ABSTRACT 
 
 
 
 
Title of Dissertation: THERMAL ACTIVATION, LONG-RANGE 
ORDERING, AND TOPOLOGICAL 
FRUSTRATION OF ARTIFICIAL SPIN ICE   
  
 Jasper Altman Drisko 
Doctor of Philosophy, 2016 
  
Dissertation directed by: Professor John Cumings 
Department of Materials 
Science and Engineering 
 
 
Frustrated systems, typically characterized by competing interactions that 
cannot all be simultaneously satisfied, are ubiquitous in nature and display many rich 
phenomena and novel physics. Artificial spin ices (ASIs), arrays of lithographically 
patterned Ising-like single-domain magnetic nanostructures, are highly tunable systems 
that have proven to be a novel method for studying the effects of frustration and 
associated properties. The strength and nature of the frustrated interactions between 
individual magnets are readily tuned by design and the exact microstate of the system 
can be determined by a variety of characterization techniques. Recently, thermal 
activation of ASI systems has been demonstrated, introducing the spontaneous reversal 
of individual magnets and allowing for new explorations of novel phase transitions and 
phenomena using these systems. In this work, we introduce a new, robust material with 
favorable magnetic properties for studying thermally active ASI and use it to 
  
investigate a variety of ASI geometries. We reproduce previously reported perfect 
ground-state ordering in the square geometry and present studies of the kagome lattice 
showing the highest yet degree of ordering observed in this fully frustrated system. We 
consider theoretical predictions of long-range order in ASI and use both our 
experimental studies and kinetic Monte Carlo simulations to evaluate these predictions. 
Next, we introduce controlled topological defects into our square ASI samples and 
observe a new, extended frustration effect of the system. When we introduce a 
dislocation into the lattice, we still see large domains of ground-state order, but, in 
every sample, a domain wall containing higher energy spin arrangements originates 
from the dislocation, resolving a discontinuity in the ground-state order parameter. 
Locally, the magnets are unfrustrated, but frustration of the lattice persists due to its 
topology. We demonstrate the first direct imaging of spin configurations resulting from 
topological frustration in any system and make predictions on how dislocations could 
affect properties in numerous materials systems. 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
THERMAL ACTIVATION, LONG-RANGE ORDERING, AND 
TOPOLOGICAL FRUSTRATION OF ARTIFICIAL SPIN ICE   
 
 
 
by 
 
 
Jasper Altman Drisko 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2016 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor John Cumings, Chair 
Professor Christopher Lobb 
Professor Richard Greene  
Professor Frederick Wellstood  
Professor Ichiro Takeuchi  
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
© Copyright by 
Jasper Altman Drisko 
2016 
 
 
 
 
 
 
 
 
 
 
 
 
ii 
 
Acknowledgements 
First and foremost, I would like to thank my advisor, John Cumings, for his 
constant support, encouragement, and belief in me during my time at Maryland. He 
always had valuable insight on my research and pushed me to become the best scientist 
I could be. He was an encyclopedia of knowledge about both experimental techniques 
and theoretical ideas. He always respected me, listened to me, and supported what I 
wanted to do. I am grateful for everything I was able to learn from him.   
I would also like to thank all of the great teachers I have been fortunate enough 
to have throughout my life who inspired me and contributed to my love of learning. 
There are too many to list, but I am especially thankful for the influence that Mark 
Powell, Sue Davenport, Biff Maier, Dr. Michael Fiveash, Todd Kresser, Nick Gould, 
Dr. Bryan Partridge, Dr. D. Barnes Boffey, Dr. Alex Zettl, Dr. Paulo Bedaque, and Dr. 
Ian Appelbaum had in my education.  
I would like to thank my committee members, Dr. Rick Greene, Dr. Chris Lobb, 
Dr. Fred Wellstood, and Dr. Ichiro Takeuchi, for their support and evaluation of my 
work. I am grateful to Ichiro for access to his sputter chamber which I used to anneal 
my thin films, without which this dissertation would not have been possible. I would 
also like to thank Rick for the experience of working with him as a TA, which I greatly 
enjoyed.  
I want to thank my undergraduate research mentors, Dr. Shane Cybart and Dr. 
Bob Dynes, for introducing me to experimental research and arming me a with a wealth 
of knowledge and skills for graduate school and beyond. I want to thank Dr. Stephen 
Wu for valuable advice about navigating graduate school.  
iii 
 
I am extremely grateful to the other members of Cumings’ group with whom I 
had the pleasure of collaborating and working with. Stephen Daunheimer worked on 
the artificial spin ice project before me and gave me priceless information, training, and 
tips about the project. I am grateful to him for teaching me the ropes and answering all 
my questions when I was starting this research. I also want to thank Thomas Marsh, a 
high school student who worked closely with me for a summer and greatly contributed 
to the topological frustration project. I am grateful to Hanna Nilsson for her friendship, 
moral support, listening skills, motivation, and useful discussions over the years. I also 
want to acknowledge valuable discussions, advice, and help from all the other members 
of our group: Kai He, Merijntje Bronsgeest, Vladimir Oleshko, Kamal Baloch, Khim 
Karki, John Bavier, Norvik Voskanian, Paris Alexander, Jeremy Ticey, Zoey Warecki, 
Eric Epstein, Konnor Kim, Ben Shapiro, and Raj Bajwa. I want to thank Kale Johnson, 
Sean Fackler, and Alex Kozen for their friendship and enlightening discussions.  
I am very grateful to all of the staff at Maryland I was able to interact with and 
who taught and helped me immensely along the way. I want to thank the AIMLab staff, 
especially Dr. Phil Piccoli and Dr. Wen-An Chiou, the FABLab staff, especially Tom 
Loughran and Jon Hummel, the CNAM staff, Brian Straughn and Doug Bensen, and 
the MSE business office staff, especially Ginette Villeneuve.  
I want to thank my bandmates in The Vico Cycle, B Side Shuffle, and 
Staycation, and everyone I was fortunate enough to play music with during my time at 
Maryland. Music served as a perfect distraction and balance to the long days in the lab.  
Finally, I want to thank my mom, dad, and sister Arielle, for unconditionally 
supporting me throughout my education and my life. I love you.   
iv 
 
Table of Contents 
 
 
Acknowledgements ....................................................................................................... ii 
Table of Contents ......................................................................................................... iv 
List of Figures ............................................................................................................. vii 
Chapter 1  Introduction and background ...................................................................... 1 
1.1 Frustration and water ice ............................................................................... 1 
1.2 Magnetism..................................................................................................... 5 
1.2.1 Conventional magnetism .......................................................................... 5 
1.2.2 Frustrated magnetism .............................................................................. 20 
1.3 Spin ice........................................................................................................ 23 
1.4 Artificial spin ice......................................................................................... 31 
1.4.1 Introduction ............................................................................................. 31 
1.4.2 Square and kagome ice ........................................................................... 33 
1.4.3 Thermal activation .................................................................................. 48 
1.4.4 Other lattices ........................................................................................... 57 
Chapter 2  Fabrication, experimental, and numerical methods .................................. 62 
2.1 FePd3 as a material for studying thermally active ASI systems ................. 63 
2.1.1 Introduction ............................................................................................. 63 
2.1.2 Film growth ............................................................................................. 68 
2.1.3 Film characterization .............................................................................. 74 
2.2 Fabrication .................................................................................................. 82 
2.2.1 Electron beam lithography ...................................................................... 85 
2.2.2 Electron beam evaporation ..................................................................... 93 
v 
 
2.2.3 Liftoff ...................................................................................................... 96 
2.2.4 Ion milling ............................................................................................... 98 
2.2.5 Common problems ................................................................................ 103 
2.3 Transmission electron microscopy ........................................................... 109 
2.3.1 Fundamentals and advantages............................................................... 109 
2.3.2 Lorentz TEM ......................................................................................... 115 
2.3.3 Lorentz image processing ..................................................................... 122 
2.4 Heating experiments ................................................................................. 125 
2.4.1 Annealing protocols .............................................................................. 125 
2.4.2 Initial results.......................................................................................... 131 
2.5 Numerical methods ................................................................................... 138 
2.5.1 Introduction and general framework ..................................................... 138 
2.5.2 Square geometry Monte Carlo .............................................................. 143 
Chapter 3  Charge-ordering in artificial kagome ice ................................................ 150 
3.1 Introduction ............................................................................................... 150 
3.2 Charge-ordering and edge effects in small kagome crystals .................... 156 
3.3 Long-range ordering in large kagome crystals ......................................... 175 
3.4 Monte Carlo modeling of large kagome crystals ...................................... 190 
3.5 Conclusions and future work .................................................................... 200 
Chapter 4  Topological frustration of artificial spin ice ............................................ 204 
4.1 Introduction ............................................................................................... 204 
4.2 Results and discussion .............................................................................. 222 
4.2.1 Main results ........................................................................................... 222 
vi 
 
4.2.2 Theoretical framework and discussion ................................................. 248 
4.2.3 Additional observations ........................................................................ 261 
4.3 Kinetic Monte Carlo simulations .............................................................. 277 
4.4 Conclusions and future work .................................................................... 281 
Glossary .................................................................................................................... 283 
References ................................................................................................................. 285 
 
 
 
vii 
 
List of Figures 
Figure 1.1 Tetrahedral structure of ice ...................................................................... 4 
Figure 1.2 The Meissner effect.................................................................................... 8 
Figure 1.3 Pauli paramagnetism................................................................................. 9 
Figure 1.4 Types of magnetism ................................................................................. 11 
Figure 1.5 Susceptibility curves ................................................................................ 12 
Figure 1.6 Magnetic hysteresis curve ....................................................................... 14 
Figure 1.7 Bloch vs. Néel domain walls .................................................................... 16 
Figure 1.8 Transverse and vortex domain walls in thin film permalloy ............... 16 
Figure 1.9 Magnetic domains in thin film Ni35Fe65 ................................................. 17 
Figure 1.10 Domain walls and sharp paramagnetic transition in thin film  
Fe35Pt65  ................................................................................................... 18 
 
Figure 1.11 Shape anisotropy flux minimization in single domain magnets ........ 19 
Figure 1.12 Flux closure domains ............................................................................. 19 
Figure 1.13 Types of frustration ............................................................................... 21 
Figure 1.14 The pyrochlore lattice and the equivalence to water ice. ................... 24 
Figure 1.15 Specific heat and entropy in Dy2Ti2O7................................................. 25 
Figure 1.16 Neutron scattering on Ho2Ti2O7. .......................................................... 26 
Figure 1.17 Spin ice ground state ............................................................................. 28 
Figure 1.18 The dumbbell picture and magnetic monopoles in spin ice ............... 29 
Figure 1.19 Artificial square ice ............................................................................... 34 
Figure 1.20 Rotation demagnetization protocols .................................................... 36 
Figure 1.21 Statistics of moment configurations after a demagnetizing protocol 37 
viii 
 
Figure 1.22 Artificial kagome ice .............................................................................. 38 
Figure 1.23 Phase diagram of artificial kagome ice ................................................ 40 
Figure 1.24 Dumbbells of magnetic charge and ordered states in artificial 
kagome ice .............................................................................................. 41 
 
Figure 1.25 Magnetic monopole motion in square ASI .......................................... 42 
Figure 1.26 Direct observation of magnetic monopoles in kagome ASI ............... 42 
Figure 1.27 Disorder in square ASI ......................................................................... 44 
Figure 1.28 Disorder in kagome ASI ........................................................................ 46 
Figure 1.29 Avalanches in artificial kagome ice...................................................... 47 
Figure 1.30 Shapes of avalanches ............................................................................. 47 
Figure 1.31 Avalanche distribution in artificial kagome ice. ................................. 48 
Figure 1.32 Ground-state domains formed during growth.................................... 52 
Figure 1.33 Direct observation of relaxation in square ice .................................... 55 
Figure 1.34 Charge ordering in kagome ice ............................................................ 56 
Figure 1.35 Shakti and tetris lattices ........................................................................ 58 
Figure 2.1 Phase diagram of the Fe-Pd system ....................................................... 65 
Figure 2.2 L12 ordered phase .................................................................................... 65 
Figure 2.3 SiN membrane substrate ......................................................................... 69 
Figure 2.4 Sputtering ................................................................................................. 71 
Figure 2.5 FePd3 sputtering target ........................................................................... 72 
Figure 2.6 Annealing of FePd3 film .......................................................................... 74 
Figure 2.7 TC characterization .................................................................................. 77 
Figure 2.8 Lorentz contrast in different FePd3 depositions ................................... 79 
Figure 2.9 Magnetization as a function of temperature ......................................... 80 
ix 
 
Figure 2.10 Hysteresis curves ................................................................................... 81 
Figure 2.11 Fabrication process overview ............................................................... 83 
Figure 2.12 Electron beam lithography ................................................................... 87 
Figure 2.13 Lithography re-write scheme ............................................................... 89 
Figure 2.14 Electron beam evaporation ................................................................... 95 
Figure 2.15 Sample holder ........................................................................................ 96 
Figure 2.16 Large square crystal ............................................................................ 101 
Figure 2.17 Large kagome crystal .......................................................................... 102 
Figure 2.18 Higher magnification square and kagome images ........................... 103 
Figure 2.19 Kagome geometry with cookies .......................................................... 105 
Figure 2.20 Sample with poor liftoff ...................................................................... 105 
Figure 2.21 Missing elements .................................................................................. 107 
Figure 2.22 Bent elements ....................................................................................... 107 
Figure 2.23 Wrong doses ......................................................................................... 108 
Figure 2.24 Spots on SiN membranes .................................................................... 109 
Figure 2.25 Schematic of a TEM ............................................................................ 112 
Figure 2.26 TEM ray diagram ................................................................................ 113 
Figure 2.27 Lorentz TEM ....................................................................................... 117 
Figure 2.28 In-focus vs. Lorentz contrast TEM images ....................................... 118 
Figure 2.29 Lorentz contrast simulation................................................................ 119 
Figure 2.30 Ambiguity in using MFM to image connected kagome structures .122 
Figure 2.31 Heating holder radiative heat loss...................................................... 126 
Figure 2.32 Tube furnace ........................................................................................ 127 
x 
 
Figure 2.33 Annealing protocol .............................................................................. 129 
Figure 2.34 Other annealing graphs ...................................................................... 130 
Figure 2.35 Square ice ground state ....................................................................... 132 
Figure 2.36 Vertex types notation .......................................................................... 134 
Figure 2.37 Square ice ground-state domains ....................................................... 135 
Figure 2.38 Square ice domains .............................................................................. 136 
Figure 2.39 Experimental temperature window ................................................... 137 
Figure 2.40 Spin state double well potential .......................................................... 142 
Figure 2.41 No disorder square Monte Carlo ....................................................... 145 
Figure 2.42 Square Monte Carlo with varying disorder ...................................... 146 
Figure 2.43 Square Monte Carlo with high disorder ........................................... 148 
Figure 2.44 Evolution of a square geometry kinetic Monte Carlo simulation ... 149 
Figure 3.1 Kagome charge domains notation ........................................................ 152 
Figure 3.2 C parameter calculation ........................................................................ 154 
Figure 3.3 Polarized kagome crystal ...................................................................... 155 
Figure 3.4 Small kagome crystal array .................................................................. 157 
Figure 3.5 Large kagome crystal ............................................................................ 158 
Figure 3.6 One-hexagon 2FE crystal ...................................................................... 159 
Figure 3.7 Two-hexagon 2FE crystals .................................................................... 160 
Figure 3.8 Charge domains in a three-hexagon 2FE crystal ............................... 162 
Figure 3.9 Charge-ordered three-hexagon 2FE crystal ....................................... 163 
Figure 3.10 Three-hexagon 2FE crystal OOMMF ............................................... 164 
Figure 3.11 Charge-ordered 4-hexagon 2FE crystal ............................................ 165 
xi 
 
Figure 3.12 Charge domains in 4-hexagon 2FE crystals ...................................... 166 
Figure 3.13 Charge-ordered 5-hexagon 2FE crystal ............................................ 167 
Figure 3.14 Charge domains in 5-hexagon 2FE crystals ...................................... 168 
Figure 3.15 Charge-ordered 7-hexagon 2FE crystal............................................. 169 
Figure 3.16 Charge domains in 7-hexagon 2FE crystals ...................................... 170 
Figure 3.17 Charge-ordered one-hexagon 1FE crystal ........................................ 171 
Figure 3.18 Charge domains in one-hexagon 1FE crystals .................................. 172 
Figure 3.19 Charge-ordered two-hexagon 1FE crystal ........................................ 173 
Figure 3.20 Charge domains in two-hexagon 1FE crystals .................................. 173 
Figure 3.21 Charge domains in three-hexagon 1FE crystals ............................... 174 
Figure 3.22 Charge domains in assorted larger crystals ...................................... 175 
Figure 3.23 Charge domain map of a 500 nm element length kagome crystal .. 178 
Figure 3.24 Charge domain map of a 450 nm element length kagome crystal .. 179 
Figure 3.25 Charge domain map of a 400 nm element length kagome crystal .. 180 
Figure 3.26 Charge domain map of a 350 nm element length kagome crystal .. 181 
Figure 3.27 Charge domain map of a 300 nm element length kagome crystal .. 182 
Figure 3.28 Charge domain map of a 350 nm element length kagome crystal .. 183 
Figure 3.29 Charge domain map of a 300 nm element length kagome crystal .. 184 
Figure 3.30 Spin map of 350 nm element length crystal ...................................... 185 
Figure 3.31 Nearest neighbor charge correlation as a function of element length 
in the kagome geometry ...................................................................... 187 
 
Figure 3.32 Higher order charge correlations ....................................................... 189 
Figure 3.33 Simulation element sizes ..................................................................... 192 
Figure 3.34 Monte Carlo results ............................................................................. 194 
xii 
 
Figure 3.35 500 nm element length simulation charge domains output ............. 195 
Figure 3.36 450 nm element length simulation charge domains output ............. 196 
Figure 3.37 400 nm element length simulation charge domains output ............. 197 
Figure 3.38 350 nm element length simulation charge domains output ............. 198 
Figure 3.39 300 nm element length simulation charge domains output ............. 199 
Figure 4.1 Dislocation in a 3D crystal .................................................................... 206 
Figure 4.2 Topological defect in a 2D crystal ........................................................ 207 
Figure 4.3 Topological frustration ......................................................................... 208 
Figure 4.4 Spring model schematic ........................................................................ 211 
Figure 4.5 Lithography pattern generation ........................................................... 212 
Figure 4.6 Patterned topological defect ................................................................. 213 
Figure 4.7 Full crystal with one topological defect ............................................... 214 
Figure 4.8 Full crystal with two topological defects ............................................. 216 
Figure 4.9 Family of T geometries .......................................................................... 217 
Figure 4.10 Family of A geometries ........................................................................ 218 
Figure 4.11 Family of R geometries ........................................................................ 219 
Figure 4.12 Other assorted geometries studied ..................................................... 220 
Figure 4.13 Topological defect after annealing ..................................................... 223 
Figure 4.14 Vertex map of a crystal with one topological defect ......................... 224 
Figure 4.15 Vertex map of a crystal with one topological defect ......................... 225 
Figure 4.16 Vertex map of a crystal with one topological defect ......................... 226 
Figure 4.17 Vertex maps of crystals with two topological defects ....................... 228 
Figure 4.18 Vertex maps of crystals with two topological defects spaced very 
close together ........................................................................................ 229 
xiii 
 
Figure 4.19 Vertex maps of crystals with two topological defects spaced very 
close together ........................................................................................ 230 
 
Figure 4.20 Vertex maps of crystals with two topological defects spaced very 
close together ........................................................................................ 231 
 
Figure 4.21 Vertex maps of crystals with two topological defects spaced very 
close together ........................................................................................ 232 
 
Figure 4.22 Vertex maps of crystals with two topological defects ....................... 234 
Figure 4.23 Vertex maps of crystals with two topological defects ....................... 235 
Figure 4.24 Vertex maps of crystals with two topological defects ....................... 236 
Figure 4.25 Vertex maps of crystals with two topological defects ....................... 237 
Figure 4.26 Vertex maps of crystals with two topological defects ....................... 238 
Figure 4.27 Vertex maps of crystals with two topological defects ....................... 239 
Figure 4.28 Vertex maps of crystals with two topological defects ....................... 240 
Figure 4.29 Vertex maps of crystals with two topological defects ....................... 241 
Figure 4.30 Vertex maps of crystals with two topological defects ....................... 242 
Figure 4.31 Vertex maps of crystals with two topological defects ....................... 243 
Figure 4.32 Vertex maps of crystals with two topological defects ....................... 244 
Figure 4.33 Probability of domain walls not connecting dislocations vs. distance 
between dislocations in crystals with two topological defects ......... 246 
 
Figure 4.34 Probability of domain walls not connecting dislocations for different 
geometry families ................................................................................. 247 
 
Figure 4.35 Multiple paths near a topological defect ........................................... 252 
Figure 4.36 [11] Burgers vector topological defect ............................................... 253 
Figure 4.37 Full crystal with a [11] Burgers vector topological defect ............... 254 
Figure 4.38 Vertex map of a [11] Burgers vector topological defect ................... 255 
Figure 4.39 Vertex map of a [11] Burgers vector topological defect full crystal 256 
xiv 
 
Figure 4.40 Vertex map of a [11] Burgers vector topological defect full crystal 257 
Figure 4.41 Pyrochlore spin ice ground state with order parameter 
q = (2п/a)[001] ...................................................................................... 258 
 
Figure 4.42 Topological frustration in pyrochlore spin ice .................................. 259 
Figure 4.43 Vertex map of a non-topological defect ............................................. 263 
Figure 4.44 Vertex map of non-topological defect crystals .................................. 264 
Figure 4.45 Repeated annealing of the same crystal............................................. 266 
Figure 4.46 Domain wall pinning sites ................................................................... 267 
Figure 4.47 Vertex map of a crystal with two topological defects showing 
meandering domain walls ................................................................... 269 
 
Figure 4.48 Vertex map of a crystal with two topological defects showing 
relatively straight domain walls ......................................................... 270 
 
Figure 4.49 Vertex map of a crystal with two topological defects showing unique 
domain wall patterns ........................................................................... 271 
 
Figure 4.50 Vertex map of a crystal with two topological defects showing unique 
domain wall patterns ........................................................................... 272 
 
Figure 4.51 Magnetization direction vertex notation ........................................... 274 
Figure 4.52 Magnetization direction vertex map .................................................. 275 
Figure 4.53 Magnetization direction vertex map .................................................. 276 
Figure 4.54 Monte Carlo simulations ..................................................................... 278 
Figure 4.55 Evolution of a topological defect kinetic Monte Carlo simulation .. 280 
 1 
 
Chapter 1  Introduction and background  
 
 
Artificial spin ice (ASI) is a highly tunable model system that can be used to gain 
unique insights into frustrated systems. In this chapter, I will introduce a variety of topics 
relevant to ASI. I will begin with a definition of frustration and describe the first crystalline 
system discovered to be frustrated, water ice. I then describe the quantum origins and 
different types of conventional magnetism including frustrated magnetism. After a brief 
survey of different classes of magnetically frustrated materials, an introduction to the 
pyrochlore spin ices will be given. Finally, I provide a thorough overview of ASI, the main 
focus of this dissertation, highlighting seminal initial work and describing some recent 
results on the thermal activation of ASI systems. 
 
1.1 Frustration and water ice 
 
 
Frustrated systems are characterized by competing interactions that cannot all be 
simultaneously satisfied. They are ubiquitous in nature and display many novel phenomena 
not found elsewhere in physics. Many physical systems tend to find unique, highly-ordered, 
low energy, and low entropy configurations as the temperature of the system is lowered 
towards 0 K. The Third Law of Thermodynamics, originally known as the Nernst heat 
theorem [1], states that as the temperature of a system tends towards 0 K, the entropy of 
the system, a measure of its disorder, should also go to 0. Frustrated systems, however, do 
not obey this law. A high number of degenerate or nearly degenerate ground states is 
 2 
 
characteristic of a frustrated system. There is built-in disorder that persists down to the 
lowest temperatures, resulting in a residual entropy.  
The first crystalline system known to exhibit frustration is water ice, which was 
described by Pauling in 1935 [2] to resolve a discrepancy in different methods of 
calculating the entropy of ice [3,4]. Entropy can be expressed as  
𝑆(𝑇) =  ∫
𝐶𝑝(𝑇
′)𝑑𝑇′
𝑇′
𝑇
0
   (1.1) 
where S is the entropy, T is the temperature, and Cp is the heat capacity at constant pressure. 
By measuring the heat capacity of a system, one can use Equation 1.1 to find the entropy. 
Giauque and Ashley did exactly this [3], using heat capacity measurements of ice at low 
temperatures to calculate the entropy of the system [5,6]. A separate method to find the 
entropy of a system is to use spectroscopic band data of a gas and knowledge of the 
corresponding molecular structure to calculate the total entropy [7–9]. Gordon calculated 
entropy in this manner for water vapor and found a value that did not match the number 
found from heat capacity measurements [4].  
Pauling showed that this discrepancy in the entropy is actually due to the degenerate 
ground state of the ice system. As temperature is lowered towards 0 K, the oxygen atoms 
form a tetrahedral lattice where each oxygen is bonded to four other oxygens around it (see 
Figure 1.1) [10]. The hydrogen atoms sit on the oxygen-oxygen bonds, but do not sit in the 
middle; each hydrogen atom is closer to one oxygen atom or the other. The closer hydrogen 
atoms are 0.95 Å from the oxygen and are covalently bonded, corresponding to an H2O 
molecule, while the further atoms are 1.81 Å apart and only hydrogen bonded [11]. Pauling 
was aware of both the structure of oxygen atoms and the oxygen-hydrogen bond lengths 
from x-ray measurements. He assumed that the lowest energy configuration, locally, has 
 3 
 
two hydrogen atoms close to each oxygen and two further away, a structure proposed by 
Bernal and Fowler and known as the “two-in, two-out” ice rules [11]. Pauling showed that 
there are 6 different ways to satisfy the ice rules out of 24 = 16 total possible configurations 
for the 4 hydrogen atoms around each oxygen. He then solved what would now be an 
undergraduate statistical mechanics problem to calculate the residual entropy in ice due to 
the 6 out of 16 possible lowest energy configurations.  
For N oxygen atoms there are 22N possible states, since each oxygen shares a bond 
with 4 other oxygens, and (6/16)N of these states obey the ice rules. Therefore the total 
number of possible ground-state configurations is  
Ω0 = [2
2𝑁 (
6
16
)
𝑁
]  (1.2) 
and the residual entropy is  
𝑆0 = 𝑘𝐵 ⋅ 𝑙𝑛(Ω0) = 𝑘𝐵 ⋅ 𝑙𝑛 ((
3
2
)
𝑁
) = 𝑁𝑘𝐵 ⋅ 𝑙𝑛 (
3
2
)  (1.3) 
For N atoms in a mole, NkB becomes the gas constant R = 1.987 cal K
-1mol-1 and then S0 = 
0.805 cal K-1mol-1, which is very close to the difference in entropy of 0.870 cal K-1 mol-1 
between the values reported in Refs. [3] and [4]. Further, careful heat capacity 
measurements reduced the measured value of the excess entropy to 0.820 cal K-1 mol-1 [6] 
and more rigorous calculations resulted in a theoretical value of 0.814 cal K-1 mol-1 [12].  It 
is interesting to note that the same amount of residual entropy per atom is present in the 
magnetic pyrochlore spin ices which I discuss in detail in Section 1.3 [13].  
 4 
 
 
Figure 1.1 Tetrahedral structure of ice. As ice is cooled towards 0 K, the oxygen atoms 
(open circles) form a tetrahedral structure where each oxygen is bonded to four others 
around it. The hydrogen atoms (filled circles) sit on the oxygen-oxygen bonds, but don’t 
sit in the middle. Each hydrogen proton is closer to one oxygen atom or the other. This 
schematic shows a lowest energy “ice rule” state, where two of the hydrogens are close to 
the center oxygen and two are further away. There are 6 different ways to satisfy this 2-in, 
2-out local ground state ordering, leading to geometric frustration. Figure adapted from 
Ref. [14]. 
 
A great number of other frustrated systems have since been discovered and the 
frustrated interactions can arise from a variety of sources. Many of the interactions are 
magnetic in nature as will be discussed at length in this dissertation. Some other examples 
of non-magnetic systems with residual entropy are molecular crystals, such as crystalline 
CO, where disorder comes from varying orientations of the molecules, and in quenched 
alloys, such as β-brass (CuZn), where rapid cooling results in disordered positions of atoms 
throughout the lattice [15].  
 
 
 
 5 
 
1.2 Magnetism 
 
1.2.1 Conventional magnetism  
 
Electric and magnetic fields are well described by Maxwell’s equations and their 
intrinsic relationship is an essential topic of classical physics. In materials, however, the 
fundamental, atomic origin of magnetism is much more complex and generally quantum 
mechanical in nature. In the most basic sense, it arises from the quantized angular 
momentum, either orbital or intrinsic (spin), of electrons around atoms and whether the 
atoms have completely filled or partially filled outer electronic shells. Different types of 
magnetism in materials and diverse magnetic behaviors originate from a variety of 
(sometimes competing) factors. In this section I provide an overview of these factors and 
a survey of the resultant bulk properties. The physics of small ferromagnetic islands with 
strong shape anisotropy will also be discussed, as these will be especially important in later 
chapters.  
 The magnetic field B inside a material is related to the external magnetic induction 
H and the magnetization of the material (the magnetic moment per unit volume) M by  
𝑩 = 𝜇0(𝑯 + 𝑴) (1.4) 
M is related to H as 
 M = χH  (1.5) 
 where χ is the magnetic susceptibility, a property which can vary greatly between materials 
and is also a function of temperature [16]. χ < 0 is defined as diamagnetism, where the 
magnetization of a material opposes an applied field. This can be understood as a 
microscopic application of Lenz’s law, in which a changing applied magnetic field induces 
 6 
 
a current in a loop of wire such that the induced field from that current will oppose the 
change in applied field. In the classical picture, when a field is applied to a material, 
electrons will begin to orbit due to Larmor precession [17]. The induced electron orbits 
will generate a magnetic field that tends to oppose the applied field. The same result can 
be similarly obtained from quantum mechanics. Interestingly, all materials have a weak 
diamagnetic contribution, though if another type of magnetism is present the diamagnetic 
part will generally be negligible. Many purely diamagnetic materials typically have 
susceptibilities around χ ≈ -10-5 [18]. This is very weak, though the application of a 
sufficiently strong magnetic field ( >10 T) can induce enough magnetism to allow many 
ordinary, seemingly non-magnetic objects to levitate in the applied field, such as a frog or 
a hazelnut as was demonstrated by Berry and Geim [18]. Bulk superconductors display 
perfect diamagnetism, the Meissner effect [19], though the microscopic origin is different 
than for the atoms discussed above. Persistent screening currents in the superconductor 
generate fields that cancel (either perfectly in type-I superconductors or partially in type-
II) the applied magnetic field, though the application of a strong enough external field will 
destroy the superconductivity in either type [20]. A schematic of the Meissner effect is 
shown in Figure 1.2.  
The net magnetic moment of an atom is determined by the sum of the orbital and 
spin angular momentum [16]. If the atom has full electron shells, it will have no net moment 
and can only display diamagnetism. If instead, a material has atoms with open shells, there 
will be a net moment that will tend to align with an applied magnetic field since that will 
lower the energy of the system. This will result in χ > 0 and is defined to be paramagnetism.  
 
 7 
 
The temperature dependence of χ for a simple paramagnet is given by the Curie law [17] 
𝜒 =
𝐶
𝑇
  (1.6) 
In metals, where conduction electrons are delocalized from the atoms and energy 
levels are described by an electronic band structure, the picture is slightly different. 
Electrons of opposite spin respond differently to the applied field, which shifts the 
dispersion relations in opposite directions for each spin orientation. Spins aligned 
antiparallel to the field have more energy, so the relative number of parallel spins will 
increase in order to keep the filled electron states at the Fermi level [17]. The preferential 
population of one of the spin species results in a net magnetization in the material. This is 
known as Pauli paramagnetism and is shown schematically in Figure 1.3.  
Both diamagnetism and paramagnetism are only present during the application of 
an external field, but atoms can also have a spontaneous magnetic moment in the absence 
of an applied field. Atoms with a spontaneous moment can also interact with each other 
through the overlap of their wavefunctions. Different materials favor parallel or antiparallel 
alignment of atomic moments through the exchange interaction. Exchange arises through 
a combination of the Pauli exclusion principle and the Coulombic repulsion between 
electrons. For conduction electrons i and j of the same spin in a metal, the probability 
electron i will be found in volume dri and electron j found in volume drj is [16]  
|𝜓𝑖𝑗|
2
𝑑𝒓𝒊𝑑𝒓𝒋 =
1
𝑉2
[1 − cos ((𝒌𝑖 − 𝒌𝑗) ⋅ (𝒓𝒊 − 𝒓𝒋))]𝑑𝒓𝑖𝑑𝒓𝑗   (1.7) 
This clearly shows that an electron of a given spin orientation cannot be in the same 
location as a second electron of the same spin as the probability for this to occur is 0. 
 8 
 
 
Figure 1.2 The Meissner effect. As a superconductor (red circle) is cooled through its 
transition temperature (TC), magnetic field lines are expelled from the material. The net 
field inside the superconductor becomes zero and it behaves like a perfect diamagnet. a. 
Above TC. b. Below TC.  
 
   
 9 
 
 
Figure 1.3 Pauli paramagnetism. An applied magnetic field induces a higher population 
of one spin species resulting in a net magnetic moment. Here, the electrons antiparallel to 
the field cost more energy, so the parallel spins increase in number while the antiparallel 
decrease so that the total energies of the two populations are equal to the Fermi level, εF. μ 
is the magnetic moment of the electrons. 
 
The exchange interaction is given as [17] 
𝑈 = −2𝐽𝑺𝑖 ⋅ 𝑺𝑗  (1.8) 
where Si are the spins of the electrons. When J > 0, the material will display ferromagnetic 
ordering, where atomic moments tend to align parallel with each other, as this will lower 
the exchange energy U. The parallel moments in the material will add up to result in a 
(often strong) net spontaneous magnetization.   
 10 
 
The strength of the exchange interaction J varies between different elements and 
can be washed out when the thermal energy of the system is comparable with the exchange 
energy. The temperature where the atomic moments no longer favor spontaneous 
alignment is known as the Curie temperature (TC). Above this temperature, the material 
will be paramagnetic with the susceptibility following the (mean-field) Curie-Weiss law 
[17] 
𝜒 =
𝐶
𝑇−𝑇𝐶
  (1.9) 
The reciprocal susceptibility of many ferromagnets is roughly linear in temperature in the 
paramagnetic regime. Below TC, the susceptibility diverges with the material showing a 
finite M for zero H.  
Ferromagnets are the simplest outcome due to an exchange interaction, but 
materials can have many other types of exchange interactions and behavior. If J < 0, the 
magnetic moments will tend to anti-align with each other, and if the moments of the atoms 
are all equal, this will result in an antiferromagnetic material. Antiferromagnetism sets in 
below the Néel temperature (TN) where the material has zero net magnetic moment. Above 
TN the material will be paramagnetic, with (mean-field) susceptibility given as [17] 
𝜒 =
2𝐶
𝑇+𝜃
  (1.10) 
Theoretically θ = TN and experimentally θ/TN is on the order of 1, though it can deviate due 
to longer range interactions.  
If the material has more than one species of atom, the interactions can become even 
more complex. A material with J < 0 and two sublattices of different atoms with unequal 
moments is known as a ferrimagnet. A schematic of different magnetic ordering is shown 
 11 
 
in Figure 1.4 and characteristic qualitative susceptibility curves for the different types of 
magnetism we have discussed are given in Figure 1.5. All of the exchange interactions 
covered so far are known as “direct exchange” since they are a consequence of direct 
interaction between the atoms. If a material has magnetic atoms separated by non-magnetic 
ones, it is possible for the non-magnetic atoms to mediate an exchange interaction between 
its neighboring magnetic atoms that will dominate the direct exchange between them. This 
is called “superexchange.” If the exchange interaction is mediated by conduction electrons 
in metals it is known as “itinerant exchange” [21].  
In the preceding discussion I covered the basic atomic origin of different types of 
conventional magnetism. I now take a step back and consider the aggregate properties of 
ferromagnetic materials which will be relevant to the study of ASI.  
 
 
 
Figure 1.4 Types of magnetism. a. Ferromagnetic coupling between spins causes them to 
point in the same direction. b. Antiferromagnetic materials have spins of alternating 
orientation with no net magnetic moment. c. Ferrimagnetic materials have magnetic 
moments that alternate orientation, but one orientation has a higher magnitude so there is 
still a net magnetic moment.    
 
 12 
 
 
Figure 1.5 Susceptibility curves. Magnetic susceptibility vs. temperature for (a) 
Paramagnets, (b) Ferromagnets, and (c) Antiferromagnets. Also shown are the general 
equations for susceptibility vs. temperature for different types of magnetism. Figure 
adapted from Ref. [17].  
 
While it is true that ferromagnetically coupled atoms will align with each other 
locally, in samples with ~1023 atoms, different overall properties can emerge. In 
considering bulk effects within ferromagnetic materials and between separate 
ferromagnets, the dipolar interaction will be important. Dipolar coupling exists between 
any two magnetic moments m1 and m2 and is given by [22]  
𝑈 =  
−𝜇0
4𝜋|𝒓|3
(3(𝒎1 ∙ ?̂?) (𝒎2 ∙ ?̂?) − 𝒎1 ∙ 𝒎2))  (1.11) 
where r = r1 – r2 is the vector between the two moments. The dipolar interaction between 
two neighboring atomic moments is generally much weaker than the exchange interaction, 
but the exchange energy falls off over a very short range, while the dipolar coupling is a 
much longer range interaction, decreasing as 
1
𝑟3
. The effect from atomic ferromagnetic 
moments can add up so that the dipolar interaction becomes significant.  
 13 
 
While iron is known to be a very strong permanent magnet, not all naturally 
occurring pieces of iron will display a net magnetization. This is because a magnetic 
material can divide into separate magnetic “domains” which can have different 
orientations. If the domains generally all point in the same direction, the sample will have 
a spontaneous moment, but if the domains are randomly oriented, the material will not have 
a strong magnetization. The highest possible magnetization for a sample occurs when all 
its domains point in the same direction and the maximum value of the magnetization in this 
configuration is known as the saturation magnetization. The higher the saturation 
magnetization of a material, the stronger the magnet it can be.  
In general, magnetic domains will tend to align with external fields. Consider a 
sample with a distribution of randomly oriented domains. As a magnetic field is initially 
applied to the sample, domains preferentially aligned with the field will begin to grow at 
the expense of unfavorably oriented domains [17]. As the magnitude of the external field 
increases, all the domains will eventually rotate to align with the field until the sample 
reaches its saturation magnetization. If the field is removed, the sample will return to a 
value known as the remnant magnetization (Mr). The applied field value that causes the 
magnetization of the sample to reach zero after it has been saturated is defined as the 
coercive field (Hc) and this value will change sign depending on the previous saturation 
state of the magnet. This phenomenon is known as magnetic hysteresis and a typical 
hysteresis curve showing the situation I just described is given in Figure 1.6. The shape of 
the hysteresis curve, or equivalently the values of Mr and Hc, give insight into a magnet’s 
utility for different applications.  
 14 
 
 
Figure 1.6 Magnetic hysteresis curve. Magnetization M vs. applied field H. An external 
field is applied to a magnet with randomly oriented domains and no net magnetization. 
Initially, domains aligned with the field grow in size at the expense of unfavorable 
domains. As the field is increased, domains will rotate to align with the field until the 
magnet reaches its saturation magnetization. When the field is removed, the magnetization 
reduces to the magnet’s remnant magnetization, Mr. The field is then applied in the opposite 
direction. The magnitude required to bring the net magnetization back to zero is known as 
the coercive field, Hc. The magnitude of the field continues to increase and saturates the 
magnet in the opposite direction. The value of the magnetization for a given value of 
applied field depends on the history of the magnet, an effect known as hysteresis.  
 
Some crystals have a preferential axis for magnetization to point along. The 
preferential axis is called the easy axis and an additional magnetic anisotropy energy arises 
due to this preference. The crystalline anisotropy originates from spin-orbit coupling 
creating asymmetric charge distributions around atoms [17]. A preferred magnetic axis will 
impact the domain configurations through the relative orientation of adjacent 
crystallographic grains and also through the added anisotropy energy competing with other 
interactions. It can greatly complicate a magnet’s behavior and in general, materials with 
very low crystalline magnetic anisotropy are better for ASI studies. Shape anisotropy of 
 15 
 
the magnet is a separate consideration from crystalline anisotropy and will be discussed in 
detail below.  
Different magnetic domains will be separated by a domain wall where the 
magnetization switches from one orientation to another. Domain walls often occur at grain 
boundaries in polycrystalline materials as these are natural locations for a transition, but 
they don’t have to. Neighboring grains can have the same magnetization or two domains 
can be present in the same, single-crystal grain. In the boundary between two separate 
domains, the magnetization does not jump discontinuously. There is a characteristic width 
of the domain wall over which the magnetization will slowly rotate, as this reduces the 
exchange interaction between the non-parallel spins in the wall. A Bloch wall is most 
common for bulk magnets, in which the magnetization rotates out-of-plane, as shown 
schematically in Figure 1.7. In magnetic thin films, if the material is thin enough it may be 
favorable for the magnetization to rotate in-plane due to external magnetic field. This is 
defined as a Néel wall and is shown in Figure 1.7(b). In thin-film samples of ASI made 
from permalloy (Ni81Fe19), other types of domain walls, such as a vortex wall, have been 
observed (see Figure 1.8) [23]. Some examples of magnetic domain patterns and domain 
walls I observed are shown in Figures 1.9 and 1.10.  
Domain walls in thin films have a characteristic width that can be tens to hundreds 
of nanometers depending on the material and the thickness of the film [24]. When a sample 
of magnetic material is small enough, it will not be able to stabilize a domain wall and will 
possess only a single magnetic domain. The exact size required for single domains depends 
on the material, but it is generally on the order of a few hundred nanometers or less [17]. 
If the island is elongated, shape anisotropy will cause the magnetization to point along 
 16 
 
 
Figure 1.7 Bloch vs. Néel domain walls. Relative magnetization orientations for (a) Bloch 
and (b) Néel domain walls in a thin film. The magnetization in a Bloch wall rotates out of 
the plane of the thin film. In a Néel wall, the magnetization rotates in plane.  There are 
different energies associated with the two different types of walls. Figure adapted from 
Ref. [17].  
 
 
Figure 1.8 Transverse and vortex domain walls in thin film permalloy. Micromagnetic 
simulations of (a) transverse and (b) vortex domain walls in thin films of permalloy 
(Ni81Fe19). The Transverse domain wall is similar to a Néel wall. The two types of walls 
are found depending on the thickness of the permalloy. Ladak and coworkers [23] found 
samples with 18 nm thick permalloy supported transverse walls, while thicker samples 
made from 36 nm films showed vortex walls. The blue and red color corresponds to 
positive and negative divergence of the magnetization, which is equivalent to magnetic 
charge density. Figure adapted from Ref. [23].  
 17 
 
 
Figure 1.9 Magnetic domains in thin film Ni35Fe65. Lorentz contrast Transmission 
Electron Microscopy (TEM) image of a magnetic thin film showing multiple magnetic 
domains. The thick bright and dark lines are domain walls in the film. The other ripple 
contrast is representative of small local variations in the magnetic order of the atoms in the 
material. See Section 2.3.2 for a discussion of Lorentz TEM.  
 
the long axis of the island. This is due to the magnet trying to minimize its external 
magnetic field energy, proportional to B2 [22]. Magnetic field lines connect the north and 
south poles of the single domain magnet by traveling through the free space around it with 
low density. If the magnetization points along the short axis of an elongated bar magnet, 
there will be more external flux energy than if the magnetization points along the long 
axis. Therefore, simple energy minimization can explain the tendency to point in these 
directions [25]. This is shown schematically in Figure 1.11. A similar consideration 
explains some domain configurations observed in larger, multi-domain magnetic samples. 
Neighboring domains will tend to arrange themselves in a configuration that minimizes  
 18 
 
 
Figure 1.10 Domain walls and sharp paramagnetic transition in thin film Fe35Pt65. 
Lorentz TEM image of a thin film heated in-situ. A temperature gradient across the film 
results in both paramagnetic and ferromagnetic regions in the image. The ferromagnetic 
region has two domain walls visible. See Figure 2.31 for an explanation of the temperature 
gradient in this image.  
 
the magnetic energy of flux outside of the sample, which often results in circular, or flux 
closure patterns [17,24]. An example of this is shown in Figure 1.12.  
Free spins or magnetic moments can point in any direction, though many times 
when spins are in a crystal they are confined by lattice geometry to point, e.g. in one of 
only two directions, either “up” or “down.” Such two-state spins are called Ising spins and 
are a good approximation to the way spins behave in many real materials [13,14,26,27]. 
Ising spins essentially behave as a two level system (they only have two readily accessible 
quantized energy levels) corresponding to the two spin directions. Other higher energy 
states may be present, but are typically so much higher in energy that the they can be 
 19 
 
 
 
Figure 1.11 Shape anisotropy flux minimization in single domain magnets. a. When 
the magnetization points along the long axis of a single-domain, elongated bar magnet, 
there are fewer magnetic field lines than when the magnetization points along the short 
axis (b). Each effective magnetic charge generates flux lines that connect to an opposite 
charge on the other side. The greater the area of effective magnetic charges, the more flux 
lines and the higher the magnetic field energy will be. In general, the bar magnet’s 
magnetization will point along the long axis in order to minimize the field energy.  
 
 
Figure 1.12 Flux closure domains. Different domain configurations in multi-domain 
samples without strong shape anisotropy can help minimize external flux and magnetic 
field energy. a. A single domain has no flux minimization. b-d. Domain patterns with 
decreasing external flux. The flux closure domain configuration in (d) has the lowest 
external flux.   
 
 
 20 
 
ignored in many situations. Single-domain, elongated, nanoscale bar magnets can act like 
giant Ising spins that can either point one way or the other along the island’s long axis. If 
Ising-like magnets or magnetic moments are arranged in certain geometries with just the 
right interactions, it is possible that pairs of spins will not be able to minimize coupling 
energy with only two states per spin and some moments will be forced into unfavorable 
arrangements. A situation where all of the pairwise couplings cannot be satisfied results in 
frustration of the system.   
1.2.2 Frustrated magnetism 
 
 
In magnetic materials, frustration is found when the pairwise interactions between 
neighboring magnetic moments cannot be simultaneously satisfied. Typically, the 
frustration arises due to either disorder in the system or its geometry. Prototypical examples 
are shown in Figure 1.13. Ising spins on a square lattice with ferromagnetic coupling are 
not frustrated (Figure 1.13(a)). On the other hand, if there is disorder in the material 
through, for example, impurities or a disordered alloy, this can create varying bond types 
and lead to frustration (Figure 1.13(b)). Antiferromagnetic exchange between spins on a 
triangular lattice causes geometric frustration since the geometry of the system does not 
allow for all bonds to be simultaneously satisfied (Figure 1.13(c)). This is similar to the 
frustration found in water ice that I discussed in Section 1.1 and is the source of the 
frustration in the magnetic analog to water ice that I will discuss in Section 1.3, pyrochlore 
spin ice. Before I explore the rich physics of spin ice, the material that inspired the ASI 
systems that are the main focus of this dissertation, I describe a few other frustrated spin 
systems with interesting properties.  
 21 
 
 
Figure 1.13 Types of frustration. a. No frustration: Ferromagnetic interactions between 
Ising spins on a square lattice are not frustrated. b. Disorder frustration: Disordered bonds 
between Ising spins on a square lattice cannot be satisfied with any spin configuration, 
leading to frustration. c. Geometric frustration: Antiferromagnetically coupled Ising spins 
on a triangular lattice are frustrated due to the geometry of the system. 
 
Spin glasses are a class of frustrated system where disordered magnetic atoms are 
randomly arranged among a lattice of non-magnetic atoms [28,29]. The magnetic moments 
couple through the RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction [30–32] which 
oscillates (changes sign and magnitude) depending on the distance between the atoms. If 
the atoms are random distances apart, they have alternating, competing interactions that 
result in a frustrated system. Well-studied spin glass systems include the dilute magnetic 
alloy CuMn [33], with Mn concentrations on the order of 1 at. %, and the insulating system 
EuxSr1-xS [34]. Due to random arrangements of atoms and random coupling between the 
atoms, these systems are frustrated and have difficulty finding an ordered state. There is a 
fundamental question about spin glasses, as well as other frustrated systems, on whether a 
unique ground state actually exists. It is predicted that there probably is a unique lowest 
energy state in any given spin glass, but it is very difficult for the system to find this state. 
Instead, there are a vast number of low lying energy states that, while not the absolute 
 22 
 
lowest energy of the system, are a local minimum in energy in the configuration space. The 
system will tend to find one of these minimums at low temperature and will not have 
enough thermal energy to move to a different local or global minimum. 
Some materials will display spin fluctuations down to extremely low temperatures 
[35–38]. Tb2Ti2O7 is one such frustrated system. Frustration preventing any long-range 
order is a fairly typical property, but it is interesting that there are still spin fluctuations 
down to as low as 70 mK [35], indicating that the energy barrier between degenerate states 
is extremely small, on the order of 6 μeV. The lack of correlations between spins at any 
distance is similar to molecules in a liquid and thus this material and others like it are 
known as “spin liquids.” Small perturbations in frustrated systems can actually induce 
ordered states. In water ice, doping with KOH will cause the lattice to find a unique ground 
state [39]. Similarly, in Tb2Ti2O7, small magnetic fields or applied pressure will result in 
spin ordering, or “crystallization” of the liquid [36].  
Many more complex materials with unique phases and orderings are possible and 
often times multiple phases and phase transitions will be present in a single material. 
Gd3Ga5O12 can display a spin glass phase, spin liquid phase, and antiferromagnetic long-
range-ordered phase depending on the temperature and applied field [40]. Neutron 
scattering experiments on Zn2Cr2O4 reveal a spin liquid phase characterized by fluctuations 
of not individual spins, but clusters of 6 antiferromagnetically coupled spins [41]. Fe2O3 
nanoparticles display a spin glass phase only in a 0.6 nm thick layer at their surface [42]. 
These examples only scratch the surface of the vast field of frustrated magnetic materials 
and there are even more purely theoretical frustrated systems that yield a plethora of new 
 23 
 
phases and physics. Overall, the consequences of frustration are diverse, but I now turn my 
attention back to the materials that motivate the main focus of this dissertation.  
1.3 Spin ice 
 
Spin ices are a class of geometrically frustrated magnetic materials with properties 
that are highly analogous to water ice. The canonical and most frequently studied materials 
are Dy2Ti2O7 and Ho2Ti2O7, although other systems such as Ho2Sn2O7 also display spin 
ice physics [13,43–46]. In these materials, the rare earth atoms form a pyrochlore lattice of 
corner sharing tetrahedra, as shown in Figure 1.14(c). The magnetic moments of the rare 
earth ions are Ising-like and are confined to point either directly in or directly out of each 
tetrahedron along the <111> crystal direction. It is believed that the coupling between the 
rare earth atoms is via a slightly antiferromagnetic superexchange, but ferromagnetic 
dipolar interactions dominate due to the extremely high atomic magnetic moment of the 
rare earth atoms [47]. However, the ferromagnetic interactions cannot all be simultaneously 
satisfied due to the tetrahedral geometry and spin confinement and thus the system is 
frustrated. The lowest energy configuration for the 4 spins on each tetrahedron has 2 
pointing in and 2 pointing out, completely analogous with the ice rules and ordering of 
hydrogen atoms in water ice. Similarly, there are 6/16 different ways to arrange the 4 spins 
that satisfy the ice rules resulting in a macroscopically degenerate ground state and residual 
entropy at zero temperature. The equivalence between spin ice and water ice is shown in 
Figure 1.14(a) and (b). Ramirez and coworkers were the first to measure the heat capacity 
of Dy2Ti2O7 and calculate the residual entropy, confirming the analogy with water ice [13].  
 
 24 
 
 
Figure 1.14 The pyrochlore lattice and the equivalence to water ice. a-b. Two-in, two-
out ice rule ordering of hydrogen atoms in water ice maps onto two-in, two-out ordering 
of rare earth magnetic moments in spin ice. c. The pyrochlore lattice of corner sharing 
tetrahedra showing the ice rule state in spin ice. Magnetic moments are indicated for one 
tetrahedron and are seen to point directly into or out of the tetrahedron. Black and white 
dots indicate the spin directions for downward facing tetrahedra and are defined in the 
tetrahedron with arrows indicated. Figure adapted from Ref. [43].  
 
They successfully match the entropy to a model  
𝑆0 = 𝑘𝐵 ⋅ 𝑙𝑛(Ω0) = 𝑘𝐵 ⋅ 𝑙𝑛 ((
3
2
)
𝑁
2
) =
𝑁
2
𝑘𝐵 ⋅ 𝑙𝑛 (
3
2
)  (1.12) 
where the extra factor of ½ comes from mapping N oxygen atoms onto N tetrahedra that 
contain 2N spins in the pyrochlores. Their measurement results are shown in Figure 1.15. 
Neutron scattering is a common technique used to gain insight into magnetic 
correlations within materials [48–50]. The technique is based on the fact that neutrons 
possess spin, but no charge, so they will diffract in a scattering experiment due to magnetic 
interactions. Neutron scattering has been used to confirm models for the interactions in 
spin ice. Figure 1.16 shows experimental neutron scattering data on Ho2Ti2O7 compared to 
simulations based on a nearest neighbor exchange model and a model with long-range 
dipolar interactions [43,46]. It can be seen that the experimental data matches much better 
with the long-range dipolar model, confirming the presence of dipolar coupling in spin ice.  
 25 
 
 
 
Figure 1.15 Specific heat and entropy in Dy2Ti2O7. a. Specific heat C divided by 
temperature T as a function of temperature measurements of Dy2Ti2O7 in zero field and H 
= 0.5 T. b. Entropy S as a function of temperature for Dy2Ti2O7 in zero field and H = 0.5 T. 
The entropy is found by integrating the specific heat. The Pauling value of R[ln(2) – 
1/2ln(3/2)] is indicated showing the analogy to water ice. Inset: Magnetic susceptibility as 
a function of temperature for Dy2Ti2O7 in an applied field of 0.2 T. Figure adapted from 
Ref. [13].   
 
 26 
 
 
Figure 1.16 Neutron scattering on Ho2Ti2O7. a. Experimental neutron scattering data on 
Ho2Ti2O7. Dark blue is low intensity, dark red is high intensity. b. Calculated neutron 
scattering pattern from a spin ice model with only nearest-neighbor exchange interactions 
at T = 0.15J. Outlined region corresponds to the experimental data in (a). c. Calculated 
neutron scattering pattern for a spin ice model with long-range dipolar interactions at T = 
0.6 K. This model does a better job of capturing all the features in the experimental data. 
Figure adapted from Refs. [43] and [46].  
 
 27 
 
Despite measurements confirming a residual entropy in spin ice and implying a 
disordered and degenerate low-temperature state, a unique ground state has nevertheless 
been predicted [51,52], due to the long-range dipolar interactions in the system, but it has 
proven difficult to observe experimentally [13,53–56]. The ground state was predicted by 
Melko, den Hertog, and Gingras (MDG) and is shown schematically in Figure 1.17 [51]. 
The state can be described by an order parameter and associated ordering vector 
q = (2π/a)<001>. MDG predict a jump in heat capacity and associated transition to the 
ground state with no residual entropy at 0.18 K. Recently, a single crystal sample of 
Dy2Ti2O7 thermally equilibrated for extremely long times (> 100 hours) showed a slight 
upturn in heat capacity and a reduction in entropy, below the Pauling value [57]. 
Additionally, a thin film sample of Dy2Ti2O7 with a high level of strain was also reported 
to find an ordered state and recovery its entropy, though this is different than the MDG 
ground state [58]. The MDG ground state has yet to be confirmed in spin ice and I will 
give some insight on a potential explanation for its elusiveness and propose some 
promising ways to reach it in Chapter 4.  
One property of the spin ice system that has attracted great interest is its excitations 
above the spin ice state. The spin ice state has perfect 2-in, 2-out ordering of magnetic 
moments in each tetrahedron, consistent with the ice rules. If we consider each spin as a 
dumbbell of magnetic charge, then this state has two positive and two negative charges in 
each tetrahedron and these add up to no net charge (Figure 1.18(c)). Excitations above this 
manifold are spin flips that create pairs of neighboring tetrahedra with 3-in, 1-out and 3-
out, 1-in arrangements of spins. In the dumbbell picture, these elementary excitations 
create pairs of magnetic monopoles, regions of excess magnetic charge inside the  
 28 
 
 
Figure 1.17 Spin ice ground state. a. Predicted MDG spin ice ground state projected down 
the z axis. +/- indicates the component of each spin in the z direction. b. Three dimensional 
schematic showing one unit cell of the ground state.  Figure adapted from Refs. [43] and 
[51].  
 
neighboring tetrahedra (Figure 1.18(d)). Once the monopoles are created, it only costs a 
Coulomb-like energy for them to separate and move throughout the lattice, leaving behind 
a Dirac string (Figure 1.18(e)) [59].  
Monopoles in spin ice were first proposed by Castelnovo et al. in 2008 as emergent 
phenomena and the elementary excitations in spin ice [59]. Shortly thereafter, monopole 
signatures were observed experimentally [55,56,60,61] and the monopole model was 
successfully used to explain previously poorly understood magnetic relaxation 
measurements on Dy2Ti2O7 [62].  Neutron scattering experiments further confirmed the 
monopole description [61] and the presence of Dirac strings in these systems [55]. Muon 
spin relaxation measurements reported a magnitude of the monopole charge to be 5 μBÅ-1 
[63], confirming a prediction in Ref. [59], although the validity of these results has been 
challenged [64,65]. Still, the monopole picture has proven to be an excellent theoretical 
model for describing the physics of spin ice and as a method to interpret and understand 
 29 
 
 
Figure 1.18 The dumbbell picture and magnetic monopoles in spin ice. The images in 
(c) and (d) correspond to replacing the spins in (a) and (b) with dumbbells where the ends 
are opposite magnetic charges. Frames (a) and (c) show the spin ice state which has two 
positive and two negative charges in each tetrahedron and thus no net magnetic charge. 
Frames (b) and (d) show an excitation of the system, where one spin is flipped, creating 
neighboring tetrahedra each with excess positive or negative magnetic charge, or 
monopoles. Once the monopoles are created, it costs no more energy other than the 
standard Coulomb interaction for them to separate and move throughout the lattice. The 
monopoles will leave behind a Dirac string along the path they take, indicated in frame (e). 
Figure adapted from Ref. [59].  
 30 
 
experimental data on these systems [66,67]. Some groups have taken the analogy to 
electricity even further, coining and proposing terms and ideas such as “magnetolyte” for 
a medium to carry magnetic charge, “magnetronics” for circuits utilizing magnetic charge, 
and “magnetricity” for monopole motion in a magnetic field as an analog to how electrons, 
electric monopoles, move in an electric field [63,68].  
It is important to note that monopoles in spin ice are monopoles in M, not B. Dirac 
predicted the existence of monopoles in the magnetic B field in the 1930’s to help explain 
the quantization of electric charge [69]. Despite a great number of experiments and 
searches for them [70–73], no isolated Dirac monopoles have been ever been confirmed. 
One study at Stanford University did see a single potential monopole event, although this 
was never duplicated, independently verified, or confirmed [70]. Dirac monopoles were 
recently observed in a synthetic magnetic field, although this does not indicate that they 
exist naturally [74]. Though the monopoles in spin ice behave in the same manner as 
Dirac’s proposed monopoles in B, their magnitude is different than a Dirac monopole and 
they do not explain the quantization of the charge of an electron. They can be thought of 
more as quasi-particle excitations that describe the physics of spin ice at low temperatures, 
similar to other particle-like descriptions of physical phenomena such as photons or 
phonons [21,22]. Dirac strings in spin ice are an observable phenomenon [55], whereas 
Dirac strings of B in the vacuum have no reported experimental signature. Overall, the 
pyrochlore spin ices are an intriguing class of materials that display rich physics and novel 
phenomena originating from the extensive frustration in the system.  
 
 31 
 
1.4 Artificial spin ice 
1.4.1 Introduction 
 
In the study of frustrated systems such as ice and spin ice, it would be ideal to have 
the ability to image exact atomic and spin configurations to better understand the effects of 
frustration and the dynamics in the system. Unfortunately, this is not possible with current 
technology. An alternative approach to studying these systems is through lithographically 
patterned, two-dimensional arrays of Ising-like, single-domain nanoscale bar magnets, 
collectively known as artificial spin ice (ASI) [75,76]. The individual bar magnets behave 
like pseudo-atoms or giant “spins” that interact in a classical manner. ASI arrays are highly 
tunable systems built by design to mimic the frustrated interactions in real materials, but 
are also amenable to a variety of direct characterization techniques such as Magnetic Force 
Microscopy (MFM) [75,77–79], Photoemission Electron Microscopy (PEEM) [80–85], 
Magneto-Optical Kerr Effect (MOKE) [86,87], x-ray resonant magnetic scattering [88,89], 
or Lorentz Transmission Electron Microscopy (LTEM) [90–95]. LTEM is the technique I 
use and will discuss in depth in Section 2.3.2. ASI presents a playground for studying 
frustrated systems, where the geometry of the system as well as the strength and character 
of the magnetic interactions are readily chosen by design. Precise information at the single 
spin level gives new and unique insight into frustrated systems and is a huge advantage of 
ASI over other experimental systems and techniques.  
Through this work, I aim to answer some long-standing and fundamental questions 
about frustrated systems. Namely, how do these systems find long-range ordered states or 
is this even possible at all in the presence of frustration? In frustrated systems, an apparent 
 32 
 
degeneracy of ground states prevents long-range order, often when detailed analysis of 
perturbations predict that an ordered state nevertheless should occur [51]. Despite decades 
of intense interest, frustrated systems still pose fundamental problems, with many 
unanswered questions, due in part to the tendency of these systems to inefficiently explore 
their configuration spaces and to lose ergodicity [96]. Monte Carlo simulations can address 
some of these issues through the introduction of more complicated basic excitations 
[97,98], but questions about the specific conditions that will allow or restrict a system from 
reaching a predicted ground state often go unanswered in a general sense.  An emerging 
approach seeks to address these issues by fabricating physical simulators of real materials 
[99–101], and ASI comprise a highly productive example. 
Materials-by-design approaches to condensed matter physics have recently 
produced noteworthy results, exemplified by photonic crystals and materials with a 
negative index of refraction [102,103]. Metamaterials are a growing trend, where "bottom 
up" methodology allows one to realize physical structures not found in nature in order to 
investigate and understand genuinely new properties and phenomena [104,105]. These 
materials systems provide unique, tunable control over parameters previously frozen in or 
set by a single choice of material or stoichiometry. They have allowed for the realization 
of old science fiction ideas like invisibility through optical cloaking by precise control over 
electromagnetic fields [106–108]. An artificially patterned lattice recently demonstrated 
the first skyrmions at room temperature without the application of a magnetic field [109]. 
Certain types of arrays of Superconducting QUantum Interference Devices (SQUIDs) are 
metamaterials which exhibit unique microwave transmission properties [110,111]. Clearly, 
 33 
 
materials-by-design systems are a fruitful route to new ideas and new physics and as we 
will see, artificial spin ices are an excellent example.  
 
1.4.2 Square and kagome ice  
 
The first ASI geometry studied was square ice [75], chosen for its analogy to spin 
ice with 4 spins that meet at each vertex. An overview of the geometry is shown in Figure 
1.19. The lowest energy states follow the two-in, two-out ice rules, although in square ASI 
the 6 ground-state configurations split into two energy levels due to the inequivalent 
interaction strengths between the four magnets at each vertex. The variation in the distance 
between adjacent and opposite spins at each vertex lifts the six-fold degeneracy of the ice 
rule state and results in 2 configurations having slightly lower energy than the other 4. The 
16 different possible spin arrangements are traditionally grouped by their energy and given 
a type I-IV nomenclature. The lowest energy, type I configurations have two spins pointing 
in and two pointing out with zero net magnetic moment. Type II vertices also obey the ice 
rules, but have a net magnetic moment and slightly higher energy than type I. Type III 
vertices are three-in, one-out or one-in, three-out arrangements, and type IV are four-in or 
four-out. Schematics of these vertex types are shown in Figure 1.19(d).  
 34 
 
 
Figure 1.19 Artificial square ice. a. Schematic of artificial square ice geometry. The 
arrows represent the directions of the spins in part (c). b. Atomic Force Microscopy image 
of a square ASI sample. c. Magnetic Force Microscopy image of a square ASI sample. The 
moment directions can be exactly determined and are shown in part (a). The outlined 
regions in pink, blue, and green correspond to vertices of type I, II, and III respectively. d. 
The 16 possible moment configurations for a single vertex, grouped into the 4 types based 
on their relative energy level. Figure adapted from Ref. [75].  
 
The ground state of artificial square ice is a perfect tiling of type I vertices [112]. 
This state is two-fold degenerate, owing to the two spin configurations that make up the 
type I energy level. These two ground states are related to each other by a global reversal 
of spins. Much of the early work in ASI used rotation demagnetization protocols in an 
attempt to minimize array energy and access ground-state configurations [75,113–116]. 
Rotation demagnetization is a process where the sample is rotated in a magnetic field that 
is alternating in polarity at some frequency and also slowly decreasing in amplitude [113]. 
A schematic of typical protocols is shown in Figure 1.20. The seminal first ASI paper by 
 35 
 
Wang et al. used rotation demagnetization on square ice samples with varying lattice 
spacings [75]. This resulted in random populations of vertex types for the larger lattice 
spacings and ice rule ordering of spins below a random population for the shorter lattice 
constant samples. Closer magnets increase the magnetostatic coupling between nearest 
neighbors and favor more ordered states. Indeed, the ordering is seen to emerge as the 
lattice spacing is decreased as in Figure 1.21. Still, even for the shortest lattice spacing, the 
samples remain far from the true ground state [75]. Rotation demagnetization was generally 
not very successful at accessing large regions of ground-state ordering or long-range 
ordered states [75,114–116], though a recent report using a linear decrease in applied field 
showed that by tuning the rotation rate of the sample as well as the rate at which the field 
is ramped down, moderately sized domains of ground-state order can be generated [117].  
Disorder is also known to play an important role in energy minimization and how 
well square geometry samples can find ordered states [118–121]. Specific effects of 
disorder and its origins in both the square and kagome lattices will be discussed later in 
this section. In general, other experimental techniques beyond rotation demagnetization are 
needed to realize long-range ordered states and these methods will be explored in depth in 
the next section and later chapters. 
The spin configurations of artificial square ice are typically frozen in place at room 
temperature, but the statistical populations of vertex types still yield interesting physical 
results. Different demagnetization protocols and lattice spacings (coupling strengths) lead 
to different relative populations of vertex types in a given sample. Using these statistical 
ensembles, a framework for effective thermodynamics and temperature has been developed 
by Nisoli and coworkers [115,116].  The probability that a system will be in a given state  
 36 
 
 
Figure 1.20 Rotation demagnetization protocols. Three possible rotation 
demagnetization protocols used in ASI studies. The amplitude of the magnetic field 
changes at a given rate and is reduced to zero. Figure adapted from Ref. [113].  
 
is described by the Boltzmann distribution and this probability is a function of the 
temperature, energy, and number of possible states of the system [122]. Since the exact 
microstate of an ASI sample can be precisely characterized and the total number of states 
fully enumerated, the Boltzmann distribution can be worked backwards to yield an 
effective temperature for a given a sample. More ordered states, closer to the ground state, 
have lower effective temperatures and other thermodynamic quantities, such as the 
configurational entropy, can be extracted as well [78,115,123].  
The artificial kagome lattice has also been extensively studied in both theory and 
experiment [78,93,95,124,125]. The kagome lattice is relevant and experimentally useful 
since it is derived from the pyrochlore lattice found in the rare earth spin ices. The plane of 
atoms perpendicular to the <111> crystal axis in the pyrochlore lattice of corner sharing 
tetrahedra is a kagome lattice of corner sharing triangles (see Figure 1.22(a)). Applying a 
magnetic field along the <111> crystal direction will isolate kagome planes since those 
spins will not couple to the external field. In artificial kagome ice, only three nanomagnets 
 37 
 
meet at each vertex and so the ice rules become “two-in, one-out” and “one-in, two-out” 
arrangements of spins (see Figure 1.22(b)). The local lowest energy configuration is six-
fold degenerate, just like the pyrochlore spin ice state, and there are two higher energy 
vertices corresponding to three-in or three-out spin arrangements.  
 
 
Figure 1.21 Statistics of moment configurations after a demagnetizing protocol. a. 
Deviation of vertex populations from a random configuration as a function of lattice 
spacing between neighboring moments. For larger spacings, the population is nearly 
random. As the spacing is decreased and the magnetostatic coupling between the spins is 
increased, the populations become no longer random and transition to a nearly ice-rule 
state. b. Average moment correlation values for different neighbors as a function of lattice 
spacing. If neighbors are in a favorable (unfavorable) configuration, the pair is assigned a 
1 (-1). The average over all pairs in the lattice is given as the correlation value. Figure 
adapted from Ref. [75].  
 38 
 
 
Figure 1.22 Artificial kagome ice. a. A plane of the pyrochlore lattice of corner sharing 
tetrahedra perpendicular to the [111] crystal axis is a kagome lattice of corner sharing 
triangles. A kagome plane is outlined at the top. These kagome layers alternate with 
triangular layers in the pyrochlore lattice. b. A kagome lattice with spins added to each 
lattice point showing ice rule ordering. The spins can either point directly in or directly out 
of each triangle and follow the one-in, two-out or two-in, one-out ice rule ordering. Figure 
adapted from Ref. [126].  
 
Rotation demagnetization experiments on kagome ice yield nearly equal 
populations of the six ice-rule vertex types [93]. While the ice-rule state in this lattice is 
easily achieved due to the extensive degeneracy of the vertices, artificial kagome ice is 
predicted to have a rich phase diagram with multiple ordered states and phase transitions 
due to long-range dipolar interactions in the system [98,125,127]. The calculated phase 
diagram is shown in Figure 1.23. For the highest effective temperatures, the system is 
disordered, paramagnetic, and does not obey the ice rules. As the temperature is lowered, 
the first phase transition and plateau in entropy corresponds to the ice-rule state, known as 
the kagome ice I state. This is still an extensively disordered state. The next phase transition 
corresponds to a charge-ordered state, which is known as the kagome ice II state. As in 
Figure 1.18, ASI bar magnets can be thought of as dumbbells of magnetic charge with the 
 39 
 
magnitude |𝑄| =
𝜇
𝑙
=
𝑀𝑉
𝑙
= 𝑀𝐴, where μ is the magnetic moment of the element, given 
by the magnetization M times the volume V. l is the length of the magnet or dumbbell and 
A is then the cross sectional area of the nanomagnet [22]. For ice-rule obeying states, there 
will be an excess +Q or –Q charge at each vertex (see Figure 1.24(a)). Configurations that 
violate the ice rule (Figure 1.24(b)) have +3Q or –3Q charges and are highly energetically 
unfavorable. The charge-ordered state has alternating +Q and –Q charges on neighboring 
vertices and a considerably lower entropy than the ice-rule state (see Figure 1.24(c)). There 
is an even further ordered state within the charge-ordered manifold which displays long-
range spin order and is known as the loop state [98,125,127]. This state has zero entropy 
as seen in Figure 1.23 and is shown schematically in Figure 1.24(d).  
Both the square and kagome ASI lattices can be used to study magnetic monopole 
excitations analogous to the ones found in the pyrochlore spin ices. Unlike in the real 
crystalline materials, the monopoles and their interactions in ASI can be imaged directly 
for a more precise understanding of their properties and behavior. The ground state of the 
square lattice is a perfect tiling of type I vertices. If one spin is excited and flipped within 
the ground-state order, it will create a pair of neighboring, oppositely charges monopole 
excitations as two type III vertices. The three-in, one-out and one-in, three out pair of 
vertices have +2Q and –2Q charges, respectively. These monopoles can separate and move 
throughout the lattice, leaving behind a Dirac string of excited type II vertices connecting 
them [76]. In the kagome lattice, there are ± Q charges at each vertex already and the 
monopole excitations are the ± 3Q ice-rule violating vertices. Monopoles have been 
proposed as the mechanism for magnetic reversals in ASI structures and have been directly  
 
 40 
 
 
Figure 1.23 Phase diagram of artificial kagome ice. Calculated phase diagram showing 
specific heat C (blue) and entropy per spin S (red) of kagome ASI as a function of effective 
temperature T. At high temperatures, the system is paramagnetic. As the system is cooled 
it first orders into an ice-rule obeying state known as the kagome I state. As it is cooled 
further, there is a predicted charge-ordered state known as kagome ice II. This state is 
shown in Figure 1.24(c). Finally, there is a transition to a spin-ordered state within the 
charge-ordered manifold. This state is shown in Figure 1.24(d). Figure adapted from Ref. 
[127].  
 
observed in both the square (see Figure 1.25) and kagome (see Figure 1.26) geometries 
[75,80,94,128].  
Disorder is known to play an important role in experimental investigations of ASI 
where it can arise from a variety of sources [94,95,118–121,130,131]. ASI samples are 
generally patterned using electron beam lithography (see Section 2.2.1). While this 
technique, produces very precise and controlled arrangements and is an essential  
 41 
 
 
Figure 1.24 Dumbbells of magnetic charge and ordered states in artificial kagome ice. 
a. ASI element modeled as a dumbbell of magnetic charge and schematics of ice-rule 
obeying moment configurations. Ice-rule obeying states in the kagome lattice have an 
excess +Q or –Q charge at each vertex. b. Schematic of excited, ice-rule violating states in 
the kagome geometry. Ice-rule violating states have an excess +3Q or –3Q at each vertex. 
c. Schematic of charge-ordering in kagome ASI. The predicted charged ordered kagome II 
state has alternating +Q and –Q charges on neighboring vertices. d. Schematic of the spin-
ordered, zero-entropy ground state in kagome ASI. There are six possible ground-state 
tilings. Three are shown here and the other three are given by a global reversal of spins. 
Part (d) adapted from Ref. [98].  
 42 
 
 
Figure 1.25 | Magnetic monopole motion in square ASI. Monopoles in the square lattice, 
type III vertex excitations connected by a Dirac string, are seen to move in the application 
of an applied field from (a) to (b). Figure adapted from Ref. [129].  
 
 
Figure 1.26 Direct observation of magnetic monopoles in kagome ASI. MFM images 
and corresponding spin maps of a kagome ASI sample in an applied magnetic field. Two 
pairs of ice-rule violating monopoles are observed to move apart from each other in the 
applied field from (a) to (b), leaving behind Dirac strings highlighted in green and blue. 
Another monopole also appears in (b) with its partner out of the field of view. Figure 
adapted from Refs. [80] and [128].  
 
fabrication tool, is still subject to some variation [75,95]. Magnets can have a distribution 
of widths or differences in edge roughness due to shot noise in the electron beam during 
lithography. There can be height variations in the material deposited for the magnets and 
different amounts of surface roughness depending on the deposition technique. Magnets 
may have slight variations in their location, not being placed in a perfectly ordered array 
 43 
 
[121]. These factors relating to the size of the magnets affect the coupling between 
neighboring ASI spins because they change the magnetic moments of the spins and thus 
the strength of the dipolar interactions between them. Besides variations in coupling 
strength, there can also be differences in the coercive field for individual magnets 
originating from variations in shape, size, or local microstructure in the magnetic material 
[94,121,132].  
While the origins of disorder in ASI are diverse, the effects are surprisingly 
consistent. Budrikis and coworkers showed in the square lattice that while there are 
different sources of disorder, their general effect is to broaden the energy levels of the 
different vertex types [121]. This is shown in Figure 1.27 where, as the amount of the 
different types of disorder is increased, the energy of each vertex spreads into a range of 
possible energies where eventually some of the ranges overlap, blurring the energetic 
distinction between vertex types. It makes ground-state ordering less energetically 
favorable and makes it easier for systems to access higher energy vertices and vertex 
configurations. Defects and disorder are also known to cause pinning sites for domain 
walls. I have observed this in my experiments and Monte Carlo simulations, as discussed 
in later chapters (see, for example, Figure 4.46).   
 44 
 
 
Figure 1.27 Disorder in square ASI. Disorder from different sources has similar effects 
on the energy landscape of the vertex types in square ASI. From top to bottom, the graphs 
show the relative energy of type IV, III, II, and I vertices due to disorder in (a) pairwise 
energy, (b) orientation, and (c) position. d. Disorder in the field required to switch a type 
II to a type III vertex and vice-versa. Figure adapted from Ref. [121].    
 
An interesting set of experiments performed in the Cumings group by Stephen 
Daunheimer on the kagome geometry gives additional insights into disorder in ASI 
systems. Daunheimer’s work involved applying a strong magnetic field to polarize kagome 
samples along one direction, rotating the sample 100°, 120°, or 180°, and then slowly 
increasing the magnetic field from zero until the sample was completely polarized in the 
other direction, capturing the state of the system at each field step. Daunhimer found that 
the individual magnets began to flip in stronger applied fields and the total magnetization 
of the crystal slowly reversed. This technique provides a measure of the disorder in the 
 45 
 
system by giving the spread in coercivities for the individual magnets and by the functional 
form of the magnetization curve. An example of these measurements is shown in Figure 
1.28. The shape of the reversal curves is different for different rotation angles due to the 
relative orientation between the applied field and the three magnetic sublattices in the 
kagome geometry. Depending on the field angle, different magnetic sublattices are more 
(or less) aligned with the applied field and couple more (or less) strongly to it. This 
technique can also be used to isolate the effects of the separate individual sublattices on 
the total magnetization so the sublattices can be studied independently. The experimental 
data can be fit with error functions so that the average coercivity and characteristic spread 
in coercivities can be extracted from the fit. The ratio of the spread in coercive fields to the 
average coercive field defines a figure of merit for the disorder in a sample. Daunheimer 
showed that fabricating kagome samples out of permalloy and in a connected geometry 
reduces the disorder compared to prior studies of kagome ASI [80,94,128].  
A noteworthy feature of the 180° magnetic reversals of artificial kagome ice is 
avalanches of reversed magnets. Avalanche phenomena are seen in a wide variety of 
physical systems including sand piles, rice piles, earthquakes, and Wolfram’s cellular 
automata [133–137]. A close inspection of the magnetization curves in Figures 1.28 and 
1.29 will reveal significant discontinuous jumps in the data. These jumps are large chains, 
or avalanches, of magnets that all reverse together in a single field step. The jumps are 
random and can take on many shapes, as seen in Figure 1.30. The avalanches are different 
every time, suggesting that there is stochastic disorder in the system [132]. The sizes of 
the avalanches follow a power law distribution with a cutoff given by the size of the finite 
system, or correspondingly, the largest possible avalanches (see Figure 1.31). Power law  
 46 
 
 
Figure 1.28 Disorder in kagome ASI. Magnetization vs. applied field for magnetic 
reversals of kagome ASI samples in three directions relative to the original polarization 
direction along one sublattice. The shape of the magnetization curves are a superposition 
of two error functions which fit characteristic spreads in the coercivities of individual 
sublattices that reverse as the applied field is increased. The spread in coercivity is a 
measure of the disorder in the system. The curves differ for the different reversal directions 
because the various sublattices couple more or less strongly to the applied field depending 
on the reversal angle. Figure adapted from Ref. [94]. 
 
behavior is characteristic in other systems that have avalanches, such as granular media 
and earthquakes, and power laws that stretch over several orders of magnitude are 
reflective of the self-organized criticality of the system [138].  
In all of the pioneering work on ASI, experiments and samples were exclusively 
athermal. That is, they showed no changes in their properties or behavior as a function of 
temperature. Samples were only studied by applying global, external magnetic fields. 
However, applying arbitrary magnetic fields only allows for certain transitions between 
different states of the system. In order to access predicted long-range ordered states, such 
as the ground state in the square lattice or the charge-ordered state in the kagome geometry,  
 47 
 
 
Figure 1.29 Avalanches in artificial kagome ice. Measured Magnetization vs. applied 
field in a 180° reversal of a kagome ASI sample. The large jumps in the magnetization data 
are due to large avalanches of magnets that all reverse at the same time. On the right are 
snapshots of the crystal at different points in the reversal process. Figure adapted from Ref. 
[132].  
 
 
Figure 1.30 Shapes of avalanches. Representative example of some of the shapes of 
avalanches observed by Daunheimer in his kagome ASI magnetic reversal experiments.  
 
 48 
 
 
Figure 1.31 Avalanche distribution in artificial kagome ice. Power law distribution of 
avalanche sizes in magnetic reversals of kagome ASI samples. Log of the probability of 
observed avalanche size vs. avalanche size. Experimental data is fit to a power law with a 
cutoff related to the size of the finite crystal studied. Figure adapted from Ref. [132].  
 
more transitions between different states of the system need to be opened up. One approach 
to achieving this is to allow for thermally activated reversals of individual magnets. 
1.4.3 Thermal activation  
 
Ice and spin ice have properties that explicitly depend on and vary greatly with 
temperature. Spins and atoms are constantly interacting and fluctuating, even down to the 
lowest temperatures. To improve artificial spin ice as a model for real systems and to 
expand on its functionality for investigating long-range ordered states in frustrated systems, 
 49 
 
thermal activation of the system has been introduced [78,79,81,82,84,85,87,95,139–141]. 
Including temperature dependent effects vastly increases ASI’s capability as an 
experimental tool. It allows for deep investigations of ordered and low energy states, as 
magnets that are allowed to interact and flip on their own will naturally try to minimize 
their energy within their local environment [78,79,85,95]. Allowing the system to relax 
and lower its energy makes it much more analogous to other physical systems that tend to 
find ground states. Thermal activation also introduces dynamics into the system – real 
timescales which can be measured and implemented in simulations to give a more realistic 
picture of what will happen in experimental systems [81,82]. This section will discuss 
techniques used to introduce thermal effects into ASI and give a survey of the noteworthy 
results thus far. The field is rapidly growing and this dissertation’s main focus in later 
chapters is on improving and adding to the techniques and knowledge of thermally active 
ASI.  
Before discussing the techniques and results of thermal flipping of magnets in ASI, 
it will be beneficial to understand the physics behind why magnets spontaneously flip 
magnetization. An elongated bar magnet has an anisotropy energy density K and volume 
V, giving it a total anisotropy energy KV. This energy must be overcome in order to flip 
the magnet, so if the thermal energy in the system is raised close to the anisotropy energy, 
the magnet will flip at a rate given by an Arrhenius law [142] 
𝜏−1 = 𝜈0𝑒𝑥𝑝(
−𝐾𝑉
𝑘𝐵𝑇
)    (1.13) 
where ν0 is a constant dependent on the material, KV is the anisotropy energy, kB is 
Boltzmann’s constant, and T is the temperature [143]. We can see that when KV is 
comparable to kBT that the rate for the magnet to flip approaches ν0. Microscopically, what 
 50 
 
is happening is thermal fluctuations are overcoming the exchange interaction energy that 
defines ferromagnet order, causing the magnet to lose a defined moment. The atoms may 
then reorder with a new magnetization direction which can be different from the initial 
direction. When the magnet is flipping often enough that it is hard to even define a 
magnetization, this is known as superparamagnetism and was first explained by Néel [144].  
In a nanomagnet without magnetocrystalline anisotropy, the anisotropy energy KV 
becomes the shape anisotropy energy of the magnet [81]. Magnets that interact with each 
other, such as in ASI arrays, are influenced by their local magnetic environment which also 
affects the flip rate. In this situation, KV is replaced by E0+ΔE, where E0 is the shape 
anisotropy energy, or the intrinsic energy barrier of the magnet and ΔE is the magnetostatic 
energy change if the magnet were to flip. The shape anisotropy energy is given by [145] 
𝐸0 =
1
2
𝑉𝐷𝑍𝑀
2  (1.14) 
where V is the volume, M is the magnetization, and Dz is the demagnetizing factor along 
the long axis of the magnet. If we consider an isolated magnet, ΔE is zero and we can see 
that if E0 is small, the flip rate for the magnet will be high. This can be accomplished with 
either a small volume, as in a very thin magnet, or a low magnetization. The intrinsic energy 
barrier for a permalloy nanomagnet is on the order of 104 K and dipolar coupling between 
the nanomagnets is also on the order of 104 K. This is why thermal fluctuations were not 
seen in the initial ASI experiments.  
A key point in considering the flip rates of nanomagnets is that the Curie 
temperature (TC) sets an energy scale for the system. As a magnetic material approaches 
TC from below, the saturation magnetization reduces continuously until it reaches zero at 
TC. A small magnetization will greatly reduce E0 and ΔE, allowing for thermal flipping. If 
 51 
 
the temperature of the magnet is below TC, it will also still have a finite magnetization and 
be able to interact with neighboring magnets. Essentially, when the thermal energy of the 
system is close to, but not above TC, the magnets will flip spontaneously in order to 
minimize their local energetic configuration. The temperature of the onset of thermal 
flipping is known as the blocking temperature (Tb) [144]. It is also well known that thin 
films of a magnetic material can have a lower TC than the bulk form [24,78].  
A seminal study by Morgan and coworkers found large domains of ground-state 
ordering in the square lattice could be formed through thermalization occurring during the 
growth of the ASI magnetic nanostructure [77]. As the magnetic material is deposited and 
the individual magnets grow in thickness, their volume and TC increases and the 
nanostructures go through a regime where they are superparamagnetic and able to flip 
spontaneously due to their temperature and local magnetic environment. This allows the 
magnets to lower their local energy configurations. By controlling the deposition rate and 
the substrate temperature, Morgan et al. allowed the system to order during growth. Once 
the magnets become thick enough, the blocking temperature becomes too high and the 
magnetization is frozen in place. The magnetic configuration was imaged immediately 
after fabrication (see Figure 1.32). This technique was successful at revealing the long-
sought square ground state, but the approach does not allow for much control over the 
samples or process. In order to efficiently and completely answer fundamental questions 
about these materials, more robust and flexible techniques are needed, allowing for more 
control over the system, the material parameters, the nanostructure geometries, the heating 
processes, as well as the ability to perform repeated experiments.   
 52 
 
 
Figure 1.32 Ground-state domains formed during growth. MFM image of a square ASI 
sample that thermalized during growth. Large regions of ground-state ordering of type I 
vertices separated by higher energy vertex domain walls. Inset shows vertex types and how 
they appear in the false color MFM contrast. Figure adapted from Ref. [77].  
 
 More recently, a new experimental approach has been devised that seeks thermal 
activation in ASI by heating close to the TC of the constituent materials 
[78,79,81,82,84,87,95,139–141]. In this technique, nanoscale magnets begin to exhibit 
spontaneous reversal, driven simultaneously by thermal fluctuations and by the local 
magnetic environment. However, the success of this approach thus far has been limited by 
the choice of material and the nanoislands’ geometries. Most studies have used permalloy 
 53 
 
(Ni80Fe20), which has a relatively high TC, approximately 800 °C in the bulk material [146].  
To accommodate such a high ferromagnetic ordering temperature, realizations utilize either 
very narrow temperature ranges, which are limited at the upper end by degradation of the 
nanostructures [78,79], or very thin films (≈3 nm) [81,82,85], which simultaneously 
reduce the onset temperature of thermally activated reversal and the desired magnetostatic 
interactions of the nanoscale bar magnets. Another technique is to use δ-doped Pd(Fe) 
monolayer stacks [84,87], which allow for a highly tunable TC, but the films are still very 
thin (~7 nm) and do not allow for strong magnetic coupling between macro-spins. A CoGd 
alloy has also been used to investigate the kagome lattice [140,141]. It has a relatively low 
TC, around 200 °C, though the material is ferrimagnetic and the magnetization density is 
still significantly lower than that of the commonly-used permalloy. Section 2.1 of this 
dissertation is dedicated to developing a material which overcomes these challenges and 
tradeoffs. 
Despite the challenges with these techniques, work along these directions has 
produced some exemplary results using them. Kapaklis and coworkers first demonstrated 
that the net magnetization of a patterned array of δ-doped Pd(Fe) ASI decreased before the 
TC of a continuous film of the same material, suggesting the nanomagnets were 
spontaneously flipping [87]. Later that year, Arnalds et al. showed thermally active 
building blocks of kagome ASI – one, two, or three hexagon rings – that ordered into 
ground-state or low energy configurations [84]. For this they used they also used δ-doped 
Pd(Fe) monolayer stacks. This system was then investigated further, describing the full 
energetic landscape of the few-hexagon samples and measuring the dynamics of the 
samples with PEEM, this time by fabricating arrays from ultrathin permalloy [81]. The 
 54 
 
material was deposited using a moving shutter method, so elements on one side of the 
sample where thinner than on the other side. The researchers then looked for regions of the 
sample that were fluctuating around room temperature and used those crystals for their 
studies. These magnets turned out to be ~3 nm thick, so they flip readily near room 
temperature, but don’t couple to each other as strongly as other systems. The same group 
also published results that directly showed the relaxation process from a polarized state to 
a nearly perfectly ordered ground state in the square geometry [82]. Snapshots of the 
process are shown in Figure 1.33. It is interesting to note how the samples relaxed towards 
the ground state. One can see chains of magnets that reverse initially, followed by the rest 
of the magnets filling in the space between the chains.  
Shortly thereafter, a study on both the square and kagome lattice was published by 
another group who used thicker permalloy nanoislands, ≈25 nm high, heated to ≈550 °C 
[78]. This work also showed large regions of ground-state order in the square geometry, 
with elements spaced closer together resulting in larger continuous ground-state domains. 
More importantly, Zhang and coworkers provided the first experimental demonstration of 
the charge-ordered kagome ice II state, shown in Figure 1.34. Similar to the square arrays, 
the size of the kagome charge-ordered domains increases when the spins are spaced closer 
together. However, the observed charge-ordered domains are fairly small and still quite far 
from the predicted perfectly charge-ordered state and the heating experiments with thicker 
permalloy are quite difficult. The substrate for these studies needs special treatment to 
reduce lateral diffusion of the permalloy. Still, there is only a very small temperature 
window where the experiments can be performed. Zhang reports magnets begin to flip at 
 55 
 
 
Figure 1.33 Direct observation of relaxation in square ice. PEEM with x-ray magnetic 
circular dichroism (XMCD) images of thermal relaxation in square ASI made from 
ultrathin permalloy. The sample first relaxes with “strings” and then forms large domains 
of ground-state order. Field of view in (a)-(h) is 20 µm, field of view in (i) is 50 µm. Figure 
adapted from Ref. [82].  
 
545 °C and structural degradation of the magnets sets in at 560 °C. This makes repeated 
experiments difficult and samples cannot last very long. Clearly, there are challenges and 
tradeoffs with all these methods for thermal activation of ASI. In Chapter 2, I introduce a 
new material for realizing thermal effects in ASI which will greatly improve the ease of 
 56 
 
these types of experiments, allow for more thorough explorations of the parameter space 
of these systems, and aid in answering fundamental questions about frustration.  
 
 
Figure 1.34 Charge ordering in kagome ice. a, c. MFM maps of thermalized kagome ASI 
crystals made from permalloy with lattice constant (a) 260 nm and (c) 490 nm. b, d. 
Corresponding charge domain maps showing incipient regions of charge-ordering. Blue 
and red dots correspond to vertices within the two degenerate charge-ordered states while 
green and yellow dots represent ice-rule violating vertices. The smaller lattice constant 
crystal shows better ordering because it has stronger magnetostatic interactions between 
neighboring elements. Figure adapted from Ref. [78].  
 
 
 
 
 57 
 
1.4.4 Other lattices  
 
In addition to square and kagome 2D arrays fabricated from magnetic materials, 
there are a variety of other materials-by-design, artificially frustrated systems that have 
been studied both experimentally and theoretically. The shakti and tetris lattices are 
derivatives of the square ASI lattice and are fabricated from nanomagnets, but with selected 
elements intentionally removed [79,85,147,148]. These systems become more complex as 
they have vertices where 2, 3, and 4 elements meet. Figure 1.35 shows the structure of the 
two lattices. The 4-element vertices have the same energy landscape as in square ASI 
samples, but the 2- and 3-element vertices are new and different. The ground state of these 
geometries typically has all of the 4-element vertices as type I and all of the 2-element 
vertices in head-to-tail configurations (type a or type α in Figure 1.35(c)).  
The frustration in the system and degeneracy of the ground state arises from the 
assignment of the 3-element vertex types. These vertices have a lowest energy 
configuration, type A in Figure 1.35(c), but it is physically not possible for all of the 3-
element vertices to be type A. In the shakti lattice, a maximum of 50% of the 3-element 
vertices can be type A, forcing type B vertices into the system and creating frustration and 
disorder from where these vertices are distributed [79]. The system cannot satisfy the 
lowest energy configuration at each vertex, thus it is frustrated, and there are many 
configurations with different locations of the type A and B vertices that all have the same 
ground-state energy. Similarly, the tetris lattice cannot have only type A 3-element vertices 
and must contain a finite number of type B vertices [85]. The frustration due to these 
geometric constraints has been termed “emergent vertex frustration” and both of these 
lattices have been investigated as thermally active systems. An interesting feature in the  
 58 
 
 
 
Figure 1.35 Shakti and tetris lattices. Scanning electron microscopy (SEM) images of (a) 
the shakti and (b) the tetris ASI lattices. Both are derived from the square lattice with some 
elements removed. c. Vertex classifications for these lattices. The type I, II, III, and IV are 
the same as the square lattice, type A, B and C are 3-moment vertices, and type a and b or 
α and β are the 2-moment vertices. Figure adapted from Refs. [79] and [85].  
 
shakti results is that there is charge-ordering in the 3-element vertices. The type A and B 
vertices have a ± Q charge on them and these charges are observed to arrange themselves 
in domains of order when the system is thermalized. Additionally, it is observed that type 
III 4-element vertices, which have a strong ± 3Q charge, attract opposite charges towards 
them. These charge-charge interactions have a strong theoretical footing, as Monte Carlo 
 59 
 
simulations show Coulomb interactions between charged vertices are required in order to 
reproduce experimental data on vertex populations [79]. 
Chern and coworkers have proposed a three dimensional ASI system which 
overcomes some of the issues with the square geometry [149]. 2D square ASI is not an 
exact analog to pyrochlore spin ice because of the inequivalent interaction strengths 
between adjacent and opposite spins on the square lattice – one magnet is further away than 
the other two. This breaks the six-fold degeneracy of the ice ground state. To restore the 
full degeneracy, Chern et al. propose a 3D structure with alternating, stacked planes of 
nanomagnets. With the correct height between layers, the interaction strength between the 
four magnets at a vertex becomes equal. This system would be very interesting to study, 
though it faces some difficult challenges to realize. First, the height between layers must 
be controlled with extreme precision or else the magnetic interactions would not be equal. 
Alternating layers need very precise alignment as well. Finally, one must use measurement 
techniques that can resolve the buried moments in different layers. Improvements in 
fabrication techniques and development and improvement of 3D imaging techniques could 
help realize this idea. 
Most experimental realizations of ASI have spins with in-plane magnetic moments 
due to the geometry of the nanomagnet and the chosen magnetic material. In many physical 
systems, however, spins are not confined to be in a plane. Many models of frustration have 
out-of-plane spins, notably the 2D Ising model, where Ising spins are perpendicular to a 
lattice of a given geometry [26]. The picture in Figure 1.13(c) illustrates a 2D Ising model 
on a triangular lattice with antiferromagnetic coupling. To investigate this and similar 
models, a version of ASI has been created by Zhang and coworkers using the perpendicular 
 60 
 
magnetic anisotropy in Co/Pt multilayer stacks patterned into nanodisks in different 
geometries [150]. This system has spins pointing out of the 2D plane of the lattice and 
displays regions of ordering depending on the geometry and coupling strength between the 
nanodisks. Other unique realizations of ASI include Penrose quasi-crystals [151,152], 
which show interesting hysteretic behavior in magnetic reversals, and actual macroscopic 
bar magnets arranged in a kagome geometry and allowed to rotate out of plane [153]. 
Finally, frustration has been studied experimentally and theoretically in artificially 
designed systems without nanomagnets. Vortices in superconductors with lithographically 
patterned double well pinning sites can be used as up/down spin analogues [130,154,155]. 
These add an additional element to the system because while normally the sites have one 
vortex in them, they can have defects with two or zero vortices, producing unphysical 
behavior analogous to a magnet with two north or two south poles [130,131]. A similar 
realization of artificially frustrated systems is with interacting colloidal particles in periodic 
optical traps or soft matter [131,156–161]. These systems can also have doubly occupied 
or empty traps. Both superconducting and colloidal systems can be thermalized during real 
temperature changes or by applying external stimulus like current passing through the 
superconductor [131,158,159].  
In this chapter, I have discussed the wide body of work which this dissertation fits 
into. There are clear challenges I aim to overcome with my work, starting with finding a 
material system for realizing thermal activation. Improvements in the material and 
thermalization technique will allow for easier experiments and more flexibility, resulting 
in more results to compare with theory. Improved experimental methods will enable this 
work to answer outstanding questions on long-range order in frustrated systems, especially 
 61 
 
in the kagome ASI geometry. Additionally, I introduce a new type of frustration, 
specifically due to topological defects, and show how these affect long-range order in the 
square geometry, an otherwise ordered and well-understood system.  
  
 62 
 
Chapter 2  Fabrication, experimental, and numerical methods 
 
This chapter will give a complete description of the fabrication process and the 
experimental and numerical methods used in this dissertation. A major aspect of my 
research was developing a process to grow thin films of magnetic material with favorable 
properties for thermally active ASI studies so that I could fabricate high quality ASI arrays 
from these films. This chapter begins with a description of our material of choice, FePd3, 
its advantages over other groups’ materials, and the process I developed for growing high 
quality films and characterizing them. I will then give a detailed description of the 
fabrication process I developed including various tips and tricks that may help future 
researchers working on this project. Next I will provide an overview and introduction to 
our most important experimental tool, the transmission electron microscope, specifically 
describing the Lorentz transmission electron microscopy (TEM) technique we use to 
characterize our samples and image exact spin configurations. I will describe the heating 
experiments I performed and highlight the advantages of our samples and experimental 
techniques over other groups’ and show how we overcome previous challenges in the field. 
In this section I will also present my results on thermally active square ASI as these help 
demonstrate the advantages of FePd3 as well as provide concrete examples as I introduce 
the experimental and numerical techniques I use. This chapter concludes with the theory 
and details of the Monte Carlo simulation techniques I employ as well as a framework for 
modeling the various different experiments discussed in this dissertation. In addition, I will 
present the results of my Monte Carlo simulations modeling the thermally active square 
 63 
 
ASI geometry. Overall, this chapter will detail the methods used throughout this research 
and the remaining chapters of this dissertation.  
 
2.1 FePd3 as a material for studying thermally active ASI systems 
 
2.1.1 Introduction 
 
 
An ideal material for thermal activation of ASI would exhibit simultaneously a high 
magnetic moment per atom and a relatively low TC. The high moment allows for strong 
magnetostatic interactions between the macro-spin bar magnets. On the other hand, the 
lower TC – ideally not far above room temperature – allows accessing thermal activation 
while avoiding high temperatures where nanostructures are prone to degradation. Unlike 
previous studies, which have utilized thin magnetic layers with weak interactions 
[81,82,84,85,87] or thick layers and high temperatures [78,79], in this chapter I report my 
work developing a materials system for ASI studies that simultaneously incorporates low 
temperatures, thick layers, and strong magnetostatic interactions, which are all highly 
desirable features for thermally active ASI.   
 I developed techniques for a FePd alloy, close to the stoichiometry of FePd3, for 
my thermally active ASI samples. To the best of my knowledge, FePd3 has not previously 
been used for such studies. According to binary alloy work [162], the FexPd1-x system has 
a local maximum in the TC, centered on the FePd3 composition, so that fabrications of films 
within a wide range near this stoichiometric ratio, about 5%, still result in uniform 
properties and high experimental utility. Nevertheless, FePd3 is a well-defined crystal 
structure, with a wide solid-solution concentration window. The phase diagram for the Fe-
 64 
 
Pd system is given in Figure 2.1 with the experimentally measured TC as a function of 
composition highlighted. At the FePd3 stoichiometry, the TC of the bulk material is reported 
as ≈260 °C [162–164].   The bulk material can also have a moment density as high as 
650 kA m-1 [165] which is comparable to permalloy and other strongly magnetic materials. 
The TC is ideal for ASI studies because the moments are stable at room temperature, but 
can be heated easily and quickly, not too hot and far from where the nanomagnets will 
begin to structurally degrade or melt, in order to see spontaneous flipping of the 
magnetization of individual magnets.   
The properties of FePd3 have some dependence of how well ordered the atoms are 
in the lattice. FePd3 is known to form a face centered cubic (FCC) L12 ordered phase, shown 
in Figure 2.2. This structure has the Pd atoms on the faces of the cube and the Fe atoms on 
the corners. A perfectly ordered alloy will have all its atoms in these locations, though this 
is generally never the case and there is always some level of randomness and atomic 
disorder in the system. The degree of disorder can affect the material’s properties because 
the different atomic magnetic moments and exchange interaction between the atoms 
depends on the exact interatomic distance [164]. As we will see, annealing films of FePd3 
to allow the atoms to find more ordered positions greatly improves the magnetic properties 
of this material for ASI studies.  
The FePd system has unique magnetic properties due to the interaction between the 
Fe and Pd atoms. Pd is not intrinsically magnetic, but coupling with the Fe atoms can 
induce a magnetic moment. This occurs through a short range exchange interaction 
between the Pd 4d and the Fe 3d electronic orbitals which results in a hybridization of these 
 65 
 
orbitals [164,168,169]. This splits the Pd 4d band into two bands, one for each spin 
orientation. The density of states for each of these bands is different, causing an imbalance 
 
 
Figure 2.1 Phase diagram of the Fe-Pd system. Temperature and composition dependent 
binary phase diagram. TC is given by the dotted line where there is a local maximum 
centered on the FePd3 composition. Figure adapted from Ref. [166].  
 
 
Figure 2.2 L12 ordered phase. Crystal structure of ordered FePd3. The lattice is FCC with 
Fe on the corners and Pd on the faces of the cube. Figure adapted from Ref. [167].   
 66 
 
in the two spin populations and an induced net magnetic moment of the Pd atom [164,170]. 
The hybridization also affects the Fe moments, with an increasing atomic Fe moment for 
higher Pd compositions. At the FePd3 composition, the Fe moment is relatively high due 
to the high concentration of Pd in the system. Elemental Fe has an atomic magnetic moment 
around 2.2 Bohr magnetons (µB) [17]. The calculated and measured Fe moments in FePd3 
agree well with each other and are approximately 3.1 µB, considerably higher than 
elemental Fe due to the interaction with the Pd. The Pd moment is ~0.4 µB, giving a total 
moment per FePd3 fundamental unit of 4.3 µB, or about 1 µB per atom [164,171,172]. 
Additionally, the unique exchange interaction between the Pd 4d and the Fe 3d orbitals is 
also responsible for the relatively low TC of the material. The short range exchange between 
the orbitals has a relatively small energy difference between the ferromagnetic and 
paramagnetic states which results in the low TC.  
The value of the magnetic moment for each atom varies with composition but also 
with the lattice constant of the material, or equivalently the spacing between neighboring 
atoms. Burzo and Vlaic report a calculated spontaneous onset of the large Fe moment and 
modest Pd moment when the lattice spacing reaches 3.4 Å and a maximum calculated 
moment for each atom at 4.3 Å [164]. Under 3.4 Å the exchange interaction causes the 
atoms to have no magnetic moment as there is a transition to antiferromagnetic exchange 
below this critical value [164]. The equilibrium lattice constant for FePd3, both calculated 
and experimentally measured, is approximately 3.85 Å [164,172]. This explains why 
annealing the FePd3 films is essential. If the atoms do not find highly ordered states with 
equilibrium lattice spacings or interatomic distances greater than 3.4 Å, the magnetic 
moment of the atoms are significantly reduced. Annealing helps the atoms move into 
 67 
 
equilibrium locations where they are spaced correctly to have favorable exchange 
interactions and strong magnetic moments.  
FePd3 has three other properties that make it favorable for thermally active ASI 
systems. First, the system has very low magnetocrystalline anisotropy. Strong crystalline 
anisotropy can compete with the shape anisotropy of the elongated bar magnets in ASI and 
reduce the success of the Ising-like behavior. A low crystalline anisotropy also makes it 
easier to fully reverse the nanomagnet and result in all magnetic grains orienting along the 
long axis, increasing the total moment of the bar magnet. FePd3 is known to have a first 
anisotropy constant K1 = -2.0 x 10
3 erg cm-3, which is very low compared to, for example, 
cobalt, which has K1 = 4.1 x 10
6 erg cm-3 [17,165]. Second, the material has a low 
coercivity, so that the spins are able to flip easily under the influence of nearby magnets 
when in a thermally active regime. The low anisotropy and low coercivity make FePd3 
what is known as a “soft” magnet. Third, the magnetization of many materials near TC is 
known to follow a behavior given by [21] 
𝑀(𝑇) ~ (𝑇𝐶 − 𝑇)
𝛽 (2.1) 
where β is a material dependent parameter. β has been measured for FePd3 and is found to 
be ~0.37 [163]. This is favorable because the magnetization will decrease slowly as T 
approaches TC, so there is a wide temperature range where the magnets will be thermally 
active and still posses a finite magnetization. If the transition at TC were very sharp, it 
would be difficult to find temperatures that allowed for thermal activation and magnetic 
relaxation in the lattice. In the following sections, I will present measurements of my FePd3 
films that confirm all of the favorable magnetic properties discussed in this section.  
 
 
 68 
 
2.1.2 Film growth  
 
The ultimate quality of the data from thermally active ASI samples depends heavily 
on the quality of the films they are fabricated from. Thus, growing high quality films of 
FePd3 is essential to success of this project. This section will give a detailed description of 
the film growth process I developed for this project. It is somewhat specific to the tools 
and resources available at the University of Maryland (UMD) and to the experimental 
characterization techniques I use as well, though researchers from other institutions should 
still find the information useful.  
I use TEM as my main experimental tool and as I will discuss in Section 2.3, TEM 
has many advantages over other typical ASI characterization techniques. In order to 
effectively use TEM to image my samples and their precise magnetic configuration, 
samples must be fabricated on a substrate that is electron transparent – that is, a substrate 
that does not have contrast of its own. The Lorentz TEM technique and associated image 
processing required to characterize exact spin directions is extremely sensitive to small 
differences in contrast in the TEM images, thus it is essential that the substrate does not 
add any additional contrast.  
The substrate of choice for my work is a SiN membrane, close to the stoichiometric 
ratio Si3N4. A Scanning Electron Microscopy (SEM) image of one is shown in Figure 2.3. 
The substrate consists of a 3 mm square Si frame, 200 µm thick with a 500 µm square 
window cut out of the middle. On top of the whole frame is a 100 nm thick, electron 
transparent SiN membrane. ASI samples are fabricated on the membrane suspended over 
the cut-out window. While the substrates are necessary for TEM studies, there are some 
challenges in working with them. The 100 nm thick membranes are very fragile and can  
 69 
 
 
Figure 2.3 SiN membrane substrate. Scanning electron microscopy image of a SiN 
substrate for TEM studies. The substrate consists of a 200 µm thick, 3 mm square Si frame 
with a 500 µm window cut out of it. On top of the whole Si frame is a 100 nm thick SiN 
electron-transparent membrane. Figure adapted from Ref. [173]. 
 
break easily if dropped or touched. Also, samples must be fabricated one at a time or in 
small batches compared to other Si wafer based processing. Still, the advantages of using 
TEM outweigh the challenges of working with the SiN substrates. 
Before the FePd3 is deposited, the substrates are prepared with thin underlayers. 
First, a 2 nm Ti adhesion layer is electron beam evaporated onto the bare SiN membrane. 
Electron beam evaporation will be discussed further in Section 2.2.2. This layer is 
important because Ti bonds strongly to the SiN and prevents the films from delaminating 
as Pd adhesion to SiN is not always very strong. Second, a 2 nm Pd wetting layer is also 
electron beam evaporated on top of the Ti. This is the layer the FePd3 will be deposited 
onto and helps facilitate diffusion of the FePd3 during annealing, which is essential to 
achieving a high quality film. The wetting layer helps control the microstructure of the 
film, reducing surface roughness and nucleating small grain sizes in the FePd3, both 
 70 
 
important factors in the film quality. Small grains help keep the magnet soft and reduce the 
effects of any magnetocrystalline anisotropy in the material.  
 After preparing the substrates, they are removed from vacuum and transferred to a 
sputtering system where the FePd3 is deposited. The sputtering process is shown 
schematically in Figure 2.4. The technique uses a vacuum chamber filled with Ar gas and 
a gun with a target of the desired deposition material mounted on it. A large RF potential 
is applied to gun relative the chamber which ionizes the Ar atoms in the gas to a +1 state, 
creating a plasma. The plasma is confined to an area near the target using strong magnetic 
fields from the gun and the charged Ar+ ions are accelerated towards the target with an 
electric potential. When the Ar+ ions hit the target, they knock atoms of the target material 
off the surface which then diffuse through the Ar gas down to the substrate mounted below 
(see Figure 2.4).  
To deposit FePd3 alloy films I made a custom sputtering target from a pure Fe target 
decorated with Pd foil. A photograph of my target is shown in Figure 2.5. Essentially, the 
ratio of the surface area of the exposed Fe to the exposed Pd dictates the composition of 
the films deposited. I made the target modular so that I could tune the composition by 
adding or removing pieces of Pd foil until the films were near the desired stoichiometry. 
To fabricate the target, I first spot welded pure Fe wire in two concentric octagons. I was 
then able to slide pieces of Pd foil under the Fe wire to secure them to the target. In the 
next section, I will discuss how I characterized the composition of the films to know when 
I had reached the FePd3 composition. 
 
 71 
 
 
Figure 2.4 Sputtering. Gas is fed into a vacuum chamber which is then ionized, creating a 
plasma, and accelerated at a source target. The sputtering gas ions knock material off the 
target which diffuses back through the gas and lands on a substrate, growing a thin film of 
the source target material. Figure adapted from Ref. [174].  
 
  The first step in my sputtering process is to load the samples into the vacuum 
chamber in the Center for Nanophysics and Advanced Materials (CNAM) at UMD and 
pump it down to base pressure. I would typically load 6-9 individual membranes with the 
Ti/Pd underlayers into the chamber for each deposition. I often also loaded small Si wafer 
dies as witness samples for the deposition. The membranes and dies were placed on a small 
Al holder which was then loaded onto the stage of the sputter chamber. I pump down the 
chamber first with a rough mechanical pump and then cross it over to a diffusion pump. 
The base pressure of the chamber varied over the years I used it, depending on what 
materials other users were depositing, and ranged from 3.0 x 10-6 Torr to 1.1 x 10-5 Torr. 
Though I always tried to get the base pressure as low as possible, I saw no correlation 
between a given pressure and the quality of the films grown at that pressure. After the 
chamber reaches base pressure, I fill it with flowing Ar gas. I adjust the flow rate and pump 
 72 
 
 
Figure 2.5 FePd3 sputtering target. My custom built target is made from a 3 inch pure Fe 
target with Fe wire welded onto it. Pieces of Pd foil are slid under the wire. The ratio of 
the surface area of the Fe to the Pd dictates the composition of the film that is deposited. 
Scale bar is 0.5 inches.  
 
rate of the diffusion pump (by closing the gate valve of the diffusion pump most of the 
way) to get an Ar pressure of 4.0-4.3 x 10-3 Torr in the chamber. I then spark the plasma 
with 200 W RF power. There is a shutter that protects the membranes while the system 
pre-sputters for a few minutes to remove any oxide layer that has formed on the target since 
the last sputtering run. The color of the plasma changes from pink to purple once the oxide 
layer is removed. I then open the shutter and deposit 23 nm material onto the membranes 
at a rate of about 1.1 Å s-1. The deposition rate, as well as the composition of the films 
 73 
 
deposited, depends on the sputtering power. I generally kept the power constant at 200 W 
for consistency.  
The deposition rate and total thickness are measured using a quartz crystal and 
crystal monitoring system which I calibrated for my target and other deposition parameters. 
I used the density and Z ratio for Pd and changed the tooling factor to account for the alloy 
composition. To calibrate, I would bake drops of resist (see Section 2.2.1) onto Si witness 
samples and deposit 100-150 nm of material onto the samples. I would then remove the 
drops of resist and measure the step height using a profilometer. Averaging over several 
steps on one sample and over several different samples generally gave an accurate 
measurement of the actual film thickness. I would then use the experimentally measured 
thickness compared to the crystal monitor reading to adjust the tooling factor and would 
repeat the whole procedure if necessary.  
As-deposited films tended to be very rough, columnar, and generally not even 
ferromagnetic. An annealing procedure was essential to making the films magnetic, 
reducing the surface roughness, and achieving a nice grain structure. Figure 2.6 shows 
before and after annealing images of the films. The sputter chamber I used to deposit the 
films did not have a heating stage, so I would remove the films from that chamber and load 
them into a different system in Professor Ichiro Takeuchi’s lab at UMD. I placed them on 
a steel holder and then positioned the holder on a heating stage in an Ultra-High Vacuum 
(UHV) chamber. I would anneal the films at 750 °C for three hours in UHV (base pressure 
< 86 10 Torr). There are numerous papers that report an improvement in FePd3 magnetic 
properties after annealing at 600 °C or higher [162–165]. After annealing, I would  
 
 
 74 
 
 
Figure 2.6 Annealing of FePd3 film. High resolution TEM images of FePd3 films (a) 
before and (b) after annealing showing the grain structure of the film. Before annealing, 
the film has a columnar structure with large gaps and cracks.  After annealing, the film is 
continuous and grains are 3-7 nm in diameter. Scale bar is 30 nm. 
 
characterize the films for their composition and magnetic properties. The characterization 
techniques I used are described in the next section. 
2.1.3 Film characterization  
 
 
Once FePd3 films are fabricated, it is essential to check their properties before 
patterning them into ASI arrays. The main characteristics I checked were the composition, 
the grain structure, the magnetization and how well they supported magnetic domains and 
domain walls, and their TC. Much of this characterization can be done with TEM, though I 
used other tools as well. During the initial stages of developing the film growth process I 
was only measuring the composition of the films in order to tweak the design of my alloy 
target to get the composition near FePd3. 
 75 
 
 Composition is characterized using Wavelength Dispersive x-ray Spectroscopy 
(WDS) on a JEOL JXA-8900 electron probe microanalyzer. WDS is a very quantitative 
and precise technique for measuring the composition of materials. A beam of electrons is 
accelerated towards the material at high energy which knocks out core electrons of the 
atoms. This leaves the atoms in a very unstable state and an outer shell electron quickly 
transitions down to fill the core shell vacancy, emitting an x-ray with an energy 
characteristic of the specific element it came from. The x-rays are collected and used to 
identify the elements present and their relative concentrations. Often, the x-ray energies 
are measured directly in a technique called Energy Dispersive x-ray Spectroscopy (EDS). 
However, this technique does not have very precise energy resolution and it can be difficult 
to distinguish different elements that have characteristic x-ray energies very close to each 
other.  
Since the wavelength of a photon is inversely proportional to its energy by hc, 
where h is Plank’s constant and c is the speed of light, the x-ray can be equally well 
characterized by its wavelength. WDS tends to be more precise than EDS in quantitative 
measurements of composition because it can better distinguish wavelengths that are close 
together. The x-rays are passed through different crystals which diffract them using 
Bragg’s law [17] 
 2dsinθ = nλ (2.2) 
A detector in the microprobe is moved to different θ to help detect the different 
characteristic wavelengths. The microprobe analyzer also has a built-in computer with 
calculated and well characterized x-ray wavelengths and emission rates for different 
elements as well as sophisticated algorithms for determining relative emission rates for the 
 76 
 
different characteristic x-rays. Before each measurement, these rates and the exact 
elemental wavelengths are re-calibrated from samples of pure Fe and Pd always kept inside 
the tool and this aids is giving highly quantitative and precise elemental information. I 
performed numerous depositions from my FePd alloy target, adding additional pieces of 
Pd foil until I was consistently getting compositions within a few atomic percent of the 
FePd3 stoichiometry.  
After I finalized my target design, the exact composition of my films still varied 
slightly from run to run due to the chaotic nature of sputtering and my nonstandard target 
design. However, in a single deposition there was generally very uniform composition, 
only varying by 2-3% across several substrates spread out over an 8 cm deposition area, 
with each 3 mm substrate being highly uniform in composition. Different runs gave slightly 
different magnetic properties and sometimes different TCs dependent on the given 
composition. For each batch of films, I would measure the TC using Lorentz TEM and a 
specialized heating holder. A schematic of the heating holder is shown in Figure 2.31 and 
this holder will be discussed further in Section 2.4.  
Figure 2.7(a) shows a Lorentz TEM image of one deposition of a 23 nm thick FePd3 
film at room temperature. The textured bright and dark contrast in the image indicates the 
magnetization of the film is relatively high as the amount of contrast is proportional to the 
magnetization in the film [175]. Figure 2.7(b) shows that when the film is heated to 140 °C, 
the Lorentz contrast disappears, indicating the material is near or above its Curie 
temperature. Some early films I made had a TC closer to 110 °C, but most of the later ones 
had a higher TC, around 140 °C. I will note that measuring the TC using this method is not 
perfectly accurate due to some considerations with the in-situ heating holder I will discuss  
 77 
 
 
Figure 2.7 TC characterization. Lorentz contrast TEM images of a 23 nm FePd3
 thin film 
at (a) room temperature and (b) 140 °C. The textured contrast in (a) indicates the film has 
a high magnetization, while the absence of contrast in (b) signifies the material is near or 
above its Curie point. Scale bar is 20 μm.  
 
in Section 2.4. The TC of the films may have been slightly higher than reported here as I 
will show with some measurements at the end of this section. 
Figure 2.6(b) in the previous section presents a high resolution TEM image showing 
the grain structure of a film. Grains in my films were typically 3-7 nm in size, which is 
favorable for thermal ASI studies since it will give rise to a relatively low coercivity. The 
grain structure is generally very consistent between different depositions and is a good way 
to check that the annealing process was successful. One of the largest variations between 
different film growths is the magnetization of the films and how well they support magnetic 
domains. A film with nicer domains and sharper domain walls in the Lorentz TEM images 
tended to have better Lorentz contrast in the individual patterned nanoscale bar magnets 
 78 
 
and yielded better data and more-ordered states than a film that only displayed ripple 
Lorentz contrast.  
An example of the two main types of contrast observed is shown in Figure 2.8. The 
films with sharper domain walls have a higher magnetization which is better for ASI 
studies and probably have less surface roughness as well. The variability of the total 
magnetization and surface roughness could depend on many factors, though most likely it 
is the sputtering process and the annealing of the films that have the greatest effect. Some 
batches of films came out better than others and I did not have as much control as I wanted 
over the quality and consistency of them. I used the best films with the best magnetization 
and sharpest domain walls for my samples and the data in this dissertation, but I made 
many other films that did not turn out as well as those. At the end of this section, I discuss 
some possible improvements in future work to overcome these challenges and increase the 
consistency of the FePd3 films.  
Finally, a group at the University of Illinois that also studies thermal ASI samples 
was generous enough to take some SQUID microscope measurements of my FePd3 films. 
Figures 2.9 and 2.10 confirm all of the favorable magnetic properties of my films discussed 
in this section and are credit to Ian Gilbert and Peter Schiffer at the University of Illinois. 
The only discrepancy between what I characterized and their measurements is the TC of the 
film which appears to be slightly higher than 140 °C, the temperature where I saw the 
Lorentz contrast disappear in the TEM images. This may indicate that there is still some 
magnetization in the film when the Lorentz contrast disappears, just not enough to give any 
noticeable contrast. Figure 2.9 shows the magnetization as it approaches TC, showing a 
nice gradual approach, though it does not go to zero at 140 °C. The hysteresis curves in 
 79 
 
Figure 2.10 confirm that the coercive field is very low and that there is no strong crystalline 
anisotropy in the films.    
 
 
Figure 2.8 Lorentz contrast in different FePd3 depositions. a-d. Lorentz TEM images 
of thin films of annealed FePd3. In (a) and (b), the films have a high magnetization and 
low surface roughness which supports magnetic domains and shows sharp, strong Lorentz 
contrast. In (c) and (d), the films only display ripple contrast, indicative of more surface 
roughness which is less favorable for fabricating high quality ASI samples. All scale bars 
are 100 µm.   
 
 80 
 
 
Figure 2.9 Magnetization as a function of temperature. Field cooled (FC) and zero field 
cooled (ZFC) magnetization M vs. temperature T for a thin film of FePd3 showing a gradual 
approach to TC. The TC appears to be slightly higher than was measured in Lorentz TEM 
heating experiments shown in Figure 2.7, though it is still relatively low.  Measurement 
and figure courtesy of Ian Gilbert and Peter Schiffer at the University of Illinois. 
 
 
In future work on this project, improving the consistency and quality of the FePd3 
films will be essential. I fabricated a number of very good films that yielded useful data, 
but I also made many films that did not have as desirable properties. I was able to work 
with the high quality films I had, but it would have been better to be able to readily produce 
the highest quality films. To accomplish this, it would be beneficial to custom order a 
uniform FePd3 alloy sputter target to get more consistent depositions. I believe that a 
careful study of the annealing temperature and time would also help produce the best 
possible films. I had to break vacuum between depositing and annealing my films and if 
this process could be accomplished in one chamber it would certainly reduce the time it 
takes to grow the film and almost certainly improve the properties by reducing the 
oxidation of the as-deposited film shown in Figure 2.6(a). I did some work to study the  
 81 
 
 
 
Figure 2.10 Hysteresis curves. a, b. Magnetization vs. applied field hysteresis curves for 
a thin film of FePd3. The measurement was taken at 50 K for both in-plane and out-of-
plane magnetization. The film shows no sign of in-plane anisotropy which is favorable for 
ASI studies. (b) shows a higher resolution detail of the same measurement in (a).   
Measurement and figure courtesy of Ian Glbert and Peter Schiffer at the University of 
Illinois. 
 
 82 
 
effects of the underlayers, and tuning the thickness of these or potentially using different 
material could help as well. Overall, a new target and a fine-tuned annealing procedure all 
in a single vacuum chamber should improve reliability and yield excellent films with 
correspondingly good data. 
 
2.2 Fabrication  
 
 
An important part of my project was to develop a process to pattern my FePd3 films 
with favorable properties for thermal activation of ASI into usable ASI samples. I built 
upon Stephen Daunheimer’s work in our group making kagome ice samples out of 
permalloy [132], but had to significantly modify the process to adapt it to the FePd3. I also 
fabricated some ASI geometries that were different from those made by Daunheimer and 
altered the process to suit the new geometries. This section will give a broad overview of 
each technique and tool used in my fabrication process as well as all of the details required 
to reproduce the process.  
An overview of my method is shown in Figure 2.11. Starting with the fully 
characterized FePd3 films on top of SiN TEM membrane substrates, I spin coat a bilayer 
polymer resist onto the membranes (Figure 2.11(c), Section 2.2.1). I perform electron 
beam lithography (Figure 2.11(d), Section 2.2.1) to define the ASI patterns and then 
electron beam evaporate an Al etch mask over the whole sample (Figure 2.11(e), Section 
2.2.2). A liftoff step is performed (Figure 2.11(f), Section 2.2.3) and finally, the pattern is 
transferred into the FePd3 with Ar ion milling (Figure 2.11(g), Section 2.2.4). Characteristic 
images of samples I made are shown in Section 2.2.4. In Section 2.2.5, I discuss some  
 83 
 
 
Figure 2.11 Fabrication process overview. The sample starts as a bare SiN TEM 
membrane substrate (a). A 2 nm Ti adhesion layer and 2 nm Pd wetting layer are electron 
beam evaporated onto the membrane. Then a 23 nm FePd3 film is deposited using RF 
magnetron sputtering (see Section 2.1.2) (b). The sample is spin-coated with a bilayer resist 
(c) and electron beam lithography is performed (d). A 16 nm Al etch mask is electron beam 
evaporated over the whole sample (e) and liftoff is performed (f). The sample is finally 
etched using Ar+ ion milling (g), transferring the pattern into the FePd3 film (h).  
 84 
 
common problems and modes of failure for the fabrication process with the intention that 
these may help future researchers intending to use the procedure I developed. 
The method for fabricating samples of FePd3 ASI needed modification from 
Daunheimer’s process [132] for permalloy samples because of fundamental differences in 
how the magnetic materials are deposited. Permalloy can easily be electron beam 
evaporated onto lithographically patterned polymer resist and lifted off to define the 
individual nanomagnets. The problem with using this technique with my FePd3 films is 
twofold. First, electron beam evaporation is a low pressure process, typically around 
3 x 10-6 Torr. The mean free path of atoms at this pressure is around 1 meter, larger than 
the distance between the evaporation source and sample inside the evaporation chamber 
(see Section 2.2.2). At this low pressure, the permalloy atoms reach the sample highly 
anisotropically and will only coat the top of the resist or land on the substrate, similar to 
the evaporated Al etch mask in Figure 2.11(e). This leaves the resist sidewalls exposed, 
allowing for a solvent to access the remaining resist to remove it and the metal on top of it 
during liftoff.  
On the other hand, sputtering is a much higher pressure process, around 
4 x 10-3 Torr of Ar as discussed in Section 2.1. The mean free path of atoms at this pressure 
is only on the order of centimeters and their angles of incidence to the sample are nearly 
isotropic and the atoms end up coating the sidewalls of the resist. This prevents the liftoff 
solvent from being able to access the remaining resist and will prohibit good liftoff from 
occurring. In addition to the pressure difference between the deposition methods, the 
sputter deposited FePd3 needs to be annealed at high temperature to achieve its favorable 
magnetic properties as was discussed in Section 2.1.2. The continuous thin films of the 
 85 
 
material benefit from high heat, but patterned nanostructures will degrade at the annealing 
temperature due to their small size. Therefore, it is the need to anneal the FePd3 at high 
temperature and the fact that sputtering is a conformal technique and not compatible with 
liftoff which motivated the development of a new fabrication process for my ASI studies.  
 
 
2.2.1 Electron beam lithography  
 
Lithography is a common process for patterning structures and circuits of all kinds 
used in both research and industrial manufacturing. There are two main types of 
lithography. The first is optical, or photolithography, which uses visible light passing 
through a semi-transparent mask to expose regions of a polymer resist sensitive to light. 
The second is electron beam lithography (EBL), which uses electrons directly traced into 
a resist sensitive to electrons [176]. Wherever the resist is exposed to light or electrons, the 
polymers in the chemical react to the stimulus, cross-linking with each other and locally 
change their structure. After exposure, the resist is developed and depending on the type of 
resist, the exposed area will be removed and everywhere else the resist will remain or vice-
versa, leaving the pattern or its negative defined on the substrate. This is depicted 
schematically in Figure 2.11(d) for the type of resist I use, known as a positive resist, where 
the exposed regions are removed.   
Photolithography is most common in industrial applications because of its high-
throughput capabilities. It is fast and reliable, but requires a static, premade mask 
specifically designed for a single pattern only. Also, the feature size of the structures that 
can be made by photolithography is limited by the wavelength of light used, typically 
 86 
 
around tens or hundreds of nanometers. There have been many improvements to the 
process, driven by modern integrated circuit and semiconductor engineering, including a 
shift towards extreme ultra-violet light which has a shorter wavelength and thus can reduce 
the minimum feature size. Nevertheless, to get the smallest feature sizes and also produce 
small features more reliably, EBL is a better method [176]. The de Broglie wavelength of 
electrons depends on their energy. A typical EBL system accelerates electrons with a 
potential of tens of kilovolts, giving the electrons a wavelength on the order of 10 pm. This 
is significantly smaller than anything achievable with optical lithography and can be used 
to make the smallest possible feature sizes and structures.  
Small and reliable feature sizes are essential for ASI studies, which makes EBL the 
better technique. EBL is also more suited for low-throughput research applications because 
it is much more flexible. New patterns, sample designs, and geometries can readily be 
created, tested, and changed. EBL is slower than photolithography, but in ASI research 
where only a few samples need to be produced at a time, EBL is the better choice. A 
schematic of an EBL tool is shown in Figure 2.12. Electron-beam writers are typically 
modified scanning electron microscopes with precise stage control and beam deflectors 
that quickly and accurately move the beam of electrons, tracing out the path of desired 
structures.  
A major advantage of EBL is that the patterns the tool writes can be readily created 
and changed. At UMD, I use a Raith e_LiNE system that has an associated proprietary 
editor for defining the lithography patterns. The patterns are first created with the computer 
program and then transferred to the system that controls the electron beam and writes them. 
The Raith system can draw both lines and 2D filled-in shapes. The lines and 2D areas in a 
 87 
 
 
Figure 2.12 Electron beam lithography. Lithography capabilities are integrated into 
existing scanning electron microscopy technology. The electron gun (1), accelerator (2), 
electromagnetic lenses (3), axis alignment coil (4), astigmatism corrector (6) are all typical 
components in a scanning electron microscope. The addition of a blanking electrode (5), 
electrostatic deflectors (7), and a laser controlled stage (8) allow for high speed and high 
accuracy patterning of structures. Figure adapted from Ref. [177].  
 
pattern have different electron dose settings that can be controlled individually. The dose 
is essentially how many electrons are accelerated into the resist per unit length or per unit 
area in order to expose it and remove the resist. There is a certain threshold that must be 
reached in order to fully expose the resist so it is able to be removed, but too high a dose 
can cause other unwanted effects. I draw all of my ASI nanomagnets as lines in the 
lithography software. The length of the line dictates the length of the nanomagnet and the 
line dose will determine how wide the magnet ends up being, with a higher dose resulting 
 88 
 
in a wider magnet. Line doses are carefully calibrated for each geometry to result in 
magnets of the correct size and these doses are given later in this section.  
ASI arrays often have thousands of individual magnets and it would be extremely 
time consuming and tedious to manually draw every line by hand. Instead, I use a 
MATLAB script to automatically generate the lithography patterns. Stephen Daunheimer 
wrote a script to produce patterns for kagome geometry samples. I utilized and modified 
his script for my kagome samples and I also made new scripts for my square geometry 
samples and for the topological defect samples discussed in Chapter 4. The MATLAB 
scripts generate a text file with a particular syntax that can be read by the Raith software. 
Since the ASI patterns are made entirely of lines, the script will specify to draw a line with 
a given start and end point, write that line to the text file, calculate the next line’s 
coordinates, and then write that line to the text file, continuing until the whole pattern is 
written. I then take the text file and import it into the Raith software, checking to make sure 
it was produced correctly.  
A single ASI array of a given size is often referred to as a “crystal.” I fabricated 
kagome crystals as small as one hexagon with six individual magnets or as large as 27 
hexagons on a side with over 7,000 elements. Example of these can be found in Chapter 3, 
for example, in Figures 3.6 and 3.23. The square samples I made were typically 40 elements 
on a side; an example is shown in Figure 2.16. In my kagome geometry samples, the 
magnets are connected in order to increase their magnetostatic interactions. In the 
lithography patterns for these samples, the lines are also connected. Daunheimer figured 
out that is it helpful to actually write each element twice, going over the resist once in each 
direction. This also helps to keep the vertices symmetric in the kagome samples which is 
 89 
 
important in reducing disorder in the lattice. A schematic of the kagome re-write scheme 
is found in Figure 2.13(a), showing the symmetry of the write algorithm. The order in 
which the MATLAB script generates the individual lines dictates the order they are written 
by the Raith pattering software, thus there is precise control over the pattern and line order 
which is essential for creating uniform elements and vertices. I found rewriting elements 
was very important in the square geometry as well to keep the magnets and the vertices 
symmetric. The scheme for the square rewrite process is shown in Figure 2.13(b).  
Many times multiple crystals of different sizes or geometries are fabricated on the 
same SiN membrane. Sometimes this is done to calibrate the correct electron dose for a 
new pattern or geometry. To do this, an array of varying dose is generated and after  
 
 
Figure 2.13 Lithography re-write scheme. The re-write scheme the EBL pattering 
software uses to write the (a) kagome and (b) square geometries. In (a), each line is written 
twice, once in each direction, and each vertex is kept symmetric by moving the electron 
beam along each of the 120° interior angles. In (b), the electron beam moves along one 
row of elements alternating writing a horizontal and a vertical element. Then the beam 
moves back along the same row, writing over all of the elements in the other direction. The 
red arrows represent lines written in the first past and the blue arrows are lines written on 
the way back. The vertical elements of the next row are also shown in lighter colors.   
 
 90 
 
fabrication, the same is examined to determine what dose yields the desired size elements. 
Otherwise, fabricating many different crystals on one sample is a good way to test multiple 
geometries and effectively carry out multiple experiments at once. A great deal of data can 
be generated from one successful sample with many different crystals on it. Typically, a 
100 µm x 100 µm array of different crystals is designed and then this array is repeated in 
a 4 x 4 grid across the membrane. This provides 16 copies of each crystal or geometry. 
Many times one part of the membrane will be dirty or have the wrong dose or bad liftoff, 
but another part will be usable. This technique helps improve the success rate and utility 
of every sample run.  
Once patterns are defined in the Raith software, the first step of EBL is to coat the 
FePd3
 films and substrates with resist (see Figure 2.11(c)). I use two layers of poly(methyl 
methacrylate) (PMMA) resist with slightly different molecular weights. The bottom layer 
has a lower weight, which makes it more sensitive to electron exposure, so that the same 
electron dose will remove a larger area than the top later. This creates an undercut as shown 
in Figure 2.11(d). The first layer is MicroChem PMMA 495 C2 resist. 495 is the molecular 
weight and C2 relates to the concentration of the polymer in solution and affects how thick 
the coating is when spun on at different rates. We use a spin coater in our lab with a special 
chuck that is specifically designed to hold the SiN TEM membrane substrates. For each 
layer of resist, I put one drop of the solution into a small groove in the chuck that helps 
hold the SiN chip in place while spinning and then put two drops directly on top of the 
membrane. I spin the PMMA 495 layer at 6000 RPM for 45 seconds, with a 5 second 
acceleration and 5 second deceleration. I estimate the thickness of the resist spun at this 
rate to be around 100 nm. The resist is then baked at 180 °C in an oven for 45 minutes. 
 91 
 
Baking the resist removes the solvent that the polymer is suspended in and leaves just the 
polymer on the samples. The next layer is MicroChem PMMA 950 C2 resist, again spun 
at 6000 RPM for 45 seconds with a 5 second acceleration and deceleration. The second 
layer is also baked at 180 °C for 45 minutes. Finally, a layer of Mitsubishi Rayon’s 
aquaSAVE conductive polymer is spin coated in the same way as the PMMA. This layer 
is not baked and is not a resist. Its purpose is to prevent the PMMA from chargeing up with 
electrons during lithography, as this would deflect the electron beam or make it more 
difficult to focus the beam. The aquaSAVE prevents charging by giving the electrons a 
path to move off of the resist surface.  
Spinning resist is one of the most variable steps in my fabrication process. Because 
of the SiN substrates’ small size, the resist often does not coat evenly or reproducibly. The 
thin SiN membranes are also flexible and can bow while spinning, which adds additional 
variability and irreproducibility. If the substrate is not perfectly centered in the chuck it 
will also cause uneven resist. It is important to inspect the substrates after spinning to make 
sure the resist is evenly coated. Sometimes the first layer of resist can be re-spun if it does 
not come out correctly. Certain failure mechanisms in sample fabrication could be caused 
by uneven or irreproducible resist. These include poor liftoff, due to there not being enough 
undercut, or individual elements being too thin or too wide due to wrong thickness of resist.  
After the layers of resist are coated and baked and the aquaSAVE is spun on, the 
sample are ready to be loaded into the vacuum chamber for lithography. The electron beam 
is carefully aligned, stigmated, and focused on the surface of the samples. I use a 10 kV 
accelerating potential, a 30 µm aperture for the beam, and a 10 mm working distance 
between the gun and sample. The accelerating potential can affect the resolution and size 
 92 
 
of elements written. Typically, higher energy means smaller features, but ~100 nm wide 
features do not require a higher potential and I found 10 kV to be sufficient. The aperture 
size sets the beam current. I always used a 30 µm aperture, but a smaller aperture might 
help reduce disorder in future work. The electron beam moves extremely fast during 
patterning. I observed some effects of elements bending or missing in finished samples that 
may be reduced if the beam moves slower. A smaller aperture will reduce the beam current 
and if the electron dose is kept the same, the beam will have to move slower to deliver the 
same dose with lower current.  
After focusing and aligning the beam, the beam current is measured and the electron 
dose must be specified. The doses for square and kagome geometries were carefully 
calibrated through dose arrays discussed before as well as trial and error. The doses are 
inherently related to and dependent on the exact geometry and design of the specific crystal 
being written. Connected kagome structures have connected lines in the digital pattern and 
require certain doses based on the length of the lines to get the widths right. In the square 
geometry, I want the elements disconnected and so the lines in the digital pattern are 
disconnected as well. In the kagome geometry, I fabricated connected samples of varying 
element length. There are proximity effects in lithography so that shorter elements need 
lower doses in order to have similar widths to the longer ones with higher doses. I tried to 
correct for this as much as possible, but sometimes shorter elements I fabricated still tended 
to be wider and this was an important consideration when modeling the kagome ordering 
behaviors discussed in Chapter 3. Typical doses for 500 nm length connected kagome 
elements that result in ~100 nm wide magnets are around 600 pC cm-1. For the shortest 
300 nm length elements, the dose was closer to 350 or 400 pC cm-1 and this resulted in 
 93 
 
elements around 130 nm wide. For the square geometry elements and topological defect 
crystals in Chapter 4 (which are derivations of the square geometry) I use a dose of 
550 pC cm-1. For the square lattices I try to get the elements as close as possible without 
touching to increase magnetostatic coupling between neighboring elements. This is done 
through a combination of the electron dose and the actual length of the line in the patterning 
software. I typically used a lattice spacing of 500 nm with a line length of 340 nm in the 
software, which resulted in magnets with an actual length around 400-420 nm long. The 
square geometry elements were typically 110-120 nm wide as well.  
After pattering, the resist needs to be developed. First, the samples are placed in de-
ionized water to remove the aquaSAVE layer. The samples are then submerged in methyl 
isobutyl ketone (MIBK) developer mixed with isopropanol (IPA) in a ratio of 1:3 
MIBK:IPA for 1 minute. The samples are then transferred to pure IPA for 1 minute to stop 
the development process. Finally, they are removed and blown dry with pure nitrogen gas.  
 
2.2.2 Electron beam evaporation 
 
Following development, the next step is to electron beam evaporate an Al etch mask 
over the patterned resist. Some of the Al will land on the resist, but wherever the electron 
beam was traced, the Al will land directly on the FePd3 as shown in Figure 2.11(e). Electron 
beam evaporation is a common and versatile method to deposit a wide range of materials. 
It is quite similar to thermal evaporation and both techniques work through the same 
principle. A material is heated inside a high vacuum chamber until the material’s vapor 
pressure becomes comparable to the chamber pressure. The vapor pressure increases with 
temperature, so as the material gets hotter, its vapor pressure rises until it equals the 
 94 
 
ambient pressure and the atoms in the material start to evaporate. Typically, materials with 
higher melting points require higher temperatures in order to start evaporating.  
Both thermal and electron beam evaporation use this process, they differ in how the 
material is actually heated. In thermal evaporation, a small pellet of desired material is 
placed in a conducting boat, typically tungsten, and current is passed through the boat 
which heats due to resistive, or joule, heating. Electron beam evaporation uses a high 
energy beam of electrons focused at a small crucible containing the desired material. The 
electrons heat the material until it melts and starts to evaporate. Electron beam evaporation 
can get materials hotter because the energy and high temperature is in a more focused 
region. This allows for the evaporation of a lot of oxides and other high melting point 
materials and makes electron beam evaporation more versatile than thermal evaporation, 
though both techniques work equally as well for some materials. Generally, for both 
techniques, the source material has to be at the bottom of the vacuum chamber in order for 
the boat or crucible to hold it and the sample is placed upside down at the top of the 
chamber, shown schematically for electron beam evaporation in Figure 2.14.  
I always used electron beam evaporation, typically in a Denton tool at UMD, to 
evaporate my films. As was discussed at the beginning of Section 2.2, evaporation is 
essential for the liftoff process I use after lithography, but it is also a quick, easy, and 
reliable way to deposit the underlayers for my FePd3
 films as well. The Denton has multiple 
material pockets so that I can deposit both underlayers consecutively without having to 
break vacuum. No matter what material I am depositing, I clip my SiN substrates into a 
custom deposition holder (see Figure 2.15) so that they can be mounted upside down inside 
 
 95 
 
 
Figure 2.14 Electron beam evaporation. Electrons are directed into a crucible with 
evaporation material inside a vacuum chamber. The materials heats up, melts, and starts 
evaporating. The atoms land on the substrates mounted upside down at the top of the 
chamber. A quartz crystal monitor measures the material deposition rate and thickness. 
Figure adapted from Ref. [178].  
 
the Denton chamber. I carefully calibrated the crystal monitor just like in the sputter 
chamber in Section 2.1.2 to evaporate precisely known thicknesses of the different 
materials.  
To deposit the Al etch mask for sample patterning (Figure 2.11(e)) I first load the 
developed samples into the vacuum chamber and pump it down. Once a base pressure of 
around 3 x 10-6 Torr is reached, the electron beam is turned on and aimed at the Al crucible. 
The beam power is increased until the metal begins to melt. It is important to turn on the 
beam modulation so that the metal melts evenly and the crucible will not be damaged. 
Once the deposition rate reaches about 0.4 Å s-1, I open the shutter covering the samples  
 96 
 
 
 
Figure 2.15 Sample holder. Optical photograph of SiN substrate samples clipped into 
deposition and milling holder. The frame is made from Al and the clips are Be/Cu. The 
holder is colored due to various materials depositions. Scale bar is 0.5 inches.  
 
and begin the deposition. 16 nm of Al is then deposited at a rate of 0.4-0.8 Å s-1. Once 
16 nm is reached, the shutter is moved back in front of the samples and the electron beam 
power is ramped down and turned off. The Al crucible is still extremely hot and must be 
allowed to cool for 5-10 minutes before the chamber is vented and the samples are 
removed. 
2.2.3 Liftoff  
 
Liftoff is a common process used in the fabrication of many types of structures. It 
works through a solvent removing the resist left on a substrate after lithography and metal 
deposition, causing the metal deposited on top of the resist to “lift off” as shown in Figure 
2.11(e)-(f). It helps to have an undercut in the resist so that the solvent can readily access 
 97 
 
the resist to remove it. Liftoff is generally easy, though using it with SiN membranes adds 
some challenges to the technique. Typically, samples are agitated with ultrasonic waves or 
“sonicated” during liftoff to assist the solvent with removing the resist and metal. The 
problem with doing this to the membranes is that the extremely thin membrane will 
immediately break if it is subjected to sonication. Instead, I developed other techniques to 
aid the liftoff process, though it is still one of the most difficult steps in my fabrication, 
especially for the kagome crystals.  
The kagome crystals are much harder to liftoff because of their connected structure. 
In the square geometry samples, the elements are disconnected, so the Al film on top of the 
resist is nominally continuous and can come off in large flakes. In a kagome crystal, every 
connected hexagon of desired nanomagnets has its own hexagonal piece of material that 
needs to be lifted off in the middle. We have named these pieces “cookies” since they look 
like little cookies on the sample. It can be extremely difficult to get these cookies to liftoff. 
Sometimes large areas of a membrane will not liftoff at all due to problems in spinning or 
developing the resist, but even if the sample has generally good liftoff, there were almost 
always a few cookies left on the sample as defects. An image of some cookies is shown in 
Figure 2.19 in Section 2.2.5.  
 To liftoff the Al on top of the PMMA I first submerge the samples in an acetone 
bath and soak them for 1 hour. The next step is a trick I have developed and perfected over 
time for the membranes. I take a spray bottle filled with acetone in one hand and grab a 
sample with tweezers in the other hand. Keeping the sample submerged at first, I begin to 
spray the top of the acetone bath and I lift the sample out of the bath and into the continuous 
spray. The continuous spray breaks the surface tension of the small droplet of acetone that 
 98 
 
normally forms on top of the membrane if you just lift it out of the bath as is. I have found 
that this trick greatly helps reduce the number of cookies left on the kagome samples and 
generally results in better liftoff than just lifting the samples out, allowing the droplet to 
form, and then spraying them. Spraying the samples with a strong stream of acetone is also 
very important and works something like sonication during liftoff. The spray helps to 
forcibly remove the resist and metal from the membrane, but without breaking the SiN 
membrane. It is important to spray from all angles to get the best results.  
After spraying, the samples are placed back in the acetone bath for 20 minutes for 
the square geometry sample and 1 hour for the kagome geometry samples. All samples are 
then sprayed again, using the same technique to break the surface tension as they are 
removed from the bath. Finally, the samples are submerged in IPA for 2-3 minutes to help 
remove any residue from the acetone. They are removed from the IPA and blown dry with 
nitrogen gas.  
 
2.2.4 Ion milling  
 
After liftoff, we are left with patterned Al elements on top of the FePd3 film as 
shown in Figure 2.11(f). The next step is to etch the entire sample using Ar+ ion milling. 
Ion milling is very similar to sputtering in that an Ar plasma is created inside a vacuum 
chamber and the Ar+ ions are accelerated to knock material off of a target. The difference 
here is that in ion milling the Ar is being accelerated towards the sample and no material 
is being deposited anywhere, only removed. The Ar atoms bombard the whole surface of 
the sample and etch the top layer which is either the patterned Al or the FePd3 film. We 
want to etch away all of the FePd3 everywhere it is not protected by the Al and leave the 
 99 
 
FePd3 that is under the Al in order to transfer the pattern of the Al into the FePd3 film, as 
shown in Figure 2.11(g)-(h).  
The Ar atoms etch the Al and FePd3 at different rates. I had to carefully calibrate 
the etch rates simultaneously with the FePd3 sputter deposition thickness calibration 
discussed in Section 2.1.2. Using an ion mill in CNAM at UMD, I would etch a film of 
known thickness with some drops of resist baked on the substrate for 1 minute at 200 W 
power. I then remove the resist and material on top of it and use a profilometer to measure 
the step height. Then I compare this height to the measured, non-etched thickness of the 
original film and use the difference in the film thicknesses to calculate an etch rate. The 
step height of the etched films tends to be difficult to measure, but by averaging over many 
samples and many points on each sample, I was able to obtain a consistent and accurate 
rate. I found the Al etches at a rate of 15.3 nm min-1 and the FePd3 at a rate of 32.5 nm min
-
1. The difference in the rates is striking, but comes from the Pd being a high higher Z 
(atomic number) material than Al, which actually etch faster than lower Z materials.  
 At 32.5 nm min-1 it should take ~42 seconds to etch through the 23 nm FePd3. I 
typically etched for slightly longer, 45-50 seconds, to make sure all of the material would 
be removed. I don’t want to etch much longer than 50 seconds because then the SiN 
membrane will start to be etched which could weaken or break it. Luckily I never saw this 
problem in my samples. I use the 45 second etch time to calculate how much Al to deposit 
during electron beam evaporation. Based on the calibrated rate, 11.5 nm of Al will be 
etched in 45 seconds, so I deposit a little more (16 nm) to make sure none of the FePd3 I 
want to keep is etched away. A few extra nm of Al on top of the FePd3 elements will also 
help protect them from oxidation and it does not affect their magnetic properties.  
 100 
 
 After ion milling, the desired pattern has been etched into the FePd3 film, shown 
schematically in Figure 2.11(h). There is a final step to electron beam evaporate a 5 nm Ti 
layer over the entire sample. This layer acts as an anti-charging layer for TEM studies. 
Since the SiN is insulating, electrons can build up on the surface, creating an electric field 
that will deflect or move the electron beam. This can cause images to shift around and 
make it impossible to take clear micrographs. The Ti layer also helps protect the FePd3 
elements from oxidation.  
Images of some finished samples are shown in Figures 2.16, 2.17, and 2.18. The 
many different samples and geometries I fabricated will be discussed in later chapters, but 
these figures provide representative examples of the two main geometries (square and 
kagome). Figure 2.16 shows a full square geometry sample of 420 nm x 110 nm elements 
with a 500 nm lattice spacing. Figure 2.17 shows a large kagome crystal along with some 
smaller kagome crystals. There are some one and two hexagon crystals in the upper right 
corner and some medium sized crystals with different edge types in the bottom right. The 
different types of edges and their relative effects will be discussed in depth in Chapter 3. 
Figure 2.18 shows higher magnification images of the square and kagome geometries 
showing more detail of the elements.  
 101 
 
 
Figure 2.16 Large square crystal. In-focus TEM image of a square geometry sample. 
Scale bar is 3 µm.  
 
 102 
 
 
Figure 2.17 Large kagome crystal. Lorentz TEM image of a large kagome crystal along 
with a few smaller crystals. The crystals in the top right are one and two-hexagon crystals 
and there are three medium sized crystals in the bottom right. Scale bar is 3 µm.  
 
 103 
 
 
Figure 2.18 Higher magnification square and kagome images. a. In-focus TEM image 
of a square geometry sample at moderate magnification. b. High-resolution, high-
magnification TEM image of square geometry elements. The grain structure of the FePd3 
is visible. c. In-focus TEM image of a kagome geometry sample at moderate magnification. 
d. High-resolution, high-magnification TEM image of connected kagome elements. Scale 
bar for (a) and (c) is 1 µm, scale bar for (b) and (d) is 200 nm.  
 
2.2.5 Common problems  
 
The most common problems I encountered during fabrication are membranes 
breaking and poor liftoff. The membranes are handled frequently during the fabrication 
process, from spin-coating the resist to development to loading and unloading the 
 104 
 
deposition holder, liftoff, and more. If the tweezer does not have a solid grip on the sample 
or if it has too strong a grip, the membrane can drop or fly out and break. The membrane 
can also be launched out of the spin coater chuck when it is spinning at high speeds. Careful 
focus and practice can help prevent most of the membrane breaking problems, though some 
loss is probably inevitable. The rest of this section discusses other common issues with 
fabrication, plausible causes for these issues, and ways to help prevent these issues in future 
work.  
 
2.2.5.1 Poor liftoff 
 
 
Poor liftoff is a difficult issue to overcome. I developed some tricks that seemed to 
help, but there are always some areas of a sample that do not lift off well. Many times 
crystals near the edges of the membrane do not liftoff as well, probably because of 
variations in the resist near the edges. There are regularly cookies in even the best kagome 
crystals. Figure 2.19(a) shows a kagome crystal with a large number of cookies on it. This 
crystal has too many cookies to be usable, though good data and insight can still be gained 
from crystals that have just a few cookies. Figure 2.20 shows a large region of a sample 
where some kagome crystals lifted off and others did not. There is some dependence on 
size and dose, as smaller element length kagome crystals have a harder time lifting off 
because the cookies are smaller, but there are still some crystals of the same size and dose 
that have different degrees of success with lifting off.  
 105 
 
 
Figure 2.19 Kagome geometry with cookies. a. Low magnification, in-focus TEM image 
of a kagome crystal with lots of cookies present, representing poor liftoff. Scale bar is 
5 µm. b. Higher magnification TEM image of the cookies showing more detail, in 
particular their hexagonal shape. Scale bar is 2 µm. 
 
 
Figure 2.20 Sample with poor liftoff. TEM image of a large region of a sample with both 
square and kagome geometry crystals. Some kagome crystals show good liftoff, some have 
poor liftoff, and others have no liftoff at all. There are also many elements missing from 
one of the square geometry crystals. Scale bar is 10 µm.  
 
 
 106 
 
2.2.5.2 Spin coating resist and electron dose 
 
 
There are a variety of problems I encountered related to the thickness and 
consistency of spin coating PMMA and local electron dose effects, though these tend to 
affect individual crystals or smaller regions of a sample more than poor liftoff or membrane 
breakage. One prevalent issue is that individual elements will be missing from a larger 
array. This tends to be more common in the square geometry samples because the elements 
are disconnected, but it is found in the kagome geometry as well. Examples are shown in 
Figure 2.21. Missing elements may be due to local defects in the resist due to artifacts from 
spin coating or from spraying the membranes too hard with acetone during liftoff. This 
could be prevented by spraying less and more gently, though there is a tradeoff because 
some spraying is generally necessary to aid in liftoff.  
Elements in the square geometry can also be bent or deformed as is shown in Figure 
2.22. This is likely due to one or more layers of resist not spinning on at a uniform 
thickness, altering the optimal electron doses for different regions of the membrane. The 
bent elements may also be caused by the electron beam charging the surface of the sample 
if the aquaSAVE layer did not spin on evenly. Sometimes dirt or chemical residue can be 
left on a crystal as well, masking or covering some of the elements. Finally, the elements 
can come out the wrong width, either too thin or too wide. This can be caused by improper 
dose calibration during lithography, an unknown change in the beam current during 
lithography, or the resist spinning on at a different thickness than during the dose testing. 
Sometimes different regions of a sample can have different widths which is likely caused 
by variations in the thickness of the resist across the sample. An example of both under- 
and overdosed elements is shown in Figure 2.23.  
 107 
 
 
 
 
Figure 2.21 Missing elements. a. In-focus TEM image of a square geometry sample with 
a region of missing elements. b. In-focus TEM image of a kagome sample with only a few 
missing or broken elements. Such defects do not substantially affect the long-range 
ordering in these samples. Scale bars are 1 µm. 
 
 
Figure 2.22 Bent elements. In-focus TEM image of a topological defect square geometry 
sample. Many elements are touching or curved. Scale bar is 3 µm.  
 108 
 
 
Figure 2.23 Wrong doses. a. In-focus TEM image of a kagome sample with too low a 
dose. The elements are extremely thin and too small. b. TEM image of a square geometry 
sample with too high a dose. The elements are supposed to be disconnected, but came out 
connected and too wide because the dose was too high. Scale bars are 1 µm.  
 
2.2.5.3 Membrane quality  
 
 
A major issue I ran into at one point in my work related to the quality of the SiN 
membranes themselves. We purchase the membranes commercially and discovered that a 
large batch we had received has dark spots in them (see Figure 2.24(a)). The membranes 
should not add any extra contrast to the TEM images and so the dark spots posed a 
significant problem. We believe that the spots are Si-rich regions within the SiN. The 
manufacturer can fabricate SiN that is close to the Si3N4 stoichiometric ratio which has no 
spots, but is more fragile, or they can make the films Si-rich, which gives them more 
strength, but also these dark spots. The main problem with the dark Si spots is that in the 
Lorentz TEM mode (see Section 2.3.2) that we use to image the precise magnetic 
configurations of the samples, the dark spots become very bright spots (see Figure 2.24(b)). 
The magnetization of each element is determined by very slight asymmetry in the contrast 
across that element. We use image processing software (see Section 2.3.3) to extract the  
 109 
 
 
Figure 2.24 Spots on SiN membranes. a. In-focus TEM image of a section of a kagome 
crystal showing SiN membrane spots. The spots appear dark when in focus. Scale bar is 
1 µm. b. Lorentz contrast TEM image of a 3-hexagon kagome crystal showing the SiN 
membrane spots. The spots appear bright in the Lorentz imaging mode. Scale bar is 
500 nm. c. Lorentz contrast TEM image of a section of a large kagome crystal also showing 
the bright spots. The spots skew the contrast asymmetry across each element that normally 
determines the magnetization direction. The bright spots cause the image processing 
software to incorrectly assign spin directions, making the data from these samples 
unusable. See Section 2.3.3 for a further discussion. Scale bar is 1 µm.  
 
magnetization direction of the individual magnets. The bright spots cause our image 
processing software to read many of the magnetization directions incorrectly because the 
bright spots skew the contrast asymmetry. This results in analysis that is wrong, not 
representative of the actual magnetic configuration, and completely unusable. After this 
problem was discovered, new samples had to be fabricated on new membranes that did not 
have the spots. 
2.3 Transmission electron microscopy  
 
2.3.1 Fundamentals and advantages  
 
The transmission electron microscope is the main experimental tool I use to image 
my ASI structures. Transmission electron microscopy (TEM) is an extremely versatile and 
powerful technique for analyzing all kinds of different materials. The microscopes work 
 110 
 
by accelerating electrons at extremely high energies down a vacuum column, transmitting 
them through a specimen, and collecting them on the other side. The specimen blocks or 
diffracts the electrons attempting to pass through and the variation in the number of the 
electrons hitting different parts of a florescent screen or CCD camera creates contrast 
differences that give information about the specimen being studied.  
There are many different TEM imaging modes that can provide different types of 
information about a specimen. High-resolution or atomic-resolution imaging is very 
common and a particular advantage of TEM. The de Broglie wavelength of an electron in 
a typical 200 kV microscope is on the order of 2.5 pm [175]. This allows for some of the 
best atomic-resolution images possible in a wide variety of materials. Lorentz TEM is the 
technique I use most and will be discussed at length in the next section, but in general it 
can provide magnetic information about a specimen. There are also multiple techniques for 
elemental analysis built into many microscopes. EDS was discussed in Section 2.1.3 and 
there are often EDS detectors inside the microscope column that can provide qualitative 
information on what elements are present in a specimen. Electron Energy Loss 
Spectroscopy (EELS) can give not only elemental information, but also specific atomic 
bonding information and can tell the difference between different molecules and molecular 
bonding in a substance. EELS can be combined with atomic resolution imaging for 
incredibly precise investigations of crystal structure and atomic arrangements. This also 
allows for novel investigations of interfaces and defects because the exact atomic positions 
and species are known. Electron diffraction is a powerful technique for revealing the crystal 
structure of many different kinds of materials. In some cases it is even better than x-ray 
diffraction because the electron wavelength is smaller than the wavelength of x-rays and 
 111 
 
so the electrons are sensitive to smaller features and more subtle differences between the 
structures of some materials or different phases of the same material. Normal TEM imaging 
is done in what is called “bright field” mode, where higher Z elements appear darker in the 
CCD images. “Dark field” imaging is also possible, which uses diffracted electrons to 
image the sample and can reveal dislocations and grain boundaries very easily, among 
other information. An important factor in all of these imaging modes is that the specimens 
have to be thin in order for electrons to transmit through them, typically around 100 nm or 
less. Thicker samples can be milled down using ion milling or Focused Ion Beam (FIB) in 
order to prepare them for TEM investigation.  
A schematic of a transmission electron microscope is shown in Figure 2.25. 
Electrons are generated at the top of a vacuum column and accelerated with an electrostatic 
potential typically of around 200 kV, though some tools are capable of imaging at as low 
as 50 kV or as high as 1 MV. The beam of electrons then passes through a series of 
apertures and lenses as well as through the sample itself before being projected onto a 
florescent screen or a CCD camera. The CCD camera exposure can be adjusted to integrate 
electrons reaching it over a period of time and provide better contrast images. The lenses 
are complex electromagnets, or solenoids, though their principles of operation are very 
similar to the glass lenses in an optical microscope. A ray diagram for the electrons is 
shown in Figure 2.26. Electrons bend in a magnetic field due to the Lorentz force, so the 
magnetic fields that the solenoids generate can spread or focus the beam of electrons as 
well as shift its position. Before imaging, the whole instrument is aligned in order to capture 
high quality images. The alignment procedures generally differ for the different imaging 
modes.  
 112 
 
 
Figure 2.25 Schematic of a TEM. Cross sectional view of a TEM column showing the 
locations and relative sizes of the parts of the microscope. Figure adapted from Ref. [179].  
 
 113 
 
 
Figure 2.26 TEM ray diagram. Schematic of electron trajectories in a TEM and how the 
various lenses and apertures affect the electron beam. Figure adapted from Ref. [180].  
 
The capabilities and resolution limits of a microscope are determined by a number 
of factors. Recently, some of the biggest improvements to imaging have come through 
aberration correctors. There are two types of aberrations, chromatic and spherical. 
Chromatic aberration is caused by the electrons in a beam not having the exact same 
energy, but instead a spread of energies, usually Gaussian, centered around the nominal 
accelerating potential. The different energy electrons will respond differently to a lens, 
 114 
 
preventing the lens from being able to perfectly focus the whole beam and limiting the 
minimum resolution. Spherical aberration arise because electrons passing through the 
middle of the lens are focused differently from electrons passing through the edge of the 
lens. In an optical microscope it is possible to correct for this by using a divergent lens, but 
for a long time the solenoid TEM lenses could only be made convergent and thus could not 
correct for spherical aberration. New spherical aberration correctors effectively act as 
divergent electromagnetic lenses and have enabled new regimes of TEM studies. New gun 
sources can create more uniform energy distributions, reducing the chromatic aberration as 
well. A final limiting factor for TEM studies is beam damage. The high energy beam of 
electrons interacts with every specimen and can cause damage, especially to biological or 
very small samples. Reducing the accelerating potential of the TEM can help alleviate this 
problem, though it can never be completely removed. Many of the newest microscopes 
combine low kV imaging, to prevent damage to sensitive samples, with aberration 
correction to achieve the best possible images. 
The microscope I used most for my analysis was a JEOL JEM 2100 LaB6 tool 
operating at 200 kV. It worked extremely well for taking all the images needed for this 
dissertation. The “LaB6” refers to how the electrons are generated in this microscope; a 
lanthanum hexaboride (LaB6) filament generates the beam of electrons. Most of the 
imaging I did was in low-magnification mode (< 2000x), which is ideal for imaging arrays 
with thousands of ASI elements, both in-focus and in Lorentz mode which will be 
discussed in the next section (see Figures 2.16 and 2.35 for examples). The LaB6 
microscope has a large gap inside the objective lens which allows for more versatile 
imaging, better low-magnification imaging, and a variety of specialized sample holders to 
 115 
 
fit inside. All members of Cumings’ group use in-situ TEM to study various nanoscale 
phenomena, apply stimuli inside the microscope, and observe what is happening on the 
nanoscale in real time. I most often used a specialized heating holder, which could heat 
samples up to 500 °C or more inside the microscope. I used this to measure the TC of FePd3 
films as well as do some of the initial heating of my patterned thermally active ASI 
samples. We also have some custom electronics that let us take over control of the objective 
lens, the most powerful lens in the microscope and the one that is directly around the 
specimen. Low-magnification imaging does not require use of the objective lens and so it 
is normally “off” in this mode. We are able to switch over control of the lens to an external 
power supply that is controlled by a LabVIEW program, allowing us to apply arbitrary 
magnetic fields in-situ or remove any remnant field of the objective lens that could bias a 
sensitive sample. The LabVIEW software can also automate some other tasks in the 
microscope such as taking images, although I did not use this often in my studies. Finally, 
I did do some high-resolution TEM imaging to both investigate the grain structure of my 
FePd3 films (see Figure 2.6(b)) as well as inspecting and measuring the size of finished, 
patterned ASI elements (see Figure 2.18(b) and (d)). Overall, the transmission electron 
microscope is an extremely powerful tool that was crucial for all parts of my dissertation 
work.  
 
2.3.2 Lorentz TEM 
 
 
Lorentz TEM is a technique that exploits the Lorentz force for gaining magnetic 
information about a sample. A schematic of the technique is shown in Figure 2.27. As 
electrons are transmitted through a material with in-plane magnetization, the electrons are 
 116 
 
deflected, due to the Lorentz force from the magnetic field inside the material, in different 
directions based on the direction of the in-plane magnetization. This can create added 
bright or dark contrast in the TEM images that provides information about the magnetic 
structure or properties of the specimen. This contrast can be seen as domain walls and 
ripple contrast in continuous thin films found in Figures 1.9, 1.10, 2.7, and 2.8. In the ASI 
nanomagnets, imaging in Lorentz mode creates a fairly subtle asymmetry in the contrast 
across each element. This can be seen in Figure 2.28(b) and (d). There is a darker black 
line on one side of the element then on the other side. The line profile in Figure 2.28(e) 
shows the asymmetry more clearly and shows that it differs for the two different 
magnetization directions.  
Yi Qi, a graduate student in Cumings’ group who worked on the ASI project before 
Daunheimer, ran some simulations of the Lorentz TEM technique. He used a contrast 
transfer function to confirm and interpret the observed contrast differences in Lorentz TEM 
images were due to a single magnetic domain oriented in different directions [181]. His 
simulation results are shown in Figure 2.29. A full explanation of the fundamental origins 
and spacial variation of Lorentz contrast in arbitrary geometries is somewhat involved and 
beyond the scope of this dissertation. Derivations can be found, for example, in Refs. [181] 
or [182]. It will suffice to note that the observed contrast is phase contrast from interference 
of electron plane waves with phase shifts due to the magnetic field inside the specimen. 
The phase shift in the plane wave of electrons due to the in-plane magnetic field is given 
by 
𝜑(𝑥) =  −
2𝜋𝑒
ℎ
𝑀𝑡𝑥  (2.3) 
 
 117 
 
 
 
Figure 2.27 Lorentz TEM. As electrons transmit through a material with in-plane 
magnetization, the electrons are deflected, due to the Lorentz force, in one direction or 
another based on the direction of the magnetization. This can lead to added bright or dark 
contrast due to interference fringes in the TEM image. Figure adapted from Ref. [175]. 
 
 
 
 
 118 
 
 
 
 
 
Figure 2.28 In-focus vs. Lorentz contrast TEM images. a. In-focus TEM image of a 
square geometry sample. b. Lorentz contrast TEM image of the same square geometry 
sample, showing ground-state order. c. In-focus TEM image of a kagome geometry sample. 
d. Lorentz contrast TEM image of the same kagome geometry sample, showing only ice-
rule ordering. e. Intensity profile along the black line in (d) with corresponding arrows 
showing how the asymmetry in contrast across each element is used to determine the 
direction of the magnetic moment. Scale bar is 1 µm.  
 
 
 119 
 
 
Figure 2.29 Lorentz contrast simulation. Contrast transfer function simulation 
confirming that the observed contrast asymmetry in Lorentz TEM images is due to 
magnetization of individual elements in different directions. Figure adapted from Ref. [93].  
 
where e is the electron charge, h is Plank’s constant, M is the magnetization, t is the 
thickness of the film, and x is the spacial coordinate in the object plane perpendicular to 
the direction of M. The contrast intensity distribution is then found to be [181] 
𝐼(𝑥′, 𝑦′) = |∬(∬exp[𝑖𝜑(𝑥)] exp[−2𝜋𝑖(𝑘𝑥𝑥 + 𝑘𝑦𝑦)] 𝑑𝑥𝑑𝑦) ∗
exp [𝜋𝑖Δ𝑧𝜆(𝑘𝑥
2 + 𝑘𝑦
2)]exp [−2𝜋𝑖(𝑘𝑥𝑥
′ + 𝑘𝑦𝑦
′)]𝑑𝑘𝑥𝑑𝑘𝑦|
2
  (2.4) 
where λ is the wavelength of the electrons and Δz is the defocus. This equation can be 
solved numerically for different specimen geometries with a given magnetization.  
An illuminating aspect of Equation 2.4 is that the contrast is adjusted by the defocus 
Δz. This means that an important part of setting up the microscope in Lorentz mode is to 
actually intentionally operate it out-of-focus. There is an optimal defocus to get the strong 
contrast asymmetry in Figures 2.28 and 2.29 and this amount of defocus will depend on 
the size of the structures being imaged and the microscope magnification. Aside from 
finding the correct defocus, the rest of the microscope alignment for Lorentz imaging is 
generally straightforward. The rest of this paragraph is intended to help someone who is 
familiar with the standard operation of a transmission electron microscope to achieve good 
Lorentz contrast images. I tended to use the largest condenser aperture to be able to image 
large crystals and large sample areas. I always wanted the brightest possible beam so I 
chose spot size 1 and made sure to align the gun so that the spot was the brightest it could 
 120 
 
be. If this gave a brightness asymmetry across the image I would then slightly adjust the 
gun or the beam itself so the whole crystal was evenly illuminated. I always used 
convergence angle setting α = 3. For large crystals I used a magnification typically 400-
600x and for the small kagome crystals I used around 1000-1200x, always in low-mag 
mode. I would adjust the condenser stigmation as usual and try to have a fairly widely 
spread beam so that the phase of the electrons was constant across the crystal. The beam 
was about 1/3 to 1/2 the diameter of the fluorescent screen. The one other main adjustment 
to improve the Lorentz contrast asymmetry is to change the intermediate lens (IL) 
stigmation if the different ASI sublattices have different amounts of asymmetry or different 
Lorentz contrast. It can be difficult to get all sublattices to look the same, especially in the 
kagome geometry. A lot of the IL stigmation adjusting is trial-and-error. I always capture 
images with CCD bin size 1, typically for 3 seconds for large crystals. Bin size 1 exposures 
for 1.5-2 seconds in search mode are helpful in adjusting the defocus and IL stigmation.  
The small contrast asymmetry across each element shown in Figure 2.28 is all that 
is necessary to determine the magnetization direction of that element. Despite the 
challenges of working with the SiN TEM membrane substrates, I developed my fabrication 
process for FePd3 ASI samples on top of the membranes in order to utilize the great 
advantages of the Lorentz TEM for imaging of the ASI magnetic configurations. Lorentz 
TEM is fast. It only takes about 3 seconds of exposure on a large crystal to capture a high 
quality Lorentz TEM image. For the smaller crystals found in Section 3.2, exposures were 
only 0.5 or 1 seconds. For the large crystals, this is about 200-400 times faster than MFM 
for a similarly sized crystal, which can take up to 10 or 20 minutes per image.  
 121 
 
Besides MFM, PEEM is the next most common imaging technique for ASI 
structures. It is fairly fast, with individual exposures on the order of 12 seconds, but 
requires highly specialized x-rays and predetermined beam time at a cyclotron facility. 
Users have limited imaging during their allotted time and the tool is not readily available. 
In contrast, the TEM it is significantly faster and it is easier to move the sample around to 
quickly look at different areas and either find the best crystals to image or take many images 
in one session. In general, Lorentz TEM has huge advantages over other techniques 
because it allows for the collection of significantly more data. I can image more crystals, 
more copies of similar geometry crystals, and the same crystals multiple times to get better 
averages and more precise measurements of ordering behavior, representative of true 
properties and statistics. The tool is readily available at UMD for frequent and versatile 
use. Lorentz TEM allowed Daunheimer to collect enough data with small enough error 
bars to fit a power law behavior for avalanche sizes in the kagome geometry shown in 
Figure 1.31 and it has enabled me to gather thousands of images that will be summarized 
and discussed throughout Chapters 3 and 4.  
Lorentz TEM allows for unambiguous imaging of connected kagome structures. 
Connecting kagome elements increases their magnetostatic coupling and has also been 
shown to reduce disorder in the samples [94], ideal for realizing the theoretically predicted 
long-range ordered states in thermally active ASI. Lorentz TEM is sensitive to the in-plane 
component of the magnetization as opposed to MFM, which is only sensitive to the out-of-
plane component. In disconnected structures, MFM can uniquely determine the exact spin 
configuration (shown, for example, in Figure 1.34), but if the elements are connected, there  
 122 
 
 
Figure 2.30 Ambiguity in using MFM to image connected kagome structures. a. MFM 
image of a connected kagome structure and corresponding proposed spin map (b) from 
Tanaka et al. [124]. c, d. Alternative valid spin maps created by reversing chains of spins, 
shown in different colors. Figure adapted from Refs. [93] and [124].  
 
is ambiguity in MFM imaging of the kagome lattice. This ambiguity is demonstrated in 
Figure 2.30. 
2.3.3 Lorentz image processing  
 
Often times in experiments I capture hundreds of Lorentz TEM images, each with 
thousands of individual magnetic elements. It is impossible to go through by hand and 
assign a direction to each element, so I use MATLAB scripts to process the images (see 
 123 
 
Appendix in Ref. [132]). The general outline of the scripts is to first find the coordinates 
of the individual elements, then go through one-by-one to analyze the asymmetry in each 
element and assign a direction to it. Then the script can output a variety of data to help 
understand the magnetic configurations. The scripts can calculate ordering parameters that 
will be discussed later, or simply output a text file with all of the information about spin 
directions and locations that can later be used to do more rigorous statistical analysis. The 
scripts can also output visual representations of the data to aid in interpreting and analyzing 
the configurations. This could mean printing colored lines or arrows on top of the Lorentz 
images to make it easier to see the individual element directions, putting labels or dots on 
the vertices to illustrate what type of vertex it is in the square geometry or what the charge 
of the vertex is in the kagome geometry, or overlaying color to represent the domains of 
ordering observed in the kagome geometry. All of these outputs will be shown later, for 
example, in Figures 2.35, 3.15, and 3.26. 
Stephen Daunheimer wrote the main analysis script, named the “crawler,” for the 
kagome geometry along with help from our advisor, John Cumings. I modified this script 
to be able to analyze the small kagome crystals found in Section 3.2 and also added vertex 
plotting, charge domain plotting, and other analysis to investigate charge-ordering. Dr. 
Cumings wrote a different crawler to analyze the square geometry samples which I also 
modified to plot arrows on the Lorentz images. I worked for one summer with a high school 
student, Thomas Marsh, who adapted the existing crawlers to suit the topological defect 
samples discussed in Chapter 4. Thomas was very successful at making the crawler highly 
versatile and adaptable to any geometry we wanted to study and also implemented the 
vertex notation found in Figures 2.36 and 4.51 as well as many other figures in Chapter 4.  
 124 
 
I can now explain why the bright spots on the SiN membranes discussed in Section 
2.2.5 were so detrimental to data collection. The crawler is extremely sensitive to the slight 
asymmetry in the contrast of individual elements in the Lorentz TEM images. If the crawler 
is not entirely sure of the moment direction, there is a mode where it allows the user to 
double check or decide the doubted elements in question. This makes it possible to correct 
some data that may not have as high an asymmetry as desired. The problem with the really 
bright spots is that they throw off the contrast asymmetry so much that instead of just being 
unsure about an element and allowing for user input, the crawler is sure that the element is 
pointing in the direction opposite from what it actually is and the crawler simply moves 
on. In normal samples without spots, it is still possible for the crawler to get a few elements 
wrong and not doubt them so I make sure to double check all of the elements in important 
images submitted for publication, included in conference presentations, and found in this 
dissertation.  
The quality of the FePd3 thin film also affects how well the crawler is able to 
process the images. If the film does not have a high magnetization, there will be little or no 
asymmetry in each element and the crawler will doubt most of the spins and get more of 
them wrong. It can be very difficult to see any symmetry sometimes. Films with higher 
magnetization result in better ordering, but the actual quality of the Lorentz data is better 
too because there is more asymmetry. I try to select films with the best Lorentz contrast 
before patterning because these will directly result in better data and more accurate analysis 
once patterned.  
 
 125 
 
2.4 Heating experiments  
 
2.4.1 Annealing protocols  
 
In order to see thermal effects in my ASI samples, I need to heat the samples. I 
investigated a wide variety of annealing protocols throughout many different experiments. 
Samples were either heated in-situ, inside the microscope with the heating holder, or ex-
situ in an Ar atmosphere inside a tube furnace. The heating holder was beneficial during 
early work and some of the first heating experiments because it provided immediate 
information on whether any elements had flipped. I could polarize a sample along one axis 
using our custom electronics, heat it, and immediately tell whether the temperature I had 
heated to was hot enough to thermalize the system and flip any elements. After some initial 
checks it became less useful to use the heating holder on the patterned samples. It was still 
good for checking the TC of thin films, but it heats the membrane very asymmetrically. The 
heating holder clamps the Si frame of the substrate around the outside and heat is 
transferred through the clamp, meaning that all the heat comes from the edges of the 
substrate. The thin SiN membrane radiates heat from the middle, so as the edges heat up, 
the middle of the membrane stays considerably cooler than the outside. This is shown 
schematically in Figure 2.31. There is a strong temperature gradient across the membrane 
with different parts of the membrane at different temperatures.  
 
 126 
 
 
Figure 2.31 Heating holder radiative heat loss. Schematic of the in-situ heating holder. 
The holder clamps onto the SiN substrates and provides heat from the edges. The thicker 
Si conducts heat well, but there is a lot of heat lost to radiative cooling from the middle of 
the thin SiN membrane. A finite element model temperature vs. position profile shows how 
the temperature drops quadratically from the edges of the membrane held at constant 
temperature. The middle of the membrane is about 40 °C colder than the edge, creating a 
very strong temperature gradient across the membrane. Cumings’ group graduate student 
Hanna Nilsson helped me with the finite element model.  
 
The heating holder has an external controller where the temperature can be set at 
will. We typically take this temperature to be the temperature at the edge of the SiN 
membrane because there is much less radiative cooling from the thicker Si frame. However, 
as can be seen in Figure 2.31, there is approximately a 40 °C temperature gradient between 
the edge of the SiN membrane and the middle. Also, even the temperature at the edge may 
not be very accurate as well. This makes it difficult to examine specific crystals because 
they all have different temperatures where they become thermally active and the real 
temperature at the crystal is not well known. The heating holder also heats and cools rather  
 127 
 
 
Figure 2.32 Tube furnace. The samples are inserted into the glass tube and positioned in 
the middle of the furnace. Ar gas flows from the cylinders on the right through the furnace 
to heat the samples without oxidizing them. To quench the samples, the glass tube can be 
quickly slid far to the right so that the samples leave the middle section where the heat 
source is.  
 
slowly and uses valuable TEM time to do the heating. It requires a lot of manual operation 
to set and ramp temperatures and is also hard to control the temperature overall. After some 
initial testing with the heating holder where thermal flipping of individual magnets was 
observed, I moved towards using a tube furnace to heat my samples. A photograph of the 
tube furnace is shown in Figure 2.32. 
The furnace has a long glass tube that sticks out on both ends from a middle section 
where the heating elements are. Ar gas flows in from a cylinder and flow controller on one 
side and flows out into a beaker filled with oil in the fume hood on the other side. I load 
my samples onto a quartz boat covered in aluminum foil and slide the boat into the middle 
of the furnace. First I flood the tube with Ar gas to get as much oxygen out as possible, 
then reduce the flow to a very slow rate. I calculate the flow rate so that the Ar will spend 
 128 
 
3-5 minutes passing through the furnace, giving it enough time to heat up and efficiently 
heat the samples inside.  
A typical tube furnace annealing protocol I used is shown in Figure 2.33. Most 
samples went through this protocol, where they are heated in Ar atmosphere above their 
TC and cooled at a rate of 1 °C min
-1 back down to room temperature. When the sample is 
above TC, the elements have no magnetization. As they are cooled below TC, they regain a 
random magnetization and are then able to interact and flip to arrange themselves in an 
energetically favorable configuration. The magnets continue this until they reach their 
blocking temperature (Tb) when they stop flipping. I cool the samples at the given rate until 
they are safely below Tb and then can cool faster to save time. The resulting magnetic 
configurations can then be imaged using Lorentz TEM.  
The annealing protocol in Figure 2.33 is typically accomplished by simply turning 
off the tube furnace and letting it cool down on its own. The cooling may not be strictly 
linear, but I have observed the cooling rate and it is close. I tried many other annealing 
protocols as well, especially in early experiments. These include heating samples to a 
given temperature, above or below TC, holding there for anywhere from 5 minutes to 
10 hours, and then quenching the sample. Quenching typically involves quickly sliding the 
glass tube far to one side, so that the boat with samples in it would move out of the middle 
of the furnace where the heat is coming from. This, in addition to rapidly turning up the 
Ar flow, helps cool the samples as fast as possible. I also tried very slow cooling, reducing 
the temperature by 1 °C every 5 minutes, 20 minutes, 1 hour, or 5 hours. In general, I  
 129 
 
 
Figure 2.33 Annealing protocol. Typical annealing protocol for most data in this 
dissertation. Samples are first heated quickly above their TC and then cooled slowly at 
1 °C min-1 back down to room temperature. The magnets are able to flip when the 
temperature is above Tb and so they are able to flip and interact in the region between TC 
and Tb on the graph. The TC and Tb indicated are for more recent samples and can vary 
slightly depending on the deposition of FePd3.  
 
found that faster cooling tends to give more ordered states. The longer anneals may work 
better if the temperature could be more precisely controlled. The controller on the furnace 
has limited programmability, and it typically oscillates by 1 °C when it should be holding 
at a constant temperature. This does not result in linear cooling. Longer anneals apparently 
give elements more time to flip into non-ordered states and get stuck there. The process of 
heating above TC, cooling at 1 °C min
-1, and quenching tends to give the best results. Most 
of the data in later chapters comes from an annealing process like that in Figure 2.33. Any  
 130 
 
 
Figure 2.34 Other annealing graphs. a. Alternative, compact representation of the 
annealing process in Figure 2.33. Samples are heated to 145 °C and cooled at a rate of 
1 °C min-1 down to 90 °C. b. Different annealing process for most small kagome crystals 
with lower TC. Samples are still heated above TC and cooled at a given rate down past the 
temperature where spins stop flipping. c. Quenching schematic. Samples were heated at 
89 °C for 14 hours and then cooled as fast as possible.  
 
image or data from a different process will be noted with a graph similar to Figure 2.34, 
but otherwise it should be assumed the data came from the typical process in Figure 2.33. 
After annealing, the samples are imaged in the transmission electron microscope. 
Before inserting samples into the microscope, a degaussing procedure is run on the 
microscope’s objective lens to remove any remnant field (< 1 gauss) at the sample that 
could influence the magnetic configuration. This is done through custom electronics to take 
control of the objective lens as well as a LabVIEW program to run the degaussing 
procedure. The program to degauss the lens was written by Stephen Daunheimer and is 
similar to Protocol 1 in Figure 1.20. The degaussing procedure is necessary because the 
lens is capable of producing extremely high fields and in order to do so, it has a soft 
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magnetic core that amplifies the field the solenoid generates. This core will have a non-
zero magnetization when the solenoid is turned off which can affect the magnetic 
configuration of annealed ASI samples and bias the data. To check that we are at < 1 gauss, 
I use a special TEM holder with a Hall bar sensor mounted on the end. The sensor measures 
the magnetic field at the exact location the ASI sample is going to be. This is important 
because the specific LabVIEW input parameters needed to degauss the lens change over 
time, so the success of the degaussing must be verified and the parameters tweaked until 
the Hall sensor measures < 1 gauss at the sample. I try to run the degaussing procedure 
with all other settings on the microscope close to what they will be during Lorentz imaging. 
This insures that stray fields from other lenses used in Lorentz imaging will be cancelled 
during the degaussing and the field will remain close to zero at the sample during imaging. 
I have tested the stability of the setup and find that after degaussing, the field at the sample 
remained at < 1 gauss for at least 7 hours.  
 
2.4.2 Initial results  
 
The foundations of the project and this dissertation are built upon thermal activation 
of ASI using a new material with strong experimental advantages. An important benchmark 
test for the FePd3 ASI system I developed is to see whether the square ice ground state can 
be observed after annealing. This ground state was previously reported in both as-grown 
samples [77] and in permalloy samples carefully heated above their TC [78,82]. In order to 
verify that FePd3 is a good material for these studies, the square ground state should readily 
be reproduced. I found that upon heating, I observe thermally activated reversal of 
individual magnets when the FePd3 ASI samples are heated to about 110 °C, though in  
 132 
 
 
Figure 2.35 Square ice ground state. Lorentz contrast TEM image (a) and corresponding 
spin map (b) of a square geometry sample that was heated above its TC and cooled back to 
room temperature showing perfect ground-state ordering of type I vertices. Scale bar is 
4 µm.  
 
some early samples thermal activation began around 88 °C due to different FePd3 film 
properties. After the annealing protocol in Figure 2.33, I observe the square geometry 
samples to readily find the ground state, often with perfect or nearly perfect ordering of 
type I vertices. The largest perfect ground-state domains we have observed are 30 μm × 
30 μm, limited only by the size of the samples studied. An example of a perfectly ordered 
square sample is shown in Figure 2.35. 
Figure 2.35 shows a Lorentz contrast image of a square geometry sample that was 
annealed as well as a corresponding spin map added by the crawler described in Section 
2.3.3. In some crystals and especially in the crystals discussed in Chapter 4, it is beneficial 
to switch to a vertex notation for the square ice. This notation is shown in Figure 2.36 
 133 
 
where a colored dot is added to each vertex to signify its energy level. For perfectly ordered 
crystals the new notation is not very enlightening, but in some crystals where domains of 
ground-state order are separated by domain walls of higher energy vertices, the vertex 
notation is very helpful at illuminating the behavior of the samples. An example of this is 
shown in Figures 2.37 and 2.38.  
To understand Figures 2.37 and 2.38, I note that the square ice ground state is two-
fold degenerate, corresponding to the two possible tilings of alternating type I vertices. 
Therefore, there can be different domains of ground-state order separated by a domain wall. 
This is found in some square geometry samples, though they still tend to be very well 
ordered and support large domains of ordering. We can quantify the degree of ordering in 
the square geometry with a correlation parameter I will call “C.” C is found by assigning a 
value of 1 to a pair of neighboring spins of they are in a head-to-tail arrangement and a -1 
to the pair if they are head-to-head or tail-to-tail arrangement, adding up the contribution 
from all pairs, and then dividing by the total number of pairs. A C value of 1 represents 
perfect ordering, 0 is disordered, and -1 is anti-ordering of all type IV vertices. The crystal 
in Figure 2.35 has C = 1, Figure 2.37 has C = 0.95, and Figure 2.38 has C = 0.92.  
I note that I fabricated square geometry samples with both three-fold edges (3FE) 
— shown in Figure 2.38 — and one-fold edges (1FE) — shown in Figure 2.37. 1FE crystals 
have elements sticking out at the edges and 3FE crystals have flat edges. We observe no 
significant differences in the degrees of ordering between the square geometry crystals with 
different types of edges. Overall, I captured over 200 Lorentz TEM images of different 
annealed square geometry crystals fabricated from 5 different samples made from different 
depositions of FePd3. Every single crystal has a C value greater than 0.85, with most 0.95 
 134 
 
or higher. These are incredibly robust results and demonstrate the reproducible and well-
understood behavior of long-range ordering in square ASI.    
 
 
Figure 2.36 Vertex types notation. Definitions of the 16 square geometry vertex types 
grouped by energy level. The colored dots are placed on top of the vertices in Lorentz TEM 
images to aid in data analysis and visualization. This vertex notation is used in Figures 2.37 
and 2.38 as well as extensively throughout Chapter 4.  
 
 135 
 
 
Figure 2.37 Square ice ground-state domains. Vertex notation added to a Lorentz TEM 
image of a square geometry crystal with 1FE that has been annealed by the process in 
Figure 2.33. The crystal shows large regions of ground-state ordering of type I vertices 
separated by domain walls made of higher energy vertices. There is one isolated domain 
as well as two domain walls that extend to the edges of the finite crystal. Domain walls are 
typically made of type II and III vertices, consistent with other experimental and theoretical 
studies of thermalized square ASI [78,119]. We note that we have never observed a type 
IV vertex in any of our samples. The correlation parameter C for this crystal is 0.95, 
showing very good ordering. Scale bar is 3 µm.  
 
 136 
 
 
Figure 2.38 Square ice domains. Vertex notation Lorentz TEM image of a square 
geometry crystal with 3FE. C = 0.92 for this crystal. Scale bar is 3 µm. 
 
The consistent observation of square ice ground-state ordering confirms FePd3 is a 
good material for studying thermally active ASI systems and it reproduces previous results, 
though the FePd3 has some major advantages over other groups. Zhang et al. report a very 
small temperature window where they can conduct experiments with permalloy samples 
[78]. They see magnets begin to flip at 545 °C, but if their samples are heated 560 °C, they  
 137 
 
 
 
 
Figure 2.39 Experimental temperature window. Experimental temperature window 
comparing FePd3 and 25 nm thick permalloy. Lower bound is the temperature where 
magnets begin to flip due to thermal activation, higher bound is limited by nanostructure 
degradation. Stars indicate the relative Curie temperatures of the two materials. Permalloy 
data reported in Ref. [78].  
 
begin to see structural degradation of the nanomagnets due to lateral diffusion. They also 
report having to specially treat the substrates of their samples to obtain even this small 
experimental temperature window. A small window makes experiments very challenging 
and makes repeated experiments extremely difficult as the samples cannot last very long. 
We have tested our samples by heating them up to 510 °C and did not detect any structural 
degradation at that temperature. Thus the FePd3 ASI system gives a much wider 
temperature window compared to permalloy and allows for greater flexibility and ease in 
performing experiments. A comparison of the experimental temperature windows for the 
two materials is shown in Figure 2.39. 
As noted above, I found that the perfectly ordered square ice ground state was 
achieved relatively easily with FePd3. The behavior was extremely robust and can be 
observed quite quickly as well. The full annealing protocol in Figure 2.33 takes around 90 
minutes to complete, but the square ground state can also be observed with quenching and 
in that case can only take 20-30 minutes to complete the heating. This is significantly faster 
than the perfect square ordering reported by Farhan et al. which took around 8 hours to 
achieve [82]. In their case, they only image one crystal, while in my work I can heat one 
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sample for 30 minutes, quench it, and capture images of tens or hundreds of crystals in less 
time than it takes them to observe one. Zhang et al. also do not report any perfectly ordered 
square crystals like the one in Figure 2.35. The ease of experiments and robust ground-
state ordering in FePd3 ASI are huge advantages that enable the great flexibility in sample 
design and the plethora of data gathered during my dissertation research. This data and all 
experiments performed in addition to the ones in this section will be presented in Chapters 
3 and 4.  
 
2.5 Numerical methods  
 
2.5.1 Introduction and general framework 
 
 
Numerical modeling and simulation are essential tools in virtually all areas of 
physics. Simulations are useful for making predictions about experimental systems as well 
as give new insights on observed experimental behavior and properties. All of the methods 
discussed thus far in this chapter have been experimental methods for fabricating and 
investigating thermally active ASI samples. This section will provide an introduction to 
and the general framework for the computational methods that I use in combination with 
these experimental methods to better understand the behavior and long-range ordering 
properties of my FePd3 ASI samples.  
I use a Monte Carlo method, as Monte Carlo simulations have proven to be a 
successful approach in understanding complex behavior in other ASI systems 
[83,119,141,183,184]. I will describe the Monte Carlo technique that I employ and give 
the details of how the method is implemented. Often, theoretical studies and numerical 
 139 
 
methods predict an experimental behavior, but upon investigation the experimental 
behavior deviates from the prediction. In this instance it can be beneficial to go back to the 
numerical model to help understand what may be causing the deviation, modify or improve 
the theoretical model and understanding, and make new predictions. There have been many 
predictions of long-range order in artificially frustrated systems which motivated some of 
my studies, especially in the kagome geometry [51,78,95,98,125]. My numerical modeling 
provides some insight as to why the ordered states have proved difficult to observe.     
Monte Carlo simulations are very common and very powerful computational 
methods that span the physical sciences [185,186]. There are many different branches and 
methods that use different algorithms suited for different applications. The Metropolis-
Hastings algorithm is one common method that is very versatile [187–189]. It can be used 
on a variety of complex systems, but as applied to ASI studies, the basic premise is as 
follows: First, randomly select a spin in a lattice and calculate the change in total energy 
of the system if that spin were to flip. If flipping the spin will lower the total energy of the 
system, flip the spin. If flipping the spin will raise the total energy of the system, flip the 
spin with a probability that depends on the temperature of the system. Higher temperatures 
are more likely to allow for excitations that raise the total energy of the system. Some 
thermal ASI studies very successfully use a Metropolis algorithm to model observed 
behavior [78,85].  
I found a kinetic Monte Carlo technique [190] to be the most useful in modeling 
my FePd3 ASI samples. The kinetic Monte Carlo technique was originally developed to 
model vacancy migration in a binary alloy, but it has also been found to be an accurate 
method for describing thermally active ASI systems [81,82,95]. The first step is to create 
 140 
 
a lattice of the ASI spins. I choose the geometry, size, and boundary conditions of the lattice 
to match those of my experimental samples. Then, an Arrhenius flip rate is assigned to 
each element given by [142,190] 
𝜏−1 = 𝜈0exp (
𝐸0+Δ𝐸
𝑘𝐵𝑇
)   (2.5) 
where E0 is an intrinsic energy barrier, T is the temperature, ν0 is a material specific 
prefactor, kB is Boltzmann’s constant, and ΔE is half the change in total energy of the 
system if that spin were to flip. ΔE can be calculated from magnetic Coulomb interactions 
or dipolar interactions depending on the geometry and the system being studied. E0 is 
known to be well-approximated by the shape anisotropy energy of the magnetic element 
and was given in Equation 1.14 [81,87,139,145]. I always use M = 200 kA m-1, T = 420 K, 
and ν0 = 10-25 s-1. E0 depends on the demagnetizing factor DZ which is a purely geometric 
term derived from the shape of the magnet and how easy it is to flip the magnetization 
along different axes. DZ is the factor along the long axis of the magnet and has been solved 
numerically for a general ellipsoid [25] and exactly for a rectangular prism [145]. We 
approximate the ASI nanomagnet as a rectangular prism for simplicity in the calculation 
and we believe this is a reasonable and good approximation. We are less concerned with 
the values of parameters T and ν0 as these are generally used to evolve time in the model 
and affect every rate in the lattice in the same manner. We are more interested in the long-
time, steady-state ordering in the system and observing how ordered the lattice can become.  
 The magnets start in a random initial condition. The flip rates in Equation 2.5 are 
calculated for each spin and all the rates are summed. The rate for each spin is then divided 
by the total, yielding a relative probability distribution for the individual magnets to flip. 
One magnet is then selected at random, weighted by the probability distribution, and 
 141 
 
flipped. The rates are recalculated and the process repeats. A visualization of the state of 
the individual magnets is given in Figure 2.40. The magnet has two possible states in an 
effective double well potential. There is some intrinsic energy barrier E0 to overcome in 
order to flip and a change in energy ΔE which can be positive or negative if the magnet 
flips. Both E0 and ΔE are typically on the order of a few eV for our experimental samples 
and simulation parameters. The benefit of the kinetic Monte Carlo technique is that spins 
that are in less favorable energetic arrangements are more likely to flip and spins that are 
in energetically favorable environments are unlikely to flip, though it is still possible. This 
is characteristic of real samples finding ordered states. Well-ordered regions are unlikely 
to thermally excite and raise the total energy, though it is still possible. This technique also 
reduces the time it takes to run the simulations. There is little processor time wasted 
randomly picking spins that are already in low energy states. The model picks the spins 
that should flip first, quickly achieves decent ordering, and then the steady-state behavior 
progresses.  
Disorder is known to play an important role in the long-range ordering of ASI 
systems and it is important to be able to incorporate disorder into the simulations. I 
discussed in Chapter 1 how different types of disorder have similar effects on the energy 
landscape of the ASI vertices (see Figure 1.27). Therefore, it is essential to include the 
effects of disorder in the system, but less important exactly how the disorder enters the 
model. As we will see in later chapters, there is a characteristic spread in the widths of 
individual magnetic elements due to artifacts from the electron beam lithography. This is  
 142 
 
 
Figure 2.40 Spin state double well potential. Schematic of the energy landscape for one 
ASI nanomagnet. There is an intrinsic energy barrier E0 to overcome in order to flip which 
is approximated by the shape anisotropy energy of the element. ΔE is the change in energy 
if the magnet were to flip. Depending on the current state of the magnet, ΔE can be positive 
or negative. E0 is always positive.  
 
a realistic source of disorder in experimental samples and is relatively easy to implement 
in the simulations. Individual spins in the simulation lattice are randomly assigned a width 
sampled from a Gaussian distribution with a characteristic mean width and standard 
deviation (σw). The given mean width and σw depend on the exact system and geometry 
being studied, but are typically on the order of 120 nm and 1 nm, respectively.  
The effects of this width disorder enters the simulations in two ways. First, the 
intrinsic energy barrier E0 depends on both the volume of the element and DZ which are 
both strong functions of the width of the magnet. This leads to a range of E0 values assigned 
 143 
 
to each spin, heavily affected by the actual width of the magnet. Second, the ΔE term will 
be a function of the magnetic moment of the individual magnet. ΔE can be calculated in 
different ways, as will be discussed in Chapters 3 and 4, but it will always depend on the 
magnetic moment, which is the product of the nanomagnet’s magnetization and volume. 
This will cause slight variations in the interaction energies across the sample. Both E0 and 
ΔE have a strong dependence on the widths of the magnetic elements which propagate the 
disorder into the energy landscape of the vertex interactions and spin flips. This can cause 
certain spins to be very favorable to flip, and others to be very unfavorable, potentially 
pinning spins and defects that break the long-range order in the system.   
 
2.5.2 Square geometry Monte Carlo 
 
 
We observe the experimental square geometry samples to readily find large 
domains of ground-state order after annealing. It is useful to simulate these experiments to 
gain greater understanding of the system. The simulations follow the basic framework laid 
out in the previous section. The square geometry samples are modeled by elements 120 nm 
wide, 23 nm thick, and 400 nm long and the simulations are the same size as experimental 
samples with 40 elements per side. We use dipolar interactions for the coupling between 
magnets to calculate ΔE, similar to other reports [82]. The formula for dipolar coupling is 
found in Equation 1.11. The magnetic moments are given by a unit vector pointing along 
the length of the magnet, the magnetization of the material 200 kA m-1, and the volume of 
the element is nominally 1.10x10-21 m3. The simulations start in a random configuration 
and we evolve them for an average of 200 flips per spin. We investigate the effects of 
 144 
 
different degrees of width disorder σw as well as different numbers of nearest neighbors to 
include in the dipolar interactions. Dipolar interactions are typically long-range, but 
including interactions between one element and all other elements in the lattice is 
computationally expensive, therefore it is common to truncate the interactions at some 
distance. We find that including only nearest neighbor interactions does an excellent job of 
reproducing our experimental data and including higher order interactions results in 
qualitatively similar results, justifying truncating the interactions for efficiency.  
With no disorder the simulations produce perfect ground-state ordering every time. 
A perfectly ordered final state from a simulation is shown in Figure 2.41. All the simulation 
images use the vertex notation defined in Figure 2.36. When including disorder, we 
generally repeat simulations a number of times with different random sampling of disorder 
because a different sample can yield different results and domain configurations. Some 
simulations show a perfectly ordered lattice with as much at σw = 1 nm disorder, though 
this is not generally the case. Figure 2.42 shows 4 typical domain patterns observed in 
simulations, produced with varying width disorder from σw = 0.5-2.5 nm. This level of 
disorder captures the experimentally observed large domains of type I vertices and it also 
reflects a realistic spread in the actual widths of the lithographically patterned arrays as 
observed with high-resolution TEM. We can calculate the correlation parameter C for the 
simulations as well. The frame in Figure 2.41 is perfectly ordered and has C = 1. In Figure 
2.42, C ranges from 0.90 to 0.96 which is still quite high.  
 145 
 
 
Figure 2.41 No disorder square Monte Carlo. With no disorder, square simulations 
produce perfect C = 1 ordering. This frame is taken after 800,000 Monte Carlo steps.    
 
 146 
 
 
Figure 2.42 Square Monte Carlo with varying disorder. Final frames of square kinetic 
Monte Carlo simulations with domain patterns that are remarkably similar to those 
observed in experiment. The spread in widths used in these runs are 0.5 nm (a), 1 nm (b), 
1.5 nm (c), and 2.5 nm (d). The ordering parameter C values are 0.96, 0.94, 0.90, and 0.92, 
respectively. Each image is the state of the system after 800,000 Monte Carlo steps, or an 
average of 200 flips per spin.  
 
Sometimes samples can have highly disordered domains and little long-range order. 
This is generally due either to the elements being too small with low moments, the FePd3 
film having a low quality from deposition, or there being strong non-Gaussian disorder in 
 147 
 
the system from lithography. Strong disorder, for example, can come from one element 
that is fabricated very wide or very thin or some other kind of non-Gaussian defect in the 
system. Vacancies typically do not greatly affect the ordering ability of experimental 
samples, but other types of quenched disorder can. Figure 2.43 shows a simulation with a 
higher degree of disorder and correspondingly smaller ground-state domains. This much 
disorder was rarely observed in real square geometry samples, but was sometimes observed 
in the topological defect samples discussed in Chapter 4. The C value for the lattice in 
Figure 2.43 is 0.80, which is still pretty high. As we will see, this is significantly higher 
than the highest observed correlation parameters for the kagome geometry. Even with what 
appears to be a relatively disordered system, there are still some very large regions of 
ground-state order.   
The simulations start in a random state and due to the benefits of the kinetic Monte 
Carlo technique, find highly ordered states very quickly. Figure 2.44 shows snapshots of 
the system throughout the simulation. The lattice becomes well-ordered after only about 
2000 Monte Carlo steps in Figures 2.44(a)-(d), having formed large domains of ground-
state order separated by domains walls made from type II and type III vertices. Once these 
domains and domain walls are formed, the domains tend to shrink slightly as seen in 
Figures 2.44(e)-(i). This takes 30,000 Monte Carlo steps, compared to 2000 for Figures 
2.44(a)-(d), showing how quickly the system can reach a relatively well ordered 
configuration. After the large domains have formed, the type III vertices will simply 
fluctuate and move around as seen in Figures 2.44(j)-(l). This behavior and the evolution 
of the system is in agreement with other computational studies of thermally active square 
ASI [119]. I often run the simulations for up to 800,000 steps to reach an average of 200  
 148 
 
 
 
Figure 2.43 Square Monte Carlo with high disorder. Final frame of a kinetic Monte 
Carlo simulation with 3.5 nm width disorder after 800,000 Monte Carlo steps. The lattice 
has many smaller domains and many domain walls and C for this frame is 0.80.  
 
flips per spin. Figure 2.44(l) shows the system after 40,000 Monte Carlo steps, but the 
behavior in Figures 2.44(j)-(l) can persist for hundreds of thousands more steps without 
providing much new information. Sometimes the lattice can become slightly more ordered 
after many steps, but generally not significantly. The C value starts at 0.24 in Figure 2.44(a) 
and increases to 0.91 in Figure 2.44(l). Overall, we are successful at modeling observed 
behavior in thermally active square ASI and have shown that disorder plays an important 
role in the long-range ordering of the system.    
 149 
 
 
 
Figure 2.44 Evolution of a square geometry kinetic Monte Carlo simulation. Snapshots 
of the state of the system after (a) 200, (b) 600, (c) 1200, (d) 2000, (e) 4000, (f) 8000, (g) 
12000, (h) 20000, (i) 30000, (j) 32000, (k) 36000, (l) 40000 Monte Carlo steps.  
 150 
 
Chapter 3  Charge-ordering in artificial kagome ice  
3.1 Introduction 
 
This chapter aims to investigate theoretical predictions of long-range charge-
ordering in kagome ASI and increase the body of knowledge surrounding this fully 
frustrated system. Unlike the square geometry, which has only a two-fold degenerate 
ground state, the kagome geometry is extensively frustrated with more degeneracy and a 
much richer predicted phase diagram. Locally, the three-island kagome vertices are six-
fold degenerate, though it is predicted that long-range dipolar interactions can break this 
degeneracy and realize more-ordered states. As was shown in Figure 1.23, the theoretically 
predicted phase diagram for the kagome system has three plateaus in entropy 
corresponding to three distinct states of varying degeneracy: the ice rule obeying kagome 
I state, the charge-ordered kagome II state, and a spin-ordered state.  
This chapter will focus on the realization, experimental investigation, and 
numerical modeling of the kagome II charge-ordered state, shown schematically in Figure 
1.24(c). The chapter is divided into three main sections. Section 3.2 looks at small kagome 
crystals with varying edge types. Section 3.3 investigates charge-ordering in large crystals 
as a function of the lattice spacing and element sizes. Section 3.4 uses the kinetic Monte 
Carlo technique described in Section 2.5 to help better understand the behavior of the large 
crystals in Section 3.3. In all experimental studies, a great deal of data was collected and 
averaged, allowing us to approach what is likely the most natural and typical behaviors of 
the system.   
 151 
 
Ice-rule ordering in the kagome geometry has been extensively observed in 
numerous studies since the introduction of the lattice as an ASI system [87,90,93,94,124]. 
Some studies have also seen ice-rule violating vertices such as those in Figure 1.26 or 1.34 
in both thermal and athermal systems [78,80]. Long-range interactions and ordered states 
have been predicted for a long time but proved difficult to observe with rotation 
demagnetization protocols [93,125,127,132,191]. With the recent introduction of thermal 
relaxation in ASI systems, regions of charge-ordering have been observed, though the 
entropy of the system and the extent of the ordering is still far from the predicted states. 
Here, we improve on the previously reported the degrees of ordering and describe why the 
systems are not reaching the perfectly ordered configurations.  
Perfect charge-ordering is predicted for the kagome II state, but this is not observed 
in practice. Instead, crystals tend to form domains of charge-ordering separated by domain 
walls. In Figure 3.1, I introduce a notation for describing the regions of ordering. The 
kagome lattice has two inequivalent lattice sites, denoted “A” and “B”, as well as two-fold 
degenerate charge-ordering – the +Q charges can be on either A or B sites. If a +Q charge 
is on an A site or a -Q charge is on a B site, this constitutes one type of domain which I will 
define to be “blue” and overlay a blue color to represent it. If a -Q charge is on an A site or 
a +Q charge is on a B site, I will define this to be a “red” domain and overlay a red color 
to identify it, as shown in Figure 3.1(b). The actual colors are formally irrelevant, as the 
important factor is the relative sizes of the different domains. Nevertheless, I keep the 
colors consistent in all figures throughout the chapter. An example of my charge domains 
overlaid on a Lorentz TEM image is shown in Figure 3.3(c). 
 152 
 
 
Figure 3.1 Kagome charge domains notation. The kagome lattice has two inequivalent 
lattice sites defined here as “A” and “B.” If a +Q charge is located on an “A” site, I will 
overlay a blue domain on that vertex. Similarly, if there is a -Q charge on a “B” site I will 
color that with a blue domain. A +Q charge on a “B” site and a -Q charge on an “A” site 
will be overlaid with a red domain. a. Six charge-ordered vertices overlaid with a blue 
domain. b. Two domains of charge-ordering. A domain wall is created when two like 
charges sit on adjacent vertices. This notation is used throughout Chapter 3. In general, the 
red or blue domain color assigned is arbitrary and they can be interchanged without losing 
any information. The relative sizes of the different domains is what matters most in 
investigating the long-range ordering behavior of the system.    
 
We can also quantify the degree of charge-ordering with a parameter I will call “C.” 
C is found by assigning a 1(-1) to each pair of nearest neighbor vertices if they have the 
opposite (same) charge, adding up the contributions from all pairs and then dividing by the 
total number of nearest neighbors (see Figure 3.2). We note that C = 1 constitutes perfect 
ordering, C = 0 is disordered, and C = -1 is anti-ordering. This correlation coefficient 
calculation is conventional for ASI studies [75,78]. We also calculate the net magnetization 
 153 
 
of all of our crystals. This is important because the charge-ordered state is necessarily also 
demagnetized, meaning the net magnetization of the sum all the elements is near zero. If a 
crystal is polarized along one sublattice, that crystal will also be perfectly charge-ordered, 
but this is trivial and is not the kagome II state we are interested in.  
A polarized and trivially charge-ordered crystal is given in Figure 3.3. The 
magnetization of each crystal can actually be calculated along a number of directions. 
These include each sublattice direction, M1, M2, and M3, as well as the Cartesian x and y 
directions, Mx and My. We typically use the x direction, the horizontal axis in the figures 
presented in this chapter. The formula is given by  
|𝑀𝑥| =
?⃑⃑? ∙𝑥
|𝑀𝑥,𝑚𝑎𝑥|
  (3.1) 
This projects the magnetization along the x axis. If the crystal is polarized along the x axis, 
|Mx| = 1 and if the sample is demagnetized, |Mx| = 0. We find that typically the |My|, |M1|, 
|M2|, and |M3| values are all approximately equal to |Mx| after annealing and so we only 
report one of them.   
The fabrication for these crystals was described in Section 2.2 and some examples 
of crystal sizes and geometries studied are shown in Figures 3.4 and 3.5. We fabricated 
crystals as small as possible, one-hexagon, up to the largest that would still fit in the TEM 
field-of-view with enough resolution to yield accurate and usable Lorentz contrast images, 
about 28 hexagons on a side or 7000 elements. We fabricated crystals with both one-fold 
edges (1FE) and two-fold edges (2FE). Examples of 2FE are shown in crystal 5 in Figure 
3.4(a). 2FE have two elements that meet at the edge. 1FE are shown, for example, in crystal 
6 in Figure 3.4(a). Here, a single element sticks out so that there is only one element on 
the edge. We found that for small crystals, these edge effects had a large impact, 
 154 
 
 
 
Figure 3.2 C parameter calculation. The charge correlation parameter C is found by 
assigning a 1 (-1) to a pair of neighboring vertices if they have the opposite (same) charge, 
adding up the contributions from all pairs and then dividing by the total number of nearest 
neighbors. We note that C = 1 constitutes perfect ordering, C = 0 is disordered, and C = -1 
is anti-ordering.  
 
but for the larger crystals, the edges did not matter. We also looked at armchair edge 
crystals shown, for example, in crystal 11 in Figure 3.4(b). These edges did not appreciably 
impact the behavior of the crystals.  
To perform experiments, we first polarize the crystals as shown in Figure 3.3. This 
helps to check that the elements were able to flip and relax and serves as a good way to 
test what temperatures to heat to in the annealing protocols. A typical annealing protocol 
is to heat above the sample’s TC and cool at a given rate down to room temperature,  
 155 
 
 
Figure 3.3 Polarized kagome crystal. a. Lorentz TEM image of a kagome crystal 
polarized toward the bottom right corner of the image. b. The same Lorentz TEM image 
as in (a) but with colored arrows added showing the directions of individual elements and 
red and blue dots added at vertices indicating the excess magnetic charge on each vertex. 
c. Charge domain map of the same crystal. The crystal is charge-ordered, but this is trivial 
because it is polarized. A true long-range ordered kagome ice II crystal is charge-ordered 
but has zero net magnetization. Charges and charge domains are defined in Figure 3.1. 
Scale bar is 3 µm.  
 
 156 
 
as shown in Figure 2.33. For small geometry crystals, the annealing protocols vary more, 
so each figure with small crystals has an individual annealing graph indicating the process 
used for those crystals. For the large kagome crystals, the protocol is always that shown in 
Figure 2.33, cooling at a rate of 1 °C min-1, just like the square geometry crystals already 
discussed in Chapter 2. After annealing, the objective lens is degaussed and Lorentz 
contrast images are captured like the one in Figure 3.3(a). These images are then processed 
and annotated, where I can add arrows on the elements and charges on the vertices, shown 
in Figure 3.3(b), as well as charge domain overlays, shown in Figure 3.3(b). We calculate 
C for each crystal as well as the net magnetization. If after annealing |Mx|, |My|, |M1|, |M2|, 
or |M3| > 0.2, we do not use the data from that crystal as it is deemed to be too polarized 
and the degree of charge-ordering could be biased.  
 
3.2 Charge-ordering and edge effects in small kagome crystals 
 
 
We begin our investigation by examining the behavior and properties of the 
smallest possible kagome crystals after thermal relaxation and work our way up to larger 
and larger geometries. Understanding the behavior on these small scales aids in the 
understanding and theory of long-rage ordering in larger crystals. This section goes through 
each small kagome geometry one by one, highlighting the typically observed spin 
configurations and properties after annealing. Some of the first papers published on 
thermally active ASI systems looked at one, two, and three hexagon kagome crystals 
[81,84]. The small kagome geometries are interesting in their own right and yield a variety 
of relevant data. All element lengths in the crystals in this section are 500 nm. We will look  
 
 157 
 
 
Figure 3.4 Small kagome crystal arrays. a-c. In-focus TEM images of various small 
kagome crystals studied in Section 3.2. The crystals are fabricated in arrays to take large 
amounts of data quickly and average results over many crystals of the same size. The 
crystals numbered are (1) 2FE one-hexagon, (2) 2FE two-hexagon, (3) 1FE one-hexagon, 
(4) 1FE two-hexagon, (5) 2FE three-hexagon, (6) 1FE three-hexagon, (7) 2FE 4-hexagon, 
(8) 2FE 5-hexagon, (9) 2FE 7-hexagon, (10) 2FE 11-hexagon, (11) armchair-edge 13-
hexagon, (12) 1FE 7-hexagon, (13) 2FE 19-hexagon, (14) 1FE 19-hexagon, (15) 2FE 37-
hexagon crystal, (16) armchair-edge 43-hexagon, (17) 1FE 37-hexagon, and (18) 2FE 61-
hexagon. Scale bars for (a)-(c) are 2 µm, 3 µm, and 3 µm, respectively.   
 158 
 
 
Figure 3.5 Large kagome crystal. In-focus TEM image of a large kagome crystal with 
500 nm element lengths. Scale bar is 3 µm.  
 
at 2FE crystals first, working our way up in size, and then compare these to 1FE crystals 
of similar sizes.  
The smallest structure that can be studied is a single 2FE hexagon. The behavior of 
these crystals is shown in Figure 3.6. After annealing, these crystals are always found in 
either a clockwise or counterclockwise, head-to-tail, flux-closure arrangement of spins. 
This behavior is extremely robust and highly reproducible. Every single 2FE one-hexagon 
crystal we studied, approximately 100 of them, was in one of these two configurations. 
This is the lowest energy arrangement of spins for this geometry. Farhan et al. observed  
 159 
 
 
Figure 3.6 One-hexagon 2FE crystal. Lorentz TEM image (a) and corresponding spin 
map (b) of a 2FE one-hexagon kagome crystal that has been annealed. These crystals 
always end up in one of two states, with either a clockwise or counterclockwise head-to-
tail, flux-closure arrangement of spins. |Mx| = 0. c. Annealing process for this crystal. Scale 
bar is 500 nm.  
 
this state, but they also observed excitations into higher energy spin configurations with 
unfavorable head-to-head or tail-to-tail arrangements of spins too [81]. We note that we do 
not observe any other states in our samples, though we are also not in exactly the same 
temperature regime as Ref. [81]. 
Two-hexagon 2FE crystals also have extremely robust and reproducible behavior. 
I always found these crystals to be charge-ordered, with over 140 crystals studied. Each 
individual hexagon is always in a clockwise or counterclockwise flux-closure arrangement 
of spins. The hexagons can both have the same chirality as in Figure 3.7(c), or they can  
 160 
 
 
Figure 3.7 Two-hexagon 2FE crystals. a, c. Lorentz TEM images of annealed 2FE two-
hexagon crystals with arrows and charges added. b, d. Lorentz TEM images with charge 
domain maps. e. Annealing process for these crystals. The 2FE two-hexagon crystals are 
always found to be charge-ordered. They also always are observed to have head-to-tail, 
flux-closure arrangements of elements on the edges. There are two qualitatively distinct 
states, shown here, the crystals are found in within this manifold. In (a), the left hexagon 
has a clockwise arrangement of spins and the right hexagon has a counterclockwise 
arrangement. In (c), both hexagons have a counterclockwise arrangement. Both crystals 
have C = 1 and |Mx| = 0. The behavior observed here is extremely robust and reproducible. 
Scale bar is 500 nm.  
 
  
 161 
 
have opposite chirality as in Figure 3.7(a). In Figure 3.7(a), the middle spin fits into the 
head-to-tail chains of both hexagons, while in Figure 3.7(c), the middle spin only fits into 
the head-to-tail chain of the left hexagon. This results in the configuration found in Figure 
3.7(a) having slightly lower energy than in Figure 3.7(b), but both states are still low 
energy, charge-ordered, and demagnetized. 
In contrast, I almost never found 2FE three-hexagon crystals to be charge-ordered. 
They have a very strong tendency for the individual hexagons to form head-to-tail, flux-
closure arrangements of spins on the edges, which is not compatible with charge-ordering 
in this geometry. In Figure 3.8(a), the spins on the edges all follow a counterclockwise 
arrangement. In order for the crystal to be charge-ordered, the three outer vertices must 
have a different charge from the inner one, but this is not possible for any configuration of 
spins that has continuous head-to-tail ordering around the edge elements of each hexagon. 
Since the crystal is generally not charge-ordered, it is most often found with the charge 
domain pattern shown in Figure 3.8(b). We have found only one three-hexagon crystal to 
be perfectly charge-ordered out of about 100 studied. This crystal is shown in Figure 3.9. 
The charge-ordering is allowed because of an energetically unfavorable head-to-head 
arrangement of spins in the lower left corner of the image. This is very uncommon and it 
is clear from this geometry that the edge effects in 2FE crystals tend to dominate other low 
energy or ordered configurations.  
 162 
 
 
Figure 3.8 Charge domains in a three-hexagon 2FE crystal. a. Lorentz TEM image of 
an annealed 2FE three-hexagon crystal with arrows and charges added. b. Charge domain 
map. c. Annealing process for this crystal. C = 0.33 and |Mx| = 0. Three-hexagon crystals 
are almost never found to be charge-ordered because the charge-ordering is not compatible 
with having flux-closure arrangements of spins on the edges. Scale bar is 500 nm.  
 
To solidify the energetics of the three-hexagon states we model the crystal using 
the Object Oriented MicroMagnetic Framework (OOMMF) package from NIST. OOMMF 
uses the Landau-Lifshitz-Gilbert (LLG) equation given by [192] 
𝑑𝑴
𝑑𝑡
= −|𝛾|𝑴 × 𝑯𝑒𝑓𝑓 +
𝛼
𝑀𝑠
(𝑴 ×
𝑑𝑴
𝑑𝑡
)  (3.2) 
where M is the local magnetization, γ is the gyromagnetic ratio, Ms is the saturation 
magnetization, Heff is the local effective magnetic field, and α is a constant. This equation 
and the OOMMF package do an excellent job of treating magnetic systems as they take 
into account the full size and domain structures of the patterned magnets. The structure is 
defined in the software along with an initial configuration. The system is then allowed to 
 163 
 
 
Figure 3.9 Charge-ordered three-hexagon 2FE crystal. a. Lorentz TEM image of an 
annealed 2FE three-hexagon crystal with arrows and charges added. b. Charge domain 
map. c. Annealing process for this crystal. C = 1 and |Mx| = 0.36. This is the only three-
hexagon crystal we observed to be charge-ordered out of ~100. It is ordered because of an 
energetically unfavorable head-to-head arrangement of spins in the lower left corner. Scale 
bar is 500 nm.  
 
relax stepwise through the LLG equation. Once the micromagnetic state has relaxed, the 
software can calculate various energies in the system. Our OOMMF results are shown in 
Figure 3.10. As expected from the experimental ASI results, the charge-ordered three-
hexagon configuration has a significantly higher total energy than the edge flux-closure 
dominated state. Thus, the tendency of the three-hexagon system to minimize its energy 
typically results in non-charge-ordered configurations.  
 164 
 
 
Figure 3.10 Three-hexagon 2FE crystal OOMMF. OOMMF simulation results showing 
the charge-ordered state (b) in a 2FE three-hexagon crystal has a higher total energy than 
the non-ordered, flux-closure state (a). The calculated energies for each configuration are 
given as well.  
 
As the size of the crystals increase, so do the variety and complexity of their 
behavior. For 4-hexagon 2FE crystals, these are able to charge-order and are found to do 
so on some occasions, but not always. There is a spectrum of ordering and domain patterns 
observed for this geometry. Figure 3.11 shows a perfectly ordered 4-hexagon crystal. We 
have observed 10 perfectly ordered crystals out of about 50 studied, though this prevalence 
may be subject to some sampling bias. We did not capture images of every 2FE 4-hexagon 
crystal we fabricated. Still, the occurrence of both perfectly ordered and non-ordered 
crystals shows that the edge effects have become less important and the system tries to 
minimize its energy and find ordered states, though this is not always possible due to the 
 165 
 
increasing number of possible configurations. These 4-hexagon crystals have 219 = 524,288 
possible states. Figure 3.12 shows representative examples of other charge domain patterns 
observed. Figure 3.12(a) is almost perfectly anti-ordered, while Figure 3.12(b) is almost 
perfectly ordered. Figures 3.12(c) and (d) are moderately ordered, but illustrate the variety 
and diversity of possible patterns.  
 
Figure 3.11 Charge-ordered 4-hexagon 2FE crystal. a. Lorentz TEM image of an 
annealed 2FE 4-hexagon crystal with arrows and charges added. b. Charge domain map of 
the same crystal. c. Annealing process for this crystal. C = 1 and |Mx| = 0. 4-hexagon 
crystals can order and are sometimes found to be perfectly charge-ordered, though the 
majority of the time they are not (see Figure 3.12). Scale bar is 500 nm. 
 
 166 
 
 
 
Figure 3.12 Charge domains in 4-hexagon 2FE crystals. a-d. Charge domain maps on 
Lorentz TEM images of 4-hexagon 2FE crystals. e. Annealing process for these crystals. 
These images are representative of typical observed charge domain patterns for the 4-
hexagon crystals. C and |Mx| are indicated for each crystal. Scale bar is 500 nm.  
 
 2FE 5-hexagon crystals are very similar to the 4-hexagon geometries. We find these 
crystals can also perfectly charge-order, but generally do not, due to the large number of 
available low-lying states with nearly the same energy. A perfectly charge-ordered crystal 
is shown in Figure 3.13. We have found only two perfectly charge-ordered crystals out of 
about 40 studied, though again there may be some sampling bias in these numbers. Some 
representative charge domain patterns are given in Figure 3.14. These show the diversity 
of the observed configurations. It is interesting to note that for Figure 3.14(a), there is only 
one red domain vertex, but this single opposite domain brings the average correlation C 
down to 0.5. The C parameters tend to be more useful when looking at larger crystals 
because the actual values are heavily affected by the number of hexagons in the small 
geometries.  
 167 
 
 
Figure 3.13 Charge-ordered 5-hexagon 2FE crystal. a. Lorentz TEM image of an 
annealed 2FE 5-hexagon crystal with arrows and charges added. b. Charge domain map of 
the same crystal. c. Annealing process for this crystal. C = 1 and |Mx| = 0. 5-hexagon 
crystals can order and are sometimes found to be perfectly charge-ordered, though the 
majority of the time they are not (see Figure 3.14). Scale bar is 500 nm. 
 
 168 
 
 
Figure 3.14 Charge domains in 5-hexagon 2FE crystals. a-c. Charge domain maps on 
Lorentz TEM images of 5-hexagon 2FE crystals. d. Annealing process for these crystals. 
These images are representative of typical observed charge domain patterns for the 5-
hexagon crystals. C and |Mx| are indicated for each crystal. Scale bar is 500 nm. 
 
 2FE 7-hexagon crystals are the largest geometry we have found to be perfectly 
charge-ordered. We have observed 5 of these to be perfectly ordered out of about 130 
studied. A perfectly ordered 7-hexagon crystal is shown in Figure 3.15. This geometry is 
compatible with perfect ordering and it is interesting to note that the edges of this crystal 
are all still in flux-closure arrangements. Some representative examples of non-perfectly 
ordered 7-hexagon crystals are shown in Figure 3.16. Figure 3.16(a) has an interesting 
domain pattern where the central vertices belong to one domain and each of the outer 
vertices belong to the other. Figure 3.16(c) has a nice two-domain pattern as well.  
 169 
 
 
Figure 3.15 Charge-ordered 7-hexagon 2FE crystal. a. Lorentz TEM image of an 
annealed 2FE 7-hexagon crystal with arrows and charges added. b. Charge domain map of 
the same crystal. c. Annealing process for this crystal. C = 1 and |Mx| = 0. 7-hexagon 
crystals can order and are sometimes found to be perfectly charge-ordered, though the 
majority of the time they are not (see Figure 3.16). 7-hexagons is the largest crystal we 
have observed to be perfectly ordered. Scale bar is 500 nm.  
 
Small 1FE crystals do not have a strong desire to form flux-closure arrangements 
of spins on the edges. Directly comparing one-hexagon one- and two-fold edge crystals 
clearly shows this difference. 2FE one-hexagon crystals always find their lowest energy 
configurations and have extremely reproducible behavior. One-hexagon 1FE crystals, on 
the other hand, can be found charge-ordered as in Figure 3.17, but typically are not 
perfectly ordered. Figure 3.18 shows some representative domain patterns for this 
geometry. We have observed 27 perfectly ordered 1FE one-hexagon crystals out of about 
200 studied. This is in stark contrast to the even edge crystals where 100% of them were  
 170 
 
 
Figure 3.16 Charge domains in 7-hexagon 2FE crystals. a-c. Charge domain maps on 
Lorentz TEM images of 7-hexagon 2FE crystals. d. Annealing process for these crystals. 
These images are representative of typical observed charge domain patterns for the 7-
hexagon crystals. C and |Mx| are indicated for each crystal. Scale bar is 500 nm.  
 
ordered. It is interesting that in Figure 3.17(a), the elements making up the inner hexagon 
are in a flux-closure arrangement. This is not always the case, however, as often times there 
can be head-to-head arrangements due to the ice-rule degeneracy of two-in/one-out or two-
out/one-in three-element kagome vertices. When there are two elements meeting at an edge 
a head-to-head arrangement is highly unfavorable, but when there are three elements there 
is no energetic preference for specific configurations between the 6 degenerate vertex 
types.  
 171 
 
 
Figure 3.17 Charge-ordered one-hexagon 1FE crystal. a. Lorentz TEM image of an 
annealed 1FE one-hexagon crystal with arrows and charges added. b. Charge domain map 
of the same crystal. c. Annealing process for this crystal. C = 1 and |Mx| = 0. 1FE one-
hexagon crystals can order and are sometimes found to be perfectly charge-ordered, though 
the majority of the time they are not (see Figure 3.18). Scale bar is 500 nm. 
 
While 2FE two-hexagon crystals are always charge-ordered and have extremely 
reproducible behavior, 1FE two-hexagon crystals have much more diverse results. They 
are able to perfectly charge-order as is shown in Figure 3.19, but typically they are not 
perfectly ordered. We have only observed two perfectly charge-ordered 1FE two-hexagon 
crystals out of about 80 studied. Figure 3.20 shows some representative examples of charge 
domain patterns for these crystals. Similar to the one-hexagon case, adding 1FE breaks the 
flux-closure arrangements on the edges and allows for many more configurations. Adding  
 172 
 
 
Figure 3.18 Charge domains in one-hexagon 1FE crystals. a-c. Charge domain maps on 
Lorentz TEM images of one-hexagon 1FE crystals. d. Annealing process for these crystals. 
These images are representative of typical observed charge domain patterns for the odd 
one-hexagon crystals. C and |Mx| are indicated for each crystal. Scale bar is 500 nm. 
 
the 1FE elements also simply increases the number of elements and thus the total number 
of possible states, making it more likely for the system to be left in a non-ordered state.  
For larger crystals it is not surprising that there are many different and complex 
domain patterns observed. It is clear, however, that as the crystals get larger, true domains 
and well-ordered regions do emerge. Figure 3.21 shows some examples of 1FE three-
hexagon crystals and Figure 3.22 shows a variety of assorted larger crystals that are 
relatively well-ordered. None of these are perfectly ordered, but some are quite close or 
have large domains of ordering. We note that as the crystals get larger, the edges tend not 
to play as much of a role in the domain patterns. This can be seen, for example, in Figures 
3.22(c) and (d) or Figures 3.22(e), (f), and (g). These groups of medium sized crystals have 
different edges, but very similar C values and similarly sized domains. Figure 3.22(a) has 
a nice 1FE 7-hexagon crystal with two distinct domains. Figure 3.22(b) is an 11-hexagon 
 173 
 
 
 
Figure 3.19 Charge-ordered two-hexagon 1FE crystal. a. Lorentz TEM image of an 
annealed 1FE two-hexagon crystal with arrows and charges added. b. Charge domain map 
of the same crystal. c. Annealing process for this crystal. C = 1 and |Mx| = 0. 1FE two-
hexagon crystals can order and are sometimes found to be perfectly charge-ordered, though 
the majority of the time they are not (see Figure 3.20). Scale bar is 500 nm. 
 
Figure 3.20 Charge domains in two-hexagon 1FE crystals. a-c. Charge domain maps on 
Lorentz TEM images of two-hexagon 1FE crystals. d. Annealing process for these crystals. 
These images are representative of typical observed charge domain patterns for the 1FE 
two-hexagon crystals. C and |Mx| are indicated for each crystal. Scale bar is 500 nm. 
 174 
 
crystal that is nearly perfectly ordered. Figure 3.22(h) has a very large blue domain that 
spans most of the crystal. Overall, these results and the results presented in this section 
show that there are a wide variety of behaviors for smaller kagome geometry crystals. 
Sometimes these crystals can be perfectly charge-ordered, but as their size increases, this 
becomes less likely and domains of ordering begin to emerge. Section 3.3 explores the 
charge-ordering and charge domain structure of large kagome crystals.  
 
 
Figure 3.21 Charge domains in three-hexagon 1FE crystals. a-c. Charge domain maps 
on Lorentz TEM images of three-hexagon 1FE crystals. d. Annealing process for these 
crystals. These images are representative of typical observed charge domain patterns for 
the odd three-hexagon crystals. No perfectly charge-ordered 1FE three hexagon crystals 
were observed. C and |Mx| are indicated for each crystal. Scale bar is 500 nm. 
 
 175 
 
 
Figure 3.22 Charge domains in assorted larger crystals. Charge domain maps on 
Lorentz TEM images of varying size crystals. The crystal sizes shown are (a) 1FE 7-
hexagon, (b) 2FE 11-hexagon, (c) armchair-edge 13-hexagon, (d) 2FE 19-hexagon, (e) 
1FE 19-hexagon, (f) 2FE 37-hexagon crystal, (g) armchair-edge 43-hexagon, and (h) 2FE 
61-hexagon. i. Annealing process for these crystals. These images represent a relatively 
well-ordered selection of larger crystals. For each size shown, other crystals of the same 
size were generally observed to be similarly ordered or less ordered. The crystals are all 
demagnetized. C and |Mx| are indicated for each crystal. Scale bars are all 1 µm.  
 
3.3 Long-range ordering in large kagome crystals  
 
In this section we investigate the charge domain and average charge-ordering 
behavior of large kagome crystals. Under the same annealing conditions, the kagome 
geometry samples do not show the same transition to a long-range ordered state as the 
 176 
 
square geometry. Instead, we see domains of charge-ordering similar to Ref. [78], albeit 
with a greater degree of ordering, which I attribute to the more relaxed temperature 
requirements and other benefits of FePd3. This is an emergence of the predicted kagome II 
state [127]. In this section, as well as the next section, we examine the degree of charge-
ordering as a function of element length and work to understand the factors that may be 
preventing the system from reaching perfectly ordered states.  
The first large kagome crystals I fabricated have element lengths of 500 nm. This 
size was chosen mainly because it is the same size that Stephen Daunhaimer used for his 
ASI research in our group and we had experience fabricating samples of this size as well 
as well-characterized electron doses for this length. After fabrication, the samples were 
heated in a variety of annealing protocols to determine where elements began to flip. Most 
protocols yield similar results and the data presented in this section all derive from the 
process shown in Figure 2.33. An example of a charge domain map for a 500 nm element 
length crystal is shown in Figure 3.23. These 500 nm length crystals show some small 
domains of ordering, but they are still very far from the theoretically predicted perfectly 
ordered state. From observations of the behavior of the 500 nm element length crystals, we 
can hypothesize that using shorter elements lengths will result in higher degrees of ordering 
due to the increased strength of the magnetostatic interactions between neighboring spins 
and the magnetic charges at each vertex. To test this hypothesis, we fabricated several 
samples with element lengths ranging from 500 nm down to 300 nm. All of the data in this 
section derives from the annealing protocol in Figure 2.33, and we perform the experiments 
and analysis using the methods discussed in Chapter 2.  
 177 
 
Charge domain maps of kagome samples at different element lengths are shown in 
Figures 3.23-3.29. Figures 3.23-3.27 show typical degrees of charge-ordering for each 
element length and representative domain patterns. For the shorter element lengths, some 
of the domains are quite large and the samples show very nice long-rage order. For the 
longer element lengths — the domains tend to be smaller, more disordered, have more 
jagged patterns, and have more point defects or few-vertex domains. There is one example 
each of domain patterns shown for the 500 nm, 450 nm, and 400 nm geometries. There are 
two examples of each of the 350 nm and 300 nm element length crystals. It is apparent 
from these figures that shorter element lengths tend to result in larger domains of ordering.  
The best ordering we have observed is around C = 0.6 in the 350 nm length samples, 
significantly higher than was previously reported in Ref. [78]. We also note that we do not 
observe any energetically unfavorable ±3 charges in our samples in contrast to other 
reports [78,79]. The data presented in Figures 3.26, 3.27, and 3.28 demonstrate the highest 
yet degree of ordering reported for the kagome geometry. Figure 3.30 also shows a spin 
map of the same crystal shown in Figure 3.26. This is an example of a typical spin map 
corresponding to any of the charge domain maps. The spins appear random and disordered, 
corresponding to a well-demagnetized state. 
 
 178 
 
 
Figure 3.23 Charge domain map of a 500 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic domain pattern for 
500 nm element length crystals. This crystal has C = 0.34 and |Mx| = 0.20. The average C 
for all 500 nm element length crystals studied is given in Figure 3.31. Scale bar is 5 µm.  
 
 179 
 
 
Figure 3.24 Charge domain map of a 450 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic domain pattern for 
450 nm element length crystals. This crystal has C = 0.42 and |Mx| = 0.18. The average C 
for all 450 nm element length crystals studied is given in Figure 3.31. Scale bar is 3 µm. 
 
 
 180 
 
 
Figure 3.25 Charge domain map of a 400 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic domain pattern for 
400 nm element length crystals. This crystal has C = 0.49 and |Mx| = 0.05. The average C 
for all 400 nm element length crystals studied is given in Figure 3.31. Scale bar is 3 µm. 
 
 181 
 
 
Figure 3.26 Charge domain map of a 350 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic domain pattern for 
350 nm element length crystals. This crystal has C = 0.60 and |Mx| = 0.05. The average C 
for all 350 nm element length crystals studied is given in Figure 3.31. Scale bar is 3 µm. 
 182 
 
 
Figure 3.27 Charge domain map of a 300 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic domain pattern for 
300 nm element length crystals. This crystal has C = 0.57 and |Mx| = 0.19. The average C 
for all 300 nm element length crystals studied is given in Figure 3.31. Scale bar is 2 µm. 
 
 
 183 
 
 
Figure 3.28 Charge domain map of a 350 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing another characteristic, but well-ordered 
and demagnetized, domain pattern for 350 nm element length crystals. This crystal has C 
= 0.57 and |Mx| = 0.02. The average C for all 350 nm element length crystals studied is 
given in Figure 3.31. Scale bar is 3 µm. 
 
 184 
 
 
Figure 3.29 Charge domain map of a 300 nm element length kagome crystal. Lorentz 
TEM image with charge domains overlaid showing a characteristic, but well-ordered and 
demagnetized, domain pattern for 300 nm element length crystals. This crystal has C = 
0.56 and |Mx| = 0.07. There are number of vacancy defects in this crystal. The average C 
for all 300 nm element length crystals studied is given in Figure 3.31. Scale bar is 2 µm. 
 185 
 
 
Figure 3.30 Spin map of 350 nm element length crystal. Lorentz TEM image with 
colored lines added indicating the direction of each individual spin for the same 350 nm 
element length crystal as in Figure 3.26. A color key for the spin directions is given in the 
upper left corner. The image shows the spins are well demagnetized and randomly 
arranged, indicative of the kagome ice II state. Scale bar is 3 µm.  
 
We have collected a number of data sets with different length elements, different 
crystals on one substrate, and different samples made from different depositions of FePd3. 
When we average our correlation data at each length, a clear trend emerges: shorter lattice 
constants display greater degrees of charge-ordering. Reducing the element length 
increases the strength of the magnetostatic interactions between elements and the effective 
 186 
 
magnetic charges at the vertices, favoring lower energy configurations. The results of the 
full data set are shown in Figure 3.31. The shortest element length crystals, 300 nm, do not 
follow exactly the same trend as the other data points and we discuss this deviation in detail 
below. For Figure 3.31 we average over 14 crystals for the 500 nm element length, 12 
crystals for the 450 nm element length, 25 crystals for the 400 nm element length, 39 
crystals for the 350 nm element length, and 6 crystals for the 300 nm element length. The 
error bars are found by taking the standard deviation of the correlation values at each length 
and dividing by the square root of the number of crystals studied at that length. Error bars 
are enlarged in Figure 3.31 for clarity.  
It is interesting to compare the 500 nm data point in Figure 3.31 to the C values in 
Figure 3.22 in the previous section. These numbers are approximately equal and within the 
error bar in Figure 3.31. The 500 nm length medium size geometry crystals tend to show 
the same behavior as their larger counterparts, which should be expected. This degree of 
ordering is characteristic of our samples for this element length and it emerges as the 
kagome crystals increase in size up to a point where they can be confidently averaged over. 
The 300 nm element length data point in Figure 3.31 deviates from the trend in the 
other 4 data points. The point is statistically not in strong disagreement with a “straight” 
line fit, but we believe this deviation may be due to higher disorder in these crystals and 
fewer overall successful crystals fabricated (6). This geometry was especially difficult to 
fabricate because of issues with the liftoff step. The 300 nm length elements have 
extremely small areas, the “cookies,” inside of the connected hexagons that need to be 
lifted off. The cookies are so small that it is difficult to get a suitable undercut in the bilayer 
resist, making it difficult for the solvent to efficiently reach underneath the cookie and  
 187 
 
 
Figure 3.31 Nearest neighbor charge correlation as a function of element length in the 
kagome geometry. Average C of all crystals studied at each element length. Error bars are 
three times the uncertainty in the average. |Mx| ≤ 0.2 for all crystals studied.   
 
 
remove it. Even when there was successful liftoff, the crystals tended to have more disorder 
than the other sizes.  
Figure 3.29 shows a 300 nm element length crystal where more vacancies are 
present than in the other crystal sizes. There also tended to be a greater size distribution of 
the widths of individual elements in the 300 nm length crystals. I will discuss the width 
variation further in the next section on Monte Carlo modeling, but recall that Budrikis et 
al. showed that disorder in ASI systems in known to have similar overall effects even when 
 188 
 
it comes from different sources [121]. The disorder in this case is affecting how well the 
system can find a charge-ordered state. Both increased numbers of vacancies and a large 
spread in widths of individual elements contribute to the lower than anticipated charge-
ordering for this size element crystals. Based on the data from the 500 nm to the 350 nm 
element length crystals, these crystals should find highly charge-ordered states. Some of 
the 300 nm element length crystal do order very well, but others have higher disorder and 
there are only 6 successful crystals at this length to study, which results in an average 
charge correlation that deviates from the trend for the other element lengths.  
We can also calculate higher order charge correlations between vertices. In Figure 
3.31, we quantified the charge-ordering using only nearest neighbors, but we can also 
calculate similar values for further charge neighbors. When doing this, most of the higher 
order correlations can be calculated in exactly the same way as the parameter C we have 
already defined, but it is important to realize that for some specific neighbor distances, this 
would yield a parameter of opposite sign from our convention. We correct for this by 
defining the correlation to be 1 if the pair of charges fits into the same charge domain and 
-1 if the charges belong to different domains.  
The results of these calculations for up to 7 nearest neighbors are shown in Figure 
3.32. It is also interesting to remember that a correlation value of 0 represents a random 
arrangement of spins and charges, so even for the furthest charge neighbors we studied in 
our samples, the 7th nearest neighbors, the correlation is generally greater than a random 
configuration. This is definitive evidence of long-range order, long-range correlation, and 
long-range interactions in this system. The magnetic charges at the vertices interact through  
 
 189 
 
 
Figure 3.32 Higher order charge correlations. Average charge correlation over all 
crystals studied as a function of neighbor number for each element length. Error bars are 
three times the uncertainty in the average for clarity.  
 
a Coulomb term with similar long-range effects and fall-off as dipolar interactions, giving 
this system long-range order highlighted by the data in Figure 3.32. 
Overall, the charge-ordering data presented here show the highest level of ordering 
yet achieved to date in an experimental kagome ASI system. Our results still deviate from 
the predicted perfectly ordered behavior and we believe this is due mainly to the disorder 
in our system. In the next section, I describe by kinetic Monte Carlo simulations that I used 
to help understand what role disorder may play in charge-ordering. With the prospect that 
 190 
 
this disorder can be overcome, we believe these results pave the way for future 
investigations of perfectly charge-ordered and even spin-ordered states in the kagome 
geometry.  
3.4 Monte Carlo modeling of large kagome crystals  
 
In this section we report our results on kinetic Monte Carlo simulations modeling 
the data presented in Section 3.3 on charge-ordering of large kagome crystals as a function 
of element length. While we have experimentally achieved the highest yet degree of 
charge-ordering so far reported in the kagome geometry, our results are still far from the 
predicted perfectly charge-ordered state or spin ordered states. In order to better understand 
our system and its deviation from theoretical predictions, we use Monte Carlo methods to 
solidify the model for our system. The framework for the kinetic Monte Carlo technique is 
given in Section 2.5.  
The major difference between my kagome Monte Carlo simulations and the square 
geometry Monte Carlo simulations presented in Section 2.5.2 is how the ΔE term is 
calculated. We found that it is more effective to calculate the change in energy from 
magnetic Coulomb interactions, similar to other reports on the kagome geometry [59,80]. 
This interaction is given by 
𝑉𝑖𝑗 = {
𝜇0
4𝜋
𝑄𝑖𝑄𝑗
𝑟𝑖𝑗
, 𝑖 ≠ 𝑗
𝜇0
4𝜋
𝑄𝑖
2, 𝑖 = 𝑗
  (3.3) 
where Qi is the total charge at each vertex, given by the sum of individual dumbbells of 
charge and rij is the distance between the two vertices. Each half-dumbbell of charge at a 
vertex is given by whether it is a +Q or –Q charge times the magnetic moment of the 
 191 
 
element divided by its length [191]. Since the magnetic moment is the magnetization times 
the volume, the total charge at each vertex simplifies to  
𝑄𝑖 = ∑ 𝑞𝑗
𝜇
𝑙
= ∑ 𝑞𝑗𝑀𝐴
3
𝑗=1
3
𝑗=1   (3.4) 
where M is the magnetization of the material and A is the cross sectional area of an element. 
When Qi and Qj are of opposite charge, this lowers the total energy which is favorable. The 
bottom term in Equation 3.3 is the self-energy of each magnetic charge. Since it is 
proportional to the charge squared, it is therefore highly energetically unfavorable to have 
a ±3Q charge, which explains why we don’t see any of these in our experimental systems.  
 In our experimental studies, we see the degree of charge-ordering improve as the 
element length decreases. In our Monte Carlo simulations, this factors into the ΔE term as 
it changes the magnitude of rij. We also need to take into account the widths of the 
individual magnets in our simulations. In our experimental samples we vary the length of 
the elements and as we do, the widths of the magnets tend to vary as well due to proximity 
effects during lithography. When the vertices connecting three elements are fabricated 
closer and closer together, this directly affects how the resist is exposed in the regions 
between the vertices. When other elements are fabricated closer to a given element, it takes 
a smaller dose to expose the same region. We attempted to adjust the dose to account for 
this, but we still find that shorter element lengths tend to result in wider magnetic elements. 
As such, we vary the element width with length in our simulations as 0.2 190w l   . This 
is shown as the line in Figure 3.33 and is representative of the widths of our real elements. 
The experimental data points in Figure 3.33 are measured from high resolution TEM 
images like the one in Figure 2.18(d). We also use these high resolution images to measure  
 192 
 
 
Figure 3.33 Simulation element sizes. Element width vs. element length measured from 
TEM images of our samples and a line showing correlated length and width input 
parameters in our simulations. Error bars are one standard deviation of measured widths. 
Note the 300 nm length elements have a larger spread in widths than the other lengths.  
 
the characteristic spread in widths of our individual elements, given by the error bars in 
Figure 3.33. 
With no disorder, the computational simulations produce perfect C = 1 ordering 
every time, but we find that when we include a small amount of random disorder we are 
able to reproduce our experimental results quite well. There have been a number of studies 
focused on how disorder affects ground state formation in artificially frustrated systems 
[118–121]. In our samples, there is a disordered spread in the width of the magnets due to 
lithography artifacts, arising from slight local variations in electron dose due to shot noise. 
 193 
 
A 4 nm standard deviation of widths is representative of the real spread in element widths 
in our 500-350 nm length samples from our microscopy observations (see Figure 3.33). 
The 300 nm length samples are very difficult to fabricate and tend to have more edge 
roughness and a larger spread in element widths, closer to 7 nm. There were also fewer 
successful crystals made at this size. This explains the deviation of the 300 nm data point 
from the model.  
We run the simulations with the same number of elements (~5700) and same 
boundary conditions as our real samples. We use a magnetization of 200 kA m-1 and evolve 
the simulation for an average of 200 flips per spin. We also repeat each point 75 times with 
different randomly distributed disorder and average over the results. As Figure 3.34 shows, 
when we assign each magnet a width randomly sampled from a Gaussian distribution with 
a standard deviation (σw) of 4 nm, we find very good agreement with our experimental data. 
By increasing the standard deviation of the distribution to 5 nm, we are able to reproduce 
the 300 nm data point and also demonstrate the same upward trend in the correlation as a 
function of decreasing element length. The results of these simulations are presented in 
Figure 3.34.  
The width disorder in our samples enters the simulation in two ways. As was 
discussed in Section 2.5, the intrinsic energy barrier E0 depends on both the volume of the 
element and DZ which are both strong functions of the width of the magnet. Disorder also 
enters through the ΔE term, as the magnitude of the half-dumbbell of charge depends on 
the cross sectional area of the individual element which is directly proportional to its width. 
Both of these effects cause slight variations in the energy landscape of the system which 
directly affect the relative flip rates for different individual spins.  
 194 
 
 
Figure 3.34 Monte Carlo results. Charge correlation as a function of element length in the 
kagome geometry for our experimental results and kinetic Monte Carlo simulations 
including two levels of width disorder. Experimental error bars are three times the 
uncertainty in the average, simulation error bars are two times the uncertainty in the 
average for clarity. The Monte Carlo results reproduce the experimental data extremely 
well.  
 
 Figure 3.34 shows excellent agreement between theory and experiment for 
quantitative charge-ordering in our system as a function of element length and 
corresponding element width. We can also create charge-domain maps from our 
simulations, just like the experimental ones found in Figures 3.23-3.29. Figures 3.35-3.39 
show characteristic charge domains for our kinetic Monte Carlo simulations as a function 
of element length and width. These are final frames from one of the 75 simulations ran at 
each data point. In these images, it is clear that the average size of the domains increases 
 195 
 
as the element length decreases, just as we observe in our experimental data. These domain 
outputs look remarkably similar to the experimental ones, strongly suggesting that we have 
a good numerical model of our system.  
 
 
 
Figure 3.35 500 nm element length simulation charge domains output. Characteristic 
charge domain pattern for 500 nm element length kinetic Monte Carlo simulation of the 
kagome geometry. This run has an element width of 90 nm and a spread in widths σw = 
4 nm, which match measured experimental samples. The charge correlation C = 0.37, 
which is approximately the average over both all experimental crystals and all simulation 
results.  
 196 
 
 
Figure 3.36 450 nm element length simulation charge domains output. Characteristic 
charge domain pattern for 450 nm element length kinetic Monte Carlo simulation of the 
kagome geometry. This run has an element width of 100 nm and a spread in widths σw = 
4 nm, which match measured experimental samples. The charge correlation C = 0.42, 
which is approximately the average over both all experimental crystals and all simulation 
results.  
 
 197 
 
 
Figure 3.37 400 nm element length simulation charge domains output. Characteristic 
charge domain pattern for 400 nm element length kinetic Monte Carlo simulation of the 
kagome geometry. This run has an element width of 110 nm and a spread in widths σw = 
4 nm, which match measured experimental samples. The charge correlation C = 0.50, 
which is approximately the average over both all experimental crystals and all simulation 
results.  
 
 198 
 
 
Figure 3.38 350 nm element length simulation charge domains output. Characteristic 
charge domain pattern for 350 nm element length kinetic Monte Carlo simulation of the 
kagome geometry. This run has an element width of 120 nm and a spread in widths σw = 
4 nm, which match measured experimental samples. The charge correlation C = 0.58, 
which is approximately the average over both all experimental crystals and all simulation 
results.  
 
 199 
 
 
Figure 3.39 300 nm element length simulation charge domains output. Characteristic 
charge domain pattern for 300 nm element length kinetic Monte Carlo simulation of the 
kagome geometry. This run has an element width of 130 nm and a spread in widths σw = 
5 nm, which match measured experimental samples. The charge correlation C = 0.55, 
which is approximately the average over both all experimental crystals and all simulation 
results.  
 
Having found a good model for our kagome geometry samples that can reproduce 
the experimental data very well, it is interesting to compare the results of the kagome 
simulations with our square geometry simulations discussed in Section 2.5.2. In the square 
geometry Monte Carlo work, we find that the same choice of parameters and approximately 
same amount of width disorder (4 nm) as the kagome simulations still produces very good 
long-range ordering with large domains of ground-state type I vertices in the square lattice, 
 200 
 
similar to the experiments shown in Figures 2.35, 2.37, and 2.38. This shows that the 
critical slowing down of spin dynamics in the kagome geometry [127] may be more 
susceptible to disorder, further preventing perfect long-range ordering in this system. We 
include disorder as a distribution of widths in our system which is reasonable given our 
high-resolution TEM observations. The actual source of disorder may be more than simply 
width disorder; as I discussed how different types of disorder have similar effects in ASI 
systems [121]. Our numerical results still capture the fundamental ideas and the overall 
behavior of the system extremely well and give a plausible explanation for the observed 
deviations from theoretical predictions [125,127].  
 
3.5 Conclusions and future work 
 
In the previous sections of this chapter, we presented results from studied on the 
charge-ordering properties of kagome geometry samples from the smallest possible 
crystals up to large arrays with thousands of individual elements. We model our large 
kagome systems using a kinetic Monte Carlo technique, and we are able to reproduce our 
experimental data extremely well by including a small amount of disorder in the 
simulations. In the smallest crystals, we find that edge effects play an important role, as 
head-to-tail flux-closure arrangements of spins compete with higher order interactions that 
favor charge-ordering. This observation is consistent with the emergent frustration of the 
system. The smallest crystals have a limited number of possible states and their finite size 
causes edge effects to dominate the effects of the frustration in some cases. For larger 
crystals, enumerating all the possible states and ground states quickly becomes intractable, 
 201 
 
the edge effects play a negligible role, and the frustration is much more apparent. We were 
able to observe perfectly charge-ordered crystals up to 7 hexagons in size and demonstrate 
extremely reproducible behavior in one, two, and three-hexagon crystals.  
In Section 3.3, we presented charge-ordering behavior as a function of element 
length for large kagome crystals. Here, we realize the highest yet degree of charge-ordering 
reported in the kagome ASI geometry by taking advantage of the benefits of FePd3 as a 
material for studying thermally active ASI systems. We find that as the length of individual 
elements decreases, the degree of charge-ordering and the size of the charge domains 
increase due to stronger magnetostatic interactions between neighboring effective 
magnetic charges. In Section 3.4, we model the results presented in Section 3.3 and 
attribute the samples’ deviations from predicted perfectly ordered states to disorder in the 
system due to lithography artifacts and fabrication. Future work may aim to achieve even 
further ordering than presented here. Small kagome crystals were only fabricated with 
500 nm element lengths and further investigations could yield new insights and larger 
perfectly ordered crystals by using shorter lattice spacings.  
A two stage ordering process is predicted for the kagome geometry, characterized 
by two plateaus in the entropy of the system and two corresponding phase transitions [98]. 
The first plateau is the magnetic charge-ordered state, followed by a further reduction of 
entropy into a spin-ordered state. Presumably, this state could be achieved by reducing the 
disorder in our samples. Careful studies of resist thickness, dose, and accelerating voltage 
in the lithography step will help reduce the spread in widths of individual elements. A move 
to higher energy lithography processes, the use of a smaller aperture to reduce beam current 
and slow beam movement, or a study of different types of resist could all improve the 
 202 
 
spread in widths we observe in our samples. Also, Ar ion milling is an inherently disordered 
technique and new ways to transfer the ASI pattern into the FePd3 films could be explored. 
In light of the success in easily achieving the square geometry ground state and our very 
well charge-ordered kagome lattices, we believe FePd3 creates a promising pathway for 
reaching the first plateau with perfect charge-ordering, paving the way for future studies 
probing fundamental kinetics of the second phase transition. The experimental observation 
of this phase transition is of great interest and could shed light on analogous phase 
transitions in real frustrated materials like the pyrochlore spin ices.  
Another major route to improve our results would be through improving the quality 
of FePd3 films we fabricate. The process I developed using a modular target was excellent 
as a proof of concept and yielded very nice and promising results. There is still a wide 
range of film qualities produced and throughout all my work it is evident that the starting 
quality of the films directly impacted both the quality of the Lorentz contrast images and 
the quality of the final data. Simply put, better thin films result in better long-range 
ordering. Future work could involve exploring ways to reduce the surface roughness and 
grain sizes in the films and how to fabricate only the highest quality films more 
consistently. Using an alloy FePd3 alloy target could likely create more uniform films and 
depositing the films in the annealing chamber could likely help as well. I had to break 
vacuum between deposition and annealing, and I did not do controlled studies of UHV 
annealing protocols for the FePd3 films. Studying the effects of different annealing 
temperatures, annealing times, quenching, and annealing during deposition could all 
impact the final favorable magnetic properties of the films. Different underlayers and 
underlayer thicknesses could be explored as well.  
 203 
 
Our square geometry samples find highly ordered states with large regions of 
ground-state order. Under the same annealing conditions, the kagome geometry samples 
do not show the same level of order, likely due to the kagome geometry being more 
frustrated. The local vertices of the kagome geometry are more degenerate than the local 
vertices in the square geometry and the kagome geometry needs higher order interactions 
to realize the charge-ordered and spin-ordered states. With the exciting results presented 
here, and the prospect that the disorder in our samples can be overcome and the quality of 
our films can be improved, our system shows great promise as a model system for future 
studies. In particular, it may allow for even higher degrees of ordering and new 
explorations into more sensitive order parameters, closer to the true kagome ground state. 
 
 
 
 
 
 
 
 
 
 204 
 
Chapter 4  Topological frustration of artificial spin ice 
 
4.1 Introduction 
 
In this chapter, we introduce controlled topological defects into square ASI lattices 
and directly observe the resulting spin configurations. This work is quite distinct from the 
experiments and theory discussed in previous chapters and the existing body of work on 
ASI systems in general. As such, this topic deserves its own thorough introduction and 
motivation. I will first discuss the idea of a dislocation as a topological defect and the 
diverse and interesting phenomena associated with these defects in different systems. I will 
then survey previous studies related to topological frustration in a variety of systems and 
define exactly how we interpret and apply this idea to ASI. With some motivation from 
crystalline systems such as the pyrochlore spin ices, I will describe how we introduce 
topological defects into our ASI samples, the fabrication details of our system, and all of 
the different defect geometries we study. I will then present extensive results from our 
system and show the diverse behaviors we observe specifically due to topological 
frustration. Finally, I develop a theoretical framework for topological frustration, 
simulations using the kinetic Monte Carlo technique, and I use the results to make 
predictions about the role of topological frustration in a variety of ordered systems.  
An ASI approach to topological frustration gives detailed experimental information 
at the single-spin level, which is generally not accessible in real materials. This opens the 
possibility for direct study of specific crystalline imperfections. Topological defects, such 
as dislocations [193], have an unavoidable presence in crystal lattices although they are 
often neglected in experimental and theoretical investigations due to their complexity and 
 205 
 
the difficulty in resolving their precise atomic arrangements. The results we present in this 
chapter demonstrate the non-trivial nature of topological defects, with implications for 
many real systems in which a typical density of dislocations could fully frustrate an 
otherwise unfrustrated system. We interpret our results in the context of unanswered 
questions about the pyrochlore spin ices [13,27,51,55,57], and point to some other 
materials where the presence of topological defects may also play a strong role. It is striking 
to note how little the role of dislocations have been considered in the extensive literature 
on ordered systems.   
Dislocations are topological defects [193] ubiquitous in crystalline materials and 
can evoke a diverse range of phenomena across vastly different systems. Examples of these 
phenomena include: theoretically predicted ferromagnetic dislocations in an 
antiferromagnetic lattice [194–197], which have recently been experimentally observed in 
antiferromagnetic NiO [198]; one-dimensional fermionic excitations in topological 
insulators [199]; plasticity in metallic alloys [200]; and a recent report on confined 
structural states at dislocations in an FeMn alloy [201]. The topological nature of 
dislocations means their presence can be measured far away from the actual defect site by 
following a path around the dislocation core, resulting in an uncompensated lattice constant 
and requiring an extra vector to complete the loop, known at the Burgers vector [17]. Figure 
4.1 shows a schematic of a topological defect in a 3D crystal and Figure 4.2 presents a 
defect in a 2D lattice and gives a definition of the Burgers vector. While non-topological 
defects such as vacancies or substitutions can impart useful and favorable properties for 
crystals in, for example, semiconductor engineering, the presence of such a defect is 
generally unknown in parts of the material far removed from it. However, the non-trivial 
 206 
 
presence of even a single dislocation can produce long-range topological effects that 
permeate the crystal. Typical dislocation densities in metals are on the order of 
105-1012 cm-2, while in ceramics they can be much lower, around 104-106 cm-2 [202]. For 
this reason, dislocations generally receive less attention in ceramic materials. 
 
 
Figure 4.1 Dislocation in a 3D crystal. Schematic of an edge dislocation in a 3D crystal. 
At the dislocation point the number of lattice constants changes from 6 to 7 and an extra 
plane of atoms appears. The conventional “T” notation for a dislocation is also shown. 
Figure adapted from Ref. [203].  
 
 207 
 
 
Figure 4.2 Topological defect in a 2D crystal. An edge dislocation like the one in Figure 
4.1 projected onto a 2D crystal. The Burgers circuit and Burgers vector (?⃑? ) are also 
indicated. The dislocation is a topological defect because any loop around the dislocation, 
no matter how big, with an equal number of lattice constants in each direction cannot be 
completed without adding an additional vector, known as the Burgers vector. 
 
 An important consideration in the relationship between structure and frustration 
that has received relatively little attention is the presence of topological structural defects. 
Topological influences on frustration have been studied in protein folding [204–208], in 
the carbon bonding of graphene nanoflakes [209,210], in nematic liquid crystals [211], and 
in spin configurations at the Cr(001) surface [212,213].  However, the level of control and 
direct characterization in these systems leaves numerous unanswered questions.  Here, we 
present a controlled and direct study of the topological frustration, specifically due to 
topological defects, in otherwise ideal crystals. We introduce controlled topological defects 
in the form of edge dislocations into our thermally active square ASI systems and directly 
observe the resulting frustrated spin configurations upon annealing. The individual 
plaquettes of the system are locally unfrustrated, but the whole structure remains frustrated 
due to its topology, leading to a high number of low lying energetic states. Figure 4.3 
 208 
 
presents a schematic representation of topological frustration as compared to other well-
studied sources of frustration. Geometric frustration has been discussed at length thus far 
in this dissertation, and geometric considerations are the source of the frustration in both 
the square and kagome ASI lattices as well as in the pyrochlore spin ices. Disorder 
frustration is like that found in spin glasses, which were discussed in Section 1.2.2.  
 
Figure 4.3 Topological frustration. a-c. Schematics of different origins of frustration. a. 
Disorder frustration: Disordered bonds between Ising spins on a square lattice cannot be 
satisfied with any spin configuration, leading to frustration. b. Geometric frustration: 
Antiferromagnetically coupled Ising spins on a triangular lattice are frustrated due to the 
geometry of the system. c. Topological frustration. Ferromagnetic spins perpendicular to a 
Mobius strip are locally unfrustrated, but the system is still frustrated due to its topology. 
 209 
 
As I discussed in Section 1.3, the pyrochlore spin ices are a class of ceramics that 
have attracted considerable recent scientific interest due to the geometric frustration 
[13,27] in the magnetic moments of rare earth atoms on the pyrochlore lattice of corner 
sharing tetrahedra, analogous to Pauling’s description of hydrogen disorder in hexagonal 
water ice at low temperatures [2]. A finite residual entropy at low temperature, resulting 
from a macroscopically degenerate ground state, is characteristic of these fully frustrated 
systems. Despite measurements confirming a residual entropy in spin ice [13] and implying 
a disordered and degenerate low-temperature state, a unique ground state has nevertheless 
been predicted, based on long-range dipolar interactions in the system [51], but it has 
proven difficult to observe experimentally [13,53–56]. Notably, one recent study did 
observe a partial recovery of entropy, below the Pauling value, for single crystal samples 
of Dy2Ti2O7 thermally equilibrated for very long time scales [57]. These considerations are 
of course influenced by the crystalline perfection of the materials, the nature and density 
of defects, and the precise configuration of spins around these defects, details that are 
challenging to experimentally determine or control. ASI presents a possible pathway to 
resolving these problems as the precise structure of the lattice may be controlled by design.  
Square ASI nanomagnet arrays were originally created to model the six-fold 
plaquette-degeneracy in pyrochlore spin ice [13,75]. The system has proven to be an 
excellent vehicle for studying frustration and its effects. Unfortunately, it falls slightly short 
of a completely analogous model to the pyrochlores due to the inequivalent interaction 
strengths between the four magnets at each vertex. The variation in the distance between 
adjacent and opposite spins at each vertex lifts the six-fold degeneracy of the ice rule state 
and leads to a ground state given by a perfect tiling of type I vertices, which is unique up 
 210 
 
to a global reversal of spins. In Chapter 2, we demonstrated perfect ground-state ordering 
in our square ASI samples (see Figure 2.35).  Our observation of large domains of ground-
state ordering in the square geometry is robust and highly reproducible, demonstrating the 
benefits of FePd3 as a material system for ASI studies. In this chapter, I discuss results on 
samples where I have modified the perfect square geometry by introducing edge 
dislocations into square ASI lattices.  
To define our lithography patterns, square lattices are first modeled as a network of 
connecting springs. The springs are all have equal spring constants and act so as to keep 
the lengths between neighboring nodes the same, shown schematically in Figure 4.4(a). 
We also include a term that aims to keep the angles between connections coming out of 
each node the same, e.g. 90° for four connections or 120° for three connections. This is 
modeled as a torsional spring attached to each pair of connections, as shown in Figure 
4.4(b). Equation 4.1 gives a mathematical description of the relaxation algorithm. The force 
on a specific node i is given by 
  𝐹 𝑖 = ∑ 𝑘(𝑥 𝑖 − 𝑥 𝑗 − 𝑥 0)
#𝑁𝑁
𝑗 + ∑ 𝜀(𝑥 𝑖 − 𝑥 𝑗)
#𝑁𝑁
𝑗    (4.1) 
where 𝑥 n are the locations of the different nodes, 𝑥 0 is the equilibrium spring length vector 
with magnitude of one lattice constant, and k and ε are constants.  
To create a defect, we remove a chain of nodes starting from a given point, doubling 
the spring lengths along the chain of removed points. We then let the system relax, iterating 
the system by slightly moving each node based on the force acting on it from the other 
nodes. A damping term (not shown) brings the velocity to zero at the end of each iteration, 
so that mass and inertial effects may be ignored. Once the lattice has equilibrated, we record 
the x and y location of all nodes. We then use these relative coordinates as a guide for our  
 211 
 
 
Figure 4.4 Spring model schematic. a. Each node is connected to its nearest neighbors by 
a linear spring of spring constant k and equilibrium length 𝑥 0. b. The connections between 
nodes have torsional springs of spring constant ε attached to them.    
 
lithography patterns. Figure 4.5 shows snapshots of a relaxation process for a lattice with 
one dislocation. John Cumings developed the relaxation algorithm in MATLAB and wrote 
the original script to define the x and y locations of the nodes for lattices with one 
topological defect. I then added to this script the ability to output a lithography pattern file 
where the element lengths and lattice constant could be controlled. Thomas Marsh, a high 
school student who worked with me over the summer of 2015, then added the capability to 
include multiple dislocations in our lattices. 
The lattice starts with nodes that are all one lattice constant apart. After the system 
has fully relaxed, lithography line lengths and locations are found by subtracting a constant 
amount from the location of the nodes. The line lengths we use here are very similar to the 
traditional square geometry samples. For the samples in this chapter, the element lengths 
directly adjacent to the dislocation points needed to be slightly shorter than the other  
 212 
 
 
Figure 4.5 Lithography pattern generation. a-d. Frames from a relaxation algorithm that 
generates the x and y coordinates of the vertices in the lithography patterns for topological 
defect samples. First, the lattice starts as a perfect square lattice with nodes connected by 
springs all of equal spring constant (a). A chain of atoms is removed starting from a given 
point (b) and the system is then allowed to relax. Frame (c) shows the node locations after 
a few iterations and (d) shows the final pattern that is used to generate the locations of the 
individual spins in each crystal. The springs attempt to keep all the bond lengths the same 
and our model also tries to keep the angles between the bonds coming out of each node 
equal as well. See Equation 4.1 in the text for a mathematical description of the model.  
 
 
 213 
 
 
Figure 4.6 Patterned topological defect. In-focus TEM image of a section of a square ASI 
crystal containing a dislocation point defect. The Burgers circuit and vector are overlaid in 
the image confirming the topological nature of the defect. Scale bar is 500 nm. 
 
 
elements because the angles around these vertices are much sharper. Additional dose tests 
were required in order to prevent these elements from touching. Figure 4.6 presents an in-
focus TEM image of a dislocation point in our samples. A Burgers circuit is drawn 
schematically around the defect site to confirm its topological nature. 
Figure 4.7 shows a completed full ASI crystal with one topological defect. I 
fabricated samples with both one and two dislocations in different positions and different 
distances apart. For the one-dislocation geometries, I will label them with the number 1 to 
indicate they have one defect and a letter, either M, C, or E to indicate the dislocation is 
located in the “Middle,” “Corner”, or “Edge” of the lattice. Within these geometries, the  
 214 
 
 
 
Figure 4.7 Full crystal with one topological defect. In-focus TEM image of a full crystal 
showing one topological defect near the edge, referred to here as a 1-Edge, or 1E crystal. 
Inset shows the schematic representation of this crystal. Scale bar is 3 µm.  
 
dislocation can also point in different crystallographic directions, though we observe no 
noticeable differences in behavior for the different directions. 
I also fabricated many geometries with two dislocations to study how the defects 
interact and affect each other. Figure 4.8 shows an example of a full crystal with two 
defects. The two dislocation geometries tended to fall into three main families where, 
within each family, the distance between the dislocation points was systematically varied. 
There were some additional patterns studied as well. The three main families are T or 
 215 
 
“Towards,” A or “Away,” and R or “Right.” T geometries have two defects pointing at each 
other along the same crystal line in the center of the lattice. These geometries are effectively 
adding an extra lattice constant along the line between the two dislocations. A geometries 
have two defects pointing away from each other along the same crystal line. This is like 
removing one lattice constant along the line connecting the defects. R geometries have the 
two defects at right angles to each other. In-focus TEM images of all the crystals belonging 
to the different families are found in Figures 4.9-4.11. In addition to a letter defining the 
overall geometry, each crystal type has a number indicating the distance between the two 
dislocations, expressed in units of a lattice constant, nominally 500 nm.  
There are a few other geometries we studied that fall outside of the T, A, and R 
families. One is shown in Figure 4.8, an OT or “Opposite-Towards” configuration. The rest 
are given in Figure 4.12. These include two OA or “Opposite-Away” geometries, as well 
as a P or “Parallel” and a D or “Diagonal” geometry. Figure 4.12 also shows an A1 
geometry, which is not actually a topological defect because it does not have an associated 
Burgers vector. This geometry represents a relaxed vacancy.  
 
 216 
 
 
Figure 4.8 Full crystal with two topological defects. In-focus TEM image of a full crystal 
showing two topological defects in an Opposite-Towards, or OT configuration. This crystal 
has the defects 27 lattice constants apart and thus the geometry is defined as OT27. The 
inset shows a schematic representation of the geometry indicating the location of the 
dislocation points, their directions, and the distance between them. Scale bar is 3 µm.  
 217 
 
 
Figure 4.9 Family of T geometries. In-focus TEM images of all of the “Towards”, or T, 
geometries studied. a. Full T4 crystal with 4 lattice constants between the defect points 
where the dislocations are pointing towards each other along the same crystal line. Inset is 
the schematic representation of the crystal showing the location of the dislocation points, 
their directions, and the distance between them. Scale bar is 3 µm. The other frames show 
a section of the crystal around the defects and the associated schematic for that geometry. 
The other geometries studied are (b) T1, (c) T2, (d) T3, (e) T5, (f) T8, and (g) T18. Scale 
bars for (b)-(e) are 1 µm, scale bars for (f) and (g) are 2 µm. 
 
 218 
 
 
Figure 4.10 Family of A geometries. In-focus TEM images of all of the “Away”, or A, 
geometries studied. a. Full A6 crystal with 6 lattice constants between the defect points 
where the dislocations are pointing away from each other along the same crystal line. Inset 
is the schematic representation of the crystal showing the location of the dislocation points, 
their directions, and the distance between them. Scale bar is 3 µm. The other frames show 
a section of the crystal around the defects and the associated schematic for that geometry. 
The other geometries studied are (b) A2, (c) A3, (d) A4, (e) A8, (f) A12, and (g) A22. Scale 
bars for (b)-(e) are 1 µm, scale bars for (f) and (g) are 2 µm. Note that (f) is actually a 
Lorentz TEM image as there were no high quality in-focus images of the A12 geometry 
readily available.  
 219 
 
 
Figure 4.11 Family of R geometries. In-focus TEM images of all of the “Right”, or R, 
geometries studied. a. Full R7 crystal with 7 lattice constants between the defect points 
where the dislocations are at right angles to each other. Inset is the schematic representation 
of the crystal showing the location of the dislocation points, their directions, and the 
distance between them. Scale bar is 3 µm. The other frames show a section of the crystal 
around the defects and the associated schematic for that geometry. The other geometries 
studied are (b) R2.8, (c) R3, (d) R3.2, (e) R3.6, (f) R4.2, (g) R5, (h) R5.7, (i) R8.5, and (j) 
R15.5. Scale bars for (b)-(i) are 1 µm, scale bar for (j) is 2 µm. Note that (j) is actually a 
Lorentz TEM image as there were no high quality in-focus images of the R15.5 geometry 
readily available. 
 
 220 
 
 
Figure 4.12 Other assorted geometries studied. In-focus TEM images of all of other 
geometries studied. Each frame shows a section of the crystal around the defects and the 
associated schematic for that geometry. The other geometries studied are (a) OA15, (b) 
OA30, (c) P20, (d) OT2, (e) A1, and (f) D2.8. Scale bars for (a)-(c) are 2 µm, scale bars for 
(d)-(f) are 1 µm. Note that (b) is actually a Lorentz TEM image as there were no high 
quality in-focus images of the OA30 geometry readily available. 
 
Similar to the thermal ASI experiments discussed in Chapters 2 and 3, samples 
discussed in this chapter were heated using the annealing protocol in Figure 2.33 and 
Lorentz TEM images were captured after degaussing the microscope’s objective lens. The 
MATLAB crawler for analyzing the Lorentz TEM images was modified by Thomas Marsh 
to work for all of the dislocation geometries. This was extremely helpful as the script was 
 221 
 
able to automatically handle any of the geometries presented to it without additional input. 
The vertex notation shown in Figure 2.36 and used extensively throughout this chapter was 
also implemented by Marsh. I will also note here that some of the TEM images in this 
chapter, both in-focus and Lorentz contrast, have a skew distortion in the image. This is 
because all of the images are taken in the low magnification mode of the microscope. In 
this mode, the focal length of the electron optics is longer, and since image distortions scale 
with the focal length of the lens, the distortion is pronounced in many of my images [182]. 
The skew is fairly small and can be corrected with digital image processing software, 
though this process is tedious and does not add any new information. Image distortion can 
be seen, for example, in Figures 4.11(a) and 4.19, as well as many other images in this 
chapter. Corrected images can be found, for example, in Figures 4.14, 4.17, and 4.45.  
In this chapter, we essentially introduce a new kind of frustration, specifically due to 
the topology of the system. We find the presence of a topological defect results in extended 
frustration within our samples. Locally, the magnets are unfrustrated, but frustration of the 
lattice persists due to its topology and strictly prevents the perfect square ground-state 
ordering found in Figure 2.35. ASI was originally developed to model real frustrated 
systems and while many interesting results and insights have been gained from studying 
ASI, it has rarely been able to make predictions about and aid in understanding real 
crystalline systems. In this chapter, we aim to accomplish this and use our samples to 
speculate on the impact of topological defects in a variety of systems. After discussing the 
behavior of our ASI lattices at the single spin level, we argue that topological defects could 
explain some unsolved problems in real systems and deserve more attention in future work 
on a range of materials with non-zero order parameters.  
 222 
 
4.2 Results and discussion  
 
4.2.1 Main results 
 
 
In this section, we present extensive results and observations on topological 
frustration of ASI, providing the first realization of domain wall pinning due to a lattice 
defect with information at the single-spin level. We discuss our observations in the context 
of crystalline materials systems with non-zero order parameters and develop a theoretical 
model for the presence of topological frustration within a system. Our main results are 
summarized in Figure 4.13. We find that crystals with dislocations show large domains of 
ground-state ordering, but they also always have a chain of type II and type III higher 
energy vertices originating from the dislocation point. These chains are required because 
of the topological nature of the dislocation. A closed path around the defect is topologically 
altered, making it impossible to support continuous ground-state order, and thus a 
frustrated chain must be present. The chains in the single-dislocation geometries always 
continue to an edge of the finite crystal, as shown by representative examples in Figures 
4.14-4.16 for the 1E, 1M, and 1C geometries. Figure 4.16 shows a nice example of a 
domain wall originating from the defect site and not taking the shortest path to the edge of 
the crystal, a behavior we observe in many different crystals. The 1C geometry has a 
dislocation very close to the edge, only 8 lattice constants away, but the vertex chain 
extends away from the closest edge. Figure 4.15 shows a very well ordered crystal with 
minimal defects and domains. This crystal is mainly split into two domains with an 
additional small one in the bottom right corner. Figure 4.14 displays a few domains in  
 223 
 
 
Figure 4.13 Topological defect after annealing. a. Lorentz contrast TEM image of the 
same defect site as in Figure 4.6 after annealing. b. The same Lorentz contrast TEM image 
in (a) with arrows overlaid indicating the direction of each macro spin. The image shows 
mainly ground-state order with a domain wall originating from the dislocation point. c. 
Vertex map of the same image in (a) and (b) representing the same information, but with 
vertex types as defined in Figure 2.36. Two and three spin vertices found in our samples 
are simply mapped onto the corresponding four-spin type, excluding the highest energy, 
type IV vertex because we do not observe any in our samples. This notation clearly shows 
a chain of type II and type III vertices originating from the defect site surrounded by a 
single domain of ground-state order. Scale bar is 2 μm.  
 
addition to the topologically required vertex chain. These crystals are representative of the 
behavior we see in one-dislocation geometries. 
The crystals shown in Figures 4.14-4.16 display the extended frustration of the 
system through long, meandering vertex chains within large regions of ground-state order. 
Other domains and domain walls may be present as are seen in our crystals of canonical 
square ASI, for example, in Figures 2.37 and 2.38, and in other previous studies as well 
[78], but a vertex chain must always begin at the dislocation point. Other than this, the 
frustrated chains and conventional domain walls are indistinguishable, exhibiting the same 
basic phenomena. The lowest energy configuration for either would be the shortest 
possible, straight chain of only type II vertices, extending in a <11> direction, but this is 
not generally observed. Type III vertices are almost always present in both the frustrated  
 224 
 
 
 
Figure 4.14 Vertex map of a crystal with one topological defect. Lorentz contrast TEM 
image with vertex types overlaid on a 1E geometry crystal. A meandering vertex chain 
originates from the dislocation among a few ground-state domains separated by traditional 
domain walls. The inset shows the schematic representation of this geometry indicating the 
location and direction of the dislocation. Scale bar is 2 μm. 
 
chains and the conventional domain walls likely because the type III vertices play an 
essential role in domain wall motion in the square ice lattice [119]. Furthermore, since a 
frustrated wall’s presence is required by topology, there is only a fractionally small change 
in its total energy to include type III vertices and take a longer, often bent or winding path  
 225 
 
 
 
Figure 4.15 Vertex map of a crystal with one topological defect. Lorentz contrast TEM 
image with vertex types overlaid on a 1M geometry crystal. A meandering vertex chain 
originates from the dislocation among a few ground-state domains separated by traditional 
domain walls. The inset shows the schematic representation of this geometry indicating the 
location and direction of the dislocation. Scale bar is 2 μm. 
 
through the lattice. It appears that both such walls are at least partly entropically driven, as 
opposed to purely energetically driven, as the lowest energy configurations do not 
dominate and the walls in general do not appear to have any tension. Additionally, we note 
here that topological frustration is distinct from vertex frustration, which is found, for  
 226 
 
 
Figure 4.16 Vertex map of a crystal with one topological defect. Lorentz contrast TEM 
image with vertex types overlaid on a 1C geometry crystal. A meandering vertex chain 
originates from the dislocation among a few ground-state domains separated by traditional 
domain walls. The inset shows the schematic representation of this geometry indicating the 
location and direction of the dislocation. Scale bar is 2 μm. 
 
example, in the shakti lattice [79] because vertex frustration is only due to the geometry of 
the system, not its topology. 
 For crystals with two dislocations, we see a wide range of behaviors. Typically, and 
especially when the defects are more than a few lattice constants apart, the two defects 
have separate domain walls that each meander to the edges of the crystal. Occasionally, a 
 227 
 
single domain wall will connect the two dislocations, as the inclusion of two topological 
defects actually allows for the possibility of a larger continuous ground-state domain. 
Examples of both of these effects are shown in Figure 4.17. In both one- and two-
dislocation geometries, the crystal is effectively a regular square geometry crystal far away 
from the defects, but the topological nature of the dislocations can still have long-range 
effects that support domain walls permeating the whole lattice. 
 We have fabricated a variety of two-dislocation geometries with varying distance 
between the defects as discussed in Section 4.1. When the two dislocations are extremely 
close to each other, they can essentially be thought of as one defect. It is straightforward to 
draw a Burgers circuit around both defects and find that no vector is required to complete 
the loop and naively conclude that there is no topological defect inside. In reality, there are 
two topological defects inside with equal magnitude but opposite direction Burgers vectors 
that cancel each other out. Taking a loop in the same direction (clockwise or 
counterclockwise) around each defect produces two Burgers vectors that add to 0. In these 
defects that are extremely close together, we still see both domain walls that connect the 
defects and separate domain walls that individually extend to the edges of the crystal. 
Examples of some well-ordered crystals that display these alternating behaviors for very 
close defects are given in Figures 4.18-4.21. It should be surprising that even at essentially 
no separation, the two dislocations can with some frequency behave as individual defects 
that appear not to interact or to know about other defects in the lattice, and they 
independently nucleate domain walls to accommodate the topological frustration in the 
system. 
 228 
 
 
Figure 4.17 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on a (a) T18 and (b) OA15 geometry crystal. In (a), 
domain walls begin at each dislocation and end at the edges of the crystal. In (b), a single 
domain wall connects the two dislocations. Insets show the location of the dislocation 
points, their directions, and the distance between them. Scale bar is 2 μm.  
 
 229 
 
 
Figure 4.18 Vertex maps of crystals with two topological defects spaced very close 
together. Lorentz contrast TEM images with vertex types overlaid on A2 geometry crystals 
showing domain walls (a) not connecting and (b) connecting the dislocations. In (a), 
domain walls begin at each dislocation and end at the edges of the crystal. In (b), a single 
domain wall connects the two dislocations. Both configurations are very well ordered 
otherwise.  The defects are still topological defects because a Burgers circuit and vector 
can be defined for each one individually, though a loop around both of them does not 
indicate the presence of any topological anomaly because they cancel each other out.  Inset 
shows the location of the dislocation points, their directions, and the distance between 
them. Scale bar is 2 μm. 
 230 
 
 
Figure 4.19 Vertex maps of crystals with two topological defects spaced very close 
together. Lorentz contrast TEM images with vertex types overlaid on OT2 geometry 
crystals showing domain walls (a) connecting and (b) not connecting the dislocations. In 
(a), a single type II vertex connects the two dislocation. In (b), domain walls begin at each 
dislocation and end at the edges of the crystals. Both configurations are very well ordered 
otherwise. Inset shows the location of the dislocation points, their directions, and the 
distance between them. Scale bar is 2 μm. 
 231 
 
 
Figure 4.20 Vertex maps of crystals with two topological defects spaced very close 
together. Lorentz contrast TEM images with vertex types overlaid on T1 geometry crystals 
showing domain walls (a) not connecting and (b) connecting the dislocations. In (a), 
domain walls begin at each dislocation and end at the edges of the crystal. In (b), a single 
domain wall connects the two dislocations. Both configurations are very well ordered 
otherwise. Inset shows the location of the dislocation points, their directions, and the 
distance between them. Scale bar is 2 μm. 
 232 
 
 
Figure 4.21 Vertex maps of crystals with two topological defects spaced very close 
together. Lorentz contrast TEM images with vertex types overlaid on D2.8 geometry 
crystals showing domain walls (a) not connecting and (b) connecting the dislocations. In 
(a), domain walls begin at each dislocation and end at the edges of the crystal. In (b), a 
single domain wall connects the two dislocations. Both configurations are very well 
ordered otherwise. Inset shows the location of the dislocation points, their directions, and 
the distance between them. Scale bar is 2 μm. 
 233 
 
 As we move the defects further away from each other, we still see domain walls 
that connect the two defects, but this tends to become less and less likely as the dislocations 
get further apart. Here we present a representative, but not exhaustive, selection of well-
ordered crystals that show a range of behaviors. Figures 4.22-4.32 display a variety of two-
dislocation geometries, each with an example of domain walls not connecting the 
dislocations and an example of a crystal where a domain wall does connect the dislocations. 
There are a lot of images that all show similar effects over many of the different geometries 
studied. Note that some of the examples here, such as Figures 4.22(a) or 4.26(a), are 
perfectly ordered except for the topologically required domain wall. Some other crystals, 
such as in Figures 4.27(a) or 4.28(b), are nearly perfect except for the required wall. We 
also see that Figure 4.32(b) has an interesting domain wall that snakes in between and 
around the defects to connect them.  
 We can conclude from these images that the inclusion of two dislocations creates 
extensive domain walls in the system, but the exact geometry and placement of the 
dislocations are less important to the overall behavior, especially when the defects are more 
than a few lattice constants apart. Behaviors for the different geometries at the closest 
spacings will be discussed in more detail below. We also note that the domain walls 
themselves do not interact with each other or form “X” or “Y” junctions, which is an 
interesting observation in its own right.  
 
 234 
 
 
Figure 4.22 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on A3 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 235 
 
 
Figure 4.23 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on A4 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 236 
 
 
Figure 4.24 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on A8 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 237 
 
 
Figure 4.25 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on R2.8 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 238 
 
 
Figure 4.26 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on R3 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 239 
 
 
Figure 4.27 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on R3.2 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 240 
 
 
Figure 4.28 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on R5.7 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 241 
 
 
Figure 4.29 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on T2 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 242 
 
 
Figure 4.30 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on T3 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 243 
 
 
Figure 4.31 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on T4 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 244 
 
 
Figure 4.32 Vertex maps of crystals with two topological defects. Lorentz contrast TEM 
images with vertex types overlaid on T5 geometry crystals showing domain walls (a) not 
connecting and (b) connecting the dislocations. In (a), domain walls begin at each 
dislocation and end at the edges of the crystal. In (b), a single domain wall connects the 
two dislocations. Both configurations are very well ordered otherwise. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 245 
 
 From our observations, we compute the probability of domain walls not connecting 
dislocations in our samples for 30 different two-dislocation geometries and plot the 
probability vs. the distance between the two dislocations in Figure 4.33. Figure 4.33(a) 
shows this probability and Figure 4.33(b) has the same probability information, but also 
includes labels for each point specifying the geometries that contributed to that point. Note 
that some points have multiple geometries averaged together. Overall, we find that the 
probability for a domain wall not connecting the two defects is typically almost 1 for 
dislocations that are spaced greater than 10 lattice constants apart. Between 5 and 10 lattice 
constants, the probability hovers around 0.8, which is still quite high. At 5 lattice constants, 
the probability starts to drop sharply and reaches about 0.3 at one lattice constant.  
 We also examine the relative behavior of the different families of T, A, and R 
geometries. Figure 4.34 shows a subset of the data given in Figure 4.33 divided into the 
three main families. For larger lattice spacings, the three geometries all seem to have very 
similar behaviors and probabilities. A close examination of Figure 4.34(b) shows at the 
very shortest inter-defect distances there are signatures of a difference between the T and 
A geometries. These are crystals like those in Figures 4.20 and 4.29 for the T geometries 
and Figures 4.18 and 4.22 for the A geometries. The T geometries tend to have a higher 
rate of a domain wall connecting the two dislocations. It is interesting that when a domain 
wall connects two dislocations in a T geometry, a lot of the time the wall is a very straight 
line of type II vertices that goes directly from one defect to the other via the three-island 
vertex of the dislocation point. This is seen in Figures 4.20, 4.29, and 4.30. For the A 
geometries, the domain walls are much more likely to extend outwards from the three-
island vertex as seen in Figures 4.18(a), 4.22, and 4.23(b). Because the walls from one 
 246 
 
dislocation tend to form and extend away from the other dislocation in the A geometries, it 
is slightly less likely for a wall to connect the two dislocations in this geometry. There may 
be similar effects in the R geometries for the same reason discussed here, but it is harder to 
make a direct comparison because the defects cannot be as close together in the R geometry. 
Examples of the R family behavior are given in Figures 4.25, 4.26, 4.27, and 4.28. Here, it 
is again easy for a very short type II vertex chain to connect the two dislocations when the 
chain extends out of a three-island vertex at the defect point.  
 
 
Figure 4.33 Probability of domain walls not connecting dislocations vs. distance 
between dislocations in crystals with two topological defects. Parts (a) and (b) present 
the same information, but in (b), letters are added indicating the geometries of the data 
points. Some data points include two or three geometries of equal distance apart. Distances 
are expressed in units of the lattice constant a ≈ 500 nm. Error bars are one standard 
deviation and are calculated from counting statistics. Each geometry includes 
approximately 40 crystals.  
 247 
 
 
Figure 4.34 Probability of domain walls not connecting dislocations for different 
geometry families. Similar data to Figure 4.33, but probability vs. distance divided into 
the different families of geometries for direct comparison. Part (b) is a close up of a section 
of the full dataset in (a). Distances are expressed in units of the lattice constant a ≈ 500 nm. 
Error bars are one standard deviation and are calculated from counting statistics. Each 
geometry includes approximately 40 crystals. 
 
 248 
 
 Overall, we find the presence of a domain wall at a dislocation to be extremely 
robust. We studied five samples fabricated from different depositions of FePd3 and 
collected over 1600 images of annealed crystals, and in every image, there is a domain wall 
originating from every dislocation and connecting to either the edge of the crystal or to 
another dislocation. For samples with two dislocations spaced more than 5 lattice constants 
apart, it is possible, but rare, for a single domain wall to connect both dislocations. It is 
much more common for the defects to behave independently and nucleate their own 
domain walls which extend throughout the system and rarely take the shortest available 
path to an edge of the crystal. We note that topological frustration, while it derives from 
crystalline disorder, is fundamentally different from standard disorder-induced frustration.  
Here, the system shows extended frustration as the topology of the defect causes long-
range effects in the lattice. In 2D ASI, the dislocation is a 0D point defect, but it promotes 
itself into a 1D line of frustration. In a 3D material, dislocations are 1D defects, and this 
would result in an extended 2D surface of frustration of the system.   
 
4.2.2 Theoretical framework and discussion  
 
 
In this section we discuss our results in the context of crystalline materials systems 
and develop a theoretical framework for the presence of topological frustration within an 
ordered system. An important question remains about how our observations relate to 
dislocations in 3D crystalline materials and the outstanding issue of whether a unique 
ground state could be realized in spin ice. We consider previous measurements of residual 
entropy in spin ice in the context of sample preparation and dislocation density. The 
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material studied in the seminal report by Ramirez et al. [13] was a pressed powder sample, 
which should result in a high density of dislocations both due to damage during compaction 
and also due to grain boundaries in the resulting structure.  This material, with a presumably 
high density of dislocations, showed a level of entropy equal to the Pauling ice value. On 
the other hand, the 2 × 1 × 1 mm3 single crystal studied by Pomaranski and coworkers 
showed less entropy, a small amount below the Pauling value [57]. Pomaranski et al. also 
studied powder samples, which showed the same qualitative behavior as the samples in 
Ref. [13]. We suspect that the topological frustration due to a high density of dislocations 
in both powder samples could be preventing ground-state ordering, while the single crystal 
in Ref. [57] could be showing a slight emergence of the ground state because it contains 
fewer topological defects. However, it is important to note that larger single crystals 
maintain lattice coherence over greater distances and are thus actually more susceptible to 
topological frustration.  As an example, a cubic crystal of side length N unit cells and 
volume N3 unit cells could need only on the order of Nd = N dislocations to be fully 
frustrated, if the emanating frustrated surfaces are non-interacting, similar to the domain 
walls we observe in our specimens. The dislocation density would be ρd = Nd/N2, so the 
density required to frustrate a system scales as 1/N, implying that larger single crystals are 
actually more likely to be impacted by the presence of dislocations than smaller crystals. 
This is counter to the conventional expectation that larger single crystals show more 
intrinsic ground-state behavior.  On the contrary, smaller single crystals, or single crystals 
with lower densities of dislocations, should be better able to reach a fully ordered ground 
state when topological frustration is present. 
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Implications of topological frustration could extend far beyond the systems 
discussed above, and we now lay out a framework for such considerations as follows. Many 
materials, simple and exotic alike, display ordered states that can generally be characterized 
by an order parameter and associated ordering vector q ≠ 0. In our square ASI, there are 
two q vectors corresponding to the doubly degenerate ground state, q = ±(π/a)[11], which 
tile type I vertices onto the lattice. Using these q vectors, spin directions in the ordered 
states can be assigned through a Bloch form  
𝑺𝑖 = 𝑢𝑖(𝒓)𝑒
𝑖𝒒⋅𝒓   (4.2) 
where q is the ordering vector, r is the coordinate of the spin, i refers to a given sublattice, 
either 1 or 2, and ui(r) acts like a key, specific to the choice of basis and origin. The q 
vectors depend on the choice of basis and origin as well. For q = ±(π/a)[11], we have chosen 
the origin at the middle of a vertex where 4 elements meet. The system has a square Bravais 
lattice with lattice constant a and we choose the basis vectors for the two spins per unit cell 
as [a/2 0] and [0 –a/2]. For the horizontal sublattice, which we will call sublattice 1, u1(r) 
specifies that a spin labeled as an “i” is in the +x direction and a spin labeled as a “-i” is in 
the –x direction. For the vertical sublattice, sublattice 2, u2(r) specifies that a spin labeled 
as a “-i” is in the +y direction and a spin labeled as an “i” is in the –y direction. These 
definitions completely specify the two possible ground states of square ASI with 
q = ±(π/a)[11] as the ordering vectors, with each vector corresponding to one of the two 
degenerate ground states. As an example using the positive q vector, consider two 
neighboring spins on sublattice 2 located at (0, a/2) and (a, a/2). If we compute S for each 
spin, we find that the first one at (0 , a/2) is assigned an i and the other spin is assigned a -i 
indicating that they are pointing in opposite directions, consistent with ground-state 
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ordering in the system. If we choose the other q vector, the values assigned to the two spins 
both switch, so they remain pointing in opposite directions. It is straightforward to show 
that this continues for both sublattices throughout the whole lattice.  
We can now state our theoretical description of topological frustration: If the 
topological Burgers vector of a defect is perpendicular to a given q, or contains a complete 
wave of the order parameter with q∙b = 2πn, where n = 0 or an integer, the system will not 
be frustrated. However, if these conditions do not hold, the system will exhibit topological 
frustration. To demonstrate this idea we now consider ordering in the square ASI described 
above. Figure 4.35 shows a schematic of a coordinate system with a dislocation. There is 
a spin of interest at location r. We can take a path x + y (as defined in Figure 4.35) to the 
spin, or alternately, we can take an equivalent path k + l + m + n to the same spin. However, 
if the second path encloses a topological defect, we will need an extra vector, the Burgers 
vector b, to reach the same spin. Now, when we compute q∙r for the two paths, we should 
get the same result because we are specifying the same spin in the ordered lattice. Since x 
+ y = k + l + m + n, these terms dotted with q give the same answer each time, but there is 
an additional q∙b term for the second path. If q∙b = 2πn, ei(q∙b) = ei(2πn) = 1, and the two paths 
have the same result for the given spin and there is no issue, but if q∙b equals anything else, 
there is a difference and the system is frustrated. If the second path encloses two topological 
defects with Burgers vectors that add to 0, there will also not be difference between the 
two paths because any additional terms will cancel out.  
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Figure 4.35 Multiple paths near a topological defect. Schematic showing how multiple 
paths can be taken to reach the same point in 2D space. Here, ?⃑?  represents the given point. 
?⃑?  + ?⃑?  = ?⃑?  + 𝒍  + ?⃑⃑⃑?  + ?⃑⃑? , but the green path requires an extra vector, the Burgers vector ?⃑? , 
to reach the point specified by r when it encloses a dislocation.   
 
All the geometries we have studied thus far have had a Burgers vector in the [10] 
or [01] direction and these will always be non-orthogonal to ordering vectors q = ±(π/a)[11] 
in square ASI, thus the system will always be topologically frustrated. Now, we 
demonstrate that a sample with a Burgers vector in the [11] direction will not be 
topologically frustrated. We have fabricated crystals with dislocations that have Burgers 
vectors in the [11] direction using the same methods described in this chapter. An in-focus 
image of a b = a[11] topological defect is shown in Figure 4.36 with the Burgers loop 
overlaid confirming the orientation of the Burgers vector. I will refer to crystals of this 
geometry as B11. A full B11 crystal is shown in Figure 4.37.  
 
 
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Figure 4.36 [11] Burgers vector topological defect. In-focus TEM image of a section of 
a crystal containing an edge dislocation with Burgers vector b = a[11]. The Burgers circuit 
and vector are overlaid in the image confirming the topological nature of the defect. Scale 
bar is 1 μm. 
 
We anneal and image the crystals in the same way as the other experiments in this 
chapter. When we do, we find that the B11 crystals indeed do not require a domain wall 
originating from the dislocation point and the crystal can support continuous ground-state 
ordering around a defect of this nature. A vertex map around a b = a[11] dislocation is 
given in Figure 4.38 and full crystals containing one of these defects are given in Figures 
4.39 and 4.40. It is interesting that in Figure 4.40, there are two domain walls that come 
out of the dislocation point. The crystal here is not topologically frustrated, so it cannot 
have exactly one domain wall originating from it, but it can have two. These images give 
experimental verification of the theoretical framework for our system. 
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Figure 4.37 Full crystal with a [11] Burgers vector topological defect. In-focus TEM 
image of a full crystal showing one topological defect with a Burgers vector b = a[11], 
referred to here as a B11 geometry crystal. Inset shows the schematic representation of this 
crystal. Scale bar is 3 µm.    
 
   
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Figure 4.38 Vertex map of a [11] Burgers vector topological defect. Lorentz TEM image 
with vertex types overlaid on a section of a crystal containing an edge dislocation with 
Burgers vector b = [11]. This is a topological defect, but because of the Burgers vector 
orientation, it can still support perfect ground-state ordering around it. Scale bar is 1 μm.  
 256 
 
 
Figure 4.39 Vertex map of a [11] Burgers vector topological defect full crystal. Lorentz 
TEM image with vertex types overlaid of a full crystal containing an edge dislocation with 
Burgers vector b = a[11]. The defect does not nucleate a domain wall, though other 
traditional square ASI domains and domain walls are present. Inset shows the schematic 
representation of this crystal. Scale bar is 3 µm.    
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Figure 4.40 Vertex map of a [11] Burgers vector topological defect full crystal. Lorentz 
TEM image with vertex types overlaid of a full crystal containing an edge dislocation with 
Burgers vector b = a[11]. The defect cannot nucleate a single domain wall, but can support 
two domain walls intersecting it. Inset shows the schematic representation of this crystal. 
Scale bar is 3 µm.    
 
Returning now to the broad and general impact of topological frustration on 3D 
crystalline systems with nonzero q vectors, we consider some concrete examples. In the 
pyrochlore spin ices, there are 6 q vectors for 6 realizations of the predicted long-range 
ordered state, q = (2π/a) <100> [51]. A schematic of the ground state with q = (2π/a)[001] 
is shown in Figure 4.41. If we assume Burgers vectors of dislocations to point along one 
of the primitive FCC lattice vectors, (a/2) <110> , then, for each possible Burgers direction, 
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4 out of the 6 ground-state q vectors will not be perpendicular to the Burgers vector and 
will thus be frustrated. This is shown schematically in Figure 4.42 with a Burgers vector in 
the (a/2)[101̅] direction as an example. This Burgers vector will not cause any structural 
mismatch away from the dislocation core [214], but it will disrupt the ground-state ordering 
in the lattice. The dislocation creates a domain wall in the q = (2π/a)[001] ordering of the 
system. The location of this domain wall is arbitrary, but it must be present. It is easy to 
calculate that q∙b ≠ 2πn, thus the system is topologically frustrated.  
 
 
Figure 4.41 Pyrochlore spin ice ground state with order parameter q = (2п/a)[001]. 
Reduced unit cell showing the locations of the rare earth atoms and their magnetic moment 
directions in the q = (2п/a)[001] ground state. This state has alternating planes of ordered 
tetrahedra, here colored red and blue, corresponding to the planes of ordering in Figure 
4.42.  
 259 
 
 
Figure 4.42 Topological frustration in pyrochlore spin ice. Schematic showing how a 
dislocation in pyrochlore spin ice could disrupt ground-state ordering. Each cube is 1/8 the 
pyrochlore unit cell shown in Figure 4.41. This example has a dislocation with a Burgers 
vector (shown as a white arrow) in the (a/2)[101̅] direction which is a primitive basis vector 
for the FCC parent structure of the pyrochlore lattice so that this does not cause any 
structural mismatch away from the dislocation. This dislocation does create a domain wall 
in the q = (2п/a)[001] ground-state order parameter, highlighted in yellow. The location of 
this domain wall is arbitrary, but it must be present. Note that q∙b ≠ 2πn and thus the system 
is topologically frustrated.  
 
If a pyrochlore crystal contains dislocations with all three FCC primitives, then all 
6 ground states will exhibit frustration, and the specimen could be prevented from finding 
its unique ground state, as is observed experimentally.  This frustration should be present 
in many other systems with q ≠ 0 order parameters. Prime examples may include spin-
density-wave materials [215], such as antiferromagnets and ferrimagnets, 
antiferroelectrics, such as SrTiO3 [216], charge-density-wave materials [217], and pair-
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density-wave superconductors [218]. In cases where the dislocation densities are low, or 
where the frustrated domain walls have stronger surface tension between dislocations, the 
frustration may not prevent the formation of the order parameter, but rather it will only 
degrade it or delay it somewhat.  In such cases, we expect topological frustration may 
thermally broaden the phase transition, degrading the formation of the q ≠ 0 order 
parameter.  Topological defects may even play a role in the frustration of ice-XI, the 
thermodynamic ground state of water not known to form under typical experimental 
conditions [10].  In general, the effects of topological frustration could be observed in a 
wide variety of materials systems, and artificial spin ice could serve as a valuable platform 
for future studies to develop these general considerations. 
We aim to argue here that dislocations should receive more attention in theoretical 
work and interpretations of experimental systems with q ≠ 0 than they currently do. A 
recent and extremely thorough study on Dy2Ti2O7 spin ice by Henelius and coworkers 
indeed does not discuss the possibility of dislocations affecting the ordered states [219]. 
The paper does suggest, however, that “random disorder” is the key underlying cause of 
the varied and unique behavior observed in this compound. The authors speculate that 
either stuffed spins – extra magnetic atoms in the lattice – or oxygen vacancies should be 
responsible for the upturn in heat capacity and corresponding reduction of entropy 
observed by Pomaranski et al. [57,219]. There have been a number of studies on the effects 
of stuffed spins in pyrochlore spin ices [220–222] as well as oxygen vacancies [223]. 
Henelius et al. rule out most plausible reasons for the reduction of entropy. They try 
including higher order terms in the dipolar spin ice model and exploring more sophisticated 
models using advanced computational techniques. Through all this work, their main 
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conclusion is that the system is extremely sensitive to disorder. Based on our results with 
topologically frustrated ASI, we argue that dislocations in the crystals are just as likely to 
be a source of random disorder and cause the range of observed behaviors, based on the 
specific density of dislocations in the materials studied. Indeed, dislocations may be better 
able to account for the observed frustration, since 0D point defects only produce 0D regions 
of frustration, whereas 1D dislocations produce 2D sheets (domain walls) of frustrated 
regions (see Figure 4.42). Overall, there are still many unresolved questions about a 
ground-state transition in spin ice and more work is needed on higher purity samples and 
at lower temperatures. ASI presents a highly tunable system that will aid in these studies 
by providing unique information at the single spin level.  
 
4.2.3 Additional observations  
 
In this section we survey a wide range of experimental observations about our 
topological defect samples. We reference some figures already presented in this chapter as 
well as introduce some more rare and specific behaviors in a variety of different 
geometries. We begin by examining edge effects in our crystals. Figure 4.14 shows a 
common feature found in many different topological defect crystals and geometries. In the 
bottom right corner there is an isolated pair of type III vertices. If a single spin flips within 
a domain of ground-state ordered type I vertices, it will create a pair of type III vertices. 
These isolated defects do not greatly affect the overall behavior of the lattice, but do 
indicate some interesting ideas. They appear somewhat rarely, though some examples are 
also seen in Figures 4.19, 4.21, 4.46, and 4.48. At the top of Figure 4.14 is a pair of type II 
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vertices on the edge of the crystal. This type of defect is again caused by flipping a single 
spin among ground-state order, but these defects located at the edges are much more 
common. This is because the edge vertices have only three elements and the energy 
difference between the type I and type II three-element vertices is much smaller than the 
corresponding energy difference for 4-element vertices. Edge elements also tended to come 
out slightly thinner than bulk elements because of proximity effects during lithography. 
This further reduces the intrinsic energy barrier for these elements to spontaneously flip 
into energetically unfavorable arrangements. Type II vertex pair defects on the edges of 
crystals are very common and are seen in many of the figures in this chapter. Some 
examples include Figures 4.20, 4.24, and 4.27. Figure 4.39 has a chain of 4 type II defects 
in a row on the left edge of the crystal. There are many examples of small domains that are 
contained very close to an edge, likely because it is easy to nucleate them close to the edge, 
but more energetically unfavorable for the domains to grow further into the lattice. Some 
of these small domains can be seen, for example, in Figures 4.22, 4.23, 4.24, and 4.30.  
We have discussed topological defects at length in this chapter, but it is important 
to think about other types of defects as well. We note that a vacancy, a single missing 
magnet without lattice distortion, is not a topological defect and does not induce frustration 
or nucleate a domain wall; a square ASI lattice can still display perfect order with a single 
spin removed. This is seen in Figure 4.43 for a simple vacancy. Here, the missing spin is 
an unintentional defect from fabrication, but it illustrates an important point. We also 
intentionally fabricated crystals with a vacancy, but allowed the system to relax in the same 
way we created the topological defect lattices. This crystal is shown in Figure 4.44 and can 
be thought of an A1 geometry, though it was not included in Figures 4.33 or 4.34 because  
 263 
 
 
Figure 4.43 Vertex map of a non-topological defect. Lorentz TEM image with vertex 
types overlaid on a square geometry sample with a vacancy. A simple vacancy, a single 
missing spin in an otherwise perfect square lattice, does not nucleate a vertex chain or result 
in topological frustration. Long-range ground-state ordering is not affected by the presence 
of a vacancy since it is not a topological defect. Scale bar is 1 μm. 
 
it is not a true topological defect. Still, this crystal displays two main behaviors shown in 
Figure 4.44. The lattice can either maintain continuous ground-state order around the non-
topological defect as in Figure 4.44(a), or the defect can have two domain walls coming 
out of it, as in Figure 4.44(b). This situation is very similar to the B11 geometry crystals 
shown in Figures 4.39 and 4.40. These examples are interesting because it shows that the 
defects can nucleate domain walls even when they are not required and they do seem to 
nucleate them more frequently than would occur in a randomly selected defect-free 
location in one of these crystals. We also find it very interesting that when we compute, 
from our data, the probability of having two domain walls originating from the non-
topological defects vs zero domain walls, for both the A1 and B11 geometries we get a 
probability of about 0.5. This is quite similar to the probabilities for the closest spaced  
 264 
 
 
Figure 4.44 Vertex map of non-topological defect crystals. Lorentz contrast TEM images 
with vertex types overlaid on A1 geometry crystals with a vacancy. In (a), there is perfect 
ground-state ordering surrounding the defect. In (b), two domain walls originate from the 
defect point and extend to the edge of the crystal. Both configurations are very well ordered 
otherwise. Inset shows the location of the effective dislocation points, their directions, and 
the distance between them. Scale bar is 2 μm. 
 
 265 
 
two-dislocation crystals to have two independent domain walls originating from the 
individual defects. 
We know that disorder plays an important role in our samples from the kagome 
crystals in Chapter 3 and the canonical square ASI samples in Chapter 2. Indeed, we will 
discuss this more in the next section, but I will highlight a few observations about disorder 
here. Figure 4.45 shows the same crystal that has been annealed twice and imaged after 
each time. We see that there is a completely different domain pattern in the two crystals. 
The behavior is qualitatively similar, but the same quenched disorder in the system gives 
different low temperature configurations after annealing. We have imaged almost every 
crystal we made multiple times (typically 2-3 times) and we have never seen a completely 
reproduced domain pattern in any sample. Figure 4.46 shows an in-focus TEM image along 
with a corresponding Lorentz contrast image of the same crystal. In this figure, we see how 
point defects in the individual elements, here as elements that came out too wide, can affect 
the domain wall patterns. We see that some of the defects, highlighted by the red circles, 
can act as pinning sites for a domain wall. We note, however, that these point defects are 
not required to pin a wall — as the topological lattice defects are — as the two defects 
highlighted in green do not have a wall associated with them. This shows that these 
fabrication defects can contribute to domain wall pinning, but they don’t always have this 
effect.  
 
 266 
 
 
Figure 4.45 Repeated annealing of the same crystal. Lorentz TEM images with vertex 
types overlaid on the same 1E geometry crystal after repeated annealing. The same crystal 
annealed twice under the same conditions results in different domain configurations. The 
quenched disorder in the sample gives qualitatively similar, but not exactly reproducible 
ordering of vertices. Scale bar is 2 μm. 
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Figure 4.46 Domain wall pinning sites. a. In-focus TEM image of a T4 geometry crystal. 
b. Lorentz Contrast TEM image with vertex types overlaid of the same crystal as in (a). 
Multiple point defects are highlighted with red and green circles in (a). Three of these 
defects, highlighted in red, appear to pin a domain wall in (b), while the other two defects, 
highlighted in green, do not. Inset shows the location of the dislocation points, their 
directions, and the distance between them. Scale bar is 2 μm. 
 268 
 
We see a range of behaviors in the domain walls of our crystals. We have already 
discussed this idea briefly, but it should be restated that the walls rarely take the shortest 
path to an edge of the crystal. Figure 4.47 shows an example of some long, meandering 
domain walls. While this behavior is slightly more common, we do see some very straight 
and short domain walls as well, shown in Figure 4.48. Both of these crystals are R 
geometries, but have completely different domain wall lengths among otherwise nearly 
perfectly ordered crystals. These behaviors are common throughout all of our experimental 
data.  
Topological defects require one domain wall originate from them, but similar to the 
observation that non-topological defects can have zero or two domain walls associated with 
them, the topological defects can also support three domain walls. This behavior is rare in 
our experimental samples, but Figures 4.49 and 4.50 show some unique examples. In 
Figure 4.49, one defect has a domain wall that connects to the other defect, but then that 
second defect has two more walls originating from it as well. In Figure 4.50, one of the 
defects has a single wall that runs to an edge of the crystal and the other defect has three 
walls that originate from it and all individually run to the edges of the crystal. Both of these 
configurations are extremely rare in experiment. For reasons not yet fully understood, this 
behavior was actually much more prevalent in our simulations, as discussed in the next 
section. In the experimental samples, we only saw three domain walls from one defect a 
handful of times and generally this occurred in lower quality crystals with more disorder. 
This suggests that the three domain wall configuration is energetically unfavorable and 
requires a sample with higher disorder, far from the ground state, to support the 
phenomenon.  
 269 
 
 
Figure 4.47 Vertex map of a crystal with two topological defects showing meandering 
domain walls. Lorentz contrast TEM image with vertex types overlaid on a R5 geometry 
crystal. A meandering vertex chain originates from each of the dislocation points and both 
take paths much longer than the shortest possible chain to the edges. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 270 
 
 
Figure 4.48 Vertex map of a crystal with two topological defects showing relatively 
straight domain walls. Lorentz contrast TEM image with vertex types overlaid on a R7 
geometry crystal. A fairly straight vertex chain originates from each of the dislocation 
points and both take nearly the shortest paths to the edges of the crystal. Inset shows the 
location of the dislocation points, their directions, and the distance between them. Scale 
bar is 2 μm. 
 
 271 
 
 
Figure 4.49 Vertex map of a crystal with two topological defects showing unique 
domain wall patterns. Lorentz contrast TEM image with vertex types overlaid on a R5.7 
geometry crystal. A domain wall originates from one defect and connects to the other defect 
which also has two more domain walls originating from it and extending to the edges of 
the crystal. Inset shows the location of the dislocation points, their directions, and the 
distance between them. Scale bar is 2 μm. 
 272 
 
 
Figure 4.50 Vertex map of a crystal with two topological defects showing unique 
domain wall patterns. Lorentz contrast TEM image with vertex types overlaid on an A6 
geometry crystal. A domain wall originates from one defect and extends to the edge of the 
crystal. The other defect has three domain walls that originate from it and extend to the 
edges of the crystal. Inset shows the location of the dislocation points, their directions, and 
the distance between them. Scale bar is 2 μm. 
 
 Our final observations in this section relate to the net magnetization directions of 
the domain walls. Both type II and type III vertices have a net magnetization associated 
with them. For these final images, we introduce a new notation that indicates the direction 
of the magnetization, shown in Figure 4.51. Type I vertices are the same blue circles and 
 273 
 
type II and III vertices have the same colors, but the higher energy vertices are represented 
as arrows in the directions of the net magnetization instead of just circles. Images of two 
crystals with this notation are given in Figures 4.52 and 4.53. Each figure shows a large 
crystal with some regions enlarged that contain interesting domain walls. We see that type 
II vertices always have a magnetization that points in the same general direction as their 
neighbors. Whole chains of type II vertices all tend to follow the same direction. If the 
domain wall extends in a <11> direction, as in part of Figure 4.52(d), the type II vertices 
are all in exactly the same direction. If the wall moves along a straighter path, as in Figure 
4.53(b), the neighboring vertices just have a component of the magnetization in the same 
direction. Figure 4.52(c) also has a nice chain of type II vertices that forms a vortex 
arrangement. At the type III vertices, the direction of the type II chain switches. This can 
be seen in Figure 4.52(b) and (d) and in Figure 4.53(c). The type III vertices are required 
to switch the direction of the type II chains. Overall, in this section we have seen a plethora 
of data and behaviors of topologically frustrated ASI crystals. In the next section, we utilize 
kinetic Monte Carlo simulations to model our experiments and gain a better understanding 
of our system.  
 
 274 
 
 
Figure 4.51 Magnetization direction vertex notation. Notation definitions for different 
vertex types used in Figures 4.52 and 4.53. Type II and type III vertices have a net magnetic 
moment for the vertex. This notation uses color to indicate the type of vertex (same scheme 
as throughout the dissertation) and an arrowhead to indicate the direction of the net 
magnetization of the element.  
 275 
 
 
Figure 4.52 Magnetization direction vertex map. Lorentz TEM image of an A6 geometry 
crystal with magnetization direction vertex types overlaid. The full crystal is shown in (a) 
with regions highlighted that correspond to the magnified sections in (b)-(d). The direction 
of the net magnetic moments of type II vertices in a domain wall align with each other and 
can switch direction at a type III vertex. Inset shows the location of the dislocation points, 
their directions, and the distance between them. Scale bars are (a) 2 μm, (b) 1 μm, (c) 
500 nm, and (d) 1 μm.  
 276 
 
 
Figure 4.53 Magnetization direction vertex map. Lorentz TEM image of a R2.8 geometry 
crystal with magnetization direction vertex types overlaid. The full crystal is shown in (a) 
with regions highlighted that correspond to the magnified sections in (b) and (c). This is 
the same crystal and configuration found in Figure 4.25(a). The direction of the net 
magnetic moments of type II vertices in a domain wall align with each other and can switch 
direction at a type III vertex. Inset shows the location of the dislocation points, their 
directions, and the distance between them. Scale bars are (a) 2 μm, (b) 1 μm, and (c) 
500 nm. 
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4.3 Kinetic Monte Carlo simulations 
 
In this section we present results on kinetic Monte Carlo simulations of our 
topological defect samples. We have already discussed the numerical methods at length in 
Chapters 2 and 3. For these simulations, we use dipolar coupling between neighboring 
spins, given by the formula in Equation 1.11, and similar to the canonical square geometry 
simulations in Section 2.5.2. We use the same procedure to define our simulation lattices 
as we use for the lithography patterns. The simulations have the same number of spins and 
same boundary conditions as the experimental samples. We include the same type of 
disorder as previously discussed in the form of a spread in widths of the individual 
elements. We found that a small amount of disorder, on the order of 1-10 Å, was able to 
reproduce our experimental results extremely well. We find remarkably similar behavior 
between our simulations and TEM data, where the dislocation points always terminate a 
domain wall, but they are surrounded by large domains of ground-state order. Monte Carlo 
results also show domain walls separating spontaneous ordinary domains and occasionally 
connecting two dislocations, consistent with our experimental observations. We show the 
steady state behavior of some of the simulations in Figure 4.54. Here, as in the square 
Monte Carlo work discussed in Section 2.5.2, a given amount of disorder can generate a 
range of domain patterns. The frames in Figure 4.54 tended to have a relatively high 
number of domain walls not required by topology, but these are mainly just to illustrate the 
variety of patterns we see. We note that in most of the frames, the domain walls do not take 
the shortest path to the edge of the crystal. Two-dislocation geometry simulations show 
domain walls both connecting (Figure 4.54(c)) and not connecting (Figures 4.54(b) and 
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(d)) the dislocations. Figure 4.54(d) is relatively well ordered aside from the required 
domain walls.  
 
Figure 4.54 Monte Carlo simulations. Final frames of kinetic Monte Carlo simulations of 
(a) 1M and (b)-(d) A22 geometry lattices. The domain wall patterns observed are 
remarkably similar to experimental data in Section 4.2. The spread in widths used in these 
runs are (a) 5 Å, (b) 6 Å, (c) 6 Å, and (d) 6 Å. Insets show the location of the dislocation 
points, their directions, and the distance between them. 
 
 279 
 
Figure 4.55 presents the evolution of the simulation shown in Figure 4.54(a). The 
lattice starts in a randomized state and achieves most of its ordering relatively quickly, first 
eliminating the highest energy type IV vertices and progressing to mainly type I ground-
state order. Once the sample has reached a ground-state-domain-dominated phase, the 
domain walls slowly fluctuate and move around, often shrinking some of the smaller 
domains. In the domain wall motion, we note fluctuations of type III vertices very similar 
to those seen in the evolution of our canonical square geometry Monte Carlo shown in 
Figure 2.44, as well as previous simulations from other groups [119]. In Figure 4.55, we 
can see two domain walls interacting in frames (h)-(j). There are two walls in the bottom 
right corner of the lattice that meet and then separate into new domains. We can imagine 
this kind of domain wall motion is common when the crystals are thermally active and 
finding ordered states. Domains tend to shrink in size, favoring more ordered states, and 
smaller domains converge to form larger regions of ground-state order. These Monte Carlo 
results exactly reproduce the behavior we see in our experimental samples. Even with no 
disorder, we still see the topologically required domain walls originating from the 
dislocations. Movies of the evolution of some of these simulations as well as a movie of 
the lattice relaxation process can be found at dx.doi.org/10.13016/M25H71.  
 
 280 
 
 
Figure 4.55 Evolution of a topological defect kinetic Monte Carlo simulation. 
Snapshots of the state of the same run as in Figure 4.54(a) after (a) 50, (b) 250, (c) 500, 
(d) 750, (e) 1250, (f) 2500, (g) 5000, (h) 15000, (i) 17500, (j) 20000, (k) 27500, and (l) 
35000 Monte Carlo steps.  
 281 
 
4.4 Conclusions and future work 
 
In this chapter, we introduce controlled topological defects into canonical square 
geometry ASI samples and observe the resulting behavior upon annealing. We find the 
presence of a topological defect results in extended frustration within the system with a 
chain of higher energy vertices required to originate from each defect and extend to an edge 
of the finite lattice or to another defect among large domains of ground-state order. We 
develop a theoretical model for topological frustration in a variety of systems with nonzero 
order parameters and discuss our results in the context of unanswered questions about the 
pyrochlore spin ices. We provide the first experimental demonstration of dislocations 
nucleating domain walls with information at the single spin level. We also simulate our 
system and find excellent agreement between our experimental and numerical results. This 
work opens the door for further investigations into defects in both ASI systems and 3D 
crystalline materials.  
In topologically frustrated ASI, there is ample more work to be done with this system. 
We can investigate more geometries with two dislocations to try to better understand how 
the dislocations behave individually and how they interact. We can fabricate larger crystals 
and observe how far the frustration extends into the system. We can compute the energy of 
different domain wall configurations to further investigate the range of behaviors we 
observe. We can fabricate more samples and take more data to get better statistics on the 
emergence of multiple dislocations not interacting with each other. We can fabricate 
crystals with one- or two-fold edges and see how that affects topological and non-
topological domain wall and domain formation. There has been recent work on imaging 
thermally active magnetic configurations in real time and this would be very interesting to 
 282 
 
compare our experimental defect crystals to our Monte Carlo movies and the evolution of 
our simulations [81,82,85]. We could also include more defects in the crystals and observe 
the effects of a higher density of dislocations in the system. The complexity of the system 
will grow quickly with the number of dislocations, so it is very important to have a 
comprehensive understanding of one- and two-defect samples before fabricating lattices 
with more.  
 The ideas presented here should be expanded on to examine the effects of other 
types of defects as well as investigate the behavior of defects in other ordered systems with 
unprecedented information at the atomic level. The work in this chapter is new and thus 
opens many routes for future studies. A potentially fruitful path would be to look for 
signatures of topological frustration in 3D crystalline materials. The pyrochlore spin ices 
are an obvious starting point with the aim of resolving the longstanding and ongoing 
debates about the ability of the system to find a ground state or whether this is even possible 
at all. Beyond the pyrochlores, topological frustration could have far reaching implications 
in a wide range of systems with nonzero order parameters. Dislocations have typically not 
been given much theoretical treatment, so it would be interesting to examine their effects 
in more detail. Moreover, our results should demand that more attention be paid to these 
defects as we have show they cannot and should not be ignored, for they can completely 
alter the properties of a system. Overall, we have expanded the field of topological 
frustration and presented data with unprecedented detail. ASI has proven again to be a 
highly tunable and useful system and will undoubtedly aid in future investigations into the 
properties of defects in ordered systems.  
 
 283 
 
Glossary 
List of acronyms  
1FE – One-Fold Edge 
2FE – Two-Fold Edge 
3FE – Three-Fold Edge 
AF – Antiferromagnetic  
ASI – Artificial Spin Ice 
CNAM – Center for Nanophysics and Advanced Materials 
Dz – Demagnetizing factor 
EBL – Electron Beam Lithography  
EDS – Energy Dispersive x-ray Spectroscopy  
EELS – Electron Energy Loss Spectroscopy  
FCC – Face Centered Cubic  
FIB – Focused Ion Beam 
FM – Ferromagnetic  
Hc – Coercive field 
IL – Intermediate Lens 
IPA – Isopropanol  
LLG – Landau-Lifshitz-Gilbert  
LTEM – Lorentz Transmission Electron Microscopy  
MDG – Melko, den Hertog, and Gingras 
MFM – Magnetic Force Microscopy  
MIBK – Methyl Isobutyl Ketone  
 284 
 
MOKE – Magneto-Optical Kerr Effect 
Mr – Remnant magnetization  
OOMMF – Object Oriented MicroMagnetic Framework 
PEEM – Photoemission Electron Microscopy  
PMMA – Poly(Methyl Methacrylate)  
RKKY – Ruderman-Kittel-Kasuya-Yosida 
SEM – Scanning Electron Microscopy  
SQUID – Superconducting QUantum Interference Device 
Tb – Blocking Temperature 
TC – Curie Temperature 
TEM – Transmission Electron Microscopy  
TN – Néel Temperature  
UHV – Ultra-High Vacuum  
UMD – University of Maryland  
WDS – Wavelength Dispersive x-ray Spectroscopy  
XMCD – X-ray Magnetic Circular Dichroism  
 
 
 
 
 285 
 
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