UNIVERSITY of CALIFORNIA, SAN DIEGO Efficient Micromagnetics For
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UNIVERSITY OF CALIFORNIA, SAN DIEGO Efficient micromagnetics for magnetic storage devices A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical Engineering (Nanoscale Devices and Systems) by Marco Antonio Escobar Acevedo Committee in charge: Professor Vitaliy Lomakin, Chair Professor Raymond A. de Callafon Professor Yeshaiahu Fainman Professor Eric E. Fullerton Professor Oleg Shpyrko 2016 Copyright Marco Antonio Escobar Acevedo, 2016 All rights reserved. The dissertation of Marco Antonio Escobar Acevedo is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To my much beloved family. iv EPIGRAPH A method is more important than a discovery, since the right method will lead to new and even more important discoveries. — Lev Landau v TABLE OF CONTENTS Signature Page.................................. iii Dedication..................................... iv Epigraph.....................................v Table of Contents................................. vi List of Figures.................................. viii List of Tables...................................x Acknowledgements................................ xi Vita........................................ xiii Abstract of the Dissertation........................... xv 1 Introduction..................................1 2 Theoretical Framework............................5 2.1 Introduction...............................5 2.2 The micromagnetic model.......................5 2.3 Finite element formulation.......................9 2.3.1 External field.......................... 10 2.3.2 Anisotropy field......................... 10 2.3.3 Exchange field.......................... 10 2.3.4 Magnetostatic field....................... 11 2.4 Mathematical preliminaries...................... 11 2.4.1 Solution of non-linear equations................ 12 2.4.2 Solution of ordinary differential equations.......... 13 2.5 Computational details......................... 14 3 Magnetization Dynamics........................... 17 3.1 Numerical solution of the Landau-Lifshitz-Gilbert equation..... 17 3.2 Perpendicular magnetic recording................... 19 3.2.1 Discretization artifacts..................... 20 3.2.2 Effects caused by the shield.................. 24 3.2.3 Full scale model......................... 24 3.2.4 Dynamics on the data layer.................. 27 3.2.5 Degaussing........................... 28 3.3 Discussion................................ 40 vi 4 Energy Minimization............................. 42 4.1 Introduction............................... 42 4.2 State of the art............................. 43 4.3 Solution schemes............................ 44 4.3.1 Newton’s method........................ 44 4.3.2 Brown’s equation........................ 47 4.3.3 Minimization via an ODE................... 47 4.3.4 High damping LLG....................... 47 4.4 Performance comparison........................ 48 4.5 Permalloy................................ 51 4.6 Spin valve................................ 53 4.7 Discussion................................ 54 5 Nudged Elastic Band Method........................ 57 5.1 Introduction............................... 57 5.2 Thermal stability............................ 58 5.3 Generic nudged elastic band method................. 60 5.4 Modifications for micromagnetics................... 60 5.4.1 Magnetic nanowire....................... 65 5.4.2 Magnetic random access memory............... 67 5.4.3 MRAM array.......................... 68 5.5 Nanodots................................ 72 5.6 Nanorings................................ 75 5.7 Discussion................................ 76 6 Summary................................... 79 Bibliography................................... 81 vii LIST OF FIGURES Figure 3.1: To solve the LLG equation using FastMag, several factors that affect the speed and accuracy must be considered......... 19 Figure 3.2: Magnetic write head schematic. A magnetic write head is typi- cally comprised of a return pole, a yoke, a main pole, a writing pole tip, a WAS, a SUL and an helical coil............ 21 Figure 3.3: Time dependence of the z-component of the magnetostatic field 12 nm under the tip for different meshes.............. 23 Figure 3.4: Effects caused by the WAS..................... 25 Figure 3.5: Zoom-in on the tapered recording pole, the minimum distance between the main pole and the WAS is 50 nm........... 26 Figure 3.6: A switching rate of 1.6 Gbit/s with a rise time of 100 ps. The maximal current is 100 mA..................... 28 Figure 3.7: Surface plots of the z-component of the magnetostatic field (µ0Hz [T]) in the data layer for several consecutive time instances for the baseline signal when switching. The shield induces magnetostatic fields near the pole tip region, such fields can lead to WATER.. 29 Figure 3.8: Response to the baseline signal and a zero-current time interval. Averaged field (< µ0Hz >) in the observer box as described in the text............................... 31 Figure 3.9: Response of the PMR head to a single frecuency degaussing signal. 32 Figure 3.10: Response of the PMR head using the signals with increasing frequency and linear decay..................... 34 Figure 3.11: Response of the PMR head using the signals with increasing frequency and exponential decay.................. 35 Figure 3.12: Response of the PMR head using the signals with decreasing frequency and linear envelope decay................ 37 Figure 3.13: Response of the PMR head using the signals with decreasing frequency and exponential envelope decay............. 38 Figure 3.14: Remanent state of the full head, and the recording tip...... 39 Figure 4.1: Minimum energy state for a cubic particle............. 49 Figure 4.2: Hysteresis loop calculation..................... 52 Figure 4.3: Schematic representation of an AP-pinned spin valve, only the magnetic materials are presented. In aquamarina the FL, in pink the RL, in yellow the Pin, and in green the AF layer.... 54 Figure 4.4: Magnetoresistance (MR) for an AP-pinned spin valve as a func- tion of the external field....................... 55 Figure 5.1: The continuous evolution of the magnetization between an initial state and a final state defines an energy path........... 61 viii Figure 5.2: Minimum energy path and energy barrier calculations for a CoCr nanowire............................... 66 Figure 5.3: The energy barrier convergence for a nanowire with different tangent orders, suscripts 1 and 2, and different methods to compute the Jacobian-matrix vector products, either using DQ or analytical Jacobian-matrix (Jv) times vector.......... 67 Figure 5.4: Minimum energy path for an MRAM............... 69 Figure 5.5: The energy barrier convergence for an MRAM with different tangent orders, suscripts 1 and 2, and different methods to compute the Jacobian-matrix vector products, either using DQ or analytical Jacobian-matrix (Jv) times vector.......... 69 Figure 5.6: Image in the MEP of an MRAM.................. 69 Figure 5.7: Minimum energy path for an array of MRAMs.......... 70 Figure 5.8: The energy barrier convergence for an array of MRAMs with different tangent orders, suscripts 1 and 2, and different methods to compute the Jacobian-matrix vector products, either using DQ or analytical Jacobian-matrix (Jv) times vector....... 70 Figure 5.9: Image in the MEP of an array of MRAMs............. 71 Figure 5.10: Comparison of the exmperimental measurement, macrospin cal- culation and NEB calculation for a 35 nm nanodot with N layers. 74 Figure 5.11: Comparison of the macrospin and NEB thermal stability of varying diameters nanodots..................... 74 Figure 5.12: Nanoring with AFC across the interface (point A, in yellow). Only one DW can exist....................... 75 Figure 5.13: Energy landscapes for different MFNRs as a function of DW location for different AFCs, MS, anisotropy and a deffect.... 77 ix LIST OF TABLES Table 3.1: Mesh discretization, small PMR head model............ 21 Table 3.2: Material properties for the full scale PMR head.......... 27 Table 3.3: Mesh details for the full scale PMR head model.......... 27 Table 4.1: Static calculations: Newton’s method............... 50 Table 4.2: Static calculations: Alternative Methods............. 50 Table 4.3: Hysteresis loop calculation: Newton’s method.......... 52 Table 4.4: Hysteresis loop calculation: Alternative methods......... 53 Table 4.5: MR calculation: AP–pinned spin valve............... 55 Table 5.1: NEB method performance metrics: Nanowire........... 65 Table 5.2: NEB method performance metrics: MRAM............ 68 Table 5.3: NEB method performance metrics: Array............. 71 x ACKNOWLEDGEMENTS I would like to express my gratitude to my adviser Professor Vitaliy Lo- makin for the opportunity to work in his research group, and to learn the field of micromagnetics. I also would like to thank my committee members: Professor Yesha- iahu Fainman, Professor Eric E. Fullerton, Professor Oleg Shpyrko and Professor Raymond A. de Callafon for their involvement in my academic development. I would like to thank all my peers at the Computational Electromagnetics and Micromagnetics Laboratory who helped me understand many of the physical phenomena in magnetism and with whom I have collaborated closely, and with whom I had an enduring friendship over the last few years. I thank the ECE Department of the University of California San Diego for the departmental fellowship