Design of width-extensional, piezoelectric RF MEMS resonators and filters

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Design of Width-Extensional, Piezoelectric

RF MEMS Resonators and Filters

8dI--by

Jonathan A. Cox ,

LIBRARIES

S.B. EE, M.I.T., 2006

Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of

Master of Engineering in Electrical Engineering and Computer Sc Mnce

MASSACHUSETTS INSTI'TITE

at the Massachusetts Institute of Technology OF TECHNOLOGQy May 2007

LIBRARIES

The author hereby grants to M.I.T. permission to reproduce an - ----to distribute publicly paper and electronic copies of this thesis document in whole

and in part in any medium now known or hereafter created.

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Author__

Department of Electrica Engineering and Computer Science

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Certified by

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Amy E. Duwel, Ph.D. Charles Stark Draper Laboratory Thesis Supervisor

Certified by

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Jin Au Kong

Profe-or of Electrical Engineering

S. Thesis Adviser

Accepted by

Arthur C. Smith Chairman, Department Committee on Graduate Theses

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Design of Width-Extensional, Piezoelectric RF MEMS Resonators and Filters by

Jonathan A. Cox

Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical

Engineering and Computer Science at the Massachusetts Institute of Technology

Abstract

Existing width-extensional, piezoelectric resonators (LBARs) suffer from high motional re-sistance and susceptibility to manufacturing disorder. Attempts to lower motional rere-sistance by connecting many LBARs electrically in parallel fail because such schemes are highly sus-ceptible to the disorder inherent in the fabrication process. The manufacturing precision, not the minimum feature size, presently limits the maximum frequency for which a resonator or filter array can be fabricated. However, the effect of disorder in a group of resonators can be reduced with mechanical coupling. Therefore, we present a novel approach that is disor-der tolerant, allowing for the fabrication of higher frequency, lower impedance LBAR-based resonators and filters. This novel resonator defeats the aspect ratio limitations imposed by the Poisson effect through stress-relieving slits. By etching narrow slits in a long bar, it is constrained to act as a single LBAR-without the spurious modes which would otherwise result. In addition, the admittance of the new array scales well with the number of unit cells, permitting the length of the array to be extended in one or two dimensions until the motional resistance is reduced to an adequate level. Finite element analysis techniques for disorder simulation and filter design are explored. Radiated acoustic power (anchor loss) is analyzed with finite element simulations with absorbing boundaries. Finally, a thorough discussion of filter design with the new resonator array, as well as a comparison of various filter topologies, is conducted.

Thesis Supervisor: Prof. Jin Au Kong

Title: Professor of Electrical Engineering and Computer Science Thesis Supervisor: Amy E. Duwel, Ph.D.

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Acknowledgments

May 25, 2007

I would like to acknowledge the generous support of the people and organizations respon-sible for making this thesis posrespon-sible. First, I owe a debt of gratitude to the Massachusetts Institute of Technology and the Department of Electrical Engineering and Computer Science for believing in my determination, interest and abilities. Second, I am extremely grateful of the support the Charles Stark Draper Laboratory has provided. It has allowed me to independently pursue learning that is of interest to me.

Most significantly, however, I would like to recognize the untold hours of assistance, guidance and instruction which have been provided by my thesis supervisor, Dr. Amy E. Duwel. Without her uncommonly vast knowledge and agreeable manner, writing this thesis would not have been possible. I especially would like to note her devotion to novel ideas, academic and practical, that make research exciting.

Not to be forgotten, I ultimately owe everything I have accomplished to my parents, Albert and Suzanne Cox. Their support, discipline, work ethic and reason has enabled my achievements-not to mention the huge financial burden they endured for four years to fund my studies.

It is with the completion of this thesis that I embark on the next leg of my journey and begin my PhD studies at MIT. I hope that with the foundation laid, the next five years here will be the best.

This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under internal funding.

Publication of this thesis does not constitute approval by Draper or the sponsoring agency of the findings or conclusions contained herein. It is published for the exchange and stimulation of ideas.

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6.2 Filter Topology ... ... 76 6.3 Filter Design ... ... .. 77 6.3.1 Filter Topologies ... ... . 79 6.3.2 Filter Comparisons ... ... 81 6.3.2.1 Half-Ladder Filter ... .... 83 6.3.2.2 Full-Ladder Filter ... .... 83 6.3.2.3 Lattice Filter ... ... 83 7 Conclusions 91 8 Appendix 93 8.1 SRA Forced Harmonic Simulation (ANSYS) . ... 93

8.2 SRA Disorder Simulation (ANSYS) . ... .... 98

8.3 SRA Disordered Impedance Scaling Simulation (ANSYS) . ... 102

8.4 SRA Modal Simulation (ANSYS) ... ... 106

8.5 One-Dimensional Sponge Layer Simulation (ANSYS) . ... 108

8.6 Two-Dimensional Sponge Layer Simulation (ANSYS) . ... .. 111

8.7 Lumped Element Resonator Simulation (ANSYS) . ... 114

8.8 BVD Fitting Routlines (MATLAB) ... .... 116

8.8.1 Main Script ... . ... 116

8.8.2 Fitting Function zfun.m ... ... 117

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List of Figures

2.1 The stress positions and directions. . ... .... 18 2.2 A strain resulting from a stress in a different direction and magnitude in an

anisotropic crystal. ... 18 2.3 Axial (a) and shear (b) strains acting on a differential cube of material. . 21 2.4 The left figure (a) shows the axes of a non-diagonalized strain tensor, while

the right figure (b) shows the principal axes of the same tensor. Notice that the axes are not orthogonal on the left, but are on the right. . ... 23 2.5 The hexagonal crystal lattice, including the four in-plane a-axes and

perpen-dicular c-axis . . . . 26 3.1 A waveguide with a discontinuity in width causing a sudden change in

im-pedance. The large section on the right represents the substrate. The forward and reflected waves are shown propagating through the waveguide. ... 32 3.2 Early Draper LBAR resonator with tapered anchors. . ... 32 3.3 A transmission line with a reactive load represented as a radiation impedance,

Zr. ... 33

3.4 A diagram of a Bragg reflector layered medium. The dotted line represents a propagating wave which was reflected off of a layer such that it is in-phase with respect to those reflected from adjacent layers. . ... 35 3.5 The three basic types of one-dimensional resonator boundary conditions for a

longitudinal wave ... 36 3.6 Shown in (a) is a thin-film LBAR resonator along with accompanying

poly-crystalline AIN orientation for the LBAR (b). . ... 38 3.7 The Butterworth-Van Dyke circuit model for a crystal resonator ... . 41 3.8 A simple mass-spring-dash pot lumped mechanical model. . ... 42 3.9 The corresponding circuit model for the mass-spring-dash pot mechanical model. 43 3.10 A lumped mechanical model representation of a second order filter. ... 43

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3.11 The corresponding circuit model for the second order filter. . ... 44 3.12 The two eigenmodes for two coupled resonators. . ... 44 3.13 A representation of the harmonic frequency response of the mass-spring

sys-tem. The amplitude could correspond to the displacement of the right most ball in response to a forcing excitation on the left, for example ... 46 4.1 A two-dimensional finite element model of an anchor terminated by a sponge

layer. The sponge layer is to the right of the undamped anchor segment. The lengths of the undamped anchor segment and the sponge layer are specified as a fraction of a wavelength. . ... ... 49 4.2 A wave propagating down a two-dimensional model of an anchor terminated

by a sponge layer. The wave rapidly decays within the sponge layer. ... 49 4.3 The reflected power spectrum of the sponge layer given in Figure 4.1 with a

homogeneous damping ratio of 15 x 10- 4 _{across the entirety of the 10Xo long}

layer ... ... 50

4.4 Damping ratio profiles for a sponge layer as a function of the slice index, j. Shown above are profiles that are both linear and hyperbolic functions of the

index. The damping ratio, (, is related to the Q by ( = ... 51

4.5 The reflected power spectrum of a 10Ao sponge layer, like the one in Figure 4.1, with a linear damping profile. The spectrum is extremely flat. ... 52 4.6 The reflected power spectrum of the sponge layer given in Figure 4.1 with

a sech2 damping profile shown in Figure 4.4. The response is not as flat as for a linear damping profile, but the performance is better by four orders of magnitude . .... ... .. ... .. 53 4.7 The reflection coefficient of the sponge layer pictured in Figure 4.1 with a

linear damping profile. The mesh density is too low (half that used before), leading to reflections at smaller wavelengths. . ... 54 4.8 The reflected power spectrum of the sponge layer given in Figure 4.1 with

a linear damping profile and half the length at only 5Ao. The length of the sponge layer is much too short, leading to reflections at low frequencies. . .. 55 4.9 A two-dimensional Finite Element Model of an anchor terminated by a sponge

layer. The sponge layer is to the right of the undamped anchor segment. The lengths of the undamped anchor segment and the sponge layer are specified as a fraction of a wavelength. ... ... 55

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4.10 The reflected power spectrum of the sponge layer given in Figure 4.9 with a linear damping profile, a length of 10Ao, but a mesh density varying linearly

between -Q and AO 20 and 2between ... 56

4.11 The finite element model for an anchor attached to a two-dimensional model of a substrate. The sponge layer surrounds the substrate on three sides, simulating an infinite half-space ... 57 4.12 A wave propagating down the finite element model of an anchor attached to

a two-dimensional model of a substrate. . ... . 57 4.13 The SWR in the anchor of the model given in Figure 4.11. The SWR converges

to approximately three, which is much lower than expected. . ... 58 5.1 A LBAR resonator which is three times longer than it is wide. Internal Poisson

stresses lead to buckling... ... 60 5.2 The impedance of the wide LBAR shown in Figure 5.1 as a function of

fre-quency. Note the large, numerous, spurious modes. . ... 60 5.3 A SEM of a 10-cell resonator array designed and fabricated by Draper

Labo-ratory. ... . . . .. . 62 5.4 The frequency response of the 10-cell resonator array shown in Figure 5.3.

Notice the spurious modes present near the primary resonance... 63 5.5 A 64-element filter array consisting of eight, eighth-order filters arranged in

parallel and designed and fabricated by researchers at the Berkeley Sensors and Actuators Center and the University of Pennsylvania. See citation for im age source [3]. ... ... ... .. ... ... . 64 5.6 The frequency response of the 64-element filter array designed and fabricated

by researchers at the Berkeley Sensors and Actuators Center and the Univer-sity of Pennsylvania. See citation for image source [3]. ... 65 5.7 A lumped-element, mechanical model of two coupled simple harmonic

oscilla-tors which are separately forced ... 65 5.8 The equivalent finite element model for the system pictured in Figure 5.7... 66 5.9 The frequency response of various configurations of the disordered filter shown

in Figure 5.8. Notice that when the spring disordered filter is forced along its eigenmode, the spurious mode disappears. The same is also true for the mass disordered system, but only if the mode shape is normalized by the mass matrix. 69

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5.10 The eigenfrequencies for a pair of coupled harmonic oscillators with disparate mass as a function of coupling strength. As the coupling strength rises, the "undesirable" anti-symmetric mode (dashed line) moves away from the sym-metric mode. The frequency of the symsym-metric mode converges to that for a single resonator with a mass and spring constant equal to that of the combined

system. For this plot, mi = 3m 2, k = k2 .... . . . . . . . . . . . ... . 70

5.11 A schematic of a one-dimensional slit-resonator array. Wa = 4 pMr, L = 8 prm, L, = 6 pm, W, = .5 pm, W, = 4 pm ... ... 70 5.12 Comparison of the effect of typical manufacturing disorder on the impedance

of a 10-cell LBAR array. The top plot shows a simulation for the impedance of 10 LBAR resonators that are electrically in parallel and mechanically un-coupled. Each resonator is 8 by 4 ±0.1 pm (tolerance applied as a Gaussian distribution). The middle plot shows impedance for 10 resonators that are electrically in parallel and mechanically coupled by a center tether. The bot-tom plot shows impedance for a slit-based array. . ... 71 5.13 A simulation of the impedance scaling for the three coupling schemes shown as

admittance verses number of cells. This is based on simulations with undisor-dered resonator arrays. Both mechanically coupled designs show useful linear scaling of admittance versus number of unit cells. . ... 72 5.14 A top view of a schematic for a the fabrication of a one-dimensional

slit-resonator array at 100 MHz. All dimensions are in micrometers. ... . 72 6.1 A diagram of a clamped-clamped beam resonator. . ... . 74 6.2 A finite element modal analysis of a slit-resonator array vibrating along the

fundamental mode. ... ... .. 75 6.3 A top view of a schematic for the fabrication of a two-dimensional slit-resonator

array at 100 MHz. ... ... 76 6.4 The parallel filter technique for reducing the insertion loss of a high order filter.

Adding replica filters in parallel reduces the insertion loss, while increasing the order of the filter increases the insertion loss. . ... 77 6.5 The parallel resonator filter topology for reducing filter insertion loss. This

topology allows the resonators to be coupled together to reduce disorder. .. 78 6.6 A schematic of the two-dimensional slit-resonator array used for the filter

design. We = 24 pm, Lc = 48 pm, VA = 24 pm, tA = 6 pm, Wsx = 16 pm,

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6.7 The results of the curve fitting routine used for computing the equivalent BVD model from a finite element simulation. The real and imaginary parts are fit separately . . . . 80 6.8 The result of the curve fitting routine used for computing the equivalent BVD

model from a finite element simulation. The correlation coefficient between the frequency response of the BVD model and the that of the finite element model is in excess of 0.9999. ... 81 6.9 The half-ladder filter topology whereby each resonator is coupled with a

cou-pling capacitor . . . . .. . . . . . . 81 6.10 The full-ladder filter topology consisting of two resonators per unit cell. . . . 82 6.11 The lattice filter topology consisting of four resonators per unit cell. ... 82 6.12 Schematics for the half-ladder filter and four disordered variants. Each variant

is a copy of the top filter, with the length of a particular resonator varied by +0.1 pm. The heavy, black box encloses the disordered resonator. Measured filter specifications and the length of each resonator are specified appropriately. 84 6.13 Plots of the frequency response for each ladder filter given in Figure 6.12. .. 85 6.14 Schematics for the full-ladder filter and four disordered variants. Each variant

is a copy of the top filter, with the length of a particular resonator varied by ±0.1 pm. The heavy, black box encloses the disordered resonator. Measured filter specifications and the length of each resonator are specified appropriately. 86 6.15 Plots of the frequency response for each full-ladder filter given in Figure 6.14. 87 6.16 Schematics for the lattice filter and three disordered variants. Each variant

is a copy of the top filter, with the length of a particular resonator varied by +0.1 pm. The heavy, black box encloses the disordered resonator. Measured filter specifications and the length of each resonator are specified appropriately. 88 6.17 The fourth disordered lattice filter variant. . ... 89 6.18 Plots of the frequency response for each lattice filter given in Figures 6.16-17. 90

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List of Tables

2.1 Relevant material properties of useful piezoelectric crystals. Note: values are for comparison purposes only [9. ... ... 26 2.2 Published values for the compliance tensor of hexagonal AIN in 109 Pascals

(GPa) [11]. ... . . ... 27 2.3 Published values for the piezoelectric tensor of hexagonal, class 6mm AlN in

C [11... ... . . ... . ... 28

3.1 The corresponding quantities and variables in the electrical and mechanical model domains [24] ... ... ... 42

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Chapter 1 Introduction

MEMS RF resonators hold promise as an enabling technology for the development of single-chip radio transceivers and atomic clocks. If the resonators, filters, amplifiers and other circuitry can be fabricated on a single chip, profound size reductions are possible. However, existing thickness-mode, piezoelectric resonators (FBARs) are not ideal for chip-scale devices since it is difficult to fabricate FBARs of disparate frequencies on a single wafer. Conse-quently, recent attention has been focused on the development of length/width-extensional mode, piezoelectric RF resonators (LBARs) [1],[2],[3]. Since the pertinent dimensions of a LBAR are defined lithographically, multiple unique filters can be integrated with electronics on a single chip.

Despite their benefits, the fabrication of LBAR resonators is inherently less accurate. Also, the resulting resonators are higher impedance than of a comparable FBAR. The reso-nant frequency of a FBAR is defined by the thickness of the thin-film layer. The thickness, and hence the resonant frequency, of these thin-film layers can be very precisely controlled, as the dynamics determining the growth of these layers is very well understood and controlled. However, this layer must in general be grown across the entire wafer. Therefore, a single wafer can have, excluding techniques for tuning across a small range, only one frequency of resonator. Since LBAR resonators are defined lithographically, this level of precision is not available. In addition, because FBARs resonate along the same axis as the applied elec-tric field, their total area can be increased indefinitely. The area of an LBAR, however, is strictly determined by the resonant frequency. As the frequency increases, the area decreases, causing the device's impedance to rise.

Consequentially, the lithographic process that defines the geometry of the LBAR, and affords considerable freedom, leads to high motional impedance and susceptibility to manu-facturing disorder. This in turn causes insertion loss and distorted filter shapes. Although

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the motional impedance of an LBAR is proportional to its width, the dimensions are fixed by the frequency of operation because the aspect ratio can be no greater than 2:1, or spurious modes will result from internal Poisson stresses. Furthermore, attempts to lower motional resistance by connecting many LBARs electrically in parallel fail because such schemes are highly susceptible to disorder inherent in the fabrication process. The manufacturing pre-cision, not the minimum feature size, presently limits the maximum frequency for which a resonator or filter array can be fabricated, before disorder becomes non-negligible. However, as described theoretically in the literature, the effective disorder of a group of resonators can be reduced with mechanical coupling [4]. Therefore, we will demonstrate a novel approach that is disorder tolerant, allowing for the fabrication of higher frequency, lower impedance, LBAR-based resonators and filters. Overcoming these two main issues is critical to the commercial success of the LBAR.

Disorder tends to produce a cluster of resonant modes. To counteract this, our novel design uses stress-relieving slits to defeat the aspect ratio limitations imposed by the Poisson effect. In this way, the LBAR can be grown to greater size, reducing impedance and also reducing the impact of disorder. By etching narrow slits in a long bar or rectangular surface, forming what we call a slit-resonator array (SRA), the resonator is constrained to act as a single LBAR, but without spurious modes near the passband.

Previous attempts at low-impedance LBAR filters have involved forming filters from mechanically coupled resonators and reducing loss by placing multiple filters in parallel electrically. However, this architecture results in poor filter shape caused by fabrication disorder, and it is limited to 50 MHz because of the need for weak mechanical coupling [3]. We instead take the opposite approach and use strong mechanical coupling to reduce effective disorder and make low-impedance resonators which can be electrically coupled to form filters at frequencies beyond 50 MHz.

The design of MEMS devices is heavily reliant on computer simulations techniques. For these simulations to be meaningful, they must reflect reality. Therefore, the tensor quantities that define the material properties deserve special attention. For this reason, Chapter 2 serves as a tutorial for the pertinent crystal physics. Mechanical modeling and acoustics, which will be useful for later chapters, are also reviewed in Chapter 3. In Chapter 4, computer modeling techniques for radiated power simulations (anchor loss) are developed. The theory and considerations relevant to disorder among resonators are discussed in Chapter 5. Finally, we conclude with a complete look at the development of RF MEMS piezoelectric filters based on the SRA, which incorporates the disorder tolerant designs and modeling techniques explained below.

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Chapter 2 Crystals and Piezoelectricity

2.1 Tensors

A crystal is a periodic structure composed of well-defined unit cells. For our purposes, an exact description of the unit-cell serves as a complete description of the crystal. However, the structure of a unit-cell in a three-dimensional crystal need not be, and is usually not, identical when viewed from all directions. The lack of perfect symmetry results in a crystal which reacts differently based upon the direction of the stimuli. While a simple cubic crystal has only three unique, orthogonal axes, which can be understood with the familiar Cartesian coordinate system and matrix algebra, most crystals do not. In fact, this produces the central problem involved in crystal physics: many quantities of interest depend on the crystal orientation in a non-orthogonal space. While some physical quantities, such as density are isotropic, or independent of direction, most others, such as the speed of wave propagation or the deformation under a mechanical stress, are anisotropic. And in non-cubic crystals, these properties cannot be easily decomposed into the familiar three-component, orthogonal vectors system [5].

A general crystal can be represented by a unit-cube, as shown in Figure 2.1, on which we apply stimuli, and observe the corresponding reaction. Since the unit-cell is assumed to be very small, stimuli on one face are uniform through the opposite face. Therefore, we can represent all possible stimuli by only three separate, orthogonal coordinate systems, as shown. On each face, stimuli may be exerted perpendicular to the face, or in two directions along each face (known as shear stimuli).

The anisotropic nature of most crystals requires tensor mathematics to describe. Physical phenomena which act in isotropic media can be easily defined with simple scalar or vector

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x2

Figure 2.1: The stress positions and directions.

relations, such as Newton's second law.

t = mb _(2.1)

Newton's law can be written out component by component as (2.2).

F₁= mal, F2 = ma2, F3 = ma3 (2.2)

Of course, it is possible to write Newton's second law as a simple scalar relation of two vectors because mass is inherently an isotropic quantity, and is independent of the direction upon which a force acts. However, the relationship between the electric field vector and the electric displacement vector, for example, in an anisotropic crystal is not so simple. A field applied in one direction can result in a displacement in a different direction, as demonstrated by Figure 2.2. The relationship between these two vectors cannot be expressed as in (2.2),

D

Figure 2.2: A strain resulting from a stress in a different direction and magnitude in an anisotropic crystal.

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but rather as (2.3).

El = allD1 + a12D2 + a13D3

E2 = u21D1 + c22D2 + a23D3 (2.3)

E3 = 31D1 + 032D2 + a3 3D3

In matrix form, this relationship between two vectors is known as a tensor of second rank. A tensor of zeroth rank is simply a scalar, like mass, while a tensor of first rank is a vector. The rank of the tensor is given by the number of indices required to specify the coefficients, for example, two in the case of oij.

For other physical phenomena, such as stress and strain, the relationship is even more complicated than is the case with the electric field. Since stress, T, and strain, S, are both second rank sensors as shown in Figure 2.1, they are related by a fourth rank tensor of 9.9 = 81 elements, as given by (2.4). However, even (2.4) is a two-dimensional representation of a fourth-rank tensor which is actually four-dimensional [6].

I \ \ / \ Tii T22 T33 T,_ '11 '12 U13 .'" 19 C21 C22 C31 C33 fr'L ("_ S₂₂ S33 Ss (2.4)

Fortunately, however, for all physical phenomena of concern to this thesis, we can assume that the stress and strain tensors are symmetric. In other words, when a stimulus is applied along a direction, such as xj, and the response is observed along, say, x2, the response

is identical to when a stimulus is instead performed along x2 and the response is observed

along xl. Please note however, that this is not in general the case and that the thermoelectric tensor for some crystals is not symmetric [5]. Since we will deal only with symmetric tensors, the convention in the literature is to represent a second rank tensor by a six-dimensional vector, as shown in (2.5), where al, 2 and a₃represent the relationship along the familiar

x, y and z Cartesian axes, respectively. For our purposes, o4, u5 and a6 correspond to a

coupling between the y - z, x - z and x - y axes, respectively, and are shear stimuli. Again, following the convention in the literature, we shall from now on refer to the Cartesian axes

r ?i

"'

• Jl fT fV fl •

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x, y and z as xl, x2 and x3 [5]. 0111 0"1 /22 a2 0"11 0U12 a 1 3 a 1 1 0'1 2 013 a33 a3 2 1 a2 2 0123 0'12 022 a023 (2.5) 2 3 a4 031 32 a3 3 a1 3 a2 3 a3 3 a13 a5 0\ _{1 2} ₆

Using this abbreviated convention for the representation of a second rank tensor, it becomes possible to write the relationship between stress and strain with a 6x6 matrix, where the Ti are stress components and the Si are strain.

I \~~~ _'1 I~. 1 " " " 16 S5 S2 S3 S4 55 (2.6) \ U 0 aOO/ r- /

2.1.1 Stress and Strain

Stress and strain are roughly the elastic analog of the force and displacement associated with a classical spring from Hook's law. In fact, as will be discussed in greater detail in later chapters, finite element computer models of an elastic medium, such as a crystal, are essentially a three-dimensional network of interlinked masses and springs. Again, the relationship between the second-rank stress and strain tensors is a forth rank compliance tensor, c.

T3j = C ijkSkl (2.7)

k I

- cijklSkl (2.8)

The relationship among these tensors can also be expressed with a more compact notation known as the Einstein summation convention. In the Einstein convention, when a particular index is repeated it is called a dummy index, and a summation over that index is implied. For example, in (2.8) it is understood that summations should occur over the dummy indices k and 1. Indices which are not repeated are simply called free indices. Furthermore, a

• • • T11 T2 T3 T4 T5 To I ~cr I

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particular dummy index must not appear instead as a free index on the other side for the definition to remain valid.

For the symmetric stress and strain tensors we deal with, the tensor notion relation simplifies to (2.9).

T = cijS, = ciS j

(2.9)

Strain is defined as a change in length per unit of a purely axial (as opposed to a shear strain), between the original and subsequent lengths of a stress, divided by the change in length [6].

Au,

Ax'

s ,-l

of original length. For the simplified case it is defined as the limit of the difference unit cube, while undergoing a compressive

x₂

AXI

Ax,

Ax'

---Figure 2.3: Axial (a) and shear (b) strains acting on a differential cube of material.

u(x

1 + Axi) - U(xi) _ Ou _(2.10)

Sai.1- lim _(2.10)

Ax1--+0 X1 ax1

Furthermore, from the definition, it is clear that strain must be a dimensionless quantity. However, for a shear strain, deformations in both directions must be accounted for. We therefore choose to define strain by (2.11).

1 (&u

j

+u

S _'22-, - _\Ox~ ₀_x 3

(2.11)

A strain defined as such must be symmetric, as this expression gives only the symmetric

por-tion of the strain tensor. Consider, for example, that Si = 2 + ax ) = 2 ( au + a-x) = Sji since addition is commutative. Then because the definition of a symmetric matrix is A = AT = Aij = _{Aji, the above defines only the symmetric portion of the strain.}

Fortu-nately, however, this thesis will always deal with symmetric strains.

Stress, on the other hand, is simply a force per unit area, similar to a pressure, but with

x

"

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direction, and is therefore a second-rank tensor just like a strain.

2.2 Principal Axis Theorem

While it is all well and good to speak of second rank tensors and shear stimuli, the problem remains that in the laboratory, we are usually restricted to the Cartesian coordinate system. Moreover, it is not particularly obvious by inspection precisely how to interpret a six-element, second rank tensor which represents, for example, a stress or a strain. Fortunately, because we deal only with symmetric tensors, it can be shown that all such tensors can be expressed in terms of their principal axes. When a tensor is expressed in terms of its orthogonal, principal axes, the symmetric tensor with six unique elements can be simplified to a readily intelligible, diagonal matrix.

('

11

012 U1

3

a1 0 0

a12 a22 a23 0 2 0 (2.12)

a13 a23 a33 0 0 a3

The meaning of (2.12) is immediately clear, as al, a2 and a3 give the component of the stress

or strain along each of the three, orthogonal, principal axes.

A real, symmetric matrix, A = AT, such as a symmetric second-rank tensor, is related to its diagonalized, similar matrix by (2.13),

A = QAQT, (2.13)

where A is a diagonal matrix of the eigenvalues of A, and Q is a matrix of the eigenvectors of A along its columns. More explicitly, the relationship expressed in (2.13) can be written out fully as (2.14).

T

A(

0 0

A= v1 v2 V3 0 A2 0 V1 v2 v3 (2.14)

0 0 A3)

This property of real, symmetric matrices is known as the spectral theorem in mathematics, but is commonly referred to as the principal axis theorem in physics and engineering [7]. In addition, the matrices A and A are similar matrices, which means that they are also related

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by the transformation given by (2.15),

A = QAQ- 1 = QAQT (2.15)

where

Q-1

= QT because Q is an orthogonal matrix. When A is transformed into its similar matrix, A, it is said to be diagonalized [8]. Moreover, this transformation is actually a change of basis with the orthogonal eigenvectors of A as the new basis for A. Therefore, by diagonalizing the stress tensor, for example, we have succeeded in simplifying a six-element tensor into a three element, diagonal matrix where the components of the matrix signify the magnitude of the stress along each of the three principal axes.

x, x,

(a) (b)

Figure 2.4: The left figure (a) shows the axes of a non-diagonalized strain tensor, while the right figure (b) shows the principal axes of the same tensor. Notice that the axes are not orthogonal on the left, but are on the right.

It is important to note that stress and strain are exogenous quantities, which is to say that they represent quantities externally imposed upon a crystal. Therefore, as is evident from the Principal Axis Theorem, a stress or a strain can be in any direction desired, as determined by the applied stress or strain tensor. However, this is not the case with tensors that represent actual physical, crystal properties, such as the dielectric or piezoelectric tensor. These tensors fundamentally describe the properties of the underlying crystal structure and its symmetry,

and therefore have principal axes which remain invariant under outside influences [5].

2.3 Piezoelectricity

The phenomena of piezoelectricity links the electrical and mechanical worlds. From an engineering perspective, the magic of piezoelectricity is multifaceted. Piezoelectric crystals

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and ceramics can be formed into high frequency transducers, from kilohertz to gigahertz. The piezoelectric effect is also capable of exerting relatively enormous stresses, albeit with typically minuscule strains. However, these minuscule strains can be used to position objects with molecular precision, as is done with an atomic force microscope (AFM). For all of their utility, we are most interested in piezoelectric materials because they can be fabricated at micrometer scales and integrated with microelectronics.

Your author is of the opinion that the best way to explain piezoelectricity is simply to introduce the tensor relations given by (2.16) and (2.17),

Ti = cijSj - eikEk (2.16)

Dk = ekjS + EkiEl (2.17)

where Ti is the strain, Sj is the stress, Dk is the displacement current', Ek is the electric field, cij is the compliance, eik are the piezoelectric coefficients and Elk is the dielectric tensor. The piezoelectric tensor, eik, is in reality a third-rank tensor with three indices. It relates a second and first-rank tensor. However, because this tensor is symmetrical in two of the indices, it is possible to reduce it to a two-dimensional, rectangular matrix with 18 elements, instead of 33 = 27. Also, notice that the piezoelectric tensor appears in two separate locations, and differs only in that one is the transpose of the other. These two coupled equations define a relation between stress, strain, displacement current and the electric field. When (2.16) and (2.17) are written out in reduced matrix form, they can be simplified to (2.18) and (2.19).

S/ \/ / S11 T2 T3 T4 T5 r p r C1 .- C16 01 S2 S3 S4 S5 Q t11 G12 G13 e21 e2 2 e2 3 e3 1 e32 e33 e4 1 e4 2 e4 3 e51 e5 2 e5 3 E2 (2.18) E3) 61\ cat c ) } 6 3 /

1The displacement current is the "missing" current discovered by James Clerk Maxwell. It can be thought of as the current apparently generated in, for example, a capacitor by a time changing electric field which

produces a corresponding magnetic field: = E9 + 1, tD = E a-, VX f +

D-• L' f n_ \

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S2

D

e

...

e

₁₆ S3

D2

S

D3 e3l e36 S5 R. 611 E12 C13 El + E21 622 632 E2

(2.19)

E31 632 C33 E3

2.3.1 Coupling Coefficient

A significant issue in the piezoelectric MEMS community is the strength of the coupling between the electrical and mechanical domains of a resonator. This ratio has profound implications for the impedance and bandwidth of MEMS resonators. Known as the coupling coefficient, it is defined by (2.20),

S/Umech (2.20)

V Ueic

where Umech is the mechanical energy stored in the resonator and Ueec is the electrical energy supplied to the resonator-per cycle. The subscript of kt denotes that this coefficient is calculated for the "thickness" mode of operation where the electric field is applied along the c-axis-a typical mode of operation among thin-film MEMS devices. The coupling coefficient is a measure of the effective energy conversion, and not an efficiency. In other words, it says nothing of energy loss. This ratio is typically expressed as k2 percent, and approaches 100% as all of the supplied electrical energy is stored in the resonator as mechanical energy. When dealing with resonators, kt is most easily determined by measuring the frequencies of the parallel and series resonances of the crystal resonator (see section 3.2.1).

2.4 Crystal Properties

Perhaps the most important factor determining the performance of a piezoelectric MEMS device is the piezoelectric material selected. The significant material properties vary dras-tically among different materials. These properties significantly affect the performance of the filter or resonator. Table 2.1 lists the pertinent properties for piezoelectric crystals com-monly employed in MEMS and bulk high-frequency applications. These values are shown for different crystal cuts and are provided as a rough comparison only. For MEMS filters, we are primarily interested in semiconductor materials with large coupling coefficients because the coupling coefficient influences the impedance and bandwidth of the filter.

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Table 2.1: Relevant material properties of useful piezoelectric crystals. Note: values are for

comparison purposes only [9].

Material Cut I k2 E_/60 cs [103m/s] Ip [kg/m] E [1010_N/m 2

A1N 00 4% 8.5 10.4 3260 36

ZnO 00 8% 8.8 6.4 5680 23

CdS 00 2% 9.5 4.5 4820 9.8

Quartz (SiO2) OoX 1% 4.6 6.1 2650 9.7

LiNbO3 X 46% 44 4.8 4640 11

LiTaO3 X 19% 41 5.9 5300 19

LiGaO2 Z 9% 8.5 6.2 4190 16

Draper Laboratory has focused on developing an Aluminum Nitride fabrication process because AIN offers great strength, compatibility with common semiconductor fabrication technology and a reasonable coupling coefficient (as compared to the classic, industrial-standard quartz crystal). Therefore, Draper's A1N process motivates a discussion of its crystal structure and measured material properties. It is important to keep in mind, however, that in practice, AIN is deposited in its polycrystalline form, which is to say that it is not a single crystal, but rather composed of small "grains". Within these grains, the crystal is uniform, but the grains are randomly rotated with respect to each other along the polar axis. Therefore, the actual observed properties of a polycrystalline layer will be an average across the grains [10].

Figure 2.5: The hexagonal crystal lattice, including the four in-plane a-axes and perpendic-ular c-axis.

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2.4.1 Properties of Aluminum Nitride

For purposes of computer simulation and design, it is useful to have an understanding of the coefficients constituting the piezoelectric equations. Although it is difficult to measure these material properties with great accuracy, and the published numbers vary considerably, their value is often known with better accuracy than they can be fabricated. It is also worth noting that most of these values will vary somewhat as a function of frequency.

A typical book or article on measured AIN properties will provide values similar to those given in Table 2.2, which were obtained from a recent journal publication.

Table 2.2: Published values for the compliance tensor of hexagonal AIN in 10' Pascals (GPa) [111.

I

Ci1

0,13

I

C33 044I

411 149 99 1389 125

However, a quick inspection of the compliance tensor demonstrates fewer than 36 coefficients. Fortunately, crystallographers have spared us out the symmetries of hexagonal crystals such as AIN. Therefore, it compliance tensor for hexagonal crystals reduces to

/ r c11 C13 C13 C3 3 0 0 0 c4 4 0 0 0 0 0 0 0 0 c44 0 1(c 2 l'1J /

and is symmetric about the main diagonal [5].

The piezoelectric tensor, a third-rank, 18-element fashion. It is given by (2.22) for hexagonal, class 6mm

eij ₌

(

0 0

e_{3 3}

that it requires no the task of working is known that the

(2.21)

tensor, can be reduced in a similar crystals, including AlN.

0 e₁₅ 0 e_{1 5} 0 0

0 0 0

(2.22)

The corresponding piezoelectric coefficients are listed in Table 2.3.

As previously mentioned, polycrystalline AIN is usually deposited such that an electric field is applied along x3direction (the so-called "polar axis" or "c-axis") so that El = E2 = 0.

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Table 2.3: Published values for the piezoelectric tensor of hexagonal, class 6mm AIN in c [11].

e33 e31 e1 5

1.39 -0.58 0.29

This is necessary because the grains are randomly rotated around x3. When the resonator is

operated in thickness mode, it will simply see an effective piezoelectric coefficient determined

by e33.

Lastly, the permittivity tensor for AIN is usually expressed in its principal coordinate system. It is therefore the diagonal matrix given by (2.23),

e11

ij = Ell , (2.23)

E33

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Chapter 3 Acoustics and Modeling

3.1 Acoustics

Acoustic waves are of interest to MEMS filter designers mainly since resonators never exist in isolation. Unlike electromagnetic waves, acoustic waves do not propagate through the vacuum of space. Instead, they require a medium, such as as a crystal. Even if a MEMS resonator design employs only a single resonator, that device must be suspended from the substrate via small anchors which propagate acoustic waves. This radiated acoustic power contributes to the total power lost per cycle in the resonator, causing increased filter insertion loss.

An acoustic compressional wave, such as a sound wave, propagates as an oscillation among pressure (potential energy) and velocity (kinetic energy). We will consider only compressional waves, although several different types of waves are also possible in most MEMS structures. The variables of interest to us are thus velocity, V, pressure, p, and the density, of the at pressure, p, and the density, p, of the

medium. We will also define the particle speed as v - IV .For simplicity, we will break these quantities apart into their static, or "DC" component, and their variational, or "AC", component. Put differently, we will not concern ourselves with a steady wind, for example, but rather the vibrations propagating through the air. These quantities can then be written according to (3.1), (3.2) and (3.3),

-UT( - -U O

a

(-u r t) _(3.1)

at

PT( , t)t = po + p(-?, t) (3.2)

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where the subscript T refers to the sum of the static and variational quantities [12].

The fundamental equations of acoustics are the two linearized acoustic relations (3.4) and (3.5), which express Newton's law and conservation of mass.

t2

8 8

poV

-

a

(, t)

-=

p(

_,

_t)

_(3.5)

at

The divergence, a vector quantity, is denoted by Vf(-), while the gradient, a scalar quantity, is denoted by V - f(-). Also, Po is the average, or static density of the medium, where

p(-?, t) is the variation in density from the static density, p(T, t) is the variation in pressure

and V (-?, t) is the displacement of the particles of the medium. Although these equations are usually written in terms of velocity, they are written here in terms of displacement for consistency with later sections.

The acoustic solid constitutive relation, (3.6), which relates pressure and density, allows us to simplify (3.4) and (3.5).

ap

- _(3.6)

ap

K

K, the bulk modulus, quantifies the incompressibility of the solid. In other words, it is a measure of its resistance to a compressive stress, like the spring constant in Hook's law. The fundamental equations of acoustics can then be simplified by eliminating the time-dependent density from the conservation of mass equation to yield

PO

a2

(T,t) = -Vp(T>,t) (3.7)

PoV -

V (

=

(a t).

(3.8)

at

K at

These are the equations that are most commonly known in solid-state acoustics. They can be combined to produce the acoustic wave equation, (3.9), just as is done to derive the

Helmholtz wave equation in electromagnetics.

V2p(V, t) = p t) (3.9)

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It is then useful to define several quantities, including: the group velocity,

vg = c = (3.10)

PO the characteristic impedance of a solid,

Zo = poK [N s, (3.11)

V m

and finally, the time-averaged, acoustic Poynting vector for a plane wave,

= Re {p - V(-?, t)*} . (3.12)

The group velocity, also known as the speed of sound, is the rate of information and power propagation. The characteristic impedance, however, is the ratio of pressure to particle speed. It is similar to the electrical analog for impedance. The time-averaged Poynting vector for a plane wave, which is in the direction of the displacement vector, is useful for calculating power flow in a device.

For a typical sample of AIN, the characteristic impedance is about

Zo = poK = poc = /3300 -(3x 1011) z 3 x 107 _[

3 . (3.13)

3.1.1 Acoustic Waveguides

Acoustic waves radiating from the anchors attached to a resonator are one of the significant causes of power loss in LBAR devices. Typically, a LBAR is connected to the substrate via rectangular anchors of small cross-section-a situation analogous to the diagram below. In this model, we generally consider the transverse dimension of the anchors, which are nothing more than rectangular waveguides, to be much smaller than a wavelength. In addition, the substrate, for all practical purposes, can be assumed to be semi-infinite.

The sudden jump in the cross-sectional area of the waveguide causes a sudden jump in the impedance of the waveguide. It is important to note that the waveguide's impedance is not the same as the characteristic impedance of the medium, but rather a property of waveguide geometry (although it is influenced by the impedance of the medium). The distributed impedance of the waveguide can be thought of like the impedance of a lumped-element

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Z

A

_r..

e -... ...

-/

£'3

Figure 3.1: A waveguide with a discontinuity in width causing a sudden change in impedance. The large section on the right represents the substrate. The forward and reflected waves are shown propagating through the waveguide.

transmission line consisting of capacitors and inductors [13]. It is given by

Z = [N .s]

A M5 (3.14)

where A is the cross-sectional area of the waveguide [14]. As the waveguide narrows, the impedance increases, meaning that a narrower waveguide, all else fixed, will have a slower particle velocity.

This remarkable fact immediately leads to one simple method of limiting radiated acoustic power in a MEMS device: use small anchors connected to a iarge substrate with a sudden discontinuity in width. At the substrate, the anchors will see a zero impedance. Interestingly, one early Draper LBAR did, in fact, have tapered anchors designed for electrical impedance matching, but which likely contributed to additional power loss in the resonator.

Figure 3.2: Early Draper LBAR resonator with tapered anchors.

On the other hand, a MEMS resonator design by researchers at the University of Michigan does attempt to exploit this phenomena [15]. Unfortunately, the paper incorrectly asserts

Ai

. I, W

...

oOO.

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that a waveguide connected to a very large substrate should see an infinite impedance. Also, it states that it is necessary to employ a quarter-wavelength long segment of anchor before the discontinuity to act as a quarter-wave impedance transformer, which transforms the infinite impedance to a zero impedance. However, an infinite impedance is every bit as rigid a boundary condition as a zero impedance. Furthermore, in practice and theory, such a discontinuity is not a zero impedance. Indeed, if it was the case, then no sound would radiate from our mouths, and we would all be relegated to using megaphones (tapered impedance matching devices) to converse. In Chapter 4, we will also explore computer models of this situation to determine the impedance for a waveguide with discontinuities of arbitrary geometry.

3.1.2 Radiation Impedance

If one peruses a more advanced reference on acoustics, it can be found that an anchor attached to a substrate is similar to the classic problem of a pipe with one end open to the air. Not surprisingly, the impedance seen at the open end of the pipe is not zero, but rather, the radiation impedance

114].

This makes intuitive sense. Any musician knows that an open pipe, like a saxophone or church organ, radiates.

Anchor

< Iz

z=-I _{z =}

Figure 3.3: A transmission line with a reactive load represented as a radiation impedance, Zr.

Since acoustic waveguides can be modeled like lumped-element transmission lines, it is possible to use all the tools of transmission line theory. In this model, the anchor, or waveguide, is represented by the transmission line (see Figure 3.3), while the substrate is represented by the radiation impedance at the load. Then, the impedance of the load as seen at any point on the transmission line is given by (3.15),

e-jkz + FLejkz

Z(z) = Ze _ , (3.15)

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where j = V/T, k is the wave number and FL is the reflection coefficient at the load. FL is related to the radiation impedance by (3.16).

L = ZoA (3.16)

z, + ZoA

Finally, we can also define the "standing wave ratio", or SWR, as

SWR Umax 1+ FL (3.17)

Umin 1 - FLI

The SWR is an empirically useful metric that we will later use to determine the magnitude of the reflection coefficient from numerical simulations. Umax and Umin are the maximum and minimum values of a given quantity, such as voltage, displacement, or pressure, which are observed along a transmission line or waveguide. By measuring the extremes of the magnitude of a standing wave, we can determine the impedance of the load.

Putting the pieces together, next we note that the radiation impedance at the load, which is the impedance at the waveguide-substrate junction, is known to be both theoretically and empirically as

Z, = A. Zo (ka)2 + jO.6ka) [N- , (3.18)

where where a is the cross-sectional length of the waveguide, for A > a. From the units, it is clear that Z, = Aeff - Zo at a surface for some effective area, Aeff. Plugging this result

into the formula for the reflection coefficient and taking the modulus squared, which yields the fraction of reflected power at the load, we can compute the transmitted power

T = 1 - R = 1- 2 + (ka)2 (ka)2, for ka < 1. (3.19)

[1 + 41 (ka)2] + (0.6ka)2

Since the wave number, k, is given by k = 1 = L, the transmitted power is a function of frequency. For example, if we consider a resonator attached to an anchor with a cross-sectional width of 1 apm operating at a frequency of 1 GHz, the power reflection coefficient for

an AIN waveguide abruptly connected to the substrate (no tapering), works out to T e 0.01.

For this example, the anchor transmits 1% of the power. Thus, we find (8 + 180j) x 104 [N"]. If the waveguide is flanged, like the early Draper resonator, the transmitted power is approximately double that for the unflanged waveguide. If the energy stored in the

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resonator is known, a lower bound for the Q (including only anchor loss) can be obtained by (3.42) [14].

3.1.3 Bragg Reflectors

One alternative method for controlling power radiated through anchors or the substrate is to implement a layered reflecting structure known as a Bragg reflector. A Bragg reflector consists of thin layers of material, a quarter-wavelength wide, with alternating characteristic impedance. The net effect of such a layer is to create a condition where each wave reflected from each layer is in phase with the wave reflected from the adjacent layers. This occurs because there is a 180 degree phase shift from reflection off of a high to low impedance interface. The reflected waves must then constructively interfere. The result is a very high quality mirror that can be constructed of virtually any materials. The reflectivity of a Bragg

Bragg Reflector

1/4

01--1 2 3 n-2 n-1 n

Figure 3.4: A diagram of a Bragg reflector layered medium. The dotted line represents a propagating wave which was reflected off of a layer such that it is in-phase with respect to those reflected from adjacent layers.

reflector with N layers, when the frequency is such that the layer spacing is exactly a quarter of a wavelength, is given by (3.20), where c is the speed of sound in the medium.

R = ca/Cb - (C/Cb) 2N 2

Ca/Cb + (Ca/Cb)2N (3.20)

It can be shown that for only a few layers of typical materials, the reflectivity nearly approaches unity [16]. Not surprisingly, Bragg reflectors are commonly employed in MEMS resonator devices. They are often placed below FBARs to reduce power loss from acoustic radiation into the substrate [17]. More recently, they have been implemented in the tethers of LBARs to control mechanical coupling [3].

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3.1.4 Acoustic Resonators

Since we are concerned with MEMS resonators, it is useful to have a general understanding of the three main types of simple acoustic resonators, as shown in the Figure 3.5. Analogs

Closed Pipe

- v(x)

--- p(x)

,/2

Half Open Pipe

p 0

X/4

Open Pipe

V2

Figure 3.5: The three basic types of one-dimensional resonator boundary conditions for a longitudinal wave.

of these three basic types of resonators are seen throughout the MEMS literature, but are usually referred to in different set of jargon. The LBAR, for example, is analogous to an open-open pipe.

The acoustic boundary conditions, (3.21) and (3.22), guarantee that pressure and the perpendicular component of velocity are continuous across a boundary [12].

Po = Pi (3.21)

vo0 = viL (3.22)

Since it is sometimes reasonable to approximate the variational pressure in an infinite space as being everywhere zero (see section 3.1.1), the standing wave in our idealized open-open and closed-closed resonators must satisfy the condition that an integer number of half-wavelengths fit inside. In the case of an open-open resonator, from the boundary conditions, the pressure of the wave at the ends must always be zero, corresponding to nodes of the standing wave. For the closed-closed resonator, we see that the velocity must be zero at the walls. Lastly, the closed-open resonator supports a standing wave of an integer multiple of

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quarter wavelengths, since the velocity must be zero on the closed end, while the pressure must be zero on the open end.

f

= f oo = m , Vm N (3.23)

2L

fO

= m (3.24)

If the geometry of the pipe at the resonant frequency does not permit the approximation that the external pressure is zero, it is possible to account for the actual radiation impedance with a simple length correction of the form Leff = L + 0.6a, A > a, (where a is the diameter of the pipe) for a closed-open pipe. This relationship can be derived by considering that the impedance at the left and right side of a boundary must be equal. Therefore, using (3.15), the impedance on the transmission line from the source can be combined with the expression for the radiation impedance, and the set of frequencies for which the imaginary component of impedance vanishes can be found (the condition for resonance or anti-resonance) [14].

3.2 The Length-extensional Resonator

The design of MEMS resonators is greatly facilitated by the development of an idealized analytical model. This thesis is concerned with thin-film resonators which operate in what is known as "length-extensional" mode, which is to say that the primary mode of vibration is along the length, and not the thickness, of the thin-film. A schematic drawing of a length-extensional resonator (LBAR) is shown in Figure 3.6. Included with Figure 3.6 is the orientation of the AIN polycrystalline and the electrodes which apply a potential across the layer. For simplicity, our idealized model neglects variations along the width of the resonator. Moreover, the electric field is applied parallel to the c-axis of the crystal since the individual crystal grains are randomly rotated about the c-axis.

Although idealized, the derivation of the voltage-displacement wave equation and sub-sequent frequency response solution is not a simple task. However, detailed derivations are available [1],[18]. The reader should refer to these works for a more thorough understanding. For our purposes, it shall suffice to say that the derivation of the wave equation begins with Newton's law for elastic media, (3.25).

a

2

VT = P ui (3.25)

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c-axis

A-V

(b)

Figure 3.6: Shown in (a) is a thin-film LBAR resonator along with accompanying polycrys-talline AIN orientation for the LBAR (b).

The form of the third-rank tensor gradient operator is known to be as shown below, where we have neglected any dependence along the width in (3.27) [19].

0 0 0

a

V ₂ 0 _8X2 0 _0X3 0 _0X1_ (3.26) 0 0 a xa 3 OaX2 a OaXa 0 0 0 _ax0 0 (3.27) 3 x1l 0 0 9 ax₃ 0 xla 0

Next, a host of approximations must be performed to reduce the number of equations of the linear system until a single, second-order partial-differential equation remains. When these approximations are performed on both (3.25) and (2.18), the remaining equations can be combined to yield a single differential equation in terms of the displacement in the x, direction and the electric field across the resonator. To briefly summarize, since we initially assumed that the resonator does not vary across the width, we can neglect any dependence on the x2 dimension, including eliminating the second row of (3.25) altogether. Additionally, we can approximate p--u 3 = 0 and T3 = T5= 0, because we are not interested in thickness

mode vibrations (recall that T5 is in fact equivalent to T13,which is a shear coupling between

the vibration of the LBAR along the length and the thickness). The result is that a single equation for Newton's law remains

02

Design of width-extensional, piezoelectric RF MEMS resonators and filters