Design and Analysis of a High G Capacitive Uniaxial MEMS ...scholar.harvard.edu/files/jgafford/files/finalpaper_repaired.pdf · and fabrication of a high-g capacitive uniaxial MEMS

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Prof. Crozier

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Abstract— This paper details the modeling, design, analysis

and fabrication of a high-g capacitive uniaxial MEMS

accelerometer. Serpentine flexures transmit the motion of the

proof mass in the desired sensing direction, and the motion is

detected by an array of differential capacitors which reject

common mode responses arising from off-axis translations. The

accelerometer is designed for fabrication via surface

micromachining processes. The sensitivity of the accelerometer

was shown to be 7.3 fF/g over a range of 0 to 200 g. Settling

time of the accelerometer is 370 µs, resulting in a device

bandwidth of 2.7 kHz which is far below system resonance.

I. INTRODUCTION

Accelerometers are used in a wide range of applications in a number of industries including aviation, consumer electronics, and automotive. Due to their mechanical nature, accelerometers are extremely prone to parasitic noise and cross-talk due to the coupling between the sensitive and non-sensitive axes. This paper explores the design and analysis of a robust uniaxial accelerometer capable of high-g operation. Mechanical and electrical cross-talk compensation negates the influence of off-axis modes, isolating the sensing mode for a true output signal.

II. ACCELEROMETER DESIGN AND ANALYSIS

A. Accelerometer Modeling

In order to optimize the performance of the

accelerometer, the physics dictating the behavior must be

adequately modeled. In addition, the dynamic performance

places some practical constraints on sensor output, including

bandwidth, response time and noise. A schematic of a

simplified accelerometer model is shown in Figure 1. A

kinematic spring and a viscous damper act in parallel to

resist motion of a proof mass, resulting in second-order

harmonic behavior as per Equation (1):

(1)

where is the linear displacement, is the proof mass,

is the viscous damping coefficient, is the serpentine

flexure effective stiffness and F is the external inertial force.

Represented in state-space, choosing and as our

generalized variables, we have:

[ ] [

⁄ ⁄ ] [

] [

] (2)

Here it is assumed that the dynamic response of the system

is dominated by the inertia of the proof mass, and as such,

spring mass is ignored. The effective spring stiffness k can

be analytically determined by assuming fixed-ended

cantilevers in series. Since we have four of these springs

acting in parallel, the effective stiffness of the system is

given by Equation (3).

[∑

]

(3)

Air damping occurs when the rotor combs move laterally

with respect to the stator combs, thus compressing the air in

between and squeezing the air out. This phenomenon, called

‘squeeze film damping,’ is described using Hagen-Pouiselle

flow theory [1]:

(

) (4)

where is the number of rotor/stator finger sets,

is the fluid viscosity of air, is comb length, is comb

thickness and is the initial gap width. Additional damping effects are incurred via ‘slide film damping’, but these are assumed to be miniscule in comparison.

B. Differential Capacitance Measurement

Consider the differential plate capacitance model

depicted in Figure 2. We can model each individual

stator/rotor/stator comb set as a differential capacitor, where

the rotor plate moves perpendicularly with respect to the

stator plates axes in a dielectric medium (air). As the rotor

plate translates, capacitance changes between each

stator/rotor are given by the following:

(

)

(5)

(

)

(6)

We subtract Equation (6) from Equation (5) to obtain a

Design and Analysis of a High-G Capacitive Uniaxial MEMS

Accelerometer

Joshua B. Gafford, HUID: 10874483

Figure 1. Simplified accelerometer model

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differential capacitance measurement [2]:

| | (

) (7)

Extending this expression to the whole system, the total

change-in-capacitance is as follows:

(

) (8)

(

) (9)

where in (9) we have substituted in for by recalling the

equation of motion in Equation (1) and assume steady-state

by setting all derivatives to zero.

C. Basic Design

The basic layout of the accelerometer is depicted in Figure 3. The accelerometer footprint is 500µm x 500 µm. A proof mass is mechanically constrained to move in a predefined sensing direction via serpentine flexural elements. Rotor electrodes move laterally with respect to stator electrodes, realizing a differential capacitor that is sensitive to inertial forces. Mechanical stops prevent catastrophic failure by limiting the maximum displacement of the proof mass.

Most commercial off-the-shelf (COTS) devices feature additional forcing combs for a self-test feature, for the purpose of analyzing the sensing capabilities of the designed device, forcing combs are left out of the analysis.

D. Geometry Selection

The inertial sensitivity of the accelerometer is inversely

proportional to the spring stiffness

(10)

Given the effective stiffness from Equation (3), we can plot

the stiffness variation as a function of various geometric

parameters for purpose of optimization.

The sensitivity as a function of beam width and beam

thickness is shown in Figure 4, with the desired design

regime highlighted. While we do want to optimize

sensitivity in the direction of interest, we also want to

minimize off-axis sensitivity by choosing and such

that ⁄ to maximize system compliance in the

sensitive direction. As such, we achieve a certain degree of

mechanical isolation between the desired sensing axis and

off-axis effects.

Figure 3. Basic geometric layout of the accelerometer featuring, (1)

serpentine flexure, (2) mechanical stop, (3) stator electrode, (4) rotor

electrode, (5) proof mass with air damping holes, (6) sensing direction.

Figure 4. (left) Accelerometer Sensitivity as a function of beam geometry, (right) Resonant Frequency as a function of beam geometry

Figure 2. Illustration of differential capacitance

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An additional consideration is that of mechanical

resonance. The resonant frequency of the system (in Hz) is

given by the following:

√

(11)

Ideally we’d select the resonant frequency to be much

beyond the intended bandwidth of the system, as depicted in

Figure 5. Not only does resonance result in significant

system nonlinearity, but it can also be mechanically

catastrophic in an underdamped system. For almost any

conceivable mechanical system, frequencies of several kHz

or higher are rarely encountered, and as such, we’ll select a

resonant frequency of around 10 kHz for significant safety

factor. A plot of resonant frequency vs. beam geometry is

given in Figure 4 (right), once again, with the design regime

highlighted.

The parameters that define the accelerometer’s geometry

are summarized in Table 1.

Table 1. Accelerometer properties

Notation Parameter Value

Proof mass side length 250 µm

Proof mass thickness 50 µm

Proof mass weight 8.6 ng

Spring beam width 2 µm

Spring beam thickness 50 µm

Spring beam major length 150 µm

Spring beam minor length 100 µm

Comb finger length 200 µm

Comb finger thickness 50 µm

Initial gap thickness 2 µm

Initial Capacitance 1.15 pF

III. FABRICATION

The accelerometer is fabricated via silicon surface micromachining. The process flow is given in Figure 6. (1) A 4”, 500 µm <1 0 0> X-Si wafer is used as the substrate material. (2) A thin layer (1 µm) of Si3N4 is deposited on the surface via LPCVD to isolate the Si substrate from the PolySi structural material. (3) A 5 µm layer of Borosilicate Glass (BSG), which will act as a sacrificial material to suspend the proof mass, is deposited using LPCVD (deposition takes place over 12 minutes at a 0.41 µm/min deposition rate) [3].

Figure 6. Fabrication process flow. Refer to text for details

Figure 5. Bode plot of dynamic system response, (top) magnitude plot,

(bottom) frequency plot.

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(4) SU8 photoresist (thickness 1.3 µm) is spun onto the wafer, and exposed via contact mask alignment to define the cavity over which the proof mass will translate. (5) HF:H2O wet etch to remove BSG from everywhere except over the suspension cavity. (6) A thick layer of epitaxial PolySilicon (50 µm thick) is deposited over the entire wafer using a process called Thick Epi-Poly Layer for Micro Actuators and Accelerometers (THELMA). THELMA enables the creation of thick Polysilicon layers relatively quickly (when compared to LPCVD), without causing significant residual stresses between the Polysilicon and the oxide layer [4] [5] [6]. (7) Photolithography defines the interior features (springs, proof mass, rotor/stator, damping holes). (8) Chlorine-based RIE etch of polysilicon layer, 50 µm deep (1 µm/min for 50 minutes) to create high aspect ratio features. (9) Another HF:H2O wet etch to dissolve the rest of the BSG sacrificial material. This results in the proof mass ‘floating’ over the suspension cavity. (10) A liftoff process defines and deposits traces over stator combs.

IV. PERFORMANCE ANALYSIS

A. Static Performance

The static performance of the device was evaluated

analytically (using Equation (9)), and computationally, using

SolidWorks Simulation. An example plot of a finite element

solution for maximum displacement is given in Figure 7.

The results of both analyses are plotted in Figure 8.

Sensitivites computed analytically and computationally are

given in Equations (12) and (13), respectively.

(

)

(12)

(

) (13)

The two methods achieve a moderate level of agreement

with eachother, and we can expect a sensitivity between 7

and 10 fF/g.

B. Dynamic Performance

The dynamic performance of the accelerometer is

important in identifying system bandwidth. As in the static

analysis, both analytical and computational tools were

employed in the determination of the dynamic footprint of

the system.

We can gain a level of intuition regarding the dynamic

response of the system by rewriting the equation of motion

in the Laplace domain and re-defining some parameters:

⁄

⁄

(14)

where we have defined the damping ratio and the natural

frequency as follows:

√

(15)

√

(16)

System response to a step in put is given in Figure 9,

and the corresponding bode plot is given in Figure 10. The

dynamic characteristics of the system as computed

analytically are given in Table 2. Note that the system

reaches a steady state in just under 0.4 ms, leading to a

system bandwidth of 2.7 kHz which is far below the first

resonant mode at 6.1 kHz as designed.

Figure 7. Finite element simulation of accelerometer under 200 g inertial

load. The displacement represented is the absolute maximum travel.

Figure 9. Step response of mechanical system, from which rise time,

settling time and percent overshoot are computed.

Figure 8. Total capacitance change as a function of input acceleration

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Table 2. Dynamic characteristics of mechanical system

Parameter Notation Value

Resonant Frequency 6.1 kHz

Rise Time 33.2 µs

Settling Time 370 µs

Bandwidth 2.70 kHz

A dynamic analysis was performed computationally

using SolidWorks Simulation. This finite element software

resolves the system modal footprint by solving for the

eigenvalues in the following matrix equation:

[ ][ ] [ ][ ] [ ] (17)

where [ ] is the mass matrix, [ ] is the stiffness matrix and [ ] is the displacement matrix.

Solving Equation (17) computationally for the

accelerometer yields the lowest four modes present in Figure

11. The mode represented in (a) is the sensing mode, which

is the lowest-occurring mode at around 7 kHz (as designed,

compare to the analytical estimate of 6.1 kHz). The next

mode, an off-axis mode in the x-direction, occurs at about 14

kHz. A twisting mode represented by (c) occurs around 15

kHz, and finally, a z-direction mode occurs around 17 kHz.

D. A Note on Common Mode

In the previous section, we discussed the first four modes of the dynamic response. The differential capacitance of the accelerometer results in an implicit rejection of common mode, which makes the signal impervious to the x-direction and z-direction translations. Consider lateral rotor displacement in the x-direction, as illustrated in Figure 12. Capacitances and are written as follows:

(

) (18)

(

) (19)

| | (20)

Thus, lateral displacements result in null outputs. The same

phenomenon occurs with z-displacements as well. As such,

our sensor is not vulnerable to lateral modes. The rotational

mode is a potential issue but compensation is beyond the

scope of this report.

V. CONCLUSION

This paper presents the design, analysis, fabrication and

Figure 10. Results of frequency analysis, (a) y-mode, (b) x-mode, (c) θ-mode, (d) z-mode

Figure 11. Analytical bode plot, showing a resonant peak at roughly 6 kHz

which is beyond device bandwidth of 2.7 kHz.

Figure 12. Lateral translation, resulting in null output

Stator 1

Stator 2

Rotor

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evaluation of a high-g uniaxial accelerometer with

differential capacitance measurement for common mode

rejection. The design was optimized using both analytical

and computational modeling, and the theoretical

performance is evaluated. The accelerometer was shown to

have a sensitivity of between 7 and 10 fF/g over an input

range of 0-200 g. The dynamic performance is suitable for

mechanically realistic frequency ranges, featuring a

bandwidth of 2.7 kHz and a response time of 370 µs.

VI. REFERENCES

[1] B. and Yang, "Squeezed Film Air Damping in MEMS,"

Sensors and Actuators, vol. 136, pp. 3-27, 2007.

[2] Sharma, Macwan, Zhang, Hmurcick and Xiong, "Design

Optimization of a MEMS Comb Accelerometer.," in

ASEE, 2008.

[3] Govindarajan and Patel, "Rapid Deposition of

Borosilicate Glass Films". United States Patent

US20040058559 A1, 25 March 2004.

[4] Owen, VanDerElzen, Peterson and Najafi, "High Aspect

Ratio Deep Silicon Etching," in IEEE MEMS, 2012.

[5] Sagazan, "MEMS Fast Fabrication By Selective Thick

Polysilicon Growth in Epitaxial Reactor," in MEMS and

MOEMS, 2005.

[6] Ayazi and Najafi, "High Aspect Ratio Polysilicon

Micromachining Technology," in Sensors and Actuators,

2007.