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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. 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