Download Phononic Crystal Waveguiding in GaAs Golnaz Azodi Aval

Phononic Crystal Waveguiding in GaAs
by
Golnaz Azodi Aval
A thesis submitted to the
Department of Physics, Engineering Physics & Astronomy
in conformity with the requirements for
the degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
November 2013
c Golnaz Azodi Aval, 2013
Copyright Abstract
Compared to the much more common photonic crystals that are used to manipulate
light, phononic crystals (PnCs) with inclusions in a lattice can be used to manipulate
sound. While trying to propagate in a periodically structured media, acoustic waves
may experience geometries in which propagation forward is totally forbidden. Furthermore, defects in the periodicity can be used to confine acoustic waves to follow
complicated routes on a wavelength scale. Using advanced fabrication methods, we
aim to implement these structures to control surface acoustic wave (SAW) propagation
on the piezoelectric surface and eventually interact SAWs with quantum structures.
To investigate the interaction of SAWs with periodic elastic structures, SAW interdigital transducers (IDTs) and PnC fabrication procedures were developed. GaAs
is chosen as a piezoelectric substrate for SAWs propagation. Lift-off photolithography
processes were used to fabricate IDTs with finger widths as low as 1.5µm.
PnCs are periodic structures of shallow air holes created in GaAs substrate by
means of a wet-etching process. The PnCs are square lattices with lattice constants
of 8µm and 4µm. To predict the behavior of a SAW when interacting with the PnC
structures, an FDTD simulator was used to calculate the band structures and SAW
wave displacement on the crystal surface. The bandgap (BG) predicted for the 8
micron crystal ranges from 180 MHz to 220 MHz. Simulations show a shift in the
i
BG position for 4µm crystals ranging from 391 to 439 MHz.
Two main waveguide geometries were considered in this work: a simple line waveguide and a funneling entrance line waveguide. Simulations indicated an increase in
acoustic power density for the funneling waveguides. Fabricated device evaluated with
electrical measurements. In addition, a scanning Sagnac interferometer is used to map
the energy density of the SAWs. The Sagnac interferometer is designed to measure
the outward displacement of a surface due to the SAW. Interferometric measurements
confirmed waveguiding in the modified funnel entrance waveguide embedded in the
4µm PnC. However, they also revealed strong dissipation of the SAW in the waveguide
due to the non-vertical sidewalls resulting from the wet-etch process.
ii
Acknowledgments
I would like to express my deepest appreciation to my supervisor, James Stotz, whose
invaluable guidance, helpful suggestions, and endless patience during the course of
my research I will never forget. It has been a privilege working with him and having
him as a supervisor.
I would also like to thank my dear friend and colleague, Aaron, whose help and
encouragements are greatly appreciated. I would like to express my appreciation
to other members of my research group Ryan, who helped me with the basics of
fabrication portion of this work when I started my work, Colin and Edward for
valuable discussion and sharing ideas.
Many thanks to Rob Knobel and his group members: Jennifer and Arnab. Their
prompt repairs of equipment in the clean room, insightful discussions, and fabrication
suggestions were greatly appreciated.
Special thanks to my dear parents. Their unconditional support and voices filled
with love always gave me energy and motivation. Last but not least, I would like to
thank Mohsen, my dear husband, for his love and never-ending support, for always
being there for me, and for having faith in me. I want to thank him for his encouragement when I was desperate or unfocused, and, most of all, for always supporting
my decisions despite the hardships they put him through.
iii
Table of Contents
Abstract
i
Acknowledgments
iii
Table of Contents
iv
List of Tables
vi
List of Figures
vii
1 Introduction
1
Chapter 2:
2.1
2.2
2.3
2.4
2.5
Surface Acoustic Waves in
Acoustic Wave Terminologies . . .
Wave Propagation Equation . . . .
Surface Acoustic Waves . . . . . . .
Interdigital Transducers . . . . . .
Device characterization . . . . . . .
Solids . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
10
11
15
17
Phononic Crystals . . . . . . . . . . . . . . .
One-Dimensional Harmonic Crystal . . . . . . . . . . .
Phononic Band Gap Structures . . . . . . . . . . . . .
Numerical Simulation of PnCs . . . . . . . . . . . . . .
FDTD Simulation Parameters . . . . . . . . . . . . . .
Phononic Crystal Waveguides . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
20
23
25
28
31
Experimentation . . . . . . . . . . . . . . . . . . . . . . . .
Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
35
Chapter 3:
3.1
3.2
3.3
3.4
3.5
Chapter 4:
4.1
iv
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
35
38
42
47
51
52
53
. . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
58
58
61
69
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
79
81
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.2
4.1.2 Sample preparation . . . . . .
4.1.3 Optical lithography . . . . . .
4.1.4 Interdigitated Transducers .
4.1.5 PnCs . . . . . . . . . . . . . .
Sagnac Interferometry . . . . . . . .
4.2.1 What do we want to measure?
4.2.2 Optical experimental setup . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter 5:
5.1
5.2
5.3
Results and Discussion
IDT Characterization . . . . . .
PnC Waveguide Design . . . . .
Sagnac optical interferometry .
.
.
.
.
Chapter 6:
6.1
6.2
v
List of Tables
4.1
IDTs features on the photomasks . . . . . . . . . . . . . . . . . . . .
vi
43
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.1
Schematic representation of particle displacement ,u, with respect to
equilibrium position. Picture taken from [3]. . . . . . . . . . . . . . .
Coordinate convention on GaAs sample. . . . . . . . . . . . . . . . .
Illustration of a Rayleigh wave. Particle motion is shown relative to
wave propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single and double finger IDTs with same pitch but different wavelength.
Schematic of a delay line (double transducer) on GaAs substrate. . .
Schematic representation of a 2-port network . . . . . . . . . . . . . .
Experimental and theoretical sonic transmission through a BG structure. Figure taken from [24]. . . . . . . . . . . . . . . . . . . . . . . .
One-dimensional harmonic crystal . . . . . . . . . . . . . . . . . . . .
Dispersion relation of 1D harmonic crystal with different mass ratios.
Mass ratio increases from left to right. On the left m1 = m2 , in the
middle m1 = 1.1 m2 and on the right m1 = 1.5 m3 . The mass difference
opens up a BG for mechanical waves. The size of the resulting BG is
proportional to the mass difference between m1 and m2 . . . . . . . .
Schematic of a square lattice phononic crystal structure (view from
top). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of computational Yee cell for numerical FDTD simulations.
Note that in practice only either T 12 or T21 is calculated not both.
Same is true for T23 and T32 , T13 and T31 . . . . . . . . . . . . . . . . .
Schematic of the 2D unit cell and the applied boundary condition for
square lattice PnC. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sonic waveguide demonstration in a PnC structure. . . . . . . . . . .
Diffraction effect on the resist profile. On the left, a positive resist is
shown, the exposed area will be removed and the remaining resist has
positive side walls. On the right the remaining resist on the sample
after developing is the exposed part and has negative side walls. Blue
is the substrate, orange is the resist, green is the exposed resist, and
yellow indicates the regions that mask block the UV light. . . . . . .
vii
7
13
14
15
16
18
20
21
23
24
27
30
33
40
4.2
4.3
4.4
4.5
4.6
4.7
5.1
5.2
5.3
5.4
Photomask design: The top left one is the IDT photomask. On the top
right is an overlay of a group of IDTs and their corresponding PnCs.
On the bottom is a close up of a single finger IDT and a line waveguide
crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Different resist profiles and corresponding deposited metals. On the
left: Enough undercut to provide a clean lift-off . On the right: Forming continuous metal film (not enough undercut) will not allow the
remover to reach the resist to have a successful lift-off. Blue is substrate, orange is resist, and gray is metal. . . . . . . . . . . . . . . . .
Overview of IDT fabrication steps. Blue is substrate, orange is photoresist, yellow is the mask, and gray is metal. . . . . . . . . . . . .
Examples of different development times. Top: underdeveloped sample, resist is not fully removed. Middle: a well-developed sample, ready
for deposition, the finger widths and the spacing between them are approximately equal. Bottom: edge quality is degraded, also the spacing
and the finger width are not equal. . . . . . . . . . . . . . . . . . . .
Schematic of isotropic and anisotropic etching. Blue is the substrate,
orange is the resist mask, and white areas represent the etched regions.
Schematic of the Sagnac interferometer. The blue lines demonstrate
the path of the beam from the source to the sample, while the red one
is for the light reflecting from sample and is going toward the detector.
The top and bottom figures represent the two beams polarizations. .
Scattering parameter S11 (left) and S21 (right) for a single finger transducer with a finger width of 2.42 µm. The measurements are for a 200
pair, 2-port, single finger delay line, with a wavelength of 9.68µm and
a resonance frequency in the transmission with a peak at 293.72 MHz.
The fundamental frequency of the transducers, obtained from S11 measurements plotted versus wave vector. Transducers are single and double fingers of aluminum on a GaAs substrate. The half width of S11
peaks is about 4 MHz and therefore the error bars are too small to be
shown on the graph. The Linear fit is shown by the solid line. Inset
shows the equation for the fitted line. . . . . . . . . . . . . . . . . . .
S11 measurement of a single device, using two different techniques.
The blue line is the probe station measurement, the red line is from
the wire-bonded sample. . . . . . . . . . . . . . . . . . . . . . . . . .
Line waveguide PnC with lattice constant 8µm and filling fraction of
0.5. The PnC waveguide is fabricated by wet-etching between a 200
pair single finger transducer with wavelength of 10.08µm. . . . . . .
viii
44
45
46
48
49
54
59
60
61
63
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
Reflection (left) and transmission (right) for a delay lines operating at
frequency within a band gap of a square crystal with lattice constant
of 8µm. The red and blue lines correspond to measurements before
and after etching the PnCs, respectively. . . . . . . . . . . . . . . . .
Band gap comparison for two different lattice constants of a = 8µm on
the top and a = 4µm on the bottom. . . . . . . . . . . . . . . . . . .
Band gap comparison for four different filling fraction. . . . . . . . .
Simulated outward surface displacement for 410 MHz SAWs incident
on square crystal with a = 4µm of filling fraction 0.55 with different
waveguide geometries. . . . . . . . . . . . . . . . . . . . . . . . . . .
Normalized intensity of a 1.72µm finger width transducer mouth taken
by Sagnac interferometer. The aluminum fingers and pad are significantly more reflective than GaAs substrate. Fabrication debris is observed around the fifth finger on the intensity image. . . . . . . . . .
A close view of reflection measurement from Fig. 5.13. The red arrow
indicates the FM depth for a typical sample that used to map with
Sagnac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured displacement map of a 317.86 MHz SAW. The SAW frequency is lower than the crystal BG. Top: normalized intensity of
reflected light obtained near the entrance to the waveguide. Bottom:
SAW displacement near the waveguide entrance taken simultaneously
with the reflection image. . . . . . . . . . . . . . . . . . . . . . . . .
Plot of y-cut displacement of the standing wave averaged inside the
waveguide shown in Fig. 5.11 . . . . . . . . . . . . . . . . . . . . . .
Reflection (on the left) and transmission (on the right) measurements of
PnC waveguide with a delay line. The image of Sagnac interferometer
for this device is shown in Fig. 5.14. . . . . . . . . . . . . . . . . . .
Measured displacement map of a 410.344 MHz SAW. The SAW frequency is within the crystal BG. Top: normalized intensity of reflected
light obtained near the entrance to the waveguide. Bottom: SAW displacement near the waveguide entrance taken simultaneously with the
reflection image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured displacement map of a 410.344 MHz SAW. The SAW frequency is within the crystal BG. Top: normalized intensity of reflected
light obtained near the entrance to the waveguide. Bottom: SAW displacement near the waveguide entrance taken simultaneously with the
reflection image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plot of y-cut displacement of the standing wave averaged inside the
waveguide shown in Fig. 5.14 . . . . . . . . . . . . . . . . . . . . . .
ix
63
65
67
68
70
71
73
74
75
76
77
78
List of Abbreviations
ABC
Absorbing Boundary Condition
Al
Aluminium
BAW
Bulk Acoustic wave
BG
Band Gap
DI
Deionized
FDTD
Finite Difference Time Domain
FM
Frequency Modulation
GaAs
Gallium Arsenide
HMDS
Hexamethyldisilazane
IDT
Interdifital Transducers
IPA
Isopropyl Alcohol
PBC
Periodic Boundary Condition
PnC
Phononic Crystal
PnCSim
Phononic Crystal simulator
PtC
Photonic Crystal
PWE
Plane Wave Expansion
RF
Radio Frequency
RIE
Reactive Ion Etching
SAW
Surface Acoustic Wave
x
Chapter 1
Introduction
Over the past two decades, a great interest in quantum information technologies has
developed. A variety of photon-based [27] and spin-based [19, 14] solutions have been
proposed for implementing different elements necessary for future quantum networks
and quantum computers [18]. Among many challenges that face this new born technology, in specific in spintronic devices, reliable transport mechanisms will be needed
to be implemented [34]. In photonic applications on the other hand, dynamical modulations of the properties of the quantum system needs to be done locally for desired
quantum device operation [10]. Quite interestingly, surface acoustic waves (SAWs)
seem to have the potential to offer solutions to both of these problems[17].
The concept of surface acoustic waves was originally introduced by Lord Rayleigh
back in 1885 [32] as he analyzed the behavior of surface waves on a homogeneous
isotropic elastic surface. SAWs, also known as Rayleigh waves, are essentially mechanical waves which propagate on the surface of an elastic medium with the particle
motion in the sagital plane 1 , and their energy is concentrated near the substrate
1
Plane containing the normal plane to the surface and the wave propagation direction.
1
CHAPTER 1. INTRODUCTION
2
surface.
However, it was not until 1965, that such wave motion was efficiently utilized for
electronic applications using metal film interdigital transducers (IDTs) on the surface of a piezoelectric substrate. White and Voltmer [41] published the first work to
generate and detect SAWs on a single device. Their experiment consisted of two aluminum IDTs on a quartz piezoelectric substrate. One IDT was generating a Rayleigh
wave which propagates along the surface of the crystal and was detected by the other
IDT. It was observed that, by means of this device, a delay and a filtering could be
obtained in a very compact package. Nowadays, SAW devices are of extensive use in
electronics and communication mobile technologies [9].
The physical phenomenon on which the SAW devices is based is piezoelectricity.
In other words, certain materials produce an electric field when mechanically strained
due to the electromechanical coupling property of the material. The most common
piezoelectric substrates are lithium niobate, lithium tantalate, and quartz. Other
materials such as gallium arsenide (GaAs) have lower piezoelectricity coefficients, but
because of compatibility in device integrations in modern technologies, GaAs can also
be considered as an alternative substrate.
In optics, using modern fabrication techniques, artificial periodic structures called
photonic crystals (PtCs) have driven great progress in controlling light in photonicon-chip integrated devices such as lasers [30], single photon sources [1] and much more
[27]. The key feature that makes photonic crystals distinguished for light manipulation in small feature sizes is the concept of the photonic bandgap (BG), a range
of frequencies for which light is not allowed to propagate through the structured periodic crystal. This photonic BG allows for engineering cavities and waveguides on
CHAPTER 1. INTRODUCTION
3
the scale of the wavelength of the operating light. These cavities and waveguides can
be building blocks of much more complicated photonic structures with application in
communication and quantum information.
In analogy to the photonic BG, people have been exploring phononic BG structures for surface acoustic waves [38, 39, 43]. Similar to PtCs, phononic crystals (PnCs)
enable us with controlling SAW propagation on integrated devices. The micron wavelength range of the SAWs makes PnC feature sizes larger than the normal feature
sizes involved in PtC structures. Therefore, less elaborate techniques can be used in
fabricating the phononic band gap structure.
The usefulness of SAWs for applications in photonics and spintronics, in addition
to the concept of the phononic bandgap, can add up to make SAWs even more powerful for future integrated devices with applications in communication and quantum
information. Specifically, implementing PnC waveguides can be useful for SAW delivery upon the region on the integrated chip where it is needed. For example, a SAW
might be needed for dynamic frequency tuning of an optical cavity embedded inside
a photonic crystal structure [2, 17].
With this regard, the goal of this thesis is to design, fabricate and characterize
SAW waveguides using PnC structures in GaAs. As mentioned before, a variety of
photonic device implementations have been done in GaAs. Therefore, fabricating
PnC waveguides in GaAs, and the potential to couple these systems, opens up a
new road to more complex integrated device fabrications. It should also be noted
that, although silicon is the primary choice for many on-chip devices, the lack of
piezoelectricity complicates SAW generation and makes silicon less appealing.
The remainder of this thesis is structured as follows. In Chapter 2, a review of
CHAPTER 1. INTRODUCTION
4
the theory of SAW propagation, followed by an overview of IDTs, is provided. In
Chapter 3, the basics of phononic crystal theory is described, and the finite difference
time domain (FDTD) numerical technique for calculating PnC band structures is
presented. Chapter 4 is devoted to fabrication methods and recipes used in this thesis
as well as to the optical interferometry technique used to image the SAW. Chapter
5 describes the electrical performance characteristics of the fabricated transducers
before and after PnC fabrication; optical interferometry provides confirmation of
waveguiding in PnC waveguides in agreement with FDTD simulations. Chapter 6
concludes with a summary of the results and an outline of the future directions.
Chapter 2
Surface Acoustic Waves in Solids
SAWs are elastic waves that propagate on the surface of a material, e.g. GaAs in this
work. Thus, the general theory of elasticity can be used to describe SAW behavior.
A brief introduction to SAWs, as elastic waves, is done in this chapter following with
some basic properties of Rayleigh waves and introducing the interdigital transducers,
which generate SAWs in this thesis. Primary source of information for the topics
covered in this chapter are text by Auld[3] , Morgan[25] and Royer[33].
2.1
Acoustic Wave Terminologies
A disturbance that propagates through space and time is known as a wave. Elastic
waves, in particular, are mechanical disturbances that propagate through a material
and causes oscillations of the particles of that material about their equilibrium positions. In such a case, an internal restoring force opposes body deformations due
to the particles displacement. Thus, in a normal mathematical treatment of these
vibrations, either localized oscillations or traveling waves, three fundamental concepts
5
6
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
need to be introduced which are the particle displacement, the material deformation,
and the internal restoring force.
Particle displacement, u, is a measure of the particle distances relative to its
equilibrium as a function of particle position; see Fig. 2.1. In general, this can be a
function of all three coordinates, {xi } = {x, y, z}. However, the particle displacement
itself is not enough to have a restoring force. No deformation occurs, when a simple
translation or rotation of material particles happens. That is why, a strain tensor,
S, needs to be defined to describe the material deformations. In fact, the strain
tensor includes information on relative movement of different particles. In linear
approximation, S can be defined as
1
Sij (r, t) =
2
∂ui ∂uj
+
∂xj
∂xi
,
(2.1)
Note that S is dimensionless. The matrix representation of the strain tensor defined
above can be written out as:



∂u1


∂x1



 S=
 1 ∂u1 +
 2 ∂x2


 
 1 ∂u1
+
2 ∂x3







∂u2
∂u2
1 ∂u2 ∂u3 
.
+
∂x1
∂x2
2 ∂x3 ∂x2 




∂u3
1 ∂u2 ∂u3
∂u3

+
∂x1
2 ∂x3 ∂x2
∂x3
1
2
∂u1 ∂u2
+
∂x2 ∂x1
1
2
∂u1 ∂u3
+
∂x3 ∂x1
(2.2)
S is symmetric and therefore has only six independent elements. The diagonal
components are normal strains and the off diagonal components are shear strains. In
response to deformations, the material generates internal forces to return particles to
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
7
Figure 2.1: Schematic representation of particle displacement ,u, with respect to equilibrium position. Picture taken from [3].
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
8
their equilibrium. The stress, the force per unit area, is a quantitative measure of the
generated internal forces defined as:
Tij = Cijkl Skl ,
(2.3)
where Cijkl are the components of the forth rank stiffness tensor. Eq. (2.3) can
be imagined as the Hooke’s law generalization for a three dimensionally extended
material. In the absence of external torques, it can be shown that T is symmetric.
Due to this symmetry and also the mentioned symmetry of S, the stiffness tensor, C,
would be also symmetric. Moreover, different components of the stiffness tensor can
be shown to satisfy the following relations:
Cijkl = Cjikl = Cijlk = Cjilk .
(2.4)
For simplicity, the following abbreviated notation can be used instead of the double
indices notation



 xx 




 yy 










 zz 


→



 yz, zy 







 xz, zx 







xy, yx

1 

2 



3 
.

4 


5 


6
(2.5)
Using this, the strain tensors can be re-written as a six-elements column matrix
9
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
instead of a 3 × 3 as below:


 S1

1
S=
 2 S6

1
S
2 5
1
S
2 6
1
S
2 5
S2
1
S
2 4
1
S
2 4
S3







S=













→

S1 

S2 



S3 
.

S4 


S5 


S6
(2.6)
The exact same convention can be used for the T tensor. According to this new
convention, the stiffness tensor elements can be indexed by two numbers instead of
four letters; e.g. Cxxxx → C11 and Cxyxy → C66 . In this notation, the stiffness tensor
reads as a 6×6 matrix. Considering the mentioned symmetries governing on Cijkl , the
stiffness tensor has at most 21 independent elements. A further reduction is possible
by choosing reference coordinate axis in appropriate way in relation to a crystal axis.
e.g. a cubic crystal with the coordinate reference axes parallel to the crystal axis will
have only 3 numbers of independent elastic constant coefficients giving:

















C11 C12 C12
0
0
C12 C11 C12
0
0
C12 C12 C11
0
0
0
0
0
C44
0
0
0
0
0
C44
0
0
0
0
0
0 

0 



0 
.

0 


0 


C44
(2.7)
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
2.2
10
Wave Propagation Equation
The fundamental dynamical equation of motion for waves in an elastic, homogeneous,
and either anisotropic (elastic properties depend on direction) or isotropic (elastic
properties independent of direction) medium is:
∂ 2 ui
∂Tij
,
ρ 2 =
∂t
∂xj
(2.8)
where ρ is the mass density of the elastic medium and ui are the already discussed
displacements in the respective co-ordinate directions. This is reminiscent of the
Newton’s laws of motion that relates a point particle acceleration to the applied net
force on it. From Eqs. (2.1) and (2.3), it can be seen that
Tij = Cijkl
∂ul
.
∂xk
(2.9)
Thus, Eq. (2.8) can be re-written as
ρ
∂ 2 ul
∂ 2 ui
=
C
,
ijkl
∂t2
∂xj ∂xk
(2.10)
which is in fact the wave equation of motion that any particle displacement within
the material medium has to follow. Therefore, by solving Eq. (2.10) with specific
boundary conditions, the elastic waves solutions for a given material of known mass
density ρ and stiffness tensor C, can be obtained.
The simplest elastic wave solution is for an unbounded material (bulk material),
when boundary conditions are placed at infinity. In such a case, the solution is a
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
11
plane-wave solution
u = u0 exp [i (ωt − k · x)] ,
(2.11)
where u0 is the displacement amplitude, ω is the elastic wave frequency and k is
the wave-vector. Depending on the particles displacement direction with respect to
propagation vector, there are two different plane-wave solutions for a bulk system:
transverse elastic waves, when the particle displacement is perpendicular to the wavevector, and longitudinal elastic waves, when the particle displacement is parallel to
the propagation wave-vector1 . By substituting particle displacement expression into
the wave equation, the velocities of the wave are determined in terms of the direction
of propagation in the solid. Generally, bulk transverse wave velocities are lower than
the bulk longitudinal modes.
2.3
Surface Acoustic Waves
The SAWs are the elastic waves that propagate along the surface of a solid material. In 1885, Rayleigh introduced waves propagating on the stress-free surface of a
semi-infinite isotropic half space medium [32]. As opposed to the bulk material, a
proper boundary condition needs to be applied on the surface of the material[42]. For
Rayleigh waves, the boundary condition results from the fact that waves propagate
on a stress-free surface. Therefore, SAWs are solution to the Eqs. (2.10) and (2.9)
with the following boundary condition applied on the solid surface:
1
In some materials particle displacements are neither exactly parallel nor perpendicular. In these
cases wave solutions are called quasi-transverse and quasi-longitudinal.
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
Ti3 |z=0 =
X
kl
∂uk Ci3kl
∂xl = 0.
12
(2.12)
z=0
In the case of stress-free boundary condition, there are two sets of transverse solutions
where the particle displacement can be orthogonal to the propagation wave-vector.
These two transverse solutions tend to have different velocities. Including a longitudinal mode, there are three types of solutions to the wave equation of motion for an
acoustic wave on the surface. The three solutions, two transverse and one longitudinal, do not propagate independently. In fact, due to the presence of the boundary
condition, a mixing of both longitudinal and transverse elastic waves occurs. Thus, a
SAW has components from longitudinal and transverse elastic waves. One component
of physical displacement is parallel to SAW propagation direction axis, and the other
one is normal to the surface. These two wave motions are 90◦ out of phase with one
another in the time domain.
Due to this wave mixing, the displacement on the surface takes an elliptical form,
because wave amplitude along the x3 -axis (perpendicular to the surface as depicted
in Fig. 2.2) is larger than along the SAW propagation axis, x1 . Depicted in Fig. 2.3,
is a demonstration of SAW propagation and the corresponding particle displacement
on the solid surface. It should be noted that, since the particles are less dense on the
surface of the solid, the SAW velocity is less than the slowest elastic waves in bulk
material, typically in the order of 5% to 13%. This provides a waveguide effect, and
helps to prevent SAWs from scattering into bulk waves[12].
Because SAWs propagate along the 2D surface instead of the whole 3D medium,
the majority of its energy is localized near the surface, normally within a depth of one
wavelength below the solid surface. Thus, external observations on the state of the
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
13
Ͳ
Figure 2.2: Coordinate convention on GaAs sample.
system are possible. Therefore, the SAW solution can be written in the form in which
the x3 -dependence is treated as a decaying wave amplitude and the x1 -dependence
(parallel to the surface) describes the oscillatory behavior. In particular
Ui = Ui0 exp (−γkx3 ) exp [i (ωt − kx1 )] .
(2.13)
Here, γ represents the decay depth into the bulk portion of the system and k and ω
are just wave-vector and frequency as normal. Due to the energy decay, as moving
further in depth, the wave amplitude reduces and therefore the elliptical particle
displacement shrinks in size; see Fig. 2.3.
In practice, for real applications, the mechanical wave must somehow be introduced to the system. This can be accomplished by employing the piezoelectric properties of the subject material. Piezoelectricity describes the coupling between the
mechanical and electrical properties of the solid medium. In other words, applying a
voltage on the system changes the mechanical displacement of the particles or conversely, a mechanical displacement of the particle can be converted to an electric
voltage. For piezoelectric materials, instead of Eq. (2.3), the following equations are
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
14
^tWƌŽƉĂŐĂƚŝŽŶŝƌĞĐƚŝŽŶ
Figure 2.3: Illustration of a Rayleigh wave. Particle motion is shown relative to wave
propagation.
responsible for describing the system :
Tij = Cijkl Skl − ekij Ek
(2.14)
Di = eikl Skl + ik Ek .
(2.15)
Here, E and D are the electric field and electric displacement. The e and are
the piezoelectricity and the permittivity tensors of the medium, respectively. It can
be seen from this set of equations how electrical and mechanical displacement are
coupled together. A wave of electric field now accompanies the elastic wave, and
the wave velocity depends upon elasticity, piezoelectricity and the material dielectric
properties. However, Eq. (2.10) will be still used to obtain displacement solutions in
the presence of the external electric field. More importantly, this electromechanical
coupling is what can be used to generate SAWs on the surface of our devices. For SAW
propagation in a piezoelectric material, it can be shown that the electromechanical
coupling coefficient, K2 , is defined in terms of the piezoelectricity coefficients, stiffness
15
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
w
w
w
w
d
λ
λ
Figure 2.4: Single and double finger IDTs with same pitch but different wavelength.
coefficients and electrical permittivity in the following form[9]:
K2 ≡
2.4
e2
.
C
(2.16)
Interdigital Transducers
As mentioned earlier, applying electric voltage on the surface of a piezoelectric material can generate mechanical displacements of solid particles. However, not any
mechanical displacement is practically useful. In 1965, White and Voltmer [41] used
IDTs both as a source and receiver of surface waves. An IDT is a periodic arrangement of deposited metal strips on the surface of the solid that can be used to generate
a mechanical wave of desired shape, when specific electric voltages are applied to it.
Normally, two sets of fingers of opposite polarity are brought together in a comb
configuration, see Fig. 2.4, in order to alternatively change the sign for the applied voltage. Correspondingly, the particles displacements will alternatingly change.
Therefore, a desired mechanical wave is introduced on the material surface where the
IDT has been fabricated.
16
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
RF
Ṽ
Figure 2.5: Schematic of a delay line (double transducer) on GaAs substrate.
In the design of any IDT, there are three important factors that need to be considered: the IDT finger width, w, the metallization ratio (which is a measure of
the surface area covered with metal to the uncoated surface), and the IDT aperture
width, d, (which is the transverse overlap of two sets of fingers). Adjusting these
three parameters, results in generating elastic waves with different frequencies and
profiles. Finger width and metallization ration determine the center frequency of the
generated wave as will be discussed later in this section.
Although introduced here as a SAW generation device, an IDT can also be used to
detect mechanical waves on the material surface as well. This is due to the electromechanical conversion property of the solid, as discussed earlier. In many applications,
as is the case throughout this thesis, IDTs are fabricated in pairs against each other
with specific separation between them depending on the device fabrication requirements; see Fig. 2.5. This allows the user to generate a SAW, send it through a system
of interest and detect the transmitted SAW at the other end of the device. Therefore, the generation and the detection is performed by exactly the same mechanism.
This type of IDT configuration is the so-called delay line as it takes well-defined time
for the wave to travel from the generation IDT to the detection IDT, which exactly
depends on the material parameters described earlier in this chapter.
In this thesis, IDTs with metallization ratio of 0.5 are designed. This results
in equal finger width and finger spacing. Two different types of IDTs, single-finger
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
17
transducers and double-finger transducers, are designed and fabricated, as depicted
in Fig. 2.4. In the case of single finger transducers, the corresponding acoustic
wavelength, λ, equals to four times of the pitch of the electrode, 4w. Normally,
single-finger transducers have a significant degree of internal reflection, and because
of that, they are so-called reflective transducers [25]. For double finger IDTs the
fingers in each side are in pair as depicted in Fig. 2.4 and the SAW wavelength is
λ = 8w. This type of transducer may also be referred to as a non-reflective transducer
[25].
2.5
Device characterization
IDT delay lines, as shown in Fig. 2.5, can be considered as a 2-port network. Thus,
scattering matrices can be used to evaluate the performance of such devices. A scattering matrix is a quantitative measure of radio frequency (RF) energy propagation
through a multi-port network which for an N-port device, containing N 2 coefficients
(S-parameters) that describe the response of the network to voltage signals at each
port. For a 2-port device, this is mathematically represented by:





 B1   S11 S12   A1 

=

,
B2
S21 S22
A2
(2.17)
where Bi and Ai are the output and incident voltages of port i, respectively, and Sij
are the scattering parameters with the first and second suffix refer to destination and
source port respectively and defined as:
Bi
Vref lected at port i Bi
Vout of port i =
Sij =
=
.
Sii =
Ai
Vejected f rom port i Aj =0
Aj
Vejected f rom port j Ai =0
(2.18)
18
CHAPTER 2. SURFACE ACOUSTIC WAVES IN SOLIDS
S21
−
→
A1
←
−
B1
←
−
A2
S11
S22
−
→
B2
S12
Figure 2.6: Schematic representation of a 2-port network
S-parameters are complex values as both the magnitude and phase of the input signal
are changed by the network. Note that, Sii are reflection coefficients and only refer
to what happens at a single port, while Sij (i 6= j) are the transmission coefficients
which describe what happens from one port to another.
Chapter 3
Phononic Crystals
Phononic crystals (PnCs) are periodic structures made of an alternating arrangement
of host and inclusion materials, such that over a specific range of frequencies, acoustic
waves are not allowed to propagate through them. The forbidden range of frequencies is called a bandgap (BG) and is due to constructive interference from multiple
reflections off the different inclusions periodically placed within the host medium [38].
The earliest work on PnCs backs to 1979 by Narayanumurti et al. [26]. The
experiment was established to investigate the propagation of high frequency phonons
through a GaAs/AlGaAs super lattice. Although not known as a PnC at the time,
later on, by introducing the concept of phononic crystals, this type of structure can
be considered as a one-dimensional PnCs. Later, in 1993, Kushuwaha published the
first calculation of a full band structure for periodic structures (cylindrical nickel
inclusions in an aluminum host) by using the plane wave expansion (PWE) technique
[20]. With increasing interest in photonic crystal materials, experimental work on
PnCs has increased as well.
One example of an experiment study of a two-dimensional PnC has been published
19
CHAPTER 3. PHONONIC CRYSTALS
20
Figure 3.1: Experimental and theoretical sonic transmission through a BG structure.
Figure taken from [24].
by Miyashita et al. in 2004 [24]. In their experiment, the PnC is a periodic structure of
acrylic cylinders placed in air forming a square lattice, as acoustic transmission data is
taken in the [100] and [110] crystal directions. The geometry of the structure is chosen
based on their numerical calculations. As depicted in Fig. (3.1), the experimental BG,
the frequency range where transmission is significantly reduced, is in good agreement
with their theoretical calculation for the BG.
3.1
One-Dimensional Harmonic Crystal
Acoustic waves are due to mechanical vibrations of the medium, thus a simple vibrational system in 1D helps to describe a concept of the phononic crystal and the
underlying concept of the BG. This classic example is presented below.
Consider a periodic 1D arrangement of two types of particles, with mass m1 and
m2 separated by distance a, as depicted in Fig. 3.2 . Let us assume that all these
21
CHAPTER 3. PHONONIC CRYSTALS
n2n−1
β
m1
n2n+1
n2n
β
β
m1
m2
a
β
m2
m1
a
Figure 3.2: One-dimensional harmonic crystal
particles are connected via springs of the same constant, β. Using Hooke’s law
Fn = −β un ,
(3.1)
the Newton’s equation of motion for odd and even labeled particles are find
d2 u2n
= β (u2n+1 − 2 u2n + u2n−1 )
dt2
d2 u2n+1
m2
= β (u2n+2 − 2 u2n+1 + u2n ) .
dt2
m1
(3.2)
(3.3)
Here, un is the nth particle displacement. Assuming solutions of the form of
u2n = A eiωt e2ikna
u2n+1 = B eiωt eik(2n+1)a ,
(3.4)
(3.5)
where ω is the corresponding frequency of the vibration and k is the wave number.
It can be shown that [13] :
s 2
1
1
1
1
4β 2
2
2
ω =β
+
± β
+
−
sin2 (ka).
m1 m2
m1 m2
m1 m2
(3.6)
Therefore, the mechanical vibration frequency does depend on the wave number
and vice versa. For given mass and spring constant values, this can be used to
illustrate the system dispersion relation, a plot of ω versus k. However, looking at
CHAPTER 3. PHONONIC CRYSTALS
Eqs. (3.4) and (3.5), for any integer m we find that
mπ = u2n (k)
u2n k +
2 mπ
u2n+1 k +
= u2n+1 (k) .
2
22
(3.7)
(3.8)
This means that, although k extends to infinity, any k > π/2 can be projected
back onto 0 ≤ k ≤ π/2, the reduced Brillouin zone. Therefore, there are multiple
frequencies allowed for a given wave number within this reduced range. Depicted
in Fig. 5.6 is plot of system dispersion over the reduced Brillouin zone. As seen,
at ka = π/2 there are discontinuity in a form of gap in dispersion. Tracking back
to all other wave numbers, there is no other allowed vibration frequency anywhere
within the reduced zone. This means that there are certain frequencies that never
get excited regardless of the excitation wave number. This is the vibrational BG (in
this simple model there are only two bands) that arises from the multiple phonon
scattering within this simple 1D system.
Using the dispersion relation, Eq.(3.6), the width of the BG can be calculated to
be
WBG
1
1
.
−
= 2β m1 m2 (3.9)
Therefore, the more mass difference between the sites, the bigger of a vibrational BG
will be seen. Moreover, when we set m1 = m2 , the BG disappears. This suggests that
the periodic mass difference is crucial in forming a BG. In fact, phonon scattering can
occur when there is a property difference in our system; in this example, it is mass.
This is similar to the traditional Bragg reflector that is used in optics as an example
of 1D photonic crystal. In that case, alternating layers of material with different
refractive indices are placed such that, for certain frequencies, all the incident light
23
4
4
3
3
3
2
1
Frequency HΩL
4
Frequency HΩL
Frequency HΩL
CHAPTER 3. PHONONIC CRYSTALS
2
1
0
1
0
Π
4
0
Wave Number HkL
Π
2
2
0
0
Π
4
Π
2
Wave Number HkL
0
Π
4
Π
2
Wave Number HkL
Figure 3.3: Dispersion relation of 1D harmonic crystal with different mass ratios.
Mass ratio increases from left to right. On the left m1 = m2 , in the
middle m1 = 1.1 m2 and on the right m1 = 1.5 m3 . The mass difference
opens up a BG for mechanical waves. The size of the resulting BG is
proportional to the mass difference between m1 and m2 .
is reflected.
3.2
Phononic Band Gap Structures
Advanced micro-fabrication techniques can be used to introduce inclusions in the host
medium by keeping the system and the procedure as simple as possible. In this work,
etching the surface in a certain pattern removes some part of the host material and
creats air inclusions, which leads to the periodic property changes of the medium,
mass density and stiffness, that is essential to produce an acoustic BG. However, a
periodic alteration of the host medium has to be done such that an overlap among
the all three BGs (corresponding to the three sets of solution that contribute to
SAW) occurs. Because of this, designing phononic crystal structures with wide BGs
24
CHAPTER 3. PHONONIC CRYSTALS
Figure 3.4: Schematic of a square lattice phononic crystal structure (view from top).
is difficult. The most common geometries used for PnCs are square and triangular
lattices. In this thesis however, we focus mainly on the square lattice types of PnCs.
An example is shown in Fig. 3.4.
In order to obtain different vibrational modes of such PnC structure, a set of
coupled equations
ρ
d2 ui
∂ 2 ul
=
C
,
ijkl
dt2
∂xj ∂xk
(3.10)
and
Tij = Cijkl
∂ul
,
∂xk
(3.11)
need to be solved, where u is the particle displacement, ρ is the mass density and
Cijkl is the medium stiffness tensor. Recall from chapter 2 that these are equations
of motion governing waves in an elastic medium.
CHAPTER 3. PHONONIC CRYSTALS
3.3
25
Numerical Simulation of PnCs
Normally, analytic solutions to wave equations (3.10) and (3.11) for structures with
complex geometries such as PnCs are not available. However, for practical applications, it is important to have such information. Therefore, numerical alternatives
must be employed to find the solutions of the wave equations. Several numerical techniques have been used to obtain PnC modes [28], among them plane wave expansion
(PWE) and finite difference time domain (FDTD) have been the most successful. Wu
et al. [43] used the PWE method to calculate band structure of a two-dimensional
phononic crystal including SAW and bulk acoustic waves (BAWs) dispersion relation.
Sun et al. [35] using FDTD method, reported dispersion relation of SAW and BAW.
An FDTD software, PnCSim, has been developed by a previous member of our
group at Queen’s, Joseph Petrus [31], that will be used in this thesis to study different
PnCs of interest. Using FDTD [37], the displacement can be calculated everywhere
within the computational volume for a given PnC geometry with proper boundary
conditions. FDTD calculated simulation results will be then analyzed to obtain useful
information such as band structure and transmission.
As is common with numerical analytics, space and time have to be discretized.
Therefore, all the partial derivatives involved in Eqs. (3.10) and (3.11) need to be
replaced by finite derivatives. However, the two mentioned equations are coupled, so
the spatial derivatives of T are needed to calculate u and vice versa. Therefore, a
computationally efficient solution to this problem is to evaluate u and T at different
locations in an interlacing configuration such that, at every point of our computational grid, either u or T needs to be calculated while the other one is calculated at
neighbor nodes. Therefore, mid point estimation for the partial derivatives of different
CHAPTER 3. PHONONIC CRYSTALS
quantities in Eqs. (3.10) and (3.11) can be used
1
1
0
f (x) ∆x = f x + ∆x − f x − ∆x .
2
2
26
(3.12)
This also has the benefit of reducing the computational error at the order of (∆x)3
compared to the normal derivative estimations that the value of the function at the
desired location and the adjacent point are being used.
With a few steps of calculations, discretized alternatives to the two coupled equations governing the acoustic waves can be obtained. For example, for the u1 it can
be seen that
un1 (i, j, k) = 2un−1
(i, j, k) − u1n−2 (i, j, k)
(3.13)
1
2
1
1
(∆t)
n−1/2
n−1/2
i − , j, k
T11
i + , j, k − T11
+
ρ (i, j, k) ∆x1
2
2
2
(∆t)
1
1
n−1/2
n−1/2
+
i, j − , k
T
i, j + , k − T12
ρ (i, j, k) ∆x2 12
2
2
2
(∆t)
1
1
n−1/2
n−1/2
+
T
i, j, k +
− T13
i, j, k −
.
ρ (i, j, k) ∆x3 13
2
2
Here, as shown in Fig. 3.5, (i, j, k) represent an arbitrary point on the computational
cell, and i ± 1/2 represents the two nearest grid points to it in x1 direction. The
superscript n represents the time step at which the calculation is being performed.
As seen, the value of u1 depends on its own values at two previous steps (because the
dynamical equation involved is of second order in time) and also the values for several
components of T at half a time-step before at different locations than u1 itself. The
counterpart equation for T evaluation can be find to be
27
CHAPTER 3. PHONONIC CRYSTALS
T32
T23
u3
T13
z
T31
(i, j, k)
y
T21
T12
u1
x
u2
T33
T22
T11
Figure 3.5: Schematic of computational Yee cell for numerical FDTD simulations.
Note that in practice only either T 12 or T21 is calculated not both. Same
is true for T23 and T32 , T13 and T31 .
n+1/2
T11
1
i + , j, k
2
n
u1 (i + 1, j, k) − un1 (i, j, k)
1
= C11 i + , j, k
(3.14)
2
∆x1

 1
1
n
n
u2 i, j + , k − u2 i, j − , k


1
2
2

+ C12 i + , j, k 


2
∆x2

un3

1
+ C13 i + , j, k 

2
1
i, j, k +
2
−
un3
∆x3

1
i, j, k −
2 
.

In practice, u and T are being calculated at different times and the results will be
fed from one into another.
The simulations performed for this thesis were executed on a computing platform
that had 4 GB of memory and a quad core AMD processor that runs at 2.8 GHz. The
operating system on the machine was Arch Linux 2009.08 (64 bit). A typical 2D band
structure simulation on the mentioned platform takes about 4 hours of calculation
time in order to obtain a meaningful result as will be discussed in the next section.
28
CHAPTER 3. PHONONIC CRYSTALS
3.4
FDTD Simulation Parameters
In order for FDTD programs to provide meaningful results, several computational
parameters must be set properly. These parameters include discretization constants,
∆x and ∆t, boundary conditions on the entire computational volume, edges, and
the initial excitation function (source). In addition, choosing the physical location at
which the source needs to be applied and the system response should be monitored.
Regarding the discretization parameter ∆x, a convergence study always has to
be performed to find the optimum grid size for a given geometry in order to obtain
physical answers. In general, finer grids give more accurate results. However, there
is always a trade off between saving computational time and the accuracy of the
calculated result. Therefore, a convergence study helps us to find the margin beyond
which ∆x produces desired results. In our FDTD simulations of square lattice type
PnCs, 60 grid points per lattice period has been verified to produce PnCs band
structures in agreement with previous available information.
Similar to the spatial discretization constant, ∆x, a finer time discretization constant, ∆t, leads to more accurate computational results. However, in this case, it
can be shown that there is a mathematical upper limit to ∆t for our solutions to be
computationally stable. That is the so called Courant condition
∆t|critical =
s
vmax
1
∆x1
1
2
+
1
∆x2
2
+
1
∆x3
2 .
(3.15)
Forcing ∆t to be less than ∆t|critical ensures the stability of the numeric, but it
does not guarantee however sufficiently accurate results. It is still recommended to
CHAPTER 3. PHONONIC CRYSTALS
29
study the ∆t dependence of the desired physical quantities.
The physical system under simulation must also have a finite computational volume in a computer. This volume includes the system and those part of the environment interacting with the system. However, the boundaries need to be set somewhere
in order to be able to run a simulation. In the study of PnCs, two types of boundary conditions are typically used: periodic boundary condition (PBC) and absorbing
boundary condition (ABC). The PBC is applied when calculating the band structures
of different PnCs. Assuming that the crystal under study extends to infinity, the perfect symmetry can be used to reduce computational volume to one unit cell of the
crystal. This is only because of the fact that any solution must follow the underlying
symmetry of the structure. Depicted in Fig. 3.6 is the unit cell of the square lattice
PnCs in GaAs and the properly applied boundary conditions in a 2D simulations.
However, if a 3D simulation of the 2D crystal with a finite height for the holes is
needed, due to lack of symmetry, a different boundary condition must be applied
along the third direction. Extra care must be taken with the boundary condition on
the third direction as it need to simulate the infinite extent of the system and not
only a sharp cut in the computational volume. In other words, for example, solutions
to the wave equations cannot simply be forced to go to zero at these non-periodic
boundaries as it results in unphysical answers due to computational reflections off
hard boundaries. The ABC has been suggested as the proper boundary condition
to be applied on a non-periodic edge as it simulates the situation as if there was no
end to the computational volume. This type of boundary condition is also called
outgoing boundary condition which can be understood from the above discussion. In
this work, only 2D computations of the band structures have been performed, thus
30
CHAPTER 3. PHONONIC CRYSTALS
PBC
PBC
PBC
PBC
Figure 3.6: Schematic of the 2D unit cell and the applied boundary condition for
square lattice PnC.
no ABC needed to be applied. However, when calculating SAW transmission through
PnCs, ABC was used as, in general, the infinite symmetry was not preserved in our
waveguide designs. Although a full 3D FDTD simulation is required to obtain accurate band structure information on SAW PnC devices, it has been reported that the
band structure of the bulk modes which can be obtained using 2D simulations are in
close relation to the SAW band structures. In fact, it has been claimed [21, 7] that
the BGs for the SAW waves and bulk waves falls on the same range of frequencies for
a given PnC structure.
Even with great choices of discretization constants, ∆x and ∆t, and boundary
conditions, it is still possible for the FDTD algorithm to produce meaningless results
if the simulated system is not initially excited properly. Depending on the nature of
the computation, a variety of sources may be used. For example, if a wide range of
frequencies is needed to be excited initially, one would choose some specific time dependence for the pulse, such as a narrow Gaussian or even a delta function, to excite
many modes in frequency space. This can be the case in a typical band structure
computation that for a given wavelength, one would be looking for all the possible
CHAPTER 3. PHONONIC CRYSTALS
31
resonant frequencies of the structure. However, if for example, one looks at transmission through a waveguide at a certain frequency, then a plane wave like function
might be best employed. In the FDTD software available in our group, there are a
few useful excitation functions that can be used depending on the need:
• Gaussian function
• Modulated Gaussian
• Delta function
• Sinusoidal function
Refer to [31] for further details.
Finally, it is possible for the computation to run and produce a physical result,
but, for some reason, one is not able to extract useful data from monitoring the
computation process. It is very important to monitor the system dynamics at a
proper location inside computational volume. For example, when calculating band
structures of the PnCs, one might want to avoid monitoring displacement at a high
symmetry point (such as the center) inside the unit cell. If not, only solutions with
specific symmetries will be detected by that monitor located at the high symmetry
point. Because of this, it might be useful to use multiple monitors at different locations
that are preferably not at high symmetry points.
3.5
Phononic Crystal Waveguides
A perfect phononic crystal can be used to exclude wave propagation in desired regions
of space over the frequency range of the crystal band gap(s). Moreover, introducing
CHAPTER 3. PHONONIC CRYSTALS
32
defects into the perfect crystal results in interesting structures such as cavities (point
defects) and waveguides (line defects). Point-defect cavities can be used to localize
waves in small region of space and line-defect waveguides can be used to transmit
waves through the PnC structure.
A nice demonstration of guided sonic waves in sonic waveguides has been done by
Miyashita et al.[24]. As shown in Fig. 3.7a a sharp bend waveguide has been created
in the sonic crystal using Acrylic pillars. Plotted in Fig. 3.7b shows the experimental
measurement in which they demonstrate sonic wave guiding through the proposed
structure. Enhancement of the detected sonic wave is seen within the BG range. In
particular, closer to the middle of the BG, the enhancement is maximized. This is
because of the BG structure surrounding the waveguide in all directions except the
guiding direction. Presence of the BG material reduces energy loss and directs more
energy to the detector.
CHAPTER 3. PHONONIC CRYSTALS
33
a) Sonic waveguide using aluminum rods in air. Picture taken from [24].
b) Transmission measurement for the sonic waveguide. Presence of the PnC waveguide changes the transmission. Taken from [24].
Figure 3.7: Sonic waveguide demonstration in a PnC structure.
Chapter 4
Experimentation
Experimental contributions for this thesis include two main parts: fabrication and
optical imaging. First, the general fabrication methods used in this research are discussed, followed by an introduction of the optical setup of the Sagnac interferometer
which is used for imaging the SAWs on the fabricated device.
4.1
Fabrication
Device fabrication in this thesis consists of two main parts: IDTs and PnCs. Transducers are made from Al fingers on GaAs samples which are made by means of lift-off
photolithography technique and the PnCs are air holes etched in the GaAs substrate.
A detailed description of he basic concepts, many of which are common for both IDTs
and PnCs, is introduced and specifically explains the details of the fabrication recipes
for the devices presented in this thesis. Several fabrication recipes were examined
within this work. Although many of them did not meet expectations, the successful
procedures will be discussed here.
34
CHAPTER 4. EXPERIMENTATION
4.1.1
35
Substrate
The main concern in choosing a substrate with regard to IDT fabrication, is the
piezoelectric characteristics of the substrate, which are required for SAW generation
as discussed in chapter two. The PnCs are fabricated on the same device by means
of etching; thus, choosing a sample that can be etched easily with the desired etching profile must also be considered. Considering these facts and the potential future
applications of PnC for semiconductor quantum devices, GaAs has been chosen for
device fabrication. GaAs is a semiconductor that is crystallized in a zinc blend structure. The GaAs wafers used in this thesis were semi-insulating, undoped and two
inches wide in diameter. The orientation of GaAs wafers was (100) with the major
flat along the [011] direction. As discussed in chapter two, SAWs must be generated
in specific directions of the piezoelectric surface, hence it is important to identify the
GaAs sample orientations. The semiconductor wafer orientation is defined by the flat
sides on the wafer. The flats are called the major and the minor flat which are slightly
different in size to make identification easier.
4.1.2
Sample preparation
The surface of the substrate plays a significant role in the fabrication process. For
successful device fabrication, a substrate free of contaminants is needed. There is
always some organic vapour in the air of a clean room and a wafer that has been
exposed to them for even a moderate amount of time can become contaminated.
The surface treatment of GaAs consist of two main parts; first is the removal of
contaminants such as organic compounds, and the second one is the removal of the
native oxide to expose the bare semiconductor for subsequent processing.
CHAPTER 4. EXPERIMENTATION
4.1.2.1
36
Cleaning
Removing the contaminants of the wafer can be performed by many different methods
based on the device applications and the source of contaminants formed on the surface
of the semiconductor. One primary source of organic contamination results from the
fabrication processes themselves, which occurs if a process needs to be restarted.
These sources of organic contaminants are typically removable by rinsing in acetone
followed by isopropyl alcohol (IPA). Rinsing in deionized (DI) water and drying must
also be considered as essential parts of any wet cleaning process. As a general strategy,
the wafer should be kept wet all along the cleaning process and minimize the number
of times when wafer is drawn from liquid to air. Since adsorbed water could itself be
called contamination, baking on a hot plate is suggested to remove water from the
wafer surface.
The general procedure used in this thesis to clean GaAs wafers is in the following
sequence:
• Rinse with DI Water.
• Immerse in acetone, sonicate in ultrasonic bath for 5 min.
• Rinse with IPA.
• Rinse with DI water.
• Nitrogen blowing to dry sample.
• put sample on hot plate at 180◦ C for 5 min.
Note that delays between these steps are not desired as drying liquids can cause
contamination on the sample surface. An extra step may be taken to the above
CHAPTER 4. EXPERIMENTATION
37
procedure when dealing with a used wafer. Oxygen plasma cleaning can be added
after baking the sample on the hot plate. Oxygen plasma can etch away the resist
scum that may be left from the previous fabrication steps on the GaAs surface.
4.1.2.2
Removing The Native Oxide
Although successful in treating organic contamination, the cleaning procedure described above will not be able to remove the native oxide layer from the sample
surface. All III-V semiconductors are oxidized by exposure to air. The surface of a
GaAs wafer that has been exposed to air for a long time is typically covered with
an oxide layer of 1-2 nm [5]. Immersion in either an acidic or basic dilute solution
etches away the native oxide and provides an oxide free sample surface. Removal of
the native oxide can be examined by the contact angle measurement technique based
on water formation on the substrate surface [4]. The oxide layer on the surface of
semiconductor is hydrophilic, and water spreads evenly on such a surface. The oxide
free semiconductor surface, is hydrophobic in contrast, and water droplets form on
the surface.
As an example, ammonium hydroxide N H4 OH is one of the base chemicals that
works well to remove native oxide from GaAs. Alternatively, dilute acidic solutions
such as HCl or H3 P O4 or H2 SO4 can be used. For this thesis, based on the availability
of chemicals and safety protocols to be followed, the basic solution is being used. The
recipe for native oxide removal is the following:
• Add 2 ml of N H4 OH to 20 ml of H2 O.
• Immerse GaAs wafer in solution for 20 min.
• Rinse with DI water.
CHAPTER 4. EXPERIMENTATION
38
• Blow dry with N2 .
However, electrical measurement of SAW transducers, fabricated on two different
samples showed that the native oxide removal process does not have a pronounced
effect on measurement results. This could be an indication of the fact that it is
unlikely to avoid GaAs sample from air exposing while fabricating on it in the clean
room.
4.1.3
Optical lithography
During optical lithography, a UV light source exposes a photosensitive film on the
substrate through a photomask and transfers a specific pattern to the film. This procedure includes a spin coating, light beam exposure and chemical development. After
the photoresist film on the sample is patterned, it then gives the opportunity to do a
variety of fabrication techniques on sample; e.g. the open area of underlying material
can be etched away or undergo thin film deposition. For this reason, photolithography is a common step for most microfabrication processes. This section discusses
the lithography process steps in the same order as it was performed to fabricate the
device used in this thesis.
4.1.3.1
Photoresist and Spin Coating
Photoresist, a light-sensitive material, is used to transfer the pattern of photomask to
a substrate. The chemical properties of the photoresist change in the areas exposed
to light leaving the rest of the surface un-exposed and therefore unchanged. Based on
the resist type, the two regions would have different solubility in a solution referred to
as the developer. Generally, the photoresists are classified into two groups, positive
CHAPTER 4. EXPERIMENTATION
39
and negative. After exposure, the resist normally becomes more or less acidic. Since
developer is an alkali solution, the more acidic the resist is therefore more easily
removed with developer. A resist that turns more acidic due to light exposure is called
“positive”. In such a case, the exposed resist is removed under chemical development.
Conversely, the resist that becomes less acidic is called “negative”. Thus, it is very
important to choose which area needs to be exposed based on the resist tone applied
on the sample and the features on the photomask.
In addition, the resist tone directly affects the resist profile, which needs to be
chosen properly depending on what device fabrication is required. Negative tone
resist tends to have a negative side-wall slope, where the top of the resist is wider
than the bottom at the wafer surface after developing process. This type of profile is
very useful for metal lift-off as will be discussed later in subsection 4.1.4.2. Perfectly
vertical side-walls are hard to achieve due to diffraction. In fact, some resist regions,
underneath the opaque area of the mask, are exposed to light as well as other regions
underneath the open areas. Based on the photoresist type, an either positive or
negative side-wall slope is being formed as shown in Fig. 4.1.
Photoresist adhesion is another parameter that plays a critical role in the outcome of the fabricated device. Even after removing native oxide layer off the wafer,
as discussed earlier, a thin layer of oxide is still left on the sample. Because of hydrophilic nature of oxidized surface, the adhesion between the wafer and resist is not
high enough to have a perfect uniform resist layer. Hexamethyldisilazane (HMDS) is
suggested to use as an adhesion primer prior resist coating [6].
Spin coating is the standard resist application method. A few milliliters of specific
resist is dispensed at the center of the wafer. Rapid acceleration of spinning spreads
CHAPTER 4. EXPERIMENTATION
40
Figure 4.1: Diffraction effect on the resist profile. On the left, a positive resist is
shown, the exposed area will be removed and the remaining resist has
positive side walls. On the right the remaining resist on the sample after
developing is the exposed part and has negative side walls. Blue is the
substrate, orange is the resist, green is the exposed resist, and yellow
indicates the regions that mask block the UV light.
41
CHAPTER 4. EXPERIMENTATION
the resist toward the edges and leaves a very uniform thin layer on the wafer. This
is important because the resolution of features depends on the resist thickness due
to diffraction. In addition, the energy needed for exposure depends on the resist
thickness. The film thickness can be controlled by viscosity (η) of the resist and spin
speed (ω) of the spin coater according to
r
t∝
η
.
ω
(4.1)
The spin coated resist contains up to 15% solvent and may contain built-in stresses
[16]. Baking on a hot plate helps to remove solvent and to improve adhesion of the
resist layer to the wafer.
4.1.3.2
Pattern Exposure
After applying the resist, the photomask and resist-covered wafer are brought into
intimate contact to expose the photoresist to the light. A mask aligner is a standard device for lithography purposes. Usually, a Mercury or Xenon-Mercury lamp
is used to provide strong spectral lines at specific wavelengths. The most common
wavelengths are 436, 405 and 365nm called, respectfully, the g-line, the h-line and the
i-line. Exposing should take place in a controlled time because exposing for a specific amount of time is necessary to have enough reaction in the exposed photoresist
regions. Underexposure may lead to no pattern transfer to the wafer. On the other
hand, exposing for longer times can expose the protected areas under the metal of
the mask. Exposure for too long or too short will change the width of the pattern
from the designed one on the photo mask, and it also affects the resist profile and
side-walls.
After exposing the resist, the soluble parts of the resist are etched away in the
CHAPTER 4. EXPERIMENTATION
42
compatible developer. The wafer should always be inspected with a microscope at
this point. If there is a problem with the pattern development, it is still possible
to develop further or strip the resist with acetone and try again. For example, if a
feature appears to be larger or smaller than the desired size, either over development
or underdevelopment has occurred. These can be caused by errors in exposure time,
prebake temperature, and development time. Post-development baking of the patterned wafer can slightly toughen the photoresist and enhance its resistance to the
subsequent etch process. This is generally done between 90◦ C and 110◦ C for a few
minutes. Baking at too high temperature may cause the resist to “reflow” and change
the side-wall profile. Post-development baking is especially beneficial for dry etching
with plasmas while it is less critical for wet processing.
4.1.4
Interdigitated Transducers
The fabrications steps described in previous sections are the ones which are similar for
fabricating both IDTs and PnCs, but after developing the pattens, IDTs and PnCs
follow different fabrication methods. The fabrication of IDTs require metallic lines
to be patterned on the GaAs surface. This section describe the fabrication method
and recipes for IDTs used in this work.
4.1.4.1
Transducer Patterns
To make a pattern by means of photolithograhy as explained earlier, a photo mask is
required. Two different photomasks, covering different frequency ranges, were used
for IDTs as outlined in table 4.1. The masks were designed with LASI CAD software
[8] and made by University of Alberta NanoFab Facility [40]. Photomasks are made
43
CHAPTER 4. EXPERIMENTATION
Crystal
(Mask1)
(Mask2)
λ (µm)
7.24 - 29.04
6.32 - 9.44
f (M Hz)
100 - 400
300 - 450
Table 4.1: IDTs features on the photomasks
on transparent glass covered with a chrome film to block the UV light in specific areas.
For this thesis, a negative tone resist is used to fabricate transducers. Therefore, the
chrome film covers the areas that metal needs to be deposited on. The general layout
of each photomasks contains a total of 148 delay lines (296 IDTs) with single and
double fingers configuration. Fig. 4.2 shows IDTs mask layout and an overlay of the
transducer and waveguide masks. Besides the transducers, each IDT mask contains
alignment marks for two purposes, first for aligning the transducers in the [011] or
[01̄1] crystal directions1 , and second for aligning marks to fabricate the PnCs exactly
in between the transducers afterwords.
4.1.4.2
Evaporation and Lift-off
Physical vapor deposition (PVD) is a process by which metal sources are heated
in high vacuum to evaporate and deposit on a wafer placed over the metal source.
Atoms can be ejected from the target by various means. The two primary types of
evaporation processes are thermal evaporation and electron-beam evaporation. The
latter had been used in this thesis to fabricate SAW transducer. In case of electron
beam evaporation, the target is heated by a localized electron beam to reach the
melting point. A high pressure vapour of metal then travels to the substrate in
a high vacuum chamber. Low melting point metals, such as gold and aluminum,
can easily be evaporated. In this work, IDTs are mainly fabricated using aluminum
1
The alignment bars in the photomasks should be parallel to the flats of GaAs wafers.
CHAPTER 4. EXPERIMENTATION
44
Figure 4.2: Photomask design: The top left one is the IDT photomask. On the top
right is an overlay of a group of IDTs and their corresponding PnCs.
On the bottom is a close up of a single finger IDT and a line waveguide
crystal.
CHAPTER 4. EXPERIMENTATION
45
Figure 4.3: Different resist profiles and corresponding deposited metals. On the left:
Enough undercut to provide a clean lift-off . On the right: Forming
continuous metal film (not enough undercut) will not allow the remover
to reach the resist to have a successful lift-off. Blue is substrate, orange
is resist, and gray is metal.
fingers.
When evaporation is complete, the resist will be removed by immersing the sample
in the compatible remover. The remover attacks the resist layer, hence the resist
and metals deposited on the areas that is still covered with resist, will lift off the
surface. Thus, metal is left wherever resist has been removed in development stage.
Of course, the thickest metal layer that can be removed is limited by the thickness
of the deposited resist itself. A negative resist is recommended for lift-off purposes.
Negative side-wall angle of this type of resist helps to provide an undercut during
the development process preventing the continuous metal deposition as shown in Fig.
4.3. This is necessary if a clean lift-off is desired.
4.1.4.3
IDTs Fabrication Procedure
Fabrication was performed at the Queen’s University Nanofabrication Facility (QFAB)
located in Jackson Hall. The diagram shown in Fig. 4.4 represents the steps taken
to make transducers on the GaAs samples.
Sample cleaning is done as described in section 1.2.1. After that, about 2 ml of
Micro-resist ma-N 405, a negative tone resist, is applied on the center of the wafer and
46
CHAPTER 4. EXPERIMENTATION
1. Clean Wafer
2. Resist Coating
4. Development
5. Metal Deposition
3. UV Exposure
6. Lift-off
Figure 4.4: Overview of IDT fabrication steps. Blue is substrate, orange is photoresist, yellow is the mask, and gray is metal.
CHAPTER 4. EXPERIMENTATION
47
then spun at 3000 rpm for 30 seconds. The resist thickness measured with an AFM
microscope, was about 0.45µm. However, the thickness of the resist is not exactly the
same on the entire sample. Resist can pile up at the wafer edges, causing an edge bead
problem that makes proper exposure during contact lithography difficult. Generally
the thickness of the resist is slightly thinner at the center areas of the substrate. A
pre-bake step is done at 95◦ C for 1 minute on a hot plate. The sample was then
exposed using an Oriel mask aligner in vacuum contact mode with a 1kW mercuryxenon lamp. The optimum exposure time for transducer fabrication was found to be
3 seconds. At this point the sample was immersed in ma-D 331S developer for about
110 seconds; examples of underdeveloped (90s) and overdeveloped (120s) samples are
shown in Fig. 4.5. A post-bake for one minute at 95◦ C on a hotplate was done
to make sample ready for metal deposition. A 40 nm thick layer of aluminum was
then deposited on the prepared sample using electron beam evaporation. At the final
step, the unwanted aluminum will be removed along with the photoresist underneath.
Acetone acts as a suitable remover for this purpose. This final lift-off process had to
be done in a sonicator bath for about 5 minutes.
4.1.5
PnCs
Once the desired IDTs are fabricated and tested, PnCs are added to the pattern.
The most common method of fabricating crystals (both photonic and phononic) is an
etching process. In the etching process the unprotected areas of substrate are attacked
and eroded chemically or physically. In particular, wet-etching is the removal of a
material by chemical reactions in a liquid chemical bath while in dry etching, plasmas
or etchant gasses remove the substrate material utilizing high kinetic energy of particle
CHAPTER 4. EXPERIMENTATION
48
Figure 4.5: Examples of different development times. Top: underdeveloped sample,
resist is not fully removed. Middle: a well-developed sample, ready for
deposition, the finger widths and the spacing between them are approximately equal. Bottom: edge quality is degraded, also the spacing and the
finger width are not equal.
CHAPTER 4. EXPERIMENTATION
49
Figure 4.6: Schematic of isotropic and anisotropic etching. Blue is the substrate,
orange is the resist mask, and white areas represent the etched regions.
beams, chemical reaction or a combination of both. One major difference between
the two methods is the lateral etch ratio value. Wet etching tends to be roughly the
same in both horizontal and vertical directions on the sample. The corresponding
lateral etch ratio is defined as
RL =
Horiontal Etch Rate
.
Vertical Etch Rate
(4.2)
This parameter for most of the wet-etch processes is RL ≈ 1, that results in an
isotropic etch profile as shown in Fig. 4.6. However, some level of anisotropy can be
introduced by adjusting the etchant concentration, temperature and crystal orientation if desired. On the other hand, in dry etching process RL ≈ 0 which results in
deep, uniform holes created in wafers with vertical side-walls. Selectivity is another
important features for etching process which is defined as the etch rate ratio between
two materials; i.e. the sample (GaAs) and the etch mask (resist). Selectivity has
normally higher value in wet-etching and that is one of the advantages of this method
over plasma etching.
Although dry etching seems the be the best choice for PnC fabrication, due to the
existence equipment of fabrication facility at Queen’s University, a wet-etch process
CHAPTER 4. EXPERIMENTATION
50
was used to create air holes on GaAs substrate.
Wet chemical etching of most semiconductors follows the same mechanism. This
process consist of two steps. First, an oxidizer chemical oxidizes the semiconductor
surface. Then an etchant (either acid or base) dissolves the oxidized layer of the
surface [22]. Hydrogen peroxide (H2 O2 ) is a common oxidizing agent to promote the
formation of the GaAs surface oxidization.
4.1.5.1
Wet Etching Procedure
There are a variety of solutions for GaAs etching. A citric acid and Hydrogen peroxide
solution is reported in several works as a candidate for GaAs wet-etching [29]. A 50%
acid solution is prepared by adding 30 grams of mono hydrate citric acid to 30 ml
of DI water. The solution needs to be completely dissolved with stirring for about
10 minutes. The ratio of citric acid and DI water volume to hydrogen peroxide can
change etch rates significantly. Through trial and error the ratio of 1:10 hydrogen
peroxide:citric acid is chosen. Thus, in the next step H2 O2 needs to be added to
the Citric acid and DI water solution. At this point, the GaAs substrate with the
photoresist mask at the desired area, provided by the discussed photolithography
process, is placed in the solution for 8 minutes.
The designed masks for PnCs contain square and triangle crystals with lattice
constants of 8µm and 4µm. A filling fraction of 0.45, the ratio of between the hole
area and unit cell area, is the same for every crystal on the photomask. However due
to the existence of undercut in wet-etch process and different resist thicknesses on the
sample, the filling fraction of fabricated devices varies between 0.5 to 0.65.
CHAPTER 4. EXPERIMENTATION
4.2
51
Sagnac Interferometry
Optical interferometry is a useful tool in measuring small quantities that are normally
difficult to otherwise measure. The basic idea behind optical interferometry is the
superposition of electromagnetic waves. Two different beams of light will be combined
such that the resulting pattern contains information on the desired physical quantity.
The interference pattern, a constructive and destructive superposition of the two
beams, is due to a phase difference between the two beams that itself is caused by
the system under study. Therefore, the crucial point is to couple the physical system
under study to the optical interferometer such that any change in the system results
in a definite phase difference and therefore definite interference pattern, which is a
physical quantity that can be measured.
A common method to introduce a phase difference between the two beams is to
have them travel different distances along two arms of the detector. The most famous
example of this approach is the Michelson interferometer that was initially used to
examine the speed of the light. Straightforward environmental noise can be different
along each of the two arms of the detector, which would cause a non-physical phase
difference and therefore false interference signal. A Sagnac interferometer minimizes
this effect by forcing the two beams to take exactly identical paths. This way, the
effect of environmental factors on the two beams is typically the same, leaving the
phase differences to be attributed to the system of interest.
Limiting environmental factors is critical to the types of experiments performed
in this thesis because of the small surface displacements and operating in the Radio
range of frequencies. Therefore, adopting a Sagnac type interferometer, is a wise
decision to make. In the following two subsections, an optical Sagnac interferometer
CHAPTER 4. EXPERIMENTATION
52
that was originally implemented by a former student of our group, Ruble Mathew
[23], will be discussed. Small but important modifications are made to enhance the
performance of the interferometer.
4.2.1
What do we want to measure?
As already discussed, SAWs are mechanical displacements of the sample surface created using electric voltages applied to an IDT. To identify SAW behavior on the
sample surface, one needs to detect mechanical displacements from equilibrium at
different locations on the sample. The mechanical displacement at each position is
itself a time dependent quantity. Therefore, if looking at the sample surface position (along z-direction, perpendicular to the surface), it will be slightly different at
different times depending on the frequency of the propagating SAW.
The Sagnac interferometer provides two beams of light with zero intrinsic phase
difference, though it has an induced phase due to travel distances within the two arms
of the interferometer. By directing these two beams of light toward the sample while
the SAW is propagating on the sample, each beam is sensitive to different phase of the
wave and will inherit different surface displacement information. This displacement
difference will cause an additional, small phase difference, between the two beams.
After returning through the Sagnac, through the opposite arms, the induced phase is
removed, and the interference pattern after superimposing the two beams is only due
to the SAW. Note that two beams can be still assumed to take identical paths as the
order of sample surface oscillations is small compared to the path lengths the beams
have to travel.
53
CHAPTER 4. EXPERIMENTATION
The phase difference between the two beams, δφ, relates to the surface displacement, δz, in the following form [36]:
δz = −
λ
4π
δφ.
(4.3)
Here, λ is the SAW wavelength. Therefore, for a known wavelength, measuring δφ
gives us the surface displacement, δz, as desired.
However, the arrival time difference of the beams has to be designed carefully in
order for the surface displacement to be detected. For example, if the induced phase
equals exactly one period of oscillation for the SAW propagating on the sample, the
two beams will travel exactly same distances, leading to no effective phase difference.
As a result, changing the relative length of the two arms of the Sagnac interferometer
enables one to effectively control the induced phase of the two beams.
To induce a phase difference, as seen in Fig. 4.7, the closed path that each one of
the two beams takes is broken down into two unequal parts (the shorter arm and the
longer arm) such that the order in which the two beams travel along the two sub-path
is different. One beam, for instance, takes the shorter path first before arriving at
the sample location and then takes the longer path when it comes back after being
reflected off the sample while the second beam does the opposite. In this way, the
phase at the sample location is different even though the two beams will eventually
have traveled identical journeys.
4.2.2
Optical experimental setup
As discussed earlier, in the Sagnac interferometer, the two beams of light are forced
to take the same path but in different orders, a clockwise arm circulation and a
counter clockwise arm circulation. The key point is to use light polarization to steer
54
CHAPTER 4. EXPERIMENTATION
Nirvana
Photodiode
M-4
Detector
M-3
QWP-2
Pol-2
M-5
M-2
PBS-2
PBS-1
NPBS
QWP-1
Pol-1
GaAs
Sample
M-6
M-1
Objective
Lens
Nirvana
Spatial
532 nm
Filter
Laser
Photodiode
M-4
Detector
M-3
QWP-2
Pol-2
M-5
M-2
PBS-2
PBS-1
NPBS
QWP-1
Pol-1
GaAs
Sample
M-6
Objective
Lens
M-1
Spatial
532 nm
Filter
Laser
Figure 4.7: Schematic of the Sagnac interferometer. The blue lines demonstrate the
path of the beam from the source to the sample, while the red one is for
the light reflecting from sample and is going toward the detector. The
top and bottom figures represent the two beams polarizations.
CHAPTER 4. EXPERIMENTATION
55
different beams of light in different directions. To understand how this works, the
polarization state of the light must be followed as the light passes through different
optical components of the setup in the order shown in Fig. 4.7.
A single beam of light is delivered by a continuous wave, 532 nm wavelength
diode pumped YAG laser is the optical source of the experiment. Immediately after
the source, the laser light passed through a spatial filter to ensure good spatial mode
quality. It then arrives at two consecutive mirrors facing each other used to handle the
beam alignment properly. At this point, no polarization manipulation has occurred,
and it can be assumed that the light polarization is the intrinsic polarization of the
light coming out of the laser.
However, a proper reference direction for the initial polarization of the beam is
desired. Ideally the light beam is split into two equal beams. Therefore, according
to the horizontal x-direction, a linear beam polarizer (Pol-1) is set at 45◦ in the
x̂ + ŷ direction. As a result, when the beam goes through the first polarized beam
splitter (PBS-1), two beams with equal intensities but different polarizations, namely
x-polarized and y-polarized, are created.
The non polarizing beam splitter (NPBS) does not operate on the polarization
state of the light and is not part of the interferometer setup. This only helps to redirect
the light coming back from the sample in a different direction than the incoming beam
so it can be collected at the detector. However, as the beam transmits through this
NPBS, half of the beam reflects out of the beam path and is wasted.
After the NPBS, the PBS-1 differentiates the different polarization components
of the light and redirects them in two different paths. As mentioned earlier, an
even distribution of intensity is achieved when Pol-1 is set at x̂ + ŷ direction. The
56
CHAPTER 4. EXPERIMENTATION
y-polarized component of the light passes directly through the PBS-1 while the xpolarized component deflect upward toward mirrors. As seen from the figure, the
upper path takes a longer time of flight so the light taking this path has acquired
an additional phase. Two mirrors on this upper path are used to redirect back the
light toward the second polarized beam splitter (PBS-2). The PBS-2 takes beams
with two different polarizations back on the same track and directs them to the
sample. Assuming that d is the distance between PBS-1 and first mirror, M-1, the
time difference, τ , between arrival of the two paths is
δτ =
2d
,
c
(4.4)
where c is the speed of light. In order to detect the maximum possible displacement
of the sample under SAW propagation, one sets this time difference equal to the half
of the period of the SAW, i.e:
δτ =
2d
1
= τSAW .
c
2
(4.5)
In general, this can be obtained by changing the mirrors position in the setup.
After being reflected from the sample, the two beams with different polarizations
have a Sagnac induced phase difference as well as a SAW-related phase difference.
However, a true Sagnac interferometry requires one to switch the paths for the two
polarizations in the return journey of the two beams. This can be accomplished using
a quarter wave plate (QWP-1) placed between the sample and the PBS-2 such that
the fast axis of the retarder is set at 45◦ with respect to the x-polarization axis.
QWP-1 changes a linear polarization to a circular polarization and vice versa. Thus,
the y-polarized (x-polarized) light is transformed onto clockwise (counter clockwise)
polarization when it first passes through the QWP-1. The two beams of light are then
CHAPTER 4. EXPERIMENTATION
57
transmitted through an objective lens to be focused on the sample and, after reflection
off the sample, will be collimated back through the same objective. Reflection off the
sample interchanges the left-hand and right-hand polarizations. Therefore, when the
two beams travel through the QWP-1 for the second time, they transform to opposite
polarizations compare to the incoming polarizations. In other words, the beam that
was initially x-polarized is now y-polarized and vice versa.
With the two beams having switched polarizations on the way back, the PBS-2
now redirects them toward the path they have not taken yet. The x-polarized beam
that came from the longer arm is now y-polarized, and therefore takes the shorter
arm in return. Conversely, the y-polarized that came from the shorter arm initially
is now x-polarized, and it is forced to take the longer arm of the Sagnac. The two
beams, then arrive at PBS-1, where they were initially split apart, and recombine
back again on the same path. At this point, the Sagnac-type preparation of the two
beams is complete.
The two combined beams now arrive at the NPBS where they are directed toward
the detector. However, one final polarization manipulation remains. A second quarter
wave plate (QWP-2) is used to create circular light that is less sensitive to the final
polarization. The fast axis for QWP-2 is set at 45◦ with respect to the x-polarization
axis.
Finally, before the detector, the two beams of light need to be superimposed. This
is done by passing the two beams through a polarizer. At this final stage, the light
is now in the same state, and the detector can detect the interferometric information
due to the phase differences picked up at the sample. The final single beam of the
light is optimized on the detector and a lock-in measurement is performed.
Chapter 5
Results and Discussion
This chapter presents results in the three distinct topics discussed in the earlier chapters. It begins with the performance characterization of the fabricated IDTs. Next,
the simulation results for the PnC waveguide design are discussed, and finally, the
optical measurements using Sagnac interferometer are presented.
5.1
IDT Characterization
Fabricated delay lines should be characterized before any further processing to best
determine their performance in the absence of a PnC. When an RF voltage from a
source is applied, a portion of the power is transmitted forward through the network
while some is reflected backward. Therefore, as discussed in Chapter 2, measuring
the scattering parameters (reflection and transmission) for a given delay line can be
used to examine the performance of the fabricated device. Fig. 5.1 shows plots of
S11 (reflection) and S21 (transmission) as a function of frequency for a single finger
IDT with a designed finger width of 2.42 µm. As is evident from the figure, a peak in
58
59
CHAPTER 5. RESULTS AND DISCUSSION
ϮϴϬ
ϮϵϬ
ϯϬϬ
ϯϭϬ
ϮϳϬ
ϯϮϬ
ͲϬ͘ϯ
ͲϯϬ
ͲϬ͘ϰ
Ͳϯϰ
^ϭϭ;ĚͿ
ͲϮϲ
ͲϬ͘ϱ
ϮϵϬ
ϯϬϬ
Ͳϯϴ
ͲϬ͘ϲ
ͲϰϮ
ͲϬ͘ϳ
Ͳϰϲ
ͲϬ͘ϴ
ϮϴϬ
ͲϱϬ
&ƌĞƋƵĞŶĐLJ;D,njͿ
ϯϭϬ
ϯϮϬ
^Ϯϭ;ĚͿ
ϮϳϬ
ͲϬ͘Ϯ
&ƌĞƋƵĞŶĐLJ;D,njͿ
Figure 5.1: Scattering parameter S11 (left) and S21 (right) for a single finger transducer
with a finger width of 2.42 µm. The measurements are for a 200 pair, 2port, single finger delay line, with a wavelength of 9.68µm and a resonance
frequency in the transmission with a peak at 293.72 MHz.
transmission and a dip in reflection both occur at 293.72 MHz. Recall from Chapter
2, for a single finger transducer with a finger width of w, the acoustic wave speed on
GaAs sample can be easily calculated as
VSAW = λν = 4wν.
(5.1)
Assuming that 9.68 µm is the true wavelength of the fabricated transducers, the
Rayleigh wave speed on the existing sample is equal to 2843.21 ms−1 , which is in a
good agreement with reported values in [15] for SAW on GaAs substrate in the h110i
direction. Therefore, this particular transducer is operating at its designed frequency.
The mask design for IDT fabrication contains 25 different delay lines corresponding
to 25 different frequencies. Depicted in Fig. 5.2 is the plot of wavelength versus
measured frequency for all fabricated delay lines on the same sample (the ones for
which scattering measurements could be performed). Linear dependence within the
60
&ƌĞƋƵĞŶĐLJ;DŚnjͿ
CHAPTER 5. RESULTS AND DISCUSSION
ϰϭϬ
LJсϮϴϰϱ͘ϴdžͲ Ϯ͘ϬϰϱϮ
ϯϲϬ
ϯϭϬ
ϮϲϬ
ϮϭϬ
ϭϲϬ
Ϭ͘Ϭϱ
Ϭ͘Ϭϳ
Ϭ͘Ϭϵ
Ϭ͘ϭϭ
Ϭ͘ϭϯ
Ϭ͘ϭϱ
tĂǀĞǀĞĐƚŽƌ;ϭͬђŵͿ
Figure 5.2: The fundamental frequency of the transducers, obtained from S11 measurements plotted versus wave vector. Transducers are single and double
fingers of aluminum on a GaAs substrate. The half width of S11 peaks
is about 4 MHz and therefore the error bars are too small to be shown
on the graph. The Linear fit is shown by the solid line. Inset shows the
equation for the fitted line.
frequency range of interest is observed which is again in agreement with previous
report and our expectations.
Initially, IDT characterization was performed using a probe station and two |Z|probes of model Z010K3N SG 500 connected to an Agilant E5071C ENA network
analyzer to measure the S-parameters. Once an appropriate delay line was identified,
the wafer was diced, and the device was wire bonded on a chip. Then, S-parameter
measurements were again performed using the wire bonded sample instead of using
the probes. Fig. 5.3 compares electrical scattering measurements on the same sample
with the two approaches. Interestingly, the scattering measurements performed using
the wire-bonded sample is significantly stronger compared to the probe station measurement while the general trend in S-parameters are the same. It is suspected that
in the first method of measurement, using the probes, there is a calibration issue that
61
CHAPTER 5. RESULTS AND DISCUSSION
ϮϮϬ
Ϭ
Ͳϭ
ͲϮ
Ͳϯ
Ͳϰ
Ͳϱ
Ͳϲ
Ͳϳ
ϮϳϬ
ϯϮϬ
ϯϳϬ
ϰϮϬ
ϰϳϬ
WƌŽďĞ
tŝƌĞŽŶĚ
&ƌĞƋƵĞŶĐLJ;D,njͿ
Figure 5.3: S11 measurement of a single device, using two different techniques. The
blue line is the probe station measurement, the red line is from the wirebonded sample.
causes this discrepancy. Another source of difference could be weak contact of the
probes with the sample. At this time, the exact origin of the discrepancy is unknown.
As a more representative measurement of the performance during optical measurements, the electrical measurements using the wire bonded sample are primarily used
to characterize the samples. However, since the sample must be diced to be then wire
bonded, taking wire-bonded S-parameter measurements before and after etching the
PnC structure between IDTs is not possible. In addition to the fact that any wire
left overs on the sample would degrade the PnC fabrication process and quality.
5.2
PnC Waveguide Design
Once the performance of the delay lines at desired frequencies is confirmed, phononic
crystal waveguides of interest can be fabricated between the transducers. The goal
CHAPTER 5. RESULTS AND DISCUSSION
62
is that the guided mode propagates along the channel while non guided modes dissipate in the PnC. The simplest waveguide structure is a line-defect waveguide, which
is made by removing a whole line of holes from the square PnC. The first designed
waveguide was a row of missing holes in square lattice crystal with lattice constant
of 8µm. The photomask was designed with a filling fraction of F = 0.45. However,
microscope images of the fabricated crystal revealed that F varied from 0.5 to 0.65
in different positions of the GaAs substrate, as discussed in Chapter 4. For the particular fabricated PnC waveguide shown in Fig. 5.4, the filling fraction of F = 0.5 is
obtained. In order to confirm the effects on the transmission due to the presence of
the waveguide, S-parameter measurements were performed before and while waveguide fabrication. Figure 5.5 represents typical S11 and S21 measurements for the IDTs
alone and when a PnC structure is fabricated between the IDTs. As seen from the
figure, the reflection is enhanced by introducing the PnC waveguide in the network
along with a decrease in the transmission. This would indicate that waveguide does
not seem to significantly alter the transmission of the network. While the increase of
S11 seems to indicate some fraction of the SAWs is being reflected back to the transducer, the remaining relatively strong transmission shows the SAW are still passing
through the PnC. There are two main aspects of this observation that need to be
discussed in more detail as follows.
First, it should be noted that the PnC fabrication is much more difficult than
IDT fabrication due to the small feature size involved and the nature of the wet-etch
process used for the PnC fabrication. In particular, the hole depth for fabricated
PnCs might not be as deep as desired. The best etch depth achieved on GaAs at the
Queens fabrication facility was around 2 microns, whereas this value is smaller than
63
CHAPTER 5. RESULTS AND DISCUSSION
Figure 5.4: Line waveguide PnC with lattice constant 8µm and filling fraction of 0.5.
The PnC waveguide is fabricated by wet-etching between a 200 pair single
finger transducer with wavelength of 10.08µm.
ϮϬϬ
ϮϬϱ
ϮϭϬ
Ϯϭϱ
ϭϵϱ
ϮϮϬ
ͲϬ͘ϭϱ
ͲϱϬ
^ϭϭ;ĚͿ
Ͳϰϱ
ͲϬ͘Ϯ
ͲϬ͘Ϯϱ
/dнWŶ
ϮϬϬ
ϮϬϱ
ϮϭϬ
Ͳϱϱ
ͲϲϬ
Ϯϭϱ
ϮϮϬ
^Ϯϭ;ĚͿ
ϭϵϱ
ͲϬ͘ϭ
/dнWŶ
ĂƌĞ/d
ĂƌĞ/d
Ͳϲϱ
ͲϬ͘ϯ
&ƌĞƋƵĞŶĐLJ;D,njͿ
&ƌĞƋƵĞŶĐLJ;D,njͿ
Figure 5.5: Reflection (left) and transmission (right) for a delay lines operating at
frequency within a band gap of a square crystal with lattice constant of
8µm. The red and blue lines correspond to measurements before and
after etching the PnCs, respectively.
CHAPTER 5. RESULTS AND DISCUSSION
64
the penetration depth of the SAWs which is on order of the SAW wavelength. As
a result, the fabricated PnC waveguide does not work effectively as a phononic BG
structure for the propagating SAW. Because of the depth limitation in the wet-etch
process, the penetration depth of the SAW must be reduced in order for the PnC
structure to be more effective. Therefore, a new design for the PnC was needed that
shifted the phononic BG to higher frequencies (smaller wavelengths). This way, with
the same wet-etch depth, the goal is to obtain better transmission through fabricated
waveguides.
FDTD simulations were performed on PnC structures with smaller hole radii to
determine the BG shifts to higher frequencies. Shown in Fig. 5.6 is the phononic
band structure for two square lattice PnCs, where the hole radius for one is twice
the other. Agreeing with intuition, the simulation confirms that reducing the hole
radius by a factor of two shifts the the resonant frequencies of the structure up by the
same factor. The phononic BG position was originally between 180 − 220 MHz for
the lattice period of a = 8 µm, while the new BG for the lattice period of a = 4 µm
is shifted to 360 − 440 MHz. With the new design, the corresponding operation
wavelengths, and therefore SAW penetration depths, can be reduced to 6 − 8 µm.
This is still larger than the maximum wet-etch depth possible, 2 µm. Fortunately
most of the energy density is localized near the surface. However, reducing the PnC
features further is not possible due to feature size limitation of lithography process.
By the present design, the smallest lateral features on the crystal structure is 0.5 µm
which was the smaller value accessible in the fabrication process. In fact, the choice
of lattice period of 4 µm seems to be optimum trade off between the small crystal
feature size and large hole depth.
CHAPTER 5. RESULTS AND DISCUSSION
65
a) Band structure for square lattice crystals with lattice constant a = 8µm simulated
using PnCSim developed in our group. The corresponding filling fraction is 0.55. The
BG ranges from 180 MHz to 220 MHz.
b) Band structure for square lattice crystals with lattice constant a = 4µm simulated
using PnCSim developed in our group. The corresponding filling fraction is 0.55.
The BG ranges from 370 MHz to 440 MHz. The BG shifts to higher frequencies and
increases in size as well.
Figure 5.6: Band gap comparison for two different lattice constants of a = 8µm on
the top and a = 4µm on the bottom.
CHAPTER 5. RESULTS AND DISCUSSION
66
As discussed previously, the fabricated hole radii varies locally on GaAs. Therefore, it is useful to have information on the BG position for different filling fractions.
Depicted in Fig. 5.7, are the band structures of a = 4 µm lattices when the filling
fraction changes from F = 0.5 to F = 0.65 based on the average values normally
obtained from sample hole radii measurements. Note that the filling fraction directly
affects the BG position.
With the simple line waveguide design, the waveguide entrance covers only a
small portion of the wave front area incident on the PnC structure, and only a small
fraction of the wave couples into the waveguide while the majority gets reflected by
the periodic crystal structure. This results in low transmission power through the
waveguide. Therefore, an alternative waveguide design with a focus on an improved
entrance coupling is needed. One possible implementation is to remove more holes
along both sides of the line-defect waveguide to funnel the SAW wave front and couple
it into the waveguide.
FDTD simulations were performed on three different waveguides in order to study
the effect of the entrance design on the coupling efficiency into the waveguide. The
initial line waveguide was modified by removing one, three and six holes at each
side of the waveguide entrance. Depicted in Fig. 5.8 is the SAW displacement as
a function of position. From the simulations, the acoustic power increases inside
the waveguide when removing more holes at the waveguide entrance. This suggests
the line waveguide with six removed holes from each side as the best candidate for
improved transmission. Ideally, one would like to take off many more holes in order
to increase wave coupling into the waveguide. However, based on the PnC design we
already had, removing more holes would change the waveguide geometry such that
CHAPTER 5. RESULTS AND DISCUSSION
67
a) Band structure for square crystals with
lattice constant a = 4 µm and filling fraction of F = 0.5. The corresponding band
gap ranges form 406 MHz to 472 MHz.
b) Band structure for square crystals with
lattice constant a = 4 µm and filling fraction of F = 0.55. The corresponding band
gap ranges form 381 MHz to 439 MHz.
c) Band structure for square crystals with
lattice constant a = 4 µm and filling fraction of F = 0.6. The corresponding band
gap ranges form 340 MHz to 405 MHz.
d) Band structure for square crystals with
lattice constant a = 4 µm and filling fraction of F = 0.65. The corresponding band
gap ranges form 307 MHz to 368 MHz.
Figure 5.7: Band gap comparison for four different filling fraction.
CHAPTER 5. RESULTS AND DISCUSSION
68
Figure 5.8: Simulated outward surface displacement for 410 MHz SAWs incident on
square crystal with a = 4µm of filling fraction 0.55 with different waveguide geometries.
the waveguide itself would be limited. If one were to design a new mask from scratch
with the freedom to increase the lattice size, removing more holes from the ends of
the waveguide would be possible.
With regard to these investigations, the final design settled upon was to reduce the
lattice scale of the PnC by a factor of 2. Also, all three modified types of waveguides
were fabricated in order to confirm our simulations in experiment. This design was
used for final fabrication and optical measurements as is discussed in the next section.
CHAPTER 5. RESULTS AND DISCUSSION
5.3
69
Sagnac optical interferometry
As introduced in Chapter 4, a Sagnac interferometer can be used for sample imaging
and SAW detection. Different reflectivity coefficients at different sample locations
change the detected intensity as the sample is scanned across, allowing the sample
to be imaged. In addition, measuring the phase difference between the two beams of
laser light with different polarizations, with a well defined phase difference between
their arrival on the sample, is the key concept in SAW mapping.
Imaging the sample due to reflectivity changes in different materials can be considered a pre-test for the actual SAW detection experiment that requires interference
type measurements. In particular, being at the right focus on the sample for the laser
light has a significant effect on interferometer measurements. An intensity map of the
device under test indicates how well the beam is focused on the sample. In practice,
a couple of quick (with low resolution) 2D scan of the sample surface need to be done
in order to obtain the best focus point for the intensity map. Fig. 5.9 shows an
intensity map of a 1.72 µm aluminum transducer finger width on GaAs sample with
spatial step of 0.5 µm in both x- and y-directions.
In addition to a best focus confirmation, an intensity image also confirms sufficient lateral spacial resolution in identifying features on the order of the transducer
fingers. Therefore, detection of the GaAs sample surface displacements on the order
of the SAW wavelength, which is itself on the order of the transducer finger width, is
possible.
Once the Sagnac interferometer is properly aligned, a 2D displacement map of
the device can be obtained. To do this, the IDTs of the delay line are powered by
an sinusoidal RF signal, using Agilent N5183A MXG signal generator, at the delay
CHAPTER 5. RESULTS AND DISCUSSION
70
Figure 5.9: Normalized intensity of a 1.72µm finger width transducer mouth taken
by Sagnac interferometer. The aluminum fingers and pad are significantly more reflective than GaAs substrate. Fabrication debris is observed
around the fifth finger on the intensity image.
71
CHAPTER 5. RESULTS AND DISCUSSION
Ͳϱ
ϰϬϴ
ϰϬϵ
ϰϭϬ
ϰϭϭ
ϰϭϮ
ϰϭϯ
&DĞƉƚŚ
^ϭϭ;ĚͿ
ϰϬϳ
Ͳϳ
Ͳϵ
Ͳϭϭ
Ͳϭϯ
&ƌĞƋƵĞŶĐLJ;D,njͿ
Figure 5.10: A close view of reflection measurement from Fig. 5.13. The red arrow
indicates the FM depth for a typical sample that used to map with
Sagnac.
line designed resonance frequency which is the dip and peak from the S-parameters
measurement.
From previous experiments, the experiment was initially setup such that the RF
signal was amplitude modulated at a low frequency that is controlled by a TTL signal
provided by the lock-in amplifier. However, due to the crosstalk between the signal
and detection cables, signal extraction was not successful in amplitude modulation
technique. To eliminate the crosstalk, frequency modulation (FM) was adopted instead. In this case, the original RF signal is frequency modulated up to a 5 MHz
range. As shown in Fig. 5.10, the FM can be tuned to turn the SAW on and off so
that a lock-in detection can be performed. While RF crosstalk will still be present,
it will not be modulated by the FM and is then removed from the detected signal.
The detected interference signal is measured using a New Focus 2007 Nirvana Silicon
photodiode which is then delivered to the lock-in for signal extraction.
Fig. 5.11 shows a Sagnac interferometer measurement for a waveguide with square
crystal structure of a = 4 µm and one missing hole on either side of the waveguide
CHAPTER 5. RESULTS AND DISCUSSION
72
entrance. The PnC is located between a pair of 200 single fingers transducers with
finger width of 2.23 µm. The resonance frequency of the transducers is 317.86 MHz,
which is determined by measuring the S-parameters. Thus, the operated frequency
of delay line in this case, is below the predicted BG of the crystal shown in Fig.
5.6. The figure shows the normalized reflected intensity from the sample as well as
the SAW displacement taken simultaneously when the delay line is excited at 317.86
MHz. Since operating below the BG, no specific influence due to the PnC structure
is expected. However, a traveling wave down the waveguide is being detected. We
believe that this periodic pattern is only a standing wave resulting from in-phase
interference of forward and backward traveling waves between the two transducers of
the delay line.
The most apparent reason that the etched array is not acting like a PnC is that
the wave pattern inside the waveguide is not any different than the wave pattern
outside the waveguide, which suggests that this is not likely a guided mode of the
line waveguide. Normally, one might expect wavelength of the guided modes inside
a waveguide to have the same period as the PnC. Depicted in Fig. 5.12 is the ycut of the standing wave along the waveguide position. The wavelength of the wave
pattern inside the waveguide is about 4.5 µm which is half of the wavelength of the
transducer and it does not follow the 4µm period from the crystal. These confirm
that the pattern is a standing wave due to reflection off the second transducer.
In a second sample, the displacement mapping with the Sagnac interferometer
is quite different for a PnC waveguide with a SAW having a frequency inside the
PnC bandgap. Fig 5.13 shows the reflection and the transmission parameters for this
device that show an operating frequency near 410 MHz. Fig. 5.14 depicts the SAW
CHAPTER 5. RESULTS AND DISCUSSION
73
Figure 5.11: Measured displacement map of a 317.86 MHz SAW. The SAW frequency
is lower than the crystal BG. Top: normalized intensity of reflected light
obtained near the entrance to the waveguide. Bottom: SAW displacement near the waveguide entrance taken simultaneously with the reflection image.
CHAPTER 5. RESULTS AND DISCUSSION
74
tĂǀĞŐƵŝĚĞŶƚƌĂŶĐĞ
Figure 5.12: Plot of y-cut displacement of the standing wave averaged inside the
waveguide shown in Fig. 5.11
propagation and interference pattern detected for the waveguide with three missing
holes on either side of its entrance. The square host crystal has a lattice period of
4 µm in GaAs. The applied RF signal to the IDT has a frequency of 410.344 MHz
that is designed to be in the middle of the BG for the 4 µm lattice period crystal with
a filling fraction of 0.55. As seen in the figure, the normalized reflected intensity shows
the PnC waveguide pattern (top) while on the bottom is the outward displacement
for the SAW pattern on the sample surface. Strong evidence of SAW interference on
the free portion of the sample (the wave-like pattern) and the waveguide entrance is
observed which is in agreement with the simulations presented earlier. In particular,
note the localized acoustic anti-nodes that are also in the simulation but absent from
Fig. 5.11. To examine the distance that the SAW travels inside the waveguide a much
longer scanning time is required. In order to reduce the time for a single scanning
processes, a longer scan of the same sample is performed with lower resolution and
75
CHAPTER 5. RESULTS AND DISCUSSION
ϰϬϬ
ϰϭϬ
ϰϮϬ
ϰϯϬ
ϯϵϬ
Ͳϲ
ͲϯϮ
^ϭϭ;ĚͿ
ͲϯϬ
Ͳϴ
ͲϭϬ
ͲϭϮ
ϰϬϬ
ϰϭϬ
ϰϮϬ
ϰϯϬ
^Ϯϭ;ĚͿ
ϯϵϬ
Ͳϰ
Ͳϯϰ
Ͳϯϲ
Ͳϯϴ
Ͳϭϰ
ͲϰϬ
&ƌĞƋƵĞŶĐLJ;D,njͿ
&ƌĞƋƵĞŶĐLJ;D,njͿ
Figure 5.13: Reflection (on the left) and transmission (on the right) measurements of
PnC waveguide with a delay line. The image of Sagnac interferometer
for this device is shown in Fig. 5.14.
is depicted in Fig. 5.15. As seen in the figure, for a few of lattice periods down the
waveguide, SAWs have been detected.
A 1D y-cut of the interference pattern is shown in Fig. 5.16. Existence of the
standing wave outside the waveguide is observed along with a wave with decreasing amplitude inside the waveguide. This is also further evidence of waveguiding in
contrast with the previous measurements. Note that the pattern inside the waveguide is quite different than the standing wave pattern outside. This confirms that
transmission is due to the waveguiding as opposed to an traveling over the crystal.
We can also qualitatively infer that the wavelength of the guided mode is different
than the transducer generated wavelength, which would again be due to the fact that
waveguiding is achieved, but the strong attenuation makes it difficult to definitely
determine the wavelength.
Unfortunately, no evidence of full waveguiding down to the other end of waveguide
was observed when scanning over a wider range of positions. This was expected as
our S-parameter measurements did not show evidence of strong transmission for these
CHAPTER 5. RESULTS AND DISCUSSION
76
Figure 5.14: Measured displacement map of a 410.344 MHz SAW. The SAW frequency is within the crystal BG. Top: normalized intensity of reflected
light obtained near the entrance to the waveguide. Bottom: SAW displacement near the waveguide entrance taken simultaneously with the
reflection image.
CHAPTER 5. RESULTS AND DISCUSSION
77
Figure 5.15: Measured displacement map of a 410.344 MHz SAW. The SAW frequency is within the crystal BG. Top: normalized intensity of reflected
light obtained near the entrance to the waveguide. Bottom: SAW displacement near the waveguide entrance taken simultaneously with the
reflection image.
CHAPTER 5. RESULTS AND DISCUSSION
78
Figure 5.16: Plot of y-cut displacement of the standing wave averaged inside the
waveguide shown in Fig. 5.14
samples. We suspect that this is most probably because of the slanted side wall of the
crystal holes due to the nature of wet-etching process. The slanted side walls results in
increased SAW scattering into the bulk material compare to perfect vertical sidewalls
[28]. Also, as mentioned earlier, deeper holes (at least in the order of SAW penetration
length) may improve propagation of guided mode along the waveguide.
Chapter 6
Conclusions
6.1
Summary
This thesis investigates the possibility of phononic crystal waveguiding on GaAs.
Using FDTD, different PnC geometries (by varying the lattice constant and filling
fraction) were examined to design the desired phononic BG. FDTD simulations of
different waveguide geometries were performed to model SAW waveguiding in PnC
structured waveguides. The most reliable waveguide designs were fabricated in the
clean room facility at Queen’s using a wet-etching process. A scanning Sagnac interferometer is used to map the SAW propagation on the sample surface and SAW
waveguiding through the PnC region.
Initially, a square lattice PnC with lattice constant of a = 8µm was chosen. A
phononic BG ranging form 190 − 210 MHz was predicted corresponding to the filling
fraction of F = 0.55. Within the frequency range of 100 − 300 MHz, SAW delay lines
were designed to be placed on sides of the crystal region for excitation and detection.
Clean room lift-off photolithography was used to fabricate acoustic transducers
79
CHAPTER 6. CONCLUSIONS
80
on piezoelectric GaAs substrates. Electrical measurements were made on bare delay
lines to evaluate the IDT performance. Our periodic PnCs consisting of air holes
were fabricated on the same GaAs sample between the IDTs using a citric acid and
hydrogen peroxide wet-etching process. The resulting holes from this etch technique
were found to significantly undercut the photoresist etch mask, thus resulting in PnCs
of a larger filling fraction than designed ones. After etching, electrical measurements
were also performed to look for any changes in S-parameters due to the existence of
the PnC structure.
Electrical measurements on a simple line-defect waveguide in our PnC structure
did not satisfy expectations regarding efficient waveguiding. To overcome the weak
SAW coupling to the line waveguide, several waveguide geometries were examined and
simulated using FDTD. Among different waveguide designs, the funneling waveguide
entrance shows the strongest SAW coupling to the waveguide. Three different geometries of such waveguides are designed with removing one, three and six holes from
either sides of the waveguide entrance. Simulations showed stronger wave coupling
to the waveguide by removing more holes from the waveguide entrance.
Due to the limitations in the wet-etch fabrication process, the maximum etch
depth that could be achieved for our PnC holes was 2µm. This was an order of magnitude smaller than the originally designed SAW wavelength. The penetration depth
for the Rayleigh waves, generated with IDTs on GaAs sample is approximately the
SAW wavelength. Therefore, reducing the SAW wavelength seemed to be an effective
solution toward having a BG structure at all depths that SAW travels. Fortunately,
this can be accommodated by reducing the lattice constant of the designed PnCs
structures. FDTD simulations show an increase in the BG frequency and BG size
CHAPTER 6. CONCLUSIONS
81
when the crystal is scaled down while keeping the filling fraction constant. The 4µm
lattice constant of the PnCs is the smallest value that can be achieved corresponding
to the smallest lateral feature size of 0.5µm on the substrate for the PnCs with filling
fraction of 0.65.
A PnC photo mask was designed that included three different funneling waveguide
entrances for the host PnC crystal. These new crystals had a lattice constant of 4µm
corresponding to a phononic BG of 300 − 480 MHz.
Finally, a Sagnac interferometer was used to image the surface displacement of
the GaAs sample due to the traveling SAW. Interferometric measurements show the
existence of PnC waveguiding for the SAWs when operating within the BG range.
When the PnC structure was excited at frequencies outside BG, unaltered standing
waves formed due to reflections off the second transducer interfering with the forward traveling SAWs inside the waveguide. When operating within the BG range,
wave interference was observed that was in agreement with our FDTD simulations
and electrical measurements as no transmission was expected. Unfortunately, strong
scattering of the SAWs into bulk waves occurred for guided waves and limited the
propagation of SAWs through the channel.
6.2
Future Work
At this point, there are several points of recommendations for further implementations
of the SAW waveguides using PnCs.
First of all, improvements in the fabricated PnC structures is necessary by adopting alternative etch techniques. The wet etched holes tend to have slopped side walls,
CHAPTER 6. CONCLUSIONS
82
which is not accounted for in our FDTD simulations, and it is not desired for optimum operation of the device. However, RIE can be used to obtain vertical side
walls. At the present time, RIE equipment is not available at Queen’s, and several
arrangements with external fabrication facilities were made to make vertical side wall
PnCs on GaAs. So far, this has not been successful, due to difficulties on obtaining
a suitable etch mask to be used in RIE process.
In order to achieve better waveguiding, broader band transducers need to be
designed. Currently, we have not exact information as to what is the best guided
mode of a fabricated PnC waveguide in terms of operating frequency. A broader
band IDT for a single PnC would enable excitations at the proper frequency to be
done, and most probably, better coupling to the waveguide will be achieved.
Finally, IDTs at higher powers can be designed in order to achieve higher signals
inside the waveguide. Linear IDTs were used in this thesis and generated plane waves.
However, curved finger IDT designs are one alternative to consider for obtaining
higher output powers from IDTs at a focal point [11].
Bibliography
[1] M. Atatre J. Dreiser E. Hu P. M. Petroff A. Imamoglu A. Badolato, K. Hennessy.
Deterministic coupling of single quantum dots to single nanocavity modes. Science, 308(5725):1158–1161, 2005.
[2] Aaron Kaim Taylor Ali. Gigahertz modulation of a photonic crystal cavity.
Master’s thesis, Queen’s University, 2013.
[3] B. A. Auld. Acoustic fields and waves in solids. Wiley Interscience, 1973.
[4] J. Taylor N. Dandekar B. Lu, O. Vladimirsky. Surface chemistry of gaas wafers
and reaction with chemically amplified resist during resist processing. Proceedings
of SPIE, (pp. 525-530):525–530, 1998.
[5] Albert G. Baca and Carol I. H. Ashby. Fabrication of GaAs Devices. The
instituition of Engineering and Technoloy, London, United Kingdom, 2009.
[6] J. Bauer, G. Drescher, and M. Illig. Surface tension, adhesion and wetting of materials for photolithographic process. Journal of Vacuum Science & Technology
B, 14(4):2485–2492, 1996.
83
84
BIBLIOGRAPHY
[7] S. Benchabane, A. Khelif, J.-Y. Rauch, L. Robert, and V. Laude. Evidence for
complete surface wave band gap in a piezoelectric phononic crystal. Physical
Review E, 73:065601, Jun 2006.
[8] Lasi CAD.
The layout system for individualists.
Available at http: //
lasihomesite. com/ index. htm .
[9] Colin K. Campbell. Surface Acoustic Wave Devices for mobile and wireless
communications. Academic Press, INC, 1998.
[10] O. D. D. Couto, S. Lazić, F. Iikawa, J. A. H. Stotz, U. Jahn, R. Hey, and P. V.
Santos. Photon anti-bunching in acoustically pumped quantum dots. Nature
Photonics, 3(11):645–648, October 2009.
[11] M. M. de Lima, F. Alsina, W. Seidel, and P. V. Santos.
Focusing of
surface-acoustic-wave fields on (100) gaas surfaces. Journal of Applied Physics,
94(12):7848–7855, 2003.
[12] M. M. de Lima and P. V. Santos. Modulation of photonic structures by surface
acoustic waves. Reports on Progress in Physics, 68(7):1639, 2005.
[13] P. A. Deymier. Acoustic Metamaterials and Phononic Crystals. Springer Heidelberg New York Dordrecht London, 2013.
[14] J. M. Elzerman, R. Hanson, L. H. W. van Beveren, B. Witkamp, L. M. K.
Vandersypen, and L. P. Kouwenhoven. Single-shot read-out of an individual
electron spin in a quantum dot. Nature, 430(6998):431–435, 2004.
BIBLIOGRAPHY
85
[15] C.M. Flannery, E. Chilla, S. Semenov, and H-J Frohlich. Elastic properties of
gaas obtained by inversion of laser-generated surface acoustic wave measurements. In Ultrasonics Symposium, 1999. Proceedings. 1999 IEEE, volume 1,
pages 501–504 vol.1, 1999.
[16] S. Franssila. Introduction to Microfabrication. John Wiley & Sons, Ltd, second
edition, 2010.
[17] D. A. Fuhrmann, S. M. Thon, H. Kim, D. Bouwmeester, P. M. Petroff, A. Wixforth, and H. J. Krenner. Dynamic modulation of photonic crystal nanocavities
using gigahertz acoustic phonons. Nature Photonics, 5(10):605–609, 2011.
[18] H. J. Kimble. The quantum internet. Nature, 453(7198):1023–1030, 2008.
[19] M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G.Abstreiter, and
J. Finley. Optically programmable electron spin memory using semiconductor
quantum dots. Nature, 432(7013):81–84, 2004.
[20] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, and B. Djafari-Rouhani.
Theory of acoustic band structure of periodic elastic composites. Physical Review
B, 49(4):2313–2322, 1994.
[21] V. Laude, M. Wilm, S. Benchabane, and A. Khelif. Full band gap for surface
acoustic waves in a piezoelectric phononic crystal. Physical Review E, 71:036607,
2005.
[22] M. V. Lebedev, E. Mankel, T. Mayer, and W. Jaegermann. Wet etching of gaas
(100) in acidic and basic solutions: A synchrotron-photoemission spectroscopy
study. Journal of Physical Chemistry C, (100):18510–18515, 2008.
BIBLIOGRAPHY
86
[23] R. Mathew. Creating and imaging surface acoustic waves on gaas. Master’s
thesis, Queen’s University, 2009.
[24] T. Miyashita. Sonic crystals and sonic wave-guides. Measurement Science and
Technology, 16(5):R47–R63, 2005.
[25] D. Morgan. Surface acoustic wave filters with application to electronic communications and signal processing. Elsevier, Ltd, second edition, 2007.
[26] V. Narayanamurti, H. L. Störmer, M. A. Chin, A. C. Gossard, and W. Wiegmann. Selective transmission of high-frequency phonons by a superlattice: The
”dielectric” phonon filter. Physical Review Letters, 43:2012–2016, 1979.
[27] J. L. O’Brien, A. Furusawa, and J. Vuckovic. Photonic quantum technologies.
Nature Photonics, 3:687, 2009.
[28] R. Olsson III and I. El-Kady. Microfabricated phononic crystal devices and
applications. Measurement Science and Technology, 20(1):012002, 2009.
[29] M. Otsubo, T. Oda, H. Kumabe, and H. Miki. Preferential etching of gaas
through photoresist masks. Journal of The Electrochemical Society, 123(5):676–
680, 1976.
[30] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus,
and I. Kim. Two-dimensional photonic band-gap defect mode laser. Science,
284(5421):1819–1821, 1999.
[31] J. A. Petrus. A computational and experimental study of surface acoustic waves
in phononic crystals. Master’s thesis, Queen’s University, 2009.
BIBLIOGRAPHY
87
[32] J. W. Rayleigh. On waves propagated along the plantar surface of an elastic
solids. Proceedings of the London Mathematical Society, 17(42), 1885.
[33] D. Royer and E. Dieulesaint. Elastic Waves in Solids I: Free and Guided propagation. Springer, 1996.
[34] J. A. H. Stotz, R. Hey, P. V. Santos, and K. H. Ploog. Coherent spin transport
through dynamic quantum dots. Nature materials, 4(8):585–8, 2005.
[35] J. Sun and T. Wu. Propagation of surface acoustic waves through sharply
bent two-dimensional phononic crystal waveguides using a finite-difference timedomain method. Physical Review B, 74(17):174305, 2006.
[36] T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B.
Wright. Scanning ultrafast sagnac interferometry for imaging two-dimensional
surface wave propagation. Review of Scientific Instruments, 77(4):043713, 2006.
[37] A. Taflove. Computational Electrodynamics: The finite Difference Time Domain
Method. Artech House, 2nd edition edition, 2000.
[38] Y. Tanaka and S. Tamura. Surface acoustic waves in two-dimensional periodic
elastic structures. Physical Review B, 58(12):7958–7965, 1998.
[39] Y. Tanaka and S. Tamura. Acoustic stop bands of surface and bulk modes in twodimensional phononic lattices consisting of aluminum and a polymer. Physical
Review B, 60:13294–13297, 1999.
[40] University of Alberta. Nanofab - canada’s premier micro and nanofabrication
facility., 2013. Available at http: // www. nanofab. ualberta. ca .
BIBLIOGRAPHY
88
[41] R. M. White and F. W. Voltmer. Direct Piezoelectric Coupling To Surface Elastic
Waves. Applied Physics Letters, 7(12):314, 1965.
[42] R.M. White. Surface elastic waves. Proceedings of the IEEE, 58(8):1238–1276,
1970.
[43] T. Wu, Z. Huang, and S. Lin. Surface and bulk acoustic waves in two-dimensional
phononic crystal consisting of materials with general anisotropy. Physical Review
B, 69:094301, 2004.