Harmonic Suppression in Nonlinear Systems

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Harmonic Suppression in Nonlinear Systems

A study of multifrequency atomic force microscopy in liquid

STEVAN SKROBIĆ 830709-0210

Master of Science Thesis at

Department of Applied Physics, Nanostructured Physics Royal Institute of Technology, Stockholm, Sweden

Supervisor: Riccardo Borgani Examiner: David Haviland

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TRITA-FYS 2015:42 ISSN 0280-316X

ISRN KTH/FYS/–15:42—SE

©Stevan Skrobić 2015

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Abstract

Atomic Force Microscopy is a highly sensitive method that uses a mi- croscopic cantilever to measure minute forces with nanometer precision.

The method is successfully used to image and characterize a variety of samples by the interaction force created between a sharp tip and surface.

When the sample is immersed in liquid one is faced with the challenge of a highly nonlinear force distribution.

This project describes and addresses a few of the challenges encountered in trying to interpret this nonlinear interaction force and thus make measurements in liquid more precise. The project builds upon and extends ongoing research to expand the applicability of AFM through faster and more accurate force measurements. The work performed focuses on implementing a structured data acquisition as well as performing and analyzing new modes of actuating the AFM cantilever.

To make force reconstruction in liquids more accurate we propose a new method of exciting the cantilever and we propose a method of mitigating the negative effect this drive scheme may have on the image quality.

Referat

Harmonisk dämpning i ickelinjära system

Atomkraftsmikroskopi är en känslig metod som använder sig av en mik- roskopisk hävstång för mätning av prov med nanometer precision. Me- toden har framgångsrikt använts för att avbilda och karaktärisera inter- aktionskrafter hos en stor mängd material genom interaktionen mellan spets och prov. Under mätningar där provet är nedsänkt i vätska står man inför utmaningen av en mycket icke-linjär kraftfördelning .

Projektet syftar till att beskriva och hantera några av de utmaning- ar som uppstår vid försök att tolka denna ickelinjära interaktion och därmed göra mätningar i vätska mer exakt. Projektet bygger vidare på och utökar pågående forskning som har det övergripande syftet att utvidga tillämpningen av AFM mätningar genom snabbare och mer ex- akta avbildningar. Det utförda arbetet fokuserar på att implementera en strukturerad datainsamling samt utföra och analysera nya sätt att driva en AFM hävstång.

För att återskapa krafter i vätskor mer exakt föreslår vi en ny metod att excitera hävstången samt ge en metod för att hantera de negativa effekter som drivningen kan ha på bildkvaliteten.

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Introduction

Since the 1590’s when Zacharias Janssen[1] begun to combine lenses to form the first microscope, there has been a continuous development towards higher resolution microscopy. When using light to probe ever smaller scales one eventually reaches a physical limit related to the wavelength of the light. Recently this limit has been extended [2] and the current resolution of optical microscopes are at the scale of 10 nm [3].

Atomic Force Microscopy (AFM) is a technique that has become a standard tool to image, manipulate and to investigate physical properties on the nanometer scale. The AFM was invented by Binnig, Quate and Gerber in 1987 [4] and it uses a cantilever equipped with a sharp tip to mechanically probe a sample. The resolution of AFM has been proven to be in the range of 0.1 nm [5].

The first AFM measurements were performed by monitoring the cantilever deflection while scanning the sample. The community then started to excite the cantilever in pursuit of faster scans and more content-rich images. This dynamic drive extended the measurements with a new observable, the phase-lag between excitation and response, which proved sensitive to changes in the material stiffness.

Another advantage with dynamic measurements is the possibility to reconstruct the tip surface interaction force. It has however been challenging to make the reconstruction accurate, mainly due to the nonlinear behavior of the interaction force which spreads the system response over a large region of the frequency domain.

A new take on this reconstruction problem was performed by the Nanostructure Physics group at the Royal Institute of Technology [6][7]. By exciting two closely spaced frequencies near resonance, many intermodulation products will form near resonance. Collecting the nonlinear response near a single resonance allows for the accurate use of a point mass model. However, such is not the case in liquid, where excessive dampening results in a low quality factor.

In liquids, the nonlinear response spreads across the frequency spectrum in the form of harmonics and it is clear that the second eigenmode of the cantilever is exited. It has been proposed that the point mass model is a poor approximation for liquid measurements away from resonance [8]. It has also been proposed that the

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CHAPTER 1. INTRODUCTION second eigenmode response may be screened by focusing the laser over the antinodes of the cantilever [9]. To address the challenges of multi-frequency AFM in liquid, new software tools had to be developed with the aim to investigate the validity of the point mass model and by measuring the full unscreened spectrum response. The result of this investigation leads to a proposal on how the harmonic response may be suppressed by adjusting the cantilever excitation.

The report is structured in the following way. In Chapter 2 I present a theoretical background on the point mass model as well as a description of the nonlinear interaction. The chapter ends with a background on AFM and the modes of operation. Chapter 3 contains a description of the system components relevant to this project and the challenges posed by the system. Chapter 4 presents the work done and the results produced from this project are analyzed in Chapter 5. Chapter 6 concludes this report.

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Chapter 2

Background

This chapter is aimed at describing the theory and concept behind Atomic Force Microscopy (AFM) as well as different measurement techniques. These concepts are not only general in the field of AFM but also relevant to this project.

The next section 2.1 contains most of the theory used in the project, while section 2.2 is meant to give general insight into the field of AFM and its different modes of operation.

2.1 Theory

This section serves as an aid in understanding the physics of oscillators, including the origin of the nonlinear response encountered in this project, which is central to understanding the main challenge encountered in the project.

2.1.1 Forced damped harmonic oscillator

At the core of this project is the point mass model which is widely used to approximate the motion of the cantilevers free end. There are however a few criteria for the approximation to be valid. The cantilever length must greatly exceed the width and every cross section of the beam must be uniformly shaped. Furthermore, the vibrations of the beam must be small compared to all physical scales of the cantilever.

Using Newtons second law it is easy to model the deflection around the equilib- rium point:

m ¨d= −c ˙d − kd + F

→ 1

mF = ¨d+ c m ˙d + k

md

In mechanical systems one usually restates the constants c and k in terms of resonance frequency, ω0, and dampening ratio, ζ. These quantities reflect the ho- mogeneous solution of two simpler systems, the damped and undamped harmonic oscillator.

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CHAPTER 2. BACKGROUND

Figure 2.1: Schematic representation of the point-mass damped and driven harmonic oscillator.

ω₀ = s

k m ζ = c

2√ mk

In the AFM community it is usual to use the quality factor, Q, instead of the dampening ratio, ζ. The quality factor is defined as:

Q= 2π E_stored E_dissipated

_ω=ω

0

= 1 2ζ

Rewriting the force equation in terms of the spring constant, resonance frequency and quality factor gives the following expression:

1

mF = ¨d+ω₀

Q ˙d + ω0²d

→ F = k 1

ω₀²d¨+ 1

Qω₀ ˙d + d

To find the steady state solution, it is easier to work in frequency space, which is accessed by using the inverse Fourier transform.

d(t) = 1 2π

Z ∞

−∞

dˆ(ω) e^iωtdω

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2.1. THEORY

Which, after some algebra gives the following expression for the force:

Fˆ(ω) = k ω₀²

ω²₀− ω²+iωω₀ Q

dˆ(ω) = ˆχ⁻¹(ω) ˆd(ω) (2.1) Eq.2.1 is the key equation in this project since (if χ is known) it relates a deflection directly to an arbitrary force. The function χ is called the linear response function and is determined by the parameters k, ω0 and Q.

2.1.2 Nonlinearities

Since this report handles nonlinear behavior it is suitable to present an insight to nonlinear theory. The standard introduction is the description of a weakly disturbed harmonic oscillator, named the duffing oscillator.











d¨(t) + ω₀²d(t) = −kd³(t) d(0) = A

˙d(0) = 0

Assuming that the solution to the nonlinear equation is periodic (since the point mass moves periodically) the solution can be approximated by using the Poincare- Lindstedt method. First, introduce a new timescale τ = ωt, where ω is an unknown frequency of the periodic solution. This gives the following equation:

ω²d⁰⁰+ ω₀²d= −kd³ (2.2)

To approximate the solution one expands ω and assumes that the solution can be asymptotically approximated by:

ω = ω0+ ω1+ O², → 0 d(τ) = d0(τ) + d1(τ) + O², → 0

Inserting the expansions into Eq.2.2 gives two separate equations each to a power in . Solving those equations with the help of initial values and adjusting constants to eliminate secular terms gives the following solution:

ω= ω0+ 3kA²

8ω0 + O², → 0

d= A cos (ωt) + kA³

32ω²₀ (cos (ωt) − cos (3ωt)) + O², → 0

From this solution it is clear that the cubic nonlinearity gives rise to a new frequency component, 3ω.

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CHAPTER 2. BACKGROUND

2.1.3 Nonlinear Force

A highly idealized but more intuitive approach to nonlinear behavior is to expand a nonlinear interaction as a deflection polynomial, to some degree N. This assumption has its foundation in newtons third law, the cantilever exerts the same force on the sample as the sample exerts on the cantilever.

Finteraction=≈

N

X

n=0

a_ndⁿ

To gain a deeper insight, one should take note of the periodic motion of the cantilever, it is thus convenient to express the deflection in terms of a Fourier series.

d(t) =

∞

X

m=−∞

c⁰_me^im∆ωt

Here ∆ω refers to the global frequency of one whole sample segment. When an oscillator is driven with multiple frequencies, the carrier wave may be modulated to form a periodic envelope or beat. The beat period Tbeat will form the basis for the Fourier expansion ∆ω = _T^2π_beat.

The final step in creating a force expression is to incorporate restrictions caused by the driving force. Based on equation 2.1 the free steady state solution of the deflection should have the same periodicity as all involved driving frequencies. It is thus desirable to only allow those frequencies (ωk)

ω1= ±m∆ω ω_k = ±(m + k − 1)∆ω

Using these assumptions, the interaction force takes the following form:

Finteraction=

N

X

n=0

a_n

K

X

k=1

c_ke^iω^k^t+ c−ke^−iω^k^t

!ⁿ

(2.3) For a system driven by one frequency (ω1) the interaction simplifies into

Finteraction=^X^N

n=0

an

c1e^iω¹^t+ c−1e^−iω¹^tⁿ

= a0+ a1

c₁e^iω¹^t+ c−1e^−iω¹^t+ a2





c²₁e^i2ω¹^t+ c²−1e^−i2ω¹^t

| {z }

first harmonic

+ 2c1c−1





+ ... (2.4) If Finteraction is viewed as a perturbation on the free deflection, it would imply that the perturbed motion contains harmonic components at integer values of the fundamental drive frequency ω1.

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2.2. ATOMIC FORCE MICROSCOPE

2.2 Atomic Force Microscope

Figure 2.2: AFM setup

AFM belongs to the Scanning Probe Microscope (SPM) branch of microscopy in which a sample is scanned with the help of a probe. The probe in this case is a sharp tip mounted on a cantilever which acts as a force transducer that translates the mechanical interaction force between a tip and sample into a measurable deflection.

A crude schematic of the AFM used in this project is depicted in Fig.2.2. In this construction, the cantilever is mounted under a polished glass block. The end of the cantilever is equipped with a sharp tip that is free to interact with the sample. The cantilever deflection is monitored with the help of a laser-detector setup and the glass block eliminates unpredictable refractive errors induced by movements of the liquid surface. The glass block of interest to this project is equipped with a shaker piezo that gives the ability to excite the cantilever. In addition to the components in Fig.2.2 the height of the glass block is adjustable by another piezo element (z-piezo).

2.2.1 Modes of operation

There are two basic modes to operate an AFM, quasi-static mode and dynamic mode. In quasi-static mode the shaker piezo is kept static. As the cantilever is swept or dragged over the sample, the laser is used to monitor the current deflection.

After setting a reference value (set-point), proportional-integral feedback is used to keep the cantilever deflection constant.

The quasi-static motion simplifies Eq.2.1 into Hook´s law (F = kd), and the static deflection thus translates to a constant load force between the tip and the sample. While scanning, this contact between tip and sample causes a rather high mechanical wear. Another standard measurement used to find material properties is to construct force curves, by approaching the sample with the cantilever and

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CHAPTER 2. BACKGROUND simultaneously monitor the deflection. This project however, falls into the scope of dynamic AFM.

Dynamic mode

In this mode the shaker piezo is biased with an AC voltage at fixed drive frequency, which excites the cantilever and oscillates it over the sample and the frequency is typically chosen near resonance. The resulting cantilever motion can be described in terms of an oscillation amplitude and phase at the drive frequency. The feedback could be set to either parameter, but in this case it is set to the amplitude. One advantage of dynamic measurements is that a periodic tap on the surface considerably reduces the risk of tip and material damage or wear. Another profound advantage is that the phase is sensitive to the sample stiffness, making it easy to distinguish between different materials. In addition, dynamic scans are generally faster but generate images of slightly lower quality.

Due to the periodic motion of the cantilever, every pixel of a scan is in effect a tip surface approach. To reconstruct the interaction force applied to the tip one uses Eq.2.1. Treating the free oscillations and the engaged as two separate cases gives the following system of equations:

Fˆ_{f ree} = ˆχ⁻¹dˆ_{f ree} Fˆ_{f ree}+ ˆF_ts = ˆχ⁻¹dˆ

A subtraction of the free force expression from the engaged, gives an expression of the tip-surface interaction force, ˆFts,

Fˆts = ˆχ⁻¹d − ˆˆ df ree

(2.5)

It is interesting that this leaves us with an expression that appears to be linear, but the nonlinearity is actually contained within the change of deflection spectrum.

Since the linear transfer function, ˆχ⁻¹, is complex, Eq.2.5 contains more information than a static force would. The real part of the tip-surface force corresponds to the conserved force experienced by the cantilever, which is easily verified by tak- ing a glance at Eq.2.1 and identifying that the real part is the equation for the undamped harmonic oscillator. Looking at the imaginary part one quickly identi- fies a dissipative force, because it contains the only reference to the quality factor, which by definition is a diffusive term.

Challenges Regarding Force Reconstruction

In the initial stages of dynamic mode, the cantilever was driven and monitored with one frequency [10], but in resent years, the AFM community has experimented with multiple drive frequencies. To see why multi-frequency drive is desirable, one should identify the weaknesses of using a single frequency to probe a nonlinearity. From Eq.2.4, it is easy to identify that as the nonlinear terms (n > 1) become stronger,

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2.2. ATOMIC FORCE MICROSCOPE

the system response spreads across a larger region in frequency space. This spread results in a myriad of challenges.

A naive extension of the single frequency AFM is to monitor the higher harmonics. At the first glance this method promises an exact reconstruction of the nonlinear interaction. Upon closer inspection it turns out that this method puts substantial strain on our point mass model. Although the point mass model has been proven to accuracy describe the motion of a cantilever beam [11], it has also been proven that this is only valid if dissipative effects in the surrounding medium are negligible [8].

Even tough the model begins to differ from the actual response in a global sense, it is reasonable to assume that if the model is normalized to the actual response it will still remain a good approximation within some nearby region. Since the fit or calibration of model to reality is done at resonance, by monitoring the thermally excited cantilever motion[12], it is desirable to collect the majority of the response in that region.

Even in a non-dissipative medium there exist challenges. If the nonlinearity is spread far from the resonance, the second eigenmode of the cantilever is likely to be excited, this means that the system must be modeled as a coupled oscillator. This is problematic, due to the lack of accurate methods of calibrating the second eigenmode. Furthermore, some of the response that occur outside the sensitive region of resonance will have a low signal to noise ratio, which will have an undesirable effect on the measurement accuracy.

Intermodulation AFM

To increase measurement accuracy and sensitivity, the nanostructure physics group at KTH started investigating the phenomenon called intermodulation which occurs when a resonator is driven by multiple closely spaced frequencies. To get a clear understanding of the involved physics, we once again turn to the derived theory. Re- stricting ourselves to a drive configurations of two drive frequencies and scrutinizing the third order (n = 3) nonlinearity of Eq.2.3

a₃c₁e^iω¹^t+ c−1e^−iω¹^t+ c2e^iω²^t+ c−2e^−iω²^t³

When the cubical term is expanded it can be grouped into four different components:

a3(A + B + C + D) Where:

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CHAPTER 2. BACKGROUND 𝐹_𝑡𝑠≈

𝑛=0 𝑁

𝑎_𝑛 𝑐₁𝑒^𝑖𝜔¹^𝑡+ 𝑐₋₁𝑒^−𝑖𝜔¹^𝑡+𝑐₂𝑒^𝑖𝜔²^𝑡+ 𝑐₋₂𝑒^−𝑖𝜔²^{𝑡 𝑛}

Two Drive Nonlinear Expansion

𝐹_𝑡𝑠

𝜔 𝐴

𝐶

𝐵

𝐷 𝐷

𝐶

Figure 2.3: Schematic representation of the force distribution, red corresponds to the contribution from the third order expansion.

A= 3 (c1c−1+ c2c−2)c₂c−2

c₁e^iω¹^t+ c−1e^−iω¹^t+ c1c−1

c₂e^iω²^t+ c−2e^−iω²^t

B= c³₁e^3iω¹^t+ c³−1e^−3iω¹^t+ c³₂e^3iω²^t+ c³−2e^−3iω²^t

C = 3c−2c²₁e^i(2ω¹^−ω²^)t+ c²−1c₂e^−i(2ω¹^−2ω²^)t+ c²−2c₁e^i(ω¹^−2ω²^)t+ c−1c²₂e^−i(ω¹^−2ω²^)t

D= 3c²₁c2e^i(2ω¹^+ω²^)t+ c−2c²₋₁e^−i(2ω¹^+ω²^)t+ c1c²₂e^i(ω¹^+2ω²^)t+ c²−2c−1e^−i(ω¹^+2ω²^)t A quick inspection shows that the terms in A correspond to the driven vibrational states and B are a nonlinear response in states of the second harmonic. The terms C and D are intermodulation or mixing products of the two fundamental frequencies, to be more specific, C terms correspond to a response centered around the drive frequencies and D are centered around the second harmonic, this is visually depicted in Fig.2.3.

If the two drive frequencies are centered close to resonance, it becomes evident that the intermodulation products are collecting nonlinear harmonic information at resonance.

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Chapter 3

System Architecture

This project is built upon and aims to extend the work of several previous projects that have resulted in extensive hardware and software development.

The next section contains a description of the hardware setup which is followed by a presentation of the involved software. Hopefully this will make findings transparent and thus reproduction of the experiments easy. This chapter ends with a description of the involved challenges encountered during the course of this project.

3.1 Hardware setup

The hardware consists of a JPK NanoWizard AFM with an additional breakout box and the Intermodulation lock-in analyzer. The breakout box allows the operator to either monitor or control all electronic components of the AFM. During experiments the lateral detector signal is monitored while the z-piezo hight (set-point trigger) and shaker piezo drive is controlled. As for the lock-in analyzer it contains multiple waveform generators that have the ability to simultaneously monitor and drive 42 different channels.

3.2 Software architecture

The software consists of the JPK SPMControl software that bundles with the NanoWizard. Unfortunately, the JPK SPMControl source code was not available, but the existing interface proved sufficient. The programming language of choice is Python and all developed code is aimed to be included into the ImAFM software suite.

The JPK SPMControl is sparsely used during measurements with the main component being the ramp designer, which allows for the creation of a z-piezo drive scheme. For this project, a ramp made up of fast sample approaches and slow retractions was chosen.

Otherwise this project relies primarily on three tools the ImAFM software suite.

The first one is the drive constructor, which is a Python based interface that enables

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CHAPTER 3. SYSTEM ARCHITECTURE the construction of arbitrary piezo shaker drive schemes. The limiting factors in the drive setup is the number of hardware enabled drive channels and the 2³² sample values that builds up the model sinus wave.

The second tool is the streamrecorder, which offer the choice of monitoring either a selection of 42 arbitrary frequencies in lockin mode or a stream of the entire deflection signal, from which the frequency spectrum can be calculated by FFT. The latter mode is limited, by data transfer speed, to a sample rate of 3.125 MHz, which according to the sampling theorem gives a Nyquist frequency of approximately 1.56 MHz.The final tool is the calibrator which uses the deflection induced by Brownian motion and the dimensions of the cantilever to calculate the resonance frequency, quality factor and the spring constant [12].

3.3 Shortcomings of the software architecture

While the Intermodulation method works excellently for imaging and force reconstruction in air, it has not yet been rigorously tested in a liquid medium. Due to stronger nonlinear interaction forces, the methods developed for force reconstruction in air do not translate well. A high harmonic response will excite the second eigenmode which is undesirable since it breaks down the harmonic oscillator model.

The focus of this project is to find a way to suppress the harmonic response. To achieve this the following software weaknesses must be remedied.

• The StreamRecorder saves data in a legacy binary file format, which is incon- sistent with the rest of the software and not the best choice for large datasets.

• The tools for analyzing the full spectrum of the StreamRecorder data does not exist.

• The software relies on a small harmonic response, and thus cannot accurately reconstruct forces in liquids.

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Chapter 4

Project Work

The following sections are meant to present all the work involved in addressing the challenges faced when using the imaging data for force reconstruction. The first section covers the preliminary work needed before actual experiments could be performed. Section 4.2 covers the experimental procedure and this chapter ends with a presentation of the analysis tool.

4.1 Data extraction

As mentioned in the previous section, the software saved streamed data in a legacy file format. This was a handcrafted file format which would be hard to manage in the long run. To fix this, an open source package named h5py is implemented to interface to the HDF5 binary format. HDF stands for Hierarchical Data Format and is designed to simplify any file structure into one file containing two types of objects.

The lowest level object is the Dataset, which is essentially a multidimensional array.

On a higher level there are Groups that act as container objects for other Groups or Datasets. The format also allows for random access directly from the hard drive as well as assigning metadata in the form of attributes attached to either Datasets or Groups. Furthermore, the HDF group provides tools for visualization and management of the data, making it easy and transparent for an outsider to access and manage all the data in a file.

4.2 Experimental Procedure

Approach and retraction measurements were performed and the entire deflection signal was monitored with the stream recorder. The cantilever was a tap-300 with resonance at f0 = 300kHz and stiffness k ≈ 40N m⁻¹ and the probed sample con- sisted of polystyrene immersed in water. The setup procedure performed before the measurements is listed below.

• Attach cantilever is to the hyperdrive glass-block.

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CHAPTER 4. PROJECT WORK

• Mount the glass block on the AFM, align laser and detector

• Make a calibration in ambient air and compare with reference data-sheet.

• Immerse sample in liquid and submerge the cantilever.

• Realign mirror and detector and recalibrate the cantilever, so that the voltage from the detector can be related to the deflection of the cantilever.

• Engage surface, and redo calibration close to the surface, to minimize hydro- dynamic effects.

• Setup the approach parameters with the JPK ramp designer.

• Setup the shaker-piezo drive with the ImAFM drive constructor.

With the setup done, it is straightforward to record the deflection using the ImAFM stream recorder and start the ramp approach from the JPK SPM Con- troller. When the ramp is completed and the stream recorder is stopped, a new drive-scheme is set using the drive constructor and the experiment is repeated.

In order to compare the results for different drives, the maximum free deflection amplitude is kept fairly constant throughout the measurements.

4.3 Data Analysis Tool

In order to make fast analyses, an interactive tool is needed. This tool was built outside the ImAFM software but with the aim of being easily integrable into the suite. The program is written in Python and structured according to the Model View Controller design pattern. The different classes are:

Model: Class containing all data and mathematical functions, such as Fourier transformations, calculation of the transfer function and force inversion.

View: Class containing a User Interface (UI), built on the wxPython wrapper library. The UI consists of a frame that holds a matlibplot canvas, slider and menu-bar. With the slider it is easy to cycle through all beats in a measurement. The menu offers the ability to open data files, change graphs and select frequency components to included in the force reconstruction.

Controller: Class containing the event handlers and matplotlib figure object. The controller also acts on user input and handles the interaction between the model and view classes.

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Chapter 5

Results & Discussion

This chapter is divided into five parts which reflect upon the work and results from this project. Section 5.1 contains an analysis of the used method and is followed by a review of the liquid and air calibration. Section 5.3 contains a discussion of the results produced by the approach curves. This is followed by a comparison between normal intermodulation and 23 frequency drive, where a force reconstruction is presented in section 5.4 and scans in section 5.5. Finally, this chapter ends with a future outlook.

5.1 Method

The method used in this project aimed to investigate the impact that different drive configurations would have on the response spectrum in liquid. This could have been done by using the lock-in analyzer in the existing software, but that method is limited to monitor 42 frequencies. To gain insight into the physics governing the response distribution accurately it is desirable to have as much of the spectrum as possible, thus new tools had to be developed. Because of this, the majority of the work invested in this project went into programming. It was time-consuming to find the way around the 78 000 lines of code contained in the ImAFM software and to get familiar with the new program libraries needed for this project. However, this approach not only presented interesting results, it also resulted in an easily expandable analysis solution.

5.2 Calibration

During the setup procedure several calibrations were made, and in Fig.5.1 on page 16, I present calibration results for both ambient air and purified water.

As expected, it shows that the resonance peak is both broadened and several orders of magnitude lower in water, which is also reflected in the quality factor, Q_air= 428 → Qliquid= 7.7.

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CHAPTER 5. RESULTS & DISCUSSION

(a) Air

(b) Purified water

Figure 5.1: Power spectrum density around resonance of a tap-300 cantilever, in

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5.3. APPROACH CURVES

It is also notable that the resonance frequency is shifted down by a factor of two, ω0,air = 322 → ω0,liquid = 148, 5 kHz but the cantilever stiffness remains fairly constant, kair= 25 → kliquid= 30 N m⁻¹.

5.3 Approach Curves

In this section I present experiments from three different drive configurations, which were all performed with the same approach setup. Initially the sample is engaged using a slow approach 0.1 µm s⁻¹followed by a very slow retraction 0.01 µm s⁻¹ when the deflection amplitude drops below 50% of the original value. The turning- point is clearly visible in the left column of Fig.5.2 on page 18, which also illustrates that all parameters except the actual drive are kept static. The free oscillation amplitude is kept around ∼ 19 − 20 nm and the engaged amplitude is ∼ 16 nm in all three cases.

When investigating the right column of Fig.5.2 we see that the single drive frequency is engaging the surface across the entire time window. For the two frequency case we see that the carry signal is heavily amplitude modulated and the motion can be split up into both a fast oscillation and a slow modulation of the envelope.

This means that the carry signal is oscillating partly free and partly engaged during a beat. The 23 frequency drive modulates the deflection signal into a sinc shaped envelope and is engaging the sample during a considerably smaller portion of the time window.

Fig.5.3 on page 19 depicts the Fourier transform of the selected beats. The images contain at least four interesting pieces of information. We first note that as the frequency components of the drive increase so does the intermodulation products around the resonance, which is consistent with Eq.2.5. There is also a weak but clustered response around the drive frequency in both Fig.5.3a and Fig.5.3b, which is caused by Brownian motion at resonance. What is more interesting is the behavior of the harmonic response. Fig.5.3a depicts a pronounced response up to the seventh harmonic. It is also notable that the response initially decreases but starts to increase at the fourth harmonic, indicating that the second eigenmode of the cantilever is excited. The response of the normal intermodulation setup in Fig.5.3b is similar, but as more frequencies are included in the drive, it becomes apparent that the harmonic response is suppressed. In Fig.5.3c we see the significantly more coupled frequency spectrum for 23 drive frequencies.

With these observations we conclude that the nonlinear response can be guided to a more coupled distribution. Using Eq.2.3 we see that the driven frequencies determine the vibrational states in which the response is confined. The determining relation on how the response is redistributed is not precisely known, but it is likely to be coupled to the shape of the linear transfer function.

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(a) One frequency drive

(b) Two frequency drive

(c) 23 frequency drive

Figure 5.2: Experimental deflection signal where left column corresponds to maximum and minimum value of each sampled time window. The right column shows the zoomed time window corresponding to the vertical marker in the time envelope.

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5.3. APPROACH CURVES

(a) One frequency drive

(b) Two frequency drive

(c) 23 frequency drive

Figure 5.3: The Fourier spectrum corresponding to the three different drives and time windows with a deflection of 16 nm presented with blue plots in Fig.5.2 on

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(a) Free spectrum. (b) Engaged spectrum.

Figure 5.4: Zoomed in portion of the 23 frequency drive Fourier spectrum around resonance.

5.4 Force reconstruction

From the previous section it is clear that the 23 frequency drive has a more coupled response, and as presented in Fig.5.4 on page 20, it is clear that these intermodulation products contain information from the nonlinear interaction. To investigate what this additional information contains, we make an force inversion using an envelope inversion algorithm based on separation of timescales. It has been shown that the slowly varying envelope can be used to completely describe the involved forces [13]. First Eq.2.5 is used to calculate the force spectrum and instead of searching for the turning points of the carrier wave, the frequency spectrum is shifted down so it is centered around ¯ω = 0. To only keep the slow variation, a low pass filter is applied and the spectrum is finally inverse Fourier transformed to yield the time domain envelope.

The result of this inversion is plotted in Fig.5.5 on page 21, where the FI curve represents the real part of the force is plotted against the deflection and the FQ curve represents the imaginary part. The different colors of the graphs represent force curves that correspond to the time domain data in Fig.5.2. Upon closer inspection of Fig.5.5a and Fig.5.5b, we notice that the drive containing two frequencies is sampled much more evenly, whereas the majority of the samples for the 23 frequency drive are in the weakly interacting regime.

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5.4. FORCE RECONSTRUCTION

(a) Two frequency drive, 90 included channels (b) 23 frequency drive, 160 included channels

Figure 5.5: Force inversion for a cantilever driven with two different drives in liquid, Colors correspond to different the levels of engagement to the surface.

Upon closer inspection of the two drives it is noticeable that the involved repulsive forces are higher in the 2 frequency drive for all measured deflections. One possible cause may be that when the oscillation amplitude is restricted due to the nonlinear interaction, the cantilevers stored energy must either dissipate or be redistributed in the available states. As some of these states are excluded in the inversion, the result may indicate a higher repulsive force than is actually experienced, it may however also be a result from the slightly larger free oscillation amplitude of this configuration (see Fig.5.2).

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(a) Two driven channels (b) 23 driven channels

Figure 5.6: Scan corresponding to the feedback signal for two different drive configurations. The scan size is 10 µm × 10 µm.

5.5 Imaging

This section acts as proof of concept for multiple frequency AFM imaging and contains two scans of a well used HarmoniX training sample which is a blend of Polystyrene and Ployolefin Elastomeres. The cantilever is a Tap-300 with oscillation amplitude set to 20 nm and a setpoint of 75%.

Fig.5.6a depicts the feedback frequency from a normal intermodulation scan that act as a reference and the equivalent image for 23 driven channels is presented in Fig.5.6b. It is evident that the 23 frequency scan is comparable and in this case, has better quality than the images produced by normal intermodulation. It is however important to recognize that this may be a result of the feedback setup and not related to the difference in drives. It is also unexpected that the two frequency

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5.6. OUTLOOK

drive appears more noisy, even though it should have a higher signal to noise ratio.

Since the ImAFM software and hardware has the ability to monitor 42 different frequency channels, it can produce the equivalent number of images. Fig.5.7 on page 24 depicts a selection of channels from normal intermodulation, where the two first figures correspond to driven frequencies and the rest is monitored intermodulation response. The corresponding response for the 23 frequency drive is presented in Fig.5.8 on page 25 and Fig.5.9 on page 26. In both cases we get a rich intermodulation response which decreases as the channels move further away from the drive. The most obvious difference between the two configurations is the large number of high quality images that is produced in the 23 frequency drive. It would be interesting to create a compound image using all this information where a crude implementation would be to form a transfer function weighted mean of each pixel. A more complex method has been proposed where a machine learning algorithm would vastly improve the image contrast and material discrimination of multifrequency scans[14].

5.6 Outlook

This project gives rise to many future possibilities. First of all, the analysis tool is to be included in the ImAFM software. Beyond that, it would be preferable to make the following modifications to the software: In the drive constructor, the drive is set as a percentage of the maximum drive. A more convenient solution is to set a desired maximum amplitude. It would also be good to have the possibility to setup the drive and setpoint directly from the drive constructor.

Since approach curves are standard measurements, a trigger could be added so that a measurement can be initiated and terminated directly from the ImAFM software. If possible one could also consider an automated approach ramp that is used to evaluate if a constructed drive gives a suitable response spectrum for force reconstruction. An automated approach curve could also be integrated in the scan panel and thus give the user a convenient way of to analyze regions of interest with more precision.

A more challenging task is to further investigate the physics, where the derived force expression (Eq.2.3) may reveal some additional information regarding the response distribution. Another possibility is to investigate if the suppression, induced by multiple drives, can be recreated in simulations. This way, the suppression can be better investigated in the absence of noise and hopefully give some indications on how to create an optimal drive scheme.

It would also be interesting to investigate if the proposed drive allows for better soft sample probing. Investigating the zoomed in time envelope in Fig.5.2c, it suggests that this drive would allow for imaging samples with longer relaxation times.

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Figure 5.7: Two driven channels and the 13 closest intermodulation products. The images are acquired simultaneously during one scan with the size of 10 µm × µm.

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5.6. OUTLOOK

Figure 5.8: 15 of the 23 driven channels closest to the cantilever resonance. The images are acquired simultaneously during one scan with the size of 10 µm × µm.

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Figure 5.9: The eight outer channels of the 23 frequency drive followed by the seven closest intermodulation products. The images are acquired simultaneously during one scan with the size of 10 µm × µm.

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Chapter 6

Conclusions

In this project I investigated the nonlinearities encountered during AFM measurements in liquids. Due to challenges that decoupled nonlinear response pose to force reconstruction, a detailed method of analysis had to be implemented. This de- manded that large amounts of measurement data could be stored efficiently, for which the HDF5 file format was chosen. In addition, a fast analysis tool based primarily on the wxPython library, was created.

The software development from this project and the previous work of our group, now makes it possible to experimentally evaluate the effect that a cantilever drive setup has on the distribution of nonlinear response. More importantly, it was possible to identify that the harmonic response can be suppressed significantly. The mechanism behind this suppression is thought to be related to the availability of vibrational states, which are created by the drive acting on the cantilever. It is also thought that the redistribution of stored energy under the influence of a tip-force perturbation is intimately related to the shape of the transfer function.

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Acknowledgments

First and foremost I would like to thank Prof. David Haviland for not only giving me the opportunity to work with this project but also for his never ending optimism and vigorous enthusiasm. In addition, my work would have been impossible to do without all the development performed by the current and all former members of the Nanostructure Physics group at KTH. I would especially want to thank my supervisor Riccardo Borgani, colleagues Per-Anders Thorén, Matthew Fielden and Daniel Forchheimer for the invaluable input and support during this thesis.

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Harmonic Suppression in Nonlinear Systems