Dzyaloshinskii-Moriya chiral magnets and boundary conditions in Skyrmion electronics

Skyrmion-based electronic devices are a subset of spintronic nanodevices based on chiral materials (1, 2). The Dzyaloshinskii-Moriya (D-M) interaction is a chiral magnetic interaction which models chiral magnetic materials showing particular promise for extending CMOS compatible Skyrmionel ectronics at scales where silicon devices can no longer compete. There are several approaches to realizing such materials in practice. One is to focus on realizing D-M interactions as a fundamental problem in materials science supported by first principles quantum field theoretic models incorporating Majorana spinnors. Another very successful approach is to extend phenomenological micro-scale models of magnetism based on the Landau-Lifschitz-Gilbert (LLG) equation to the nanoscale by incorporating spin-torque coupling. However, this phenomenological approach obscures ties to more fundamental physics and the resulting boundary conditions can be a mystery. The present work uses well established mathematical techniques to show how Majorana spinnors and Skyrmions can appear in phenomenological models. There are three key aspects in this geometric/topological approach: • The first are Weitzenboeck identities and the Gaffney inequality (3). In electromagnetic theory, they enable us to study the distinction between Maxwell and Lame eigenmodes of cavity resonators; in micromagnetics they enable us to rewrite exchange energy in terms of fewer squares. • The second set of tools is familiar from the investigation of instantons; namely the identification of suitable divergence terms which enable one to rewrite a Hamiltonian in terms of the fewest number of squares. It is in this later step that the Majorana spinnors emerge without considerations of quantum mechanics and the Skyrmion solutions become apparent in a broader geometric context than the customary thin film scenarios. • Third, is the geometric observation that the LLG equation projects the magnetization vector so as to leave its length invariant. This enables us to consider the Hamiltonian of the system modulo the rescaling of the magnetization vector. As a result of this geometric reformulation, a clearer understanding of the use of the LLG equation at the nanoscale emerges as well as a more geometric connection to the underlying quantum phenomena. Finally, the role of chirality emerges more cleanly and it points to the role of topology in the possibility of near reversible computing generating a minimum of entropy and heat (4, 5, 6).First author draf

Chiral magnet models and boundary condition geometry in Skyrmion electronics

Field theoretic techniques are used to relate (i) the Landau-Lifschitz approach to Skyrmion devices based on Dzyaloshinskii-Moriya (D-M) chiral magnets, and (ii) the mathematical approaches to quantum magnetism. This results in a geometric understanding of micromagnetic singularities and boundary conditions without the usual thin-film assumptions.First author draf

Spin-Hall Nano-Oscillator Simulations

A spin-Hall nano-oscillator (SHNO) is a type of spintronic oscillator that shows promising performance as a nanoscale microwave source and for neuromorphic computing applications. Within such nanodevices, a non-ferromagnetic layer in the presence of an external magnetic field and a DC bias current generates an oscillating microwave voltage. For developing optimal nano-oscillators, accurate simulations of the device’s complex behaviour are required before fabrication. This work simulates the key behaviour of a nanoconstriction SHNO as the applied DC bias current is varied. The current density and Oersted field of the device have been presented, the magnetisation oscillations have been clearly visualised in three dimensions and the spatial distribution of the active mode determined. These simulations allow designers a greater understanding and characterisation of the device’s behaviour while also providing a means of comparison when experimental results are generated