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NXP Semiconductors

Optimized design for frequency

invariance of MEMS resonators

T.P. Luiten

DCT 2009.031

NXP restricted

Traineeship report

Coach(es): dr. ir. C. van der AvoortNXP Semiconductors

Supervisor: dr. ir. R.H.B. Fey

Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Technology Group

Eindhoven, April, 2009

Contents

Abstract v

Samenvatting vii

1 Introduction 11.1 Research goal and approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Eigenfrequency of MEMS resonators 32.1 Uniform symmetric clamped-free resonator . . . . . . . . . . . . . . . . . . . . . . 32.2 Composite symmetric clamped-free resonator . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Temperature aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Rayleigh approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Frequency variability of Dogbone-type resonators 113.1 Modeling mismatch in dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Improving reproducibility of Dogbone-type resonators . . . . . . . . . . . . . . . . 133.3 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Temperature aspects of Dogbone-type resonators 234.1 Extended analysis of the Rayleigh approach . . . . . . . . . . . . . . . . . . . . . . 234.2 Modeling the application of silicon-dioxide . . . . . . . . . . . . . . . . . . . . . . . 264.3 Improving temperature dependency and robustness of Dogbone-type resonators . . 284.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conclusions and recommendations 395.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Bibliography 43

A Resonator properties 45A.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 Symmetric clamped-free resonator dimensions . . . . . . . . . . . . . . . . . . . . . 46A.3 Dogbone-type resonator dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B Determination of effective material properties 49B.1 Composite symmetric clamped-free resonator . . . . . . . . . . . . . . . . . . . . . 49B.2 Composite Dogbone-type resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

iii

iv CONTENTS

Abstract

Modern handheld and wireless electronics highly depend on frequency control devices that provideprecise time and frequency. It is often required to produce a signal whose frequency is very stableand exactly known. Normally, this is realized by means of a so-called oscillator. Since the 1920s,when quartz was first utilized to realize crystal resonators in oscillators [3, 5], the applicationsof devices based on quartz crystals have expanded dramatically. Their high quality factor and,for the high-end components, low temperature drift, provide high stability and exceptional preci-sion. However, their size and integrability with IC-technology are major drawbacks. Due to theircompact size and feasibility of integration with IC-technology, MEMS (micro-electro-mechanicalsystem) resonators provide an interesting and promising alternative. However, it is the need forhigh reproducibility (robust eigenfrequency with respect to dimension errors) and low tempera-ture dependence that is expected to be challenging in the MEMS resonator design process. Inthis project we derived compact analytical relations that define a reproducible Dogbone-type res-onator design and we significantly improved temperature dependence while guaranteeing a robusteigenfrequency.First a symmetric clamped-free resonator with a uniform cross-sectional area is studied. Onlythe essentially one-dimensional resonators are considered and the models are designed for a vi-bration in an extensional mode. Using Newton’s second law of motion, the general description ofsuch beams subjected to axial loading is formulated with a partial differential equation. Togetherwith the necessary boundary conditions a boundary value problem is derived, which is solved tofind the eigenfrequencies and corresponding eigen-modes. From the analysis it is found that theeigenfrequencies are described by a factor defined by geometry and a factor defined by the mate-rial properties. Moreover, the geometry-factor contains only the length of our resonator, whereasthickness and width are irrelevant. As a result the reproducibility of this symmetric clamped-freeresonator is very low.Because of the opposite temperature dependence in the Young’s modulus of silicon-dioxide com-pared to silicon, it is expected that compensation for the temperature dependence in the stiffnessis possible by oxidation of the silicon resonator. To analyze the temperature aspects and the effectof the oxidation on the eigenfrequency the composite symmetric clamped-free resonator is studied.Note that it is assumed that resonator dimensions are temperature-independent, because this effectis significantly smaller than temperature dependence in the stiffness. Moreover, effective materialproperties are taken into account, defined as a weighted average with respect to the surface areaof the different materials. The thermal oxidation alters the temperature coefficient TC, definedas 106 times the quotient of the change in frequency per degree Kelvin and the original frequency,and the eigenfrequency of the resonator. An optimal oxide thickness τTC is found to compensatefor the temperature drift, but this also results in a non-robust eigenfrequency with respect to theinaccuracy (5%) in the oxidation process. Nevertheless it is shown that an optimal oxide thick-ness τf which results in a robust eigenfrequency does exists, i.e. ∂f

∂τis as small as possible, but

that this result is physically impossible and significantly different from optimal thickness τTC. Allresults of the analysis of the symmetric clamped-free resonator are used in the modeling of theDogbone-type resonator.We derived compact analytical expressions which define the geometry of a reproducible Dogbone-type resonator. Therefore we analyzed the eigenfrequency which is expressed as a function of thebasic dimensions of the Dogbone-type resonator as well as a "growth"-parameter δ representing

v

vi ABSTRACT

the geometric uncertainty. The model is based on the assumption that growth and shrinkage ofthe intended dimensions will occur during production and that this effect is equal in perpendic-ular directions. Moreover the Dogbone-type resonator is modeled as a coupling of two uniformbeam structures with different cross-sectional areas and only vibration in an extensional mode isconsidered. The original boundary value problem is not exactly solvable for the eigenfrequency.However, with a new solution approach where robustness with respect to dimension errors is ob-tained by equating the first and second derivatives of the frequency with respect to growth to zero,we managed to find these analytically exact solutions. For known resonator head width b0 andspatial wavelength β (hence the eigenfrequency) all other resonator dimensions follow from theserelations. The initial spread of the Dogbone-type resonator design is reduced by a factor 100 toapproximately 10 ppm after geometric optimization.The composite Dogbone-type resonator is modeled with effective material properties taken intoaccount. The starting dimensions before oxidation are defined by the geometrically optimizedresonator design. It is shown that it is possible to compensate for the temperature drift with auniform oxide thickness along the whole resonator, but that robustness with respect to variationsin this oxide thickness is not guaranteed. Therefore, we analyzed the Rayleigh approach, whichdetermines the angular eigenfrequency by the quotient of the potential and kinetic energy, whichare related to mode-shape and time derivative of mode-shape, respectively. Moreover, the poten-tial energy depends also on the stiffness of the resonator, whereas the kinetic energy is influencedby the resonator’s mass. The addition of silicon-dioxide to a silicon resonator has an effect onthe distributed stiffness and mass hence the eigenfrequency of the composite resonator. To com-pensate for this significant frequency drop two different oxide thicknesses along the length of theresonator are considered. It is assumed that the stepwise change in oxide thickness is situated onthe resonator springs and that the two different oxide thicknesses are related by an oxide growthrate amplification n. The composite Dogbone-type resonator is therefore modeled as a couplingof three uniform beam structures with different cross-sectional areas. Closed form solutions ofthe corresponding boundary value problem do not exists. Results are derived by analysis of theTaylor series approximation of the spatial wavelength β (hence the eigenfrequency). Due to thedistinction in oxide thickness along the length of the resonator we managed to shift the optimaloxide thickness τf (hence a frequency invariant resonator design) from a physically impossiblesolution for uniform oxide thickness to a physically possible solution close to the optimal oxidethickness τTC, which compensates for the temperature dependency in the stiffness. By optimiza-tion of the position of the stepwise change in oxide thickness we found a unique resonator designwith oxide thickness τ1 = τTC = τf resulting in an eigenfrequency that is temperature-invariantand robust with respect to production-induced variations in dimensions and the 5% accuracy atwhich the oxidation process could be controlled. To ensure that this frequency corresponds to theintended eigenfrequency, the initial design frequency and corresponding resonator dimensions areslightly adapted. Note that all results are geometry dependent.

Samenvatting

Moderne compacte en draadloze elektronica is erg afhankelijk van frequentie-regelaars die zor-gen voor exacte tijdindicaties en frequenties. Vaak is het nodig om een signaal te producerenwaarvan de frequentie stabiel en exact bekend is. In de meeste gevallen wordt dit gerealiseerddoor middel van een zogenaamde oscillator. Sinds de jaren twintig, toen quartz voor het eerstgebruikt werd voor de vervaardiging van kristal resonatoren in oscillatoren, is het toepassings-gebied van apparaten die gebaseerd zijn op deze quartz kristallen enorm uitgebreid. De hogekwaliteit-factor en de lage temperatuur afhankelijkheid van de quartz kristallen zorgen voor eenhoge stabiliteit en een uitzonderlijke precisie. Echter, de grootte en de slechte integratie mogelijkhe-den met IC-technologie zijn belangrijke nadelen. MEMS (micro-elektrisch-mechanisch-systeem)resonatoren scoren beteren op deze aspecten, en vormen daarom een veelbelovend alternatief.Echter, het produceren van meerdere resonatoren die ondanks kleine onderlinge maatafwijkin-gen eenzelfde eigenfrequentie bezitten (reproduceerbaarheid) en het realiseren van temperatuurongevoelig gedrag is een enorme uitdaging in het ontwerpproces. In dit project zijn compacteanalytische relaties opgesteld, die de reproduceerbaarheid van een zogenaamde Dogbone resonatorgaranderen. Vervolgens introduceren we een oplossing om een temperatuur-invariant ontwerp terealiseren. Hierbij zorgen we er tegelijkertijd voor dat de robuustheid van de eigenfrequentie nietwordt aangetast.Voordat we de beoogde Dogbone resonatoren bestuderen is eerst een frequentie-analyse van eensymmetrisch geklemd-vrije resonator met uniforme dwarsdoorsnede uitgevoerd. In deze studie ishet eendimensionale gedrag van de resonatoren onderzocht; de modellen zijn ontworpen voor eenaxiale trilling van de resonator. Met behulp van de tweede wet van Newton is de algemene beschrij-ving van dergelijke balkmodellen onderworpen aan axiale belasting geformuleerd in de vorm vaneen partiële differentiaal vergelijking. Samen met de benodigde randvoorwaarden is een eigen-waarde probleem opgesteld dat vervolgens is opgelost om de eigenfrequenties en corresponderendeeigenmodes te vinden. Het blijkt dat de eigenfrequenties afhankelijk zijn van de materiaaleigen-schappen en de geometrie van de resonator. Met betrekking tot de geometrie is alleen de lengtevan de resonator van belang. Als gevolg hiervan kan de reproduceerbaarheid van de symmetrischgeklemd-vrije resonator niet worden gegarandeerd.Vanwege een tegengestelde temperatuur afhankelijkheid in de Young’s modulus van silicium-dioxide vergeleken met silicium, verwachten we dat de temperatuur afhankelijkheid van de stijf-heid kan worden gecompenseerd door middel van oxidatie van de silicium resonator. Om detemperatuur aspecten en het effect van de oxidatie op de eigenfrequentie te analyseren is dezesamengestelde resonator bestudeerd. Hierbij is aangenomen dat de afmetingen van de resonatortemperatuur ongevoelig zijn, omdat dit effect vele malen kleiner is dan de temperatuur afhanke-lijkheid van de stijfheid. Bovendien is gebruik gemaakt van effectieve materiaal eigenschappen.Deze zijn gedefinieerd als een gewogen gemiddelde met betrekking tot de grootte van het opper-vlak van de verschillende materialen. De thermische oxidatie verandert de temperatuur coeffi-cient (TC), gedefinieerd als 106 maal het quotient van de frequentie verandering per graad Kelvinen de originele frequentie van de resonator, en de eigenfrequentie van de resonator. Een optimaleoxide dikte τTC die compenseert voor de temperatuur drift is gevonden, maar resulteert ook in eenniet-robuuste eigenfrequentie met betrekking tot de onnauwkeurigheid (5%) in het oxidatie proces.Desondanks is aangetoond dat een dergelijk optimale oxide dikte τf , resulterend in een robuusteeigenfrequentie, theoretisch wel degelijk bestaat, maar dat deze fysisch onmogelijk is en significant

vii

viii SAMENVATTING

verschilt van de optimale oxide dikte τTC. Alle resultaten van de analyse van de symmetrischgeklemd-vrije resonator zijn gebruikt in het modelleren van de Dogbone resonator.Compacte analytische relaties zijn opgesteld die de reproduceerbaarheid van de Dogbone resonatorgaranderen. Daarvoor is de eigenfrequentie, uitgedrukt in de basis dimensies van de Dogboneresonator en een "groei"-parameter δ die de geometrische onzekerheid representeert, geanalyseerd.Het model is gebaseerd op de aanname dat tijdens productie groei en krimp in gelijke mate op-treedt in de richtingen loodrecht op het resonatoroppervlak. Daarnaast is de Dogbone resonatorgemodelleerd als een koppeling van twee uniforme balk structuren met elk een verschillende dwars-doorsnede. Het originele eigenwaarde probleem is niet exact oplosbaar voor de eigenfrequentie.Toch zijn we er in geslaagd om deze analytisch exacte oplossing te vinden. Hiervoor is een nieuweaanpak gebruikt, waarin robuustheid met betrekking tot maatafwijkingen is gegarandeerd doormiddel van het gelijkstellen aan nul van zowel de eerste als de twee afgeleide van de frequentiemet betrekking tot groei. Wanneer zowel de resonator kopbreedte b0 en de ruimtelijke golflengte β(en dus de eigenfrequentie) bekend zijn, liggen alle andere afmetingen van de resonator eenduidigvast. De initiële spreiding in de eigenfrequentie van de Dogbone resonator is na geometrischeoptimalisatie verminderd met een factor 100 tot ongeveer 10 ppm.De samengestelde Dogbone resonator (bestaande uit silicium en silicium-dioxide) is gemodelleerdmet behulp van effectieve materiaal eigenschappen. De initiële afmetingen vóór de oxidatie wor-den bepaald door het geometrisch geoptimaliseerde resonator ontwerp. Het is aangetoond dat hetmogelijk is om de temperatuur drift te compenseren met een uniforme oxide dikte over de heleresonator, maar dat dit samengaat met een enorme afname in de robuustheid van de eigenfrequen-tie met betrekking tot variaties in de oxidedikte. Om dit probleem op te lossen is de Rayleighbenadering geanalyseerd. Hierbij wordt de eigenhoekfrequentie bepaald door het quotient vande potentiële en de kinetische energie, gerelateerd aan respectievelijk de eigenmode en de tijds-afgeleide van de eigenmode. Bovendien is de potentiële energie ook afhankelijk van de stijfheidvan de resonator, terwijl de kinetische energie is gerelateerd aan de massa. De toevoeging vansilicium-dioxide aan een silicium resonator heeft een effect op de uiteindelijke stijfheid en massadus de eigenfrequentie van de samengestelde Dogbone resonator. Om te compenseren voor hetverschil in frequentie maken we gebruik van twee verschillende oxidediktes langs de lengte van deresonator. We nemen aan dat de stapsgewijze verandering in oxidedikte is gesitueerd op het veergedeelte van de resonator en dat beide oxidediktes aan elkaar zijn gerelateerd door middel van eenoxide groeisnelheid factor n. De samengestelde Dogbone resonator is daarom gemodelleerd als eenkoppeling van drie uniforme balk structuren met elk een verschillende dwarsdoorsnede. Een ge-sloten uitdrukking voor de eigenfrequentie volgend uit het corresponderende eigenwaarde probleembestaat niet. Resultaten zijn verkregen door numerieke analyse van een Taylor reeks benaderingvan de ruimtelijke golflengte β (en dus de eigenfrequentie). Door het onderscheid in oxidediktezijn we in staat de optimale oxidedikte τf (dus een frequentie-invariant resonator ontwerp) teverschuiven van een fysisch onmogelijke oplossing voor een uniforme oxidedikte, naar een fysischeoplossing dicht bij de optimale oxidedikte τTC, die compenseert voor de temperatuur afhanke-lijkheid van de stijfheid. Door te optimaliseren naar de positie van de stapsgewijze verandering inoxidedikte hebben we een uniek resonator ontwerp gevonden met een oxidedikte τ1 = τTC = τf .Het resultaat is een eigenfrequentie die temperatuur-invariant is en robuust met betrekking totzowel variaties in dimensies als de 5% nauwkeurigheid waarmee het oxidatie proces kan wordengecontroleerd. Om er voor te zorgen dat deze eigenfrequentie overeenkomt met de gewenste eigen-frequentie, zullen de initiële ontwerp frequentie en de corresponderende resonator afmetingen ietsmoeten worden aangepast. Alle resultaten in het temperatuur probleem zijn afhankelijk van degeometrie van de Dogbone resonator.

Chapter 1

Introduction

Modern handheld and wireless electronics highly depend on frequency control devices that provideprecise time and frequency. It is often required to produce a signal whose frequency is very stableand exactly known. Normally, this is realized by means of a so-called oscillator. An oscillator,producing some sort of periodic electrical output, is relatively simple to construct, but to producemany of precise frequency and high stability is not trivial.

Since the 1920s, when quartz was first utilized to realize crystal resonators in oscillators [3,5],the applications of devices based on quartz crystals have expanded dramatically. Their highquality factor and, for the high-end components, low temperature drift, provide high stabilityand exceptional precision. Note, a higher quality factor means less frequency shift for a change inoscillator load capacitance and less shift due to other external factors such as oscillator supply volt-age. However, their size and integrability with IC-technology are major drawbacks. Due to theircompact size and feasibility of integration with IC-technology, MEMS (micro-electro-mechanicalsystem) resonators provide an interesting and promising alternative, an additional niche marketinaccessible for quartz.

The high quality factor and beneficial form factor of a MEMS resonator alone is not enough tomake it a candidate for most applications. The long built standard of quartz crystals has resultedin specifications that are hard to attain with MEMS resonators. One of these is the temperaturedependent value of the eigenfrequency. In this research, theoretical results are presented, thatdefine MEMS resonator designs which show an almost invariant eigenfrequency with respect to acertain level of geometric uncertainty and temperature variations.

1.1 Research goal and approach

As mentioned above, it is the need for frequency accuracy that is expected to be challengingin the MEMS resonator design process. Therefore, the goal of this research is twofold. Firstly,it is desired to derive analytical expressions which define the geometry of a specific resonator,which eigenfrequency is insensitive to production-induced variations in dimensions (reproducibil-ity). Secondly, these geometrically optimized resonator designs are used to find resonators with atemperature-invariant and robust eigenfrequency.

In this report, only essentially one-dimensional resonators are considered, i.e. only axial vi-bration in extensional mode will be studied. The resonators consist of one single material, beingsilicon (Si), with non-uniform cross-sectional area. To be more specific, the so-called Dogbone-type resonator model as presented in [1] will be studied. Using Newton’s second law of motion, thedynamic behavior of such resonators subjected to axial loading is modeled. Based on the equationsof motion the analytical expressions describing the desired geometry of a specific resonator will

1

2 CHAPTER 1. INTRODUCTION

be derived. The eigenfrequency will be expressed as a function of these basic dimensions (fourdegrees of freedom) as well as a "growth"-parameter. The model is based on the assumptionthat growth and shrinkage of the intended dimensions will occur during production and that thiseffect is equal in directions perpendicular to the resonator surfaces. A desired frequency is relatedto the dimensions, whereas frequency robustness with respect to dimension errors is obtained byequating the first and second derivatives of the frequency with respect to the growth parameterto zero.

In order to find resonators with a robust and temperature-invariant eigenfrequency, compositeresonators, i.e. resonators consisting of both silicon and silicon-dioxide, are studied. Because ofthe negative temperature coefficient (TC) of the Young’s modulus of silicon, the eigenfrequency ofthe resonator has a negative TC as well. Silicon-dioxide (SiO2) shows the opposite temperaturedependence and can be used in combination with silicon to reduce the temperature dependenceof the resonator. The addition of silicon-dioxide, the result of an oxidation process applied toa silicon resonator, has an effect on the distributed stiffness and mass of this resonator. Theresonant mode-shape might be affected as well. The optimum addition of silicon-dioxide to arriveat a temperature-invariant eigenfrequency can be related to the geometry of a specific resonator.The robustness of the found eigenfrequency with respect to variation around the optimum amountof added oxide is investigated. Again, only one-dimensional resonators subjected to axial loadingare considered.

1.2 Outline

In Chapter 2, a symmetric clamped-free resonator with a uniform cross-sectional area will beconsidered. The behavior of the eigenfrequency and corresponding eigen-modes will be studiedfor both the single material resonator and the composite resonator. The results of the analysis inthis chapter will be used to study reproducibility and temperature aspects of the Dogbone-typeresonator in Chapter 3 and Chapter 4, respectively. In both chapters the modeling of the res-onator under consideration will be presented. In addition, the solution approach for the frequencyaccuracy problem will be elucidated and theoretically results will be given. Note that startingdimensions of the Dogbone-type resonator in Chapter 4 will be defined by the geometrically op-timized resonator design from Chapter 3. Finally in Chapter 5, conclusions will be drawn andrecommendations for further research will be given.

Chapter 2

Eigenfrequency of MEMS resonators

In order to gain insight in the behavior of the eigenfrequency and the corresponding eigen-modesof the Dogbone-type resonators with respect to production-induced variations in dimensions andtemperature dependency, several models of resonators are studied. Here, only the essentially one-dimensional resonators are considered. In this chapter, we restrict ourselves to a resonator witha uniform cross-sectional area. This symmetric clamped-free resonator design is equivalent tothe free-free beam model, presented in [1] and is depicted in Figure 2.1. The resonator actuallyconsists of two opposite clamped-free resonators, since the resonator in theory is symmetric withrespect to x = 0. To illustrate the approach to express the eigenfrequency as a function of theresonator dimensions, first, the resonator model consists of one material (model 1). To be ableto study the temperature aspects, an additional oxidation step in the production process of theresonator is taken into account. Therefore, the first model is extended to a composite resonatorthat comprises two different materials (model 2). All resonator models are designed for vibrationin an extensional mode. The material properties and the typical dimensions of the resonatorsunder consideration, can be found in Appendix A. The results of the analysis in this chapter areused to study reproducibility and temperature aspects of the intended Dogbone-type resonator inChapter 3 and Chapter 4, respectively.

2.1 Uniform symmetric clamped-free resonator

For simplicity, first a resonator that consists of one material with a uniform cross-sectional area(model 1) is considered. Using Newton’s second law of motion, the general description of suchbeams subjected to axial loading can be formulated by the partial differential equation

∂

∂x

[

EA(x)∂u(x, t)

∂x

]

= m(x)∂2u(x, t)

∂t2, (2.1)

Figure 2.1: The symmetric clamped-free resonator.

3

4 CHAPTER 2. EIGENFREQUENCY OF MEMS RESONATORS

where E is the Young’s modulus of the used material and A(x) and m(x) describe the cross-sectional area and mass per unit of length of our resonator, respectively. Moreover, u(x, t) de-scribes the motion of the resonator due to the axial loading, i.e. the actuation applied over gap g.Because it is desired to find eigenfrequencies and corresponding eigen-modes of the resonator,u(x, t) is defined as

u(x, t) = u(x) · f(t), with f(t) = cos(ωt). (2.2)

Combining Eq.(2.1) and Eq.(2.2), and using a constant cross-sectional area A(x) = A and constantmass per unit of length m(x) = ρsiA for 0 ≤ x ≤ L0, i.e. using a single material, it is found thatfor the resonator under consideration, the following boundary value problem (BVP) has to besolved to find the eigenfrequencies and corresponding eigen-modes,

∂2u(x)∂x2 + α2u(x) = 0,

(1) u(0) = 0,

(2) ∂u(x)∂x

|x=L0= 0.

(2.3)

Here, α2 = ρsiω2

Esiand u(x) = c1cos(αx) + c2sin(αx) describe the general solution of the partial

differential equation in Eq.(2.3). With the boundary conditions the solution of BVP (2.3) is foundto be

ui(x) = c2sin(αix), with αi =(2i − 1)π

2L0for i ∈ Z

+0 . 1 (2.4)

Therefore, the corresponding angular eigenfrequencies (in radians per second) equal

ωi =(2i − 1)π

2L0

√

Esi

ρsi

for i ∈ Z+0 , (2.5)

As can be seen from this result, the amplitude c2 can be chosen freely in order to satisfy Eq. (2.3),since eigen-modes are defined apart from a scaling factor, a property that will be used later.Moreover, the amplitude c2 does not influence the eigenfrequencies. The latter are describedby a factor defined by geometry and a factor defined by the material properties. Moreover, thegeometry-factor contains only the length of our resonator, thickness and width are irrelevant.The eigen-modes corresponding to the four lowest eigenfrequencies of the clamped-free resonatormodel, with amplitude c2 = 1 µm, are depicted in Figure 2.2. The corresponding eigenfrequenciesare presented in Table 2.1.

Table 2.1: Eigenfrequencies corresponding to the first four eigen-modes of the clamped-free res-onator model 1.

Mode Eigenfrequency fi [MHz]

1 93.72762 281.18293 468.63824 656.0935

1Z

+

0is the set of strictly positive whole numbers, thus with exception of 0; zero corresponds to the trivial solution

at stationary equilibrium.

2.2. COMPOSITE SYMMETRIC CLAMPED-FREE RESONATOR 5

0−1

−0.5

0

0.5

1

L0/5 2L0/5 3L0/5 4L0/5 L0Dis

pla

cem

ent

ui[µ

m]

Position x [µm]

(a) Mode 1 (solid) and mode 2 (dashed)

0−1

−0.5

0

0.5

1

L0/5 2L0/5 3L0/5 4L0/5 L0Dis

pla

cem

ent

ui[µ

m]

Position x [µm]

(b) Mode 3 (solid) and mode 4 (dashed)

Figure 2.2: Eigen-modes corresponding to the four lowest eigenfrequencies of the clamped-freeresonator model 1, with amplitude c2 = 1 µm.

2.2 Composite symmetric clamped-free resonator

To illustrate the effect of oxidation on the eigenfrequency and the additional robustness problem,in this section the clamped-free resonator with two different materials (model 2) is studied. Byoxidizing the silicon resonator, the desired silicon-dioxide grows with controllable thickness τaround the silicon core. During this oxidation process, the silicon is consumed with constant ν, thefraction of the oxide thickness that lies within the starting dimensions, as depicted in Figure 2.3(a).In the analysis effective stiffness and effective mass per unit of length are considered. Therefore,the general description from Eq. (2.1), transforms into

∂

∂x

[

(EA)eff(x)∂u(x, t)

∂x

]

= meff(x)∂2u(x, t)

∂t2, (2.6)

where(EA)eff = (w∗h∗)Esi + (2τw∗ + 2τh∗ + 4τ2)Eox,

meff = (ρA)eff = (w∗h∗)ρsi + (2τw∗ + 2τh∗ + 4τ2)ρox,(2.7)

are the effective stiffness and effective mass, respectively. The reader is referred to Appendix B.1for the complete determination of these effective properties and the explanation of the resonatordimensions.The composite clamped-free resonator is modeled with a lumped mass M , representing the oxidelayer at the boundary L∗. This model is presented in Figure 2.3(b), with corresponding boundaryconditions

(1) u(0) = 0,

(2) (EA)eff∂u(x,t)

∂x|x=L∗ + M ∂2u(x,t)

∂t2|x=L∗ = 0.

(2.8)

Using the results from Section 2.1, Eq.(2.6) - Eq.(2.8), the clamped-free resonator can be analyzed.Referring to the goals stated in Chapter 1, the reproducibility and the temperature aspects of theclamped-free resonator will be studied in Section 2.2.1 and Section 2.2.2, respectively.


Before oxidationAfter oxidation L∗

ντ

τ

(a) Application of silicon-dioxide

After oxidation

(EA)eff , (ρA)eff

Lumped massL∗

M

(b) Model for analysis

Figure 2.3: Composite clamped-free resonator, model 2.

2.2.1 Reproducibility

Facing the problem of reproducibility, it is desirable to find a geometry for a specific resonatorthat, with respect to eigenfrequency is insensitive to production-induced variations in dimensions.Analysis of the clamped-free resonator model showed that the eigenfrequency of such a resonatoronly depends on 1

L0and a factor defined by the material properties. Therefore, the clamped-free

resonator is not a good candidate when demands on reproducibility are set high. This observationcan be confirmed by Figure 2.4, where the frequency error with respect to a dimension error inL0 = 20 µm is depicted. A dimension error δ of −50 to +50 nm can be expected. Althoughthis seems to be a minor change in dimension, it results in nearly 5000 ppm change in frequency,whereas we seek for a few hundred ppm error at the most.

−100 −50 0 50 100−6000

−4000

−2000

0

2000

4000

6000

Dimension error δ [nm]

Fre

quen

cyer

ror

[ppm

]

Figure 2.4: Frequency error due to variations in dimensions of the clamped-free resonator withL0 = 20 µm.

2.2. COMPOSITE SYMMETRIC CLAMPED-FREE RESONATOR 7

2.2.2 Temperature aspects

In the analysis of the temperature aspects of frequency accuracy of specific resonators, the fo-cus is on the temperature dependency of the stiffness, whereas it is assumed that the resonatordimensions are temperature-independent. This assumption is justified by realizing that the lin-earized temperature-dependent Young’s moduli for silicon and silicon-dioxide expressed in GPaare described by

Esi,T = 131 · (1 − 60 · 10−6∆T ),

Eox,T = 73 · (1 + 196 · 10−6∆T ),(2.9)

and the linearized thermal expansion for both materials are defined by

Lsi,T = Lsi,T0· (1 + 2.5 · 10−6∆T ),

Lox,T = Lox,T0· (1 + 0.5 · 10−6∆T ),

(2.10)

with initial resonator dimensions known and ∆T = (T − T0). Note that according to Eq. (2.5)the eigenfrequency globally scales with 1

Land

√E. From other studies [2, 4] it is known that

the temperature change in combination with a certain level of oxidation alters the temperaturecoefficient (TC) and nominal frequency of a resonator. The linear temperature coefficient or driftin ppm/K is defined as the change in frequency per degree over the original frequency times 106,

TC =ωT − ωT0

ωT0(T − T0)

· 106. (2.11)

The temperature dependency of the eigenfrequency follows from the solution of the partial differ-ential equation (2.6) in combination with the boundary conditions in Eq.(2.8). It is not possible togive an exact analytical expression for this result. However, one could use a Taylor Series approx-imation for α (hence the eigenfrequency) or perform a numerical analysis with iteration processesin order to find approximated results. Here, it is tried to find an optimum addition of silicon-dioxide to arrive at a robust and temperature-invariant eigenfrequency, using a numerical analysis.

In Figure 2.5, the temperature coefficient, as defined by Eq.(2.11), is depicted for two differentclamped-free resonator designs with length L0,1 = 20 µm and L0,2 = 22 µm, respectively. Theresonators are studied in a temperature range [T0 T ] = [298 398] K. Note that the resonator widthis 10 µm for both designs, see Appendix A.2. It is shown that for different lengths almost the samerelation is found between the linear TC and the oxide thickness τ . Based on the fact that for theseresonators the vibration is mostly one-dimensional this is as expected. The optimal thickness τTC

to compensate for the temperature dependence in the stiffness of the two clamped-free resonatordesigns under consideration equals 290 nm, i.e. τTC1

= τTC2= τTC, as depicted in Figure 2.5.

This optimal oxide thickness is a realistic result. Note that because of the consumption of siliconthere is a maximum realizable oxide thickness τ . According to Appendix B.1, a resonator with aninitial silicon thickness h0 = 1.5 µm is totally consumed for an oxide thickness τ = 1705 nm.

In Figure 2.6, the frequency change in ppm (at 298 K) for the two resonator designs is pre-sented as function of the oxide thickness. This frequency change is evaluated around the optimaloxide thickness τTC = 290 nm, which is found to be equal for both designs. From experimentspresented in [4], it is known that the accuracy at which the oxidation process could be controlledhas a tolerance of roughly 5% on the desired oxide thickness. Thus, when compensating for thetemperature dependent value in the eigenfrequency, another frequency error is introduced whichis significantly larger than the initial error of -30 ppm/K from Figure 2.5. Here a temperaturerange ∆T = 100 K with T0 = 298 K is assumed. From Figure 2.6, we find for the designs un-der consideration with length L0,1 and L0,2, a frequency change of 233 ppm/nm oxide and 236ppm/nm oxide, respectively. Therefore, both temperature-invariant designs provide non-robusteigenfrequency, with respect to the accuracy at which the oxidation process could be controlled.


0 50 100 150 200 350 400−35

−30

−25

−20

−15

−10

−5

0

5

10

15

L0,1 = 20 µmL0,2 = 22 µm

τTC1= τTC2

Oxide thickness τ [nm]

Lin

ear

TC

[ppm

/K

]

Figure 2.5: Linear temperature coefficient TC versus oxide thickness for two clamped-free res-onator designs with different lengths. Temperature range [T0 T ] = [298 398] K.

250 260 270 280 300 310 320 330 340 350

−90k

−85k

−80k

−75k

−70k

−65k

5%

L0,1 = 20 µmL0,2 = 22 µm

τTC


Fre

quen

cych

ange

[ppm

]

Figure 2.6: Frequency change (at 298 K) versus oxide thickness for different clamped-free resonatordesigns, with the reference frequency equal to the initial frequency before oxidation.The 5% variation in oxide thickness results in non-robust eigenfrequency.

2.3. THE RAYLEIGH APPROACH 9

In spite of this result, it is shown in Figure 2.7 that such a robust design still exists. Takinginto account the 5% inaccuracy in the oxidation process, the optimum addition of silicon-dioxidewhich will result in resonator designs with robust eigenfrequency, should be at the extremum (inthis case the minimum) of the frequency change versus oxide thickness, i.e. ∂f

∂τmust be as small

as possible, as depicted in Figure 2.7. It is found that the trends in absolute frequency versusoxide thickness are different for the resonator designs under consideration, i.e. τf1

6= τf2. The

optimum addition of silicon-dioxide τf to find a resonator design with robust eigenfrequency isgeometry dependent. In addition, it can be seen that although there exists a numerical optimumwith respect to the 5% inaccuracy in oxide thickness for these designs, the results are physicallyimpossible and differ significantly from τTC. Apparently, for the current designs, with a uniformoxide thickness τ along the total length of the resonator, it is not possible to find a robust andtemperature-invariant resonator design, i.e. compensated for the temperature drift and stable withrespect to eigenfrequency. In [4], this observation is also found from experiments for the Dogbone-type resonator designs. To solve for this problem, later a non-uniform oxide thickness along thelength of the resonator in combination with the Rayleigh approach will be considered.

2.3 The Rayleigh approach

In order to determine the eigenfrequencies of different beam structures, it is also possible to use theRayleigh approach. To validate the result from Eq.(2.5) and to elucidate how energy is distributedalong the resonator during vibration, it is interesting to use this approach. The Rayleigh approachuses the quotient of the potential and kinetic energy to determine the angular eigenfrequency. TheRayleigh quotient is given by

ω2 =

∫ L0

0 E(x)A(x)(

du(x)dx

)2

dx∫ L0

0ρ(x)A(x)u(x)2 dx

, (2.12)

0 5.000 20.000 25.000

−350k

−300k

−250k

−200k

−150k

−100k

−50k

0

L0,1 = 20 µmL0,2 = 22 µm

τf1τf2


Fre

quen

cych

ange

[ppm

]

Figure 2.7: Frequency change (at 298 K) versus oxide thickness for different clamped-free resonatordesigns, with the reference frequency equal to the initial frequency before oxidation.Around τf an 5% variation in oxide thickness results in relatively robust eigenfre-quency.


00

0.5

1

L0/4 L0/2 3L0/4 L0

‘pot’ : cos2(

πx2L0

)

‘kin’ : sin2(

πx2L0

)

x [m]

Ener

gy

dis

trib

ution

Figure 2.8: Scaled potential en kinetic energy distribution for the first eigen-mode u1(x) of theclamped-free resonator, with E(x)A(x) = ρ(x)A(x) = 1.

which relates angular frequency to the mode-shape and the time derivative of the mode-shape,as these constitute kinetic and potential energy, respectively. Since BVP (2.3) is solved already,ui(x) is described by Eq. (2.4), A(x) = A, E(x) = Esi and ρ(x) = ρsi for 0 ≤ x ≤ L0. Togetherwith Figure 2.8, it becomes clear that also with the Rayleigh approach an angular eigenfrequencyequal to Eq. (2.5) is found. Figure 2.8 shows the potential and kinetic energy distribution for thefirst eigen-mode u1(x) of the clamped-free resonator. At x = 0, the potential energy is maximal,whereas at the end, i.e. at x = L0, the potential energy equals zero. For the kinetic energy thisholds the other way around. These observations are inline with the boundary conditions fromBVP (2.3).

In the analysis of the composite resonator models the Rayleigh approach will be a helpful andillustrative relation to gain insight in the behavior of a resonator design. Note, the addition ofsilicon-dioxide to a silicon resonator has an effect on the distributed stiffness (potential energy)and mass (kinetic energy) of the resonator. Therefore, it is the evaluation of this Rayleigh quotientwhich resulted in the solution to use a "split"-factor to distinguish two different oxide thicknessesalong the length of the resonator. The result is a resonator design with a robust eigenfrequencywith respect to the accuracy (5%) at which the oxidation process could be controlled. This willbe further explicated in Chapter 4.

2.4 Discussion

In this chapter, several models of clamped-free resonators with a uniform cross-sectional area arestudied. The eigenfrequency is found to be a function of a factor defined by geometry and a factordefined by the material properties. The eigenfrequency of the composite clamped-free resonator(consisting of Si and SiO2) is studied with respect to temperature dependency and geometricuncertainty of resonator dimensions and oxide layer dimensions. Apparently, it is not possible tofind a robust and temperature-invariant resonator design, i.e. compensated for the temperaturedrift and stable with respect to eigenfrequency, with a uniform oxide thickness τ along the totallength of the resonator. This observation holds for both the clamped-free resonator and theDogbone-type resonator, although in this report only the former has been studied explicitly. Itis expected that this problem can be solved by using two different oxide thicknesses, where the"split", i.e. the location of the stepwise change in oxide thickness, is free to choose along thelength of the resonator. Therefore, we analyzed the Rayleigh approach, which will be furtherexplicated in Chapter 4. The results from the analysis in this chapter are taken into account in themodeling of the Dogbone-type resonator. First in Chapter 3, these are used to find a reproducibleDogbone-type resonator design, i.e. a design stable eigenfrequency with respect to production-induced variations in dimensions. Secondly in Chapter 4, the results are used to find Dogbone-typeresonator designs with relatively robust and temperature-invariant eigenfrequency. Here, robustis with respect to the accuracy (5%) at which the oxidation process could be controlled.

Chapter 3

Frequency variability of Dogbone-

type resonators

In Chapter 2, it is shown for the clamped-free resonator that the eigenfrequency is extremelysensitive for production-induced variations in dimensions. Moreover, when compensating for thetemperature dependence in the stiffness of the resonator design using a uniform oxide layer thick-ness and an inaccuracy of 5% on that amount of oxide, an even larger frequency variation is founddue to the altered effective stiffness. These observations are also found in the analysis of specificDogbone-type resonators [4]. A clear visualization of these variations is the overall frequency bud-get. When produced in mass-production, one should be able to pick any resonator that is producedand operate it at any temperature in the specified range. For that reason, all origins of frequencydeviations can be summed to render the frequency budget applicable to that resonator’s designand oxidation treatment. In [4], the overall frequency budget is visualized as a function of theoxide thickness for Dogbone-type resonators currently under study, here depicted in Figure 3.1.Several sources can be distinguished that provide a spread in frequency. One of them is the spreadafter fabrication in silicon, which on the base of experimental results is set at a constant 1000ppm error (initial spread). It can also be seen that the thermal drift diminishes to the desiredvalue of zero, whereas the frequency spread due to the altered effective stiffness by oxidation andthe inaccuracy of 5% on the amount of oxide (absolute frequency range) increases with thickness.

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Absolute frequency rangeOther effects, e.g. 2nd order effects

Thermal drift over 100 KInitial spread


Fre

quen

cybudget

[ppm

]

Figure 3.1: Frequency budget for a specific Dogbone-type resonator.

11

12 CHAPTER 3. FREQUENCY VARIABILITY OF DOGBONE-TYPE RESONATORS

W / 2

x

Figure 3.2: The Dogbone-type resonator.

Thus, in order to find a Dogbone-type resonator design with relatively robust and temperatureindependent eigenfrequency, we have to compensate for the initial spread and absolute frequencyrange in an 5% range around the thickness for which the temperature coefficient is zero. Thesetwo problems are studied in this chapter and Chapter 4, respectively. Hereto, the Dogbone-typeresonator as depicted in Figure 3.2 is used [1]. We distinguish a spring part with length L andwidth w and a resonator head with length a and width b. In the modeling resonator symmetryand only vibration in an extensional mode are taken into account.

3.1 Modeling mismatch in dimensions

As mentioned in the introduction of this chapter, the initial spread in the eigenfrequency due toproduction-induced variations in dimensions of Dogbone-type resonators currently under study isequal to 1000 ppm. In this chapter it is tried to compensate for this frequency deviation. Analyticalexpressions, which define a geometry that is relatively insensitive to production-induced variationsin dimensions are derived. The eigenfrequency is expressed as a function of the basic dimensionsof the Dogbone-type resonator as well as a "growth"-parameter δ. Initially, the model is based onthe assumption that growth and shrinkage of the intended dimensions will occur during productionand that this effect is equal in perpendicular directions. As a consequence, the design could beconsidered to be consisting of two coupled uniform beam structures with different cross-sectionalareas, see Figure 3.3. Here a constant number of springs ns = 2 is assumed, which in y-directionresults in a dimension error in the spring part of the Dogbone-type resonator that is twice as largeas the corresponding dimension error in the resonator head.

Design dimensions

Uncertain dimensions

δδ

L0

w0/2

a0

b0

(a) Uncertain dimensions

Design dimensions

Uncertain dimensionsCoupling

δ

2δ

1 2


Figure 3.3: Dogbone-type resonator model with production-induced variations in dimensions.

3.2. IMPROVING REPRODUCIBILITY OF DOGBONE-TYPE RESONATORS 13

Referring to the clamped-free resonator model from Chapter 2, a general description in theform of Eq. (2.1) holds for each of the uniform beam structures in Figure 3.3(b), with ui(x, t),i = {1, 2} defined similar to Eq. (2.2). As a result the boundary value problem which is used tosolve for the reproducibility problem of the Dogbone-type resonator is described by

∂2u1(x)∂x2 + β2

1u1(x) = 0, 0 ≤ x ≤ Ls,

∂2u2(x)∂x2 + β2

2u2(x) = 0, Ls ≤ x ≤ Ls + as,

(1) u1(0) = 0,

(2) u1(Ls) = u2(Ls),

(3) As,1∂u1(x)

∂x|x=Ls

= As,2∂u2(x)

∂x|x=Ls

,

(4) ∂u2(x)∂x

|x=Ls+as= 0,

(5) u2(Ls + as) = 1.

(3.1)

Boundary conditions (1) and (4) represent the clamped-free ends of the resonator, respectively.The coupling of the two beam structures is described by boundary conditions (2) and (3). In orderto solve for the four unknown dimensions and the eigenfrequency, boundary condition (5) is taken

into account, which only scales the eigen-modes. Hence, β1 = β2 = β =√

ρsi

Esiω are the spatial

wavelengths, which are equal since production before oxidation is considered, i.e. single material.The Dogbone-type resonator dimensions are defined as in Figure 3.3,

Ls = L0 − δ,as = a0 + 2δ,As,1 = wsh = (w0 + 4δ)h,As,2 = bsh = (b0 + 2δ)h,

(3.2)

where h is the thickness of the resonator, which eventually will not show up in the final expressions.As mentioned before, at first instance it is assumed that production-induced variations are equalin all dimensions, i.e. δ. Later on, a distinction between variations in x-direction and y-directionis made. In all situations the thickness of the resonator remains unaffected.

3.2 Improving reproducibility of Dogbone-type resonators

Just as in Section 2.1, boundary value problem Eq. (3.1) should be solved to find an expressionfor the form factor β, hence the eigenfrequency ω. The general solution of the coupled BVP (3.1)is given by

u1(x) = c1cos(βx) + c2sin(βx),u2(x) = c3cos(βx) + c4sin(βx).

(3.3)

With boundary conditions (1), (2), (3) and (5), the eigen-modes can be solved for the unknownconstants ci, i = {1, 2, 3, 4}. Substituting these results in the last boundary condition (4), thiswill result in

ys = As,1cos(βLs)cos(βas) − As,2sin(βLs)sin(βas) = 0, (3.4)

which should be solved for β as function of the resonator dimensions and the dimension error δ.Note, that ys = 0 is the necessary and sufficient condition to ensure that boundary condition (4)is satisfied. Unfortunately, a closed form solution when solving for β does not exist. As a conse-quence, the dependence of the dimension error δ on the spatial wavelength β, hence the eigenfre-quency ω, is unknown and additional steps in the solution approach must be taken.


Actually, the problem to solve in order to ensure high reproducibility is to find Dogbone-typeresonator dimensions such that β(L0, a0, w0, b0, δ) satisfies the conditions

β(L0, a0, w0, b0, 0) =√

ρsi

Esiω,

∂β(L0,a0,w0,b0,δ)∂δ

< ǫ, for δ ∈ [−100, 100] nm,

(3.5)

with ǫ as small as possible. From experiments it is found that dimension errors of ±50 nm couldbe expected. Therefore, in Eq. (3.5) the dimension error δ is chosen in a reasonable range. Usingys(β, L0, a0, w0, b0, δ) as defined in Eq. (3.4), the conditions from Eq. (3.5) can be translated in thesolvable problem

ys(β, L0, a0, w0, b0, δ)|δ=0 = 0, (3.6)

∂ys(β, L0, a0, w0, b0, δ)

∂δ|δ=0 = 0, (3.7)

∂2ys(β, L0, a0, w0, b0, δ)

∂δ2|δ=0 = 0. (3.8)

With this new solution approach to improve reproducibility of Dogbone-type resonators we searchfor a combination of resonator dimensions, such that ys(β, L0, a0, w0, b0, δ = 0) is a saddle pointof the function ys(β, L0, a0, w0, b0, δ), see Figure 3.4. If such a geometry exists, a robust eigen-frequency of the Dogbone-type resonator model with respect to production-induced variations indimensions is guaranteed, as illustrated with Figure 3.10. The theoretical proof of the appliedproblem translation is an aspect for further research. Here, it is assumed that the intended eigen-frequency ω hence β is known. Therefore, we are left with three relations Eq. (3.6) - Eq. (3.8),solvable for three unknown dimensions as a function of the fourth dimension.

−100 −80 −60 −40 −20 0 20 40 60 80 100

0

−100 −80 −60 −40 −20 0 20 40 60 80 100

0

−100 −80 −60 −40 −20 0 20 40 60 80 100

0

y s∂

ys

∂δ

∂2y

s

∂δ2


Figure 3.4: Visualization of solution approach Eq. (3.6)-Eq. (3.8).

3.2. IMPROVING REPRODUCIBILITY OF DOGBONE-TYPE RESONATORS 15

When solving Eq. (3.6) - Eq. (3.8) for the unknown resonator dimensions, the following steps aretaken:

1. Solve Eq. (3.8) for spring length L0. This will give one possible solution.

2. Substitute the solution of the spring length L0 in Eq. (3.7) and solve for the resonator headlength a0. This will give two possible solutions, which only differ from sign. So only thepositive solution of a0 has a physical meaning.

3. Substitute the solution of the spring length L0 and the physically possible solution of theresonator head length a0 in Eq. (3.6) and solve for the spring width w0. This will give five

possible solutions. Note that these are not all physically possible, as illustrated in Section 3.3.

4. Choose one of the physically possible solutions of the spring width w0 and substitute inbackward order to find the resonator dimensions only as a function of the last dimension b0

and β.

Although we are left with five possible solutions describing the unknown dimensions L0, a0 andw0 as function of the fourth dimension b0 and the known intended eigenfrequency ω, hence β,there is only one unique realistic solution for a robust Dogbone-type resonator design (for chosenb0 and β). This will be illustrated by the case study to be presented in Section 3.3. In caseof uniform dimension error δ, a Dogbone-type resonator geometry, which is highly frequencyinsensitive with respect to production induced variations in dimensions, is described by

w0 = 12b0,

a0 = atan(

b0β4

)

1β,

L0 = atan(

2b0β

)

1β.

(3.9)

In other words, a geometry described by Eq. (3.9) results in an eigenfrequency, which is highlyreproducible. Note that relations (3.9) only hold for ns = 2 springs and a resonator head consistingof nh = 1 convex part. In the following paragraphs these parameters ns and nh are assumed tobe variable. Moreover, non-uniform dimension errors are taken into account. Although this givesnice results, in the remainder of this project ns = 2, nh = 1 and a uniform dimension error δ willbe considered.

Variable number of resonator springs and resonator head parts

Referring to the robustness problem of composite resonators (see Section 2.2 and later in Chap-ter 4) and the analysis of the Rayleigh approach to determine eigenfrequencies of different beamstructures (see Section 2.3 and later in Section 4.1), it could be an advantage to have multiplespring parts and resonator head parts. This variability gives an extra degree of freedom to controllocal stiffness and mass of the Dogbone-type resonator after oxidation, i.e. to control the eigen-frequency. Note that there will be more places on the resonator where oxidation of the siliconcore is possible, as depicted in Figure 3.5. Here we used edged trenches of equal size. However,resonator designs with a total area that could oxidize equal to the corresponding area for the casewith edged trenches could also be described by the to be presented relations. One could think ofa resonator design with edged holes along the surface area of the resonator.

During the study of this variability it is found that the solution approach presented in this sec-tion, only gives analytically exact solutions if the number of springs ns and the multiple resonatorhead parts nh are related as

ns = 2 · nh. (3.10)


ns = 2

nh = 1

(a) Standard Dogbone-type resonator

ns = 2

nh = 3

(b) Adapted Dogbone-type resonator

Figure 3.5: Variable number of resonator springs and resonator head parts.

With condition (3.10) and a uniform dimension error δ considered, a reproducible Dogbone-typeresonator design is described by

w0 = 12b0,

a0 = atan(

b0β2ns

)

1β,

L0 = atan(

ns

b0β

)

1β.

(3.11)

From Eq.(3.11) it can be seen that while increasing the number of springs ns, the spring length L0

also increases, whereas the resonator head length a0 decreases. In the case study presented inSection 3.3, it is found that the resonator head length a0 is the smallest resonator dimension.Therefore, the resonator head length a0 is the restrictive dimension in the unique realistic (physi-cally possible) solution, i.e. there exists a upper bound for the number of springs ns, which dependson the resonator head width b0 and the intended eigenfrequency ω, hence β.

Non-uniform dimension errors

From experiments in earlier studies, it is found that dimension errors after production in siliconare not uniform for all dimensions of the Dogbone-type resonator. There is a small but significantdistinction between dimension errors in x-direction and those in y-direction. Therefore, δx and δy

are introduced, whereδy = kδx, with 1/2 < k < 3/2. (3.12)

When this distinction and the variable number of springs ns from Eq.(3.10) are taken into account,a Dogbone-type resonator design with robust eigenfrequency with respect to production-inducedvariations in dimensions is described by

w0 = 12b0,

a0 = atan(

b0β2nsk

)

1β,

L0 = atan(

nskb0β

)

1β.

(3.13)

Again, a compact solution for the reproducibility problem of the Dogbone-type resonator is found.However, in the remainder of this research, for simplicity the original Dogbone-type resonator withns = 2, nh = 1 and uniform (k = 1) dimension error δ will be considered.

3.3. AN ILLUSTRATIVE EXAMPLE 17

3.3 An illustrative example

In this section, an illustrative example with respect to the reproducibility problem of the eigen-frequency of a specific Dogbone-type resonator is presented. It is desired to find a geometry of aDogbone-type resonator, that has a first extensional eigenfrequency of f = 48 MHz, which shouldbe highly invariant for production-induced variations in the four dimensions L0, a0, w0 and b0.As mentioned before, when solving Eq. (3.6) - Eq. (3.8) for these unknown dimensions, more thanone solution will be found. This is depicted in Figure 3.6 - Figure 3.8.For Eq. (3.8), just one solution for L0 exists, which after backward substitution of the other di-mensions results in L0(b0) depicted in Figure 3.6. For a0 on the other hand, two solutions followfrom Eq. (3.7) with again the other dimensions substituted. Since only solution 1 leads to positivedimensions a0, this is the unique physically possible solution, see Figure 3.7. Finally from Eq.(3.6)and the solutions of both the dimensions L0 and a0 substituted, five solutions for the resonatorspring width w0 are found, see Figure 3.8. Two of them are physically impossible because of theirnegative values. Solutions 1, 3 and 5 are all physically possible solutions. However, in this researchit is desired to have a resonator design known as the Dogbone-type resonator, because it has thelargest electrode area for actuation. From Figure 3.8 and Figure 3.9 it can be seen that only so-lution 5 satisfies the Dogbone-type resonator design. Solution 1 and 3 result in resonator designswith a resonator spring width w0 larger than the resonator head width b0. Therefore, solution 5gives the unique robust geometry description of the Dogbone-type resonator, as a function of theresonator head width b0 and the intended eigenfrequency ω, hence β.

0 5 10 15 20 25 30 35 40 45 50

20

25

30

35

40

Resonator head width b0 [µm]

Res

onato

rsp

ring

length

L0

[µm

]

Figure 3.6: Solution for resonator spring length L0 versus resonator head width b0, according toEq. (3.8) ensuring a zero second derivative.


0 5 10 15 20 25 30 35 40 45 50−20

−15

−10

−5

0

5

10

15

20

1

2

Resonator head width b0 [nm]

Res

onato

rhea

dle

ngth

a0

[nm

]

Desired solution (Solution 1)

Solution 2

Figure 3.7: Solutions for resonator head length a0 versus resonator head width b0, according toEq. (3.7) ensuring a zero first derivative and considering the solution of the resonatorspring length L0 from Figure 3.6.

0 5 10 15 20 25 30 35 40 45 50−50

−40

−30

−20

−10

0

10

20

30

40

50

5

4

3

1

2

A

B

C


Res

onato

rsp

ring

wid

thw

0[µ

m]

Solution 1

Solution 2

Solution 3

Solution 4Desired solution (sol 5)

Boundary w0 = b0

Figure 3.8: Solutions for resonator spring width w0 versus resonator head width b0, so that theintended eigenfrequency is met and both first and second derivative are zero, i.e.solutions of L0 and a0 substituted.


A CB

Figure 3.9: Physically possible solutions for resonator designs with improved reproducibility. Thethree solutions A, B and C are indicated in Figure 3.8. All designs have a similarresonator head, whereas the resonator spring width is different. Only design C satisfiesthe Dogbone-type resonator design. Therefore solution 5 is the unique desired solution.

Now, the frequency error in ppm versus uniform dimension error δ for the proposed illustrativeexample with b0 = 20 µm will be analyzed. In Table 3.1 the resulting robust Dogbone-type dimen-sions are presented, as described by Eq. (3.9). Using the exact dimensions found for this specificdesign, the frequency spread as a function of the dimension error δ is given in Figure 3.10. Indeed,the frequency spread due to this uncertainty is reduced significantly. Where at the beginning ofthis chapter, a frequency spread of 1000 ppm was assumed, now a frequency spread smaller than1 ppm is found.

The necessary and sufficient condition ys = 0, as defined in Eq. (3.4), to ensure that thefinal boundary condition (4) from BVP (3.1) is satisfied, is depicted in Figure 3.11 as functionof the frequency f . Indeed, the first extensional eigenfrequency of our Dogbone-type resonatordesign equals 48 MHz. A visualization of the robust Dogbone-type resonator design is given inFigure 3.12.

Dimension accuracy

With the current mask manufacturing, resonator designs can be produced with an accuracy of10 nm. Note, that masks with a even finer grid exist, but that these are more expensive. Therounding of the exact resonator dimensions as defined by Eq. (3.9), will affect the frequencyerror due to production-induced variations in dimensions. In Figure 3.13 this frequency spread isdepicted and found to be of order 10 ppm, i.e. ten times worse than for exact dimensions. Still,the result is significantly better than initially assumed. Random and uncorrelated deviations of δin all directions are not to be expected. Rounding at 10 nm and allowing a slight deviation δx 6= δy

is the most realistic check in our approach.

Table 3.1: Robust Dogbone-type resonator dimensions with eigenfrequency f = 48 MHz.

Dogbone-type dimension value [µm]

b0 20.0000w0 10.0000a0 4.9342L0 29.5455


−100 −80 −60 −40 −20 0 20 40 60 80 100

−0.15

−0.1

−0.05

0

0.05

0.1

0.15


Fre

quen

cyer

ror

[ppm

]

Figure 3.10: Frequency error versus uniform dimension error δ, for f = 48 MHz, b0 = 20 µm andexact dimensions as defined by Eq. (3.9).

0 20 40 60 80 100 120 140

−2

−1.5

−1

−0.5

0

0.5

1

1.5

x 10−11

Frequency f [MHz]

y s[m

]

Figure 3.11: The necessary and sufficient condition ys, as defined in Eq. (3.4), to ensure that thefinal boundary condition (4) from BVP (3.1) is satisfied. Here presented as functionof the frequency f . The eigenfrequency is indicated with a ∗.


0

b02

−b02

w0+g2

−w0−g2

L0 L0 + a0

Figure 3.12: Visualization of the robust Dogbone-type resonator design with trench g = 0.8 µm,b0 = 20 µm and eigenfrequency f = 48 MHz.

−100 −80 −60 −40 −20 0 20 40 60 80 100

−4

−2

0

2

4

6


Fre

quen

cyer

ror

[ppm

]

Figure 3.13: Frequency error versus uniform dimension error δ, for f = 48 MHz, b0 = 20 µm androunded dimensions. Rounding is because of the current mask manufacturing withan accuracy of 10 nm.


3.4 Discussion

In this chapter the reproducibility of the first extensional eigenfrequency of a Dogbone-type res-onator with respect to uncertainties in geometrical dimensions is analyzed. Using higher orderderivatives, very compact relations are found that define the geometry of a Dogbone-type res-onator design with a highly reproducible eigenfrequency. The spread in eigenfrequency for thisoptimal design (with exact dimensions) is of order 1 ppm and is a significant improvement as com-pared to other designs. With the current mask manufacturing resonator designs can be producedwith an accuracy of 10 nm. With these rounded dimensions a frequency error of order 10 ppm isfound. Referring to the frequency budget presented in Figure 3.1, the initial spread is significantlyimproved, see Figure 3.14. However, the frequency spread due to the altered effective stiffnessby oxidation and the inaccuracy of 5% on the amount of oxide (absolute frequency range) is stillsignificantly large at the oxide thickness for which the temperature coefficient is zero. Therefore,in the next chapter the geometrically optimized Dogbone-type resonator design is used to analyzethe temperature aspects in order to reduce this absolute frequency spread.

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Absolute frequency rangeOther effects, e.g. 2nd order effectsThermal drift over 100 KInitial spread


Fre

quen

cybudget

[ppm

]

Figure 3.14: Frequency budget for a specific Dogbone-type resonator with improved reproducibil-ity (initial spread).

Chapter 4

Temperature aspects of Dogbone-

type resonators

In order to find a Dogbone-type resonator design with relatively robust and temperature inde-pendent eigenfrequency, we have to compensate for the "initial spread" and "absolute frequencyrange" in an 5% range around the oxide thickness for which the temperature coefficient is zero, asillustrated with the frequency budget depicted in Figure 3.1. In the previous chapter a Dogbone-type resonator design with robust eigenfrequency with respect to production-induced variations indimensions (initial spread) is presented. The initial frequency spread is reduced by a factor of 100,as depicted in Figure 3.14. However, the frequency spread due to the altered effective stiffnessby oxidation and the inaccuracy of 5% on the oxide thickness (absolute frequency spread) is stillsignificantly large around the intended oxide thickness (for which TC = 0).In Section 2.2.2, temperature aspects of the clamped-free resonator were studied. For differentsymmetric clamped-free resonator designs, almost the same relation between the linear tempera-ture coefficient and the uniform oxide thickness τ was found. On the other hand, the trends ineigenfrequency as function of this oxide thickness were different and gave unrealistic results, i.e. theoxide thickness was larger than the material available. Moreover, the optimal oxide thickness τf

for frequency invariance with respect to 5% oxide thickness uncertainty was significantly differentfrom optimal thickness τTC, which compensates for temperature-dependence in stiffness. As men-tioned above and illustrated with Figure 3.1, these observations are also found for Dogbone-typeresonators [4]. To solve for this problem, in this chapter again the Rayleigh approach from Sec-tion 2.3 will be considered. It will be tried, using the results from Chapter 3, to find a compositeDogbone-type resonator design which is frequency-invariant with respect to temperature and withrespect to 5% uncertainty in oxide thickness. As a result, the final Dogbone-type resonator designwill have a first extensional eigenfrequency that, in a range around the intended oxide thickness,is relatively temperature independent and robust with respect to variations in dimensions and theinaccuracy in the oxidation process.

4.1 Extended analysis of the Rayleigh approach

In Section 2.3, the Rayleigh quotient was introduced to illustrate a different approach to deter-mine eigenfrequencies and corresponding eigen-modes of different beam structures. The Rayleighapproach uses the quotient of the potential and kinetic energy, which are related to the derivativeof the mode-shape and the mode-shape, respectively. Moreover, the potential energy dependsalso on the stiffness distribution of the resonator, whereas the kinetic energy is influenced by theresonator’s mass distribution. As mentioned before, the addition of silicon-dioxide to a siliconresonator may result in a temperature independent design, but also will have an effect on thedistributed stiffness and mass, hence the eigenfrequency of the composite resonator. Togetherwith the accuracy at which the oxidation process can be controlled this results in a significantlylarge frequency spread around the optimal thickness τTC which compensates for the temperaturedependency in the stiffness.

23

24 CHAPTER 4. TEMPERATURE ASPECTS OF DOGBONE-TYPE RESONATORS

As mentioned before, to solve for this problem the Rayleigh approach will be considered. Forsimplicity, the first eigen-mode of the symmetric clamped-free resonator from Chapter 2 will bestudied. With Eq. (2.4) and the scaling factor c2 = 1 µm this first eigen-mode is described by

u1(x) = 10−6sin(πx

2L). (4.1)

Recalling the Rayleigh quotient from Eq.(2.12) and the effective material properties from Eq.(2.7),the first extensional angular eigenfrequency of the clamped-free resonator is defined as

ω21 =

∫ L

0(EA)eff(x)

(

du1(x)dx

)2

dx∫ L

0 (ρA)eff(x) u1(x)2 dx, (4.2)

with(EA)eff(x) = (w∗h∗)Esi + (2τw∗ + 2τh∗ + 4τ2)Eox,

(ρA)eff(x) = (w∗h∗)ρsi + (2τw∗ + 2τh∗ + 4τ2)ρox.(4.3)

Here, the resonator dimensions depend on the oxide thickness τ , as defined in Appendix B.1.The different oxide thicknesses are depicted in Figure 4.1. Initially, we are interested in theresonator before oxidation is considered, i.e. τ = 0. In Section 2.3 it was already shown that thescaled potential and kinetic energy, related to the derivative of the mode-shape and the mode-shape respectively, are equal in magnitude if all material properties are chosen equal to one, i.e.the scaling factors of the mode-shape and the derivative of the mode-shape both equal one, seeTable 4.1. This is also depicted in Figure 4.2(a), where the surface area below both functionsis the same. In a second case, a uniform oxide thickness τ as presented in Figure 4.1(a) isconsidered. Oxidation of the silicon resonator has an effect on the distributed stiffness (potentialenergy) and mass (kinetic energy). In Figure 4.2(b) it is shown that due to the oxidation thekinetic energy increases more than the potential energy, which according to Eq. (4.2) results in afrequency drop. This observation is in agreement with the frequency change depicted in Figure 2.6from Section 2.2.2. To compensate for this frequency drop a "split"-factor s to distinguish twodifferent oxide thicknesses along the length of the resonator is considered. This stepwise change inoxide thickness is depicted in Figure 4.1(b). With two different oxide thicknesses, where τ1 > τ2,the effective material properties hence the potential and kinetic energy change as presented inFigure 4.2(c). Together with Figure 4.2(d) we see that due to this distinction in oxide thicknesswe gain more in potential energy than we do for kinetic energy. As a result the frequency dropfrom Figure 4.2(b) is reduced. It is expected that by optimization of the split-factor s and bothoxide thickness τ1 and τ2, we are able to compensate for the frequency drop due to the alteredeffective stiffness by oxidation and the inaccuracy of 5% on that amount of oxide. The result willbe a Dogbone-type resonator design with an eigenfrequency insensitive for geometrical uncertainty(in dimensions and oxide thickness) and temperature changes.

Before oxidationAfter oxidation

L∗

ντ

τ

(a) Uniform oxide thickness τ

τ1τ2

sL∗ L∗

(b) Two different oxide thicknesses τ1 and τ2

Figure 4.1: Different oxide thickness along the length of the resonator to compensate for thefrequency drop due to oxidation.

4.1. EXTENDED ANALYSIS OF THE RAYLEIGH APPROACH 25

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2 PotentialKinetic

Position x [µm]

Ener

gy

dis

trib

ution

[a.u

.]

(

du1(x)dx

)2u2

1(x)

(a)

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1


Position x [µm]

Ener

gy

dis

trib

ution

[a.u

.]

(b)

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1


Position x [µm]

Ener

gy

dis

trib

ution

[a.u

.]

(c)

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1


Position x [µm]

Ener

gy

dis

trib

ution

[a.u

.]

(d)

Figure 4.2: Evaluation of the Rayleigh quotient with different oxide thicknesses along the lengthof the resonator. The potential en kinetic energy of each case study is defined asin Table 4.1. As a result of the oxidation process in (b) a frequency drop is found.However, in (c, d) it is shown that due to a stepwise change in oxide thickness we areable to compensate for this frequency spread, as we gain more in potential energy thanin kinetic energy.

Table 4.1: Potential and kinetic energies of the different case studies from Figure 4.2. A visual-ization of the different oxide thicknesses is depicted in Figure 4.1.

Case study Energy Definition Oxide thickness

a potential∫ L

0

(

du1(x)dx

)2

dx τ(x) = 0 for 0 ≤ x ≤ L

kinetic∫ L

0 u1(x)2 dx τ(x) = 0 for 0 ≤ x ≤ L

b potential∫ L

0(EA)eff(x)

(

du1(x)dx

)2

dx τ(x) = τ for 0 ≤ x ≤ L∗

kinetic∫ L

0 (ρA)eff(x) u1(x)2 dx τ(x) = τ for 0 ≤ x ≤ L∗

c, d potential∫ L

0 (EA)eff(x)(

du1(x)dx

)2

dx τ(x) = τ1 for 0 ≤ x ≤ sL∗

τ(x) = τ2 for sL∗ ≤ x ≤ L∗

kinetic∫ L

0 (ρA)eff(x) u1(x)2 dx τ(x) = τ1 for 0 ≤ x ≤ sL∗

τ(x) = τ2 for sL∗ ≤ x ≤ L∗


4.2 Modeling the application of silicon-dioxide

In Section 3.1 the Dogbone-type resonator was modeled for the production induced mismatch indimensions. Only the vibration in the lowest extensional mode was studied and the resonatordesign was considered to be consisting of two coupled uniform beam structures with differentcross-sectional areas, see Figure 3.3. In the modeling of the application of silicon-dioxide to asilicon resonator the same concept is used. Note that starting dimensions of the Dogbone-typeresonator are defined by the geometrically optimized design from Chapter 3. As in the analy-sis of the composite symmetric clamped-free resonator from Section 2.2, again effective materialproperties are taken into account. Moreover, the oxide layers at the boundaries L∗ and L∗ + a∗

are modeled as lumped masses M1 and M2, respectively. Together with the solution approachfrom the previous section, where different oxide thicknesses along the length of the resonator com-pensate for the frequency drop due to the oxidation, the Dogbone-type resonator is modeled aspresented in Figure 4.3. Note that the dimensionless split-factor s is defined as a part of the totallength of the resonator. It is chosen that the stepwise change in oxide thickness is situated on theresonator springs, i.e. s ∈ [0, L∗

L∗+a∗]. Appendix A.3 gives a complete overview of the Dogbone-type

dimensions and in Appendix B.2 the effective material properties are discussed.

An import observation for the composite Dogbone-type resonator with two different oxidethicknesses along the length of the resonator must be made here. From the assumption thatthe resonator width w∗ is much larger than the silicon thickness h∗ of our resonator, it can beconcluded that the effect of the silicon-dioxide contributes most in the resonator’s thickness h∗

(z-direction). The silicon thickness along the length of the resonator can be described by

h∗

1 = h0 − 2τ1ν, for 0 ≤ x ≤ s(L∗ + a∗),

h∗

2 = h0 − 2τ2ν, for s(L∗ + a∗) ≤ x ≤ L∗,

h∗

3 = h0 − 2τ2ν, for L∗ ≤ x ≤ L∗ + a∗,

(4.4)

with the different oxide thicknesses related to each other as τ1 = nτ2, where n ≥ 1. Thus,the silicon thickness is now very important. From other studies it followed that with an oxidegrowth rate amplification n ≈ 4, temperature dependence could be reduced while robustness withrespect to 5% uncertainty in oxide thickness is still guaranteed. However, it is not known whatamplification factors could be physically realized, while the position of the stepwise change can becontrolled very well. Therefore, in this research the factor n is slightly changed around four, whileoptimizing for the location of the stepwise change in oxide thickness.

Before oxidationAfter oxidationLumped mass

s(L∗ + a∗) L∗ L∗ + a∗

nsτ1 nsτ2

τ2


After oxidationLumped mass

s(L∗ + a∗) L∗ L∗ + a∗

(EA)eff,1

(ρA)eff,1

(EA)eff,2

(ρA)eff,2

(EA)eff,3

(ρA)eff,3M1 M2


Figure 4.3: Dogbone-type resonator model with application of silicon-dioxide and distinction inthis oxide thickness along the length of the resonator. Number of springs ns = 2.

4.2. MODELING THE APPLICATION OF SILICON-DIOXIDE 27

For each of the three uniform beam structures in Figure 4.3(b), a general description in theform of Eq. (2.1) holds, with ui(x, t), i = {1, 2, 3} defined similar to Eq. (2.2). The boundary valueproblem of the composite Dogbone-type resonator can now be described by

∂2u1(x)∂x2 + β2

1u1(x) = 0, 0 ≤ x ≤ s(L∗ + a∗),

∂2u2(x)∂x2 + β2

2u2(x) = 0, s(L∗ + a∗) ≤ x ≤ L∗,

∂2u3(x)∂x2 + β2

3u3(x) = 0, L∗ ≤ x ≤ L∗ + a∗,

(1) u1(0) = 0,

(2) u1(s(L∗ + a∗)) = u2(s(L

∗ + a∗)),

(3) u2(L∗) = u3(L

∗),

(4) (EA)eff,1∂u1(x,t)

∂x|x=s(L∗+a∗) = (EA)eff,2

∂u2(x,t)∂x

|x=s(L∗+a∗),

(5) (EA)eff,2∂u2(x,t)

∂x|x=L∗ + M1

∂2u2(x,t)∂t2

|x=L∗ = (EA)eff,3∂u3(x,t)

∂x|x=L∗ ,

(6) (EA)eff,3∂u3(x,t)

∂x|x=L∗+a∗ + M2

∂2u3(x,t)∂t2

|x=L∗+a∗ = 0,

(7) u3(L∗ + a∗) = 1,

(4.5)

with β2i =

(ρA)eff,i(EA)eff,i

ω2, i = {1, 2, 3} the spatial wavelengths, which differ slightly per section due to

varying effective material properties. Boundary conditions (1) and (6) represent the clamped-freeends of the resonator, respectively, and the coupling of the three beam structures is described byboundary conditions (2), (3), (4) and (5). Again an additional scaling of the eigen-modes (7) isused to make the boundary value problem solvable for the unknown eigenfrequency ω and the sixunknown constants cj, j = {1, 2, 3, 4, 5, 6} of the general solution

u1(x) = c1cos(β1x) + c2sin(β1x),

u2(x) = c3cos(β2x) + c4sin(β2x),

u3(x) = c5cos(β3x) + c6sin(β3x).

(4.6)

When solving for the unknowns in the general solution Eq. (4.6) it is found that, as encounteredbefore, a analytically closed form solution for the last unknown, the eigenfrequency ω, couldnot be derived. However, the problem can be solved numerically. Using this approach the lastboundary condition is approximated with a Taylor series expansion of order four, by which it ispossible to find an (approximate!) relation for the unknown eigenfrequency ω. Analysis of thisapproximate model is much faster than the numerical analysis of the original problem. Therefore,the fourth order Taylor series expansion is used to determine trends and approximated results ofthe eigenfrequency of the composite Dogbone-type resonator. With these results, the numericalgrid for the initial (exact) problem can be reduced, and (almost) exact solutions can be foundefficiently. However, in this research only the approximated results of the Taylor series expansionare presented. Nevertheless, this will give pretty accurate results.


4.3 Improving temperature dependency and robustness of

Dogbone-type resonators

The initial temperature and robustness problem of Dogbone-type resonators currently under studyis clearly visualized with the frequency budget (again) presented in Figure 4.4(a). In Chapter 3 wefound a geometrically optimized resonator design, which significantly improved robustness withrespect to production-induced variations in dimensions, as (again) depicted in Figure 4.4(b). Withthe solution strategy presented in Section 4.1 and the composite resonator model from Section 4.2 itis tried in this section to find composite Dogbone-type resonator designs, which have temperature-independent eigenfrequency that is also robust with respect to the accuracy (5%) at which theoxidation proces could be controlled. This frequency spread is indicated as the absolute frequency

range in Figure 4.4.

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000



Fre

quen

cybudget

[ppm

]

(a) Initial Dogbone-type resonator design

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000



Fre

quen

cybudget

[ppm

]

(b) Geometrically optimized Dogbone-type resonator design


Both the temperature dependency and corresponding robustness problem can also be visualizedwith Figure 4.5. Here, we analyze a specific Dogbone-type resonator with initially a uniform oxidethickness along the whole resonator. The starting dimensions before oxidation are defined by thegeometrically optimized design presented in Table 3.1 with b0 = 20 µm. The frequency change as aresult of the oxidation process is for this design indicated with a square in Figure 4.5. Note that theinitial frequency change does not equal zero due to the Taylor series approximation as explainedlater. In agreement with other results a significant frequency drop is found. To compensate forthis frequency drop a design with a stepwise change in oxide thickness is considered (frequencychange indicated by triangle and TC by circle). The two different oxide thicknesses are relatedby an oxide growth rate amplification n = 4. The optimal oxide thickness τf , which will result ina robust eigenfrequency with respect to the variation in oxide thickness, i.e. the oxide thicknesswhere ∂f

∂τis as small as possible (see Section 2.2.2), is for this resonator design found near the

optimal thickness τTC which compensates for the temperature drift. Thus by using two differentoxide thicknesses along the length of the resonator we manage to shift the optimal thickness τf

from a physically impossible (square, not indicated because out of range) to a physically possibleoxide thickness (triangle). This physically possible solution is also indicated in Figure 4.5 andequals τf = 370 nm. For the compensation of the temperature drift (circle) an optimal thicknessτTC = 310 nm is found. So the different optimal oxide thicknesses do still not result in a uniquedesign with temperature-independent and robust eigenfrequency.

Therefore it is desired to find an oxide thickness τ1 = τTC = τf , which does result in such aunique resonator design. This can be realized by optimizing for the location of the stepwise changein oxide thickness, i.e. for split-factor s. The optimization approach is presented in Figure 4.6,where both optimal oxide thicknesses τTC and τf are shown as function of the split-factor s. Fromthis figure it can be concluded that for this specific resonator indeed a composite resonator de-

4.3. IMPROVING TEMPERATURE DEPENDENCY AND ROBUSTNESS 29

sign with relatively temperature invariant and robust eigenfrequency exists (vertical line indicatedby 2). The optimal design is described by s = 0.4230 and τTC = τf = 326 nm. In addition, theinitial design (vertical line indicated by 1) from Figure 4.5 with s = 0.4700, τTC = 310 nm andτf = 370 nm is shown. For the optimized design (2) the temperature drift and frequency changedue to oxidation is presented in Figure 4.7 and Figure 4.8. The latter shows the frequency changearound the optimal oxide thickness τ1 = 326 nm for a 5% inaccuracy in the oxidation process.

0 100 200 300 400 500−35

−30

−25

−20

−15

−10

−5

0

5

−90k

−75k

−60k

−45k

−30k

−15k

0

15k

τTC

τf

Oxide thickness τ1

Lin

ear

TC

[ppm

/K

]

Fre

quen

cych

ange

[ppm

]

Figure 4.5: Linear TC and frequency change as function of the oxide thickness for a specificDogbone-type resonator (b0 = 20 µm and s = 0.4700). The frequency change (withreference frequency f = 48 MHz) for a resonator with uniform oxide thickness (n = 1)is represented with a �, whereas for the design with two different oxide thicknesses(n = 4) a △ is used. Both frequency changes correspond to the right vertical axis. Forthe latter design (n = 4) the linear TC is shown, indicated with a ◦ and correspondingto the left vertical axis. Optimal oxide thicknesses τTC = 310 nm and τf = 370 nm donot result in a unique design with temperature-independent and robust eigenfrequency.

0.2 0.25 0.3 0.35 0.4 0.45 0.5250

300

350

400

τTC

τf

0.2 0.25 0.3 0.35 0.4 0.45 0.5

−100

0

100

200

2 1

Split position s [-]

Oxid

eth

icknes

sτ 1

[nm

]τ T

C-τ

f[n

m]

Figure 4.6: Optimizing for split-factor s to find relatively temperature-independent and robusteigenfrequency for geometrically optimized design with b0 = 20 µm and n = 4. Theinitial design from Figure 4.5 is indicated with (1), whereas the optimized designis indicated with (2). The temperature drift and frequency change of the optimaldesign (2) is depicted in Figure 4.7.


0 100 200 300 400 500−35

−30

−25

−20

−15

−10

−5

0

5

0 100 200 300 400 500

−90k

−75k

−60k

−45k

−30k

−15k

0

15k

τTC

= τf

Oxide thickness τ1 [nm]

Lin

ear

TC

[ppm

/K

]

Fre

quen

cych

ange

[ppm

]

Figure 4.7: Linear TC and frequency change as function of the oxide thickness for the optimizedDogbone-type resonator from Figure 4.5 (b0 = 20 µm and s = 0.4230). The frequencychange for a resonator with uniform oxide thickness (n = 1) is represented with a �,whereas for the design with two different oxide thicknesses (n = 4) a △ is used. Bothfrequency changes correspond to the right vertical axis. For the latter design the linearTC is shown, indicated with a ◦ and corresponding to the left vertical axis. Optimaloxide thickness τTC = τf = 326 nm results in a unique design with temperature-independent and robust eigenfrequency.

−88k

−84k

−80k

−76k

300 305 310 315 320 325 330 335 340 345 350

−2.36k

−2.34k

−2.32k

−2.30k

5%


Fre

quen

cych

ange

[ppm

]Fre

quen

cych

ange

[ppm

]

Figure 4.8: Frequency change as function of the oxide thickness from Figure 4.7 with the 5% inac-curacy bounds of the oxidation process. In the top figure the uniform oxide thicknessdesign (n = 1) is shown (�), whereas in the bottom figure the frequency change ofthe optimized Dogbone-type resonator design (b0 = 20 µm, n = 4 and s = 0.4230)with two different oxide thicknesses is shown (△). In the 5% range around the optimaloxide thickness the absolute frequency spread is reduced from 3000 ppm to 6 ppm.


The presented results (Figure 4.5 - Figure 4.8) showed that due to a stepwise change in oxidethickness along the length of the resonator and by optimization of the location of this distinction inoxide thickness, a Dogbone-type resonator design with relatively temperature invariant and robusteigenfrequency is theoretically guaranteed. Considering the accuracy (5%) at which the oxidationprocess could be controlled, we managed to reduce an absolute frequency range of approximately3000 ppm for a resonator design with uniform oxide thickness to a minor absolute frequency spreadof approximately 6 ppm for a design with two different oxide thicknesses along the length of theresonator, as depicted in Figure 4.8.

Adaptation of the initial design frequency

From the lower plot in Figure 4.8 it can also be concluded that in spite of the significant improve-ment we are still left with an absolute frequency drop of approximately 2360 ppm for optimal oxidethickness τf . This frequency change as function of the oxide thickness τ1 is also represented by theshaded area in Figure 4.9. Note that the frequency spread is related to the intended eigenfrequencyof f = 48 MHz and that we distinguish two sources responsible for this frequency spread. Firstly,the frequency change can be related to the application of silicon-dioxide to the silicon resonator.Due to the oxidation resonator dimensions and effective material properties change, resulting inan unavoidable frequency drop. For the Taylor approximation this frequency change is indicatedby eO in Figure 4.9. In addition, the frequency spread is caused by the Taylor approximationof the spatial wavelength β itself. This can also be seen in Figure 4.9, where for the Taylor ap-proximation the intended eigenfrequency of f = 48 MHz is not realized for an oxide thicknessτ1 = 0, indicated by eT . For increasing oxide thickness the error eT due to the approximationalso increase in magnitude. As a result the optimal oxide thickness τf of the original problem isdifferent from the optimal oxide thickness τf of the model based on Taylor approximation. Notethat by using Taylor series expansions of higher order this theoretical error source eT can be re-duced and in practice will not be present. To compensate for the total frequency spread and toguarantee a Dogbone-type resonator design with a relatively temperature-independent and robusteigenfrequency equal to the intended f = 48 MHz we have to adjust the initial resonator designhence the design frequency.

0 50 100 150 200 250 300 350 40047,4

47,5

47,6

47,7

47,8

47,9

48

48,1

48,2

48,3

48,4

= τf


Eig

enfr

equen

cyf

[MH

z] eO

eT

Figure 4.9: Different sources for the frequency change of the Dogbone-type resonator. The eigen-frequency is shown as function of the oxide thickness τ1. The numerical analysis of theoriginal problem is presented with a dashed line, whereas for the analysis of the Taylormodel a solid line is used. For both problems a frequency drop due to the oxidationis found. For the Taylor problem this source is indicated by eO. In addition the errordue to the Taylor approximation is indicated by eT . The shaded area represents thefrequency spread of the Taylor model.


The solution approach to find such a Dogbone-type resonator design is elucidated here forthe model based on Taylor series approximation. The presented results theoretically guarantee arelatively temperature-independent and robust eigenfrequency f = 48 MHz for the approximatedmodel. The same concept could be used to find the adapted design frequency of the originalproblem, but again this would be time-consuming.To determine the initial design frequency and corresponding resonator dimensions such that thefinal composite resonator design has a relatively temperature-independent and robust eigenfre-quency f = 48 MHz, again Figure 4.9 is considered. From this figure it can be seen that for thisproblem a frequency range of approximately ±0.5 MHz around the intended eigenfrequency hasto be taken into account. It is desired to shift the optimal oxide thickness τf for the approximateTaylor model (dot on solid line) to the intended eigenfrequency f = 48 MHz. Therefore the ap-proximate Taylor model is evaluated for an initial design frequency in the range f0 = [47.5 , 48.5]MHz, with constant resonator head width b0 = 20 µm, an optimal split-factor s = 0.4230 withcorresponding optimal oxide thickness τ1 = τTC = τf = 326 nm and an oxide growth rate amplifi-cation n = 4. This is illustrated with Figure 4.10, where the frequency change with respect to theintended eigenfrequency for optimal oxide thickness τf is depicted as function of the initial designfrequency f0. It is found that an initial design frequency f0 = 48.12 MHz results in a temperature-invariant and robust eigenfrequency of f = 48 MHz. Small variations in this frequency f0 resultin a large spread in frequency f (indicated by the large slope). However with the resonator designconcept from Chapter 3, a Dogbone-type resonator with an eigenfrequency that is invariant toproduction-induced variations in dimensions is theoretically guaranteed. The initial and adaptedDogbone-type resonator designs are both presented in Table 4.2. The corresponding frequencyspread of each design is depicted in Figure 4.11 as function of the oxide thickness τ1. Figure 4.12shows the frequency spread of the adapted design with the 5% inaccuracy bounds of the oxidationprocess. Indeed a Dogbone-type resonator design with relatively temperature-invariant and robusteigenfrequency f = 48 MHz is found.

47,5 47,6 47,7 47,8 47,9 48 48,1 48,2 48,3 48,4 48,5

−12k

−9k

−6k

−3k

0

3k

6k

9k

Design frequency f0 [MHz]Fre

quen

cych

ange

for

optim

alox

ide

thic

knes

sτ f

[ppm

]

Figure 4.10: Frequency change with respect to the intended eigenfrequency f = 48 MHz for op-timal oxide thickness τf = 326 nm as function of the initial design frequency f0.The resonator head width b0 = 20 µm, split-factor s = 0.4230 and oxide growth rateamplification n = 4 are kept constant.


Table 4.2: Adaptation of initial design frequency f0 and corresponding resonator dimensions suchthat a Dogbone-type resonator design with relatively temperature-independent androbust eigenfrequency f = 48 MHz is theoretically guaranteed. See Figure 4.11 for thecorresponding frequency spread of both designs.

Parameter Initial design Adapted design Dimension

f0 48.00 48.12 MHzb0 20.00 20.00 µmw0 10.00 10.00 µma0 4.93 4.93 µmL0 29.55 29.45 µm

s 0.4230 0.4230 −τ1 326 326 nmn 4 4 −

0 50 100 150 200 250 300 350 400

−2k

0

2k

4k

6k

8k

10k

5%


Fre

quen

cych

ange

[ppm

]

Figure 4.11: Frequency change with respect to the intended eigenfrequency f = 48 MHz for bothdesigns presented in Table 4.2. The initial design is depicted with a dashed line,whereas the adapted design is shown with a solid line. The latter design results ina relatively temperature-invariant and robust eigenfrequency f = 48 MHz aroundoptimal oxide thickness τ1 = τTC = τf = 326 nm.


300 305 310 315 320 325 330 335 340 345 3500

10

20

30

40

50

60

5%


Fre

quen

cych

ange

[ppm

]

Figure 4.12: Frequency change with respect to the intended eigenfrequency f = 48 MHz for theadapted design presented in Table 4.2. The 5% inaccuracy bounds of the oxidationprocess are shown. The design results in a relatively temperature-invariant and robusteigenfrequency f = 48 MHz around optimal oxide thickness τ1 = τTC = τf = 326 nm.

Variations in resonator head width and oxide growth rate amplification

In the previous chapter it was found that a Dogbone-type resonator design with an eigenfrequencythat is relatively insensitive for production-induced variations in dimensions is completely deter-mined by resonator head width b0 and the spatial wavelength β, hence the eigenfrequency ω, seeEq. (3.9). Therefore we analyze in this paragraph the solution approach to improve the temper-ature dependency and robustness of Dogbone-type resonators for different dimensions b0. Theintended eigenfrequency f = 48 MHz is assumed to be constant. In addition, it is not known whatoxide growth rate amplification n could be physically realized. In this section it was already shownthat for a factor n = 4 the temperature-dependency and robustness of the eigenfrequency couldbe improved. Here, we also analyze Dogbone-type resonator designs with different oxide growthrate amplifications n.The results of these variations in both parameters are presented in Figure 4.13 and Figure 4.14.The first figure shows the optimal split-factor s as function of the resonator head width b0, whereasthe latter shows the corresponding optimal oxide thickness τ1 = τTC = τf . The oxide growth rateamplification is varied in a range n = [2.5 , 5.0]. In Figure 4.13 we see that for larger n there isless deviation in optimal split-factor s as function of the resonator head width b0. It is expectedthat there exists an oxide growth rate amplification n were this deviation is almost zero. As aresult one single split-factor s will be the optimal solution for all dimensions b0 = [5 , 40] µm.For even larger amplifications the opposite behavior of the smaller amplifications is expected, i.e.increasing optimal split-factor s for increasing dimensions b0. In addition, it can be seen that forsmall n = {2.5 , 3.0} a unique resonator design with an eigenfrequency that is both temperature-independent and robust does not exist for all dimensions b0 = [5 , 40] µm. Looking at Figure 4.14we see that the behavior of the optimal oxide thickness τ1 = τTC = τf as function of the resonatorhead width b0 changes in almost a linear relation if the oxide growth rate amplification n becomessmaller. Taking into account a trench width of 800 nm between the two resonator springs, alower bound for the maximal realizable oxide thickness equals 400 nm. As a result there exists alower-bound for the resonator head width b0 that depends on the oxide growth rate amplificationn. This lower-bound is also depicted in Figure 4.13 by the dashed line. All solutions above thisline are realizable.


5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5


Split-

fact

or

s[-]

Figure 4.13: Optimal split-factor s for different dimensions of the resonator head width b0, differentoxide growth rate amplifications n and corresponding optimal oxide thickness τ1

depicted in Figure 4.14. The lower-bound for the resonator head width b0 (dashedline) is the result from Figure 4.14.

5 10 15 20 25 30 35 40150

200

250

300

350

400

450

500

550

600

650


Optim

alox

ide

thic

knes

sτ 1

[nm

]

Figure 4.14: Optimal oxide thickness τ1 = τTC = τf for different dimensions of the resonator headwidth b0, different oxide growth rate amplifications n and corresponding optimal splitfactor s depicted in Figure 4.13. The trench width between the two springs results ina lower bound for the maximal realizable oxide thickness τ1 = 400 nm. This results ina lower-bound for the resonator head width b0 that depends on the amplification n,indicated by the dashed line and also shown in Figure 4.13.


4.4 Discussion

In order to find a Dogbone-type resonator design with relatively robust and temperature-invarianteigenfrequency, we have to compensate for the initial spread and absolute frequency range in a 5%range around the thickness for which the temperature coefficient is zero, as illustrated with thefrequency budget depicted in Figure 4.15. In the previous chapter a Dogbone-type resonator designwith robust eigenfrequency with respect to production-induced variations in dimensions (initialspread) is presented. The initial frequency spread is reduced by a factor of 100, as depicted inFigure 4.16. However, the frequency spread due to the altered effective stiffness by oxidationand the inaccuracy of 5% on the amount of oxide (absolute frequency range) is still significantlylarge around the intended oxide thickness (for which TC = 0), in Figure 4.16 at τ1 = 300 nm.Therefore we studied in this chapter the temperature aspects of Dogbone-type resonators. Anextended analysis of the Rayleigh quotient resulted in the solution approach to use two differentoxide thicknesses along the length of the resonator. By optimization of the position of the stepwisechange in oxide thickness and an adaptation of the initial design frequency we managed to deriveDogbone-type resonator designs which all result in an intended eigenfrequency that is temperature-invariant and robust with respect to production-induced variations in dimensions and the 5%inaccuracy in the oxidation process. This is depicted in Figure 4.17 at τ1 = 330 nm. The analysisis based on a Taylor series approximation of the spatial wavelength β, hence the eigenfrequency ω,because analytically exact solutions could not be derived and numerical analysis of the originalproblem is too time-consuming.

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Absolute frequency range

Other effects, e.g. 2nd order effects

Thermal drift over 100 KInitial spread


Fre

quen

cybudget

[ppm

]


4.4. DISCUSSION 37

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000



Fre

quen

cybudget

[ppm

]

Figure 4.16: Frequency budget for a specific Dogbone-type resonator with improved reproducibil-ity (initial spread).

0 50 100 150 200 250 300 350 4000

2000

4000

6000

8000

10000

12000

14000



Fre

quen

cybudget

[ppm

]

Figure 4.17: Frequency budget for a specific Dogbone-type resonator with improved reproducibil-ity (initial spread) and where intended eigenfrequency is temperature-invariant androbust with respect to 5% inaccuracy in the oxidation process.

Chapter 5

Conclusions and recommendations

In this chapter, the conclusions of this project are stated. Furthermore, recommendations forfurther research are presented.

5.1 Conclusions

The goal of this project is twofold. Firstly, it is desired to derive analytical expressions which definethe geometry of a Dogbone-type resonator that is insensitive to production-induced variations indimensions with respect to the eigenfrequency (reproducibility). Secondly, these geometrically op-timized resonator designs are used to find resonators with a temperature-invariant eigenfrequency,which also is insensitive to 5% variation in oxide thickness.

In order to gain insight in the behavior of the eigenfrequencies and the corresponding eigen-modes of the Dogbone-type resonator (with non-uniform cross-sectional area), first a symmetricclamped-free resonator with a uniform cross-sectional area is studied. Only the essen-tially one-dimensional resonators are considered so the models describe vibrations in extensionalmodes. Using Newton’s second law of motion, the dynamic behavior of such beams subjected toaxial loading is formulated by a partial differential equation. Together with the necessary bound-ary conditions a boundary value problem is derived, which is solved to find the eigenfrequenciesand corresponding eigen-modes. As expected it is found that the amplitude of the eigen-modesis free to choose and that it does not influence the eigenfrequencies. The latter are described bya factor defined by geometry and a factor defined by the material properties. Moreover, for theclamped-free resonator the geometry-factor contains only the length of our resonator; thicknessand width are irrelevant. As a result the reproducibility of this clamped-free resonatoris very low.To analyze the temperature aspects and the effect of the oxidation on the eigenfrequency the com-posite (silicon and silicon-dioxide) symmetric clamped-free resonator is studied. Because of theopposite temperature dependence in the Young’s modulus of silicon-dioxide compared to silicon,it is expected to compensate for the temperature dependence in the stiffness of the resonator.Note that it is assumed that resonator dimensions are temperature-independent (because of muchsmaller degree). Moreover, effective material properties are taken into account, defined as aweighted average with respect to the surface area of the different materials. The thermal oxi-dation alters the Temperature Coefficient TC (ppm/K), defined as the change in eigenfrequencyper degree over the original frequency times 106, and the eigenfrequency of the resonator. Anoptimal oxide thickness τTC is found to compensate for the temperature drift, butthis also results in a non-robust eigenfrequency with respect to the inaccuracy (5%)in the oxidation process. Nevertheless it is shown that an optimal oxide thickness τf which

results in a robust eigenfrequency does exists, i.e. a design for which ∂f∂τ

is as small as possible, butthat this result is physically impossible to realize (very thick oxide) and is significantly differentfrom optimal thickness τTC. All results of the analysis of the symmetric clamped-free resonatorare used in the modeling of the Dogbone-type resonator.

39

40 CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS

The initial spread in the eigenfrequency due to production-induced variations in dimensions ofDogbone-type resonators currently under study is equal to 1000 ppm. Here we derived compactanalytical expressions which define the geometry of a reproducible Dogbone-typeresonator. Therefore we analyzed the lowest eigenfrequency which is expressed as a functionof the basic dimensions of the Dogbone-type resonator as well as a "growth"-parameter δ. Themodel is based on the assumption that growth and shrinkage of the intended dimensions will occurduring production and that this effect is equal in perpendicular directions. Moreover the Dogbone-type resonator is modeled as a coupling of two uniform beam structures with different cross-sectional areas and only vibration in an extensional mode is considered. The original boundaryvalue problem is not exactly solvable for the eigenfrequency. However, with a new solutionapproach where robustness with respect to dimension errors is obtained by equatingthe first and second derivatives of the last boundary condition with respect to growthto zero, we managed to find these analytically exact solutions. For known resonator headwidth b0 and spatial wavelength β, hence the eigenfrequency ω all other resonator dimensions followfrom these relations. The initial spread of the geometrically optimized Dogbone-type resonatordesign is reduced by a factor 100 to approximately 10 ppm for δ = ±50 nm. Note that withthis solution approach also two other physically possible and geometrically optimized resonatordesigns are found, but that these designs do not match with the Dogbone-type resonator design.The results are subsequently used to study temperature aspects and the effect of the oxidation onthe eigenfrequency of the Dogbone-type resonator.

The composite (silicon and silicon-dioxide) Dogbone-type resonator is also modeledusing effective material properties. The starting dimensions before oxidation are defined by theresonator design optimized for production induced variations in geometry. It is shown that it ispossible to compensate for the temperature drift with a uniform oxide thickness along the wholeresonator, but that robustness with respect to variations in this oxide thickness is not guaran-teed. Therefore, we used the Rayleigh approach, which determines the angular eigenfrequencyby the quotient of the scaled potential and kinetic energy, which are related to the derivative ofthe mode-shape and the mode-shape, respectively. Moreover, the potential energy depends on thedistributed stiffness of the resonator, whereas the kinetic energy is influenced by the resonator’sdistributed mass. The addition of silicon-dioxide to a silicon resonator has an effect on the dis-tributed stiffness and mass (stiffness 5 times less than mass), hence on the eigenfrequency ω ofthe composite resonator. To compensate for this significant frequency drop two differ-ent oxide thicknesses along the length of the resonator are considered. It is assumedthat the location of the stepwise change in oxide thickness is situated on the resonator springsand that the two different oxide thicknesses are related by an oxide growth rate amplificationn. The composite Dogbone-type resonator is therefore modeled as a coupling of three uniformbeam structures with different cross-sectional areas. Closed form solutions of the correspondingboundary value problem do not exist. Results are derived by analysis of the Taylor series approx-imation of the spatial wavelength β, hence the eigenfrequency. Due to the distinction in oxidethickness along the length of the resonator we managed to shift the optimal oxide thickness τf

from a physically impossible solution for uniform oxide thickness to a physically possible solutionclose to the optimal oxide thickness τTC, which compensates for the temperature dependency inthe stiffness. By optimization of the position of the stepwise change in oxide thicknesswe found unique resonator designs with oxide thickness τ1 = τTC = τf resulting in aneigenfrequency that is temperature-invariant and robust with respect to production-induced variations in dimensions and the 5% accuracy at which the oxidation processcan be controlled. To ensure that this frequency corresponds to the intended eigenfrequency,the initial design frequency and corresponding resonator dimensions are slightly adapted. Notethat all results are geometry dependent.

5.2. RECOMMENDATIONS 41

5.2 Recommendations

The most important assumption in this project is that only essentially one-dimensional resonatorswith a vibration in extensional mode are considered. Depending on the geometry of the Dogbone-type resonator, it is recommended to extend the study to a two-dimensional analysis. For a specificresonator currently under study, with dimensions presented in Appendix A.3, the one-dimensionalanalysis results for the first extensional eigenfrequency in a frequency error of several MHz (f = 19MHz from experiments and f = 22 MHz from 1-D model). This frequency error is mainly causedby the ignorance of the second in-plane movement (y-direction) between resonator spring and res-onator head. However, for Dogbone-type resonators with very width springs and a small resonatorhead the effect of this motion is less and the one-dimensional model will give more accurate results.

In order to find a Dogbone-type resonator design with an eigenfrequency that is insensitivefor production-induced variations in dimensions (reproducibility), a problem transformation isapplied in Section 3.2. Robustness with respect to dimension errors is obtained by equating thefirst and second derivatives of the last boundary condition with respect to growth to zero. This isillustrated by the numerical example presented in Section 3.3. The theoretical proof of the appliedproblem translation is an aspect for further research.

During the study of the temperature aspects and the effect of the oxidation on the eigenfre-quency, numerical analysis of the Taylor series approximation of the spatial wavelength β hencethe eigenfrequency is performed. This approximation results in an additional error source. Inorder to find better results, a numerical analysis of the exact but complicated model could beperformed. However, this process will be too time-consuming if possible at all, e.g. solutions forclamped-free beam are not available in close form.

To avoid this time-consuming analysis one could think of a more effective solution approach.Just as for the reproducibility problem their might be an other solution approach, which allowsfor analytically optimization of the temperature problem.

In order to shift the optimal oxide thickness τf to a physically possible solution that resultsin a robust eigenfrequency with respect to variations in the oxide thickness, two different oxidethickness along the length of the resonator are used. In the presented analysis, the stepwise changein oxide thickness is assumed to be rectangular. However, with the current manufacturing thesesharp distinctions could never be realized. Therefore it is very important to know how variationsin the optimal split-factor s, which defines the position of the stepwise change in oxide thicknesson the resonator springs, effect the eigenfrequencies of the Dogbone-type resonator. In furtherresearch a sensitivity analysis could be performed.

42 CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS

Bibliography

[1] J. J. M. Bontemps, “Mems resonators; research concept of market product?” Stan Acker-mans Instituut, centrum voor technologisch ontwerpen and Philips Semiconductors Nijmegen,Eindhoven, Tech. Rep. 2006/016, April 2006.

[2] R. Melamud, B. Kim, S. A. Chandorkar, M. A. Hopcroft, M. Agarwal, C. M. Jha, and T. W.Kenny, “Temperature-compensated high-stability silicon resonators,” 2007, Applied PhysicsLett. 90, 244107.

[3] D. Salt, Hy-Q Handbook of Quartz Crystal Devices. UK: Van Nostrand Reinhold Co. Ltd.,1987.

[4] C. van der Avoort, “The effect of thermal oxidation of mems resonators on temperature driftand absolute frequency,” May 2008, technical note NXP-R-TN 2008/00164.

[5] J. R. Vig and A. Ballato, Ultrasonic Instruments and Devices. Academic Press, Inc., 1999,ch. 7: Frequency Control Devices.

43

44 BIBLIOGRAPHY

Appendix A

Resonator properties

In this appendix, the material properties (Appendix A.1) and typical dimensions of the resonatorsunder consideration are presented. We distinguish the symmetric clamped-free resonator and theDogbone-type resonator in Appendix A.2 and Appendix A.3, respectively.

A.1 Material properties

The uniform material resonators which have been studied in this research, consist of silicon,indicated by the index si. The composite resonators comprise two different materials, beingsilicon and silicon-dioxide. Here the index ox is used to indicate the silicon-dioxide properties.Throughout the report the term "oxide" is often preferred above the term "silicon-dioxide". SeeTable A.1 for the material properties.

Table A.1: Material properties of the resonators under consideration

Property Symbol Value Dimension

Young’s modulus Esi 131 GPaEox 73 GPa

Density ρsi 2330 kg/m3

ρox 2200 kg/m3

Silicon consumption ν 0.44 −

Here, the constant ν describes the consumption of silicon during the oxidation process. It repre-sents the fraction of the oxide thickness τ that lies within the starting dimensions, as depicted inFigure A.1.

Before oxidation

After oxidationντ

τ

Figure A.1: Silicon consumption due to oxidation.

45

46 APPENDIX A. RESONATOR PROPERTIES

A.2 Symmetric clamped-free resonator dimensions

The typical dimensions of the symmetric clamped-free resonator, which is analyzed in Chapter 2,are presented in Table A.2. The dimensions are also indicated in Figure A.2. Note that onlyvibration in an extensional mode (x-direction) is considered.

Table A.2: Typical dimensions of the symmetric clamped-free resonator


length (x-direction) L0 20 µmwidth (y-direction) w0 10 µmthickness (z-direction) h0 1.5 µm

X

Y

Z

w0

L0

Y

Y′

(a) Dimensions in xy-direction

w0

h0

(b) Cross-sectional area Y Y′

(yz-direction)

Figure A.2: Visualization of the typical dimensions of the symmetric clamped-free resonator.

A.3 Dogbone-type resonator dimensions

The typical dimensions of a Dogbone-type resonator currently under study, which is analyzed inChapter 3 and Chapter 4, are presented in Table A.3. We distinguish a resonator spring part anda resonator head. The dimensions are also indicated in Figure A.3 and Figure A.4. Note that onlyvibration in an extensional mode (x-direction) is considered.

Table A.3: Typical dimensions of a Dogbone-type resonator currently under study


number of springs ns 2 −spring length (x-direction) L0 20 µmtotal spring width (y-direction) w0 20 µmspring thickness (z-direction) h0 1.5 µm

head length (x-direction) a0 35 µmhead width (y-direction) b0 64 µmhead thickness (z-direction) h0 1.5 µm

A.3. DOGBONE-TYPE RESONATOR DIMENSIONS 47

X

Y

Z

w0/2

L0

b0

a0

Y1

Y′

1

Y2

Y′

2

Figure A.3: Visualization of the typical dimensions of a Dogbone-type resonator currently understudy.

w0/2

h0

(a) Cross-sectional area Y1Y′

1(yz-direction)

b0

h0

(b) Cross-sectional area Y2Y′

2(yz-direction)

Figure A.4: Different cross-sectional areas along the length of the Dogbone-type resonator fromFigure A.3.

48 APPENDIX A. RESONATOR PROPERTIES

Appendix B

Determination of effective material

properties

In this appendix, the determination of the effective material properties is elucidated. Theseeffective material properties are used in the analysis of the composite resonators in Section 2.2and Chapter 4. We consider effective stiffness (EA)eff and effective mass (ρA)eff per unit of length,both determined analogously.

B.1 Composite symmetric clamped-free resonator

The composite symmetric clamped-free resonator model, which is analyzed in Section 2.2, is againdepicted in Figure B.1. The resonator dimensions before oxidation hence the dimensions withsubindex 0 were already presented in Appendix A.2. Due to the oxidation the silicon is consumedby an amount equal to ντ in each direction, while a silicon-dioxide layer with thickness τ isgrown around the silicon core. The effective stiffness and effective mass per unit of length ofthe resonator are determined with these dimensions after oxidation. Therefore we use a weightedaverage with respect to the cross-sectional surface area of the different materials hence silicon andsilicon-dioxide. From Figure B.1 we find the dimensions of the silicon core after oxidation equalto

L∗ = L0 − ντ,

w∗ = w0 − 2ντ,

h∗ = h0 − 2ντ.

(B.1)

X

Y

Z

Before oxidation

After oxidation

L0

w∗

L∗

ντ

τ

Y

Y′


w0

h0

w∗

h∗

ντ τ

(b) Cross-sectional area Y Y′

(yz-direction)

Figure B.1: Composite symmetric clamped-free resonator model from Section 2.2.

49

50 APPENDIX B. DETERMINATION OF EFFECTIVE MATERIAL PROPERTIES

As a result the surface area (per unit of length) of the silicon equals Asi = w∗h∗ for 0 ≤ x ≤ L∗.From the cross-sectional area as depicted in Figure B.1(b) also the surface area (per unit of length)of the silicon-dioxide can be derived, Aox = (2τw∗ + 2τh∗ + 4τ2) for 0 ≤ x ≤ L∗. Finally, bothresults are combined and effective stiffness and effective mass per unit of length can be defined as

(EA)eff = (w∗h∗)Esi + (2τw∗ + 2τh∗ + 4τ2)Eox,

(ρA)eff = (w∗h∗)ρsi + (2τw∗ + 2τh∗ + 4τ2)ρox,(B.2)

for 0 ≤ x ≤ L∗ and with the material properties presented in Table A.1 in Appendix A.1. Now, weare only left with the oxide layer at the boundary L∗. In Section 2.2, this oxide layer is modeledas a lumped mass M , again depicted in Figure B.2. Taking into account the resonator’s thicknessthis mass M is described by

M = V ρox = [(w∗ + 2τ)(h∗ + 2τ)(τ)]ρox. (B.3)

Before oxidationAfter oxidation L∗

ντ

τ


After oxidation

(EA)eff , (ρA)eff

Lumped massL∗

M


Figure B.2: Composite symmetric clamped-free resonator model from Section 2.2.

B.2 Composite Dogbone-type resonator

The composite Dogbone-type resonator model is studied in Chapter 4 and again depicted inFigure B.3. Both the number of springs ns and different oxide thicknesses along the length of theresonator are taken into account. The model is designed to study the vibration in an extensionalmode. As a result the resonator model is considered to be consisting of three uniform beamstructures with different cross-sectional areas. Resonator dimensions before oxidation (index 0)are presented before in Appendix A.3, see also Figure B.4. Due to the consumption of siliconand the growth of a silicon-dioxide layer around the silicon core, the composite Dogbone-typeresonator is modeled with effective stiffness (EA)eff and effective mass (ρA)eff per unit of length.As in Appendix B.1 we use a weighted average with respect to the surface area of the differentmaterials hence silicon and silicon-dioxide. To derive these effective material properties for eachsection, Figure B.5 is used, which depicts the different cross-sectional areas along the length of

B.2. COMPOSITE DOGBONE-TYPE RESONATOR 51

the resonator. The dimensions of the silicon core after oxidation are described by

L∗ = L0 + τ2ν,

a∗ = a0 − 2τ2ν,

w∗

1 = w0 − 2nsτ1ν, for 0 ≤ x ≤ s(L∗ + a∗),

h∗

1 = h0 − 2τ1ν, for 0 ≤ x ≤ s(L∗ + a∗),

w∗

2 = w0 − 2nsτ2ν, for s(L∗ + a∗) ≤ x ≤ L∗,

h∗

2 = h0 − 2τ2ν, for s(L∗ + a∗) ≤ x ≤ L∗,

b∗ = b0 − 2τ2ν, for L∗ ≤ x ≤ L∗ + a∗,

h∗

3 = h0 − 2τ2ν, for L∗ ≤ x ≤ L∗ + a∗,

(B.4)

with constant ν representing the fraction of the oxide thickness that lies within the startingdimensions and where the different oxide thickness are related by τ1 = nτ2 with n the oxidegrowth rate amplification. As a result the effective material properties of the three uniform beamstructures are defined as

EAeff,1 = (w∗

1h∗

1)Esi + (2τ1w∗

1 + 2nsτ1h∗

1 + 4nsτ21 )Eox, for 0 ≤ x ≤ s(L∗ + a∗),

(B.5)ρAeff,1 = (w∗

1h∗

1)ρsi + (2τ1w∗

1 + 2nsτ1h∗

1 + 4nsτ21 )ρox, for 0 ≤ x ≤ s(L∗ + a∗),

EAeff,2 = (w∗

2h∗

2)Esi + (2τ2w∗

2 + 2nsτ2h∗

2 + 4nsτ22 )Eox, for s(L∗ + a∗) ≤ x ≤ L∗,

(B.6)ρAeff,2 = (w∗

2h∗

2)ρsi + (2τ2w∗

2 + 2nsτ2h∗

2 + 4nsτ22 )ρox, for s(L∗ + a∗) ≤ x ≤ L∗,

EAeff,3 = (b∗h∗

3)Esi + (2τ2b∗ + 2τ2h

∗

3 + 4τ22 )Eox, for L∗ ≤ x ≤ L∗ + a∗,

(B.7)ρAeff,3 = (b∗h∗

3)ρsi + (2τ2b∗ + 2τ2h

∗

3 + 4τ22 )ρox, for L∗ ≤ x ≤ L∗ + a∗,

with the material properties presented in Table A.1 in Appendix A.1. The remaining oxide layersat the boundaries L∗ and L∗ + a∗ are modeled as lumped masses M1 and M2, respectively. FromFigure B.3 - Figure B.5 we find that these are described by

M1 = V1ρox = [(b∗ + 2τ2 − w∗

2 − 2nsτ2)(h∗

3 + 2τ2)(τ2)]ρox,

M2 = V2ρox = [(b∗ + 2τ2)(h∗

3 + 2τ2)(τ2)]ρox.(B.8)

Before oxidationAfter oxidationLumped mass

s(L∗ + a∗) L∗ L∗ + a∗

nsτ1 nsτ2

τ2


After oxidationLumped mass

s(L∗ + a∗) L∗ L∗ + a∗

EAeff,1

ρAeff,1

EAeff,2

ρAeff,2

EAeff,3

ρAeff,3M1 M2


Figure B.3: Dogbone-type resonator model with application of silicon-dioxide and distinction inthis oxide thickness along the length of the resonator. Number of springs ns = 2.

52 APPENDIX B. DETERMINATION OF EFFECTIVE MATERIAL PROPERTIES

X

Y

ZX

w∗

1 w∗

2

L∗

L0a0

a∗

b∗

s(L∗ + a∗)

Before oxidation

After oxidation

nsτ1 nsτ2

τ2

Y1

Y′

1

Y2

Y′

2

Y3

Y′

3

Figure B.4: Composite Dogbone-type resonator model from Chapter 4.

w0

h0

τ1

w∗

1

h∗

1

nsτ1ν nsτ1

(a) Cross-sectional area Y1Y′

1

w0

h0

τ2

w∗

2

h∗

2

nsτ2ν nsτ2

(b) Cross-sectional area Y2Y′

2

h0

b0

b∗

h∗

3

τ2ν

τ2

τ2

(c) Cross-sectional area Y3Y′

3

Figure B.5: Different cross-sectional areas along the length of the composite Dogbone-type res-onator model from Figure B.4.

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