Novel Micromechanical Bulk Acoustic Wave Resonator Sensing

Degree project in

Novel Micromechanical Bulk Acoustic Wave Resonator Sensing Concepts for

Advanced Atomic Force Microscopy

STEFAN WAGNER

Stockholm, Sweden 2012

XR-EE-MST 2012-003

Microsystem Technology

Second level, 30 HEC

KTH Electrical Engineering

Master thesis

Master’s programme in Nanotechnology

Carried out in the department Microsystem Technology at KTH

Novel micromechanical bulk acoustic wave

resonator sensing concepts for advanced atomic

force microscopy

STEFAN WAGNER

Supervisor

Umer Shah

Examinor

Joachim Oberhammer

External advisor

David Haviland, Nanostructure Physics, KTH

2012

Abstract

This thesis investigates novel concepts of micromechanical bulk acoustic wave sensors for advanced

atomic force microscopy (AFM), using micromachined silicon resonators, which are analyzed with

regard to their performance as compared to conventional AFM sensors. Conventional AFM systems use

a cantilever resonator for sensing the surface forces of the sample. Since a laser is used to detect the

cantilevers movement, a certain minimum area for reflecting the laser beam is required. These restrictions

in the cantilever scalability limits the resonant frequency and quality factor of the resonator and thus the

overall performance and resolution of a conventional AFM system. To overcome these limitations and

to improve atomic force microscopy, new sensor concepts are proposed. First an analysis of different

extensional mode resonator geometries is conducted to determine the dependency of the shape and

dimensions on the stiffness, resonant frequency and displacement for the use as AFM sensor. Based

on that a two resonator system is introduced, consisting of a flexural mode resonator acting as sensing

unit with design specifications oriented on an AFM cantilever and of an additional bulk mode resonator

with high resonant frequency and high quality factor detecting the movement of the first resonator. They

are coupled electrostatically, where a DC potential between the two resonators and a variation in gap

width, caused by the oscillation of the flexural mode resonator, modulate the pre-stress in the bulk mode

resonator resulting in a frequency shift, which is then detected capacitively. Finite-element simulations

are conducted to determine the sensitivity of the system for different resonator geometries, dimensions

and DC potentials between the two resonators, as well as the thermal noise and thus the detection limit of

the system. As a second design, a variation of the existing sensor is proposed using a mechanical spring

system to couple the two resonators. The benchmark criteria of these novel concepts is that it should be

possible to detect a force in the range of the Brownian motion at room temperature. Summarizing the

results it can be concluded that a ring shaped geometry is the most suitable for a single bulk acoustic

wave resonator sensor for AFM applications, as it achieves highest displacement for a given size, resonant

frequency and quality factor. These findings can also be used to improve the electrostatically coupled

resonator system and to reduce the DC potential needed between the two resonators to avoid pull-in and

at the same time still achieve good sensor sensitivity.

3

Acknowledgement

I wanted to thank everyone who helped and supported me in the 6 months of my master thesis at

the microsystem technology department of kungliga Tekniska hogskolan. Special thanks goes to my

supervisors associate professor Joachim Oberhammer and Ph.D. student Umer Shah for the opportunity to

write this thesis, also for so many fruitful discussions, the help with occurring problems and for always

having an open ear for questions. Another big thank you goes to David Haviland for the very fruitful

meetings and providing of valuable information and his expertise with atomic force microscopy. Also

thanks to everyone else in the MST department for the help, the nice lunch time and fika discussions.

Another big thanks goes to my wife Michelle for motivating me and giving me strength in some stressful

times.

4

Contents

Contents

List of Figures 7

List of Tables 10

1. Introduction 11

1.1. MEMS for sensor applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2. Motivation for the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3. Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Background: Mechanical resonators and the atomic force microscope 13

2.1. Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1. Quality factor and losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2. Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3. Mechanical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2. Flexural mode resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3. Extensional mode resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1. Longitudinal mode resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2. Lame mode resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3. Wine-glass mode resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4. Resonance excitation and sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1. Electrostatic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2. Electrostatic actuation of resonators . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3. Capacitive sensing of resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5. Applications of mechanical resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1. Sensing principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.2. Example Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6. The atomic force microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.1. Working principle and operation modes . . . . . . . . . . . . . . . . . . . . . . 28

2.6.2. Key data of a conventional AFM system . . . . . . . . . . . . . . . . . . . . . . 30

2.6.3. Improvements of the atomic force microscope utilizing bulk-mode resonators . . 31

3. General analysis of bulk acoustic resonators for AFM 33

3.1. Analysis of disk and ring shaped resonators . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1. Set-up and simulation of wine-glass mode resonators . . . . . . . . . . . . . . . 34

3.1.2. Results of wine-glass mode resonators . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.3. Conclusion of wine-glass mode resonators . . . . . . . . . . . . . . . . . . . . . 40

3.2. Analysis of longitudinal mode resonators . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1. Set-up and simulation of longitudinal resonators . . . . . . . . . . . . . . . . . 41

3.2.2. Results of longitudinal resonators . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3. Conclusion of longitudinal resonators . . . . . . . . . . . . . . . . . . . . . . . 46

3.3. Conclusion for wine-glass and longitudinal resonators . . . . . . . . . . . . . . . . . . . 46

5

Contents

4. Two electrostactically coupled resonators as AFM force sensor 47

4.1. Concept and design ideas for an electrostatically coupled force-to-frequency transducer . 47

4.1.1. AFM transducer of the electrostatically coupled sensor system . . . . . . . . . . 49

4.1.2. BAW detector of the electrostatically coupled sensor . . . . . . . . . . . . . . . 51

4.2. Calculations and Simulation of the electrostatically coupled force sensor . . . . . . . . . 52

4.2.1. AFM transducer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2. BAW detector sensitivity simulation . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.3. BAW detector geometry simulations . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.4. AFM transducer extended stable range simulation . . . . . . . . . . . . . . . . . 55

4.2.5. Mechanical noise calculations for the sensor . . . . . . . . . . . . . . . . . . . . 55

4.3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4. Conclusion of the electrostatically coupled force sensor . . . . . . . . . . . . . . . . . . 62

5. Two mechanically coupled resonators as AFM force sensor 64

5.1. Concept of the mechanically coupled force sensor . . . . . . . . . . . . . . . . . . . . . 64

5.1.1. AFM transducer of the mechanical coupled system . . . . . . . . . . . . . . . . 64

5.1.2. BAW detector of the mechanical coupled system . . . . . . . . . . . . . . . . . 65

5.1.3. Coupling spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2. Simulation for the mechanical coupled force sensor . . . . . . . . . . . . . . . . . . . . 66

5.2.1. Mechanical coupling efficiency simulation . . . . . . . . . . . . . . . . . . . . 67

5.2.2. BAW detector simulation for mechanical coupling . . . . . . . . . . . . . . . . 68

5.2.3. Simulation of the complete mechanical coupled system . . . . . . . . . . . . . . 68

5.3. Results and discussion of the mechanical coupled system . . . . . . . . . . . . . . . . . 69

5.4. Conclusion of the mechanically coupled force sensor . . . . . . . . . . . . . . . . . . . 71

6. Conclusion 72

7. Future work 73

A. Appendix 74

A.1. COMSOL simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

References 83

6

LIST OF FIGURES

List of Figures

1. a) Spring system in maximum position with restoring force; b) initial position; c) maxi-

mum position with restoring force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2. Bandwidth (∆f ) measured at -3dB points in resonance curve with low and high quality

factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. Flexural mode of a clamped-clamped, clamped-free and free-free beam up to the 4th mode. 18

4. a) Torsional mode beam; b) Flexural mode beam anchored with a torsional mode beam . 19

5. Simulation of different extensional mode resonators with decompression and compression

state: a) longitudinal mode, b) Lame mode and c) wine-glass mode. . . . . . . . . . . . 20

6. Forces and important parameters in parallel plate system with one fixed and one movable

plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7. Plot of Fel and Fs versus xg between the two plates, indicating stable and pull-in region. 23

8. Set-up of a BAW resonator with electrostatic actuation and sensing integrated in a device. 25

9. Working principle of chemical sensor for detection of NOx. . . . . . . . . . . . . . . . . 26

10. Working principle of a Biosensor to detect the concentration of antibodies in a sample.

Drawing based on [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11. Design of a IR sensor. Drawing based on [21, 64]. . . . . . . . . . . . . . . . . . . . . . 27

12. Exploded view of a resonator gyroscope and its working principle. Drawing is based on

[37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

13. Working principle and set-up of an atomic force microscope. . . . . . . . . . . . . . . . 29

14. Piezoelectric actuated longitudinal resonator as AFM transducer. Drawing based on [28]. 31

15. Capacitive actuated ring resonator as AFM transducer. Drawing based on [1]. . . . . . . 32

16. All shapes with dimensions used for the simulations. . . . . . . . . . . . . . . . . . . . 33

17. Stiffness in dependency of diameter, respectively length of the disk and square shape. . . 36

18. Stiffness of ring and frame shape with variation of width at a certain diameter (length). . 37

19. Resonance frequency of ring and frame shape with variation of width at a certain diameter

(length). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

20. Logarithmic scale displacement of a disk shaped resonator with different diameters against

the air gap width between resonator and electrode. . . . . . . . . . . . . . . . . . . . . . 38

21. Dependency between dimension and displacement for the ring shaped resonator. . . . . . 39

22. Dependency between dimension and displacement for the ring frame resonator. . . . . . 40

23. Stiffness development in dependency of length and width of a longitudinal resonator. . . 42

24. Resonance frequency distribution of a longitudinal resonator depending on its length and

width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

25. Influence of the anchor points on side displacement for a) length >width; b) length =

width; c) length <width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

26. Logarithmic plot of the dependency between air gap width and displacement with a 12 V

AC signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

27. Displacement distribution in dependency of length and width of a longitudinal resonator. 44

28. Dependency between the anchor dimensions and the frequency of the resonator. . . . . . 45

29. Set-up of the complete system with AFM transducer and BAW detector. . . . . . . . . . 47

30. Device flow chart of influencing factors of the different parts and the complete system. . 48

7

LIST OF FIGURES

31. Set-up and working principle of the AFM transducer with important parameters. . . . . . 49

32. Working principle of the counter electrode. . . . . . . . . . . . . . . . . . . . . . . . . 51

33. Set-up of the BAW detector with important parameters. . . . . . . . . . . . . . . . . . . 52

34. Bandwidth (Displacement of the AFM transducer in dependency of beam length and

stiffness of the spring system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

35. a) Sensitivity of the system with different DC potential on the AFM transducer electrode

and with a 60 µm diameter disk shaped BAW detector; b) Logarithmic plot of the trendlines. 58

36. Sensitivity comparison of disk resonator for the BAW detector with different radii. . . . 58

37. Sensitivity comparison of disk, square, ring and frame shape as resonator for the BAW

detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

38. Comparison between electrostatic force over electrode gap width with and without counter

electrode at pull-in distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

39. Comparison between the system with and without counter electrode related to the air gap

width in dependency of the applied potential. . . . . . . . . . . . . . . . . . . . . . . . 61

40. Sensitivity of the system with different DC potential on the AFM transducer electrode

acting on a ring shaped resonator as BAW detector. . . . . . . . . . . . . . . . . . . . . 62

41. Set-up of the complete system with AFM transducer, BAW detector and mechanical

coupling system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

42. Set-up and working principle of the AFM transducer with important parameters for the

mechanically coupled system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

43. Set-up of the BAW detector with important parameters for the mechanical coupled system. 66

44. Simulation set-up with the force caused by the Brownian motion on one side and on the

other side a counter acting force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

45. Simulation set-up to determine the effect of the mechanical coupling system on the quality

factor of the BAW resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

46. Resulting force and coupling efficiency of the mechanical coupling system with an initial

force of 100 pN depending on spring beam stiffness. . . . . . . . . . . . . . . . . . . . 69

47. Frequency change df from undisturbed resonator resulting from applied forces on the disk

resonator at different diameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

48. Quality factor of the BAW resonator in dependency of the mechanical coupling stiffness. 71

49. COMSOL start menu, for selection of a) dimension; b) physics; c) study type. . . . . . . 74

50. a) COMSOL model library; b) Geometry creation. . . . . . . . . . . . . . . . . . . . . . 75

51. a) Setting parameter for geometry; b) Material selection. . . . . . . . . . . . . . . . . . 76

52. a) Setting the thickness for the geometry; b) Parameter sweep. . . . . . . . . . . . . . . 77

53. Display of result with selection of parameter and mode. . . . . . . . . . . . . . . . . . . 77

54. a) Allocation of the surrounding medium; b) Define the solid mechanic structure for the

simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

55. a) Set damping coefficients; b) Electrical potential defined on the structure. . . . . . . . 79

56. Defining the values for the sweep with two parameters. . . . . . . . . . . . . . . . . . . 79

57. Defining the sweep over the gap width interval for pre-stress of the resonator. . . . . . . 80

58. a) Pre-stressed, Frequency domain case with AC and DC voltage part; b) Frequency-

Domain with only AC actuation part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8

LIST OF FIGURES

59. Adding static boundary force for a stationary study type. . . . . . . . . . . . . . . . . . 81

9

LIST OF TABLES

List of Tables

1. Summary of the key data for a conventional AFM system. . . . . . . . . . . . . . . . . . 31

2. Parameter change for ring and frame shape simulations. . . . . . . . . . . . . . . . . . . 34

3. Summary of the most important design parameters. . . . . . . . . . . . . . . . . . . . . 50

10

1 INTRODUCTION

1. Introduction

Technological process is driven among others by the need for devices with always increasing performance,

but at the same time decreasing resource, space and power consumption. The best example for this

development is chip fabrication described by Moore’s law, which states that every 18 month the number

of transistors on integrated circuits is doubled. This leads to a doubling in chip performance, however,

with constant occupied space, which is made possible by a decrease in size, power consumption and used

resources for every single transistor, but at the same time an increase in performance.

Moore’s law can not only be applied to computer processors, but also for example to memory capacity,

the pixel size and number of a digital camera or for sensors. A few decades ago it was still possible

for a single person to assemble, for example, a vacuum tube, a camera or a sensor without any special

tools. Today it is not even possible anymore to see a single transistor with the naked eye, which is able to

distinguish structures down to 100 µm. Even with magnifying glasses or a simple light microscope this is

not possible, because their feature size have decreased to several nanometers, which is close to the range

of single atoms. In order to fabricate, characterize and interact with such small structures special tools,

sensors and other devices are needed.

1.1. MEMS for sensor applications

Some of these specialized tools are Micro-Electromechanical Systems (MEMS). As the name already

suggests these systems have a feature size in the micrometer range with a mechanical part in combination

with electrical functionalities. For example a device consisting of a diaphragm, which is displaced in

reaction on acting pressure and an electrical circuit to determine the displacement and thus the pressure

acting on the device. MEMS are not only sensor systems, but are used for a great variety of applications

ranging from high frequency systems like GHz antennas for mobile phones, over microfluidic devices,

like pumps or valves, to different sensor systems, like accelerometers, gyroscopes, biosensors or pressure

sensors just to name a few applications. The advantages of MEMS as sensors in comparison to conventional

systems are the decreased dimensions and mass, as well as higher performance, higher reliability, lower

power consumption and rapid response times. It is easy to combine them with integrated circuit design,

because both can be fabricated with well established CMOS fabrication processes, which makes the

production of MEMS components very cheap and a high yields are possible. [16, 38]

Sensors use very different detection methods, for example piezoelectric, capacitive and optical, to measure

the parameter of interest. Another possibility is to use resonators, which change their resonance frequency

caused by the acting measurand. This has the additional advantage that the sensor output signal has

already digital signal characteristics and can be directly used by the evaluation electronics. Here more and

more bulk acoustic wave resonators are used providing better performance of the sensors.

1.2. Motivation for the thesis

Atomic force microscopes use such a resonating detection method to map surfaces on the atomic level.

For conventional AFM a cantilever with an attached tip is used and a laser as readout system to detect its

movement. This system set-up has several limitations, which prevent an increase in speed of measurement,

sensitivity and resolution. One disadvantage are the dimensions of the cantilever, because the laser beam

needs a certain surface area to be reflected properly. The laser unit also needs a lot of space and optical

11

1.3 Structure of the thesis

components to direct the beam onto the cantilever and the reflected beam to the detector. This hinders the

reduction in size of the AFM system. Another disadvantage is the limitation in resonance frequency and

quality factor of the AFM cantilever, because of high losses to the surrounding medium of this flexural

mode resonator.

To overcome these drawbacks a investigation is conducted to determine if the conventional AFM system

can be replaced by a bulk acoustic wave resonator, which can reach much higher resonance frequencies

and quality factors to increase the performance of an AFM system. Furthermore can bulk resonators also

fulfill the function of the detector and replace the laser unit for more flexibility in the design and reduction

in size of the system.

1.3. Structure of the thesis

For conducting this investigation a literature search for background information of sensors, bulk acoustic

wave resonators and AFM have to be conducted and important parameters, working principles, and

applications have to be analysed, which is described in chapter 2. In chapter 2.6 conventional AFM

systems are described, the important performance parameters are identified, as well as some existing

investigations on improving the design are described.

Subsequent in chapter 3 the most promising geometries for electrostatically actuated bulk acoustic wave

resonators are analysed on basis of their use in AFM systems. Disk, square, ring and frame shapes are

investigated in chapter 3.1 to determine their behaviour in matters of stiffness, resonance frequency and

displacement for different dimensions. Longitudinal mode resonators are described in chapter 3.2 and

also the dependency between the dimensions and stiffness, resonance frequency and displacement of the

resonating structure are analysed.

That is followed by the main chapter 4, where a design for a new kind of AFM sensor is proposed

consisting of two resonators, which are electrostatically coupled and non-linear effects of an electrostatic

force acting between the two electrodes are used to detect the movement of the tip by a bulk acoustic wave

sensor. In addition the results from the general analysis of bulk acoustic wave resonators are utilized to

maximize the performance of the sensor. The concept is described and simulations are conducted in order

to determine the important parameters and limitations of this sensor for the use in AFM and to compare

these to conventional AFM systems.

In chapter 5 a design variation of the previous sensor concept is proposed, where the two resonators are

coupled mechanically, not electrostatically, to overcome some critical drawbacks of the previous design.

In the final chapter 6 and 7 all results are summarized and conclusions are drawn, as well as future work

and investigations are proposed.

12

2 BACKGROUND: MECHANICAL RESONATORS AND THE ATOMIC FORCE

MICROSCOPE

2. Background: Mechanical resonators and the atomic force

microscope

2.1. Resonance

Mechanical oscillation of an object can be modelled by a simple mass with two springs, where one end is

fixed and the other can be driven with a certain frequency, as can be seen on figure 1.

a)

b)

c)

Frestoring

Frestoring

Fig. 1: a) Spring system in maximum position with restoring force; b) initial position; c) maximum

position with restoring force.

If a frequency is forced upon a system it oscillates with the same frequency, as soon as its steady state

is reached. There is one special case, however, when the actuation frequency is equal to the natural

frequency of the system. This case is called resonance, in which the amplitude of the oscillation would in

theory grow infinitely high for a loss-less system under continuous excitation. In practice there are losses,

like damping and friction preventing an infinitely high amplitude. [18]

From Newton’s second law the equation for a damped harmonic oscillator can be found.

F = −kcx− cdx

dt= m

d2x

dt2(1)

Where kc is the spring constant, x the position at time t, c is the viscous damping coefficient and m is the

proof mass. The undamped angular frequency of the oscillator ω0 and the damping ratio ζ described in

the following can be used to rewrite equation 1.

ω0 =

√

kcm

and ζ =c

2√mkc

(2)

withd2x

dt2+ 2ζω0

dx

dt+ ω2

0x = 0 (3)

If the damping ratio ζ is larger than 1 the system returns to its equilibrium state with an exponential decay

without oscillating, which is called overdamped. In a critically damped system the damping ratio equals 1

that means the equilibrium is reached as fast as possible without oscillating. In the underdamped system

the damping ratio is smaller than 1 and the system oscillates with a decreasing amplitude. [47]

For an increasing number of coupled masses in the system, an increasing number of resonance frequencies

can be found, which are called harmonics. All objects can be seen as mass-spring systems with an infinite

number of coupled masses representing the atoms and therefore a theoretically infinite number of natural

frequencies exist. To find these harmonics of a system the excitation is done by sweeping through all

13

2.1 Resonance

frequencies, e.g. a sinusoidal signal sweep is used similar to determining the resonance frequency of RF

filters. Another method is to excite the system with an impulse containing a wide frequency spectrum.

Out of these the system selects its natural resonance frequencies and starts to resonate. [18]

One example is the excitation of a tunic fork by knocking it against the edge of a desk. The geometry of

the tuning fork is designed to resonate with one specific frequency, for example 440 Hz corresponding to

the note A. The way of excitation provides a wide range of frequencies, but only the designed 440 Hz

create a resonance in the tuning fork. [18]

An example of a so called resonance catastrophe is breaking a wine glass only with the help of the voice.

Here the energy of the acoustic waves, created by the vocal cords of the singer, is coupled into the glass

body. If the voice has a high enough amplitude and a frequency close to the resonance, the glass starts

oscillating in the so called wine-glass mode. The smaller the difference between the excitation frequency

and the natural frequency, the more violently the glass oscillates and the amplitude increases until the

glass reaches its elasticity limit and breaks. [18]

These are examples for mechanical systems. Besides mechanical resonators many other types exist, like

electrical, optical, acoustic and magnetic systems. Besides these, many resonators are coupled, like

electromechanical, electromagnetic or optomechanical systems. All these, however, are based on the same

principles. In some cases it is possible to substitute one with the other, e.g. electrical resonators can be

used to simulate mechanical resonance systems. [33]

The following chapters will emphasize on mechanical resonator systems, explain them and give examples.

2.1.1. Quality factor and losses

The quality factor, also Q-factor or simply Q is a dimensionless parameter describing the damping of

a resonating system. It is the ratio between the energy stored in the oscillating system and the energy

dissipated per cycle or according to Crowell [18], it is defined as the number of oscillation cycles needed

for the energy to fall off by a factor of e2π.

A resonating device will be considered of high-quality, if it has a high quality factor that means the system

looses very little energy per cycle. Therefore it oscillates for a long time until the better part of its energy

is lost and it also needs less energy to maintain a constant amplitude. Thus it can be written with the

following equation. [4, 18]

Q = 2π ·stored energy

dissipated energy per cycle(4)

The quality factor can also be expressed by setting damping, the spring constant and the effective mass in

relation to each other. Here a damping ratio ζ between the damping coefficient c and the critical damping

c0 is used, which is expressed in the following equation:

ζ =c

c0=

1

2 ·Q(5)

with

c0 = 2 ·√

kc ·m (6)

The quality factor can be written as

Q =

√kc ·m

c(7)

14

2.1 Resonance

Where kc is the spring constant and m the mass. [11, 43]

Another way to describe the quality factor is the ratio between the resonance frequency f0 and the

bandwidth BW , defined as the frequency difference ∆f between the 3dB points of the resonance curve

f0.

Q =f0∆f

(8)

In figure 2 two resonance frequencies of the same oscillator with different quality factors can be seen. At

-3 dB from the peak value the bandwidth or ∆f can be found. The smaller the bandwidth, the larger is the

Q-factor and the narrower is the resonance curve. A better defined resonance frequency with high quality

factor means an improvement of the resolution and performance of the device. It also indicates that the

influence of surrounding factors on the system are minimized. [4]

Resonance high Q

Resonance low Q

Frequency

Vib

rati

on

ampli

tude

BW (low Q)

-3 dB point (low Q)

-3 dB point (high Q)PeaklowQ

PeakhighQ

BW (high Q)

Fig. 2: Bandwidth (∆f ) measured at -3dB points in resonance curve with low and high quality factor.

There are four dominating mechanisms damping the system and limiting the quality factor. Only the

damping effect with the highest losses is most relevant to this work and is described.

1

Q=

1

Qviscous+

1

QTED+

1

Qsurface+

1

Qclamp(9)

The different factors of this equation are:

• 1/Qviscous is the damping arising from the surrounding fluids

• 1/QTED stands for the thermal-elastic dissipation

• 1/Qsurface are the losses occurring on the surface

• 1/Qclamp are the losses caused by the suspension of the resonator

All these factors have to be minimized in order to achieve a high Q-factor. [4, 41]

Viscous damping The most energy is potentially lost to the surrounding environment of the device

(1/Qviscous), which will mostly be gases or fluids. At low pressure, between 1 and 100 Pa the molecules

15

2.1 Resonance

move independently of each other and the medium has gas form, called gas damping. Viscous drag occurs

when the pressure is higher then 100 Pa and the molecules behave like fluid moving over the surface of

the resonator. [4, 7]

The damping effect is caused by collision between the molecules in the surrounding medium and the

surface of the resonator. Here kinetic energy is transferred between the molecules and the resonator by

exchange of momentum corresponding to the relative velocities. It is directly proportional to the pressure

of the medium and can also be influenced by several other factors, like close proximity of solid objects

to the resonator, which will increase the damping effect. Furthermore the losses depend on the gas or

liquid composition, temperature and pressure. This source of damping can be minimized by packaging

the resonator and operating it preferably in vacuum or at a pressure below 0.1 Pa, where the gas damping

becomes negligible. [4, 7]

Thermal-elastic dissipation Losses originating at room temperature in the resonator material itself

called thermal-elastic damping (1/QTED). This is caused by scattering of acoustic phonons with thermal

phonons, while the resonator is taken out of its equilibrium state and the material is elastically compressed

and decompressed. In the process a thermal gradient is created increasing the entropy of the system and

leading to energy dissipation, because the compressed areas heats up and the decompressed areas of the

resonator cools down. [4, 61]

This damping mechanism is influenced by the material, its impurity level and grain boundaries. It can

only be controlled in a limited way by choosing the appropriate material with certain impurity level for

the desired application. Also temperature influences the resonance frequency, so cross sensitivities can

occur and temperature compensating measures have to be taken if this effect is not wanted. [4, 62, 66]

Surface damping With increasing surface-to-volume ratio of small resonators the surface effects

increase causing energy dissipation (1/Qsurface). This mainly results from atoms and molecules on the

surface of the resonator interacting with the surrounding medium. This damping mechanism counts as

internal and is also material related, like thermal-elastic dissipation. This effect can be influenced by

treating the surface of the resonator to have desired properties to minimize the surface damping effect.

[4, 66]

Clamping damping Structural losses (1/Qclamp) are caused by damping effects in the coupling between

the resonator structure and the surrounding solid. To minimize this factor several measures can be taken,

such as decoupling of the resonator and its supporting structure, using the nodal points to anchor the

resonator and to design a balanced system. Another measure is to excite the resonator in a higher mode

and actuate the system electrostaticly in order to sustain the resonance for a longer time and have a higher

quality factor in comparison to other methods like piezoelectric actuation. The clamping losses can also

be minimized by several design methods to fabricate the suspension of resonators, like t-shaped tethers for

bulk mode resonators described by Lee et al. [40, 41] or a stem fabricated on the inside of a disk resonator,

described by Wang et al. [67]. Minimizing the structural losses provides also the advantages of having a

good frequency resolution, a good immunity against environmental influences and therefore an improved

long-term stability of the system. [4, 33, 44, 54]

16

2.1 Resonance

2.1.2. Sensitivity

Sensitivity is one of the important factors to evaluate the performance of a sensor. Several different

definition of sensitivity can be found in the literature, which also was referred to as responsivity. These

two terms were used as synonyms, but in recent years their definitions changed.

To avoid confusion the definition that will be used in the report is the definition of sensitivity, which is

widely used in the industry and in most reference books. Charr et al. [12] defined it as ”the minimum

input of a physical parameter that will create a detectable output change”.

2.1.3. Mechanical noise

At high sensor output levels, the signal decreases proportional to the input value. If it would be possible to

continue this correlation an infinitely small change in the input value could be measured. In reality at low

level, however, the output signal reaches a lower limit regardless of the input value level. This lower limit

is called the noise floor, which is caused by several different sources originating from thermal motion of

the atoms or quantized current flow in the circuits. These noise components add to the output signal and

consist of random fluctuation. As soon as the value of the signal and the noise become similar, the output

signal will be so distorted that it can not be analysed anymore. The goal for applications is to keep the

noise as low as possible in order to be able to detect the smallest change in the input by analysing the

output signal. The noise floor is one of the limiting factors in sensor applications and has to be analysed

in order to determine the limits and performance of a device. [28, 63]

For mechanical resonating systems several noise sources are dominating and add to the noise floor.

Thermal noise Above absolute zero, atoms are subject to thermal motion. Especially at room temper-

ature a considerable random fluctuation in the output signal is caused, for example by moving atoms in a

mechanical system. This noise can not be avoided at a given temperature so it is a fundamental limitation

for sensing and the precision of applications. In resonance systems for example the thermal noise excites

the oscillator with a energy of kBT , with the Boltzmann-constant kB and the temperature T caused by

the Brownian motion, which creates a significant noise floor in the system. The fluctuation dissipation

theorem is given by the following equation:

SFF = 2kBTmeffγ0 (10)

Where kB is the Boltzmann-constant, T the temperature, meff is the effective mass and γ0 the damping

coefficient. This equation can be transformed using the following.

γ0 =2ω0

Q,m =

kcω20

and ω0 = 2πf0 (11)

With the angular resonance frequency ω0, the quality factor Q, the spring constant kc and the resonance

frequency f0 . With equations (10) and (11) the thermal noise can be expressed by the following equation.

√

SFF =

√

2kBTkcπf0Q

(12)

[28, 49, 56, 63]

17

2.2 Flexural mode resonators

Deflection and frequency detector noise Measuring the displacement or the frequency of a

resonator is also subject to noise, because the detection methods do not have a infinite precision. Therefore

the driving frequency varies slightly around the resonance frequency and creates a constant deflection

or frequency detector noise, which indicates the precision of the detection methods. The noise itself

depends very much on the bandwidth of the sensor and therefore on the quality factor. The equation for

the deflection detector noise is the following.

δktsdet =

√

8

3

kcnqBW 3/2

f0A(13)

Where kc is the spring constant, nq the deflection detector noise density, BW the bandwidth, f0 the

resonance frequency and A the amplitude. [28]

2.2. Flexural mode resonators

One of the most simple flexural mode resonators is a rope, which is fixed on one side to a wall and the

other side is held in hand to move it up and down for excitation. Another example is a violin string, which

is fixed at both ends of the violin and is excited by rubbing the bow over it. Like that the resonator can

oscillate in their natural frequency and also in higher harmonics, depending on the excitation frequency

and energy. [18]

These systems are called clamped-clamped flexural mode resonators (figure 3a), because the two ends are

fixed and can not move. In addition there are clamped-free systems (figure 3b), where one end is fixed

and one end is loose and free-free systems (figure 3c), where the two ends can oscillate freely. All the

different resonators are compared in figure 3, here also several higher harmonics are displayed to show

the difference between the systems. The excited resonator has the shape of a standing sinusoidal wave

and forms so called nodes, which are not subject to displacement. These nodes can be used to anchor the

resonator with least losses for the system. In figure 3 the nodes can easily be seen as blue areas. The blue

area in the figures stands for low displacement of the oscillating resonator. [30, 33]

n=1

n=2

n=3

n=4

clamped-clamped clamped-free free-free

Fig. 3: Flexural mode of a) a clamped-clamped, b) a clamped-free and c) a free-free beam up to the 4th

mode.

Flexural mode resonators are only used for low frequency applications up to 100 MHz. To reach high

frequency with a reasonable quality factor a vacuum environment is preferred. In air and in liquids the

damping becomes so high that the quality factor will decrease significantly. For increasing frequencies the

system has to be excited in a higher harmonic or the structure has to be decreased in size. With small

dimensions the fabrication tolerances become large and also the product uniformity is not guaranteed

anymore. Besides that the anchor losses become too high and the power handling capability is also limited.

18

2.3 Extensional mode resonators

Noise becomes an important factor, due to thermal fluctuation and surface roughness. In addition to all

that the quality factor is dropping significantly for small dimensions. All these reasons prevent flexural

mode resonators from reaching frequencies in the GHz-range. For high frequencies extensional mode

resonators have to be used to overcome the drawbacks of flexural mode resonator.[46]

In MEMS systems mostly clamped-clamped beams and clamped-free cantilevers are used in devices.

Some of the many applications of flexural mode resonators are atomic force microscope cantilevers,

RF-switches, pressure sensors, chemical sensors and biological sensors.

Another type of flexural displacement is the torsional mode, where the system oscillates in a torque

motion as can be seen in figure 4a.

a) b)

Torsional anchor

Flexural beam

Electrode

Fig. 4: a) Torsional mode beam; b) Flexural mode beam anchored with four torsional mode beams

These resonators have a higher quality factor and are able to reach higher frequencies compared to the

other flexural mode systems. Some examples for torsional mode resonators are special tuning forks and

resonance densitometers, described by Enoksson et al. [19]. The principle of torsional mode resonators is

also used for fixing free-free flexural mode resonators at their nodes for minimizing anchor losses, which

can be seen in figure 4b. [19, 45]

For frequencies higher than 100 kHz and at the same time high quality factors flexural and torsional

mode resonators can only be used if they are scaled down to nano-size. The high force constant is the

reason that a high power level is needed for excitation in order to receive a appreciable response from the

system. This has a negative effect on the power consumption, dynamic range and the quality factor of the

system, also reducing the tuning capabilities. In addition the surface and anchor losses of these resonators

become very high, because of a high surface-to-volume and a small length-to-width ratio. Besides that

the fabrication technology needed is very complex. In contrast the extensional mode resonators can be

fabricated with normal production methods. These are therefore preferred for high frequency applications

instead of flexural mode resonators. [11, 30]

2.3. Extensional mode resonators

The flexural mode can be seen as a transverse standing wave, where the displacement is orthogonal to the

bending stress in the structure. Whereas shear stress causing rotational displacement is responsible for

the torsional mode oscillation in devices. In case of bulk acoustic wave (BAW) resonators a longitudinal

standing acoustic wave is created inside the bulk material and leads to an in-plane displacement. [11, 14]

In extensional mode more mass is vibrating, which means the effective mass is higher and the damping is

lower, resulting in higher maximum energy in the system. Also extensional mode resonators are orders of

19


magnitude stiffer than flexural mode resonators and the losses are much smaller because of the smaller

surface-to-volume ratio as the same resonance frequency can be achieved for larger dimensions. Therefore

frequencies above 100 MHz with a much higher quality factor can easily be reached, even frequencies in

the GHz-range are possible. [9, 30, 33, 48]

Besides the higher stiffness, bulk resonators have several other advantages over flexural mode resonators.

For the resonator dimensions smaller than 50 µm are possible, because of the compact geometry. They

also have the capability of storing vibrational energy is orders of magnitude higher than in flexural mode

resonators. It has to be kept in mind, however, that the smaller the resonator becomes the lower is the

energy storage and power handling capabilities. If electrostatically driven, non-linearity is a big issue

for both resonator types to reach a sufficiently good energy and signal-to-noise ratio, which makes them

unsuitable for filter applications, in comparison to large quartz crystals, which can store enough vibrational

energy without being operated in the non-linear region. But extensional mode resonators are still not as

susceptible to non-linear effects as flexural mode resonators. [36, 58]

Another advantage of extensional over flexural mode resonators is that they are less susceptible to

environmental influences like pressure change.

Depending on the geometry and the aspect ratio, actuation frequency and energy transferred to the system,

several different higher resonance modes can be promoted. The most commonly used geometries are

rectangular, square and disk shaped resonators, which have a high symmetry in order to be actuated and

resonate homogeneously with least losses possible. These shapes are also much easier to fabricate than

resonators with irregular and or complex 3-dimensional geometries. [14]

Decompression Compression

b) Lame mode

c) Wine-glass mode

a) Longitudinal mode

Fig. 5: Simulation of different extensional mode resonators with decompression and compression state:

a) longitudinal mode, b) Lame mode and c) wine-glass mode (red means larger displacement

and blue less displacement).

In figure 5 the most common extensional modes for mechanical resonators are displayed in the two

maximum positions (compression and decompression).

20


2.3.1. Longitudinal mode resonator

A standing wave is created if bulk material is excited in the longitudinal mode, which causes an in-plane

length extension and contraction, as it can be seen in figure 5a.

In the middle of the structure a nodal point appears, which is used as anchor and is only subject to very

small displacement. To promote the free-free longitudinal mode, a symmetric geometry is preferred,

where the dimensions and weight are equally distributed around the central plane in order to receive a good

longitudinal movement. In comparison to flexural mode resonators the total displacement is much smaller,

but the frequency and the quality factor are higher for the same dimensions, which makes the longitudinal

mode resonator interesting for applications with frequencies over 100 MHz and large displacement in one

direction. [68]

2.3.2. Lame mode resonator

Lame mode and wine-glass mode are very similar, which can be seen in figure 5b and c. In many articles

both are used as synonyms and no difference between these two modes are made. Often the wine-glass

mode for a square shaped resonator is also referred to as Lame mode and for a disk shape resonator as

wine-glass mode to make a difference between these two most common shapes, which can be seen in

figure 5c. According to Chandorkar et al. [14], however, the Lame mode resonates is a higher order

harmonic than the wine-glass mode and therefore has a different shaped resonance pattern, where the

motion preserves the volume of the resonator, which can be seen in figure 5b. This pattern received its

name from the french mathematician Gabriel Lame who first discussed it in 1817. [29]

The geometry of the structure is either square or disk shaped, but the displacement is very small in case of

the the disk structure. The nodal points are situated in the middle of the faces, which hardly move and are

used to anchor the resonator. The corners in the square shape and the area between the anchors in the disk

shape are subject to the largest displacement. [14]

This mode is suited for high frequency applications, but is very similar to the wine-glass mode and because

of the impractical anchor placement of the square geometry it is not used much in this configuration.

2.3.3. Wine-glass mode resonator

The name wine-glass mode comes from the example mentioned before, where a opera singer manages

to break a glass only with the help of the voice. The vibration caused by the voice creates an elliptic

displacement of the round shape of the glass body. For bulk acoustic wave resonators the wine-glass mode

is the most used oscillation type. In the square shape the nodal points appear at the corners, so these can

be used as anchor points. Also in the middle of the disk or square a region of very low displacement

appears, that means this area can also be used for anchoring the resonator with a stem, which can be seen

in figure 5c. The faces have the largest displacement, where the motion also preserves the volume of the

resonator. [9, 30]

In the wine-glass mode, the resonance frequency is inversely proportional to the disk radius or side length

of the square shape, which means the smaller the dimensions of the resonator, the higher is the frequency.

But for decreasing size, also the quality factor is decreasing, which sets a limit to the scalability of bulk

acoustic wave resonators. [53]

21

2.4 Resonance excitation and sensing

2.4. Resonance excitation and sensing

Mechanical resonators can be excited in many different modes and each one has again many overtones.

These resonance frequencies are dependent on their geometry and the material used. The excitation can

be electrostatic, piezoelectric, with laser, through mechanical vibrations and with a magnetic field. The

piezoelectric and electrostatic actuation are the most common methods for excitation of a resonator. In

this report, however, only the electrostatic principle is investigated and will be explained in detail for

excitation of a bulk acoustic wave mode resonator. [33, 35]

2.4.1. Electrostatic forces

An electrostatic force is created by applying an electric potential between two electrodes, which are

separated by a gap filled with a dielectric material or vacuum. In most cases this material is air, but Bhave

et al. [6] describe that the gap can also be filled with a low Young’s modulus high-κ-dielectric material

instead of air.

A simplified model consists of two parallel plates, where one plate is fixed and the other one movable

with an attached spring for restoring force. The movable plate is deflected out of its initial position (gray

plate) by the electrostatic force caused by the potential between the two plates, as can be seen in figure 6.

Fs

Fel V0

gxg

fixed plate

movable plate

initial position

Fig. 6: Forces and important parameters in parallel plate system with one fixed and one movable plate.

The restoring spring force Fs has a linear characteristic and is defined with the following equation

Fs = kc · (g − xg) (14)

Where kc is the spring constant, g the initial gap distance at unstressed spring and xg the actual gap

distance. The term (g − xg) describes the spring deflection from the initial position.

The electrostatic force Fel on the other hand has a non-linear characteristic and is define as

Fel = −ǫ0ǫrAU2

2x2g(15)

Where ǫ0 is the vacuum permittivity, ǫr relative permittivity, A the electrode area, U the electric potential

and xg the actual gap distance. From equation (15) it can be seen that the electrostatic force is inversely

proportional to x2g. This relationship is the reason, why at some point, usually around 2/3 of the initial gap

width g, the electrostatic force will be stronger than the spring force and the system will become unstable

and a so called pull-in occurs.

22


FsF

orc

eFs,−Fel

Gap x

unstable points

Fel

always unstable

one solution

stable points

d0 2/3 · d

stable regionunstable region

actuation voltage

Fig. 7: Plot of Fel and Fs versus xg between the two plates, indicating stable and pull-in region.

In figure 7 can be seen the linear spring force and the non-linear electrostatic force with different applied

potentials. The intersection between the curves indicates a stable solution, if the gap is still in the stable

region of the device, which means a equilibrium is created between the two forces. At any other points,

either the spring force is stronger the gap becomes wider or if the electrostatic force is stronger, pull-in

occurs and the gap becomes 0. The two electrodes would touch creating a short circuit, which has to be

avoided. Either stoppers or a insulating layer on the electrode surface is used to prevent short-circuit. A

device should be operated in the stable region, except the pull-in is desired. [70]

2.4.2. Electrostatic actuation of resonators

Electrostatic actuation has several advantages over other methods. First of all the technique can be

completely realised in silicon, which makes the fabrication cheap, easy and CMOS integrable, because

well established CMOS processes can be used. The devices have a small size, are very stable and have

a high resistance against shock and vibration, because of the lower mass. It is also easy to compensate

for frequency shifting effects and fine tune the device by applying a DC bias (pre-stress). In comparison

to piezoelectric actuation the resonator has a higher quality factor, because no physical contact with the

resonator is needed, which creates increased structural losses. Furthermore the material used is more

homogeneous than the combination of materials needed for piezoelectric excitation and therefore provides

higher power storage capabilities. [3, 9, 34, 54]

But there are also some drawbacks, which have to be mentioned. In comparison to for example piezoelec-

tric crystals the motion resistance and the maximal energy storage capability are lower, because of smaller

size of the system. Also the frequency stability is not as good, due to variation of the DC bias, higher

fabrication tolerances and larger temperature drift of electrostatic actuators. [9, 54]

The resonators should be excited with a frequency similar to its resonance. To accomplish that an AC

signal has to be applied between the electrodes. With the sinusoidal rising and falling of the voltage,

the resonator is subject to periodic deformation caused by the electrostatic force between the electrodes.

23


Depending on the geometry and the anchoring points the resonator starts to oscillate either in one of the

flexural modes or in one of the extensional modes. The actuation electrodes have to be placed in the

area with the highest displacement of the resonator in order to be most effective and create the highest

vibrational amplitude. The displacement at resonance frequency for an AC signal excitation can be

calculated by the displacement at constant applied DC potential times the quality factor of the resonator.

[9, 35, 44, 51]

A DC voltage also has to be applied in addition to the AC potential in order to charge the capacitance

between electrode and the resonator and act as a current amplifier for the AC potential, creating an output

current. Without the DC bias the resonator can only be excited to the second harmonic. An increase in

DC pre-stressing results in an increase of the resonator amplitude. [15, 17, 60, 67]

Additionally with an applied DC bias to the electrode the frequency can be tuned, which cannot be

done with piezoelectric actuation. This bias can be used for example to compensate for temperature

effects, material impurities, fabrication tolerances or frequency shifts caused by resonator packaging. By

applying the DC bias the resonator becomes pre-stressed and the stiffness change in the resonator material

influences the resonance frequency. The higher the DC pre-stressing is, the more will the frequency of the

resonator increase. [9, 34]

The main parameter, which influences the performance of the resonator is the air gap between the driving

electrode and the resonator. Especially in case of the BAW resonator it is important to have a very narrow

gap. The smaller the gap, the better is the coupling efficiency, because of the non-linear dependency of

electrostatic force and gap width, which was described in equation (15).[9]

With traditional fabrication methods, like deep reactive ion etching (DRIE), it is difficult to fabricate gaps

smaller than 1 µm, but some methods exist, where gap width down to 90-100 nm can be fabricated, which

for example is reported by Pourkamali et al. [52, 55]. Another method is to move the driving electrodes

closer to the resonator by using comb drive actuators, which is described by Galayko et al. [25], for

achieving a narrow gap and with good coupling efficiency. This, however, has some drawbacks, because

of the side wall roughness created by the DRIE method.

2.4.3. Capacitive sensing of resonators

In order to use resonators as sensors or other applications, a method has to be found to detect the frequency

of the oscillating device. External influences cause a change in stiffness, mass or shape of the resonator

and shift its resonance frequency. This shift is detected and used in sensors to determine the quantity of

the external factors acting on the resonator. The dependencies and applications will be explained later on.

[4, 44]

There are also as many detection methods as there are actuation methods. In case the resonator is already

electrostatically actuated, the easiest way will be capacitive sensing. The sensing detects the change

in capacity between the sensing electrode and the resonator, the output signal of the resonator will be

received in form of an AC signal and frequency change, phase shift or amplitude change can be analysed

by a measurement system. [3, 4, 44]

The advantage of capacitive sensing of a resonator is the frequency output, which is more immune to

noise and therefore a high resolution can be achieved. Furthermore the output can easily be converted into

a digital format, that is compatible to integrated CMOS technology. [4]

24

2.5 Applications of mechanical resonators

A set-up of a BAW resonator, which is used as a sensor can be seen in figure 8.

Resonator (grounded) (blue)

Actuation electrodes (AC+DC voltage) (green)

Sensing electrodes (yellow)

Bulk material

Underetch for freestanding resonator

Suspension and anchor

Air gap between electrode and resonator

Fig. 8: Set-up of a BAW resonator with electrostatic actuation and sensing integrated in a device.

Here a disk resonator is shown, which is actuated by two driving electrodes with applied AC voltage and

DC biasing (pre-stress). The sensing is done by another set of electrode with the same narrow gap as the

driving electrodes. The resonator itself resonates in the wine-glass mode and therefore can be anchored at

the four nodal points of the disk.

2.5. Applications of mechanical resonators

Using the principle of flexural or extensional mode resonators a lot of applications can be designed. Most

of them are different kinds of sensors, which utilize various methods to manipulate the output signal in

order to analyse the measurand. The first step is to understand the principles and external parameters of

how and how much the output signal can be affected. The second step is to use these dependencies and

create an application in a way to detect only the parameter of interest.

2.5.1. Sensing principles

In order to shift the resonance frequency of a resonator two different approaches can be utilized, either a

change of mass, or a change in stiffness of the resonator. This is based on the following equation

ω0 =

√

kcm

(16)

Where kc is the spring constant or stiffness, ω0 the angular resonance frequency and m is the mass of

the resonator. All physical quantities, which want to be measured have to manipulate one of these two

parameters. If a sensor is designed for a task to measure one specific parameter, there are often other

parameters, which can influence the measurement results. For example while measuring the frequency

shift caused by a mass, also temperature change can influence the resonance frequency of the resonator.

This phenomenon is called cross sensitivity and is a big issue in designing a sensor for a specific task.

To avoid this, all other parameters have to be kept constant throughout the measurement or possible

influences on the results due to other parameters have to be compensated. [4, 18]

Mass change As can be seen in equation (16) an increase of the resonators mass leads to a decrease

of the resonance frequency and the other way around. This has to be taken into account by choosing the

material for the resonator and can be used to create sensors to count particles, analyse mass and volumes

25


of objects or determine layer thickness of depositions. Depending on the resonator design, the detection

limit can be low enough to measure single molecules or even atoms. [4, 20, 39, 44]

Stiffness change The other possibility of shifting the resonance frequency is, according to equation

(16), to vary the stiffness of the resonator. Where an increase of stiffness results in an increase of

frequency. This can be done by inducing stress or strain in the resonator material, by DC biasing,

temperature, pressure, shape change, torque and other forces acting on the resonator. [4, 20, 35, 44]

2.5.2. Example Applications

A lot of different applications can be designed using these sensing principles to achieve a shift of the

resonance frequency . In the following, four examples will be introduced and their working principle will

be explained shortly.

Chemical sensor The principle of sensing mass can be used to realize a chemical sensor. It is possible

to detect the quantity of specific molecules or compounds, like Hydrogen or NOx. In order to do that the

surface of the resonator is functionalized by applying a layer of a specific component, which binds to the

measurand.

It is reported by Seh et al. [57] that for NOx (orange spheres) sensing, the surface of the resonator is

coated with a BaCO3 film (blue layer). Only NOx molecules are creating a compound with the layer and

are bound to the surface. This process can be seen in figure 9.

Particle flow

Coating

Cantilever

Particles on surface

Fig. 9: Working principle of chemical sensor for detection of NOx.

The additional weight from the measurand on the resonator causes a resonance frequency shift, which is

detected and gives information about the quantity of the particles. Recently more and more bulk acoustic

wave resonators are used, because they have increased sensitivity compared to flexural mode resonators.

They also can be used in fluid environment due to their high quality factor, which guarantees good results

even with high viscous damping.

There are also chemical sensor arrays possible, which detect several compounds at the same time. This is

accomplished through many cantilevers next to each other, where each of them is coated with a different

compound interacting with other measurands. [7, 10]

26


Biosensor The same principle of sensing the resonance frequency shift caused by a change in mass

is used in bio sensors. This can be used to detect biological compounds in samples, like cells, viruses,

proteins, DNA, antibodies or enzymes. [20]

Carrascosa et al. [13] describe a sensor for detecting antibodies (orange spheres). Here the resonator

surface is functionalized with the antigens, which just binds to one specific antibody (yellow structure).

AntigenAntibodyCantilever

CoatingParticle flow

Fig. 10: Working principle of a Biosensor to detect the concentration of antibodies in a sample. Drawing

based on [13].

In figure 10 it can be seen how the antigens are sitting on the resonator surface and bind to on specific

antibody. The mass increase caused by the measurand can be detected and the quantity of antibodies in a

sample can be measured.

For this kind of application it is also very important to have a high quality factor of the resonator, because

most biological compounds cannot be detected in air, but need to be suspended in fluid. [7, 10]

Infra-red sensor A resonating cantilever can be used to detect infra red radiation. Here the cantilever

is coated with an infra red absorbing material, which heats up and expands if hit by infra red light of

a certain wavelength. The expansion causes a strain in the material of the resonator and changes the

stiffness, which results in a shift of the resonance frequency. [21, 64]

IR-RaysBulk material

IR absorbing material (blue)

Thermal isolated area (yellow)

Cantilever

Electrode

Fig. 11: Design of a IR sensor. Drawing based on [21, 64].

Ono et al. [64] describe such a sensor, which can be seen in figure 11. It consists of a silicon cantilever,

which is coated by the IR absorbing material NiCr (blue layer) and is thermally isolated from the bulk

(yellow structure). The stiffness change from the thermal stress and the temperature change cause the

resonance frequency of the cantilever to shift, which can be detected.

27

2.6 The atomic force microscope

Gyroscope To detect movement in 3-dimensional space gyroscopes are used. These find application

for example in infotainment sector and in mobile phones. Most of them are based on a resonator, which

is actuated in one specific direction. If now a force acts perpendicular to that oscillation direction, the

resonator is deflected to a third direction because of the Coriolis force. [37]

Resonator

Actuator and sensing electrodes

Bulk material

Initial resonance mode (yellow)

Changed resonance mode (red)

Rotation vector

Fig. 12: Exploded view of a resonator gyroscope and its working principle. Drawing is based on [37].

Keymeulen et al. [37] describe a disk shaped resonator (bright blue structure) fixed with a stem in the

middle with an underlying matching electrode structure. This electrode excites in the inner part of the

disk and senses the deflection in the outer part of the disk (dark blue structure). The disk is driven in

wine-glass mode in the x-y-plain (yellow). In figure 12 it can be seen that as soon as a rotational motion

around the z-axis appears, the disk changes its vibration pattern (red), which is detected through changes

in the capacitance in the underlying electrode.

2.6. The atomic force microscope

Another application of flexural mode resonators is the atomic force microscope (AFM), which was

invented 1986 by Gerd Binnig and Heinrich Rohrer at the IBM research center in Zurich. They won the

nobel prize in 1986 for the scanning tunnelling microscope (STM), the predecessor of the AFM.[8, 27]

2.6.1. Working principle and operation modes

With this measurement instrument it is possible to scan a surface with a resolution in the range of single

atoms. In figure 13 the working principle of an AFM is explained.

A cantilever with a tip is moved over a sample in x- and y-direction and scans a defined area line by line.

In close proximity to the surface of the sample the tip with a radius of 2 nm - 5 nm interacts with surface

forces and the cantilever bends. This displacement in z-direction is measured by a laser reflected off

the cantilever. The signal is used in a feedback loop to adjust the piezoelectric stack, which moves the

cantilever in z-direction. Combining data form the x-, y- and z-direction a 3-dimensional picture of the

topography of the sample surface is created. [8, 27]

Several operating modes exist, where the three most important ones are the contact, non-contact and

tapping mode. [50]

28


DetectorLaser

Laser beam

Y-direction

Z-direction

Rotation

Cantilever

TipSample

Stage

Piezo stack

X-direction

Fig. 13: Working principle and set-up of an atomic force microscope.

Contact mode In the first mode the tip has contact to the sample surface. Repulsive short-range

atomic forces on the surface acting on the tip deflects the cantilever upwards. This deflection is kept

constant by using a feedback loop to move the cantilever according to the surface topography. This

movement of the cantilever in z-direction is transferred into the height information. Together with the

movement in the x-y-plain a 3-dimensional topography image is created. In order to decrease noise and

drift soft cantilevers are used, which deflect more easily to achieve a better signal from the deflection.

[26, 27]

The contact mode, however, damages the sample surface and the tip is subject to wear, because it is in

direct contact with the surface and basically is dragged over it. That is also the reason, why the contact

mode is not suitable for samples with soft surfaces. [26, 27]

Non-contact mode The cantilever is oscillated close to its resonance frequency and brought near the

surface, but does not come in contact with it. In a range between 1 nm - 10 nm above the surface the van

der Waals forces and other long-range forces, interact with the AFM tip and attract it. This attracting

force creates a stress in the cantilever material and as a result the frequency shifts. As soon as a change

in frequency or amplitude is detected a feedback loop adjusts the distance to the surface by moving the

cantilever in z-direction to keep the frequency or amplitude of the oscillation constant. This information

together with the in-plane movement over the sample is used to create a 3-dimensional image of the

surface topography. [26, 27]

The advantage here is that neither the sample, nor the cantilever tip are damaged in the process, because

they do not get in contact with each other. Also it is possible to measure soft surfaces, but it is not possible

to measure samples inside a liquid environment, because the cantilever hovers above the surface. A big

problem for non-contact mode sensing are liquid meniscus layers, which develop on the surface of most

samples. The layer thickness is in the range of the forces interacting with the cantilever tip. The problem

is to keep the tip close enough to still be affected by these short-ranged forces and at the same time

preventing the tip from being drawn down to the surface by the liquid meniscus covering the surface.

[26, 27]

Tapping mode The tapping mode was developed in order to overcome the problem the liquid meniscus

layers pose for the non-contact mode sensing. This mode combines the advantages of contact and non-

29


contact mode. Here the cantilever is oscillating close to its resonance frequency, the oscillation amplitude

is higher than in the non-contact mode and the tip comes nearly in contact with the surface. That means

also the shorter ranged forces like electrostatic forces, dipole-dipole interaction, as well as van der Waals

forces act on the tip and increase the frequency shift. A feedback loop uses this data to adjust height of the

cantilever in order to keep the amplitude of the oscillation constant. This information together with the

in-plane movement is used to create the topographical image of the sample surface. With this method it is

possible to analyse soft surfaces as well as samples in a liquid environment. Also in contrast to the contact

mode, the tip is not subject to wear as it never comes completely in contact with the surface. [26, 27]

2.6.2. Key data of a conventional AFM system

In order to receive the best possible measurement result a few important parameters have to be understood

and optimized. These parameters also show the physical limitation of the AFM system. This can be used

as starting point to improve the conventional AFM.

Spring constant Depending on the operation mode and sample the spring constant is between 10 N/m

and 100 N/m. Also, it should not be much softer than the surface force gradient of the probed structure,

otherwise the oscillation dynamics will be strongly non-linear and hard to analyse. [27, 49, 59]

Amplitude For most samples the range of interest is around 10 nm from the contact point of the sample

surface. In this range all important surface forces for the measurement can be found and their variation is

detected, especially the range of a few nanometers from the contact point the forces change rapidly.

For samples with a long range magnetic or electrostatic field, as well as in liquid medium with free-ions,

the range of interest can change influenced by these additional forces, which have to be taken into account.

[49, 59]

Resonance frequency The dimensions and the material determine the resonance frequency of a

cantilever. Usually the cantilever of an AFM is driven close to its resonance frequency, which typically is

between 60 kHz and 300 kHz. [27, 59]

Quality factor It is desirable to have a high quality factor, which has the advantage of good results in

air measurements and more importantly for measuring in a fluid environment.

Standard AFM cantilevers have usually a quality factor of around 500 in air. This low quality factor results

form the bending motion of the AFM cantilever, which has to move the surrounding medium in order to

osciallate creating high viscous damping. That is also the reason, why conventional AFM are rarely used

to analyse samples inside a fluid medium, because the damping caused by the surrounding medium would

degrade the quality factor very much along with the sensitivity of the whole system. [49, 59]

Thermal noise limit The detection method of a conventional AFM system is sensitive enough to

determine the thermal fluctuation of a certain moving effective mass point in air at room temperature and

therefore is limit by the thermal noise limit. [49]

In case of a conventional AFM the cantilevers have a thermal noise limit between 25 fN/√

Hz and 5 fN/√

Hz.

[49]

30


Resolution of the laser unit The laser system is used to detect the movement of the cantilever that

also limits the minimum size for the oscillator in order to provide enough width for the focused laser beam

to be reflected back to the detection unit. The sensitivity range of available laser units is between 50 and

150 fm/√

Hz.

Table 3 shows the most important parameters for the conventional AFM.

Parameter Values for conventional AFM

Spring constant 10 – 100 N/m

Amplitude 10 nm

Resonance frequency 60 – 300 kHz

Quality factor 500

Thermal noise limit 5 – 25 fN/√

Hz

Laser sensitivity 50 – 150 fm/√

Hz

Tab. 1: Summary of the key data for a conventional AFM system.

2.6.3. Improvements of the atomic force microscope utilizing bulk-mode resonators

Even though the AFM has a resolution down to the atomic level, there is still much room for improvement

of resolution sensitivity, noise reduction, as well as the analysis speed of the measurement. In the

following, two methods are described, which redesign the resonator and sensing part of the AFM in order

to reduce the size and at the same time increase the resolution, sensitivity and reduce the noise in the

resonator system.

Longitudinal mode resonator sensing unit Heike et al. [31], An et al.[2], Giessibl et al. [28]

and several others examined the possibility to replace the oscillating cantilever and the laser system of a

conventional AFM with a longitudinal mode resonator, where a piezoelectric actuator excites and senses

the movement of the oscillator.

AFM tip

Longitudinal resonator

Piezo electrodes

Anchor

Bulk material

Fig. 14: Piezoelectric actuated longitudinal resonator as AFM transducer. Drawing based on [28].

As seen in figure 14 the resonator is fixed at the nodal points in the middle of the bulk material, where

the piezoelectric actuator is placed. The resonator itself has a length of 1340 µm, a width of 136 µm and

a thickness of 70 µm. A tip for analysing the sample surface is glued to both ends of the bulk material

31


to distribute the weight symmetrically on both sides of the structure. The resonance frequency of the

longitudinal mode resonator can be increased to 1 MHz with a quality factor of 15000 at room temperature

and in air. The system has a high stiffness of 1 MN/m and resonates at an amplitude of around 100 pm. All

this increases the sensitivity of the AFM, reduces the noise and increases the operation speed. In addition

to that in the scanning tunnelling microscope (STM) mode the bending of the cantilever at changing

voltage can be prevented by the use of this system. [28, 31]

Ring resonator sensing unit Another design is described by Faucher et al. [1, 22, 65, 69]. Here a

ring resonator is used as sensing unit for the AFM to overcome the fabrication drawbacks of the piezo

crystal actuated device proposed by Heike et al. [31]. It is actuated and sensed by capacitive transducers,

where the electrodes are in the area of the largest displacement and have an AC actuation potential of

500 mV and a DC biasing of 12 V. The four nodal points of the actuated wine-glass mode are used for

anchoring the resonator and the AFM tip is added at one side. The tip will negatively influence the quality

factor, because the structure is not symmetric anymore.

Bulk material

Ring resonator

Actuation electrode

Actuated ring resonator (blue)

Sample surface

Oscillation in z-direction

Fig. 15: Capacitive actuated ring resonator as AFM transducer. Drawing based on [1].

The ring itself has a diameter of 250 µm and a ring width of 50 µm. With electrostatic actuation a resulting

peak amplitude for the oscillation of 2.3 nm can be reached at a resonance frequency of around 1.1 MHz.

This device is used in air and was also tested in fluid environment. The quality factor under atmospheric

pressure air is only 500, which results probably from the asymmetrically added AFM tip. For this system

force resolution is 0.2 nN/√

Hz and it is claimed that this value is two orders of magnitude higher than for

high-performance AFM probes. [1]

32

3 GENERAL ANALYSIS OF BULK ACOUSTIC RESONATORS FOR AFM

3. General analysis of bulk acoustic resonators for AFM

Bulk resonators can be used for many different applications. Examples for longitudinal and wine-glass

mode resonators used in AFM are described by Giessibl et al. [28] and by Faucher et al. [1].

In order to design a novel device a general investigation of extensional mode resonators has to be done.

The relation between the important parameters for the specific case of an AFM have to be found and

understood with the help of simulations.

The shape of the resonator is defined first and in what mode it is supposed to resonate in order to determine

the anchoring system, as well as the electrode placement for actuation and sensing. Figure 16 shows the

most common and practical shapes used, which promise the best performance.

Air gap width

Air

gap

wid

th

Air

gap

wid

th

Air

gap

wid

th

Diam

eter

Diam

eter

Length

Len

gth

Width Width

Width

Length

Air

gap

wid

th

Displacement

Displacement Displacement Displacement Displacement

Longitudinal Disk Ring Square Frame

Fig. 16: All shapes with dimensions used for the simulations.

The rectangular shape is resonating in the longitudinal mode and therefore has only two nodal points,

where anchoring is possible. The dimensions are determined by the length and width, because the

thickness for all shapes is considered constant at 10 µm due to the device layer thickness of th available

SOI wafer.

All the other shapes in figure 16 are resonating in a wine-glass mode and therefore have four nodal points

to anchor the free etched structure to the bulk material. The dimensions of the disk and ring resonator are

determined by the diameter and in case of the ring also by a ring width, which is defined as

ring width = outer radius− inner radius (17)

A square and frame shape are also possible, with the dimensions being determined by the side length for

the square and in addition the frame width for the frame.

All resonators are actuated electrostatically and therefore electrodes are used, which are separated from

the resonator by a certain air gap. The electrodes are placed in the area with the largest displacement. To

receive the best performance two electrodes are used for symmetry reason. In case of the longitudinal

resonator only one electrode is used at the top, because the bottom side with large displacement should be

kept free in order to have the possibility of attaching an AFM tip. To guarantee efficient actuation the

electrode area is adjusted to the resonator size in order to receive the largest possible area and therefore

have the largest possible electrode capacitance. The actuation voltage on the electrodes stay constant at 12

33

3.1 Analysis of disk and ring shaped resonators

V.

All the dimensional parameters can be varied and their effect on the stiffness, frequency and displacement

have to be investigated. The stiffness in combination with the dimensions is important for the quality factor

of the system. The absolute frequency or the frequency shift is detected capacitively and is analysed as the

output signal of the system. If the resonator is used directly as sensing unit for an AFM the displacement is

also important to determine the amplitude of the tip to get an idea about which surface forces are possible

to measure.

In the following the different shapes are analysed according to these parameters. This will be done with

the help of simulating different set-ups in COMSOL. A description about COMSOL and the different

models used can be found in appendix A.

3.1. Analysis of disk and ring shaped resonators

All wine-glass mode shapes have a similar behaviour, because of their four anchors and the same symmetry

in the system especially between the two filled and the two not filled shapes. To find the most suitable

shape for the AFM application, all these geometries are simulated and their performance is compared

based on several parameters, which are important for this specific application.

3.1.1. Set-up and simulation of wine-glass mode resonators

In order to determine the important relations a few simulations are performed for the disk, ring, square

and frame shaped resonators.

Stiffness in relation to dimensions First it is important to know how the stiffness changes with

variation of radius and ring width. For that a simple simulation is used, where a constant static force

is applied in the area of the electrodes. The stiffness is then calculated from the applied force and the

resulting displacement. Here the following equation can be used.

kc =F

x(18)

Where kc is the stiffness, F the applied force and x the displacement of the resonator resulting from the

force.

The diameter of the disk resonator and the side length of the square resonator are varied from 40 µm to

200 µm and all other parameters are kept constant. The values used for the diameter in case of the ring

shape and the length in case of the frame shape with their corresponding change in width are shown in the

following table.

Ring diameter Ring width variation Frame length Frame width variation

80 µm 5 µm – 20 µm 80 µm 5 µm – 20 µm

120 µm 5 µm – 40 µm 120 µm 5 µm – 40 µm

160 µm 5 µm – 60 µm 160 µm 5 µm – 60 µm

200 µm 5 µm – 80 µm 200 µm 5 µm – 80 µm

Tab. 2: Parameter change for ring and frame shape simulations.

There are also several different ways to compare the resonators, for example tuning the dimensions in a

34


way to keep the resonance frequency constant. Another way would be to use the same distance between

the anchor points for all resonators. But in this case the resonators are compared according to their size in

order to occupy the same area and use the same package size.

For a high ratio between width and diameter respectively length problems in the fabrication can occur,

especially for narrow rings or frames with a width of 5 µm or smaller. Also the stability of the resonator

can be critical for a high ratio, like 200 µm diameter and 5 µm width. The simulations are done to cover

all cases, whether each specific case is reasonable to fabricate has to be determined in a separate step.

Resonance frequency in relation to dimensions The next simulation step is to find out how the

resonance frequency of a resonator changes with a variation in dimensions. For this the Eigenfrequency

analysis in COMSOL is used and the resonance frequency is determined while the diameter of the ring

shape and the side length of the square resonator are varied, as well as the ring and frame width, the values

used for the different shapes can be seen in table 2. Here the same considerations for the fabrications have

to be taken into account as in the last simulation.

Displacement in relation to air gap width A third series of simulations is done to determine how

the amplitude of the resonator is affected by the gap between the actuation electrodes and the resonator.

Here only the disk resonator is tested, because the gap width will have the same effect on all other

structures. The air gap is varied between 0.1 µm and 1 µm. This interval corresponds to the current state

of the art technology, where 1 µm can still be fabricated with the DRIE process. This process however has

the disadvantage of scalloping of the side-walls and can not be used for smaller gap widths, because of

non uniform distribution of the electric field. Here a different technique has to be used, where a very thin

sacrificial layer is deposited in the fabrication process and is removed at the end to achieve small gaps up

to 0.1 µm. An AC voltage of 12 V is applied on the two electrodes and the resonator structure has ground

potential. Furthermore damping is applied to achieve a constant Q·f product on the tested resonators of

6.6·1011 Hz, which for example corresponds to a disk with diameter of 80 µm and a resonance frequency

of 66 MHz to achieve a quality factor of 10000. The Q·f product is used as standard comparison method

for resonators in literature.

Displacement in relation to dimensions To determine the displacement caused by the change

in dimensions a last set of simulations are performed. Here a constant gap width of 0.5 µm is chosen,

because it is a relatively easy to fabricate and at the same time distinct non-linear effect of the electrostatic

force can be seen. Also for smaller gap widths the simulations in COMSOL take exponentially more

time, if the results converge at all, so it was decided to use 0.5 µm gap width as a compromise instead of

0.1 µm. In this simulation also a constant AC voltage of 12 V is applied to the electrodes and the Q·f

product is kept constant for all tested resonators. The parameter variation in case of the ring and frame

shape can be seen in table 2. With each change in dimension the resonance frequency also changes that is

why the frequencies found for each dimension are used in this simulation in order to find the maximum

displacement with the corresponding resonance frequency. Considerations from the fabrication point of

view have to be taken, while analysing the results from this simulation.

35


3.1.2. Results of wine-glass mode resonators

The results from calculations and simulations should help compare disk and square, respectively ring

and frame shaped resonators and determine, which is the more suitable for this specific application of

replacing cantilevers and laser detection unit of a conventional AFM.

First the stiffness of these shapes is simulated. In figure 17 the stiffness of disk and square shape is plotted

depending on the diameter and the side length of the square resonator. This results in a different resonator

area, different distance between the anchoring points and a different resonance frequency for each length

in comparison to the diameter. But this way of comparing the shapes guarantees the same device size and

package for the application.

50 100 150 200

2,0x106

2,5x106

3,0x106

3,5x106

Sti

ffnes

s[N

/m]

Diameter (Length) [µm]

DiskSquare

Fig. 17: Stiffness in dependency of diameter and length of the disk and square shape respectively.

The difference in area is the reason for the different stiffness between disk and square shape, shown in

figure 17. It also can be seen that for increasing resonator size, the stiffness also increases. The curves

have parabolic character, which means that there is a square dependency between the resonator dimension

and the stiffness, which results from the area of the oscillating structure.

This result can also be seen in figure 18, where ring and frame shaped resonators are compared, for

different diameter and length respectively with the fill ratio, which is defined as

ring fill ratio =ring width

outer radiusand frame fill ratio =

frame width

frame length 2. (19)

As fill ratio close to 0 means the width goes to 0 and therefore the ring is not filled with bulk material at

all. A fill ratio of 1 means the width is as large as the radius and therefore the ring is filled completely and

becomes a solid disk.

A result can be concluded that with increasing width and increasing diameter or length, the stiffness also

increases, which means the closer the resonator comes to a complete filled disk or square shape and the

larger it is and the higher is the stiffness. Comparing ring and frame shape it can be said that the frame

shape, like the square shape, has a higher stiffness at the same dimensions, because of the larger resonator

area.

36


0

50

100

150

200

00.5x10

61.0x10

6

1.5x106

2.0x106

2.5x106

3.0x106

0

0.2

0.4

0.6

0.81.0

Ring D = 80 µmRing D = 120 µmRing D = 160 µmRing D = 200 µmFrame L = 80 µmFrame L = 120 µmFrame L = 160 µmFrame L = 200 µm

Sti

ffnes

s[N

/m]

Fillra

tioDiameter (Length) [µm]

Fig. 18: Stiffness of ring and frame shape with variation of width at a certain diameter (length).

According to equation 18 this means that square and frame shape should have a higher resonance frequency

at the same dimensions as disk and ring shape, because their stiffness is higher. But the larger area results

in an increase in mass of the resonator, which decreases the resonance frequency. This effect is stronger

than the effect of the stiffness and as can be seen in figure 19 the resonance frequency of the frame shape

is lower in comparison to the ring shape. This effect will be the same for the disk in comparison to the

square shape, which can be seen as a ring (frame) with a large fill ratio.

The resonance frequency of all shapes decreases slightly non-linear with increasing width and diameter or

length. The curve slopes of the square and frame shaped resonators are not as steep as the slopes of their

round counterparts. Usually the resonance frequency increases with decreasing mass, which is the case

for decreasing the fill ratio. But this is accompanied by a decrease in stiffness, which dominates so the

resonance frequency decreases even with decreasing mass. These are non-linear effects so it is hard to

predict the exact behaviour of a structure with both effects combined.

This could also be one possible explanation for the counter intuitive result of an increase in resonance

frequency of the ring shaped resonator after a minimum at a small width of around 6 µm. Here the effect

of the decreasing mass can start to dominate over the stiffness decreasing effect, which results in the rise

of resonance frequency. But what can not explained completely is the fact that this occurs only for the

ring shape, but not for the frame shape.

Another explanation could be found in the equations used in COMSOL for calculating the simulation. A

detailed analysis of the way COMSOL calculates this problem could reveal the reason for increase of in

resonant frequency close to a fill ratio of 0.

Now that the resonance frequency is known for the different shapes, the resonator can be actuated

electrostatically with a fitting AC signal. The air gap width between the electrode and the resonator has a

non-linear influence on the actuation force (Fel) created by the AC signal on the electrode, as can be seen

in the following equation 20.

Fel = −ǫ0ǫrAU

2

2x2g(20)

37


040

80

120

160

200

5

10

15

20

25

30

0.0

0.2

0.4

0.6

0.8

1.0

Ring D = 80 µ

Ring D = 120 µm

Ring D = 160 µm

Ring D = 200 µm

Frame F = 80 µm

Frame F = 120 µm

Frame F = 160 µm

Frame F = 200 µm

Res

onan

tfr

equen

cy[M

Hz]

Fillra

tioDiameter (Length) [µm]

Fig. 19: Resonance frequency of ring and frame shape with variation of width at a certain diameter

(length).

Where ǫ0 is the vacuum permittivity, ǫr relative permittivity, A the electrode area, U the electric potential

and xg the actual gap distance.

0 0.2 0.4 0.6 0.8 1.00.01

0.1

1

10

100

Dis

pla

cem

ent

[nm

]

Air gap width [µm]

Disk D = 80 µmDisk D = 120 µmDisk D = 160 µmDisk D = 200 µm

Fig. 20: Logarithmic scale displacement of a disk shaped resonator with different diameters against the

air gap width between resonator and electrode.

As can be seen in the logarithmic plot in figure 20 the displacement increase is highly non-linear, caused

by the increase in electrostatic force, with decreasing air gap width. This influence is the same for all

shapes and dimensions, because it does not change the mass or stiffness of the resonator and therefore

does not influence the resonance frequency. That is why only a disk shaped resonator is simulated to show

this dependency. The displacement, however, depends on the size of the resonator, which can be seen in

figure 20. This dependency between the resonator dimensions and the displacement is of major interest

38


for the application in AFM and is now investigated in detail and the result of the simulations for the ring

and frame shape are shown.

The results for the ring shape resonator can be seen in figure 21, where the resonance frequency in

dependency of a variation in fill ratio for several different diameters is shown. Instead of the ring width a

fill ratio is used like in the graphs before to be able to compare all shapes.

040

80

120

160

200

0

0.03

0.06

0.09

0.12

0.15

0

0.2

0.4

0.6

0.8

1.0

Ring D = 80 µm

Ring D = 120 µm

Ring D = 160 µm

Ring D = 200 µm

Dis

pla

cem

ent

[µm

]

Fillra

tioDiameter [µm]

Fig. 21: Dependency between dimension and displacement for the ring shaped resonator.

The ring shaped resonator shows a distinct maximum in displacement for a fill ratio between 40% and

50%, corresponding to a width between 30 µm and 40 µm. This maximum is more pronounced the more

the ring diameter increases, and shows a displacement of up to 120 nm for the simulated dimensions.

The frame shape in comparison has one distinct maximum of displacement close to 0% fill ratio, which

can be seen in figure 22. This is not taken into account, because it requires structures smaller than the

minimum feature size and therefore can not be fabricated. But a second maximum appears at around 60%

fill ratio of the frame with length 120 µm and 160 µm with a displacement at this point of around 1.5

nm. The displacement of the frame shaped resonator, is around two orders of magnitude lower than the

displacement of the ring maximum.

There was no definite explanation found for why these distinct maxima occur at a specific fill ratio, but

one possible explanation could be that at specific fill ratio there is an energetic optimum, where most of

the energy received by the actuation can be transferred into deformation of the material. Here the ratio

between the stiffness decreasing with decreasing fill factor and the energy storage capability increasing

with increasing fill factor create a maximum in displacement.

These results, however, have to be regarded with care, because using a very steep resonance curve of a

high quality factor resonators, a small change in the frequency can have a large effect on the displacement.

The simulations are done by using the resonance frequency calculated by the eigenfrequency solver of

COMSOL. The received values are bound to rounding errors, which will have a small influence on the

simulation of displacement.

39


020

4060

80100

120140

160180

200

0

0.0015

0.0030

0.0045

0.0060

0

0.2

0.4

0.6

0.8

1.0

Frame L = 80 µm

Frame L = 120 µm

Frame L = 160 µm

Dis

pla

cem

ent

[µm

]

Fillra

tio

Length [µm]

Fig. 22: Dependency between dimension and displacement for the frame shaped resonator.

3.1.3. Conclusion of wine-glass mode resonators

In summary it can be stated that the square and frame shape resonators have a higher stiffness than the

disk and ring shaped ones, because of their overall larger volume due to the way of comparison of these

different geometries in this investigation. Independent of the shape the stiffness increases with the size

and the amount of bulk material the resonator consists of.

The resonance frequency, however, is lower and the curve slopes are not as steep for the square and

frame shape resonator, which can be explained by the fact that these resonators are stiffer than the round

counterparts, but mass increase negatively influences the resonance frequency.

This can also be seen in displacement of the resonators, where the disk and ring shapes have a much

higher displacement than the square and frame shape. For the AFM application it can be concluded that

disk and ring shapes are preferred over square and frame shapes, because a higher resonance frequency, as

well as a higher displacement can be realized with these shapes.

Comparing filled shapes and shapes with a hole it can be concluded that the disk and square shapes have a

higher resonance frequency and stiffness, but ring and frame structures have a much higher displacement.

Depending on the requirements of the resonator, different shapes can be chosen. If the goal of an

application is to have high displacement with a wine-glass mode resonator, as narrow gap as possible

between the electrode and the resonator is recommended, as well as a ring shaped resonator with a fill

factor of around 40%, depending on the diameter.

A disk or square resonator, however, will be the shape of choice, if high resonance frequency is required

and a displacement of a few nanometers is sufficient. The displacement can always be improved by

reducing the gap between the electrode and the resonator to receive a higher excitation force.

40

3.2 Analysis of longitudinal mode resonators

3.2. Analysis of longitudinal mode resonators

The second kind of resonators which is investigated is the longitudinal mode resonator. The length and

width are varied in the simulations with otherwise constant simulation condition in order to determine the

best shape for the AFM application.

3.2.1. Set-up and simulation of longitudinal resonators

For the longitudinal resonator similar simulation as for the wine-glass mode resonators have to be

performed to evaluate its performance and make a prediction on its potential.

Stiffness in relation to dimensions A constant force is set to act on both sides of the resonator

causing a displacement of the structure. The stiffness can be determined by calculating the displacement

for an applied force with equation 18. For the simulation the length is varied from 50 µm to 300 µm and

the width from 5 µm to 150 µm. In this case again as in the ring and frame structure the designs with a

large ratio between length and width have to be assessed in terms of fabrication.

Resonance frequency relation to dimensions The resonance frequency of a resonator is de-

pending on its stiffness and the mass. The dimensions of the structure influences both parameters and

change the resonance frequency, which is determined by simulation. Firstly the width is varied from 5

µm to 150 µm for a length of 100 µm, 150 µm, 200 µm, 250 µm and 300 µm. This is followed by a

simulation where the length is varied between 50 µm and 300 µm for width of 10 µm, 60 µm, 110 µm

and 160 µm. This is done with the eigenfrequency solver in COMSOL.

Displacement in relation to air gap width For the longitudinal resonator it is also simulated how

the displacement influences the change in gap width between actuation electrode and oscillator. It is

expected that the effect will be similar to the wine-glass mode resonators, but not as strong, because

only one electrode is used for actuation in case of the longitudinal resonator, which means half of the

electrostatic force is acting on the structure in comparison to the other group of resonators.

For the simulation a resonator with a length of 300 µm, a width of 20 µm, a gap width of 0.5 µm and a

actuation voltage of 12 V is used. The displacement is measured at the small side of the structure on the

side with no electrode. Also damping is applied to achieve for a constant Q · f product.

Displacement in relation to dimensions For the last simulation the resonance frequencies found

for the different dimensions are used to find the displacement of this structure with a gap between electrode

and resonator of 0.5 µm, an applied AC voltage of 12 V on the electrodes as well as a constant Q · f

product. For the simulation itself first the width is varied from 5 µm to 150 µm for a length of 100 µm,

150 µm, 200 µm, 250 µm and 300 µm. This is followed by a simulation where the length is varied

between 50 µm and 300 µm for width of 10 µm, 60 µm, 110 µm and 160 µm.

3.2.2. Results of longitudinal resonators

Longitudinal resonators are also a good candidate for use in an AFM application. The question is how to

find the best dimensions for the application. The thickness is already defined by the device layer thickness

41


of the SOI wafer used for fabrication. That leaves to find the best ratio between length and width of the

resonator.

First the stiffness is determined in dependency of length and width of the longitudinal shape.

050

100

150

200

250

300

0

2.5x106

5.0x106

7.5x106

10x106

0

50

100

150

Sti

ffnes

s[N

/m]

Wid

th[µ

m]

Length [µm]

Fig. 23: Stiffness development in dependency of length and width of a longitudinal resonator.

As can be seen in figure 23 the stiffness increases with decreasing length and increasing width. The

decrease in length alone increases the stiffness non-linearly and an increase in width increases the stiffness

linearly. From this the following relationship can be found

kc ∼w

L3(21)

Where w is the width and L is the length of the resonator. In this equation the width contributes linearly,

whereas the length is accounted for in the third power. The increasing stiffness with increasing width

can be explained with the anchor points, which are fixed to the surrounding bulk material. According

to Poisson’s ratio the volume of the resonator stays constant while oscillating that means if the length

of the structure is extended, the width is shortened and the other way around. With increasing width the

displacement increases in x-direction, but the anchors prevent a free oscillation and thus strain is build up

in the structure and increases its stiffness.

As can be seen in figure 24 the resonance frequency increases with decreasing length and width.

Analysing the results in more detail it can be seen that the behaviour of the resonator changes when the

dimensions approach a square shape, which means the length and width are equal. This trend continues

with the length becoming smaller than the width. This can be seen for example at constant length with

increasing width. First the resonance frequency stays constant, but with approaching square shape the

frequency starts to decrease. A similar effect can be seen with constant width and decreasing length. At

first the resonance frequency increases non-linear, but after the dimensions come close to square shape the

inclination of the slope starts to become much smaller.

42


50

100

150

200

250

300

0

20

40

60

80

100

150

120

90

60

30

0

Res

onan

tfr

equen

cy[M

Hz]

Wid

th[µ

m]

Length [µm]

Fig. 24: Resonance frequency distribution of a longitudinal resonator depending on its length and

width.

widthwidth

width

length

length

length

Length > width Length = width Length < widtha) b) c)

Anchor length

Anchor width

Fig. 25: Influence of the anchor points on side displacement for a) length >width; b) length = width; c)

length <width.

This unexpected change in resonance frequency is caused by the resonance pattern of the longitudinal

mode and the anchoring at the two nodal points. As explanation, the Poisson’s ratio also has to be taken

into account. If the length is longer than the width the largest displacement is on the two free ends at the

top and the bottom of the resonator as can be seen in figure 25a. The displacement on the sides is very

low, because of the small width in comparison to the length. As the value of the dimensions approach

each other the displacement on the sides increases, but is hindered by the two anchors, and the structure

can not oscillate as freely anymore, which can be seen in figure 25b. The larger the width becomes in

comparison to the length the more pronounced is the effect, which means the displacement in the free

direction to the top and bottom is very small and to the side it increases, but is blocked by the anchors,

which are fixed to the surrounding bulk structure, as can be seen in figure 25c.

43


The longitudinal resonator is actuated, like the wine-glass mode resonators, with an AC signal applied to

an electrode located in the area with the most displacement. In case of the longitudinal resonator only one

electrode is used, which decreases the effect on the displacement in comparison to the two electrodes in

case of the wine-glass resonators.

0,0 0,2 0,4 0,6 0,8 1,01E-4

1E-3

0,01

0,1D

ispla

cem

ent

[nm

]

Air gap width [µm]

Longitudinal L = 100 µm, W = 50 µm

Fig. 26: Logarithmic plot of the dependency between air gap width and displacement with a 12 V AC

signal.

A still strong non-linear dependency between the gap width of the electrode and the resonator in depen-

dency to the displacement as can be seen on the logarithmic plot in figure 26. The slope, however, is not

as steep as in case of the two electrode of the wine-glass mode resonators, which in comparison can be

seen in figure 20.

In order to determine the displacement of a longitudinal resonator at certain length and width the resonance

frequency simulated previously is used. The result can be seen in figure 27.

030

6090

120150

50100

150200

250300

0

0.1

0.2

0.3

0.4

0.5

Dis

pla

cem

ent

[nm

]

Length [µm] Wid

th[µ

m]

Fig. 27: Displacement distribution in dependency of length and width of a longitudinal resonator.

44


In total the displacement increases with increasing length and decreasing width of the resonator. For

constant width and variation in length, as well as for constant length and variation in width there exists a

non-linear dependency between the dimensions and the displacement. The increasing displacement with

decreasing width can be explained by the Poisson’s ratio, where the volume of the structure is bound to

remain constant during the displacement. In figure 25 it can be seen that the displacement of the structure

in x-direction as well as in y-direction increases with increasing width. This increases the restricting effect

of the anchoring system fixed to the surrounding bulk material and so the resonator can not oscillate freely

in x-direction. The effect is stronger the more the width of the resonator increases. That means the longer

and thinner the resonator is, the higher is its displacement and the less influence has this effect on the

resonator, because the displacement at the sides becomes very small and therefore the resonator is less

restricted in its movement.

In order to investigate the effect of the anchors on the resonance frequency of the system, the dimensions

of the longitudinal resonator stay constant and the width and length of the anchors are varied.

05

10

15

20

25

21.8

22.0

22.2

22.4

22.6

0

2

4

6

810

Fre

quen

cy[M

Hz]

Wid

thof one an

chor [µ

m]

Length of one anchor [µm]

Fig. 28: Dependency between the anchor dimensions and the frequency of the resonator.

It can be seen in figure 28 that the resonance frequency of the system is influenced by the dimensions of

the anchors. The shorter and the wider the anchors are, the higher is the frequency of the resonator. For

the same dimensions of the oscillator an increase in resonance frequency means an increase in stiffness,

which leads to a decreased displacement of the system, as can be seen in the previous simulation results.

The maximum displacement reached for the total system is determined to be in the range of 500 pm with

one actuation electrode and a air gap width of 0.5 µm. This value, however, should be regarded with care,

because of the short comings of the COMSOL simulations as described in the results for the wine-glass

mode shapes.

45

3.3 Conclusion for wine-glass and longitudinal resonators

3.2.3. Conclusion of longitudinal resonators

In summary it can be said that the longitudinal resonator scales nicely with the dimensions. The smaller

the resonator, the higher is the resonance frequency and the longer and narrower the structure is, the higher

the displacement. That means also that the performance in case of resonance frequency and displacement

is only limited by minimum feature size and therefore the fabrication processes. For example a resonator

with a length of 300 µm and a width of 2µm has a high displacement, but will be very unstable and very

difficult to fabricate. For the length a trade-off has to be made between the displacement and the desired

resonance frequency, whereas the width has to be minimized in both cases and depends on the minimum

feature size.

The anchors also have a strong influence on the performance of the device, which becomes even larger

with increasing width of the resonator, especially when dimensions approach the square shape with equal

length and width and even further when the width becomes larger than the length of the resonator.

In order to increase the displacement the gap width between the actuation electrode and the resonator has

to be decreased. This has to be done in order to achieve a reasonable displacement for use in an AFM

system.

3.3. Conclusion for wine-glass and longitudinal resonators

Considering all simulation results of the tested geometries it can be said that for an AFM application the

ring shape with around 40% fill ratio will be the shape of choice. It has the highest displacement and also

a high enough resonance frequency in order to replace the cantilever and the laser unit of a conventional

AFM system. The exact ratio between diameter and width have to be determined in further simulations in

order to find the desired displacement and frequency.

The longitudinal resonator has a much higher resonance frequency compared to the wine-glass mode

resonators, but at the same time a much smaller displacement. To improve the displacement either the air

gap between actuation electrode and the resonator has to be as small as possible or a different actuation

method should be used to achieve a higher displacement.

As alternative to the tested electrostatic method piezoelectric actuation could be used, which would help

to increase the displacement of the resonator with keeping the same resonance frequency. But at the same

time the quality factor will decrease, because different materials are combined to create the piezoelectric

effect and therefore cause an inhomogeneity in the resonator, which decreases the energy storage capacity.

For piezoelectric actuation, however, the devices have to be simulated again in order to determine all

parameters and compare the performance to the eletrostatic actuation.

46

4 TWO ELECTROSTACTICALLY COUPLED RESONATORS AS AFM FORCE SENSOR

4. Two electrostactically coupled resonators as AFM force sensor

Using the knowledge gained from studying the improvements for AFM described by Giessibl et al. [28]

and Faucher et al. [1, 22, 65, 69] and the results gained by simulating the behaviour of different resonator

shapes in the previous chapter, a new kind of force sensor for use in an AFM is introduced, described and

investigated.

4.1. Concept and design ideas for an electrostatically coupled

force-to-frequency transducer

The concept consists of two different electrostatically coupled resonator. One resonator is designed

with parameters close to a conventional AFM cantilever resonator and oscillates in flexural mode. The

second one is a bulk acoustic wave resonator oscillating in wine-glass mode and is used to detect the

movement of the first resonator, thus replacing read-out function of a laser in a conventional AFM system.

Applying DC potential on the flexural mode resonator and the variation in air gap width between the two

resonators influence the stiffness and thus the frequency of the bulk resonator, which is detected and used

as information in z-direction creating a topographical image of the scanned surface by an AFM, together

with the information about the sample movement in x- and y- direction.

In the following the flexural mode resonator (see figure 29) with attached tip for detecting long and short

range atomic forces on the surface is referred to as AFM transducer, and the bulk acoustic wave resonator

(see figure 29) detecting the movement of the AFM transducer caused by interaction with surface forces is

referred to as BAW detector.

wine-glass resonance mode

BAW disk resonator

Flexural resonance mode

AFM flexural resonator

AFM transducer

BAW detector

Fig. 29: Set-up of the complete system with AFM transducer and BAW detector.

The surface forces interacting with the AFM tip happen in the range of up to 10 nm from the sample

surface. Thus this range is the most interesting for the device. Therefore the AFM transducer is designed

for an amplitude in that range, as shown in figure 29, and the AFM system is oscillating close to its

resonance frequency. Forces acting on the system will shift the frequency of the AFM transducer, which is

similar to the working principle of a conventional AFM cantilever. In addition the structure is biased with

a DC voltage to cause an electromechanical pre-stress in the BAW detector, which is separated by a small

47

4.1 Concept and design ideas for an electrostatically coupled force-to-frequency transducer

air gap from the AFM transducer. These two mechanisms cause a shift in the resonance frequency of the

BAW detector depending on the oscillation position of the AFM transducer, resulting from the variation

in air gap distance between the two resonators. The gap and the DC bias voltage are used to couple the

two mechanical systems electrostatically, which enables sensing of the AFM resonator by frequency

shift of the BAW detector. The electrostatic force created by the DC voltage over the air gap is used to

pre-stress the bulk resonator material, which changes its stiffness and its resonance frequency changes.

This oscillation change in the BAW detector is detected and can be used to create a 3-dimensional image

together with the sample movement in x- and y-direction.

The most important aspect for the sensor performance is the coupling between the two resonators. To

achieve a high effective coupling the air gap has to be designed as small as feasible, which is limited by

the fabrication process. With state-of-the-art technology a gap width of 100 nm can be realized. With

the AFM transducer oscillating with an amplitude of 10 nm, the frequency shift resulting from the DC

pre-stress in a gap width of 90 to 110 nm has to be large enough to result in a high enough sensitivity for

detecting the smallest change in surface force acting on the AFM-transducer.

Figure 30 summarizes the sensitivity consideration of the complete device with all input and output signals

of the whole system as well as the different parts.

∆g

∆F

∆f

∆g

AFM transducer BAW detector

∆F ∆f

∆f

∆Fg

∆g

Fig. 30: Device flow chart of influencing factors of the different parts and the complete system.

A change in surface forces ∆F acting on the oscillating AFM transducer results in a change in gap width

∆g between the two resonators. The sensitivity of this first transducer is given as gap width change per

force change ∆g/∆F . The frequency shift of the BAW detector ∆f is created by the gap width change

∆g and is controlled by the initial gap width g. The sensitivity of the BAW detector can be expressed as

frequency change per gap width change ∆f/∆g. That results in an overall sensitivity of ∆f/∆F , where the

surface force change of the AFM transducer ∆F is the input signal and the frequency change of the BAW

detector ∆f is the output signal of the complete sensor system.

In order to determine the performance of the system the different noise sources have to be investigated.

The most important noise sources are the mechanical thermal noise of both resonators and the electrical

noise of the DC-biasing on the AFM transducer electrode. Other sources like noise from the AC excitation

and noise created by the frequency read-out equipment will also have influence on the performance of the

system, but are not considered here, as they either can be neglected or can not be influenced by the device

design itself and therefore have to be investigated separately.

48


4.1.1. AFM transducer of the electrostatically coupled sensor system

Several design parameters have to be considered in order to achieve the same or better performance as

compared to conventional AFM resonators.

Amplitude

Gap width

Actuation Electrodes

Suspension initial position

Suspended structure

Etch holes

Anchor

Spring length

Fig. 31: Set-up and working principle of the AFM transducer with important parameters.

In figure 31 the function of the AFM transducer is shown. Important parameters, which have to be matched

by the AFM transducer, are the spring constant, amplitude, resonance frequency, quality factor and the

thermal noise limit of the cantilever for conventional AFM systems. [49]

Spring constant The spring constant of conventional AFM cantilevers is between 10 N/m and 100

N/m. This value is used as the starting point for designing the suspension of the AFM transducer. It is

determined by the beam length and width of the spring system, whereas the thickness is already given by

the device layer thickness of the SOI wafer. [49]

Amplitude The 10 nm region around the sample surface is of most interest, which is why the AFM

transducer beam length is tuned to have a displacement of 10 nm and therefore an amplitude of 20 nm

with a spring constant of 100 N/m is required in order to provide with good performance. [49]

Quality factor In contrast to a conventional AFM cantilever, the AFM transducer has no limitations

in dimensions for providing an area for a laser beam to be reflected. With smaller size of the resonator,

viscous losses are reduced and the quality factor increases. To resemble a conventional AFM system

the quality factor is designed to be at least 500. The higher the quality factor is the better will be the

performance of the sensor. [49, 59]

Resonance frequency In order to achieve a high resolution the following condition has to be

fullfilled.f0, BAW

QBAW>

f0, AFM

QAFM(22)

Where f0, BAW is the resonance frequency of the BAW detector resonator, QBAW is the quality factor of

the BAW resonator, f0, AFM is the resonance frequency of the AFM transducer resonator and QAFM is

the quality factor of the AFM resonator. These terms show the bandwidth of the resonator system. The

49


larger the bandwidth the faster is the response time of the system. The BAW resonator can not follow

the frequency of the AFM resonator if its response time is larger than the one of the AFM resonator,

which would result in a decrease in time resolution and thus scanning speed of the sensor. That is why a

resonance frequency of 300 kHz and a quality factor of 500 are chosen for the AFM resonator, which

resembles the parameters of a conventional AFM cantilever. [27]

Thermal noise limit The question that has to be answered first for the total system is, if the detection

method is sensitive enough to determine the thermal fluctuation of a certain moving effective mass

point in air at room temperature. If this is the case then the measurement will be limited by the ther-

mal noise, which means the thermal noise limit for the AFM transducer needs to be as low as possible. [49]

In table 3 the most important design parameters are summarized, which are taken as starting point

for the design of the AFM transducer. [49]

Parameter Values for conventional AFM

Spring constant 100 N/m

Amplitude 10 nm

Resonance frequency 300 kHz

Quality factor 500

Thermal noise limit 41.63 fN/√

Hz

Minimum feature size 3 µm

Silicon layer thickness 10 µm

Tab. 3: Summary of the most important design parameters.

For designing the AFM transducer the above parameters have to be taken into account. From those

parameters the dimensions and mass can be calculated. In order to define the geometry of the AFM

transducer several factors have to be considered. What a first concept could look like figure 31, where the

top side of the transducer acts as DC biasing electrode for the BAW detector. The shape of the electrode

depends on the radius of the BAW disk resonator and the active area has to be as large as possible to

ensure good pre-stress of the disk resonator with as low DC bias as possible. On the other hand the AFM

tip is fixed to the structure for analysing the sample surface, which has a tip diameter of around 2 nm. The

suspension of the AFM transducer structure consists of a spring on each side for symmetry reason and

anchoring the free etched resonator to the bulk material. The stiffness of the springs is chosen to be 100

N/m. To reach that value a guided cantilever beam is used. The thickness for all structures is given by the

device layer thickness of the SOI wafer used, which is 10 µm in this case.

The AFM transducer needs to have a potential difference to the BAW detector. This is done by applying a

DC voltage to the structure creating an electrostatic force, due to the small gap between the BAW detector

and AFM transducer. The smaller the gap, the larger is the acting force and the smaller can be the potential.

But if the electrostatic force is too high, pull-in can occur, which has to be prevented. The fact that spring

forces scale linearly and electrostatic forces scale non-linearly, generates a stable region in which the

AFM transducer can safely operate. If the so created stable region is not large enough for the applied

pre-stress DC potential it has to be extended. This phenomenon is investigated by simulations and the

stable region for the device is determined in this thesis.

50


0

x

Am

pli

tude

FBAW

FBAW

FCounter

FCounter

FSpring

Fig. 32: Working principle of the counter electrode.

In order to extend the stable region, the spring constant can be increased, but this will change the prede-

termined design parameters. Another method is to use a counter electrode with the same potential, the

same distance and the same active electrode area. Such a set-up is shown in figure 32, which is used to

counteract the attracting electrostatic force between the two resonators. At the initial position of the AFM

transducer the forces created by the potential on the electrodes should cancel out each other for the system

to be in equilibrium. When oscillated the structure is attracted to the closer electrode, but the force is

counteracted by the spring force and the counter electrode. At both sides the danger of a pull-in exists and

has to be prevented by choosing the parameters accordingly in order to extend the stable region in a way

to use high enough DC potential to achieve a good sensitivity of the sensor.

Applying a DC bias on the structure means also that the tip has to be electrically isolated from the rest

of the structure, otherwise the sample surface and the tip will interact electrically. To accomplish that

stoppers can be used or a dielectric isolation layer is deposited in the area of the tip.

Besides these design rules the choice of the main structure geometry is free and can be adopted to the

needs of the system. For big bulk structures etch holes have to be included to guarantee the free etching of

the resonating structure. Such holes decrease the total mass and have to be considered in the simulation

and design. The lighter the structure, the higher will be the frequency of the system and with that the

scanning rate can be increased, but at the same time the quality factor will be reduced.

4.1.2. BAW detector of the electrostatically coupled sensor

The second part of the sensor detects the movement of the AFM transducer and translates it into an output

signal, which can be analysed to obtain topographical data of the sample surface. In its function it will

replace the conventionally used laser detection unit in AFM systems.

The resonance frequency of this resonator has to be much higher than the frequency of the AFM tranducer

to prevent interference and influence of these two systems. To fulfill the requirements a bulk acoustic wave

resonator is preferred, because of its high frequency and quality factor. This guarantees high sensitivity and

low noise. Commercially available optical detectors for AFM systems have a sensitivity range between 50

and 150 fm/√

Hz [59], which is already very high and will be hard to improve, but the fact that this new

system is very space efficient and that it allows for AFM resonator designs not possible with conventional

laser readout makes it an attractive alternative.

The resonance of the BAW detector is excited by electrostatic force created by an AC-potential between

the actuation electrodes (green) and the grounded resonator, which can be seen in figure 33. For the

resonator itself it was decided to use a disk shaped resonator with a diameter of 60 µm in order to achieve

a high resonance frequency of 66 MHz and a high quality factor of around 10000 for the first design, due

51

4.2 Calculations and Simulation of the electrostatically coupled force sensor

to the analysis done with the wine-glass mode resonators in the previous chapters. The gap width between

the electrodes and the resonator has to be as small as possible to achieve optimal coupling. With available

fabrication technology a gap down to 100 nm can be created.

Sensing electrode

Etch holes

BAW disk resonator

Actuation electrode

Anchor

Air gap

Radius

Fig. 33: Set-up of the BAW detector with important parameters.

As can be seen in figure 33 the sensing is performed by an electrode on the top of the structure (yellow).

The resonator is driven in the wine-glass mode in order to have four nodal points to anchor the structure

to the bulk. The thickness of the resonator is defined by the silicon device layer thickness of the SOI

wafer used, for example 10 µm, the minimum line width is 3µm and a air gap of 1 µm is possible for the

available fabrication process. Also etch holes have to be included in order to free-etch large structures,

which have to be taken into account in the calculations and simulations.

The most important relationship to be determined is how different DC potentials on the AFM transducer

electrode influence the resonance frequency of the BAW resonator depending on the gap width variation

generated by the oscillation of the AFM transducer. From these relations the sensitivity of the system can

be determined.

In order to achieve a good performance, a high quality factor and high sensitivity in detecting the movement

of the AFM transducer, the results from the previous analysis has to be taken into account and different

sizes and geometries of the resonators have to be simulated. In this case the wine-glass mode resonator

geometries disk, square, ring and frame resonators will be investigated.

The response time of the BAW detector has to be lower than the one from the AFM transducer that

the BAW resonator is able to follow the AFM resonator for a good resolution of the sensor. A high

displacement of the disk resonator is required for fast sensing and a good capacitive detection efficiency

with low noise from the sensing electrode and the signal analysis of the output signal.

4.2. Calculations and Simulation of the electrostatically coupled force sensor

To verify the results of calculations and to determine non-linear correlations of the device, simulations

have to be done. For these the simulations the MEMS-module in COMSOL is used.

The complete system cannot be simulated as one, because of its complexity and the limitation of computa-

tion resources. That is why the system has to be divided into several smaller models, which include the

parameters of interest and use external parameters and assumptions on the intersecting area between the

52


parts in this way the interesting models and their behaviour to certain parameters are simulated easily and

time-efficient.

4.2.1. AFM transducer simulation

First the mass and volume of the structure has to be calculated. This was done by targeting a resonance

frequency of 300 kHz and a stiffness of 100 N/m, which are the minimum requirements of the system. With

the following equation the total mass can be calculated.

ω0 =

√

kcm

and ω0 = 2πf0 (23)

m =kc

4π2 · f20

(24)

Where kc is the spring constant, ω0 the angular resonance frequency and f0 the resonance frequency. With

that a suitable mass was found to be 28.1 ng. The mass of the spring structure also has to be taken into

account in the mass calculation, but here this can be neglected due to the very small contribution to the

total mass. With a density of silicon 2329 kg/m3 at room temperature [23], a volume of 1.21·10−5 mm3 is

found. [42]

In order to determine the length of the spring arms the following formula for a guided in-plane movement

beam can be used.

L = 3

√

2 · Etw3

kc(25)

Where L is the beam length of one side of the suspension, kc is the given stiffness of 100 N/m, E is the

Young’s Modulus of silicon 169 GPa [32], t is the thickness of 10 µm determined by the silicon device

layer thickness of the SOI wafer and w is the width of 3 µm given by the minimum feasible feature size

for fabrication with the process used.

The first simulation determines the length of the suspension beams using the predefined stiffness and veri-

fies the calculations done before. Furthermore a possible geometry of the AFM transducer is investigated

in order to fulfill all parameters defined by a conventional AFM cantilever.

The model used for the simulation consists of the resonator, which is suspended using two none folded

beams connected to the bulk structure. The width and thickness of the beams are predefined by the

fabrication processes. That means only the spring length has to be determined by the simulation to

resemble a stiffness of 100 N/m.

To find the right spring length a static force of 1 µN is set to act on the structure to achieve a displacement

of 10 nm, which resembles the oscillation amplitude of the resonating AFM transducer in the device.

While keeping the force constant, the length is varied until the structure has a displacement of 10 nm. For

this simulation the geometry of the transducer structure and mass does not need to be taken into account,

because it has no effect.

In the second simulation the geometry is tuned to have a resonance frequency of 300 kHz in the flexural

mode. This is done by modifying the dimensions of the resonator structure and conduct an eigenfrequency

analysis.

As some of the geometry is already predefined, like the electrode area, the rest of the structure is adjusted

in order to achieve the desired resonance frequency. The etch holes also have to be taken into account,

53


because a decrease in mass increases the resonance frequency of the structure. 300 kHz is only the

minimum requirement and a possible increase of the frequency can further boost the performance of the

AFM transducer.

4.2.2. BAW detector sensitivity simulation

The next point of interest is how the DC bias on the AFM transducer electrode and the gap width between

AFM transducer and BAW detector influence the resonance frequency of the bulk resonator. This relation

resembles the sensitivity of the system, which is defined as the frequency change per gap width change

against the electrode gap width for different DC bias.

The following equation shows that the electrostatic force created between the AFM transducer electrode

and the BAW disk resonator changes non-linearly with the gap width as well as with the applied biasing

voltage.

Fel = −ǫ0ǫrAU2

2x2g(26)

Where ǫ0 is the vacuum permittivity, ǫr relative permittivity, A the electrode area, U the applied electrical

potential and xg the actual gap.

Simulations are needed to find the complex relationship between the static force created by the DC bias

from the oscillating AFM transducer acting on the actuated disk resonator and how the frequency is

influenced.

The simulation model consists of a disk resonator oscillating with the resonance frequency and a DC

biased electrode separated by an air gap form the disk. The gap width between the resonator and the

electrode is varied from 500 nm to 100 nm in 10 nm steps. For every simulation parameter the resonance

frequency of the disk is calculated with a constant potential and the given gap width. This is done for

different DC bias levels by applying a DC potential on the electrode.

In these simulations a disk with diameter d = 60 µm is used with a quality factor Q = 100000, which

has a resonance frequency f0 = 66.63 MHz of the unbiased system. The simulation results, however, are

independent of the quality factor. The reason for using a disk shaped resonator for these simulations are the

good performance, high frequency and quality factor, as well as a well pronounced wine-glass mode and

easy to fabricate structure of a disk shape as found in previous chapter. In follow up simulations different

geometrical shapes and different sizes are simulated and it is investigated how these are influenced by a

DC biasing to be able to find the optimal geometry for this BAW resonators. The simulations done in the

previous chapter serve as basis for this analysis.

4.2.3. BAW detector geometry simulations

To determine the best geometry and to analyse how their resonance frequency is influence by a certain DC

biasing, two variations of the disk resonator geometry are simulated. The parameters chosen for these

simulations are found from the general analysis of resonator shapes conducted in the previous chapter.

Disk radius variation The first one is variation of the disk radius, while the other parameters stay

constant. Here the radius is changed from 20 µm up to 60 µm in 10 µm steps. With changing resonator

dimensions the resonance frequency changes. The DC potential of V = 40 V on the electrode stays

54


constant and the the gap width varies from 500 nm to 100 nm in 10 nm steps as in the simulation before.

In this way the sensitivity of the disk shape with different radii can be determined.

Shape variation The next step is to compare different geometrical shapes according to their sensitivity

as resonator in the BAW detector. For an effective comparison only the geometry is changed and all other

parameters are kept constant, like the unbiased resonance frequency f0 = 66.63 MHz, DC potential V =

40 V on the electrode and the interval of the gap width variation from 500 nm to 100 nm in 10 nm steps.

The tested geometries have to resonate in the wine-glass mode and it should be possible to anchor them at

the four nodal points to the bulk material. That is why disk, ring, square and frame shape are selected for

testing, which is shown in figure 16. The resonator dimensions have to be adapted in order to keep the

resonance frequency constant. The eigenfrequency mode is used to simulate the frequency of the shapes

with variation of the size. In case of ring and frame shape a 40% fill ratio is kept.

4.2.4. AFM transducer extended stable range simulation

An electrostatic force is created between the DC biased electrode of the AFM transducer and the disk

resonator of the BAW detector, because of the small air gap of around 100 nm and a high DC potential on

the electrode of 10 V to 60 V. The device should be designed in a way to avoid pull-in of the electrode and

in order for the resonators to work in a stable operation region.

The first step is to determine this stable region and the electrostatic force occurring between the structures.

This is done by using a representative disk shaped BAW detector resonator with a diameter d = 30 µm

and a fitting AFM transducer electrode with a spring constant of 100 N/m, an applied DC bias and a gap

width of 100 nm. The bias will be varied to determine the stable point of the system. A display of the

acting forces in the initial position was already shown earlier in figure 32a and in the maximum position

of 10 nm in figure 32b.

The electrode area is determined by the device layer thickness of 10 µm and the length of the curved

electrode of 20.94 µm, which corresponds to an electrode area of 628.2 µm2.

4.2.5. Mechanical noise calculations for the sensor

In order to determine the physical detection limit of this sensor a thermal noise limit calculation has to

be done for both resonators. The fluctuation dissipation theorem SFF = 2kBTmeffγ0 can be used for

calculating the thermal noise limit [49]:

√

SFF =√

2kBTmeffγ0 (27)

Where kB is the Boltzmann constant, T the temperature in Kelvin, meff the effective mass of the resonator

and γ0 the damping coefficient. Using the following equations the effective mass meff and the damping

coefficient γ0 can be replaced in equation 27. [49]

γ0 =2ω0

Q, meff =

kcω20

and ω0 = 2πf0 (28)

Where ω0 is the angular resonance frequency, Q the quality factor, kc the stiffness of the resonator and

f0 the resonance frequency of the resonator. These equation can be used with 27 to make the following

55

4.3 Results and discussion

equation, which will be used to calculate the thermal noise limit of the two different resonators.

√

SFF =

√

2kBTkcπf0Q

(29)

The following parameters for the AFM transducer flexural resonator are used: the room temperature T is

296 K, a stiffness kc of 100 N/m, a resonance frequency f0 of 300 kHz and a quality factor Q of 500.

For the BAW detector wine-glass mode disk shaped resonator are used: room temperature T of 296 K, the

stiffness kc is 2.6·106 N/m, the resonance frequency f0 is 66.63 MHz and a quality factor Q is 10000.

As a last critical noise source, the electrical noise limit of the DC potential for pre-stressing the BAW

detector resonator has to be found. This noise should not dominate the noise in the system that means the

electrical noise limit for the bandwidth of the AFM transducer resonator should not be higher than the

highest mechanical noise source as calculated above. To accomplish that the frequency variation caused

by the mechanical noise is determined. This is used to compare it with the frequency variation caused

by the electrical noise of the DC potential at the bandwidth of the AFM resonator. A simulation is done

to find this maximum electrical noise limit at the point, where the mechanical noise still dominates the

system in order to achieve the best possible resolution for the sensor.

4.3. Results and discussion

By analysing the results of the simulations and calculations, the relations of the different parameters

to each other can be found and a sensor system is designed fulfilling all the defined requirements and

functions.

The results of the simulations for the AFM transducer define the boundary conditions for the simulations

and design of the BAW detector. Here especially the suspension is of importance, to determine the

geometrical shape, stiffness, amplitude, resonance frequency and quality factor for spring system in the

AFM transducer.

As can be seen in figure 34, the displacement has a non-linear dependency on the spring length. A beam

length of around 97 µm was found for a stiffness of 100 N/m and a displacement of 10 nm, which verifies

the calculations done beforehand.

In figure 34 the dependency between the displacement and the spring constant can be seen with a constant

displacement of 10 nm. Using figure 34 the beam length of the suspension is determined, also parameters

for a possible design variation with different stiffness or amplitude can be found. The resulting dimensions,

however, will be influenced by the fabrication process and fabrication tolerances have to be taken into

account as they are effecting the stiffness of the spring system. The etching process of these structures,

if not compensated for, will create a slight underetch and therefore decrease the dimensions and thus

stiffness.

Using these results the AFM transducer can now be fabricated. To determine the best design, several

different device solutions have to be fabricated and tested.

Most crucial is the sensitivity of the system expressed by how the change in gap width and the variation

of the DC bias affects the resonance frequency of the BAW detector system. From a series of simulations

with variation in gap width and DC bias the change in frequency per changes in gap width is determined,

which can be seen in figure 35a. In figure 35b the trend lines for the previous results are plotted in a

56


0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

0

0.02

0.04

0.06

0.08

0.10

Dis

pla

cem

ent

[µm

]

Beam length [µm]

Spring stiffness 50 N/mSpring stiffness 100 N/mSpring stiffness 150 N/mSpring stiffness 200 N/mSpring stiffness 250 N/m

Fig. 34: Displacement of the AFM transducer in dependency of beam length and stiffness of the spring

system for a constant for of 10 µN.

logarithmic scale to better illustrate the change in sensitivity with the gap width change.

The results of the simulations show a strong non-linear dependency of the sensitivity on the gap width

between the electrode and the disk resonator. The sensitivity df/dg has a square dependency on the biasing

DC voltage applied to the AFM transducer electrode, this can also be seen in equation 26. The electrostatic

force itself, however, shows only a linear dependency on the change in resonance frequency of the BAW

resonator. It should be noted that results are independent of the disk resonator’s quality factor. At a gap

width of 200 nm the non-linearity is already very pronounced and the smaller the air gap gets, the stronger

is the effect. Looking at different gap widths with a DC voltage of 60 V the sensitivity is 0.0015 Hz/pm at a

gap of 500 nm, but already 1 Hz/pm at 100 nm and the sensitivity would even be around 3 Hz/pm at a 50 nm

gap width, which can be inferred from figure 35b.

To analyse the influence of the resonator geometry different disk radii are simulated with a DC bias of 40

V, a disk thickness of 10 µm and a gap width change between 500 nm and 100 nm to be able to compare

the results. The data found can be seen on figure 36a, with the sensitivity df/dg against the gap width for

different disk radii.

From the simulations it can be concluded that the sensitivity is independent of the disk radius. This

counterintuitive result can be explained with the fact that the effect of increasing frequency with decreasing

resonator size is compensated by the decreasing electrode surface, because the AFM transducer electrode

is scaled with the resonator size. Therefore the change in frequency over the change in gap width is

independent of the dimensions. Another simulation is conducted, where in comparison to the previous

one the electrode surface is kept constant for different sized resonators. It can be seen in figure 36b that at

constant electrode surface the sensitivity increases with decreasing resonator size, which shows that the

increasing electrode surface with increasing disk size compensates for the decreasing resonance frequency

57


100 200 300 400 500

0

100

200

300

400

500

600

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Electrode gap width [nm]

10V20V30V40V50V60V

100 200 300 400 5000,01

0,1

1

10

100

1000

10000

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Electrode gap width [nm]

trendline 10Vtrendline 20Vtrendline 30Vtrendline 40Vtrendline 50Vtrendline 60V

Logarithmic scale

a) b)

Fig. 35: a) Sensitivity of the system with different DC potential on the AFM transducer electrode and

with a 60 µm diameter disk shaped BAW detector; b) Logarithmic plot of the trendlines.

a) b)

0 100 200 300 400 5000

100

200

300

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Air gap width [nm]

With adjusted electrodedisk D = 80 µmdisk D = 120 µmdisk D = 160 µmdisk D = 200 µm

0 100 200 300 400 5000

100

200

300

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Air gap width [nm]

Without adjusted electrodedisk D = 80 µmdisk D = 120 µmdisk D = 160 µmdisk D = 200 µm

Fig. 36: Sensitivity comparison of disk resonator for the BAW detector with different radii a) with AFm

transdcuer electrode adjusted to disk radius; b) without electrode adjusted to disk radius.

58


for larger resonators.

In order to determine the best possible geometry for this device disk, square, ring and frame struc-

tures are compared with the help of a simulation using the previous result of the general resonator analysis

as basis. The results can be seen in figure 37. For a reasonable comparison a DC bias of 40 V and the

variation of the gap distance between 500 nm and 100 nm is kept constant for all simulations.

100 200 300 400 500

0

200

400

600

800

1000

1200

1400

1600

1800

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Electrode air gap width [nm]

disk (D = 60µm, 40 V)

square (a = 60µm, 40 V)

ring (D = 60µm, 40 V)

frame (a = 60µm, 40 V)

Fig. 37: Sensitivity comparison of disk, square, ring and frame shape as resonator for the BAW detector.

Here the sensitivity df/dg of the different geometries is set in relation to the gap width of the biased electrode

for the different geometrical shapes. It can be seen that ring and frame shape have a much higher sensitivity

in comparison to the disk and square shape, if a lower resonance frequency can be accepted for that

application. This is in accordance with the results found in the general analysis found in the previous

chapter and can be explained by lower stiffness and therefore higher displacement capability of these

structures in comparison to the solid geometries. It has to be taken into account that the ring shape in

comparison to the disk has a lower resonance frequency and mass at the same outer dimensions. That

means that the sensitivity of the different shapes should only be compared by their outer dimensions and

not by their resonance frequency or quality factor.

Due to the high DC voltage needed for the pre-stressing of the BAW detector a high electrostatic

force between the two electrodes is created, which can be calculated depending on the gap width or

in this case the displacement of the AFM transducer. The initial gap width is set to 100 nm in order

to receive a good coupling and sensitivity between the two resonating systems. The suspension has a

linear dependency between force and displacement, whereas the electrostatic component has a non-linear

relation depending on the voltage as additional parameter. Pull-in will therefore occur at 2/3 of the initial

gap width, which will at a displacement of 33.3 nm from the initial position at a gap width of 100 nm,

59


which corresponds to a gap width of 66.6 nm. Here the linear spring force will be equal to the non-linear

electrostatic force and can be seen in figure 38. The system can only be operated in the stable region,

where the spring force is higher than the electrostatic force. The highest sensitivity of the sensor can be

achieved by maximizing the DC-potential on the AFM transducer electrode resulting in an electrostatic

force displacing the spring system close to pull-in range.

0 20 40 60 80 100

0

2

4

6

8

10

12

14

16

18

20

Forc

e[µ

N]

Eletrode air gap width [nm]

Fs

Fel without counter at 2.3 VFel with counter at 2.65 V

unstable region

stable region

initial position

66.6

Fig. 38: Comparison between electrostatic force over electrode gap width with and without counter

electrode at pull-in distance.

It can be seen that the stable point close to a gap width of 66.6 nm is reach at a voltage of 2.3 V for

the system without counter electrode, here the retracting force of the spring system and the attracting

electrostatic force are in equilibrium. For this potential exactly one stable point exists in that system. If the

voltage is increased beyond this point the system becomes unstable and pull-in will occur. Unfortunately

the maximum voltage for a stable system is much lower than what is needed to receive a good sensitivity

of the frequency detection by the BAW detector through the pre-stressing of the resonator.

In figure 38 can also be seen that the stable region can be extended to 2.65 V before pull-in occurs with the

introduction of a counter electrode to the system, because its force will counteract the electrostatic force

from the AFM transducer electrode. This is an improvement of 13% over the system without counter

electrode at the maximum displacement of the AFM transducer of 33.3 nm.

The suspension system of the AFM transducer, however, is design for a displacement of 10 nm, which

is inside the stable region. Tuning the system to this displacement a potential of only 1.8 V can be used

without a counter electrode, which can be seen in figure 39. At the system with the counter electrode a

voltage of 3 V can be applied in order to reach the equilibrium position at a air gap width of 90 nm, which

corresponds to a AFM transducer displacement of 10 nm and is an improvement of 40% compared to the

system without counter electrode. The more the displacement of the AFM transducer decreases the higher

can the potential be on the electrodes to create a stable operation point, which can be seen in figure 39.

In theory an infinite high voltage can be applied to the electrodes at initial position when the distance

60


0 20 40 60 80 100

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Volt

age

nee

ded

for

equil

ibri

um

in[V

]

Electrode air gap width in [nm]

Without counter electrode

With counter electrode

stable regionunstable region

66.6

initial position

90

Fig. 39: Comparison between the system with and without counter electrode related to the air gap width

in dependency of the applied potential.

between the two electrodes is equal and there is no spring force. But as soon as the system is moved out

of its initial position, even at a femtometer, 3.1 V are enough to create a electrostatic force much higher

than the spring force and pull-in will occur, as can be seen in figure 39.

But even with the extension of the stable region the potential is still lower than what is needed in order to

receive a good sensitivity of the complete system. Another disadvantage is that the DC potential has to be

adjusted exactly, because of the flat slope for the voltage curve of the system with a counter electrode,

which can be seen in figure 39. That means for a small change in potential either pull-in occurs or the

oscillation amplitude of the AFM transducer will decrease, which can be seen in figure 39.

In order to decrease the DC biasing needed for receiving good sensitivity and for operating the sys-

tem in the stable region, i.e. to avoid pull-in of the AFM transducer, the disk resonator is exchanged

for a ring shaped resonator with a diameter of 200 µm and a ring width of 40 µm. It has a resonance

frequency of 6 MHz, which is around a factor of 10 lower than the disk. The displacement, however, is

much higher, because of the increased size and the decrease in stiffness of the ring structure. Its sensitivity

is determined with the help of a simulation in dependency on the electrode gap width, like it was done

already with the disk resonator before.

As can be in figure 40 the system behaves the same way as the disk resonator (see figure 35), but for a

reduction by a factor 2 in DC voltage the same sensitivity can be achieved. Therefore the system can be

operated at a much lower DC potential on the AFM transducer electrode for pre-stressing of the BAW

detector resonator. For example at a potential of 2 V, which is inside the stable region, a sensitivity of

around 0.003 Hz/pm can be found at a gap width of 100 nm.

To make a statement about the noise in the system the thermal noise limit of the two mechanical resonators

61

4.4 Conclusion of the electrostatically coupled force sensor

0 100 200 300 400 500

0

50

100

150

200

250

Sen

siti

vit

ydf/

dg

[Hz/

nm

]

Electrode gap width [µm]

2 V

5 V

10 V

20 V

Fig. 40: Sensitivity of the system with different DC potential on the AFM transducer electrode acting

on a ring shaped resonator as BAW detector.

and the electrical noise limit have to be calculated. These three noise sources define the physical limit of

the sensor.

The thermal noise limit of the AFM transducer system is found to be√SFF = 41.63 fN/

√

Hz. This value

can still be reduced by optimizing the parameters of the AFM transducer system.

For the BAW detector disk resonator a thermal noise limit of√SFF = 101 fN/

√

Hz can be found. In case

of reducing the DC potential for pre-stressing a ring resonator with a diameter of 200 µm, a ring width of

40 µm and a resonance frequency of 6 MHz is used. For this geometry the thermal noise limit can be

found to be√SFF = 147.2 fN/

√

Hz.

For the electric noise limit it is found that the level can increase up to 0.556 V/√

Hz until it would dominate

the noise floor in the system over the thermal noise limit of the BAW detector resonator. For the ring

shaped resonator a value of 0.0011 V/√

Hz is found. It is very unlikely that the electrical noise limit is so

high for the DC-biasing, which means that the thermal noise limit of the BAW detector is the dominating

noise floor in this device.

4.4. Conclusion of the electrostatically coupled force sensor

Summarizing this investigation it can be said that even if the overall concept is very elegant and the

potential possibilities of the system to measure surface forces with the uncoupled sensing and detection

resonators are large, the system as proposed is not feasible or only with large limitations.

The advantage is the separation of sensing and detecting unit, where the high non-linear influence of the

gap width is used to detect small surface forces with a high sensitivity. Another advantage is the high

scalability of the sensor, because the AFM transducer is not limited in dimensions in comparison to the

conventional AFM cantilever and the laser unit is replaced by BAW detector, which has a small diameter

in the range of 100 µm.

A drawback is the relatively high thermal noise limit of the system dominated by the BAW resonator, which

is factor 5 larger than the thermal noise limit of a conventional AFM cantilever. Another disadvantage is

62

4.4 Conclusion of the electrostatically coupled force sensor

that a DC potential of 60 V is needed for a high sensitivity, which is very high for such small systems. To

receive normal power capability the potential on between the AFM transducer electrode and the BAW

detector resonator should be decreased to between 5 V and 12 V. But with that also the sensitivity of the

system will decrease exponential and the sensing time will increase. In order to reach a good sensitivity

with lower DC pre-stress a different resonator has to be used, for example a ring shape with a larger

diameter, which provided an improvement over the disk resonator. This increases the susceptibility of the

resonance frequency to the DC bias with the change in the gap width and at the same time will increase the

displacement of the oscillating BAW resonator, which will increase the sensitivity of the sensing electrode

and provide a faster detection and a decreased noise of the system.

The most crucial drawback, however, is that the maximum possible DC potential for the extended stable

region before pull-in is at least one order of magnitude lower than the potential needed to receive a good

sensitivity of the total system. With the use of a ring resonator the voltage can be reduced by half to

receive the same sensitivity. But this measure is still not enough to keep the device in a stable working

region and at the same time have a high sensitivity, because at potentials over 3 V pull-in occurs. In order

to make the sensor work, either a different method of pre-stressing has to be found, a low sensitivity has

to be accepted for the sensor or the coupling system between the two resonators has to be redesigned in a

non electrostatic way to avoid the drawbacks created with such a system.

63

5 TWO MECHANICALLY COUPLED RESONATORS AS AFM FORCE SENSOR

5. Two mechanically coupled resonators as AFM force sensor

The second force sensor design avoids the drawbacks of the electrostatically coupled resonators by

coupling them mechanically with a soft spring construction, with otherwise only minor changes in the

design. This concept will be introduced and investigated on feasibility and performance with the help of

COMSOL simulations.

5.1. Concept of the mechanically coupled force sensor

The design is very similar to the one explained in the previous section. It consists of two separate parts, an

AFM resonator with fixed tip for surface sensing and a BAW resonator for detecting the motion of AFM

resonator. These are coupled with a soft mechanical spring, as can be seen in figure 41.

AFM transducer

BAW detector

mechanical coupling

Wine-glass resonance mode

BAW disk resonator

Coupling spring system

Flexural resonance mode

AFM flexural resonator

AFM tip

Fig. 41: Set-up of the complete system with AFM transducer, BAW detector and mechanical coupling

system.

The AFM part will still be referred to as AFM transducer and the sensing part will be referred to as BAW

detector.

The resonance frequency of the AFM transducer is shifted with the change in surface forces acting on

the tip. The oscillation of the AFM transducer is transferred as a force acting on the BAW detector via

the spring system, which couples the resonators. A force acting on the BAW resonator creates strain

in the material and changes its resonance frequency. This resonance shift is capacitively detected by

an electrode close to the bulk resonator. These different parts will be described in greater detail in the

following section.

5.1.1. AFM transducer of the mechanical coupled system

The design parameters of a conventional AFM system also apply for this AFM transducer. This means

that it is designed to resonate with a frequency of 300 kHz. The suspension has a stiffness of 100 N/m and

oscillates with an amplitude of 20 nm. Also the quality factor is around 500. The thickness of 10 µm is

64

5.1 Concept of the mechanically coupled force sensor

predefined by the device layer thickness of the SOI wafer used. A minimum feature size of 3 µm defined

by the available fabrication processes also restricts the design.

Suspended structure

Suspension initial position

Actuation electrodes

Anchor

AmplitudeAFM tip

Fig. 42: Set-up and working principle of the AFM transducer with important parameters for the me-

chanically coupled system.

As can be seen in figure 42 the AFM transducer only consists of the two suspension beams, a region where

the tip is attached and electrodes for excitation. The large electrode area required for electrostatically

coupled system is not needed here, instead the mechanically coupled spring structure is attached to the

AFM transducer. This means that the mass and volume for the AFM transducer decreases in comparison

to the previous design and the performance can be improved for example by increasing the resonance

frequency for faster detection. Also no etch holes have to be considered, because the maximum feature

size of the structure stays below the critical dimension, where etch holes are needed to completely free

etch the suspended parts.

5.1.2. BAW detector of the mechanical coupled system

The sensing unit is a resonator oscillating in the wine-glass mode and anchored on the four nodal points to

the bulk material. It is actuated with an AC excitation voltage and a DC potential applied between the

two electrodes next to the resonator. A third electrode is used for capacitive sensing of the resonance

frequency. The set-up of this design can be seen in figure 43.

The attached coupling spring system will have an effect on the resonance frequency, as well as the quality

factor of the BAW resonator. This effect is one of the limiting factors for the design and has to be

investigated. If the quality factor decreases too much, this design can not be used as force sensor.

The geometry of the resonator has to be chosen in a way that the shift in the resonance frequency of

the BAW detector caused by a force from the AFM transducer is maximized in order to achieve a high

sensitivity for the system. On the other hand, however, the displacement of the resonator should not be

too low, because the lower it is, the longer takes for the capacitive detection. Another parameter that has

to be considered is the resonance frequency of the BAW resontor in comparison to the AFM resonator.

Here the ratio between resonance frequency and quality factor has to be higher than that of the AFM

transducer, otherwise the BAW resonator would not be able to follow the motion of the AFM transducer,

65

5.2 Simulation for the mechanical coupled force sensor

Sensing electrode

Etch holes

BAW disk resonator

Actuation electrode

Anchor

Spring as mechanical coupling

Radius

Air gap

AFM tip

Fig. 43: Set-up of the BAW detector with important parameters for the mechanical coupled system.

which results in decreased resolution. The sensitivity of the system should be high enough so that it is

possible to detect the effect of the Brownian motion on the AFM tip.

The use of etch holes for free etching of the large resonator structure also has to be included in the design

considerations.

5.1.3. Coupling spring system

The two resonator systems are coupled mechanically by a spring system. Its purpose is to provide the

output signal of the AFM transducer as an input signal to the BAW detector. So the most effective

design would be a direct rigid connection, like a beam connecting the two resonators. This, however, is

not practical, since the resonators have very different frequencies and quality factors. That means the

connection between the two systems has to be designed as stiff as possible with a high coupling efficiency,

but at the same time soft enough that the two resonators do not influence each other by reduction in the

quality factor of the resonators. It has to be possible to transfer the effect of the Brownian motion on the

AFM tip to the BAW resonator with high enough efficiency in order to have a detectable output signal

from the BAW detector.

For designing the spring system the limiting factors are the minimum feature size of the available

fabrication process and the thickness of the beams defined by the device layer of the SOI wafer used.

For simulation purpose a simple system is used to find the right design parameters for the spring. Here

the limitations of the fabrication process are not taken into account. This is done in a next step after the

specifications have been found.

5.2. Simulation for the mechanical coupled force sensor

To determine all design parameters, simulations have to be done. Some parts are divided up and

investigated separately to reduce the complexity of the system for the COMSOL simulations.

66


5.2.1. Mechanical coupling efficiency simulation

Firstly the Brownian motion acting on the AFM transducer has to be calculated in order to know the

minimum force, which should still be detectable by the BAW detector. Using the spectral density of the

fluctuating force related to any mechanical resistance. [5, 24]

F =√

4kBTD in [N/√Hz] (30)

Where kB is the Boltzman constant, T is the temperature in Kelvin, D is the damping coefficient and

m is the proof mass. With the use of the following two equations damping coefficient and the angular

resonance frequency ω0 can be substituted in equation 30.

Q =mω0

Dand ω0 =

√

kcm

(31)

From the resulting equation the force of the Brownian motion can be determined.

F =

√

4kBTBW

Q

√

kcm (32)

A temperature T of 296 K is used, 300 kHz for the bandwidth BW , 10 ng for the weight, 500 for the

quality factor and 100 N/m for the stiffness is used for calculation. This results in a force of around 100

pN on the AFM transducer caused by the Brownian motion at room temperature. With this the coupling

efficiency and the resulting change in frequency of the BAW detector can be found.

FCounter

equal to force on BAW detector resonator

FBrownian motion

Fig. 44: Simulation set-up with the force caused by the Brownian motion on one side and on the other

side a counter acting force.

The set-up for the simulation is shown in figure 44, where the force caused by the Brownian motion

(FBrownian motion) acts from the AFM transducer side and a counter force (Fcounter) at the other end

of the coupling mechanism is applied against the initial force. This counter force is equal in value to

the force acting on the BAW detector resonator. It is determined by increasing the counter force until it

cancels out the force acting from the AFM transducer side. The ratio between these two forces gives the

efficiency of the coupling mechanism. The efficiency can be adjusted by changing the dimensions of the

beams. But for these simulations only the width is varied between 0.5 µm and 1 µm. The length and

thickness are kept constant at 200 µm and 10 µm respectively.

In order to fabricate the coupling mechanism, it has to be redesigned to fulfill the criteria of minimum

67


feature size available with the fabrication processes. For that the stiffness of the selected simplified

coupling system is determined from simulation and is used to redesign the structure according to the

fabrication rules.

5.2.2. BAW detector simulation for mechanical coupling

The resulting value of the counter force from the previous simulations is applied to the BAW detector

resonator to determine the resonance frequency change. The resonator disk has a diameter of 60 µm.

The first simulation investigates the effect of an applied static force on the resonance frequency of the disk

resonator. Here a point load is applied, which is varied from 1 to 100 pN to mimic the force transferred

from the AFM transducer via the coupling mechanism.

In a second simulation the size of the resonator is varied from a diameter of 40 µm to 200 µm. This

simulation is used to find the most suitable resonator geometry for this application with the decision based

on the highest frequency change calculated between the initial unstressed resonance case and the case

with an applied force.

5.2.3. Simulation of the complete mechanical coupled system

In a next step a simulation is done to find out how the quality factor is influenced by the attached coupling

spring system with increasing stiffness and where the limit for a good resolution of the sensor lies. As can

be seen in the set-up in figure 45 the structure is fixed at the connection point to the AFM transducer and

the other end of the coupling mechanism is attached to the BAW resonator.

Fixed

Actuated with 12 V AC signal

Wine-glass mode resonance pattern

BAW detector resonator

Mechanical coupling spring system

Fig. 45: Simulation set-up to determine the effect of the mechanical coupling system on the quality

factor of the BAW resonator.

The disk is excited by applying a 12 V AC voltage on the actuation electrodes. A disk resonator with

diameter of 60 µm is used with a resonance frequency of 66.63 MHz in the initial unstressed case without

attached mechanical coupling structure. The damping of this system is kept constant and the quality factor

is determined with the help of the following equation.

Q =f0∆f

(33)

Where f0 is the resonance frequency and ∆f is the 3dB bandwidth.

The simulation will be done for different stiffness of the coupling mechanism by varying the beam width

68

5.3 Results and discussion of the mechanical coupled system

from 0.5 µm to 1µm with constant length of 200 µm and thickness of 10 µm.

5.3. Results and discussion of the mechanical coupled system

To determine the performance and feasibility of the sensor with the mechanical coupling between the two

resonators the results of the simulations are analysed.

For a good performance of the sensor the mechanical coupling needs a high efficiency, which means a

high stiffness, but on the other hand it has to be soft enough not to decrease the quality factor too much.

A high quality factor is important for a better read out of the BAW detector with the sensing electrode.

Figure 46 shows the resulting efficiency simulation of the coupling mechanism, where the dependency of

the spring stiffness on the resulting force on the BAW detector resonator and the coupling efficiency of

the mechanism is shown.

0 0.5 1.0 1.5 2.0 2.50

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Initial force F = 100 pN

Spring beam stiffness [N/m]

Res

ult

ing

forc

e[p

N]

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Coupli

ng

effi

cien

cy[%

]

Fig. 46: Resulting force and coupling efficiency of the mechanical coupling system with an initial force

of 100 pN depending on spring beam stiffness.

The initial force on the AFM tip is 100 pN resembling the force created by Brownian motion. The force

transferred via the coupling mechanism will act on the BAW resonator to change its resonance frequency.

This force depends on the stiffness of the coupling spring system and the coupling efficiency is given by

the ratio between this force and the initial force. The stiffer the spring system is, the higher the efficiency

of the mechanism and the higher is the resulting force acting on the disk resonator. Here 100 % efficiency

would correspond to infinitely high stiffness or in approximation to a direct connection between the AFM

transducer and the BAW detector without any softening in the coupling mechanism.

This result is used in the next simulation to determine the resonance frequency shift of the BAW detector

resonator in reaction to the acting force. After analysing the results of several simulations with the

same parameters, it is found that there is a small deviation in the results each time the simulation is run.

Concluding from the fact that the forces are very small and are assumed to cause only a small change in

resonance frequency of the disk, the standard deviation of the build-in deviation of frequency in COMSOL

has to be calculated. The resonance frequency of an undisturbed disk resonator has an average standard

69

5.3 Results and discussion of the mechanical coupled system

deviation of 0.427·10−3 Hz.

This fact has to be kept in mind for the simulation to determine the change in resonance frequency between

the undisturbed case and with a force acting on the resonator.

0 50 100 150 200

0,0000

0.0005

0,0010

0.0015

0.0020

0.0025

0.0030

Fre

quen

cych

ange

df

[Hz]

Diameter [µm]

F = 0.1 nN

F = 1 nN

F = 10 nN

Fig. 47: Frequency change df from undisturbed resonator resulting from applied forces on the disk

resonator at different diameters.

As can be seen in figure 47 with an applied force of 100 pN the frequency change is in the range of the

calculated standard deviation, also with a force of 1 nN. That means these results are not sufficient, as

the change in frequency is so small that it can not be distinguished from the standard deviation in the

simulation and also very difficult to detect capacitively. A force of 10 nN, however, shows a non-linear

increase in resonance frequency change for a decreasing disk diameter. For a disk with a diameter of

40 µm and a applied force of 10 nN the resonance frequency change in this case is 0.003 Hz, which is

still very low compared to the undisturbed resonance frequency of 100 MHz. To reach a larger change in

resonance frequency and thus a higher sensitivity of the sensor, either a smaller resonator has to be used

or a higher force acting on the resonator is needed.

It is expected that the attached mechanical coupling structure reduces the quality factor of the BAW

detector resonator, because the symmetry is lost by adding these structures. With increasing stiffness of

the coupling spring the quality factor decreases to a point to result in a reasonable good resolution of the

sensor. Additional simulations are done to determine the dependency between the reduction in quality

factor in reaction to the stiffness of the coupling system.

The results, however, proved inconclusive, as can be seen in figure 48 and thus no statement on the

dependencies can be made. Resulting from the high quality factor of the BAW resonator it is difficult to

determine the exact peak value and the -3 dB values and therefore to calculate the quality factor.

70

5.4 Conclusion of the mechanically coupled force sensor

0 0.5 1.0 1.5 2.0 2.5

9340

9360

9380

9400

9420

9440 Resonator D = 60 µm

Qual

ity

fact

or

of

BA

Wre

sonat

or

Stiffness of mechanical coupling [N/m]

Fig. 48: Quality factor of the BAW resonator in dependency of the mechanical coupling stiffness.

5.4. Conclusion of the mechanically coupled force sensor

The goal was to design a sensor detecting forces in the range of the Brownian motion. This was done by

tuning the mechanical coupling mechanism between the AFM transducer and the BAW detector to be stiff

enough to still detect such a small force, but at the same time be weak enough that the quality factor of the

BAW resonator does not drop too much to still receive a good resolution of the sensor.

That goal was not reached, as even a force two orders of magnitude higher does not result in a high enough

resonance frequency shift of the BAW resonator to detect it with a sufficient sensitivity. Also the smaller

the dimensions of the resonator, the higher is its frequency change, but at the same time the displacement

decreases and is more difficult to detect by the sensing electrode. If the force at the BAW resonator has to

be more than two orders of magnitude higher than the force created by a Brownian motion to be detected,

then the actual force acting on the tip of the AFM transducer has to be even higher, because of the low

efficiency of the mechanical coupling mechanism needed to preserve a high quality factor of the BAW

resonator. In summary it can be said that a sensor with two mechanically coupled resonators can not be

used to replace a conventional AFM unit consisting of sensing cantilever and detecting laser unit, because

its performance will be much lower that this does not compensate for the advantage of reduced size of this

sensor.

71

6 CONCLUSION

6. Conclusion

To improve the sensor set-up of a conventional AFM with the use of bulk mode resonators two different

approaches were found in the literature. Based on these findings a general analysis of different extensional

mode resonators was done for the use in AFM. Stiffness, resonance frequency and displacement according

to the dimensions for different geometrical shaped resonators were investigated. In conclusion, it can be

said that for the application in AFM the ring shape is preferred with a maximum in displacement at a

certain ratio between diameter and ring width and still high resonance frequency and good quality factor.

For even higher quality factor and higher frequency, but lower displacement, either a disk shape or a

longitudinal resonator can be used.

On basis of these previous investigations a force sensor for the use in AFM was proposed, which consists

of an electrostatically coupled flexural mode resonator acting as sensing unit and a bulk mode resonator

detecting the movement of the other resonator. For the sensor a relatively low thermal noise limit was

calculated, which is dominated by the noise of the bulk resonator. For a high DC potential between the

AFM transducer electrode and the BAW resonator and for a small air gap the sensor has a good sensitivity.

A big disadvantage is that pull-in between the two units will occur at DC potential larger than 3 V even

with counteracting measures. This strongly limits the sensitivity of this sensor design.

With an embodiment which couples the resonators of the introduced force sensor mechanically it is

possible to avoid the drawbacks occurring with the electrostatically coupled resonators. But for such a

system a high force acting in the tip of the AFM transducer is required in order to result in a detectable

change in resonance frequency of the BAW resonator, because the coupling has only a linear dependency

between the force at the AFM tip and the detected change in frequency of the BAW resonator in contrast

to the non-linear dependency of the electrostatically coupled system. Also the quality factor of the bulk

mode resonator is lower, because of the attached mechanical coupling structure. Even though the big

disadvantage of the electrostatically coupled system could be overcome by the design variation, this does

not outweigh the inferior performance of the mechanical coupled system.

With further improvements of the electrostatically coupled force sensor this drawback can probably be

resolved. In a first attempt can the thermal noise limit and sensitivity of the sensor be improved by the use

of a ring instead of a disk shaped resonator. The dimensions for the ring shaped resonators were chosen

from the results of the previous general analysis of different resonator geometries. With this measure it is

possible to half the potential needed to receive the same sensitivity of the sensor. Further improvements

can help to increase the performance of the electrostatically coupled force sensor to be a viable option for

the use in AFM systems.

Further investigations should be conducted with the simpler and more promising one resonator sys-

tems investigated in the general analysis and in devices described by Giessibl et al. [28] and Faucher et

al. [69]. These combine sensing and detection in one unit and are therefore very compact, can be scaled

down easily, have a high resonance frequency and high quality factor. The low displacement may pose

some problems, but maybe also opens up new possibilities for detecting the sample topography, because

of the very close proximity to the surface the forces will be high. Here especially a smaller version of the

design proposed by Giessibl et. al [28] could be of great interest for the use in AFM systems, because of

the high resonance frequency and quality factor.

72

7 FUTURE WORK

7. Future work

The next step will be to further improve the design of the electrostatically coupled force sensor for the

use in AFM. Measures have to be investigated to reduce the thermal noise of the BAW resonator to reach

the thermal noise limit of a conventional AFM system. More importantly, measures to reduce the DC

potential between the AFM transducer and the BAW resonator have to be found to avoid pull-in and at

the same time have a high sensitivity of the sensor. As first step this can be done by further optimize the

geometrical shape of the bulk mode resonator. Furthermore can the design of the AFM transducer be

improved by increasing the quality factor and minimizing its area and mass to reduce losses.

As soon as these problems are solved different designs of the device should be fabricated for charac-

terization to determine its performance and limitations outside the simulations. After the sensor has to

be packaged to preserve its functionality and protect the resonators against environmental influences or

reduce the losses caused by the surrounding medium.

New and more detailed investigations need to be for bulk mode resonator used for sensing and detecting at

the same time. Here especially longitudinal and ring shaped resonators are in the focus of interest, because

of their high scalability, high resonance frequency and high quality factors. Piezoelectric excitation has to

be investigated and compared to the results from electrostatic excitation to find the most effective and best

performing method.

73

A APPENDIX

A. Appendix

A.1. COMSOL simulations

For all simulations the FEM program COMSOL in the version 4.2 a is used. In the following chapters the

different simulation physics and study types are described and how they are used to receive reasonable

results. The modi will only be explained specific on the work with selected examples, because the func-

tionality of COMSOL is very extensive and contains many different physics for all kinds of simulations,

which can not all be discussed here. For answers to specific questions a search in the official COMSOL

forum provides valuable answers and tips. Also more elaborate questions and complex problems can be

addressed to the COMSOL support directly. But most valuable information and help can be found from

the example simulations in the model library and the describing model PDFs included in COMSOL itself.

These examples can also be used as basis for own simulations.

COMSOL simulation physics and study types

The space dimensions, the desired physics and the study type have to be selected when a new simulation

is started. The calculations for the simulations take a lot of time that is why the model should be designed

as simple as possible. It helps also to divide a complex model into different, simpler parts, simulate them

separately and use these output values to receive the final result of the complete model. For example

using a 2-dimensional approach saves valuable calculation time in comparison to a 3-dimensional one.

Therefore using a 2-D simulation is preferred if possible. A quasi third dimension can be specified later

inside the simulation itself. If a symmetric design is used, also only half of it can be simulated with the

selection axisymmetric, which again simplifies the model and therefore takes less time for calculation.

The menu for the dimension selection and the different available options can be seen in figure 49a.

a) b) c)

Fig. 49: COMSOL start menu, for selection of a) dimension; b) physics; c) study type.

After specifying the dimensions the desired physics has to be defined, which can be seen in figure 49b.

74

A.1 COMSOL simulations

For the simulations done in the thesis only two physics are used. The first one is the solid mechanics

physics located in structural mechanics and the other one is the electromechanics physics, which can also

be found under structural mechanics.

Solid mechanics is used to simulate only mechanical cases, like applying mechanical forces on the

structure and to receive results on stress, strain, displacement and many other information about the

structure itself.

Electromechanics is a combination of solid mechanics and electrostatics and is part of the MEMS module

of COMSOL. Its functions are specifically designed for the simulation of electrostatic actuated MEMS

systems. Here electrostatic forces and there result on solid structures can be simulated.

The next step is to choose the study type, here several different methods are possible, which can be seen

in figure 49c.

Eigenfrequency Every solid structure has several modes it can resonate. Each mode has a specific

resonant frequency. This study calculates these frequencies for the different modes.

With the eigenfrequency study in the electromechanics physics no results could be found, because it was

not possible to find the right options and properties for this study to make it work.

a) b)

Fig. 50: a) COMSOL model library; b) Geometry creation.

To avoid the problem, the example biased resonator 2d modes from the Model Library under MEMS

Module and Actuators is used, which can be seen in figure 50a. This example simulates the eigenfrequency

modes for a structure with the electromechanics physics. The existing geometry can be replaced by a new

design and then reasonable results are achieved with the simulation, which can be seen in figure 50b. This

method is just a workaround, it should also work on the conventional way in building up the model from

the start, but in the short time there could not be found a solution, instead the workaround had to be used.

In the solid mechanics physics the eigenfrequency study is used without problems and normal results are

received. With this set-up all resonant frequencies of the different shapes and geometries are determined.

The next step is to create the geometry, where parameters are used for each value, which has to be

varied in the simulation. For example in case of the disk resonator the radius is set as parameter. All other

structures, which depend on the size, like the anchors are set in relation to the radius. The length of the

75


anchors for example are set to 2 · radius+ 1

2· radius in order to change with a variation in radius, which

can be seen in figure 51a.

a) b)

Fig. 51: a) Setting parameter for geometry; b) Material selection.

After the geometry is defined the material has to be chosen from the Material Browser. In the simulations

only Si(c) was used as material in order to simulate the device fabricated out of the top silicon layer of a

SOI wafer, which can be seen in figure 51b.

The thickness as third dimension of a 2-dimensional structure can be set in Solid Mechanics under Thick-

ness, which can be seen in figure 52a. In the eigenfrequency analysis the result is independent of thickness,

but in other studies it is very important to set the thickness correctly. After that Fixed Constraints is used

to fixate structures to the bulk, for example the ends of the anchors in case of the disk resonator.

The next step is to choose the amount of resonance modes displayed under Study1 and Step 1: Eigenfre-

quency in Desired number of eigenfrequencies. It can also be search for eigenfrequencies around a certain

frequency if this is desired.

Before starting the simulation a parametric sweep is created in order to simulate multiple parameter values

in one run. The parameter can be varied from a start to an end value either in a fixed step width or in a fix

number of steps, which can be seen in figure 52b.

After the simulation is finished the modes are dispalyed in the Results section under Mode Shape (solid).

To view all available modes the different simulated radii can be switched under the drop-down menu

Parameter value and the according frequencies in the drop-down menu Eigenfrequency, which can be

seen in figure 53.The wine-glass mode is usually the fourth frequency, but depending on the radius it can

also change sometimes to a higher or lower mode. Here care is advised not to use the wrong resonant

frequency for further simulations, especially in case of the ring, frame and longitudinal resonator this is

critical.

At the end the received data can be displayed in a plot and exported as text file for further use in other

programs.

Frequency Domain A given structure can be excited by a specific frequency or a sweep over an

interval of frequencies. This functionality is covered by the frequency domain study type. Here the set

value of a static force applied to a structure is represents the peak value of a sinusoidal acting force with

applied frequency. For example a electrical potential value set to 12 V represents the peak value of a 8.32

76


a) b)

Fig. 52: a) Setting the thickness for the geometry; b) Parameter sweep.

Fig. 53: Display of result with selection of parameter and mode.

77


V AC voltage in the frequency domain type of the electromechanics physics.

The frequency study is used to determine the displacement for a certain geometry and a corresponding fre-

quency. For electrostatically actuated structures this is done with the electromechanics physics. After the

geometry is created an additional rectangle has to be drawn, which includes all structures and represents

the surrounding medium. In the material section the medium has to be define, which can be seen in figure

54a in case of air.

a) b)

Fig. 54: a) Allocation of the surrounding medium; b) Define the solid mechanic structure for the simu-

lation.

Then the solid mechanics structure has to be defined, which is the geometry affected and deformed by the

electrostatic forces and can be seen in figure 54b. In this simulation a damping effect of the surrounding

medium is considered. This damping coefficient can be specified by the use of Damping under Linear

Elastic Mechanical Material Model with the two coefficients α as mass damping and β as material

damping. These two parameters have no connection to the physics. The way they are defined can be found

in the Model library under MEMS Module and Actuators in the model PDF of biased resonator 2d freq,

which can be seen in figure 55a.

The structure has to be fixed with Fixed Constraint and afterwards the electrical potential is defined, as can

be seen in figure 55b. In this case the electrodes surrounding the resonator are set to a peak potential of 12

V, which resembles a AC voltage of 8.32 V. The resonator is set to a potential of 0 V and the surrounding

medium is automatically set to a charge of zero.

In order to do a frequency sweep over several different resonator sizes, the resonant frequencies have to

be found out with the eigenfrequency solver of COMSOL, as described before. The data then is used for

a parameter sweep with two parameters. In general x and y are used as parameters and the data in the

Parameter Values has to be set in the following way: x1 y1 x2 y2 x3 y3 and so on. This way to each value

x the according value of y is used, which are for example the parameters radius and frequency for the

simulation in figure 56.

After setting all values and parameters the simulation can be started. The process takes more time than

simple simulations, because of the combination of electrostatic and solid mechanic physics. The result is

displayed in a graph and all values can be viewed and exported for later use.

78


a) b)

Fig. 55: a) Set damping coefficients; b) Electrical potential defined on the structure.

Fig. 56: Defining the values for the sweep with two parameters.

79


Prestressed Analysis, Eigenfrequency This study type is very similar to the already described

eigenfrequency type, except that a pre-stressing force can be applied. That means a static force can be set

and is applied before the eigenfrequency analysis is started, so all simulation results will be influenced by

this pre-stress of the structure.

This study type is used to determine the dependency of the resonant frequency of an oscillator to the

gap width between two electrode with different potential. For this example the geometry is created and

material is allocated for the structure and the surrounding medium. A parameter for the gap width is

defined and the resonator is fixed to the bulk material, as well as an defined constant electric potential is

set for the moving electrode and 0 V for the resonator and the surrounding medium.

Fig. 57: Defining the sweep over the gap width interval for pre-stress of the resonator.

Both structures are defined as moving solid mechanic structures. No damping is set, because it has no

effect in an eigenfrequency analysis. Then a sweep is defined over a defined interval of gap width, as can

be seen in figure 57a. The force created by the electric potential between the electrode and the resonator

acts as pre-stress for the resonator and changes its resonant frequency. The simulation is complex, because

of the combination of electrostatics and solid mechanics and therefore takes a longer time to be calculated,

especially with gap width smaller than 0.1 µm. Sometimes the program is not able to find a solution

for small gap widths that is why it was decided to use a gap width, where a solution for all simulated

geometries could be found. The results of the effect of the gap width on the disk resonant frequency can

be displayed and exported for later use.

Prestressed Analysis, Frequency domain Pre-stressing can also be used for the frequency do-

main study type. Here a static force can be set in addition to the sinusoidal acting force to simulate a

pre-stress on the structure. This is for example necessary to simulate a electrostatically actuated bulk

resonator. Here a AC actuation voltage and DC pre-stressing is used. The DC potential creates a static

force acting on the structure, which is then actuated with the AC voltage over the electrodes.

The study type is not used in the simulations of this work, but it can be of great interest, because for

example electrostatic actuated resonators can be simulated, which need an AC actuation voltage and a DC

bias for pre-stressing the structure.

80


a) b)Prestressed Analysis, Frequency domain Frequency domain

Fig. 58: a) Pre-stressed, Frequency domain case with AC and DC voltage part; b) Frequency-Domain

with only AC actuation part.

As can be seen in figure 58a in the case of the electromechanics physics, the AC and DC part of the

actuation can be defined at Electric potential for the DC pre-stressing and under Harmonic Perturbation

for the AC actuation voltage. This is counter intuitive, because in the Frequency Domain study type the

value for the AC voltage has to be set in Electric Potential and Harmonic Perturbation has no function in

this study type, which can be seen in figure 58b.

Using this study type some problems occurred simulating a structure modelled with the electromechanics

physics. As workaround the example biased resonator 2d freq in the model library under the MEMS

module and Actuators is used, which includes the pre-stress functionality in the frequency domain study.

To use this example the existing geometry is replaced. The working principle of this model is also

explained in the appendant PDF file.

Stationary In this simplest study type the forces applied are constant and the simulation result show

the stationary case, for the solid mechanics, as well as for the electromechanics physics.

Fig. 59: Adding static boundary force for a stationary study type.

In the simulations this study type was used in many different ways for all cases, where a stationary

analysis is enough. The simulations are very fast, because not many different cases have to be calculated

by the program in contrast to frequency analysis. This study type was especially used to determine the

81


beam length at a certain stiffness and displacement of the it and to receive the stiffness of the different

geometries for the resonators.

As can be seen in figure 59 static forces can be applied on geometries, like in that case a Boundary Load

is applied on the surface. Here a force over the surface, pressure or a total force can be selected. In case

of total force a force is specified, which then acts on the marked surface and is adjusted to the surface

size. Also a parametric sweep can be used to test variation of certain parameters in one simulation and the

results can be exported for later use.

82

REFERENCES

References

[1] E. Algre, Z. Xiong, M. Faucher, B. Walter, L. Buchaillot, and B. Legrand, “Mems ring resonators

for laserless afm with sub-nanonewton force resolution,” Microelectromechanical Systems, Journal

of, no. 99, pp. 1–13. 7, 32, 33, 47

[2] T. An, T. Eguchi, K. Akiyama, and Y. Hasegawa, “Atomically-resolved imaging by frequency-

modulation atomic force microscopy using a quartz length-extension resonator,” Applied Physics

Letters, vol. 87, p. 133114, 2005. 31

[3] L. Baxter, “Capacitive sensors,” Ann Arbor, vol. 1001, p. 48109, 2000. 23, 24

[4] S. Beeby, MEMS mechanical sensors. Artech House on Demand, 2004. 14, 15, 16, 24, 25, 26

[5] J. Bernstein, R. Miller, W. Kelley, and P. Ward, “Low-noise mems vibration sensor for geophysical

applications,” Microelectromechanical Systems, Journal of, vol. 8, no. 4, pp. 433–438, 1999. 67

[6] S. Bhave and R. Howe, “Internal electrostatic transduction for bulk-mode mems resonators,” in Solid

State Sensor, Actuator and Microsystems Workshop (Hilton Head 2004). Citeseer, 2004, pp. 59–60.

22

[7] R. Bhiladvala and Z. Wang, “Effect of fluids on the q factor and resonance frequency of oscillating

micrometer and nanometer scale beams,” Physical Review E, vol. 69, no. 3, p. 036307, 2004. 16, 26,

27

[8] G. Binnig, C. Quate, and C. Gerber, “Atomic force microscope,” Physical review letters, vol. 56,

no. 9, pp. 930–933, 1986. 28

[9] K. Caekenberghe, “Modeling rf mems devices,” Microwave Magazine, IEEE, vol. 13, no. 1, pp.

83–110, 2012. 20, 21, 23, 24

[10] A. Cagliani and Z. Davis, “Bulk disk resonator based ultrasensitive mass sensor,” in Sensors, 2009

IEEE. IEEE, 2009, pp. 1317–1320. 26, 27

[11] H. Campanella, Acoustic wave and electromechanical resonators: concept to key applications.

Artech House Publishers, 2010. 15, 19

[12] J. Carr and J. Brown, Introduction to biomedical equipment technology. Prentice Hall New Jersey,

2001, vol. 4. 17

[13] L. Carrascosa, M. Moreno, M. Alvarez, and L. Lechuga, “Nanomechanical biosensors: a new

sensing tool,” TrAC Trends in Analytical Chemistry, vol. 25, no. 3, pp. 196–206, 2006. 7, 27

[14] S. Chandorkar, M. Agarwal, R. Melamud, R. Candler, K. Goodson, and T. Kenny, “Limits of quality

factor in bulk-mode micromechanical resonators,” in Micro Electro Mechanical Systems, 2008.

MEMS 2008. IEEE 21st International Conference on. Ieee, 2008, pp. 74–77. 19, 20, 21

[15] W. Chang and C. Zorman, “Electrical characterization of microelectromechanical silicon carbide

resonators,” Sensors, vol. 8, no. 9, pp. 5759–5774, 2008. 24

83

REFERENCES

[16] W. Chuang, H. Lee, P. Chang, and Y. Hu, “Review on the modeling of electrostatic mems,” Sensors,

vol. 10, no. 6, pp. 6149–6171, 2010. 11

[17] J. Clark, M. Abdelmoneum, and C. Nguyen, “Uhf high-order radial-contour-mode disk resonators,”

in Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency

and Time Forum, 2003. Proceedings of the 2003 IEEE International. IEEE, 2003, pp. 802–809. 24

[18] B. Crowell, Vibrations and waves. Light and Matter, 2000, vol. 3. 13, 14, 18, 25

[19] P. Enoksson, G. Stemme, and E. Stemme, “A silicon resonant sensor structure for coriolis mass-flow

measurements,” Microelectromechanical Systems, Journal of, vol. 6, no. 2, pp. 119–125, 1997. 19

[20] K. Eom, H. Park, D. Yoon, and T. Kwon, “Nanomechanical resonators and their applications in

biological/chemical detection: Nanomechanics principles,” Physics Reports, 2011. 26, 27

[21] M. Esashi, “Resonant sensors by silicon micromachining,” in Frequency Control Symposium, 1996.

50th., Proceedings of the 1996 IEEE International. IEEE, 1996, pp. 609–614. 7, 27

[22] M. Faucher, B. Walter, A. Rollier, K. Seguini, B. Legrand, G. Couturier, J. Aime, C. Bernard,

R. Boisgard, and L. Buchaillot, “Proposition of atomic force probes based on silicon ring-resonators,”

in Solid-State Sensors, Actuators and Microsystems Conference, 2007. TRANSDUCERS 2007.

International. IEEE, 2007, pp. 1529–1532. 32, 47

[23] K. Fujii, A. Waseda, and N. Kuramoto, “Development of a silicon density standard and precision

density measurements of solid materials by hydrostatic weighing,” Measurement Science and

Technology, vol. 12, p. 2031, 2001. 53

[24] T. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” Elec-

tron Devices, IEEE Transactions on, vol. 40, no. 5, pp. 903–909, 1993. 67

[25] D. Galayko, A. Kaiser, L. Buchaillot, B. Legrand, D. Collard, and C. Combi, “Design, realization and

testing of micro-mechanical resonators in thick-film silicon technology with postprocess electrode-

to-resonator gap reduction,” Journal of Micromechanics and Microengineering, vol. 13, p. 134, 2003.

24

[26] R. Garcıa and R. Perez, “Dynamic atomic force microscopy methods,” Surface science reports,

vol. 47, no. 6, pp. 197–301, 2002. 29, 30

[27] F. Giessibl, “Advances in atomic force microscopy,” Reviews of modern physics, vol. 75, no. 3, p.

949, 2003. 28, 29, 30, 50

[28] F. Giessibl, F. Pielmeier, T. Eguchi, T. An, and Y. Hasegawa, “Comparison of force sensors for

atomic force microscopy based on quartz tuning forks and length-extensional resonators,” Physical

Review B, vol. 84, no. 12, p. 125409, 2011. 7, 17, 18, 31, 32, 33, 47, 72

[29] N. Gridgeman, “Lame ovals,” The Mathematical Gazette, vol. 54, no. 387, pp. 31–37, 1970. 21

[30] Z. Hao, S. Pourkamali, and F. Ayazi, “Vhf single-crystal silicon elliptic bulk-mode capacitive disk

resonators-part i: design and modeling,” Microelectromechanical Systems, Journal of, vol. 13, no. 6,

pp. 1043–1053, 2004. 18, 19, 20, 21

84

REFERENCES

[31] S. Heike and T. Hashizume, “Atomic resolution noncontact atomic force/scanning tunneling mi-

croscopy using a 1 mhz quartz resonator,” Applied physics letters, vol. 83, p. 3620, 2003. 31,

32

[32] M. Hopcroft, W. Nix, and T. Kenny, “What is the young’s modulus of silicon?” Microelectrome-

chanical Systems, Journal of, vol. 19, no. 2, pp. 229–238, 2010. 53

[33] W. Hsu, “Recent progress in silicon mems oscillators,” DTIC Document, Tech. Rep., 2008. 14, 16,

18, 20, 22

[34] W. Hsu and A. Brown, “Frequency trimming for mems resonator oscillators,” in Frequency Control

Symposium, 2007 Joint with the 21st European Frequency and Time Forum. IEEE International.

IEEE, 2007, pp. 1088–1091. 23, 24

[35] W. Hsu and C. Nguyen, “Stiffness-compensated temperature-insensitive micromechanical resonators,”

in Micro Electro Mechanical Systems, 2002. The Fifteenth IEEE International Conference on. IEEE,

2002, pp. 731–734. 22, 24, 26

[36] V. Kaajakari, T. Mattila, A. Oja, and H. Seppa, “Nonlinear limits for single-crystal silicon mi-

croresonators,” Microelectromechanical Systems, Journal of, vol. 13, no. 5, pp. 715–724, 2004.

20

[37] D. Keymeulen, C. Peay, D. Foor, T. Trung, A. Bakhshi, P. Withington, K. Yee, and R. Terrile,

“Control of mems disc resonance gyroscope (drg) using a fpga platform,” in Aerospace Conference,

2008 IEEE. IEEE, 2008, pp. 1–8. 7, 28

[38] G. Kovacs, Micromachined transducers sourcebook. WCB/McGraw-Hill New York, NY, 1998. 11

[39] J. Lee, B. Bahreyni, Y. Zhu, and A. Seshia, “Ultrasensitive mass balance based on a bulk acoustic

mode single-crystal silicon resonator,” Applied Physics Letters, vol. 91, no. 23, pp. 234 103–234 103,

2007. 26

[40] J. Lee, J. Yan, and A. Seshia, “Low loss hf band soi wine glass bulk mode capacitive square-plate

resonator,” Journal of Micromechanics and Microengineering, vol. 19, p. 074003, 2009. 16

[41] ——, “Study of lateral mode soi-mems resonators for reduced anchor loss,” Journal of Microme-

chanics and Microengineering, vol. 21, p. 045010, 2011. 15, 16

[42] R. Legtenberg, A. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacements,”

Journal of Micromechanics and microengineering, vol. 6, p. 320, 1996. 53

[43] V. Lindroos, M. Tilli, A. Lehto, and T. Motooka, Handbook of silicon based MEMS materials and

technologies. William Andrew, 2009. 15

[44] K. LYNN, “Performance parameters of micromechanical resonators,” 2010. 16, 24, 26

[45] Y. Naito, P. Helin, K. Nakamura, J. De Coster, B. Guo, L. Haspeslagh, K. Onishi, and H. Tilmans,

“High-q torsional mode si triangular beam resonators encapsulated using sige thin film,” in Electron

Devices Meeting (IEDM), 2010 IEEE International. IEEE, 2010, pp. 7–1. 19

85

REFERENCES

[46] C. Nguyen, “Vibrating rf mems for next generation wireless applications,” in Custom Integrated

Circuits Conference, 2004. Proceedings of the IEEE 2004. IEEE, 2004, pp. 257–264. 19

[47] K. Ogata, System dynamics. Pearson/Prentice Hall, 2004. 13

[48] H. Pakdast and Z. Davis, “Development and characterization of high frequency bulk mode resonators,”

DTU Nanotech, June, 2008. 20

[49] D. Platz, D. Forchheimer, E. Tholen, and D. Haviland, “The role of nonlinear dynamics in quantitative

atomic force microscopy,” Arxiv preprint arXiv:1202.5217, 2012. 17, 30, 49, 50, 55

[50] D. Platz, E. Tholen, D. Pesen, and D. Haviland, “Intermodulation atomic force microscopy,” Applied

Physics Letters, vol. 92, p. 153106, 2008. 28

[51] S. Pourkamali, High frequency capacitive single crystal silicon resonators and coupled resonator

systems. ProQuest, 2006. 24

[52] S. Pourkamali and F. Ayazi, “Soi-based hf and vhf single-crystal silicon resonators with sub-

100 nanometer vertical capacitive gaps,” in TRANSDUCERS, Solid-State Sensors, Actuators and

Microsystems, 12th International Conference on, 2003, vol. 1. IEEE, 2003, pp. 837–840. 24

[53] ——, “High frequency capacitive micromechanical resonators with reduced motional resistance

using the harpss technology,” in Silicon Monolithic Integrated Circuits in RF Systems, 2004. Digest

of Papers. 2004 Topical Meeting on. IEEE, 2004, pp. 147–150. 21

[54] S. Pourkamali, Z. Hao, and F. Ayazi, “Vhf single crystal silicon capacitive elliptic bulk-mode disk

resonators-part ii: implementation and characterization,” Microelectromechanical Systems, Journal

of, vol. 13, no. 6, pp. 1054–1062, 2004. 16, 23

[55] S. Pourkamali, A. Hashimura, R. Abdolvand, G. Ho, A. Erbil, and F. Ayazi, “High-q single crys-

tal silicon harpss capacitive beam resonators with self-aligned sub-100-nm transduction gaps,”

Microelectromechanical Systems, Journal of, vol. 12, no. 4, pp. 487–496, 2003. 24

[56] P. Saulson et al., “Thermal noise in mechanical experiments,” Physical Review D, vol. 42, no. 8, pp.

2437–2445, 1990. 17

[57] H. Seh, T. Hyodo, and H. Tuller, “Bulk acoustic wave resonator as a sensing platform for no¡ sub¿

x¡/sub¿ at high temperatures,” Sensors and Actuators B: Chemical, vol. 108, no. 1, pp. 547–552,

2005. 26

[58] L. Shao, T. Niu, and M. Palaniapan, “Nonlinearities in a high-q soi lame-mode bulk resonator,”

Journal of Micromechanics and Microengineering, vol. 19, p. 075002, 2009. 20

[59] D. Smith, “Limits of force microscopy,” Review of scientific instruments, vol. 66, no. 5, pp. 3191–

3195, 1995. 30, 49, 51

[60] S. Sridaran and S. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Arxiv

preprint arXiv:1006.2783, 2010. 24

86

REFERENCES

[61] Y. Sun, D. Fang, and A. Soh, “Thermoelastic damping in micro-beam resonators,” International

journal of solids and structures, vol. 43, no. 10, pp. 3213–3229, 2006. 16

[62] K. Sundaresan, G. Ho, S. Pourkamali, and F. Ayazi, “Electronically temperature compensated silicon

bulk acoustic resonator reference oscillators,” Solid-State Circuits, IEEE Journal of, vol. 42, no. 6,

pp. 1425–1434, 2007. 16

[63] J. Vidkjær, “Noise and distortion,” 2005. 17

[64] H. Wakamatsu, T. Ono, and M. Esashi, “Parametrically amplified resonant sensor with pseudo-

cooling effect,” in Micro Electro Mechanical Systems, 2005. MEMS 2005. 18th IEEE International

Conference on, jan.-3 feb. 2005, pp. 343 – 346. 7, 27

[65] B. Walter, M. Faucher, E. Algre, B. Legrand, R. Boisgard, J. Aime, and L. Buchaillot, “Design

and operation of a silicon ring resonator for force sensing applications above 1 mhz,” Journal of

Micromechanics and Microengineering, vol. 19, p. 115009, 2009. 32, 47

[66] D. Wang, T. Ono, and M. Esashi, “Thermal treatments and gas adsorption influences on nanome-

chanics of ultra-thin silicon resonators for ultimate sensing,” Nanotechnology, vol. 15, p. 1851, 2004.

16

[67] J. Wang, Z. Ren, and C. Nguyen, “1.156-ghz self-aligned vibrating micromechanical disk resonator,”

Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on, vol. 51, no. 12, pp.

1607–1628, 2004. 16, 24

[68] D. Weinstein and S. Bhave, “Internal dielectric transduction in bulk-mode resonators,” Microelec-

tromechanical Systems, Journal of, vol. 18, no. 6, pp. 1401–1408, 2009. 21

[69] Z. XIONG, B. WALTER, E. MAIRIAUX, M. FAUCHER, L. BUCHAILLOT, and B. LEGRAND,

“Conception and fabrication of silicon ring resonator with piezo-resistive detection for force sensing

applications.” 32, 47, 72

[70] M. Younis, “Sensing and actuation in mems,” MEMS Linear and Nonlinear Statics and Dynamics,

pp. 57–96, 2011. 23

87

Documents

Novel Micromechanical Bulk Acoustic Wave Resonator Sensing