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University of Southampton Research Repository Eprints Soton University of Southampton Research Repository ePrints Soton Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk UNIVERSITY OF SOUTHAMPTON Faculty of Engineering and the Environment Computational Engineering and Design Group Computational Methods in Micromagnetics by Dmitri Chernyshenko Thesis for the degree of Doctor of Philosophy June 2016 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING AND THE ENVIRONMENT Doctor of Philosophy COMPUTATIONAL METHODS IN MICROMAGNETICS by Dmitri Chernyshenko With the continued growth of computational power, computational modelling has become an increasingly important part of modern science. The field of micromagnetism has benefited from the increase of computational power, leading in recent decades to the development of many important micromagnetic methods. This thesis aims to address some computational challenges relevant to the field of micromagnetism today. The computation of the demagnetising field is often the most time-consuming part of a micro- magnetic simulation. In the finite difference method, this computation is usually done using the Fourier transform method, in which the demagnetising field is computed as the convolution of the magnetisation field with the demagnetising tensor. An analytical formula for the demagnetising tensor is available, however due to numerical cancellation errors it can only be applied for close distances between the interacting cells. For far distances between the interacting cells other ap- proaches, such as asymptotic expansion, have to be used. In this thesis, we present a new method to compute the demagnetising tensor by means of numerical integration. The method offers improved accuracy over existing methods for the intermediate range of distances. In the finite element method, the computation of the demagnetising field is commonly done using the hybrid FEM/BEM method. The fast multipole method offers potential theoretical advantages over the hybrid FEM/BEM method particularly for the current and future generations of computing hardware. In micromagnetics, it has been applied to compute the demagnetising field in the finite difference setting and to compute the magnetostatic interaction between nanoparticles, however no implementation of the fast multipole method in finite elements is yet available. As one of the steps towards it, in this thesis we develop a new formula for the energy of the magnetostatic interaction between linearly magnetized polyhedrons. This formula can be used to compute the direct interaction between finite element cells in the fast multipole method. Ferromagnetic resonance is a popular experimental technique for probing the dynamical properties of magnetic systems. We extend the eigenvalue method for the computation of resonance modes to the computation of the FMR spectrum, and apply it to compute ferromagnetic resonance for a proposed FMR standard reference problem. Contents 1 Introduction1 1.1 Thesis outline....................................2 1.2 Software.......................................2 2 Background3 2.1 Micromagnetics...................................3 2.1.1 Brown’s equations.............................4 2.1.2 Derivation of Brown’s equations......................5 2.2 Micromagnetic simulation.............................6 2.2.1 Finite Differences..............................6 2.2.2 Finite Elements...............................7 2.2.3 Effective field and the box method for finite elements...........7 2.3 Demagnetising field.................................8 2.3.1 Demagnetising field: overview.......................8 2.3.2 Demagnetising field: finite difference method............... 10 2.3.3 Demagnetising field: finite element method................ 12 2.3.4 Fast multipole method........................... 14 2.4 Time integration of the Landau-Lifshitz-Gilbert equation............. 16 2.4.1 The Jacobean of the Landau-Lifshitz-Gilbert equation.......... 17 2.5 Ferromagnetic resonance.............................. 18 2.5.1 Eigenvalue approach............................ 18 2.5.1.1 Linearized equation without damping.............. 19 2.5.1.2 Linearized equation with damping: perturbative analysis.... 20 2.5.1.3 Linearized equation with damping: numerical solution..... 20 2.5.1.4 Linearized equation with damping: spectrum computation... 20 3 Demagnetizing tensor in finite difference micromagnetics 22 3.1 Introduction..................................... 22 3.1.1 Micromagnetics............................... 22 3.1.2 Numerical accuracy of N(r) evaluation.................. 24 3.2 Method....................................... 24 3.2.1 Integrand.................................. 26 3.2.1.1 6d method............................ 26 iii 3.2.1.2 4d method............................ 26 3.2.2 Error estimation.............................. 27 3.3 Results........................................ 27 3.3.1 Overview.................................. 27 3.3.2 Sparse grid integration parameters and execution performance...... 29 3.3.3 Comparison of 4d and 6d integration.................... 29 3.3.4 Performance................................ 30 3.3.5 Single point floating precision....................... 31 3.3.6 Forward and backward Fast Fourier Transform.............. 32 3.3.7 Other high accuracy methods........................ 32 3.4 Summary...................................... 33 4 Computation of the magnetostatic interaction between linearly magnetized polyhe- drons 34 4.1 Introduction..................................... 34 4.2 Formulation of the problem............................. 36 4.3 Method....................................... 36 4.3.1 Auxiliary functions............................. 37 4.4 Analytical derivation — reduction to a surface integral.............. 38 4.5 Analytical result — integrand and the primitive terms Ik ............. 39 4.5.1 Evaluation of Ik .............................. 39 4.6 Numerical results.................................. 40 4.6.1 Test problem formulation.......................... 40 4.6.2 Test results................................. 41 4.7 Derivations..................................... 42 4.7.1 Derivation of I2 ............................... 42 4.7.2 Derivation of η1 .............................. 43 4.7.3 Derivation of λ1 .............................. 43 4.7.4 Vector calculus identities.......................... 44 4.8 Summary...................................... 44 5 Ferromagnetic resonance and normal modes 45 5.1 Introduction..................................... 45 5.2 Selection and definition of standard problem.................... 47 5.2.1 Problem definition............................. 47 5.2.2 Problem Selection............................. 48 5.2.3 Data Analysis................................ 49 5.2.3.1 Spectrum Sy(f) of spatially averaged magnetization...... 49 5.2.3.2 Local power spectrum and S~y(f) ................ 49 5.2.3.3 Phase information........................ 50 5.3 Results and discussion............................... 52 iv 5.3.1 Standard Problem Simulation Results................... 52 5.3.2 Eigenvalue method results......................... 53 5.3.3 Falsification Properties........................... 54 5.3.3.1 Damping Parameter....................... 56 5.3.3.2 Relaxation Time......................... 56 5.3.3.3 Relaxation Stage Perturbation Angle.............. 57 5.3.3.4 Spatial Discretization...................... 57 5.3.4 Comparison of Simulation Methodologies................. 59 5.4 Simulations without Demagnetization....................... 61 5.5 Nmag tolerances.................................. 61 5.6 Summary...................................... 62 6 Joule heating in nanowires 63 6.1 Introduction..................................... 63 6.2 Method....................................... 64 6.3 Simulation results.................................. 65 6.3.1 Case study 1: Uniform current density................... 65 6.3.2 Case study 2: Constrictions........................ 66 6.3.3 Case study 3: Nanowire with a notch on a silicon nitride substrate membrane 68 6.3.4 Case study 4: Nanowire with a notch on a silicon wafer substrate..... 71 6.3.5 Case study 5: Zigzag-shaped nanowire on a silicon wafer substrate.... 72 6.3.6 Case study 6: Straight nanowire on diamond substrate.......... 73 6.4 Analytical models.................................. 75 6.4.1 Model by You, Sung, and Joe for a nanowire on a (3d) substrate..... 75 6.4.2 Analytic expression for nanowire on a membrane (2d substrate)..... 78 6.5 Perpendicular nanowires.............................. 79 6.6 Summary...................................... 80 7 Summary and conclusions 82 v List of Figures 3.1 The relative error η in the computation of the demagnetizing tensor as a function of the distance between the interacting cuboids................... 27 3.2 Comparison of numerical
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