Defect engineering of silicon based 2 d phononic crystals

Specifically, we design and fabricate three types of PnC resonators based on partially modified scattering air holes, i.e., resonators with central-hole defects, defects of reduced centr

DEFECT ENGINEERING OF SILICON

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2013

DECLARATION

I hereby declare that this thesis is my original work and it has been written by

me in its entirety I have duly acknowledged all the sources of information

which have been used in the thesis

This thesis has also not been submitted for any degree in any university

previously

Wang Nan

July 7, 2013

ACKNOWLEDGEMENTS

With the completion of this thesis, a wonderful four-year journey has come to

an end, with a lot of fond memories left behind I would like to express my sincere gratitude to the following people who have helped me during this four-year journey

Firstly, I would like to thank my thesis advisors, Prof Lee Chengkuo, Prof Moorthi Palaniapan and Prof Kwong Dim-Lee My greatest thanks go to Prof Lee Chengkuo, my main advisor, for his patience and valued guidance throughout my research He has graciously given some of his valuable time up for consultation and discussion, to impart his knowledge and valuable insight towards the progress of my research work He has also provided many valuable references and research tips which have helped me immensely This thesis would never be possible without his guidance and support Many thanks also go to my co-advisors, Dr Moorthi Palaniapan and Prof Kwong Dim-Lee, for their continuous support in the completion of my Ph.D research work

Special thanks go to Prof Hsiao Fu-Li, for inspiring me to enter the world of phononic crystals with his knowledge on the fundamental theory, for his valuable help on various numerical calculations and modelling, and for his

suggestions and discussions when replying to the reviewers‟ comments during the journal manuscript submission process

Special thanks also go to Dr Tsai Ming-Lin, for his help, discussions and guidance on various complicated microfabrication processes

I would like to express my deep thanks and appreciation to Mr Soon Bo Woon, for his time and help on the microfabrication of the batches of devices reported in this thesis The days and nights we spent together in the cleanrooms will be an everlasting part of my memory

I would also like to acknowledge the fabrication support from the Institute of Microelectronics (IME), A*STAR, Singapore and the help from all my dear group mates from the Centre for Integrated Circuit Failure Analysis and Reliability (CICFAR), NUS, as well as people who helped me in one way or another

Last but not least, my special thanks go to my family: my loving parents and wife, who have always been supporting me during the entire course of my four-year journey

CONTENTS

DECLARATION i

ACKNOWLEDGEMENTS ii

CONTENTS iv

SUMMARY ix

LIST OF TABLES xii

LIST OF FIGURES xiii

LIST OF SYMBOLS AND ABBREVIATIONS xxvi

CHAPTER 1: INTRODUCTION 1

1.1 General introduction 1

1.2 Theoretical background 3

1.2.1 The Bravais Lattice and the unit cell 3

1.2.2 The Reciprocal Lattice and Brillouin Zone 4

1.2.3 Bloch theorem and the energy band theory 5

1.3 Literature review 9

1.3.1 PnC with different material compositions 9

1.3.1.1 Solid inclusions in solid background 9

1.3.1.2 Air inclusions in solid background 11

1.3.1.3 Vertical pillars on top of a substrate 13

1.3.2 PnC with different geometry 14

1.3.2.1 3-D PnC substrate 15

1.3.2.2 2-D PnC slab 17

1.3.2.3 1-D PnC strip 17

1.3.3 Applications of various PnC devices with defects introduced 18

1.3.3.1 Waveguides 18

1.3.3.2 Resonators 20

1.4 Motivation and objective 21

1.5 Organization of thesis 23

CHAPTER 2: METHODS FOR NUMERICAL CALCULATION28 2.1 Introduction 28

2.2 Introduction on finite-element-method (FEM) modelling techniques 29

2.3 Numerical methods of pure PnC band gap calculation using COMSOL Multiphysics software 31

2.3.1 Governing equations 31

2.3.2 Subdomain settings 33

2.3.3 Boundary conditions 36

2.3.4 Meshing and equation solving 37

2.4 2-D PnC slab band gap optimization 38

2.5 Numerical methods for modelling defected PnC using COMSOL Multiphysics software 40

2.5.1 Calculation of the defected band structure of the PnC resonator with linear defects introduced in an otherwise perfect PnC 44

2.5.2 Calculation of the transmission spectra of the PnC resonator with linear defects introduced in an otherwise perfect PnC 47

2.5.3 Calculation of the steady-state displacement profiles of the PnC resonator with linear defects introduced in an otherwise perfect PnC 50

2.6 Conclusions 53

CHAPTER 3: MICROFABRICATION PROCESS AND

TESTING SETUP 55

3.1 Introduction 55

3.2 Detailed fabrication steps for devices in current work 56

3.2.1 AlN deposition and patterning 58

3.2.2 Top Al electrode deposition and patterning 61

3.2.3 PnC structure formation by DRIE through the silicon device layer 62 3.2.4 Backside release by DRIE of silicon substrate and RIE of the BOX layer 64

3.3 Processing outcome 66

3.4 Testing setup and procedure 74

3.5 Conclusions 79

CHAPTER 4: PHONONIC CRYSTAL WITH FABRY-PEROT TYPE OF DEFECTS 81

4.1 Introduction 81

4.2 Design approach 85

4.3 SEM images of the microfabricated cavity-mode PnC resonator based on Fabry-Perot types of defects 86

4.4 Testing results 88

4.5 Numerical simulations and discussions 91

4.6 Conclusions 96

4.7 Discussions on improved designs 97

CHAPTER 5: PHONONIC CRYSTAL WITH

CENTRAL-HOLE DEFECTS 101

5.1 Introduction 101

5.2 Design approach 102

5.3 SEM images of the fabricated PnC with central-hole defects 105

5.4 Testing results 107

5.5 Numerical simulations and discussions 112

5.6 Conclusions 121

CHAPTER 6: PHONONIC CRYSTAL WITH DEFECTS OF REDUCED CENTRAL-HOLE RADII 122

6.1 Introduction 122

6.2 Design approach 124

6.3 SEM images of the fabricated PnC with defects of reduced central-hole radii 127

6.4 Testing results 129

6.5 Numerical simulations and discussions 136

6.5.1 Calculation of the defected band structure 136

6.5.2 Calculation of steady-state displacement profiles 143

6.6 Conclusions 149

CHAPTER 7: PHONONIC CRYSTAL WITH ALTERNATE-HOLE DEFECTS 151

7.1 Introduction 151

7.2 Design approach 153

7.3 SEM images of the fabricated PnC with defects of reduced central-hole radii 157

7.4 Testing results 158

7.5 Numerical simulations and discussions 164

7.6 Conclusions 176

CHAPTER 8: CONCLUSIONS AND FUTURE WORK 178

8.1 Conclusions on current work 178

8.2 Recommendations for future work 181

BIBLIOGRAPHY 183

LIST OF PUBLICATIONS 194

SUMMARY

The propagation of acoustic waves or elastic waves in phononic crystals (PnCs) has been extensively studied during the past two decades PnCs are the acoustic wave equivalent of the well-known photonic crystals (PhCs), whereby the propagation of acoustic waves which falls into a certain frequency range (acoustic band gap) are completely forbidden in the PnC structure, which consists of a PnC lattice formed by a periodic array of scattering inclusions located in a homogeneous background material Moreover, by strategically engineering defects through removal or distortion

of the scattering inclusions on an otherwise perfect PnC material, devices of different functionalities like resonators and waveguides can also be realised

However, up to date, the defect engineering on an otherwise perfect PnC to form resonators is mostly focused on the line defects in the form of Fabry-Perot resonant cavities, in which the line defects are engineered by completely removing several rows of the scattering air holes at the centre of the PnC and in the directions perpendicular to the direction of wave propagation In this type of resonators, the performance in terms of the Q factors and the insertion loss cannot be good, due to the significant mismatch

in the acoustic impedance of the central defected region and the surrounding PnC region

In this thesis, we aim to overcome the challenge faced by PnC resonators of Fabry-Perot defects by reducing the mismatch of acoustic impedance through partially modifying the scattering air holes instead of a complete removal Before engineering the partially modified scattering air holes, we first vary the length of the cavity of the PnC resonators with Fabry-Perot defects, which is the number of rows of scattering air holes being removed from the centre of the PnC region, to study the effect of the cavity length on the performance of the PnC resonators Then, we investigate other types of defects engineered by partially modified scattering air holes Specifically, we design and fabricate three types of PnC resonators based on partially modified scattering air holes, i.e., resonators with central-hole defects, defects of reduced central-hole radii,

as well as alternate-hole defects In addition to reducing the mismatch of acoustic impedance, the latter two types of defects are also aimed to further enhance the resonators‟ performance by reducing the energy leakage along the lateral direction of the resonant cavity, with the help of the slow sound effect existed in these two types of defects

All the design processes are done by self-developed FEM methods using

commercial software to calculate the band structure, the transmission spectra,

as well as the steady-state displacement profiles of the resonators The devices are realised by a customized fabrication process Besides enhancing the resonators‟ performance, the testing results also show that the PnC resonators with properly engineered defects can be a potential candidate to overcome the trade-off existed in conventional capacitive-based and piezoelectric-based devices: the trade-off between the Q factor and the insertion loss, demonstrating promising acoustic characteristics to be further optimized for applications in RF communications

LIST OF TABLES

Table 4-1: Performance parameters of various cavity-mode resonators with different cavity lengths 91Table 5-1: Performance parameters of phononic crystal with central-hole

defects of different cavity lengths (L) and central-hole radii (r’) 111

Table 6-1: Performance parameters of phononic crystal with defects of

reduced central-hole radii of different cavity lengths (L) and central-hole radii (r’) 134

Table 7-1: Performance parameters of PnC resonator with alternate-hole

defects of different cavity lengths (L) and central-hole radii (r’) 162

LIST OF FIGURES

Figure 1.1: (a) SEM image of a fabricated PnC structure with solid inclusions in solid background (b) Theoretically predicted (solid line) and experimentally measured (dotted line) transmission data of a PnC reported in [5] This figure is reproduced from [5] 10Figure 1.2:(a) SEM image of the top view of the fabricated PnC device reported in [7] with the hexagonal PnC lattice structure located in the centre

of the device and the transducer electrodes located on the two sides and (b) cross sectional view of the device This figure is reproduced from [7] 12Figure 1.3: Experimentally measured transmission data of the fabricated PnC structure shown in Figure 1.2 This figure is reproduced from [7] 12Figure 1.4: Schematic drawing of a PnC device reported in [39], which is made of an array of finite cylinders arranged in a square lattice and deposited

on top of a regular plate This figure is reproduced from [39] 14Figure 1.5: SEM image of a PnC of square lattice reported in [4] formed by embedding air inclusions in a lithium niobate substrate This figure is reproduced from [4] 15Figure 1.6: Experimentally measured transmission data of the reference devices (dotted line) and the PnC devices (solid line) reported in [4] The top figure shows the transmission along ΓM direction, whereas the bottom figure shows the transmission along ΓY direction This figure is reproduced from [4] 16Figure 1.7: Numerically simulated transmission coefficient in the unit of dB

of the 1-D PnC strip reported in [72] The schematic drawing of the designed 1-D PnC strip is depicted in the inset This figure is reproduced from [72] 17Figure 1.8: (a) pure PnC structure without any defect in [25] with integrated AlN electro-acoustic couplers (b)PnC based waveguides formed by removing one row of scattering W rods along the horizontal direction from the perfect PnC structure as shown in (a) Acoustic waves travel along the horizontal direction of the device This figure is reproduced from [25] 18Figure 1.9: Measured transmission spectra of the fabricated devices reported

in [25] The blue line, the red line, and the green line represent the transmission spectra of the pure background matrix without any PnC structure, the PnC structure without any defects and the waveguide structure, respectively This figure is reproduced from [25] 19Figure 1.10: (a) Schematic drawing of the PnC resonator structure reported

in [94] (b) SEM image of the top view of a fabricated PnC resonator formed

by completely removing air holes at the centre of the PnC region This figure

is reproduced from [94] 20Figure 1.11: Normalized transmission spectra of the PnC resonator structure for the (a) first, (b) second mode for the structure with three periods (12 rows) of scattering air holes on each side of the cavity, (c) first, and (d) second mode for the structure with only two periods (eight rows) of scattering air holes on each side of the cavity This figure is reproduced from [94] 20Figure 2.1: (a) Schematic drawing of the unit cell of the structure for band gap calculation The ideal PnC structure can be considered as an infinite

repetition of the unit cell along x and y directions, when periodic boundary

conditions are applied along these two directions (b) Schematic drawing of the first Brillouin zone Shaded area indicates the irreducible part of the first Brillouin zone (c) The band structure of the phononic crystal structure with

d=10μm, r=8μm and a=18μm, which gives r/a=0.45 and d/a=0.55 (d)-(h):

mode shapes of the first six modes as labelled in Figure 2.1(c) 35

Figure 2.2: Band gap optimization by (a) keeping d/a=0.5 and varying r/a (b) keeping r/a=0.475 and varying d/a d is fixed at 10μm for both cases 39

Figure 2.3: (a) Schematic drawing of a typical PnC resonator with linear

defects introduced along y direction at the centre of the PnC region (b) A

supercell of the typical PnC resonator which includes the introduced defects and is used in the numerical calculation It simulates the ideal PnC resonator which is formed by an infinite repetition of the supercell when periodic

boundary conditions are applied along y direction (on the two faces which are parallel to the x-z plane) Acoustic waves travel along x direction in

numerical modelling and in actual microfabricated devices 42Figure 2.4: An example of the band structure of the PnC resonator with linear defects It can be easily seen that the original band gap which exists in Figure 2.1 (c) disappeared 46

Figure 2.5: An example of the steady-state displacement profiles The ux, uyand uz represent the displacement vector components in x, y, and z directions,

respectively The colour bar indicates the amplitude of displacements in an arbitrary unit, with extreme red represents the maximum displacement (maximum displacement amplitude in positive direction) and extreme blue represents the minimum displacement (maximum displacement amplitude in negative direction) 52

Figure 3.1: The schematic drawing (not to scale) of a typical PnC resonator structure with linear defects introduced after all the fabrication process Inter digital transducers (IDTs) are formed on the two sides of the phononic structure and the device is formed on a free-standing membrane, which is released from the underlying silicon substrate 56Figure 3.2: Microfabrication process flow (not drawn to scale) starting from SOI wafer: (i) AlN deposition and patterning (ii) Top Al electrode deposition and patterning (iii) Phononic crystal formation by DRIE through the silicon device layer (iv) Back side release by DRIE of the silicon substrate handle layer and RIE of the BOX layer 58Figure 3.3: SEM image of the perfect PnC structure with air holes in a silicon background Inter digital transducers (IDTs) are formed by the piezoelectric AlN layer and the Al IDT electrodes on the two sides of the phononic structure The scale bar represents 100µm 66Figure 3.4: Close-up view of the central phononic region with air holes drilled in a silicon background The scale bar represents 10 µm The design diameter of the air holes is 16µm The deviation of fabricated value from the designed is mainly due to the drift during the fabrication process 67Figure 3.5: More zoom-in view of the air holes after the silicon device layer DRIE process, in order to examine the effects of over etching and micro loading 69Figure 3.6: Surface profiles measured by Veeco Wyko Non-contact Profilometer for (a) the free-standing silicon slab without any PnC structure (b) the optimized PnC structure 72

Figure 3.7: Picture of the testing setup, including a calibrated vector network analyser (VNA), which generate and detect AC signal, and a probe station, which provides electrical interface between the VNA and the IDTs, along with the low impedance RF cables 75Figure 3.8: Measured transmission spectra of the free-standing silicon slab without any PnC structure 77Figure 3.9: Measured transmission spectra of the optimized perfect PnC structure 77Figure 4.1: Fabry-Perot etalon The image is reproduced from reference [111] 82Figure 4.2: Schematic drawing (Top view) of the supercell of the PnC

resonators with Fabry-Perot type of defects introduced of (a) L=2a (b) L=3a (c) L=4a Elastic waves travel along x direction L represents the length of

the cavity, which means the number of rows of air holes being removed The grey regions represent silicon background and the white circles represent air

holes The supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 85

Figure 4.3: SEM images (Top view) of cavity-mode PnC resonators with

Fabry-Perot type of defects introduced of (a) L=2a (b) L=3a (c) L=4a Elastic waves travel along x direction due to the piezoelectric effect The scale bar in

each of the SEM image represents 100μm Areas in solid square represents supercells for FEM Areas in dashed squares are the region to be zoomed in and the close-up views are shown in Figure 4.4 86Figure 4.4: Close-up views (Top view) of the central defected regions of the

cavity-mode PnC resonators with Fabry-Perot type of defects introduced of

(a) L=2a (b) L=3a (c) L=4a The scale bar in each of the SEM image

represents 20μm 88

Figure 4.5: Measured transmission spectra of the cavity-mode PnC

resonators with Fabry-Perot type of defects introduced of (a) L=2a (b) L=3a (c) L=4a 89

Figure 4.6: Simulated transmission spectra of the cavity-mode PnC

resonators with Fabry-Perot type of defects introduced of (a) L=2a (b) L=3a (c) L=4a 92

Figure 4.7: Steady-state displacement profiles of the cavity-mode PnC

resonators with Fabry-Perot type of defects introduced of (a) L=2a (b) L=3a (c) L=4a 94

Figure 5.1: Schematic drawings (Top view) of the supercells of the PnC

resonators of L=2a with central-hole defects with radii (r’), (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm L represents the length of the cavity,

which means the number of rows of air holes being removed r’ represents

the radii of the air holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The

supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 102

Figure 5.2: Schematic drawings (Top view) of the supercells of the PnC

resonators of L=3a with central-hole defects with radii (r’), (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm L represents the length of the cavity,

which means the number of rows of air holes being removed r’ represents

the radii of the air holes in the central defect region The grey regions

represent silicon background and the white circles represent air holes The

supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 103

Figure 5.3: SEM images (Top view) of the PnC structure with central-hole

defects and with different L and r’ (a) L = 2a and r’ = 4μm, and (b) L = 3a and r’ = 2μm L represents the length of the cavity, which means the number

of rows of air holes being removed r’ represents the radii of the central air

holes in the silicon background IDTs, which are formed by Al electrodes and piezoelectric AlN layer, are on the two sides of the PnC region,

responsible of launching and detecting acoustic waves along x direction The

area enclosed by solid red square represents the supercell which is used for later FEM calculation For each sub-figure, the picture on the left is the SEM image of the entire structure and the picture on the right is the close-up view

of the central-hole region as enclosed by the dashed squares For each L, four different values of r’, i.e., r’=2μm, r’=4μm, r’=6μm, and r’=8μm are included in the study but only two devices, one from each group of L, are

shown here for simplicity and illustration purpose 105Figure 5.4: Measured transmission spectra of the PnC structure with

central-hole defects and with L = 2a and different r’ (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm 107

Figure 5.5: Measured transmission spectra of the PnC structure with

central-hole defects and with L = 3a and different r’ (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm 109

Figure 5.6: Simulated steady-state displacement profiles of the PnC structure

with central-hole defects and with L = 2a and different r’, under their

respective resonant frequencies (a) r’=2μm, (b) r’=4μm, (c) r’=6μm, and (d)

r’=8μm The u x, uy and uz represent the displacement vector components in x,

y, and z directions, respectively The colour bar indicates the amplitude of

displacements in an arbitrary unit 113Figure 5.7: Simulated steady-state displacement profiles of the PnC structure

with central-hole defects and with L = 3a and different r’, under their respective resonant frequencies (a) r’=2μm, (b) r’=4μm, (c) r’=6μm, and (d)

r’=8μm The u x, uy and uz represent the displacement vector components in x,

y, and z directions, respectively The colour bar indicates the amplitude of

displacements in an arbitrary unit 114Figure 6.1: (a) Schematic drawings (Top view) of the supercell of the PnC

resonators with Fabry-Perot type of defects introduced of L = 2a (b) – (d) Schematic drawings of the supercells of the PnC resonators of L=2a with defects of various reduced central-hole radii (r’), (b) r’=2μm, (c) r’=4μm, and (d) r’=6μm L represents the length of the cavity, which means the number of rows of air holes being removed r’ represents the radii of the air

holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The supercells repeat

themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 124

Figure 6.2: (a) Schematic drawings (Top view) of the supercell of the PnC

resonators with Fabry-Perot type of defects introduced of L = 3a (b) – (d) Schematic drawings of the supercells of the PnC resonators of L = 3a with defects of various reduced central-hole radii (r’), (b) r’=2μm, (c) r’=4μm, and (d) r’=6μm L represents the length of the cavity, which means the number of rows of air holes being removed r’ represents the radii of the air

holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The supercells repeat

themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 125

Figure 6.3: SEM images (Top view) of the air/silicon PnC structure with

defects of reduced central-hole radii and with different L and r’ (a) L = 2a and r’ = 4μm, and (b) L = 3a and r’ = 4μm IDTs, which are formed by Al

electrodes and piezoelectric AlN layer, are on the two sides of the PnC

region, responsible of launching and detecting acoustic waves along x

direction The area enclosed by solid red square represents the supercell which is used for later FEM calculation Pictures on the left are SEM images

of the full structure and pictures on the right are the close-up view of the

central defected region as enclosed by the dashed squares For each L, three different values of r’, i.e., r’=2μm, r’=4μm, and r’=6μm are included in the study but only two devices, one from each group of L, are shown here for

simplicity and illustration purpose 127Figure 6.4: Measured transmission spectra of the PnC structures with defects

of reduced central-hole radii and with L = 2a and different r’ (a) r’=2μm, (b)

r’=4μm, and (c) r’=6μm 129

Figure 6.5: Measured transmission spectra of the PnC structures with defects

of reduced central-hole radii and with L = 3a and different r’ (a) r’=2μm, (b)

measured resonant peaks shown in Figure 4.5(a) and Figure 6.4 are labelled with red arrows 136

Figure 6.7: Calculated band structures of the PnC resonators with L = 3a and with defects of different reduced central-hole radii (r’): (a) r’=0μm, (b)

r’=2μm, (c) r’=4μm, and (d) r’=6μm The bands which correspond to the

measured resonant peaks shown in Figure 4.5(b) and Figure 6.5 are labelled with blue arrows 140Figure 6.8: Simulated steady-state displacement profiles the PnC structures

with defects of reduced central-hole radii and with L = 2a and different r’ (a) r’=2μm, (b) r’=4μm, and (c) r’=6μm The ux, uy and uz represent the displacement vector components in x, y, and z directions, respectively The

colour bar indicates the amplitude of displacements in an arbitrary unit 143Figure 6.9: Simulated steady-state displacement profiles the PnC structures

with defects of reduced central-hole radii and with L = 3a and different r’ (a) r’=2μm, (b) r’=4μm, and (c) r’=6μm The ux, uy and uz represent the displacement vector components in x, y, and z directions, respectively The

colour bar indicates the amplitude of displacements in an arbitrary unit 146Figure 7.1: Schematic drawings (Top view) of the supercells of the PnC

resonators of L=2a with alternate-hole defects with radii (r’), (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm L represents the length of the cavity,

which means the number of rows of air holes being removed r’ represents

the radii of the air holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The

supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 153

Figure 7.2: Schematic drawings (Top view) of the supercells of the PnC

resonators of L=3a with alternate-hole defects with radii (r’), (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm L represents the length of the cavity,

which means the number of rows of air holes being removed r’ represents

the radii of the air holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The

supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 154

Figure 7.3: Schematic drawings (Top view) of the supercells of the PnC

resonators of L=4a with alternate-hole defects with radii (r’), (a) r’=2μm, (b)

r’=4μm, (c) r’=6μm, and (d) r’=8μm L represents the length of the cavity,

which means the number of rows of air holes being removed r’ represents

the radii of the air holes in the central defect region The grey regions represent silicon background and the white circles represent air holes The

supercells repeat themselves in y direction in the microfabricated devices and the acoustic waves travel along x direction 155

Figure 7.4: SEM images (Top view) of (a) the perfect PnC structure (b) the

PnC resonator with alternate-hole defects of L=2a and r’=4μm, (c) the PnC resonator with alternate-hole defects of L=3a and r’=4μm, (d) the PnC resonator with alternate-hole defects of L=4a and r’=4μm L represents the

length of the cavity, which means the number of rows of air holes being

removed r’ represents the radii of the alternate central air holes in the silicon

background The insets of (b), (c) and (d) depict the close-up views of the resonant cavities of the three cases, respectively (e) Schematic drawing of

one of the designed devices For each L, four different values of r’, i.e.,

r’=2μm, r’=4μm, r’=6μm, and r’=8μm are included in the study but only

three devices, one from each group of L, are shown here for simplicity and

illustration purpose 157Figure 7.5: Measured transmission spectrum the PnC resonator with

alternate-hole defects of L=2a and with (a) r’=2μm, (b) r’=4μm, (c) r’=6μm, and (d) r’=8μm The measured transmissions correspond to the cavity modes

of ky=0 158Figure 7.6: Measured transmission spectrum the PnC resonator with

alternate-hole defects of L=3a and with (a) r’=2μm, (b) r’=4μm, (c) r’=6μm, and (d) r’=8μm The measured transmissions correspond to the cavity modes

of ky=0 159Figure 7.7: Measured transmission spectrum the PnC resonator with

alternate-hole defects of L=4a and with (a) r’=2μm, (b) r’=4μm, (c) r’=6μm, and (d) r’=8μm The measured transmissions correspond to the cavity modes

of ky=0 160Figure 7.8: Calculated band structure of (a) the perfect PnC structure (b) of

the PnC resonator with alternate-hole defects of L=4a and with r’=8μm 164

Figure 7.9: The band structure which corresponds to the resonant peak (bottom figure), the first order derivative of frequency against wave vector (middle figure), and the simulated transmission field distributions of the displacement profile under the resonant frequency (top figure) of the PnC

resonator with alternate-hole defects of L=2a and with (a) r’=2μm (b)

r’=4μm (c) r’=6μm (d) r’=8μm The simulated field distributions correspond

to the cavity modes of ky=0 The colour bar represents the amplitude and sign of the displacement 168

Figure 7.10: The band structure which corresponds to the resonant peak (bottom figure), the first order derivative of frequency against wave vector (middle figure), and the simulated transmission field distributions of the displacement profile under the resonant frequency (top figure) of the PnC

resonator with alternate-hole defects of L=3a and with (a) r’=2μm (b)

r’=4μm (c) r’=6μm (d) r’=8μm The simulated field distributions correspond

to the cavity modes of ky=0 The colour bar represents the amplitude and sign of the displacement 169Figure 7.11: The band structure which corresponds to the resonant peak (bottom figure), the first order derivative of frequency against wave vector (middle figure), and the simulated transmission field distributions of the displacement profile under the resonant frequency (top figure) of the PnC

resonator with alternate-hole defects of L=4a and with (a) r’=2μm (b)

r’=4μm (c) r’=6μm (d) r’=8μm The simulated field distributions correspond

to the cavity modes of ky=0 The colour bar represents the amplitude and sign of the displacement 170Figure 7.12: Graph of Q factor against the standard deviation of the band that corresponds to the measured resonant peak for all designed PnC resonators with alternate-hole defects 173

LIST OF SYMBOLS AND ABBREVIATIONS

r Location vector of lattice point

R Translation vector

n Arbitrary integers

b Fundamental translation vectors

K Reciprocal lattice vector

u x Displacement vector in x direction

u y Displacement vector in y direction

u z Displacement vector in z direction

ω Eigenfrequency of the governing equation

k Wave vector in the irreducible Brillouin zone

G Two-dimensional reciprocal lattice vector

CG-G‟ Fourier transforms of C(r)

ρ G-G‟ Fourier transforms of ρ(r),

ε Relative permittivity

kx Wave vector in the irreducible Brillouin zone in x direction

ky Wave vector in the irreducible Brillouin zone in y direction

a Lattice constant (pitch) of the PnC, which means the distance

between the centres of two adjacent holes in the square lattice

d Thickness of the 2-D PnC slab

r Radius of the air holes of the PnC structure

f 0 Resonant frequency

Q Quality factor

L Length of the cavity, which means the number of rows of air

holes to be completely removed or partially modified

Δf3dB 3dB width in the transmission spectra

ABG Acoustic band gap

AC Alternating current

Al Aluminium

AlN Aluminium nitride

BOX Buried oxide

CD Critical dimension

CVD Chemical Vapour Deposition

DRIE Deep Reactive Ion Etching

ECR Electro Cyclotron Resonance

FEM Finite element methods

GSG Ground-Signal-Ground

HCl Hydrochloride

IABG Inverse acoustic band gap

IC Integrated circuit

ICP Inductively Coupled Plasma

IDT Inter digital transducer

KOH potassium hydroxide

LPCVD Low Pressure Chemical Vapour Deposition MEMS Microelectromechanical Systems

PECVD Plasma Enhanced Chemical Vapour Deposition

PhC Photonic crystal

PnBG Phononic band gap

PnC Phononic Crystal

PR Photoresist

PVD Physical Vapour Deposition RIBE Reactive Ion Beam Etching RIE Reactive Ion Etching

CHAPTER 1: INTRODUCTION

1.1 General introduction

During the past two decades, propagation of acoustic waves in phononic crystals (PnCs) has received a great deal of research attention because of their renewed physical properties and potential applications [1-11] PnCs are also referred as phononic band gap (PnBG) materials or acoustic band gap (ABG) materials, whereby a periodic array of scattering inclusions located in

a homogeneous background material leads to the formation of PnBG or ABG, prohibiting elastic waves of certain frequencies from traveling in any direction

One of the key components in radio frequency communication devices is frequency reference oscillator Due to the incompatibility of quartz crystals, which currently most reference oscillators are based on, with IC fabrication, silicon-integrated micromechanical oscillators are gaining more research interest due to the ability to be integrated with electronics [12] Currently, researchers are mainly focusing on two types of microresonator technology, namely capacitive-based [13-18] and piezoelectric-based [19-21] devices However, there is a trade-off between these two types of devices, which is known as the trade-off between Q factor and motional impedance For silicon

micromechanical reference oscillators which are based on capacitive MEMS resonators, researchers have already demonstrated an f-Q product as high as 2×1013 [22] However, the motional impedance is also very high due to weak capacitive electro-acoustic coupling On the other hand, although piezoelectric-based microresonators can have their motional impedance to be below 50Ω, the Q factor cannot be very high due to high loss in the piezoelectric materials [21]

PnC is a potential candidate to overcome the trade-off between the Q factor and the motional impedance as PnC can store elastic energy in microcavity made on high Q materials such as silicon [23] PnCs are the acoustic wave equivalent of photonic crystals (PhCs), which consist of periodically arranged scattering centres embedded in a homogeneous background matrix From the aspect of elastic properties, PnCs are inhomogeneous materials with periodic variations Thus the dispersion characteristics of the PnCs lead to the existence

of phononic band gaps, in which the propagation of elastic waves within a certain frequency range is prohibited in any direction PnCs with properly engineered band gaps can be the basis of realizing a variety of functionalities such as acoustic waveguides, cavities and filters We can obtain such functionalities by modifying portions of the PnC structure for various applications in wireless communication and sensors

1.2 Theoretical background

As mentioned previously, PnCs are formed by a periodic array of scattering inclusions located in a homogeneous background material, forming a lattice structure This lattice structure is analogous to the crystal lattice existed in the crystalline solid Therefore, the theoretical background of the PnC structure is similar to the theoretical background of the crystalline solid, which starts with the basic knowledge on lattice structure in solid state physics [24]

1.2.1 The Bravais Lattice and the unit cell

In a crystalline solid, the atoms or group of atoms are arranged in a periodic pattern The form of a crystal is such that the identical building blocks were added continuously, in which the building blocks are atoms or groups of atoms, so that a crystal is a three-dimensional periodic array of atoms In general, the periodic translational symmetry of a space lattice may be

described in terms of three fundamental translation vectors, b 1 , b 2 , b 3, defined in such a way that any lattice point r (n1, n2, n3) can be generated from any other lattice point, r (0, 0, 0), in the space by translational operation

r(n ,n ,n ) = r(0,0,0)+R (1.1)

Bravais lattice and the parallelepiped spanned by the three basis vectors, b 1,

b 2 , b 3, is called the unit cell of the Bravais lattice, which is the basic building block of a crystal, whereas a primitive cell is the smallest unit cell that can be repeated to form the lattice

1.2.2 The Reciprocal Lattice and Brillouin Zone

The Bravais lattice is a space lattice, which has the translational symmetry in the real space However, the motion of electrons in a crystal is usually

described in both the real space and the momentum space (or k-space) The

spatial properties of a periodic crystal can be described by the sum of the

components in the k-space, or the reciprocal space For a perfect crystal, the

reciprocal lattice in the reciprocal space consists of an infinite periodic three-dimensional array of points whose spacing is inversely proportional to the distance between lattice planes of a Bravais lattice

In a reciprocal lattice, a set of reciprocal basis vectors, b 1 *, b 2 *, and b 3 *, can

be defined in terms of the basis vectors, b 1 , b 2 , and b 3, of the direct lattice in the real space by equation (1.3) below

where h, k, and l are Miller indices The unit cell in the reciprocal lattice is

called the Brillouin Zone, while the first Brillouin zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice

1.2.3 Bloch theorem and the energy band theory

The energy band structure of a solid can be constructed by solving the Schrödinger equation for electrons in a crystalline solid which contains a large number of interacting electrons and atoms However, it is extremely difficult

to solve the Schrödinger equation for the many-body problem in a crystal without simplification Therefore, one-electron approximation, which neglects the effects that arise from motion of atomic nuclei and assumes that each electron sees only some average potential due to the charge distribution

of the rest of the electrons in the crystal in addition to the potential of the fixed charges (i.e., positive ions), is adopted to simplify the calculation

The Schrödinger equation shown in equation (1.5) describes the electrons in the semiconductor

periodicity, R, as described in equation (1.6) R is the translational vector in

the direct lattice as defined in equation (1.2)

The solutions of the Schrödinger equation for a periodic potential as described

in equation (1.6) must be of a special form, as proved by F Bloch:

Therefore, the eigenfunctions of the wave equation for a periodic potential are

the product of a plane wave exp(ik⋅r) times a function uk(r) with the periodicity

of the crystal lattice

A restricted proof of the Bloch Theorem will be discussed here, starting from

assuming that Ψk(r) is non-degenerate, which means there is no other wave

function with the same energy and wave vector The potential energy is

periodic in a, with V(x) = V(x + sa), where s is an integer, when considering N identical lattice points on a ring of length Na We look for solutions of

Schrödinger equation such that

Because Ψ(x) must be single valued, C is one of the N roots of 1, or

With k = 2πs/Na we have the Bloch function (1.7)

As such, by Bloch Theorem, the wavefunction of an electron in a periodic potential can be expressed in the form of a periodic function Analogically, the acoustic phonons in an environment with periodic elasticity can also be expressed by a periodic function, according to Bloch Theorem As a result, the methods to calculate the phononic band structure is analogical to the methods

to calculate the electron energy band structure The detailed methods to calculate various phononic band structures will be covered in Chapter 2

1.3 Literature review

So far, various types of acoustic and elastic wave propagation have been studied in different classes of PnC structures Currently, works on PnC are mainly classified according to two criteria: material compositions and geometry

1.3.1 PnC with different material compositions

From the view of the material compositions, PnCs can be classified into three main groups, i.e., solid scattering inclusions in solid background, air scattering inclusions in solid background, and vertical pillars on top of a substrate

1.3.1.1 Solid inclusions in solid background

Phononic band gaps can be formed when an array of solid scattering inclusions are located in a homogeneous background of another solid of different elasticity [5, 10, 25, 26], i.e., solid/solid configuration In this type

of PnCs, a relatively wider band gap can be obtained as for the same designed parameters as compared to the configuration of air inclusions in solid background (air/solid configuration) This is because of the higher contrast in the materials‟ elasticity for solid/solid configuration than that of air/solid configuration

Figure 1.1: (a) SEM image of a fabricated PnC structure with solid inclusions in solid background (b) Theoretically predicted (solid line) and experimentally measured (dotted line) transmission data of a PnC reported in [5] This figure is reproduced from [5]

For example, El-Kady, I., et al reported, both theoretically and

experimentally, a PnC band-gap structure whose operating frequency is in the MHz range As shown in Figure 1.1 (a), the PnC studied in their work