## 國立臺灣大學工學院應用力學研究所 碩士論文

### Institute of Applied Mechanics College of Engineering

### National Taiwan University Master Thesis

## 拉福波於聲子晶體之波傳現象研究與應用 A Study of Love Waves in a Phononic Structure

## 劉庭瑋 Ting-Wei Liu

## 指導教授：吳政忠 教授

## Advisor: Tsung-Tsong Wu, Professor 中華民國 102 年 12 月

## December 2013

## 誌 誌謝 謝

能完成這篇論文，首先要感謝恩師 吳政忠教授的教導與鼓勵。在超聲波實驗室研究 期間，吳老師細心不倦地在研究上之困難給予指導。同時老師對於學生的信任與容錯，也 培養了學生獨立思考、勇於嘗試的研究態度。此外，吳老師於日常言談中展現出對學生關 心社會國家大事的期許，更讓學生在碩士期間獲益良多，在此謹向吳老師獻上由衷的感激。

承蒙大同大學 陳永裕教授 長庚大學 孫嘉宏教授於百忙之中撥冗擔任學生的論文 口試委員。對於本論文的不吝指正，以及許多精闢見解及寶貴意見，使本論文更臻完備，

在此表達誠摯的感謝。

在研究期間感謝 327 夥伴們平日之照顧與協助。感謝嘉宏、永裕學長及偉姍學姊在研

究期間不厭其煩的幫助與指導。彥廷、智偉、山松以及峻偉、家浩、純劭學長們在研究上 的指導，夢禎、駱爺平常的陪伴、切磋、打氣與鼓勵，以及學弟妹儒璋、思伶、泰成、東 柏、辰峰、佩羚的關心與支持。與實驗室夥伴相處的點點滴滴，使我的碩士班生涯精采而 豐富，這份情誼我會永遠珍惜而銘記於心。

在日本東北大學交換的半個年頭，最要感謝的就是育菁老師以及燿全學長，除了實驗 上大力幫助以及政治上的庇護之外，育菁老師提供宿舍使我有棲身之所，燿全學長更打點 我的吃喝玩樂和食衣住行，此恩實在難報！另外實驗上也感謝佑莨學長協助入門帶練。也 感謝實驗室開放的環境，小野研、田中（秀、徹）研、羽根研的成員們，忍受我一再的使

用英文叨擾，大家不厭其煩的指導，容忍一MEMS 素人的搗亂及破壞。

另外也感謝所有的留學生及非留學生夥伴們（礙於篇幅無法逐一列出）的照顧，在玩 樂中讓我又找回實驗挫折所損失的能量，也使我在人生地不熟的異鄉有個溫暖的避風港，

非常高興認識大家，使我在仙台度過充實又燦爛的半年，寫下人生中精采的一頁。

感謝我的母親 洪錦華女士、父親 劉兆宏先生和哥哥庭碩對我的鼓勵與支持，細心 提攜我長大至今，給我無微不至的關懷，使我能更放心而努力學習。

最後感謝上帝賜給我對真理的渴望以及好奇心，使我在研究的路上樂此不疲。

謹將此論文獻給他們。

## 中

## 中文 文摘 摘要 要

聲子晶體泛指材料性質或幾何形狀週期排列之彈性結構。近二十年，由於聲子 晶體具有特殊波傳效應，如完全頻隙、負折射、聲波聚焦等，已引起相當多學者 的興趣與投入。由於壓電晶圓較難進行精密蝕刻，目前文獻大都以矽基聲子晶體 元件為主，但矽不具壓電特性，須將氧化鋅或氮化鋁等壓電薄膜鍍於叉指電極部 分以激發表面波或板波，元件插入損失較高

另一方面，在層狀介質的表面波研究中，除了偏振方向與波傳矢狀平面(sagittal

plane)平行的雷利表面波外，亦存在有偏振方向與矢狀平面垂直的拉福波(Love wave)。拉福波存在於表面鍍上橫波波速小於基底材料橫波波速材料之層狀半無限 域結構，大部分波的能量局限於表面的薄層中，形成一理想的橫波波導。在壓電 晶體表面亦存在各類偏振方向與矢狀平面垂直的水平剪切式表面聲波，其發生原 理與拉福波相似，可歸類為廣義拉福波。此類聲波常具有較雷利波為高的機電耦 合係數及波速，因此廣泛用於高頻、寬頻濾波器用途。

基於拉福波之特性，如能於不易蝕刻的壓電基板鍍上一具聲子晶體結構之薄 層，除具有高激發效率之波源外，亦可經由聲子晶體之頻隙特性製做高效能之濾 波器或感測器。有鑒於此，本文提出具聲子晶體薄鍍層於壓電基材上之結構，使

用有限元素法搭配布洛赫理論(Bloch theorem)，分析此類結構之頻散特性，證實此

類結構具有拉福波的可傳頻帶及頻隙，並藉由複數波數頻散關係進一步分析頻隙 內聲波之衰減現象。此外，本文也探討頻隙內聲波由均勻介質進入聲子晶體介面 之反射、透射及散射現象，並設計一聲子晶體共振器結構。

實驗部分則是以奈米機電系統製程，利用電子束微影系統產生次微米級之叉指 電極及聲子晶體圖案，並以反應離子蝕刻法製作出聲子晶體結構。使用斜叉指電

極激發及接收寬頻聲波驗證聲子晶體之頻溝，發現在頻溝處有超過40 分貝之插入

損失，且其頻率範圍與理論之預測相當吻合。而設計於1.25 GHz 之共振器也實現，

並具有約400 之 Q 值。

關

關鍵鍵字字：：聲聲子子晶晶體體、、拉拉福福波波、、奈奈米米機機電電系系統統製製程程、、共共振振器器

**ABSTRACT **

A phononic crystal (PC) is a structure whose mechanical properties are periodical- ly arranged. 2-D PCs started to attract great attentions two decades ago. A lot of concern has been focused on their acoustic reflecting phenomenon due to the band gap. However the PC can also be an acoustic conductor, which its conductance is determined by the band structure. Novel properties such as negative refraction or acoustic lensing effect can be achieved with PCs.

Among the many applications, in the area of ultra-high frequency (UHF) acoustic wave devices, complete or partial band gap of air/silicon PC was utilized as the reflec- tive gratings to reduce the devices’ size. In those devices, a piezoelectric thin film has to be deposited on the silicon substrate to generate acoustic waves for silicon is non-piezoelectric. Although the fabrication process of the silicon based phononic device has the CMOS compatible advantage, the insertion loss of such a device is high rela- tively. The other more straightforward way of making a phononic acoustic wave device is to construct periodically micro-holes directly on a piezoelectric substrate. Although the electro-acoustic conversion is higher, it suffered from the anisotropic nature of lith- ium niobate that lead to a complicated fabrication process.

On the other hand, SH-type SAWs have several advantages compared to conven- tional Rayleigh-type SAWs. For example SH-type SAWs possess larger piezoelectricity than Rayleigh-type SAWs on the same substrate material. Also they are faster than Rayleigh-type SAWs therefore desirable for high frequency applications. And in liquids, SH-type SAWs lose less energy than Rayleigh-type SAWs do due to their polarization, also they are sensitive to surface loadings, so they are desirable for (bio-) sensing appli-

cations. Love wave, being one of the SH-type SAWs, shares similar physical character- istics with other SH-type SAWs. Investigating the propagating of Love waves in PCs will expand the PC applications to SH-type SAWs that have bright outlook.

In this study, we investigate Love waves propagating in a piezoelectric substrate coated with a phononic guiding layer. The phononic layer is consisted of a thin layer with periodic machined holes. It is worth noting that the thin phononic layer can be non-piezoelectric which makes the fabrication process relatively simple. And since most energy is trapped in the guiding layer it may serve as efficient reflective gratings in Love wave devices. The method proposed in this study is suitable for other SH-type SAWs also.

**Keywords：**

：**Phononic crystal, Love wave, NEMS, Resonator **

**CONTENTS **

口試委員會審定書 ... #

誌謝 ... i

中文摘要 ... ii

ABSTRACT ... iii

CONTENTS ... v

LIST OF FIGURES ... viii

LIST OF TABLES ... xiv

**Chapter 1
Introduction ... 15
**

1.1 Motivation ... 15

1.2 Literature Review ... 17

1.3 Contents of the Chapters ... 18

**Chapter 2
Love Wave Dispersions of a Phononic SiO**

**2**

**/Quartz Layered Structure19
**

2.1
The Mathematical Model of Acoustic Waves in Piezoelectric solids ... 20
2.1.1 Piezoelectric acoustic wave equations ... 20

2.1.2 Coordinate transformation for material constants ... 22

2.1.3 Materials ... 23

2.2 Waves in periodic structures ... 25

2.2.1 Direct and reciprocal lattices ... 25

2.2.2 Bloch’s theorem ... 28

2.3 Love wave dispersion of a layered half-space ... 29

2.3.1 Numerical methods ... 30

2.3.2 Determination of the SiO2 film thickness ... 34

2.4 Love wave dispersion in a SiO2/IDT/quartz structure. ... 36

2.4.1 Background ... 36

2.4.2 Dispersion relation ... 37

2.4.3 Frequency-pitch relation ... 38

2.5 Love wave dispersion in phononic layered structures ... 38

2.5.1 Geometries ... 38

2.5.2 M-Y-Γ-X-M-Γ band structures ... 39

2.5.3 More on the Γ-X band structures ... 43

**Chapter 3
Design of a One-port Resonator ... 65
**

3.1 Design parameters of a one-port resonator ... 65

3.2 Reflection on the PC border ... 66

3.3 Optimization for the delay distance ... 68

3.4 Resonator evaluation ... 69

3.5 Experiment setup ... 70

**Chapter 4
Fabrications ... 83
**

4.1 Brief of EB lithography ... 83

4.1.1 Why EB lithography ... 84

4.1.2 Anti-charging ... 84

4.1.3 Dose determination and the proximity effect ... 84

4.2 Making the alignment marks for EB lithography ... 86

4.2.1 Background ... 86

4.2.2 Process and parameters ... 87

4.3 Fabrication of the aluminum IDT ... 90

4.4 Fabrication of the aluminum wire ... 90

4.4.1 Background ... 90

4.4.2 Process and parameters ... 92

4.5 Deposition of SiO2 film ... 93

4.6 Revealing of the contact pad ... 94

4.7 Fabrication of the PC structure ... 96

**Chapter 5
Experiment Results ... 131
**

5.1 Transmission of PCs ... 131

5.1.1 SFIT without PC —verification of SFIT designs ... 131

5.1.2 SFIT with PC ... 133

5.2 One-port resonator ... 133

**Chapter 6
Conclusions and Future Works ... 150
**

6.1 Conclusions ... 150

6.2 Future works ... 151

**Appendix Determination of the Elastic Stiffness of the PECVD SiO**

**2**

**...152 **

REFERENCE ... 163

**LIST OF FIGURES **

Fig. 2.1 The specimen axes. ... 48

Fig. 2.2 Euler angles of the three rotations of the coordinate system, where (X, Y, Z)
*stand for the crystal axes while (x, y, z) stand for the specimen axes. ... 48
*

*Fig. 2.3 ST-cut and 90°-z-rotated ST cut. ... 49
*

Fig 2.4 The bases of the direct and reciprocal lattices and the unit cells of a 2-D lattice.50
Fig 2.5 The Wigner-Seitz cell and the 1^{st} Brillouin zone of a 2-D square lattice. ... 50

*Fig 2.6 Love wave dispersion (thick red lines) obtained by a thickness-fixed (10h) plate *
model (left) and a thickness-varying (1.6 l) model (right). The blue dashed line
and the purple dot-dashed line indicate the SH bulk wave of the substrate (90ST
quartz) and the S wave velocity of the guiding layer (CVD SiO2), respectively.51
Fig 2.7 Determination of the SH component. ... 52

*Fig 2.8 Slowness of the bulk waves in 90°-rotated ST-cut quartz along its xy-plane ... 52
*

Fig 2.9 Love modes selecting ... 53

Fig 2.10 (Quasi-) Love waves’ dispersions along 0° (upper), 45° (mid) and 90° (lower)
*off the x-axis. ... 54
*

Fig 2.11 Electromechanical coupling coefficient ... 55

Fig 2.12 Penetration depth of the Love waves ... 55

Fig 2.13 DOE of the Love waves ... 55

Fig 2.14 Love wave dispersion of the SiO2/IDT/quartz structure, with SiO2 thickness = 0.9 µm, IDT aluminum thickness = 100nm, IDT pitch = 3 µm. ... 56

Fig 2.15 Two resonant modes on the lower (a) and the upper (b) edge of the stop band.56 Fig 2.16 The admittance of the IDT consists of infinite pairs around the two modes. ... 56

Fig 2.17 The relation of the excited frequency vs. IDT pitch of the SiO2/IDT/quartz, with SiO2 thickness = 0.9 m m, IDT aluminum thickness = 100nm, IDT pitch =

3 µm. ... 57

Fig 2.18 A schematic drawing of the phononic SiO2/quartz layered structure ... 58

Fig 2.19 The unit cell of the phononic SiO2/quartz layered structure ... 58

Fig 2.20 The sound lines ... 59

*Fig 2.21 (unfiltered) calculated band structures of phononic crystals with (a) r/a = 0.1, *
*(b) r/a = 0.2, (c) r/a = 0.4. ... 60
*

*Fig 2.22 the color marked band structures of phononic crystals with (a) r/a = 0.1, (b) r/a *
*= 0.2, (c) r/a = 0.4. ... 61
*

Fig 2.23 (a) Dispersion of a homogeneous medium. (b) Dispersion of a periodic structure with vanishingly small modulation (empty lattice). (c) Dispersion of a periodic structure, mode coupling happens when two modes are close and not orthogonal. ... 62

*Fig 2.24 Unfiltered dispersion data of PCs with r/a = 0.1 (left) and 0.2 (right). ... 63
*

*Fig 2.25 Colorized dispersion data of PCs with r/a = 0.1 (left) and 0.2 (right). ... 63
*

*Fig 2.26 The complex band structure of the PC with r/a = 0.2. ... 64
*

Fig 3.1 A schematic drawing of the layout of the one-port resonator ... 73

Fig 3.2 Geometry of the FEM simulation investigating PC reflection ... 73

Fig 3.3 interfered displacement amplitude ... 74

Fig 3.4 A schematic drawing shows the reflection at the effective plane ... 74

*Fig 3.5 Interfered amplitude vs. d/wavelength with different a. ... 75
*

Fig 3.6 interfered displacement amplitude at different depth ... 75 Fig 3.7 Reflection simulation on 3 rows of PC. (a), (c): destructive interference. (b), (d):

*constructive interference. (a), (b): colors indicate the y-displacement in linear *

*scale. (c), (d): colors indicate the y-displacement in log scale. ... 76
*

Fig 3.8 Reflection simulation on 5 rows of PC. (a), (c): destructive interference. (b), (d):
*constructive interference. (a), (b): colors indicate the y-displacement in linear *
*scale. (c), (d): colors indicate the y-displacement in log scale. ... 76
*

Fig 3.9 Reflection simulation on 5 rows of PC. (a), (c): destructive interference. (b), (d):
*constructive interference. (a), (b): colors indicate the y-displacement in linear *
*scale. (c), (d): colors indicate the y-displacement in log scale. ... 77
*

Fig 3.10 Geometry of the FEM simulation ... 78

Fig 3.11 interfered displacement amplitude (with mechanical loading effect of IDT) ... 78

Fig 3.12 Geometry of the FEM simulation (resonator) ... 79

Fig 3.13 Resonant displacement amplitude (with mechanical loading effect of IDT) .... 79

Fig 3.14 Simulated admittance of the resonator ... 80

Fig 3.15 Transverse displacement amplitude at resonance ... 80

Fig 3.16 Mechanical power flux at resonance ... 81

Fig 3.17 Mechanical power flux on the model boundary at resonance ... 81

Fig 3.18 Experiment setup of PC transmission experiment. ... 82

Fig. 4.1 A schematic picture of the device (not to scale) ... 99

Fig. 4.2 A section view of the device (not to scale) ... 100

Fig. 4.3 The layout design ... 101

Fig. 4.4 Fabrications of the alignment mark ... 102

Fig. 4.5 Fabrications of the IDT ... 103

Fig. 4.6 Fabrications of the wire (i) ... 104

Fig. 4.7 Fabrications of the wire (ii) ... 105

Fig. 4.8 Fabrications of the contact pad window (i) ... 106

Fig. 4.9 Fabrications of the contact pad window (ii) ... 107

Fig. 4.10 Fabrications of the PC structure ... 108

Fig. 4.11 A schematic drawing of the electron charging effect ... 109

Fig. 4.12 The writing strategy of a Gaussian vector scan lithography system [56] ... 110

Fig. 4.13 Back-scattering electrons expose the resist as well. [56] ... 111

Fig. 4.14 Simulation of the proximity effect ... 112

Fig. 4.15 Developed patterns without (upper) and with (lower) coating of ESpacer .... 113

Fig. 4.16 Circle size changing due to proximity effect. ... 114

Fig. 4.17 A wafer mark after fabrication of IDT (Al remains on scanned area) ... 114

Fig. 4.18 The chip mark becomes unusable. ... 115

Fig. 4.19 SEM images of a developed alignment mark pattern ... 116

Fig. 4.20 SEM images of a developed IDT pattern ... 117

Fig. 4.21 Microscope pictures of fabricated IDTs ... 118

Fig. 4.22 A variety of Au-Al intermetallic ... 119

Fig. 4.23 A Pt/Al contact pad was damaged during buffered HF wet etching ... 120

Fig. 4.24 Problems in wet etching ... 121

Fig. 4.25 Development is carried out in a 23°C water bath. ... 122

Fig. 4.26 The EB evaporator ... 123

Fig. 4.27 The strategy of scanning a mark (upper) and the detected signals (lower) .... 124

Fig. 4.28 The P-TEOS CVD system ... 125

Fig. 4.29 The ellipsometer ... 126

Fig. 4.30 The RIE system ... 127

Fig. 4.31 PC patterns being written by the EB lithography system ... 128

Fig. 4.32 a SEM image of the PC structure ... 128

Fig. 4.33 more SEM images of the PC structure ... 129

*Fig. 4.34 Complex index of refraction (𝑛𝑛 + iκ) of P-TEOS SiO*2 vs. wavelength ... 130

Fig 5.1 VNA Agilent E5071C ... 136

Fig 5.2 Cascade Microtech probes with the platform ... 136

Fig 5.3 Impulse response of SFIT #2 on ST-cut quartz. ... 137

Fig 5.4 Original and gated signal in frequency domain. ... 137

Fig 5.5 Impulse response of SFIT #2 on 90°-rotated ST-cut quartz. ... 138

Fig 5.6 Original and gated signal of SFIT/90°-rotated ST-cut quartz in frequency domain. ... 138

Fig 5.7 Impulse response of SiO2/SFIT #1/90°-rotated ST-cut quartz ... 139

Fig 5.8 Original and gated signal of SiO2/SFIT #1/90°-rotated ST-cut quartz in frequency domain. ... 139

Fig 5.9 Impulse response of SiO2/SFIT #2/90°-rotated ST-cut quartz ... 140

Fig 5.10 Original and gated signal of SiO2/SFIT #2/90°-rotated ST-cut quartz in frequency domain. ... 140

Fig 5.11 Impulse response of SFIT #1 with 10 rows of PC ... 141

Fig 5.12 Frequency response of SFIT #1 with 10 rows of PC ... 141

Fig 5.13 Impulse response of SFIT #2 with 10 rows of PC ... 142

Fig 5.14 Frequency response of SFIT #2 with 10 rows of PC ... 142

Fig 5.15 Impulse response of SFIT #1 with 15 rows of PC ... 143

Fig 5.16 Frequency response of SFIT #1 with 15 rows of PC ... 143

Fig 5.17 Impulse response of SFIT #2 with 15 rows of PC ... 144

Fig 5.18 Frequency response of SFIT #2 with 15 rows of PC ... 144

Fig 5.19 Impulse response of SFIT #1 with 20 rows of PC ... 145

Fig 5.20 Frequency response of SFIT #1 with 20 rows of PC ... 145

Fig 5.21 Impulse response of SFIT #2 with 20 rows of PC ... 146

Fig 5.22 Frequency response of SFIT #2 with 20 rows of PC ... 146

Fig 5.23 Admittance signals of resonators containing 10.5 IDT pairs with (a) 10, (b) 15, (c) 20 rows PC as reflectors ... 147 Fig 5.24 Admittance signals of resonators containing 30.5 IDT pairs with (a) 10, (b) 15, (c) 20 rows PC as reflectors ... 148 Fig 5.25 Admittance signals of resonators containing 30.5 IDT pairs with (a) 10, (b) 15, (c) 20 rows PC as reflectors ... 149

**LIST OF TABLES **

**Table 2.1 Material properties of a-quartz w.r.t. crystal axes ... 44
**

**Table 2.2 Material properties of ST-cut quartz w.r.t. specimen axes ... 45
**

**Table 2.3 Material Properties of 90°-rotated ST-cut quartz w.r.t. specimen axes ... 46
**

**Table 2.4 Material properties of the PECVD SiO**

2 and aluminum. ... 47
**Table 3.1 SFIT parameters. ... 72
**

**Table 3.2 IDTs for resonators. ... 72
**

**Table 4.1 Dependence of required dose on some factors ... 98
**

**Chapter 1 ** **Introduction **

**1.1 Motivation **

A phononic crystal (PC) is a periodic elastic structure comprised of two or more elastic materials which differ in mechanical properties, or, with periodic geometry. And the resultant acoustic dispersion exhibits a band nature analogous to the electronic band structure in the solid. Examples of one-dimensional (1-D) PCs are the metal strip or grooved grating structures that are widely used in surface acoustic wave (SAW) devices.

2-D PCs have attracted great attentions in the past two decades. A lot of concern has been focused on their acoustic reflecting phenomenon in the frequency band gap. How- ever the PC can also be an acoustic conductor, which its conductance is determined by the band structure. Novel properties such as negative refraction or acoustic lensing ef- fect can be achieved with PCs [1]-[5].

Among the many applications, in the area of ultra-high frequency (UHF) acoustic wave devices, complete or partial band gaps of air/silicon PC were utilized as the reflec- tive gratings to reduce the devices’ size [7]-[10]. In those devices, a piezoelectric thin film has to be deposited on the silicon substrate to generate acoustic waves for silicon is

non-piezoelectric. Although the fabrication process of the silicon based phononic device has the CMOS compatible advantage, the insertion loss of such a device is high rela- tively. The other more straightforward way of making a phononic acoustic wave device is to construct periodically micro-holes directly on a piezoelectric substrate [11], [12].

Although the electro-acoustic conversion is higher than that of the piezoelectric film type device mentioned previously, it suffered from the anisotropic nature of lithium ni- obate that lead to a complicated fabrication process.

On the other hand, SH-type SAWs have several advantages compared to conven- tional Rayleigh-type SAWs. For example sometimes SH-type SAWs possess larger pi- ezoelectricity than Rayleigh-type SAWs do on the same substrate material. Also they are faster than Rayleigh-type SAWs therefore desirable for high frequency applications.

And in liquids, SH-type SAWs lose less energy than Rayleigh-type SAWs do due to their polarization, also they are sensitive to surface loadings, so they are desirable for (bio-) sensing applications. Love wave, being one of the SH-type SAWs, shares similar physical characteristics with other SH-type SAWs. Investigating the propagating of Love waves in PCs will expand the PC applications to SH-type SAWs that have bright outlook.

In this study, we investigate Love waves propagating in a piezoelectric substrate coated with a phononic guiding layer. The phononic layer is consisted of a thin layer with periodic machined holes. It is worth noting that the thin phononic layer can be amorphous material which makes the fabrication process relatively simple. And since most energy is trapped in the guiding layer it may serve as efficient reflective gratings in Love wave devices. The method proposed in this study is suitable for other SH-type SAWs also.

**1.2 Literature Review **

In 1911, A. E. H. Love published his treatise about a study of shear surface wave that particles move perpendicular to the sagittal plane of wave propagation [13]. This wave, which is called Love wave, exists when the half space is covered with a guiding layer of material with lower bulk shear wave velocity. Although pure SH-type SAWs do not exist on the half-space of an ordinary solid, in the late 1960s a pure SH-SAW was discovered to exist on some piezoelectric solids and is called the

“Bleustein-Gulyaev-Shimizu wave (BGS wave)” [14], [15]. A similar but not pure SH-type “leaky SAW (LSAW),” exist in some piezoelectric solids. These SH-type SAWs are being widely used for practical SAW devices for their advantages mentioned before [16]. In 1977, Lewis demonstrated that the surface skimming bulk wave (SSBW) can exist on some piezoelectric substrate and can be used as an alternative to a SAW for a device [17]. And it is reported that the SSBW could be transformed to a “Love wave”

by depositing thick electrodes on the substrate.

The research of PCs launched at the beginning of 1990s [18]. The studies of acous- tic wave propagation in PCs can be classified according to the bulk acoustic wave [19]-[21], surface acoustic wave [22]-[25], and Lamb wave [26]-[29]. To analyze band structures of PCs, there are several methods exist such as the plane wave expansion (PWE) method [19], the finite difference time domain (FDTD) method [28], multiple scattering theory (MST) [30] and the finite element method (FEM).

*As for SH-type SAWs in 1-D periodic structure, Auld et al. first proposed the for-*
mation of periodic corrugations on substrate surface of quartz to trap SSBW or LSAW
and is called the “surface transverse wave (STW).” [31] The stop band of the structure
was also studied. As well as corrugations, heavy or thick metal strips can also trap the

SSBW or LSAW with similar slowing mechanism. It is sometimes called STW also [32], or Love wave [33], or grating modes [34], depends on the researchers’ preferences.

There are also structures proposed with a guiding layer covering the gratings on the substrate [35]. A lot of application devices were made with these kind of SH-type SAWs and most of them are resonators or resonator based filters [33]-[37] since the waves concentrate near the surface only with the periodic structures, and are likely res- onant in the periodic structure for the strong periodic modulation.

In this study we investigate the ordinary Love wave that is formed by SSBW trapped by a slower guiding layer deposited on the substrate. And we study the phono- nic crystal that is consisted of a 2-D periodic structure (micro-machined hole array) on the guiding layer. Both numerical and experimental investigations were conducted. The partial band gap was calculated and verified. We further utilized the reflection phe- nomenon to design and fabricate one-port Love wave resonators.

**1.3 Contents of the Chapters **

There are six chapters in this thesis. Chapter 1 is the introduction, with the research motivation and literature review. In Chapter 2 we show the method to calculate propa- gating eigenmodes in piezoelectric substrate covered with a phononic guiding layer, with a detailed discussion of their band structures. Chapter 3 is the design method of the one-port Love wave resonator. Also the physics of Love wave incident to the PC is studied. Chapter 4 is the fabrications of the experiment devices. The process of how the devices were made are shown in detail. Chapter 5 shows the experiment results with discussion. And Chapter 6 is the conclusions and future works.

**Chapter 2 **

**Love Wave Dispersions of a Phononic ** **SiO** _{2} **/Quartz Layered Structure **

_{2}

In this chapter, the numerical methods for calculating the Love wave dispersions in three different structures are proposed. They are (1) CVD SiO2/quartz layered piezoe- lectric half space, (2) CVD SiO2/aluminum electrodes grating/quartz structure, with consideration of the electrodes’ mechanical loading effect, which is a 1-D periodic structure, and (3) phononic SiO2/quartz, a 2-D periodic structure.

First the governing equations of the “quasi-static” approximated model of piezoe- lectric solid are derived, and then the theory of waves in periodic structure is introduced.

After we have the governing equations and boundary conditions the finite element method is employed to solve the differential equations to find the dispersion relation of the 3 structures.

**2.1 The Mathematical Model of Acoustic Waves in Piezo-** **electric solids **

### 2.1.1 Piezoelectric acoustic wave equations

The mechanical system and the electrical system of the piezoelectric solid are cou-
pled together in their constitutive relations. Meanwhile, Newton’s 2^{nd} law of motion

* *∇ ⋅

**T + X = ρ ∂**

^{2}

**u**

*∂t*^{2} , (2.1)

and the Maxwell equations

** ∇⋅D = ρ**^{f} (2.2)

** ∇ ⋅B = 0** (2.3)

* *∇ ×

**E = − ∂B**

*∂t* (2.4)

* *∇ ×

**H = J**

_{f}

**+ ∂D**

*∂t* (2.5)

**are still satisfied, where T, X, and u are the Cauchy stress tensor, the body force density **
**vector, and the displacement vector, respectively, while D, B, E, H,**_{ ρ}_{f},and_{ J}_{f}are the
electric displacement field, the magnetic flux density, the electric field, the magnetic
field strength, the free charge density, and the free current density, respectively. To
simplify the mathematical model, some fair assumptions are made as the following.

Body forces such as gravity are ignored (** _{ X = 0}**). The piezoelectric material is as-
sumed to be dielectric, i.e., no free charge or free current exist (

_{ ρ}_{f}

**= 0, J**

_{f}= 0). And the

“quasi-static” approximation is introduced, that is, the electric field and the magnetic field are assumed to be static (

** **

**∂B**

*∂t* **= 0, ∂D**

*∂t* = 0). For the reason that we are focusing on
the “quasi-acoustic waves,” and the frequencies of the “quasi-electromagnetic waves”

are much higher under the same wavenumber so that the electromagnetic field can be assumed to be static. Therefore the electric field and the magnetic field become decou- pled; moreover, the magnetic field is decoupled from the piezoelectric system. Thus the governing equations of the piezoelectric solid are reduced to the following two equa- tions (with indicial notation)

* **T** _{ij, j}*=

*ρ!!u*

*, (2.6)*

_{i}* **D** _{i,i}*= 0, (2.7)

with the constitutive equations which couple the two systems together:

*T** _{ij}*=

*c*

_{ijkl}

^{E}*S*

*−*

_{kl}*e*

_{kij}*E*

*, (2.8)*

_{k}*D*

*=*

_{i}*e*

_{ikl}*S*

*+ε*

_{kl}

_{ij}

^{S}*E*

*, (2.9)*

_{j}where *S** _{kl}* is the strain tensor satisfying

* **S** _{kl}*= 1

2*(u** _{k,l}*+

*u*

*), while*

_{l,k}

_{ }*c*

_{ijkl}*,*

^{E}* *ε_{ij}* ^{S}*, and

*e*

*represent the elastic stiffness tensor under a constant electric field, the permittivity ten- sor with a constant strain, and the piezoelectric coupling tensor, respectively. Substitut- ing equations (2.8) and (2.9) into equations (2.6) and (2.7), with the electric field-potential relation*

_{kij}

_{ }*E*

*= −φ*

_{i}*, and the strain-displacement relation*

_{,i}* **S** _{kl}*= 1

2*(u** _{k,l}*+

*u*

*), we obtain the governing equations of piezoelectricity with variables*

_{l,k}*u and φ, *

* *

### (

*c*

_{ijkl}

^{E}*u*

*+*

_{k,l}*e*

*φ*

_{kij}

_{,k}### )

_{, j}^{=}

^{ρ!!u}

^{i}^{,}

^{(2.10) }

* *

### (

*e*

_{ikl}*u*

*−ε*

_{k,l}

_{ij}*φ*

^{S}

_{, j}### )

_{,k}^{= 0.}

^{(2.11) }

In the case of time harmonic motion (when calculating propagating eigenmodes or simulating the frequency response of a device), the displacement vector and the electric

potential can be represented as

** u(x, y,z,t) = ˆu(x, y,z)e**^{i}^{ω}* ^{t}*, (2.12)

*φ(x, y,z,t) =*

*ˆφ(x, y,z)e*

^{i}^{ω}

*, (2.13) where ω is the angular frequency. Hence equations (2.10) and (2.11) can be rewritten as*

^{t}* *

### (

*c*

_{ijkl}

^{E}*ˆu*

*+*

_{k,l}*e*

*ˆφ*

_{kij}

_{,k}### )

*, j*= −ρω

^{2}

*ˆu*

*, (2.14)*

_{i}* *

### (

*e*

_{ikl}*ˆu*

*−ε*

_{k,l}

_{ij}*ˆφ*

^{S}

_{, j}### )

_{,k}^{= 0}

^{. }

^{(2.15) }

*With proper boundary conditions (BCs), the solutions of u and φ can be found. In this *
thesis, the rest jobs of solving the differential equations are leaved to a commercial fi-
nite element method (FEM) software COMSOL Multiphysics.

### 2.1.2 Coordinate transformation for material constants

All piezoelectric materials are anisotropic; their properties depend on the frame of
reference, i.e., the crystal cut and the orientation of wave propagation, so they should be
first correctly obtained. Usually the material properties with respect to (w.r.t.) the crys-
tal axes (denoted by X, Y, Z, defined by crystal symmetry) are found in books. As for
some commonly used material with specific crystal cuts such as ST-cut quartz,
128°-YX lithium niobate, the material properties can also be found in literatures. How-
ever with different definitions of the specimen axes, they are easily confused and mis-
*used. In this thesis, the specimen axes (x, y, z) are defined as follows. The axis normal *
*to the substrate’s top surface is defined as the z-axis, with the wave propagating direc-*
*tion defined as the x-axis, the y-axis is then obtained by the right-hand rule, as shown in *
Fig. 2.1. According to Euler’s rotation theorem, any rotation can be decomposed into
three successive rotations with three angles (Φ, Θ, Ψ), w.r.t. the axes in a certain order.

However the decomposition is not unique and is order-dependent. In the thesis the

Z-X’-Z” convention is adopted, that is, the specimen axes is rotated from the crystal ax-
*is first with an angle Φ w.r.t. the Z-axis, then rotated with an angle Θ w.r.t. the current *

*x-axis (the X’-axis), and finally rotated with an angle Ψ w.r.t. the current z-axis (the * *Z”-axis). Namely, a rotation rule (ZXtlt) Φ/Θ/Ψ following IEEE standard on piezoelec-*

tricity [39], is applied, with the plate length direction coincident with the wave propa-
gating orientation.
*Let V*i* and V’*j** denote the components of a vector V w.r.t. the crystal coordinates **
*and the specimen coordinates respectively, there exist a transformation rule between V*i

*and V’*j,

*V** _{i}*′=β

_{ij}*V*

*, (2.16)*

_{j}where β*ij* can be expressed in the matrix form as

* *
β_{ij}

### ( )

^{=}cosΨ sinΨ 0

−sinΨ cosΨ 0

0 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

1 0 0

0 cosΘ sinΘ 0 −sinΘ cosΘ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

cosΦ sinΦ 0

−sinΦ cosΦ 0

0 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ . (2.17)

Similarly, the permittivity, the piezoelectric coupling tensor, and the elastic stiffness tensor, which are tensors of order 2, 3, and 4, respectively, have the transformation rules as

* *

ε* _{ij}*′ =β

*β*

_{ik}*ε*

_{jl}*,*

_{kl}*e*

*′ =β*

_{ijk}*β*

_{ip}*β*

_{jq}

_{kr}*e*

*,*

_{pqr}*c*

*′ =β*

_{ijkl}*β*

_{ip}*β*

_{jq}*β*

_{kr}

_{ls}*c*

*.*

_{pqrs}(2.18)

### 2.1.3 Materials

To support Love wave propagation, it is important choosing the material combina- tion of the layered half-space structure. First the SH wave velocity in a virtual un- bounded medium of the layer material must be slower than that of the substrate material, for it is the condition of existence of Love waves. [40], [41] Second, the substrate mate-

rial (with specific crystal cut) should have strong piezoelectric coupling to SH type
wave (SH bulk wave, SSBW, SH LSAW, or BGS wave), but with zero piezoelectric
coupling to Rayleigh modes or other (P or SV) bulk wave modes to prevent the spurious
response. Several crystal cuts of lithium niobate (LN, LiNbO3), lithium tantalate (LT,
LiTaO3) and quartz are found desirable for such requirements. [42] Eventually,
*90°-z-rotated ST-cut quartz (henceforth 90ST quartz) was chosen for there is a fast *
(5000m/s) SSBW on it and ST-cut quartz is commercially available with a lower price
than LT or LN. But there is a disadvantage that is the poor piezoelectric coupling of
quartz. On the other hand, the layer material was chosen to be amorphous silicon diox-
ide (SiO2, silica henceforth), which is the most frequently used dielectric material in in-
tegrated circuit (IC) manufacturing hence the deposition and etching techniques for sil-
ica are well developed. In this study the silica was deposited by plasma enhanced
chemical vapor deposition (PECVD) method. Its shear wave velocity is estimated to be
3438 m/s.

The elastic constants of quartz used in this study were from [43], with the kind
suggestion from Prof. Yook-Kong Yong. The properties of quartz with respect to its
**crystal axes are listed in Table 2.1. The ST-cut quartz, in words of wafer manufacturers, **
is a singly rotated (42°45’ along X-axis) Y-cut quartz and the corresponding Euler an-
gles (defined in the previous subsection) are_{ Φ,Θ,Ψ}

### ( )

= 0,132°45', 0### ( )

. As for the*“90°-z-rotated ST-cut quartz,” it is physically no difference from a ST-cut quartz sub-*
*strate, but with the wave propagation direction parallel to the y-axis of ST-cut quartz, *
*and perpendicular to the X-axis of quartz (or the x-axis of ST-cut quartz). So actually *
we demand ST-cut quartz wafer from the supplier, and make a 90°-rotated layout. The
Euler angles of 90ST quartz are

### (

Φ Θ Ψ =^{, ,}

### ) (

0,132 45', 90° °### )

. The properties of ST-cut**quartz and 90°-rotated ST-cut quartz w.r.t. their specimen axes are detailed in Table 2.2 **
**and Table 2.3, respectively. It can be seen that the difference between them is that the **
*components related to axes x or y are exchanged. ST-cut quartz is usually used as a *
Rayleigh wave device, while on a 90ST quartz SH type SSBW can be generated but
Rayleigh wave is decoupled with the applied electric field from the interdigital trans-
ducer (IDT). A schematic drawing is shown in Fig. 2.3.

The amorphous silica layer in this study is deposited by a plasma-enhanced chem-
ical vapor deposition (PECVD) method, which is described in Chapter 4. The elastic
**properties are listed in Table 2.4, where the elastic stiffness constants were obtained by **
a SAW method. By measuring the Love wave and Rayleigh wave velocities on the sili-
ca film/quartz substrate, the two elastic constants can be obtained. The method is de-
tailed in the Appendix . It shows a significant difference from the bulk SiO2 (fused
quartz).

**2.2 Waves in periodic structures **

### 2.2.1 Direct and reciprocal lattices

* Considering a 1-D, 2-D or 3D lattice, a basis {a}, {a*1

**, a**2

**} or {a**1

**, a**2

**, a**3} can be chosen by selecting a lattice point as the origin, and the nearest points define the basis vector, such that the vector coordinate of any lattice points can be given by linear com- binations of the basis with integer coefficients. Note that the basis is not unique. A 2-D

**case is shown in Fig 2.4, where {a**1

**, a**2

**} or {a’**1

**, a’**2} both can be the basis but here the

**grid lines are drawn w.r.t {a**1

**, a**2}.

A “primitive unit cell” can be used to represent the lattice, as it is the smallest per- iod of the lattice. Various primitive unit cells can be chosen but the Wigner-Seitz cell is

the most common and maybe the most useful unit cell. The Wigner-Seitz cell can be chosen by constructing a domain (an interval, a polygon, or a polyhedron stand for 1-D, 2-D or 3-D lattices, respectively) about a lattice point. The boundaries (end points, bor- derlines or faces) of the cell are points, lines of planes that are perpendicular bisectors of the lines joining the origin with neighboring lattice points [44]. A Wigner-Seitz cell in the 2-D case is shown schematically in Fig 2.4.

* Until here the discussion is about the lattice defined by {a}, {a*1

**, a**2

**} or {a**1

**, a**2

**, a**3}

*basis. We refer to it as the direct lattice. For each lattice we may define a reciprocal lat-*

**tice that has basis {b}, {b**

1**tice that has basis {b}, {b**

**, b**2

**} or {b**1

**, b**2

**, b**3} given by the equation

** ****a*** _{i}*⋅

**b**

*= 2πδ*

_{j}*, (2.19) where δ*

_{ij}*ij*is the Kronecker delta symbol, defined by

* *

δ* _{ij}*=

*1, i = j*

*0, i ≠ j*

⎧⎨

⎪

⎩⎪

⎫⎬

⎪

⎭⎪. (2.20)

**In other words, b**1** is perpendicular to a**1** but has reciprocal components on a**2**, a**3 (with a
scaling factor (2π)^{−1}). To find the basis of the reciprocal lattice, we first build two 2×2
**(3×3) matrices containing the Cartesian components of the bases {a**1**, a**2**} and {b**1**, b**2}
**({a**1**, a**2**, a**3**} and {b**1**, b**2**, b**3}), as, in the 2-D case,

** **

**A =** **a**_{1}^{T}
**a**_{2}^{T}

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= **a**_{1x}**a**_{1y}**a**_{2x}**a**_{2 y}

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟,

**B = b**

### (

_{1}

**b**

_{2}

### )

^{=}

^{⎛}

_{⎝}

^{⎜}

_{⎜}

_{b}^{b}

^{1x}

_{1y}

^{b}_{b}

^{2x}

_{2 y}^{⎞}

_{⎠}

^{⎟}

_{⎟}

^{,}

^{ }

^{(2.21) }

or in the 3-D case,

** ****A =**

**a**_{1}^{T}
**a**_{2}^{T}
**a**_{3}^{T}

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

=

**a**_{1x}**a**_{1y}**a**_{1z}**a**_{2x}**a**_{2 y}**a**_{2z}**a**_{3x}**a**_{3y}**a**_{3z}

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟ ,

**B = b**

### (

_{1}

**b**

_{2}

**b**

_{3}

### )

^{=}

^{b}^{b}

^{1y}

^{1x}

^{b}^{b}

^{2 y}

^{2x}

^{b}^{b}

^{3x}

^{3y}**b**_{1z}**b**_{2z}**b**_{3z}

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟ .

(2.22)

So equation (2.19) can be rewritten as

** A ⋅B = 2πI,** (2.23)
**Where I is the identity matrix, therefore it follows that **

** B = 2πA**^{−1}. (2.24)
Hence the basis of the reciprocal lattice is obtained from a given direct lattice. And the
reciprocal lattice in 1-D case can be simply represented as

* **b = 2π*
*a* .

**The wave vector (k) for a wave propagating in the direct lattice is drawn in the re-**
**ciprocal lattice. The direct lattice gives the periodicity of the medium (x-space), while **
the reciprocal lattice gives the periodicity of the frequency of the waves propagating
**through the medium (k-space) [44]. **

**Since the reciprocal lattice shows the periodicity in k-space, just like the direct lat-**
**tice in x-space, a primitive unit cell can be found to describe the whole reciprocal lattice. **

That is the first Brillouin zone, obtained by constructing a Wigner-Seitz zone in the re- ciprocal lattice. It is worth noting that discontinuous frequency spectrum often occurs on the boundary of the first Brillouin zone that can be realized as Bragg’s reflection condition

* k ⋅a = nπ,* (2.25)

*where n is an integer. Or the level repulsion due to coupling with diffracted waves.*

Fig 2.5 shows the direct lattice and the reciprocal lattice of a 2-D square lattice, as

well as the Wigner-Seitz cell of the direct lattice and the first Brillouin zone in the re-
ciprocal lattice. Some special points Γ, X, M, and Y, on the boundary of the Brillouin
*zone are labeled. And the irreducible Brillouin zone is the first Brillouin zone reduced *
by all of the symmetries of the lattice. The irreducible Brillouin zone of a square lattice
consists of points in the space, is the triangular gridded by the three points Γ, X, and M.

But if we consider lattices containing anisotropic materials, the irreducible Brillouin zones may be different. In this study the lattice is constructed on a 90ST quartz sub- strate, the

### ( )

^{1 1}

^{ and }

^{( )}

^{11}rows of the square lattice are no longer lines of mirror symmetry while rows

### ( )

^{10}

^{ and }

### ( )

^{01}still remain, therefore the irreducible Brillouin zone should be the square surrounded by the four points Γ, X, M, and Y.

### 2.2.2 Bloch’s theorem

The special form of the solutions of differential equations with periodic coeffi-
cients is given by Bloch’s theorem. It dates back to 1883, Floquet [45] first showed that
*the solutions of a 1-variable n × n system of linear differential equations with the co-*
*efficients being a piecewise continuous periodic function with period a *

** **

′ ψ

ψ

### ( )

*x*

^{=}

^{A x}### ( )

^{ψ}

^{ψ}

### ( )

^{x}

^{, x ∈R}**A x + a**

### ( )

^{=}

^{A x}### ( )

^{,}

⎧⎨

⎪

⎩⎪ (2.26)

have the form

* *
ψ

ψ_{k}

### ( )

*x*

^{=}

^{e}

^{ikx}^{ηη}

*k*

### ( )

*x*

^{,}ηη

_{k}### ( )

*x + a*

^{= ηη}

*k*

### ( )

*x*

^{,}

⎧⎨

⎪

⎩⎪

(2.27)

where ηη*k** is a periodic function with period a. This result can be extended and applied to *
higher order differential equations by reducing the equation(s) to first order equations
with increasing the dimension of the system. It is known as Floquet’s theorem. And in

1929, Bloch generalized the theorem for three dimensions in his work on the electron in
a crystal [46]. We adopt his result and we have that the eigenmodes of a wave equation
in a periodic medium (1-D, 2-D, or 3-D) may be written as the product of a plane wave
function _{ e}*^{ik⋅x}* and a periodic function

_{ ηη x}### ( )

that has the same periodicity as the me- dium, that is,* *
ψ

ψ_{k}

### ( )

**x**

^{=}

^{e}

^{ik⋅x}^{ηη}

**k**

### ( )

**x**

^{,}ηη

_{k}### (

**x + T**

### )

^{= ηη}

**k**

### ( )

**x**

^{,}

⎧

⎨⎪

⎩⎪ (2.28)

**where T is the translation vector of the crystal that is a linear combination of the basis **
**vectors of the direct lattice with integer coefficients. We can also define G as the trans-**
lation vector in the reciprocal lattice.

Applying the Bloch’s theorem we obtain the boundary conditions of a Wig-
**ner-Seitz cell. Then we search the eigenmodes with wave vectors k’s located in the ir-**
reducible Brillouin zone, than the Bloch waves can be completely characterized. We
**make the dispersion relation (the ω-k relation, where ω is the eigenfrequency) plotted **
**with k varying along the boundary of the irreducible Brillouin zone, which is call the **

*band structure, adopted from the terminology of solid state physics. The frequency *

*bands that contain no propagating modes in the band structure are called the band gaps,*in which no wave can propagate in the periodic structure.

**2.3 Love wave dispersion of a layered half-space **

In this section we conduct the calculation of the acoustic dispersion relation of the layered structure comprises a silica film deposited on a 90ST quartz substrate. It was accomplished by applying the Bloch’s theorem and the finite element method (FEM).

And a proper thickness of the silica film is proposed for the experiment devices.

### 2.3.1 Numerical methods

We are considering the Love wave propagating on the layered structure that is ho-
*mogeneous along the x-y plane. But if we treat it as a periodic structure with “vanish-*
ingly small variation,” the Bloch’s theorem can be applied to calculate the dispersions.

Since the structure is actually homogeneous, the choice of the unit cell size can be
*whatever we like. Let the cell have a length a on the x-direction (i.e., have a “period” a), *
*and a piezoelectric acoustic wave with wavenumber k is propagating through this 1-D *

“lattice,” then the Bloch wave function can be represented as

* *
ψ

ψ_{k}

### ( )

*x*

^{=}

^{e}

^{ikx}^{ηη}

*k*

### ( )

*x*

^{,}ηη

_{k}### ( )

*x + a*

^{= ηη}

*k*

### ( )

*x*

^{,}

⎧⎨

⎪

⎩⎪ (2.29)

or

* ψ*ψ_{k}

### ( )

*x + a*

^{= ψ}

^{ψ}

*k*

### ( )

*x*

^{⋅}

^{e}

^{ika}^{.}

^{ }

^{(2.30) }Where here

_{ ψ}^{ψ}

*can be the displacement field, velocity field, strain field, stress field, electric potential, electric displacement, etc. Therefore we can write down the boundary*

_{k}**conditions of this cell with the dependent variable u and φ**

** **

**u**_{+}=**u**_{−}⋅*e** ^{ika}*,
φ

_{+}=φ

_{−}⋅

*e*

*,*

^{ika}⎧⎨

⎪

⎩⎪ (2.31)

where the subscription + and – stand for the properties on the right and left boundaries
of the cell respectively. This is sometimes called Floquet periodicity. Generally a con-
tinuous condition requires constraints on both the displacement and traction (electric
potential and electric displacement) on the identity pair boundaries. But since we know
* ηη*^{k}*(x)*** in equation (2.29) for u and φ are smooth functions, we set the constraint order **
of boundary conditions (2.31) to be 2 as the quadratic Lagrangian element is used in the
FEM. Therefore the traction and electric displacement also satisfying condition (2.29)

**for they are just results of linear operation on u and φ. Thus they are automatically con-**
strained similar to boundary conditions (2.31). And with the governing equations of the
domain (2.14), (2.15), the dispersion relation can be obtain by finding the eigenfre-
*quency ω with varying the wavenumber k. *

It is expected that all the dispersion information _{ k ∈ 0, ∞}

### ( )

is completely deter-mined in the irreducible Brillouin zone that is a finite interval * k ∈ 0, π a*⎡⎣ ⎤⎦. As a result
we see the dispersion curves fold back when they touch the boundary of the irreducible
*Brillouin zone. It is a natural result for e*^{ika}* is a periodic function of k with period 2π/a, *
and there is no difference in velocity between the right- or left-propagating waves. So
*we have a dispersion that is periodic and even function of wavenumber k. We may also *
consider the boundary conditions (2.31) as “wave propagating boundary conditions”

*that a wave with wavenumber k propagating through a medium with a length a, experi-*
*ences a phase shift ka, so the wave function on the right boundary is that on the left *
*boundary being multiplied by e** ^{ika}*. Consider propagating waves with wavenumber

### (

^{2 1}

### )

2 ,or ,*n* *n*

*k* *k*

*a* *a*

π ^{−} π

+ − + * where n is an integer, the boundary conditions of them are *
all identical to (2.31) for the periodicity and of the exponential function.

We would like to postpone the folding on the boundary of the irreducible Brillouin zone to make the dispersion curve look simpler. It can be achieved by selecting a smaller unit cell, therefore the irreducible Brillouin zone, which is a unit cell in the re- ciprocal lattice, becomes larger, and the folding is postponed.

Ideally the model is a layered half-space, that is, the substrate is infinite deep
*along –z-direction. However we cannot build a model with infinite depth in FEM calcu-*
lation. Also the perfect matching layer (PML), and some absorption or low reflecting
boundaries are not suitable in the eigenfrequency analysis. Therefore instead, a fi-

**nite-depth model is used with the bottom boundary set to be fixed (u = 0) and zero **
*charged (D*z = 0). As a result, we are actually calculating the dispersion of a “plate.” But
the SAW behavior can be found on the plate as the wavelength becomes much smaller
than the plate thickness, or

*kh*

_{substrate}>> [47] Thus, if the thickness of the model’s 1.

substrate is thick enough, we can still obtain the SAW dispersion with such model. Be-
ing a surface wave, the Love wave has a displacement field in the substrate exponen-
*tially vanishing along the –z-direction as *

* *

*u*

*=*

_{y}*Ae*

^{bz}*e*

^{ik(x−c}^{Love}

*,*

^{t)}*b = k 1−* *c*

_{Love}

*c*

_{SH, 90ST}

⎛

⎝⎜ ⎞

⎠⎟

⎡ 2

⎣

⎢⎢

⎤

⎦

⎥⎥

12

.

⎧

⎨

⎪⎪

⎩

⎪⎪

(2.32)

*[40] We can see the vanishing factor b depends on both wave number k and the phase *
*velocity of the Love wave c*Love*. And the phase velocity c*Love always sets off from the
*substrate’s shear wave velocity c*SH, 90ST, and approaches to the shear wave velocity of
*the film material c*S, silica* as k increasing. A thickness-varying (with k) model and a fixed *
thickness model are employed to calculate the dispersion of the Love wave, and they
show similar results (Fig 2.6) after mode selection (described in the next paragraph).

Considering the penetration depth of Love waves is not only wavelength dependent
*(check “b” in equation (2.32)), and the thickness-varying model requires huge compu-*
*tation in the small k region which we do not really concern, so henceforth fixed thick-*
ness models with sufficient thickness are used instead of the thickness-varying models.

To pick out the modes standing for Love waves among numerous plate modes in the dispersion relation calculated by the finite-depth model, two conditions are set for mode selecting. We first define the depth of energy (DOE) as

* *
DOE =

1

2

*T*

_{ij}*S*

_{ij}^{*}

### ( )

−*z* ^{dx dy dz}

^{dx dy dz}

unit cell

## ∫∫∫

1

2

*T*

_{ij}*S*

_{ij}^{*}

*dx dy dz*

unit cell

## ∫∫∫

^{, }

^{(2.33) }

which is the depth of the strain-energy-weighted centroid of the unit cell. The star sym- bol “*” indicates the complex conjugate value of the calculated mode shape. Since the stress and strain are always in phase, the amplitude of the strain energy can be expressed

as ^{1} ^{1} ^{*}.

2*T S** ^{ij ij}* =2

*T S*

*By setting a threshold of DOE (for example 0.2λ), the SAW modes whose energy concentrate near the free surface filters through. The selected SAW modes contain generalized Rayleigh waves (particle motion parallel to the sagittal plane), Love waves (particle motion perpendicular to the sagittal plane), and modes lie somewhere in between. We make a threshold of polarization to execute a dichotomy between them. First the displacement component normal to the sagittal plane (i.e. the*

^{ij ij}*shear horizontal component) of a wave with the propagation vector lies on the xy-plane*can be expressed as

SH * _{x}*sin

*cos*

_{y}*u*

= −*u* θ

+*u* θ

(2.34)
*where θ represents the angle between the wave vector and the x-axis. (See Fig 2.8.) *
Then we define a value called ratio of transverse polarization (ROT) as

SH SH* unit cell

* * *

unit cell

ROT .

*x x* *y y* *z z*

*u u dx dy dz* *u u u u u u dx dy dz*

= + +

## ∫∫∫

## ∫∫∫

^{ }

^{(2.35) }

It is a ratio of the norm of the component perpendicular to the sagittal plane, to the total displacement amplitude. When the value is greater than 0.5 it means the perpendicu- lar-to-the-sagittal-plane component is greater than the parallel one, and is classified as the (quasi-) Love wave. On the contrary if the ROT is less than 0.5, the mode will be

classified as the (quasi-) Rayleigh wave.

*From the quasi-SH wave slowness of the xy-plane of 90°-rotated ST-cut quartz *
(Fig 2.8) we found the velocities in all directions are higher than the S wave velocity of
the guiding layer material, so we may think non-leaky quasi-Love waves will exist in all
*directions. But however we found leaky behavior on the directions off the x-axis. This is *
due to the fact that the quasi-SH wave is the fast shear wave (faster than the quasi-SV
wave), and the quasi-Love wave, in order to satisfy the free boundary condition on the
surface and the continuity condition between quartz and SiO2, the polarization is NOT
coincident to the quasi-SH wave. Having a component on the slower quasi-SV wave
and being faster than quasi-SV wave, it becomes leaky, energy leaks into the substrate
*in the form of quasi-SV wave. Only when the wave vector is parallel to the x-axis, a *
ROT value of 1 is obtained and the non-leaky pure-SH Love wave exists. This is due to
the existence of a pure-SH bulk wave of quartz in this direction, the polarization of the
pure-SH Love wave is then orthogonal to that of the quasi-SV wave, so there is no
mode conversion and therefore no attenuation. We also found that the quasi-SH wave
slowness of the 90°-rotated ST-cut quartz is extremely similar to the slowness of the SH
LSAW (but the LSAW is slightly slower), and the attenuation of the LSAW also be-
*comes zero when the propagation direction is parallel to the x-axis (pure-SH). [48] *

These results show that various types of SH SAWs can be considered as SH bulk waves
trapped by some slow mechanism near the surface. Fig 2.10 shows the dispersion of
*Love waves along 0° (x-axis), 45° and 90° (y-axis) of the layered structure. *

### 2.3.2 Determination of the SiO

2### film thickness

As we will further design a resonator, we have to determine a proper thickness for the guiding layer which affects the performance significantly. The first thing we con-