# MANA 2016 - Micromagnetics:

Analysis, Numerics, Applications

TU Wien, February 18-19, 2016

## Abstracts

### Invited talks

Lubomir Banas (Bielefeld University, Germany)Numerics for thermally assisted magnetodynamicsMagnetization dynamics in ferromagnetic materials at low temperatures can be described by the Landau-Lifshitz-Gilbert (LLG) equation. The classical LLG model holds for constant temperatures that are sufficiently far from the Curie temperature. Due to increasing miniaturization of magnetic devices and recent applications, such as the thermally-assisted magnetic recording, it has become necessary to account for temperature effects in the model. We review two most common approaches currently used for the modelling of thermally activated dynamics in ferromagnetic materials. The first approach includes thermal fluctuations via a random term and leads to a stochastic mesoscopic model, the so-called Stochastic-LLG equation. The second approach includes temperature effects via an additional term in the LLG equation and leads to a deterministic macroscopic model. We propose structure preserving finite element based numerical approximations for the respective equations and show numerical experiments to demonstrate some interesting features and connections between the two models. |

Gilles Carbou (University of Pau and Pays de l'Adour, France)Walls dynamics in ferromagnetic nanotubesFerromagnetic nanotubes are proposed as an alternative to ferromagnetic nanowires for data-storage applications. We consider a two-dimensional model of such devices and we establish the stability of moving walls in the Walker regime when the tube is subject to a small magnetic field. |

Michael J. Donahue (NIST, USA)Quantitative effect of cell size on precision of micromagnetic energy calculationMicromagnetic simulations present a trade-off between precision and resources devoted to the task. Understanding the relative origins of errors is needed to allocate resources to the most efficient production of any desired precision. In this talk I will present work aimed at quantitative determination of errors in numerical simulations as compared to the mathematically smooth solutions arising from the canonical micromagnetic model. To this end very finely meshed 3D finite difference (OOMMF) simulations are run on several magnetization configurations (stripe domains, vortex, pseudo-single-domain). With a nominal exchange length of 5 nm, the several hundred million 0.1 cubic cells used in the simulations each span a mere one fiftieth of the exchange length. Each equilibrium state (torque < 10^-6 A/m) is lifted to coarser resolutions and the component energy values (anisotropy, exchange, demagnetization, Zeeman) are re-computed on the coarser meshes. The coarse sub-samplings are selected so that the center of each coarse cell coincides with the center of a cell at the finest resolution, so the magnetization is obtained on the coarser meshes without interpolation. Each component value is then seen to converge as the mesh cell size is reduced, and the limit value, or the value at the finest resolution, can be used to discern the magnitude of the error at the coarser resolutions that are more representative of practical micromagnetic simulations. |

Carlos J. García-Cervera (UC Santa Barbara, USA)A mean-field model for spin dynamics in multilayered ferromagnetic mediaMagnetic storage devices rely on the fact that ferromagnetic materials are typically bistable, and that it is possible to switch between different states by applying a magnetic field. The discovery of the Giant Magneto-Resistance effect has enabled the use of layered ferromagnetic materials in magnetic devices, such as magnetic memories (MRAMs). Even in the absence of thermal effects, there are limitations in the storage capacity of such devices due to the fact that as the size is decreased, the magnitude of the switching field increases, due to an increase in shape anisotropy. Given that magnetic fields have long range interactions, the density of such devices is limited. A new mechanism for magnetization reversal in multilayers was proposed by Slonczweski and Berger. In this new mechanism, an electric current flows perpendicular to the layers. The current is polarized in the first layer, and the polarization travels with the current to the second layer, where it interacts with the underlying magnetization. Since currents are localized in each cell, long range effects can be reduced. In this talk we will discuss the connection between several models for the description of the spin transfer torque at different physical scales. Specifically, we connect the quantum and kinetic descriptions with the help of the Wigner transform, and the kinetic and diffusion models by a specific parabolic scaling. Numerical examples will presented to illustrate the applicability and limit of the different models. This is joint work with Jingrun Chen and Xu Yang at UCSB. |

Gino Hrkac (University of Exeter, UK)Nanoscale switch for vortex polarization mediated by Bloch core formation in magnetic hybrid systemsAdding new functionality to devices, which often requires precise control at the nanoscale, has always been a driving force in research. We investigated magnetic hybrid structures and found, after predicting it from simulations, in a collaboration with scientists at the Paul Scherrer Institute (PSI) and the ETH Zurich, Switzerland and Manchester, UK, a new way to locally switch the orientation of a magnetic feature that is smaller than 20 nm. This discovery provides added control for future technological applications such as data processing, microwave sources or magnetic sensors. |

Martin Kružík (The Czech Academy of Sciences, Czech Republic)Thermodynamically-consistent mesoscopic model of the ferro/paramagnetic transitionA continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermomagnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that fine spatial oscillations of the magnetization are modelled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization. We will also present some numerical computations showing qualitative agreement with experiments. This is a joint work with B. Benesova (Würzburg), T. Roubicek, and J. Valdman (Prague). |

Christof Melcher (RWTH Aachen University, Germany)Topological solitons in chiral magnetismMagnets without inversion symmetry are a prime example of a solid state system featuring topological solitons on the nanoscale. We shall prove the existence of isolated chiral skyrmions minimizing a ferromagnetic energy in a non-trivial homotopy class. In contrast to the classical Skyrme mechanism from nuclear physics, the stabilization is due to an antisymmetric exchange (Dzyaloshinskii-Moriya) interaction term of linear gradient dependence, which breaks the chiral symmetry. We shall also discuss the dynamic stability and effective dynamics of chiral skyrmions subject to external currents. |

Andreas Prohl University of Tübingen, Germany)Stochastic ferromagnetismI start with a survey of convergent discretizations for the Landau-Lifshitz-Gilbert equation. The second part is on showing convergence for a fully discrete scheme of the stochastic counterpart. The last part studies the role of noise via considering long-time distributions of solutions in the case of finitely many particles. This is joint work with L. Banas (Bielefeld University), Z. Brzezniak (University of York), and M. Neklyudov (University of Pisa). |

Thomas Schrefl (Danube University Krems, Austria)Computational challenges for micromagnetics of permanent magnetsPermanent magnets are essential for sustainable technologies. They form the key building blocks for energy conversion. High performance magnets are used in wind power generators and in hybrid and electric vehicles. Using numerical micromagnetics we want to compute basic properties such as coercivity or energy product. However, the difference in the length scales involved is a major challenge for developing accurate numerical solvers. (1) The grain size is a few hundert nanometers or a few micrometers depending on the production method. (2) The grain boundary phases or defect layers have a thickness of 2nm or smaller. (3) The ferromagnetic exchange length is around 2nm for hard magnetic materials. In my talk I will discuss three methods to treat this problem: (a) Massively parallel computation with a fine finite element mesh. (b) Introducing an effective field term that mimics the effect of a soft magnetic defect layer. (c) Applying the Stoner Wohlfarth model at a representative point in the grain. For each method I will present numerical results and compare them with experimental data. This work is supported by EU FP7 (Romeo), Austrian Science Fund (F41), METI (MagHEM), and JST (CREST). |

Claudio Serpico (University of Naples Federico II, Italy)Nonlinear oscillations in nanomagnetsIn the analytical study of spin-torque nano-oscillators, both those based on uniformly magnetized free layer and those with free layer in a magnetic vortex state, one is naturally led to the derivation and the use of simplified models of self-oscillating systems. Such models can be derived by appropriate simplification of the Landau-Lifshitz equation based on time scale separation between the fast precessional motion and slow dynamics due to nonconservative effects such as damping and spin transfer torque. These oscillator models include universal features such as nonlinear dependence of frequency with respect to the amplitude of oscillations, and presence of positive and negative damping effects. In this respect, by using an appropriate change of variables, nonlinear oscillator models can be put in standard (normal) forms. In this work, we propose a normal form for the study of nonlinear self-oscillating systems subject to external AC excitation. By using combined numerical and analytical techniques based on bifurcation theories we show that the deceivingly simple structure of the nonlinear oscillator equation actually encompasses a very rich and nontrivial bifurcation patterns which include both local (saddle-node and Hopf) and nonlocal (homoclinic connection) bifurcations. In this respect, while the role of local bifurcations has been studied in detail, the important role of homoclinic phenomena in synchronization problems have been largely overlooked. In this work, the full range of possible bifurcations of the standard nonlinear oscillator model is studied. The various regimes in the control plane (frequency, AC magnitude) are identified by means of a bifurcation diagram. This enables one to have a comprehensive understanding of phase-locking and nonlinear resonance phenomena. The relevance of this analysis in the area of spin-torque nano-oscillators is discussed. |

Franck Sueur (University of Bordeaux, France)On the weak solutions to the Landau-Lifshitz equationsIn this talk I will deal with weak solutions to the Maxwell-Landau-Lifshitz equations. First these solutions satisfy some weak-strong uniqueness property. Then I will investigate the validity of energy identities. In particular I will give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. This condition corresponds to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally I will examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes. This is a joint work with Eric Dumas. |

Thanh Tran (UNSW Sydney, Australia)Solutions to the stochastic Maxwell-Landau-Lifshitz-Gilbert equationsThe Landau-Lifshitz-Gilbert equation (LLGE) is a well-known model for magnetisation. In a ferromagnetic material, magnetisation is created or affected by external electro-magnetic fields. It is therefore necessary to augment the Maxwell system with the LLGE, which describes the influence of ferromagnet. Thermal noise plays a significant role as it can reverse magnetisation. There is a need to study the stochastic LLGE coupled with the Maxwell system. In this talk we will discuss different approaches to tackle the coupled system. |