Topological Patterns and Dynamics in Magnetic Elements and in Condensed Matter

Domain walls in permalloy strips may exhibit various equilibrium micromagnetic structures depending on the width and thickness of the strip. We show that the equilibrium phase diagram of domain wall structures displays in addition to the previously found structures (symmetric and asymmetric tranverse wall, vortex wall) also double and triple vortex walls for large enough strip widths and thicknesses [1]. We analyze the field driven dynamics of such domain walls in permalloy strips of widths from 240 nm up to 6 $\mu$m, using the known equilibrium domain wall structures as initial configurations. Our micromagnetic simulations show that the domain wall dynamics in wide strips is very complex, and depends strongly on the geometry of the system, as well as on the magnitude of the driving field. We discuss in detail the rich variety of the dynamical behaviors found, including dynamic transitions between different domain wall structures, periodic dynamics of a vortex core close to the strip edge, transitions towards simpler domain wall structures of the multi-vortex domain walls controlled by vortex polarity, and the fact that for some combinations of the strip geometry and the driving field the system cannot support a compact domain wall [2]. [1] V. Estevez and L. Laurson, Phys. Rev. B {\bf 91}, 054407 (2015). [2] V. Estevez and L. Laurson, Phys. Rev. B {\bf 93}, 064403 (2016).

As a model system for understanding the annihilation of topological solitons in magnets, we studied the behavior of two domain walls in a one-dimensional easy axis ferromagnet. This is a system in which one can obtain many analytic results. It is well known that a single domain wall (soliton) can be characterized by two collective coordinates: the position of the center (X) and the azimuthal angle (\Phi). Two-soliton configurations are not stable in a one dimensional magnet, because they do not form local extrema of the energy functional. This means we cannot have a static two-soliton solution as a starting point for our analysis. To find an alternative starting point, we used the fact that our system has two conserved quantities: linear and angular momentum and looked at extrema of energy with additional constraints of having fixed (linear and angular) momenta. As a first step, we found the local extremum of the energy functional that has a given fixed angular momentum. This yielded a stationary configuration of two solitons with the same \Phi that precesses with a constant angular velocity but otherwise preserves its shape in the absence of dissipation. The angular velocity depends on the relative separation between the solitons. We chose boundary conditions at \pm\infty that yields a configuration with a total winding number zero, i.e. one which can be continuously deformed to the uniform state. Adding dissipation to the system causes the two solitons to move towards each other and eventually annihilate. We were able to find an analytic description of this dynamics using two collective coordinates: the separation between the domain walls (\zeta) and the azimuthal angle (\Phi), which are canonically conjugate to each other. We then studied the configuration that extremizes the energy functional for a fixed linear momentum and a fixed angular momentum. This yields a stationary state of two solitons with different azimuthal angles \Phi_1 and \Phi_2. This state has both a constant linear and a constant angular velocity, the linear velocity depending on the relative azimuthal angle \chi = \Phi_1 - \Phi_2. Like before, adding dissipation causes the solitons to attract each other and eventually annihilate. An analytic description of this dynamics requires two more (canonically conjugate) collective coordinates, in addition to the separation \zeta and the average azimuthal angle \Phi = (\Phi_1 + \Phi_2)/2. They are the position of the center of the two-soliton system (Z) and the relative azimuthal angle (\chi). We think this work will be a good starting point for studying the annihilation of more complicated topological solitons like vortices and skyrmions in two dimensions.

We carry out large-scale micromagnetic simulations that demonstrate that due to topological constraints, Bloch lines within extended domain walls are more robust than domain walls in nanowires. Thus, the possibility of spintronics applications based on their motion channeled along domain walls emerges. Bloch lines are nucleated within domain walls in perpendicularly magnetized media concurrent with a Walker breakdown-like abrupt reduction of the domain wall velocity above a threshold driving force, and may also be generated within pinned, localized domain walls. We observe fast field and current driven Bloch line dynamics without a Walker breakdown along pinned domain walls, originating from topological protection of the internal domain wall structure due to the surrounding out-of-plane domains.

The Dzyaloshinskii-Moriya interaction in chiral magnets like MnSi, FeGe or Cu$_2$OSeO$_3$ stabilizes a magnetic helix with a magnetization that is periodic along the pitch vector. We consider defects of this helimagnetic ordering and their properties. Similar to cholesteric liquid crystals, there exist disclination defects around which the pitch vector rotates by $\pi$. A dislocation is formed by combining a $\pi$ and $-\pi$ disclination whose distance is directly related to the Burgers vector. We find that dislocations fall into two classes with vanishing or a finite topological skyrmion density depending on whether the length of their Burgers vector is a half-integer or integer multiple of the helix wavelength, respectively. As a result, only the latter will couple to spin currents via Berry phases. Dislocations have been recently identified to be important for the relaxation of helimagnetic ordering in FeGe [1]. [1] Local dynamics of topological magnetic defects in the itinerant helimagnet FeGe, A. Dussaux et al. arXiv:1503.06622

This poster illustrates some symmetry issues for divergence-free vector fields minimizing some Aviles-Giga type functionals. This is an important problem, in particular in micromagnetics where it is well known that very complex structures can appear. We bring to light a new calibration method that allows to identify some conditions on the potential which insure that any global minimizer is 1D.

In the last few years, magnetic whirls with integer winding number, so-called 'skyrmions', have gained a lot of attention due to their thermal (topological) stability, nanometer scale size, and the ability to be controlled at ultra low current densities. These properties make skyrmions promising candidates for future logic devices and in particular magnetic memory devices. As the most prominent example, the skyrmion racetrack memory has been proposed. Using numerical and analytical calculations, we investigated the interaction of a single skyrmion with different kinds of single defects. We will present the possible phases of interaction [1]. Finally we will motivate an alternative to the skyrmion racetrack, that is supposed to take care of most of its disadvantages and be a realistic candidate for skyrmion memory devices. [1] J. Müller and A. Rosch, Phys. Rev. B 91, 054410

O. V. Pylypovskyi$^1$, D. D. Sheka$^1$, V. P. Kravchuk$^2$, K. V. Yershov$^{2,3}$, D. Makarov$^{4,5}$, Y. Gaididei$^{2}$ $^1$ Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine $^2$ Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, 03680 Kyiv, Ukraine $^3$ National University of “Kyiv-Mohyla Academy”, 04655 Kyiv, Ukraine $^4$ Helmholtz-Zentrum Dresden-Rossendorf e. V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany $^5$ Institute for Integrative Nanosciences, IFW Dresden, 01069 Dresden Germany Effective anisotropies and Dzyaloshinskii-Moriya interaction emerge in nanomagnets with nonzero curvature $\kappa$ and torsion $\tau$ [1, 2]. The simplest object with constant $\kappa$ and $\tau$ is a helix wire. We consider helices with easy-tangential anisotropy. The equilibrium magnetization is tilted by the angle $\psi \propto \kappa \tau$ inside the tangent-binormal surface by the effective easy-axis anisotropy and easy-surface anisotropies. The effective Dzyaloshinskii-Moriya interaction significantly influences to the structure of domain wall: (i) The coupling between domain wall type (head-to-head or tail-to-tail) and its magnetization chirality appears; (ii) The azimuthal angle of magnetization is a linear function of coordinate along a wire with a slope $Y \propto \tau$. In contrast to planar systems, where head-to-head domain walls cannot be moved by spin-orbit torques in parallel current injection geometry [3], the head-to-head domain wall can be efficiently moved under the action of the Rashba torque in helix wires. Altering of the ground state results in the existence of a Rashba field component along magnetization in one of the domains. It pushes domain wall with velocity, proportional to $\sin\psi/(1+Y^2)$ [4]. All analytical predictions are confirmed by micromagnetic [5] and spin-lattice simulations [6]. [1] D. D. Sheka, V. P. Kravchuk, Y. Gaididei, J. Phys. A: Math. Theor. 48, 125202 (2015). [2] D. D. Sheka, V. P. Kravchuk, K. V. Yershov, Y. Gaididei, Phys. Rev. B. 92, 054417 (2015). [3] A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, A. Fert, Phys. Rev. B. 87, 020402 (2013). [4] O. V. Pylypovskyi, D. D. Sheka, V. P. Kravchuk, K. V. Yershov, D. Makarov, Y. Gaididei, arXiv:1510.04725 (2015). [5] T. Fischbacher, M. Franchin, G. Bordignon, H. Fangohr, IEEE Trans. Magn. 43, 2896-2898 (2007). [6] SlaSi spin-lattice simulations package http://slasi.rpd.univ.kiev.ua.

This abstract is from our paper submitted to PRB: We study the dynamics of topological defects in the magnetic texture of rectangular Permalloy thin film elements during relaxation from random magnetization initial states. Our full micromagnetic simulations reveal complex defect dynamics during relaxation towards the stable Landau closure domain pattern, manifested as temporal power-law decay, with a system-size dependent cut-off time, of various quantities. These include the energy density of the system, and the number densities of the different kinds of topological defects present in the system. The related power-law exponents assume non-trivial values, and are found to be different for the different defect types. We discuss details of the processes allowed by conservation of the winding number (topological charge) of the defects, underlying their complex coarsening dynamics. We reproduce the relaxation using dynamics predicted by Thiele's equation accompanied with some simplifications and approximations.

The nonlinear Landau-Lifshitz-Gilbert (LLG) equation models the dynamics of the magnetization in ferromagnetic materials. Numerical challenges arise from strong nonlinearities, a nonconvex pointwise constraint which enforces length preservation, and the possible nonlinear coupling to other PDEs. We discuss numerical integrators, based on lowest-order FEM in space that are proven to be (unconditionally) convergent towards a weak solution of the problem. Emphasis is put on an effective numerical treatment, where the time-marching scheme decouples the numerical integration of the coupled equation. As an example, we consider the nonlinear coupling of the LLG equation and a diffusion equation which models the evolution of the spin accumulation in magnetic multilayers in the presence of electric current.

Materials such as ferromagnets, liquid crystals, and granular media involve orientation degrees of freedom. Mathematical descriptions of such materials involve fields of nonlinear objects such as unit vectors, rotations matrices, or unitary matrices. Classical numerical methods like the finite element method cannot be applied in such situations, because linear and polynomial interpolation is not defined for such nonlinear objects. Instead, a variety of heuristic approaches is used in the literature, which are difficult to analyze rigorously. We present nonlinear generalizations of the finite element method that allow to treat problems with orientation degrees of freedom in a mathematically sound way. This allows to construct numerical methods that are more stable, more efficient, and more reliable. We use the technique to calculate stable configurations of chiral magnetic skyrmions, and wrinkling patterns of a thin elastic polyimide film.

Magnetic skyrmions are topologically protected smooth magnetic whirl textures, which are energetically stabilized in chiral magnets by the Dzyaloshinskii-Moriya interaction. Skyrmions can be manipulated by ultra-low electric current densities, which makes them promising candidates for novel spintronic applications. In insulating chiral magnets, it turns out that skyrmions are sensitive to tiny magnon currents, induced, for example, with the help of a temperature gradient [1]. In turn, the scattering of spin-waves off skyrmions results in an emergent Lorentz force that leads to a topological magnon Hall effect. Based on our previous work [2,3], we discuss thermal transport of magnons in the presence of a dilute gas of skyrmions in a two-dimensional chiral magnet. Using the Boltzmann equation, we derive the thermal transport coefficients for the magnons as well as the effective equation of motion for the skyrmion positions. [1] Mochizuki et al., Nature materials 13.3 (2014): 241-246. [2] C. Schütte and M. Garst, Phys. Rev. B 90, 094423 (2014). [3] S. Schroeter and M. Garst, Low Temperature Physics, 41, 817-825 (2015).

Shape Memory Alloys are materials which, after having been deformed at a low temperature, are able to recover their previous shape upon heating. This effect is a result of the fact that at low temperature the crystal lattice has several distinct stress-free variants. Using these so-called martensite variants, the material can form microstructures. We present our current progress on characterizing the macroscopic geometry of a class of microstructures called habit planes in shape memory alloys undergoing cubic-to-tetragonal transformations. Starting from a geometrically linear variational model we describe the patterns formed by the habit planes under some assumptions on the regularity of the macroscopic configuration.

Single crystal MnSi is a prototypical material system that carries the skyrmion lattice phase, which is well described by the Ginzburg-Landau theory taking thermal fluctuation into account. The thin-downed MnSi thin films from the bulk have extended skyrmion phase region, as the uniaxial magnetic anisotropy is favoured by the skyrmion lattice formation in the micromagnetic model. This seems to suggest the fact that the skyrmions may be more stable for a grown MnSi films. However, the existence of the skyrmion lattice phase, and the magnetic structures in epitaxial MnSi thin film systems are still an unknown issue, or at least is under controversial debates currently. I would like to present our recent findings from the perspectives of the growth and characterisation of this system, which infer a non-skyrmion conclusion. In fact, several fundamental differences between the bulk and films, both structurally and magnetically, may be the reason why the skyrmion state is not favoured in films. These are also general problems for all the grown B20 thin films.