MEMS Fabrication Introduction - 微壓電陀螺儀設計、分析與製作

1 Introduction

1.4 MEMS Fabrication Introduction

Micro electro mechanical systems (MEMS) is the technology of very small devices. The fabrication of MEMS is from the process technology in semiconductor device fabrication, and the basic techniques are deposition of material layers, pattern

stress

by photolithography and etching to produce the required shapes. MEMS technology, due to the advantages of being small in size and cheap in fabrication, has received more and more attention in specific applications.

Chapter 2 Gyroscope

2.1 Scientific literature review

There are two main fields in motion sensors, one is accelerometer, which is used for measuring speed changing, the other one is gyroscope used for measuring angular velocity. The first appearance of gyroscope can be tracked to the year 1852, Leon Foucault (1819-1868), a 19th-century French experimental physicist, who used the

“gyroscope” apparatus to investigate the rotation of the earth. After that, along with the technologies advances, there are multiple variances in the development of gyroscopes. Gyroscopes have been extensively used to measure the angular velocity in many applications in our daily lives, such as vehicle navigation, vehicle rollover stability, digital camera image stabilization, and even in advanced military applications like aircrafts and satellites. In recent years, due to the regular need for better sensitivity, the development of gyroscopes are still in process and still attract attentions from the researchers in various fields. In addition, in order to the surge of virtual reality, and other applications as well, the development of gyroscope moves towards sensitive, reliable and miniature size for mass production.

According to their working principles, gyroscopes can be roughly classified to mechanical gyroscopes (rotor gyroscopes, vibratory gyroscopes… etc.) and optical gyroscopes (fiber gyros and laser gyros etc.). They all have theirs pros and cons and been utilized in various applications.

2.2 Introduction of Gyroscope

2.2.1 Traditional Gyroscope

Gimbals is the basic of gyroscope, gimbals was first described by the Greek inventor Philo of Byzantium (280 – 220 BC) [1][2][3][4]. A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of the rotation of its support (cf. vertical in the first animation). For example, on a ship, the gyroscopes, shipboard compasses, stoves, and even drink holders typically use gimbals to keep them upright with respect to the horizon despite the ship's pitching and rolling. Figure 2.1 shows a simple two-axis gimbal set.

Figure 2.1: A simple two-axis gimbal set

The first vibratory gyroscope was produced in early 1980s, it was made with quartz in fork-shaped. The basic idea of the vibratory gyroscope is that when the mass is oscillating along one axis (driving axis) and experiencing the angular velocity

simultaneously, the Coriolis force will produce a force along the axis (detecting axis) that is orthogonal to its original oscillating axis. Furthermore, the amplitude of the detecting axis is proportional to the applied angular velocity.

2.2.2 Micro Gyroscope

The first silicon MEMS gyroscope was produced in 1991 in Charles Stark Draper Laboratory; in the same year, a small vibratory piezoelectric gyroscope was made from Fujishima team [5]. After that, vibratory gyroscope became popular.

Compares to traditional mechanical gyroscope, the most characteristic of MEMS gyroscope is the micro size. Furthermore, in mass production, the cost is very low.

However, because of the super mini size and the uncertainties in fabrications, these would cause a large influence to structure parameters, for instance, coefficient of elasticity, coefficient of dampness, oscillator mass… etc. Besides, defects in sensing circuits, such as, noise, parasitic capacitance, and non-idealities of amplifier… etc. All these will cause performance substantially be reduced. Hence, the most characteristics, micro-size, is also the most challenge of how to promote higher performance.

There are many methods by using electricity to drive gyroscope, electromagnetic, electrostatic, and piezoelectric …etc. Electromagnetic gyroscope could work in harsh environments, but it consumes power highly, further, the input circuit is so great that hard to control and not easily produce in MEMS fabrication, these made electromagnetic gyroscope can’t be used widely.

Electrostatic gyroscope generally measures values change of changeable capacities to estimate the oscillator mass deflection, which is caused from Coriolis force when it is rotating. The characteristics of consuming low power, stable, and input voltage is easy to be controlled. Yet, the extreme request of gaps accuracy and weak signal output are its biggest defects. Even though those defects, but the high

integrating with MEMS fabrication, it is still the most common type of MEMS gyroscope now. Figure 2.2 [6] shows an example of electrostatic MEMS vibratory gyroscope; the gyroscope needs an input signal to drive it under a fixed frequency to resonate it in drive direction, then when the device is in rotation, Coriolis force will push the active mass shifting along sense direction; and further, gaps distance in sense capacities, at two ends changed, then values of capacities will change lately.

According to the capacity differences the Coriolis force and angular velocity could be measured.

Figure 2.2: MEMS vibratory gyroscope

Figure 2.3 [7] and Figure 2.4 [8] are another electrostatic MEMS vibratory gyroscopes, both they need an input signal at theirs designed resonating frequencies to oscillate the main masses to resonate along the driving axis, then when they are rotating, Coriolis force will cause the changing the values of changeable capacities, those comb-shaped structure. By counting the values, Coriolis force and angular velocity can be counted.

Figure 2.3: Conceptual illustration of the Distributed-Mass Gyroscope with 8 symmetric drive-mode oscillators

Figure 2.4: Structure schematic of linear vibration micro-gyroscope

Piezoelectric gyroscope has great output, responses fast and structure is uncomplicated, but piezoelectric materials performance is easily affected by

temperature and hard to produce in MEMS fabrication. So the purpose of this study is proposing a design with great output signals and can be combined with MEMS fabrication.

2.2.3 Piezoelectric Gyroscope

Piezoelectric gyroscope has many generations, but all principles are the same.

Operator gives an input to oscillate main mass, when rotation occurs, Coriolis force cause transform on their structure, piezoelectric materials would produce corresponding electric signal output. Measuring the output can count the Coriolis force and angular velocity out.

The first generation is only like a beam (Figure 2.5) [9], oscillate in driving direction, when it is rotating, Coriolis force will curve it to the direction which is perpendicular to the driving direction, then by measuring the curving, then Coriolis force and angular velocity would be counted out. But there’s capacity effect due to the driving electrode is too close to the sensing electrode. Capacity effect means when two electrodes are charged, if they are too close, there will occur a capacity between two electrodes. This capacity will affect both electrodes and cause measuring errors.

Besides, in this design the two modes, driving mode and sensing mode, are easily interfered by each others, this would made a huge error of output, hence fork-shaped gyroscope was proposed.

Figure 2.5: Beam-shaped piezoelectric gyroscope

Fork-shaped gyroscopes (Figure 2.6(a)) [9], and its related version, H-shaped gyroscopes (Figure 2.6(b)) [10], are both important designs in vibratory gyroscope, in order to lower influence of capacity effect, fork-shaped gyroscope separates driving electrodes and sensing electrodes at two divided arms. The driving arm is driven under the resonating frequency, and then the other one will resonate. When Coriolis force curve these arms the sensing electrode on sensing arm can provide the output from piezoelectric material, then angular velocity can be measured. However, fabrication error will made the two arms not the same, therefore, the resonating frequencies of two arms are different that would make this gyroscope performs not as well as its design.

Figure 2.6: (a) Fork-shaped piezoelectric gyroscope (b) H-shaped piezoelectric gyroscope

H-shaped gyroscope is improved configuration since it can reduce the coupling effects between driving and sensing axis. For separating driving electrodes and sensing electrodes, the driving part is on top half and sensing part is at the other half to lower the electrostatic coupling and this symmetrical structure could also reduce the error from mechanical coupling. Wakatsuki [11][12] proposed the prototype of

H-shaped LiTaO3 Piezoelectric Gyroscope in 1997 and claimed that it had a better ability to suppress the leakage couplings than that of the fork gyroscope. Moreover, the H-type vibratory gyroscope using LiTaO3 single crystal has large electromechanical coupling coefficients, which can sharply suppress the “leakage”

output. He also suggested that using a transducer attached to the driving axis to monitor and compensate the temperature drifting effect, which existed in most of the piezoelectric materials [13]. In 2001, an H-type gyroscope made of LiNbO3 with an oppositely polarized single crystal plate had been reported [14]. Due to the electromechanical coupling factor of LiNbO3 is larger than that of LiTaO3 for the flexural vibratory mode, this type of gyroscope is more suitable for miniaturization, due to the fact that they can produce the same amount of vibration amplitude in a more compact size, and achieving the same level of resolution.

Many shapes of piezoelectric gyroscopes had been developed, such as tri-fork-shaped (Figure 2.7) [15]. All designs are in order to reduce capacity effect, electrostatic coupling and mechanical coupling…etc, and try to get as much stronger signals as they could.

Figure 2.7: Tri-fork-shaped piezoelectric gyroscopes

2.3 Mathematic Model of Gyroscope

Coriolis acceleration could be counted out from relative motion equation as below.

Total force on mass (m) is below (relative to a rotating frame):

f⃑ = m ∙ a�⃑ = m �^𝑑_𝑑𝑡²^𝑅�⃑₂ +^𝑑_𝑑𝑡²^𝑟⃑₂+^𝑑𝛺_𝑑𝑡^��⃑× 𝑟⃑ + 𝛺�⃑ × �𝛺�⃑ × 𝑟⃑� + 2𝛺�⃑ ×^𝑑𝑟⃑_𝑑𝑡� (2.1)

Figure 2.8: Chart of mass in rotating frame and fixed frame

Figure 2.8 shows a chart of a mass in rotating frame and fixed frame at the same time. The f �⃑ is total force on mass (m); a�⃑ is defined as the absolute acceleration of mass; the position vector 𝑅�⃑ is relative to rotating frame from fixed frame. Denoting the rotating frame by

F

for short; 𝑟⃑ is the position vector of mass relative to

F

denoting by 𝛺�⃑ the angular velocity of the

F

The first term in (2.1) is the linear acceleration of

F

relative to fixed frame, the second term is the linear acceleration of mass relative to

F

, the third term is the linear acceleration of mass relative to

F

^{, the}

fourth term is the centripetal acceleration of mass relative to

F

, and the last term is

called the complementary acceleration, or Coriolis acceleration, after the French mathematician de Coriolis (1792-1843), it’s also the value that gyroscope wants to measure. Notice that, on purpose of getting Coriolis acceleration, the last term in (2.1) must be existed, and the vector is cross product of angular velocity (𝛺�⃑) and the linear velocity (^𝑑𝑟⃑

𝑑𝑡), of mass relative to

F

, hence Coiolis acceleration is vertical to both terms.

If the terms of location vectors and angular velocity in (2.1) are replaced by Cartesian coordinates, it can be rewritten as:

{

F

^{} = {}𝑒̂𝑥 𝑒̂𝑦 𝑒̂𝑧} (2.2)

(2.3) (2.4) (2.5) also can be replaced by Cartesian coordinates as below:

𝑓𝑆

The d and k represents the coefficient of dampness and elasticity separately; u is the

input.

Generally, mass of single-axis vibratory gyroscope is fixed on a plane. In this study, it assumes motion of the mass is restricted within x-y plane, but only detects the angular velocity on z-axis. In this assuming (2.1) can be rewritten into 2 axes dynamic equations by (2.2) (2.3) (2.4) (2.5) and (2.6) (2.7) (2.8):

m𝑎_x+ mẍ + 𝑑₁ẋ + �k₁− m�Ω_y²+ Ω_z²�� x + m�Ω_xΩ_y− Ω̇_z�y = 𝑢_𝑥+ 2𝑚Ω_𝑧𝑦̇ (2.9) m𝑎_y+ mÿ + 𝑑₂ẏ + �k₂− m(Ω_x²+ Ω_z²)�y + m�ΩxΩ_y− Ω̇_z�x = 𝑢_𝑦− 2𝑚Ω_𝑧𝑥̇ (2.10)

The two forces, _2𝑚Ω_𝑧𝑥̇ and 2𝑚Ω_𝑧𝑦̇ are resulted from Coriolis effect, from here, if wants the accuracy of angular velocity measuring more accurate the two Coriolis forces need to as greater as they can be. That depends on the mass and linear velocity.

However the most characteristic of MEMS device is the micro-size, small size comes along with slight mass, about 10⁻⁶ ~ 10⁻⁸ kg. That also results in the difficulty of high accuracy of measuring.

Vibratory MEMS gyroscope used to be operated under the range of frequency from thousands to tens thousands hertz, hence the terms of angular velocities multiplying are far smaller than resonating frequency can be ignored. Therefore, (2.9) (2.10) can be simplified as:

mẍ + d1ẋ + k1x = ux+ 2mΩzẏ (2.11)

mÿ + d₂ẏ + k₂y = u_y− 2mΩ_zẋ (2.12)

Equation (2.11) (2.12) describes single-axis gyroscope dynamic motions. From (2.11) (2.12), in ideality, there’s only dynamic coupling, in other words, excepting

itself rotation (_Ω_z≠0) gyroscope only affected by its own structure design, characteristics. Basic on this equation, angular velocity could be counted from dynamic coupling status.

Chapter 3 Theory of Piezoelectric Material

3.1 Conspectus of Piezoelectric Material

In 1824, Brewster found that tourmalines generate electrical charge when heated, or what is called “pyroelectricity”.

In 1880, the brothers Pierre Curie and Jacques Curie discovered piezoelectricity.

They showed that crystals of tourmaline, quartz, and Rochelle salt (sodium potassium tartrate tetrahydrate) generated electrical polarization from mechanical stress.

There are many categories of piezoelectric material, and this study classifies them into 7 crystal systems and 32 point groups. 7 crystal systems include triclinic crystal system, monoclinic crystal system, orthorhombic crystal system, tetragonal crystal system, trigonal crystal system, hexagonal crystal system and cubic crystal system.

Point groups are classified according to the structure and crystal symmetry. Crystal structure is directly related to piezoelectricity. For example, when one material is crystallized around a point of symmetry, the material is definitely not piezoelectric material. However, all non-center of symmetry point group, except for 432 point group, are piezoelectric materials. Now a widely used material, piezoelectric ceramics i.e. PZT, is a hexagonal crystal system and in the 6mm point group.

Piezoelectric ceramics are crystallized under high temperature. After polarization, it is like a single crystal with directionality. Another physical characteristic is the dielectric effect. When piezoelectric material is put in an electric field, electric charges are affected by the electric field and move, becoming electric dipoles, which

causes polarization - hence the dielectric effect. After that, under electric field or mechanical stress influences, piezoelectric materials generally will have a linear response.

Piezoelectricity is an interaction between electrical and mechanical systems.

Piezoelectric materials have two main effects. One is direct piezoelectric effect, and the other is converse piezoelectric effect. The direct piezoelectric effect occurs when piezoelectric materials produce electric polarization by forcing mechanical stress on the material. On the contrary, the converse piezoelectric effect occurs when piezoelectric materials produce deformation by applying electric field on the material.

The electromechanical coupling effect of piezoelectric materials is used widely in engineering. With the two effects, many kinds of actuators and sensor can be made.

However, there are some disadvantages worthy of concern in the application of piezoelectric materials. For example, it usually produces unnecessary coupling effects in any direction perpendicular to driving axis in the process of applying voltage.

In improving the accuracy and stability of piezoelectric devices, the null signal suppression of leakage couplings is significant. These null couplings have two main sources: mechanical and electromechanical couplings. The former is produced from manufacture defects, and the latter is because of the piezoelectric property between the driving and detecting electrode. These cases of leakage couplings can be improved either by changing the designed structure or dealing with control stratagems, in order to improve the resolution of the gyroscopes.

In general, piezoelectric materials have advantages of high strength, high bandwidth, short response time, and are used widely in vibratory gyroscopes.

3.2 Piezoelectric Constitutive Equations

Piezoelectric Constitutive Equation is described mathematically as below:

d-form:

�𝑆⁶¹(𝑇, 𝐸) = 𝑠₆₆^𝐸 𝑇61+ 𝑑63𝐸31

𝐷31(𝑇, 𝐸) = 𝑑36𝑇61+ 𝜀66𝑇 𝐸31 (3.1)

The four state variables (S, T, D, and E) can be rearranged arbitrarily to give an additional 3 forms for a piezoelectric constitutive equation by mathematics operation.

It is possible to transform piezoelectric constitutive data from one form to another form. In addition to the coupling matrix d, they contain the other coupling matrices e, g, or h in another 3 forms. What follows another 3 piezoelectric constitutive equations and their mutual transformations：

T, S, E, D, d, g, h, 𝑘_𝑑² and e represent respectively stress, strain, electric field, electric displacement, piezoelectric strain/coefficient of electric charge, piezoelectric strain/coefficient of voltage, piezoelectric stress/ coefficient of voltage, coefficient of mechanical-electric coupling and piezoelectric stress/ coefficient of electric charge.

Those codes, such as 𝑠^𝐸 with superscript means when the coefficient of function of S with constant variable E. 𝑠^𝐷, 𝜀^𝑇, 𝛽^𝑇, 𝛽^𝑠, 𝐶^𝐸 represent respectively the coefficient of softness with constant electric displacement, coefficient of dielectric with constant stress, coefficient of converse dielectric with constant stress, coefficient of converse dielectric with constant strain and coefficient of stiffness with constant electric field.

In order to determine the piezoelectric material’s status, it needs 45 independent coefficients of material, but most piezoelectric materials have characteristic of symmetry in structure, so there are not as many coefficients as the hypothesis. The following diagram discribes piezoelectric ceramic as example, which is polarized in

x3 direction.

(3.5)

The T, D, S, E, 𝐶^𝐸 in (3.5) are stress, electric displacement, strain, electric field and coefficient of stiffness with constant electric field respectively 𝜀^𝑇, e, are

coefficients of dielectric and piezoelectric with constant strain. Table 3.1 shows parameters of PZT-5, which is also the material used in this study.

Parameters Values

Table 3.1: Parameters of piezoelectric ceramic (PZT-5)

Chapter 4 Design and Simulation Analysis

4.1 Shape Design and Vibration mode Analysis

Existing piezoelectric gyroscopes have various designs - beam-shaped, fork-shaped etc. This study proposes a design that separates the driving and detecting part at two end of the device (Figure 4.1) to reduce the capacity effect. Based on the fork-shaped gyroscope and H-shaped gyroscope, this one is designed in a flat shape (Figure 4.2) in order to reduce installing-volume in vertical direction. Four hammer-shaped masses at four ends of driving and detecting arms are for coefficient of stiffness and resonating frequencies. At the ends of two stretched out parts from center mass are fixed points.

Figure 4.1: Chart of gyroscope this study proposes

Figure 4.2: Flat piezoelectric gyroscope

This gyroscope uses piezoelectric effects to drive (Figure 4.3). When there is input of positive volt at the outer electrode (PZT) of the driving arm and negative volt at the inner electrode of the driving arm, the outer one will extend and the inner part will contract, then the driving arm will curve (Figure 4.4).

Figure 4.3: Flat piezoelectric gyroscope working principle

Figure 4.4: Input voltages on electrodes (PZT)

Meanwhile, if the gyroscope is driven under resonating frequency then the left half part will resonate in opposite directions (Figure 4.5).

Figure 4.5: Detecting part resonates in opposite phase

Furthermore, if the gyroscope is rotating around the Z-axis, then the detecting arm will curve (Figure 4.6) and then the deformation on detecting arm will make PZT electrodes output signals. Depending on the signals, Coriolis force and angular velocity can be measured.

In figure 4.1, the driving arm is for the driving input. The detecting driving arm is used to sense the amplitude of driving arm of being driven under resonating frequency. When the driving arm is driven under resonating frequency, the detecting driving arm will resonate in opposite direction with same amplitude. The detecting arm is for sensing. The feedback driving part will be introduced in section 4.2.

Figure 4.6: Flat piezoelectric gyroscope is driven and rotating at angular velocity 20 rad/sec.

After the prototype shape has been decided, vibration modes were close to being determined. Figure 4.7 shows 10 vibration modes. Sizes of every part are still important parameters to decide structure characteristics i.e. coefficient of stiffness, natural resonating frequency and performance etc.

Figure 4.7: (a) 1120 Hz Figure 4.7: (b) 1149Hz

Figure 4.7: (c) 1161 Hz Figure 4.7: (d) 1179Hz

Figure 4.7: (e) 1222 Hz Figure 4.7: (f) 2125Hz

Figure 4.7: (g) 3433 Hz Figure 4.7: (h) 4654 Hz

Figure 4.7: (i) 4674 Hz Figure 4.7: (j) 5784 Hz Figure 4.7: 10 vibration modes of gyroscope

This study picks figure 4.7(h) and figure 4.7(i) as the driving and detecting mode respectively. As the driving mode and detecting mode resonating frequencies become much closer, the device will produce a greater signal output. These two modes

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