3 Theory of Piezoelectric Material
3.2 Piezoelectric Constitutive Equations
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Piezoelectric Constitutive Equation is described mathematically as below:
d-form:
�𝑆61(𝑇, 𝐸) = 𝑠66𝐸 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：
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T, S, E, D, d, g, h, 𝑘𝑑2 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 𝑠𝐸 with superscript means when the coefficient of function of S with constant variable E. 𝑠𝐷, 𝜀𝑇, 𝛽𝑇, 𝛽𝑠, 𝐶𝐸 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, 𝐶𝐸 in (3.5) are stress, electric displacement, strain, electric field and coefficient of stiffness with constant electric field respectively 𝜀𝑇, e, are
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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)
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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
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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
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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).
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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.
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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
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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
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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 resonating frequencies are 4654 and 4674, so the input frequency in this study used is 4664 Hertz.
Silicon makes up the whole structure, and compared to PZT thickness, the thickness of silicon affects performance much more. Table 4.1 shows the displacement when it is driven under operation frequency and its resonating frequencies. The displacement is an objecting point on driving arm, and this is a harmonic simulation with damping ratio 0.005.
The first row of table was the very first idea of design, and this table shows that the thicknesses of silicon and PZT does not affect resonating frequencies as much, but performance (the displacement of driving) is strongly affected by thickness of the silicion. In addition, it shows linear tendency. The thicker the silicon, the poorer the performance, while the thicker the PZT, the better the performance.
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Table 4.1: Performance and resonating comparisons of different thickness
After a lot of tests and optimization of every part sizes, and taking into
consideration the cost, the final thickness and resonating frequency decided are shown in Table 4.2, the silicon basement is 300μm and PZT is 1.875μm.
Input voltage
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Table 4.2: Performance and operation frequency of flat piezoelectric gyroscope
Detail sizes are showed in figure 4.8:
Figure 4.8: Detailed sizes of flat piezoelectric gyroscope Input voltage
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4.2 Harmonic Simulation Analysis
In this study, harmonic simulation inputs a sinusoidal signal, which is within the resonating frequency to drive gyroscope. This section shows the extent of the deformation it could have. When the gyroscope is put it in a rotating frame, the experiment can also simulate how heavy the Coriolis force is and how much signal it can output.
Figure 4.9 was driven by a sinusoidal signal ranged from 20 ~ -20 volt:
Figure 4.9: Flat gyroscope is driven under operating frequency (4664 Hz)
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The structure of design is on one side of wafer, and hence when it is operated, there may be structure deflection along out-plane axis caused by stress concentration.
Table 4.3 shows 3 axes displacements of 4 nodes in figure 4.9. In consideration of fabrication process, the PZT is only designed on one side, hence the unsymmetrical structure may cause z-axis deformation, so the displacement in z-axis is a key point needed to observe. Table 4.3 shows that the displacement in z-axis is much smaller than two other axes, thus this displacement is insignificant. Damping ratio here is 0.005
Node Ux (m) Uy (m) Uz (m)
29524 5.80963E-6 -3.26782E-7 1.31608E-9
34374 5.73231E-6 5.39366E-7 1.27389E-9
13634 -5.73259E-6 -5.38841E-7 1.13422E-9
17531 -5.80965E-6 3.27264E-7 1.04556E-9
Table 4.3: Displacements of 4 nodes on gyroscope arms
In figure 4.1 there is one part of the gyroscope named ‘feedback driving arm’, which is used for feedback driving. Because of its micro size, if the device undergoes a small velocity condition, the Coriolis force would be very weak, and in this condition, the feedback driving arm can be used. Input signal on the feedback driving arm can help the detecting arm reach the predicted amplitude, then, subtracting the amplitude from feedback input, the amplitude caused from Coriolis force can be derived. Figure 4.10 shows the diagram with 2 nodes that be observed.
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Figure 4.10: Driving on feedback driving arm under operation frequency (4664Hz)
Table 4.4 shows displacement of 2 nodes in figure 4.10. Apparently, the displacements in y-axis of 2 nodes are almost the same, hence once the gyroscope is driven on feedback driving arm, the left part can resonate in opposite phase. In addition, displacements in z-axis are much smaller, so the displacement of z-axis can be ignored. Damping ratio here is 0.005.
Node Ux (m) Uy (m) Uz (m)
32907 2.63623E-9 -8.77733E-7 9.81261E-10
17827 -2.66178E-9 8.77252E-7 9.97343E-10
Table 4.4: Displacement of 2 nodes by driving on feedback driving arm
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4.3 Motion Simulation Analysis
4.3.1 Displacement and Electric Potential Simulation
The harmonic simulation in section 4.2 without considering Coriolis effect, but in this section, Coriolis effect will be considered.
Figure 4.11 shows that, when gyroscope is driven on right arm then left arm will resonate in opposite phase, when it is in rotation frame (undergo angular velocity), Coriolis force will occur then bend detecting arm.
Figure 4.11: Flat piezoelectric gyroscope undergo angular velocity 20 rad/sec and Coriolis force chart
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After Coriolis force bend the detecting arm, because of PZT electrodes deformation then they will output signals. Figure 4.12 shows that, when detecting arm is bended by Coriolis force, the PZT electrodes on detecting arm will deform. The below PZT electrode will extend a little bit and the one above will be contract slightly.
Once deformation occurs, piezoelectric material will output electric signal. Hence according to the signals, Coriolis force and angular velocity can then be measured.
Figure 4.12; Bending arm chart
Because the bending curve quantity is small, it is assumed that the bending curve can be seen as linear displacement. Following Table 4.5 takes this assumption.
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Figure 4.13 and table 4.5 show that, there is linear relation between displacement of the end of detecting arm and electric potential. Damping ratio here is 0.005:
Figure 4.13: Diagram of displacement – electrical potential
Voltage (v) 1.11E-04 2.22E-04 5.56E-04 1.11E-03 2.22E-03 5.56E-03 1.11E-02 Displacement
(m) 6.99E-09 1.40E-08 3.50E-08 6.99E-08 1.40E-07 3.50E-07 6.99E-07 Table 4.5: Data of displacement of the end of detecting arm and electric potential
4.3.2 Relation between Angular Velocity and Electric Potential
After determining the linear relation between deformation and output electric potential, it is important to find out the link between angular velocity and output voltage.
Table 4.6 shows voltage outputs of flat piezoelectric gyroscope undergo different angular velocities:
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ω (°/sec) 1 2 5 10 20 50
Voltage
(volt) 1.11E-04 2.22E-04 5.56E-04 1.11E-03 2.22E-03 5.56E-03
ω (°/sec) 100 150 200 250 300 350
Voltage
(volt) 1.11E-02 1.67E-02 2.22E-02 2.77E-02 3.32E-02 3.87E-02 Table 4.6: Angular velocity – Output voltage
Figure 4.14 plots the table 4.6 data to a diagram:
Figure 4.14: Diagram of angular velocity – output voltage
Table 4.6 and figure 4.14 show the linear tendency between output voltage and angular velocity.
4.3.3 Coriolis force simulation
Another important purpose for the gyroscope is to determine how great the
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Coriolis force is. Hence this study tries to address this. In steady state condition, inputting a constant electric field on PZT will cause deformation. Table 4.7 shows the displacements of Node 29524 (Figure 4.15) by input different voltages on driving arm:
Figure 4.15: Nodes for observation
Input voltage (v) Node 29524 x-axis displacement (m)
0 0
1 -1.12E-08
2.5 -2.80E-08
5 -5.60E-08
7.5 -8.40E-08
10 -1.12E-07
15 -1.68E-07
20 -2.24E-07
30 -3.36E-07
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Table 4.7: Table of input voltage – displacement
After that, a force is input on Node 29524 to match the displacement. Table 4.8 shows the conclusion:
Input force (N) Node 29524 x-axis displacement (m)
Table 4.8: Force – displacement chart
Tables 4.7 and 4.8 is combined into table 4.9:
Node 29524 x-axis Input
voltage Displacement (m) Force/voltage N/V
Input
force Displacement (m) K- driving (N/m)
Table 4.9: Relations of voltage, displacement and force on driving arm
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Because the displacements are small, this study assumes the bending displacement can be seen as linear displacement, and according to basic hook’s law:
𝐹 = 𝑘𝑥, k is coefficient of stiffness, x is displacement, hence the coefficient of stiffness (K-driving, in table 4.9) of driving arm can be derived.
In the same way, the same process was done on Node 32907 then arranged data to table 4.10:
Node 32907 y-axis Input
voltage Displacement (m) Force/voltage N/V
Input
force Displacement (m) K- detecting (N/m)
Table 4.10: Relations of voltage, displacement and force on detecting arm
Tables 4.9 and 4.10 show the coefficients of two arms, and the coefficient for the transformation between force and voltage. That means when detecting arm outputs a signal, it can be measured then transformed to an equal force, which is seemed apply on observing node. Therefore, the Coriolis force can then be measured.
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Chapter 5
Fabrication Process
5.1 Fabrication process
The fabrication process will be discussed in this chapter step by step. During the fabrication process underwent changes twice because of unforeseen problems. The problems will also discussed in this chapter.
In order to simplify the process, PZT is only deposited on one side of silicon wafer, hence a single polished silicon wafer is used in this study. PZT is the actuating and detecting material, it is deposited by spin-coating [16] [17] [18] [19] [20] [21] [22]
[23] [24] [25]. However, PZT is not an electric conductor, so it needs other metal layer as its electrodes. Platinum (Pt) is the most compatible candidate with PZT, and their crystal sizes are similar. As for electrodes, platinum has low electronic resistance, high melting point and high chemical stability. Because of the high chemical stability, that leads to challenges when patterning Pt by wet-etching, therefore, in order to pattern Pt, lift-off is used as the solution to this problem [25] [26]. Lift off process is a method of patterning of a target material on the surface of a substrate (ex. wafer) using a sacrificial material (ex. Photoresist). It is an additive technique as opposed to the more traditional subtracting technique like etching.
5.2 Experiments of Fabrication
5.2.1 First failed fabrication process The first process is shown below (Figure 5.1):
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Figure 5.1: Fabrication process I
At very beginning, a layer of silicon nitrite is deposited by PECVD (plasma-enhanced chemical vapor deposition) as a seed layer, then a layer of photoresist(FH-6400L) is spin coated for lift-off, in order to pattern platinum (bottom electrode). Next 0.1μm thick platinum is evaporating deposited on it. Right after that, the whole wafer is dipped into acetone and an ultra sonic cleaner is used to remove photoresist. After these steps, the Pt is patterned on wafers.
The following chart shows the steps to spin coat PZT. (Figure 5.2):
Figure 5.2: Steps for spin-coating PZT
The purpose of baking at 150℃ is to evaporate water molecules. Baking at 350℃
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is to evaporate organic compounds. Baking at 700℃ is to crystallize PZT. The was smooth, but PZT layer was unable to function.
In figure 5.3, it is obviously can seen that, where the bottom is platinum (Pt), PZT could crystallize well on it, but apparently not on silicon nitride (Si3N4).
Figure 5.3: PZT sample by microscope
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During the process, a main issue that surfaced is where PZT cannot be crystallized on a silicon nitride well, and cavities grow between the crystals. The rough surface then gradually grows and extends to a smooth surface (Figure 5.4). In the end, smooth surface will be covered by a rough surface. (Figure 5.5)
Figure 5.4: Sample of PZT - I
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Figure 5.5: Sample of PZT - II
In this condition, the process definitely will fail, hence the fabrication process need to be changed.
5.2.2 Second Failed fabrication process
The improved fabrication process is shown in figure 5.6:
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Figure 5.6: Fabrication process II
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The core idea of the 2nd generation fabrication process is based the ease of PZT crystallizing on platinum. In the 2nd process, the Pt layer (bottom electrode) does not need to be patterned. The 1st process top electrodes and bottom are at the same layer, but once the bottom electrode is separated on the whole wafer surface in 2nd process, the top electrodes are needed to pattern on another layer. The study proposes an idea, which is depositing a silicon oxide (SiO2) layer on PZT to isolate the top Pt layer and the bottom Pt layer.
The first few steps are retained - silicon nitride (Si3N4) is deposited by PECVD, Pt is deposited by evaporative deposition, and PZT is deposited in the same steps by spin-coating. In addition, the PZT thickness per layer by the steps of spin-coating is about 0.075μm. Figure 5.7 is plotted by ET-4000, which is an instrument used for plotting surface sketch. The sample of PZT plotted by ET-4000 is spin-coated into 16 layers
Figure 5.7: PZT surface plot by ET 4000
The PZT etchant BHF* is BHF with other buffered solutions [27] [28] and the photoresist (PR) for protecting PZT from wet etching is AZ4620. The steps and recipe of PZT etchant is shown:
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1. Etchant, BHF: HCI: NH4Cl: H2O = 1:2:4:4 (Volume ratio) 2. Dip in HNO3: H2O (2:1) solution for 10-15 secs
3. Immersing into DI water for 3- 5 mins
In this case, the average 1.2μm thick sample needs 55~60secs to be etched out, and the rate of etching is at about 0.02μm/sec and according to the experiment the undercutting ratio is almost 1:1.25 (Deepness : Sideness).
The next step is depositing silicon dioxide. It is deposited by PECVD for 0.5μm thick. After the step, patterning is done by wet etching. The etchant is BHF [29] [30].
The etchant is diluted 1:5. 0.5μm thick SiO2 is etched for 80~90 seconds. During the wet etching process, BHF etches PZT layer as well( Figure 5.8).
Figure 5.8: Sample of wet etching by BHF
In order to confirm the etching rate of the etchant on PZT, A separate experiment was done to record the etching status every 10 seconds. (Figure 5.10)
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Figure 5.10: Experiment of PZT wet etching by BHF
The surface of PZT is rough after undergoing wet-etching for over 20 seconds. In addition, the PZT sample is about 1.2μm. When it has been etched for about 60 seconds, almost all the PZT is gone, so the etching rate is about 0.02μm/sec.
Compared to SiO2, 0.5μm thick silicon dioxide can bear about 90 seconds, and the etching rate of SiO2 is about 0.005μm/sec.
BHF etches PZT more strongly than silicon dioxide. Figure 5.11 shows that after a few seconds, the PZT was unable to maintain its smooth surface. Over-etching is hard to avoid in the process of wet etching, but in this experiment, once that happens, BHF will hurt the surface of PZT. This would cause irreversible damage, hence this
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process was also unsuccessful.
Figure 5.11: Comparison of 20sec etching sample and deeply hurt sample
5.3 Fabrication process improvement
After the above two unsuccessful processes, another improving fabrication process is proposed as below (Figure 5.12):
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Figure 5.12: Fabrication process improvement
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The PZT layer in this improved fabrication process is still deposited on Pt layer to keep it crystallizing well. Because of over-etching from 2nd process, in this process, another Pt layer is deposited as an etching mask before silicon dioxide is deposited.
This Pt layer will be a protecting mask for the wet-etching of silicon dioxide.
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Chapter 6
Conclusion and Future work
6.1 Conclusion
6.1.1 Simulation analysis
Various types of simulations have been analyzed in this study in chapter 4. As mentioned, flat shaped designs could help vertical volume to be minimized, and the difference of resonating frequency between driving mode and detecting mode are only 20Hz. In addition, the relationship between displacement-electric potential and angular velocity-output voltage are both linear tendencies. Most importantly, this gyroscope can still output 10−3 level volt output in small angular velocity condition.
The coefficient of transfer between voltage and force is also derived.
The steps to derive Coriolis force are also described at the end of chapter 4. In simulations, this design should have many advantages.
6.1.2 Fabrication process
Although the first two fabrications were unsuccessful, they both still provided important fabrication information.
Firstly, even though silicon nitride is a delicate material, and is widely used as a seed layer in MEMS fabrication processes, but in this case, that would cause fabrication failure. Because of spin-coating PZT layer by layer, it would cause the rough part to extend to smooth parts in the end, which is mentioned in section 5.2.
Hence, spin-coating PZT on the whole surface with Pt is the best defense against
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fabrication process failure.
Secondly, these experiments were able to provide data on etching rates, and conclusions about the thickness of PZT per layer by spin coating and etchant recipes etc.
Thirdly, silicon dioxide has a strong ability of electric isolation, but in this case, the etchant, BHF, for silicon dioxide etches PZT much quicker than the target material, SiO2. Furthermore, wet etching is a common technique in MEMS fabrications and over etching is unavoidable, therefore, if any fabrication needs to pattern SiO2 on PZT, they need to avoid wet etching silicon dioxide, if the SiO2 layer is directly deposited on PZT.
Lastly, an improved fabrication of PZT related device for MEMS fabrication process is proposed.
6.2 Future work
This study had made some simulations, but if the factors in mathematics model
This study had made some simulations, but if the factors in mathematics model