Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction

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), because 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. We then use the methods of {$\Gamma$}-convergence to analyze the asymptotics of the two-dimensional mag- netization patterns in samples of large spatial extent in the presence of weak applied magnetic fields

Geometrically induced phase transitions in two-dimensional dumbbell-shaped domains

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One-dimensional in-plane edge domain walls in ultrathin ferromagnetic films

We study existence and properties of one-dimensional edge domain walls in ultrathin ferromagnetic films with uniaxial in-plane magnetic anisotropy. In these materials, the magnetization vector is constrained to lie entirely in the film plane, with the preferred directions dictated by the magnetocrystalline easy axis. We consider magnetization profiles in the vicinity of a straight film edge oriented at an arbitrary angle with respect to the easy axis. To minimize the micromagnetic energy, these profiles form transition layers in which the magnetization vector rotates away from the direction of the easy axis to align with the film edge. We prove existence of edge domain walls as minimizers of the appropriate one-dimensional micromagnetic energy functional and show that they are classical solutions of the associated Euler-Lagrange equation with Dirichlet boundary condition at the edge. We also perform a numerical study of these one-dimensional domain walls and uncover further properties of these domain wall profiles

Walker solution for Dzyaloshinskii domain wall in ultrathin ferromagnetic films

We analyze the electric current and magnetic field driven domain wall motion in perpendicularly magnetized ultrathin ferromagnetic films in the presence of interfacial Dzyaloshinskii-Moriya interaction and both out-of-plane and in-plane uniaxial anisotropies. We obtain exact analytical Walker-type solutions in the form of one-dimensional domain walls moving with constant velocity due to both spin-transfer torques and out-of-plane magnetic field. These solutions are embedded into a larger family of propagating solutions found numerically. Within the considered model, we find the dependencies of the domain wall velocity on the material parameters and demonstrate that adding in-plane anisotropy may produce domain walls moving with velocities in excess of 500 m/s in realistic materials under moderate fields and currents.Comment: 6 pages, 2 figure

Variational principles of micromagnetics revisited

We revisit the basic variational formulation of the minimization problem associated with the micromagnetic energy, with an emphasis on the treatment of the stray field contribution to the energy, which is intrinsically non-local. Under minimal assumptions, we establish three distinct variational principles for the stray field energy: a minimax principle involving magnetic scalar potential and two minimization principles involving magnetic vector potential. We then apply our formulations to the dimension reduction problem for thin ferromagnetic shells of arbitrary shapes

Reduced energies for thin ferromagnetic films with perpendicular anisotropy

We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material's lateral edges, we carry out an asymptotic analysis of the energy by means of $\Gamma$-convergence. Depending on the scaling assumptions on the size of the material domain vs. the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample's lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.Comment: 29 pages, 1 figur

Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree k 2 for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points

Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals

We investigate prototypical profiles of point defects in two dimensional liquid crystals within the framework of Landau-de Gennes theory. Using boundary conditions characteristic of defects of index $k/2$, we find a critical point of the Landau-de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, $b^2$ small, we prove that this critical point is the unique global minimiser of the Landau-de Gennes energy. We investigate in greater detail the regime of vanishing elastic constant $L \to 0$, where we obtain three explicit point defect profiles, including the global minimiser.Comment: 15 pages, 16 figure

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Γ-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy