing widths, and as a result, an initial temperature sweep showed roughly 35 parts per million (ppm) difference between two TCf curves over -20˚C to 80˚C. In an inclinometer test over the temperature range, the differential signal to ±1g acceleration, which correlates directly with the device sensitivity, decreased with increasing temperature as shown in Fig. 4.

Proceedings of E240, Winter 2015 DD SHIN, RN Haksar, BJ Galligan and J Baek

Figure 4. Results of inclinometer test of original device which shows a decrease in sensitivity as temperature increases.

Figure 5. Side-view illustration of the standard Epi-seal process for a standard resonator on a silicon wafer.

Fabrication. The epi-seal encapsulation process was developed by a close collaboration between Robert Bosch Research and Technology Center in Palo Alto and Stanford University. The resonator is sealed in an 1100˚C high temperature hydrogen environment by a layer of epitaxial silicon cap immediately after vapor HF releasing the device from the surrounding oxide. The high temperature hydrogen annealing process provides an ultra-clean, native-oxide-free device surface, while smoothing out the sidewall scallops through silicon migration. A near-vacuum cavity pressure (<1 Pa) is achieved by diffusing out the residual hydrogen.

The standard Epi-seal process is illustrated in Fig. 5 [12]. Step (1): The MEMS structure is defined by etching trenches in the device layer of a silicon-on-oxide (SOI) wafer using deep reactive ion etching (DRIE). A sacrificial oxide layer is deposited to fill the trenches and to provide an oxide spacer layer above the device. Step (2): Contacts are etched in the oxide and a silicon layer is deposited in an epitaxial reactor. This results in crystalline silicon where there is a crystalline seed and epitaxial polysilicon where there is oxide. Step (3): Top electrode isolation trenches are etched and refilled with

nitride. After patterning the nitride, an intermediate epi-poly cap layer is deposited. Step (4): Vents are then etched into this cap and vapor HF is used to etch oxide to release the device. Step (5): After a high temperature hydrogen bake in an epitaxial reactor to remove contaminants and native oxide, a second layer of epitaxial silicon is deposited, sealing the device in a clean cavity. Step (6): Electrical isolation and contacts are defined. The pressure in the cavity is further lowered by diffusing hydrogen in the cavity out, achieving a low pressure environment in the cavity.

Performance Analysis. The sensitivity of the resonant accelerometer due to a differential frequency shift can be calculated by,

𝑆!! = 0.14𝐿𝑤

!

𝑓! 2𝜖!"#$!! 𝐻𝑧𝑔

(1)

where 𝐿 and 𝑤 are the length and width of the resonant beam, respectively, 𝑓! is the resonant frequency of the beams, and 𝜖!"#$!! is the strain in the beams due to an applied acceleration of 1g. Since the beam’s no longer have matching resonant frequencies, the average value is used, which is 638 kHz. The beam strain is also estimated from previous work [4]. With these numbers, the calculated sensitivity is approximately 374 Hz/g.

The resonant beams used in the design can be analyzed as fixed-guided structures. The equivalent beam stiffness is calculated by,

𝑘 =𝐸!!""!𝑤!𝑡

4𝐿! (2)

where 𝐸!!""! is the Young’s modulus of the beam material, and 𝑤, 𝑡, and 𝐿 are the beam’s width, thickness, and length, respectively. The proposed device will be oriented in the <100> direction in a single crystal silicon wafer.

Using the beam stiffness, the resonant frequency can be calculated by,

𝑓! =12𝜋

𝑘𝑚=

12𝜋

𝐸𝑤!𝑡4𝐿!𝜌𝑤𝑡𝐿

=12𝜋

𝐸𝑤!

4𝜌𝐿! (3)

where m is the mass of the beam and 𝜌 is the density of single crystal silicon. Using the beam width and length and material properties of silicon, the resonance frequency of the modified beam is found to be 612 kHz.

Therefore, it is possible to modify the beam’s resonant frequency by changing the dimensions. The desired frequency difference between the two beams can be calculated based on the desired dynamic range. Given the desired dynamic range of ±50g with a safety factor of 2, the beam frequency split is calculated to be about 50 kHz. The unmodified beam has a frequency of 665.4 kHz, so this design meets the desired frequency split.

Proceedings of E240, Winter 2015 DD SHIN, RN Haksar, BJ Galligan and J Baek

The relationship between modified beam geometry and TCf, suggested by Melamud et. al. [13], examines how the parameters in Equation 3 change with temperature. The temperature dependence can be expanded as,

𝑇𝐶𝑓 =1𝑓𝜕𝑓𝜕𝑇

(4a)

𝑇𝐶𝑓 =1𝛽𝜕𝛽𝜕𝜎

𝜕𝜎𝜕𝑇

+𝑇𝐶𝐸2

+𝛼2

(4b)

where 𝛽 is the beam mode constant, TCE is the temperature coefficient of change for the Young’s modulus, and 𝛼 is the linear thermal coefficient of expansion. For a beam with a constant rectangular cross section and fixed-guided boundary conditions, the mode constant is 𝛽 = 3.92. The mode constant 𝛽 has a nonlinear dependence on the axial stresses in the resonator and can be solved for numerically. An approximation can be found by considering the rate of change of beta about the zero stress state, which is valid for small stress changes. The resulting linear fit found by the prior work suggests that the relationship has the following dependencies,

1𝛽𝜕𝛽𝜕𝜎

= 𝑓 𝐿! ,𝑤! , 𝑡! ,𝐸! (5)

Therefore, modifying the beam geometry will modify the TCf of a resonant beam. Since this relationship is highly dependent on the beam geometry and boundary conditions, simulations were used to propose a beam design that would meet the performance requirements.

Publicly available simulation tools based on COMSOL [14] were used to iterate on a beam design; the final version is shown in Fig. 6. The TCf values over a 100 ˚C temperature sweep for each beam were simulated and the curves are shown in Fig. 7. The difference between the curves was found to be at most 0.06 ppm, which is well within the necessary range to reduce the temperature effects by more than factor of 500.

Figure 6. Final beam design generated in COMSOL showing modified geometry; the colors denote displacement of the resonant mode.

Uncertainty Analysis. To determine the uncertainty of the device sensitivity, it was necessary to first determine the uncertainty in the natural frequency of the beam. Applying Holman’s uncertainty method [15] in Equations 1 and 3, the uncertainties in natural frequency and in sensitivity can be estimated by,

Figure 7. Simulated TCf curves for the proposed device with inset showing the difference between the curves. The difference is close enough for effective compensation.

𝑤!!𝑓!

=12𝑤!𝐸

!

+𝑤!𝑤

!+

2𝑤!𝐿

!

!!

(6a)

𝑤!!!𝑆!!

=2𝑤!𝐿

!

+2𝑤!𝑤

!

+𝑤!!𝑓!

!! (6b)

where 𝑤! represents the uncertainty in variable x. Table 1 explains the anticipated errors and their sources. The calculated uncertainty in natural frequency is 100 kHz and the corresponding uncertainty in sensitivity is 130 Hz/g; this corresponds to a 30% error in the sensitivity. The major source of uncertainty is from the uncertainty in the beam width.

Table 1. Estimated errors used in the uncertainty analysis and their sources.

Variable Estimated Error Source of Error

Young’s modulus, E 0.844 GPa Assume an alignment error of 1º off of the <100> direction

Beam width, w 0.3 µm Expected error due to fabrication

Beam length, L 0.3 µm Expected error due to fabrication

Note that this analysis relies on the following assumption. Since the increased width in the center of the modified beam is relatively small compared to the length and thickness of the beam. From this, it is reasonable to assume that an analysis for beams of constant rectangular cross-section is valid for our design.

Conclusions. Resonant accelerometers are an attractive alternative to capacitive transduction methods, but previous designs have been unable to provide the necessary performance for high-end applications, especially with regard to temperature effects. The proposed design in this paper will maintain the high

Proceedings of E240, Winter 2015 DD SHIN, RN Haksar, BJ Galligan and J Baek

sensitivity of the previous work while effectively compensating for temperature changes, which will provide consistency over changing operating environments. The analysis of the redesign shows that the beam geometry modification is a feasible method to meet the desired TCf specifications, and the desired changes are verified by simulations. However, since the major source of uncertainty is from the fabrication process, multiple devices may need to be fabricated to achieve the desired geometry. Testing, such as the inclinometer test, will also need to be used to determine if performance specifications are met. Finally, iteration of either the beam design or fabrication method may be required to make batch fabrication more consistent and reliable.

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