Investigation of Switching Characteristics of Nanomagnets ...peter/theses/collins.pdf ·...

Investigation of Switching Characteristics

of Nanomagnets via Magnetic Force Microscopy

Sean CollinsDepartment of Physics

McGill University

Montreal, Quebec

Canada

A Thesis submitted to the

Faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of

Master of Science

c© Sean Collins, 2004

Contents

Abstract xi

Resume xii

Acknowledgments xv

1 Introduction 1

2 Principles & Instrumentation of Magnetic Force Microscopy 52.1 Principles of MFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Stray Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Contrast formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Imaging and Operating Modes . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Other Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 van der Waals Forces . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Electrostatic Forces . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.3 Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 MFM in Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Intstrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7.1 Force Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7.2 Frequency Modulation Detector . . . . . . . . . . . . . . . . . 18

2.7.3 Electromagnets and Hall Probe . . . . . . . . . . . . . . . . . 19

2.7.4 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7.5 Piezoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Simulations of Magnetism on the Nanoscale 243.1 Energy Terms in Magnetism . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Magnetostatic Energy . . . . . . . . . . . . . . . . . . . . . . 25

3.1.3 Crystalline Anisotropy Energy . . . . . . . . . . . . . . . . . . 26

3.1.4 Zeeman Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Magnetic Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Nanomagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 OOMMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Particle Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Single Domain Simulations . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Particle Coercivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iii

iv Contents

4 Experimental Results 424.1 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Characteristics of 750 nm Particles . . . . . . . . . . . . . . . . . . . 44

4.2.1 Imaging in Magnetic Field . . . . . . . . . . . . . . . . . . . . 454.2.2 Ensemble Hysteresis Loop . . . . . . . . . . . . . . . . . . . . 484.2.3 Switching Field Distribution . . . . . . . . . . . . . . . . . . . 484.2.4 Comparison with Previous Study . . . . . . . . . . . . . . . . 514.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Characteristics of 250 nm Particles . . . . . . . . . . . . . . . . . . . 524.3.1 Imaging in Magnetic Field . . . . . . . . . . . . . . . . . . . . 534.3.2 Ensemble Hysteresis Loop . . . . . . . . . . . . . . . . . . . . 554.3.3 Switching Field Distribution . . . . . . . . . . . . . . . . . . . 554.3.4 Comparison with Simulations . . . . . . . . . . . . . . . . . . 594.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Conclusions & Outlook 625.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Appendix 66A.1 The Frequency Modulation Detector and PLL . . . . . . . . . . . . . 66A.2 Final Notes on OOMMF . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 71

List of Figures

1.1 A schematic of an “and” gate in a magnetic quantum cellular automatasystem. The signal is propagating from left to right. In order for chain3 to switch from its “left” configuration to the “right” state, the strayfields from the right most particles in both chains 1 and 2 must beinfluencing the first particle in chain 3. In a) and b) the stray fieldsare insufficient to induce switching in chain 3. In c), the combinedstray field is enough to induce switching in chain 3. Now that the firstelement in chain 3 has switched, it will switch the particle adjacent toit, and so on, propagating the signal to the right. . . . . . . . . . . . 3

2.1 The principle of magnetic force microscopy. A magnetic tip is scannedacross a magnetic sample. The attraction or repulsion between the tipand the stray field of the sample causes the status of the cantilever tochange, and this change is detected using optical techniques. Adaptedfrom [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The geometry used for calculating the force and force gradient on themagnetic tip. Adapted from [1]. . . . . . . . . . . . . . . . . . . . . . 7

2.3 The effect of a damaged tip on image quality. The image on the leftwas taken with an undamaged tip. The image on the right is of thesame area and was obtained with identical parameters, but the tipwas damaged. Note that individual particles cannot be resolved in theimage on the right. Image was taken in frequency shift mode withsample voltage acting as the servo. Sample was permalloy disks withdiameters of 1 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 In the slope-detection method, the cantilever is driven at a frequency,ωd. A change in the force gradient causes a shift in the resonancefrequency (ω0 → ω′0), and the resulting change in amplitude, ∆A, isdetected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Scanning electron microscope image of a single crystal silicon can-tilever, similar to the one used in this study. From [2]. . . . . . . . . 17

2.6 A block diagram of the frequency detector used in the study. From [3]. 19

2.7 The hysteresis of the electromagnets. Top panel shows the full hys-teresis loop, whereas the bottom panel shows a section of the full loopto show the hysteresis in the electromagnets when the applied voltageis low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8 A piezoelectric crystal when not under mechanical stress. The semi-mobile ion (black) is in the center of the crystal and there is no netelectric field on the crystal. . . . . . . . . . . . . . . . . . . . . . . . . 22

v

vi List of Figures

2.9 A piezoelectric crystal when under a mechanical stress. The semi-mobile ion (black) is no longer in the center, which creates an electricfield that polarizes the crystal. . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Illustration showing a Bloch wall (left) and Neel wall (right). In aBloch wall the magnetization rotates out of plane, and in a Neel wallthe magnetization rotates in-plane. From [4]. . . . . . . . . . . . . . . 27

3.2 The general particle shape of interest, whose outline resembles a peanut.Its characteristic lengths are labelled as a, b and c. For this particlea = 750 nm, b = 300 nm and c = 220 nm. . . . . . . . . . . . . . . . 29

3.3 A schematic flowchart of the OOMMF program. The path outlined inred is the function performed by iterations, and the path outlined ingreen is the function performed by stages. . . . . . . . . . . . . . . . 30

3.4 A graphic representation of the Landau-Lifshitz-Gilbert equation. Themagnetization vector M precesses around the effective applied field,H, and it also tends to align itself with H. The damping coefficient, α,determines how quickly the magnetization lines up with the effectiveapplied field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 The transition from an SEM image of a particle in the real-world,a), to a black and white image, b), that OOMMF can then initialize,c). The arrows and their colors in c) represent the direction of themagnetization orientations of the cells. The direction of the arrowsshow the direction of the magnetization in the plane of the page andtheir color represent the direction of the magnetization out of the planeof the page. Red represents out of the page and blue represents intothe page. Particle is 750 nm long with minimum aspect ratio of 2.63. 33

3.6 After having the magnetization of each cell randomized, as in Fig-ure 3.5 c), the particles then relaxed into either a a) single vortexstate, b) double vortex state, or c) an “S” state. The frequency ofthese resultant states was dependent on particle length and aspectratio. Arrows show the in-plane direction of the magnetization. Back-ground also shows the in-plane direction of the magnetization. Redrepresents areas where the magnetization lies along the ±x axis andgreen/blue represents areas where teh magnetization lies along the ±yaxis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7 Magnetization along the y-axis versus simulation time for a particlethat was assigned random initial magnetization 10 times and allowedto relax. Each curve shows the time evolution of the magnetizationalong the y direction for one trial. Five trials resulted in saturation inthe +y direction, four resulted in saturation in the −y direction, andone did not saturate. Instead, it adopted one of the states in Figure 3.6.Particle was a 250 nm permalloy particle with cell size of 6 nm. . . . 36

List of Figures vii

3.8 The percentage of particles that relaxed into single domain states, plot-ted as a function of their aspect ratio. The rate reached 90% whenthe aspect ratio was 2.63. Note that the particle often adopted ametastable state before finally saturating. Calculation was performedon 250 nm permalloy particle with cell size of 6 nm. . . . . . . . . . . 37

3.9 The y magnetization of a particle as a function of simulated time. Attime T0 an applied field was simulated in the +y direction. The fieldwas turned off at time T1 and the reverse field of the same magnitudewas turned on at time T2. The top curve shows the magnetization ofthe particle switching from saturation in the −y direction to saturationin the +y direction in an applied field of 775 Oe. The bottom curveshows a trial where the particle did not switch to the +y direction,with the applied field only 740 Oe. The particle was a 250 nm longpermalloy particle with maximum width 102 nm and a cell size of 6 nm. 39

3.10 The switching phase diagram for a 250 nm long permalloy particle,with damping coefficient α = 0.25 and cell size 8 × 8 × 8 nm3. Asparticle widths (aspect ratios) increase, coercive fields decrease. . . . 40

4.1 A cross-section view of the fabricated sample before lift-off was per-formed. The lower molecular mass of the bottom layer makes it morereactive, creating an undercut. When a material, such as permalloy, isthen deposited (black) it is not in physical contact with either layer soit will not be affected by lift-off. . . . . . . . . . . . . . . . . . . . . . 43

4.2 The result of a preliminary attempt at fabricating the particles of inter-est. Particle is approximately 750 nm long and approximately 500 nmacross at its widest point. . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 The array of 750 nm particles after being saturated in a field of -375 Oe.The single “dark-light” contrast for each particle shows they are in asingle domain state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Images taken while obtaining the ensemble hysteresis loop for the arrayof 750 nm particles. The image on the left (imaged at remenance aftera field of -33 Oe was applied) shows the array with some particlesappearing to be “missing”. These particles appeared to vanish becausethey had adopted flux closure states. The middle figure (imaged atremenance after a field of - 65 Oe) shows most particles in the arrayhad entered the flux closure state, though some have already switchedto the opposite single domain state, and some were still in their originalstates. The image on the right (imaged at remenance after a field of- 87 Oe) shows the array when almost all particles had switched to theopposite single domain state. The particles in the image on the rightare distorted because of a damaged tip (See Figure 2.3). . . . . . . . 46

4.5 Typical hysteresis loop for a particle with a two stage switching process.The particle demagnetizes before switching to either saturated state.Adapted from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

viii List of Figures

4.6 The array of 750 nm particles after being saturated in a field of -375 Oe.The single “dark-light” contrast for each particle shows they are in theopposite single domain state than that in Figure 4.3. The scanned areais the same as that in Figure 4.3 and Figure 4.4, but the scan area wasrotated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7 A 750 nm particle in a single domain state. For the purposes of count-ing the magnetization of the array, this particle has a value of +1. . . 48

4.8 The ensemble hysteresis loop for the array of 750 nm particles. . . . . 49

4.9 The switching distribution for the array of 750 nm particles, with Gaus-sian fits. The two distributions indicate the two-step switching processof the particles. The cartoons at the top show the most common stateof particles in the array in that range of applied field. Below Hd theyare in a single domain state (“up”), as on the left. Between Hd andHs they tend to be in a closure state, as in the middle picture, andabove Hs the particles tend to be in the switched single domain state(“down”), as on the right. The peaks are located at Hd = 60 Oeand Hs = 130 Oe. The FWHM of the distributions are 16 Oe and56 Oe, respectively. For the red curve R2 = 0.812 and for the bluecurve R2 = 0.683. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.10 SEM image of a section of the 250 nm particle array. Imaged area is3 µ m× 3 µ m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.11 MFM image of array of 250 nm particles before any magnetic fieldswere applied. Particles are in single domain configurations, confirmingthe prediction made in Section 3.6. Imaged area is 4 µ m× 4 µ m. . . 54

4.12 The array of 250 nm particles after being saturated in a field of -580 Oe.Imaged area is 4 µ m× 4 µ m. . . . . . . . . . . . . . . . . . . . . . . 55

4.13 Typical image in the construction of the ensemble hysteresis loop forthe 250 nm particles. The number of particles in each configuration iscounted to calculate the magnetization of the ensemble. Imaged areais 4 µ m× 4 µ m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.14 The hysteresis loop for the ensemble of 250 nm particles. When theshape of this loop is compared with that of the hysteresis loop for the750 nm particles (Figure 4.8) it is clear that this loop is wider and thetransition sharper. The abruptness of the transition is indicative ofboth the one-step switching process of the 250 nm particles and thatthe particles tend to switch more uniformly than the 750 nm particles. 57

4.15 The switching field distribution of the 250 nm particles, with Gaussianfit. The peak of the Gaussian curve is at 490 Oe, the FWHM is 40 Oeand R2 = 0.726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.16 The switching phase diagram of Figure 3.10 is reproduced here with theexperimentally determined point for the ensemble added as the greenpentagon. Experimental results and calculated values agree within error. 60

List of Figures ix

A.1 The front panel of the frequency modulation detector with componentsnumbered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

List of Tables

3.1 Table showing the discrepancies in coercive field as a function of thedamping coefficient, α, and cell size. H6 and H8 are the coercive fieldfor cell sizes 6 × 6 × 6 nm3 and 8 × 8 × 8 nm3, respectively. . . . 41

x

Abstract

Magnetic quantum cellular automata (MQCA) have been proposed as an alternate

computing architecture. Single domain magnetic particles represent “1” or “0”; their

stray field interaction controls the propagation and manipulation of information. An

inherent requirement for an MQCA system is to know the conditions under which

nanomagnets switch between the purely “up” (1) and the purely “down” (0) state,

and to control this reproducibly.

As a first step to study this, arrays of two types of permalloy particles were de-

signed, simulated, fabricated and imaged, and their switching distributions ascer-

tained. Individual particles were “peanut”-shaped, to investigate the effect of a shape

anisotropy for an elliptical particle. Particles had long axes of 750 nm and 250 nm,

but had identical aspect ratios.

Particles were simulated with a public domain software package, Object Oriented

Micromagnetic Framework (OOMMF), fabricated by electron beam lithography with

standard lift-off techniques in the fabrication facility in Sherbrooke, Canada, and im-

aged in vacuum using a custom built magnetic force microscope in constant height

mode with an in plane, in-situ magnetic field. Ensemble hysteresis loops were ob-

tained as was the average switching fields for both arrays.

The 750 nm particles were found experimentally to have a two-step switching

process. The first switch occurred at 60 ± 16 Oe and the second at 130 ± 56 Oe.

These results were nominally better than those obtained in a previous study on sim-

ilarly sized ellipses.

Simulations on the 250 nm particles predicted that particles of that size would

have the single domain configuration as their virgin state, and would have a one-

step switching process. The switching field of a typical particle was calculated to

be 550 ± 30 Oe. This was confirmed experimentally, where the switching field distri-

bution had its peak at 490 ± 40 Oe. Thus, theory and experiment are in agreement,

within error.

xi

Resume

Les automates cellulaires magnetiques quantiques (ACMQ) ont etes proposes comme

nouvelle architecture de computation. Ces particules a domaine unique peuvent

representer soit l’etat “1” soit l’etat “0”, l’interaction des champs magnetiques des

particules entre elles permet la propagation et la manipulation de l’information. Une

des conditions fondamentales d’un tel systeme est la connaissance des conditions sous

lesquelles les “nano-aimants” passent d’un etat a l’autre, soit de l’etat purement “0”

a l’etat purement “1” et le controle reproductible de cet echange.

De facon a etudier cette interaction, deux types de particules en permalloy ont

ete concues, simulees, fabriquees et imagees, et la distribution des valeurs auxquelles

l’echange de leurs etats se produit a ete verifiee. Les particules etudiees etaient

en forme d’arachide, ce qui a permis d’etudier l’effet d’une deviation dans la forme

d’objets ellipsoidaux. Leurs axes primaires mesuraient 750 nm et 250 nm avec des

dimensions proportionnelles.

Le comportement des particules a ete simule par un programme accessible au pub-

lic, “Object Oriented Micromagnetic Framework” (OOMMF). Elles ont ete fabriquees

par lithographie a faisceau d’electrons utilisant une technique de pelage standard a

Sherbrooke et imagees sous vide a l’aide d’un microscope a force magnetique (MFM)

fabrique sur place, en mode de hauteur constante avec un champ magnetique dans le

plan in-situ. Dans chacun des cas, des courbes d’hysterese ainsi que la valeur normale

d’echange ont ete obtenues.

Les particules ayant un axe principal de 750 nm ont montre experimentalement un

comportement d’echange a deux phases. Le premier echange s’est produit a 60 ± 16 Oe

et le second a 130 ± 56 Oe. Ces resultats ont ete meilleurs que ceux obtenus lors

d’une etude anterieure visant des particules elliptiques de taille comparable.

Les simulations faites sur les particules de 250 nm ont predit l’obtention d’un do-

maine magnetique unique a l’etat vierge et un echange de la direction de magnetisation

en une seule etape. La valeur du champ a laquelle l’echange se produit a ete cal-

culee et une valeur de 550 ± 30 Oe a ete obtenue. Cette valeur a ete confirmee

experimentalement, la valeur d’echange ayant son maximum a 490 ± 40 Oe. Ainsi,

les valeurs experimentales et theoriques sont en accord, dans les limites de l’incertitude.

xii

Acknowledgments

First, I would like to thank my supervisor Prof. Peter Grutter for introducing me to

the exciting field of scanning probe microscopy. His unending patience and enthusi-

asm, even in the face of delays and dead ends, have helped make these last two years

a true learning experience.

I would like to thank Dr. Jean Beerens in Sherbrooke for allowing me access to the

e-beam facilities and for his helpfulness while I was there. Here at McGill, I would

like to thank Robert Gagnon for his assistance with the sputtering machine and Prof.

Zaven Altounian for his kind interest in my project and for his assistance.

Special thanks goes to Dr. Xiaobin Zhu for getting me up to speed on the MFM,

and for answering my numerous trivial emails.

A huge debt is owed to Dr. Mark Roseman, for helping me with the MFM system,

for encouraging me to submit abstracts to conferences, for offering a sympathetic ear

in frustrating times and, most importantly, for his friendship.

I would also like to thank all of my colleagues in the SPM lab for making my

experience that much more enjoyable, and especially Patricia Davidson and Mehdi

El Ouali for translating my abstract. In particular, a special thank you must go to

Sarah Burke, for actively sympathizing with me that the first year of grad school is

tough on most people.

To my friends back in Windsor, and especially the other “cogs” of “The Machine,”

I thank you for getting me to grad school in the first place, for your emails, and for

reading my numerous inconsequential ones.

Also, I would like to thank my family for their encouragement throughout the

entire course of my education.

Finally, I would like to thank Caitlin St. John. Her unwavering support, patience

and affection have made my world a much happier and vastly more interesting place.

xv

1

Introduction

Our reliance on computers has increased drastically since their inception; computers

are now ubiquitous. While this has unquestionably influenced human society, the

days of computers, as we know them, are numbered. Within the working lives of this

generation of scientists and engineers a new computing architecture will be needed as

the end of Moore’s Law rears its ugly head. It predicts that the number of transistors

on a chip doubles every 18 months, providing a corresponding increase in the abil-

ity to process information, allowing for a near-constant improvement in computing

technology for the last 30-plus years [6]. For decades, we have reaped the benefits of

Moore’s Law’s promise of better, faster, more powerful computers. Soon we will have

to confront the fact that elements of conventional semiconductor industry fabrication

will either become too small to reliably produce, or so small that the new paradigm

of quantum mechanics will dominate. Clearly, then, another method of computation

is required.

An alternate method of computing is cellular automata, which uses cells to imple-

ment logic functions. The fundamental principle of the cellular automata architecture

is that the status of a cell is determined by the status of its neighboring cells. The

most familiar example of cellular automata is Conway’s “Game of Life” [7]. The game

consists of a square grid, with each square able to adopt one of two states: “alive” (1)

or “dead” (0). The status of one cell is determined by how many of its surrounding

cells are alive.

One possible method of realizing a cellular automata system has been suggested

1

2 1 Introduction

by Cowburn [8], which he called magnetic quantum cellular automata (MQCA). This

arrangement involves individual magnets, the average sizes of which are on the scale

of nanometers. On this length scale it becomes energetically favorable for a magnet

to behave like a giant dipole, the word “giant” being very relative in this case. All

the dipoles of all the atoms point in the same direction. This dipole can point in

either one of two directions, which can be arbitrarily called “up” and “down” or “1”

and “0,” making this a binary system. In principle, then, it is possible to construct

a cellular automata system using the stray field coupling of these binary particles to

propagate signals and perform logic operations.

To propagate a signal, a chain of particles could be used if the coupling between

the particles was strong enough. The first particle in the chain would change from its

“up” state to “down” which would cause its neighbor to switch to the “down” state,

which would in turn cause its neighbor to switch, and so on. The obvious analogy to

this would be dominoes. Each domino is tipped by its neighbor on one side, and in

turn tips the neighbor on the other side.

Logic gates can also be made with these particles. Consider the system in Fig-

ure 1.1, where all chains function in the manner described above. With proper engi-

neering to ensure the right degree of coupling, the particles in Chain 3 will propagate

the signal if and only if the far right elements of Chains 1 and 2 have their magne-

tizations pointing simultaneously to the right. Logic gates in cellular automata are

discussed further in [9].

Clearly, having single domain particles that switch reliably is crucial for an MQCA

system. Each particle must switch consistently or signals will not propagate, and

logic gates will not function. It has been shown that permalloy particles that have

been simultaneously fabricated, and should therefore be identical in behavior, do

not uniformly switch at a certain applied field [5]. The origin of this distribution is

unclear. It may be that fabrication techniques give rise to particles that have different

shape anisotropies; that sputtering permalloy leaves different particles with slightly

3

Figure 1.1: A schematic of an “and” gate in a magnetic quantum cellular automata system. Thesignal is propagating from left to right. In order for chain 3 to switch from its “left” configurationto the “right” state, the stray fields from the right most particles in both chains 1 and 2 mustbe influencing the first particle in chain 3. In a) and b) the stray fields are insufficient to induceswitching in chain 3. In c), the combined stray field is enough to induce switching in chain 3. Nowthat the first element in chain 3 has switched, it will switch the particle adjacent to it, and so on,propagating the signal to the right.

4 1 Introduction

different compositions of iron and nickel; that each individual particle switches at a

different applied field in each trial, giving rise to an intrinsic switching distribution;

or the origin of the distribution arises from a combination of these or other sources.

This study is being conducted to investigate the effects of shape anisotropies on the

switching field distribution of arrays of magnetic nanoparticles.

A very useful tool for tracking the magnetic state of such small particles is the

Magnetic Force Microscope (MFM). The MFM is a specific type of Atomic Force

Microscope (AFM), which was invented in the 1980s, soon after the Scanning Tun-

nelling Microscope was developed. This family of microscopes, known as Scanning

Probe Microscopes (SPMs), provides information about surface topography, elec-

tronic structure, friction, and more.

In the case of the MFM, magnetic properties of micron and submicron samples for

over 15 years now [10, 11]. Because of its high resolution, low cost to implement, ease

in use and ability to function in an external magnetic field, it is particularly well-suited

to investigate a magnetic-based computation system. What’s more, with judicious

use, the MFM could be used as a read/write head for such a system, becoming not

only a tool for investigation, but a critical part of a new computing paradigm.

Its role in magnetic based computation is just the latest potential area for MFM

to show its usefulness, helping to ensure that it will remain a powerful tool in years

to come.

The goal of this thesis is two-fold: first is the challenge of making viable samples

within the NanoQuebec network. The second goal of this thesis seeks to explore if

the results of micromagnetic modelling agree with experimental results.

2

Principles & Instrumentation of Magnetic Force Microscopy

Scanning Probe Microscopy (SPM) has developed into a field of invaluable tools for

the emerging disciplines of nanoscience and nanotechnology. The SPM field was

essentially founded in 1982, with the invention of the Scanning Tunneling Microscope

at IBM Zurich by Gerd Binnig and Heinrich Rohrer [12]. The device consisted of a

small, sharp conducting tip being held in close proximity to a flat conducting sample.

The tip was sufficiently close to the sample to allow for quantum mechanical tunneling

of electrons from the atoms of the sample to the tip, which was then measured as

a current. The success of the technique earned Binnig and Rohrer a Nobel Prize in

1986. More importantly, it encouraged further development of SPM techniques.

The method most relevant to the current discussion is the Scanning Force Micro-

scope (SFM), developed by Binnig, Quate and Gerber in 1986 [13]. In SFM, a sharp

tip is fixed to a flexible cantilever arm and mechanically scanned across a sample (or

a sample is scanned under the tip). At small tip-sample separations, forces change

the cantilever status. This change could be in the form of a physical deflection of the

cantilever, or a change in its resonance frequency. The change in cantilever status is

ultimately what is measured to create an image. One of the first forces investigated

using scanning force techniques was magnetism; that is, Magnetic Force Microscopy

(MFM) was developed [10, 11].

5

6 2 Principles & Instrumentation of Magnetic Force Microscopy

Figure 2.1: The principle of magnetic force microscopy. A magnetic tip is scanned across a magneticsample. The attraction or repulsion between the tip and the stray field of the sample causes thestatus of the cantilever to change, and this change is detected using optical techniques. Adaptedfrom [1].

2.1 Principles of MFM

The basic concept of magnetic force microscopy is illustrated in Figure 2.1. An

analogy can be drawn between MFM and an audio record player. In a record player,

a tip moves across a surface while in contact with it; the grooves in the vinyl record

move the tip up and down. This movement is converted into an electrical signal,

which is then converted into an audio signal, namely, music. In MFM the tip is

influenced by the magnetic force between tip and sample, not the mechanical contact

between record and needle. This interaction is converted into an electrical signal,

which is then converted into an image.

2.2 Stray Fields

The stray magnetic field from a ferromagnetic sample can be calculated with [1, 14]

H(r) = −∫

Vs

∇Ms(r′′) · r − r′′

|r − r′′|3dV ′′ +∫

As

n ·Ms(r′′)

r − r′′

|r − r′′|3dA′′ (2.1)

where the integrations are over the particle volume and particle surface, respectively

and n is the normal vector pointing outward from the particle surface; see Figure 2.2.

2.3 Contrast formation 7

Figure 2.2: The geometry used for calculating the force and force gradient on the magnetic tip.Adapted from [1].

Note that the magnetization can be written as a sum of two parts

M = Mdiv + Mcurl

Such that ∇Mcurl = 0 and ∇×Mdiv = 0.

Equation 2.1 shows that only the curl-free term contributes to the stray field. It

can be concluded that there are an infinite number of magnetizations that would give

the same stray field and, thus, the same stray field gradient and MFM contrast. This

emphasizes the need for reliable modelling for probing the true magnetization of the

sample.

2.3 Contrast formation

The general expression for the force acting on the tip due to the stray field from a

magnetic sample is [14, 1]

F (x, y, z) = µ0

∫ ∞

−∞Mtip(x

′, y′, z′) · ∇Hsample(x + x′, y + y′, z + z′)dV ′ (2.2)


Figure 2.3: The effect of a damaged tip on image quality. The image on the left was taken withan undamaged tip. The image on the right is of the same area and was obtained with identicalparameters, but the tip was damaged. Note that individual particles cannot be resolved in theimage on the right. Image was taken in frequency shift mode with sample voltage acting as theservo. Sample was permalloy disks with diameters of 1 µm.

If only the z component is examined, as is often the case in MFM, the force on the

tip can then be written as

Fz(x, y, z) = µ0

∫ ∞

−∞Mtip(x

′, y′, z′) · ∂

∂zHsample(x + x′, y + y′, z + z′)dV ′ (2.3)

In both Eq. 2.2 and Eq. 2.3 the integrations are over the primed coordinate system,

the coordinate system of the tip.

2.4 Imaging and Operating Modes

There are two imaging modes: contact and non-contact. As its name suggests, contact

mode involves the cantilever tip physically touching the sample. This mode is not

ideal for MFM. First, there is the possibility of damaging the tip and/or sample while

the two are in physical contact. A damaged tip can lead to artifacts and reduces tip

resolution, as in Figure 2.3. This risk is present in any AFM application.

A second danger, unique to MFM, is the possibility of induced domain distortions

in either the tip or sample. Due to its intrinsic shape anisotropy, the tip is more likely

to retain its magnetization than the sample. In addition, the stray field at the end

of the tip is large, increasing the possibility for magnetic distortion of the sample.

Ultimately, either scenario can occur [15, 16, 17]. These distortions can be reversible

(i.e., the distortion appears when the tip and sample are close and disappears when

2.4 Imaging and Operating Modes 9

the tip and sample are moved far away) or irreversible (i.e., the distortion appears

when the tip and sample are close and remains even after the tip and sample are

separated). For an in-depth discussion about induced distortions, see [5].

The other imaging mode, non-contact, is therefore the one most commonly used

in our MFM. In non-contact mode typical tip-sample separations are between 10 nm

and 200 nm. In this range there are several forces acting on the cantilever including

van der Waals and electrostatic forces. See Section 2.5 for more details.

In general, there are two categories of operating modes, static (DC) and dy-

namic (AC). When operating in the DC mode, the forces between sample and tip

cause the cantilever to bend according to Hooke’s law,

∆z =Fn

kc

(2.4)

where ∆z is the deflection of the cantilever, Fn is the component of the force normal

to the cantilever and kc is the spring constant of the cantilever. The advantages of

this mode are its intuitively simple operation and that output is directly related to

the tip-sample forces. The drawback is that measurements taken in the static mode

are more susceptible to noise sources such as vibrations and electric 1f

noise.

In the AC mode the cantilever is oscillated and the changes in its resonant fre-

quency are detected. To a first approximation the cantilever can be modelled as a one

dimensional damped driven harmonic oscillator. This assumption is valid for small

perturbations of the cantilever, as is usually the case for MFM. Under this assump-

tion, the AC mode is sensitive to force gradients, not the forces themselves and the

force gradient changes the effective spring constant of the cantilever according to

kceff= kc − F ′ ⇒ ω0 =

√kceff

m?(2.5)

where m? is the effective mass of the cantilever and F ′ is given by

F ′ =dFn

dn= n · ∇Fn = n · ∇(n · F ) (2.6)


Figure 2.4: In the slope-detection method, the cantilever is driven at a frequency, ωd. A change inthe force gradient causes a shift in the resonance frequency (ω0 → ω′0), and the resulting change inamplitude, ∆A, is detected.

where n is the unit vector normal to the plane of the cantilever and ddn

is the derivative

in the direction of n.

There are multiple ways to detect the shift of the cantilever’s resonant frequency.

One method that has become less popular is the “slope-detection” technique. In this

technique the cantilever is driven at a frequency near its resonance frequency. As

the tip interacts with the sample, the cantilever resonance frequency is altered. The

amplitude (or phase) of the deflection sensor signal will thus change accordingly, as in

Figure 2.4. More thorough discussions of the slope-detection technique can be found

in various sources [10, 5, 1, 18, 19].

Albrecht et al. [20] developed a method that uses frequency modulation (FM)

detection to directly observe the frequency shift of the cantilever due to F ′, the tip-

sample interaction. The oscillation of the cantilever is maintained by a feedback

loop using the signal from the deflection detector. The oscillation amplitude of the

cantilever is controlled by an amplifier with automatic gain control. Any changes

in F ′, i.e., a force gradient, immediately cause a change in the oscillation frequency

2.4 Imaging and Operating Modes 11

of the cantilever. This change can be measured by a frequency counter, an FM

discriminator or, in our case, a phase locked loop (PLL).

If the sensitivity of a system is thermally limited slope detection and FM detec-

tion methods have the same detection limits within a factor of√

2 [20]. However,

FM detection allows for an increase in the minimum detectible force gradient by

using a cantilever with a higher Q-factor without sacrificing bandwidth. Dynamic

mode measurements are more than a factor of ten more sensitive than static mea-

surements [20]. This improvement makes the compromise on detecting the gradient

of the force, rather than the force itself, acceptable. See [1] for more details.

To obtain an image, an MFM is most commonly used in one of three modes:

constant frequency shift mode, constant height mode or tapping/lift mode.

In constant frequency shift mode the frequency of cantilever vibration is kept con-

stant by changing the tip-sample separation. Since the interaction between magnetic

tip and sample can be attractive or repulsive, the servo force needs to be monotonic

to ensure feedback stability. While van der Waals forces can be used for this, it is

common to apply a DC voltage between tip and sample.

Alternatively, constant height mode consists of lifting and maintaining the tip at a

predetermined height and scanning. A small (or zero) voltage is applied between tip

and sample to compensate for the contact potential difference (CPD) between them.

Doing so minimizes the sample topography in the image, giving just the magnetic

contrast of the sample. The CPD has a dependency on tip and sample shape and

this is one way to monitor tip deformations due to crashes, etc. If the CPD changes,

a tip deformation may have occurred, yielding lower quality images.

In constant height mode the feedback system is not required and can be effectively

turned off. Without feedback, scanning rates can be increased significantly, though

caution must be exercised in order to not crash the tip into the sample. Typical lift

heights can be anywhere from 20 nm to 150 nm, depending on the size of the features

on the sample, the stray field of the sample, and the size of the scan area. Plane


subtraction hardware is used to ensure that the sample is not tilted relative to the

tip. Even with this hardware, precision is needed to ensure a flat image. For example,

if the desired scan area is 12 µm on a side, and the lift height is only 20 nm, the tilt

of the sample must be less than 0.1 degrees or the tip will crash, while a scan of an

area 500 nm on a side at a lift height of 20 nm will not have a tip crash even if the

misalignment is 2 degrees. In any case, tilt should be eliminated as much as possible

since a tilted sample yields an inferior image, even if it does not create a tip crash.

Other deciding factors in lift heights could be sample roughness and cleanliness.

The most common of the three modes is tapping/lift mode, developed by Digital

Instruments [21]. Tapping/lift mode is a two stage mode that de-convolves topogra-

phy and magnetic contrast. In the first stage (“tapping”), the sample topography is

obtained using the root mean square of the cantilever’s oscillation amplitude as feed-

back [14]. The second stage (“lift”) involves re-scanning that topography at a user

controlled height and measuring the frequency shift or phase shift of the cantilever.

This is the most common mode because of its popularity in commercial systems. Care

must be taken when using this technique because during the first stage (tapping) the

sample can be subjected to a large, localized stray field from the tip. Also, it is tacitly

assumed that there is no thermal drift during a scan or any piezo creep or nonlinear-

ities, since the second stage would not be re-scanning the same area mapped out in

the first stage. Tapping/lift mode is the most popular method of imaging because it

is the most common mode found in commercial systems.

2.5 Other Forces

Although care has been taken to ensure that images arise from only magnetic forces,

other forces may play a role in image formation.

2.5 Other Forces 13

2.5.1 van der Waals Forces

The van der Waals force originates from dipole-dipole interactions. It arises because

although the time average dipole of an atom is zero, its instantaneous dipole is, in

general, finite. This non-zero dipole induces a dipole in neighboring atoms. At a

tip-sample separation of more than a few nanometers van der Waals forces become

quite small, less than 1% of other forces [22].

2.5.2 Electrostatic Forces

If both tip and sample are conducting and at different electric potentials, then the

tip-sample system can be treated as a capacitor with capacitance C and the force

between them as [14]

Fel =∂C

∂z(Ubias − UCPD)2 (2.7)

where Ubias is the applied bias between tip and sample and UCPD is the contact poten-

tial difference due to the difference in work functions. As discussed in Section 2.4, Ubias

is used to create a stable monotonic servo force. The electrostatic force is strongly

dependent on tip shape as is the ∂C∂z

term. Saint Jean et al. [23] modelled the tip

as a truncated cone and half sphere and saw that for small distances the dominant

electrostatic force term is

Fel = πε0R

z(Ubias − UCPD)2 (2.8)

Electrostatic forces are a much longer range force than the van der Waals force;

not only can the electrostatic force between the sample and the tip be relevant, the

electrostatic force between the sample and the cantilever may need to be considered

as well [14].

2.5.3 Capillary Forces

If measurements are performed in ambient conditions capillary forces need to be

accounted for if the radius of the contact is less than the Kelvin radius. Below this


dimension vapors (usually water) condense into the contact area. The Kelvin radius

is given by

rK =γV

RT log(p/ps)(2.9)

where γ is the surface tension, R is the universal gas constant, T is the temperature.

V is the molar volume and ps is the saturation vapor pressure. Due to the large tip-

sample separations and because the experiments were carried out in vacuum, capillary

forces are assumed to be negligible in this study.

2.6 MFM in Context

Magnetic force microscopy is but one of many tools available to the researcher in-

terested in investigating magnetism on a small scale. Techniques such as alternating

gradient magnetometry (AGM), vibration sample magnetometry (VSM), supercon-

ducting quantum interference device magnetometry (SQUID) and imaging via the

magneto-optical Kerr effect (MOKE) allow for the collective magnetic behaviors of

arrays of nanomagnets to be characterized. However, these techniques lack nanometer

resolution and thus are not ideal when studying individual nanomagnets.

To study individual elements of an array there exist several techniques with spa-

tial resolution capable of characterizing individual nanomagnets. These techniques

include magnetic force microscopy (MFM) [10, 11, 1, 24], Lorentz electron microscopy

(LEM) [25], scanning electron microscopy with polarization analysis (SEMPA) [26],

spin polarized scanning tunneling microscopy (SPSTM) [27, 28], spin polarized low

energy electron microscopy (SPLEEM) and MOKE microscopy [29]. To fully charac-

terize nanomagnets and ensembles of nanomagnets, a combination of these techniques

should be used.

The principle behind LEM is the Lorentz force which describes the motion of an

electron through a magnetic field, classically given by F = qv×B. The full classical

and quantum mechanical explanation of the mechanism is given in [30]. Electrons are

incident on the sample and the image is constructed based on how the sample deflects

2.6 MFM in Context 15

the electrons. LEM gives high resolution and is a direct measurement of the magne-

tization of the sample. A disadvantage of this technique is that special preparation

is required to ensure that the sample is electron transparent. Another drawback of

this technique is the difficulty associated with applying an in-situ magnetic field, as

this field would change the path of the incident electrons.

With SEMPA (sometimes called spin polarized SEM), the sample surface is bom-

barded with electrons, causing it to release secondary electrons. When secondary

electrons emanate from a magnetic sample they are polarized antiparallel to the local

magnetization vector at their point of origin. These secondary electrons are collected

and analysis yields direct information about the magnetization of the sample. The

resolution of SEMPA is also high, on the order of 10 nm. The drawbacks of SEMPA

are the requirement of ultra high vacuum (UHV) conditions, the sample must be a

clean conducting surface and SEMPA can only probe magnetic properties to a depth

on the order of a nanometer. The last two restrictions arise because polarization of

the secondary electrons is lost due to scattering in unclean or thick samples.

SPSTM also makes use of the fact that electrons in magnetic atoms are polarized.

A typical setup has an antiferromagnetic tip scanned across a ferromagnetic sample.

The tunneling current is dependent on the relative orientation of the magnetization

vectors of the tip and sample. When a finite bias is applied between the tip and

sample the spin resolved density of states (DOS) need to be taken into account. The

spin resolved DOS is very sensitive to the bias voltage, possibly even changing sign.

More details about SPSTM can be found in [31].

SPLEEM takes advantage of the different reflection coefficients for electrons of

different polarizations. Electrons of both polarities are sent incident on the sample,

and they will have different reflectivities depending on the sample magnetization. This

technique does not directly measure the sample magnetization. Rather, it measures

the projection of the magnetization vector onto the polarization vector of the incoming

electron beam [32].


The magneto-optical Kerr effect (MOKE) notes that the polarization of light will

be changed after reflection from a magnetic surface. MOKE microscopy focuses laser

light onto a sample and detects the polarization shift of the reflected beam relative

to the incident beam. It has excellent time resolution but its spatial resolution is

limited to the wavelength of the laser light, which is in the hundreds of nanometers,

and laser spot size.

These techniques and others are discussed with excellent clarity in a review article

by Freeman [33].

MFM allows for investigation of both individual particles and ensembles of parti-

cles. It is simple to implement, does not necessarily require UHV conditions or special

sample preparation and is therefore cost effective. Scan times are usually on the order

of minutes, depending on feedback settings. However, it lacks time resolution on a

scale appropriate for studies regarding switching mechanisms in nanomagnets, and

does not directly measure the sample magnetization. Nevertheless, it remains a useful

tool for characterizing magnetic nanoparticles.

2.7 Intstrumentation

In the gathering of the data for this thesis a custom built vacuum magnetic force

microscope was used. Relevant components of the microscope will be discussed in the

following section. Other relevant aspects of the system and its electronics have been

discussed previously [5, 18, 19, 22])

2.7.1 Force Sensor

The force detectors in our system are commercially available cantilevers from NanoSen-

sors [34] and MikroMasch [2], similar to those in Figure 2.5. The position of the

cantilever was monitored with an interferometer [35]. Both of these manufacturers

also offer tips designed specifically for MFM, but these are to be avoided. The com-

mercially designed MFM tips have large stray fields which yield large signal-to-noise

2.7 Intstrumentation 17

Figure 2.5: Scanning electron microscope image of a single crystal silicon cantilever, similar to theone used in this study. From [2].

ratios (SNR) but are also more prone to induce magnetic distortions in the sample.

These commercial MFM probes are popular because most commercial MFM systems

are operated in air, where the increased SNR is needed to compensate for decreased

sensitivity due to the decreased Q-factor of the cantilever.

Our system operates in vacuum, increasing the Q-factor of the cantilever and de-

flection sensitivity. This allows for use of a lower moment tip which is less prone

to inducing distortions in the sample, but has a lower SNR. It has been found pre-

viously [5] that coating cantilevers with Co71Pt12Cr17 offers an appropriate balance

between signal-to-noise performance and possible distortions. The standard coating

thickness was 20 nm, deposited by sputtering. A 5 nm layer of gold can be deposited

on top of the magnetic layer to prevent oxidation and thus increase the shelf life of

the cantilever. An electromagnet was used to magnetize the cantilever along its z axis

prior to the experiments. Because of the shape anisotropy of the tip it is assumed

that the magnetic structure of the coating is a single domain configuration.

The cantilevers used in this study are single crystal silicon cantilevers with typical

spring constants on the order of 1 N/m and resonance frequency between 65 kHz

and 90 kHz. Cantilevers can be purchased with a coating on the backside to increase

reflectivity but this is not necessary for our system since the deflection detector, the


cleaved end of an optical fiber, is positioned close enough to the cantilever that there

is sufficient reflection without coating.

Another relevant parameter of a cantilever is its Q-factor. The Q-factor is a

dimensionless quantity defined as the resonance frequency of the cantilever divided

by the bandwidth, or full width at half maximum (FWHM):

Q =ω0

∆ω(2.10)

To calculate the bandwidth the cantilever was driven at its resonance frequency, ω0,

and the peak-to-peak amplitude from the interferometer, Vres, was noted. The driving

frequency was changed such that the cantilever signal had a magnitude of Vres√2

. There

was one such frequency greater than ω0 and one less than ω0, denoted by ω> and ω<,

respectively. The bandwidth, ∆ω, is the difference between ω> and ω<. Thus, the

Q-factor can be calculated with

Q =ω0

ω> − ω<

(2.11)

Typical Q-factor values for the single crystal silicon cantilevers described above are

about 200 in air and about 40 000 in vacuum.

2.7.2 Frequency Modulation Detector

A phase locked loop (PLL) is used to directly monitor the shift of the resonance

frequency of the cantilever. Two PLLs were used in this study. One is a commercial

system from NanoSurf [36] while the other was designed by Dr. Kei Kobayashi of the

International Innovation Center, Kyoto University, Japan and was assembled here

at McGill University by Dr. Yoichi Miyahara. The design is based on the Kyoto

Instruments KI2000 XEL FM Detector [3].

The detection bandwidth of the latter can be switched between 1 kHz, 4 kHz

and 10 kHz. The indeterminate frequency (IF ) is 4.5 MHz. The local oscillator

frequency is fosc = IF − fres. The input frequency range is quoted as being 10

kHz to approximately 10 MHz. However, we found that it was not possible to lock


Figure 2.6: A block diagram of the frequency detector used in the study. From [3].

signals coming from cantilevers with resonance frequencies of fres ≤ 55 kHz. When

the cantilever signal is fed into the detector, the mixer creates both IF + fres, and

IF −fres. See Figure 2.6. Since fres is typically two orders of magnitude less than IF

(tens of kilohertz versus megahertz), the these waveforms have similar frequencies.

The signals are then sent through the band pass filter (BPF). For cases where fres ≤ 55

kHz, the frequency of the signals are too close together for the band pass filter to

resolve them, and both are sent to the PLL, preventing locking. Therefore, only the

rectangular single crystal silicon cantilevers can be used with this PLL. The triangular

silicon nitride cantilevers already in the lab have resonance frequencies too low for

the FM detector.

See Appendix A.1 for operating details of the FM detector.

2.7.3 Electromagnets and Hall Probe

A pair of in-situ electromagnets were used to apply an in-plane magnetic field to the

sample. They function both in air and in vacuum. A Hall probe has been attached

on the pole of one of the cores of the elctromagnet in an attempt to give in-situ


measurements of the applied field. However, as has been reported previously [22] the

drop-off from the tip of the core to the centre of the air gap where the sample is

located can be substantial, as much as 35% [22].

Instead, an external F.W. Bell Model 4048 Gauss/Tesla Meter (probe model 1451)

was used to calibrate the measured applied magnetic field versus the applied voltage

from a Kikusui POW 35-5 bipolar power supply. In doing so, the hysteresis of the

electromagnets was measured and plotted in Figure 2.7. Depending on whether the

electromagnets were saturated positively or negatively, the actual measured applied

fields can differ by 20 Oe or more for the same applied voltage. To compensate for

this, the electromagnets were calibrated using an external Hall probe, and before the

sample is inserted into the system the electromagnets were saturated to either the

positive or negative state. The particular branch of the hysteresis curve was then

known and applied field magnitudes could then be quoted with confidence.

2.7.4 Vacuum

All experiments were conducted under high vacuum conditions. The vacuum pump

is a two stage Balzers TCP 121 turbomolecular pump. Preliminary images (to ensure

that sample positioning is acceptable, for example) can be taken when the pressure

is of the order of 10−3 mbar. More detailed images can be taken at pressures of the

order of 10−5 mbar.

Pumping times and final chamber pressures can be reduced by ensuring that all

wing nuts on the chamber are hand tight, but no tighter as this can deform a gasket,

causing a leak. Keeping the inside edge of the chamber clean can improve vacuum

conditions as well, as this is where the two portions of the chamber meet, making it

the junction most prone to leaks. To clean this part of the chamber, a kimwipe with

acetone is wiped around the edges. It is crucial to allow the edge of the chamber to

dry or to dry it with another kimwipe to ensure that the o-ring of the gasket does

not come in contact with the acetone. Acetone tends to cause dryness and eventual


-40 -30 -20 -10 0 10 20 30 40

-400

-300

-200

-100

0

100

200

300

400

Mag

netic

Fie

ld (

Oe)

Applied Voltage (V)

-15 -10 -5 0 5 10 15-200

-150

-100

-50

0

50

100

150

200

-15 -10 -5 0 5 10 15-200

-150

-100

-50

0

50

100

150

200

Applied Voltage (V)

Mag

netic

Fie

ld (

Oe)

Figure 2.7: The hysteresis of the electromagnets. Top panel shows the full hysteresis loop, whereasthe bottom panel shows a section of the full loop to show the hysteresis in the electromagnets whenthe applied voltage is low.


Figure 2.8: A piezoelectric crystal when not under mechanical stress. The semi-mobile ion (black)is in the center of the crystal and there is no net electric field on the crystal.

cracking in the o-ring.

It is important to note that when pumping down or venting no voltage should be

applied to the sample piezo tube. At pressures of the order of 10−1 mbar there is a

possibility that a plasma could form, as in a fluorescent light, which could damage

the piezo tube.

2.7.5 Piezoelectrics

Piezoelectric materials are crucial to scanning probe microscopies because they al-

low for mechanical manipulation at the nanometer scale. Piezoelectric crystals have

noncentrosymmetric unit cells; that is, they have no center of symmetry and possess

a semi-mobile ion which has several possible quantum states within the crystal [37].

When isolated, the unit cell has no polarization, as in Figure 2.8. When the unit cell

is mechanically deformed the semi-mobile ion is no longer in the center of the crystal,

as in Figure 2.9. The displaced ion creates an electric field in the crystal, which in-

duces a charge on the surface of the crystal. Thus, there is an overall polarization on

the crystal. The direction of polarization is dependent on the direction of mechanical

deformation. A property of piezoelectrics is that the converse of this, with applied

voltages causing deformations, is true as well.

One common family of piezoelectric materials are the lead zirconate titanate


Figure 2.9: A piezoelectric crystal when under a mechanical stress. The semi-mobile ion (black) isno longer in the center, which creates an electric field that polarizes the crystal.

(PZT) ceramics. It is a solid solution of lead zirconate, PbZrO3 and lead titanate,

PbTiO3 [38]. In our system PZT piezos are used for raster scanning the sample under

the tip, to position the cantilever in the middle of the interferometer fringe and to

walk the sample, if necessary.

3

Simulations of Magnetism on the Nanoscale

The discovery of magnetism dates to antiquity. Today, magnets are ubiquitous in

society, from computers to security devices to refrigerator decorations. However, ex-

ceedingly few people can explain the origins of the phenomenon. For the student, it

is an unfortunate fact that magnetism is often glossed over or neglected in Electric-

ity & Magnetism courses. This seems to occur simply because the chapters covering

magnetism tend to be at the back of textbooks. It is therefore common to still not

feel confident about discussions involving magnetism even after conducting research

probing it.

Throughout this thesis, a reference to “magnetism” implies ferromagnetism, as

ferromagnets are the type of magnets used in this study. Information on diamag-

netism, paramagnetism, antiferromagnetism and ferrimagnetism is readily found in

solid state physics and materials engineering books [4, 39].

In 1907 [40] Pierre Weiss introduced a theory in an attempt to explain ferromag-

netism. He postulated that ferromagnetism arose from an internal “molecular field”

that was sufficiently strong to spontaneously magnetize a magnet. Obviously it is

possible to find a piece of iron, for example, that is unmagnetized. Weiss explained

this by making a second assumption, that ferromagnets divided themselves into do-

mains. Each domain spontaneously magnetized itself in a certain direction but these

domains could macroscopically cancel each other out giving a net unmagnetized sam-

ple. Conceptually, both of these arbitrary assumptions are true, though the actual

physical explanations are more complicated.

24

3.1 Energy Terms in Magnetism 25

3.1 Energy Terms in Magnetism

The total energy of a magnetic particle in an applied magnetic field is the sum of the

exchange energy, magnetostatic energy, crystalline anisotropy energy and Zeeman

energy. The energy density is therefore

Etot = Eex + Ems + Eanis + EH (3.1)

and the total free energy is

F =∫

d3r(Eex + Ems + Eanis + EH). (3.2)

3.1.1 Exchange Energy

Weiss’ molecular field is not an actual field. Heisenberg later identified it as the

quantum mechanical exchange interaction [41]. The exchange interaction describes

the force between two electrons, which depends only on their relative spin orientations:

Eex = −2JexSiSjcos(φ) (3.3)

where Si and Sj are the spin angular momenta of the atoms, φ is the angle between

spins and Jex is the exchange integral. The condition for ferromagnetism is that

Jex > 0. It follows from Eq. 3.3 that if the exchange energy is to be negative (that is,

for ferromagnetism to be energetically favorable), the cos(φ) term must be positive.

That implies that the electron spins must be parallel rather than antiparallel for

ferromagnetism to occur. A more rigorous treatment of the exchange interaction can

be found in [42].

3.1.2 Magnetostatic Energy

The magnetostatic energy, also called the stray field energy, is the energy of the

magnetic field produced by the particle itself.

26 3 Simulations of Magnetism on the Nanoscale

3.1.3 Crystalline Anisotropy Energy

Crystalline anisotropy energy arises from the orientation of individual spins relative

to the crystal lattice, and to neighboring spins. The origin of anisotropy energy is

spin-orbit coupling. Crystalline anisotropy energy was neglected because all samples

in this study were made of permalloy, which is polycrystalline.

3.1.4 Zeeman Energy

Zeeman energy arises from applying an external magnetic field on a magnetic sample.

The Zeeman energy density is

EH = −MHext(r) ·m(r) (3.4)

where M is the sample magnetization, Hext(r) is the external field applied at point r

and m(r) is the normalized magnetization of the element of volume containing r.

3.2 Magnetic Domains

Magnetic domains in ferromagnetic materials are a result of the minimization of the

energy in Eq. 3.2. As seen in Section 3.3, the exchange energy prefers to have spins

aligned parallel. However, the magnetostatic energy is minimized by orienting spins

in an antiparallel configuration. The compromise is the formation of domains. The

spins rotate from one direction to another to minimize the magnetostatic energy, but

they do so gradually, to keep the energy cost (from the increase in exchange energy),

as small as possible. This region of gradual transition is known as a domain wall.

The width of a domain wall in the absence of an external field is given by [43]

w0 = (2JS2π2/K1a)12 (3.5)

where J is the exchange constant between adjacent spins, K1 is the anisotropy

constant and a is the unit cell dimension of the sample. Domain wall widths are

usually of the order of hundreds of nanometers [4, 44], but this width is dependent on

3.3 Nanomagnets 27

Figure 3.1: Illustration showing a Bloch wall (left) and Neel wall (right). In a Bloch wall themagnetization rotates out of plane, and in a Neel wall the magnetization rotates in-plane. From [4].

the anisotropy energy of the sample. The two most common types of domain walls

are Bloch walls and Neel walls, which are illustrated in Figure 3.1. In a Bloch wall

the magnetization rotates out of the plane occupied by the initial and final domain

magnetizations. Conversely, the magnetization in a Neel wall in the plane occupied

by the initial and final domain magnetizations. Neel walls are found only in very thin

films, where the thickness of the film is much less than the domain wall width [4],

and occasionally in samples that are in applied fields [29].

3.3 Nanomagnets

Brown’s fundamental theorem [45] states that if a ferromagnetic particle has dimen-

sions of the order of domain wall width, the energy cost of incorporating a domain

wall will be too large, and the particle will adopt a single domain configuration. This

condition is highly dependent on particle shape and composition, but this size is gen-

erally between several tens to a few hundred nanometers. In this arrangement all

dipoles in the particle are pointed in the same direction, effectively creating a single

spin comprised of many thousands of atoms. These particles are binary: the atomic

spins can either align all in the “up” configuration or all in the “down” configuration,

and can therefore be used to store one bit (binary digit) of information.

Particles need to be characterized to determine what particle dimensions will yield

single domain particles and at what applied fields the single domain particles will

switch configurations. Micromagnetic simulations were used to predict sample be-


havior.

Cowburn’s system relied primarily on circular ferromagnetic disks. Rings are an-

other possible particle shape [46, 47] because the vortex state is eliminated, which

narrows the switching distribution. This is because, once it has been formed, the en-

ergy cost of moving a vortex is very small, so the vortex tends to seek out the “best”

(most energetically favorable) path out of the particle. This searching out process

causes a broadening in the switching field distribution. Furthermore, rings also have

potential for use in magnetic random access memory (MRAM) drives [48, 49, 50], so

research on rings is ongoing. Ellipses also tend not to adopt vortex states, provided

that their aspect ratio is sufficiently high [51, 52, 53].

Koltsov [54] examined how indentations influenced magnetic structure of permalloy

squares. While squares have been the subject of some theoretical work [55], squares

are not ideal candidates for a magnetostatic based system because there are large

stray fields at corners. This stray field could be large enough to unintentionally

influence the magnetic state of neighboring particles.

While squares are not ideal for the current study, characterizing the magnetic be-

havior of particles based on shape anisotropies is important if more elaborate MQCA

architectures are to be fabricated. The effect of constricting and stretching square

particles has been studied by Koltsov, and the characteristics of stretched circular

particles (ellipses) are known [5]. The characteristics of compressed elliptical parti-

cles have not been studied. The shape of interest is illustrated in Figure 3.2, which

resembles a “peanut”. These particles were simulated to find if their equilibrium state

was single domain (and, by corollary, their suitability for use in an MQCA system)

and what their simulated coercivity was. These results were then tested by fabri-

cating the particles and conducting MFM measurements with an in-situ magnetic

field. Ideally, the calculated coercivity would be the center of the experimentally

constructed switching field diagram and the distribution would be narrow.

The particles have three characteristic dimensions: the length of the long axis of

3.4 OOMMF 29

Figure 3.2: The general particle shape of interest, whose outline resembles a peanut. Its characteristiclengths are labelled as a, b and c. For this particle a = 750 nm, b = 300 nm and c = 220 nm.

the particle, a, the width of the widest portion of the short axis of the particle, b,

and the width of the “waist” of the particle, c. Particle lengths a = 750 nm and

a = 250 nm were chosen as set parameters. The other two parameters were varied in

the simulations, but their ratio, bc

, was kept constant.

Throughout the rest of this thesis, particles will be referred to by the dimension

of their longest axis.

3.4 OOMMF

All micromagnetic simulations were performed using the Object Oriented Micromag-

netic Framework (OOMMF) code from the National Institute of Standards and Tech-

nology [56]. OOMMF is portable, extensible and is in the public domain. The Tcl/Tk

scripting language is required to execute OOMMF.

OOMMF runs simulations according to a hierarchy, as illustrated in Figure 3.3.

The simulation as a whole is run in large increments called stages. Individual stages

are run in small increments called iterations or steps.

• Iterations are run by evolvers, which update the magnetic state of the sample

from one step to the next. The evolvers, in turn, are controlled by drivers.


Figure 3.3: A schematic flowchart of the OOMMF program. The path outlined in red is the functionperformed by iterations, and the path outlined in green is the function performed by stages.

3.4 OOMMF 31

• Stages are run by drivers, which coordinate how the simulation evolves as a

whole. New conditions (the magnitude of an applied magnetic field, for ex-

ample) are introduced at the beginning of stages. The evolvers then update

magnetization of the sample. The length of a stage is controlled by the user,

who defines stopping criteria in the driver. Once the stopping criteria have been

met, a new stage begins. Depending on the stopping criteria, a stage can have

as few as one iteration to as many as hundreds of thousands of iterations, and

more. Therefore, prudent choices for stopping criteria are critical.

For a certain driver to be used its corresponding evolver must be used in conjunction.

There are two types of driver-evolver pairs. The first type is the minimization driver-

evolvers. These locate local energy minima through direct minimization techniques.

See [57] for more information about the algorithms. For this study the minimization

approach was inadequate since the global minimum was sought rather than local min-

ima. That is, the simulated particle often evolved into metastable states or unstable

equilibria.

The second pair is the time driver-evolver. These track the time evolution of the

magnetization according to the Landau-Lifshitz-Gilbert (LLG) equation:

dM

dt= −γM×Heff − γα

Ms

M× (M×Heff) (3.6)

where M is the magnetization, Heff is the effective field, γ is the Landau-Lifshitz

gyromagnetic ratio, and α is the damping constant.

The LLG equation describes the general behavior of individual dipoles in an ap-

plied magnetic field. See Figure 3.4. Use of the right hand rule shows the first term

in the LLG equation describes the precession of the magnetization vector around the

applied field vector. The second term tends to align the magnetization vector with

the applied field. This is clear from Figure 3.4 when the right hand rule is used twice

and the negative sign in front of the term included.

The damping coefficient α determines how quickly the magnetization vector aligns

itself with the applied field. It differs between materials and must be user defined.


Figure 3.4: A graphic representation of the Landau-Lifshitz-Gilbert equation. The magnetizationvector M precesses around the effective applied field, H, and it also tends to align itself with H.The damping coefficient, α, determines how quickly the magnetization lines up with the effectiveapplied field.

Quoted values for bulk permalloy range between 0.005 to 0.01 [58, 59]. When the

sample thickness become small this value becomes dependent on the thickness; as

sample thicknesses decrease the damping parameter tends to increase [60].

The default value for α set by OOMMF is 0.5, which is much larger than real values.

A large damping coefficient decreases computation times since individual dipoles align

themselves with the applied field quickly. In addition, since our patterned particles

were simulated to be only 20 nm thick the damping coefficient should be at least

nominally larger than the bulk value. Thus, simulations were carried out with α = 0.5

and α = 0.25. Simulations with a damping value of 0.10 were also run, though below

α = 0.10 computation times became prohibitively long. The tendencies of the particles

in each case were similar.

3.5 Particle Initialization

OOMMF allows for real world objects to be inserted into simulations. For this, a

scanning electron microscope (SEM) image was taken and saved in .gif format, as in

Figure 3.5 a). The image was then imported to any image processing program and

3.5 Particle Initialization 33

a) b) c)

Figure 3.5: The transition from an SEM image of a particle in the real-world, a), to a black andwhite image, b), that OOMMF can then initialize, c). The arrows and their colors in c) representthe direction of the magnetization orientations of the cells. The direction of the arrows show thedirection of the magnetization in the plane of the page and their color represent the direction of themagnetization out of the plane of the page. Red represents out of the page and blue represents intothe page. Particle is 750 nm long with minimum aspect ratio of 2.63.

the contrast adjusted to give a purely black and white image, as in Figure 3.5 b). This

black and white image was then imported as a .gif file into the OOMMF environment,

as in Figure 3.5 c).

Particles were configured so that their long axes were along the y axis. It is impor-

tant to note that a given set of conditions always yields the same final configuration

and that final configuration is always reached via the same path. There are no ran-

dom variations due to the thermal agitation of spins, for example. However, the Curie

point of permalloy is about 600 degrees Celsius, well above the ambient temperature

that experiments were conducted. Thus, thermal effects are expected to be minimal.

To simulate a three dimensional object OOMMF discretizes the object into rect-

angular blocks. The dimensions of the blocks are user defined. In this thesis all cells

were chosen to be cubes, and cell sizes throughout this thesis will be referred to by

one side of this cube. The size of a cell should be chosen such that each edge of a


cell is less than the exchange length of the simulated material. The exchange length

is given by

lex =(

Aex

4πMs

) 12

(3.7)

where Aex is the exchange constant for the given material and Ms is the saturation

magnetization for the material. For ferromagnets the exchange length is approxi-

mately 10 nm [61, 62, 63]. As is standard practice in the literature, cell sizes were

chosen to be smaller than the exchange length. In this study, cube sizes ranged from

5 nm to 8 nm on a side, all of which are below the exchange length. Smaller cells

result in more cells for a sample of a set size, which increases simulation run time.

Particle widths were defined by adjusting the width of the grid on which the

particle is initialized. OOMMF stretches or compresses the .gif image automatically.

To simulate a given material, the appropriate parameters must be included in the

code. For permalloy the exchange constant is 1.3× 10−6 erg/cm and the saturation

magnetization is 860 emu/cm3.

3.6 Single Domain Simulations

Simulations were performed to determine what eventual sample dimensions should be

attempted for fabrication. The simulations were meant to determine which particles

were single domain particles and to predict what the typical coercivity of one of these

elements in an array would be.

To be classified as being a single domain particle, the virgin state of a sample is

examined. The virgin state of a particle is the state the particle is in immediately

after fabrication, before any external fields have been applied. The virgin state of

a single domain particle should not contain any domain walls or vortices. This was

done by assigning a random orientation to the magnetization of each cell and allowing

the magnetization to relax to its equilibrium state. The trials that did not result in a

single domain configuration evolved into either a single vortex state as in Figure 3.6 a),

a double vortex state as in Figure 3.6 b) or an “S” state as in Figure 3.6 c). In the

3.6 Single Domain Simulations 35

a) b) c)

Figure 3.6: After having the magnetization of each cell randomized, as in Figure 3.5 c), the particlesthen relaxed into either a a) single vortex state, b) double vortex state, or c) an “S” state. Thefrequency of these resultant states was dependent on particle length and aspect ratio. Arrows showthe in-plane direction of the magnetization. Background also shows the in-plane direction of themagnetization. Red represents areas where the magnetization lies along the ±x axis and green/bluerepresents areas where teh magnetization lies along the ±y axis.

regime where the aspect ratio of the 750 nm particles were similar to those already

studied experimentally [5] the simulated 750 nm long particles always adopted one of

the configurations in Figure 3.6, indicating that the preferred state for a particle of

that size was not the single domain state.

For the 250 nm particles, the probability of relaxing into a single domain state

was much higher. Particles 250 nm long but with different widths were initialized

with random spin orientations and allowed to relax into their equilibrium state. This

process was repeated a minimum of eight times for each particle. Figure 3.7 shows

the results of all trials for one particle. The probability of saturation in the positive

or negative configurations is the same.


Figure 3.7: Magnetization along the y-axis versus simulation time for a particle that was assignedrandom initial magnetization 10 times and allowed to relax. Each curve shows the time evolutionof the magnetization along the y direction for one trial. Five trials resulted in saturation in the+y direction, four resulted in saturation in the −y direction, and one did not saturate. Instead, itadopted one of the states in Figure 3.6. Particle was a 250 nm permalloy particle with cell size of6 nm.

3.6 Single Domain Simulations 37

Figure 3.8: The percentage of particles that relaxed into single domain states, plotted as a functionof their aspect ratio. The rate reached 90% when the aspect ratio was 2.63. Note that the particleoften adopted a metastable state before finally saturating. Calculation was performed on 250 nmpermalloy particle with cell size of 6 nm.

The number of trials resulting in single domain configurations is plotted as a

percentage of the total number of trials run in Figure 3.8. This shows the probability

that an element fabricated to those dimensions would be a single domain particle.

There is a large jump in probability beginning when the aspect ratio is about 2.4,

tapering off when the aspect ratio is approximately 2.6. The goal was that a fabricated

particle had a 90% probability to be a single domain particle. As seen in Figure 3.8

this probability is achieved when the aspect ratio of the particle is 2.63. Therefore,

a goal to fabricate particles with maximum widths of 95 nm was set for the 250 nm

long particles. The same aspect ratio was maintained for the 750 nm long particles,

giving particles with maximum widths of 285 nm.


3.7 Particle Coercivity

Once the dimensions for single domain particles were determined, a theoretical pre-

diction of the magnetic coercivity could be calculated. Particles were initialized as

being saturated in the −y direction. An external magnetic field was applied in the

+y direction. Results are seen in Figure 3.9. As seen in the top curve of Figure 3.9,

when the applied field is larger than the coercivity of the particle the magnetic state

switches abruptly, on the order of 1 ns which is comparable to previously reported

results [64]. When the applied field is insufficient to induce switching the particle can

either relax back into saturation in the −y direction or into a vortex state, as seen in

the bottom curve of Figure 3.9. The latter occurs when the field was large enough

to induce significant magnetization change, but still less than the coercive field. The

simulated coercive field was assumed to be halfway between the highest applied field

that did not induce switching and the lowest applied field that did induce switching.

Although ideal particle dimensions were found, variations in the sample due to

the fabrication process are inevitable. To ensure that particles with slightly different

aspect ratios would still yield relevant results, 250 nm particles having an aspect

ratio of 2.63 were simulated along with particles whose widths (and, therefore, aspect

ratios), were slightly larger. 750 nm particles were not simulated for this as simulation

times became prohibitively long. Several applied fields were simulated for particles

of all dimensions. Whether the particle switched magnetic states was noted, and a

particle switching phase diagram was constructed in Figure 3.10. The phase diagram

gives a theoretical prediction of the switching characteristics of the particles before

they are fabricated. The coercive field for a given aspect ratio was found within

as little as 5 Oe, giving a well defined boundary between switched and unswitched

particles.

The 750 nm particles were not simulated for a switching phase diagram in part

because simulation times became prohibitively long. More significant was that there

was already existing data from our system [5] from similarly sized elliptical particles

3.7 Particle Coercivity 39

T0 T1 T2

Figure 3.9: The y magnetization of a particle as a function of simulated time. At time T0 anapplied field was simulated in the +y direction. The field was turned off at time T1 and the reversefield of the same magnitude was turned on at time T2. The top curve shows the magnetization ofthe particle switching from saturation in the −y direction to saturation in the +y direction in anapplied field of 775 Oe. The bottom curve shows a trial where the particle did not switch to the+y direction, with the applied field only 740 Oe. The particle was a 250 nm long permalloy particlewith maximum width 102 nm and a cell size of 6 nm.


Figure 3.10: The switching phase diagram for a 250 nm long permalloy particle, with dampingcoefficient α = 0.25 and cell size 8× 8× 8 nm3. As particle widths (aspect ratios) increase, coercivefields decrease.

which could be used as a means of comparison.

For 250 nm particles with the same aspect ratio, smaller cell sizes tended to de-

crease the value of the coercive field. The deviations in coercive field values increased

as the particle aspect ratio increased. This trend was not considered to be significant

because of the relatively few data points (only (8 nm)3 and (6 nm)3 sized cells were

often used), and because the fabrication process was precise enough that only small

deviations of aspect ratio were present, so these calculated deviations would not be

relevant.

A more relevant variation arises from the damping coefficient, α. The coercivity

was calculated for 250 nm long particles with a maximum width of approximately

105 nm, which was close to the desired particle width. Separate trials were run

with particles cell sizes of 8 nm and 6 nm. For each of these cell sizes the damping

coefficient was varied between α = 0.5, α = 0.25 and α = 0.10. Table 3.1 shows

the percentage difference in coercive fields between the 8 nm cell particles and the

3.7 Particle Coercivity 41

α H6 H8 Difference DifferenceH8

(Oe) (Oe) (Oe) %

0.5 755 605 150 25

0.25 590 545 45 8.25

0.10 565 515 50 9.70

Table 3.1: Table showing the discrepancies in coercive field as a function of the damping coefficient,α, and cell size. H6 and H8 are the coercive field for cell sizes 6 × 6 × 6 nm3 and 8 × 8 × 8 nm3,respectively.

6 nm cell particles as a function of α, the damping coefficient. The discrepancy in

the coercive field as a function of cell size decreased dramatically when α was less

than 0.5.

Furthermore, the calculated coercive fields for α = 0.10 and α = 0.25 are

within approximately 5% of each other, whereas the fields for α = 0.5 differ from

the others by as much as 30%. A lower damping coefficient is more realistic and

so the convergent tendency seen at lower values of α was expected to give a valid

prediction of the coercive field of the particles. If the coercivity is averaged between

the 6 nm and 8 nm trials for α = 0.10 and α = 0.25, the coercivity is predicted to

be 550 ± 30 Oe.

In conclusion, real-world permalloy particles were simulated using publicly avail-

able micromagnetic code which tracked the time evolution of the magnetization using

the Landau-Lifshitz-Gilbert (LLG) equation. Results of the simulations were found

to have a slight dependence on the cell size used, but a much stronger dependance on

the value of the damping coefficient in the LLG equation. Particles with a length of

250 nm and aspect ratio of 2.63 were determined to be good candidates to be single

domain particles and were expected to have a coercive field of 550 ± 30 Oe.

4

Experimental Results

For the 250 nm particles, once the simulations had been run and the switching phase

diagram was constructed, particle arrays were fabricated to test the theoretical cal-

culations. For the 750 nm particles, particle arrays were fabricated to compare the

switching characteristics to similarly sized elliptical particles studied previously with

this microscope [5].

4.1 Electron Beam Lithography

Samples were fabricated by electron beam lithography (EBL). Sample fabrication

was conducted at Universite de Sherbrooke in Sherbrooke, Quebec, Canada with the

assistance of Dr. Jean Beerens and under the supervision of Dr. Jacques Beauvais.

Electron beam lithography is a technique that allows for sample fabrication at the

submicron scale. It involves scanning a beam of elctrons over a sample covered with

an electron-sensitive film to create a pattern. Electron beam lithography is capable

of achieving high resolution over areas tens of micrometers on a side [65] because

small electron wavelengths can be produced. The movement of the electron beam is

controlled by a computer aided design (CAD) program, so EBL can pattern virtually

any desired shape. A major drawback of EBL is that it is not a parallel process, unlike

optical lithography techniques, which allow for the fabrication of many samples at

one time. Electron beam lithography is also slower than optical lithography, and

an EBL system is expensive, with the electron beam writer often costing millions

42

4.1 Electron Beam Lithography 43

Figure 4.1: A cross-section view of the fabricated sample before lift-off was performed. The lowermolecular mass of the bottom layer makes it more reactive, creating an undercut. When a material,such as permalloy, is then deposited (black) it is not in physical contact with either layer so it willnot be affected by lift-off.

of dollars [66].

4.1.1 Fabrication

The sample substrate was silicon, and polymethyl methacrylate (PMMA) served as

the electron resist. PMMA is a positive resist; when irradiated by electrons, chemical

bonds in the polymer structure are broken, leaving fragments of lower molecular mass.

These fragments, if they are below certain mass, are selectively washed away with

an appropriate solvent developer. A bi-layer resist structure was adopted, with two

layers of PMMA being spin coated onto the substrate: a slightly more sensitive low

molecular mass (approximately 500 000 amu) layer was coated first, followed by a high

molecular mass (approximately 950 000 amu) layer. This was followed by electron

irradiation. The result is a slight undercut, as in Figure 4.1, since the lower mass

resist is the more reactive of the layers.

The pattern was then transferred to the substrate via a liftoff process. This con-

sisted of depositing the desired material (in our case, depositing permalloy via sput-

tering) onto the substrate and patterned resist. Because the lower layer of resist was

slightly undercut when material was deposited, it took on the dimensions of the upper

layer of the resist. Note that the deposited material is, ideally, not in physical contact

44 4 Experimental Results

Figure 4.2: The result of a preliminary attempt at fabricating the particles of interest. Particle isapproximately 750 nm long and approximately 500 nm across at its widest point.

with either layer of resist. The resist was then washed away with acetone, leaving

the patterned arrays on the substrate. The bi-layer resist configuration ensured that

the particles were not damaged during the liftoff process because they were not in

mechanical contact with the resist that was being dissolved.

For the scope of this study it was desirable to have as many particles as possible in

a given area. This had to be balanced against the requirement that particles not be

magnetostatically coupled, i.e., close enough that the stray field of a particle would

not influence its neighbors. In elliptical particles the stray field is found primarily at

the ends of the long axis, so the distance between the particles along their long axis was

at least twice the separation along the short axis. If particles are magnetostatically

coupled, they will tend to switch with adjacent particles.

4.2 Characteristics of 750 nm Particles

Arrays of “peanut”-shaped particles with aspect ratio of about 2.6 were fabricated via

EBL. Experiments were performed in a vacuum of 2.0× 10−5 mbar with an in-plane,

in-situ magnetic field and with a commercial silicon cantilever with typical spring

constant of 1 N/m, resonance frequency of 74.9 kHz and Q-factor of approximately

40 000 coated with 20 nm of Co71Pt12Cr17. A phase locked loop (PLL) was used to

4.2 Characteristics of 750 nm Particles 45

Figure 4.3: The array of 750 nm particles after being saturated in a field of -375 Oe. The single“dark-light” contrast for each particle shows they are in a single domain state.

detect cantilever frequency shifts. Maximum frequency shifts were typically about

0.6 Hz. The pitch (a2→ a

2separation between particles) on the long axis was approxi-

mately 1.5 µm, and the pitch on the short axis ( c2→ c

2spacing between particles) was

approximately 0.75 µm. Images were taken in constant height mode (see Section 2.4),

with typical lift heights of 110 nm, and a voltage of 0.2 V was applied between the

tip and sample to minimize electrostatic interactions.

4.2.1 Imaging in Magnetic Field

The corner of the array was located and imaged to ensure that the same section of

the array was present in each image. This was needed to fight the effects of piezo

creep and thermal drift, the latter being enhanced due to the heat generated by the

electromagnets, which does not dissipate well in vacuum. Since the particles of the

array were designed to be magnetostatically uncoupled, particles on the edge of the

array should not have had different switching characteristics than particles in the

middle of the array. Relatively few particles could be imaged at once because of their

large size, which requires a large scan size. This is also due to the challenge of finding

a large area free of dust or parasitic EBL liftoff remnants.

A magnetic field of -375 Oe was applied along the long axis of the particles to the

sample to saturate the array as in Figure 4.3. The presence of a single “dark-light”


Figure 4.4: Images taken while obtaining the ensemble hysteresis loop for the array of 750 nmparticles. The image on the left (imaged at remenance after a field of -33 Oe was applied) shows thearray with some particles appearing to be “missing”. These particles appeared to vanish becausethey had adopted flux closure states. The middle figure (imaged at remenance after a field of- 65 Oe) shows most particles in the array had entered the flux closure state, though some havealready switched to the opposite single domain state, and some were still in their original states.The image on the right (imaged at remenance after a field of - 87 Oe) shows the array when almostall particles had switched to the opposite single domain state. The particles in the image on theright are distorted because of a damaged tip (See Figure 2.3).

pairing in each particle indicates that there are no vortices or domains in the particle

sand they are in a single domain configuration.

Magnetic fields were then applied in the opposite direction, with increasing mag-

nitude. After each application of a field, the sample was imaged at remenance (zero

applied field). As seen in Figure 4.4, particles did not generally switch directly from

one single domain state to the other. Rather, the particles tended to adopt a two-

stage switching process, going from a single domain to a flux closure state and then

to the other single domain state. Switching occurs through vortex formation (to the

flux closure state) and vortex expelling (to the opposite single domain state).

The effort to ensure that the particles would be magnetostatically uncoupled was

successful. If the particles were coupled together, they would tend to switch in chains

or small groups; that is, they would have displayed the basic idea of the magnetic

quantum cellular automata system by influencing their neighbors’ states. Instead,

particles switched individually, and there was no evidence that particles switched

with their neighbors with any undue frequency.


Figure 4.5: Typical hysteresis loop for a particle with a two stage switching process. The particledemagnetizes before switching to either saturated state. Adapted from [5].

Figure 4.6: The array of 750 nm particles after being saturated in a field of -375 Oe. The single“dark-light” contrast for each particle shows they are in the opposite single domain state than thatin Figure 4.3. The scanned area is the same as that in Figure 4.3 and Figure 4.4, but the scan areawas rotated.

A hysteresis curve for a particle with a two-step switching process has the general

appearance of Figure 4.5. As the name suggests, particles in flux closure states

minimize their energy by reducing or eliminating their stray fields. Since MFM detects

the gradients of stray fields, particles in flux closure states provide little contrast; they

appear to vanish.

Once the array had been saturated in the opposite single domain configuration, as

in Figure 4.6, the process was reversed. Each particle displays hysteresis individually,

and the array as a whole shows an ensemble hysteresis.


Figure 4.7: A 750 nm particle in a single domain state. For the purposes of counting the magneti-zation of the array, this particle has a value of +1.

4.2.2 Ensemble Hysteresis Loop

Because MFM does not directly measure the magnetization of a sample (see Sec-

tion 2.4), a system of arbitrary units was required to quantify the ensemble magne-

tization. Single domain particles with a single “light-dark” contrast as in Figure 4.7

were assigned a value of +1. Particles with the opposite (“dark-light”) contrast were

assigned a value of -1. Particles in any other state were assigned a value of 0. After

each image was taken the “magnetization” of the ensemble could be simply counted

off. This quantity was normalized by dividing by the maximum possible magneti-

zation of the sample, equal to the number of particles in the scan area. This was

performed after each application of an external magnetic field. Figure 4.8 shows the

resulting ensemble hysteresis loop for the 750 nm particles.

4.2.3 Switching Field Distribution

By tracking the magnetic states of the individual particles, their switching fields could

be monitored as well. For the 750 nm particles, switching was generally a two step

process so there were two relevant applied fields; Hd, the field at which particles

switched from the single domain state to the demagnetized closure state, and Hs,

the field at which particles completed switching by evolving from the closure state to

the opposite single domain state. Therefore, separate tallies had to be kept for Hd

and Hs, giving rise to two distinct particle switching distributions. At each different


Figure 4.8: The ensemble hysteresis loop for the array of 750 nm particles.

magnitude of applied field the resulting image was examined and the total number of

particles that were in the closure state was noted, as was the total number of particles

that were in the other “switched” single domain configuration. These were plotted

against the applied field and Gaussian curves fit to each distribution in Figure 4.9.

Gaussian curves were used to fit the data because each of the particles was designed

to be magnetostatically uncoupled (independent) from its neighbors. Thus, according

to the Central Limit Theorem their coercivities should be distributed in a Gaussian

distribution.

Figure 4.9 shows the average field for a particle to switch to the flux closure

state was 60 Oe. The full width at half maximum (FWHM) of the distribution was

16 Oe, or 27% of the average switching field. The average field for a particle to

switch from the flux closure state was 130 Oe. The FWHM of the distribution was

56 Oe, or 43% of the average switching field. The transition from the flux closure

state is obviously much broader than the transition to the flux closure state, and

the Gaussian fit for this transition (blue) is of poorer quality, with R2 = 0.683,

compared to R2 = 0.812 for the virgin-to-closure state distribution (red). This is


Figure 4.9: The switching distribution for the array of 750 nm particles, with Gaussian fits. Thetwo distributions indicate the two-step switching process of the particles. The cartoons at the topshow the most common state of particles in the array in that range of applied field. Below Hd theyare in a single domain state (“up”), as on the left. Between Hd and Hs they tend to be in a closurestate, as in the middle picture, and above Hs the particles tend to be in the switched single domainstate (“down”), as on the right. The peaks are located at Hd = 60 Oe and Hs = 130 Oe. TheFWHM of the distributions are 16 Oe and 56 Oe, respectively. For the red curve R2 = 0.812 andfor the blue curve R2 = 0.683.


because the vortex a flux closure state is very mobile and can easily be trapped,

leading to a broad distribution, reflecting a distribution of these traps in the particle.

The two distributions of Figure 4.9 also have a slight overlap. This means that while

some particles had yet to enter the flux closure state, others had already adopted the

opposite single domain state. This was seen in the middle panel of Figure 4.4. These

characteristics have been noted previously in elliptical particles of similar size [5].

4.2.4 Comparison with Previous Study

The particles studied in [5] were ellipses with an aspect ratio of 4 : 1, with a width

of 200 nm. The peaks of this distribution were at approximately Hd = 65 Oe, and

Hs = 110 Oe, giving a separation of 45 Oe between peaks. The width of the second

peak is 65 Oe, or 59% of the field required to fully switch from the flux closure state

to the opposite single domain state.

Our particles had an aspect ratio of approximately 2.6 : 1 at their widest point and

3.6 : 1 at their narrowest point (see Figure 4.2). The peaks, as shown in Figure 4.9

were at Hd = 60 ± 16 Oe and Hs = 130 ± 56 Oe, so the separation between the

peaks was 70 Oe. The width of the first peak was comparable to that of the previous

study. Our second peak was 10 Oe narrower than that of previous study.

4.2.5 Conclusion

Thus, the particles in this study had switching distributions which were spaced further

apart and were marginally narrower than the previous study, despite having a lower

aspect ratio. This implies that the particles of current interest switch with slightly

better characteristics than those of previous study. The current data represents only

one trial and figures could be quoted with more confidence if more trials examining

more particles were conducted. The aspect ratios of 4 : 1 (previous study) and 3.6 : 1

(the largest aspect ratio of the particles in this study) are comparable, and it may

also be fruitful to investigate if the smallest dimension of a particle determines its

magnetic switching characteristics.


Based on the experimental data in can be concluded that particles with a length of

750 nm are poor candidates for a magnetic quantum cellular automata system. This

is because of their two step switching process. Since the particle can adopt a flux

closure state as part of its switching process, it is not binary and therefore cannot

contain a bit of data. Smaller particles would be less prone to adopting a closure state

during switching, and so should be better candidates for an MQCA system. However,

we have also seen that engineering “defects,” such as the narrow waist in this study,

can improve the switching field distribution by helping to nucleate reversal.

4.3 Characteristics of 250 nm Particles

Arrays of “peanut”-shaped particles with aspect ratio of about 2.6 were fabricated via

EBL. Experiments were performed in a vacuum of 2.5× 10−5 mbar with an in-plane,

in-situ magnetic field and with a commercial silicon cantilever with typical spring

constant of 1 N/m, resonance frequency of 70.0 kHz and Q-factor of approximately

45 000 coated with 20 nm of Co71Pt12Cr17. A phase locked loop (PLL) was used

to detect cantilever frequency shifts. Maximum frequency shifts were typically only

0.3 Hz. Thus, these measurements are approaching the limit of sensitivity for MFM.

The pitch along the long axis was approximately 500 nm, and the pitch on the short

axis was approximately 250 nm. Images were taken in constant height mode (see

Section 2.4) with typical lift heights of 100 nm, and a voltage of -0.1 V was applied

between the tip and sample to minimize electrostatic interactions between tip and

sample. An SEM image of a section of the patterned array is shown in Figure 4.10.

Because of the small length scales involved in the fabrication of this sample, exact

particle dimensions cannot be produced. While they were designed to be 250 nm

in length, the particles in the array were found to be approximately 270 nm long

after fabrication. The extra 20 nm is less than 10% of the particle length and is not

expected to alter the switching characteristics of the particles significantly.

The predictions arising from the simulations in Section 3.6 were tested. The pre-


Figure 4.10: SEM image of a section of the 250 nm particle array. Imaged area is 3 µ m× 3 µ m.

diction was that at least 90% of particles with these dimensions would be in a single

domain state. As seen in Figure 4.11, all particles in view were in single domain states,

as indicated by their single dark-light contrast. Note that either single domain state

can be achieved.

4.3.1 Imaging in Magnetic Field

The corner of the array was located and imaged so that the same section of the ar-

ray was present in every image, ensuring that the data in each scan was consistently

taken from the same particles. Since the particles of the array were designed to be

magnetostatically uncoupled, particles on the edge of the array should not have had

different switching characteristics than particles in the middle of the array. Signifi-

cantly more particles could be imaged for the 250 nm particles than could be imaged

for the 750 nm particles, allowing for a more statistically significant sampling.

The resolution of the images is significantly less than is generally possible for

MFM. This is a result of multiple factors. One is that a large area was scanned,

in order to image as many particles in each scan as possible to improve statistics.

One way to counterbalance this would be to increase the number of points in each

image. However, the memory required to store such images is large enough that it

is not practical for a study that requires dozens of images, like the current one. The


Figure 4.11: MFM image of array of 250 nm particles before any magnetic fields were applied.Particles are in single domain configurations, confirming the prediction made in Section 3.6. Imagedarea is 4 µ m× 4 µ m.

time required to capture such images also increases substantially. Aside from the in-

convenience of long data acquisition times, the long scan times increase the possible

interference from effects like piezo creep and thermal drift. Another possible cause for

decreased resolution is a damaged tip. When scanning large areas, it becomes more

probable that parasitic debris would be found on a sample. Also, the issue of sample

tilt comes into play, along with the issues described in Section 2.4. Ultimately, the

decreased resolution of the following measurements is inconsequential. The measure-

ments needed only to supply information about the magnetic state of the particles

and the resolution is sufficient for definite statements to be made with confidence.

A magnetic field of -580 Oe was applied along the long axis of the particles to

the sample to saturate the array as in Figure 4.12. The presence of a single “dark-

light” pairing in each particle indicates that there are no vortices or domains in the

particle and it is in a single domain configuration. Magnetic fields of increasing

magnitude were then applied in the opposite direction. After each application of

a field, the sample was image again at remenance. Unlike the 750 nm particles

studied in Section 4.2, the 250 nm particles were never observed in a flux closure


Figure 4.12: The array of 250 nm particles after being saturated in a field of -580 Oe. Imaged areais 4 µ m× 4 µ m.

state; switching occurred directly from one single domain state to the other. This is

because these smaller particles do not have enough electrostatic energy to necessitate

the formation of a domain, indicating that the dominant energy term in the particles is

the exchange energy. A typical image taken during the construction of the hysteresis

loop is Figure 4.13.

As in the case for the 750 nm particles, the effort to ensure that the 250 nm

particles would be magnetostatically uncoupled was successful. They did not tend to

switch based on the state of adjacent particles.

4.3.2 Ensemble Hysteresis Loop

The system of arbitrary units outlined in Section 4.2.2 for quantifying the array’s

magnetization was again employed. Figure 4.14 shows the resulting ensemble hys-

teresis loop for the 250 nm particles.

4.3.3 Switching Field Distribution

While the 750 nm particles have a two step switching process, the 250 nm particles

switch directly from one single domain state to the other, with no intermediate config-


Figure 4.13: Typical image in the construction of the ensemble hysteresis loop for the 250 nmparticles. The number of particles in each configuration is counted to calculate the magnetizationof the ensemble. Imaged area is 4 µ m× 4 µ m.

uration. Therefore, there is only one relevant field, Hs, the switching field. It follows

that there is only one switching distribution associated with the 250 nm particles.

Figure 4.15 shows the switching distribution of the sample going from saturation in

the negative direction to saturation in the positive direction. The switching field

distribution in the other direction is similar. The peak is seen to be at approxi-

mately 490 Oe, and therefore the ensemble coercivity is 490 Oe. The FWHM of the

distribution is approximately 40 Oe, or 8% of the array’s switching field.

While it was hoped that certain particles would be noted to switch far from the

peak of the switching distribution, specific particles were not found to consistently

have this distinction. That is, the first few particles to switch states during the

construction of a hysteresis loop were not the same from trial to trial. This seems

to indicate that there is an intrinsic distribution to each particle. The investigation

of this intrinsic distribution should be the subject of further study if MQCA systems

are to be realized.


Figure 4.14: The hysteresis loop for the ensemble of 250 nm particles. When the shape of this loopis compared with that of the hysteresis loop for the 750 nm particles (Figure 4.8) it is clear thatthis loop is wider and the transition sharper. The abruptness of the transition is indicative of boththe one-step switching process of the 250 nm particles and that the particles tend to switch moreuniformly than the 750 nm particles.


Figure 4.15: The switching field distribution of the 250 nm particles, with Gaussian fit. The peakof the Gaussian curve is at 490 Oe, the FWHM is 40 Oe and R2 = 0.726


4.3.4 Comparison with Simulations

Section 3.6 predicted that 250 nm long particles would be in single domain states even

before an external field was applied. This was confirmed in Section 4.3, specifically

in Figure 4.11.

In Section 3.7 the coercivity of an individual particle was calculated. This calcu-

lated value can be compared with the average coercivity of an array of these particles,

where the average coercivity of the array was found experimentally.

As seen in Figure 4.10, the largest particle width is approximately 115 nm, and

the ensemble coercivity is 490 ± 40 Oe. This figure coincides with the prediction in

Figure 3.10 for particles of this width, shown graphically in Figure 4.16. The green

pentagon shows the experimentally determined point on the phase diagram. While

the current study only yielded one experimentally determined point on the switching

phase diagram, the point indicates that the results of the simulations and experiments

agree within error.

4.3.5 Conclusion

An array of 250 nm long particles was patterned with EBL. Its magnetic switching

characteristics were then examined using a magnetic force microscope in vacuum.

The virgin state of the particles was predicted by simulation to be a single domain

configuration, which was confirmed by experiment.

The switching distribution of the array was constructed experimentally and com-

pared to simulated results. The peak of the experimentally determined switching dis-

tribution was 490 ± 40 Oe, which coincided with the calculated value of 550 ± 30 Oe

quoted in Section 3.7. This resulted in one experimentally confirmed point on the

switching phase diagram, Figure 4.16. The width of the peak was 8% of the switching

field of the array. For these particles there was only one switching distribution peak

because particles underwent a one step switching process, unlike the larger 750 nm

particles.


Figure 4.16: The switching phase diagram of Figure 3.10 is reproduced here with the experimentallydetermined point for the ensemble added as the green pentagon. Experimental results and calculatedvalues agree within error.


Therefore, particles with a length of 250 nm are possible candidates for an MQCA

system. They are intrinsically single domain particles, have a one step switching

process and the width of their switching field distribution is relatively narrow. This

narrow distribution implies that most particles switch at approximately the same

applied field and so switching behaviors across a system should be uniform. In the

context of an MQCA system, this largely uniform behavior decreases the need for

error correction algorithms in an MQCA architecture.

One possible drawback to the 250 nm particles is their large switching field of

approximately 500 Oe. For use in an MQCA system, these particles must have stray

fields in excess of this value. Calculations to determine the stray field of these particles

as a function of position should be carried out in order to determine whether particles

of this size possess stray fields large enough to induce switching in it neighbors. The

stray field of a particle scales with its magnetization, which in turn scales with its

volume. Thus, the desired particle would have a low coercivity, be small enough that

it would not adopt a closure state during switching, but large enough that it would

have a stray field larger than its coercivity and, finally, an ensemble of these particles

would have a narrow switching distribution.

5

Conclusions & Outlook

5.1 Conclusions

Arrays of “peanut”-shaped permalloy particles were designed, simulated, fabricated,

imaged and their magnetic switching distributions were found. Their shape was

chosen to investigate how compression on the short axis of ellipses affected their

switching distribution. Particles were of two different lengths, 750 nm and 250 nm,

but had the same aspect ratio. The particles were then simulated using the Object

Oriented Micromagnetic Framework simulator. The simulations determined which

particles would have single domain virgin states, and the coercivities of such particles.

These particles were then fabricated by electron beam lithography and imaged using

a magnetic force microscope in vacuum in constant height mode.

The simulations predicted that 250 nm particles with an aspect ratio of about

2.6 would have the single domain state as their virgin state, which was confirmed by

experiment. Simulations also gave the coercivity of a particle with these dimensions

as 550 ± 30 Oe. The ensemble coercivity of an array of such particles was found to

be 490 ± 40 Oe. Thus, there is agreement between theory and experiment, within

experimental and computational error. We conclude that OOMMF can be used to

theoretically investigate the switching field, Hs, of particles that do not have the

standard elliptical, ring, disk or rectangular shapes.

The 250 nm particles are considered good candidates for a magnetic quantum cel-

lular automata (MQCA) system because they switch directly from one single domain

62

5.2 Outlook 63

state to the other and their switching distribution was less than 10% of their switching

field.

The 750 nm particles had a two-step switching process, going to a flux closure state

before adopting the opposite single domain state. This gave rise to two separate

switching distributions, one for each step. This was seen in a previous study in

elliptical particles of similar size. The distribution for the transition to the closure

state had its peak at 60 ± 16 Oe and the distribution for the transition from the

closure state to the fully switched state had its peak at 130 ± 56 Oe. The switching

distributions for our particles were separated further apart and were narrower than

those of previous study. Because of their two-step switching process 750 nm particles

were determined to be poor candidates for an MQCA system.

Finally, we have established procedures to fabricate the samples with required

dimensions within the NanoQuebec network.

5.2 Outlook

Samples with arrays of “peanut”-shaped particles had arrays of ellipses of identical

length and aspect ratio patterned on them, as well. Comparing the switching dis-

tribution of these arrays would give a more controlled comparison between particles

with and without the indentation on their side, since they were fabricated under the

same conditions. Also on these samples were chains of the “peanut”-shaped particles.

Experiments can be conducted to try and induce MQCA-type switching behavior in

them by using the stray field of the tip to trigger the first flip.

For nearly 10 years our room temperature magnetic force microscope has been

collecting data and giving insight into magnetic phenomena at the nanometer length

scale. While the success of the current configuration cannot be questioned, improve-

ments can be made.

First, the microscope head should be redesigned. The current setup using a sta-

tionary fiber and the cantilever mounted on a bimorph is inefficient. To replace a

64 5 Conclusions & Outlook

broken or damaged cantilever silver paint is used as a conductive adhesive. Unfortu-

nately, the silver paint requires several hours (usually overnight) to dry completely,

otherwise the cantilever will drift as it dries. Another potential problem with this

technique occurs if slightly too much silver paint is used to fix the cantilever to the

bimorph. The silver paint can then run over the side of the bimorph, shorting it.

Using a bimorph also requires an extra electronic circuit to be maintained.

A more practical setup would be one similar to that employed by the low temper-

ature atomic force microscope in our group. The cantilever is held in place mechan-

ically, allowing for quick removal and replacement. The optical fibre is then moved

toward the cantilever using a walking mechanism designed by Roseman [67]. Recently,

a second mechanism was suggested to allow full three dimensional manipulation of

the fiber relative to the cantilever. This would also be an advantage over the current

system of set screws that are used for fine positioning adjustments, which are tedious.

A redesigned microscope head should also have an integrated light source to assist in

positioning a sample below a tip.

The other major redesign project needed for the system is the upgrading of the

data acquisition hardware software. In the last two years, the scanning software has

become much more prone to crashing without cause. Also, the PC that runs the

acquisition software (a 486) has started to show its age, occasionally not starting

when turned on.

Replacing or simply upgrading the computer hardware requires a financial invest-

ment, but is a simple task. However, replacing the software will require a redesign

of the feedback system, which was designed specifically for this software. A similar

task was successfully undertaken by the low temperature group in our lab recently,

showing that such a renovation is possible.

And, of course, since the MFM is a custom built system, as opposed to a commer-

cial system, upkeep and replacements are required as well.

The magnetic force microscope here at McGill University has been an invaluable

5.2 Outlook 65

tool this past decade, and with some modifications, should continue to provide infor-

mation about magnetism at nanometer length scales.

Appendix

A.1 The Frequency Modulation Detector and PLL

The basic opperation of the frequency modulation detector described in Section 2.7.2

is described here. The following is a black box treatment, but should suffice to aid in

its operation.

First, the resonance frequency, fres, of the cantilever should be found, using either

the Standford Research Systems SR770 FFT Network Analyser or by driving the

bimorph and examining the interferometer signal on an oscilloscope. Once fres is

found, the interferometer signal should be filtered through the Krohn-Hite 3382 Filter.

The filtering window should be set such that the resonance frequency is centered in

a 10 kHz bandwidth. The filtered signal can then be sent to the detector.

Figure A.1 is a photograph of the detector, with all features numbered. Most

settings do not change from scan to scan. The following is a reference for “standard”

settings on the detector and, in some cases, a brief description of their function. This

is meant as a guide only; these settings were appropriate for the measurements in this

thesis; future experiments may require different settings.

1. The interferometer signal, usually filtered and amplified, is connected here.

2. The local oscillator signal, IF − fres, is connected here, where IF = 4.5 MHz.

3. This allows for the input signal to be attenuated, if needed. Standard setting

is x1.

4. When in the on position, the detector works in tracking mode. When in the off

position, the detector works in self oscillating mode. Standard setting is on.

66

A.1 The Frequency Modulation Detector and PLL 67

Figure A.1: The front panel of the frequency modulation detector with components numbered.

5. This LED is lit when the PLL is not locked. Thus, one wants this light to be

off.

6. Standard setting is H.

7. At the time of this writing, only the exc setting was functioning.

8. Standard setting is L

9. Standard setting is L

10. There is no standard setting for this feature. It allows the Phase setting to run

between 0◦ and 180◦.

11. This displays the frequency of the excitation signal being sent to the cantilever.

12. This has no standard setting and is variable from one set of measurements to

the next. This is discussed further below.

13. Standard setting is passive

14. Standard setting is on

15. Standard setting is H

68 A Appendix

16. This coarse gain control setting has no standard setting and is also variable

from one set of measurements to the next. When the fibre-tip alignment is very

good and the vacuum is high, the cantilever can be overdriven, and in this case

the L or M settings are probably appropriate. Conversely, when the signal is

small, H may be the best setting.

17. Standard setting is exc.

18. This shows the relative amplitude of the excitation signal being sent to the

cantilever. A reading of 327 indicates that the detector has saturated. The

value is variable from one set of measurements to the next, but the driving

amplitude should always be at a local minimum.

19. Standard setting is neg. This setting should not be changed.

20. This fine gain control setting has no standard setting and is variable from one

set of measurements to the next. It provides adjustment of the gain between

the settings of setting #16.

21. Standard setting is on. This setting should not be changed.

22. This has no standard setting, but should not be changed. This, along with #21

and #19 are meant to change the offset of the frequency shift signal, and also

how the detector reacts to increases or decreases of cantilever frequency. There

is an external circuit that is already employed to control these parameters, and

any changes should be made on that circuit.

23. This controls the amplitude of the excitation signal. Care should be taken to

ensure that the oscillation of the excitation signal (and in turn, the cantilever)

should not be too large; otherwise, there is a risk of crashing the tip into the

sample and damaging the tip.

24. This port usually remains unconnected.

A.2 Final Notes on OOMMF 69

25. Standard setting is L.

26. This signal is fed directly to the ADC for measurement of the dissipation of the

system during scanning.

27. This signal is the relative frequency shift of the cantilever during scanning.

28. This signal is the signal used to drive the cantilever. It is a square wave because

the phase shifter in this detector is digital. The extra harmonics that included

in the square wave should not effect the measurements because the resonance

peak of the cantilever is very sharp, so the possible excitations of the higher

harmonics of the cantilever will be negligible compared to the fundamental.

29. This turns the detector on and off. Standard setting is, of course, on.

30. This LED indicates whether the detector is on or off. Its standard state is on.

Under normal operation only the Coarse Gain, Fine Gain, Amplitude, Polarity and

Phase settings are adjusted. The signal on the oscilloscope should be sinusoidal. If

not, adjust the Phase. If adjusting the Phase (the full 360◦) does not give a sinusoid,

increase or decrease the coarse gain and begin again. Eventually a sinusoid should

appear. Once it has, the phase should be adjusted such that the amplitude readout

on feature #18 is minimized. Equivalently, the dissipation signal from port #26 can

be viewed on the oscilloscope as well, and the phase adjusted to minimize its DC

value. The LED of feature #4 should be off, showing that the detector is ready for

use.

A.2 Final Notes on OOMMF

• OOMMF is straightforward and users with even a rudimentary programming

background will quickly adapt to its environment, and quirks. Once installed, it

provides the new user with numerous sample programs. Because of its object-

based setup, “cannibalizing” these sample files is delightfully simple.

70 A Appendix

• OOMMF requires input files to be in .mif format. A .mif file can be created by

typing code into a text editor (Microsoft Wordpad, for example), saving it, and

then manually changing the extension of the text file.

• It is often advantageous to use more than one stage stopping criterium. For

example, for a certain simulation, having a stage end every 0.1 ns or when

none of the dipoles of any of the cells has an angular speed of more than

1 x 10−3 degrees/ns. The latter condition allows for the simulation to move

forward to the next stage if the magnetization of the sample is not changing

substantially. If appropriately chosen, having more than one stopping criterium

keeps simulation run times down. It is important that the stopping criteria are

not too strict, so that they will never be met and the simulations runs indefi-

nitely. Conversely, if they are too lax, so that the criteria will be too easily met

and the simulation would progress too quickly, not allowing the interactions

that determine real-world behavior to be properly simulated.

• When your code just won’t compile, take a break and come back in a while.

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