INFORMATION TO USERS The negative microfilm copy of this dissertation was prepared and inspected by the school granting the degree. We are using this film without further inspection or change. If there are any questions about the content, please write directly to the school. The quality of this reproduction is heavily dependent upon the quality of the original material. The following explanation of techniques is provided to help clarify notations which may appear on this reproduction. 1. Manuscripts may not always be complete. When it is not possible,to obtain missing pages, a note appears to indicate this. 2. When copyrighted materials are removed from the manuscript, a note ap pears to indicate this. 3. Oversize materials (maps, drawings, and charts) are photographed by sec tioning the original, beginning at the upper left hand comer and continu ing from left to right in equal sections with small overlaps. 4. 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This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TUNABLE MICROMACHINED MICROWAVE DEVICES A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the requirements for the Degree of Doctor of Philosophy by Arturo Alejo Ay6n May 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Arturo Alejo Ayon 1996 ALL RIGHTS RESERVED Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIO G R A PH IC A L SK ETC H Arturo Alejo Ay6n Ballesteros was bom in the sunny state of Sonora, Mexico on July 17th, 1958. Even though he left his native town, Navojoa, in 1969, he is still looking for an opportunity to return to the Sonoran Desert, and become a high-tech Hacendado. During his undergraduate years, he started working for Burroughs (now Unisys) as a technician and learned that even state of the art equipment fails once in a while in a catastrophic manner. He finished his coursework in Electronic Engineering in January, 1983, and IBM hired him in June, 1983, only to send him on International Assignment 5 months later to another sunny and hospitable place: Rochester, Minnesota. While in the Midwest, life was generous to him bringing several yards of snow, memorable low temperatures, engines that refused to start in the morning, trips to Mexico, Canada, Yellowstone Park and Hawaii, and two daughters: Maria Alejandra and Nancy Yvette. His 1985 homecoming welcome was a management position at IBM, very warm weather and plenty of sunshine. The arrival of his third daughter, Dianne, in 1986, was an additional source of joy. Entrepreneurial opportunities arose and left IBM in 1986 to explore the wonderful world of business and stress. Trying to stay away from collapsing, he and his family crisscrossed Mexico twice and visited all archeological sites on the map. When pyramids started to appear in his dreams he decided to visit the Pacific Ocean every other month. However, he missed White Christmases so much, that he arrived in Ithaca in 1989. He received the Masters of Science Degree in 1992, and in the spring of 1996, the degree of Doctor of Philosophy. As a prelude of what 1995 would bring, his son Arturo Alessandro was bom in December 8th, 1994. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To the memory of my father, with deep admiration and gratitude To my wife and children, for the future iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACK NO W LEDG M ENTS My wife and children have always provided for an oasis of serenity and understanding at home. I am particularly lucky for having them, and for enduring with me the hardships of graduate school. This thesis could not have been completed without the guidance and financial support of my advisor, Professor Noel C. MacDonald. I thank him for keeping me focused when my mind began to wander but also allowing me to think on my own feet. I will need both skills in the future. I also thank the chairman of my Special Committee, Professor David A. Hammer who introduced me to the fascinating world of Plasma Physics. He has always been a source of inspiration and support, especially in the most difficult periods while staying at Cornell. I deeply appreciate his sound advice, both personal and professional and deserves a special thanks for his patience in reading through this material and giving valuable advice. I also thank the other members of my committee Drs. David C. Clark and Clifford Pollock. Dave Clark continues to pique my interest in Nuclear Engineering. Clifford Pollock awakened my interest in optoelectronics and lasers, his lessons on how to tackle engineering problems will always be helpful. Among my colleagues I enjoyed the conversations and discussions with Alex Atwood, John Mercier, Brian Oliver, the late Gilberto Barreto, David Hong, Trent Huang, Dan Haronian, Nick Kolias, Taher Saif, John Chong, Robert Mihailovich, Scott Miller, Sean O’Keefe, Wolfgang Hofmann and Hercules Neves. Finally, I want to extend my appreciation for the technical support of the staff of the Cornell Nanofabrication Facility at Cornell University. Without their help, I would not be able to have accomplished my experimental work. This research has been partially funded by ARPA and NSF. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Chapter 1 Introduction 1.1 MEMS Technology. . . . . . 1 1.2 Thin Film Technology versus High Aspect Ratio Structures 3 1.3 Other Schemes. . . . . . . 7 1.4 Applications. . . . . . . 10 . . . 10 . . . 12 1.5 Synopsis. . 1.6 References. . . . . . . Chapter 2 Transmission Line Theory 2.1 Transmission Line Equations. . . . . 17 2.2 The Field-Cells Approach. . . . . 22 . 28 . 36 2.3 Thin Plane Conductors. . . 2.4 Quasi-Static Approximation. . . . . . 2.5 Resonance and Q-factor. . . . . . 41 2.6 Scattering Matrices. . . . . 47 . . 51 2.7 References. . . . . . Chapter 3 Fabrication Techniques 3.1 SCREAM Process. . . . . . 56 3.2 Process Overview.. . . . . . 58 3.3 Alternative Approaches: Fully Oxidized Structures. 70 3.4 Coplanar Waveguides. 79 . . . . . vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 References. 82 C hapter 4 M easured Performance 4.1 Test Setup and De-Embedding. 4.2 On-Wafer Probing. . 4.3 Measured Performance. . . . . . . . . 86 . . 89 . 4.4 Electrically Thin Substrates. . . . . . 4.5 Surface Roughness and Other Considerations. 4.6 Impedance Variation. . . . 4.7 Electromechanical Tuning. . . . . 97 . 99 . . 103 4.8 Comparison with Conventional Structures. . . . 104 4.9 References. . . . . . 112 . . . 113 . 114 . . . 92 . . 91 . . . . . . 107 C hapter 5 Half-Wavelength Micromachined Dipole Antennas 5.1 Introduction. . . . 5.2 Basic Antenna Parameters. . . 5.3 Radiation Caracteristics of Linear Dipoles. 5.4 Microfabrication. . . . . . 5.5 Test Setup and Measured Performance . 5.6 References. . . . . . . . . 126 . . 132 . . 137 Chapter 6 Conclusions and Future W ork 5.1 Conclusions. . . . . . . . 141 5.2 Future Work. . . . . . . . 142 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix I Other Matrix Representations. . . . . . . 143 Appendix II Other Considerations in Deep Trench etches.. . . . . 145 . . . . 148 Appendix III Equations related to CPW and CPWG . Appendix IV Surface Roughness. . . . . . . . . viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 LIST OF TABLES Table 4.1: Etch rates for different silicon planes. ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure 1.1: Cross section o f a slotline, the thickness of the metallic plates is many times smaller than the thickness of the substrate.. . . 4 Figure 1.2: Microstrip, this geometry is well suited for certain applications such as filters and periodic structures. . . . . . 4 Figure 1.3: Ccplanar Waveguide (CPW), geometry useful for amplifiers. . 4 Figure 1.4: Schematic view of tunable transmission lines. 5 . . Figure 1.5: Cross section o f the geometry presented in this work. . . . 6 . 8 Figure 1.7: Top view of an electromechanically tunable transmission line. . 9 Figure 1.6: Layout of an electrically tunable coplanar transmission line. Figure 2.1: Equivalent circuit of an infinitesimal length Ax of a transmission line 19 Figure 2.2: All space being considered is divided in unit cells of length /. . 23 Figure 2.3: Conductor carrying a uniform current J. . 24 . . . Figure 2.4: The conductor is divided in unit cells o f length /. Figure 2.5: Conducting strips carrying a current/. Figure 2.6: Field-cell transmission line. . . . Figure 2.7: Profile of the sputtered metallic film. 24 . . . . . . . Figure 2.8: Coordinate system of parallel-plate transmission line. . 25 . 26 . 30 . . 3T Figure 2.9: Z0 versus h/s, 377 s/h is the impedance o f a semi-infinite line includ ed for comparison purposes. . . Figure 2.10: Scattering waves in a two-port network. . . . . . 39 . Figure 3.1: Overview of the general processing approach presented in this work. x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 57 Figure 3.2: Surface Roughness for a PECVD film deposited @ 240°C. . 59 Figure 3.3: Surface Roughness for a PECVD film deposited @ 300°C. . 60 Figure 3.4: Surface Roughness for a PECVD film deposited @ 360°C. . 60 Figure 3.5: SEM micrograph showing the effect of leaving residues on top o f the mesas which are subsequently physically sputtered and produce micromasking. . . . . . . . 61 Figure 3.6: SEM micrograph showing the micromasking effect of contaminants gathered during photolithography. . . . . 62 Figure 3.7: Auger analysis o f a wafer with abundant presence of “grass” revealed the presence of aluminum, oxygen and silicon. . 63 . . 64 Figure 3.9: Chlorine chemistries provide for an excellent anisotropy. . 65 Figure 3.8: SEM micrograph showing the presence of trenching. Figure 3.10: SEM micrograph showing the characteristics sought in deep silicon etches. . . . . . . . . 65 Figure 3.11: SEM micrograph showing the passivation layer that has to be removed before releasing the structures. . . . . 66 Figure 3.12: Removal o f the passivation layer provides for smooth uniformly released structures. . . . . Figure 3.13: SF6 chemistries provide for isotropic etches. . . 66 . . . 68 Figure 3.14: SEM micrograph showing a released structure.. . . 68 Figure 3.15: Sputtering at a pressure of 9 mT the surface roughness and stress increase. Noticeable lumps of aluminum form at the rim o f the structures. . . . . . . . 69 Figure 3.16: Annealing of an aluminum film to change the stress from compressive to tensile. Courtesy of J. Drumheller. . . 69 Figure 3.17: SEM micrograph showing a detail of the cantilevered actuators. 70 Figure 3.18: Picture presenting the growth of the structures during oxidation. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 Figure 3.19: After a 4-hour oxidation 0.8 pm beams are fully oxidized. Figure 3.20: After a 4-hour oxidation there is still a silicon spike in beams 1 pm wide. . . . . . . . 73 . Figure 3.21: For a pitch too small, the oxidation step closes the gap at the top before the beams are fully oxidized. . . . 74 . Figure 3.22: The right pitch permits the full oxidation of the beams. 74 Figure 3.23: The MIE step damages the structures when the pitch is too wide. 75 Figure 3.24: The pitch is crucial to achieve full oxidation of the anchors such that 76 they become a single unit and able to survive an MIE step. Figure 3.25: Large structures can be suspended with the anchor technique. . 76 Figure 3.26: SEM micrograph presenting a detail of the actuators cantilevered and fully oxidized. . . . . . 77 . Figure 3.27: Detail of the cantilevered fully-oxidized parallel-plate lines. . Figure 3.28: Detail of meandering lines fully oxidized and cantilevered. 77 78 Figure 3.29: SEM micrograph presenting the effect of using PECVD oxide during oxidation. . . . . . . 79 Figure 3.30: Layout for a conventional coplanar waveguide (CPW). 81 Figure 3.31: Layout for a grounded coplanar waveguide (CPWG). . 81 Figure 4.1: Overview of the test setup. . . . . . Figure 4.2: Modeling the shunt impedance of the launch pads. 87 89 Figure 4.3: Comparison of reflection and transmission scattering parameters for a line and a model including a shunt impedance. 90 Figure 4.4: Theoretical and experimental attenuation. 92 Figure 4.5: Extracted and theoretical phase in units of Rad/mm. 93 Figure 4.6: Extracted attenuation for lines fabricated on substrates with different xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. thicknesses. . . . . . . . 96 Figure 4.7: Extracted phase constant for different substrate thicknesses. . 96 Figure 4.8: Extracted impedance for different substrate thicknesses. . 97 Figure 4.9: SEM micrograph showing the surface roughness of the sputtered aluminum film deposited at a pressure of 1.6 mT. . 98 Figure 4.10: Impedance variation with plate height, for a plate separation of 4.7 pm. . . . . . . . . 100 Figure 4.11: Impedance variation with plate separation, for a plate height of 12.7 pm.. . . . . . . . 100 Figure 4.12: Measured and theoretical impedance for a plate height of 12.7 pm. 101 Figure 4.13: Measured and theoretical impedance for a plate height of 10 pm. 101 Figure 4.14: Measured and theoretical impedance for a plate height of 7 pm. 102 Figure 4.15: Measured and theoretical impedance for a plate separation of 8.9 pm. . . Figure 4.16: Electromechanical . . tuning, . . impedance . . variation 102 can be accomplished either by opening the plates (Zo increases) or by closing them (Zo decreases). . . . . 104 Figure 4.17: Electromechanical tuning, when the transmission line array is perfectly matched permits the optimization of the working point. 105 Figure 4.18: Comparison of transmission parameters s21 for a CPW and one of the lines presented in this work as measured, before de-embedding 106 Figure 5.1: Coordinate system for an infinitesimal linear dipole of total length 1 115 Figure 5.2: Coordinate system for a linear dipole. . . . . 119 Figure 5.3: The radiation pattern for a half-wavelength linear dipole. . 122 Figure 5.4: The variation of radiation resistance with dipole length.. . 124 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.5: The reactance as a function of dipole length for two different dipole radii. . . . . . . . . 125 Figure 5.6: Schematic cross section of the silicon wafer with the alignment marks electrochemically etched.. . . Figure 5.7: SEM micrograph of an alignment mark. . . . . 127 . . . 128 Figure 5.8: Schematic cross section of the masking wafer showing the alignment marks and the standing pads. . . . . . 128 Figure 5.9: Schematic cross section of the masking wafer showing the alignment marks, standing pads and antenna well. . . . . 129 Figure 5.10: Antenna structure microfabricated as described in Chapter 3. . 130 Figure 5.11: After removing the silicon substrate we have the microfabricated structure on a thin silicon membrane. . . . . 130 Figure 5.12: SEM micrograph showing the alignment mark and the antenna after the silicon substrate has been removed . . . 131 Figure 5.13: SEM micrograph showing the electrochemical etch underneath the silicon beams. . . . . . . . 131 Figure 5.14: After removing the silicon nitride membrane, the structure is freestanding with no substrate underneath. . . . . 132 Figure 5.15: Test setup for antennas, only one probe is needed to measure the reflection coefficient. . . . 133 Figure 5.16: Schematic view o f the structure tested in this experiment. . 134 Figure 5.17: SEM micrograph showing the finished device . . 135 Figure 5.18: Measured reflection Coeficient.. . . . . . Figure AIL 1: SEM micrograph showing the characteristics usually sought in deep trenches including smooth and steep sidewalls, no trenching Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 and absence of grass. . . . . . . 145 Figure AII.2: SEM micrograph showing the effect on an isolated feature. . 146 Figure AII.3: SEM micrograph showing the effect on a wall. . 146 . Figure AII.4: SEM micrograph showing the effect only where the walls have a clearance larger that the “new feature size”. . . . xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 Chapter 1 INTRODUCTION 1.1 MEMS Technology MicroElectroMechanical Systems Technology is the developing field of micro devices and structures utilized mainly as sensors, actuators and transducers. It is common to refer to this technology as one of the most promising in both the short and the long range. In fact, it has been identified by the National Science Foundation as an area of national importance [1]. The ability to manufacture moving and sensing mechanisms with feature size in the micrometer regime presents challenges in 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 fabrication techniques, software tools, reliability and repeatability. On the other hand, it opens the door to new ideas, and the opportunity to test theoretical predictions at a scale not found in our everyday experience. The number o f applications of MEMS is growing rapidly and the explosion of novel designs now includes microrobots with elastic hinges [2], diaphragms [3], micromotors [4], chromatography applications [5], microvalves [6], piezoelectric structures for optical applications [7], thermally driven devices [8], microgimbals for disk drives [9], loading devices [10], object imaging schemes [11] and tunable transmission lines [12], along with actuators, sensors, gears, transformers, inductors, pumps, switches and accelerometers [13]-[19]. The number o f fields in science and engineering that benefit from this technology continues to grow. Commercially available sensors to monitor acceleration, pressure, temperature, flow, angle, light intensity and magnetic fields, testify to the versatility and enormous potential of these devices. Other areas of active research are those of chemical and biological applications [20]-[24]. In these fields disposable structures are required, especially in those instances where fluids are directly in contact with the sensing device [20]. This characteristic, however, presents an additional challenge not only from a cost competitiveness point of view, but more importantly, because of the problem of determining and selecting the packaging for these structures. Pressure sensors and accelerometers are sealed in a package and isolated from the environment. While this protection increases stability and reliability, it demands new packaging paradigms for biosensors. Because of the large number of applications in many different fields the realm o f MEMS technology has already left its imprint in large scale manufacturing due to its large array of possibilities. For example in most automobiles manufactured today, there are close to fifty sensors measuring a variety of parameters from oil pressure and oxygen content of the exhaust to air intake velocity [25], As the drive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 for better, faster and cheaper devices continues, this MEMS will carve an even larger niche in our everyday life. 1.2 Thin Film Technology versus High Aspect Ratio Structures Figures 1.1 through 1.3 show some of the conventional thin film microwave planar geometries: Slotline, Microstrip and Coplanar Waveguide (CPW). The thickness of the metallic layers is usually of the order of 1 pm while the thickness of the substrate is on the order of 350 pm. In all three cases the metallic plates are deposited on the wafer and the horizontal dimension is larger than the vertical dimension. Thin film technology has proven valuable in many applications, however, the main disadvantage of using thin films for structural materials is that the internal stress of the films limits the thickness of those films to a few micrometers. Since the vertical stiffness o f a structure is proportional to its height, for a thin film structure the limitation in thickness represents a limitation on the vertical stiffness. This fact places a stringent constraint on the length that thin film structures can span. On the other hand, High Aspect Ratio Structures (HARS) (Figure 1.4) are characterized by larger vertical stiffness compared to thin film technology and, if needed, it can be increased even more by increasing the aspect ratio of the micromachined devices. Furthermore, HARS also present large surface areas of the sidewalls and are, therefore, capable of producing large electrostatic forces. This thesis exploits the aforementioned advantages of HARS and describes a new micromachined microwave application of cantilevered structures, namely, a voltage-tunable transmission line coupled to a dipole antenna. We use comb-like actuators to move the plates of a parallel-plate waveguide. Since the characteristic impedance of a parallel-plate transmission line is a function of the plate separation, modifying the separation we are able to vary the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 METALLIC PLATES SUBSTRATE Figure 1.1: Cross section of a slotline, the thickness of the metallic plates is many times smaller than the thickness of the substrate. SIGNAL PLATE Substrates GROUND PLATE Figure 1.2: Microstrip, this geometry is well suited for certain applications such as filters and periodic structures. GROUND SIGNAL GROUND SUBSTRATE Figure 1.3: Coplanar Waveguide (CPW), geometry useful for amplifiers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LAUNCH PAD LAUNCH PAD Figure 1.4: Cross section of the geometry presented in this work. The height of the plates is o f the order of 13 pm and the separation is of the order of 5 pm. A waveguide as presented is compatible with slotlines and microstrips. For a CPW two lines back-to-back (in parallel) are required. The out-of-plane parallel-plate transmission lines presented here are, therefore, compatible with all thin-film geometries but their high aspect ratio permit the electromechanically tunability. characteristic impedance of the waveguide. This scheme, active change of the therefore, permits the electrical characteristics of the waveguide and allows the introduction o f other applications, such as n-way switching, resonators of variable length, etc. Variable matching networks could represent an excellent application for this device. Under this scheme, it is necessary to micromachine transmission lines only in the vicinity of the intended operating point and use the actuators to find the optimum performance. In the case of a directly coupled resonator, the resonator length determines the resonant frequency, but the possibility to optimize the network presents the advantage of minimizing the reflected signal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 Figure 1.5 presents a schematic view of a voltage-tunable waveguide. The launch pads form a coplanar waveguide (CPW) configuration and have very large horizontal dimensions in comparison to those needed for the vertically-oriented Electrostatic Actuators Ground Launch Pads Parallel-Plate Transmission Lines Signal Ground Electrostatic Actuators Figure 1.5: Schematic view of tunable transmission lines. parallel-plate transmission lines. In this work the dimensions of the ground-pads were 2 2 150 x 200 pm , and those of the middle or signal conductor were 100 x 200 pm . The width of the original CAD design of the parallel-plates was 1 pm with the separation normally fixed at ~5 pm. The work presented in this thesis, in order to be compatible with coplanar waveguide geometry, requires the arrangement of two parallel-plate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 transmission lines in parallel. Therefore the total width needed for a parallel-plate transmission line is < 20 pm, much less than the 400 pm required for the CPW. Each transmission line consists of two plates, one is the ground plate and the other, the signal plate. The actuators move the ground plate. It is not necessary for the motors to be electrically attached to the plates. If needed an additional mask can be used to remove the metallic film connecting plate and actuator. Another option is to physically disconnect plate and actuator and allow them to be mechanically in contact when the device is being operated. The electrical connection of the motors to the transmission lines causes additional losses at those frequencies where the extra electrical length permits radiation. Therefore, it is always advisable to have the actuators electrically disconnected from the transmission lines. 1.3 Other Schemes The tunability concept has been applied in different schemes, such as, depositing superconducting layers of YBa2Cu3C>7-x (YBCO) and dielectric layers of SrTi03 (STO) on LaA103 substrates [26]. Figure 1.6 shows the layout o f the electrically tunable coplanar transmission line resonator. STO is a material chosen in the fabrication of tunable transmission lines mainly because of its large electric field tunability of the dielectric constant and its chemical and structural compatibility with YBCO. Device operation is based on the modulation of the phase velocity and the attenuation constant of microwaves propagating along the transmission lines through the dc electric field induced changes in the dielectric constant of the STO layer, superconducting YBCO electrodes serving only to limit conductor losses. However, the temperatures required are of the order of 80° K, and the Q's obtained are in the vicinity o f 200. This low temperature regime, is still out of the ordinary for most Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 lr J 1 J 3 < t i > '-f Substrate STO, Dielectric YBCO, Superconductor Figure 1.6: Layout for an electrically tunable coplanar transmission line. applications. With electromechanical tuning, however, the characteristic impedance can be varied, and we can tune to a prescribed resonant frequency. Additionally, with a wise selection o f periodic structures the phase velocity can be selected to match that of an electron beam, for instance, in a traveling wave tube. Figure 1.7 presents a top view of a micromachined voltage-tunable transmission line. The structure includes two transmission lines in parallel and sets of capacitive microactuators to move the ground plates. In this particular design, use of both banks of microactuators permits a large change in impedance. Smaller variations are possible by moving only one bank of microactuators. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 The design presented in this work has only one requirement: high resistivity silicon substrates <100>. The process, though, is compatible with standard industry practices, does not require high temperature steps, and the geometry is a natural extension of that of coplanar waveguide structures. Therefore, low-loss, compact devices with superior performance characteristics are achieved without stringent processing or material requirements. Although surface micromachining [27]-[30] is the current standard technology, this work presents bulk micromachining as an alternative approach and has an additional dimension of freedom. The specific needs of the user will determine which option is the most appropriate. The commercial viability in the long run is not a function of approach, but is more a function o f the compatibility of the structure with the rest of the device into which it must fit. Figure 1.7: Top view of an electromechanically voltage-tunable transmission line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 1.4 Applications The number of applications of voltage-tunable transmission lines is very large. A tunable transmission line can be utilized to match a load, or to characterize new structures and applications such as tunable low-pass filters, and, by cascading several in series, a variable length stub becomes possible. Also, an array of these devices, in combination with a splitter, provides for n-way switching. Tunable resonators can be used as filters, switches and detectors. Variable phase velocity periodic structures find applications in traveling wave tubes, free electron lasers [31]-[33] and micro cyclotrons. With the implementation of active devices it is also feasible to fabricate video detectors and amplifiers. It is therefore evident that tunable micromachined transmission lines offer an excellent alternative to conventional planar circuits [27]-[30]. 1.5 Synopsis Chapter 2 provides the theoretical background for this work, starting with standard transmission line equations both from the electric circuit theory and from the electromagnetic field theory viewpoints. The quasi-static approximation is presented, followed by the theory of electromagnetic components that exhibit a resonant behavior. The problem of on-wafer probing is also addressed in this chapter, along with the theory of scattering matrices. Chapter 3 describes the processing of voltage-tunable micromachined microwave devices. The general approach is to follow the SCREAM technology developed at Cornell [34]. In this chapter the process is reviewed in detail and all modifications explained within the scope of the research presented in this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Chapter 4 introduces the concept of electromechanical tuning and presents the analysis o f the data gathered with a network analyzer. The performance of the micromachined transmission lines is compared with the theoretical predictions for parallel plate transmission lines presented in chapter 2. Chapter 5 applies the ideas presented in all previous chapter to develop a new process to microfabricate dipole antennas where a voltage-tunable waveguide is used as matching network. Chapter 6 concludes this dissertation by summarizing the work which has been accomplished, and outlines future applications and research areas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 1.6 References [1] K. Gabriel, J. Jarvis and W. Trimmer, editors, "A report on the emerging field o f microdynamics", Workshop on Microelectromechanical Systems Research, Salt Lake City, Utah, July, 1978 [2] K. Suzuki, I. Shimoyama and H. Miura, "Insect-model based microrobot with elastic hinges", J-MEMS, Vol. 3, No. 1, March 1994, pp. 4-9. [3] P. R. Scheeper, W. Olthuis and P. Bergveld," The design, fabrication and testing of corrugated silicon nitride diaphragms", J-MEMS, Vol. 3, No. 1, March 1994, pp. 36-42. [4] K. Deng, M. Mehregany and A. S. Dewa, "A simple fabrication process for polysilicon side-drive micromotors", J-MEMS, Vol. 3, No. 4, Dec. 1994, pp. 126133. [5] R. R. Reston and E. S. Kolesar, "Silicon-micromachined gas chromatography system used to separate and detect ammonia and nitrogen dioxide", parts I and II, JMEMS, Vol.3, No. 4, Dec. 1994, pp. 134-154. [6] P. Barth, "Silicon microvalves for gas flow control", 8th International Conference on solid-state sensors and actuators, and eurosensors IX, Stockholm, Sweden, June 25-29,1995, pp. 276-279. [7] H. Toshiyoshi, H. Fujita and T. Ueda, "A piezoelectrically operated optical chopper by quartz micromachining", J-MEMS, Vol. 4, No. 1, March 1995, pp. 3-9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 [8] P. L. Bergstrom, J. Ji, Y. Liu, M. Kaviany and K. Wise, "Thermally driven phasechange microactuation", J-MEMS, Vol. 4, No. 1, March 1995, pp. 10-17. [9] V. Temesvary, S. Wu, W. Hsieh, Y. Tai and D. Miu, "Design, fabrication and testing of silicon microgimbals for super-compact rigid disk drives", J-MEMS, Vol. 4, No. 1, March 1995, pp. 18-27. [10] M. T. Saif and N. C. MacDonald, "A milli newton micro loading device", Proceedings of the SPIE's Smart Structures and Materials Conference, 26 Feb-3 March, 1995, San Diego, CA. [11] E. Kolesar and C. S. Dyson, "Object imaging with a piezoelectric robotic tactile sensor", J-MEMS, Vol. 4, No. 2, June 1995, pp. 87-96. [12] A. A. Ay6n, N. Kolias and N. C. MacDonald, "Tunable, micromachined parallelplate transmission lines", 15th Biennial IEEE/Cornell University Conference on Advanced Concepts in High Speed Semiconductor Devices and Circuits", Ithaca, New York, August 7-9,1995. [13] L. S. Fan, Y. C. Tai and R. S. Muller, "IC-process electrostatic micromotors: design, technology and testing", IEEE Micro Robots Mechanical Systems Workshop, Salt Lake City, Utah, February, 1988, pp. 1-6. [14] W. S. N. Trimmer and K. J. Gabriel, "Design considerations for a practical electrostatic micromotor", Sensors and Actuators, 11,1987, pp. 189-206. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 [15] W. Trimmer and R. Jebens, "Harmonic electrostatic motors", Sensors and Actuators, 20,1989, pp. 17-24. [16] H. Fujita and A. Omodaka, "Electrostatic actuators for micromechanics", IEEE Micro Robots and Tdeoperators Workshop, Hyannis, MA, November, 1987. [17] L. S. Fan, Y. C. Tai and R. S. Muller, "Pin joints, gears, springs, cranks and other novel micromechanical structures", IEEE Solid State Sensors and Actuators, Tokyo, Japan, June, 1987, pp. 849-852. [18] R. Jebens, W. Trimmer and J. Walker, "Microactuators for aligning optical fibers", Sensors and Actuators, 20,1989, pp. 65-73. [19] J. J. Clark, "CMOS magnetic sensor arrays", IEEE Solid State Sensors and Actuators Workshop, Hilton Head Island, SC, June, 1988, pp.72-75. [20] J. Bryzek and W. McCulley, “Micromachines on the march”, IEEE Spectrum, May 1994, pp. 20-31. [21] S. A. Boppart, B. C. Wheeler and C. S. Wallace, “A flexible perforated microelectrode array for extended neural recordings”, IEEE Trans, on Biomedical Eng., Vol. 39, No. 1, January 1992, pp. 37-42. [22] J. M. Corey, B. C. Wheeler and G. J. Brewer, “Compliance of hippocampal neurons to patterned substrate networks”, Journal of Neuroscience Research, Vol. 30, 1991, pp. 300-307. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 [23] J. L. Novak and B. C. Wheeler, “Multisite hippocampal slice recording and stimulation using a 32 elemental microelectrode array”, J. of Neuroscience Methods, Vol 23,1988, pp. 149-159. [24] B. C. Wheeler and G. J. Brewer, Multineuron Patterning and Recording, in Enabling Technologies for Cultured Neural Networks, Academic Press, 1994. [25] P. J. Hesketh and D. J. Harrison, "Micromachining, the fabrication of microstructures and microsensors", Interface, Vol. 3, No. 4, Winter, 1994, pp. 21-26. [26] A. T. Findikoglu, Q. X. Jia, H. Campbell, X. D. Wu, D. Reagor, C. B. Mombourquette and D. McMurry, " Electrically tunable coplanar transmission line resonators using YBa2Cu307 -x/SrTiC>3 bilayers", Appl. Phys. Lett., Vol. 66, No. 26, June 1995, pp. 3674-3676. [27] J. Helszajn, “Microwave Planar Passive Circuits and Filters”, John Wiley and Sons (1994). [28] T. Edwards, “Foundations for Microstrip Circuit Design”, Second Edition, John Wiley and Sons (1992). [29] E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications, John Wiley and Sons, 1988. [30] K. C. Gupta, R. Garg and I. J. Bahl, Microstrip Lines and Slotlines, Artech, 1979. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 [31] A. Gover and P. Sprangle, "A unified approach of magnetic Bremsstrahlung, electrostatic Bremsstrahlung, Comptron-Raman scattering, and Cerenkov-SmithPurcell free-electron lasers", IEEE J. Quantum Electron., Vol. QE-17, No. 7, July, 1981, pp.l 196-1215. [32] S. J. Smith and E. M. Purcell, "Visible light from localized surface charges moving across a grating", Phys. Rev., Second Series, Vol. 92, No. 4, pp. 1069. [33] T. C. Marshall, Free-Electron Lasers, Macmillan Publishing Co., 1985 [34] Z. Lisa Zhang and N. C. MacDonald, "A RIE process for submicron, silicon electromechanical structures", J. Micromech. Microeng., 2(1992), pp. 31-38. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 TRANSMISSION LINE THEORY 2.1 Transmission Line Equations A transmission line can be rigorously defined as any structure that guides a propagating electromagnetic wave between points A and B. It is, therefore, to a theoretician a set of boundary conditions to Maxwell's equations that allow the description o f a propagating wave between those points [1]. Since we are transporting energy from A to B, we desire that the energy lost by the transmission line be kept at a minimum [2], We also prefer single-mode propagation and small attenuation in the transmission line. The theory can be developed either from the viewpoint of 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 electromagnetic field theory [3], or from the viewpoint of electric circuit theory [4]. It is the latter approach we take first in this work. We will also present the theory of parallel plate transmission lines and make a comparison with quasi-static results. In order to derive the equations that govern transmission lines, we have to keep in mind that they differ from ordinary electric networks in the essential feature o f their dimensions ranging from a substantial fraction of a wavelength to many wavelengths long. On the other hand, the physical dimensions of electric networks are much smaller than the operating wavelength [5]. Thus, whereas the ordinary electric circuit consists o f lumped elements, the transmission line has to be treated as a distributed parameter network. To analyze a transmission line in terms of ac circuit theory, we must obtain the equivalent circuit of the line, which, being passive, can only be composed o f combinations of resistive, capacitive and inductive elements [6]. An assumption throughout this exercise, is that line is uniform [7], i.e., cross sections of the line in planes normal to the power flow are the same for all points of the line. Consider a differential length Ax of a transmission line that is described by four parameters; the resistance R and the inductance L, which are series elements, with units ohms per meter and henrys per meter, respectively, and the conductance G and the capacitance C, which are shunt elements, with units mhos per meter and farads per meter, respectively. Figure 2.1 shows the equivalent circuit of this line segment. The quantities V(x,t) and V(x+Ax,t) denote instantaneous voltages at x and x+Ax, respectively. Similarly, l(x,t) and I(x+Ax,t) denote the instantaneous currents at x and x+Ax, respectively. Applying Kirchhoff s voltage law to this circuit, we obtain (x,0 - L h x 4 ~ Kx,t) - RAxI(x,t) - V(x + Ax,0 = 0 o t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.1) 19 Figure 2.1: Equivalent electric circuit o f an infinitesimal length Ax of a transmission line. In a similar manner, applying Kirchhoffs current law to the node N in Figure 2.1, we have I(x,t) - CAx— V(x + Ax,/) - GAxV(x + Ax,/) - 7(x + Ax,/) = 0 d t (2.2) On the limit A x -» 0 , equations (2.1) and (2.2) become, ~ V ( x , t ) = L-^-I(x,t) + RI(x,t) ox at (2.3) and - ^ - I ( x A = C^-V(x,f) + GV(xj) d x d t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.4) 20 respectively. Equations (2.3) and (2.4) are the general transmission-line equations, also known as telegrapher's equations. Assuming harmonic time dependence for V and I, phasors can be used to simplify these equations to ordinary differential equations. Consider, for example F (x ,0 = R e[v (x )e^'] (2.5) I(x,t) = Re[z(x) eJa,\ (2.6) from which we obtain from equations (2.3) and (2.4), dv(x) = (R + jeo L) i(x) dx (2.7) = (G + jo> C) v(x) (2.8) We may obtain a single second-order differential equations containing either the voltage or the current by differentiating either equation with respect to x and combining with the other equation, namely d = (R + jeo L^ G + jeo ^ = y 2 v(x) = (R + j<oL )(G +jo)Q i(x ) = r 2 i(x). In equations (2.9) and (2.10), y is the propagation constant, given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.9) (2.10) 21 y = tJ(R+jeo L)(G+ ja> C) - a + jfi (2.11) where a is the attenuation constant in nepers per meter, and P is the phase constant in radians per meter. It should be noted at this point, that the field penetration occurring in nonperfect conductors causes a phase shift that can be represented by an inductance [8], which is the internal inductance of the transmission line. This inductance is usually a small fraction of the total inductance. Therefore, for the purpose of this work, when no reference to external or internal inductance is made, it is to be assumed that the external inductance is being considered. The solutions of equations (2.9) and (2.10) are v(x) = v+(x) + v"(x) = vj e~r x + vj er x (2.12) i(x) = i+(x) + i'(x) = %e~r x + i^er x. (2.13) In these equations, the plus and minus signs denote waves traveling in the +x and -x directions, respectively. The constants v j, v^, i£ and are arbitrary amplitude constants for those waves. Consider now a matched line, i.e., without standing waves. This is the equivalent of an infinite line, and, therefore, we only require the solution traveling in one direction. Under these circumstances, applying equations (2.7) and (2.8) we obtain vo _ v~0 *o io R + jw L ^ r y (2.14) G + jwC This is called the characteristic impedance of the line. As was the case with the propagation constant, Z0 depends only on R, L, G, C and co and not on the length of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the line. It is important to observe the possibility of obtaining the distributed parameters o f a transmission line with the knowledge of the propagation constant and the characteristic impedance [9], R = Re[y Z0] (2.15) (2.16) L = j - l m l r Z„] CO G = Re[-£-] A) (2.17) (2.18) We will review this topic once more when scattering parameters be introduced. 2.2 The Field-Cells Approach We can arrive at equation (2.9) directly from Maxwell's equations, using the concept o f field cells as presented by J. D. Kraus [10]. From this viewpoint, space is regarded as an array o f field-cell transmission lines in which the upper and lower surfaces are considered conducting strips of width w and infinite length in the direction of propagation of the wave, i.e., the x-direction. With this approach L = p. = inductance per unit length in H/m, (2.19) C = e = capacitance per unit length in F/m, (2.20) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 G = <7 = conductance per unit length in mhos/m. (2 .21) We can review briefly how to arrive at these results. In the case o f the equipotential lines in a parallel plate capacitor where A is the area and s is the separation between the plates the capacitance (neglecting flinging) is given by C= eA (2.22) where s is the dielectric constant of the medium. If we now consider a square cross section equation o f side b and let the separation between the plates be also b (Figure 2 .2) b 1r b Figure 2.2: All space being considered is divided in unit cells o f length /. then (2.22) simplifies, to read C=s I (2.23) with all the space under consideration divided into field cells, for each cell the capacitance per unit depth is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 (2.24) 1 =£ and this is the significance of the value of &: it is the capacitance per unit depth of a field-cell capacitor. Let us consider now a uniform current density in a conductor shaped like a bar whose conductivity, ct, is known (Figure 2.3), Figure 2.3: Conductor carrying a uniform current J. Once more, we divide the side of the bar into square areas that represent the end surface of a conductor cell (Figure 2.4), 1 / / w Figure 2.4: The conductor is divided in unit cells of length /. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 In this case, the resistance o f a cell is h 1 R = -------7 = — a wl a l (2.25) from equation (2.25) we can easily obtain (2.26) j = a. The conductivity, therefore, is the conductance per unit depth of a conductor cell. Finally, consider two flat parallel conducting strips of height h and separation s, each carrying a current I (Figure 2.5), I s Figure 2.5: Conducting strips carrying a current I. Again, let us divide the line into a number of field-cell transmission lines arranged in parallel, o f square cross section (Figure 2.6). The current, /, in each cell is i=~I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.27) 26 t i s Figure 2.6: Field-cell transmission line, and the inductance is given by A juH sl L = — = —~ — = / / / i Hs (2.28) where A is the magnetic flux. From equation (2.28) we finally obtain (2.29) = ju. This result allow us to conclude that the permeability can be looked upon as the inductance per unit length of a transmission line cell filled with the medium of permeability p. To apply the previous results, consider an electromagnetic wave with the electric field polarized in the y-direction, the magnetic field in the z-direction, and propagation in the x-direction. Then Maxwell's equations become „ TT T d D d E V x H = J + —r - = c E + s dt at aH 2 _ dEy - = c r E v+ e — - ax y at and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.30) 27 „ „ <?H dEy dH2 V x E = -------- = - u ------ => — - = - / / ---- 2-. f d t d x d t (2.31) We can express equations (2.30) and (2.31) in phasor form to obtain d x = - (<r + j a s ) E (2.32) and I? — £- = -j< O fi H .. o x (2.33) Differentiating equation (2.32) respect to x and using equation (2.31) we obtain d 1 £„y d x‘ = j <DH(cr + j o > £ ) E . (2.34) At this point, we can introduce definitions (2.19) - (2.21), and equation (2.34) becomes 32^ —— y = j co L(G + j o ) C ) E y. ox y (2.35) Upon integration between the upper and lower strips we obtain the potential difference: *K* } = j a > L ( G + ] a > C ) \ { x ) . ox Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.36) 28 However, we still have to remove the lossless condition for the conducting strips. Introducing a finite resistance per unit length, we recover equation (2.9) ax = (R + ja> L)(G + jeo Qv(*) = Y 2 v(x). (2.9) We, therefore, observe that the same transmission line wave equations are obtained either applying circuit theory or following a field approach. 2.3 Thin Plane Conductors The previous two sections presented the general transmission line equations, both from the circuit theory viewpoint (Section 2.1) and from the standpoint o f field theory (Section 2.2). However, equations 2.11 and 2.14 do not actually relate y to the dimensions o f the line. We actually need this relationship in order to compare any measured response to that predicted by Maxwell’s equations. In order to do this, we have to apply those equations to our specific geometry and consider the boundaries as well. Our reference point can be the simple theory of semi-infinite plates found in elementary textbooks of electromagnetic theory. In this case, the characteristic impedance is directly proportional to the ratio of the plate separation to the plate height [5],[7]. In our case, the plates are not semi-infinite, and their thickness is not many times greater than the skin-depth even at the highest frequency covered in this work o f 40 GHz. In order to obtain the right equations we start from basic principles and take into consideration the dimensions of the plates as well as the thickness o f the deposited metallic film. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 The measured average metallic film thickness on the plates is of the order of 0.25 pm for a sputtering step of 60 minutes (Figure 2.7), but this number can be larger, provided the sputtering time is longer or another technique to deposit the aluminum, or any other metallic film, is applied. Therefore, the theory for conductors of any thickness is outlined. Consider the geometry presented in Figure 2.8, two strip conductors o f thickness t, having a separation b and height h, (h » b), have a current applied o f the form JeJ0**. It is also assumed that only the vector components Jx, Hy, Ex and Ez exist, and that they do not have a y-dependence. For this configuration, Maxwell's equations read (2.37) V xH = J + — dt (2.38) (2.39) V»B = 0. (2.40) We also need the constitutive relations [11], D = sE (2.41) B=/*H (2.42) and Ohm's law, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 r:'5,0t:V S -Q .S p tt #081 i Figure 2.7: Profile of the sputtered metallic film. J = cr E and assume s, p and a are constants. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.43) 31 Taking the curl of equation (2.37), applying equations (2.38) and (2.42), and considering that there is no storage of charge density, we obtain y X ------ 1 z s Figure 2.8: Coordinate system of parallel-plate transmission line. V 2 E = m — ( o -E + s — ) dt dt (2.44) or using phasor notation V 2 E = (y*y//cr - c o 2 /ue) E. (2.45) Similarly, taking the curl of equation (2.38), and applying equations (2.37), (2.40) and (2.41), we get Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 V 2 H = 0 ' a j u c r - a 2f i s ) U (2.46) Having obtained the equations we need, we start the analysis in the dielectric region between the plates, where a = 0. Then the x-component of equation (2.45) becomes d 2 Ex d 2 Ex 2 <2-47) Since Ex can only be harmonic or exponential, we can observe from Figure 2.8 that it has to be positive at z = b/2 and negative and of equal amplitude at z = -b/2, this can only happen if it varies sinusoidally, therefore Ex = Ate ~yx Sin K,z (2.48) Substituting equation (2.48) into equation (2.47) we obtain There is, however, only a y-component of H, which from equation (2.38), z - ^ - x - j j = j a s (xEx + z E x ). (2.50) From the x-component of equation (2.50) we obtain 3H a — JL = j a s Ex => Hy = j a s e ' yx —^-CojK,z. (/ z K, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.51) 33 Meanwhile, equating the z-component of equation (2.50) yields dH > = j ■a s E —— ox (2.52) and from equations (2.51) and (2.52) we find E; = ~ Y e ~r* A Cos K, z. (2.53) This equation completes the analysis in the dielectric. In the conductor we assume that the conduction current is much larger than the displacement current as « (2.54) <j . Using equation (2.45) we obtain the x-component of the E-field from f-J O M vH '. (2.55) The general solution for this differential equation is Ex = e-]'x ( A 2 e~K^ + A3 eK22). (2.56) Substituting this value into equation (2.55) we obtain / 2 + K 22 = j co fxa . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.56a) 34 From equation (2.38) we find (2.57) Equating the x-components in equation (2.57) we obtain (2.58) Finally, equating the z-components in equation (2.57), the result is dHy dx = <jE2 => Ez = - y e - y x ( ^ - e - K' z - ^ - e K' 2 ) K_2 1C (2.59) We can now evaluate the constants by applying the appropriate boundary conditions. Since h » b, the H field external to the line is 0 because the fields from currents in the +x-direction cancel the fields of the currents in the opposite direction, even though they add at internal points. Therefore, at z = t + b/2, Hy = 0 and we obtain from equation (2.58) (2.60) On the other hand, at z = b/2 we have a dielectric-conductor boundary. Exploiting the continuity of the tangential component o f E, Ex at z=b/2, equations (2.48) and (2.56), together with equation (2.60) gives Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 ASin( K , £ ) A3 =( e 2Kj' + l ) e (2.61) K,tb ' 2 Similarly, from the continuity o f the tangential component of H, Hy, equations (2.51) and (2.58) and employing equation (2.60) we get 4 A xj< a z K2 Cos ( Kj —) IT a K , ( e z**f - l ) e (2-62) equating equations (2.61) and (2.62), we obtain [12], Tan( K, - ) = 2 c — Coth K2 1. K, (2.63) In equation (2.63) the hyperbolic term tends to 1 as the conductor thickness tends to infinity. In that limit, we recover the better known expression for thick parallel-plate conductors, = 2 (2.64) CF i\ | Equation (2.63) can be solved for y numerically and represents the general case of propagation in a transmission line with thin conductors. We will have need of equation (2.63) along with equations (2.49) and (2.56a) to compare the predicted attenuation and the measured attenuation. Exploiting the concept of surface impedance [13], we can obtain the effective surface resistance and the internal inductance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 Ex ( z = b! 2 ) Z^ “ = HE, K, ^ C° ,h ^ ' ( 2 ' 6 5 ) result based on the assumption that h » b . In reality, the field is not totally confined within the plates and we need another scheme to account for all the field throughout space. This is accomplished using conformal mapping. 2.4 Quasi-Static Approximation Even though section 2.3 addressed the problem of relating the characteristic impedance as well as the propagation constant to the dimensions of the plates and the conductivity o f the metallic film involved, conformal mapping can provide for a quick and rather accurate estimate for the characteristic impedance. The main drawbacks o f this approach are the conspicuous absence o f frequency dependence, and the difficulty o f applying this approach to many arbitrary given geometries. However, the parallel plate capacitor is a well-known problem that has been solved by several authors and it is outlined here for comparison purposes with the more general theory developed in section 2.3. The fundamental problem of conformal mapping is to find an analytic function w = f(z) that maps a given region of the z-plane [14]-[16] into some particular region of the w-plane. If the function f(z) = u(x,y) + iv(x,y) is analytic in a region S, then u(x,y) and v(x,y) satisfy the Cauchy-Riemann conditions [17], [18] £ “ d x d y p. 66) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 d u (xy) h o y — dv(xy) o-------• ox (2.67) If we differentiate equation (2.66) with respect to y and equation (2.67) with respect to x, based on the continuity of the partial derivatives of second order, we obtain the Laplace equation for v(x,y) lj a l-0 oy ox (2.68) similarly, differentiating equation (2.66) with respect to x and equation (2.67) with respect to y, we obtain the Laplace equation for u(x,y), * d X1 J. d y‘ (2.69) Harmonic functions, i.e., those satisfying Laplace's equation, are normal to each other [19], as can be seen if we multiply equation (2.66) by dv/ dx, equation (2.67) by dv/dy and add the results. We obtain ■+ — = Vu»Vv = 0. d xd x d y d y (2.70) Therefore, the curves u = constant and v = constant are mutually perpendicular. Finally, the transformation maintains the angle of intersection o f the curves, i.e., it is conformal. Conformality only fails where the function w(z) or its inverse z(w) is not analytic. However, these methods are limited to this particular partial differential equation and to problems that can be reduced to two dimensions [20]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Among the methods employed, the Schwarz-Christoffel transformation maps the real axis in the z-plane into a general polygon in the w-plane with the upper half of the z-plane mapping into the region interior to the polygon [21]. In the case of a parallel-plate arrangement, the capacitance per unit length is given by [22], [23] c = s K^ farad s K meter (2 J)) where K and K1are the complete elliptic integrals of the first kind of modulus k and k1, respectively. The ratio of height to separation, h/s, is related to k through the equation where E(<j»,k') is the incomplete elliptic integral of the second kind of modulus k', and F(4»,k') is the incomplete elliptic integral of the first kind of amplitude § and modulus k. The value of <j>can be found by applying V/ _ EV Sin2 $ = — . v (1 —& )K ' (2.73) The evaluation of elliptic integrals can be avoided by the use o f the approximate formula « h r, s , 2 tv C = s - [ 1 + — —(1 + In------- )] S TV h s and the characteristic impedance can be found by applying Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.74) Figure 2.4 presents a graph of Z0 versus the ratio of height to separation. As can be seen in Figure 2.9, the fringe effect increases the capacitance and lowers the characteristic impedance of the line with respect to the ideal semi-infinite transmission line. Additional details can be found in the literature [24]-[ 30]. H. A. Wheeler [31]-[32] has developed relations in terms of ordinary functions, i.e., exponential and hyperbolic. R. S. Elliott [33] arrived at a similar expression as equation (2.74). His approximation, however, provides for a lower bound value. 250 200 150 Zo 7 s/h 100 50 0 1 2 3 4 5 6 h/s Ratio Figure 2.9: Z0 versus h/s, 377s/h is the impedance of a semi-infinite line included for comparison purposes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Inspection of Figure 2.9 reveals not only the relevance of the fringe capacitance but the expected variation of impedance as the ratio of height/separation changes. Evidently, the impedance departs more from the ideal semi-infinite plates as the ratio gets smaller, i.e., as the separation increases or the height decreases. Finally, Sato and Ikeda [34], have included in their analysis the thickness of the plates. The capacitance is still given by equation (2.71), but k, the modulus o f the elliptic integrals, and the dimensions of the plates are now related by h /(!,< ? ,) J / ( 0, 1) T K 8 x, S 2 ) (2.76) (2.77) s “ 2 / ( 0, 1 ) (2.78) n s 2, k - x) where (2.78a) the 8j are points are found when mapping the z-plane into the auxiliary w-plane, and T is the thickness of the plates. Their analysis show that the characteristic impedance is lowered by the thickness of the plates. For the purpose of this work, however, the ratio of thickness to separation is very small, usually less than 0.1 which implies a correction of the order of 5% or less. Therefore, the thin plate approximation will suffice. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 2.5 Resonance and Q-factor Resonant devices find applications in filters, oscillators, tuned amplifiers, phase equalizers and as frequency meters [35]. Even though with our current technology it is feasible to fabricate resonant circuit with extremely small lumped elements, this approach has some drawbacks: dielectric, ohmic and radiative losses can be beyond tolerance and, lumped parameter circuits usually have severe power handling restrictions. On the other hand, structures with dimensions comparable to wavelengths provide a suitable alternative most of the time. We are, therefore, mainly concerned with short and open circuited sections of transmission lines. Consider an open-circuited transmission line of length 2n-l quarter-wavelengths at the resonant frequency and characteristic impedance Z0, in which the input port is located at x = -1 and the open output at x = 0. Then using equations (2.12) and (2.13), the expressions for the mode voltage and current are (x) = Ae~rx + B e rx (2.79) (2.80) where A and B are the complex amplitudes of the incident and reflected waves and y is the propagation constant y = a + j p. To satisfy the boundary condition at x = 0 one must set A = B. Then equations (2.79 ) and (2.80) become (x) = 2 A Cosh yx (2.81) I (x) = —— Sink y x . 4 (2.82) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 From equations (2.81 ) and (2.82 ), we obtain the input impedance Z, /(-/) = Za C o t h ( a + j f i I) = Z0 ?-anh( a l ) J Co t ( P l ) ° \ - j Ta n h ( a l ) C o t ( p I) (2 .83) K ' Expression (2.83) can be put in a more tractable form, taking into consideration the use o f low-loss materials, for which a l « 1, this implies Z * Z a l J C o t ( P 1') . °1 - j a l C o t ( p i ) (2.84) K J Consider now (2.85) co- cor + Aco where tor is the angular resonant frequency and Aco/cor « 1, then for a propagating TEM mode we have yg/ = ( 2n - l ) ^ - ^ + Afi> = ( 2» - l ) - ^ ( l + — ) 4 Xr vr 2 cor (2.86) on the other hand C of [ ( 2 » i - l ) ^ ( i + ^ ® ) ] = _ 3 r a » [ ( 2 » - l ) f — ]. 2 cor 2 oor (2.87) At this point we will assume that the argument in this equation is small enough to obtain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 m ft A(0 -.71 A<0 Tan[(2n —1)--------- ]« (2k- 1 ) -------- . 2 0)r 2 0)r r / A (2.88) The validity o f the last step comes from the fact that the bandwidth is related to the quality factor Q through the simple relation Q=— =1 ^ * BW <or (2.89) where Q indicates the sharpness of the resonance. Usually Q is required as high as possible but open geometries like the ones presented in this thesis do not achieve Q larger than a few hundred under optimum conditions. We will go back to the Q factor in a few lines. We, therefore, require 2n-l « Q/10 to ensure the argument is much less than 1. Equation (2.84) has now reduced to , . .. ^ A6) a l + j (2n - l ) — — Z,=Z„ ---------------------- - —C - r—. i ^ n n A© 1+ j a I (2n - 1) — — 2 <o.r (2.90) The term in the denominator is second order in a and Aco/cor, therefore, to first approximation, equation (2.90) can be expressed as ^ r i ^ A . ( 2 h - 1) n Aco.. Z , = Z 0 [ a l + j (2« —1) — — ] = a IZ0 [ 1 + ; -------— ]. 2 CO. Ct I 2 CD. ,.nn (2.91) In order to shed some light on equation (2.91), we recall the theory of a series resonant circuit RLC, in which the impedance is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Z = R + j ( a L — — ). a t (2.92) Very close to resonance, i.e., using condition (2.85), the impedance can be expressed as Z = R + j to. L + j AcoL-----— + / - ^ r — + ... = /?+j A<o L + j (or C to; C a; C (2.93) where we have employed the equality (2.94) = (Dr C Applying (2.94) to equation (2.93) we obtain Z = R + jZ 4 ^ = R(l+A - a; C (or R C (2-95) which resembles equation (2.91). We can further illuminate the meaning of equation (2.95) by defining the quality factor Q as - time - average energy stored in the system .. Q - a ) . --------------- 2------- —-----------------------------------------------------------------energy loss per unit time in the system where all factors are computed at the resonance frequency. The Q-factor is understood to describe the sharpness of the resonance [36] or selectivity of a resonant circuit. Equation (2.102) will show that circuits with low losses have higher Q's. In the capacitor the peak energy stored is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 1 Ja W- m 2 l ? C (Z 9 7 ) and the average power lost in the resistor is Using equations (2.97) and (2.98), the unloaded Q-factor can be expressed as (2-"> Now we can apply this result to express the impedance as „ _ . 2 A<» . 2 AG)/a>r . _ A©. Z = R + j - r - = R ( l + j ------- - ~ ^ ) = R ( l + j 2 Q ). co; C eor R C (or (2 .100) Utilizing this result in equation (2.91), we can represent the open-circuited transmission line as a series resonant RLC-citcuit, in which R = a l Z 0 = ^ j ^ a A r Z0 (2.101) _ _ { 2 n - \ ) n _ 7C P 4a l ~ aX~Ya (2.102) a>r 4 a>r c = —7 7 = 77 77------ — a>‘ L ( 2 w - l ) ncor Z0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.104) 46 where \ is the wavelength of the resonant frequency. Equations (2.101) - (2.104) show that the resonance frequency has a dependence on the characteristic impedance and, therefore, changing the impedance implies a shift in the resonant frequency. However, what is actually measured is [37] the loaded-Q of the circuit, which is the total Q o f the circuit. In equation (2.105), p is called the coupling factor (this P should not be confused with that appearing in y = a + jP) and is related to the reflection coefficient p at the resonance frequency by B - l + P (<cor ) 1-p(a>r ) (2.106) In order to obtain the unloaded-Q , we first determine the loaded-Q of the circuit [38] via the equation (2.107) where the denominator is the 3dB bandwidth o f the resonator; with the measured reflection coefficient we obtain p, and applying equation (2.105) we determine the unloaded-Q. At this point it is obvious that the interest in determining Q0, is that it is directly related to the attenuation coefficient. Therefore, the variables to which we have access through measurement, resonant frequency, reflection coefficient and 3dB bandwidth, permit the calculation of a via the unloaded Q. Reference [36] has a complete discussion o f different techniques for determining Q. In addition, reference Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 [38] also presents a least-squares method, that can be applied when there is only a sampling of the resonance region. 2.6 Scattering Matrices Scattering parameters are defined in terms of incident and reflected waves, and they require transmission lines terminated in their characteristic impedance as the boundary condition [1]. This is particularly important at high frequencies at which measuring current, for instance, can disturb the circuit under consideration. In the scattering matrix approach, however, the measurable quantities are the amplitudes and phase angles o f the waves relative to those of the incident wave. It is this normalization that provides the symmetry of the scattering matrix [35] (Note: for the geometry studied in this work Sn = S22 and S2i = S12). Consider a to be the incident vector and b the reflected vector. Then, in this formulation they are related by [39] [41] b=Sa (2.108) For a two-port network, a and b can be expressed as a= b= (2.109) bx (2.110) h Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 and the scattering matrix is 5= 5„ 5,2 (2.111) 5 2, S 22 where the diagonal elements are reflection coefficients and the off-diagonal elements are transmission coefficients [10]. According to equation (2.108), the relationship between the incoming and outgoing waves is described by (2.112) 6, —a, 5„ + a 2 5,2 b2 = a, 5 2, + a 2 5 22 (2.113) The arrangement can be seen in Figure 2.10. The S-parameter responses measured from a lossy unmatched transmission line with parameters y and Z in a Z0 impedance system [42] are ( Z 2- Z 02 ) S i n h ( r I) 2ZZ0 2ZZo ( Z 2 - Z 2) Sinh ( y I) (2.114) where A = 2 Z Z a Cosh( y l) + ( Z 2 +Z02 ) S i n h ( y I) (2.115) Notice that in equations (2.114) and (2.115) Z is the characteristic impedance of the line we are actually measuring, but Z0 is the known characteristic impedance of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 line used to apply the signal to the device under test. We can convert matrix (2.114) to the equivalent ABCD matrix [43] to obtain Cosh(y I) [ABCD]= Z S i n h ( y I) (2.116) Co s h ( y I) —Sink ( y /) S 2i ---------------- > \f >i S 11 „ 22 < -------------- Figure 2.10: Scattering waves in a two-port network. Furthermore, the S-parameters and the ABCD matrix are related [43] via A = ( l + Sn - S n - A S ) / 2 S n B = ( \ + SH+S22+ A S ) / 2 S 2i C = (1-<S'I1- S 22+ A S ) / 2 S 2i Z0 D = - Sxl + S22 - A S ) / 2 S2l where A S —Su S22 S2i Si2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.118) 50 Using equations (2.114) - (2.118) we can express the parameters y and Z of the line under consideration as e ^ = 1 — 'l— (2.119) IS.21 where I ( Sjj - S 2, +1) - ( 2 S n ) ‘ i (2 5 21) 2 (2.120) and 2= 2 0 ± 5 iIz ^ L (2.121) ° ( l - S , ? ) 2- ^ furthermore, applying equations (2.15) - (2.18) we can also find the equivalent network for the line in consideration. From a known ABCD matrix we can find the Sparameters using g _ A Z0 + B - C Z g - D Za A Z „ + B + C Z„ + DZ„ S,-, —■ S2l —■ A Z 0 + B + CZ„ + DZ0 - A Z0 + B —C Z 2 + D Z0 S22 — A Z 0 + B + C Z l + DZ0 2 (AD -BC )Z0 a z 0+ b + c z 2 0 + dz0 Additional relations can be found in Appendix I. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.122) 51 2.7 References [1] L. N. Dworsky, Modem Transmission Line Theory and Applications, John Wiley & Sons, 1979. [2] R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill Book Co., 1966. [3] R. E. Collin, Field Theory of Guided Waves, Second Edition, IEEE Press, 1991. [4] S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields, Addison-Wesley Publishing Co., 1971. [5] D. K. Cheng, Field and Wave Electromagnetics, Second Edition, Addison-Wesley Publishing Company, 1990. [6] R. E. Matick, Transmission Lines for Digital and Communication Networks, McGraw-Hill Book Co., 1969. [7] R. L. Liboff and G. C. Dalman, Transmission lines, Waveguides and Smith Charts, Macmillan Publishing Co., 1986. [8] D. J. Griffiths, Introduction to Electrodynamics, Second Edition, Prentice Hall, Inc., 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 [9] R. B. Marks and D. F. Williams, “Characteristic impedance determination using propagation constant measurement”, IEEE Microwave and Guided Wave Letters, Vol. l,No. 6, June 1991, pp. 141-143. [10] J. D. Kraus, Electromagnetics, Fourth Edition, McGraw-Hill, Inc., 1992. [11] J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, 1975 I [12] H. Y. Lee and T. Itoh, "Wideband conductor loss calculation of planar quasiTEM transmission lines with thin conductors using a phenomenological loss equivalence method", IEEE MTT-S Digest, 1989, pp. 368-370 [13] S. A. Schelkunoff, "The impedance concept and its application to problems of reflection, refraction, shielding and power absorption", Bell Systems Technical Journal, Vol 17, 1938, pp. 17-48. [14] M. D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, Inc., 1978 [15] R. A. Silverman, Introductory Complex Analysis, Dover Publications, Inc., 1972 [16] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Publishing Co., Part 1, 1953. [17] E. Butkov, Mathematical Physics, Addison-Wesley Publishing Co., 1968. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 [18] G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Hod Books, 1983. [19] J. Matthews and R. L. Walker, Mathematical Methods of Physics, Second Edition, Addison-Wesley Publishing Co., 1970. [20] G. Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press, Inc., 1985. [21] M. A. Evgrafov, Analytic Functions, Dover Publications, Inc., 1966. [22] A. E. H. Love, "Some electrostatic distributions in two dimensions", Proc. London Math. Soc., (2), 22 (1924), pp. 337-369. [23] H. Palmer, "The capacitance of a parallel-plate capacitor by the SchwarzChristoffel transformation", AIEE Trans., Vol. 56, March, 1937, pp. 363. [24] F. Bowman, "Notes on two-dimensional electric fields problems", Proc. London Math Soc., (2), 41 (1936), pp. 205-215. [25] E. Weber, Conformal Mapping Applied to Electromagnetic Field Problems, Construction & Applications o f conformal maps, National Bureau of Standards, Applied Mathematics Series, No. 18,1952, pp. 59-69. [26] W. H. Chang, "Analytical IC metal-line capacitance formulas", IEEE Trans. MTT, Vol. MTT-24, Sept., 1976, pp. 608-611. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 [27] W. Hilberg, " From approximations to exact relations for characteristic impedances", IEEE Trans. MTT, Vol. MTT-17, No. 5, May, 1969, pp. 259-265. [28] H. R. Kaupp, "Characteristics of microstrip transmission lines", IEEE Trans. Electr. Comp., Vol. EC-16, No. 2, April, 1967, pp. 185-193. [29] G. Warner and R. Anderson, "Numerical conformal mapping for undergraduates", Int. J. Elect. Enging. Educ., Vol. 18,1981, pp. 359-373. [30] R. Schinzinger and P. A. A. Laura, Conformal Mapping: Methods and Applications, Elsevier Science Publishers B. V., 1991. [31] H. A. Wheeler, " Transmission-line properties o f parallel wide strips by a conformal- mapping approximation", IEEE Trans. MTT, Vol. MTT-12, May, 1964, pp. 280-289. [32] H. A. Wheeler, " Transmission-line properties of parallel-strips separated by a dielectric sheet", IEEE Trans. MTT, Vol. MTT-13, March, 1965, pp. 172-185. [33] R. S. Elliott, Electromagnetics, McGraw-Hill Book Co., 1966. [34] R. Sato and T. Ikeda, Line Constants, Microwave Filters and Circuits, Edited by A. Matsumoto, Chapter V, Academic Press, 1970, pp. 129-131. [35] R. S. Elliott, An Introduction to Guided Waves and Microwave Circuits, Prentice-Hall, Inc., 1993. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 [36] M. Sucher, Measurements of Q, in Handbook of Microwave Measurements, Third Edition, Edited by M. Sucher and J. Fox, Polytechnic Press, Vol. II, Chapter VIII. [37] R. C. Compton, Microwave Integrated Circuits, notes for course EE 433, Fall, 1993. [38] N. Kolias, Design of Millimeter Wave Integrated Active Antenna Arrays, M. S. Thesis, Cornell University, August, 1993. [39] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communications Electronics, John Wiley & Sons, Inc., 1965. [40] H. J. Carlin and A. B. Giordano, Network Theory, Prentice-Hall, Inc., 1964. [41] J. Helszajn, Microwave Planar Passive Circuits and Filters, John Wiley & Sons, 1994. [42] W. R. Eisenstadt and Y. Eo, "S-parameter-based IC interconnect transmission line characterization", IEEE Trans, on Components, Hybrids and Manufacturing Technology, Vol. 15, No. 4, August, 1992, pp. 483-490. [43] K. C. Gupta, R. Garg and R. Chadha, Computer-Aided Design of Microwave Circuits, Artech House, Inc., 1981. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 FABRICATION TECHNIQUES 3.1 SCREAM Process We follow the general guidelines of the SCREAM process [1] developed at Cornell. SCREAM stands for Single Crystal Reactive Etching and Metallization, and this methodology has proven to be extremely flexible and readily applicable, with some appropriate modifications, even to GaAs substrates [2], The basic process is described in detail elsewhere[3] and a large number of applications has proven the effectiveness of this method including transmission lines [4], and loading devices [5], 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 among many others. A schematic overview of the general process can be seen in Figure 3.1. photorwitf Single crystal SIBcon st£sirate(SCS) ■ « rtamnmniiiw v iw v r f v v v w a w (•) Pattern transfer (b) SP6 Release (0 Metal Sktewal oxkte deposition (PECVO) (d) MetaKzation by sputtering Figure 3.1: Overview of the general processing approach presented in this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 3.2 Process Overview 1.) SILICON OXIDE DEPOSITION: The process begins with the deposition of a Plasma Enhanced Chemical Vapor Deposition (PECVD) film of Si02 @ 300°C, chamber pressure o f 450 mT, 15 seem o f SiH4 and 50 seem of N 2O with the power set at 50 Watts. We use high resistivity (p > 2000 Q-cm) silicon (<100>) substrates (nominal thickness 380 ± 20 pm) which have been widely used in other microwave applications. The deposition time is related to the etch time in a subsequent step, because this film will be used as the mask to etch the silicon substrate. Even though the selectivity is normally greater than 20:1 (silicon to oxide), the film thickness must account for other aspects of dry etching such as faceting. Therefore, there are added advantages to keeping the film thicker than strictly required; 0.5 pm in addition to the thickness needed is considered enough. The temperature of deposition is relevant because the quality of the film, as well as the compressive stress of the film, increase with temperature; 90 MPa, 110 MPa and 120 MPa are typical for films deposited at 240°C, 300°C and 360°C, respectively. Figures 3.2 through 3.4 show the surface roughness taken with an Atomic Force Microscope. Each sample was scanned in three different places, and each scan was obtained sweeping 10 pm of the surface. The definitions of all roughness parameters, Ra, Rp, Rt, Rpm and Rtm can be found in appendix IV. The coefficient of thermal expansion of silicon is » 4.24 times [2] that of silicon oxide, providing for thermal stresses. Furthermore the flow temperature for the oxide is 960°C and the deposition is carried at a lower temperature, which, in turn, provides for intrinsic stresses; therefore, after film deposition the surface of the wafer is convex [6], representing a possible problem especially for thin membranes. If required, the stress can be decreased with a 30-minute annealing cycle @ 700°C [7]. S i0 2 is often chosen not only because o f its stability and to provide a base for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 resist adhesion and imaging [8], but, more importantly, because it can provide for electrical passivation as well as isolation and protection o f the silicon substrate [9]. 2.) PHOTOLITHOGRAPHY STEP: After the oxide deposition, photosensitive material (KTI 895i, 34 cs) is spun on the wafer to achieve a thickness of 2.4 pm (2500 rpm), baked on using a 90°C hot-plate for 1 minute, exposed to light of wavelength 365 nm with a 10:1 stepper (0.46 seconds, focus = 249), and developed following the manufacturers recommendation of a 3 minute immersion in KTI 945. Figure 3.2: Surface roughness for a PECVD film deposited @ 240°C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Figure 3.3: Surface Roughness for a PECVD film deposited @ 300°C. Figure 3.4: Surface roughness for a PECVD film deposited @ 360°C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 3.) SILICON OXIDE ETCH: The topography is now transferred to the oxide layer using a Magnetic Ion Etching (MIE) machine: 30 seem of CHF3 with 1 kW power and chamber pressure < 3 mT. 4.) PHOTORESIST STRIPPING: Afterwards, the photoresist is removed with an oxygen plasma in a barrel etcher (fully isotropic etch). Subsequent steps are very sensitive to photoresist organic residues left on the wafer (Figures 3.5 and 3.6), because the chlorine etch can bum the residues which serve as micromasking sites when sputtered during plasma etching. Figure 3.5 SEM micrograph showing the effect of leaving residues on top of the mesas which are subsequently physically sputtered and produce micromasking. The formation of grass can be catastrophic in the event of an air leak in the chamber or the use of tap water instead of deionized water during rinsing (wafers are rinsed after development). In the case of tap water, the large number of impurities Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 serve as micromasking sites. An auger analysis was performed on a sample presenting heavy formation of grass due to a leak (Figure 3.7). The presence of silicon and oxygen was certainly expected but the results also show a significant presence of aluminum. The cathode of the Reactive Ion Etcher (RIE) is made of aluminum and it is believed that the presence of air permitted its physical sputtering. 5.) DEEP SILICON ETCH: The oxide layer is then used as a mask to etch the silicon substrate with a chemistry rich in chlorine [10] with a small amount of BCI3 to prevent trenching [11] (Figure 3.8). This etch is performed with a RIE machine. It should be noted that BCI3 also lowers the etch selectivity [12] between silicon and silicon oxide, therefore, it must be kept at a minimum. Some authors recommend the removal o f native oxide before etching the actual deep silicon trench [1], [3]. The [I & ill ILIA; tisib lltiil! )■ im ii. i i — X456 ■ 1 c» O \r f a 25.0k V «" i ■■■<' NNF #0 0 Figure 3.6: SEM micrograph showing the micromasking effect of contaminants deposited on the sample during photolithography. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 stoichiometry of native oxide has been reported [13] to be SiO with an even thinner layer o f Si0.7O0.3 at the interface with the silicon substrate. It is also noteworthy to point out the possible promotion of formation of native oxide with the use of HF [13]. ru a CD •rH CD AL2 LU SI2 "O TD 1 201 401 601 601 1001 1201 E l e c t r o n Energy 1401 1601 1801 (eV) Figure 3.7: Auger analysis of a wafer with abundant presence of “grass” revealed the presence o f aluminum, oxygen and silicon. The flows of Cl2 and BCI3 were set at 60 seem and 1.5 seem respectively. Chlorine chemistries follow reasonably well relations developed by Zarowin [12], [14]-[17]. According to his approach the anisotropy of the trench is proportional to the ratio of electric field strength to density (Figure 3.9). Figure 3.10 shows some of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 characteristics sought during deep silicon etches: smooth and steep sidewalls and the absence of grass or trenching. Ratios of self-bias to chamber pressure of -11.88 V/mT [3] have been reported. In an effort to provide for high quality parallel-plates, a ratio of -15 V/mT was chosen in the same piece of equipment, i.e., self-bias = - 450V, chamber pressure = 30 mT. With this selection the measured etch rate is 12 pm/hour. 6 .) REMOVAL OF PASSIVATION LAYER: There is always a residue left on the walls o f the trench after the chlorine etch which disturbs the subsequent release of the structures. An RCA (Radio Corporation o f America) clean as proposed in the original SCREAM paper [1] eliminates the problem. Figure 3.8: SEM micrograph showing the presence of trenching. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Figure 3.9: Chlorine chemistries provide for excellent anisotropy. Figure 3.10: SEM micrograph showing the characteristics sought in deep silicon etches. A 30 minute immersion in nanostrip [18] followed by a 10 minute clean in bubbling deionized water can be used, but this step usually does not provide the superior Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 results o f the RCA step. Figures 3.11 and 3.12 show released structures with and without a passivation layer, respectively. Figure 3.11: SEM micrograph showing the passivation layer that has to be removed before releasing the structures. Figure 3.12: Removal of the passivation layer provides for smooth uniformly released structures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 7.) WALL PROTECTION W ITH SILICON OXIDE: Prior to releasing the structures, an additional 15 minute conformal deposition of PECVD oxide is needed to protect the walls. This is also done at 300°C, followed by a MIE step to remove the oxide from the floor of the trenches. 8.) RELEASE STEP: The structure can now be released. We use 120 seem SF6 and 150W power in an RIE machine. Figures 3.13 and 3.14 show the results after a release step on a wall and on a beam, respectively. With these settings the etch rate is 1.1 |im/minute. It should be noted that the effect of the overhang obtained during release counters the effect of the top silicon oxide. Thus, it is feasible to obtain straight beams adjusting the thickness of the oxide film and the height o f the overhang [19]. 9.) ALUMINUM SPUTTERING: After release, the devices are conformally coated with aluminum in a DC magnetron sputtering system with 2 kW power and by adjusting the Ar gas flow to obtain a chamber pressure of 1.6 mT. With this low pressure, the compressive stress of the film is kept a t « 50 MPa [20]. Operating at a higher pressure, i.e., 9 mT, the thickness of the sidewall film can be increased but the surface roughness increases considerably (Figure 3.15) and the stress is also much higher. Once more, this stress can be made tensile with an annealing cycle [20], [21]. Figure 3.16 presents the change in stress from compressive to tensile for an aluminum film. 10.) PROBE PROTECTION: Finally, a thin film (100 A) of Cr as an adhesion layer and 450 A o f Au are evaporated to protect the microwave probes. Figure 3.17 presents a detail o f the actuators cantilevered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 •'A t< . • • \ • C 2 •? ; 5. 0 I. V . ■ ft y r< = 0 0 1 4 Figure 3.13: SF6 chemistries provide for isotropic etches. Figure 3.14: SEM micrograph showing a released structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Figure 3.15: Sputtering at a pressure of 9 mT the surface roughness and stress increase. Noticeable aluminum lumps form at the rim of the structures. 200 lO O O So iS o 200 Figure 3.16: Annealing o f an aluminum film to change the stress from compressive to tensile. Courtesy of J. Drumheller. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 % Figure 3.17: SEM micrograph showing a detail of the cantilevered actuators. 3.3 Alternative Approaches: Fully Oxidized Structures Other options have been explored that can provide for additional flexibility in the manufacture o f these devices and other similar structures. We start depositing a film o f Low Temperature Oxide (LTO) or Low Pressure Chemical Vapor Deposition (LPCVD) instead of PECVD oxide, even though we have also used layers of thermal oxidation with equally good results. The LPCVD oxide deposition is accomplished with 104 seem of SiH4 and 75 seem of 0 2 @ 400 °C and chamber pressure of 275 mT. This deposition is very sensitive to the separation between the wafers. Therefore, we can obtain samples with various oxide thickness in the same run, flexibility that other processes do not have. The thermal oxidation or steam oxidation, is done using 0.4 1/min of C2H3CI3 Trichloroethane (TCA), 4.5 lt/min of H 2 and 2.5 lt/min of O2 at atmospheric pressure having the temperature as a means to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 control the oxidation rate. With the temperature set @ 1150 °C beams 0.8 pm wide are fully oxidized and mesas and walls develop an oxide layer 1.4 pm thick in 4 hours. Even though the selectivity in chlorine chemistry does not vary significantly when using PECVD, LTO or Thermal oxide, the films differ in their compressive stress, which varies from 90 MPa for PECVD deposited at 240 °C to 300 MPa [22] for thermal oxide grown at 1100 °C. The compressive stress of LTO films vary from 90 MPa to 250 Mpa [23] depending of the position of the wafer in the deposition furnace. The index o f refraction measured at a light wavelength o f 633 nm, reflecting the composition of the film, varies from 1.46 for thermal (1.47 for LTO) to 1.51 for PECVD. Regardless o f which method of oxidation is used, the photolithography, MIE transfer, photoresist strip and trench etch steps remain the same. After the chlorine etch, we proceed with an RCA-cleaning step. This process involves a basic (5 parts deionized water, 1 part H20 2 and 1 part NH4OH at 75°C) and an acidic bath (6 parts deionized water, 1 part HC1 and 1 part H20 2 also at 75°C, followed by 10 parts deionized water and 1 part HF at room temperature) with bubbling deionized water bath steps after each one. Once the wafers are clean, we proceed to thermally oxidize the structures, as previously described. During thermal oxidation silicon atoms combine with oxygen atoms to form amorphous silica, i.e., a random version of crystalline quartz [24] with a similar index o f refraction of 1.462 and dielectric constant of 3.9. Since the molar volume of oxide is 120% larger than the molar volume of Si [9], the structures grow in all directions and allowances have to be made to have this growth included (Figure 3.18). In fact, for every thickness x of oxide formed, 0.45x of silicon is consumed. The exact nature and charge of the diffusion species have not yet been fully identified but kinetic models based on the steps of transport to the surface, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 diffusion through the oxide layer and reaction at the silicon surface, have been proposed and tested, with reasonably good accuracy [25]. It was found that structures 0.8 pm wide, can be fully oxidized after 4 hours @ 1150°C (Figure 3.19) while structures 1 pm wide have a spike of silicon left inside the beams (Figure 3.20). Furthermore, the spacing of the grids expected to provide the anchors to support the released structures structures is critical in the process. Openings « 2 pm during the thermal oxidation close first at the top (Figure 3.21) rendering a smooth surface of oxide. (This effect could be used for making vessels for the transport of liquids for instance). Openings > 2 pm can accommodate the growth of the beams and permit the full oxidation (Figure 3.22). Figure 3.18: Picture presenting the growth of the structures during oxidation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 Figure 3.19: After a 4-hour oxidation 0.8 pm grid-beams are fully oxidized. Figure 3.20: After a 4-hour oxidation there is still a silicon spike in beams 1pm wide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 xg5S8 : I 1• lO v n ■ 2 5.0k V , ' S flyon a0058 Figure 3.21: For a grid separation too small, the oxidation step closes the gap at the top before the grid-beams are fully oxidized. Figure 3.22: The right grid spacing permits the full oxidation of the grid-beams. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 After the thermal oxidation step, the beams are fully oxidized, but the floor also has oxidized as well. An MIE step is necessary to remove the oxide from the floor of the trenches. However, it was found that the grid spacing left after oxidation is crucial in this step as can be seen in Figure 3.23 for a spacing considered too wide. A combination o f the microloading effect [26], [27] and ions bouncing on the walls explain the necessity to choose the opening carefully. Therefore, to avoid forming mesas o f oxide and still be able to remove the oxide on the floor, the opening was finally fixed at 2 pm. With this selection, it is possible to fully oxidize all beams and still form anchors to hold the structure after it has been cantilevered, as can be seen in Figures 3.24 and 3.25. Figures 3.26 and 3.27 present additional details o f these structures. Because of the growth of the beams, it is not possible to fully oxidize them and still have straight features. With curved beams, however, this can be accomplished, and this case is portrayed in Figure 3.28. Figure 3.23: The MIE step damages the structures when the grid spacing is too wide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.24: The grid spacing is crucial to achieve full oxidation of the anchors such that they become a single unit and are able to survive an MIE step. x79O 25.0k V P ^ o n ' 94 «90 16 Figure 3.25: Large structures can be suspended with the anchor technique. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.26: SEM micrograph presenting a detail of the actuators cantilevered and fully oxidized. Figure 3.27: Detail o f cantilevered and fully-oxidized parallel-plate lines. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Figure 3.28: Detail of meandering lines fully oxidized and cantilevered. At this point it is worth mentioning why it is not possible to use PECVD oxide as a mask in thermal oxidation processes, namely, large blisters form on the surface (Figure 3.29) and many sections peel off. This phenomenon is believed to be due to the presence of hydrogen in the PECVD oxide film since SiH4 and N 2O are the compounds utilized in the glow discharge plasma. After the oxidation and MIE steps, the structures are released and the release time depends only on the size of the anchors. Once more the structures are conformally metalized with aluminum, layers of Cr and Au are evaporated and the samples tested. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 i' x595 i ii ■tOPpw. 2 5 .9 k V ■ ' ” ' I AVON *0012 Figure 3.29: SEM micrograph presenting the effect of using PECVD oxide during oxidation. 3.4 Coplanar Waveguides In order to compare the performance of these novel structures, standard coplanar waveguides (CPW's), as well as CPW's with lower ground planes were prepared using the same high resistivity silicon wafers as substrates. CPW's are not only well understood, but they are production-proven commercially valuable devices. Equations describing them have been developed using conformal mapping methods [28]-[29] and can be found in Appendix III. Figures 3.30 and 3.31 present both layouts. The only difference between them is the metal layer required in the latter case to avoid the electric field from reaching into the substrate. Therefore, the fabrication descriptions are the same. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 We begin depositing an oxide layer on the silicon wafer. The thickness of this film is o f relevance because the attenuation decreases as the thickness of this dielectric layer increases; some results on this will be presented in Chapter 4. We use a conventional image reversal and lift-off process to form the structures on the wafer, which we now briefly describe. Photosensitive material is spun on the wafer, baked before exposure, exposed and instead of developing, it is image-reversed. This is accomplished by exposing the wafer to an ambient rich in ammonia which diffuses into the exposed resist. Then the ammonia binds with the indene carboxilic acid generated by exposure to light and makes the area insoluble in developer solution. The whole wafer is then floodexposed and the photoresist that was not originally exposed can dissolve during development. The advantage of this process is that the side wall slope can be tailored to give an undercut profile for lift-off. In other words, positive resist acts like negative resist [30], With the photoresist walls presenting an undercut, aluminum is evaporated, as well as a thin adhesion layer of Cr and finally Au. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 81 4 l 1 1 ,v v v ... . 21 A ■■ '• .’ *■ • r- ' f Substrate Dielectric Metal Layer Figure 3.30: Layout for a conventional Coplanar Waveguide (CPW). Substrate Metal Layer Dielectric Figure 3.31: Layout for a grounded coplanar waveguide (CPWG). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 3.5 References [1] Z. Lisa Zhang and N. C. MacDonald, "A RIE process for submicron, silicon electromechanical structures", J. Micromech. Microeng., 2 (1992), pp. 31-38. [2] Z. Lisa Zhang, G. A. Porkolab and N. C. MacDonald, "Submicron, movable Gallium Arsenide mechanical structures and actuators", Micro Electro Mechanical Systems '92, Travemunde, Germany, February 4-7,1992, pp. 72-77. [3] K. Shaw, Scream I: a single crystal silicon, single mask, reactive ion etching process for microelectromechanical systems, M.S. Thesis, Cornell University, 1993. [4] A. A. Ayon, N. J. Kolias and N. C. MacDonald, "Tunable micromachined parallel-plate transmission lines", 15th Biennial IEEE/Cornell University Conference on Advanced Concepts in High Speed Semiconductor Devices and Circuits, Ithaca, NY, August 7-9,1995. [5] M. T. Saif and N. C. MacDonald, "A milli newton micro loading device", Proceedings o f the SPIE's Smart Structures and Materials Conference, 26 Feb-3 March, 1995, San Diego, CA. [6] F. J. Blatt, Principles of Physics, Second Edition, Allyn and Bacon Inc., 1986. [7] D. Haronian, Private Communication. [8] D. J. Elliott, Integrated Circuit Fabrication Technology, McGraw-Hill Book Company, 1982. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 [9] E. A. Irene, Silicon oxidation: a process step for the manufacture o f integrated circuits, in Integrated Circuits: Chemical and Physical Processing, edited by P. Stroeve, American Chemical Society, 1985. [10] N. C. MacDonald and A. Jazairy, "Single crystal silicon: application to microoptico-electromechanical devices", SPIE, Vol. 2383, pp. 125-135,1995. [11] J. Maa, H. Gosenberger and L. Hammer, "Effects on sidewall profile of Si etched in BCI3/CI2 chemistry", JVST, Vol. 8B, 1990, pp. 581-585. [12] P. VanDerVoom, Y. Chieh and P. Krusius, “Cl2- and BC13- based two-step ultrafine-line gate polysilicon etch process”, 15th Biennial IEEE/Cornell University Conference on Advanced Concepts in High Speed Semiconductor Devices and Circuits, Ithaca, NY, August 7-9, 1995. [13] C. R. M. Grovenor, A. Cerezo and G. D. W. Smith, Atom probe analysis of native oxides and the thermal oxide/silicon interface, in Layered Structures, Epitaxy, and Interfaces, edited by J. M. Gibson and L. R. Dawson, Materials Research Society, 1985. [14] C. B. Zarowin and R. S. Horwath, “Control of plasma etch profiles with plasma sheath electric field and RF power density”, J. Electrochemical Soc, Vol. 129, No. 11, Nov. 1982, pp. 2541-2547. [15] C. B. Zarowin, "Plasma etch anisotropy- theory and some verifying experiments relating ion transport, ion energy and etch profiles", J. Electrochemical Soc., Vol. 130, No. 5, May, 1983, pp. 1144-1152. [16] C. B. Zarowin, "Relation between the RF discharge parameters and plasma etch rates, selectivity, and anisotropy", JVST, Vol. 2A, pp. 1537-1549,1984 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 [17] A. Kassam, C. Meadowcroft, C. A. T. Salama and P. Ratnam, “Characterization of BC13-C12 silicon trench etching”, J. Electrochem. Soc., Vol. 137, No. 5, May 1990, pp. 1613-1617. [18] K. Shaw, Private Communication. [19] M. T. A. Saif and N. C. MacDonald, “Deformation of large MEMS due to thermal and intrinsic stresses”, SPIE, Vol. 2441, pp. 329-340,1995. [20] J. Drumheller, Private Communication. [21] M. Saif, Private Communication. [22] W. Hofmann, Private Communication. [23] P. Infante, Private Communication. [24] B. E. Deal, The thermal oxidation of silicon and other semiconductor materials, in Semiconductor Materials and Process technology Handbook for VLSI and ULSI, edited by G. E. McGuire, Noyes Publications, 1988. [25] W. E. Beadle, J. C. C. Tsai and R. D. Plummer, Editors, Quick Reference Manual for Silicon Integrated Circuit Technology, John Wiley & Sons, 1985. [26] M. Sato, S. Kato and Y. Arita, "Effect of gas species on the depth reduction in silicon deep-submicron trench reactive ion etching", Jap. J. Appl. Phys., Vol. 30, pp. 1549-1555,1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 [27] D. Chin, S. H. Dhong and G. J. Long, “ Structural effects on a submicron trench process”, J. Electrochemical Soc., Vol. 132, No. 7, 1985, pp. 1705-1707. [28] G. Ghione and C. Naldi, "Analytical formulas for coplanar lines in hybrid and monolithic MIC's", Electronic Letters, Vol. 20, No. 4, pp. 179-181, 16 February, 1984. [29] G. Ghione and C. Naldi, "Parameters of coplanar waveguides with lower ground plane", Electronic Letters, Vol. 19, No. 18k, pp. 734-735,1 September, 1983. [30] Notes for YES vacuum oven, Cornell Nanofabrication Facility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 MEASURED PERFORMANCE 4.1 Test Setup and De-Embedding Process Scattering parameters were measured using a Network Analyzer (HP8510C) with nominal frequency range from 45 MHz to 50 GHz. A set of coaxial cables connected the analyzer to the air coplanar probes (Figure 4.1). The cables can operate up to 50 GHz, but the Cascade Microtech probes can only operate up to 40 GHz. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 HP8510C Network Analyzer Cables (50 GHz) HP8517A S-Param eter Test Set — Iw K * C ascade Microtech Probe Station Figure 4.1: Overview of the test setup. Therefore the frequency limitation in the probe operation determined the highest frequency to be used. The probes had a characteristic impedance of 50 Q and a pitch of 150 pm in a coplanar configuration (Ground-Signal-Ground). The Device Under Test (DUT) was placed on a Cascade Microtech Probe Station where the probes can be moved in three orthogonal directions. A vacuum pump held the samples on the platform. The mechanical system includes additional hardware to level the coplanar probes whenever the three points do not touch the sample at the same time. It is very important to mention that the measurement (Figure 1.1) includes the launch pads as well as the out-of-plane parallel-plate transmission lines. We will need to de-embed the performance of the out-of-plane transmission lines. We will go back to this problem shortly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 Before a precise measurement is made it is necessary to calibrate the entire setup. This is necessary because even small temperature variations change the operating point of the equipment and measurements done on different days with the same calibrations will not be self-consistent. The calibration is done with a Cascade Microtech Impedance Standard Substrate (ISS), which contains short circuits, transmission lines and 50 Q loads all of them on a CPW geometry. This Line-ReflectMeasurement (LRM) calibration defines a reference point at the tip of the coplanar probes. The ISS consists o f highly accurate microwave reference devices made of gold on an alumina substrate. The transmission lines presented in this work, are too small to be measured directly (< 4 pm). For this reason launch pads with dimensions of 120 pm x 200 pm are included during the test. As has been reported in the literature, they can be seen as shunt impedances [l]-[3] that can be subtracted if we measure the launch pads standing alone. Therefore, all devices always included an additional set of launch pads standing alone and as close as possible to the devices to be tested. In other words we measure first a Device Under Test (DUT, which includes the launch pads) and then a set o f launch pads standing alone. In order to extract the impedance and propagation constant the data obtained from the network analyzer is converted to PC format, from here the scattering parameters S are obtained and converted to Y parameters. We then use [2], Y EXTRACTED DUT = Y OUT ' Y LAUNCH PADS (4-1) We convert once more to S parameters and extract Z 0 and y as was outlined in Chapter 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 4.2 On-Wafer Probing The problem of on-wafer probing has been addressed by several authors and different techniques have been developed based on their analyses [4]-[16j. Extensive consideration has also been given to modeling and interpreting interconnections [17][18] and transitions [19]-[21], as well as to the problems of de-embedding and unterminating [22]-[23]. Algorithms for error correction can also be found in the literature [24]-[25]. The method based on the subtraction of the effect of the launch pads [l]-[3] was thought to be appropriate because of the geometry of the arrangement and also because the transition from quasi-coplanar to parallel-plate can be modeled as a shunt capacitance [26]-[28]. To prove this assertion we can consider Figure 4.2, where the pads are being modeled as shunt capacitance in parallel with a resistance [29]. Between the pads there is a slab of high resistivity silicon, which like any other dielectric, can be modeled by lumped circuit equivalents. For this exercise, we considered a resistance value of 1500 Q, and a capacitance value of 0.08 pF. Figure 4.3, presents the measured and the modeled values of transmission and reflection parameters for a 1000 pm transmission line. The comparison was done with PUFF ™ considering a microstrip line with silicon as substrate. LOSSY LINE C T Figure 4.2: M odeling the shunt impedance o f the launch pads. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 In figure 4.3 we can see that the predicted values of reflection and transmission scattering parameters follow those of the measured sample. The successful application of the de-embedding procedure applied in this work hinges in the presence of these shunt impedances. It is worth mentioning that some authors [4], in comparing different on-wafer measurements, have reported the highest accuracy for the arrangement of large pads-small devices. M odeling o f Performance 0 » -5 H -10 -15 t> ‘SSV. £ -2 0 -25-30-35 -40 0 ■at SI 1 S21 SI 1 S21 5 10 M odel M odel Sam ple Sam ple 15 20 25 30 35 Frequency (GHz) 40 Figure 4.3: Comparison o f reflection and transmission scattering parameters for a line and a model including a shunt impedance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 4.3 Measured Performance Figure 4.4 presents the measured and the theoretical attenuation of a parallelplate transmission line with thin conductor as derived in Chapter 2. As can be seen, the measured values follow the expected ones very closely. The excess attenuation is believed to be due to surface waves trapped in the silicon substrate with nominal thickness of 380 ±20 pm. The predicted values are obtained using equation (2.63) h * Tan ( K, - ) = 2 cr K — Coth K2 1 K, (2.63) According to this expression, the attenuation has a very strong dependence on film thickness and only a mild dependence on the conductivity of the film. Therefore, it would be advantageous to be able to grow films much thicker than the average 0.25 pm on each wall presented in this experiment. Aluminum sputtering, however, has the significant drawback that it is thicker at the top (1.3 pm after sputtering 60 minutes) of the structure and thinner on the walls (0.25 pm average thickness) as can be seen in Figure 2.2. Furthermore, lumps of aluminum tend to form at the top of the structure and preclude the possibility of just extending the sputtering time until the film thickness on the walls is considered enough. The electroless plating technique could be successfully applied in this case, even though this particular experiment is not reported in this work. Electroless plating can provide for film thicknesses o f 1 pm and beyond. Transmission lines fully cantilevered but not expected to move, could even accommodate films several microns thick, bringing the attenuation to a fraction o f the values reported in this work, for example 0.17 dB/mm at 10 GHz for a 1 pm film, compared to 0.6 dB/mm found in this experiment. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 The aforementioned equation (2.63) is also used to obtain the predicted phase for the microwave signal, and this is presented in Figure 4.5, along with the measured values. The lines show little dispersion as can be desired for an interconnect or simple transmission line. Extracted Attenuation Theory Measured - 0.2 -0.4 - 0.8 0 5 10 15 20 25 30 35 Frequency (G H z) 40 Figure 4.4: Theoretical and experimental attenuation. 4.4 Electrically Thin Substrates All testing was done with transmission lines on silicon substrates that are 380 ± 20 pm thick. It is feasible to think that surface waves are being trapped by this high dielectric substrate and contributing to the attenuation. In order to determine whether or not electrically thin substrates are the best alternative [30]-[31], an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 experiment was setup involving wafers that were etched back 80, 190 and 290 pm respectively. An additional sample was etched back 330 pm but it was found that the electrical testing could not be performed because the thin membrane Extracted Phase Theory M easured 0 .8- 0 .2 - 0 5 10 15 20 25 30 Frequency (GHz) 35 40 Figure 4.5: Extracted and theoretical phase in units of Rad/mm. bends when the microwave probes touch it. Another technique must be applied for these extremely thin membranes. The etch-back procedure is done depositing 3000 A of a low tensile stress LPCVD (Low Pressure Chemical Vapor Deposition) silicon nitride film («150 MPa ) [32] to be used as a mask in a subsequent wet etch. This deposition is performed with 10 seem of NH3 and 47 seem of SiH2Cl2 @ 850°C and a chamber pressure of 150 mT. The stress of the film can be altered by changing the flows and the temperature of the deposition; the standard, high stress nitride deposition («800 MPa) [33], is performed with flows of 90 seem o f NH 3 and 30 seem Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 of SiH2Cl2 @ 800°C. Besides their different stress, these nitride films also differ in their index o f refraction which is 2.0 (measured @ 633 nm) in the low stress case and 2.15 (measured @ 633 nm) for the standard film. It should be noted that PECVD (Plasma Enhanced Chemical Vapor Deposition) nitride films of comparable thickness, do not have an acceptable masking performance in a KOH solution. The PECVD films have pinhole defects and etching does occur even on the side of the wafer that the film is supposed to protect. After the nitride film has been deposited, with a photolithography step, we open windows o f 10 mm^ on the back of the wafer and immerse the sample in a solution of 440 g KOH + 1200 ml deionized water @ 80°C. This solution provides for an etch rate o f « 2pm /minute (Table 3.1). The final opening on the other side of the wafer is not 10 mm^ but smaller due to the fact that the KOH etch is not fully anisotropic and proceeds at an angle of 54.7° respect to the surface of the wafer. Empirical equations have been worked out [34] to determine the final opening as a function of wafer thickness: A f ~ A 0- 2 2 T (4.2) where A f is the final aperture, A0 is the original opening and T is the thickness of the wafer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 Table 4.1: Etch rates for different silicon planes. Plane Etch Rate (pm/hour) 100 122 111 0.85 331 182 After this step, the nitride film can be removed using a MIE machine with 30 seem CHF3 and 1 kW power and chamber pressure < 3 mT. The nitride film when not removed from the wafer can be used to counterbalance the compressive stress of the oxide film required as a mask to etch silicon. In general, there is an added benefit in not removing this film. With substrates made electrically thin, we proceed with the rest of the process as discussed in Chapter 3. Figures 4.6 through 4.8 present the measured impedance, attenuation and phase constant for devices fabricated on the aforementioned thin substrates. According to Figures 4.6 and 4.7, the extracted values of attenuation and phase constant follow very closely the predicted values. Furthermore, Figure 4.7 also suggests that the extra attenuation located around 20 GHz is not related to the thickness of the substrate. All three figures lead to the conclusion that for the purpose o f cantilever interconnects and tunable microwave devices, the thickness of the substrate plays only a minor role in the final performance of these structures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 Extracted Attenuation Etch Back — 0.2 •••■ 80 urn 190 um 290 um 0.4 0.8 0 5 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.6: Extracted attenuation for lines fabricated on substrates with different thicknesses. Extracted Phase Etch Back jim 600- 200 80 - 0 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.7: Extracted phase constant for different substrate thicknesses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 Extracted Impedance 140 Etch Back fim 120 --------- _100 80 190 290 S 80 N 60 40 20 0 0 5 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.8: Extracted impedance for different substrate thicknesses. 4.5 Surface Roughness and other Considerations The effect of surface roughness on TEM modes has been covered in detail by several authors [35]-[38] and the results have been analyzed using perturbation methods that give approximate solutions for surfaces of arbitrary waveshape, as well as using Bessel-series methods which provide for exact solutions for sinusoidal surfaces. In general, the effect of surface roughness can be understood as a surface displacement that modifies all transmission line parameters. Roughness, however, has been shown to have a second order effect on surface impedance [35] and equation (2.65) is therefore accurate enough, except for very demanding applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 The modification o f the characteristic impedance cannot be so easily neglected, and it can be proved that taking surface roughness into consideration, to the first order in surface displacement, the characteristic impedance is given by [35], (4.3) where Z00 is the characteristic impedance for a smooth surface, ti 0 is the impedance of free space; Ad is the total surface displacement, and p is the perimetral length of the rough conductor. The ratio o f (y\J2p) for our plates range from 3.67 Q/pm for plates 12.7 pm tall to 6.28 Q/pm for plates 7 pm tall and this ratio times the equivalent surface displacement is the change in the characteristic impedance for a smooth surface. The surface roughness measured is < 0.15 pm. Therefore, we can expect a correspondingly small increase in the value of the characteristic impedance and the maximum correction < 2 Q (Figure 4.9). X 14 1 e.e •' • ■2 S.flk’v # $ 6 1 .1 Figure 4.9: SEM micrograph showing the surface roughness of the sputtered aluminum film. This is a top view of one of the plates in a transmission line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 The main role on impedance modification is played by fringing. In equation (2.74) it was assumed a large ratio of height to separation, but as the ratio becomes smaller, theoretical values should depart more and more from the measured values and this will be seen in the following section. 4.6 Impedance Variation Conventional transmission lines present a definite impedance, and this value is selected beforehand and set during processing. In this sense, our devices can also be fabricated to meet impedance specifications as can be seen in Figures 4.10 and 4.11. In Figure 4.10 the separation is chosen and the etch time is adjusted to obtain a working point for the device. Alternatively, the height can be fixed and the separation adjusted to achieve a pre-determined value for the characteristic impedance as can be seen in Figure 4.11. Very high impedance values can be obtained by evaporating a metal film instead o f sputtering once all processing has been done (in this case a thin layer is deposited only on top of the structures, i.e., the separation to height ratio is very large). Values exceeding 100Q for a single transmission line have been measured using this approach. All values obtained are compared to those predicted by h r , s 2 C = e - [ 1 + — - (1 + In i S TV h tv /i., )] » _ (2.74) s equation (2.74) which takes fringing into consideration, and this is seen in Figures 4.12 through 4.15. It is evident that the measured values drift from those predicted as the ratio of height to separation decreases. This tendency is expected because of the growing fringing importance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Separation = 4.7 pm Height in |tm 150 120 - 12.7 90 6030 0 5 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.10: Impedance variation with plate height, for a plate separation of 4.7 pm. Plate H eight = 12.7 pm Sep. in pm 150- 4.7 120 90 60 30 0 5 10 15 20 25 30 35 Frequency (G H z) 40 Figure 4.11: Impedance variation with plate separation, for a plate height of 12.7 pm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 Plate H e i g h t = 12.7 pm 150 ; Sep = 4.7 pm 1 : 1 1 t 120 j Theory 1 1 1 1 k S 90 T O 11 ! i j V N 60 I 30 ; ..... ; ..... r ..... i 0 i 1 0 5 10 15 2 0 25 30 F re q u en cy (G H z ) 35 40 Figure 4.12: Measured and theoretical impedance for a plate height of 12.7 pm. Plate H eigh t = 1 0 pm Sep = 4.7 pm 180 Measured Theory 150 G120 60 30 0 0 5 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.13: Measured and theoretical impedance for a plate height of 10 pm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Plate H eig h t = 7 pm Sep = 4.7 nm 150- 120 Theory Measured - 906030- 15 20 25 30 35 Frequency (GHz) 5 40 Figure 4.14: Measured and theoretical impedance for a plate height of 7 pm Plate H eig h t = 12.7 pm Sep = 8.9 pm 180- ------------------- 1 5 0 -i Measured Theory G120 !» ■ 60H 30 0 0 10 15 20 25 30 35 Frequency (GHz) 40 Figure 4.15: Measured and theoretical impedance for a plate separation of 8.9 pm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 4.7 Electromechanical Tuning The most remarkable aspect of this experiment is the electromechanical tunability o f the structures. To this effect, banks of finger capacitors are added to the structure to move the plates, thus, changing the impedance o f the transmission line arrangement, as can be seen in Figure 4.16. The results reported in Figure 4.16, correspond to two different transmission line arrays: in one o f them the plates are opened by application o f the voltage (Z0 increases), in the other the plates are brought closer by applying the voltage (Z0 decreases). For this reason, the 0 V line and the 70 V line of decreasing impedance match perfectly because they correspond to the same device, whereas the line of 70 V of increasing impedance has a slightly different performance. Evidently the lines were not perfectly identical. It is worth mentioning that the number o f finger capacitors determines the potential required to achieve a determined variation for the characteristic impedance [39]-[40], and it is feasible to operate with single digit voltages provided that the number o f finger capacitors is large enough. This, in turn, implies additional space for the structure. If necessary a compromise has to be worked out between the area that can be allotted to the device, and the maximum impedance variation expected from the device. Finally, it is necessary to underline once more that the tested structure actually comprises two lines in parallel. Therefore, the total impedance change-per-line is twice that reported in the previous figures. This large impedance variation is thought to suffice for most applications. There is room, however, for larger variations. Since the spacing between finger capacitors also plays a role in the voltage required for a specific change, in general, it is necessary to maintain this gap as small as practically possible [39]. With the metal-sputtering approach, the gap has to be maintained larger than strictly required due to the lumps of aluminum forming on the rims of the beams. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 In the case o f a directly coupled resonator, the length determines the resonant frequency but we can still optimize the performance of the circuit by moving the plates until the point of perfect matching is encountered as can be seen in Figure 4.17. It should be noted that this figure also permits the observation that even extremely small variations can be achieved with the techniques covered in this work. 4.8 Comparison with Conventional Structures In order to compare the performance of the micromachined transmission lines presented in this work with conventional structures, we also fabricated Coplanar Waveguides on high resistivity silicon wafers. We started depositing 2.3 pm of E lectro m ech a n ica l T uning 100 — Open (70 V ) — R ef. (0 V) 80- — C lo se (70 V ) a 60N 4020 - 0 5 10 15 2 0 25 30 35 F requ en cy (G H z) 40 Figure 4.16: Electromechanical Tuning, impedance variation can be accomplished either by opening the plates (Z0 increases) or by closing the plates (Z0 decreases). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 R eso n an t F req u en cy - 10 - 10 v 15 v -5 0 -6 0 2 5 2 5 .5 2 6 2 6 .5 2 7 2 7 .5 2 8 2 8 .5 2 9 F re q u e n c y (G H z) Figure 4.17: Electromechanical Tuning, when the transmission line array is perfectly matched permits the optimization of the working point. PECVD Si0 2 @ 240°C. The CPW lines were then made using the lift-off process described in Chapter 3. The thickness of the thermally evaporated aluminum was fixed at 0.457 pm. We also evaporated 100 A of Cr and 480 A of Au to protect the probes from collecting unwanted aluminum residues. The equations describing CPW and CPWG can be found in appendix III. As can be seen in Figure 4.18, the transmission parameters S2i are comparable even though the film thickness in our work is only 0.25 pm. According to equation (2.63) with only twice the film thickness, the performance our devices would be better almost over the entire range of frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 T R A N S M I S S I O N IN dB — CPW — LINE -0 .5 « rT3 -1 .5 - 2 - -2.5 0 5 10 15 20 25 30 35 Frequency (G H z) 40 Figure 4.18: Comparison o f transmission parameter S21 for a CPW and one of the lines presented in this work as measured, before de-embedding. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 4.9 References [1] W. R. Eisenstadt and Y. Eo, “S-parameter-based IC interconnect transmission line characterization”, IEEE Trans, on Components, Hybrids and Manufacturing Technology, Vol. 15, No. 4, August 1992, pp. 483-490. [2] P. J. van Wijnen, H. R. Claessen and E. A. Wolsheimer, “A new straightforward calibration and correction procedure for “on-wafer” high frequency S-parameter measurements (45 MHz -1 8 GHz)”, IEEE 1987 BCTM, pp. 70-73. [3] H. Cho and D. E. Burk, “A three-step method for the de-embedding o f highfrequency S-parameter measurements”, IEEE Trans, on Electron Devices, Vol. 38, No. 6, June 1991, pp. 1371-1375. [4] A. Fraser, R. Gleason and E. W. Strid, “GHz on-silicon-wafer probing calibration methods”, IEEE 1988 Bipolar Circuits and Technology Meeting, pp. 154-157. [5] M. Roos, “A measurement and calibration technique for accurate measurement of amplifier S parameters”, 1987 IEEE MTT-S Digest, pp. 449-451. [6] C. A. Hoer and G. F. Engen, “Calibrating a dual six-port or four-port for measuring two-ports with any connectors”, 1986 IEEE MTT-S Digest, pp. 665-668. [7] G. F. Engen and C. A. Hoer, “”Thru-Reflect-Line”: an improved technique for calibrating the dual six-port automatic network analyzer”, IEEE Trans. MTT, Vol. MTT-27, No. 12, December 1979, pp.987-993. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 [8] J. Pla, W. Struble and F. Colomb, “On-wafer calibration techniques for measurement o f microwave circuits and devices on thin substrates”, 1995 IEEE MTTS Digest, pp. 1045-1048. [9] J. S. Kasten, M. B. Steer and R. Pomerleau, “Enhanced Through-Reflect-Line characterization of two-port measuring systems using free-space capacitance calculation”, IEEE Trans. MTT, Vol. 38, No. 2, February 1990, pp. 215-217. [10] G. Dawe and L. Raffaelli, “Characterization of active and passive millimeterwave monolithic elements by on-wafer probing”, 1989 IEEE MTT-S Digest, pp. 413415. [11] E. W. Strid, “26 GHz wafer probing for MMIC development and manufacture”, Microwave Journal, August 1986, pp. 71-82. [12] K. E. Jones, E. W. Strid and K. Reed Gleason, “mm-wave wafer probes span 0 to 50 GHz”, Microwave Journal, April 1987, pp. 177-183. [13] M. Fossion, I. Huynen, D. Vanhoenacker and A. Vander Vorst, “A new simple calibration method for measuring planar lines parameters up to 40 GHz”, 22nd European Microwave Conference, Espoo, Finland, 24-27 August, 1992, pp. 180-185. [14] D. Williams and R. B. Marks, “Accurate Transmission Line characterization”, IEEE Microwave and Guided Wave Letters, Vol. 3, No. 8, August 1993, pp. 247-249. [15] Y. Shih, “Broadband characterization of conductor-backed coplanar waveguide using accurate on-wafer measurement techniques”, Microwave Journal, April 1991, pp. 95-105. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 [16] D. Williams and R. B. Marks, “On-wafer impedance measurement on lossy substrates”, IEEE Microwave and Guided wave Letters, Vol. 4, No. 6, June 1994, pp. 175-176. [17] J. R. Brews, “Transmission line models for lossy waveguide interconnections in VLSI”, IEEE Trans, on Electron Devices, Vol. ED-33, No. 9, September 1986, pp. 1356-1365. [18] S. B. Goldberg, M. B. Steer, P. D. Franzon and J. S. Kasten, “Experimental electrical characterization of interconnects and discontinuities in high-speed digital systems”, IEEE Trans, on Components, Hybrids and Manufacturing Technology, Vol. 14, No. 4, December 1991, pp. 761-765. [19] D. F. Williams and T. H. Miers, “A coplanar probe to microstrip transition”, IEEE Trans, on MTT, Vol. 36, No. 7, July 1988, pp. 1219-1223. [20] S. R. Pennock, C. M. D. Rycroft, P. R. Shepherd and T. Rozzi, “Transition Characterization for de-embedding purposes”, 17th European Microwave Conference, Rome, Italy, 7-11 September, 1987, pp. 355-360. [21] K. C. Gupta, R. Garg and R. Chadha, Computer-Aided design of Microwave Circuits, Artech, 1981. [22] D. F. Williams and T. H. Miers, “De-embedding coplanar probes with planar distributed standards”, IEEE Trans. MTT, Vol. 36, No. 12, December 1988, pp. 18761880. [23] R. F. Bauer and P. Penfield, “De-embedding and unterminating”, IEEE Trans. MTT, Vol. MTT-22, No. 3, March 1974, pp. 282-288. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 [24] A. A. M. Saleh, “Explicit formulas for error correction in microwave measuring sets with switching-dependent port mismatches”, IEEE Trans. Instrumentation and Measurement, Vol. IM-28, No. 1, March 1979, pp. 67-71. [25] O. J. Davies, R. B. Doshi and B. Nagenthiram, “Correction of microwavenetwork analyzer measurements of 2-port devices”, Electronic Letters, Vol. 9, No. 23, November 1973, pp. 543-544. [26] J. R. Whinnery and H. W. Jamieson, “Equivalent circuits for discontinuities in transmission lines”, Proc. IRE, February 1944, pp. 98-114. [27] R. N. Simons and G. E. Ponchak, “Modeling of some coplanar waveguide discontinuities”, 1988 IEEE MTT-S Digest, pp. 297-300. [28] R. Sato and T. Ikeda, Line Constants, in Microwave Filters and Circuits, Edited A. Matsumoto, Chapter V, Academic Press, 1970, pp. 129-131. [29] A. R. V. Hippel, Dielectrics and Waves, John Wiley and Sons, 1954, pp. 86-91. [30] N. J. Kolias, Design of millimeter wave integrated active antenna arrays, M.S. Thesis, Cornell University, 1993. [31] J. F. Luy and P. Russer, Editors, Silicon-Based Millimeter-Wave Devices, Springer-Verlag, 1994. [32] Phil Infante, Private Communication. [33] Phil Infante, Private Communication. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill [34] A. Amar, R. L. Lozes, Y. Sasaki, J. C. Davis and R. E. Packard, "Fabrication of submicron apertures in thin membranes of silicon nitride", JVST, Vol. B 11(2), Mar/Apr 1993, pp. 259-262. [35] A. E. Sanderson, “Effect o f surface roughness on propagation of the TEM mode” in Advances in Microwaves, edited by L. Young, Academic Press, 1971, pp. 1-57. [36] S. P. Morgan, Jr., “Effect of surface roughness on eddy current losses at microwave frequencies”, Journal of Applied Physics, Vol. 20, April 1949, pp. 352362. [37] A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance”, Proc. Inst. Elec. Eng., Part B, Vol. 102,1955, pp. 698-708. [38] H. M. Barlow, “High-frequency impedance of a practical metal surface and the effect of a thin coating of dielectric over it”, Electronic Letters, Vol. 6, No. 13, June 1970, pp. 413-415. [39] W. Hofmann, L. Cheng and N. C. MacDonald, “Design and fabrication of micromachined electron guns (MEGS) using multiple-level planar tungsten process”, SPIE 1995 symposium on Micromachining and Microfabrication, 23-24 October, 1995. [40] A. A. Ay6n, N. Kolias andN. C. MacDonald, "Tunable, micromachined parallelplate transmission lines", 15th Biennial IEEE/Cornell University Conference on Advanced Concepts in High Speed Semiconductor Devices and Circuits", Ithaca, New York, August 7-9, 1995. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Half-Wavelength Micromachined Dipole Antennas 5.1 Introduction We now apply the microfabrication tools described in previous chapters to the design and microfabrication of half- wavelength dipole antennas. An antenna is a structure designed to radiate and receive energy effectively [1]. It is a transducer between a free-space wave and a guided wave and vice-versa [2]. In order to optimize the performance of an antenna we have to understand its electrical characteristics because antennas always are part of a complete electronic 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 circuit. This fact places constraints on the antenna design because of limitations on the size and shape o f the surface area available for the radiating aperture, the size and shape of the volume available for the antenna [3], the presence of obstacles and other conducting elements, interference from other sources of electromagnetic energy in the microcircuit, etc. These constraints are reflected in the physical design of the antenna, but the physical geometry determines the performance of the antenna. Therefore, some of the desired performance specifications have to be compromised. 5.2 Basic Antenna Parameters All antenna types involve the basic principle that radiation is produced by accelerated charges [4], The outgoing electromagnetic wave has a radiation pattern that depends on the geometry of the antenna. This radiation pattern has a dependence in both 6 and <|>and the purpose of the design is to obtain a pattern as close as possible as that prescribed by the specific application. In general it is not feasible to tailor radiation patterns precisely, but excellent approximations can be made. Directivity is also associated with the radiation pattern and is defined as the ratio o f the maximum power density to its average value over a sphere [5]. Directivity is also called maximum directive gain by some authors. For an antenna radiating the same power in all directions (isotropic) the directivity has a value of 1. Gain is also widely used to account for ohmic losses (which heat the antenna) in the efficiency of an antenna. For a lossless antenna the gain equals the directivity, in general [6] gain and directivity are related according to G = C0D Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.1) 114 where 0<Co<l is the efficiency factor o f the antenna. The radiation patterns usually have several lobes, and special techniques have to be applied to maintain the main lobe much higher than all the others. Before proceeding any further it is necessary to point out the basic assumptions made in this work. First, the dipole antenna in this experiment of rectangular cross section with cross section (cross section = width x height = 2.5 pm x 12 pm and X is 10 000 pm) dimensions « X can be approximated to a cylindrical dipole antenna o f radius equal to one-fourth of the height [7], [8]. Second, the current distribution along this symmetrical dipole antenna of small radius (length/radius > 1000) can be assumed to be sinusoidal [4], [9], [10]. We now proceed to derive the necessary expressions for a dipole antenna. 5.3 Radiation Characteristics o f Linear Dipoles Consider an infinitesimal linear wire (total length « X) oriented along the zaxis, positioned symmetrically at the origin (Figure 5.1) and carrying a constant current represented by 7(z ')= /0 z' (5.2) Then the vector potential at the point of observation (x, y, z) is given by [11] (5.3) where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 (5.4) r = J x 2+ y 2+ z 2 \i0 is the permittivity o f free space and k is the wavenumber. z X Figure 5.1: Coordinate system for an infinitesimal linear dipole of total length /. We can express equation (5.3) in spherical coordinates using [11] = Sin 9 Cos<f> Sin 9 Sin p Cos 9 Cos 9 Cos<f> Cos9 Sinp - S i n 9 1---1 w Ao -Sin ip Cos<j> 0 4 Ay A . to obtain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 u I le~Jkr Ar = ^ - 2 Cos e 4 it r (5.6) u I le~Jkr Ag= - Mo Sin 6 4n r (5.7) A^O. (5.8) We can now apply [12] B = //0 H = V x A (5.9) to obtain Hr = H g= 0 h k l a l Sin G 1 .. *r = J — An a r ( I +j kt trt ) 6 (5.10) (5-n ) Finally, applying Ampere's law in phasor form [12], and when J - 0, E = — ^— V x H ja>e0 we obtain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.12) 117 (5, 4) ] _ )e- , r E v= 0 (5.15) \Mo k where rja = I— is the impedance of free space, and co~ 4 m0^0 The complex Poynting vector associated with these fields is given by S = i(E x H -)= i(£ „ « ;r-£ rH ;e) (5.16) integrating equation (5.16) over a sphere of radius r we obtain the total complex power, given by p = # S«ds=770^ h i (1 -7 __ 1_ ( kr) (5.17) The imaginary part in equation (5.17) determines the reactive power of the antenna while the real part is the time-average power radiated [13]. We can conclude, therefore, that the loss o f power by radiation is due to the resistive part and this explains the name radiation resistance, R rad [14], Therefore, P = Eradiated + ha clive and we can equate [15] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.18) 118 (5.19) P.radiated and using equations (5.17) and (5.19) we obtain P-md = 8 0 ;r 2 (-jj- ) 2 Q (5.20) According to equation (5.20) the radiation resistance of an infinitesimal dipole is too low. For instance, for I = A/60 the predicted radiation resistance is only 0.22 Q. This is the reason the use of electrically small antennas is limited to applications where there are space limitations [16]. It is also worth mentioning that C. Balanis [13] calculated the radiation resistance for small dipoles of lengths A/50 < I < A/10 assuming a triangular current distribution and found that the radiation resistance is given by = 20 * 2 ( j ) 2 n (5.21) Even in this regime the predicted radiation resistance is too small, for / = A/10 the radiation resistance is only 1.97 Q. Apparently as the length of the antenna increases the radiation resistance increases quadratically. However in the previous derivation we assumed / « X. We can go back and remove this constraint to extend the results to antennas with sizes comparable to the wavelength. In this case the current is given by / = / „ S i n [ k { - * z ’) ] e J,0(,-slc) 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.22) 119 the retarded values are needed because the antenna can be seen as composed of a large number o f infinitesimal dipoles and the phase difference of all their contributions must be taken into account (Figure 5.2). Then in the far field region where kr » 1 the distance to the observation point is given by s = tJx 2 + y 2 + (z-z')2 (5.23) and the fields are given by ,„ .60 7 tl Sin 0 _iks , , d E0= j e J dz’ sX 1/2 Figure 5.2: Coordinate systems for a linear dipole. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.24) Evidently it is only necessary to calculate either Eg or H^. Therefore, we now have j o»»io nJ 0>t Sink(--z') H i —y ° { f-------* J OX 1J (5.27) o Sink(—+z') f --------2-------e~ja ,s ,c dz'} + - S 1,2 and from Figure 5.2 we can use (5.28) s=r-z'C os0 to obtain / Smff p j w r1^ j { J e ^ ^ ' ‘Sink(L-z’)dz’ 0 + | - in j en*>cose)z'/c (5.29) Sink(-+z')dz'} we can apply to equation (5.29) the relations [17] QU f e au Sin(b+cu) d u = —t — =-[a5/n(&+CM)-cCoj(h+cn)] J a +c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.30) 121 to obtain from the contribution of the integrals 2 kl kl Integrals = — [ Cos (— CosQ ) - C o s — ] k Sin 0 2 2 (5.31) and finally J e J a( l - r l c ) C0S{ kl kl COSQ ) - C o S ---- H d= j —------------- [------- 2------------------ 2 i * 2nr L Sin 0 i (532) K Applying equation (5.25) we obtain kl kl 6 0 / e J<0(-, ~rlc) ^ 0,s'(— C o s Q ) - C o s — E . = j - ^ - r-----1— 2— ------------- * - ] (5-33) The first factor gives the instantaneous magnitude of the fields as functions of distance and antenna impedance while the term in brackets gives the far-field pattern. For instance, for / = ?J2 the pattern factor, shown in Figure 5.3, is simply [18] cosy-cos e ] F<' 0 ) = v ~ a9 Sin (5-34) As was previously done with the infinitesimal dipole we can determine the radiation resistance by applying equations (5.16) and (5.17) to obtain ?=<£[ S»ds=<$ —(E xH *)»ds s s 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.35) 122 Figure 5.3: The radiation pattern for a half-wavelength linear dipole. we can use equations (5.32) and (5.33) to obtain .kl„ ^ ,kL,2 n [C oj(— CosQ ) - C o s ( — )]' /> = 3 0 /2 J ---------2----2----- dQ J Sin 9 (5.36) Using equation (5.19), we obtain n * - = 60 / ■k l „ [Cos(— CosQ ) - C o s ( — )]" — '— a re — ~ d e (5 3 7 ) finally with the substitutions x = Cos Qand dx = - Szw QdQ [19] it can be proved that the radiation resistance is given by [20]-[22] And = 60{y + L o g k l - C i (kl) +^ Sin (k l) [S, ( 2 k l ) - 2 S , (kl)] 1 kl + ± C o s(k l)[y+ L o g (-j)+ C l (2kl)-2C i (kl)]} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.38) 123 where y = 0.577 is Euler's constant, (5.39) and (5.40) 00 Figure 5.4 presents the computed values for the radiation resistance as a function of dipole total length. For the specific case of a half-wavelength dipole the radiation resistance is 73.13 Q. It can be readily seen that the radiation resistance decreases rapidly as the length of the dipole becomes smaller. Finally to obtain the directivity of a small dipole we can use equation (5.16) to obtain the maximum power density and equation (5.17) to obtain the average power radiated over a sphere o f radius r. From these, we obtain The directivity of a half-wavelength dipole can be obtained in a similar way. Using equations (5.32) and (5.33) we determine S, then along with equation (5.36) we obtain [13] D half - wavelength 4 7? Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.42) 124 Dipole Radiation Resistance 250- 0 1 2 3 4 Normalized Dipole Length (length/A) 5 Figure 5.4: The variation of radiation resistance with dipole length. We can now conclude that both the higher radiation resistance and improved directivity of a half-wavelength dipole make it more suitable for our purposes, relative to a small dipole. There is also a reactance associated with a half-wavelength dipole which can be extracted using the methodology previously outlined and is found to be [19], [23] j 42.5 Q. The general expression for the reactance is X = — [2S, ( k l ) + C o s ( k l ) { 2 S , ( k l ) - S t ( 2 kl)} 4n - S i n ( k l ) { 2 C , ( k l ) - C , (2 k l ) - C , (5.43) 2 k R2 and this equation is presented in Figure 5.5 for different values of the equivalent radius R. Evidently the reactance for a half-wavelength linear dipole is not very Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 sensitive to variations in the equivalent radius of the structure. This particular effect can be seen as providing for flexibility during processing. This inductive reactance of a half-wavelength linear dipole is eliminated by shortening the dipole to about 0.475 X [18]. At this length the radiation resistance is only 67 Q. At this point it is already evident what the advantages of a half-wavelength dipole are: 1.) the radiation resistance is higher than that of any smaller dipole, 2 .) the directivity is better compared to that of any smaller dipole and 3.) the reactance is insensitive to the radius of the dipole. Dipole Reactance 300 200 100- « o - 100 - -200 0 2 3 4 Normalized Dipole length (length/A) 1 5 Figure 5.5: The reactance as a function of dipole length for two different dipole radii. The larger fluctuating curve corresponds to a radius of .03 X while the other curve corresponds to a radius of .0003 X. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 The advantages in radiation resistance and directivity are evident, but the advantage as far as the reactance is concerned is extremely important because we can concentrate the fabrication effort in controlling the impedance of the lines without having to worry about additional electrical implications that could have a deleterious effect on the performance o f the antenna. Finally, we proceed to describe the processing in detail. 5.4 Microfabrication The processing is based in the work presented in previous chapters. There is, however, one additional consideration: silicon has a high dielectric constant and in order to improve radiation efficiency we need to remove all silicon underneath the antenna structure. This implies that the processing being done on both sides o f the silicon wafer has to be perfectly aligned. In order to accomplish this objective we require alignment marks that are visible from either side o f the wafer and this is the starting point for this new scheme. (a.) Alignment marks. We begin with an RCA cleaning procedure prior to depositing on a blank wafer a 3000 A layer of LPCVD low-stress silicon nitride which will serve as a mask for electrochemical etching. This step is followed by a photolithography step to transfer the topography to the photoresist. This is done using KTI 895i, 50 cs spun at 4000 rpm (2.4 pm), baked on a hot-plate @ 90°C for 60 seconds, exposed (6 seconds, focus = 251) and developed 3 minutes in OCG 945. The alignment marks are then transferred to the nitride layer using an RIE machine with 30 seem CHF3, 1 seem 0 2 at 30 mT and 90W power. The photoresist is stripped and the sample is etched electrochemically using 660 g of KOH in 1400 ml of deionized water @ 85°C (Figures 5.6 and 5.7). The alignment marks start out as squares of 1000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 pm per side. When the etch is done the squares measure 460 pm per side. The measured etch rate is 2 pm/minute. H H Silicon Hi Silicon Nitride Figure 5.6: Schematic cross section of the silicon wafer with the alignment marks electrochemically etched. The thickness of the silicon nitride membrane is 3000 A while the wafer is 360 pm thick. (b.) Masking Wafer: Standing Pads. In order to obtain the radiation resistance predicted by equation (5.39) it is necessary to avoid the formation of conducting planes during aluminum sputtering. This implies that the metal being sputtered has to be deposited only on the antenna structure and not on tire rest of the wafer. In order to accomplish this, a second wafer can be used as a mask. This shadowing wafer is going to be placed on top of the wafer with the antenna, and in order to eliminate the possibility of the surface of the shadowing wafer damaging the antenna, we electrochemically etched standing pads. The height of the standing pads determines the separation between the two wafers (Figure 5.8). The standing pads are squares that start out measuring 9000 pm per side. (c.) M asking Wafer: Antenna Wells. The final step in the preparation of the shadowing wafer is to fabricate the openings that will allow the sputtering of the antenna structure. This is done electrochemically. Therefore we proceed with a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 Figure 5.7: SEM micrograph of an alignment mark. The electrochemical etch proceeds until there is only a silicon nitride membrane left. HZ) Silicon ■ Silicon Nitride Figure 5.8: Schematic cross section of the masking wafer showing the alignment marks and the standing pads. The height of the pads was fixed at 120 pm second RCA cleaning procedure and deposit an additional layer of LPCVD lowstress silicon nitride to be used as a mask during the KOH etch. The photolithography step follows the description in (a), the only difference being the utilization o f the alignment marks to place the antenna wells in the proper Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 locations. We then transfer the topography to the silicon nitride layer, strip the resist and electrochemically etch the antenna wells (Figure 5.9). Silicon Figure 5.9: Schematic cross section of the masking wafer showing the alignment marks, the standing pads and the antenna well. The height of the pads was fixed at 120 pm. (d.) Antenna Structure. Having placed alignment marks and having prepared a shadowing wafer we can now proceed to microfabricate the antenna structure. The process for this was presented in Chapter 3: deposit a layer of PECVD silicon oxide followed by a photolithography step, transfer the topography to the oxide, strip the resist, make the deep trench in silicon with chlorine chemistry, remove the passivating layer using nanostrip followed by an RCA cleaning procedure, deposit a thin layer of PECVD oxide and remove the silicon oxide on the floor (Figure 5.10). At this point we depart from the process presented in Chapter 3. Instead of releasing the structure, we deposit an additional layer of LPCVD silicon nitride (after the corresponding RCA cleaning procedure) and then electrochemically remove the silicon substrate underneath the antenna structure (Figure 5.11). At the end of this step we have fabricated a structure that is 12 pm high sitting on a silicon nitride membrane that is 0.3 pm thick (Figure 5.12). Figure 5.13 shows how the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 electochemical etch proceeds into the beams of the structure but stops when the etch fronts meet. Figure 5.10: Antenna structure microfabricated as described in Chapter 3. x l9 2 , . i O'C1>.»'<*•M 25.8k V a f l v o n ' 9 6 »0886. Figure 5.11: SEM micrograph showing the antenna sitting on the nitride membrane. The total thickness of the wafer can be readily seen as a clear band across the picture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure S.]2: SEM micrograph showing the alignment mark and the antenna after the silicon substrate has been removed. Figure 5.13: SEM micrograph showing the electrochemical etch underneath the silicon beams. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.14: After removing the silicon nitride membrane, the structure is free standing with no substrate underneath. (d.) Free-Standing Structures (no Substrate). We now proceed to etch the silicon nitride membrane using an RIE step with 30 seem CHF3, 1 seem Oz at 30 mT and applying 90 W power (Figure 5.14). An additional RIE isotropic step (SFg) is required in order to electrically disconnect the launch pads. The antenna structure is now substrate-free. We place the masking wafer on top of the wafer with the structures and sputter aluminum with the settings described in Chapter 3. After this step the masking wafer is fully covered with aluminum but the wafer with the corresponding antenna structures receives the aluminum only on the structure of interest. We finish the process depositing 100 A of Cr and 450 A of Au to protect the microwave probes. Testing is done after removing the masking wafer. 5.5 Test Setup and Measured Performance The test setup is similar to the one used for transmission lines. However, in this case we need only one probe (Figure 5.15) to measure the reflection coefficient which is related to the load impedance [24]-[26] according to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 (5.44) where Z 0 is the characteristic impedance of the transmission line. Prior to performing the test we proceed with a conventional Through-Reflect-Line (TRL) calibration. The geometry of the transmission lines selected for this experiment consists of a single parallel-plate waveguide. This is, therefore, comparable to microstrip or slotline geometries. Figure 5.16 shows the general layout for the structure in this experiment and Figure 5.17 an SEM micrograph of the finished device. HP8510C Network Analyzer HP8517A S-Parameter Test Set DUT C ascade Microtech Probe Station Figure 5.15: Test setup for antennas, we only need one probe to measure the reflection coefficient. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 Electrostatic Actuators Launch Pads Dipole Electrostatic Actuators Figure 5.16: Schematic view (not at scale) of the structure used in this experiment: the launch pads measure 120 x 230 pm, the length of the waveguide used as matching network (BC) is 1000 pm, the length between the launch pads and the antenna (AE) is 3500 pm and the total dipole length (d) is 5000 pm. The pads at the end of the dipole antenna are squares of 50 pm per side. Upon calibration of the equipment we measured the response. Figure 5.18 shows the performance in the frequency range of interest, as well as the values predicted by equations (5.38) and (5.43) which take into consideration the equivalent radius of the antenna [27]-[33]. As can be seen, the measured results follow closely the predicted values. It should be noted that special care has to be given to the length p of the waveguide that feeds the antenna. The original antenna design had a p length of 3.5 mm but in the final experiment the p-length was fixed at 2 mm and a larger number of antenna structures withstood all processing. This is due to the fact that the vertical stiffness of a suspended structure is inversely proportional to the cube of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 length it spans, therefore, a shorter p produces a larger vertical stiffness. There is, however, an added advantage in keeping p > 2 mm: standard dipole antenna theory predicts a lower radiation resistance as the dipole antenna gets closer to the area occupied by the actuators where the presence of the sputtered metal modifies the boundary conditions in the solution of Maxwell’s equations. Specific applications will determine the length p that can be tolerated. Figure 5.17: SEM micrograph showing the finished device. In the bottom o f the picture we can see the voltage-tunable waveguide and the arrays of interdigitated capacitors similar to those presented in Figure 1.7. After the matching network a section of parallel-plate transmission line feeds the dipole antenna. The ends of the antenna are being held in place by pads measuring 50 pm x 50 pm. After removing the silicon nitride membrane a (see page 129) short RIE step with SF6 is required to permit the antenna end pads to be electrically disconnected from the silicon substrate. The bending o f section p can be controlled by using higher aspect ratio structures and by minimizing the thickness of the silicon oxide film. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 Reflection Coefficient 0 Measured Theoiy -1 -7 -8 15 20 25 30 Frequency (GHz) 35 Figure 5.18: Measured performance of the antenna, the reflection coefficient Sn is given in dB. With an applied voltage of 15 V, Su at resonance decreases 1.5 dB. This work can now be readily extended to include antenna arrays for omnidirectional reception. In order to accomplish this, we can microfabricate 7 (or more) dipole antennas positioned around a circle. Another alternative is to microfabricate an array of dipoles in parallel within a fully oxidized frame to avoid the high dielectric constant of silicon. This work can also find radar applications. In this case the structure has to be stiff enough to withstand the electrostatic pressure, therefore, the waveguides would not be voltage-tunable, but we can still use the mechanical resonance frequency to operate the device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 5.6 References [1] D. K. Cheng, "Field and Wave Electromagnetics", Second Edition, AddisonWesley Publishing Company, 1990. [2] J. D. Kraus, "Electromagnetics", Fourth Edition, McGraw-Hill, Inc., 1992. [3] D. R. Rhodes, "Synthesis of Planar Antenna Sources", Clarendon Press, Oxford, 1974. [4] J. D. Kraus, "Antennas", Second Edition, McGraw-Hill, Inc., 1988. [5] S. Ramo, J. R. Whinnery and T. Van Duzer, "Fields and Waves in Communication Electronics", John Wiley and Sons, Inc., 1965. [6] R. W. P. King, H. R. Mimno and A. H. Wing, "Transmission Lines Antennas and Waveguides", McGraw-Hill Book Company, Inc., 1945. [7] Y. T. Lo, "A note on the cylindrical antenna of noncircular cross section", J. Appl. Phys., Vol. 24, pp. 1338-1339,1953. [8] C. W. Harrison and R. King, “The radiation field of a symmetrical centerdriven antenna of finite cross section”, Proc. IRE, Vol. 31, Dec. 1941, pp. 693-697. [9] R. King and C. W. Harrison, "The distribution of current along a symmetrical center-driven antenna", Proc. IRE, Vol. 31, pp. 548-566, October 1943. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 [10] "The Handbook of Antenna Design", A. W. Rudge, K. Miine, A. D. Olver and P. Knight, Editors, Peter Peregrinus, Ltd., 1983. [11] D. J. Griffiths, "Introduction to Electrodynamics", Second Edition, Prentice Hall, 1989. [12] J. D. Jackson, "Classical Electrodynamics", Second Edition, John Wiley & Sons, 1975. [13] C. A. Balanis, "Antenna Theory Analysis and Design", Harper & Row Publishers, 1982. [14] S. A. Schelkunoff and H. T. Friis, "Antennas Theory and Practice", John Wiley & Sons, 1952. [15] J. R. Wait, "Introduction to Antennas and Propagation", Peter Peregrinus, Ltd., 1986. [16] J. T. Bolljahn and R. F. Reese, "Electrically small antennas and the lowfrequency aircraft antenna problem", IRE Trans. Antennas and Propagation, October 1953. [17] G. B. Thomas and R. L. Finney, "Calculus and Analytic Geometry", AddisonWesley Publishing Co., 1979. [18] L. V. Blake, "Antennas", Artech House, Inc., 1984. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 [19] R. W. P. King, "The theory o f Linear Antennas", Harvard University Press, 1956. [20] R. Bechmann, "Calculation of electric and magnetic field strengths of any oscillating straight conductors", Proc. IRE, Vol. 19, No. 3, pp. 461-466, March, 1931. [21] A. A. Pistolkors, "The radiation resistance of beam antennas", Proc. IRE, Vol. 17, No. 3, pp. 562-579, March, 1929. [22] P. S. Carter, "Circuit relations in radiating systems and applications to antenna problems", Proc. IRE, Vol. 20, No. 6, pp. 1004-1041, June, 1932. [23] H. Jasik, Editor, "Antenna Engineering Handbook", McGraw-Hill Book Co., 1961. [24] W. L. Weeks, Antenna Engineering, McGraw-Hill Book Co., 1968. [25] Microwave Antenna Measurement, Edited by J. S. Hollis, T. J. Lyon and L. Clayton, Scientific Atlanta, 1969. [26] R. E. Collin and F. J. Zucker, Antenna theory, McGraw-Hill Book Co., 1969. [27] R. King, “Coupled antennas and transmission lines”, Proc. IRE, Vol. 31, Nov. 1943, pp. 626-640. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 [28] R. C. Hansen, “Fundamental limitations in antennas”, Proc. IEEE, Vol. 69, No. 2, Feb. 1981, pp. 170-182. [29] R. King and C. W. harrison, “The impedance of short, long, and capacitively loaded antennas with a critical discussion of the antenna problem”, J. Applied Physics, Vol. 15, Feb. 1944, pp. 170-185. [30] R. King and F. G. Blake, “The self-impedance of a symmetrical antenna”, Proc. IRE, Vol. 30, July 1942, pp. 335-349 [31] R. W. P. King, R. B. Mack and S. S. Sandler, Arrays of cylindrical dipoles, Cambridge University Press, 1968. [32] S. A. Schelkunoff, “Theory of antennas o f arbitrary size and shape”, Proc. IRE, Vol. 29, Sep. 1941, pp. 493-521 [33] S. A. Schelkunoff, “Antenna theory and experiment”, J. Applied Physics, Vol. 15, Jan. 1944, pp. 54-60. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 CONCLUSIONS AND FUTURE WORK 6.1 Conclusions We have reviewed in the previous chapters, the performance of high-aspectratio voltage-tunable microwave devices as well as the corresponding processing approaches. Even though thin film schemes, i.e. coplanar waveguides, microstrips and slotlines, have already carved out their own niche in current technology, they are only well suited for certain applications, while large structures fully cantilevered require MEMS-compatible devices, and this is the arena where our work fits in. The work 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 presented provides an alternative with good performance, ease o f manufacture, and compatibility with presently used microwave devices. Fully oxidized structures are an additional option when small attenuation is required. The results presented are a direct consequence o f high-aspect-ratio silicon structures being aggressively exploited because HARS produce large forces, they can span lengths on the order of millimeters and help to decouple vibrational modes. The list of achievements is already very large as was mentioned in Chapter 1. 6.2 Future Work The devices explored in this thesis are only the working principle for a family o f microwave structures that may now be fabricated, such as: filters, detectors, amplifiers and antenna arrays with omnidirectional reception. They will find direct application in advanced military and commercial devices due to their performance, functionality and low. cost. The work presented here fills the gap for devices with electrical characteristics that can be electromechanically changed. They are also sturdy and reliable because silicon has an excellent array of mechanical properties making it suitable for microelectromechanical solutions. The growing number of applications testifies to the enormous versatility of this material..............and we shall see more in the near future. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX I Other Matrix Representations Even though scattering parameters are used in this work, there are other equivalent matrix arrangements, among them impedance and admittance. ABCD parameters have been mentioned. They have the advantage of permitting the analysis of cascaded elements simply by multiplying their individual ABCD matrices. This direct matrix multiplication is not feasible with scattering matrices because part of the variables at a port are independent and the others are dependent. Instead of using ABCD matrices we can arrange all parameters at port 1 to be dependent on the variables at port 2. In this case we obtain similar properties to ABCD. We accomplish this by defining a new set of parameters called transfer scattering or T-parameters, which are related to S-parameters through 7Ji = ( —Su S22 + Sn S21) / S2] (2.123) and *^11 = ^12^ ^22 = (^11 T22 —Ti2 T2]) / T22 521 = 1 / T22 (2.124) 522 = —T2i / r 22 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 T-parameters are usually chosen instead of ABCD because o f the simplicity o f the equations involved. Z and Y matrices on the other hand, provide for a standard visualization o f the network in terms o f conventional impedance and admittance devices. The necessary equations for both cases are included here: _ (*11 1 ) ( *22 + 1 ) ”•*12 *21 s„ = (*11 + 1 ) (*22 + 1 ) ~ *12 *21 2 z2l (*11 + 1 ) (z22 + 1 ) “ *12 *21 621 ( 1+ s , . ) ( i - s 22) + 5 12 S* ( 1- Sn ) ( l - S 22) ~ s a S21 2 5 2, ( l - 5 „ ) ( l - 5 22 ) - 5 ,2 52, 2z“ (* 11+ 1 ) ( z 22 + 1 ) Z I2 Z2l J O _ ( *11 + ^ ) ( * 2 2 — 1 ) — *12 *21 On —---------------------------------(*11 + 0 (*22 + 1 ) *12 *21 12 — 25” ( l - S u ) ( l - S 22) - S l2S2l _ ( 1+ S22 ) (1 —l^n ) + Sy2 S2i 222 (1 - 5 „ ) ( I - S 22 ) - S ]2 S2t (A.I-4) and S _ (1-^11 ) U + >22 )+>12>21 (1 + ^11 ) ( l + y22 ) - y , 2 y21 S = _______ ~2y,2_______ 12 (1 + ^u )(1+^22)-^12 >21 S s = ________________ - 2 21 yn v y* 0 y 21 + > » ) ( 1 + ^ 2 2 ) ~ > 1 2 >21 ( l + S22) ( l - S u) + S l2S2l (1 + Sl{) (1 + S22) - 5 12 52, ... U 22 1 -^ 2 2 ) + > 1 2 >21 + > ll ) 0 + > 2 2 ) “ >12 >21 -2 5n y '2 (1 + 5„) (1 + 5 22) - 5 ,2 52, —2 5 2,_________ (1 + 5„) (1 + S22) - 5,2 Sn 0 + > |1 ) ( —( 1 ~ y22 ^ + + ^ 12 ^ 21 (1 + 5„) (1 + S22) - 5,2 52, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (AI 5) APPENDIX II Other considerations in Deep Trench Etches Deep trenches are conventionally done without varying the bias-to-pressure ratio. This approach (Figure AII.l) permits the micromachining of very steep sidewalls with smooth transitions at the floor where they join the substrate. Figure AII.l: SEM micrograph showing the characteristics usually sought in deep trenches including smooth and steep sidewalls, no trenching and absence of grass. These transitions usually have a small radius of curvature. When drastic changes occur during the etch process, the formation of tapered steps becomes feasible, as is shown in Figures AII.2 and AII.3. The effect presented was achieved by changing the working point from -450 V/30 mT to -140 V/140 mT. According to the Zarowin model, when going from the first working point with a bias/pressure ratio of -15 V/mT to the second one with a ratio of only -1 V/mT, we are increasing the isotropy of the etch. Whether or not this particular effect can be controlled to achieve 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 a prescribed radius of curvature at the bottom remains an open question and additional experiments would be needed to settle this matter. With the new working point, we continued the etch for 10 minutes, evidently the ion bombardment 2 5 .0 k V HHF- «080 Figure AII.2: SEM micrograph showing the effect o f drastically changing the etching operating point on an isolated feature. HHH j HH ■ HH HI ^H H H H H Figure AII.3: SEM micrograph showing the effect on a wall. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 adjusted to a “new feature size” before continuing the etch. This effect can be seen on isolated features as well as at the end of mesas. Finally it is shown in Figure AIL4 that open areas larger than the “new feature size” or larger than the new radius of curvature of the smooth transition are needed to observe this effect. In Figure AII.4, the outermost walls show the effect but the interior walls do not. Figure AII.4: SEM micrograph showing the effect only where the walls have a separation larger than the “new feature size”. As can be seen in Figure AII.4, the distance in the topographic features can be a limitation in the application of this effect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX III EQ U A TIO N S R ELA TED TO CPW A N D CPW G The impedance of a conventional CPW can be determined using Zc, „ = (AIII.l) where, , er - l K(k' )K(k, ) eEFF= 1+ - Lz V ' EFF 2 K(k)K(k{) (AIII.2) v also (AIII.3) k=- and S ,n « * 1 ) *> = ( ^ I L 4) T T S in h (^ ) In equations (AIII.l) through (AIII.4) s is the width of the signal conductor, p is the opening between the ground and the signal conductors, and er and t are the dielectric constant and the height of the substrate respectively. K is the complete elliptic integral 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 constant and the height o f the substrate respectively. K is the complete elliptic integral of the first kind and k and k ’ are related via k' =V( l - k 2 ) (AIII.5) Additional equations applicable to Coplanar Striplines can be found in the original paper by Ghione and Naldi (Electronic Letters, Vol. 20, No. 4,1984, pp.179-181). For a Coplanar Waveguide with lower ground plane (CPWG) the impedance is given by 7 60 7t ■‘CPWG - I 1 v tu \ K(k) { K W K ( t ) AT(Jfef) tr \ t/U U .O J where u K(k’) K ( k l ) i _ W ) W j ) - u K ( t ) W , ) m l1 ) K ( k ) K( k {) also T anh< f > T a n h (^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (AHI.8) 150 k is still given by equation (AIII.3). The reader is referred to the original paper by Ghione and Naldi for a complete discussion o f the derivation of these equations: Electronic Letters, Vol. 19, No. 18,1983, pp. 734-735. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX IV Surface R oughness We can start defining the arithmetic mean H as the sum of all height values Hi divided by the number of data points n in the profile (AIV.l) 1=1 ±H , n 1=0 Ra is then the arithmetic mean of the deviations in height from the profile mean value which can be expressed as (AIV.2) Rt is the maximum peak to valley height in the profile: D 17 _ JJ ^max ^min (AIV.3) Rp is the maximum height o f the profile above the mean line: (AIV.4) RP = Hmax - H Rtm and Rpm are the mean values of R{ and Rp respectively: 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 *» = - £ ( * , ) , m -r i =l (AIV.5) 1 m (AIV.6) « /.l and are usually considered more representative o f the complete profile. Additional information can be found in the User’s Manual for TMX 2000 Discoverer Scanning Probe Microscopy, TopoMetrix, Version 3.05. Reproduced with permission of the copyright owner. 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