Universidad Tecnológica Nacional

Transcription

1 Universidad Tecnológica Nacional Facultad Regional Haedo CONCEPTO Y DISEÑO DE UN DISPOSITIVO PARA LA ADMISIÓN Y ACOPLAMIENTO DE COMPONENTES PARA ESCÁNER DE MICROSCOPÍA POR FUERZA ATÓMICA Tesis presentada para obtener el grado de INGENIERO MECÁNICO Tesista: Mariano Nicolás Metallo Director: Ing. Hugo E. Garbuglia Co-director: Ing. Antonio N. Lotorto 2012

2 Technische Universität Ilmenau Fakultät für Elektrotechnik und Informationstechnik Institut für Mikro- und Nano-Elektronik Diplomarbeit Konzeption und Entwurf einer Vorrichtung zur Aufnahme und Koppelung der Komponenten eines Scanners für die Rasterkraftmikroskopie Verfasser: Mariano Nicolás Metallo Verantwortlicher Hochschullehrer: Betreuer: Prof. Dr.-Ing. habil. Ivo W. Rangelow Dipl-Ing. Sindy Hauguth-Frank Dipl-Ing. Joszef Krol 2012

3 Contents

4 Summary... 4 Basic Principles of AFM Introduction Components of the Microscope Tip Surface Interaction Lennard-Jones potential Ideal force-distance curve Modes of Operation Contact Mode Hertz problem definition and its solution in a general form Non-Contact Mode Measurement Principle Tapping Mode (Intermittent Contact Mode) Comparison of Modes The Scanner and its related artifacts Mechanics of Cantilevers The tip of the cantilever Deflection under forces acting in XYZ Deflection under Fz Deflection under Fy Deflection under Fx Effective mass and Eigen frequency of the cantilever Measuring Systems Optical Beam Deflection (OBD) Components of the Stage Mechanical Strength of Silicon Wafers Strength of Silicon Wafers with micro cracks Analysis of Si Wafer and FEM Simulations Design of the Support Characteristics of the fastening structure Specifications Motion and Positioning of Piezoactuators Mechanics from flexure-based amplification... 47

5 Working principle of flexure mechanism Bridge-type amplification mechanism Designs and FEM Simulations Primary design Secondary design Third design Preloading of the Piezoactuator Conclusions Bibliography Summary

6 AFM (Atomic Force Microscopy) is an extremely precise microscope that images a sample by rapidly moving a probe with a nanometer-sized tip across its surface. This is quite different from an optical microscope which uses reflected light to image a sample. An AFM probe offers a much higher degree of resolution than an optical microscope because the size of the probe is much smaller than the finest wavelength of visible light. In an ultra-high vacuum, an atomic force microscope can image individual atoms. The AFM uses a micro scale cantilever with a probe tip whose size is measured in nanometers. An AFM operates in one of two modes: contact (static) mode and dynamic or non-contact (oscillating) mode. In static mode, the probe is kept still, while in dynamic mode it oscillates. When the AFM is brought close to or contacts the surface, the cantilever deflects. Usually, on top of the cantilever is a mirror which reflects a laser. The laser reflects onto a photodiode, which precisely measures its deflection. When the oscillation or positions of the AFM tip changes, it is registered in the photodiode and an image is thus produced. AFM allows non-destructive testing of conductive and nonconductive surfaces with nanometer resolution. To fulfill these measuring duties in shorter time or to observe measuring objects on a real-time basis, the AFM technology puts growing demands to the conceptual-mechanical construction of an AFMs. This diploma thesis focuses in the design of a suitable device for the storage and coupling of the individual components of a silicon-based scanning system and the characterization of a prototype. The basic principles for the design, the simulations it underwent and the instrumentation will all be topics that will be discussed in this document. Assignment Analysis of the components of the scanner system (FEM Simulations, analysis of the piezoelectric actuators) Design and construction of a device for the admission and coupling of the components (CAD Model, calculations, technical designs) Validation of simulations through measurements Documentation Chapter 1

7 Basic Principles of AFM 1.1 Introduction AFM can be operated in a number of modes, such as phase imaging microscopy, magnetic force microscopy (MFM) or electrostatic force microscopy. In general, these imaging modes are divided into two groups, static (or contact) modes and dynamic (or non-contact) mode, where the cantilever vibrates over the sample. Fig General Operation of an AFM system

8 a) b) Fig a) Principle of STM, b) Principle of AFM Fig Scan of an area of 5x5 μm. Picture courtesy of Technical University Ilmenau Fig Image showing tip-sample interaction at an atomic scale level.

9 1.2 Components of the Microscope The resolution of AFM depends mainly on the sharpness of the tip which can be currently manufactured with an end radius of a few nanometers. A close enough inspection of any AFM tip reveals that it is rounded off; therefore tips are generally evaluated by determining their end radius. In combination with tip-sample interaction effects, this end radius generally limits the resolution of AFM. As such, the development of sharper tips, i.e. nano-tubes, is currently a major concern. Atomic resolution is easily obtained on relatively robust and periodic samples. Soft samples particularly biological samples provide a more difficult surface to image because the forces exerted by the tip during imaging can cause deformation of the sample. The problem involved with imaging soft samples has been overcome to a large extent by the introduction of tapping mode AFM imaging. Instead of maintaining a constant tip-sample distance of a nanometer or so, the cantilever is oscillated in a direction normal to the sample resulting in only intermittent contact with the surface. This greatly reduces the lateral forces being applied in the plane of the sample which are responsible for most of the damage as the tip is scanned. The AFM is capable of better than 1 nm lateral resolution on ideal samples and of 0.01 nm resolution in height measurement. Fig Principle of Scanning: Protrusions appears wider and depressions narrower than they are in reality If the interaction decay length κ << R, the gap s between sample and tip, s << R, the tip geometry determines the resolution; in the most simple approximation we imagine a sphere of radius R to roll over the surface S, and the path A of its center to be recorded. The non-linearity of this process is evident: the image of a protrusion appears wider, but the image of a depression appears narrower than the object. The practical resolution, however, is also determined by the sensitivity of the height detector, i. e. the noise level. Provided, that the mechanical and electronic stability of the microscope allows detection of i.e nm height differences, it is estimated that the tip radius R needs to be smaller than 3 nm to resolve spheres of 0.5 nm diameter.

10 Fig Variables that determine the resolution of the scan The equation that provides us with the diameter FMWH goes as follows: FMWH = 2 d(r + d 4 ) 1.3 Tip Surface Interaction When the tip is brought close to the sample, a number of forces may operate. Typically the forces contributing the most to the movement of an AFM cantilever are the Coulomb and van der Waals interactions. Coulomb interaction: This strong, short range repulsive force arises from electrostatic repulsion by the electron clouds of the tip and sample. This repulsion increases as the separation decreases. Van der Waals interactions: These are longer range attractive forces, which may be felt at separations of up to 10 nm or more. They arise due to temporary fluctuating dipoles. The combination of these interactions results in a force-distance curve similar to that below:

11 Fig Plot of force against distance As the tip is brought towards the sample, van der Waals forces cause attraction. As the tip gets closer to the sample this attraction increases. However at small separations the repulsive coulomb forces become dominant. The repulsive force causes the cantilever to bend as the tip is brought closer to the surface. There are other interactions besides coulomb and van der Waals forces which can have an effect. When AFM is performed in ambient air, the sample and tip may be coated with a thin layer of fluid (mainly water). When the tip comes close to the surface, capillary forces can arise between the tip and surface Lennard-Jones potential A pair of atoms or molecules is hold to two different forces at the limit of a great separation and of a small one: an attractive force acts at great distances (Van der Waals, or dispersion force) and a repulsive force acts at short distances (the result of the superposition of electronic orbitals, known as the Pauli repulsion force). The Lennard-Jones potential is a mathematical model that represents this phenomenon Fig Lennard-Jones potential This can be represented through the following equation Where is the potential depth, is the distance (finite) in which the potential between particles is zero and r is the distance between particles

12 These parameters can be adjusted to reproduce experimental data or can be deduced from precise calculations from quantic chemistry. The term r -12 describes repulsion and r -6 describes attraction Ideal force-distance curve Idealized force-distance curve describing a single approach-retract cycle of the AFM tip, which is continuously repeated during surface scanning. The AFM tip is approaching the sample surface (A). The initial contact between the tip and the surface is mediated by the attractive van der Waals forces (contact) that lead to an attraction of the tip toward the surface (B). Hence, the tip applies a constant and default force upon the surface that leads to sample indentation and cantilever deflection (C). Subsequently, the tip tries to retract and to break loose from the surface (D). Various adhesive forces between the sample and the AFM tip, however, hamper tip retraction. These adhesive forces can be taken directly from the force-distance curve (E). The tip withdraws and loses contact to the surface upon overcoming of the adhesive forces (F). Fig Cantilever over a surface in Contact mode. 1.4 Modes of Operation Contact Mode In contact mode AFM, the tip from the cantilever remains in a constant physical contact with the surface of the sample. The tip follows the topography of the surface and forms an image.

13 Fig Cantilever over a surface in Contact mode. The cantilever is deflected due to strong atomic forces, and this is measured by its refraction. High scan speeds are feasible due to the fact that the feedback loops do not require much signal processing. The continuous contact of the tip from the cantilever results in a subsequent deformation of the tip due to elastic forces, leading to poor image quality or even destruction of the sample (when referring to soft samples). Fig Soft samples are more easily deformed when the interaction is permanent In addition to the repulsive van der Waals force, two other forces are generally present at contact mode: a capillary force exerted by the thin water layer often present in an ambient environment and the force exerted by the cantilever itself. The capillary force arises when water wicks its way around and applying a strong attractive force (about 10-8 N) that holds the tip in contact with the surface. The magnitude of the capillary force depends upon the tip-to-sample separation.

14 Fig Graph showing the interaction between the tip and the sample During contact of the tip of the cantilever and the surface of the sample, the elastic repulsion force is predominant. An approximation to the way this force acts on the system is called Hertz Model and the solution it brings will be discussed in the following point Hertz problem definition and its solution in a general form When two solids are in point contact between each other (assuming that these solids are the point of the tip from the cantilever and the sample itself) and a force is applied, we would have a deformation that we could represent with the Hertz Model. Let us adopt the following simplifying assumptions [4]: 1. Bodies are filled with uniform isotropic linearly elastic media characterized by Young's module E, E and Poisson ratios μ, μ. 2. The surfaces curvature weakly affects the mode of deformation. 3. Boundary surfaces are interchanged by the elliptic parabolic. 4. The point of contact is not the singular point, the contact area is the simply connected domain and its contour is ellipse.

15 Fig Two bodies contact before deformation. Fig Deformation two bodies. Surfaces before deformation are shown by dotted line, and squeezed surfaces full line. The characters z and z denote lengths, which are determined by equations (1) and (2). When a load is applied, the two contacting bodies would no longer be having a single contact point but rather a contact area. As the problem symmetry is axial, the area is circular, und we could denote its radius by a. Let us introduce the following convenient quantities: 1 = and effective Young's modulus of R r r the given pair of materials: 1 = 3 K 4 (1 μ μ2 ) [1] E E At small deformations (assumption 3) the following geometric relation between penetration depth h and contact circle radius a is valid: h = a2 R [2] The Hertz problem solution relates the loading force F and the penetration depth h: F = Ka3 = R Kh3 2R 1 2 [3] Accordingly, the pressure is the following function of the force: P = F = 1 3 πa 2 π FK2 R 2 [4] The given solution for the case of two spherical bodies contact includes one important special case of the flat sample contact with the tip having curvature radius R (r = R, r = ).

16 Let us depict the Hertz problem solution, i.e. the dependence of the penetration depth (horizontal axis) upon the loading force (vertical axis) for positive h. In Fig. 3, the rising branch corresponds to the Hertz problem solution. The Hertz problem solution relates the deformation and applied load. Penetration h is proportional to the compressing force as F 2 3. Fig Force F depending on penetration depth h (graph of the Hertz problem solution) Non-Contact Mode In non-contact AFM, the cantilever is placed in a way where it can vibrate over the sample at its free resonance frequency, and as it approaches the sample its frequency is changed and thus its amplitude. Fig Cantilever oscillating over a surface in Non-Contact mode.

17 The changes are measured and an image of the topography from the surface is made. As there is no contact between the surface of the tip and the sample, no deformation is produced. Fig Graph showing the different forces applied (attractive or repulsive) between the tip and the sample as the distance d is changed. Because of the attractive force between the probe tip and the surface atoms (Van der Waals forces), the cantilever vibration at its resonant frequency near the sample surface experiences a shift in spring constant from its intrinsic spring constant (k 0 ). This is called the effective spring constant (k eff ), and the following equation holds: k eff = k 0 F (1) When the attractive force is applied, k eff becomes smaller than k 0 since the force gradient F = F z is positive. Accordingly, the stronger the interaction between the surface and the tip (in other words, the closer the tip is brought to the surface), the smaller the effective spring constant becomes. This alternating current method (AC detection) makes more sensitive responds to the force gradient as opposed to the force itself. The AFM system detects changes in the resonant frequency or vibration amplitude as the tip comes near the sample surface. The sensitivity of this detection scheme provides sub-angstrom vertical resolution in the image, as with contact AFM. On the other hand, the spring constant affects the resonant frequency (f 0 ) of the cantilever, and the relation between the spring constant (k 0 ) and the resonant frequency (f 0 ) in free space is shown as in (2). f 0 = k o m (2)

18 As in (1), since k eff becomes smaller than k 0 due to the attractive force, f eff too becomes smaller than f 0. If one vibrate the cantilever at frequency f 1 (a little larger than f 0 ) where a steep slope is observed in the graph representing free space frequency vs. amplitude, the amplitude change ( A ) at f becomes very large even with a small change of intrinsic frequency caused by atomic attractions. Therefore, the amplitude change measured in f 1 reflects the distance change ( d) between the probe tip and the surface atoms. If the change in the effective resonance frequency, f eff, resulting from the interaction between the surface atoms and the probe, or the change in amplitude ( A) at a given frequency (f 1 ) can be measured, the Non-Contact mode feedback loop will then compensate for the distance change between the tip and the sample. By maintaining a constant amplitude (A 0 ) and distance (d 0 ), Non- Contact mode can measure the topography of the sample surface by using the feedback mechanism to control the Z-scanner movement following the measurement of the force gradient represented (1). Fig Resonant frequency of a cantilever Fig Resonant frequency shift as the tip approaches the sample surface. Figure Tip-sample distance vs. amplitude change as the tip approaches sample surface Measurement Principle As seen previously, the tip is moved up and down periodically as the surface is scanned. The tip stays within the range of the atomic interaction forces and only the points at which the deflection of the cantilever is zero are used as a trigger for the readout of the interferometers. The measurement data from the interferometers determines the samples topography. The reason to use this measurement principle, called zero-cross-system, are the following: Nonlinear effects, for example the nonlinearity of the cantilevers stiffness are avoided. Calibration of these effects is not needed.

19 Moving the sample up and down instead of keeping the sample at its position within 1nm results in reduced control requirements Fig Cantilever deflection with zero-cross-system Tapping Mode (Intermittent Contact Mode) In tapping mode the cantilever oscillates at or slightly below its resonant frequency. The amplitude of oscillation typically ranges from 20 nm to 100 nm. The tip lightly taps on the sample surface during scanning, contacting the surface at the bottom of its swing. Because the forces on the tip change as the tip-surface separation changes, the resonant frequency of the cantilever is dependent on this separation. Oscillation is also damped when the tip is closer to the surface. Hence changes in the oscillation amplitude can be used to measure the distance between the tip and the surface. The feedback circuit adjusts the probe height to try and maintain CONSTANT amplitude of oscillation (opposing the non-contact mode that maintains a constant FREQUENCY). This value is the amplitude set point.

20 1.4.4 Comparison of Modes Advantage Disadvantage Contact Mode High scan speeds Lateral (shear) forces may distort features in the image Rough samples with extreme changes in vertical topography can sometimes be scanned In ambient conditions may get strong capillary forces due to adsorbed fluid layer more easily Combination of lateral and strong normal forces reduce resolution and mean that the tip may damage the sample, or vice versa Tapping Mode Lateral forces almost eliminated Slower scan speed than in Higher lateral resolution on most samples Lower forces so less damage to soft samples or tips contact mode Non-contact Mode Both normal and lateral forces are minimised, so good for measurement of very soft samples Can get atomic resolution in a UHV environment In ambient conditions the adsorbed fluid layer may be too thick for effective measurements Slower scan speed than tapping and contact modes to avoid contacting the adsorbed fluid layer 1.5 The Scanner and its related artifacts Hysteresis The piezoelectric s response to an applied voltage is not linear. This gives rise to hysteresis. Since the scanner makes more movement per volt at the beginning of a scan line than at the end, this can

21 cause artefacts in the images, especially at large scan sizes. This is overcome by using a non-linear voltage waveform calculated during a calibration procedure. Fig Example of a Voltage waveform calibrated to overcome hysteresis Scanner creep If the applied voltage suddenly changes e.g. to move the scanning position, then the piezoscanner response is not all at once. It moves the majority of the distance quickly, and then the last part of the movement is slower. If this is done during scanning, then the slow movement will cause distortion. This is known as creep. a) b) Fig a) When a change in x-offset is applied, features are distorted in the x-direction. b) Features distorted in the Y- direction

22 Fig Scan size is changed abruptly, and features are distorted Bow and tilt Because of the construction of the piezo-scanner, the tip does not move in a perfectly flat plane. Instead its movement is in a parabolic arc, as shown in the image below. This causes the artifact known as scanner bow. Also the scanner and sample planes may not be perfectly parallel, this is known as tilt. Both of these artifacts can be removed by using post-processing software. Fig Scheme depicting curved motion of probe

23 Chapter 2 Mechanics of Cantilevers The cantilever is the most widely used sensor in AFM. The information is acquired through the means of mechanical deflection of the cantilever beam which are detected by different methods (i.e. optical system). These can be used both in contact (static) or non-contact (dynamic) modes.

24 Fig. 2.1 Cantilever used in AFM systems Silicon is the common material for cantilevers, as it has a high stiffness and offers a regular crystal structure. This structure offers very good operation at high frequencies and when high a Q-factor is needed. Polymers are also used as they have a very low Young s modulus (1-10 GPa) and thus can deflect more widely under bending moments. They also have very good thermal and electrical insulation. Fig. 2.2 Rectangular cantilever with a length l, thickness t, width w and a tip of length l tip at its free end. The lateral stress σ L = R R of the cantilever is indicated by the following equation: σ L = R R = β 3π L(1 ν) (σ t 1 σ 2 ) Where: π L is the piezoresistive coefficient from Si (110), σ 1 is the longitudinal stress, σ 2 is the transverse stress, t is the thickness, ν the Poisson s ratio and β the piezoresistor s lage. When a force F is applied and the free end of the cantilever beam, it deflects. This displacement generates a tensile stress in the upper edge and a compressive one in the lower edge, which are equal and opposite between each other. A hole is generated in this place, to allow the stress to be

25 concentrated into one area, due to the smaller surface. This way, by placing the sensors the values measured would be higher, and thus would allow for a better resolution. a) b) c) Fig. 2.3 a) Cantilever s deflection, b) Stress concentration, c) Closer looks of the full bridge + Fig. 2.4 Images showing piezo-resistive sensor and its full bridge. PRONANO, Ilmenau. 2.1 The tip of the cantilever In order to detect local forces or closely related physical quantities the sharp probe scanning the sample surface at same distance has to be liked to some sort of force sensor. A convenient way to precisely measure forces is to convert them into deflections of a spring according to Hooke s law: z = f k c Where the deflection Δz is determined by the acting force ΔF and the spring constant kc.

26 The resonant frequency of a spring with spring constant k c and lumped effective mass m is given by π 0 = k c m Because of the Hook s law it is desirable to have a low spring constant in order to achieve maximum force sensitivity. This is contradicted by three aspects: The spring constant should be a maximum in order to achieve a maximum resonant frequency, and thus, a minimum vibrational sensitivity and a maximum scan rate. The ultimate sensitivity of the force measurement is restricted by thermal excitation of the cantilever. The latter quantity can be determined from the equipartition theorem (Δ z ) rms = k BT k C Where is (Δ z ) rms the rms displacement amplitude of the end of the cantilever due to thermal excitation. If the cantilever is subject to a long-range attractive force, and this will almost always be the case upon probe-sample approach, its position becomes unstable if the magnitude of the force gradient equals the cantilever s spring constant. Thus, a certain minimum spring constant is needed in order to approach the sample sufficiently closely without a jump to contact. In order to estimate the order of magnitude which the spring constant of the cantilever could have, it is straightforward to matc k C to the respective constant of interatomic coupling in solids. Taking m = kg and ϖ 0 = 1013 Hz for atomic masses and vibrational frequencies are arrives at k C = 10N/m. Even smaller spring constants can be easily obtained by minimizing the cantilever s mass. Commercial cantilevers have a typical spring constant in the range of 10-2 N/m k C 102 N/m, typical resonant frequencies in the range of 1 khz ϖ khz, a radius of curvature of the probing tip as small as 10 nm, and are usually fabricated of Si, SiO2 or Si3N4. If one again takes the above estimate for the interatomic coupling (c = 10 N/m) for a rough estimate of the resulting deflection of a cantilever which is subject to an interatomic interaction, one finds that a force of 1 nn causes a deflection of 1Å, while thermal rms raise amounts to above 20% of this value. Thus, the task is to precisely measure cantilever deflections being smaller than 1Å.

27 2.1 Deflection under forces acting in XYZ Deflection under F z Here we can see how the force F z is acting on the cantilever beam and how z-type deflection is produced. This is the result of a vertical bending force action. Fig Vertical deflection in the z-axis For reactions on a portion of a beam we consider the deformation of the beam having a length l between the two cross sections. As we can see in Fig , the inner edge of the material is compressed due to atoms being closer to each other, and the other edge is stretched in tension. As the two deformations are but the same, there is a neutral plane of zero stress between the two surfaces. Fig Section of the bent beam Fig Cross-section of the beam

28 As we can see the longitudinal extension is proportional to distance z from the neutral plane: l l = z R, R being the beam curvature radius. According to Hooke s law, if we consider the force acting on a unit area in a small strip near z then we have: df = Ez ds R The forces are seen to be acting over the neutral surface and in the other direction below it. This produces a bending moment M z with respect to the neutral axis: M z = s zdf = E R J z The axial moment of inertia J z is given by: J z = s z 2 df = wt3 12 We can denote the deflection of the beam point at the distance y from the fixed end in the z- direction by using u(y). This curvature of u(y) at small bends (du/dy 1) is given by 1 R = d 2 u dy 2. Therefore, the bending moment M z can be expressed as: M z = EJ z d 2 u dy 2 But M z, is a bending moment composed of two forces acting in respect to point y, F z (M z = F z (l y)) and the beam s own weight (M mg = mg l l pdp = mg 2l (l2 y 2 )). y It is known that the weight for a typical cantilever is just a fraction of an angstrom and the first term is of a magnitude hundreds of times more, we can neglect it. The deformation is small when compared to minimal detected displacement. And by setting as boundary conditions u asdasd we arrive to the following equation: And then the beam end deflection Z is: We know that Z = c F z and that c is the inverse stiffness (so 1/c characterizes the cantilever stiffness). We can rewrite this equation for a cantilever with a rectangular cross-section as follows: And for the beam deflection angle we would have:

29 2.1.2 Deflection under F y In the figure below, it s possible to see how the force F y produces a moment M = F y l tip that results in a deformation of the y-type. Deflection not only occurs in the longitudinal but also in the vertical direction. It is a result of an axial bending force action. Fig Vertical deflection of the y-type The equation describing y-type bending is the following: As boundary conditions are the same as before u y=0 = 0 and du = 0 we can say that the dy y=0 deflection in the z-axis is as follows: The angle of the beam end deflection is given by the following formula: And so, we can arrive to the equation ruling the tip deflection due to the force F y.

30 2.1.3 Deflection under F x Here we will determine the magnitude and direction of the deformation produced by the transverse force F x. As we can see in the following figures, the resulting deformation will be a combination of simple bending and twisting. Fig Simple bending Fig Torsion The stiffness (1/c) for simple bending is analogous to the vertical bending of z-type with the difference that we have to interchange beam width with its thickness. Solution to the problem of the rectangular beam torsion is more complicated to solve. The moment applied by a force F x acting on a tip of length l tip is M = F x l tip. The tip lateral deflection is given by the following equation: Where Beta is the torsion angle And so the inverse stiffness coefficient reads: Where G is the shear modulus and is given by G = E 2(1+v) majority of materials) and so the torsional coefficient is calculated as:, the Poisson s Ratio is v~1,3 (for the

31 The resulting tip deflection we just have to sum the corresponding deflections (assuming that deformations are small) Thus, the resulting inverse stiffness is a sum of the simple bending and torsion inverse stiffness, too: 2.2 Effective mass and Eigen frequency of the cantilever As we saw before in Chapter 1, there are other ways of conducting a scan using AFM. One is using a dynamic (or non-contact) mode, where the cantilever will vibrate at its natural frequency and by maintaining the amplitude constant and through feedback and use of piezo-actuators, form an image of the topography. At first we have to determine the kinetic and potential energy of the cantilever. Considering the beam element to have a length dy at a distance y from the fixed end. Kinetic energy would be given by the following: Where u(τ, y) is the displacement from the beam at a distance y in an amount of time t. Using previous formulas, we arrive to the following equation: We change this expression for u(τ, y) and integrate over the beam length, resulting in: The force F only acts on the free end and E nom is equal to the work at distance u(τ, y). Potential energy it thus:

32 If systems vibrations are considered to occur without total energy W dissipation, i.e. W = E kuh + E nom = const, then when we differentiate W with respect to time, we get the equation of the cantilever free end move: Therefore, the effective mass would be: Then, by knowing the effective mass and the coefficient of stiffness 1/c, the Eigen frequency of the cantilever oscillation can be expressed as the following: Where ρ is the cantilever density, E the Young s Modulus and π o is inversely as the square of the beam length. The cantilever eigenfrequency must be as high as possible; otherwise its natural oscillations will be readily excited due to the probe trace-retrace move during scanning or due to external vibrations influence. Chapter 3 Measuring Systems Several measurement systems are considered to measure the deflection of the cantilever. In this chapter a measurement system is selected, after which a more detailed explanation of the systems working principle is given. The mechanical design of the components will follow from the requirements and evaluations of different designs. System selection

33 The three common deflection detection systems and their major advantages and disadvantages are: 1. Capacitance detection: The cantilever is one side of a capacitor; deflection is measured as change in capacitance. Advantages Relatively simple and compact Low noise Disadvantages Electrostatic gradient can cause snap-in of cantilever to capacitor plate Drift in tip-sample capacitance Stray capacitance 2. Optical Beam Deflection (OBD): A laser is pointed on the backside of the cantilever under an angle and is reflected on a spot displacement detector, at some distance. Deflection and thus bending of the cantilever is measured as laser spot displacement on the detector. Because of the optical path length required, the system is sensitive to angle deviations of the laser source [14],[15]. Advantages High resolution easily achievable Easy to operate Torsion and deflection of cantilever can be measured No influence optical path elongation Disadvantages Medium laser power needed Tuning of optics required 3. Interferometry: A laser beam is split into a reference and measurement beam. The measurement beam is pointed perpendicular on the backside of the cantilever; the reference beam travels an equal path length. At cantilever deflection the path length of the measurement and reference beam differs, the difference can be measured as a phase shift of the optical waves in the laser. Differences in the speed of light between the reference and measurement path, due to temperature gradients, are also measured as displacements. Advantages Disadvantages Traceability wavelength Complicated system Large bandwidth High laser power needed Not influenced by cantilever shape or Sensitive to temperature gradients size

34 3.1 Optical Beam Deflection (OBD) The components of the OBD measurement system are listed below in the order in which they are discussed in the next sections. Cantilever Module (CLM) holds the cantilever. Photo Diode Module (PDM) contains the spot displacement detector. Laser Diode Module (LDM) contains the laser source with beam-shaping optics. The lay-out of the OBD is shown in Figure 3.1. The figure shows the optical axis of the laser beam with and without cantilever deflection. The laser beam is focused on the cantilever by the LDM and hits the detector at distance s. The relation between the vertical cantilever deflection z and spot displacement a on the detector is given by: a = 3s l z In which l[m] is the cantilever length and s[m] is the cantilever to detector distance. The vertical resolution of the OBD can be increased by: Selecting a cantilever with a smaller length l CL, Increasing the cantilever to detector distance s, Selecting a detector with a small minimal detectable spot displacement a. Fig. 3.1 Basic lay-out of OBD

35 To detect the spot displacement two types of detectors are considered: segmented and lateral Position Sensing Detectors (PSD). Segmented PSD The left image of Figure 3.11 shows a schematic drawing of a segmented PSD. Segmented PSD s are substrate photodiodes divided into segments, separated by a gap. A symmetrical spot generates equal photocurrents in all segments, when it is positioned at the center. The difference in currents from the diodes is used to describe the x and y displacement of the spot, as shown in Equation (3.5) and (3.6). The signal is normalized by the total signal; thereby the detector is not sensitive for uniform changes in optical power. Segmented PSD s can offer the required 0.1nm resolution due to the match in photosensitivity between the segments. x = (i B + i D ) (i A + i C ) i A + i B + i C + i D y = (i A + i B ) (i C + i D ) i A + i B + i C + i D Fig. 3.2 Schematic drawing of Segmented PSD, gray area = laser spot. Chapter 4 Components of the Stage

36 This project is currently under development in the Technische Universität Ilmenau and receives the name Dynamic nano-positioner device (DNPD). Its main points due its manufacturing under a monocrystalline Si-Wafer (001) are: No plastic deformation No Creep effect Piezo resistive sensors integrated Fig. 4.1 Setup at FG MNES (Mikro- und nanoelectronische Systeme) Fig. 4.2 Setup at Department of Micro and Nanotechnology Systems 4.1 Mechanical Strength of Silicon Wafers

37 Due to testing conducted in [11], the relationship between breakage force F and thickness has been found to be linear instead of quadratic as theory predicts. Therefore, thinner wafers tolerate higher force than expected. These wafers bend and stretch due to increased flexibility and redistribute the stress inside the wafer. Fig Graph showing the measured fracture force as function of the wafer thickness As for the stage a Si wafer (100) with a thickness of 1000 μm will be used, it s of relevance this relationship as the breakage force expected by increasing the thickness will not follow the theoretical relationship σ F t 2 A Si Wafer in (100) orientation was reported to may break at lower stress than a (111) does, but as it was measured by [12] fracture stress depends more on the surface condition or the length and number of micro cracks than on the surface orientation. The fracture stress of an ideal material is theoretically expressed as: σ m = ( Eε 0 a ) Where E is the Young s Modulus, ε 0 the surface energy of the material and a is the inter-atomic distance. For a (111) surface, for an ideal case we would obtain σ m = 32 GPa. The real material reacts in a different way, and is typically almost two orders of magnitude more fragile. This is caused mostly to micro-defects in the specimens. σ F = ( 2Eε 0 l C )

38 Where l C is the crack length. Since a = m and l C Strength of Silicon Wafers with micro cracks Residual micro cracks are one of the major causes for reducing the strength of Si Wafers, and thus resulting in a breakage of the material at lower stresses than expected. These are generated by cutting and wafering procedures and are not removed by subsequent etching of the wafers. According to [9] we can relate the strength of a brittle piece to its volume, area or length to the failure mode that we are calculating. So, the probability F V (σ) survives load σ is given by: F V (σ) = exp ( V ( σ γ ω V V ) dv ) α V And respectively for damages on the surface and on the edges: F A (σ) = exp ( A ( σ γ ω A A ) da ) α A F L (σ) = exp ( L ( σ γ ω L L ) dl ) α L By combining the three failure modes we can yield the effective probability of survival for the entire wafer as the following: F(σ) = F V (σ). F A (σ). F L (σ) F(σ) = exp [ V ( σ γ π V V ) α V dv A ( σ γ π A A ) α A da L ( σ γ π L L ) α L dl] This will follow a Weibull distribution and according to tests conducted by [10], we can see it being represented in this way:

39 Fig Predicted wafer strength distribution for a statistical sample of 100 wafers. 4.2 Analysis of Si Wafer and FEM Simulations The first part of this research was to simulate the stress applied to the Silicon stage and realize its limitations as well as other physical characteristics. That way, it was found that the stage has an eigenfrequency of 1.9 KHz without any sort of clamping (being in the order of 36 KHz considering all factors)[a], and that the maximum force that can be applied to it without it breaking is in the order of 5-10N. The maximum travel range of the stage was found to be 11um, so that means that the range provided by the PZT should not exceed this amount. In the Catalogue from PI it was found that the best option fitting this need was the P with a price of 149 (as of 2011) and a 6,5um +-20% travel at 100V (dimensions are 3x2x9mm). It has also a blocking force of 190N and a stiffness of 24 N/um. This range would then be amplified to a magnitude that will suffice the needs of the system. This would be described later on. These Silicon wafers have a (001) distribution, meaning that it has its crystals organized in the (Z) direction. Silicon it s a very stiff material having a 129,5 Gpa E modulus and also a limit stress of about MPa.

40 Fig ANSYS Modal Simulation of the eigenfrequency (2KHz) of the Silicon Stage. As we can see from Fig , the eigenfrequency is seamlessly low and for high frequency scanning it should be increased. The structure designed for the support should be above this natural frequency (or at least 50% of it) so that it would not resonate before this does. We can also note that from the different resonant modes (20 modes were found below 10KHz), the area showing around the moving part of the stage is always reducing this f R. This area should be clamped in order to increase the f R. In Fig. 2, we can see how reducing the surface of the Si-Stage, and clamping it can increase the f R in an order of more than 2. It should be noted though, that only the borders were clamped for the simulation and not the upper and under parts (for example, being fixed to the surface of the support with some adhesive). This leaves a margin for error, but gives an idea of values to perceive.

41 Fig ANSYS Modal Simulation giving an eigenfrequency of 4.8 KHz for the clamped Silicon Stage. Fig ANSYS Static Simulation showing the stress applied to omega-shaped elements when trying to give a preload to the PZT. Here is shown with 2 forces applied on opposite directions and of a magnitude of 10N. These 2 forces represent the PZT in a static manner (meaning that the stage is giving a preload to them). Maximum applied stress is 360 MPa. The omega-shaped elements act as springs for the preload of the PZTs. This function but up to a certain degree, because according to previous testing and from data from the PI Catalog, the preload should be on the order of 20 to 30N, and this element is shown (Fig ) not to behave properly. Fig shows us the maximum force that can be applied to the moving part of the stage, where the tip of the cantilever would be actuating. The mode used will be that of a non-contact mode so the tip would not be pressing down against the surface, but nevertheless it is an important fact to notice as this tip has to be positioned before starting the scanning and thus some degree of force would be generated for first approaches.

42 Fig ANSYS Static Simulation showing the Stress applied to the moving plate of the Stage with a F=5N. Maximum applied stress is 417 MPa. Fig shows us what happens when the force F is applied directly on the sides of the stage and not on the springs. We can see that the amount of force that the stage is able to withstand is greatly diminished having now with half the force applied, 16% more Stress. Fig ANSYS Simulation showing the maximum physical travel of the moving part of the stage (285um). Maximum travel was later found to be 11um.

43 Chapter 5 Design of the Support The purpose of this support is to host the piezo-actuators and also provide the means of fastening the structure altogether. That means, it must dampen mechanicals vibrations from the system and also be mechanized from within a defined dimension and weight, not to produce any side effects on the rest of the structure. PZT (Piezoactuators) are widely used in precision instruments due to its major advantages of large blocking force, high stiffness, fast response, and compact size. Comparing with other types of linear actuators, the main drawback of PZT lies in its small travel stroke (between 8-30um from the PI Catalog). If the stroke of the adopted PZT cannot meet application requirements, a proper amplification mechanism will be exploited to suit the needs. 5.1 Characteristics of the fastening structure The Si-wafer should be fixed to the surface of the support, from which the options of fabrication (material) are Steel, Invar, Zerodur or Borofloat Glass. Even though Aluminium is widely used for applications in micro and nanopositioning, is not going to be used because of the grade of uncertainty it brings, as it elongates 40um every meter and that is not acceptable in the nanoscale. The means of fastening are yet be found. Being 3 ways possible: 1. Si-stage fixed to support, leaving the support fixed to ground. 2. Si-stage fixed to ground, and support fixed to it 3. Both fixed to ground separately (not likely to be used, as it would leave the Si-stage under defined). Some design changes are also suggested, such as this one shown in Fig :

44 Fig Pictoric drawing of suggested design for interaction between PZT and Si-stage The main idea is not changed, and that would be of a moving plate allowing fast scanning. Omegashaped springs would not be used as the pre-load of the PZTs is done by another mean (metal cage, springs, structural, etc). 2 PZTs could be used in the 2 Axis (X-Y) and 2 position sensors on the opposite side. The downside of this is the reduced speed that could be achieved. That way, the idea of using 4 PZTs leaves no other way than to have these piezoresistive sensors from the Sistage accurately giving out the position of the system, as when it would not have been that way the whole system would have been in an open loop not being possible to know precisely where it was positioned Specifications Choice of Material The choice for the material of the Support is between Steel, Aluminium, Invar, Zerodur and Borofloat Glass. Aluminium will not be used as it produces a displacement of 40 um per meter, and that is not acceptable in the nanoscale for an error. Invar is widely used for precision machinery, by its properties of low thermal expansion. The downside is that it has a tendency to creep. Zerodur is develop by Schott and displays a nearly zero thermal expansion ( /K at 0 C 50 C) with outstanding 3D homogeneity.

45 Borofloat Glass is manufactured by Schott, and is a highly chemically resistant borosilicate glass with low thermal expansion that is produced using the float method. Other key properties of Borofloat are its mirror like surface quality, high thermal resistance, and excellent transmission. Precise motion needs guidance, not friction As friction is one of the main issues in nanopositioning, all devices with ball, roller or sliding bearings are not used, leaving air bearings and flexures. A flexure is a frictionless, stictionless hingelike device that relies upon the elastic deformation (flexing) of a solid material to permit motion. Air bearings are ideal for long travel ranges, but they are usually bulky, high inertia and expensive to operate (require clean air supply). They have another major disadvantage: they do not work in a vacuum, as required by an ever-increasing number of nanopositioning applications. Flexures, work over short travel ranges as needed in nanopositioning. Refer to Fig for an example of a flexure design, that provides trajectory control with excellent straightness and flatness, exhibit no wear and can be designed in multi-axis arrangements. They are also maintenance free and have no operating costs. These characteristics make flexures the guiding mechanism of choice in nanopositioning. Fig Example of flexure design for single-axis nanopositioning Choice of drives Drives producing friction is not acceptable. Leadscrews, ballscrews, even ultrasonic linear piezo motor drives (friction based) cannot surpass submicron precision. Electromagnetic linear motors, voice coil drives and solid state piezo actuators are the most commonly used frictionless drives. The first two are fine for larger distances, but have the disadvantages of magnetic fields (not tolerable in ebeam lithography and many other applications), heat generation and only moderate stiffness and acceleration, resulting in a low bandwidth.

46 Piezoelectric (Fig. 3) often called PZTs, are limited to small distances but are extremely stiff and achieve very high accelerations (up to 10,000 g), a prerequisite for millisecond or sub-millisecond step and- settle and high scanning rates (today, the best piezo-driven flexure-guided stages have resonant frequencies of 10 khz). Fig Lately PZT actuators are ceramic-insulated rather than polymer-insulated, offering extended lifetime and better behaviour in vacuum applications. PZTs neither produce magnetic fields, nor are they influenced by them. A recent breakthrough in production technology now eliminates the need for polymer insulation, bringing the benefits of zero outgassing in vacuum applications, insensitivity to humidity and increased lifetime even under extreme conditions. 5.2 Motion and Positioning of Piezoactuators Positioning mechanism with a precision capability in submicron range is needed in such applications as semiconductor manufacturing equipment and STM. Compared with other kinematic pairs, the elastic hinges have many advantages, such as low friction, no clearance, same expansion and no assembly problem. It is a better way in submicron positioning. [21] In order to make the mechanism get the resolution of 0.01um or less, its nature frequency should be higher than 500Hz. At that point, its stiffness must be increased and its weight must be decreased remarkably. But the weight and the stiffness can t be changed limitless for the limited driving force of PZT. Besides, the displacements ranges of PZT will mark reduce in the rigid structure. So the mechanism should be optimized to obtain maximum displacement gain, the needed nature frequency and keep the driving force of PZT within limits.

47 [22] The natural frequency of piezoactuators is generally 7-8 KHz which is much larger than that from the micro-stage. But the high voltage DC which drives the piezoactuators comes from commercial power, so the 50Hz and 100Hz AC harmonic components can be introduced to the power supply. Besides, the environmental vibrations (2-3Hz) and the man-made vibrations (1-100Hz) could be transmitted to the micro-stage through the support structure. When the natural frequency of the micro-stage is near to the driven power harmonic frequency or the environmental vibrations frequency, the micro-stage will have resonance and influence over motion and position precision. This means that we must consider the dynamic performance of the micro-stage when we design it and increase the natural frequency to enhance the resistance ability to environment vibration and driven harmonic frequency. The PZTs should be positioned within 1um from the Omega-springs so they don t apply any added force to the structure, because as we saw before, in its current form it can only withstand a limited amount of pressure. A whole system must be designed so that it can position the PZTs as well as to give them a sufficient preload for them to work properly. Previous designs were related to a metal cage where the PZTs would be preloaded and it would then integrate the rest of the structure. This ended up being counterproductive as the metal cage would limit the travel of the piezos to a degree where it would not be enough to make good use of the system. It also does not solve the problem of parallel guidance, as there is no practical way of knowing whether the force is applied normal to the structure (in this case the omega-shaped spring) or not. In Fig we can see how the Si-stage is fixed to a support system, not taking into consideration i.e. points denoted in 1) such as the fact that the area around the moving plate concentrate most of the resonant frequency modes of the complete stage. The electrical contacts displayed on the Sistage are only to be found on the upper part.

48 Fig Photo property of Prof. Iwo Rangelow. Taken at the MNES Department. 5.3 Mechanics from flexure-based amplification The mechanics behind this flexure-based system are as follows: a)

49 b) Fig a) Working principle of a flexure mechanism, b) Schematic of bridge-type amplification mechanism with ideal pivots. Working principle of flexure mechanism First, only the compliances of the flexure hinges are considered in the PRB model. That is, it is assumed that each flexure hinge has 1-DOF rotational compliance arising from the rotational deformation, and other elements are all considered as rigid bodies. The free-body diagram of one amplifier leg is shown in Fig (a). Under the equilibrium status, the equation of moments at point A1 can be derived as follows. Fig a) Local coordinate and parameters of a right circular flexure hinge, where the rotation Oz around the z-axis is the desired motion, while the bending Oy and torsional Ox deflections are parasitic motions; (b) one quarter PRB model of the displacement amplifier and its parameters. (3)

50 With the moment (4) Where K R and α denote the rotational stiffness and deformation of a notch hinge, respectively. Differentiating both sides of the displacement relation l y = l a sin α (where l y and α are variable during the operation) with respect to time, allows the generation of: (5) Fig a) Free-body diagrams of the amplifier leg and amplifier input rod, (b) and the parameters of one quarter, (c) of the displacement amplifier. With the previous equation plus the following [21] we can define the amplification ratio For a XY stage with given parameters, by varying the parameters L X and L Y while keeping other parameters constant, the variation tendency of the stage amplification ratio obtained by the previous shown equation is displayed in Fig. 12. It is observed that the amplification ratio changes with the increase of L X and decrease of L Y. Meaning that the stage amplification ratio S = f ( L X LY )

51 Fig Variation tendency of the amplification ratio As2 versus the lengths of lx and ly. Bridge-type amplification mechanism Simplified ideal displacement amplification ratio using the kinematic theory: R amp = cot (α) Where α is the initial angle of lever arm Fig Elastic model of single arm of bridge-type flexure hinge R amp = l a 2 sin(α) cos (α) 4 K α K l (cos(α)) 2 + l a 2 (sin(α)) 2

52 Where l a is the length of the arm; α the initial arm length; K α the rotational stiffness of the flexure pivot; K l translational stiffness of the flexure pivot. 2 K α α threshold = α = arctan K l dramp =0 dt l a ( ) Fig Displacement amplification ratio vs. initial angle α 5.4 Designs and FEM Simulations Primary design The primary idea was to develop an inexpensive system to position the PZT in a way that they would continue to have this distance of 1um between them and the omge-shaped springs. As friction it s a very important subject in nanopositioning, it has to be eliminated. This rules out all devices with ball, roller or sliding bearings, leaving air bearings and flexures. A flexure is a frictionless hinge-like device that relies upon the elastic deformation of a solid material to permit motion. Air bearings are ideal for long travel ranges, but they are usually bulky, they exhibit high inertia and don t work in a vacuum Flexures on the other hand show no friction losses, no need for lubrication, no hysteresis, no clearance and no wear. One issue is that they also exhibit a limited motion, but that is not a problem for the travel range that we are working with.

53 The resonant frequency of the PZTs is between 70 and 135 KHz, so they are very high. The stage as we saw before is around 2-6 KHz, and it can also be added that the high voltage DC power that drives the piezos comes from commercial power, so the 50 and 100 Hz AC harmonic components can be introduced to the power supply. Besides, the environmental vibrations (2-3 Hz) and manmade vibrations (1-100 Hz) could be transmitted to this stage through the support structure (this could be changed by using some active damping for the system). When the natural frequency of the stage is near to the driven power harmonic frequency or the environmental vibrations frequency, the stage will have resonance and influence over motion and position precision. This means that we must consider the dynamic performance of the stage when we design it and increase the natural frequency to enhance the resistance ability to environment vibration and driven harmonic frequency. The primary idea was to develop a positioning system for the PZTs by using micrometer screws. Fig Standa (PL) Catalog for Screws. As we can see, they offer a good sensitivity but still not good enough. Some reduction system must be designed to allow for it to be used. Also, we can see that its dimensions far exceed that required for the support when placed in the same axis of movement as the piezo. The support system must be maximum 100x100 mm, and as we saw in the catalog from Standa (Fig. 8), when 2 of them are positioned opposite to each other, there would not be any room left for the PZT and the moving plate.

54 Fig Pictoric drawing showing movement for micrometer screws. For this reason, it was thought to position the micrometer screw normal to the piezo, resulting in less space being used but achieving the same displacement. Fig D Model of flexure-based amplification system. The end ball represents the tip of the micrometer screw. The system was originally thought to be used as an amplification system, but it can also function the other way around allowing us to reduce the movement of the micrometer screw obtaining more precision. For the mechanics behind this flexure-based system please refer to point 5.3 Mechanics from flexure-based amplification.

55 Problems found in this model: 1. There is no feedback from the movement of the screw. An optical encoder must be used to know the exact position of the system, and that would add complexity to this solution. The resolution of the ADC should also be in the order of 14 bits to allow for good precision 2. The screw has backslash. This in an important issue and cannot be helped. 3. Friction between the surface from the balltip of the micrometer screw and the surface of the flexure hinge. This can be seen in more detail in Fig This system is not giving a solution to the problem of preloading the PZTs. 5. Not giving a solution to the preload of the piezos. Conclusion: Another solution has to be found Secondary design Another approach is attempted. This time, it would not be about positioning the piezos but of placing them in a way that they occupy less space and also that they have sufficient travel. This is done by using the same principle described before but this time amplifying the movement generated by the PZTs. The piezo would then be placed in a horizontal way (or X-direction) and the output movement would be on a vertical way (Y-direction). It can be seen that there also exist some commercial solutions that also apply this concept: Fig Photos property of Dynamic Structures (DSM, USA)

56 Factors to be taken into account: 1. Resonant frequency must be higher than that of the stage 2. Easy to manufacture 3. Sufficient stiffness 4. Within dimension parameters With these factors into consideration can we see in the following image a first approach to this idea: Fig D model of amplification device for PZT. Hole is left to apply a preload to the PZT through the use of a screw. ANSYS Simulations:

57 Fig Resonant Frequency of 3 KHz for the first model. In Fig can be seen how the amplification system works. The simulation in ANSYS shows that when we move 4,8um in each direction through the X-axis (meaning that the piezo is actually expanding 9,6um) and we obtain a final movement in the Y-axis of 21,88um (the under part of the model is fasten with some M1.4 DIN916 Screws). Meaning that we ended up with an amplification ratio of about 2,27. With a travel for the P piezo of 6,5um that means that we can achieve a maximum travel of 14.75um. The resolution would then be 14.75/2 14 (14 bits ADC converters are used) giving us almost 1nm. a)

58 b) Fig a) ANSYS Simulations of displacement in X Direction, Y Direction Fig Applied stress to the structure The structure will likely be made out of Steel, so special attention must be given to the stress suffered by the flexure hinges and its possibility of being mechanized, though methods such as EDM (Electrical Discharge Machining). There are three problems regarding this structure: 1. The resonant frequency must be increased and that means increasing the stiffness by changing parameters such as width, length or thickness, or by changing the geometry.

59 2. There is a maximum travel of about 11um so the amplification ratio could be reduced, thus increasing the resolution. 3. The structure must be able to stand the stress applied at the weakest section of the flexure hinges. The stiffness of the flexure hinge is determined by the following equation: k α = 2Ebt5/2 9πR1/2 [1] Where E is the Young Modulus, b the width, t the minimum hinge thickness, and R the cutting radius. As we can conclude, the stiffness is strongly dependent upon the thickness t and width b, and in a smaller manner from radius R. Under an ideal condition, the flexure hinge should rotate around its center point c (shown in Figure 21b) and the center point should have a steady position. However, the flexure hinge has an offset value from its geometrical center when moment M is applied. a) b) Fig a) Geometry from flexure hinge, b) Deformation of flexure hinge The bending angle will be given by the following equation: α = M k α [2]

60 Using the example of a four-bar linkage mechanism we can see to which parameters is the displacement dependent. a) b) Fig a) Mechanism, b) Geometric diagram of mechanism Using the EBM (Equivalent Beam Methodology) we can see that the displacement formula can be given by Hooke s law: d = F k [3] With d being the displacement, F the applied force and k the stiffness of the double parallel fourbar mechanism, which derives as follows: k = 8k α L 2 [4] Where L is the length of the crank and k α the stiffness of the flexure hinge. Substituting equations [3] and [4] we end up having the following: d = FL2 8k α [5] And we can conclude the displacement is exponentially bonded to the length of the crank and is inversely proportional to k α. Input and output stiffness is related to the transitional and rotational stiffness of flexure pivots, initial angle of lever arm, and length of lever arm.

61 One problem that still remains is that of the resonant frequency. As we can see from the following formula: f R = 1 2π k m eff [6] f R is dependent upon the stiffness k from the structure and the effective mass (known to be around m eff = (33 140) m). So the geometry of the mechanism has to be changed in order to increase this frequency and avoid it affecting our system. Different approaches were simulated and are shown in Fig Fig Resonant Frequency of 0.7 KHz

62 Fig Resonant Frequency of 2.5 KHz Fig Resonant Frequency of 3.2 KHz

63 Fig Resonant Frequency of 8 KHz Fig Resonant Frequency of 16 KHz As we can see, although the geometry was changed we still could not reach the required resonant frequency of 36 KHz (Refer to point [a]). A different approach is seeked, and that is of placing the PZT in the same axis as the movement and using one with more travel range (this would account for what was shown back in Fig. 7). No amplification would then occur and it should be easier to mechanize.

64 Fig Simulation showing PZT placed inside of support (shown with a red color as well) and displaying a resonant frequency of 15 KHz. Not meeting the requirements. ANSYS Simulations from complete stage support Fig Resonant Frequency of 6.2 KHz. Not meeting the requirements. The resonance is concentrated in the centered area.

65 Fig Reduction of movement, instead of amplification. Fig Although a parallel moving guidance is used, some tilting occurs. Amplification Ratio is 0.8

66 Fig The parallel movement is improved, but still shows tilting. The ratio of amplification is reduced to 0.73 Fig The resonant frequency is improved to 6.4 KHz. Problems found in this design: 1. More complex to manufacture 2. Low resonant frequency

67 3. When being used at the same time (meaning having 4 amplifications models, instead of 2), ends up reducing the movement and bringing the opposite result. This could be fixed by placing them independently but then again, that would leave the fact that when the stage moves forces applied from a direction normal to the direction of movement would not be applied on i.e. the center of the omega-shaped spring but lightly displaced to one side, producing a moment from which its behavior and effects on the final measurements needs to be taken into account. Advantages: 1. Monolithic 2. Flexure based, so no friction losses 3. Amplification ratio can be varied 4. Different approaches can be taken Third design Also based on a flexure based mechanism, the amplification ratio can be increased or decrease by changing the length L relating the flexure hinges with each other. In Fig we can see a 2D and 3D model of the mechanism

68 Fig Theorical representation from the system Fig D flexure based amplification model. The model is fixed on the blue parts and receives an input in the central hinge delivering an amplified output.

69 Fig ANSYS Simulation showing an amplification ratio of about 2 Fig ANSYS Simulation showing a resonant frequency of 1.3 KHz Problems found in this third design: 1. Low resonant frequency Advantages: 1. Amplification ratio can be easily changed by changing L. 2. Parallel guidance

70 5.5 Preloading of the Piezoactuator For the purpose of preloading the Piezoactuators, two methods will be used: The spring wire that runs across the actuator frame on both sides applies a preload that keeps the piezo ceramic in compression. The preload allows the actuator to function in contraction and expansion even when the dominant motion of the PZT stack causes expansion. The spring preload works to restore the actuator frame to its original position. When voltage on the piezo is reduced, the piezo contracts. The spring preload pulls the end blocks of the amplification frame towards one another, causing the actuator output to contract. Fig. 5.3 Spring applying preload to PZT Or we can use the system previously demonstrated and apply some pressure springs on the sides for making the preload. Figure 38. Preload given by spring wire.