Skyrmions are topological spin textures which hold great promise as nanoscale bits of information in memory and logic devices [1].
Although room-temperature ferromagnetic skyrmions and their current-induced manipulation have been demonstrated [2,3], their velocity has been limited to about 100 meters per second [3,4]. In addition, their dynamics are perturbed by the skyrmion Hall effect, a...
We present an asymptotic analysis of the micromagnetic energy in a ultrathin ferromagnetic material with strong uniaxial anisotropy and easy axis perpendicular to the film plane. For subcritical dipolar strenghts, we show that, in the limit, the energy renormalizes the perimeter. Moreover, for critical dipolar strenghts we identify the next order $\Gamma$-limit. Lastly, we will focus on...
In extremely thin ferromagnetic films, an additional interaction, the so-called Dzyaloshinskii-Moriya interaction (DMI), arises in the micromagnetic energy. In such materials, topoligically nontrivial, point-like configurations of the magnetization called magnetic skyrmions are observed, which are of great interest in the physics community due to possible applications in high-density data...
We explore the energy landscape of ferromagnetic multilayer heterostructures that feature magnetic skyrmions in each magnetic layer. Such magnetic heterostructures have been recently pursued as possible hosts of room temperature stable magnetic skyrmions suitable for the next generation of low power information technologies and unconventional computing. The presence of stacked skyrmions in the...
The quest for predicting optimally stable and compact isolated magnetic skyrmions suitable for information technology applications relies on solving the micromagnetic equation. Numerical techniques have been successfully applied to find these solutions. The fundamental nature of these topological defects makes the development of skyrmion theory a very exciting topic from the point of view of...
It has been discovered that the Landau-Lifshitz energy with certain interaction terms can accurately describe the formation of stable vortex-like magnetization configurations,known as chiral magnetic skyrmions.
Accordingly, mathematical communities have been paying increasing attention to give rigorous proofs to support and deepen this understanding.
In this talk, we consider the...
Domain walls are natural topological structures which can be observed for the spin configuration on a ferromagnet.
I will present some stability and interaction results for domain walls on a nanowire. They evolve under the Landau-Lifschitz-Gilbert flow related to an energy which takes into account the Dzialochinski-Morya interaction. I will also discuss the effect of inhomogeneity in the...
Magnetic vortices and skyrmions are typically characterized by distinct topological invariants corresponding to elements of homotopy groups of different spaces. At the same time, intermediate forms of these states - merons - exist and are well studied. This term was introduced in the 1970s, and today an elementary magnetic meron is typically understood to be a planar vortex where the core is...
Antiferromagnetic spin systems are magnetic lattice models in which the exchange in-teraction between two spins favors anti-alignment. A system is said to be geometrically frustrated if, due to the lattice geometry, no spin configuration can simultaneously minimize all pairwise interactions. This frustration can lead to ground states with nontrivial patterns and unconventional magnetic order....
When brought to sufficiently low temperature, matter generally orders, as in the case of a crystalline solid. Some systems remain disordered, though, in the manner of a liquid, even at the lowest temperatures achievable experimentally. Entropy, a thermodynamic quantity that characterizes the degree of disorder, is then very useful to describe the low-temperature properties of matter. While...
We consider energies defined on maps from a 2-dimensional lattice to the unit sphere in R^3, consisting of a (discrete) exchange energy term competing with a (discrete) Dzylashinksii- Moriya interaction term, and assuming suitable boundary conditions outside a regular set $\Omega$.
Under additional constraints we prove the asymptotics of these energies to a confined model of magnetic...
We introduce nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions.
We analyze the asymptotic behavior of the nonlinear elastic energy, as the lattice spacing (in the purely discrete model) and the core-radius (in the semi-discrete framework) vanish. We work within the dilute regime, corresponding to a finite...
We will discuss the flat flow solution for the surface diffusion equation via a discrete minimizing movements scheme proposed in1994 in a celebrated paper by J.W. Cahn and J.E. Taylor. We will show that in dimension three the scheme converges to the unique smooth solution of the equation, provided the initial set is sufficiently regular.
In this talk I will present striped pattern formation results in presence of competing short-range/long-range interactions with crystalline symmetries. This result represents an intermediate step in passing from discrete symmetry breaking to continuous symmetry breaking (i.e. for isotropic interactions). This is a joint work in collaboration with Eris Runa and Francesco Paolo Maiale (GSSI).
I will report on a joint work with Peter Sternberg and a very recent joint work with Lia Bronsard and Peter Sternberg, where we try to describe minimizing entire solutions to the vector Allen-Cahn equation with three wells in two space dimensions. It is known that the Gamma-limit by blow-down of this problem is a minimal partition problem with weights for which minimal cones are easily...
Weak solutions of the 2D eikonal equation correspond to unit vector fields $m$ with zero divergence in the sense of distributions. They arise naturally as sharp interface limits of bounded energy configurations in micromagnetics, elasticity or liquid crystal models (e.g. Aviles-Giga). For a given weak solution $m$, entropy productions are distributions which carry information about...
We study the existence, uniqueness, and symmetry of variational domain-wall traveling waves for the LLG equation. The model is based on an energy functional obtained in a suitable asymptotic regime of the micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane.
I will present recent results on the free boundary problem for an incompressible, irrotational liquid drop of nearly spherical shape with capillarity. Some results are of technical nature, such as reduction to a problem on the boundary, the Hamiltonian structure and a linearization formula for the Dirichlet-Neumann operator. The main result is the existence of travelling waves, which are...
Magnetic singularities known as Bloch points (BPs) present a fundamental challenge for micromagnetic theory due to the divergence of the effective field. To overcome this problem, we propose a regularized micromagnetic model in which the magnetization vector is defined on the S3-sphere, while, similar to quantum systems, only three of the four vector components correspond to measurable...
