Master Thesis

POLITECNICO DI TORINO
FACOLT𝐴̀ DI ELETTRONICA
Laurea Magistrale (𝚰𝚰 Level Degree) in Nanotecnologie Per Le ICTs
(Nanotechnologies For ICTs)
Modeling of the Ferromagnetic Resonance
Behavior of Magnetic Antidot Arrays
Relatori – Tutors
Prof. Dr. Mario Chiampi
Dr. Alessandra Manzin (INRiM)
Candidato- Candidate
Ali Asghar Fathi
March 2016

To my parents, Yadollah and Soraya,
to all brave women and men,
who have been martyred in the imposed war of Iran
specially my dear uncle Asghar Fathi
to my brothers, Ahmad,
Mohammad and Behrouz

Acknowledgments
It is a great pleasure to have the possibility to thank the people, to whom I will be forever
grateful, who have introduced me to do my thesis research in micromagnetics and its
applications.
First of all, I would like to thank very much my supervisor Alessandra Manzin for his precious
teachings, spreading from advanced mathematics to electromagnetism and micromagnetics, but,
above all, for his strong support in every moment, fine or less fine, of my research activity in
these six months. It has been always challenging, but also funny, working together.
I am also very grateful to my supervisor Mario Chiampi, who always supported and guided me
like a kind father. Moreover, I would like to thank again both my supervisors for giving me the
possibility to work with them and learn from them a lot of precious things which are like a torch
for my scientific future.
Special thanks go to Enrico Simonetto, PhD student at Istituto Nazionale di Ricerca Metrologica
(INRiM), who helped me in simulations and helped me for doing my thesis much better.
Last but not least, I would like to thank other PhD students, Arash Hadadian and Mohammad
Modarres from Istituto Nazionale di Ricerca Metrologica (INRiM), for supporting and helping
me during these six months.

Contents
Acknowledgments…………………………………………………………………………...iii
1 Introduction............................................................................................................................. 1
2 Physical Model........................................................................................................................ 3
2.1 Micromagnetics................................................................................................................ 3
2.1.1 Static Micromagnetics .............................................................................................. 3
2.1.2 Dynamic Micromagnetics......................................................................................... 7
3 Description of the used micromagnetic solver...................................................................... 14
3.1 Time Discretization of the Landau-Lifshitz-Gilbert Equation....................................... 14
3.2 Spatial Discretization of Exchange Field....................................................................... 17
3.2.1 Finite Difference Method on Unstructured Meshes............................................... 18
3.3 Spatial Discretization of Magneto-Static Field.............................................................. 21
4 Introduction of the Simulations ............................................................................................ 24
4.1 Description of Sample Materials and Geometries.......................................................... 24
4.2 Description of External Sources and Simulation Procedure .......................................... 25
5 Results of Simulations .......................................................................................................... 30
5.1 Square Antidot Lattice ................................................................................................... 31
5.1.1 Influence of Bias Field Orientation......................................................................... 31
5.1.2 Influence of Thickness............................................................................................ 33
5.1.3 Influence of Diameter ............................................................................................. 35
5.2 Rhombic Antidot Lattice................................................................................................ 40
5.2.1 Influence of Bias Field Orientation......................................................................... 40
5.2.2 Influence of Thickness............................................................................................ 44
6 Conclusion ............................................................................................................................ 46
7 Bibliography ......................................................................................................................... 47

1
1 Introduction
With the advances in nanofabrication and characterization techniques, magnetic nanostructures
have become one of the leading candidates for a wide range of applications including high
density storage and data throughput, nonvolatile memory, spin logic and frequency-based
detection of magnetic particles. In this regard, magnetic antidot nanostructures, which consist of
a grid of nonmagnetic “holes” in a continuous magnetic film, have been proposed as a candidate
for ultrahigh density storage media. While the static properties of magnetic antidot arrays have
been extensively studied in recent years, it is also important to have control of the dynamic
magnetic response, since reading and writing speeds in magnetic storage devices are getting
closer to the time scale of spin dynamics. Ferromagnetic resonance (FMR) has been used earlier
to characterize the dynamic properties of magnetic antidot structures (1). FMR is a powerful tool
for investigating spin-wave spectra. In particular, the dynamic response of magnetic antidot
arrays has been investigated as a function of lattice symmetry and orientation of the applied field
and a strong dependence was observed. In this thesis, I present a computational investigation of
the spin dynamics in large-area 𝑁𝑖80 𝐹𝑒20 antidot arrays with square and rhombic lattice with
fixed pitch but varying film thickness and hole diameters. In particular, I have investigated how
the spin wave spectrum is modified by rotating the in-plane direction of the applied bias field. I
observed that the frequency of all FMR modes can be systematically tuned by varying the antidot
diameter (2).

2
In chapter 2 the physical model, i.e. the micromagnetic model based on the Landau-Lifshitz-
Gilbert (LLG) equation, will be introduced to describe magnetization phenomena in
ferromagnetic systems. First, an approach in terms of the free energy associated with the
magnetic body will be presented to derive the static equilibrium conditions for magnetization
vector field. Then, the dynamic effects due to the gyromagnetic precession will be introduced.
In chapter 3, I will present a brief description of the used micromagnetic solver, which was
previously developed at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Torino. First,
the time discretization of the Landau-Lifshitz-Gilbert equation will be presented. Then, the
spatial discretization of the exchange field with finite difference method based on unstructured
meshes and also spatial discretization of magneto-static field will be described.
Chapter 4 is devoted to the introduction of the problem, the description of materials and
geometries used in the study and then frequency and spatial character of the spin wave modes
will be addressed. At the end of this chapter, external sources which are applied to the studied
antidot array films will be present.
In chapter 5 the results of simulations, aimed at the computational investigation of the dynamic
properties of Permalloy antidot arrays in square and rhombic lattice with fixed pitch but variable
film thickness and hole size will be analyzed.
At the end, some conclusions about the obtained results will be drawn.

3
2 Physical Model
2.1 Micromagnetics
Micromagnetics is a theory of physics which deals with the behavior of ferromagnetic materials
at fine length scales, enabling the description of magnetization processes at the Nano-scale level.
It was introduced in 1963 when William Fuller Brown Jr. published his paper on antiparallel
domain wall structures (3). In recent years, a lot of attention has been paid to the implementation
of micromagnetic numerical codes for the study of magnetic domain configurations in magnetic
nanostructured systems. Micromagnetic theory includes static micromagnetics, based on the
determination of equilibrium magnetization states through the minimization of magnetic free
energy, and dynamic micromagnetics, which enables the determination of magnetization
processes through the solution of a time-dependent dynamical equation, namely the Landau-
Lifshitz-Gilbert equation (4). The simulations analyzed in this thesis have been performed
following the second approach.
2.1.1 Static Micromagnetics
The purpose of static micromagnetics is for determination of the spatial distribution of the
magnetization vector M at equilibrium. It is assumed that the amplitude of the magnetization
vector M is equal to the material saturation magnetization 𝐌 𝐬 everywhere when the temperature
is much lower than the Curie temperature. Then, the magnetization direction vector m=𝐌 𝐌 𝐬⁄ ,
(normalized magnetization) represents the magnetization spatial orientation. The magnetic
domain configuration at equilibrium can be determined by minimizing the magnetic free energy
(4),

4
E= 𝐄 𝐞𝐱𝐜𝐡 + 𝐄 𝐚𝐧𝐢𝐬 + 𝐄 𝐙 + 𝐄 𝐝𝐞𝐦𝐚𝐠 , subject to the constraint |M|=𝐌 𝐬 or |m|=1.
In the previous equation,
𝐄 𝐞𝐱𝐜𝐡: corresponds to the exchange energy, 𝐄 𝐚𝐧𝐢𝐬: to the anisotropy energy, 𝐄 𝐙: to the zeeman
energy (associated to the applied external field), and 𝐄 𝐝𝐞𝐦𝐚𝐠: to the energy of the demagnetizing
field.
In the following, each energy term is briefly described (4) (5):
2.1.1.1 Exchange Energy
The exchange interaction and associated energy in ferromagnetic materials depend on quantum
mechanics phenomena and are related to the atomic spin-spin interactions. The exchange energy
is written as:
𝐄 𝐞𝐱𝐜𝐡 = 𝑨 ∫ ((𝛁𝒎 𝒙) 𝟐
𝐕
+ (𝜵𝒎 𝒚) 𝟐
+ (𝜵𝒎 𝒛) 𝟐
) 𝒅𝑽 Eq. 2-1
where A is the exchange constant connected with the exchange length (in the order of 5-10 nm),
𝒎 𝒙, 𝒎 𝒚 and 𝒎 𝒛 are the components of m; and the integral is over the volume of the sample.
This energy is minimized when the magnetization in the sample is perfectly uniform.
2.1.1.2 Anisotropy Energy
Magneto-crystalline anisotropy is created because of the interaction between spin magnetic
moment of each atom and structure of the lattice. It can be generally written as:
𝐄 𝐚𝐧𝐢𝐬= ∫ 𝐅𝐚𝐧𝐢𝐬(𝐦)𝒅𝑽𝐕
Eq. 2-2

5
where 𝐅𝐚𝐧𝐢𝐬, is the anisotropy energy density, which is a function of the orientation of the
magnetization with respect to the easy and hard axes of the crystalline lattice . Energy directions
of 𝐅𝐚𝐧𝐢𝐬 correspond to axes when this energy is minimized. Under the hypothesis of uniaxial
anisotropy (with easy axis defined by unit vector 𝒖 𝒂𝒏𝒊𝒔):
𝐅𝐚𝐧𝐢𝐬(m,𝒖 𝒂𝒏𝒊𝒔) =𝑲 𝒂𝒏𝒊𝒔[𝟏 − (𝒎. 𝒖 𝒂𝒏𝒊𝒔) 𝟐] Eq. 2-3
where 𝑲 𝒂𝒏𝒊𝒔 is the magnetocrystalline anisotropy constant.
Fig. 2-1 First magnetization curves for a single crystal of iron along different direction (6)
2.1.1.3 25BZeeman Energy
The Zeeman energy is the interaction energy between the magnetization and any external applied
field. It is written as:

6
𝐄 𝐙 = −𝛍 𝟎 ∫ 𝐌 . 𝐇 𝐚 𝐝𝐕𝐕
Eq. 2-4
where 𝐇 𝐚 is the external applied field and 𝛍 𝟎 is the vacuum permeability.
2.1.1.4 Energy of the demagnetizing field
The demagnetizing field, also called the stray field is the magnetic field created by the magnetic
sample upon itself. The demagnetizing field is responsible for the domain structure formation in
the magnetic film. The energy contribution is written as:
𝐄 𝐝𝐞𝐦𝐚𝐠 = −
𝛍 𝟎
𝟐
∫ 𝐌 . 𝐇 𝐝 𝐝𝐕𝐕
Eq. 2-5
where 𝐇 𝐝 is the demagnetizing field. This field depends on the magnetic domain configuration,
and it can be found by solving:
𝛁 . 𝐇 𝐝 = −𝛁 . 𝐌 Eq. 2-6
𝛁 × 𝐇 𝐝 = 𝟎 Eq. 2-7
The solution of these equations is:
𝑯 𝒅 = −
𝟏
𝟒𝝅
∫ 𝛁 . 𝑴
𝒓
𝒓 𝟑
𝒅𝑽𝑽
Eq. 2-8
where 𝒓 is the vector going from the integration point to the point where 𝐇 𝐝 is being calculated.

7
Fig. 2-2 Example of micromagnetic configuration. Compared to a uniform state, the energy of the demagnetizing
field is decreased by closure flux.
2.1.2 Dynamic Micromagnetics
The purpose of dynamic micromagnetics is to describe the time dependence of magnetization
vector with respect to external magnetic fields, enabling the study of magnetization processes at
the Nano-scale level.. This is done by solving the Landau-Lifshitz-Gilbert equation (7) (8),
which is a partial differential equation describing the evolution of the magnetization in term of
the local effective field, whose main contributions are briefly introduced in the following (9).
2.1.2.1 Effective Field
The effective magnetic field is the functional derivative of the magnetic energy density with
respect to the magnetization m, it is written as:
𝐇 𝐞𝐟𝐟 = −
𝟏
𝛍 𝟎 𝐌 𝐬
𝐝 𝟐 𝐄
𝐝𝐦 𝐝𝐕
Eq. 2-9
where 𝐝𝐄 𝐝𝐕⁄ is the energy density.

8
dE = − 𝝁 𝟎 𝑴 𝒔 ∫ (𝒅𝒎) . 𝑯 𝒆𝒇𝒇𝑽
𝒅𝑽 Eq. 2-10
We need to pay attention to this, since m is a unit vector, dm is always perpendicular to m.
In point of magnetic energy, the effective field can be written as:
𝐇 𝐞𝐟𝐟 =
𝟐𝑨
𝝁 𝟎 𝑴 𝒔
𝛁 𝟐
𝒎 −
𝟏
𝝁 𝟎 𝑴 𝒔
𝝏𝑭 𝒂𝒏𝒊𝒔
𝝏𝒎
+ 𝑯 𝒂 + 𝐇 𝐝 Eq. 2-11
2.1.2.2 Gyromagnetic precession
In physics, the gyromagnetic ratio of a particle or system is the ratio of its magnetic dipole
moment to its angular momentum.
From quantum mechanics:
𝛍 = − 𝛄𝐋; 𝛄 =
𝐠|𝐞|
𝟐 𝐦 𝐞
=
𝐠 𝛍 𝐁
ℏ
Eq. 2-12
where 𝛍 is the electron magnetic dipole moment or spin magnetic moment, L, is the electron
angular moment, 𝛄, is the absolute value of the gyromagnetic ratio, g, is the Landé splitting
factor which is approximately equal to 2, e, is the electron charge which is equal to 1.6 . 10−19
C, 𝐦 𝐞, is the electron mass which is 9.1 . 10−31
Kg.

9
Fig. 2-3 Different states of spin
From momentum theorem, the torque exerted on the electron by a magnetic field 𝐇 can be
expressed as:
𝐝𝐋
𝐝𝐭
= 𝛍 × 𝐇 Eq. 2-13
𝐝𝛍
𝐝𝐭
= − 𝛄𝛍 × 𝐇 ; 𝐟 𝐋 =
𝛄𝐇
𝟐𝛑
Eq. 2-14
where 𝐟 𝐋 is the Larmor frequency or precession frequency.
According to the continuum gyromagnetic precession model, it follows that:
𝛛𝐌
𝛛𝐭
= − 𝛄𝐌 × 𝐇 Eq. 2-15
2.1.2.3 Landau‐Lifshitz Equation
The first dynamical model for the magnetization precessional motion was proposed by Landau
and Lifshitz in 1935 (7):
𝛛𝐌
𝛛𝐭
= − 𝛄𝐌 × 𝑯 𝒆𝒇𝒇 Eq. 2-16
The features of the Landau-Lifshitz (LL) equation are listed in the following :

10
● 𝛛𝐌 𝛛𝐭 → 𝟎⁄ ⟹ equilibrium condition⟹ first Brown’s equation
● boundary condition ⟹ second Brown’s equation
𝝏𝑴
𝝏𝒏
|𝝏𝛀 = 𝟎
● Hamiltonian equation ⟹ energy conservation when no dissipation terms are included.
To obtain a realistic description of magnetization processes, dissipation terms have to be
included, since:
Atomic‐level dynamics involves interactions between magnetization, electrons and phonons
(10). Magnetization damping can occur through energy transfer (relaxation) from an electron's
spin to:
● Itinerant electrons
● Lattice vibrations (spin‐phonon relaxation)
● Spin waves, magnons
● Impurities in the lattice
Fig. 2-4 Damped gyromagnetic precession

11
The modified equation in the presence of damping phenomena results to be:
𝛛𝐌
𝛛𝐭
= − 𝜸𝑴 × 𝑯 𝒆𝒇𝒇 −
𝝀
𝑴 𝒔
𝑴 × �𝑴 × 𝑯 𝒆𝒇𝒇� ; |𝑴| = 𝑴 𝒔 Eq. 2-17
where 𝜆 > 0 is phenomenological parameter characteristic of the material.
2.1.2.4 Landau‐Lifshitz‐Gilbert Equation
An alternative approach to include damping was proposed by Gilbert in 1955 (8) and consists in
the introduction of the following term:
𝛂
𝐌 𝐬
𝐌 ×
𝛛𝐌
𝛛𝐭
Eq. 2-18
Where 𝛂 = 0.001÷ 0.1 is the Gilbert damping constant (material parameter).
It results that:
𝛛𝐌
𝛛𝐭
= − 𝛄𝐌 × 𝐇 𝐞𝐟𝐟 +
𝛂
𝐌 𝐬
𝐌 ×
𝛛𝐌
𝛛𝐭
Landau‐ Lifshitz‐ Gilbert (LLG) equation) Eq. 2-19
Equivalence between two above equations:
𝛛𝐌
𝛛𝐭
= −
𝛄
𝟏+𝛂 𝟐
𝐌 × 𝐇 𝐞𝐟𝐟 −
𝛄𝛂
(𝟏+𝜶 𝟐)𝑴 𝒔
𝐌 × (𝐌 × 𝐇 𝐞𝐟𝐟) ; �
𝛄 𝐋 =
𝛄
𝟏+𝛂 𝟐
𝛌 =
𝛄𝛂
𝟏+𝜶 𝟐
Eq. 2-20
2.1.2.5 31BDifference between the two Equations
The two equations (LL and LLG) are equivalent from a mathematical point of view, but under
extreme conditions they describe different physics. They are identical only in the limit of
vanishing damping, so:𝛂, 𝛌 ⟶ 𝟎.

12
In the limit of infinite damping (𝛂, 𝛌 ⟶ ∞) (11) (12) :
● 𝛌 ⟶ ∞ (𝐋𝐋 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧) ⟹
𝛛𝐌
𝛛𝐭
⟶ ∞ ⟹ Not realistic
● 𝛂 ⟶ ∞ (𝐋𝐋𝐆 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧) ⟹
𝛛𝐌
𝛛𝐭
⟶ 𝟎 ⟹ Realistic!
A very large damping should generate a very slow motion.
2.1.2.6 Normalized LLG Equation
By dividing both sides of LLG equation by ( 𝜸𝑴 𝒔
𝟐
), it results that:
𝟏
𝛄𝐌 𝐬
𝟐
𝛛𝐌
𝛛𝐭
= −
𝟏
�𝟏+𝛂 𝟐�𝐌 𝐬
𝟐 𝐌 × 𝐇 𝐞𝐟𝐟 −
𝛂
�𝟏+𝛂 𝟐�𝐌 𝐬
𝟑 𝐌 × (𝐌 × 𝐇 𝐞𝐟𝐟) Eq. 2-21
where 𝐦 =
𝐌
𝐌 𝐬
; 𝐡 𝐞𝐟𝐟 =
𝐇 𝐞𝐟𝐟
𝐌 𝐬
By measuring the time in units of (𝛄𝐌 𝐬
𝟐
)−𝟏
𝛛𝐦
𝛛𝐭
= −
𝟏
�𝟏+𝛂 𝟐�
𝐦 × 𝐡 𝐞𝐟𝐟 −
𝛂
�𝟏+𝛂 𝟐�
𝐦 × (𝐦 × 𝐡 𝐞𝐟𝐟) Eq. 2-22
or equivalently
𝛛𝐦
𝛛𝐭
= −𝒎 × 𝒉 𝒆𝒇𝒇 + 𝜶𝒎 ×
𝝏𝒎
𝝏𝒕
Eq. 2-23
2.1.2.7 Properties of Magnetization Dynamics
● Conservation of magnetization amplitude
By scalar multiplying both sides of the LLG equation by m:
𝐝
𝐝𝐭
�
𝟏
𝟐
|𝐦| 𝟐
� = 𝟎 ⟹ |𝐦(𝐭, 𝐫)| = |𝐦(𝐭 𝟎, 𝐫)| ∀ 𝐭 𝟎, 𝐫, 𝐭 ∈ 𝛀 Eq. 2-24

13
Any magnetization motion, at a given location r, evolves on the unit sphere.
● Lyapunov structure
𝛛𝐦
𝛛𝐭
= −𝐦 × (𝐡 𝐞𝐟𝐟 − 𝛂
𝛛𝐦
𝛛𝐭
) By scalar multiplying both sides of the LLG equation by
(𝐡 𝐞𝐟𝐟 − 𝛂
𝛛𝐦
𝛛𝐭
) ⟹
𝛛𝐦
𝛛𝐭
. �𝐡 𝐞𝐟𝐟 − 𝛂
𝛛𝐦
𝛛𝐭
� = 𝟎
Considering this expression for the time derivative of free energy g:
𝐝𝐠
𝐝𝐭
= � �
𝛅𝐠
𝛅𝐦
.
𝛛𝐦
𝛛𝐭
+
𝛅𝐠
𝛅𝐡 𝐚
.
𝛛𝐡 𝐚
𝛛𝐭
� 𝐝𝐯 = � (−𝐡 𝐞𝐟𝐟 .
𝛛𝐦
𝛛𝐭
− 𝐦 .
𝛛𝐡 𝐚
𝛛𝐭𝛀
) 𝐝𝐯
𝛀
By integrating over the sample volume :
𝐝𝐠
𝐝𝐭
= − ∫ 𝛂 �
𝛛𝐦
𝛛𝐭
�
𝟐
𝐝𝐯 − ∫ 𝐦 .
𝛛𝐡 𝐚
𝛛𝐭
𝐝𝐯𝛀𝛀
Energy Balance Equation Eq. 2-25
●When
𝛛𝐡 𝐚
𝛛𝐭
= 𝟎 ⟹
𝐝𝐠
𝐝𝐭
= − ∫ 𝛂 �
𝛛𝐦
𝛛𝐭
�
𝟐
𝐝𝐯𝛀
Lyapunov structure Eq. 2-26
The free‐energy is a non‐increasing function of time, since ≥ 𝟎 .
● When 𝛂 = 𝟎 ⟹ 𝐠(𝐭) = 𝐠(𝒕 𝟎) ∀ 𝒕 𝟎, 𝒕

14
3 Description of the used micromagnetic solver
3.1 Time Discretization of the Landau-Lifshitz-Gilbert
Equation
The numerical integration of the Landau-Lifshitz-Gilbert (LLG) is performed in computational
micromagnetics for studying magnetization processes at nanometer length scales. Specific care
has to be considered for the time discretization scheme, which should allow the time step
increase with a limited number of effective field updates, without affecting the magnetization
amplitude and the Lyapunov structure of the LLG equation. In this thesis, the spatial and time
variation of the magnetization is computed by using a micromagnetic solver previously
developed at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Torino (9). In the
following Sections, a brief description of the solver and the applied techniques will be given.
Specifically, this micromagnetic code adopts a finite difference scheme for unstructured meshes
for the exchange field calculation (13), and a Cayley transform based scheme for the time
discretization of the LLG equation (14). A multipole expansion technique is implemented in the
code to speed-up the computation of the magnetostatic field in large-scale patterned magnetic
films, like the magnetic antidot arrays studied in this thesis (15). The INRIM micromagnetic
code is also suitable for handling complex lattice structures (magnetic nanostructures with
irregularly curved boundaries) and heterogeneous material composition. In the micromagnetic
solver developed at INRIM the magnetic sample is discretized into a mesh of T hexahedra,
where the magnetization M is assumed to be uniform and it is updated by time integrating a
generalized form of the LLG equation (14).

15
𝝏𝑴
𝝏𝒕
= 𝝎(𝑴) × 𝑴 Eq. 3-1
𝑴̇ (𝒏)
= 𝑨(𝑴(𝒏)
) × 𝑴(𝒏)
LLG equation for 𝒏th mesh element Eq. 3-2
where 𝝎 = 𝑨(𝑴) + 𝝈𝑴 ; 𝑨 = 𝜸 𝑮 �𝑯 𝒆𝒇𝒇 + � 𝜶
𝑴 𝒔
� � �𝑴 × 𝑯 𝒆𝒇𝒇�� /(𝟏 + 𝜶 𝟐
)
In the above equations A: is the generator ; 𝜶: is the damping constant ; 𝜸 𝑮: is the absolute
value of the gyromagnetic ratio (2.21 × 105
𝑚𝐴−1
𝑆−1
) ; 𝑴 𝒔: is the saturation magnetization ;
𝑯 𝒆𝒇𝒇: is the effective field (sum of the exchange, magneto-static, anisotropy and applied fields).
The time evolution described by Eq. (3-2) is not altered by the addition of a term parallel to M;
thus, we generalize the generator by introducing a vector quantity 𝝎 = 𝑨(𝑴) + 𝝈𝑴, where 𝝈 is
assumed to be constant in time and space.
The dynamics of vector 𝑴(𝒏)
evolves on a sphere described by smooth time-dependent curves
𝑄(𝑡) in the Lie group of (3×3) rotation 𝐺 = 𝑆𝑂(3), satisfying:
𝑴̇ (𝒏)
(𝒕) = 𝑸̇ (𝒕)𝑴(𝒏)
(𝟎) Eq. 3-3
where 𝑴(𝒏)
(𝟎) is the initial magnetization value in mesh element 𝑛 (16). By introducing the Lie
algebra of (3×3) skew symmetric matrices, the following family of differential equation derives:
𝑸̇ (𝒕) = 𝒔𝒌𝒆𝒘 �𝝎 �𝑸(𝒕) 𝑴(𝒏)
(𝟎)�� 𝑸(𝒕) Eq. 3-4
where 𝑠𝑘𝑒𝑤 𝜔 = �
0
𝜔3
−𝜔3 𝜔2
0 −𝜔1
−𝜔2 𝜔1 0
� .
Different approaches have been proposed in the literature to improve time discretization
efficiency:

16
● Semi-analytical Schemes: In semi-analytical time stepping schemes, the magnetization
dynamics is formulated and solved analytically at 𝐭 𝐢 + 𝛅𝐭 by introducing in each mesh element, a
local (u, v, w) coordinate system with the u-axis parallel to 𝐇 𝐞𝐟𝐟(𝐭 𝐢) (17) (18).
● Runge-Kutta Algorithms: The problem is based on the solution of a set of N coupled first
order differential equations (𝐲𝐢, 𝐢 = 𝟏, 𝟐, … , 𝐍) in the following form (19):
𝐝𝐲𝐢(𝐱)
𝐝𝐭
= 𝐟𝐢(𝐭, 𝐲 𝟏, … , 𝐲 𝐍), 𝐢 = 𝟏, … , 𝐍, Eq. 3-5
where 𝐟𝐢 are functions. Runge-Kutta Methods give a solution over an interval by combining the
information from a number of Euler-style steps, each of them linking an evaluation of the right
hand 𝐟𝐢
‘
s in Eq. (3-1). That information is considered to match a Taylor series expansion.
● Projection Methods: In this method, first we solve the LLG equation by a standard method
and then project the solution onto a unit sphere to maintain the constrain |m|=1 (20).
● Cayley Transform based schemes: The time discretization of Eq. (3-3) requires a geometric
integration scheme that preserves the magnetization amplitude. Algorithms of arbitrary order can
be constructed by using the Cayley transform, which is a second order approximation of the
algorithmic exponential for the rotation group (14):
𝒄𝒂𝒚(𝒗) = (𝑰 + 𝒔𝒌𝒆𝒘[𝒗 𝟐])⁄ (𝑰 − 𝒔𝒌𝒆𝒘[𝒗 𝟐])⁄ −𝟏
Eq. 3-6
where I is the identity matrix. The application of the Cayley transform allows a rotational update
of the magnetization.
By using a first-order scheme, based on Euler algorithm, the magnetization at the (𝑖 + 1)𝑡ℎ
instant is written as:

17
𝑴𝒊+𝟏
(𝒏)
= 𝒄𝒂𝒚�∆𝒕 𝝎𝒊
(𝒏)
�𝑴𝒊
(𝒏)
Eq. 3-7
where generator 𝝎(𝒏)
is calculated at the previous instant.
We can also calculate the magnetization at the (𝑖 + 1)𝑡ℎ instant by using a second-order scheme
based on Heun algorithm. It is written as:
𝑴𝒊+𝟏
(𝒏)
= 𝒄𝒂𝒚�∆𝒕 𝝎�(𝒏)
�𝑴𝒊
(𝒏)
Eq. 3-8
where generator 𝝎�(𝒏)
is written as:
𝝎�(𝒏)
= �𝝎∗
(𝒏)
�𝑴∗
(𝒏)
� + 𝝎𝒊
(𝒏)
�𝑴𝒊
(𝒏)
� 𝟐⁄ �
with:
𝑴∗
(𝒏)
= 𝒄𝒂𝒚�∆𝒕 𝝎𝒊
(𝒏)
�𝑴𝒊
(𝒏)
At each time step the effective field 𝑯 𝒆𝒇𝒇 is estimated once for the Euler algorithm and twice for
the Heun one.
The micromagnetic code developed at INRIM adopts a Cayley transform based scheme for the
time integration of the LLG equation: both Euler and Heun algorithms are implemented. In this
thesis, the simulations have been performed by using the more efficient Heun algorithm
3.2 Spatial Discretization of Exchange Field
The finite difference method for regular grids (FD-Reg.) is a technique which is widely applied
in the literature for computing exchange field in large ferromagnetic media. By this technique,
the exchange field computation is reduced to a set of independent problems. Nevertheless,

18
adopting regular spatial grids imposes severe restrictions for the object shape. Moreover, local
inaccuracies can arise in proximity of the boundaries, determining vortex pinning artifacts. To
avoid these critical aspects, an original finite difference method suitable for arbitrarily irregular
grids (FD-Irr.) was previously implemented in the INRIM micromagnetic solver for the
exchange field calculation (13). This method, used in this thesis, is based on Taylor series
expansion around the computational point, and determines second-order derivatives by solving
an over-determined set of linear equations.
3.2.1 Finite Difference Method on Unstructured Meshes
The description is here made for thin film structures, where 2-D approximation can be
considered. Under this assumption, the magnetization M is kept vary on the 𝑥𝑦-plane and
assumed to be uniform along the 𝑧-axis (𝝏𝑴 𝝏𝒛⁄ = 𝟎). In the generic point (𝑥. 𝑦) the
magnetization M is approximated by a second order Taylor series expansion around
point(𝑥0, 𝑦0):
𝐌 ≅ 𝐌 𝟎 + (𝐱 − 𝐱 𝟎)
𝛛𝐌 𝟎
𝛛𝐱
+ (𝐲 − 𝐲 𝟎)
𝛛𝐌 𝟎
𝛛𝐲
+
(𝐱 − 𝐱 𝟎) 𝟐
𝟐
𝛛 𝟐
𝐌 𝟎
𝛛𝐱 𝟐
+
(𝐲−𝐲 𝟎) 𝟐
𝟐
𝛛 𝟐 𝐌 𝟎
𝛛𝐲 𝟐
+ (𝐱 − 𝐱 𝟎)(𝐲 − 𝐲 𝟎)
𝛛 𝟐 𝐌 𝟎
𝛛𝐱𝛛𝐲
Eq. 3-9
We assume that the 2-D magnetic domain Ω lies on plane 𝑥𝑦, and it is discretized into an
irregular grid of elements, and vector M is assumed to be constant in each element. Associating
computational points to element barycentres, the five unknown derivatives in Eq. (3-11) can be
determined by considering the contribution of at least five surrounding elements N (Fig. 3-1 a).
Anyway, to avoid ill-conditioned problems, a higher number of interacting elements N has to be

19
considered (e.g. N~8). If the computational element is located on the domain boundary, the
boundary condition (𝝏𝑴 𝟎 𝝏𝒏⁄ = 𝟎) is imposed by introducing fictitious points outside the
domain with the same magnetization value (see Fig. 3-1 b).
Fig. 3-1 Computational element with barycentre (𝒙 𝟎, 𝒚 𝟎) and magnetization 𝑴 𝟎: (a) interacting elements around the
considered computational element; (b) fictitious points around an element located on the domain boundary to
impose boundary conditions on M. The fictitious points are mirror images with respect to the boundary (13).
The use of a higher number of interacting elements leads to an over-determined set of linear
equations, which can be solved by norm minimization of the following functional:
𝑭 = ∑ ��𝑴 𝟎 − 𝑴𝒊 + 𝒉𝒊
𝝏𝑴 𝟎
𝝏𝒙
+ 𝒌𝒊
𝝏𝑴 𝟎
𝝏𝒚
+
𝒉 𝒊
𝟐
𝟐
𝝏 𝟐 𝑴 𝟎
𝝏𝒙 𝟐
+
𝒌𝒊
𝟐
𝟐
𝝏 𝟐 𝑴 𝟎
𝝏𝒚 𝟐
+ 𝒉𝒊 𝒌𝒊
𝝏 𝟐 𝑴 𝟎
𝝏𝒙𝝏𝒚
�
𝟏
∆𝒊
𝟑�
𝟐
𝑵
𝒊=𝟏 Eq. 3-10
where 𝑖 is the generic adjacent element of barycentre coordinates (𝑥𝑖, 𝑦𝑖) and magnetization 𝑴𝒊
and

20
𝒉𝒊 = 𝒙𝒊 − 𝒙 𝟎 ; 𝒌𝒊 = 𝒚𝒊 − 𝒚 𝟎 ; ∆𝒊= �𝒉𝒊
𝟐
+ 𝒌𝒊
𝟐
The following 5 × 5 algebraic system of equations derives from Eq. (3-12):
[𝑪][∆𝑴 𝟎] = [𝑫] Eq. 3-11
where the transpose of [∆𝑴 𝟎] is:
[∆𝑴 𝟎] 𝑻
= �
𝝏𝑴 𝟎
𝝏𝒙
𝝏𝑴 𝟎
𝝏𝒚
𝝏 𝟐
𝑴 𝟎
𝝏𝒙 𝟐
𝝏 𝟐
𝑴 𝟎
𝝏𝒚 𝟐
𝝏 𝟐
𝑴 𝟎
𝝏𝒙𝝏𝒚
�
The elements of arrays [C] and [D] are:
𝒄 𝒎𝟏 = �
𝟐
∆𝒊
𝟑
𝒉𝒊 𝒑 𝒎
𝑵
𝒊=𝟏
; 𝒄 𝒎𝟐 = �
𝟐
∆𝒊
𝟑
𝒌𝒊 𝒑 𝒎
𝑵
𝒊=𝟏
𝒄 𝒎𝟑 = �
𝒉𝒊
𝟐
∆𝒊
𝟑
𝒑 𝒎
𝑵
𝒊=𝟏
; 𝒄 𝒎𝟒 = �
𝒌𝒊
𝟐
∆𝒊
𝟑
𝒑 𝒎
𝑵
𝒊=𝟏
𝒄 𝒎𝟓 = �
𝟐
∆𝒊
𝟑
𝒉𝒊 𝒌𝒊 𝒑 𝒎
𝑵
𝒊=𝟏
; 𝒎 = 𝟏, … , 𝟓
with
𝒑 𝟏 = 𝒉𝒊 ; 𝒑 𝟐 = 𝒌𝒊 ; 𝒑 𝟑 =
𝒉𝒊
𝟐
𝟐
; 𝒑 𝟒 =
𝒌𝒊
𝟐
𝟐
; 𝒑 𝟓 = 𝒉𝒊 𝒌𝒊
and
𝒅 𝟏 = − � 𝑮𝒉𝒊
𝑵
𝒊=𝟏
; 𝒅 𝟐 = − � 𝑮𝒌𝒊
𝑵
𝒊=𝟏
; 𝒅 𝟑 = − � 𝑮
𝒉𝒊
𝟐
𝟐
𝑵
𝒊=𝟏
; 𝒅 𝟒 = − � 𝑮
𝒌𝒊
𝟐
𝟐
𝑵
𝒊=𝟏
; 𝒅 𝟓 = − � 𝑮𝒉𝒊 𝒌𝒊
𝑵
𝒊=𝟏

21
with
𝑮 =
(𝑴 𝟎 − 𝑴𝒊)
∆𝒊
𝟑
Solving Eq. (3-11) second order derivatives are computed and exchange field is written as:
𝑯 𝒆𝒙 = 𝒌 𝒆𝒙 �
𝝏 𝟐 𝑴 𝟎
𝝏𝒙 𝟐 +
𝝏 𝟐 𝑴 𝟎
𝝏𝒚 𝟐 � , 𝒌 𝒆𝒙 being the exchange constant.
The method can be applied independently on the types of mesh elements (quadrangular or
triangles), either considering the degrees of freedom placed in the element barycentres or in the
element nodes. For 3-D problems, functional Eq. (3-10) is extended including the derivatives
with respect to 𝑧-coordinate. In this case, the minimum number of surrounding element has to be
nine.
3.3 10BSpatial Discretization of Magneto-Static Field
In detail, the patterned medium is decomposed into a macroscale grid of cells, in turn discretized
into a non-structured mesh having average size comparable to the exchange length. Locally, the
magnetostatic field is the sum of a short-range term, computed by discretizing Green integral
equation over a limited region, and a long-range term, which represents the contributions of far
macrocells as a set of multipole moments.

22
Fig. 3-2 Scheme of a magnetic antidot array with square lattice, composed of 𝟏𝟓 × 𝟏𝟓 macrocells, illustrating the
“short-range” region, associated with the computation of magnetostatic interactions through the discretization of the
Green integral equation (21).
In the following, a brief description of the methodology is given.
The domain Ω is a patterned magnetic film, an array of antidots (holes) of embedded in a
continuous magnetic medium (Fig. 3-2). In the 𝑥1 𝑥2 plane, the domain is decomposed into a grid
of 𝑵 𝒄𝒆𝒍𝒍 macrocells, representing the spatial period, in turn discretized into a mesh of T
hexahedra. In each of the (𝑵 𝒄𝒆𝒍𝒍. 𝑻) elements the magnetization is assumed to be uniform.
The magnetostatic field 𝑯 𝒎 in the element centered in 𝒓 𝟎, located in a macrocell with grid
location (𝒊, 𝒋) and barycentre 𝑶 𝟎 (Fig. 3-2), and is decomposed into a short- and a long-range
term. The first term, which represents the interactions with a limited number 𝑵 𝑺𝑹 = (𝟐𝝀 + 𝟏) 𝟐
of adjacent macrocells, is the numerical solution of the integral equation obtained from Green’s
theorem.

23
𝑯 𝒎
𝑺𝑹(𝒓 𝟎) =
𝟏
𝟒𝝅
∑ ∑ ∑ ∫ 𝑴(𝒓 𝒆)𝝏𝛀 𝒆
𝑻
𝒆=𝟏
𝒋+𝝀
𝒋−𝝀
𝒊+𝝀
𝒊−𝝀 . 𝒏 𝒆
(𝒓 𝟎−𝒓 𝒆)
‖𝒓 𝟎−𝒓 𝒆‖ 𝟑 𝒅𝒔 Eq. 3-12
where 𝝀 is the number of the adjacent macrocell layers (Fig. 3-2), and 𝝏𝛀 𝒆 is the surface of the
𝑒-th hexahedron with normal unit vector 𝒏 𝒆. The second term describes the interactions with the
remaining macrocells and it is calculated as:
𝑯 𝒎
𝑳𝑹(𝒓 𝟎) = −
𝟏
𝟒𝝅
∑ ∑ ∑ 𝑨 𝒇,𝒋
𝒌
(𝑶 𝟎)𝒋
𝒌=−𝒋
𝒑
𝒋=𝟎
𝑵 𝒄𝒆𝒍𝒍−𝑵 𝑺𝑹
𝒇=𝟏 𝛁𝑹𝒋
𝒌
(𝒓 𝟎 − 𝑶 𝟎) Eq. 3-13
where 𝑹𝒋
𝒌
(𝒘) = 𝒓𝒋
𝑷𝒋
|𝒌|
(𝒄𝒐𝒔𝜽)𝒆−𝒊𝒌𝝋
/(𝒋 + 𝒌)!, having introduced a local spherical frame
𝒘 = (𝒓, 𝜽, 𝝋) centered in 𝑶 𝟎. In Eq. (3-16) 𝑷𝒋
|𝒌|
(. ) is the associated Legendre function of degree
𝑗 and order 𝑘, 𝒊 is the imaginary unit and 𝑝 is the order of the multipole expansion. Function
𝑨 𝒇,𝒋
𝒌
(𝑶 𝟎) is written as:
𝑨 𝒇,𝒋
𝒌
(𝑶 𝟎) = ∑ ∑ 𝑸𝒋,𝒏
𝒎,𝒌
�𝑶 𝒇 − 𝑶 𝟎�𝑴 𝒏
𝒎
(𝑶 𝒇)𝒏
𝒎=−𝒏
𝒑
𝒏=𝟎 Eq. 3-14
Where 𝑴 𝒏
𝒎
(𝑶 𝒇) is one of the (𝒑 + 𝟏) 𝟐
multipole moments defined on the macrocell centered in
𝑶 𝒇,
𝑴 𝒏
𝒎
�𝑶 𝒇� = ∑ ∫ 𝑴(𝒓 𝒆)𝛀 𝒆
. 𝛁𝑹 𝒏
𝒎
�𝒓 𝒆 − 𝑶 𝒇�𝒅𝒗𝑻
𝒆=𝟏 Eq. 3-15
The operator 𝑸𝒋,𝒏
𝒎,𝒌
transposes the multipole moments from𝑶 𝒇 to 𝑶 𝟎.

24
4 Introduction of the Simulations
4.1 Description of Sample Materials and Geometries
The attention is focused on 20 nm thick Permalloy antidot array films. Permalloy which was
invented in Bell Laboratories is a soft magnetic alloy (with about 80% nickel and 20% iron) with
exceptionally high magnetic permeability. The alloy has a low coercivity, near zero
magnetostriction, and significant anisotropic magnetoresistance. The saturation magnetization
𝑴 𝒔 of Permalloy is 860 𝑘𝐴𝑚−1
, its exchange constant 13 𝑝𝐽𝑚−1
, and its magnetocrystalline
anisotropy is negligible.
The considered antidot arrays, with finite size along 𝑥 and 𝑦 directions, are characterized by a
periodic distribution of circular holes or amagnetic inclusions in the 𝑥𝑦 plane. The modeling
analysis is performed by varying the hole diameter 𝑑, considering a square and a rhombic
arrangement, with geometrically equivalent directions every 45°
. In particular, three different
hole diameters have been considered in the simulations, 𝑑 = 200,300,400 𝑛𝑚, fixing the length
of the unit cell or spatial period to600𝑛𝑚. The magnetic region Ω in the unit cell is discretized
into a mesh of hexahedra, whose edge size is comparable to the exchange length, which is in the
order of 5 nm for Permalloy (22).

25
Fig. 4-1 Scheme of the antidot array with rhombic symmetry and corresponding unit cell.
4.2 Description of External Sources and Simulation
Procedure
Magnetization dynamics in the magnetic antidot arrays is calculated by considering two in-plane
external sources: a constant and uniform bias field 𝑯 𝒃𝒊𝒂𝒔 plus a uniform excitation pulse applied
orthogonally to 𝑯 𝒃𝒊𝒂𝒔, after the determination of the ground state . In particular, the ground state
is obtained by imposing a sufficiently high bias field with magnitude of 150 𝑘𝐴 𝑚⁄ , which leads
the samples to quasi-saturation. With this external source condition, the magnetization time
evolution is computed until the equilibrium state, which is assumed to be reached when the
maximum nodal local value of the misalignment between magnetization and effective field

26
(�𝑴 × 𝑯 𝒆𝒇𝒇� 𝑴 𝒔
𝟐⁄ ) is lower than a fixed threshold 𝜏. In particular, for the determination of the
ground state the threshold 𝜏 is set at 10−8
, the damping coefficient 𝛼 is fixed to 0.1 and the
parameter 𝜎 appearing in the generalized version of the LLG equation is assumed equal to 9𝛾 𝐺,
in order to accelerate equilibrium state reaching. In this way, time steps in the order of 2.5 𝑝𝑠
can be adopted. To avoid numerical stiffness problems, we consider irregular or non-structured
meshes. Meshes are generated over the entire unit cell without any symmetry properties (23).
The uniform excitation pulse, applied after the reaching of the ground state, is a Gaussian pulse
whose time-function is reported in Eq. (4-1) together with its parameter values.
𝑯 𝒑𝒖𝒍𝒔𝒆(𝒕) = 𝒂𝒆𝒙𝒑 �− �
𝒕−𝒕 𝟎
𝝉
�
𝟐
� Eq. 4-1
where 𝒂 = 𝟏𝒌𝑨𝒎−𝟏
, 𝝉 = 𝟓𝒑𝒔
After applying the aforementioned external sources, we simulate the time evolution of the
magnetization vector by solving the LLG equation. In this case, the calculation is performed by
considering a damping coefficient 𝛼 = 0.01 (realistic value for Permalloy) and setting parameter
𝜎 at zero in order to not artificially perturb magnetization dynamics. After the determination of
the micromagnetic solution, we compute the time evolution of the spatial average of the
magnetization component parallel to the excitation field and then, we calculate the relative Fast
Fourier Transform (FFT) power spectrum. The relevant FFT graph has several peaks
corresponding to specific eigenfrequencies or ferromagnetic resonance frequencies, for each of
which we can deduce the surface plot of the magnitude of Fourier coefficients (24). These plots
correspond to spin wave mode profiles or, in other terms, to the spatial distributions of mode spin

27
precession amplitude. Peaks in these maps are associated with the highest spin precession
amplitude of the relative mode.
As described in the following chapter of the thesis, ferromagnetic resonance frequencies and
associated spin wave mode profiles are strongly influenced by the lattice geometry of the antidot
arrays.
Fig. 4-2 Time evolution of the spatial average of the magnetization component parallel to the excitation pulse 〈𝑴 𝒖〉
and associated power spectrum.

28
Fig. 4-3 Surface plot of the magnitude of Fourier coefficients for a specific eigenfrequency or ferromagnetic
resonance frequency (24).
The calculated spin wave modes, have been classified on the basis of their frequency and spatial
symmetry. The low-frequency mode corresponds to the edge mode. This is confined close to the
holes edges, where the demagnetizing field is opposed to the bias field and reaches the highest
values. At higher frequency, the quasi-uniform or extended modes are found. They are,
characterized by quasi-uniform spin precession amplitude along stripes orthogonal to the bias
field comprised between rows of holes. They appear in correspondence of regions where the
demagnetizing field is again opposite to the bias field, but it is reduced and can locally tend to
zero. Finally, there is the localized mode, having the maximum amplitude of spin precession in
between neighboring holes, where the demagnetizing field changes sign (25).
In the frequency range between the extended and localized modes, several other peaks of low
intensity, corresponding to order modes, are observed. Fast-Fourier-transform (FFT) imaging of
simulation results has been used to illustrate the distributions of spin precession amplitudes for
different modes.

29
Fig. 4-4 Respectively left to right: Edge Mode- Extended Mode- Localized Mode for a square lattice (26).
Fig. 4-5 Scheme of a square antidot film and of the corresponding macrocell with microscale mesh.
Finite size antidot arrays have been simulated with side length in the order of 5 − 10 𝜇𝑚, to take
into account possible shape anisotropy effects appearing at the film edges. To model these large-
scale samples, we use the parallelized micromagnetic solver previously developed at INRIM
designed to run on a CPU-GPU High-Performance-Computing (HPC) cluster (9). This exploits
the Fast Multipole Method for the calculation of the demagnetizing field. This was described in
detail in the previous chapter.

30
5 Results of Simulations
Frequency of all resonance modes can be systematically tuned by varying the shape and size of
holes in the antidot array. In this thesis, I simulated and analyzed square and rhombic Permalloy
antidot arrays containing circular holes or amagnetic inclusions with different diameters.
Dynamic response of antidots has been investigated as a function of lattice symmetry and
orientation of the applied bias field (𝛼) and a strong dependence was observed. It has also been
observed that multiple resonances occur in antidot arrays for in-plane magnetic fields. However,
when the hole size is small enough, the spin precession amplitude of relevant mode becomes
non-uniform due to the confinement around the hole edges of an inhomogeneous internal or
demagnetizing field. The external bias magnetic field (𝑯 𝒃𝒊𝒂𝒔) is applied at angle 𝛼 from the 𝑥-
direction. Different angles have been considered, 0°
, 15°
, 30°
, 45°
(27). In order to investigate the
spin wave mode profiles corresponding to each resonant peak, we have calculated the surface
plots of the magnitude of Fourier coefficients for the ferromagnetic resonant (FMR) modes by
using the numerical code which has been built and developed at INRIM (Istituto Nazionale di
Ricerca Metrologica, Torino) (23). Localized, edge, and extended modes are the key to
understand a wide range of phenomena in magnonics, including how spin waves propagate in
one-dimensional magnetic stripes, and details of FMR in individual microstructures. Generally,
edge modes are localized in a demagnetized region on the edges of the antidot array holes. This
mode usually difficult to be observed in large antidot lattices because edge roughness and size
distribution of antidot holes would significantly broaden the associated resonance peak. For edge
modes, the relative frequency is highly dependent upon the applied DC field direction and the
lattice type of the antidot array (28). In following, I will present the results of simulations that
have been done for square and rhombic antidot lattices.

31
5.1 13BSquare Antidot Lattice
The dynamic response of Permalloy antidot arrays with circular holes in square lattice geometry
has been systematically investigated as a function of “hole” diameter and orientation of the
applied bias field.
5.1.1 18BInfluence of Bias Field Orientation
Fig. 5-1 Time evolution of the spatial average of the magnetizaion component parallel to the excitation field (applied
along an in-plane direction perpendicular to the bias field) for different orientations of applied bias field with respect
to 𝒙-axis. The simulations were made for a 𝟐𝟎𝒏𝒎 thick square antidot array with hole diameter of 𝟐𝟎𝟎𝒏𝒎,
considering a bias field of 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
.
The time evolution of the spatial average of the magnetization component parallel to the
excitation field (a Gaussian pulse applied along an in-plane direction orthogonal to the bias field)
is reported in Fig. 5-1 as a function of the bias field orientation with respect to the 𝑥-axis. It is

32
possible to observe that the system reaches the equilibrium configuration (the magnetization
vector remains constant in time) after a time interval of about 6 ns.
Fig. 5-2 FFT power spectra of the average magnetization component parallel to the excitaion field (applied
along an in-plane direction perpendicular to the bias field) for different orientations of the applied bias field with
respect to 𝒙-axis. The simulations were made for a 𝟐𝟎𝒏𝒎 thick square antidot array with hole diameter of 𝟐𝟎𝟎𝒏𝒎,
.
After calculating the time evolution of the spatial average of the magnetization component
parallel to the excitation field, we compute the FFT power spectra of the signal itself. By
analyzing Fig. 5-2, it is possible to deduce that by increasing the orientation of applied bias field,
the FFT spectra is going to shrink in both low and high frequency range.

33
5.1.2 Influence of Thickness
Fig. 5-3 Time evolution of the spatial average of the magnetizaion component parallel to the excitation field (along
𝒚-axis) for different thicknesses of the antidot array, with hole diameter of 𝟐𝟎𝟎𝒏𝒎. The simulations were made by
applied along 𝒙-axis.
In the present section, we analyze the role of the film thickness on the FMR behavior.

34
Fig. 5-4 FFT power spectra of the spatial average of the magnetization component parallel to the excitation field
(along 𝒚-axis) for different thicknesses of the antidot array, with hole diameter of 𝟐𝟎𝟎𝒏𝒎. The simulations were
made by considering a bias field of 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
(along 𝒙-axis).
As you can see, by increasing the thickness of the antidot array film in the mid frequency range
the spectra are going to shrink. The relevant spectra are going to expand from each other when
we increase the thickness of the sample, with a decrease in the frequency of the first FMR mode
and an increase in the frequency of the last FMR mode.

35
5.1.3 Influence of Diameter
𝒚-axis) for different diameters of the antidot array holes. The simulations were made by considering a bias field of
𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
(applied along 𝒙-axis).

36
(along 𝒚-axis) for different diameters of the antidot array holes. The simulations were made by consideringa a bias
field of 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
(along 𝒙-axis).
In this section we analyze the influence of the diameter of the antidot array holes. After the
determination of the micromagnetic solution, we compute the time evolution of the spatial
average of the magnetization component parallel to the excitation field and then, we calculate the
relative FFT power spectrum. We can observe from Fig. 5-6 that in the mid frequency range the
spectra are going to shrink by increasing the diameter of circular holes.

37
Fig. 5-7 Surface plots of the magnitude of Fourier coefficients for the FMR modes observed in Fig. 5-6 for the mid
frequency range, calculated for different diameters of circular holes.
Diameter Spin Wave Mode Frequency (GHz)
200𝑛𝑚 (a) Extended 14.3
300𝑛𝑚 (b1) Extended 13.7
300𝑛𝑚 (b2) Edge 14.2
400𝑛𝑚 (c) Extended 12.9
Table 5-1 Related frequencies of the FMR modes in mid frequency range shown in Fig. 5-6, calculated for different
diameters of circular holes.

38
Fig. 5-8 Surface plots of the magnitude of Fourier coefficients for the FMR modes observed in Fig. 5-6 for high
frequency range, calculated for different diameters of circular holes.
Diameter Spin Wave Mode Frequency (GHz)
200𝑛𝑚 (a) Localized 16.2
300𝑛𝑚 (b) Localized 17.3
400𝑛𝑚 (c) Localized 18.4
Table 5-2 Related frequencies of the FMR modes in high frequency range shown in Fig. 5-6, calculated for different
diameters of circular holes.

39
In the high frequency range the spectra are going to expand by increasing the diameter of circular
holes.
Fig. 5-9 Spatial distribution of magnetostatic field calculated in a unit cell of permalloy antidot arrays with square
lattice geometry. Antidot arrays with different hole diameters are considered.

40
5.2 Rhombic Antidot Lattice
The dynamic response of Permalloy antidot arrays with rhombic lattice geometry has been
systematically investigated as a function of antidot arrays thickness and orientation of applied
bias field. Circular holes are considered.
5.2.1 Influence of Bias Field Orientation
Fig. 5-10 Time evolution of the spatial average of the magnetizaion component parallel to the excitation field
(applied in-plane orthoghonal to the bias field) for different orientations of applied bias field (amplitude equal
to 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
). Antidot arrays with hole diameter of 𝟐𝟎𝟎𝒏𝒎 and thickness of 𝟐𝟎𝒏𝒎 are considered.

41
(applied in-plane perpendicular to the bias field) for different orientations of applied bias field (amplitude equal
to 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
). Antidot arrays with diameter of 𝟐𝟎𝟎𝒏𝒎 and thickness of 𝟐𝟎𝒏𝒎 are considered.
After the determination of the time evolution of the spatial average of the magnetization
component parallel to the excitation field, we calculate the relative FFT power spectrum.

42
Fig. 5-12 Surface plots of the magnitude of Fourier coefficients for the FMR modes observed in Fig. 5-11 in the low
frequency range, calculated for different orientations of applied bias field with respect to 𝒙-axis.
Orientation of Bias Field Spin Wave Mode Frequency (GHz)
0°
(a) Edge 13.7
30°
(b) Extended 12.6
45°
(c) Extended 12.1
Table 5-3 Related frequencies in low frequency range shown in Fig. 5-11, calculated for different orientations of
applied bias field.

43
Fig. 5-13 Surface plots of the magnitude of Fourier coefficients for the FMR modes indicated in Fig. 5-11 in the
high frequency range, calculated for different orientations of applied bias field.
Orientation of Bias Field Spin Wave Mode Frequency (GHz)
0°
(a) Localized 16.3
30°
(b) Localized 14.2
45°
(c) Localized 19.7
Table 5-4 Related frequency in the high frequency range shown in Fig. 5-11, calculated for different orientations of
applied bias field.

44
5.2.2 Influence of Thickness
𝒚-axis) for different thicknesses of rhombic antidot arrays with diameter of 𝟐𝟎𝟎𝒏𝒎, calculated for a bias field of
𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
(along 𝒙-axis).

45
Fig. 5-15 FFT power spectra of the spatial average magnetization component parallel to the excitaion field (along 𝒚-
axis) for different thicknesses of rhombic antidot arrays with diameter of 𝟐𝟎𝟎𝒏𝒎, calculated for a bias field of
𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏
(along 𝒙-axis).
After the simulation of magnetization time evolution in Fig. 5-14, we calculate the FFT power
spectra for different thicknesses of rhombic antidot array films. As you can see, the spectra are
going to expand in frequency by increasing the film thicknesses.

46
6 Conclusion
A micromagnetic numerical tool has been applied to study the FMR properties of Permalloy
(𝑁𝑖80 𝐹𝑒20) antidot arrays with square and rhombic lattice, considering different thicknesses of
antidot array, different diameters of circular holes and different orientations of the applied bias
field. The study has demonstrated that by changing the geometrical parameters, you can obtain a
modification of the dynamic response of the system and thus of FMR frequencies and associated
spin wave profiles (23). The frequency of the edge localized mode and its topological character
are both highly dependent on the shape and size of the holes or amagnetic inclusions of the
antidot array. Circular holes are less sensitive to in-plane field angle, with the dynamical
magnetization localized at the edge of the hole. The edge mode in the square antidot array is very
sensitive to the in-plane field angle, regarding both the frequency and the character of the
dynamical magnetization. By engineering the shape and size of the antidot array holes, and by
varying the field angle may enable the development of new spin wave based devices, like
magnonic waveguides (28). Moreover, the FMR spectra strongly depend on the orientation of the
external bias field relative to the antidot array.

47
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