Magnetic Domain Wall Motion: Numerical Simulation and Collective Coordinate Modeling

Manipulating magnetic domain walls in nanostructures has been linked with applications in spintronic logic, sensing and storage devices. Recent studies of domain wall motion have focused on perpendicular magnetic anisotropy heterostructures of ultrathin ferromagnets sandwiched between a heavy metal layer and an oxide, in which spin-orbit coupling and broken inversion symmetry can dominate domain wall motion. Specifically, chiral domain walls are stabilized in these systems due to the Dzyaloshinskii-Moriya interaction, and current-driven domain wall motion is enhanced due to the spin Hall effect. The chirality of the domain walls in such systems may be partially influenced by the application of external in-plane magnetic fields. Such magnetic fields are used in bubble expansion experiments to assess the strength of the Dzyaloshinskii-Moriya interaction. In addition, bombarding the ferromagnetic layer with heavy metal ions can induce local changes in material properties such as magnetic anisotropy which could be used to manipulate local pinning properties. While computational micromagnetic simulations can help elucidate the behavior of domain walls, their computational cost prohibits extensive studies. As such, assessing the strength of the Dzyaloshinskii-Moriya interaction, extracting material parameters and understanding the behavior of the domain wall to an extent depends on simpler models of domain wall motion based on collective characteristics of the domain wall, and derived from applying model reduction methods to the more complex micromagnetic model. Several Lagrangian-based collective coordinate models exist to describe domain wall motion, namely the $q-\phi$, $q-\phi-\Delta$, and $q-\phi-\chi$ models. While these models can describe domain wall motion with acceptable accuracy, they fail to replicate results of micromagnetic simulations specially for domain wall motion under the application of in-plane fields in heterostructures of interest. Moreover, recent advances in domain wall motion such as pinning due to irradiation have not been included in these models. In this work, we will first present the process for developing Lagrangian-based collective coordinate models, culminating in the derivation of a four collective coordinate model for domain wall motion (the $q-\phi-\chi-\Delta$ model). We show how this model can be extended for cases where in-plane magnetic fields are present to correctly account for the physics; this extension involved introducing the canting induced by the in-plane fields in the domains. We also extend these models to describe the dynamics of magnetic bubbles. In-plane field cases are specifically studied to help identify specific conditions which could help measure properties of the magnetic material. We also compare the equations derived using our Lagrangian-based approach to another reduced model developed through the application of statistical methods to the LLG equation, shedding light on the shortcomings of our approach. The work culminates with a summary of how these models may be made more realistic, through the inclusion of pinning and thermal effects within the model.