Finite Element Analysis of Surface Acoustic Wave Resonators A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Electrical Engineering University of Saskatchewan Saskatoon, Saskatchewan by Thirumalai Kannan c Copyright Thirumalai Kannan, June 2006. All rights reserved. (cid:13) PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this Uni- versity may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or in part should be addressed to: Head of the Department of Electrical Engineering University of Saskatchewan Saskatoon, Saskatchewan, Canada S7N 5A9 i ABSTRACT Surface Acoustic Wave (SAW) devices are key components in RF and IF stages of many electronic systems. A Surface Acoustic wave is a mechanical wave, which is excited on the surface of a piezoelectric substrate, when an alternating electric voltage is applied through a comb-like interdigital transducer (electrodes) patterned on it. Most SAW applications to date have been in the sub-2GHz region, but emerg- ing applications require SAW devices at higher frequencies. The traditional models are inadequate to account for pronounced second order effects at the GHz range and also new microfabrication techniques are required to obtain quality devices as the critical dimensions shrink into the nano-scale range at these frequencies. The finite element method (a numerical method of solving differential equations) has the potential to account for these effects and ever increasing sub-micron process- ing capabilities of LIGA (X-ray lithography) present a promising outlook for high frequency SAW device modeling and fabrication respectively. A finite element model has been developed using commercial software ANSYS for one port SAW resonators and is presented in this thesis. The one port SAW resonators are generally connected in form of ladder networks to form low-loss SAW filters. The spacing between the electrodes and the velocity of the SAW deter- mine the frequency of operation of these devices. A finite element model has been developed forthree different types of SAWdevices namely Rayleigh, leaky andlongi- tudinal leaky SAW (LLSAW). The LLSAW has higher velocity as compared to other two types and hence considered in this work as a good prospect for high frequency SAW devices. A full finite element model could not be solved due to high computing require- ments andhence someassumptions were madeandtheresults werevalidatedagainst published resultsintheliterature. Theresultsindicatethatevenwithsimplifying as- ii sumptions and approximations FE model provides reasonably accurate results, that can be used in device design. Some of the simulations (in LLSAW based devices) in this work were also done with a view towards using LIGA (X-ray lithography) for fabrication of high frequency devices as they have the capability for high aspect ratios. iii ACKNOWLEDGEMENTS Iwould like to thank my supervisor, Dr. David M.Klymyshyn for providing me this wonderful opportunity to do my Masters degree and conduct some useful research. His unwavering support, invaluable guidance and suggestions in performing this research and presenting the thesis are greatly appreciated. I would also like to thank TRLABS for providing research equipment and nice work environment without which smooth conduct of research work would not have been possible. In particular, I would like to thank Jack Hanson for helping me with computing resources whenever I approached him with a problem. Technical inputs from Anton Kachayev and Venkat Subramanian needs a special mention and their help was vital for this project. Also,IwouldonceagainliketoexpressmygratitudetoDr. DavidM.Klymyshyn and the University of Saskatchewan for providing me with financial assistance for this project. Finally, I would like to thank my parents and siblings for their constant encour- agement and support during this course. I am greatly indebted to Subramanian maama for his special efforts in helping me prepare documents required for Cana- dian visa/study permit. I would also like to extend my thanks to all my friends who kept me in good spirits during my stay here. iv TABLE OF CONTENTS PERMISSION TO USE i ABSTRACT ii ACKNOWLEDGEMENTS iv TABLE OF CONTENTS v LIST OF FIGURES viii LIST OF TABLES xi LIST OF ABBREVIATIONS xii 1 Introduction 1 1.1 Surface Acoustic Wave Devices - An Introduction . . . . . . . . . . . 1 1.1.1 Classification of Acoustic Devices . . . . . . . . . . . . . . . . 2 1.1.2 Features of SAW and PSAW Devices . . . . . . . . . . . . . . 3 1.1.3 SAW Devices - Applications . . . . . . . . . . . . . . . . . . . 4 1.2 Research Motivation and Focus . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Materials for Higher Velocities and Performance . . . . . . . . 8 1.2.2 Modes for Higher Frequencies . . . . . . . . . . . . . . . . . . 9 1.2.3 Sub-micron Fabrication for High Frequency SAW Devices . . . 10 1.2.4 Need for Better Models . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Thesis - Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 v 2 Surface Acoustic Waves - Physics 16 2.1 Elastic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Acoustic Wave Terminologies . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Wave Propagation Equation . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Wave Solutions in Unbounded Anisotropic Media . . . . . . . . . . . 21 2.5 Surface Wave Solutions in Bounded Media . . . . . . . . . . . . . . . 22 2.5.1 Surface Boundary Conditions . . . . . . . . . . . . . . . . . . 23 2.5.2 Displacement Profile for Surface Waves . . . . . . . . . . . . . 25 2.6 Surface Waves With Piezoelectric Effects . . . . . . . . . . . . . . . . 28 2.7 Leaky Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8 Need for Numerical Solvers . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Modeling of SAW Devices 35 3.1 Early SAW Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Delta Function Model . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . 38 3.2 Current SAW models . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Coupling-of-Modes (COM) Model . . . . . . . . . . . . . . . . 41 3.2.2 P-matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Finite Element Analysis (FEA) for SAW Devices . . . . . . . . . . . 47 3.3.1 ANSYS as Simulator . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.2 One Port SAW Resonator . . . . . . . . . . . . . . . . . . . . 53 3.3.3 Limitation of the Finite Element Method . . . . . . . . . . . . 56 3.3.4 Assumptions for Reduced Finite Element Model . . . . . . . . 57 3.3.5 Crystal Cuts - Conventions and Implementation . . . . . . . . 58 3.3.6 Modeling of Periodic Structures . . . . . . . . . . . . . . . . . 60 3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 FEM Results for One Port SAW Resonator and Validation of AN- SYS Simulator 66 4.1 SAW Propagation on a Free Surface . . . . . . . . . . . . . . . . . . . 66 vi 4.1.1 Rayleigh SAW on YZ-LiNbO . . . . . . . . . . . . . . . . . . 66 3 ◦ 4.1.2 Rayleigh SAW on YX-128 LiNbO . . . . . . . . . . . . . . . 70 3 4.2 Finite Element Results for Periodic Structure . . . . . . . . . . . . . 72 4.2.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 SAW Resonator Response Using 3D Finite Elements . . . . . . . . . 79 4.4 Leaky SAW Based Resonators . . . . . . . . . . . . . . . . . . . . . . 82 4.4.1 Issues in Modeling . . . . . . . . . . . . . . . . . . . . . . . . 82 ◦ 4.4.2 36 YX- Lithium Tantalate Resonator . . . . . . . . . . . . . 84 4.4.3 Mass Loading in Leaky Resonators . . . . . . . . . . . . . . . 89 4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Longitudinal Leaky SAW for Higher Frequencies 92 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Displacement Profile of LLSAW . . . . . . . . . . . . . . . . . . . . . 93 5.3 Effect of Depth on LLSAW Resonator Model . . . . . . . . . . . . . . 95 5.4 LLSAW resonator on YZ-LiNbO . . . . . . . . . . . . . . . . . . . . 96 3 5.4.1 Effect of Metallization Ratio and Electrode Height . . . . . . . 98 5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6 Summary, Conclusions and Future Work 103 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3 Recommendation for Future Work . . . . . . . . . . . . . . . . . . . . 106 A Transformation of Matrices with Euler Angles 114 vii LIST OF FIGURES 1.1 IDT in a SAW device . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 IDT periodicity and wavelength . . . . . . . . . . . . . . . . . . . . . 4 1.3 Bidirectional IDT: Delay line and transversal filter . . . . . . . . . . . 5 1.4 SAW resonator with reflectors . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Lift-off process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Elastic waves - Types [23] . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Co-ordinate system for surface wave solutions [25] . . . . . . . . . . . 22 2.3 Normalized phase velocity as a function of direction in XY plane of LiNbO [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 2.4 Longitudinal and vertical displacement variation in isotropic media [25] 26 2.5 Longitudinal and vertical displacement variation with depth. Propa- gation in [100] axis of anisotropic nickel [25] . . . . . . . . . . . . . . 27 2.6 Rayleigh wave with displacements in the sagittal plane [23] . . . . . . 27 2.7 Phase velocity of leaky wave, Rayleigh wave propagating on LiNbO 3 substrate as a function of angle from x-axis [26] . . . . . . . . . . . . 31 3.1 Delta function model - Electric field lines approximated as delta func- tion sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Sample IDT filter response with 19 input IDT pairs using delta func- tion modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Electric field - Real case and approximation . . . . . . . . . . . . . . 38 3.4 Mason equivalent circuit used by Smith et al. for SAW equivalent circuit model [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Equivalent circuit for a SAW IDT using crossed-field model . . . . . . 40 3.6 Counter-propagating waves in COM model . . . . . . . . . . . . . . . 42 viii 3.7 P-matrix representation of an IDT . . . . . . . . . . . . . . . . . . . 46 ◦ 3.8 Admittance ofthe YX-36 LiTaO resonator using COM model show- 3 ing the importance of particular solution over homogeneous solu- tion(Period = 3µm, Length = 200λ) . . . . . . . . . . . . . . . . . . 48 3.9 Discretization of problem domain in FEM . . . . . . . . . . . . . . . 49 3.10 Field approximation by linear and quadratic elements . . . . . . . . . 50 3.11 SAW resonator and equivalent circuit [37] . . . . . . . . . . . . . . . 53 3.12 Typical admittance curve of one port SAW resonator . . . . . . . . . 55 3.13 Metallization ratio and electrode height . . . . . . . . . . . . . . . . . 55 3.14 Dimensions of a SAW resonator . . . . . . . . . . . . . . . . . . . . . 56 3.15 Reduction of model to 2 dimensions . . . . . . . . . . . . . . . . . . . 57 3.16 Periodic structure in a resonator . . . . . . . . . . . . . . . . . . . . . 58 3.17 Different crystal cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.18 Euler angles - Crystal cut rotations . . . . . . . . . . . . . . . . . . . 59 3.19 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.20 FE model of periodic structure . . . . . . . . . . . . . . . . . . . . . 64 3.21 Electric field vectors in a periodic structure . . . . . . . . . . . . . . . 65 4.1 Percentage of error in velocities for differing mesh densities . . . . . . 69 ◦ 4.2 Displacement and potential for SAW in YX-128 LiNbO at resonant 3 frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Phase difference between displacement components for Rayleigh wave 72 4.4 Mode Shapes for SAW propagation . . . . . . . . . . . . . . . . . . . 74 ◦ 4.5 Admittance of YX-128 LiNbO resonator calculated using FEM and 3 COM model (with MR-0.5 and h/λ=3 %) . . . . . . . . . . . . . . . 76 4.6 U displacement contour showing minimum and maximum displace- x ment regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7 U displacement contour showing minimum and maximum displace- y ment regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.8 Potential contour showing maximum and minimum potential regions 79 4.9 3D mesh of YZ-LiNbO resonator . . . . . . . . . . . . . . . . . . . . 80 3 ix