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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 Faculty of Engineering and the Environment Computer simulation studies of complex magnetic materials by Weiwei Wang Thesis for the degree of Doctor of Philosophy Supervisors: Prof. Hans Fangohr, Dr. Ian Hawke October 2015 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF ENGINEERING AND THE ENVIRONMENT Doctor of Philosophy COMPUTER SIMULATION STUDIES OF COMPLEX MAGNETIC MATERIALS by Weiwei Wang With the development of both computing power and software engineering, computer simulation of the micromagnetic model or atomistic spin model, has become an impor- tant tool for studying a wide range of different complex phenomena in magnetic ma- terials. Meanwhile, the rapid improvement of advanced measurement techniques has allowed the probing of ultrafast magnetization dynamics, as well as the magnetic phe- nomena involving charge current, heat and light. The simulation of magnetism is now moving towards a multiphysics method. Therefore, fast, user-friendly, and extensible codes with accurate algorithms are helpful in understanding the physics and designing novel magnetic devices on the nanoscale. In the preparation of this thesis we have developed Fidimag, which is a Python/C sim- ulation tool supporting both micromagnetic and atomistic spin models. The software has also been extended to support the Landau-Lifshitz-Baryakhtar (LLBar) equation. Using Fidimag, we have performed simulations to study the domain-wall motion and spin-wave decay with the LLBar equation. We also explain the exchange damping in the LLBar equation as the phenomenological nonlocal damping by linking it to spin pump- ing, therefore, LLBar equation can be considered as a phenomenological equation of the nonlocal damping. We studied magnon-induced domain-wall motion in the presence of Dzyaloshinskii- Moriya interaction (DMI) numerically and theoretically. We find that the presence of DMI and easy-plane anisotropy can drive the domain wall very effectively and that the domain-wall velocity depends on the sign of DMI constant. While the negative velocity is considered as a result of angular momentum conservation, we attribute this fast domain-wall motion to linear momentum transfer between magnons and the domain wall. By numerically solving the Landau-Lifshitz-Gilbert equation with a classical spin model on a two-dimensional system, we show that both magnetic skyrmions and skyrmion lattices can be moved with microwave magnetic fields. The mechanism is enabled by breaking the axial symmetry of the skyrmion with a static in-plane external field. Contents Declaration of Authorship xi List of publications xii Acknowledgements xiii 1 Introduction1 1.1 Background...................................1 1.2 Structure of this thesis.............................4 2 Micromagnetic and Atomistic modeling5 2.1 Magnetic moments...............................6 2.2 Equation of Motion...............................6 2.3 Interactions...................................7 2.3.1 Exchange interaction..........................8 2.3.2 Dzyaloshinskii-Moriya interaction (DMI)..............8 2.3.3 Dipolar interaction........................... 10 2.3.4 Anisotropy................................ 10 2.3.5 Zeeman energy............................. 11 2.4 Atomistic spin model.............................. 11 2.5 Micromagnetics................................. 13 2.5.1 Exchange energy............................ 14 2.5.2 Magnetostatic energy......................... 14 2.5.3 Dzyaloshinskii-Moriya Energy.................... 15 2.6 Landau-Lifshitz-Gilbert (LLG) equation................... 16 2.6.1 Spherical form of LLG equation.................... 17 2.6.2 Spin transfer torque.......................... 18 2.6.3 Nonlocal spin transfer torque..................... 19 2.7 Finmag and Fidimag.............................. 20 3 Fidimag 21 3.1 Introduction................................... 21 3.2 Fast Summation of Dipolar interactions................... 22 3.2.1 Fast Summation............................. 23 3.2.2 Dipolar interaction in triangular lattice............... 23 3.3 Landau-Lifshitz-Baryakhtar equation.................... 24 3.4 Eigenvalue Method............................... 25 3.5 Verification I................................... 27 v vi CONTENTS 3.5.1 A magnetic moment under an external magnetic field....... 27 3.5.2 Domain-wall motion under charge currents............. 27 3.5.3 Magnetic skyrmions.......................... 30 3.5.4 Normal modes of a spin chain.................... 32 3.6 Stochastic LLG equation............................ 34 3.7 Verification II.................................. 35 3.7.1 A magnetic moment.......................... 35 3.7.2 Equilibrium distribution of a nanoparticle............. 36 3.7.3 Magnon temperature.......................... 36 4 Phenomenological description of nonlocal damping 39 4.1 Introduction................................... 39 4.2 Basic equations................................. 41 4.3 Spin-wave decay................................. 45 4.4 Domain-wall motion.............................. 48 4.4.1 Parallel relaxation............................ 51 4.4.2 Nonlocal damping........................... 53 4.5 Ferromagnetic resonance (FMR)........................ 54 4.6 Summary..................................... 56 5 Magnon-driven domain-wall motion with Dzyaloshinskii-Moriya interaction 57 5.1 Introduction................................... 57 5.2 The system.................................... 58 5.3 Domain-wall profile and Spin-wave excitation............... 59 5.4 Domain-wall motion.............................. 61 5.5 Two types of domain walls........................... 65 5.6 Summary..................................... 67 6 Driving magnetic skyrmions with microwave fields 69 6.1 Introduction................................... 69 6.2 The system and asymmetric skyrmions................... 70 6.3 Spin waves modes................................ 72 6.4 Skyrmion motion................................ 73 6.5 Summary..................................... 77 7 Conclusion and Outlook 79 7.1 Conclusion.................................... 79 7.2 Outlook...................................... 80 A 81 A.1 Section A..................................... 81 A.2 Section B..................................... 82 A.3 Section C..................................... 82 A.4 Section D..................................... 83 B Using discontinuous Galerkin Methods 85 B.1 Introduction................................... 85 B.2 Laplace operator using CG method...................... 85 CONTENTS vii B.3 Background................................... 86 B.3.1 H(div) space............................... 86 B.3.2 BDM space............................... 87 B.3.3 Piola mapping.............................. 87 B.4 Using BDM space................................ 88 B.5 MFMFE...................................... 89 B.6 A Test....................................... 89 C Treecode for Boundary Element Method 91 C.1 General problem................................ 91 C.2 Treecode..................................... 91 C.2.0.1 Particle-cluster interactions................. 92 C.2.0.2 Recurrence relations of Taylor coefficients........ 93 C.2.0.3 Implementation of Treecode................ 94 C.2.1 Combine Treecode with FEM/BEM................. 95 C.3 Numerical Tests................................. 96 C.4 Improvement.................................. 97 C.5 Summary..................................... 99 Bibliography 101 List of Figures 2.1 Three heat reservoirs (spin, lattice and electron) model........... 11 3.1 Structure of Fidimag............................... 21 3.2 Triangular lattice embedded in rectangular mesh.............. 24 3.3 The procession motion of a magnetic moment................ 28 3.4 Domain-wall motion driven by spin transfer torque............. 29 3.5 Two types of skymrions............................. 30 3.6 Belavin-Polyakov skyrmion........................... 32 3.7 Normal modes of a spin chain......................... 33 3.8 Average magnetization of a magnetic moment................ 35 3.9 Equilibrium distribution of a nanoparticle.................. 36 3.10 Equilibrium magnetization of a spin chain.................. 37 4.1 The spin-wave amplitude decay along a rod................. 46 4.2 Frequency-dependent spin-wave decay.................... 47 4.3 Domain-wall velocities with different susceptibilities............ 49 4.4 Magnetization length difference of a domain wall.............. 50 4.5 Domain-wall velocities for different exchange dampings.......... 53 4.6 Dynamic susceptibility of an elliptical-shaped nanomagnet........ 55 4.7 Fitted damping from dynamic susceptibility................. 55 5.1 Domain-wall profile in the presence