Coherent Structures in Nanomagnetism

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There has been a genuine revolution in nanofabrication of advanced magnetic materials in the last 5-10 years. Swift advances in the field of spintronics - an emergent discipline that aims to take advantage of the intrinsic spin of electron in addition to its electric charge for computing and data storage - present new challenges for mathematical modeling, analysis and simulations of these materials, which need to keep pace with the progress in material design. The aim of this project is to provide efficient theoretical treatments of the next generation of magnetic materials within the general micromagnetic modeling framework down to nanoscale. Its focus on magnetic skyrmions and other coherent structures is strongly motivated by the ongoing development of new spintronic hardware for neuromorphic computing inspired by biological neural networks and particularly suited for deep learning. The nonlinear, nonlocal and multiscale nature of the problems make it into a formidable problem of the 21st century applied mathematics. Nevertheless, both the need to develop new mathematical and computational tools to tackle the genuine complexity of these systems and their potential to revolutionize computer technologies make these fundamental challenges very exciting. The investigator undertakes a combination of modeling, analytical, asymptotic and computational studies of layered ferromagnetic materials of current interest to spintronic applications. Due to the increased dominance of interfacial effects at nanoscale, the three-dimensional micromagnetic modeling framework must incorporate new types of boundary terms. This new physics often results in surprising effects near the material boundaries and gives rise to novel types of edge magnetization structures such as edge-curling walls, edge vortices and chiral bobbers. The investigator plans to formulate micromagnetic models of ferromagnetic films that exhibit interfacial magnetic anisotropy and Dzyaloshinskii-Moriya interaction, derive the reduced two-dimensional models appropriate for thin films and develop efficient computational tools to simulate current and stochastically driven systems. Next, within the obtained class of the reduced models, the investigator plans to undertake a study of several types of coherent structures supported by these materials, such as magnetic skyrmions in extended and patterned ultrathin films, spin textures in nanowires, and various edge structures. The questions of existence and asymptotic properties of these coherent structures give rise to challenging problems of energy-driven pattern formation and calculus of variations. A key component of the project is the involvement of a new generation of applied mathematicians into this highly interdisciplinary area of research. As part of this process, the investigator develops courses in applied sciences and takes part in interdisciplinary training of mathematics and engineering graduate and undergraduate students and postdocs. It is hoped that the project will also help foster better interactions between applied mathematicians and experimentalists.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.