Mathematical Analysis of Domain Wall Motion in Nanowires

Ferromagnetism is a striking and subtle phenomenon. Observable on the macroscopic scale, its origins lie outside the scope of classical physics, and are consequences of two quintessentially quantum mechanical properties of matter, namely electron spin and the Pauli exclusion principle. The quantum mechanical origin of ferromagnetism accounts for the existence of ferromagnetic domains -- regions of distinct and nearly uniform magnetisation -- on microscopic, indeed nanometer scales. The size of ferromagnetic domains, along with the modest energy required to manipulate (i.e., read and write) them has led to far-reaching applications in information technology, proceeding over the last half-century from magnetic tape drives to the current frontier, for example race-track memory: a three-dimensional memory on which domains -- ''bits'' -- may be read, moved, and written around nanowire loops.

The last two decades have witnessed a revolution in micromagnetics, both in fundamental science as well as consequent technological breakthroughs. It has long been understood how ferromagnetic domains can be controlled through external magnetic fields. More recent is the discovery of a wholly new mechanism for domain wall dynamics through the interaction of magnetisation and spin-polarised currents.

For length scales down to tens of nanometers, there is a well established and extremely successful continuum theory of micromagnetism, namely the micromagnetic variational principle and its dynamic counterpart, the Landau-Lifshitz-Gilbert equation. The theory is mathematically complex (the equations are both nonlinear and nonlocal), and encompasses a range of diverse regimes. These regimes can be separately investigated analytically using modern techniques from the calculus of variations and partial differential equations. Models on the scale of individual atoms are necessarily quantum mechanical, and require additional physical concepts and mathematical apparatus.

The first aim of this project is an analytic study of domain-wall motion in nanowires and nanotubes induced by currents and applied magnetic fields. We will establish mathematically the existence and wide range of physical behaviours, and derive formulas which describe their properties. The second aim is to elucidate the underlying quantum mechanical mechanisms of domain-wall propagation through a semiclassical analysis of the electron dynamics, including spin, in a ferromagnetic medium.

The proposed research is interdisciplinary, and therefore has the potential for impact on three distinct academic communities: applied analysis, micromagnetics and spintronics, and soft condensed matter/liquid crystals. The impact may be direct, via application of the results we obtain and transfer to new problems, as well as indirect, through facilitating communication between researchers in different communities. Impact on industry and technology, in particular the burgeoning development of nanoscale magnetic and spintronic devices, will be pursued through collaborations with colleagues in theoretical and experimental physics who work closely on engineering applications. The PI has a strong track record of industrial collaboration. A fourth area of impact is through interdisciplinary training of young researchers, who will carry out subsequent independent research in micromagnetics and related fields.

Publications. Publication is, of course, the traditional route for disseminating scientific results. However, our research team is unusual in its track record of publication in internationally leading journals across a very broad spectrum of disciplines, including the Archive for Rational Mechanics, Calculus of Variations and Partial Differential Equations, and Compte Rendus Mathematique (analysis); Proceedings of the Royal Society A and Nonlinearity (applied mathematics and theoretical physics); Physical Review Letters and Physical Review (theoretical and experimental physics); Journal of Chemical Physics and Advances in Quantum Chemistry (physical/quantum chemistry). Research will be published in mathematics and physics journals to reach a wide and diverse audience. This will facilitate awareness in both communities of their respective research agendas and engender cross-fertilisation of ideas.

Conferences and workshops. Particularly in interdisciplinary research, successful transfer of ideas depends on making new connections with researchers outside one's immediate areas of acquaintance. Grant funding will facilitate the research team's participation in leading international conferences on micromagnetics and spintronics. We will also apply for additional funding to organise international workshops at leading research institutes for mathematicians and theoretical physicists on problems in spintronics and micromagnetics related to domain wall motion.

Visits. Dissemination of results will be enhanced by extended interactions, funded by the grant, with leading scientists from a range of disciplines in mathematics and theoretical physics.

Training. The postdoctoral researcher will receive interdisciplinary training in applied analysis, micromagnetics and semiclassical analysis. They will be exposed to a broad range of research activity in mathematics, physics and engineering through seminars in the School of Mathematics and throughout the University, participation in international meetings, and interactions with visitors. In the third year of the programme, the postdoctoral researcher will be invited to deliver a short series of postgraduate-level lectures. This will provide the RA with valuable lecturing experience while introducing a new cohort of future researchers to the field. We will seek to recruit additional students by sponsoring joint research projects with experimental colleagues in Physics and Engineering in the Bristol Centre for Functional Nanomaterials, one of the University's Doctoral Training Centres. We will also involve undergraduate students through the summer research bursary schemes funded by the EPSRC and Nuffield Foundation.