# MANA 2018 - Micromagnetics:

Analysis, Numerics, Applications

TU Wien, November 8-9, 2018

## Abstracts

### Invited talks

François Alouges, Ecole Polytechnique, France |

Homogenization in micromagnetics |

Several magnetic material are composed of different compounds mixed together at the micro-scale. Among them are the nanocristalline soft ferromagnets and the spring magnets which are built from hard and soft materials intimately assembled. The mathematical theory for such systems relies on the homogenization of Brown energy for the statics and Landau-Lifshitz equation for the dynamics. The talk plans to make a small overview on recent results in both directions, in the periodic and stochastic settings where the grains in the microscopic structure are periodically or randomly distributed respectively. This is a joint work with G. Di Fratta (TU Wien), L. Nicolas, and A. de Bouard (Ecole Polytechnique). |

Sergio Conti, University of Bonn, Germany |

Modeling and simulation of magnetic shape memory composites |

We study composite materials in which small ferromagnetic shape-memory particles are embedded in a polymer matrix. Magnetic shape-memory materials react with a large change of shape to the presence of an external magnetic field. The macroscopic properties of the material depend strongly on the type of microstructure and the characteristics of the particles and the polymer. Based on a model which combines micromagnetism and elasticity, we aim at optimizing the shape of the particles in order to obtain the largest possible macroscopic deformation and work output for the composite material. We present numerical computations for the unit cell, within a homogenization framework, both for the static (variational) case and for evolution, studying rate-independent macroscopic hysteresis. This talk is based on joint work with M. Lenz and M. Rumpf (U Bonn). |

Massimiliano d'Aquino, Parthenope University of Naples, Italy |

Efficient computation of magnetization eigenmodes in complex micromagnetic systems |

The study of the normal oscillation modes of ferromagnetic nano-particle systems is a fundamental issue for its applications to the analysis of magnetization dynamics under microwave applied fields. In typical experimental situations, a small magnetic particle is saturated by applying a sufficiently strong DC magnetic field along a given direction. Small magnetization motions around this state are then excited by applying a small (compared to the DC component) radio-frequency (RF) applied field. In this condition, the ferromagnetic resonance curve is obtained by slowly varying either the frequency or the amplitude of the RF field and measuring the power absorbed by the particle. From the observation of the peaks in this curve, one determines the frequency values corresponding to the excitation of certain magnetization normal modes. This problem was theoretically analyzed by Brown and Aharoni. Their approach was based on the use of appropriate analytical techniques and was limited to particles of special shapes (spheres, ellipsoids). Another scenario where the normal oscillations around an equilibrium play a fundamental role is in the modelling of thermal fluctuations. In fact, thermal agitation tends to slightly perturb the equilibrium magnetization and therefore, from the analysis of the resonant response of the micromagnetic system, one can retrieve insightful information about fluctuation and dissipation processes. For this reason, considerable research has been recently focused on numerical computations of normal modes for particles with generic shapes and experimental observations involving spatially non uniform equilibrium magnetization configurations. In this work, a general formulation of this problem is presented. The small oscillation modes in complex micromagnetic systems around an equilibrium are numerically evaluated in the frequency domain by using a novel formulation, which naturally preserves the main physical properties of the problem. The Landau-Lifshitz-Gilbert (LLG) equation, which describes magnetization dynamics, is linearized around a stable equilibrium configuration and the stability of micromagnetic equilibria is discussed. Special attention is paid to take into account the property of conservation of magnetization magnitude in the continuum as well as discrete model. The linear equation is recast in the frequency domain as a generalized eigenvalue problem for suitable self-adjoint operators connected to the micromagnetic effective field. The generalized eigenvalue problem may be conveniently discretized by finite difference or finite element methods depending on the geometry of the magnetic system. The spectral properties of the eigenvalue problem are derived in the lossless limit. Perturbation analysis is developed in order to compute the changes in the natural frequencies and oscillation modes arising from the dissipative effects. It is shown that the discrete approximation of the eigenvalue problem obtained either by finite difference or finite element methods has a structure which preserves relevant properties of the continuum formulation. This approach has several advantages as far as the numerical computation of the normal modes is concerned: (1) It may circumvent the problems of time-domain analysis for relatively low-frequency oscillations which are relevant, for instance, in magnetic vortex dynamics. (2) The discretized operator can be assembled by using the classical exchange and magnetostatic operators implemented in both finite differences and finite elements micromagnetic codes; (3) The discretized version of this problem is a standard self-adjoint matrix eigenvalue problem which can be solved with well-established techniques of numerical linear algebra. The solution of this problem gives directly all the resonant frequencies and the normal modes. The results obtained by using this approach will be presented for several examples of complex micromagnetic systems. |

Riccardo Hertel, University of Strasbourg, France |

Fast domain wall dynamics in cylindrical nanowires:
Curvature-induced magnetochirality and Cherenkov-type spin wave emission |

Over the past years, as a result of freely available and powerful codes, micromagnetic simulations have evolved from a specialized scientific topic to a commonly used tool in research in magnetism. This growth in popularity has also shifted the general context in which micromagnetic simulation are applied. Originally used as an instrument of fundamental research, their main task today consists in complementing experimental studies. But theoretical studies based on micromagnetic simulations can still serve as an original means to discover unforeseen physical effects, and they may thereby inspire new experimental research activities. As an example for this, I will present studies on the magnetic domain wall dynamics in nanotubes and nanowires based on finite-element simulations. We find that the dynamic properties of magnetic domain walls in nanotubes and in cylindrical nanowires can be significantly different from the well known domain wall dynamics in thin films and in flat thin strips. More specifically, the simulations reveal that surface curvature can lead to a chiral symmetry breaking in the magnetization dynamics, and that one may thereby obtain magnetic domain walls that are stable against the usual Walker breakdown. Owing to their stability, these domain walls can propagate at velocities which are much higher than the usual domain wall speeds. Our simulations show that the ultrafast motion of magnetic domain walls can even lead to the spontaneous excitation of spin waves in a process that is analogous to the Cherenkov effect. In the case of solid cylindrical wires, the domain wall can contain a micromagnetic point singularity (Bloch point). Since these structures cannot be treated reliably with micromagnetic continuum theory, we employ dynamic multiscale modeling with atomistic-continuum coupling to simulate their field-driven. |

Stavros Komineas, University of Crete, Greece |

Dynamics of solitons in ferromagnets with broken chiral symmetry |

We show that chiral symmetry breaking enables dynamical behavior of solitons that can give steady states in ferromagnets with Dzyaloshinskii-Moriya interaction (DMI). An example is given by a traveling domain wall solution for the conservative Landau-Lifshitz equation in uniaxial ferromagnets with DMI. A second example is steady state rotation of a skyrmion in a realistic nanodisc geometry. This generates magnetization oscillations and it is obtained by a very simple setup, i.e., when we apply constant spin transfer torque with uniform spin polarization. |

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Germany |

The magnetization ripple: A non-local SPDE-perspective |

The ripple in a thin-film is the magnetization's response to polycrystallinity. The randomly oriented grains lead to an easy axis that is a random field, a source of quenched noise. Heuristic arguments lead to a reduced model that zooms in on the (different) longitudinal and transversal scales of the ripple. It is a two-dimensional, non-local variational model formulated in terms of the transversal magnetization component. Because the grains are smaller than the characteristic ripple scales, the random easy axis acts as a transversal field of white-noise character. The Euler-Lagrange equation is a non-local elliptic SPDE. The regularizing property of the leading-order symbol on the white noise is too weak to give a classical sense to the nonlinearity. We apply ideas from the work of DaPrato-Debussche to develop a well-posedness theory. This is joint work with R. Ignat (U Toulouse III). |

Andreas Prohl, University of Tübingen, Germany |

Stochastic optimal control in ferromagnetism |

I discuss the optimal control via an exterior field of a ferromagnetic chain immersed into a heat bath. For its numerical realization, I compare an approach which rests on Pontryagin's maximum principle with another one which is based on the dynamic programming principle. This is joint work with T. Dunst, C. Schellnegger (U Tübingen), A. K. Majee (IIT Delhi), M. Jensen (U Sussex), and G. Vallet (U Pau). |

Michele Ruggeri, University of Vienna, Austria |

Recent developments in tangent plane integrators for the Landau-Lifshitz-Gilbert equation |

Tangent plane integrators are well-established methods for the numerical integration of the Landau-Lifshitz-Gilbert equation (LLG). In the first part of the talk, we discuss an IMEX-type tangent plane scheme, which is unconditionally convergent, (almost) second-order in time, and based on an implicit-explicit treatment of the effective field contributions, designed in such a way that, e.g., only one expensive stray field computation per time-step needs to be carried out. Then, we discuss effective solution and preconditioning strategies for the arising constrained linear systems. We conclude by considering the extension of the tangent plane approach for LLG in the presence of the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. The talk is based on joint work with joint work with G. Di Fratta, C.-M. Pfeiler, D. Praetorius, and B. Stiftner (TU Wien). |

Valeriy V. Slastikov, University of Bristol, UK |

Edge domain walls in ultrathin exchange-biased films |

We present an analysis of edge domain walls in exchange biased ferromagnetic films appearing as a result of a competition between the stray field at the film edges and the exchange bias field in the bulk. We introduce an effective two-dimensional micromagnetic energy that governs the magnetization behavior in exchange biased materials and investigate its energy minimizers in the strip geometry. In a periodic setting, we provide a complete characterization of global energy minimizers corresponding to edge domain walls. In particular, we show that energy minimizers are one-dimensional and do not exhibit winding. We then consider a particular thin film regime for large samples and relatively strong exchange bias and derive a simple and comprehensive algebraic model describing the limiting magnetization behavior in the interior and at the boundary of the sample. Finally, we demonstrate that the asymptotic results obtained in the periodic setting remain true in the case of finite rectangular samples. |

Dieter Suess, University of Vienna, Austria |

Coupled micromagnetic and spin transport simulations for the design of memory and sensor applications |

Within this talk I will review our activity for the solution of the coupled micromagnetic and spin transport simulations. The relation of the developed full self-consistent model with the Slonczewski model and the Zhang-Li model will be presented. Various physical examples will be shown where the developed model leads insights to the underlying physical properties such as relation of field like and damping like term, the spin torque efficiency in MRAM structures, understanding the back-hopping mechanism in MRAM devices and minimizing the critical currents in spin torque oscillators. Finally, the model will also be applied to magnetic confined objects such as vortex structure and skyrmions for the calculation of the magnetization dynamics and thermal stability. |