Design, characterization and application of mems resonators with high performance in liquid media

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(1)Universidad de Castilla-La Mancha Doctoral Thesis. Design, characterization and application of MEMS resonators with high performance in liquid media. Author: Víctor Ruiz Díez. Supervisors: Dr. Jorge Hernando García Dr. José Luis Sánchez de Rojas Aldavero. November 19, 2018.

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(3) iii. Dedicated to my family.

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(5) v. UNIVERSIDAD DE CASTILLA-LA MANCHA. Abstract Escula Técnica Superior de Ingenieros Industriales de Ciudad Real Dep. de Ingeniería Eléctrica, Electrónica, Automática y Comunicaciones Doctor of Philosophy Design, characterization and application of MEMS resonators with high performance in liquid media by Víctor Ruiz Díez.

(6) vi Micro-electro-mechanical systems (MEMS) comprise an emerging platform for many different applications, such as telecommunication, biosensing, or automotive. There is an increased interest in the use of miniaturized resonant devices to monitor liquid properties, with an impact in many different fields, such as automotive industry, biology or food analysis. In all these applications, the device might be immersed in high viscous fluids and its vibration is hindered mainly by the hydrodynamic loading. Therefore, a requisite for an efficient performance in liquid media is a vibration mode with reduced energy losses. The idea behind a MEMS structure is that a physical, chemical o biological stimulus triggers a change in its mechanical properties, either in its resonant movement when it operates in dynamic conditions or either in in its deformation, when it operates in static conditions, while this change can be detected using different electric, magnetic or optical techniques. The most common MEMS structures are the micro-cantilevers (equivalent to a suspended beam with a fixed end, but of micrometric size), the micro-bridges (suspended beams with the two fixed ends), as well as suspended membranes (in the latter the anchoring possibilities in its perimeter are multiple), which can vibrate in either outof-plane and in-plane modes. Out-of-plane modes are characterized by particle displacement normal to those surfaces of the structure with larger area while, for the in-plane modes, particle displacement is parallel to the large area surfaces of the structure. In this work, different combinations of structures and mode shapes will be investigated, in search of the best performance in-liquid media. The resonators interaction with the surrounding media will be modelled using analytical developments and finite element method (FEM) simulations, from which the representative figures of merit from the in-liquid performance will be extracted. These models will be contrasted against experimental results from fabricated devices, particularly designed to optimally recreate the mode shapes under investigation. Among the different modes of vibration, special attention will be paid to in-plane lateral and length extensional modes, as well as a high ordered out-of-plane mode, the roof tile-shape. The study will be completed with other less studied modes of vibration, such as in-plane contour modes or degenerated out-of-plane modes, with promising results..

(7) vii Los sistemas microelectromecánicos (MEMS) se han convertido en una plataforma con soporte para muchas y diversas aplicaciones, en ámbitos tales como las telecomunicaciones, los biosensores o la automoción. Existe un creciente interés en el uso de dispositivos resonantes miniaturizados con el fin de monitorizar las propiedades de los líquidos, con un alto impacto en muchos campos diferentes, como la industria automotriz, la biología o el análisis de alimentos. En todas estas aplicaciones, se requiere que el dispositivo se sumerja en fluidos de alta viscosidad, por lo que su funcionamiento se ve obstaculizado principalmente por la carga hidrodinámica. Por lo tanto, un requisito para un rendimiento eficiente del dispositivo en medios viscosos es operar un modo de vibración con pérdidas de energía reducidas. La idea detrás del uso de estructuras MEMS es aprovechar el hecho de que un estímulo físico, químico o biológico va a desencadenar un cambio en sus propiedades mecánicas, ya sea en su movimiento resonante cuando opera en condiciones dinámicas o en su deformación, cuando opera en condiciones estáticas. Este cambio va a poder ser detectado utilizando diferentes técnicas: eléctricas, magnéticas u ópticas. Las estructuras MEMS más comunes son las micro-palancas (equivalentes a una viga suspendida con un extremo fijo, pero de tamaño micrométrico), los micro-puentes (vigas suspendidas con los dos extremos fijos), así como las membranas suspendidas (en este último las posibilidades de anclaje en su perímetro son múltiples), que pueden vibrar en modos fuera del plano y en el plano. Los modos fuera del plano se caracterizan por un desplazamiento o deformación de la estructura perpendicular a la superficie de esta con un área más grande, mientras que, en los modos en el plano, el desplazamiento es paralelo a dicha superficie. En este proyecto de tesis doctoral, se han investigado diferentes combinaciones de estructuras y formas modales, en busca de la mejor combinación para operar en medio líquido, teniendo en cuenta una serie de figuras de mérito. La interacción de los resonadores con el medio se ha modelado utilizando desarrollos analíticos y simulaciones por el método de elementos finitos (FEM), de los cuales se han extraído las figuras de mérito del movimiento resonante en medio líquido. Estos modelos se han contrastado con resultados experimentales, obtenidos de dispositivos fabricados, utilizando un diseño adecuado para reproducir de manera óptima las formas modales que se pretendían investigar. Entre los diferentes modos de vibración, se prestó especial atención a los modos extensionales o longitudinales y los modos de deformación lateral, ambos dos modos en el plano. Dentro de los modos fuera del plano, se encontró que un tipo de modo de alto orden y mayor complejidad que los habituales a flexión.

(8) viii o torsión, demostró ser el mejor candidato para operar en medio líquido. El estudio se completó con otros modos de vibración menos habituales, como son los modos de contorno en el plano o los modos fuera de plano degenerados, con resultados prometedores..

(9) ix. Acknowledgements I first want to express my gratitude to my doctoral supervisors, Dr. Jorge Hernando García and Prof. José Luis Sánchez-Rojas, since this thesis would not have been possible without their full support. I was lucky to have both of them as supervisors, and without any of them, I could not have been able to go this far. I have to thank Prof. José Luis Sánchez-Rojas for giving me the opportunity to join the Microsystems, Actuators and Sensors Group at Ciudad Real and for opening my mind to new solutions for the problems that emerged in this long journey. I would like to give a special thank to my main supervisor, Dr. Jorge Hernando in who I found a close and encouraging treatment, which supposed a great impulse for the day-to-day work. Special mention deserves the good atmosphere in the laboratory, due to the companionship and friendship of all the co-workers with whom I have had the luck to share these years. Thanks for that to María Jesús, Tomás, Marta, Javier Toledo, Kiko, Javier Vázquez, Graciela and Alex. Thanks also to the people from the School of Industrial Engineering that accompanied me in the coffee breaks, which helped me clear my mind daily. It is my due to acknowledge the academic collaboration, in the fabrication of devices, of Prof. Helmut Siedel and Dr. Abdallah Ababneh from the Saarland University; and Prof. Ulrich Schmid, Dr. Martin Kucera, Dr. Achim Bittner and Elisabeth Wistrela from the Vienna University of Technology. My fruitful discussions with Professor John Sader from the University of Melbourne are also acknowledged. I also have to thank the Spanish Ministerio de Ciencia, Innovación y Universidades for the financial support to this work by projects DPI2009 07497, DPI2012 - 31203 and my grant FPU-AP2010-6069. The Spanish Junta de Comunidades de Castilla-La Mancha also supported this work with project number PEIC11-0022-7430. Outside work, my first thanks is for my parents, Victoriano and Mari Carmen, who gave me the chance to study at the University of Castilla-La Mancha. I want to recognize their emotional and economical support since I started studying in Ciudad Real in 2005. And of course, I cannot fail to mention my sisters, Verónica and Carmen Ángela, since great part of who I am is due to them. Thanks to my friends — from Malagón, Ciudad Real, Manzanares, Puertollano and Toledo — for making possible a huge amount of those unrepeatable moments that one never forgets. Last but not least, my most sincere gratitude to my partner Mari Carmen, whose love, patience and support are a continuous motivation. Thank.

(10) x you for always being there, tanking care of me and our relationship while I was busy..

(11) xi. Contents Abstract. vi. Acknowledgements. ix. 1. . . . . . .. 1 1 2 3 4 5 7. 2. Piezoelectric resonator design 2.1 Piezoelectric resonator modelling . . . . . . . . . . . . . . . . 2.2 Conductance modelling . . . . . . . . . . . . . . . . . . . . . 2.3 Modal optimization . . . . . . . . . . . . . . . . . . . . . . . .. 9 9 12 14. 3. Fluid-Structure interaction 3.1 Introduction . . . . . . 3.2 Fluid forces . . . . . . 3.3 Analytical models . . . 3.4 CFD models . . . . . .. . . . .. 17 17 19 23 28. . . . . . . . .. 33 33 33 36 40 44 44 46 46. 4. Introduction 1.1 Micro-electro-mechanical systems 1.2 State of the art . . . . . . . . . . . . 1.2.1 Out-of-plane modes . . . . 1.2.2 In-plane modes . . . . . . . 1.2.3 Acoustic wave modes . . . 1.3 Motivation and objectives . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . .. Characterization techniques 4.1 Optical characterization . . . . . . . . . . . . 4.1.1 Laser Doppler vibrometer . . . . . . . 4.1.2 Optical profilometer . . . . . . . . . . 4.1.3 Speckle pattern-based interferometer 4.2 Electrical characterization . . . . . . . . . . . 4.2.1 Electrical impedance analyser . . . . 4.3 Fluid properties . . . . . . . . . . . . . . . . . 4.3.1 Viscometer . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . .. . . . . . .. . . . .. . . . . . . . ..

(12) xii 5. 6. 7. In-plane modes 5.1 Introduction . . . . . . . . . . 5.2 Lateral mode . . . . . . . . . . 5.2.1 Introduction . . . . . . 5.2.2 Analytical modelling . 5.2.3 Simulation modelling 5.2.4 Results and discussion 5.3 Extensional mode . . . . . . . 5.3.1 Introduction . . . . . . 5.3.2 Analytical modelling . 5.3.3 Simulation modelling 5.3.4 Results and discussion 5.3.5 Tapered extensional . 5.4 Contour modes . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 49 49 49 49 50 54 59 79 79 79 86 88 97 104. Out-of-plane modes 6.1 Flexural modes . . . . . . . . 6.1.1 Introduction . . . . . . 6.1.2 Analytical models . . Mechanics . . . . . . . Fluid . . . . . . . . . . 6.1.3 FEM Validation . . . . 6.2 Roof tile-shaped modes . . . 6.2.1 Introduction . . . . . . 6.2.2 Analytical modelling . Mechanics . . . . . . . Fluid . . . . . . . . . . Electric . . . . . . . . . 6.2.3 Simulation modelling 6.2.4 Results and discussion 6.3 Degenerated modes . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 109 109 109 109 109 110 112 113 113 113 113 115 117 118 120 130. Conclusions 137 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . 140. A Microdevices A.1 Fabrication processes . . . . . A.1.1 Saarbruecken process A.1.2 Vienna process . . . . A.1.3 MEMSCAP process . . A.2 Encapsulation . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 143 143 143 144 145 148.

(13) xiii B Material properties 151 B.1 Structural materials . . . . . . . . . . . . . . . . . . . . . . . . 151 B.2 Liquid media . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C Scientific production 159 C.1 Publications in peer-reviewed journals . . . . . . . . . . . . . 159 C.2 Contributions to international conferences . . . . . . . . . . 162 Bibliography. 165.

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(15) xv. List of Figures 2.1 2.2 2.3. 2.4. 2.5. 3.1 3.2. 3.3 3.4 3.5 3.6. Butterworth-Van-Dyke equivalent electrical circuit for a piezoelectric resonator. . . . . . . . . . . . . . . . . . . . . . . . . Modified Butterworth-Van-Dyke equivalent electrical circuit for a piezoelectric resonator. . . . . . . . . . . . . . . . . . . . (a) Modal shape with a colour scale given by S11 + S22 , (b) Plot of S11 + S22 for a middle line along the length. (c) Sign of S11 + S22 for the first three dilation-type modes (length extensional) of a 1000x500µm2 plate with anchors in the middle of its long side. . . . . . . . . . . . . . . . . . . . . . Modal shapes with a colour scale given by S11 + S22 and calculated electrode layouts (black and white drawings) for the major axis flexure modes for a plate with anchors in the middle of its two short sides. Black and white areas correspond to regions with opposite sign of charge. . . . . . Modal shapes with a colour scale given by S11 + Sy 22 and calculated electrode layouts (black and white drawings) for the diagonal-shear modes for a plate with anchors in the middle of all the sides. Black and white areas correspond to regions with opposite sign of charge. . . . . . . . . . . . . . Domains in the FSI problem. The fluid forces are calculated on the contour of the structure ` . . . . . . . . . . . . . . . . Free body diagram with velocities on a differential element d` of the contour `. The forces are computed from the fluid stresses in the y- and z-directions. . . . . . . . . . . . . . . . Schematic representation of a solid sphere executing small amplitude movements in an infinite extent fluid at rest. . . . Schematic representation of a fixed sphere surrounded by a moving fluid with a constant velocity u at infinity. . . . . . . Flow characteristics depending on the Reynolds number. . . Flowsheets explaining the process in the different approaches for the FSI numerical solving: direct and iterative coupled and uncoupled solutions. . . . . . . . . . . . . . . . . . . . .. 9 10. 15. 16. 16. 19. 20 23 24 26. 29.

(16) xvi 4.1. Setup for out-of-plane modes characterization, including the Doppler effect vibrometer, the adapter circuitry and the fluid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Example of Q approximation when analysing the frequency spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Equipment for the in-plane modes identification and characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Stroboscopic system schematics. . . . . . . . . . . . . . . . . 4.5 Major steps in the image processing algorithm for in-plane movement detection. . . . . . . . . . . . . . . . . . . . . . . . 4.6 Symtem setup with the Optonor MEMSMap for the 3D optical characterization of the mode shapes. . . . . . . . . . . . . 4.7 Schematic for the in-plan and out-of-plane configurations in the MEMSMap system and corresponding picture of the measurement area. . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Measured modal shape of a 1000x250x20µm3 cantilever’s first lateral mode at 255 KHz. The coloured scale represents the displacement amplitude of the structure. . . . . . . . . . 4.9 Picture of the impedance analyser Agilent 4294A. A device is connected through a PCB that acts as interface. . . . . . . 4.10 Picture of the density meter Anton Paar DMA 4100 widh microviscometer module Lovis 2000. . . . . . . . . . . . . . . 4.11 Density determination with U-tube in the Anton Paar DMA 4100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Viscosity determination procedure. The glass capillary, which forms and angle α with the horizontal is filled with the substance under test. A steel ball is let rolling down inside the capillary and the time spent passing time between the magnetic detectors is recorded. . . . . . . . . . . . . . . . . . 5.1. 5.2. Representation of the first order lateral mode in a rectangular cantilever. The combined x- and y-axis displacements scale is unit-normalized. . . . . . . . . . . . . . . . . . . . . . . . . Representation of the simplified lateral mode. A circularshaped domain represents the fluid model cut at an arbitrary cantilever length. . . . . . . . . . . . . . . . . . . . . . . . . .. 34 35 36 37 39 41. 42. 43 45 47 47. 48. 50. 51.

(17) xvii 5.3. Representation of the Stokes’ second problem. An infinite extent solid domain performs harmonic horizontal displacements with velocity U (t) = U0 cos(ωt), fully covered by a viscous fluid. The velocity profile in the fluid Vy (z, t) is expected to develop in the Z-axis only. No other velocity components might be present in the fluid due to the symmetry of the problem. . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Resonant frequencies in vacuum for various 500µm long silicon cantilevers with widths varying from 85 to 2000µm. The results from both the analytical model and the FEM simulations are depicted. . . . . . . . . . . . . . . . . . . . . 5.5 Ratio of the resonant frequencies in vacuum from the analytical model (eq. (5.1)) and the FEM modal analysis as a function of the width-to-length ratio of the cantilever structure. 5.6 Representation of transverse cut at an arbitrary position of the cantilever length, including the surrounding fluid. The main stresses considered in the FEM analysis are represented at each face of the laterally oscillating cantilever, with harmonic velocity U (t). . . . . . . . . . . . . . . . . . . . . . 5.7 Optical micrograph of the microcantilever (500x125µm2 ) for in-plane excitation. . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Velocity spectrum (averaged over all the scanned points on the device surface) of microcantilevers using laser vibrometry. The excitation source is a 3 V periodic chirp signal. The purpose of these spectrums is to reveal the existence of a peak response (IP1) near the in-plane frequency predicted by the FEM analysis. . . . . . . . . . . . . . . . . . . . . . . . 5.9 Displacement spectrum of the in-plane movement for the 500x125µm2 sample for one electrode and two electrodes actuation. The latter is done in phase. The stroboscopic system was used to obtain these data. . . . . . . . . . . . . . 5.10 Evolution of the resonant frequency in time for the 500x125µm2 sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Schematic of the cantilever. (a) Top view (b) cross-sectional view, including a visualization of the electrical field. . . . .. 51. 55. 56. 57 59. 61. 62 63 64.

(18) xviii 5.12 Frequency response characteristics of the basic (1x) cantilever resonating in the first in-plane mode in different surrounding media. The upper plot shows the conductance peaks in air and vacuum whereas the lower plot shows the in-plane resonance phenomenon in isopropanol, gained with an Optonor MEMSMap 510. The data represent average values of 10 measurement cycles. . . . . . . . . . . . . . . . . . . . . . . 5.13 Verification of the in-plane mode of a 25x-scaled cantilever in air and in isopropanol. The plots show the out-of-plane (left axis, lower curve) as well as the conductance spectra (right axis, upper curve). . . . . . . . . . . . . . . . . . . . . . 5.14 Verification of the in-plane mode of a 200x-scaled cantilever in air and isopropanol. The plots show the out-of-plane (left axis, lower curve) as well as the conductance spectra (right axis, upper curve) . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Conductance peak characteristics for the 25x and 200x-scaled cantilevers. The dashed line represents the calculated baseline in the conductance, being raised by the series resistance of the feed lines Rs in combination with the parallel capacitance C p . A vertical offset was adjusted for each curve for clarity purposes. The Q-factors are indicated in the legend. 5.16 Sensing characteristics of the in-plane mode for differently sized cantilevers having a thickness of T = 20µm in various liquids. The plots show the Q-factor, the conductance peak ∆G as well as the ratio ∆G/Q. The evaluation of the resonance peak for the 100x and 200x-scaled cantilevers in a N35 reference medium is difficult, due to a poor ratio between the raising conductance baseline and the resonance peak. These data points are given in brackets and not considered for the linear fitting procedure. . . . . . . . . . . . . . . . . . 5.17 Comparison of the Q-factor as well as the ∆G/Q ratio for different cantilever thicknesses for the first and second inplane mode. No second in-plane mode was detected at the 55µm and 90µm thick cantilevers. The dashed lines represent a linear and 1/T fits, respectively, corresponding to the analytical model. . . . . . . . . . . . . . . . . . . . . . 5.18 Estimated evolution of the Q-factor (dotted line) and electrical conductance ∆G (dashed line) in isopropanol for different sized cantilevers with a fixed resonant frequency of 1MHz. In order to have it fixed, for each cantilever width, the corresponding length is shown in the top axis. . . . . . . . . . . .. 67. 68. 69. 72. 73. 74. 76.

(19) xix 5.19 Estimations of the Q-factor and resonant frequency for a 2017x1272x20µm3 cantilever immersed in simulated media of densities in the range of 670 to 1000 kg/m3 and viscosities from 0.4 to 60 mPa · s. Contour lines for constant values of the two figures of merit are represented in the graph. Blue round markers are used to indicate the viscosity and density location of the common media. . . . . . . . . . . . . . . . . . 5.20 Computed sensitivities of the Q-factor and resonant frequency to the viscosity and density. The graph on the left shows the sensitivity evolution with the fluid viscosity at constant density, while the graph on the right shows the evolution of the sensitivities with the liquid density at constant viscosity. A solid line represents the mean value of the sensitivity and a banded shade represents the variation within the range of the fixed fluid property. . . . . . . . . . . 5.21 Representation of the extensional mode in rectangular plate. The x-axis displacement scale is unit-normalized. . . . . . . 5.22 Cross-sectional view of the fluid-structure interaction model. 5.23 Cross-sectional view of the fluid-structure interaction model. 5.24 Representation of the adapted Stokes’ second problem to the extensional plate. An infinite extent solid domain performs harmonic horizontal displacements with velocity U ( x, t) = x U0 sin 2λn L cos(ωt), fully covered by a viscous fluid. The velocity profile in the fluid Vx ( x, z, t) is expected to develop in both the X- and Z-axis. No other velocity components might be present in the fluid due to the symmetry of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.26 Optical micrograph of a 2mm-long, 250µm-wide, and 93µmthick microplate. . . . . . . . . . . . . . . . . . . . . . . . . . 5.27 Device schematic showing the layer structure of the fabricated devices. The length (L), width (W) and thickness (T) were varied from one device to another, as well as the gap to the substrate frame. . . . . . . . . . . . . . . . . . . . . . . . . 5.28 Optical micrograph of a silicon die containing some of the plate resonators. The die was glued and bonded inside the cavity of the package, which was completely filled with liquid and covered with a thin glass for the experiments. . . 5.29 Cross-sectional schematic representation of the structure and the fluid cell. The distance to the glass cover is ht = 800µm, to the DIP package is hb = 350µm, and to the substrate frame is Gap = 350µm. . . . . . . . . . . . . . . . . . . . . . . . . . .. 77. 78 79 81 81. 82 88. 89. 89. 90.

(20) xx 5.30 Quality factor of the extensional mode for different plates in isopropanol. The experimental measurements are plotted against the length of the plate for three different thicknesses, together with the simulation results from this work and the analytical expressions given in [11]. Results similar to those of 57µm-thick devices were obtained for 123µm-thick devices. 91 5.31 Normalized amplitude and phase displacement spectrum from the 2D FEM FSI multitone simulation of a 500x250x7µm3 plate in isopropanol. . . . . . . . . . . . . . . . . . . . . . . . 93 5.32 Q-factors for the length-extensional mode of 22µm thick microplates with different lengths in isopropanol. Results from the electrical impedance measurements (points), the FEM (blue solid lines) and the analytical (red solid lines) models. Viscous (dotted lines) and acoustic (dashed lines) contributions to the Q-factors. . . . . . . . . . . . . . . . . . . 94 5.33 Q-factors from the analytical models for different lengths and thicknesses in isopropanol. The thicknesses are indicated over each line. Contribution to the Q-factor from viscous losses (dotted lines) and acoustic losses (dashed lines) are depicted separately. . . . . . . . . . . . . . . . . . . . . . . . . 95 5.34 Measured and simulated Q-factors for 93µm thick microplates with different lengths in isopropanol. . . . . . . . . . . . 95 5.35 Simulated Q-factors for the 3 mm long and 93µm thick structure in isopropanol for different gap lengths in the FEM model with frame (upper-right corner). The value for the simulation without frame is represented by a dashed line. . 96 5.36 Resonant frequencies and quality factors in different liquids for a 1000x250x22µm3 extensional plate. Both the experimental and the simulated results (2D CFD model) are plotted versus the inverse of the square root of the viscosity-density product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.37 Representation of the extensional mode in a tapered plate, where L1 is the length of the half rectangular part, L2 , the length of the triangular-shape part, and α, the taper angle. The x-axis displacement scale is unit-normalized. . . . . . . 98 5.38 Top view schematic representation of one quarter of a tapered extensional plate. The force acting on the tapered face can be decomposed in a force parallel and perpendicular to the involved surface. Only the force perpendicular to the tapered surface contributes to the acoustic losses. . . . . . . 100.

(21) xxi 5.39 Cross-sectional view of the fluid-structure interaction model, with the applied velocity field (U), proportional to the lengthextensional mode, and the main components of the fluid forces: pressure force (P) and shear force (τ) . . . . . . . . . 100 5.40 Schematic representation of the designed devices used in the experimental study of the extensional mode in tapered plates. The device layout consisted of a stacked structure of silicon, AlN and Al, with thicknesses of 20, 1, 0.5µm respectively. The plates were anchored to the supporting frame using T-shaped structures for in reduced anchor losses configuration.101 5.41 Top-view micrograph of a tapered micro-plate with top area of 1000x250µm2 and taper angle of 102◦ . Brighter zones represent the Al electrodes. . . . . . . . . . . . . . . . . . . . 101 5.42 Measured modal shape of the same device at 4.184 MHz in air. The coloured scale represents the X-axis displacement amplitude of the structure. . . . . . . . . . . . . . . . . . . . 102 5.43 Simulated, analytical and experimental quality factors in isopropanol for tapered micro-plates with a top area of 1000x250µm2 versus the taper angle. . . . . . . . . . . . . . . 103 5.44 Evolution of the sources losses (viscous and acoustic) when the tapered devices are immersed in isopropanol. . . . . . . 103 5.45 (a) Optical micrograph of the 1000x500µm2 plate. Brighter zones represent the top electrode. (b) Simulated modal shape of the first order major axis flexural mode at 2.51 MHz. (c) Measured modal shape of the first order major axis flexural mode at 2.57 MHz in air. The coloured scale represents the amplitude of displacement. . . . . . . . . . . . . . . . . . . . 104 5.46 Simulations (upper images) and measurements in air (bottom images) of the Lamè mode shape for the 1mm2 square plate. Modal displacements in the (a) X-direction and (b) Y-direction. (c) Modal shape. . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.47 Simulations (upper images) and measurements in air (bottom images) of the barrel mode shape for the 1mm2 square plate. Modal displacements in the (a) X-direction and (b) Y-direction, and (c) modal shape. (d) Calculated electrode layout and (e) optical micrograph of the fabricated device are also provided. . . . . . . . . . . . . . . . . . . . . . . . . . 106.

(22) xxii 5.48 Simulations (upper images) and measurements in air (bottom images) of the first order square extensional mode shape for the 1mm2 square plate. Modal displacements in the (a) Xand (b) Y-direction, and (c) modal shape. (d) Calculated electrode layout and (e) optical micrograph of the fabricated device are also provided. . . . . . . . . . . . . . . . . . . . . . 107 5.49 Simulations (upper images) and measurements in air (bottom images) of the second order square extensional mode shape for the 1mm2 square plate. Modal displacements in the (a) X-direction and (b) Y-direction, (c) modal shape and (d) calculated electrode layout. . . . . . . . . . . . . . . . . . . . 107 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. Representation of a roof tile-shaped mode in a rectangular cantilever. According to [37], these modes come from the superposition of two orthogonal bending modes, in the Xand Y-axis. The z-axis displacement scaled is normalized. . 3D representation of the cantilevers used for the roof tileshaped modes study. The blue dashed area represents the 2D cross section used in the FEM modelling. . . . . . . . . . Schematic representation of the 2D FEM model. The structure (omitted in the fluid model) is represented by a rectangular shape Ω. A velocity load v proportional to the modal shape, 14-mode in this case, is applied along the contour l. The blue area represents the surrounding fluid. . . . . . . . Optical micrograph of four 10µm thick cantilevers from the MEMSCAP fabrication processAppendix A.1.3, with a length of 500µm and four different widths of 300, 500, 700 and 900µm. Each cantilever has four top electrodes, which can be accessed individually, and a common bottom electrode, which can be accessed from the metal pad in the top right corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical micrograph of a pair of 20µm thick cantilevers with a length of 1mm and a width of 1mm from the Vienna fabrication process Appendix A.1.2 . . . . . . . . . . . . . . . . . Modal shapes of the first four roof tile-shape modes and their optimum electrode design (black and white drawings) superimposed. Black and white areas correspond to regions with opposite sign of charge. . . . . . . . . . . . . . . . . . . Optical micrograph of a silicon die containing several cantilever resonators. The die was glued and bonded inside the cavity of the package, which was completely filled with liquid and covered with a thin glass for the experiments. . .. 114. 119. 119. 120. 121. 121. 122.

(23) xxiii 6.8. 6.9. 6.10. 6.11. 6.12. 6.13. Average displacement per unit voltage from the optical characterization of the 500x300x10µm3 cantilever in isopropanol using (a) even actuation and (b) odd actuation. The measured modal shape and estimated quality factor are shown next to each peak. . . . . . . . . . . . . . . . . . . . . . . . . . Measured conductance and susceptance spectra in air and isopropanol of a 500x900x10µm3 cantilever using (a) even and (b) odd actuation. The quality factor and motional conductance are given for the detected roof tile-shaped modes in this frequency range. The susceptance has been normalized to the frequency for purposes of clarity. . . . . . . . . . . . . Resonant frequencies and quality factors as a function of the mode for (a) the 500µm long, 10µm thick cantilevers and (b) the 1mm long, 20µm thick cantilevers, deduced from the electrical impedance measurements in isopropanol. Error bars were calculated from the measurements of different devices with the same nominal dimensions. The coincidence point (nc ) is also given for each width. . . . . . . . . . . . . . Resonant frequencies and quality factors as a function of the mode for a 2524x1274x20µm3 cantilever in isopropanol from: the electrical impedance measurements (black squares), 2D FEM simulations (red circles), incompressible analytical model eq. (6.4) (blue solid line) and compressible analytical model eq. (6.13) (green dashed line). Error bars were calculated from the measurements of different devices with the same nominal dimensions. . . . . . . . . . . . . . . . . . . . . Resonant frequencies and Q-factors in different liquids for a 2524x1274x20µm3 cantilever vibrating in the first roof tileshaped mode. The results of the experimental characterization, previous analytical expressions, and the 2D CFD simulations are shown versus the inverse of the square root of the viscosity-density product. . . . . . . . . . . . . . . . . Simulation results for the 2524x1274x20µm3 cantilever resonating in the firs order roof tile-shaped mode (12−mode) in isopropanol. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123. 124. 125. 127. 128. 129.

(24) xxiv 6.14 Measured and calculated motional conductance, normalized to the quality factor and resonant frequency as a function of the resonator top area, for the 12−, 13−, 14− and 15−modes in the (a) 10µm thick and (b) 20µm thick cantilevers in isopropanol. Error bars were calculated from the measurements of different devices with the same nominal dimensions. Optimum electrodes were used. . . . . . . . . . . . . . . . . . . 6.15 Optical micrograph of a 1 mm square plate with four Tshaped anchors. The top electrode layer (brighter areas) has been optimized for an efficient actuation of the (8,0)+(0,8) mode. Stripes 1 and 2 allow access to areas with opposite piezoelectric charge associated with modal deformation. . . 6.16 Synthesis of degenerate out-of-plane modes from the phase and antiphase superposition of two orthogonal modes in a square plate. The modes are labelled following the naming conventions for normal modes of plates [60], where a pair of indices represents the number of nodes in two orthogonal directions. The sign of the out-of-plane deformation is represented by different colours. . . . . . . . . . . . . . . . . 6.17 From left to right: measured modal shape in air, simulated modal shape, calculated optimal electrode layouts and optical micrograph of the 1mm2 clamped square plate under study. The designation of the modes has been done according to nomenclature in [60]. Coloured scale represents the out-of-plane displacement and black and white areas correspond to regions with opposite sign of charge. . . . . . 6.18 Quality factors and resonant frequencies of the degenerated out-of-plane modes in isopropanol from the electrical characterization. Both the (n,0)-(0,n) and (n,0)+(0,n) modes have almost the same resonant frequency but a different behaviour in the quality factor as the order of the mode increases. . . .. 130. 131. 133. 135. 136. A.1 Flow chart of the fabrication of the devices under the Saarbruecken process . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.2 Schematic layout of the Vienna process, including a visualization of the electrical field. . . . . . . . . . . . . . . . . . . . 145 A.3 3D representation of the cantilevers used for the roof tileshaped modes study, showing the layer structure of the MEMSCAP fabrication process. The length L, width W and thickness T were varied from one device to another, as well as the patterning in the top metallisation. . . . . . . . . . . . 147.

(25) xxv A.4 Optical micrograph of a silicon die containing a two cantilever resonators. The die was glued and bonded inside the cavity of the package. A glass cover completes the cell. . . . 148 A.5 Optical micrograph of a silicon die containing several cantilever resonators. The die was glued and bonded inside the cavity of the package, which was completely filled with liquid and covered with a thin glass for the experiments. . . 149 A.6 Photograph of the Kulicke&Soffa 4526 Wedge bonder system.149 A.7 Cross-sectional schematic representation of the structure and the fluid cell. The distance to the glass cover is ht = 800µm, to the DIP package is hb = 350µm, and to the substrate frame is Wg = 350µm. . . . . . . . . . . . . . . . . . . . . . . . . . . 150.

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(27) xxvii. List of Tables 1.1. 5.1 5.2 5.3. 5.4 5.5. 5.6 5.7. 5.8. Summary of the best results in terms of resonant frequency, f n and Q-factor for micrometre-sized resonators in liquid media (viscosities similar to the water). Data is coming from the state of the art and the present work. . . . . . . . . . . . Fitted values of the electrical parameters for the first in-plane mode of the two cantilevers. . . . . . . . . . . . . . . . . . . . Results for the in-plane mode using the optical vibrometer, the stroboscopic system and the impedance analyser. . . . . Geometrical dimensions of the scaled cantilevers, determined using the beam equation and the in-plane resonance frequency of the reference beam (1x). The scaling factor refers to the active piezoelectric AlN area. . . . . . . . . . . . Electrical parameters of the cantilever set in air. T = 20µm. Electrical parameters for the smallest cantilever when exposed to a variety of different media. The values for air and vacuum are measured by impedance spectroscopy whereas the Q-factor in isopropanol was determined with the Optonor MEMSMap 510. The conductance peak ∆G for isopropanol is predicted using the mean ∆G/Q ratio gained by measurements in vacuum and in air. . . . . . . . . . . . . . . Electrical parameters of the different cantilevers when exposed to isopropanol. T = 20µm. . . . . . . . . . . . . . . . . . Dimensions of the scaled cantilevers. The dimensions are determined using the beam equation and the in-plane resonance frequency of the basic beam (1x). n.d. stands for "not detected", whereas impedance spectroscopy is abbreviated as IS and optical interferometry as OI. . . . . . . . . . . . . . Experimental and simulated quality factors for the extensional mode of devices with different lengths (L) and thicknesses (T) in isopropanol. Computer simulation times are also given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 60 62. 64 66. 66 70. 71. 92.

(28) xxviii 5.9. Equivalent circuit parameters from the electrical impedance measurements for the first order major axis flexural mode of the 1000x500µm2 plate in various fluids. . . . . . . . . . . . . 105 5.10 Quality factors and resonant frequencies from the electrical characterization of the in-plane modes in vacuum, air and isopropanol. The simulated resonant frequency by a Finite Element Method (FEM) is also given . . . . . . . . . . . . . . 108. 6.1. 6.2. 6.3. 6.4 6.5. Resonant frequencies and quality factors of the first bending mode of the microcantilever C1 in different fluids. Comparison between the experimental and analytical data of [12] and the 3D simulation in [1], and the results obtained with the developed 2D FEM model. (n.a. = not available). . . . . Resonant frequencies and quality factors of the first bending mode of the microcantilever C2 in different fluids. Comparison between the experimental and analytical data of [12] and the 3D simulation in [1], and the results obtained with the developed 2D FEM model. (n.a. = not available). . . . . Numerically computed values for the coefficient Kn as a function of the order of the roof tile-shaped mode when optimum electrodes are used. The corresponding eigenvalues λn are also given. . . . . . . . . . . . . . . . . . . . . . . . . . Actuation strategies for the different 1n-modes using a top electrode with four strips. . . . . . . . . . . . . . . . . . . . . Quality factors (Q) and resonant frequencies (f) obtained from the electrical characterization of the out-of-plane modes in low vacuum and air. The simulated resonant frequency by a Finite Element Method (FEM) is also given. . . . . . . .. 112. 113. 118 122. 134. B.1 Properties for structural materials used in the analytical and simulation modelling. . . . . . . . . . . . . . . . . . . . . . . 152 B.2 Rheological properties for heptane at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . . . . . . . . . . 153 B.3 Rheological properties for methanol at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . . . . . . . 154.

(29) xxix B.4 Rheological properties for ethanol at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . . . . . . . . . . B.5 Rheological properties for isopropanol at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . . . . . . . B.6 Rheological properties for de-ionised water at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . . . . . . . B.7 Rheological properties for viscosity standard D5 at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . B.8 Rheological properties for viscosity standard N10 at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . B.9 Rheological properties for viscosity standard N10 at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . . B.10 Rheological properties for viscosity standard N10 at different temperatures. The viscosity and density at 20, 25 and 30◦ C were measured and the rest were interpolated. The bulk modulus were interpolated from the values in [69]. . . . . .. 154. 155. 155. 156. 156. 157. 157.

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(31) xxxi. List of Abbreviations AFM MEMS FEM CFD FBAR SAW. Atomic Force Microscope Micro Electro Mechanical Systems Finite Element Method Computational Fluid Dynamics Film Bulk Acoustic Resonator Surface Acoustic Wave.

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(33) 1. Chapter 1. Introduction 1.1. Micro-electro-mechanical systems. Since late 80s, after the invention of the atomic force microscope (AFM), there was a rapid evolution in the field of the microelectromechanical systems (MEMS). The idea behind a MEMS structure is that a physical, chemical o biological stimulus triggers a change in its mechanical properties, either in its resonant movement when it operates in dynamic conditions or either in in its deformation, when it operates in static conditions[25, 26, 55]. This change can be detected using different electric, magnetic or optical techniques. The most common MEMS structures are the micro-cantilevers (equivalent to a suspended beam with a fixed end, but of micrometric size), the micro-bridges (suspended beams with the two fixed ends), as well as suspended membranes (in the latter the anchoring possibilities in its perimeter are multiple). The micro-cantilever is the tool used in the AFM as the scanning element of the surface to be scanned (the most common simile is to compare the microlever with the finger of a blind man reading Braille). And its wide use in AFMs has led to its application in many other fields[5, 16, 26, 86]. Given the special importance of health sciences in today’s society, the use of resonant MEMS structures as a transducer element for biological applications is of great interest[35, 87]. A typical application would be, for example, the detection of the affinity interaction between a biomolecule and an analyte, case of an antibody and the corresponding antigen. The principle of operation is as follows. The resonance frequency of the microstructure is determined by its parameters, such as its mass and the interaction affinity between the biomolecule and an analyte, which would cause a change in the mass of the structure and the consequent change in the resonance frequency, which could be related to the analyte / biomolecule interaction. Since the mass of the structure can be very small, of the order of ng, the sensitivity to the mass of the structure allows detecting mass changes of the order of pg, hence the interest in this type of structure.

(34) Chapter 1. Introduction. 2. [49, 52, 74]. This application requires the operation of the MEMS structure in liquid medium, since the natural environment for a biosensor is liquid. Another field of application that requires operation in liquid medium is the field of rheology, for the determination of the properties of liquids such as density and viscosity[50]. However, if we focus our attention on the most extended MEMS structure, the micro-cantilever, as well as in its most studied form of vibration, flexural type (perpendicular to the plane of the structure), to date it is not usual to find micro-cantilevers in a liquid medium as transducer element. The main reason for this is that the interaction of the resonant structure with the surrounding fluid causes a great damping of the vibration. The quality factor Q (quotient between the accumulated energy and the energy lost per cycle) is usually used as a parameter to estimate this damping. Specifically, the higher Q, the less damping. The value of Q can reach 1000 in air for the first flexural mode of vibration of the usual micro-cantilevers, and decreases to the order of 10 in liquids such as water. The higher the Q-factor, the smaller the minimum detectable mass. Consequently, the performance of the micro-cantilevers is seriously affected by the liquid immersion. The importance of the interaction of the fluid with the microcantilever is reflected in the different publications that address it from a theoretical point of view, as well as theoretical-experimental[83]. Mention that some authors have analytically predicted the quality factors of micro-cantilevers, but they are limited to structures with a very large length to width ratio, as well as for pure torsional and flexural modes. Therefore, for structures that do not meet these design conditions, or different modes, it there are no practical analytical approximations, needing simulation through finite elements. Given this situation, there is great interest in the field of MEMS for the search for resonant structures with better liquid performance than those of the flexural modes of the micro-cantilevers, which generally translates into a higher quality factor Q. The most common approach to achieve this goal is the use of structures with movements parallel to the plane of the structure.. 1.2. State of the art. As it was in the previous lines, due to the high practical interest in such diverse areas, the state of the art for MEMS resonators operating in liquid media is quite extensive. There are countless combinations of geometries and mode shapes but in practice all of them can be classified into two discernible families: out-of-plane or in-plane vibration modes. In the.

(35) 1.2. State of the art. 3. following lines, the most relevant results are going to be introduced, while a brief summary can be found in table 1.1. This would lead to the motivation of the present thesis.. 1.2.1. Out-of-plane modes. Out-of-plane modes are characterized by a greater deformation of the vibrating structure normal to the surface with the larger area. This comprises the bending or flexure of the structure in either directions, the torsion of the plate, or even a combination of both. Aside from these family names, the different out-of-plane modes can be rigurosly identified thanks to the work by Leissa [37, 38]. Any out-of-plane mode can be named using a two indexes < n >, < m >, being the first one n, the number of nodal lines in across the length/larger dimension and the second one, m, the number of nodal lines in the orthogonal direction. Using this nomenclature, the pure bending of a plate might be designated as {(1, 0), (2, 0), . . . (n, 0)} when it develops along the plate length or {(0, 1), (0, 2), · · · (0, m)}, when it bends along the width. Compared to the bending modes, torsional ones have an additional nodal line in the orthogonal direction, so they can be named as {(1, 1), (2, 1), · · · (n, 1)} and from here on, any other combination can be named using the general expression. Out-of-plane bending modes are from prior interest in AFM (atomic force microscopy) applications [70], but are additionally used for (bio)chemical sensing [30]. In recent years several groups published an analytical description for the vibrational behaviour of cantilever-based sensors in liquid media excited in the out-of-plane bending mode [9, 14, 20, 23, 66], focusing especially on the theoretical prediction of the Q-factor. The work by [30] give a comprehensive overview by comparing the performance of cantilever based microstructures in liquid media of more than 50 studies, showing that quality factors do not exceed Q ∼ 44 for totally exposed micromachined resonator of different geometries operating in out-of-plane modes up to the third order. In the work by [4], it was reported for out-ofplane modes a significant drop of about 50% in the resonance frequency compared to the operation in air when immersing the cantilever in liquid media. As it was demonstrated, the resonant characteristics of the flexural and bending out-of-plane mode families when immersed in a liquid media make them less than suitable for applications in which a high quality factor or reduced fluid loading is sought. However, there are still other families that deserve a more in depth analysis such as higher order torsional modes or maybe more exotic out-of-plane resonances..

(36) Chapter 1. Introduction. 4. 1.2.2. In-plane modes. In-plane vibration modes, i.e. modes that involve a displacement parallel to the larger surface of the structure, are expected to present higher Q in liquid media, as predominantly shear instead of compressive forces are transferred to the fluid [4, 44, 62, 68, 79]. Due to a reduced viscous loading, the added mass are that low that the resonant frequency remains unaltered going from vacuum or air to a viscous media. Several analyses of in-plane modes are presented: viscous damping is analyzed in [13][46] and an analytical estimation of the deflection from a piezoelectric actuated cantilever is presented in [36]. In water high quality factors of Q ∼ 67 and 86 are reported for the first in-plane bending mode by [30] and by [3], a Q factor of about 94 for a disk shaped resonator [68] and a Q ∼ 100 for a extensional mode resonator. In [62] it was not possible to detect the first inplane mode electrically with a self-actuated and self-sensing piezoelectric cantilever in isopropanol due to a considerably reduced Q-factor. In this context, [24] stated that the actuation and deflection measurement of the first in-plane bending mode is more difficult than any out-of-plane bending mode based approach. Two of the most relevant families for in-plane vibration emerge from the preferable direction in which the harmonic deformation occurs, i.e. longitudinal modes and lateral modes. Regarding lateral modes, [3] reported on the design of microcantilevers for mass sensing with the first laterally vibrating mode, with Q-factors as high as 90 and resonant frequencies around 1 MHz. On the other hand, [11] studied the electrical detection of the first longitudinal mode of a millimetre-sized cantilever in liquid (Q ∼ 75 for a viscous fluid similar to water); and in [44] the first extensional mode of a rectangular microplate was reported in liquids of different viscosity and density, with a Q-factors around 100 in water. Despite the fact that these modes have proven to be good candidates for liquid media applications, with higher quality factors than their out-of-plane counterparts, they have not been studied in too much depth. Within in-plane modes, the contour modes comprises a particular family of modes. The pioneer work by [22] reported different types of contour modes for free rectangular plates, whose deformation and the resonant frequencies are determined by the contour dimensions. For micro devices, these modes are usually found above the megahertz, what allows to achieve high quality factors. For example, using shear mode film bulk acoustic resonators pushes the Q-factor to 150, but due to the mode shape, the resonance frequency is at 790 MHz [82]..

(37) 1.2. State of the art. 1.2.3. 5. Acoustic wave modes. Special mention is made to the acoustic wave-based resonators. In the work by [85], 2 µm -thick, millimetre-sized AlN resonators were studied in their Lamb wave mode. The performance of the resonators was measured while exciting them by means of both interdigital (IDT) and longitudinal wave transducers using lateral field excitation (LW-LFE), reaching Q-factors as high as 3000 and resonant frequencies about 885 MHz. Similar Q-factors were obtained by [39] for lamb wave resonators but at 500 MHz, while in the most recent work by [40], using a quasi-symmetric third-order lamb wave mode, Q-factors as high as 5500 with a resonant frequency near 3 GHz. All these results come from in-air measurements, though the in-liquid operation is also possible. Film bulk acoustic resonators (FBAR) vibrating in shear mode and operating at around 830MHz were reported in [41] to exhibit Q-factors up to 200 in water solutions. Similar values could be found in a later work by [84], in which, by integrating a microfluidic channel to a longitudinal-mode FBAR, a Q-factor of up to 150 was achieved with direct liquid contacting. Surface acoustic wave (SAW) resonators has also been widely utilised for liquid media applications [19, 80], due to a relatively high quality factor, a low insertion loss, a distorsionless and a linear-phase response . The work by [48] reported a shear horizontal SAW resonator with a Q-factor of 400 and at 30 MHz in a water solution. However, as it happen in air, the lamb wave resonators showed the highest Q-factor values among the acoustic wave resonators. For example, in the recent work by [81], the researchers reported a high quality factor of 1909 and linear frequency response to the square root of viscosity-density product of the liquid for a Lamb wave based resonator, partially immersed in water solutions. Acoustic wave modes are a promising alternative for high quality factor and super high resonant frequency applications, but due to the complexity of the mode shapes, achieving an efficient actuation of the modes is not trivial. In most of the cases, IDT electrodes are used, with the subsequent increased in the parasitic capacity, or complex actuation techniques such as lateral electrodes..

(38) Chapter 1. Introduction. 6. Table 1.1: Summary of the best results in terms of resonant frequency, f n and Q-factor for micrometre-sized resonators in liquid media (viscosities similar to the water). Data is coming from the state of the art and the present work.. Out-of-plane. Mode Flexural (3,0)-mode Torsional (1,1)-mode Roof tile-shaped (1,6)-mode Degenerated (8,0)-(0,8)-mode. Acosutic. In-plane. Lateral Longitudinal Shear Contour Lamè FBAR SAW Lamb. f n (kHz). Q-factor. 40. 44. [10]. 160. 40. [47]. 930. 150. [61]. 3630. 144. [64]. 480 300 65 1360 790000. 86 38 75 145 150. [3] [32] [11] [63] [82]. 3880. 70. [64]. 830000 30000 477000. 200 400 1910. [41] [48] [81]. Reference.

(39) 1.3. Motivation and objectives. 1.3. 7. Motivation and objectives. The present work would be developed in three phases. In the design phase, 3D commercially available finite element simulators, such as ANALYZER and BIOCHIP DEVELOPER from Coventor© and ADINA, will be used to include the different types of dissipative forces (viscous fluid-structure interaction, anchor losses, intrinsic losses) in the analyses. Different structures will be studied: cantilevers, bridges, membranes, discs, and the different modes of vibration will be compared, with special emphasis on the modes with movement in the plane. The figures of merit to be optimized will be the quality factor Q, the sensitivity to the mass, the electrical conductance, and the resonance frequency together with the modal form. For the optimization process, the dimensions and anchoring of the structures will be varied, as well as the mechanical properties of the materials used. A fundamental part when designing resonant MEMS is actuation technique, as well as the detection of the target mode. For example, the combination of external mechanical actuation with optical detection, inherited from the AFM, is widespread. In this thesis, given the group’s previous experience, special attention would be paid to structures with integrated piezoelectric materials. The use of this type of material allows both the actuation and detection of electrical form, which facilitates the subsequent integration of this type of structures in portable systems. Likewise, as it has already been published by the group, this type of material allows to optimize the actuation/detection efficiency of a certain resonance mode through the design of the upper electrode of the structure, in such a way that this would be another optimization step to take into account when designing the structures. As the piezoelectric material, aluminium nitride (AlN), characterized by its compatibility with silicon technology processes, as well as its good mechanical, chemical and thermal stability, would be used. In any case, the use of other piezoelectric materials is not ruled out; as well as other approaches to action and detection, such as electrostatics. Regarding the manufacture of MEMS structures with piezoelectric layers of AlN once designed, at present the research group has two highly fruitful collaborations: on the one hand, the group of Professor Seidel of the University of Saarland in Germany, and on the other hand, the group of Professor Schmid of the Vienna University of Technology in Austria. Both groups will constitute a safe source of material for the different designs that emerged throughout the thesis. Likewise, the group maintains a permanent collaboration with the group of Professor Iborra of the Polytechnic University of Madrid, a group also dedicated to manufacturing processes. Another possibility, especially for the manufacture of electrostatic structures, is to resort to foundries such as MEMCAPS..

(40) 8. Chapter 1. Introduction. Once the samples have been fabricated, the next step will be to characterize their behaviour, especially the quality factor, and compare simulation and experiment. The measurements will be made both in air and in different liquid media, from organic solvents to aqueous solutions. First, the potential of the optical detection techniques available in the laboratory will be used. It will actuated electrically taking advantage of the piezoelectric layers of the structures, but the detection of the movement will be optical. For this, two systems will be used. On the one hand, the group has a Doppler vibrometer characterization system with laser scanning, from Polytec. This equipment allows measurements up to 20 MHz but it only detects movements perpendicular to the plane of the structure. In order to overcome with the limitations of this system, regarding the direction of the detectable modes, the group has also a Veeco interferometric system with stroboscopic detection that will be used in the characterization of structures with movement in the plane. This system makes measurements up to 1 MHz, and a new equipment from the Optonor AS company, that will allow measurements up to 250 MHz for modes in the plane. For measurements in liquid media, cells will be designed that, apart from integrating the MEMS structure, can be adapted to the optical measurement systems, allow the electrical excitation of the devices, as well as an inflow and outflow of liquid. On the other hand, in order to avoid short circuit problems between the metallic contacts and conductive liquids, a final layer of insulating material will be deposited in the devices, preventing the metal from being exposed to the liquid. This layer could be silicon oxide, silicon nitride, or even the active piezoelectric material of the structures, AlN. The optical measurements will be complemented with electrical measurements, especially once optimized designs are available, given the difficulties to detect the usual modes of the literature, such as the first flexural of micro-cantilevers, in liquid medium in an electrical way. For the electrical characterization of the resonators, an impedance analyser will be used, with which the module and phase information of the response of the device will be obtained, in a wide range of frequencies. The frequency spectra will be adjusted to an equivalent electrical circuit to obtain parameters such as the quality factor from an electrical point of view, as well as relevant information for the subsequent electrical conditioning of the structure in electronic circuits that allow collecting the piezoelectric load associated with the movement of the structure..

(41) 9. Chapter 2. Piezoelectric resonator design 2.1. Piezoelectric resonator modelling. Since a piezoelectric layer is integrated in all the devices, the mechanical resonances can be detected as a change in the impedance response. So, it is possible to obtain the figures of merit that describe the mechanical response by using an appropriate model for the electrical behaviour. The Butterworth-Van-Dyke (BVD) equivalent circuit (see fig. 2.1) is a wellestablished model for piezoelectric resonators [27, 45, 64]. The most basic form of this circuit model consists of two branches in parallel: • A capacitor C0 , that models the capacitance of the dielectric film. • A Rm − Lm − Cm arm, that represents the motional behaviour of the resonator.. Rm. C0. Lm. Cm. Figure 2.1: Butterworth-Van-Dyke equivalent electrical circuit for a piezoelectric resonator..

(42) Chapter 2. Piezoelectric resonator design. 10. The BVD model assumes an ideal dielectric and perfect zero-resistance electrodes but in practise this assumptions are not usually satisfied. The fabrication process itself introduces defects in the devices that need to be included in the model. The modified Butterworth-van-Dyke circuit model depicted in fig. 2.2, includes these effects with additional passive elements: • A series resistor Rs , that models the resistance of the metal paths, pads, bonding and wirings, from the structure’s electrodes to the equipment probes. • A parallel resistor R p , that represents the intrinsic losses in the piezoelectric film. From these electrical parameters, it is possible to obtain the resonant frequency, quality factor and electrical conductance of each mode, using the following expressions [61]: 1 1 √ 2π Cm Lm r 1 Lm Q= Rm Cm. fn =. ∆G =. (2.1). (2.2). 1 Rm. (2.3). Rs Rm. C0. R0. Lm. Cm. Figure 2.2: Modified Butterworth-Van-Dyke equivalent electrical circuit for a piezoelectric resonator.. Although the quality factor and resonant frequency of each mode can be determined from the electrical characterization, they are figures of.

(43) 2.1. Piezoelectric resonator modelling. 11. merit that quantify the performance of the particular resonance in the surrounding medium. They are determined by the fluid and mechanical dynamics and completely independent from the actuation technique, so they can be estimated using optical techniques, as it will be seen in Section 4.1, and they will be modelled in detailed in Chapter 3. However, regarding the electrical conductance, it is an assessment of the performance of the electro-mechanical energy conversion, so having a better comprehension of this parameter would help in having good sensing capabilities or optimally actuation of the modes of interest, with a direct impact on the performance of the resonators..

(44) Chapter 2. Piezoelectric resonator design. 12. 2.2. Conductance modelling. Since all the devices are actuated/sensed taking advantage of the integrated piezoelectric layer, modelling the electric response by means of the electric conductance, ∆G might be useful too. The development of an analytical expression for ∆G is based on the work by [21]. In this, the electric conductance ∆G can be calculated from the expression: η1 η2 ωQ , (2.4) k where ω and Q are the resonant frequency and quality factor in the actual media, k is the dynamic modal stiffness, and η1 , η2 are the input and output electromechanical coupling coefficients. The input electromechanical coupling coefficient, η1 is the ratio of induced force to input voltage while the the output coupling coefficient η2 is the ratio of output current to velocity, or, by integration, the collected charge to the deformation: ∆G =. F V q η2 = e. η1 =. (2.5a) (2.5b). In a one port configuration, using the same electrode to either actuation and sensing, η1 = η2 . The same can be applied to two port configurations whenever the electrodes are symmetrically allocated. In either case, the input and output electromechanical coefficients are equal and can be calculated any of the above definitions. For example, taking the definition from eq. (2.5b), and considering the plane stress approximation (all stress components but S11 and S22 are neglected) the total charge collected by an electrode over the microplate surface for an arbitrary mode can be expressed by [67]: q=. ZZ Ωe. (d31 S11 + d32 S22 )dΩ ,. (2.6). where Ω represents the design region (i.e. top area of the structure), Ωe is the area covered by the top electrode, d31 and d32 are the piezoelectric coefficients and S11 , S22 the stresses in the x − and y− axis in the piezoelectric layer. For a tetrahedral piezoelectric crystal such as the Aluminium nitride (AlN), used in all the fabricated devices in the present work, d31 = d32 . This results in the following general expression for the electrical conductance in a thin resonator with an AlN piezoelectric film:.

(45) 2.2. Conductance modelling. 13. RR. (S11 + S22 ) dΩ . (2.7) k This expression can be particularized for different modes and electrodes, as it will be done in Sections 5.2.2 and 6.2.2, by defining the stresses, equivalent spring constant, frequency and quality factor generated by those particular modes with the designed electrodes. ∆G =. ωQd31. Ωe.

(46) Chapter 2. Piezoelectric resonator design. 14. 2.3. Modal optimization. As it was presented in the previous Section, the piezoelectric charge collected by a piezoelectric transducer has the form described in eq. (2.6). The design region Ωe can be split into two subregions Ω+ and Ω− which are defined as the areas of Ω where the function inside the integral in eq. (2.6) is positive and negative, respectively. Ωe =⊂ Ω = Ω+ + Ω− ,. (2.8). With this decomposition, it is simple to see that to maximize the collected charge, which is equivalent to maximize the displacement, one should choose either Ωe = Ω+ or Ωe = Ω− , taking only the positive or the negative subregion for the electrode. Thereby the optimal layout of a single top electrode covers the area in which the sum of in-plane stresses has the same sign. This is illustrated in fig. 2.3 for some particular in-plane modes: the length-extensional modes up to third order. The sum of modal stresses in the plane is represented in a colour map accompanying the modal shape. fig. 2.3(b) shows how the in-plane stresses change along the length of the plate, keeping the sign in the white and black areas indicated in figure fig. 2.3(c), which define the shape of the electrodes that integrated the maximum charge. Alternatively, the top surface may be covered by two separated electrodes, featuring Ω+ and Ω− , enabling either a two-port configuration or a differential excitation with opposite voltages (with respect to a common ground electrode below the piezoelectric film) which provides twice as much actuation area as in the case of one single electrode covering one subregion. Using this approximation, optimal top electrode layers can be calculated from a finite element modal analysis by representing the sign of the sum of the stresses in the plane. Some electrode designs are shown in figs. 5.46, 5.47 and 6.17. In some devices, the final electrode designs were simplified before the fabrication process for two reasons: parallel capacitance and restrictions of the fabrication process. On the one hand, looking for a compromise between excitation and parallel capacitance, the area of the electrode was reduced to the region where the 90% of the charge is generated and on the other hand, disjoint regions were eliminated. The pioneer work by [22] reported different types of contour modes for free rectangular plates. These modes are quite difficult to excite without an appropriate electrode layout. To illustrate the design protocol presented, another two contour mode families are used: diagonal-shear modes and dilation type-modes. It is important to notice that, in order to fabricate the suspended structures, anchors are required and they can be located at.

(47) 2.3. Modal optimization. 15. (a). +1 +0.1 +0.001 -0.001 -0.1 -1. (b). (c). First order. Third order. Fith order. Figure 2.3: (a) Modal shape with a colour scale given by S11 + S22 , (b) Plot of S11 + S22 for a middle line along the length. (c) Sign of S11 + S22 for the first three dilation-type modes (length extensional) of a 1000x500µm2 plate with anchors in the middle of its long side.. the nodal points of the modes, so that the modal shapes of the fabricated anchored structures may be comparable to those of the free structures. For the major axis flexure modes, the anchors were located at the centre of the short sides of the structure. For the diagonal-shear modes, anchors were located at the middle of all sides of the structures. While for the dilationtype modes in the length direction (length-extensional modes), anchors were located at the centre of the long sides. figs. 2.4 and 2.5 show the lowest three modal shapes of these anchored plates with the corresponding optimal electrode layout. A 1000µm long, 500µm wide rectangular plate was considered in all the cases..

(48) Chapter 2. Piezoelectric resonator design. 16. +1 +0.1 +0.001 -0.001 -0.1 -1. First order. Second order. Third order. Figure 2.4: Modal shapes with a colour scale given by S11 + S22 and calculated electrode layouts (black and white drawings) for the major axis flexure modes for a plate with anchors in the middle of its two short sides. Black and white areas correspond to regions with opposite sign of charge.. +1 +0.1 +0.001 -0.001 -0.1 -1. First order. Second order. Third order. Figure 2.5: Modal shapes with a colour scale given by S11 + Sy 22 and calculated electrode layouts (black and white drawings) for the diagonal-shear modes for a plate with anchors in the middle of all the sides. Black and white areas correspond to regions with opposite sign of charge..

(49) 17. Chapter 3. Fluid-Structure interaction 3.1. Introduction. The complex fluid-structure interaction problem is the result of the physical interaction of two different entities: an elastic solid and a viscous fluid. Since both of them can be modelled as a continuous mass rather than as discrete particles, their dynamics are governed by the continuum mechanics, i.e. solid mechanics and fluid mechanics, respectively. The solid mechanics is the study of the physics of continuous materials with a defined shape at rest that can undergo deformations when stresses are applied on them. In the particular case of the elastic solids, the initial shape can be recovered once the stresses are removed, which is not in the case of a plastic solid, which can be permanently deformed under a sufficiently high stress. Solid mechanics has been widely studied, since the early beginning of the XVI century with the contributions of Leonardo da Vinci, followed by Robert Hooke, Isaac Newton, Leonhard Euler, Daniel Benoulli or Stephen Timoshenko in the last century. In the particular case of elastic solids, all their contributions allowed for the creation of analytical models that completely describe the motion of simple shapes, as well as a more advanced tool, the finite-element method, which can be used to study the kinematics and dynamics of solids with any shape. Fluid mechanics has been also largely studied since the early days of ancient Greece, with the Archimedes’ principle, but it was not until Claude-Louis Navier and George Gabriel Stokes developed the so called Navier-Stokes equation, that the problem of the motion of a viscous fluid was fully described. The existence and uniqueness of the solution of the Navier-Stokes equations is still an open problem, and an active field of research. Only a few problems, with spherical or cylindrical symmetry or unidimensional fluxes have been solved with the help of calculus, while others can only be studied by numerical methods. This is where the computational fluid dynamics (CFD) works as an approach to solve complex fluid mechanics problems..

(50) 18. Chapter 3. Fluid-Structure interaction. The fluid-structure problem implies an additional degree of difficulty, since both physics have to be solved at once. Only a few of these problems can be analytically study, whereas most of them would require numerical approaches and computer power. In the following sections, the FSI problem will be studied for practical problems, from the point of view of MEMS resonators for liquid media application, and a methodology to estimate the figures of merit of the motion of a resonating solid immersed in a viscous fluid will be presented. This methodology would intensively be used in the following chapters..