Thermal Stability of Metastable Magnetic Skyrmions (Springer Theses)

The energy cost associated with modern information technologies has been increasing exponentially over time, stimulating the search for alternative information storage and processing devices. Magnetic skyrmions are solitonic nanometer-scale quasiparticles whose unique topological properties can be thought of as that of a Mobius strip. Skyrmions are envisioned as information carriers in novel information processing and storage devices with low power consumption and high information density. As such, they could contribute to solving the energy challenge. In order to be used in applications, isolated skyrmions must be thermally stable at the scale of years. In this work, their stability is studied through two main approaches: the Kramers' method in the form of Langer's theory, and the forward flux sampling method. Good agreement is found between the two methods. We find that small skyrmions possess low internal energy barriers, but are stabilized by a large activation entropy. This is a direct consequence of the existence of stable modes of deformation of the skyrmion. Additionally, frustrated exchange that arises at some transition metal interfaces leads to new collapse paths in the form of the partial nucleation of the corresponding antiparticle, as merons and antimerons. Supervisors’ Foreword Abstract Publications related to the thesisL. Desplat, C. Vogler, J.-V. Kim, R. L. Stamps, and D. Suess. Path sampling for lifetimes of metastable magnetic skyrmions and direct comparison with Kramers’ method. Physical Review B, 101(6):060403(R), 2020.L. Desplat, J.-V. Kim, and R. L. Stamps. Paths to annihilation of first- and second-order (anti)skyrmions via (anti)meron nucleation on the frustrated square lattice. Physical Review B, 99(17):174409, 2019.L. Desplat, D. Suess, J.-V. Kim, and R. L. Stamps. Thermal stability of metastable magnetic skyrmions: Entropic narrowing and significance of internal eigenmodes. Physical Review B, 98:134407, 2018. Acknowledgements Contents Abbreviations 1 Introduction References 2 Topological Solitons in Magnetic Systems 2.1 The Classical Hamiltonian Model 2.1.1 Exchange Interactions 2.1.2 Zeeman Energy 2.1.3 Magnetic Anisotropy 2.1.4 Summary of Atomistic and Micromagnetics Quantities 2.2 Ferromagnetic Domain Walls 2.2.1 Characteristic Wall Profile 2.2.2 Effect of Chiral Couplings 2.2.3 Winding Number and Topology 2.3 Magnetic Skyrmions 2.3.1 General Characteristics 2.3.2 Topological Charge Definition on the Atomistic Spin Lattice 2.3.3 Chiral Skyrmions in Non-centrosymmetric Magnets 2.3.4 Skyrmions in Frustrated Magnets 2.3.5 Skyrmion Bubbles in Dipolar Magnets 2.4 Summary References 3 Langer's Theory and Application to Magnetic Spin Systems 3.1 Introduction 3.1.1 Preamble: On the Theory of Brownian Motion 3.1.2 Reaction Rate Theory 3.2 Langer's Theory in the IHD Regime 3.2.1 Framework for the Derivation of the Rate Expression 3.2.2 Ratio of Energy Curvatures: Interpretation and the Case of Goldstone Modes 3.3 Application to Magnetic Spin Systems 3.3.1 Construction of the Energy Hessian in Spherical Coordinates 3.3.2 Determination of the Dynamical Prefactor λ+ 3.4 The Geodesic Nudged Elastic Band Method 3.4.1 Projections onto the Tangent Space to the Manifold 3.4.2 The GNEB Method 3.4.3 Evaluation of the Tangent to the Path 3.4.4 Initialization of the Path Along Geodesics 3.4.5 Search for the First-Order Saddle Point: The CI-GNEB Scheme 3.4.6 Variable Spring Constants 3.5 Implementation of Langer's Theory on an Atomistic Spin Lattice 3.5.1 Atomistic Simulations and Search for Local Minima 3.5.2 Implementation of GNEB 3.5.3 Computation of the Rate Prefactor 3.5.4 Numerical Tests 3.6 Summary References 4 Thermal Stability of Chiral Magnetic Skyrmions from Langer's Theory 4.1 Introduction 4.2 Simulated System 4.3 Paths to Annihilation 4.3.1 Collapse Mechanisms 4.3.2 Escape Through a Boundary 4.3.3 Paths to Collapse and Evolution of the Topological Charge 4.4 Transition Rates from Langer's Theory 4.4.1 Overview 4.4.2 The Thermal Role of Internal Eigenmodes 4.4.3 Entropic Contribution to Skyrmion Stability 4.5 Discussion References 5 Skyrmion Collapse Rate Computation via Forward Flux Sampling and Comparison with Langer's Theory 5.1 Introduction 5.2 Langevin Dynamics 5.2.1 Principle 5.2.2 Numerical Implementation 5.2.3 Numerical Tests 5.3 The Forward Flux Sampling Method 5.3.1 Principle 5.3.2 Error Estimation and Optimization 5.3.3 Numerical Implementation 5.4 Application to Skyrmion Collapse and Comparison with Langer's Theory 5.4.1 Interface Definition 5.4.2 Results 5.5 Discussion References 6 Paths to Annihilation of First- and Second-Order (Anti)skyrmions Under Frustrated Exchange 6.1 Introduction 6.2 Coexistence of Skyrmion and Antiskyrmion Solutions 6.3 Paths to Annihilation 6.3.1 Skyrmion 6.3.2 Antiskyrmion 6.3.3 Second-Order Skyrmion 6.3.4 (Anti)meron Nucleation and Evolution of the Topological Charge 6.4 Langevin Simulations 6.5 Thermal Stability 6.6 Discussion References 7 Conclusion and Outlook References Appendix A Numerical Accuracy in Langer's Theory Implementation A.1 Checking for Numerical Noice: Influence of the Lattice Size A.2 Influence of the Tolerance on the GNEB Force