LEAD ZIRCONATE TITANATE PIEZOELECTRIC
CANTILEVERS FOR MULTIMODE VIBRATING
MICROELECTROMECHANICAL SYSTEMS
by
XUQIAN ZHENG
Submitted in partial fulfillment of the requirements
For the degree of Master of Science
Thesis Advisor: Dr. Philip Feng
Department of Electrical Engineering and Computer Science
CASE WESTERN RESERVE UNIVERSITY
May, 2015
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis of
Xuqian Zheng
Candidate for the Degree of Master of Science
Committee Chair
Philip Feng
Committee Member
Soumyajit Mandal
Committee Member
Francis Merat
Date of Defense
March 27, 2015
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Table of Contents
List of Figures ...... v List of Tables ...... vii
Abstract ...... 1
Acknowledgement ...... 2
Chapter 1 Introduction ...... 3 1.1 Motivation ...... 3 1.2 Research Object and Thesis Structure...... 5
Chapter 2 Review of Related Literature and Studies ...... 7 2.1 Piezoelectric Materials ...... 7 2.1.1 Quartz...... 7 2.1.2 Polyvinylidene Fluoride ...... 7 2.1.3 Zinc Oxide...... 8 2.1.4 Aluminum Nitride ...... 8 2.1.5 Lead Zirconate Titanate ...... 9 2.2 PiezoMEMS Device Structures and Applications ...... 9 2.2.1 Actuators ...... 9 2.2.2 Energy Harvesters ...... 10 2.2.3 Sensors ...... 11 2.3 Future of PiezoMEMS ...... 12
Chapter 3 Device Structure and Fabrication Process ...... 13 3.1 Device Fabrication Process ...... 13 3.2 Device Structures ...... 15 3.2.1 Fabrication Induced Positive Curvature ...... 15 3.2.2 Stacking of Composite Beam ...... 16
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Chapter 4 Static Mechanical and Piezoelectric Characteristics ...... 18 4.1 Mechanical Characteristics ...... 18 4.1.1 Mechanical Properties of Composite Cantilever ...... 18 4.1.2 Pre-Stress Induced Curvature of Cantilever...... 19 4.1.3 Pre-Stress Analysis ...... 23 4.1.4 Radius of Curvature Measurement ...... 25 4.2 Piezoelectric Characteristics ...... 26 4.2.1 Piezoelectric Coefficients ...... 26 4.2.2 Piezoelectric Coefficients Extracting ...... 28
Chapter 5 Dynamic Mechanical and Piezoelectric Characteristics...... 32 5.1 Cantilever Dynamics ...... 32 5.1.1 Resonance Frequencies ...... 32 5.1.2 Simulation ...... 34 5.1.3 Vibrational Energy ...... 36 5.2 Dynamic Piezoelectric Characteristics ...... 37 5.2.1 Equivalent Circuit ...... 37 5.2.2 Electrical Energy ...... 38 5.3 Measurements and Results ...... 38 5.3.1 Optical Measurement ...... 38 5.3.2 Insertion Loss Measurement ...... 42 5.4 Conclusions ...... 44
Chapter 6 Multimode Resonant Energy Conversion...... 46 6.1 Reverse Piezoelectric Effect ...... 46 6.1.1 Electrical Power Analysis ...... 46 6.1.2 Mechanical Energy Analysis...... 46 6.2 Direct Piezoelectric Effect ...... 48 6.2.1 Measurement System ...... 48 6.2.2 Measurement Results ...... 50
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Chapter 7 Conclusions and Future Work ...... 52 7.1 Conclusions ...... 52 7.2 Future Work ...... 54 References ...... 55
iv
List of Figures
Figure 3-1. SEM images of PZT cantilevers with positive curvature induced by pre-stress in fabrication process. Scale bars in (a) and (b) are 100µm and 400µm, respectively. ... 15 Figure 3-2. Composite cantilever structure. (a) SEM image of composite cantilever tip. Scale bar is 5μm. (b) Illustration of composite cantilever structure stacking with thickness of each layer...... 16 Figure 4-1. (a) Schematic cantilever with positive curvature induced by a moment on tip. (b) Infinitesimal of the curved cantilever...... 20 Figure 4-2. Longitudinal strain distribution along y axis in the composite beam...... 21 Figure 4-3. (a) Schematic infinitesimal of the simplified cantilever beam composite structure with approximately defined pre-stress-induced forces. (b) Schematic infinitesimal of pre-stress induced curvature with force and moment analysis on the right ...... 24 Figure 4-4. (a) Optical microscopic image of Device #1 and #2 with color lines indicating the sections of profile measurements. Scale bar is 100µm. (b) SEM image of the same devices. Scale bar is 100µm. (c) & (d), curvature profile of relaxed Device ... 26 Figure 4-5. Schematic illustration of piezoelectric effects. (a) Direct piezoelectric effect. (b) Reverse piezoelectric effect...... 27 Figure 4-6. Photos of white light profilometer with static voltage supply system for cantilever bending curvature measurement. (a) Whole system. (b) Device supported by a customized holder under profilometer...... 29 Figure 4-7. Relation of cantilever radius of curvature and voltage applied between PZT layer. (a) Device #1. (b) Device #2...... 30 Figure 5-1. Lumped model of damped simple harmonic oscillator...... 33 Figure 5-2. Shape of first 7 transverse resonance modes of Device #1 from COMSOL Multiphysics simulation...... 35 Figure 5-3. Equivalent circuit of PZT cantilever as an unloaded resonator...... 38 Figure 5-4. (a) Scheme of optical measurement system. (b) Photo of part of the optical measurement system...... 39 Figure 5-5. Geometry of Device #1 and the laser spot of optical measurement. Inset: optical image of laser spot on cantilever (contrast adjusted to enhance the appearance of laser spot)...... 40 Figure 5-6. Device #1 electrically driven frequency domain response using optical detection. (a) Wide spectrum of all resonance frequency peaks measured. (b-g) Zoomed- in measurement of each resonance peak and fitting of resonance frequency f and quality factor Q...... 41
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Figure 5-7. Device #1 electrically driven frequency and time domain response using optical detection. (a) & (b), frequency domain response of first 2 resonance modes with fitting. (c) Oscilloscope display of time domain response to a 1 V, 30 µs pulse...... 42 Figure 5-8. Electrical measurement results of Device #1...... 43 Figure 5-9. Electrical measurement results of Device #2...... 43 Figure 6-1. (a) Schematic of cantilever harvesting capability measurement system. (b) Illustration of connection for frequency domain measurement (network analyzer) and time domain measurement (oscilloscope). (c) Image of chip mounted on the package w with a PZT plate actuator underneath. The scale bar is 1 cm...... 49 Figure 6-2. Device #2 energy harvesting capability measurement results. (a) Frequency domain electrical response of 1st mode with fitting. (b) Time domain electrical response of 1st mode with different burst cycles to ring-up the cantilever. (c) Fully ringed-up time domain response of 1st mode with 600 burst cycles and optical signal as reference. (d) Fitting of ring-down curve in CH2 of (c). (e) Frequency domain electrical response of 2nd mode with fitting...... 50
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List of Tables
Table 4-1 Summary of Geometry and Material Property of Each Layer ...... 19 Table 4-2 Summary of Parameters from Piezoelectric Coefficient Calculation...... 31 Table 5-1 Summary of Eigenvalues and Theoretical Resonance Frequencies of Device #1 ...... 34 Table 5-2 Summary of Resonance Frequencies of Device #1 by COMSOL Multiphysics ...... 36 Table 5-3 Summary of Two-Port Electrical Measurement of Device #1 ...... 44 Table 5-4 Summary of Resonance Frequencies Acquired by Different Methods (kHz) .. 45 Table 7-1 Summary of Representative Piezoelectric Energy Harvesters ...... 53
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Abstract
Lead Zirconate Titanate Piezoelectric Cantilevers for
Multimode Microelectromechanical Systems
by
Xuqian Zheng
Among the significant advances microelectromechanical systems (MEMS) have enabled in transforming portable low-power electronic devices and integrated energy- efficient systems, piezoelectric MEMS (PiezoMEMS) technologies take an important share of the contributions, especially the ones based on lead zirconate titanate (PZT).
In this work, we study the PZT-based PiezoMEMS cantilevers on their mechanical and piezoelectric properties for potential applications. We first introduce the cantilever fabrication process used to achieve optimized piezoelectric properties and designed device structures. Then, we discuss the composite stacking of the cantilevers and the modeling of bending moment and stress distribution. We extract the piezoelectric
2 coefficient of the PZT layer, which is e31 ≈ -5 C/m . Next, we evaluate multimode resonances of the cantilevers using theoretical modeling, finite element simulation, and optical and electrical measurements. Finally, we characterize the energy conversion properties of such cantilevers and achieve multimode mechanical-to-electrical energy conversion in ultrasonic frequency ranges.
1
Acknowledgement
I am grateful for the opportunity of working with a group of talented people during my master thesis study. From them, I have learned not only the technical skills, but also research strategies accumulated through discussions. I would like to thank them all, but here I wish to acknowledge some that have been particularly helpful and supportive.
I would like to express my great gratitude to my master thesis advisor, Dr. Philip
Feng, for his inspirations and encouragements. I appreciate his passion, patience and kindness throughout my graduate study.
I would like to thank Dr. Soumyajit Mandal and Dr. Francis Merat for serving in my master thesis committee. They provided insightful and valuable suggestions on my master thesis.
I am also grateful to my group colleagues for their tremendous help, Tina He, Hao
Jia, Hao Tang, Peng Wang, Zenghui Wang, and Rui Yang, throughout my master thesis study. We built great friendship and I would like to thank for their support and help.
Finally, I would like to thank my dear parents back in China for their unconditional support of my studying abroad.
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Chapter 1 Introduction
1.1 Motivation
Nowadays, we are experiencing explosive emergence of portable electronics and wearable devices [1]. While development of silicon (Si) technology is driving integrated circuits to attain lower power consumption, smaller in size and denser in integration [2], there is also a strong demand on sensors, which can retain the present performance or even achieve higher sensitivity with smaller volume and lower power consumption.
The vision and ideas of microelectromechanical systems (MEMS) technology could be traced back to the famous lecture by Richard Feynman 55 years ago [3]. After tens of years of development, sensors using MEMS technology are emerging in the past decade, competitive for their low power consumption, small volume and compatibility with Si technology. As pioneers in this area, STMicroelectronics and Nintendo added 3D MEMS accelerometers in the Nintendo Wii gaming console for motion control in 2006 [4].
Recently, MEMS technology has a strong appearance both in our daily life and industrial applications [5].
Piezoelectric MEMS (PiezoMEMS), as a major component of MEMS technology, plays an important role not only in making state-of-the-art sensors, but also realizing compact actuators and energy harvesters [5]–[9]. Because the piezoelectric properties possessed by the piezoelectric materials provide a direct transduction mechanism to convert signals from mechanical to electrical domains and vice versa [10].
Since battery is also one of the major obstacles in nowadays mobile electronic device development, PiezoMEMS enables another possibility in the situations that conventional battery or other energy harvesting methods are not feasible or sufficient
3 enough, for example, medical implants, wearable devices, and wireless sensors [11].
And in principle, piezoelectric harvester is a potentially infinite energy source for some of the applications.
Among all the materials used for PiezoMEMS, lead zirconate titanate (PZT) is most widely used for its relatively high piezoelectric coefficient [7], [10], [12]. Also, PZT devices can be grown with a variety of methods, which give it a promising potential for monolithic integration with Si technology. Even with the good potential, commercialized applications of PZT-based PiezoMEMS are still limited and various areas remain to be researched.
Singly clamped cantilever and doubly clamped beam are two most common structures of PZT-based PiezoMEMS devices. The structures are simple and easy to control which are sufficient for a great number of applications. Doubly clamped beam, as a resonator, is easy to be tuned due to its sensitiveness to mass loading, stiffness change and inner stress. Therefore, it is suitable for gas sensing applications [13]. However, this property is preventing the structure from applications that require environmentally stable resonance frequencies.
The stress in singly clamped cantilever is released by the free end of the beam, resulting in minor change due to stress and environmental instability. The structure is of our interest for careful study. Because deep understanding on this structure for PZT- based PiezoMEMS devices can easily lead to valuable applications, since less parameters need to be taken care of during operation.
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1.2 Research Object and Thesis Structure
In this work, we conduct careful and detailed analyses of static and dynamic properties of PZT-based PiezoMEMS cantilevers followed by building test systems to conduct precise measurements on device and extract essential device parameters. The experiments are designed based on theoretical analysis to acquire crucial values and evaluate device capabilities. On the basis of theoretical analyses and sufficient experiments, potential applications of the devices are stated based on our perspectives.
Here follows the structure of this thesis:
Chapter 1 states the motivation of this work and structure of thesis. The motivation part gives an overview of recent position of MEMS technology and the role PiezoMEMS plays in it. The promising future of PZT-based PiezoMEMS is forecasted and author is passionate in exploring this area.
Chapter 2 reviews a few common piezoelectric materials along with their standing- out properties based on literatures. Then, several applications of piezoelectric materials are summarized in actuator, energy harvester and sensor domains. Finally, potential challenges and future approaches of PZT-based PiezoMEMS are addressed.
Chapter 3 demonstrates the fabrication process of PZT cantilevers done in U.S.
Army Research Laboratory, followed by the illustration and explanations of cantilever’s positive bending curvature and stacking structure of composite beam.
Chapter 4 investigates the static properties of PZT cantilevers. We first analyze the mechanical properties and demonstrate the stress distribution and pre-stress-to-bending radius relation of relaxed PZT cantilever. Then, we derive to obtain the piezoelectric coefficient extraction method based on the stress analysis. Furthermore, we conduct
5 curvature radius calibration measurements to complete the evaluation of cantilever stress distribution and piezoelectric coefficient.
Chapter 5 discusses the dynamic properties of PZT cantilevers. We evaluate the transverse resonance frequencies of the cantilevers using theoretical derivation and calculation, finite element simulation, optical detection and electrical measurement.
Chapter 6 introduces the analysis and experiments on PZT cantilever’s actuating and energy converting capabilities. We also discuss the potential applications of these devices.
Chapter 7 concludes the results in this work and makes a comparison with existing energy harvesting devices and designs. We also address the future work could be done following this thesis.
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Chapter 2 Review of Related Literature and Studies
A number of materials have been discovered possessing piezoelectric effect [5].
The different properties of these materials can determine their fields of applications. In this chapter, common piezoelectric materials are reviewed with their major properties.
Useful piezoelectric device structures are illustrated together with their potential applications. And challenges and promising future of PiezoMEMS devices are depicted based on our perspective.
2.1 Piezoelectric Materials
2.1.1 Quartz
Quartz is the most conventional material known with piezoelectricity. Although it is found naturally, most quartz in practical use is synthetic, since crystalline quartz is needed to show piezoelectric property [5]. To achieve high resonance frequencies, quartz need to be in single crystal form and the thickness of it must be minimized. Thus, special cutting techniques are needed for manufacturing quartz resonators [14]. Thin film synthesis of quartz is seldom used [15] and most of the quartz-based sensors are fabricated by micromachining technique.
2.1.2 Polyvinylidene Fluoride
Polyvinylidene fluoride (PVDF) is a kind of polymer and organic material with piezoelectric effect [16]. As a polymer, it has a very low Young’s modulus around
7
6.9Gpa. Comparing to other piezoelectric materials, it is more flexible and stretchable, which makes it suitable for low-frequency, large-deflection energy scavenging applications [17]. With this property, it has comparable electromechanical coupling factor comparing to normal piezoelectric materials [18]. Also, as an organic material, it is chemically stable and biocompatible [19], which allows PVDF to be used in biological systems. However, only β phase of PVDF are able to generate piezoelectricity, so the material need to be treated to obtain piezoelectric property [20].
2.1.3 Zinc Oxide
Zinc oxide (ZnO) is a material with a wurtzite crystal structure [21]. It has much smaller piezoelectric coefficients than PZT, but it has been used for bulk acoustic wave
(BAW) applications [22], [23]. Its property is similar to aluminum nitride, however conventionally it was more widely used for it is easier to produce. It can be produced in room temperature by reactive sputtering of zinc in oxygen environment with a followed high temperature anneal [24]–[26]. But, ZnO has a major shortcoming for its relatively small bandgap ~3 eV, which introduce a situation of smaller breakdown voltages than insulating materials [5].
2.1.4 Aluminum Nitride
Aluminum nitride (AlN) is similar to ZnO. It also has a wurtzite crystal structure with similar piezoelectric coefficients with ZnO. Furthermore, it has excellent mechanical properties and chemical resistance. As a non-ferroelectric material, it has intrinsic piezoelectric properties even at elevated temperatures [27]. So, AlN has a good
8 potential for harsh environment applications. AlN can be sputter-deposited on wafer level and also compatible with Si technology. Since it is non-ferroelectric, like ZnO, the deposition of AlN need to be well-engineered to achieve well-oriented c-axis [28].
2.1.5 Lead Zirconate Titanate
Lead zirconate titanate (PZT) is one of the most widely used piezoelectric materials and it is often used in MEMS applications [5], [7], [10], [29]–[31]. The strong electro- mechanical coupling of PZT material, better than most of other piezoelectric materials, makes it very attractive for actuating and energy harvesting applications [32], [33]. The material can be synthesized by a variety of methods, like sputtering, sol-gel, MOCVD and laser ablation processes [5], [34], [35]. PZT MEMS fabrication is compatible with silicon technology, which gives it the potential for monolithic integration. However, PZT is very brittle with low strain limit, which has restricted its fields of applications as certain device structures.
2.2 PiezoMEMS Device Structures and Applications
2.2.1 Actuators
Reverse piezoelectric effect, which convert electric signal into mechanical signal, are used in PiezoMEMS actuators.
A group from U.S. Army Research Laboratory demonstrated PZT actuators for lateral, vertical and torsional motions [10], [36]. The actuators show great potential in realizing bio-inspired micro-robots for its capability of mimicking the motions of different types of joints.
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Another group from University of California, Berkeley demonstrated a design by using lateral piezoelectric actuators to mimic the flapping of insect wings [37].
Thus, the piezoelectric effect in actuator applications most lies in the regime of actuating micro-scale robotics [7]. And the reverse piezoelectric effect can be also used in sensing technologies, which will be discussed later.
2.2.2 Energy Harvesters
Piezoelectric materials can be made into energy harvesters using the direct piezoelectric effect. It can convert mechanical energy into electrical energy.
An energy harvester fabricated by direct-write of PVDF nanofibers with in situ mechanical stretch and electrical poling characteristics was introduced by Prof. Liwei
Lin’s group from University of California, Berkeley [17]. The diameters of these nanofibers range from 500 nm to 6.5 μm with the length from 100 to 600 μm. The typical electrical output tested within more than 50 of this nanogenerators were 5-30 mV and 0.5-3.0 nA by stretching the fibers at a frequency of 2 Hz. The energy conversion coefficient is an order of magnitude higher than those made of PVDF thin films. The nanofibers also have potential for sensing and actuation applications.
Professor Zhong Lin Wang’s group from Georgia Institute of technology were focusing on making ZnO-based energy harvesters by bonding two ends of ZnO nanowire
(NW) on the substrate surface for harvesting low-frequency energy [38], [39]. The nanowires have diameters of 100-800 nm and length of ~100-500 μm. For a single NW generator, the output voltage could be up to 25 mV and current more than 150 pA. With the device placed on human finger, the short-circuit current and open-circuit voltage can
10 reach ~500pA and ~50-100 mV, respectively. The harvester can be used for biomedical device energy harvesting for converting muscle movement into electricity.
Professor Albert Pisano’s group from University of California, Berkeley developed
AlN piezoelectric harvesters called AlN/SiC diaphragm energy harvester [40]. The device is not small with a 3000 μm diameter diaphragm made of 4 μm thick AlN/SiC composite. The device achieved a power density of 50 μW/cm2, which is not impressive.
But it can operate at elevated temperature up to 350 ºC as tested by his group.
A new way of fabricating flexible and stretchable PZT ribbons on silicone rubber for energy harvesting was demonstrated by Prof. McAlpine’s group from Princeton
University [41]. PZT material is usually know to be brittle. The fabrication technique has expand the area of applications of this widely used high piezoelectric coefficient material.
2.2.3 Sensors
Piezoelectric material devices with specially designed structures can be used for different types of sensing applications.
A group from University of California, Berkeley designed two types of biomimetic gyroscopes using foils of stainless steel [42]. Both devices are driven by piezoelectric actuators and Coriolis force is detected using strain gages. The devices achieve low power and high sensitivity of sensing angular velocities.
Also, doubly clamped piezoelectric beams can be used for chemical components, especially gas, sensing when coated with reactive chemicals [13]. The beam is excited on its resonance frequency during operation. When gas molecular stick on the beam and
11 react, the effective mass and inner stress of the resonator will change, causing the resonance frequency to shift. By monitoring the resonance frequency, the device can detect the appearance or even the concentration of a certain kind of gas.
2.3 Future of PiezoMEMS
Different piezoelectric materials preserve different properties. Thus, they all have their own pros and cons. For example, AlN has good mechanical property and chemical stability for harsh environment applications and it is compatible with on chip integration, but its piezoelectric coefficient is low comparing to other common piezoelectric materials.
PZT has good piezoelectric coefficient and variety ways of fabrication, but it is toxic and requires special packaging for biomedical applications.
Therefore, correct material needs to be chosen for specific application. Also, needs of each application should be accommodated by the design of device structure. So, for
PiezoMEMS technology development, it is demanding to achieve deep understanding in piezoelectric materials and MEMS device structures. Researches have been done for decades, but commercialized applications are still limited for everyday life and industrial purposes.
Another major obstacle is that on-chip fabrication of PiezoMEMS devices is still difficult to achieve for its complicated fabrication process, which, in most of circumstance, not mature enough for mass production at a reasonable cost.
In conclusion, even though for sure, PiezoMEMS technology has a promising future, much research and designs remain to be done to fully develop the technology.
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Chapter 3 Device Structure and Fabrication Process
The devices tested and analyzed in this work were fabricated in U.S. Army
Research Laboratory. In this chapter, we will demonstrate the fabrication process of the devices and illustrate the device structures and stacking of the composite.
3.1 Device Fabrication Process
In this study, the material used as active layer of the PiezoMEMS device is lead zirconate titanate (Pb(ZrxTi1-x)O3, PZT). Its electro-mechanical coupling effect which known as piezoelectric effect is used to directly convert signal from mechanical to electrical domains and vice versa. The piezoelectric coefficient is the key parameter representing the signal transduction efficiency so that high piezoelectric coefficient is desired. The coefficient is inherently linked to its crystalline quality which reaches magnitude at morphotropic phase boundary (MPB) [43]. PbZr0.52Ti0.48O3, or PZT (52/48), locate approximately at MPB and the composite has a maximum of both dielectric permittivity and piezoelectric coefficients [44].
An optimized fabrication process was used for the fabrication of composite cantilevers using PZT (52/48) as active layer [10]. The fabrication was done in U.S.
Army Research Laboratory.
The fabrication process began with thermal growth of silicon dioxide (SiO2) and deposition of silicon nitride (Si3N4) on a silicon wafer. A 3 µm thick SiO2-Si3N4 composite was formed as elastic layer of the cantilever composite.
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Then, a layer of Ti was sputter-deposited onto the SiO2-Si3N4 composite using a
Unaxis Clusterline 200 (CLC) deposition system. And the wafer was moved into a Bruce
Technologies tube furnace for oxygen annealing of Ti.
The former step has convert Ti into TiO2, which can act as a seed layer for {111} platinum (Pt) nucleation. Using CLC, bottom electrode Pt was deposited under the temperature of 500 ºC and the grown Pt layer is highly {111}-textured, providing a perfect template for {001} PZT texture growth.
Chemical solution deposition (CSD) was used to deposit the piezoelectric thin film onto the Pt layer, which consists of deposition of a single lead titanate (PbTiO3, PTO) seed layer ~17nm thick and PZT (52/48). The CSD solutions were prepared using a process modified form the processes described by Zhou et al. [45]. CSD allows for stoichiometry control to achieve reasonable cost for mass production. The approach achieved highly {001}-textured PZT (52/48) to enhance the piezoelectric coefficients.
Then, the same CLC was conducted to deposit the top layer of Pt electrode, followed by patterning of entire piezoelectric material, electrodes and passive layer stacks by etching. Thus, patterns of cantilevers were formed.
Further etching of removing the bottom silicon was performed, left the cantilevers suspended over the silicon substrate [7].
The final steps of the fabrication are etching to expose certain area of bottom Pt layer, metal deposition to create metal pads on the Pt layers, and wirebonding from the pads to Pt layers.
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This optimized PZT fabrication process has led to fairly impressive improvement in
Lotgering factor, which represents the ratio of desired orientation of crystal in the material [46].
3.2 Device Structures
3.2.1 Fabrication Induced Positive Curvature
Figure 3-1 shows scanning electron microscopic (SEM) images of the fabricated chip. Cantilevers with different clamping, connection and geometry can be seen in the images. But, it is easy to notice that regardless of different clamping and geometry, all the cantilevers have positive curvature when relaxed. This phenomenon is expected from the fabrication process.
(a) (b)
1Figure 3-1. SEM images of PZT cantilevers with positive curvature induced by pre-stress in fabrication process. Scale bars in (a) and (b) are 100µm and 400µm, respectively.
As we can see in the last section, different fabrication steps are taking place at different temperatures. Since the thermal expansion coefficients of different materials are
15 different, once the temperature of composite cantilever drop to room temperature, stress will be formed between different layers. This stress is usually known as the pre-stress induced by fabrication process. With the pre-stress, the flat cantilever beam is not in force equilibrium. So the cantilever must bend to a certain curvature to release partial of the pre-stress and use the moment induced by bending to achieve balance in the structure.
These bending curvature and its relation between bending moment and pre-stress will be analyzed in detail in the following chapter.
3.2.2 Stacking of Composite Beam
Figure 3-2 shows the SEM image of a cantilever tip with an illustration of its composite stacking, which provides detailed information on thickness of each layer.
Pt 50nm PZT 1μm Pt 100nm
TiO2 35nm
SiO2-Si3N4 3μm
(a) (b)
2Figure 3-2. Composite cantilever structure. (a) SEM image of composite cantilever tip. Scale bar is 5μm. (b) Illustration of composite cantilever stacking structure with thickness of each layer.
The both Pt layers are electrode layers. When PZT layer is stretched, charge will be generated on its surfaces and Pt layers can collect this charge and conduct it to outer circuit for measurement purpose or as voltage signal. Also, voltage can be applied on the
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Pt layers to form electric field through the PZT layer and PZT layer can generate stress due to this field.
The SiO2 and Si3N4 composite is serving as the elastic layer of cantilever in order to move the neutral plane of the composite out of the active PZT layer. Thus, once a voltage is applied between the surfaces of the PZT layer, the longitudinal stress induced by the layer will form a moment to bend the cantilever. Or, once the cantilever is bended, a strain in PZT layer will be induced by changing of the curvature and generate electricity on the two surfaces of the PZT layer. Without the elastic layer, the cantilever can only generate force or subject to the stress in longitudinal direction. And for resonator/oscillator applications, longitudinal resonance frequencies are too high to be made use of.
Pt and TiO2 layers are designed to be very thin in order to reduce their effect on beam structure mechanical property and simplify the bending moment and stress analysis.
Neutral plane of the composite is engineered far away from the center of PZT layer.
Thus, more stress can be induced in PZT layer by bending the cantilever or larger moment will be generated when applying voltage between PZT layer surfaces.
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Chapter 4 Static Mechanical and Piezoelectric
Characteristics
In section 3.2.1, we notice the interesting fact that the PZT cantilevers form positive curvature in relaxed condition. As explained in the former chapter, this positive curvature is caused by the pre-stress in different layers induced during the fabrication process. In this chapter, by carefully analyzing the stress and strain distribution along with the curvature profile, we demonstrate a damage-free experimental way to extract the piezoelectric coefficient of the PZT material in the cantilever stacks. At the same time, fundamental static material mechanics of the cantilever are illustrated for pre-stress extraction and to provide basics for further dynamic analysis.
4.1 Mechanical Characteristics
4.1.1 Mechanical Properties of Composite Cantilever
As mentioned in section 3.2.1, the cantilever has a positive curvature due to the pre- stress induced during fabrication process. The stress distribution can be evaluated by analyzing the curvature of the relaxed cantilevers. By leveraging the stress, we can further determine the piezoelectric coefficients of the PZT material, which will be demonstrated in section 4.2.
To analyze the stress in the cantilever beam, detailed composite geometry and material properties of each layer are needed. Table 4-1 is a summary of the parameters of each layer needed for static mechanical analysis of the cantilever. Layer 5 is a layer with several silicon dioxide (SiO2) and silicon nitride (Si3N4) layers stacking together. To
18 simplify the analysis process, we have evaluated and simplified this composite layer with normalized material parameters.
1Table 4-1 Summary of Geometry and Material Property of Each Layer
Young’s Layer Thickness Mass Density Poisson’s Material Modulus E Number (nm) Y ρ (g/cm3) Ratio (GPa) m 1 Pt 50 141.4 21.45 0.38 2 PZT 1000 70.1 7.50 0.39 3 Pt 100 141.4 21.45 0.38 4 TiO2 35 214 4.00 0.28 5 SiO2-Si3N4 3000 85.2 2.29 0.18
4.1.2 Pre-Stress Induced Curvature of Cantilever
The positive curvature of the cantilever is caused by different strain levels of the composite layers, which result in an evenly distributed moment among the x axis of the cantilever, causing the bending to happen. The situation here is similar to a basic bending type in mechanics of materials called pure bending [47].
Figure 4-1 demonstrates the pure bending of a singly clamped cantilever. The bending of the cantilever is induced by an anti-clockwise moment M0 on the tip (Fig. 4-1
(a)). By taking an infinitesimal of the cantilever and neglecting gravity (Fig. 4-1 (b)), we can see that the moment is evenly distributed throughout the beam and there is no shear stress. Also, it is obvious that the bending curvature of the cantilever is an arc.
In the infinitesimal, we can take a longitudinal line ab in the deflected beam and its distance from neutral plane ss’ is y. Thus, we can write the length L of line ab after bending as y L ab () y d dx dx (4-1) where ρ is the radius of curvature, dx is the original length of ab before deformation, and
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dx also we have d .
y
M R 0 dθ M 0 a b M0 s s’ x y . (a) (b)
3Figure 4-1. (a) Schematic cantilever with positive curvature induced by a moment on tip. (b)
Infinitesimal of the curved cantilever.
Therefore, as the strain is defined as the elongation divided by the initial length dx, we can write the longitudinal strain-curvature relation as: y y (4-2) x where κ is the curvature.
To simplify the analysis of our cantilever, we can assume the situation to be the same as above, where a moment at the tip is causing the cantilever to bend. Even though, indeed, the curvature is caused by the strain difference in each layer of the composite beam, we can use this assumption to extract the bending moment that evenly distributed in the beam.
By changing the pre-stress induced bending into pure bending, we can determine the pure bending longitudinal strain distribution among y direction. In Fig. 4-2, we can see that strain is zero at neutral plane (x-z plane) of the cantilever and follows the Eq. (4-2).
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(Note that this strain distribution is not the real strain distribution in the composite beam, but a component of the distribution for analysis.)
The following equation can be used to determine the neutral plane of the composite beam by setting the sum of the resultant axial force on the cross section equal to zero:
dA 0 (4-3) i xi, n where each integral is integrated with respect to the coordinate system in Fig. 4-2, i denotes the layer number of the material, σx,i is the longitudinal stress of the ith layer, and dA w dy , where w is the width of the beam.
y y
A εA
c
z x
B εB
4Figure 4-2. Longitudinal strain distribution along y axis in the composite beam.
The longitudinal stress of each material equals to the strain times Young’s modulus.
Along with Eq. (4-2), its distribution can be written as:
x,,,, iE Y i x i E Y i y (4-4)
where EY,i is the Young’s modulus of the material of layer i, and an approximation, that each layer has the same radius of curvature, is made.
Therefore, by substituting Eq. (4-4) into Eq. (4-3), we can acquire the position of the
21 neutral plane using the parameters in Table 4-1, and the distance between top surface and neutral plane of the composite is c = 2057 nm (Fig. 4-2).
Until now, let us make an argument about this stress distribution.
According to the pure bending assumption, the distribution derived is not the real bending stress distribution of the composite beam. But, by ignoring the cross section area change due to the elongation or compression, we can state that approximately, the elastic property of the composite beam remain unchanged throughout bending. Then, it is safe for us to argue that this stress distribution is a stress distribution on top of the stress distribution of the flat cantilever, thus provide a moment so as to bend the cantilever as demonstrated in former sections.
Therefore, we can calculate this bending moment:
M ydA ydA E y2 dA (4-5) 0Ax i x , i Y , i i nn where we can substitute in the area moment of inertia of each layer I y2 dA , and the i i moment can be written as:
(4-6) MEI0, Y i i n 1 where curvature κ is defined as , where radius of curvature ρ is the radius R shown in Fig. 4-1 (a) for the composite cantilever. And this moment, even though not acquired from the real bending model, is the evenly distributed moment throughout the real cantilever.
The area moment of inertia of each layer corresponding to the neutral plane can also be calculate by the following equation:
I I A d 2 (4-7) i ic i i
22 which is known as the parallel-axis theorem, where I is the area moment of inertia of a ic
3 2 wti layer with respect of its own central axis ( I y dA ), Ai is the cross sectional ic ic, 12
area of ith layer ( Aii w t ), and di is the distance between the layer center axis and the neutral plane of the composite beam.
4.1.3 Pre-Stress Analysis
In the previous section, we have acquired the relationship between bending moment and bending curvature using pure bending model. But, this model is not suitable to acquire the pre-stress level in the layers of interest. Thus, a further simplified model need to be introduced to approximately evaluate the stress level in PZE material layer and composite elastic layer.
To acquire the simplified model, Pt and TiO2 layers are neglected for their relatively small thickness. The beam is reduced into a bimorph composite with only PZT layer and composite elastic layer.
In Fig. 4-3 (a), we can notice that two forces caused by the pre-stress in both layers introduce a moment in the infinitesimal of a beam. Here, we made an approximation that the longitudinal pre-stress is evenly distributed in each layer. To balance out this moment, the beam need to be bended to provide a pair of repulsive moments equal to the moment caused by the stress (Fig. 4-3 (b)). Using Newton’s law we have
PP 0 12 (4-8) t MMP1 2 2 0 2 where t is the thickness of the composite which is 4µm. So that the Eqs. (4-8) can be
23 derived as 2 PPPMM (4-9) 1 2t 1 2 where M1 and M2 can be defined with respect to each layer’s own center axis to simplify the calculation:
tt/2EEI /2 MPZT y y dA Y,, PZT PZT y2 dA Y PZT PZT (4-10) 1 tt/2PZT /2 PZT PZT PZT RR11
t /2 EI MElastic y y dA Y, Elastic Elastic (4-11) 2 t /2 Elastic Elastic R2 by substituting M1 and M2 into Eq. (4-9) with RRR12, we can get: 2 PEIEI Y,, PZT PZT Y Elastic Elastic (4-12) Rt where P is the same force generated by pre-stress in PZT layer and elastic layer.
P Therefore the pre-stress in both layers can be calculated using PZT and wt PZT
P Elastic respectively. wt Elastic
(a) (b)
P PZT Layer 1 P1
R M1 P1’ P2 P2 Elastic Layer t P2’ M2
5Figure 4-3. (a) Schematic infinitesimal of the simplified cantilever beam composite structure with approximately defined pre-stress-induced forces. (b) Schematic infinitesimal of pre-stress induced curvature with force and moment analysis on the right half of the figure.
24
4.1.4 Radius of Curvature Measurement
As discussed in the former two sections, to analyze the inertial bending moment and pre-stress in the cantilever, the only missing parameter is the radius of curvature of the beam, which can be extracted using experimental method.
We use a white light optical profilometer (zygo NewView 7300) to scan through the interested device area in order to generate topography maps of the cantilevers and their surrounding areas. By post-processing of these topography maps, we can extract the curvature profile of the cantilevers.
Figure 4-4 (a) and (b) show the optical microscopic image and SEM image of the interested devices, respectively. Two devices have the same composite stacking, which is shown in Fig. 3-2, and same width of 25µm. The length of Device #1 is 404µm where it is 505µm for Device #2. We generate the topography map of the two devices and extract the curvature profile (Fig. 4-4 (c) & (d)). The curvatures can be well fit in a circular curve defined as follows:
y R R2 x A2 B (4-13) where y and x are the position variables, R is the radius of circle, and A and B are coefficients defining the position of the arc.
The fittings show that two devices have similar radii of curvature around 1.3 mm.
According to Eq. (4-6) and Eq. (4-12), both inertial bending moment and pre-stress are irrelevant with the length of the cantilever but are inversely proportional to the radius of the curvature, proving the observed fact that both devices should have the same radius of profile curvature.
By substituting the extracted radius of curvature into Eq. (4-6) and Eq. (4-12), we
25 can calculate the inertial bending moment and pre-stress in the cantilever beam. The pre- stress in the devices are evaluated to be: for Device #1, σPZT = 177 MPa and σElastic = -59
MPa, and for Device #2, σPZT = 175 MPa and σElastic = -58 MPa.
60
m)
40
20 R=1.295mm
Deflection ( Profile Fitting 0 0 50 100 150 200 250 300 350 (a) (c) Position (m) 60 Device #1
m)
40
Device #2 20 R=1.305mm
Deflection ( Profile Fitting 0 (b) (d) 0 50 100 150 200 250 300 350 Position (m)
6Figure 4-4. (a) Optical microscopic image of Device #1 and #2 with color lines indicating the sections of profile measurements. Scale bar is 100µm. (b) SEM image of the same devices.
Scale bar is 100µm. (c) & (d), curvature profile of relaxed Device #1 and #2, respectively. The insets show the 3D topography maps of the devices.
4.2 Piezoelectric Characteristics
4.2.1 Piezoelectric Coefficients
Piezoelectric effect couples the material’s mechanical deformation and electrical behavior. Figure 4-5 illustrates both direct piezoelectric effect and reverse piezoelectric effect of {3-1} mode. The directions of coordinate system is defined by the PZE material
26 crystal orientation. Direct piezoelectric effect happens when you apply forces to generate longitudinal stress in the PZE material, positive and negative net surface charges will be formed on the top and bottom side of the PZE material, respectively, inducing a positive voltage between two surfaces (Fig. 4-5 (a)). While reverse piezoelectric effect serves in the situations that you apply a voltage between bottom and top surface of the material or conduct an electric field in the transverse direction (3 direction) of the material, the material will generate a negative stress in longitudinal direction (1 direction). These two effects can be mathematically expressed using the following two equations [5], [48]–[50].
(4-14) Dd3 31 1 1 (4-15) 1eE 31 3 3 where σ1 is the mechanical stress in 1 direction, D3 is the electric displacement (charge density), E1 is the electric field in 1 direction, and d31 and e31 are the piezoelectric coefficients of {3-1} mode.
3
F 2 1 + + + +
– – – – (a) F
3
F 2 1 V
(b) F
7Figure 4-5. Schematic illustration of piezoelectric effects. (a) Direct piezoelectric effect. (b)
27
Reverse piezoelectric effect.
4.2.2 Piezoelectric Coefficients Extracting
As described in the former section, by applying a constant voltage between top and bottom electrodes of the PZT layer, a stress will be generated by reverse piezoelectric effect. The stress is in x direction and we can regard it as constant along y axis in the
PZT layer. If certain value of voltage is applied, we can achieve a piezoelectric stress to partially compensate the bending caused by pre-stress in the composite beam. Therefore, we can write the moment induced by this pre-stress
M ydA ydA 2 wt y (4-16) PZTPZT PZE,,, x PZE x PZT PZE x PZT PZT
where PZE, x is the piezo-stress induced by voltage applied and is constant along y direction in the layer, w is the width of the cantilever, tPZT is the thickness of PZT layer and yPZT is the distance between neutral plane of PZT layer and the neutral plane of the whole composite.
With the definition of electric potential
EV (4-17) along with Eq. (4-15), we can acquire the expression of , which can be written as
eV31 PZT, x (4-18) tPZT where ΔV is the electric potential difference in y direction in this case.
The remaining bending moment after applying the voltage in the cantilever M can be written as
MMM0 PZT (4-19) whose direction is defined as the same as M0 in Fig. 4-1 and can be defined in the same
28 expression as Eq. (4-6). Thus, we can write an equation describing the relation between the voltage applied across the electrodes ΔV and the radius of curvature after the voltage applied R: 11 EI EI 2 wyPZT e31 V (4-20) RR0 where R0 is the original radius of curvature before applying a voltage, and we define
EI EY, i I i to simplify the equation. Then, we change term R into R = R0 + ΔR, and n rearrange Eq. (4-20). Therefore, we can get EI dR (4-21) e31 2 2wyPZT R0 dV where dR R0 . And we find that e31 is proportional to the ratio of radius of curvature change and voltage change.
Thus, we can extract the piezoelectric coefficient e31 by measuring the different radius of cantilever curvature upon different DC voltage applied between the electrodes.
Figure 4-6 shows the profilometer we used to calibrate the relaxed bending curvature of cantilevers with a DC power supply system.
(a) (b) Multi- Package meter Device under Profilometer
Device DC Power Supply Chip
Connecting Cables
8Figure 4-6. Photos of white light profilometer with static voltage supply system for cantilever bending curvature measurement. (a) Whole system. (b) Device supported by a customized holder under profilometer.
29
We use a DC power supply (Agilent E3631A) as DC voltage supplier and a multi- meter (Fluke 115) to calibrate the DC voltage applied. The bottom Pt electrode is connected to positive output of the power supply while top Pt electrode to the negative one. The cantilever chip is mounted on a package with wirebonding and BNC cables are used for connection. We make a customized holder to support the package, enable cable feed-through. The holder is stuck to the profilometer stage to prevent slipping (Fig. 4-6).
As shown in Fig. 4-7, we apply various voltages between electrodes of Device #1 and Device #2 along y direction of the cantilevers and calibrate the cantilever radius of bending curvature for each voltage. Using the data, we observe clear linear relation between voltage applied and curvature radius change, which satisfy perfectly what is predicted in Eq. (4-21), since e31 should be constant for a certain material.
1.50
1.45 1.35
1.40 1.30 1.35
1.30 1.25
Radius(mm)
Radius (mm) 1.25 Radius Radius Linear Fitting 1.20 1.20 Linear Fitting (a) -2 -1 0 1 2 3 (b) -2 -1 0 1 2 Voltage (V) Voltage (V)
9Figure 4-7. Relation of cantilever radius of curvature and voltage applied between PZT layer surfaces. (a) Device #1. (b) Device #2.
Therefore, by using linear fitting of the data in Fig. 4-7, we complete the calculation of reverse piezoelectric coefficient in Eq. (4-21) for Device #1 and Device #2. Table 4-2 shows the parameters used from former results and extracted piezoelectric coefficients for both devices.
30
2Table 4-2 Summary of Parameters from Piezoelectric Coefficient Calculation
Normalized Cantilever Relative Radius of R-V Slope Piezoelectric Device Bending Rigidity Width Distance Curvature dR dV Coefficient Number 2 2 EI (N∙m ) w (µm) yPZT (nm) R0 (mm) (μm/V) e31 (C/m ) 1 1.264×10-11 25 1507 1.295 50.88 -5.09 2 1.264×10-11 25 1507 1.305 42.29 -4.17
31
Chapter 5 Dynamic Mechanical and Piezoelectric
Characteristics
In the former chapter, we have studied the static properties of PZT cantilevers. But, in terms of applications, the dynamic characteristics of the PZT cantilever is of more interests. For example, using the PZT cantilevers as energy harvesters, the cantilevers are used to pick up vibrations and convert vibrational energy into electrical energy [5], [7],
[48]. Also, in sensing technology, resonators are widely used to detect various physical parameters [13]. Moreover, if the frequency ranches several kilo-hertz range, the cantilevers can even be used for radio frequency applications [51].
In this chapter, we will first use a lumped simple harmonic oscillator (SHO) model to analyze the dynamics properties of the cantilevers. Then, we will relate this model with an equivalent circuit model and depict an intuitive relation between mechanical model and electrical model. Finally, we will show the experiment results to support the analytical analyses.
5.1 Cantilever Dynamics
5.1.1 Resonance Frequencies
Upon finishing the static analysis of cantilever mechanics, dynamic analysis is needed for further understanding the cantilever behavior. The model of lumped SHO model is often used for dynamic analysis.
Figure 5-1 shows the lumped model with damping. In the figure, m is the effective mass, k is equivalent spring constant, and ν is the damping coefficient. By applying a harmonic disturbance F(t) to the mass, a driven damped SHO model is formed. And we
32 can write down its differential equation of motion: d2 x dx m kx F t (5-1) dt2 dt by solving this equation, we can acquire the resonance frequency of the model:
22k (5-2) m where . And the quality factor (Q) of the system can be defined as 2m m Q (5-3) which is a parameter related to the damping rate of the oscillator [50], [52].
ν k
x(t) m
10Figure 5-1. Lumped model of damped simple harmonic oscillator.
From text book [52], we can find the solution of transverse mode resonance frequency of a singly clamped cantilever as a SHO: 2 kl EI f i (5-4) i 2l 2 A where fi is the ith mode resonance frequency, kil is the ith mode eigenvalue which can be calculated approximately by k l i 1 , EI is the transverse rigidity of the i 2 composite beam defined in section 4.2.2, and A is the equivalent cross-sectional mass, which is
n AA i ii (5-5)
33 where ρi is the mass density of ith layer and Ai is the cross-section area of ith layer. Note that here we have ignored the pre-stress in the cantilever to simplify the analysis. Since the stress is already released to form the bending structure of the cantilever, it will not play an important role in tuning the device resonance frequencies.
Thus, using Eq. (5-4), we can calculate the resonance frequencies of our two devices. And Table 5-1 summarizes the eigenvalues and resonance frequencies of the first 7 transverse modes of Device #1. The values of Device #2 will be shown in the conclusion of this chapter.
3Table 5-1 Summary of Eigenvalues and Theoretical Resonance Frequencies of Device #1
Mode #1 Mode #2 Mode #3 Mode #4 Mode #5 Mode #6 Mode #7 Eigenvalue 2 3.516 22.033 61.701 120.912 199.855 298.564 416.976 (kil) Resonance Frequency 18.31 114.74 321.31 629.65 1040.75 1554.78 2174.41 fi (kHz)
Here, we are only interested in transverse modes of the cantilever. Since the disturbance provided by PZT layer is in longitudinal direction with transverse by-product, it cannot provide lateral or torsional forces to form the corresponding modes. Even though the disturbance can be in longitudinal direction. The longitudinal modes resonance frequencies could be very high based on the device geometry. So that the lateral, torsional and longitudinal modes are not of our interest in this paper.
5.1.2 Simulation
To further investigate the transverse mode resonance frequencies of the cantilevers, we conduct finite element method (FEM) to analyze the cantilever resonance frequencies both with and without pre-stress in the cantilever beam using COMSOL Multiphysics.
34
To prevent from taking days to acquire the resonance frequencies for a single device, we first simplify the composite structure by combining middle Pt layer (Layer 3) and
TiO2 layer (Layer 4) into one 135nm thick Pt layer, as Pt and TiO2 have similar material mechanical properties and these two layers are very thin comparing to the composite thickness (Table 4-1). Figure 5-2 shows the simulated shapes of first 7 transverse modes of Device #1. And Table 5-2 shows the simulated results of resonance frequencies of
Device #1.
Mode #1
Mode #2
Mode #3
Mode #4
Mode #5
Mode #6
Mode #7
11Figure 5-2. Shape of first 7 transverse resonance modes of Device #1 from COMSOL
Multiphysics simulation.
Furthermore, by moving and adding the top Pt layer (Layer 1) into the middle Pt layer (Layer 3) to form a 185 nm thick Pt layer, the simulated resonance frequencies of
35
Device #1 do not change much comparing to the former simulation configuration (Table
5-2). So that we can introduce the pre-stress acquired in section 4.1.4 to this even more simplified model of simulation. The results show no change comparing to the one without pre-stress.
4Table 5-2 Summary of Resonance Frequencies of Device #1 by COMSOL Multiphysics
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Method (kHz) (kHz) (kHz) (kHz) (kHz) (kHz) (kHz) Simulation (With 18.05 113.04 316.33 619.47 1023.19 1526.92 2130.08 Top Pt Layer) Simulation (Without 17.85 111.81 312.93 612.96 1012.73 1511.90 2110.05 Top Pt Layer) Simulation (With 17.85 111.81 312.93 612.96 1012.73 1511.90 2100.05 Pre-Stress)
From FEM results of Device #1, we can conclude that the pre-stress itself cannot change the dynamic mechanical property of the cantilever. In the conclusion of this chapter, we will further discuss about the pre-stress effect and the FEM results of Device
#2.
5.1.3 Vibrational Energy
It is essential to calculate the vibrational energy of the cantilever for both actuating and harvesting capability evaluation purposes of the devices. When the cantilever is vibrating at a stable frequency without changing of amplitude, the vibrational energy of the cantilever can be defined as the potential energy when it reach its magnitude position.
1 This potential energy can be expressed as E kx2 for a spring-mass system. But, in P 2
36 the circumstance of our devices, the driving disturbance is a moment, so that we can modify the function into 1 E E k 2 (5-6) Vib P2 tip
EI where k is the equivalent rotational spring constant extracted from the cantilever l
l pure bending model [47], l is the length of the cantilever, is the magnitude of tip2 tip deflection of the cantilever tip, and θtip is the corresponding angle of rotation at the cantilever tip.
Eq. (5-6) demonstrates a way to evaluate the mechanical energy of a vibrating cantilever. Thus, we can use it to calculate the mechanical energy of vibrating cantilevers, which can be further used to calculate the energy conversion capability of PZT cantilever devices.
5.2 Dynamic Piezoelectric Characteristics
5.2.1 Equivalent Circuit
With the mechanical model of the cantilever known, further investigation on relating its piezoelectric effect to mechanical property should be made. Figure 5-3 shows the equivalent circuit model of on-resonance PZT cantilever. Rc and C0 are the contact resistance and static capacitance of the device, respectively. Ri, Ci and Li are the electrical devices representing the dynamic property of the cantilever related to the motional parameters of ith mode. Ri is proportional to damping of the cantilever, Ci is inversely proportional to the stiffness of the cantilever, and Li is proportional to the
37 effective (modal) mass of the cantilever [13], [50].
Therefore, we have depicted an intuitional circuit model to demonstrate the mechanical property of the cantilevers in an electrical perspective, where similar resonance frequency results can be acquired using LRC circuit analysis.
Rc Ri Ci Li i
v C0
12Figure 5-3. Equivalent circuit of PZT cantilever as an unloaded resonator.
5.2.2 Electrical Energy
The electric power of an alternating current in the circuit can be written as 1 V 2 PVIcosp cos (5-7) AC22 p p Z where Vp and Ip are peak amplitude of voltage and current, respectively, θ is the phase difference between voltage and current, and Z is the impedance of the circuit. When the cantilever is on resonance, the phase θ is around zero, which can be observed later in the experimental data.
5.3 Measurements and Results
5.3.1 Optical Measurement
In order to confirm the results we get from analytical calculations in section 5.1.1 and simulations in section 5.1.2. We conduct a series of experiments to detect the
38 resonance frequencies of the two devices and compare the experimental data to analytical and simulated results.
Figure 5-4 shows the measurement system, which we use to optically measure the resonance frequency of the cantilevers. The chip of devices are mounted in a customized vacuum chamber with optical window and the air pressure is kept in 20mTorr to 30mTorr range during the measurement. The cantilever under measurement is connected to a network analyzer (Hewlett Packard 3577A) through the feed-through connector of the chamber.
(a) (b) 633nm Laser
Photodetector Beam Splitter
Microscope Network Analyzer Objective Output Input
PZT Cantilever Vacuum Chamber SiO2 Si Vacuum Chamber
13Figure 5-4. (a) Scheme of optical measurement system. (b) Photo of part of the optical measurement system.
The input of network analyzer is connected to a photo detector (PD) which detects the intensity of reflected laser light from the cantilever. The laser light is provided by a
632.8nm He-Ne laser and focused on the PZT cantilever by a system of lenses. The
39 optical measurement system is much alike the Michelson interferometer, but the physical basis is different for the intensity change of reflected light is not due to interference but to different reflect angle of laser light. While the cantilever is moving, the incidence angle of incoming laser is changing, causing the reflected angle to change. Therefore, the laser spot reflected on the PD is moving so as to induce intensity change of laser light on the
PD. PD generated voltage signal is read by network analyzer and shown in frequency spectrum. Figure 5-5 shows the laser spot position on Device #1.
50µm
25µm
404µm
14Figure 5-5. Geometry of Device #1 and the laser spot of optical measurement. Inset: optical image of laser spot on cantilever (image contrast is adjusted to enhance the appearance of laser spot).
With the optical measurement system, we can use network analyzer to sweep through the frequency ranges and see the cantilever response by optical detection. Figure
5-6 shows the electrically driven, optically detected response of Device #1 in 10 kHz to
2.2 MHz range. We measure the zoomed-in spectrum of each resonance frequencies and fit the spectra to extract resonance frequencies and quality factors (Qs). The device Q is ranging from 383 to 577. We notice that the 5th mode is missing in the optically measured results. A reasonable speculation can be that certain defects in the cantilever or
40 components in the driving circuit have damped or compensated the resonance mode, making the resonance spectrum peak too small to detect.
Using the same measurement system, we also measured the first 2 resonances of
Device #2 (Fig. 5-7 (a) & (b)). In order to further investigate the time domain response of the cantilever, we measure the ring-down performance of the cantilever. We use a function generator, instead of the network analyzer, to connect to the cantilever in order to drive the cantilever with pulse signal. And PD is connect to an oscilloscope to read the time domain
1 2 3 4 6 7 (a)
Signal Amplitude
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Frequency (MHz)
)
V 250 140 50
( 1 (b) 120 2 (c) 3 (d) 200 40 100 150 80 30 100 60 20 40 50 20 10 0 0 0 f=17.06kHz Q=383 -20 f=105.0kHz Q=437 f=294.9kHz Q=577 Signal Amplitude -50 -10 16.6 16.8 17.0 17.2 17.4 103 104 105 106 107 290 292 294 296 298 Frequency (kHz) Frequency (kHz) Frequency (kHz)
) 30 5 V 8
( 4 6 7
(e) ) (f) 4 (g)
20 V
6
( 3
10 4 2 2 1
Voltage 0 0 f=577.2kHz Q=570 0 f=1.420MHz Q=484 f=1.978MHz Q=485
Signal Amplitude -1 570 575 580 585 1.410 1.415 1.420 1.425 1.96 1.97 1.98 1.99 2.00 Frequency (kHz) Frequency (MHz) Frequency (MHz) 15Figure 5-6. Device #1 electrically driven frequency domain response using optical detection.
(a) Wide spectrum of all resonance frequency peaks measured. (b-g) Zoomed-in measurement of each resonance peak and fitting of resonance frequency f and quality factor Q.
41 response of the cantilever. In Fig. 5-7 (c-e), by applying 1 V in amplitude 30 μs long pulses to the cantilever, we can clearly see smooth time domain ring-down response through the oscilloscope. And by fitting the curve, the results (Fig. 5-7 (e)) show good agreement with frequency domain results (Fig. 5-7 (a)). The measurement perfectly demonstrates the first resonance frequency of the cantilever. Interestingly speaking, by changing the pulse width to 50 μs, the ring-down curve contains a higher frequency component, which, by counting its period, has a frequency around 60 kHz to 70 kHz and should be the second mode resonance frequency of Device #2.
1.0
) ) 6 0.8 f=10.87kHz (a) CH1 (c) 4 Q≈420 (e)
mV
mV
( ( 2 0.6 Q≈442 0 0.4 -2 -4 0.2 f=10.86kHz
Voltage Voltage Voltage Voltage -6 0.0 -8 10.7 10.8 10.9 11.0 11.1 CH3 4.0 4.2 4.4 4.6 4.8 5.0 Frequency (kHz) Time (s)
) f=66.74 kHz (b) CH1 (d) CH1 (f)
mV 0.2 ( Q≈457
0.1
Voltage Voltage CH3 CH3 0.0 66.0 66.5 67.0 67.5 30μs 50μs Frequency (kHz) 16Figure 5-7. Device #1 electrically driven frequency and time domain response using optical detection. (a) & (b), frequency domain response of first 2 resonance modes with fitting. (c)
Oscilloscope display of time domain response to a 1 V, 30 µs pulse. (d) Zoomed-in time domain response. (e) Fitting of time domain response. (f) Zoomed-in time domain response of a 1 V, 50
μs pulse.
5.3.2 Insertion Loss Measurement
In the former section, we use optical detection scheme to measure the resonance frequencies of the two devices. To better understand the resonances, we can also conduct
42 measurement pure electrically, using two-port measurement scheme.
We directly connect the input and output of network analyzer to the top and bottom electrodes of the cantilever, respectively. By sweeping the frequency with the same voltage amplitude of the optical measurement, we can acquire the insertion loss, in other words negative part of forward voltage gain, of the devices.
Figure 5-8 and 5-9 show the pure electrical measurement results of Device #1 and
Device #2, respectively. Table 5-3 summarizes the parameters, which are extracted from the electrical measurements of Device #1.
-40 -30 30 -25.5 10 60 -45 -32 20 -26.0 40 5
) 20 -34 10 -26.5
dB -50
(
| 0 -36 0 0 -27.0
21 -55 -20 -38 |S -10 -27.5 -5 -60 -40 -40
-20 Phase (degree) (a) 1 -60 (b) 2 -28.0 (c) 3 -65 -42 -30 -10 17.0 17.5 103 104 105 106 107 290 292 294 296 298 Frequency (kHz) Frequency (kHz) Frequency (kHz) -20 -12.8 4 4 0.6 -13.0 -10.50 2 0.4 ) 2 -13.2 0.2
dB
(
| -21 0 -13.4 0 -10.55 0.0
21 -0.2 |S -2 -13.6 -2 -0.4
-13.8 -10.60 Phase (degree) (d) 4 -4 (e) 6 (f) 7 -0.6 -22 -14.0 -4 570 575 580 585 1.41 1.42 1.43 1.97 1.98 1.99 Frequency (kHz) Frequency (kHz) Frequency (kHz) 17Figure 5-8. Electrical measurement results of Device #1.
150 -36 -45 20 -50 100
) -38 10 -55 50
dB ( -60 | 0 0 21 -65 -40
|S -50 -70 -10 -100 -42 Phase (degree) -75 (a) (b) -20 -80 -150 10.8 11.0 11.2 66.5 67.0 67.5 68.0 Frequency (kHz) Frequency (kHz) 18Figure 5-9. Electrical measurement results of Device #2.
43
With the insertion loss on resonance known, we can use the following equation to calculate the impedance of the device on resonance:
ILi 20 ZRi 20 10 1 (5-8) where Zi is the impedance of the device on ith resonance mode, R0 is the internal output resistance of network analyzer, and (IL)i is the insertion loss of the resonance. The impedance results of Device #1 are listed in Table 5-3. We can use these impedances to calculate the electrical power in next chapter.
5Table 5-3 Summary of Two-Port Electrical Measurement of Device #1
Parameter Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Resonance 17.02 104.56 294.38 576.25 - 1417.65 1974.60 Frequency (kHz) Insertion Loss (dB) 43.80 32.32 25.94 20.32 - 12.99 10.52
Phase (Degree) 10.3 10.8 0.4 -1.8 - 0.5 -0.2 Impedance on 15.38 4.03 1.88 0.94 - 0.35 0.24 Resonance (kΩ)
In addition, by looking at the on-resonance phase in Table 5-3, we find that all the phases measured are around 0 degree, which proves our assumption after Eq. (5-7).
5.4 Conclusions
In this chapter, we have analyzed and measured the dynamic property of two devices by analytical calculation, FEM simulation, optical detection and electrical transmission. Table 5-4 summarizes the resonance frequency results acquired by the methods along this chapter.
In the table, we can find that analytical and simulated results agree well with each
44 other, while electrically driven measurements with both optical and electrical detection show fine agreement. However, the two non-experimental analyses results are slightly higher than experimental ones.
The consequence should be explained by the different clamping conditions in calculations and experiments. Since we assume perfect clamping in analytical calculation and FEM simulation, in reality, the cantilevers are clamped by the ledge which could be partially moving with the cantilever, causing more damping. This is called clamping loss, which can shift the resonance frequency of the cantilever down.
Another assumption is that the difference may be caused by the pre-stress in the cantilevers. Even though we have done the simulation about pre-stress effects on cantilevers dynamic property in section 5.1.2, we have not taken the geometry change due to the pre-stress into account. We know that a positive curvature is generated by the pre-stress in the cantilever, but in the analytical calculation and FEM simulation, we are evaluating the flat cantilevers. This geometry difference may cause the shift of resonance frequencies in calculation and simulation.
6Table 5-4 Summary of Resonance Frequencies Acquired by Different Methods (kHz)
Analytical COMSOL Optical Electrical Device Mode Calculation Simulation Measurement Measurement 1 18.31 17.85 17.1 17.02 2 114.74 111.81 105.0 104.56 3 321.31 312.93 294.9 294.38 #1 4 629.65 612.96 577.2 576.25 5 1040.75 1012.73 - - 6 1554.78 1511.90 1420 1417.65 7 2171.41 2110.05 1978 1974.60 1 11.72 11.42 10.87 11.00 #2 2 73.43 69.39 66.74 67.17
45
Chapter 6 Multimode Resonant Energy Conversion
In the former chapters, static and dynamic properties of our PZT cantilevers have been explored. In this chapter, we are focusing on the applications of the PZT cantilevers.
In the first part, we demonstrate simple analysis on specifications of the cantilevers as actuators. In the other part, we illustrate the follow-up experiments on testing the energy harvesting capability of the cantilevers.
6.1 Reverse Piezoelectric Effect
6.1.1 Electrical Power Analysis
In section 5.2.2, we provide Eq. (5-7) to calculate the power in the electrical equivalent circuit of the cantilever. And in section 5.3.2, we have calculated the impedance on each resonance from experimental results. So that we can calculate the electrical energy we have input on the PZT cantilever to be 13.00 nW on the first resonance mode of Device #1. We use the output amplitude (20 mV) of network analyzer in optical measurement as Vp and first mode impedance (15.35 kΩ) in Table 5-3 as |Z|. The electrical power of other modes of both devices can be calculated using this method.
6.1.2 Mechanical Energy Analysis
To investigate the power conversion efficiency using reverse piezoelectric effect of the PZT cantilevers, mechanical power of the vibration is needed.
46
According to the differential equation of motion in section 5.1.1 (Eq. (5-1)), we can derive: F a 2 (6-1) k m jm 0 / Q where a(ω) is the amplitude of lumped mass when frequency is ω, k is the spring constant, F(ω) is the harmonic disturbance force, ω0 is the resonance frequency, and Q is the quality factor of that resonance.
If we set ω = 0, which is the static condition, then the equation is reduced to
Hooke’s law: F 0 a0 (6-2) k while if we set ω = ω0, then it represents the situation on resonance, and the equation is becoming: FQFQ 00 (6-3) a0 2 jm0 jk and we notice that the amplitude on resonance is proportional to the quality factor Q.
But, in our case, the disturbance is a moment. Thus, we can modify Eq. (6-3) into MQ 0 (6-4) tip 0 jk where M 0 is the periodic moment with resonance frequency generated by
EI piezoelectric effect, k is the rotational spring constant, and θtip is amplitude angle l of rotation at the cantilever tip.
For the static part, similarly we have M 0 tip 0 (6-5) k Therefore, we are able to derive the relation between tip amplitude of vibrating cantilever and tip deflection of statically bended cantilever:
47
tip 00 tip MQ 0 (6-6) tip0 tip 0 M 0 which means the ratio of deflections is the ratio of the moments times the resonance quality factor.
According to Eq. (4-16) and Eq. (4-18), M is proportional to the voltage applied, so we can write: VQ tip00 p (6-7) tip 00 V where Vp is the amplitude of driving voltage and V(0) is the voltage applied in the static bending measurement.
Using the results in section 4.2.2, by applying voltage of ±2 V between electrodes of
cantilevers, the resulting change of tip rotational angle will be tip,2 V 0.025 and
tip,2 V 0.025 , respectively.
Therefore, the mechanical energy when using a 20 mV amplitude power to drive the cantilever at 1st resonance mode is 2 1 VQ E E k p 0.143nJ Vib P tip,2 V (6-8) 20V
6.2 Direct Piezoelectric Effect
6.2.1 Measurement System
We use a similar measurement system as the one used for optical measurement in chapter 5, by adding a PZT plate under the device chip as a mechanical actuator (Fig. 6-
1). Both optical detection by photo detector (PD) and electrical signal of voltage between two electrodes of the cantilever are used to monitor the vibration of the cantilever. Since the signal picked by PD is due to a mix of cantilever vibration and bottom PZT plate
48 vibration, we are only using the optical data as a reference and analyze the results from electrical measurement.
Frequency domain measurement is conduct by using network analyzer to drive the
PZT plate and detect the voltage signal between two cantilever electrodes.
(a) 633nm Laser (b) Network Analyzer A Output Input C B Beam Splitter Oscilloscope Photodetector CH1 CH2 CH3 Microscope A B C Objective B Vacuum (c) Chamber PZT Plate PZT Cantilever
SiO2 Cantilever Chip Si C PZT actuator
19Figure 6-1. (a) Schematic of cantilever harvesting capability measurement system. (b)
Illustration of connections for frequency domain measurement (network analyzer) and time domain measurement (oscilloscope). (c) Image of chip mounted on the package with a PZT plate actuator underneath. The scale bar is 1 cm.
In time domain measurement, we use a function generate as voltage source to drive the PZT plate. And the optical signal (Channel 1), electrical signal (Channel 2) and driving source (Channel 3) are monitored by an oscilloscope at the same time. We are interested in ring-down response of the cantilever electrical output for it is the result
49 purely from cantilever and the cantilevers are the only vibrating object at that time.
6.2.2 Measurement Results
We choose Device #2 to conduct the experiment for its larger volume. It may be more easily to generate noticeable signal than Device #1. Figure 6-2 shows the results of frequency and time domain measurement of Device #2 by mechanical driving at Vp = 500 mV using the bottom PZT plate.
32
)
V f=10.87kHz ) 50 (a) (b) f=66.57kHz (e)
V ( 30
Q≈452 ( 40 Q≈347 28 10 20 50 CH2 30 Voltage Voltage 26 cycles cycles cycles
Voltage Voltage 10.7 10.8 10.9 11.0 11.1 20 Frequency (kHz) CH3 66.2 66.4 66.6 66.8 67.0 (c) Frequency (kHz)
) (d) CH1: Optical Signal 2 Q≈488
mV
( 1 0 CH2: Electrical Signal -1
Voltage Voltage f=10.88kHz 600 -2 cycles CH3:Driving Signal 0 5 10 15 20 Time (ms)
20Figure 6-2. Device #2 energy harvesting capability measurement results. (a) Frequency domain electrical response of 1st mode with fitting. (b) Time domain electrical response of 1st mode with different burst cycles to ring-up the cantilever. (c) Fully ringed-up time domain response of 1st mode with 600 burst cycles and optical signal as reference. (d) Fitting of ring- down curve in CH2 of (c). (e) Frequency domain electrical response of 2nd mode with fitting.
Frequency domain electrical detection from the cantilever (Fig. 6-2) shows the same
1st mode resonance frequency as the results in section 5.3.1. But, the fitting of the ring- down curve in time domain measurement can be more accurate by eliminating the effect
50 from the moving PZT plate. Time domain measurement plots (Fig. 6-2 (b) & (c)) show that, mainly according to the optical reference, the cantilever is fully ringed-up at around
400 burst cycles. So we use 600 burst cycles, which should be sufficient to ring-up the cantilever, to do the measurement. And by fitting its ring-down curve, we extract a resonance frequency at 10.88 kHz and a quality factor of 488. The result shows a fairly good agreement with the data measured in sections 5.3.1 and 5.3.2. Also, frequency domain result of 2nd resonance mode is shown in Fig. 6-2 (e).
Therefore, it is safe for us to state that the cantilever can be used as energy harvester at its resonance frequency.
But in terms of energy conversion, it is essential to know how much the electrical energy power it can generate. By driving the bottom PZT plate very hard at Vp = 10 V for the first two resonances, we read out the output amplitude of ~100 mV for 1st mode and ~80 mV for the 2nd mode. By further calculation, we extract the power of 0.156 μW for 1st mode and 0.432 µW for 2nd mode. And the active layer power density is at very large 12.4 mW/mm3 and 34.2 mW/mm3, respectively.
Therefore, by connecting tens of this cantilevers in series, the power is sufficient to power up an ultra-low power circuit.
In conclusion, the PZT cantilevers we demonstrated have a promising future as power source for small size circuit chips. Even though the frequency range they place in may not be suitable for environmental energy harvesting, they are sufficient to serve as energy converter used to convert energy wirelessly transport by ultrasound or electro- magnetic wave. The applications contain but not limited to body implantation devices, wireless sensor nodes and normal wireless charging devices.
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Chapter 7 Conclusions and Future Work
7.1 Conclusions
In this work, we have demonstrated detailed analyses and measurements on two
PZT cantilevers with the same width (25µm), same composite stacking, and different length (404μm and 505µm, respectively).
We have developed the static and dynamic models of the cantilevers and used the models to analytically evaluate the bending and stress of the cantilevers. Pre-stress of
PZT layer was extracted for both devices, which is around ~175MPa. The reverse
2 2 piezoelectric coefficient e31 was measured to be around -4.2 C/m to -5.1 C/m .
We have acquired the resonance frequencies of two devices by the means of calculation, simulation and measurements. The experimental data matches well with the theoretical estimation with slightly difference mainly due to the clamping approximation.
Six resonance modes of Device #1 and two resonance modes of Device #2 are measured.
Most of the resonance frequencies lie in ultrasonic frequency range and show quality factors in 300-600 range.
We have further tested Device #2 by mechanically driving it. Same resonances are observed both in time and frequency domain as in former measurements. And by mechanically driving the device very hard, we achieve an AC electrical output with power of 0.432 μW and an extraordinarily impressive power density of 34.2 mW/mm3.
Table 7-1 shows some representative piezoelectric material energy harvesters and their major technical specs. Comparing to the other devices in this table, our device lies in a much higher frequency range of application. The comparison tells us that the cantilevers tested in this work as harvesters may not be suitable for ordinary low
52 frequency energy harvesting. But they show good potential for converting mechanical energy transmitted by ultrasound into electrical energy to practically achieve fast wireless charging or actuating using ultrasound.
7Table 7-1 Summary of Representative Piezoelectric Energy Harvesters
Group Volume Active Operation Area Power (Year) Footprint Power Power Material Frequency Density [Ref] Density Maenaka l: 25 mm 1.43 (2010) PVDF w: 16 mm 14 Hz 16µW - µW/mm2 [53] t: 205 µm Lin (2010) d: 0.5-6.5 µm PVDF 0.5-4 Hz 0.5pW - - [17] l: 100-600 µm Wang d: 100-800 nm (2009) ZnO ~10 Hz 25pW - - l: ~100-500 μm [38] Wang d: 100-800 nm (2010) ZnO ~100 Hz 12.5pW - - l: 100-500 µm [39] Schaijk L : 1.7 mm b 48 (2011) AlN W : 3.0 mm 1011 Hz 489µW - b mW/mm3 [54] Lm: 3.0 mm b: 2400 µm l : 250 µm Lin (2011) 1 AlN l : 750 µm 853 Hz 0.17µW - - [55] 2 t: 680 µm d: 4.68 µm Pisano d: 3000 µm ~1µW (2013) AlN 17 kHz 50 µW/cm2 - t: 4 µm level [56] Pisano R: 1000 μm ~10 µW (2014) AlN r : 375 μm 1 kHz 87 µW/cm2 - b level [40] t: 4.54 μm Roundy (2002) PZT ~1 cm3 100 Hz 70 µW - 70 µW/cm3 [57] Gu (2012) 2 cm × 2 cm × 30.8 PZT 4-16.2 Hz 123 µW - [58] 1 cm µW/cm3 Feng l: 7.1 mm 173.6 (2014) PZT w: 4 mm 120 Hz 250µW - µW/mm3 [59] t: 50 µm l: 505 µm This 34.2 34.2 PZT w: 25 µm 66.7 kHz 0.432µW Work µW/mm2 mW/mm3 t: 4 µm
53
7.2 Future Work
The vibrational energy levels harvested by ordinary piezoelectric energy harvesters lie in the range from several Hertz to hundreds of hertz. Our cantilevers operate in a much higher frequency range, so the major pathway to further develop the cantilevers used in this thesis of study is not to use it in energy harvesting of low frequency vibrations or pulses, but to apply them in higher frequency situations. The ultrasound wireless charging and actuating mentioned in the former section provides good guidance for its potential applications. And the technique is a good fit for biomedical implanted devices [60], [61].
Therefore, one major part of future work could be to investigate the capability of the cantilever to convert mechanical energy carried by ultrasound into electrical energy.
Also, further investigation could be done on finding the proper frequency for ultrasound wireless charging and reliable packaging of the harvesters for biomedical compatibility.
Also, as we can see from chapter 3, we have a variety of different geometries and dimensions of cantilevers. The ones with paddles would be interesting for further investigation because the paddle could be served as a proof mass to tune the resonance frequency of the cantilever down and achieve a higher energy harvesting efficiency.
54
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