Claas Abert

3D print of polymer bonded rare-earth magnets, and 3D magnetic field scanning with an end-user 3D printer

3D print is a recently developed technique, for single-unit production, and for structures that have been impossible to build previously. The current work presents a method to 3D print polymer bonded isotropic hard magnets with a low-cost, end-user 3D printer. Commercially available isotropic NdFeB powder inside a PA11 matrix is characterized, and prepared for the printing process. An example of a printed magnet with a complex shape that was designed to generate a specific stray field is presented, and compared with finite element simulation solving the macroscopic Maxwell equations. For magnetic characterization, and comparing 3D printed structures with injection molded parts, hysteresis measurements are performed. To measure the stray field outside the magnet, the printer is upgraded to a 3D magnetic flux density measurement system. To skip an elaborate adjusting of the sensor, a simulation is used to calibrate the angles, sensitivity, and the offset of the sensor. With this setup, a measurement resolut...

Numerical methods for the stray-field calculation: A comparison of recently developed algorithms

Different numerical approaches for the stray-field calculation in the context of micromagnetic simulations are investigated. We compare finite difference based fast Fourier transform methods, tensor-grid methods and the finite-element method with shell transformation in terms of computational complexity, storage requirements and accuracy tested on several benchmark problems. These methods can be subdivided into integral methods (fast Fourier transform methods, tensor-grid method) which solve the stray field directly and in differential equation methods (finite-element method) which compute the stray field as the solution of a partial differential equation. It turns out that for cuboid structures the integral methods, which work on cuboid grids (fast Fourier transform methods and tensor-grid methods), outperform the finite-element method in terms of the ratio of computational effort to accuracy. Among these three methods the tensor-grid method is the fastest for a given spatial discretization. However, the use of the tensor-grid method in the context of full micromagnetic codes is not well investigated yet. The finite-element method performs best for computations on curved structures.

Heat-assisted magnetic recording of bit-patterned media beyond 10 Tb/in2

The limits of areal storage density that is achievable with heat-assisted magnetic recording are unknown. We addressed this central question and investigated the areal density of bit-patterned media. We analyzed the detailed switching behavior of a recording bit under various external conditions, allowing us to compute the bit error rate of a write process (shingled and conventional) for various grain spacings, write head positions, and write temperatures. Hence, we were able to optimize the areal density yielding values beyond 10 Tb/in2. Our model is based on the Landau-Lifshitz-Bloch equation and uses hard magnetic recording grains with a 5-nm diameter and 10-nm height. It assumes a realistic distribution of the Curie temperature of the underlying material, grain size, as well as grain and head position.

A Fast Finite-Difference Method for Micromagnetics Using the Magnetic Scalar Potential

We propose a method for the stray-field computation of ferromagnetic microstructures via the magnetic scalar potential. The scalar potential is computed using the convolution theorem and the fast Fourier transform. For the discrete convolution an analytical expression for the scalar potential of a uniformly magnetized cuboid is presented. A performance gain of up to 55% compared to common simulation codes is achieved and the memory consumption is reduced by 30%. Since the stray-field computation is the most time consuming part of micromagnetic simulations, this performance gain strongly influences the overall performance. The low memory consumption allows simulations with a high number of simulation cells. This enables simulations of large systems like arrays of coupled magnetic vortices or simulations with high spatial resolution. In conjunction with modern hardware, simulations of microstructures with atomic resolution become feasible.

Spin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator

We propose and analyze a decoupled time-marching scheme for the coupling of the Landau–Lifshitz–Gilbert equation with a quasilinear diffusion equation for the spin accumulation. This model describes the interplay of magnetization and electron spin accumulation in magnetic and nonmagnetic multilayer structures. Despite the strong nonlinearity of the overall PDE system, the proposed integrator requires only the solution of two linear systems per time-step. Unconditional convergence of the integrator towards weak solutions is proved.

magnum.fe: A micromagnetic finite-element simulation code based on FEniCS

We have developed a finite-element micromagnetic simulation code based on the FEniCS package called magnum.fe. Here we describe the numerical methods that are applied as well as their implementation with FEniCS. We apply a transformation method for the solution of the demagnetization-field problem. A semi-implicit weak formulation is used for the integration of the Landau–Lifshitz–Gilbert equation. Numerical experiments show the validity of simulation results. magnum.fe is open source and well documented. The broad feature range of the FEniCS package makes magnum.fe a good choice for the implementation of novel micromagnetic finite-element algorithms.

Landau-Lifshitz-Bloch equation for exchange-coupled grains

Heat assisted recording is a promising technique to further increase the storage density in hard disks. Multilayer recording grains with graded Curie temperature is discussed to further assist the write process. Describing the correct magnetization dynamics of these grains, from room temperature to far above the Curie point, during a write process is required for the calculation of bit error rates. We present a coarse grained approach based on the Landau-Lifshitz-Bloch (LLB) equation to model exchange coupled grains with low computational effort. The required temperature dependent material properties such as the zero-field equilibrium magnetization as well as the parallel and normal susceptibilities are obtained by atomistic Landau-Lifshitz-Gilbert (LLG) simulations. Each grain is described with one magnetization vector. In order to mimic the atomistic exchange interaction between the grains a special treatment of the exchange field in the coarse grained approach is presented.

Reduction of critical current density for out-of-plane mode oscillation in a mag-flip spin torque oscillator using highly spin-polarized Co2Fe(Ga0.5Ge0.5) spin injection layer

We study spin torque oscillators comprised of a perpendicular spin injection layer (SIL) and a planar field generating layer to reveal the influence of the spin polarization of SIL material on the critical current density, JC, to induce microwave oscillation. Two systems with different SIL are compared: one with a highly spin-polarized Heusler alloy, Co2Fe(Ga0.5Ge0.5) (CFGG), and the other a prototypical Fe2Co alloy. Cross sectional scanning transmission electron microscopy observations show the B2-ordered structure in a 3-nm-thick CFGG SIL, a prerequisite for obtaining half-metallic transport properties. Current induced microwave oscillations are found at frequencies of ∼15 GHz for both systems. However, the current needed to cause the oscillations is ∼50% smaller for films with the CFGG SIL compared to those of the Fe2Co SIL. These results are in accordance with micromagnetic simulations that include spin accumulation at the SIL.

A three-dimensional spin-diffusion model for micromagnetics.

We solve a time-dependent three-dimensional spin-diffusion model coupled to the Landau-Lifshitz-Gilbert equation numerically. The presented model is validated by comparison to two established spin-torque models: The model of Slonzewski that describes spin-torque in multi-layer structures in the presence of a fixed layer and the model of Zhang and Li that describes current driven domain-wall motion. It is shown that both models are incorporated by the spin-diffusion description, i.e., the nonlocal effects of the Slonzewski model are captured as well as the spin-accumulation due to magnetization gradients as described by the model of Zhang and Li. Moreover, the presented method is able to resolve the time dependency of the spin-accumulation.

A self-consistent spin-diffusion model for micromagnetics.

We propose a three-dimensional micromagnetic model that dynamically solves the Landau-Lifshitz-Gilbert equation coupled to the full spin-diffusion equation. In contrast to previous methods, we solve for the magnetization dynamics and the electric potential in a self-consistent fashion. This treatment allows for an accurate description of magnetization dependent resistance changes. Moreover, the presented algorithm describes both spin accumulation due to smooth magnetization transitions and due to material interfaces as in multilayer structures. The model and its finite-element implementation are validated by current driven motion of a magnetic vortex structure. In a second experiment, the resistivity of a magnetic multilayer structure in dependence of the tilting angle of the magnetization in the different layers is investigated. Both examples show good agreement with reference simulations and experiments respectively.