Master Thesis

POLITECNICO DI TORINO

FACOLT�̀� 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 𝑁𝑖80𝐹𝑒20 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 𝐌𝐬 everywhere when the temperature

is much lower than the Curie temperature. Then, the magnetization direction vector m=𝐌 𝐌𝐬⁄ ,

(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= 𝐄𝐞𝐱𝐜𝐡 + 𝐄𝐚𝐧𝐢𝐬 + 𝐄𝐙 + 𝐄𝐝𝐞𝐦𝐚𝐠 , subject to the constraint |M|=𝐌𝐬 or |m|=1.

In the previous equation,

𝐄𝐞𝐱𝐜𝐡: corresponds to the exchange energy, 𝐄𝐚𝐧𝐢𝐬: to the anisotropy energy, 𝐄𝐙: to the zeeman

energy (associated to the applied external field), and 𝐄𝐝𝐞𝐦𝐚𝐠: 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:

𝐄𝐞𝐱𝐜𝐡 = 𝑨∫ ((𝛁𝒎𝒙)𝟐𝐕 + (𝜵𝒎𝒚)𝟐 + (𝜵𝒎𝒛)𝟐) 𝒅𝑽 Eq. 2-1

where A is the exchange constant connected with the exchange length (in the order of 5-10 nm),

𝒎𝒙, 𝒎𝒚 and 𝒎𝒛 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:

𝐄𝐚𝐧𝐢𝐬= ∫ 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 𝒖𝒂𝒏𝒊𝒔):

𝐅𝐚𝐧𝐢𝐬(m,𝒖𝒂𝒏𝒊𝒔) =𝑲𝒂𝒏𝒊𝒔[𝟏 − (𝒎.𝒖𝒂𝒏𝒊𝒔)𝟐] Eq. 2-3

where 𝑲𝒂𝒏𝒊𝒔 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

𝐄𝐙 = −𝛍𝟎 ∫ 𝐌 . 𝐇𝐚 𝐝𝐕𝐕 Eq. 2-4

where 𝐇𝐚 is the external applied field and 𝛍𝟎 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:

𝐄𝐝𝐞𝐦𝐚𝐠 = − 𝛍𝟎𝟐

∫ 𝐌 . 𝐇𝐝 𝐝𝐕𝐕 Eq. 2-5

where 𝐇𝐝 is the demagnetizing field. This field depends on the magnetic domain configuration,

and it can be found by solving:

𝛁 . 𝐇𝐝 = −𝛁 . 𝐌 Eq. 2-6

𝛁 × 𝐇𝐝 = 𝟎 Eq. 2-7

The solution of these equations is:

𝑯𝒅 = − 𝟏𝟒𝝅

∫ 𝛁 . 𝑴 𝒓𝒓𝟑

𝒅𝑽𝑽 Eq. 2-8

where 𝒓 is the vector going from the integration point to the point where 𝐇𝐝 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:

𝐇𝐞𝐟𝐟 = − 𝟏𝛍𝟎 𝐌𝐬

𝐝𝟐𝐄𝐝𝐦 𝐝𝐕

Eq. 2-9

where 𝐝𝐄 𝐝𝐕⁄ is the energy density.

8

dE = − 𝝁𝟎 𝑴𝒔 ∫ (𝒅𝒎) . 𝑯𝒆𝒇𝒇𝑽 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:

𝐇𝐞𝐟𝐟 = 𝟐𝑨𝝁𝟎 𝑴𝒔

𝛁𝟐𝒎 − 𝟏𝝁𝟎 𝑴𝒔

𝝏𝑭𝒂𝒏𝒊𝒔𝝏𝒎

+ 𝑯𝒂 + 𝐇𝐝 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 𝐦𝐞

= 𝐠 𝛍𝐁ℏ

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, 𝐦𝐞, 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:

𝐝𝐋𝐝𝐭

= 𝛍 × 𝐇 Eq. 2-13

𝐝𝛍𝐝𝐭

= − 𝛄𝛍 × 𝐇 ; 𝐟𝐋 = 𝛄𝐇𝟐𝛑

Eq. 2-14

where 𝐟𝐋 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 × 𝑯𝒆𝒇𝒇 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 × 𝑯𝒆𝒇𝒇 − 𝝀𝑴𝒔

𝑴 × �𝑴 × 𝑯𝒆𝒇𝒇� ; |𝑴| = 𝑴𝒔 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:

𝛂𝐌𝐬

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 × 𝐇𝐞𝐟𝐟 + 𝛂𝐌𝐬

𝐌 × 𝛛𝐌𝛛𝐭

Landau‐ Lifshitz‐ Gilbert (LLG) equation) Eq. 2-19

Equivalence between two above equations:

𝛛𝐌𝛛𝐭

= − 𝛄𝟏+𝛂𝟐

𝐌 × 𝐇𝐞𝐟𝐟 − 𝛄𝛂(𝟏+𝜶𝟐)𝑴𝒔

𝐌 × (𝐌 × 𝐇𝐞𝐟𝐟) ; �𝛄𝐋 = 𝛄

𝟏+𝛂𝟐

𝛌 = 𝛄𝛂𝟏+𝜶𝟐

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 ( 𝜸𝑴𝒔𝟐), it results that:

𝟏𝛄𝐌𝐬

𝟐 𝛛𝐌𝛛𝐭

= − 𝟏�𝟏+𝛂𝟐�𝐌𝐬

𝟐 𝐌 × 𝐇𝐞𝐟𝐟 −𝛂

�𝟏+𝛂𝟐�𝐌𝐬𝟑 𝐌 × (𝐌 × 𝐇𝐞𝐟𝐟) Eq. 2-21

where 𝐦 = 𝐌𝐌𝐬

; 𝐡𝐞𝐟𝐟 = 𝐇𝐞𝐟𝐟𝐌𝐬

By measuring the time in units of (𝛄𝐌𝐬𝟐)−𝟏

𝛛𝐦𝛛𝐭

= − 𝟏�𝟏+𝛂𝟐�

𝐦 × 𝐡𝐞𝐟𝐟 − 𝛂�𝟏+𝛂𝟐�

𝐦 × (𝐦 × 𝐡𝐞𝐟𝐟) Eq. 2-22

or equivalently

𝛛𝐦𝛛𝐭

= −𝒎 × 𝒉𝒆𝒇𝒇 + 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:

𝐝𝐝𝐭

�𝟏𝟐

|𝐦|𝟐� = 𝟎 ⟹ |𝐦(𝐭, 𝐫)| = |𝐦(𝐭𝟎, 𝐫)| ∀ 𝐭𝟎, 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 × (𝐡𝐞𝐟𝐟 − 𝛂 𝛛𝐦𝛛𝐭

) By scalar multiplying both sides of the LLG equation by

(𝐡𝐞𝐟𝐟 − 𝛂 𝛛𝐦𝛛𝐭

) ⟹ 𝛛𝐦𝛛𝐭

. �𝐡𝐞𝐟𝐟 − 𝛂 𝛛𝐦𝛛𝐭� = 𝟎

Considering this expression for the time derivative of free energy g:

𝐝𝐠𝐝𝐭

= � �𝛅𝐠𝛅𝐦

.𝛛𝐦𝛛𝐭

+ 𝛅𝐠𝛅𝐡𝐚

.𝛛𝐡𝐚𝛛𝐭

� 𝐝𝐯 = � (−𝐡𝐞𝐟𝐟 . 𝛛𝐦𝛛𝐭

−𝐦 . 𝛛𝐡𝐚𝛛𝐭𝛀

) 𝐝𝐯𝛀

By integrating over the sample volume :

𝐝𝐠𝐝𝐭

= −∫ 𝛂 �𝛛𝐦𝛛𝐭�𝟐

𝐝𝐯 − ∫ 𝐦 . 𝛛𝐡𝐚𝛛𝐭

𝐝𝐯𝛀𝛀 Energy Balance Equation Eq. 2-25

●When 𝛛𝐡𝐚𝛛𝐭

= 𝟎 ⟹ 𝐝𝐠𝐝𝐭

= −∫ 𝛂 �𝛛𝐦𝛛𝐭�𝟐𝐝𝐯 𝛀 Lyapunov structure Eq. 2-26

The free‐energy is a non‐increasing function of time, since ≥ 𝟎 .

● When 𝛂 = 𝟎 ⟹ 𝐠(𝐭) = 𝐠(𝒕𝟎) ∀ 𝒕𝟎, 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

�̇�(𝒏) = 𝑨(𝑴(𝒏)) × 𝑴(𝒏) LLG equation for 𝒏th mesh element Eq. 3-2

where 𝝎 = 𝑨(𝑴) + 𝝈𝑴 ; 𝑨 = 𝜸𝑮 �𝑯𝒆𝒇𝒇 + �𝜶 𝑴𝒔� � �𝑴 × 𝑯𝒆𝒇𝒇�� /(𝟏 + 𝜶𝟐)

In the above equations A: is the generator ; 𝜶: is the damping constant ; 𝜸𝑮: is the absolute

value of the gyromagnetic ratio (2.21 × 105 𝑚𝐴−1𝑆−1) ; 𝑴𝒔: is the saturation magnetization ;

𝑯𝒆𝒇𝒇: 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:

�̇�(𝒏)(𝒕) = �̇�(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:

�̇�(𝒕) = 𝒔𝒌𝒆𝒘 �𝝎�𝑸(𝒕) 𝑴(𝒏)(𝟎)��𝑸(𝒕) Eq. 3-4

where 𝑠𝑘𝑒𝑤 𝜔 = �0𝜔3

−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 𝐭𝐢 + 𝛅𝐭 by introducing in each mesh element, a

local (u, v, w) coordinate system with the u-axis parallel to 𝐇𝐞𝐟𝐟(𝐭𝐢) (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, 𝐲𝟏, … , 𝐲𝐍), 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 𝑯𝒆𝒇𝒇 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 ≅ 𝐌𝟎 + (𝐱 − 𝐱𝟎)𝛛𝐌𝟎

𝛛𝐱+ (𝐲 − 𝐲𝟎)

𝛛𝐌𝟎

𝛛𝐲+

(𝐱 − 𝐱𝟎)𝟐

𝟐𝛛𝟐𝐌𝟎

𝛛𝐱𝟐+

(𝐲−𝐲𝟎)𝟐

𝟐𝛛𝟐𝐌𝟎𝛛𝐲𝟐

+ (𝐱 − 𝐱𝟎)(𝐲 − 𝐲𝟎) 𝛛𝟐𝐌𝟎𝛛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 (𝝏𝑴𝟎 𝝏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 (𝒙𝟎,𝒚𝟎) and magnetization 𝑴𝟎: (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 = ∑ ��𝑴𝟎 −𝑴𝒊 + 𝒉𝒊𝝏𝑴𝟎𝝏𝒙

+ 𝒌𝒊𝝏𝑴𝟎𝝏𝒚

+ 𝒉𝒊𝟐

𝟐𝝏𝟐𝑴𝟎𝝏𝒙𝟐

+ 𝒌𝒊𝟐

𝟐𝝏𝟐𝑴𝟎𝝏𝒚𝟐

+ 𝒉𝒊𝒌𝒊𝝏𝟐𝑴𝟎𝝏𝒙𝝏𝒚

� 𝟏∆𝒊𝟑�𝟐

𝑵𝒊=𝟏 Eq. 3-10

where 𝑖 is the generic adjacent element of barycentre coordinates (𝑥𝑖,𝑦𝑖) and magnetization 𝑴𝒊

and

20

𝒉𝒊 = 𝒙𝒊 − 𝒙𝟎 ; 𝒌𝒊 = 𝒚𝒊 − 𝒚𝟎 ; ∆𝒊= �𝒉𝒊𝟐 + 𝒌𝒊𝟐

The following 5 × 5 algebraic system of equations derives from Eq. (3-12):

[𝑪][∆𝑴𝟎] = [𝑫] Eq. 3-11

where the transpose of [∆𝑴𝟎] is:

[∆𝑴𝟎]𝑻 = �𝝏𝑴𝟎

𝝏𝒙𝝏𝑴𝟎

𝝏𝒚𝝏𝟐𝑴𝟎

𝝏𝒙𝟐𝝏𝟐𝑴𝟎

𝝏𝒚𝟐𝝏𝟐𝑴𝟎

𝝏𝒙𝝏𝒚�

The elements of arrays [C] and [D] are:

𝒄𝒎𝟏 = �𝟐∆𝒊𝟑𝒉𝒊𝒑𝒎

𝑵

𝒊=𝟏

; 𝒄𝒎𝟐 = �𝟐∆𝒊𝟑𝒌𝒊𝒑𝒎

𝑵

𝒊=𝟏

𝒄𝒎𝟑 = �𝒉𝒊𝟐

∆𝒊𝟑𝒑𝒎

𝑵

𝒊=𝟏

; 𝒄𝒎𝟒 = �𝒌𝒊𝟐

∆𝒊𝟑𝒑𝒎

𝑵

𝒊=𝟏

𝒄𝒎𝟓 = �𝟐∆𝒊𝟑𝒉𝒊𝒌𝒊𝒑𝒎

𝑵

𝒊=𝟏

; 𝒎 = 𝟏, … ,𝟓

with

𝒑𝟏 = 𝒉𝒊 ; 𝒑𝟐 = 𝒌𝒊 ; 𝒑𝟑 =𝒉𝒊𝟐

𝟐 ; 𝒑𝟒 =

𝒌𝒊𝟐

𝟐 ; 𝒑𝟓 = 𝒉𝒊𝒌𝒊

and

𝒅𝟏 = −�𝑮𝒉𝒊

𝑵

𝒊=𝟏

; 𝒅𝟐 = −�𝑮𝒌𝒊

𝑵

𝒊=𝟏

; 𝒅𝟑 = −�𝑮𝒉𝒊𝟐

𝟐

𝑵

𝒊=𝟏

; 𝒅𝟒 = −�𝑮𝒌𝒊𝟐

𝟐

𝑵

𝒊=𝟏

; 𝒅𝟓 = −�𝑮𝒉𝒊𝒌𝒊

𝑵

𝒊=𝟏

21

with

𝑮 =(𝑴𝟎 −𝑴𝒊)

∆𝒊𝟑

Solving Eq. (3-11) second order derivatives are computed and exchange field is written as:

𝑯𝒆𝒙 = 𝒌𝒆𝒙 �𝝏𝟐𝑴𝟎𝝏𝒙𝟐

+ 𝝏𝟐𝑴𝟎𝝏𝒚𝟐

� , 𝒌𝒆𝒙 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 𝑥1𝑥2 plane, the domain is decomposed into a grid

of 𝑵𝒄𝒆𝒍𝒍 macrocells, representing the spatial period, in turn discretized into a mesh of T

hexahedra. In each of the (𝑵𝒄𝒆𝒍𝒍.𝑻) elements the magnetization is assumed to be uniform.

The magnetostatic field 𝑯𝒎 in the element centered in 𝒓𝟎, located in a macrocell with grid

location (𝒊, 𝒋) and barycentre 𝑶𝟎 (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 𝑵𝑺𝑹 = (𝟐𝝀 + 𝟏)𝟐

of adjacent macrocells, is the numerical solution of the integral equation obtained from Green’s

theorem.

23

𝑯𝒎𝑺𝑹(𝒓𝟎) = 𝟏

𝟒𝝅∑ ∑ ∑ ∫ 𝑴(𝒓𝒆) 𝝏𝛀𝒆

𝑻𝒆=𝟏

𝒋+𝝀𝒋−𝝀

𝒊+𝝀𝒊−𝝀 . 𝒏𝒆

(𝒓𝟎−𝒓𝒆)‖𝒓𝟎−𝒓𝒆‖𝟑

𝒅𝒔 Eq. 3-12

where 𝝀 is the number of the adjacent macrocell layers (Fig. 3-2), and 𝝏𝛀𝒆 is the surface of the

𝑒-th hexahedron with normal unit vector 𝒏𝒆. The second term describes the interactions with the

remaining macrocells and it is calculated as:

𝑯𝒎𝑳𝑹(𝒓𝟎) = − 𝟏

𝟒𝝅∑ ∑ ∑ 𝑨𝒇,𝒋

𝒌 (𝑶𝟎)𝒋𝒌=−𝒋

𝒑𝒋=𝟎

𝑵𝒄𝒆𝒍𝒍−𝑵𝑺𝑹𝒇=𝟏 𝛁𝑹𝒋𝒌(𝒓𝟎 − 𝑶𝟎) Eq. 3-13

where 𝑹𝒋𝒌(𝒘) = 𝒓𝒋𝑷𝒋|𝒌|(𝒄𝒐𝒔𝜽)𝒆−𝒊𝒌𝝋/(𝒋 + 𝒌)!, having introduced a local spherical frame

𝒘 = (𝒓,𝜽,𝝋) centered in 𝑶𝟎. 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

𝑨𝒇,𝒋𝒌 (𝑶𝟎) is written as:

𝑨𝒇,𝒋𝒌 (𝑶𝟎) = ∑ ∑ 𝑸𝒋,𝒏

𝒎,𝒌�𝑶𝒇 − 𝑶𝟎�𝑴𝒏𝒎(𝑶𝒇)𝒏

𝒎=−𝒏𝒑𝒏=𝟎 Eq. 3-14

Where 𝑴𝒏𝒎(𝑶𝒇) is one of the (𝒑 + 𝟏)𝟐 multipole moments defined on the macrocell centered in

𝑶𝒇,

𝑴𝒏𝒎�𝑶𝒇� = ∑ ∫ 𝑴(𝒓𝒆)𝛀𝒆

.𝛁𝑹𝒏𝒎�𝒓𝒆 − 𝑶𝒇�𝒅𝒗𝑻𝒆=𝟏 Eq. 3-15

The operator 𝑸𝒋,𝒏𝒎,𝒌 transposes the multipole moments from𝑶𝒇 to 𝑶𝟎.

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

𝑴𝒔 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 𝑯𝒃𝒊𝒂𝒔 plus a uniform excitation pulse applied

orthogonally to 𝑯𝒃𝒊𝒂𝒔, 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 × 𝑯𝒆𝒇𝒇� 𝑴𝒔𝟐⁄ ) 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 9𝛾𝐺,

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.

𝑯𝒑𝒖𝒍𝒔𝒆(𝒕) = 𝒂𝒆𝒙𝒑 �−�𝒕−𝒕𝟎𝝉�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 ⟨𝑴𝒖⟩

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 (𝑯𝒃𝒊𝒂𝒔) 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,

considering a bias field of 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏 .

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

considering a bias field of 𝟏𝟓𝟎 𝒌𝑨𝒎−𝟏 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

(𝑁𝑖80𝐹𝑒20) 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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Master Thesis

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