Novel Micromechanical Bulk Acoustic Wave Resonator Sensing Concepts for Advanced Atomic Force Microscopy

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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

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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

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Abstract

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Acknowledgement

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Contents

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 4thmode. 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) Lam´e 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 ofFelandFsversusxgbetween 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

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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 60µm 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

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LIST OF FIGURES

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LIST OF TABLES

List of Tables

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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 100µm. 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

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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.

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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

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)

Wherekcis 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 oscillatorω0 and the damping ratioζ described in the following can be used to rewrite equation 1.

ω0= r kc m andζ = c 2√mkc (2) with d 2 x dt2 + 2ζω0 dx dt + ω 2 0x = 0 (3)

If the damping ratioζ 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]

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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 ofe2π_.

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 ·_{dissipated energy per cycle}stored energy (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 ratioζ 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 ·pkc· m (6)

The quality factor can be written as

Q = √

kc· m

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2.1 Resonance

Wherekcis the spring constant andm the mass. [11, 43]

Another way to describe the quality factor is the ratio between the resonance frequency f0 and the bandwidthBW , 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 V ibration amplitude 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 QT ED + 1 Qsurf ace + 1 Qclamp (9)

The different factors of this equation are:

• 1_/_Q_viscous_{is the damping arising from the surrounding fluids}

• 1_/_Q_{T ED} _{stands for the thermal-elastic dissipation}

• 1_/_Q_{surf ace}_{are the losses occurring on the surface}

• 1_/_Q_clamp_{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]

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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_/_Q_{T ED). 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_/_Q_{surf ace). 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]

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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 ofkBT , with the Boltzmann-constant kBand the temperatureT caused by the Brownian motion, which creates a significant noise floor in the system. The fluctuation dissipation theorem is given by the following equation:

SF F = 2kBT mef fγ0 (10)

WherekBis the Boltzmann-constant,T the temperature, mef f is the effective mass andγ0the damping coefficient. This equation can be transformed using the following.

γ0 = 2ω0 Q , m = kc ω2 0 andω0= 2πf0 (11)

With the angular resonance frequencyω0, the quality factorQ, the spring constant kcand the resonance frequencyf0. With equations (10) and (11) the thermal noise can be expressed by the following equation.

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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 = r 8 3 kcnqBW3/2 f0A (13) Where kc is the spring constant,nq the deflection detector noise density, BW the bandwidth, f0 the resonance frequency andA 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.

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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

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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 50µm 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) Lam´e 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) Lam´e mode and c) wine-glass mode (red means larger displacement and blue less displacement).

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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. Lam ´e mode resonator

Lamé 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 Lamé 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 Lamé 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 Lamé 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]

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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 g xg 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 forceFshas 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 − x}g) describes the spring deflection from the initial position.

The electrostatic forceFelon the other hand has a non-linear characteristic and is define as

Fel= −

ǫ0ǫrAU2 2x2

g

(15)

Whereǫ0is the vacuum permittivity,ǫrrelative permittivity,A the electrode area, U the electric potential andxg the actual gap distance. From equation (15) it can be seen that the electrostatic force is inversely proportional tox2

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2.4 Resonance excitation and sensing Fs F orce Fs ,− Fel Gap x unstable points Fel always unstable one solution stable points d 0 2_/₃_{· d} stable region unstable region actuation voltage

Fig. 7: Plot ofFelandFsversusxgbetween 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]

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2.4 Resonance excitation and sensing

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 1µm, 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]

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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= r kc

m (16)

Wherekc is the spring constant or stiffness,ω0 the angular resonance frequency andm 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]

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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 NOxmolecules 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.

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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).

Antigen Antibody Cantilever Coating Particle 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-Rays Bulk material

IR absorbing material (blue) Thermal isolated area (yellow)

Cantilever

Electrode

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

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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]

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2.6 The atomic force microscope Detector Laser Laser beam Y-direction Z-direction Rotation Cantilever Tip Sample 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]

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non-2.6 The atomic force microscope

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 10N_/_m and 100N_/_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 25fN_/√

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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 150fm_/√

Hz.

Table 3 shows the most important parameters for the conventional AFM. Parameter Values for conventional AFM Spring constant 10 – 100N_/_m

Amplitude 10 nm

Resonance frequency 60 – 300 kHz Quality factor 500

Thermal noise limit 5 – 25fN_/√_Hz Laser sensitivity 50 – 150fm_/√

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].

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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 1MN_/_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 250µm 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 is0.2 nN_/√

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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

g ap width Air g ap width Air g ap width Diameter Diameter Length Length Width Width Width Length Air g ap width 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 10µm 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.

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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)

Wherekcis 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 40µm to 200µm 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

80µm 5µm – 20 µm 80µm 5µm – 20 µm

120µm 5µm – 40 µm 120µm 5µm – 40 µm

160µm 5µm – 60 µm 160µm 5µm – 60 µm

200µm 5µm – 80 µm 200µm 5µm – 80 µm

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

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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 5µm or smaller. Also the stability of the resonator can be critical for a high ratio, like 200µm 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 0.1µm and 1 µm. This interval corresponds to the current state of the art technology, where 1µm 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 0.1µm. 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 6.6·1011Hz, which for example corresponds to a disk with diameter of 80µm 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.

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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 Stif fness [N/m] Diameter (Length) [µm] Disk Square

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 f ill ratio = ring width

outer radius andf rame f ill ratio =

f rame width

f rame 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.