Micromagnetics is a field of study considering the magnetization behavior in magnetic materials and devices accounting for a wide set of interactions and describing the magnetization phenomena from the atomistic scale to several hundreds of microns. Micromagnetic simulations are essential in understanding the behavior of many magnetic systems. Modeling complex structures can require a significant computational time and in some cases, the system complexity can make simulations prohibitively long or require a prohibitively large memory.

In this thesis, we present a set of methods and their implementations that resulted in high-performance numerical micromagnetic tools for modeling highly complex magnetic materials and devices. The focus of the dissertation is on solving Landau-Lifshitz-Gilbert (LLG) equation efficiently, both with numerical methods and advanced hardware acceleration.

To understand the numerical problem to be solved, the introduction Chapter 1 addresses the LLG equation and the governing interactions involved as well as numerical modeling basics on the Finite Difference Method (FDM) and the Finite Element Method (FEM). Chapter 1 also presents a versatile micromagnetic framework, referred to as FastMag, which implements some of these methods.

Chapter 2 provides a detailed description of computing based on Graphics Processing Units (GPUs). The history of GPU programming model and the programming tips serve as the basis for understanding parallel computing on GPUs. It presents applications of GPUs on various platforms to demonstrate the current mainstream usage of GPUs and their promising future development direction. Chapter 2 also summarizes applications of GPUs in micromagnetics.

Chapters 3 and 4 address two essential aspects of micromagnetic solvers: fast algorithms for computing the key interaction components and efficient time integration methods. Chapter 3 introduces a non-uniform Fourier transform (NUFFT) method, a scalar potential method, and sparse matrix-vector multiplication (SpMVM) algorithms implemented on GPUs to accelerate the magnetostatic and exchange interactions. Chapter 4 addresses basics of the time integration methods used in FastMag as well as a preconditioner to further accelerate the time integration process.

Chapter 5 presents a numerical model for the current state-of-art magnetic recording system using advanced algorithms and GPU implementations described in Chapters 2-4.

Magnetic materials are vital components of many existing and future applications, from data storage and spin-logic devices, to Terahertz sensors and artificial synapses from neuromorphing computing. Driven by the need to faster responses and high-density storage, the focus of this work is the modeling of thermal and optical excitation of magnetic materials by an external laser source.

Many models focus on the use of fields or current as the primary driving force behind the change in magnetization and the models, without taking in account the optical contribution of the light, which has been shown to produce changes in magnetizing on a faster timescale than the ones observed with the use of either current or field.

Moreover, granular media are usually modelled using simplistic finite difference approach or numerically intensive finite-elements approach to model every grain. These approach leads to either an unrealistic description of the media (finite-difference) or to an over-sampling of the nodes of the problem, increasing significantly the computational time required to run the simulations.

This dissertation improves upon the state of the art of micromagnetic modelling by introducing a Voronoi tessellation model to simulate realistic granular structures at elevated temperature for high anisotropy materials. This approach considers the geometry of the grains for compute the far-field contribution. This approach has been proven effective in modeling realistic media for heat assisted magnetic recording and perpendicular media in general. The model presented it also introduces in the dynamic of the magnetizing the optical contribution and helps to describes complex phenomena like the ultrafast-demagnetization and the helicity dependent optical reversal of magnetic material subjected to an external optical source.

While the model provides a qualitative interpretation of the experiments, additional data is required to evaluate the quantitative contribution of the optical excitation and the correctness of the thermal fluctuations.

This dissertation is structured as follow. In Chapter 1 and 2 introduce key concepts of magnetism and the basic Micromagnetic model that is going to be used as the basis for the numerical simulations.

In Chapter 3 I introduce a micromagnetic code based on Voronoi tessellation and the non-uniform fast Fourier transform (NUFFT) method. The code is capable of efficiently and accurately simulating magnetization dynamics in large and structurally complex granular systems, such as multilayer granular media used for perpendicular magnetic recording, bit patterned media, granular nanowires, and read heads. In these systems the granular microstructure and distributions in grain and interface properties play an important role in device performance. The presented Voronoi simulator allows comprehensive studies to be performed as it accounts for the detailed granular microstructure and distributions that characterize true systems. Simulation time is greatly reduced by a non-uniform fast Fourier transform algorithm and implementation on graphics processing units (GPUs). Simulations of conventional magnetic recording, heat-assisted magnetization reversal, domain wall dynamics in granular nanowires, and particulate tape recording are presented.

Chapter 4 explores the generation of electrical field signals in the terahertz frequency (THz) range using antiferromagnets (AFM). Using micromagnetic model simulation, we investigated a potential mechanism for laser-induced THz signals in the AFM phase of FeRh/Pt bilayer films. In the simulations, the FeRh film is modelled as two Fe-sublattices coupled via intra-lattice exchange field and subjected to a sub-picosecond thermal pulse. Our simulation exposes a partial canting between the magnetizations of two Fe-sublattices, within the first picosecond after the excitation. This short-lived state relaxes abruptly into the initial AFM phase, injecting a spin current into the Pt layer via spin pumping, which will eventually be converted into charge current oscillating at THz frequency.

Chapters 5 and 6 discuss the phenomenon of all-optical switching of the magnetization in magnetic nanostructures. While all-optical switching of the magnetization in magnetic nanostructures by femtosecond circularly polarized laser pulses without an external magnetic field has been demonstrated in several systems, a theoretical framework that convincingly explain the phenomenon is still missing. In Chapter 5 we propose a theory where the ferromagnetic macrospin ground state is optically excited by the circularly polarized light to a spin reversed state, which is then “Coulomb collapsed” to the magnetization reversed ground state. The optical excitation lasts for the duration of the laser pulse and the system relaxes at a fast rate due to the electron-electron interaction. In Chapter 6 we present a computational model based on this theory. We construct a three-state model for the magnetization dynamics, the Landau- Lifshitz-Lambda (LLL) model, as an ensemble of such states to account for the temperature effects. After the optical excitation lapses, the LLL model reduces to the Landau-Lifshitz-Bloch formulation, allowing to consider the magnetization relaxation dynamics at elevated temperatures. We apply the theory to simulate AOS in FePt films subject to multiple femtosecond circular polarized laser pulses. The simulation results demonstrate characteristic AOS features and agree with recent experiments.

Chapter 7 identifies problems in the performance of the established stochastic model in micromagnetics in modeling the thermal fluctuations of longitudinal and transverse components of the magnetization at elevated temperature. A correct estimation of the thermal fluctuation is paramount to develop multiscale atomistic-micromagnetic models. The chapter presents a consistent solution for the diffusion coefficients that satisfy the corresponding Fokker-Planck equation and provide the correct equilibrium magnetization at elevated temperature.

Magnetic materials and nanoscale devices are remaining at the center of data storage technologies, from well-known magnetic hard drives to future spintronic devices such as magnetoresistive or racetrack memory technologies. Because of the constantly increasing demand for more storage space and performance, the focus has shifted from magnetic field-based applications to spin current-based applications. Spintronic devices are taking advantage of interactions between magnetic materials and electric currents. Spin transfer torque magnetic random access memory devices utilizing perpendicular magnetic tunnel junctions are the most promising candidates for future memory applications. This thesis discusses some of the challenges faced when studying interactions between electrical spin polarized currents and magnetic tunnel junctions for magnetic random access memory application. It starts with a short introduction to micromagnetics and the description of the main magnetic interactions. It then focuses specifically on the study of the perpendicular magnetic tunnel junction and its important properties, such as size dependence, thermal stability, critical switching current density and overall device efficiency. The next chapters are dedicated to magnetic tunnel junctions’ optimal design study, including a study of composite free-layers, surface defects and roughness effects, granularity, and parameters distributions. The study of switching probability or write error rates under non-zero temperature is also present and briefly discussed. This work also includes a separate study of novel spin transfer torque nano oscillator based on synthetic antiferromagnets. All these chapters contain the numerical results and simulations performed by using the FastMag finite element micromagnetic simulation software, developed by the members of Professor Vitaliy Lomakin’s research group. Some general ideas and details on the computational methods are also provided with the intention to help understand better the engineering and analytical aspects of the devices based on magnetic tunnel junctions.

Nanoscale magnetic materials and devices are at the heart of memory and recording technologies ranging from magnetic hard drives to spintronic devices, such as magnetic random access memory (MRAM) and spin transfer torque oscillators. Advanced development of these technologies requires comprehensive computational tools. This dissertation presents a theoretical and micromagnetic study of challenges faced when considering interactions between applied fields and spin-polarized currents with nanomagnetic materials. The study is about solving the generally non-linear Landau-Lifshitz-Gilbert (LLG) equation using its linearized version. The approaches include using a linearized eigenvalue framework, solving a source-excited linearized LLG equation, and using a harmonic balance approach for the study of the higher-harmonic generation in weakly-nonlinear magnetization dynamics problems. The dissertation starts with an introduction to micromagnetics and modeling of spin-torque-driven devices. The following chapters present the eigenvalue based micromagnetic framework for spin-torque-driven devices. It presents an analysis related to the MRAM switching properties, including the critical current, switching time, and magnetization time evolution. It also introduces an optimization approach based on the eigenvalue analysis to reduce the critical current in MRAM. It then extends the eigenvalue analysis to the Fokker-Planck equation framework to study of non-switching probability, namely write error rate, under finite temperature. Next, the dissertation presents a solver for the linearized LLG equation with under time-harmonic applied fields, describing its formulation, numerical implementation, results, and analysis. The linearized LLG equation solver is finally extended to create a harmonic balance solver, which represents the solution as a set of multiple frequency components with an iterative process, which allows computing the excitation coefficients of these components. All the codes are developed in the finite element method framework, which is flexible in handling complex materials and devices, and it is integrated with the high-performance micromagnetic FastMag framework.

The development of the spintronic-based data storage devices such as spin transfer torque magnetoresistive random access memory (STT-MRAM) is being driven by the surging data consumption and demand for faster data processing. The advantages of nonvolatility, higher data processing speed, lower power consumption and scalability hold the promise of the popularity of STT-MRAM in the future, of which spin transfer torque (STT) effect is the key. This thesis develops a spin diffusion model to study the spin dynamics in nanomagnetic structures and the corresponding STT effect acting on local magnetic moments. Chapter 2 provides an introduction to micromagnetic modeling, dominant magnetic interactions, and domain walls. Chapter 3 presents spin diffusion model, in which two approaches are discussed for handling the boundary conditions and we demonstrate their good performance in solving spin diffusion equation in finite element models. Chapter 3 also shows solutions for the spin accumulation in multi-layered magnetic structures at equilibrium and in dynamics. It also studies the case of spin transfer torque in magnetic nanostructures with the Néel wall, comparing it to the simplified Zhang & Li model. At the end of the chapter 3 we simulate the magnetization dynamics under STT effect using the FastMag micromagnetic simulation software coupled with the spin diffusion model.

Magnetic materials and nanoscale devices are at the heart of data storage technologies, from decades-old hard drives to future technologies such as magnetoresistive random access memory or racetrack memory. Driven by the constantly increasing demand for more storage space and performances, the focus has shifted from magnetic field-based applications to spin current-based applications. This thesis discusses some of the challenges faced when studying interactions between electrical currents and magnetic materials. It starts with an introduction to micromagnetics and the description of the main magnetic interactions. It then focuses on racetrack memories and details the advantages of antiferromagnetically coupled nanowires. The next chapters are dedicated to spin-valves, including a study of composite free-layers, the influence of Dzialoshinskii-Moriya interactions on the performance of memory cells, and the forward flux sampling method to predict the switching probability under non-zero temperature. All of these chapters contain work done on the FastMag micromagnetic simulation software, developed in-house. Some general ideas and details on the implementation of magnetic fields and methods for simulation are also provided.

High-contrast imaging Fiber Bundles (FB) have been used in recent years to integrate high resolution, compact and wide field-of-view imagers into small volumes. Imaging fiber bundles are composed of arrays of individual high-contrast optical fibers that are fused together to form a uniform and reliable image-transfer medium. Monocentric lenses have been designed to demonstrate superior diffraction-limited performance in the visible spectrum with few optical elements which enables integration of robust imagers into a small volume. However the curved image plane formed by these lenses can not be detected with conventional flat image sensors. Additional optical elements can be introduced to flatten the image plane but it would lead to the complexity of the system design and loss of compactness. One can instead take advantage of imaging fiber bundles to overcome this limitation. FBs can be milled and polished to form a curved input facet and a flat output facet to map the spherical image plane onto the flat image sensor plane. Proper understanding of imaging fiber bundles performance is therefore crucial for designing a low cross-talk image-transfer medium.

In this dissertation the Rigorous Coupled Wave Analysis (RCWA) method is used for 2D (1D+propagation) modeling of deep periodic dielectric gratings (straight imaging fiber bundles) and a numerical method is presented to efficiently calculate the electromagnetic beam transmission through the arrays of fibers. The Scanning Electron Microscope (SEM) image of the straight FB cross-section is captured and processed to extract the actual core boundaries. The measured core boundaries are then used for 3D (2D+propagation) modal analysis of fiber bundles. The effect of irregularity is also investigated in both 2D and 3D models and it is shown to improve the light confinement and image transfer fidelity in optical fiber bundles. The straight fiber bundles with various pitches are then experimentally characterized to measure the performance under different illumination conditions. A performance metric is used to quantify cross-talk in fiber bundles and demonstrate the poor performance in the fibers with core pitches of only a few wavelengths of the guided light. The experimental measurements are then compared to numerical modeling results and are shown to be in good agreement. The numerical and experimental modeling is followed by an experimental measurement of a fiber-coupled imager (FCI) in order to restore the lost resolution due to problems associated with fiber-sensor coupling. The Point Spread Function (PSF) of the strongly shift-variant FCI is first measured as a function of input location with sub pixel spatial resolution. Various computational image reconstruction methods are then used to recover the captured image resolution to that of the fiber bundle pitch.

This work is concerned with the important connection that exists between micromagnetism and high frequency electromagnetism. As micromagnetic solvers have become important tools in the study and engineering of magnetic devices, and as these devices are increasingly operated at high frequencies, it is important to understand both how micromagnetic simulations can be used in the modeling of magnetic materials for high-frequency applications and how micromagnetic models can be impacted by high-frequency electromagnetic effects.

After an introduction to the theory behind micromagnetism and micromagnetic solvers, the dissertation is divided into two main themes. The first one is the modeling and characterization through micromagnetic simulations of ferromagnetic materials for high frequency applications. A particular class of materials, namely ferromagnetic nano-granular materials, are studied since they have important technological applications as soft magnetic materials which also exhibit high saturation magnetizations and low electrical resistivities. The extraction of properties such as the hysteresis loop and frequency dependent permeability tensor from micromagnetic simulations is examined. Also, the anisotropy averaging mechanism responsible for the soft magnetic properties of this class of materials is studied and some theoretical results are obtained and presented, such as a derivation of the residual exchange energy between groups of exchange coupled nano-particles and a generalized Stoner-Wohlfarth hysteresis model which accounts for exchange interactions between ferromagnetic particles with random uniaxial anisotropies.

The second theme is the modeling of eddy currents in micromagnetic simulations. Eddy currents, also known as Foucault currents, arise in conductive materials due to rapid variations of the magnetic field and magnetization. A method coupling the Landau-Lifshitz-Gilbert equation of micromagnetics with the magnetoquasistatic Maxwell equations is presented. Based on this method, two coupled micromagnetic-electromagnetic solvers are presented, one based on an integral equation formulation, the other based on the finite element method. A test problem with a known analytical solution is suggested and used to validate the implemented solvers. Also, results concerning the bounds of validity for the magnetostatic approximation to theMaxwell equations, in which case eddy currents are neglected, as well as for the magnetoquasistatic Maxwell equations, where eddy currents are accounted for but electromagnetic wave propagation is neglected, are given. It is found that the magnetoquasistatic approximation is excellent for the vast majority of micromagnetic simulations that are executed today.