Some recent talks
Existence range for stray field-stabilized magnetic skyrmions in thin ferromagnetic films
Compact magnetic skyrmions are known to be stabilized by a variety of physical mechanisms, most prominently the Dzyaloshinskii-Moriya interaction (DMI). Yet the role the stray magnetic field plays in stabilizing non-collinear spin textures is often poorly understood due to the non-local nature of the stray field interaction. In fact, stray field can stabilize compact magnetic skyrmions under complete absence of DMI, suggesting that compact skyrmions must be a generic feature exhibited by classical ferromagnetic materials with perpendicular magnetocrystalline anisotropy. Yet the experimental verification of this fact has been elusive. In this talk, I will use the micromagnetic modeling framework and rigorous mathematical analysis techniques to resolve this apparent paradox by identifying the parameter regimes in which stray field-stabilized magnetic skyrmions can be observed in thin ferromagnetic films. The analysis reveals that existence of such skyrmions requires that the magnetocrystalline anisotropy be rather tightly tuned to be suffciently close to the value at the reorientation transition and quantities the existence range of skyrmions solutions against either collapse or bursting instabilities.
Slides
Magnetic skyrmions: an overview
I will present an overview of the results on existence and asymptotic properties of magnetic skyrmions — particle-like topologically nontrivial two-dimensional spin textures that have been envisioned as bit encoding states in the emergent field of spintronics. Mathematically these are defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.
Video Slides
An interplay between dimensionality and topology in thin ferromagnetic films
This talk presents an overview of modelling and analytical challenges associated with the studies of topological solitons in thin film ferromagnetic materials. These materials have been recently demonstrated to support a variety of topologically non-trivial spin textures, including magnetic skyrmions - local swirls of spins that exhibit particle-like behavior, nanometer size and room temperature stability. The above properties of magnetic skyrmions, as well as a possibility of their control by electric fields and currents make them attractive as possible information carriers in a new generation of magnetic memory and spintronic logic.
Video Slides
Ferromagnetism at nanoscale
Advances in nanofabrication offer an unprecedented degree of precision and control in manufacturing novel types of magnetic materials and heterostructures. As the dimensions of the constituent components go down to the atomic scale, the magnetic properties of these materials begin to be dominated by the interfaces between the adjacent material layers. This leads to the emergence of a plethora of new phenomena, including those that give rise to topological magnetism, whereby the observed spin textures acquire non-trivial topolgical characteristics. Examples of such textures include chiral domain walls and magnetic skyrmions, with promising applications in spintronics --- an emergent field of microelectronics that takes advantage of both the charge and the spin degrees of freedom of an electron. This talk will overview the chalenges and opportunities offered by modeling and analysis of the magnetic properties of these novel materials, focusing on applications of rigrorous asymptotic techniques of calculus of variations and PDE analysis.
Video Slides
One-dimensional domain walls in thin film ferromagnets: an overview
Ferromagnetic materials offer a prime example of systems exhibiting a
rich variety of spatial patterns driven by minimization of energy. A
competition between the short-range exchange and the long-range
magnetostatic interaction in ferromagnets often gives rise to the
emergence of magnetic domains, in which the magnetization remains
nearly constant and aligned along particular preferred directions in
extended regions of space. These regions are punctuated by sharp
transition layers referred to as domain wall, in which the
magnetization abruptly rotates between different preferred
orientations. In this talk, I will give an overview of the current
state of the art in the understanding of the domain wall solutions in
thin ferromagnetic films in which the magnetization is constrained to
lie in the film plane. This setting leads to a number of challenging
problems in the analysis of nonlinear PDEs involving fractional
Laplacian.
Video Slides
Magnetic skyrmions in the conformal limit
We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree ±1 harmonic maps from the two-dimensional sphere to itself.
Video Slides
Chiral domain walls and domain wall tilt in ferromagnetic nanostrips
Recent advances in nanofabrication make it possible to produce multilayer nanostructures composed of ultrathin film materials with thickness down to a few monolayers of atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic materials begin to exhibit unusual properties, such as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii-Moriya interaction (DMI), due of the increased importance of interfacial effects. The presence of surface DMI has been demonstrated to fundamentally alter the structure of domain walls. Here we use the micromagnetic modeling framework to analyse the existence and structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy functional. We explicitly construct the minimizers in the one-dimensional setting, both for the interior and edge walls, for a broad range of parameters. Using varitional methods we analyze the asymptotics of the two-dimensional magnetization patterns in samples of large spatial extent in the presence of weak applied magnetic fields and present an analytical theory of domain wall tilt. We show that under an applied field the domain wall remains straight, but tilts at an angle to the direction of the magnetic field that is proportional to the field strength for moderate fields and sufficiently strong DMI.
Video Slides
Variational models for charged drops
In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. Mathematically, these problems all share a common feature that the equilibrium shape of a charged drop is determined by an interplay of the cohesive action of surface tension and the repulsive effect of long-range forces that favor drop fragmentation. More generally, these problems present a prime example of problems of energy driven pattern formation via a competition of long-range attraction and long-range repulsion. In the talk, I will focus on two classical models - Gamow's liquid drop model of an atomic nucleus and Rayleigh's model of perfectly conducting liquid drops. Surprisingly, despite a very similar physical background these two models exhibit drastically different mathematical properties. I will discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models, and present the current state of the art for this class of geometric problems of calculus of variations.
Video Slides
Gamma-convergence for pattern forming systems with competing interactions
I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions. This is joint work with D. Goldman and S. Serfaty.
Video Slides
Pattern Formation by Energy Minimization
Video